INTRODUCTION
TO THE SERIES
The aim of the Handbooks in Economics series is to produce Handbooks for various branches of economics, each of which is a definitive source, reference, and teaching supplement for use by professional researchers and advanced graduate students. Each Handbook provides self-contained surveys of the current state of a branch of economics in the form of chapters prepared by leading specialists on various aspects of this branch of economics. These surveys summarize not only received results but also newer developments, from recent journal articles and discussion papers. Some original material is also included, but the main goal is to provide comprehensive and accessible surveys. The Handbooks are intended to provide not only useful reference volumes for professional collections but also possible supplementary readings for advanced courses for graduate students in economics. KENNETH J. ARROW and MICHAEL D. INTRILIGATOR
PUBLISHER'S
NOTE
For a complete overview of the Handbooks in Economics Series, please refer to the listing at the end of this volume.
CONTENTSOF
VOLUME
THE HANDBOOK
1A
PART 1 - EMPIRICAL AND HISTORICAL PERFORMANCE Chapter 1 Business Cycle Fluctuations in US Macroeconomic Time Series JAMES H. STOCK and MARK W. WATSON
Chapter 2 Monetary Policy Shocks: What Have we Learned and to What End? LAWRENCE J. CHRISTIANO, MARTIN EICHENBAUM and CHARLES L. EVANS
Chapter 3 Monetary Policy Regimes and Economic Performance: The Historical Record MICHAEL D. BORDO AND ANNA J. SCHWARTZ
Chapter 4 The New Empirics of Economic Growth STEVEN N. DURLAUF and DANNY T. QUAH PART 2 - M E T H O D S O F D Y N A M I C A N A L Y S I S
Chapter 5 Numerical Solution of Dynamic Economic Models MANUEL S. SANTOS
Chapter 6 Indeterminacy and Sunspots in Macroeconomics JESS BENHABIB and ROGER E.A. FARMER
Chapter 7 Learning Dynamics GEORGE W. EVANS and SEPPO HONKAPOHJA
Chapter 8 Micro Data and General Equilibrium Models MARTIN BROWNING, LARS PETER HANSEN and JAIVIES J. HECKMAN
vii
Contents of the Handbook
viii
PART 3 - MODELS
OF ECONOMIC
GROWTH
Chapter 9 Neoclassical Growth Theory ROBERT M. SOLOW
Chapter 10 Explaining Cross-Country Income Differences ELLEN R. McGRATTAN and JAMES A. SCHMITZ, Jr.
VOLUME
1B
PART 4 - CONSUMPTION
AND INVESTMENT
Chapter 11 Consumption ORAZIO E ATTANASIO
Chapter 12 Aggregate Investment RICARDO J. CABALLERO
Chapter 13 Inventories VALERIE A. RAMEY and KENNETH D. WEST
PART 5 - MODELS
OF ECONOMIC
FLUCTUATIONS
Chapter 14 Resuscitating Real Business Cycles ROBERT G. KING AND SERGIO T. REBELO
Chapter 15 Staggered Price and Wage Setting in Macroeconomics JOHN B. TAYLOR
Chapter 16 The Cyclical Behavior of Prices and Costs JULIO J. ROTEMBERG and MICHAEL WOODFORD
Chapter 17 Labor-Market Frictions and Employment Fluctuations ROBERT E. HALL
Chapter 18 Job Reallocation, Employment Fluctuations and Unemployment DALE T. MORTENSEN and CHRISTOPHER A. PISSARIDES
Contents of the Handbook
VOLUME 1C PART 6 - F I N A N C I A L M A R K E T S A N D T H E M A C R O E C O N O M Y
Chapter 19 Asset Prices, Consumption, and the Business Cycle JOHN Y. CAMPBELL
Chapter 20 Human Behavior and the Efficiency of the Financial System ROBERT J. SHILLER
Chapter 21 The Financial Accelerator in a Quantitative Business Cycle Framework BEN S. BERNANKE, MARK GERTLER and SIMON GILCHRIST PART 7 - M O N E T A R Y A N D F I S C A L P O L I C Y
Chapter 22 Political Economics and Macroeconomic Policy TORSTEN PERSSON and GUIDO TABELLINI
Chapter 23 Issues in the Design of Monetary Policy Rules BENNETT T. McCALLUM
Chapter 24 Inflation Stabilization and BOP Crises in Developing Countries GUILLERMO A. CALVO and CARLOS A. VI~GH
Chapter 25 Government Debt DOUGLAS W. ELMENDORF AND N. GREGORY MANKIW
Chapter 26 Optimal Fiscal and Monetary Policy V.V CHARI and PATRICK J. KEHOE
ix
PREFACE TO THE HANDBOOK
Purpose The Handbook of Macroeconomics aims to provide a survey of the state of knowledge in the broad area that includes the theories and facts of economic growth and economic fluctuations, as well as the consequences of monetary and fiscal policies for general economic conditions.
Progress in Macroeconomics Macroeconomic issues are central concerns in economics. Hence it is surprising that (with the exception of the subset of these topics addressed in the Handbook of Monetary Economics) no review of this area has been undertaken in the Handbook of Economics series until now. Surprising or not, we find that now is an especially auspicious time to present such a review of the field. Macroeconomics underwent a revolution in the 1970's and 1980's, due to the introduction of the methods of rational expectations, dynamic optimization, and general equilibrium analysis into macroeconomic models, to the development of new theories of economic fluctuations, and to the introduction of sophisticated methods for the analysis of economic time series. These developments were both important and exciting. However, the rapid change in methods and theories led to considerable disagreement, especially in the 1980's, as to whether there was any core of common beliefs, even about the defining problems of the subject, that united macroeconomists any longer. The 1990's have also been exciting, but for a different reason. In our view, the modern methods of analysis have progressed to the point where they are now much better able to address practical or substantive macroeconomic questions - whether traditional, new, empirical, or policy-related. Indeed, we find that it is no longer necessary to choose between more powerful methods and practical policy concerns. We believe that both the progress and the focus on substantive problems has led to a situation in macroeconomics where the area of common ground is considerable, though we cannot yet mmounce a "new synthesis" that could be endorsed by most scholars working in the field. For this reason, we have organized this Handbook around substantive macroeconomic problems, and not around alternative methodological approaches or schools of thought.
xi
xii
Preface
The extent to which the field has changed over the past decade is considerable, and we think that there is a great need for the survey of the current state ofmacroeconomics that we and the other contributors to this book have attempted here. We hope that the Handbook of Macroeconomics will be useful as a teaching supplement in graduate courses in the field, and also as a reference that will assist researchers in one area of macroeconomics to become better acquainted with developments in other branches of the field. Overview The Handbook of Macroeconomics includes 26 chapters, arranged into seven parts. Part 1 reviews evidence on the Empirical and Historical Performance of the aggregate economy, to provide factual background for the modeling efforts and policy discussion of the remaining chapters. It includes evidence on the character of business fluctuations, on long-run economic growth and the persistence of crosscountry differences in income levels, and on economic performance under alternative policy regimes. Part 2 on Methods of Dynamic Analysis treats several technical issues that arise in the study of economic models which are dynamic and in which agents' expectations about the future are critical to equilibrium determination. These include methods for the calibration and computation of models with intertemporal equilibria, the analysis of the determinacy of equilibria, and the use of "learning" dynamics to consider the stability of such equilibria. These topics are important for economic theory in general, and some are also treated in the Handbook of MathematicalEconomics, The Handbook of Econometrics, and the Handbook of Computational Economics, for example, from a somewhat different perspective. Here we emphasize results - such as the problems associated with the calibration of general equilibrium models using microeconomic studies - that have particular application to macroeconomic models. The Handbook then turns to a review of theoretical models of macroeconomic phenomena. Part 3 reviews Models of Economic Growth, including both the determinants of long-run levels of income per capita and the sources of cross-country income differences. Both "neoclassical" and "endogenous" theories of growth are discussed. Part 4 treats models of Consumption and Investment demand, f r o m the point of view of intertemporal optimization. Part 5 covers Models of" Economic Fluctuations. In the chapters in this part we see a common approach to model formulation and testing, emphasizing intertemporal optimization, quantitative general equilibrium modeling, and the systematic comparison of model predictions with economic time series. This common approach allows for consideration of a variety of views about the ultimate sources of economic fluctuations and of the efficiency of the market mechanisms that amplify and propagate them. Part 6 treats Financial Markets and the Macroeconomy. The chapters in this part consider the relation between financial market developments and aggregate economic
Preface
xiii
activity, both from the point of view of how business fluctuations affect financial markets, and how financial market disturbances affect overall economic activity. These chapters also delve into the question of whether financial market behavior can be understood in terms of the postulates of rational expectations and intertemporal optimization that are used so extensively in modern macroeconomics-an issue of fundamental importance to our subject that can be, and has been, subject to special scrutiny in the area of financial economics because of the unusual quality of available data. Finally, Part 7 reviews a number of Monetary and Fiscal Policy issues. Here we consider both the positive theory (or political economics) of government policymaking and the normative theory. Both the nature of ideal (or second-best) outcomes according to economic theory and the choice of simple rules that may offer practical guidance for policymakers are discussed. Lessons from economic theory and from experience with alternative policy regimes are reviewed. None of the chapters in this part focus entirely on international, or open economy, macroeconomic policies, because many such issues are addressed in the Handbook of International Economics. Nevertheless, open-economy issues cannot be separated from closed-economy issues as the analysis of disinflation policies and currency crises in this part of the Handbook of Macroeeonomics, or the analysis of policy regimes in the Part I of the Handbook of Maeroeconomics make clear.
Acknowledgements Our use of the pronoun "we" in this preface should not, of course, be taken to suggest that much, if any, of the credit for what is useful in these volumes is due to the Handbook's editors. We wish to acknowledge the tremendous debt we owe to the authors of the chapters in this Handbook, who not only prepared the individual chapters, but also provided us with much useful advice about the organization of the overall project. We are grateful for their efforts and for their patience with our slow progress toward completion of the Handbook. We hope that they will find that the final product justifies their efforts. We also wish to thank the Federal Reserve Bank of New York, the Federal Reserve Bank of San Francisco, and the Center for Economic Policy Research at Stanford University for financial support for two conferences on "Recent Developments in Macroeconomics" at which drafts of the Handbook chapters were presented and discussed, and especially to Jack Beebe and Rick Mishkin who made these two useful conferences happen. The deadlines, feedback, and commentary at these conferences were essential to the successful completion of the Handbook. We also woutd like to thank Jean Koentop for managing the manuscript as it neared completion. Stanford, California Princeton, New Jersey
John B. Taylor Michael Woodford
Chapter 1
BUSINESS
CYCLE
FLUCTUATIONS
US MACROECONOMIC
TIME
IN
SERIES
JAMES H. STOCK Kennedy School o f Government, Harvard University and the NBER
MARK W WATSON Woodrow Wilson School, Princeton University and the NBER
Contents Abstract Keywords 1. Introduction 2. Empirical methods of business cycle analysis 2.1. Classical business cycle analysis and the determination of turning points 2.2. Isolating the cyclical component by linear filtering 3. Cyclical behavior of selected economic time series 3.1. The data and summary statistics 3.2. Discussion of results for selected series 3.2.1. Comovementsin employmentacross sectors 3.2.2. Consumption, investment,inventories,imports and exports 3.2.3. Aggregate employment,productivity and capacity utilization 3.2.4. Prices and wages 3.2.5. Asset prices and returns 3.2.6. Monetary aggregates 3.2.7. Miscellaneous leading indicators 3.2.8. International output 3.2.9. Stability of the predictive relations 4. Additional empirical regularities in the postwar US data 4.1. The Phillips curve 4.2. Selected long-run relations 4.2.1. Long-run money demand 4.2.2. Spreads between long-term and short-term interest rates 4.2.3. Balanced growth relations Acknowledgements Appendix A. Description of the data series used in this chapter A. 1. Series used in Section 1 Handbook o f Macroeconomics, Volume 1, Edited by JB. Taylor and M. Woodford © 1999 Elsevier Science B.V. All rights reserved 3
4 4 5 8 8 10 14 14 39 39 40 41 42 43 44 44 45 45 46 46 50 50 52 54 56 56 56
4
J.H. Stock and M. W. Watson
A.2. Series used in Section2 A.3. Additionalseries used in Section4 References
56 60 61
Abstract This chapter examines the empirical relationship in the postwar United States between the aggregate business cycle and various aspects of the macroeconomy, such as production, interest rates, prices, productivity, sectoral employment, investment, income, and consumption. This is done by examining the strength of the relationship between the aggregate cycle and the cyclical components of individual time series, whether individual series lead or lag the cycle, and whether individual series are useful in predicting aggregate fluctuations. The chapter also reviews some additional empirical regularities in the US economy, including the Phillips curve and some longrun relationships, in particular long run money demand, long run properties of interest rates and the yield curve, and the long run properties of the shares in output of consumption, investment and government spending.
Keywords economic fluctuations, Phillips curve, long run macroeconomic relations J E L classification: E30
Ch. 1:
Business Cycle Fluctuations in US Macroeconomic Time Series
1. I n t r o d u c t i o n
This chapter summarizes some important regularities in macroeconomic time series data for the Uriited States since World War II. Our primary focus is the business cycle. In their classic study, Burns and Mitchell (1946) offer the following definition o f the business cycle: A cycle consists of expansions occurring at about the same time in many economic activities, followed by similarly general recessions, contractions, and revivals which merge into the expansion phase of the next cycle; this sequence of changes is recurrent but not periodic; in duration business cycles vary from more than one year to ten or twelve years; they are not divisible into shorter cycles of similar character with amplitudes approximating their own. Burns and Mitchell, 1946, p. 3.
Figure 1.1 plots the natural logarithm of an index o f industrial production for the United States from 1919 to 1996. (Data sources are listed in the Appendix.) Over these 78 years, this index has increased more than fifteen-fold, corresponding to an increase in its logarittma by more than 2.7 units. This reflects the tremendous growth o f the US labor force and o f the productivity o f American workers over the twentieth century. Also evident in Figure 1.1 are the prolonged periods of increases and declines that constitute American business cycles. These fluctuations coincide with some o f the signal events o f the US economy over this century: the Great Depression o f the 1930s; the subsequent recovery and growth during World War II; the sustained boom o f the 1960s, associated in part with spending on the war in Vietnam; the recession o f 1973-1975, associated with the first OPEC price increases; the disinflationary twin recessions of the early 1980s; the recession o f 1990, associated with the invasion o f Kuwait by Iraq; and the long expansions of the 1980s and the 1990s. To bring these cyclical fluctuations into sharper focus, Figure 1.2 plots an estimate
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J.H. Stock and M.W. Watson
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Fig. 1.2. Business cycle componentof industrial productionindex. of the cyclical component of industrial production. (This estimate was obtained by passing the series through a bandpass filter that isolates fluctuations at business cycle periodicities, six quarters to eight years; this filter is described in the next section.) The vertical lines in Figure 1.2 indicate cyclical peaks and troughs, where the dates have been determined by business cycle analysts at the National Bureau of Economic Research (NBER). A chronology of NBER-dated cyclical turning points from 1854 to the present is given in Table 1 (the method by which these dates were obtained is discussed in the next section). Evidently, the business cycle is an enduring feature of the US economy. In the next two sections, we examine the business cycle properties of 71 quarterly US economic time series. Although business cycles have long been present in the US, this chapter focuses on the postwar period for two reasons. First, the American economy is vastly different now than it was many years ago: new production and financial technologies, institutions like the Federal Reserve System, the rise of the service and financial sectors, and the decline of agriculture and manufacturing are but a few of the significant changes that make the modern business cycle different from its historical counterpart. Second, the early data have significant deficiencies and in general are not comparable to the more recent data. For example, one might be tempted to conclude from Figure 1.2 that business cycles have been less severe and less frequent in the postwar period than in the prewar period. However, the quality of the data is not consistent over the 78-year sample period, which makes such comparisons problematic. Indeed, Romer (1989) has argued that, after accounting for such measurement problems, cyclical fluctuations since World War II have been of the same magnitude as they were before World War I. Although this position is controversial [see Balke and Gordon (1989), Diebold and Rudebusch (1992) and Watson (1994a)], there is general agreement that
Ch. 1:
Business Cycle Fluctuations in US Macroeconomic Time Series
Table 1 NBER business cycle reference dates Trough
Peak
December
1854
June
1857
December
1858
October
1860
June
1861
April
1865
December
1867
June
1869
December
1870
October
1873
March
1879
March
1882
May
1885
March
1887
April
1888
July
1890
May
1891
January
1893
June
1894
December
1895 1899
June
1897
June
December
1900
September
1902
August
1904
May
1907
June
1908
January
1910
January
1912
January
1913
December
1914
August
1918
March
1919
January
1920
July
1921
May
1923
July
1924
October
1926
November
1927
August
1929
March
1933
May
1937
June
1938
February
1945
October
1945
November
1948
October
1949
July
1953
May
1954
August
1957
April
1958
April
1960 1969
February
1961
December
November
1970
November
1973
March
1975
January
1980
July
1980
July
1981
November
1982
July
1990
March
1991
aSource: National Bureau of Economic Research.
J.H. Stock and M. W. Watson
comparisons of business cycles from different historical periods is hampered by the severe limitations of the early data. For these reasons, this chapter focuses on the postwar period for which a broad set of consistently defined data series are available, and which is in any event the relevant period for the study of the modern business cycle. There are other important features of the postwar data that are not strictly related to the business cycle but which merit special emphasis. In the final section of this chapter, we therefore turn to an examination of selected additional regularities in postwar economic time series that are not strictly linked to the business cycle. These include the Phillips curve (the relationship between the rate of price inflation and the unemployment rate) and some macroeconomic relations that hold over the long run, specifically long-run money demand, yield curve spreads, and the consumptionincome and consumption-investment ratios. These relations have proven remarkably stable over the past four decades, and they provide important benchmarks both for assessing theoretical macroeconomic models and for guiding macroeconomic policy.
2. Empirical methods of business cycle analysis 2.1. Classical business cycle analysis and the determination of turning points
There is a long intellectual history of the empirical analysis of business cycles. The classical techniques of business cycle analysis were developed by researchers at the National Bureau of Economic Research [Mitchell (1927), Mitchell and Burns (1938), Burns and Mitchell (1946)]. Given the definition quoted in the introduction, the two main empirical questions are how to identify historical business cycles and how to quantify the comovement of a specific time series with the aggregate business cycle. The business cycle turning points identified retrospectively and on an ongoing basis by the NBER, which are listed in Table 1, constitute a broadly accepted business cycle chronology. NBER researchers determined these dates using a two-step process. First, cyclical peaks and troughs (respectively, local maxima and minima) were determined for individual series. Although these turning points are determined judgementally, the process is well approximated by a computer algorithm developed by Bry and Boschan (1971). Second, common turning points were determined by comparing these seriesspecific turning points. If, in the judgment of the analysts, the cyclical movements associated with these common turning points are sufficiently persistent and widespread across sectors, then an aggregate business cycle is identified and its peaks and troughs are dated. Currently, the NBER Business Cycle Dating Committee uses data on output, income, employment, and trade, both at the sectoral and aggregate levels, to guide their judgments in identifying and dating business cycles as they occur [NBER (1992)]. These dates typically are announced with a lag to ensure that the data on which they are based are as accurate as possible. Bums, Mitchell and their associates also developed procedures for comparing cycles in individual series to the aggregate business cycle. These procedures include measuring leads and lags of specific series at cyclical turning
Ch. 1:
Business Cycle Fluctuations in US M a c r o e c o n o m i c Time Series
points and computing cross-correlations on a redefined time scale that corresponds to phases o f the aggregate business cycle. The classical business cycle discussed so far refers to absolute declines in output and other measures. A n alternative is to examine cyclical fluctuations in economic time series that are deviations from their long-run trends. The resulting cyclical fluctuations are referred to as growth cycles [see for example Zarnowitz (1992), ch. 7]. Whereas classical cycles tend to have recessions that are considerably shorter than expansions because o f underlying trend growth, growth recessions and expansions have approximately the same duration. The study o f growth cycles has advantages and disadvantages relative to classical cycles. On the one hand, separation o f the trend and cyclical component is inconsistent with some modern macroeconomic models, in which productivity shocks (for example) determine both long-run economic growth and the fluctuations around that growth trend. From this perspective, the trend-cycle dichotomy is only justified i f the factors determining long-run growth and those determining cyclical fluctuations are largely distinct. On the other hand, growth cycle chronologies are by construction tess sensitive to the underlying trend growth rate in the economy, and in fact some economies which have had very high growth rates, such as postwar Japan, exhibit growth cycles but have few absolute declines and thus have few classical business cycles. Finally, the methods o f classical business cycle analysis have been criticized for lacking a statistical foundation (for example Koopmans (1947)]. Although there have been some modern treatments o f these nonlinear filters (for example Stock (1987)], linear filtering theory is better understood 1. Modern studies o f business cycle properties therefore have used linear filters to distinguish between the trend and cyclical components o f economic time series 2. Although we note these ambiguities, in the rest o f this chapter we follow the recent literature and focus on growth recessions and expansions 3.
I A linear filter is a set of weights {ai, i=0,:t:1, -t-2, ... } that are applied to a time series Yt; the filtered version of the time series is ~ i ~_ oo aiYt i. If the filtered series has the form ~i°°O aiy t i (that is, a i = 0, i < 0), the filter is said to be one-sided, otherwise the filter is two-sided. In a nonlinear filter, the filtered version of the time series is a nonlinear fimction of {Yt, t = 0, 4-1, ±2 . . . . }. 2 See Hodrick and Prescott (1981), Harvey and Jaeger (1993), Stock and Watson (1990), Baekus and Kehoe (1992), King and Rebelo (1993), Kydland and Prescott (1990), England, Persson and Svensson (1992), Hassler, Lundvik, Persson and S6derlind (1992), and Baxter and King (1994) for more discussion and examples of linear filtering methods applied to the business cycle. 3 This discussion treats the NBER chronology as a concise way to summarize some of the most significant events in the macroeconomy. A different use of the chronology is as a benchmark against which to judge macroeconomic models. In an early application of Monte Carlo methods to econometrics, Adelnaan and Adelman (1959) simulated the Klei~Goldberger model and found that it produced expansions and contractions with durations that closely matched those in the US economy. King and Plosser (1994) and Hess and Iwata (1997) carried out similar exercises. Pagan (1997) has shown, however, that a wide range of simple time series models satisfy this test, which indicates that it is not a particularly powerful way to discriminate among macroeconomic models. Of course, using the NBER dating methodology to describe data differs from using it to test models, and the low power of the test of the Adelmans simply implies that this methodology is better suited to the former task than the latter.
10
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2.2. Isolating the cyclical component by linear filtering Quarterly data on the logarithm of real US GDP from 1947 to 1996 are plotted in Figure 2.1. As in the longer index of industrial production shown in Figure 1.1, cyclical fluctuations are evident in these postwar data. Without further refinement, however, it is difficult to separate the cyclical fluctuations from the long-run growth component. Moreover, there are some fluctuations in the series that occur over periods shorter than a business cycle, arising from temporary factors such as unusually harsh weather, strikes and measurement error. It is therefore desirable to have a method to isolate only those business cycle fluctuations of immediate interest. I f the long-run growth component in log real GDP is posited to be a linear time trend, then a natural way to eliminate this trend component is to regress the logarithm of GDP against time and to plot its residual. This "linearly detrended" time series, scaled to be in percentage points, is plotted in Figure 2.2. Clearly the cyclical fluctuations of output are more pronounced in this detrended plot. However, these detrended data still contain fluctuations of a short duration that are arguably not related to business cycles. Furthermore, this procedure is statistically valid only if the long-run growth component is a linear time trend, that is, if GDP is trend stationary (stationary around a linear
11
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4 The NBER chronology in Table 1 lists 30 completecycles since 1858. The shortest full cycle (peak to peak) was 6 quarters, and the longest 39 quarters; 90% of these cycles are no longer than 32 quarters.
J.H. Stock and M.W. Watson
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.
.
. 2.4
.
.
.
. 2s
sp
Ircqucncy
Fig. 2.4. Filter gains. these fluctuations but would eliminate all other fluctuations, both the high frequency fluctuations (periods less than six quarters) associated for example with measurement error and the low frequency fluctuations (periods exceeding eight years) associated with trend growth. In other words, the gain of the ideal linear filter is unity for business cycle frequencies and zero elsewhere 5. This ideal filter cannot be implemented in finite data sets because it requires an infinite number o f past and future values of the series; however, a feasible (finite-order) filter can be used to approximate this ideal filter. Gains of this ideal filter and several candidate feasible filters are plotted in Figure 2.4. The first-differencing filter eliminates the trend component, but it exacerbates the effect o f high frequency noise, a drawback that is evident in Figure 2.3. Another filter that is widely used is the Hodrick-Prescott filter [Hodrick and Prescott (1981)]. This filter improves upon the first-differencing filter: it attenuates less of the cyclical component and it does not amplify the high frequency noise. However, it still passes much of the high frequency noise outside the business cycle frequency band. The filter adopted in this study is Baxter and King's bandpass filter, which is designed to mitigate these problems [Baxter and King (1994)]. This feasible bandpass filter is based on a twelve-quarter centered moving average, where the weights are chosen to minimize the squared difference between the optimal and approximately optimal filters,
5 The spectral density of a time series xt at frequency e) is sx((o)=(2Jr ) 1 ~)o~ ow Z~(j)exp(i~oj), where yx(j)=cov(xt,xt j). The gain of a linear filter a(L) is ]A(co)I,where A((o) = }-~V~oo ajexp(Roj). The spectrum of a linearly filtered series, Yt = a(L)xt, with L the lag operator, is sy(co) = ]A(co)l2 Sx(fO). See Hamilton (1994) for an introduction to the spectral analysis of economic time series.
Ch. 1:
13
Business Cycle Fluctuations in US Macroeeonomic Time Series
Pt
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72
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rl ] rl I !rl
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82
g7
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Fig. 2.5. Bandpass-filteredGDP (business cycle). subject to the constraint that the filter has zero gain at frequency zero 6. Because this is a finite approximation, its gain is only approximately flat within the business cycle band and is nonzero for some frequencies outside this band. The cyclical component of real GDP, estimated using this bandpass filter, is plotted in Figure 2.5. This series differs from linearly detrended GDR plotted in Figure 2.2, in two respects. First, its fluctuations are more closely centered around zero. This reflects the more flexible detrending method implicit in the bandpass filter. Second, the high frequency variations in detrended GDP have been eliminated. The main cyclical events of the postwar period are readily apparent in the bandpass filtered data. The largest recessions occurred in 1973-1975 and the early 1980s. The recessions of 1969-1970 and 1990-1991 each have shorter durations and smaller amplitudes. Other cyclical fluctuations are also apparent, for example the slowdowns in 1967 and 1986, although these are not classical recessions as identified by the NBER. During 1986, output increased more slowly than average, and the bandpass filtered data, viewed as deviations from a local trend, are negative during 1986. This corresponds to a growth recession even though there was not the absolute decline that characterizes an NBER-dated recession. This distinction between growth recessions and absolute declines in economic activity leads to slight differences in official NBER peaks and local maxima in the bandpass filtered data. Notice from Figure 2.1 that output slowed markedly before the absolute turndowns that characterized the 1970, 1974, 1980 and 1990 recessions. Peaks in the bandpass filter series correspond to the beginning of these stowdowns, while NBER peaks correspond to downturns in the level of GDE The bandpass filtering approach permits a decomposition of the series into trend, cycle and irregular components, respectively corresponding to the low, business cycle, and high frequency parts of the spectrum. The trend and irregular components are
6 To obtain filtered values at the beginning and end of the sample, the series are augmentedby twelve out-of-sample projected values at both ends of the sample, where the projections were made using forecasts and backcasts from univariate fourth-order autoregressivemodels.
14
Jt-L Stock and M.W. Watson
/ ~o
r- 47
52
57
62
67
72
77
82
87
92
Date
Fig. 2.6. Bandpass-filteredGDP (trend).
q~vvvv -V wvv
vv "v"Vv
47
52
57
62
67
12
v
V vv'v "
17
82
87
92
97
Date
Fig. 2.7. Bandpass-filteredGDP (irregular). plotted in Figures 2.6 and 2.7; the series in Figures 2.5-2.7 sum to log real GDR Close inspection of Figure 2.6 reveals a slowdown in trend growth over this period, an issue of great importance that has been the focus of considerable research but which is beyond the scope of this chapter.
3. Cyclical behavior of selected economic time series 3.1. The data and summary statistics
The 71 economic time series examined in this chapter are taken from eight broad categories: sectoral employment; the National Income and Product Accounts (NIPA); aggregate employment, productivity and capacity utilization; prices and wages; asset prices; monetary aggregates; miscellaneous leading indicators; and international output. Most of the series were transformed before further analysis. Quantity measures (the NIPA variables, the monetary aggregates, the level of employment, employee hours, and production) are studied after taking their logarithms. Prices and wages are transformed by taking logarithms and/or quarterly difference of logarithms (scaled to
Ch. 1:
15
Business Cycle Fluctuations in US Macroeconomic Time Series
be percentage changes at an annual rate). Interest rates, spreads, capacity utilization, and the unemployment rate are used without further transformation. The graphical presentations in this section cover the period 1947:I-1996:IV The early years of this period were dominated by some special features such as the peacetime conversion following World War II and the Korean war and the associated price controls. Our statistical analysis therefore is restricted to the period 1953:I1996:IV Three sets of empirical evidence are presented for each of the three series. This evidence examines comovements between each series and real GDR Although the business cycle technically is defined by comovements across many sectors and series, fluctuations in aggregate output are at the core of the business cycle so the cyclical component of real GDP is a useful proxy for the overall business cycle and is thus a useful benchmark for comparisons across series. First, the cyclical component of each series (obtained using the bandpass filter) is plotted, along with the cyclical component of output, for the period t947-1996. For series in logarithms, the business cycle components have been multiplied by 100, so that they can be interpreted as percent deviation from long run trend. No further transformations have been applied to series already expressed in percentage points (inflation rates, interest rates, etc.). These plots appear in Figures 3.1-3.70. Note that the vertical scales of the plots differ. The thick line in each figure is the cyclical component of the series described in the figure caption, and the thin line is the cyclical component of real GDR Relative amplitudes can be seen by comparing the series to aggregate output.
co ~¢
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Fig. 3.1. Contract and constructionemployment.
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,, ii 87
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16
JH. Stock and M.W. Watson
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Fig. 3.3. Finance, insurance and real estate employment.
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Fig. 3.4. Mining employment.
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Ch. 1:
17
Business Cycle Fluctuations in US Macroeconomic Time Series
N
I
I
'tt'//
47
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I
,llll tk// ,V ~
,,,,,,,//
~10
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..,~s II 52
,~ II
I
,k~ k ( k - 1)/2. However, the order and sign conditions are not sufficient for identification. For example, w h e n l = k ( k - 1)/2 underidentification c o u l d o c c u r for two reasons. First, a n e i g h b o r h o o d o f a g i v e n Ao E Qv n Qr N Qs c o u l d contain other m a t r i c e s b e l o n g i n g to Qv N Qr n Qs. This possibility can be ruled out by v e r i f y i n g a simple rank condition, n a m e l y that the matrix derivative w i t h respect to A0 o f the equations defining (3.8) is o f full rank. 9 In this case, we say we have established local identification. A second possibility is that there m a y be other matrices b e l o n g i n g to Q v n Q~ n Qs but w h i c h are not in a small n e i g h b o r h o o d o f A0. ~0 In general, no k n o w n simple conditions rule out this possibility. I f we do m a n a g e to rule it out, we say the system is globally identified. 11 In practice, we use the rank and order conditions to v e r i f y local identification. Global identification m u s t be established on a case by case basis. S o m e t i m e s , as in our discussion o f B e r n a n k e and M i h o v (1995), this can be done analytically. M o r e typically, one is limited to b u i l d i n g confidence in global identification by c o n d u c t i n g an ad hoc n u m e r i c a l search through the p a r a m e t e r space to d e t e r m i n e i f there are other e l e m e n t s
in Qv n Qr n Qs. T h e difficulty o f establishing global identification in the literature we survey stands in contrast to the situation in the traditional simultaneous equations context.
9 Here we define a particular rank condition and establish that the rank and order conditions are sufficient for local identification. Let a be the k(k + 1)/2 dimensional column vector of parameters in A0 that remain free after imposing condition (3.9), so that Ao(a) C Qr for all a. L e t f ( a ) denote the k(k + 1)/2 dimensional row vector composed of the upper triangular part of A0(a ) 1 iAo(a) its_ V. Let F(a) denote the k(k + 1)/2 by k(k + 1)/2 derivative matrix o f f ( a ) with respect to a. Let a* satisfy f(a*) - O. Consider the following rank condition: F(a) has full rank for all a E D(a*), where D(a ~) is some neighborhood of a*. We assume thatf is continuous and that F is well defined. A straightforward application of the mean value theorem (see Bartle (1976), p. 196) establishes that this rank condition guarantees f(a) ~ 0 for all a C D(a*) and a ,~ a*. Let gL : [.eL,eLI - - + Rk(k+l)/2 be defined by gL(e) = f(a* + re), where ~ is an arbitrary non-zero k(k + 1)/2 column vector, and _eL and 2t are the smallest and largest values, respectively, of e such that (a* + be) E D(a*). Note that g~(e) - trF(a* + te) and e L < 0 < et. By the mean value theorem, gL(e) = gL(0) + g~(y)e for some 7 between 0 and e. This can be written gt(e) = t~F(a * + ~e)e. The rank condition implies that the expression to the right of the equality is nonzero, as long as e ~ 0. Since the choice of t e 0 was arbitrary, the result is established. 10 A simple example is (x - a) (x - b) = 0, which is one equation with two isolated solutions, x = a and
x-b. 11 We can also differentiate other concepts of identification. For example, asymptotic and small sample identification correspond to the cases where V is the population and finite sample value of the variance covariance matrix of the VAR disturbances, respectively. Obviously, asymptotic identification could hold while finite sample identification fails, as well as the converse.
Ch. 2: Monetary Policy Shocks." What Haue we Learned and to What End?
77
There, the identification problem only involves systems o f linear equations. Under these circumstances, local identification obtains i f and only i f global identification obtains. The traditional simultaneous equations literature provides a simple set o f rank and order conditions that are necessary and sufficient for identification. These conditions are only sufficient to characterize local identification for the systems that we consider. 12 Moreover, they are neither necessary nor sufficient for global identification. We now describe two examples which illustrate the discussion above. In the first case, the order and sign conditions are sufficient to guarantee global identification. In the second, the order condition and sign conditions for identification hold, yet the system is not identified. In the first example, we select z- so that all the elements above (alternatively, below) the diagonal o f Ao are zero. If, in addition, we impose the sign restriction, then it is well known that there is only one element in Q v A Qr A Qs, i.e., the system is globally identified. This result is an implication o f the uniqueness o f the Cholesky factorization o f a positive definite symmetric matrix. This example plays a role in the section on identification o f monetary policy shocks with a recursiveness assumption. For our second example, consider the case k = 3 with the following restricted Ao matrix:
[al, 0 a13] Ao =
0 0
a22 a23/ ' a32 a 3 3 J
where aii > 0 for i = 1,2, 3. Since there are three zero restrictions, the order condition is satisfied. Suppose that Ao c Qv, so that Ao E Q v c-I Q~ cl Qs. Let W be a block diagonal matrix with unity in the (1, 1) element and an arbitrary 2 x 2 orthonormal matrix in the second diagonal block. Let W also have the property that WAo has positive elements on the diagonal. Then, g ~ = I , and WAo E Qv N Qr N Qs. 13 In this case we do not have identification, even though the order and sign conditions are satisfied. The reason for the failure o f local identification is that the rank condition does not hold. I f it did hold, then identification would have obtained. The failure o f the rank condition in this example reflects that the second and third equations in the system are indistinguishable.
12 To show that the rank condition is not necessary for local identification, consider f(x) = (x For this function there is a globally unique zero at x - a, yetf~(a) = 0. 13 To see that this example is non empty, consider the case all - 0.70, a13 = 0.40, a22 - 0 . 3 8 , a 2 3 -a32 = 0.83, a33 - 0.71 and let the 2 x 2 lower block in W be
I
0.4941 0.8694 I ' 0.8694 -0.4941
It is easy to verify that WAo satisfies the zero and sign restrictions on A0.
a) 2 .
0.50,
78
L.J. Christiano et al.
It is easy to show that every element in Qv n Qr n Qs generates the same dynamic response function to the first shock in the system. To see this, note from Equation (3.5) that the first column of Ao 1 is what characterizes the response of all the variables to the first shock. Similarly, the first column of (WAo) ~ controls the response of the transformed system to the first shock. But, the r e s u l t (WA0) -1 = Ao 1 W t, and our definition of W imply that the first columns of (WA0)-1 and ofAo 1 are the same. So, if one is only interested in the dynamic response of the system to the first shock, then the choice of the second diagonal block of W is irrelevant. An extended version of this observation plays an important role in our discussion of nonrecursive identification schemes below.
4. The effects of a monetary policy shock: a recursiveness assumption
In this section we discuss one widely used strategy for estimating the effects of a monetary policy shock. The strategy is based on the recursiveness assumption, according to which monetary policy shocks are orthogonal to the information set of the monetary authority. Section 4.1 discusses the relationship between the recursiveness assumption and VARs. Section 4.2 describes three benchmark identification schemes which embody the recursiveness assumption. In addition, we display estimates of the dynamic effects of a monetary policy shock on various economic aggregates, obtained using the benchmark identification schemes. Section 4.3 reviews some results in the literature regarding the dynamic effects of a monetary policy shock on other economic aggregates, obtained using close variants of the benchmark schemes. Section 4.4 considers robustness of the empirical results contained in Section 4.2. Section 4.5 discusses various critiques of the benchmark identification schemes. Finally, Section 4.6 investigates the implications of the benchmark schemes for the volatility of various economic aggregates. 4.1. The recursiveness assumption and VARs
The recursiveness assumption justifies the following two-step procedure for estimating the dynamic response of a variable to a monetary policy shock. First, estimate the policy shocks by the fitted residuals in the ordinary least squares regression of St on the elements of £2t. Second, estimate the dynamic response of a variable to a monetary policy shock by regressing the variable on the current and lagged values of the estimated policy shocks. In our analysis we find it convenient to map the above two-step procedure into an asymptotically equivalent VAR-based procedure. There are two reasons for this. First, the two-step approach implies that we lose a number of initial data points equal to the number of dynamic responses that we wish to estimate, plus the number of lags, q, in g2t. With the VAR procedure we only lose the latter. Second, the VAR methodology provides a complete description of the data generating process for the elements of g2t.
Ch. 2:
Monetary Policy Shocks." What Have we Learned and to What End?
79
This allows us to use a straightforward bootstrap methodology for use in conducting hypothesis tests. We now indicate how the recursiveness assumption restricts A0 in Equation (3.2). Partition Zt into three blocks: the kl variables, Xlz, whose contemporaneous values appear in g2t, the k2 variables, X2t , which only appear with a lag in £2t, and St itself. Then, k = kl + k2 + 1, where k is the dimension of Zt. That is:
z'=/% We consider kl, k2 > 0. To make the analysis interesting we assume that if kl = 0, so that Xlt is absent from the definition of Zt, then k2 > 1. Similarly, if k2 = 0, then kl > 1. The recursiveness assumption places the following zero restrictions on Ao • all (kl xkl)
Ao =
a21
0 a22
(lxk~) (1×1) 17/3l (k2 ×kl)
0
(kl x 1) (k~ xk2) 0 (lxk2)
(4.1)
a32 a33 (k2 × 1) (k2 × k2)
Here, expressions in parentheses indicate the dimension of the associated matrix and a22 = 1/G, where G > 0. The zero block in the middle row of this matrix reflect the assumption that the policy maker does not see X2t when St is set. The two zero blocks in the first row of Ao reflect our assumption that the monetary policy shock is orthogonal to the elements in Xlt. These blocks correspond to the two distinct channels by which a monetary policy shock could in principle affect the variables in Xlt. The first of these blocks corresponds to the direct effect of St on X w The second block corresponds to the indirect effect that operates via the impact of a monetary policy shock on the variables in X2~. We now show that the recursiveness assumption is not sufficient to identify all the elements of Ao. This is not surprising, in light of the fact that the first kl equations are indistinguishable from each other, as are the last k2 equations. Significantly, however, the recursiveness assumption is sufficient to identify the object o f interest: the dynamic response of Zt to a monetary policy shock. Specifically, we establish three results. The first two are as follows: (i) there is a nonempty family of Ao matrices, one of which is lower triangular with positive terms on the diagonal, which are consistent with the recursiveness assumption [i.e., satisfy Equation (4.1)] and satisfy Ao 1 (A01) ' = V; and (ii) each member of this family generates precisely the same dynamic response function of the elements of Zt to a monetary policy shock. Result (iii) is that if we adopt the normalization of always selecting the lower triangular Ao matrix identified in (i), then the dynamic response o f the variables in Zt are invariant to the ordering of variables in J(lt and X2t.
L.J. Christiano et al.
80
To prove (i)-(iii) it is useful to establish a preliminary result. We begin by defining some notation. Let the ((kl + 1)k2 + kl) × k 2 matrix r summarize the zero restrictions on A0 in Equation (4.1). So, Qr is the set o f A0 matrices consistent with the recursiveness assumption. Let Qv be the set o f A0 matrices defined by the property that Aol(Aol) / [see Equation (3.8)]. In addition, let W=
0 0
1 0 0 W33
,
(4.2)
where W is partitioned confonrlably with Ao in Equation (4.1) and WH and W33 are arbitrary orthonormal matrices. Define Q~0 = {A0 : A0 = WA0, for some W satisfying (4.2)}. Here A0 is a matrix conformable with W. We now establish the following result:
Q~o = Qv n Q~,
(4.3)
where ~]0 is an arbitrary element o f Qv n Qr. It is straightforward to establish that A0 E Q~0 implies A0 E Qv n Qr. The result, A0 E Qv follows from orthonormality o f W and the fact, ~]0 E Qv. The result, A0 C Qr, follows from the block diagonal structure o f W in Equation (4.2). Now consider an arbitrary A0 C Qv n Qr. To show that A0 E Q~0, consider the candidate orthonormal matrix W = A0~]o1, where invertibility o f ~]0 reflects A0 E Qv. Since W is the product o f two block-lower triangular matrices, it too is block-lower triangular. Also, it is easy to verify that WW / = /. The orthonormality o f W, together with block-lower triangularity imply that W has the form (4.2). This establishes A0 E Q~0 and, hence, Equation (4.3). We now prove result (i). The fact that Qv n Qr is not empty follows from the fact that we can always set A0 equal to the inverse of the lower triangular Cholesky factor of V. The existence and invertability o f this matrix is discussed in Hamilton (1994, p. 91). 14 To see that there is more than one element in Qv n Qr, use the characterization result (4.3), with A0 equal to the inverse of the Cholesky factor o f V. Construct the orthonormal matrix W ~ I by interchanging two o f either the first kl rows or the last k2 rows of the k-dimensional identity matrix. 15 Then, W~]0 ¢ ~]0. Result (i) is established because W~]0 E Qv N Qr. 14 The Cholesky factor of a positive definite, symmetric matrix, V, is a lower triangular matrix, C, with the properties (i) it has positive elements along the diagonal, and (ii) it satisfies the property, CCI = V. 15 Recall, orthonormality of a matrix means that the inner product between two different columns is zero and the inner product of any column with itself is unity. This property is obviously satisfied by the identity matrix. Rearranging the rows of the identity matrix just changes the order of the terms being added in the inner products defining orthonorrnality, and so does not alter the value of column inner products. Hence a matrix obtained from the identity matrix by arbitrarily rearranging the order of its rows is orthonormal.
Ch. 2:
Monetary Policy Shocks: What Have we Learned and to What End?
81
We now prove result (ii). Consider any two matrices, Ao, ~4o E Qv N Qr. By Equation (4.3) there exists a W satisfying Equation (4.2), with the property -40 = WAo, so that z]01 = Ao I W t.
In conjunction with Equation (4.2), this expression implies that the (kl + 1)th column of ~]o1 and Ao I are identical. But, by Equation (3.6) the implied dynamic responses of Zt+i, i = 0, 1. . . . to a monetary policy shock are identical too. This establishes result (ii). We now prove (iii) using an argument essentially the same as the one used to prove (ii). We accomplish the proof by starting with a representation of Zt in which A0 is lower triangular with positive diagonal elements. We then arbitrarily reorder the first kl and the last k2 elements of Zt. The analog to A0 in the resulting system need not be lower triangular with positive elements. We then apply a particular orthonormal transformation which results in a lower triangular system with positive diagonal elements. The response of the variables in Zt to a monetary policy shock is the same in this system and in the original system. Consider Zt = D Z , where D is the orthonormal matrix constructed by arbitrarily reordering the columns within the first kl and the last k2 columns of the identity matrix. 16 Then, Zt corresponds to Zt with the variables in Xlt and X2t reordered arbitrarily. Let Bi, i = 1. . . . . q and V characterize the VAR of Zt and let A0 be the unique lower triangular matrix with positive diagonal terms with the property Ao 1 (Aol) ~ = V. Given the Bi's, Ao characterizes the impulse response function of the Zt's to et [see Equations (3.4)-(3.6)]. The VAR representation of Zt, obtained by suitably reordering the equations in (3.1), is characterized by DBiD ~, i = 1. . . . . q, and
DVD'. Iv Also, it is easily verified that (AoD') -t [(AoD')-l] ' = DVD', and that given the DBiD ~'s, AoD ~ characterizes the impulse response function of the Zt's to et. Moreover, these responses coincide with the responses of the corresponding variables in Zt to et. Note that AoD ~ is not in general lower triangular. Let -40 = AoDq
f40=
[
nil 0 0 ] ~2l ~22 0 , a31 a32 a33
where aii is full rank, but not necessarily lower triangular, for i = 1, 3. Let the QR decomposition of these matrices be ?tu = QiRi, where Qi is a square, orthonormal
16 The type of reasoning in the previous footnote indicates that permuting the columns of the identity matrix does not alter orthonormality. 17 To see this, simply premultiply Equation (3.1) by D on both sides and note that BiZ t i = BiOtDZt-i, because D~D = L
L . J Christiano et aL
82
matrix, and Ri is lower triangular with positive elements along the diagonal. This decomposition exists as long a s ~lii , i = 1, 3, is nonsingular, a property guaranteed by the fact Ao E Qv N Qr [see Strang (1976), p. 124]. 18 Let
W=
[100] 10 0 Q 3'
.
Note that WW ' = I, (W.40) -a [(W~]0)-~] ' = DVD', and WA0 is lower triangular with L
~
positive elements along the diagonal. Since (W.~0) -1 = Ao 1W I, the (kl + 1)th columns of~]o 1W / and .~o I coincide. We conclude that, under the normalization that A0 is lower diagonal with positive diagonal terms, the response o f the variables in Zt to a monetary policy shock is invariant to the ordering o f variables in Xlt and X2t. This establishes (iii). We now summarize these results in the form o f a proposition.
Proposition 4.1. Consider the sets Qv and Qr. (i) The set Qv N Qr is nonempty and contains more than one element. (ii) The (kl + 1)th column of 7i, i = O, 1, ... in Equation (3.6) is invariant to the choice of Ao E Qv n Qr. (iii) Restricting Ao E Qv n Qr to be lower triangular with positive diagonal terms, the (kl + 1)th column of 7i, i = O, 1, ... is invariant to the ordering of the elements in Xlt and X2t. We now provide a brief discussion of (i)-(iii). According to results (i) and (ii), under the recursiveness assumption the data are consistent with an entire family, Qz n QT, o f A0 matrices. It follows that the recursiveness assumption is not sufficient to pin down the dynamic response functions o f the variables in Zt to every element o f et. But, each Ao E Qv n QT does generate the same response to one o f the et's, namely the one corresponding to the monetary policy shock. In this sense, the recursiveness assumption identifies the dynamic response o f Zt to a monetary shock, but not the response to other shocks. In practice, computational convenience dictates the choice o f some Ao E Qv n Qr. A standard normalization adopted in the literature is that the A0 matrix is lower triangular with nonnegative diagonal terms. This still leaves open the question of how to order the variables in Xlt and X2t. But, according to result (iii), the dynamic response o f the variables in Zt to a monetary policy shock is invariant to this ordering. At
is Actually, it is customary to state the QR decomposition of the (n × n) matrix A as A = QR, where R is upper triangular. We get it into lower triangular form by constructing the orthonormal matrix E with zeros everywhere and 1's in the (n + 1 - i, i)th entries, i = 1 , 2 ..... n, and writing A = (QE) (E~R). The orthonormal matrix to which we refer in the text is actually QE.
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the same time, the dynamic impact on Zt of the nonpolicy shocks is sensitive to the ordering of the variables in Xlt and Xzt. The recursiveness assumption has nothing to say about this ordering. Absent further identifying restrictions, the nonpolicy shocks and the associated dynamic response functions simply reflect normalizations adopted for computational convenience.
4.2. Three benchmark identification schemes We organize our empirical discussion around three benchmark recursive identification schemes. These correspond to different specifications of St and g2t. In our first benchmark system, we measure the policy instrument, St, by the time t federal funds rate. This choice is motivated by institutional arguments in McCallum (1983), Bernanke and Blinder (1992) and Sims (1986, 1992). Let Yt, Pt, PCOMt, FFt, TRt, NBRt, and Mt denote the time t values of the log of real GDP, the log of the implicit GDP deflator, the smoothed change in an index of sensitive commodity prices (a component in the Bureau of Economic Analysis' index of leading indicators), the federal funds rate, the log of total reserves, the log of nonborrowed reserves plus extended credit, and the log of either M1 or M2, respectively. Here all data are quarterly. Our benchmark specification of g2t includes current and four lagged values of Y , Pt, and PCOMt, as well as four lagged values ofFFt, NBRt, TRt and Mr. We refer to the policy shock measure corresponding to this specification as an F F policy shock. In our second benchmark system we measure St by NBRt. This choice is motivated by arguments in Eichenbaum (1992) and Christiano and Eiehenbaum (1992) that innovations to nonborrowed reserves primarily reflect exogenous shocks to monetary policy, while innovations to broader monetary aggregates primarily reflect shocks to money demand. We assume that f2t includes current and four lagged values of Yt, Pt, and PCOMt, as well as four lagged values of FFt, NBRt, TRt and Mr. We refer to the policy shock measure corresponding to this specification as an NBR policy shock. Note that in both benchmark specifications, the monetary authority is assumed to s e e Yt, Pt and PCOMt, when choosing St. 19 This assumption is certainly arguable because quarterly real GDP data and the GDP deflator are typically known only with a delay. Still, the Fed does have at its disposal monthly data on aggregate employment, industrial output and other indicators of aggregate real economic activity. It also has substantial amounts of information regarding the price level. In our view the assumption that the Fed sees Yt and Pt when they choose St seems at least as plausible as assuming that they don't. 20 Below we document the effect of deviating from this benchmark assumption.
19 Examples of analyses which make this type of information assumption include Christiano and Eichenbaum (1992), Christiano et al. (1996a, 1997a), Eichenbaum and Evans (1995), Strongin (1995), Bernanke and Blinder (1992), Bernanke and Mihov (1995), and Gertler and Gilchfist (1994). 2o See for example the specifications in Sims and Zha (1998) and Leeper et al. (1996).
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Notice that under our assumptions, Yt, Pt and PCOMt do not change in the impact period o f either an F F or an NBR policy shock. Christiano et al. (1997b) present a dynamic stochastic general equilibrium model which is consistent with the notion that prices and output do not move appreciably in the impact period of a monetary policy shock. The assumption regarding PCOMt is more difficult to assess on theoretical grounds absent an explicit monetary general equilibrium model that incorporates a market for commodity prices. In any event, we show below that altering the benchmark specification to exclude the contemporaneous value o f PCOMt from g2t has virtually no effect on our results. 21 In the following subsection we display the time series of the two benchmark policy shock estimates. After that, we study the dynamic response of various economic time series to these shocks. At this point, we also consider our third benchmark system, a variant o f the NBR policy shocks associated with Strongin (1995). Finally, we consider the contribution o f different policy shock measures to the volatility o f various economic aggregates.
4.2.1. The benchmark policy shocks" displayed We begin by discussing some basic properties o f the estimated time series o f the F F and NBR policy shocks. These are obtained using quarterly data over the sample period 1965:3-1995:2. Figure 1 contains two time series of shocks. The dotted line depicts the quarterly F F policy shocks. The solid line depicts the contemporaneous changes in the federal funds rate implied by contractionary NBR policy shocks. In both cases the variable Mt was measured as M l t . Since the policy shock measures are by construction serially uncorrelated, they tend to be noisy. For ease o f interpretation we report the centered, three quarter moving average o f the shock, i.e., we report (eT+l + e7 + el_ 1)/3. Also, for convenience we include shaded regions, which begin at a National Bureau of Economic Research (NBER) business cycle peak, and end at a trough. The two shocks are positively correlated, with a correlation coefficient o f 0.5 I. The estimated standard deviation o f the F F policy shocks is 0.71, at an annual rate. The estimated standard deviation of the NBR is 1.53% and the standard deviation o f the implied federal funds rate shock is 0.39, at an annual rate. In describing our results, we find it useful to characterize monetary policy as "tight" or "contractionary", when the smoothed policy shock is positive, and "loose" or "expansionary" when it is negative. According to the F F policy shock measure, policy was relatively tight before each recession, and became easier around the time of the trough. 22 A similar pattern is observed for the movements in the federal funds rate
21 This does not mean that excluding lagged values from £2~ has no effect on our results. 22 In Figure 1, the beginning of the 1973~4 recession appears to be an exception to the general pattern. To some extent this reflects the effects of averaging since there was a 210 basis point FF policy shock in 1973Q3.
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Monetary Policy Shocks." What Have we Learned and to What End?
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implied by the NBR shocks, except that in the 1981-1982 period, policy was loose at the start, very tight in the middle, and loose at the end of the recession. 4.2.2. What happens after a benchmark policy shock? 4.2.2.1. Results for some major economic aggregates. Figure 2 displays the estimated impulse response functions of contractionary benchmark F F and NBR policy shocks on various economic aggregates included in g2t. These are depicted in columns 1 and 2, respectively. Column 3 reports the estimated impulse response functions from a third policy shock measure which we refer to as an NBR/TR policy shock. This shock measure was proposed by Strongin (1995) who argued that the demand for total reserves is completely interest inelastic in the short run, so that a monetary policy shock initially only rearranges the composition of total reserves between nonborrowed and borrowed reserves. Strongin argues that, after controlling for movements in certain variables that are in the Fed's information set, a policy shock should be measured as the
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Fig. 2. The estimated impulse response functions of contractionary benchmark FF and NBR policy shocks on various economic aggregates included in f2t (columns 1 and 2). Column 3 reports the estimated impulse response functions from a third policy shock measure which we refer to as an NBR/TR policy shock. The solid lines in the figure report the point estimates of the different dynamic response functions. Dashed lines denote a 95% confidence interval for the dynamic response functions.
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innovation to the ratio of nonborrowed to total reserves. We capture this specification by measuring St as N B R and assuming that g2t includes the current value of TR. With this specification, a shock to e[ does not induce a contemporaneous change in TR. All three identification schemes were implemented using M1 and M 2 as our measure of money. This choice turned out to have very little effect on the results. The results displayed in Figure 2 are based on a system that included M1. The last row of Figure 2 depicts the impulse response function of M 2 to the different policy shock measures, obtained by replacing M 1 with M 2 in our specification of £2t. The solid lines in the figure report the point estimates of the different dynamic response functions. Dashed lines denote a 95% confidence interval for the dynamic response functions. 23 The main consequences of a contractionary F F policy shock can be summarized as follows. First, there is a persistent rise in the federal funds rate and a persistent drop in nonborrowed reserves. This finding is consistent with the presence of a strong liquidity effect. Second, the fall in total reserves is negligible initially. But eventually total reserves fall by roughly 0.3 percent. So according to this policy shock measure, the Fed insulates total reserves in the short run from the full impact of a contraction in nonborrowed reserves by increasing borrowed reserves. 24 This is consistent with the arguments in Strongin (1995). Third, the response of M1 is qualitatively similar to the response o f TR. In contrast, for the M 2 system, the F F policy shock leads to an immediate and persistent drop in M2. Fourth, after a delay o f 2 quarters, there is a sustained decline in real GDE Notice the 'hump shaped' response function with the maximal decline occurring roughly a year to a year and a half after the policy shock. Fifth, after an initial delay, the policy shock generates a persistent decline in the index of commodity prices. The GDP deflator is flat for roughly a year and a half after which it declines.
23 These were computedusing a bootstrap Monte Carlo procedure. Specifically,we constructed 500 time r denote the vector of residuals from the estimated VAR. series on the vector Z t as follows. Let { t}t=l We constructed 500 sets of new time series of residuals, {~t(j)}r 1, j = 1.... ,500. The tth element of {~t(J)}T 1 was selected by drawing randomly,with replacement, from the set of fitted residual vectors, r For each {~t(j)}tr_l, we constructed a synthetic time series of Zt, denoted {Zt(.J)}Tl, using { *},=1. the estimated VAR and the historical initial conditions on Zt. We then re-estimated the VAR using {Zt(j)}tr_l and the historical initial conditions, and calculated the implied impulse response functions forj = 1, ..., 500. For each fixed lag, we calculated the 12th lowest and 487th highest values of the corresponding impulse response coefficients across all 500 synthetic impulse response functions. The boundaries of the confidenceintervals in the figures correspond to a graph of these coefficients.In many cases the point estimates of the impulse response functions are quite similar to the mean value of the simulated impulse response functions. But there is some evidence of bias, especially for Y, M2, NBR and FE The location of the solid lines inside the confidenceintervals indicates that the estimated impulse response functions are biased towards zero in each of these cases. See Killian (1998) and Parekh (1997) for different procedures for accommodatingthis bias. 24 A given percentage change in total reserves corresponds roughly to an equal dollar change in the total and nonborrowed reserves. Historically,nonborrowedreserves are roughly 95% of total reserves. Since 1986, that ratio has moved up, being above 98% most of the time.
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Before going on, it is of interest to relate these statistics to the interest elasticity of the demand for NBR and M1. Following Lucas (1988, 1994), suppose the demand for either of these two assets has the following form: Mt = fM(g2t) - q)FF, + el, where e/denotes the money demand disturbance and M denotes the log of either M1 or NBR. Here, q~ is the short run, semi-log elasticity of money demand. A consistent estimate of q~ is obtained by dividing the contemporaneous response of Mt to a unit policy shock by the contemporaneous response of FFt to a unit policy shock. This ratio is just the instrumental variables estimate of q~ using the monetary policy shock. The consistency of this estimator relies on the assumed orthogonality of e~' with eta and the elements of g2t. 2s Performing the necessary calculations using the results in the first column of Figure 2, we find that the short run money demand elasticities for M1 and NBR are roughly -0.1 and -1.0, respectively. The M1 demand elasticity is quite small, and contrasts sharply with estimates of the long run money demand elasticity. For example, the analogous number in Lucas (1988) is -8.0. Taken together, these results are consistent with the widespread view that the short run money demand elasticity is substantially smaller than the long run elasticity [see Goodfriend (1991)]. We next consider the effect of an NBR policy shock. As can be seen, with two exceptions, inference is qualitatively robust. The exceptions have to do with the impact effect of a policy shock on TR and M 1. According to the F F policy shock measure, total reserves are insulated, roughly one to one, contemporaneously from a monetary policy shock. According to the NBR policy shock measure, total reserves fall by roughly one half of a percent. Consistent with these results, an NBR policy shock leads to a substantially larger contemporaneous reduction in M1, compared to the reduction induced by an F F policy shock. Interestingly, M2 responds in very similar ways to an F F and an NBR policy shock.
25 To see this, note first the consistency of the instrumental variables estimator:
Cov(M. eD -qJ
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Note too that: Cov(Mt, el) : cpMa~, Cov(FFt, e;) : q)Rae2, where q~M and q~R denote the contemporaneous effects of a unit policy shock on Mt and FFt, respectively, and o 2 denotes the variance of the monetary policy shock. The result, that the instrumental variable estimator coincides with q~m/cpR, follows by taking the ratio o f the above two covariances. These results also hold if Mr, FFt, and g2t are nonstationary. In this case, we think of the analysis as being conditioned on the initial observations.
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From column 3 of Figure 2 we see that, aside from TR and M1, inference is also qualitatively similar to an N B R / T R policy shock. By construction TR does not respond in the impact period of a policy shock. While not constrained, M 1 also hardly responds in the impact period of the shock but then falls. In this sense the N B R / T R shock has effects that are more similar to an F F policy shock than an N B R policy shock. A maintained assumption of the N B R , F F and N B R / T R policy shock measures is that the aggregate price level and output are not affected in the impact period of a monetary policy shock. On a priori grounds, this assumption seems more reasonable for monthly rather than quarterly data. So it seems important to document the robustness of inference to working with monthly data. Indeed this robustness has been documented by various authors. 26 Figure 3 provides such evidence for the benchmark policy shocks. It is the analog of Figure 2 except that it is generated using monthly rather than quarterly data. In generating these results we replace aggregate output with nonfarm payroll employment and the aggregate price level is measured by the implicit deflator for personal consumption expenditures. Comparing Figures 2 and 3 we see that qualitative inference is quite robust to working with the monthly data. To summarize, all three policy shock measures imply that in response to a contractionary policy shock, the federal funds rate rises, monetary aggregates decline (although some with a delay), the aggregate price level initially responds very little, aggregate output falls, displaying a hump shaped pattern, and commodity prices fall. In the next subsection, we discuss other results regarding the effects of a monetary policy shock. We conclude this subsection by drawing attention to an interesting aspect of our results that is worth emphasizing. The correlations between our three policy shock measures are all less than one (see, for example, Figure 1). 27 Nevertheless, all three lead to similar inference about qualitative effects of a disturbance to monetary policy. One interpretation of these results is that all three policy shock measures are dominated by a common monetary policy shock. Since the bivariate correlations among the three are less than one, at least two must be confounded by nonpolicy shocks as well. Evidently, the effects of these other shocks is not strong enough to alter the qualitative characteristics of the impulse response functions. It is interesting to us just how low the correlation between the shock measures can be without changing the basic features of the impulse response functions. A similar set of observations emerges if we consider small perturbations to the auxiliary assumptions needed to implement a particular identification scheme. For example, suppose we implement the benchmark F F model in two ways: measuring Mt by the growth rate of M2 and by the tog of M1. The resulting policy shock measures 26 See for example Geweke and Runkle (1995), Bernanke and Mihov (1995) and Christiano et al. (1996b). 27 Recall, the estimated correlation between an FF and NBR shock is 0.51. The analog correlation between anNBR / TR shock and anFF shock is 0.65. Finally, the correlation between anNBR/TR shock and an NBR shock is 0.82.
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Fig. 3. Evidence for benchmark policy shocks. Analog of Figure 2, but using monthly rather than quarterly data. have a correlation coefficient o f only 0.85. This reflects in part that in several episodes the two shock measures give substantially different impressions about the state o f
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monetary policy. For example in 1993Q4, the M1 based shock measure implies a 20 basis point contractionary shock. The M2 growth rate based shock measure implies an 80 basis point contractionary shock. These types o f disagreements notwithstanding, both versions of the benchmark F F model give rise to essentially the same inference about the effect o f a given monetary policy shock. We infer from these results that while inference about the qualitative effects o f a monetary policy shock appears to be reliable, inference about the state o f monetary policy at any particular date is not. 4.3. Results f o r other economic aggregates
In the previous section we discussed the effects o f the benchmark policy shocks on various economic aggregates. The literature has provided a richer, more detaited picture of the way the economy responds to a monetary policy shock. In this section we discuss some o f the results that have been obtained using close variants o f the benchmark policy shocks. Rather than provide an exhaustive review, we highlight a sample o f the results and the associated set of issues that they have been used to address. The section is divided into two parts. The first subsection considers the effects o f a monetary policy shock on domestic US economic aggregates. In the second subsection, we discuss the effects o f a monetary policy shock on exchange rates. The papers we review use different sample periods as well as different identifying assumptions. Given space constraints, we refer the reader to the papers for these details. 4.3.1. US domestic aggregates
The work in this area can be organized into two categories. The first category pertains to the effects o f a monetary policy shock on different measures of real economic activity, as well as on wages and profits. The second category pertains to the effects of a monetary policy shock on the borrowing and lending activities o f different agents in the economy. 4.3.1.1. Aggregate real variables, wages and profits. In Section 4.2.2 we showed that aggregate output declines in response to contractionary benchmark F F and NBR policy shocks. Christiano et al. (1996a) consider the effects o f a contractionary monetary policy shock on various other quarterly measures o f economic activity. They find that after a contractionary benchmark F F policy shock, unemployment rises after a delay of about two quarters. 28 Other measures o f economic activity respond more quickly to the policy shock. Specifically, retail sales, corporate profits in retail trade
28 Working with monthly data Bernanke and Blinder (1992) also find that unemployment rises after a contractionary monetary policy shock. The shock measure which they use is related to our benchmark FF policy shock measure in the sense that both are based on innovations to the Federal Funds rate and both impose a version of the recursiveness assumption.
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and nonfinancial corporate profits immediately fall while manufacturing inventories immediately rise. 29 Fisher (1997) examines how different components of aggregate investment respond to a monetary policy shock [see also Bernanke and Gertler (1995)]. He does so using shock measures that are closely related to the benchmark F F and N B R policy measures. Fisher argues that all components of investment decline after a contractionary policy shock. But he finds important differences in the timing and sensitivity of different types of investment to a monetary policy shock. Specifically, residential investment exhibits the largest decline, followed by equipment, durables, and structures. In addition he finds a distinctive lead-lag pattern in the dynamic response functions: residential investment declines the most rapidly, reaching its peak response several quarters before the other variables do. Fisher uses these results to discuss the empirical plausibility of competing theories of investment. Gertler and Gilchrist (1994) emphasize a different aspect of the economy's response to a monetary policy shock: large and small manufacturing firms' sales and inventories. 3° According to Gertler and Gilchrist, small firms account for a disproportionate share of the decline in manufacturing sales that follows a contractionary monetary policy shock. In addition they argue that while small firms' inventories fall immediately after a contractionary policy shock, large firms' inventories initially rise before falling. They use these results, in conjunction with other results in their paper regarding the borrowing activities of large and small firms, to assess the plausibility of theories of the monetary transmission mechanism that stress the importance of credit market imperfections. Campbell (1997) studies a different aspect of how the manufacturing sector responds to a monetary policy shock: the response of total employment, job destruction and job creation. Using a variant of the benchmark F F policy shock measure, Campbell finds that, after a contractionary monetary policy shock, manufacturing employment falls immediately, with the maximal decline occurring roughly a year after the shock. The decline in employment primarily reflects increases in job destruction as the policy shock is associated with a sharp, persistent rise in job destruction but a smaller, transitory fall in job creation. Campbell argues that these results are useful as a guide in formulating models of cyclical industry dynamics. We conclude this subsection by discussing the effects of a contractionary monetary policy shock on real wages and profits. Christiano et al. (1997a) analyze various measures of aggregate real wages, manufacturing real wages, and real wages for ten 2 digit SIC level industries. In all cases, real wages decline after a contractionary benchmark F F policy shock, albeit by modest amounts. Manufacturing real wages
29 The qualitative results of Christiano et al. (1996a) are robust to whether they work with benchmark NBR, FF policy shocks or with Romer and Romer (1989) shocks.
3o Gert|er and Gilchrist (1994) use various monetarypolicy shock measures, including one that is related to the benchmarkFF policy shock as well as the onset of Romer and Romer (1989) episodes.
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fall more sharply than economy-wide measures. Within manufacturing, real wages fall more sharply in durable goods industries than in nondurable good industries. Christiano et al. (1997a) argue that these results cast doubt on models of the monetary transmission mechanism which stress the effects of nominal wage stickiness per se. This is because those types of models predict that real wages should rise, not fall, after a contractionary monetary policy shock. To study the response of real profits to a monetary policy shock, Christiano et al. (1997a) consider various measures of aggregate profits as well as before tax profits in five sectors of the economy: manufacturing, durables, nondurables, retail and transportation and utilities. In all but two cases, they find that a contractionary F F policy shock leads to a sharp persistent drop in profits. 31 Christiano et al. (1997a) argue that these results cast doubt on models of the monetary transmission mechanism which stress the effects of sticky prices per se but don't allow for labor market frictions whose effect is to inhibit cyclical movements in marginal costs. This is because those types of models predict that profits should rise, not fall, after a contractionary monetary policy shock. Finally, we note that other authors have obtained similar results to those cited above using policy shock measures that are not based on the recursiveness assumption. For example, policy shock measures based on the identifying assumptions in Sims and Zha (1998) lead to a qualitatively similar impact on wages, profits and various measures of aggregate output as the benchmark F F policy shock. Similarly, Leeper, Sims and Zha's (1996) results regarding the response of investment are quite similar to Fisher's. 4.3.1.2. Borrowing and lending activities. Various authors have investigated how a monetary policy shock affects borrowing and lending activities in different sectors of the economy. In an early contribution, Bernanke and Blinder (1992) examined the effects of a contractionary monetary policy shock on bank deposits, securities and loans. Their results can be summarized as follows. A contractionary monetary policy shock (measured using a variant of the benchmark F F policy shock) leads to an immediate, persistent decline in the volume of bank deposits as well as a decline in bank assets. The decline in assets initially reflects a fall in the amount of securities held by banks. Loans are hardly affected. Shortly thereafter security holdings begin climbing back to their preshock values while loans start to fall. Eventually, securities return to their pre-shock values and the entire decline in deposits is reflected in loans. Bernanke and Blinder (1992) argue that these results are consistent with theories of the monetary transmission mechanism that stress the role of credit market imperfections. Gertler and Gilchrist (t993, 1994) pursue this line of inquiry and argue that a monetary policy shock has different effects on credit flows to small borrowers (consumers and small firms) versus large borrowers. Using a variant of the benchmark F F policy
31 The two exceptions are nondarable goods and transportation and utilities. For these industries they cannot reject the hypothesis that profits are unaffected by contractionarypolicy shock.
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shock, they find that consumer and real estate loans fall after a contractionary policy shock but commercial and industrial loans do not [Gertler and Gilchrist (1993)]. In addition, loans to small manufacturing firms decline relative to large manufacturing firms after a contractionary monetary policy shock. In their view, these results support the view that credit market imperfections play an important role in the monetary transmission mechanism. Christiano et al. (1996a) examine how net borrowing by different sectors of the economy responds to a monetary policy shock. Using variants of the F F and NBR benchmark policy shocks, they find that after a contractionary shock to monetary policy, net funds raised in financial markets by the business sector increases for roughly a year. Thereafter, as the decline in output induced by the policy shock gains momentum, net funds raised by the business sector begin to fall. Christiano et al. (1996a) argue that this pattern is not captured by existing monetary business cycle models. 32 Christiano et al. (1996a) also find that net funds raised by the household sector remains unchanged for several quarters after a monetary policy shock. They argue that this response pattern is consistent with limited participation models of the type discussed in Christiano et al. (1997a,b). Finally, Christiano et al. (1996a) show that the initial increase in net funds raised by firms after a contractionary benchmark F F policy shock coincides with a temporary reduction in net funds raised (i.e., borrowing) by the government. This reduction can be traced to a temporary increase in personal tax receipts. After about a year, though, as output declines further and net funds raised by the business and household sectors falls, net funds raised by the government sector increases (i.e., the government budget deficit goes up). Taken together, the above results indicate that a contractionary monetary policy shock has differential effects on the borrowing and lending activities of different agents in the economy. Consistent with the version of the Lucas program outlined in the introduction to this survey, these findings have been used to help assess the empirical plausibility of competing theories of the monetary transmission mechanism. 4.3.2. Exchange rates and monetary policy shocks
Various papers have examined the effects of a monetary policy shock on exchange rates. Identifying exogenous monetary policy shocks in an open economy can lead to substantial complications relative to the closed economy case. For example, in some countries, monetary policy may not only respond to the state of the domestic economy but also to the state of foreign economies, including foreign monetary policy actions. At least for the USA, close variants of the benchmark policy shock measures continue to give reasonable results. For example, Eichenbaum and Evans (1995) consider variants of the benchmark F F and NBR/TR policy shock measures in which some
32 Christiano et al. (1996a) and Gertler and Gilchrist (1994) discuss possible ways to account for this response pattern.
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foreign variables appear in the Fed's reaction ftmction. A maintained assumption of their analysis is that the Fed does not respond contemporaneously to movements in the foreign interest rate or the exchange rate. Eichenbaum and Evans use their policy shock measures to study the effects of a contractionary US monetary policy shock on real and nominal exchange rates as well as domestic and foreign interest rates. 33 They find that a contractionary shock to US monetary policy leads to (i) persistent, significant appreciations in US nominal and real exchange rates and (ii) persistent decreases in the spread between foreign and US interest rates, and (iii) significant, persistent deviations from uncovered interest rate parity in favor of US investments. 34 Under uncovered interest rate parity, the larger interest rate differential induced by a contractionary US monetary policy shock should be offset by expected future depreciations in the dollar. Eichenbaum and Evans' empirical results indicate that the opposite is true: the larger return is actually magnified by expected future appreciations in the dollar. Eichenbaum and Evans discuss the plausibility of alternative international business cycle models in light of their results. While variants of the benchmark F F identification scheme generate results that are consistent with traditional monetary analyses when applied to the USA, this is generally not the case when they are used to identify foreign monetary policy shocks. For example, Grilli and Roubini (1995) consider policy shock measures for non-US G7 countries that are closely related to Eichenbaum and Evans' measures. Using these measures, they find that a contractionary shock to a foreign country's monetary policy leads initially to a depreciation in the foreign country's currency. Grilli and Roubini argue that this result reflects that the measured policy shocks are confounded by the systematic reaction of foreign monetary policy to US monetary policy and expected inflation. This motivates them to construct an alternative policy shock measure which is based on the recursiveness assumption and a measure of St equal to the spread between foreign short term and long term interest rates. With this measure, they find that a contractionary shock to foreign monetary policy leads to a transitory appreciation in the foreign exchange rate and a temporary fall in output. In contrast to Grilli and Roubini, authors like Cushman and Zha (1997), Kim and Roubini (1995), and Ctarida and Gertler (1997) adopt identification schemes that do not employ the recursiveness assumption. In particular, they abandon the assumption that the foreign monetary policy authority only looks at predetermined variables when setting its policy instrument. Cushman and Zha (1997) assume that Bank of Canada officials look at contemporaneous values of the Canadian money supply, the exchange rate, the US foreign interest rate and an index of world commodity prices when setting a short term Canadian interest rate. Kim and Roubini (1995) assume that the reaction
33 The foreign countries which they look at are Japan, Germany,Italy, France and Great Britain. 34 Sims (1992) and Grilli and Roubini (1995) also analyze the effect of a monetary policy shock on US exchange rates using close variants of the FF benchmark policy shock. They too fred that a contractionary policy shock leads to an appreciation of the US exchange rate.
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function o f foreign central bankers includes contemporaneous values o f the money supply, the exchange rate and the world price o f oil (but not the federal fund rate). Clarida and Gertler (1997) assume that the Bundesbank's reaction function includes current values o f an index o f world commodity prices, the exchange rate, as well as the German money supply (but not the US federal funds rate). 35 In all three cases, it is assumed that the money supply and the exchange rate are not predetermined relative to the policy shock. As a consequence, monetary policy shocks cannot be recovered from an ordinary least squares regression. Further identifying assumptions are necessary to proceed. The precise identifying assumptions which these authors make differ. But in all cases, they assume the existence o f a group o f variables that are predetermined relative to the policy shock. 36 These variables constitute valid instruments for estimating the parameters in the foreign monetary policy maker's reaction function. We refer the reader to the papers for details regarding the exact identifying assumptions. 37 With their preferred policy shocks measures, all three o f the above papers find that a contractionary foreign monetary policy shock causes foreign exchange rates to appreciate and leads to a rise in the differential between the foreign and domestic interest rate. 38 In this sense, their results are consistent with Eichenbaum and Evans' evidence regarding the effects o f a shock to monetary policy. In addition, all three papers provide evidence that a contractionary foreign monetary policy shock drives foreign monetary aggregates and output down, interest rates up and affects the foreign price level only with a delay. In this sense, the evidence is consistent with the evidence in Section 4.2.2 regarding the effect o f a benchmark F F policy shock on the US economy. 4.4. Robustness o f the benchmark analysis In this subsection we assess the robustness o f our benchmark results to various perturbations. First, we consider alternative identification schemes which also impose the recursiveness assumption. Second, we consider the effects o f incorporating information from the federal funds futures market into the analysis. Finally, we analyze the subsample stability o f our results.
35 Clarida et al. (1998) provide a different characterization of the Bundesbank's reaction function as well as the reaction functions of five other central banks. 36 For example in all these cases it is assumed that a measure of commodity prices, foreign industrial production, the foreign price level and the federal funds rate are predetermined relative to the foreign monetary policy shock. 37 Clarida and Gall (1994) use long run identifying restrictions to assess the effects of nominal shocks on real exchange rates. 38 Consistent with the evidence in Eichenbaum and Evans (1995), Cushman and Zha (1997) find that a contractionary foreign monetary policy shock induces a persistent, significant deviation from uncovered interest parity in favor of foreign investments.
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4.4.1. Excluding current output and prices from f2t The estimated one-step-ahead forecast errors in Yt and FFt are positively correlated (.38), while those in Yt and NBRt are negatively correlated (-.22). Any identification scheme in which St is set equal to either the time t federal funds rate or nonborrowed reserves must come to terms with the direction of causation underlying this correlation: Does it reflect (a) the endogenous response of policy to real GDP via the Fed's feedback rule, or (b) the response of real GDP to policy? Our benchmark policy measures are based on the assumption that the answer to this question is (a). Under this assumption we found that a contractionary monetary policy shock drives aggregate out-put down. Figure 4 displays the results when the answer is assumed to be (b). Specifically, columns 1 and 3 report the estimated impulse response functions of various economic aggregates to policy shock measures that were computed under the same identification assumptions as those underlying the F F and NBR policy shocks except that Yt is excluded from f2t. The key result is that under these identifying assumptions, a contractionary policy shock drives aggregate output up before driving it down. In other respects, the results are unaffected. It might be thought that the initial response pattern of output could be rationalized by monetary models which stress the effects of an inflation tax on economic activity, as in Cooley and Hansen (1989). It is true that in these models a serially correlated decrease in the money supply leads to an increase in output. But, in these models this happens via a reduction in anticipated inflation and in the interest rate. Although the candidate policy shock is associated with a serially correlated decrease in the money supply, it is also associated with a rise in the interest rate and virtually no movement in the price level. This response pattern is clearly at variance with models in which the key effects of monetary policy shocks are those associated with the inflation tax. We do not know of other models which can rationalize a rise in output after a contractionary monetary policy shock. Absent some coherent model that can account for the response functions in columns 1 and 3 of Figure 4, we reject the underlying identifying assumptions as being implausible. We suspect that the resulting shock measures confound policy and nonpolicy disturbances. Columns 2 and 4 of Figure 4 report the estimated impulse response functions to policy shock measures computed under the same identification assumptions as those underlying the F F and NBR policy shocks except that Pt is excluded from f2t. As can be seen, the benchmark results are virtually unaffected by this perturbation. 4.4.2. Excluding commodity prices from g-2t: The price puzzle On several occasions in the postwar era, a rise in inflation was preceded by a rise in the federal funds rate and in commodity prices. An example is the oil price shock in 1974. Recursive identification schemes that set St equal to FFt and do not include the commodity prices in f2t as leading indicators of inflation in the Fed's feedback rule sometimes imply that contractionary monetary policy shocks lead to a sustained rise in
98
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Fed F u n d s Model, Y after MP
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Ch. 2: Monetary Policy Shocks: What Have we Learned and to What End?
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the price level. 39 Eichenbaum (1992) viewed this implication as sufficiently anomalous relative to standard theory to justify referring to it as "the price puzzle". 4o Sims (1992) conjectured that prices appeared to rise after certain measures o f a contractionary policy shock because those measures were based on specifications o f g2t that did not include information about future inflation that was available to the Fed. Put differently, the conjecture is that policy shocks which are associated with substantial price puzzles are actually confounded with nonpolicy disturbances that signal future increases in prices. Christiano et al. (1996a) and Sims and Zha (1998) show that when one modifies such shock measures by including current and lagged values o f commodity prices in Dr, the price puzzle often disappears. It has now become standard practice to work with policy shock measures that do not generate a price puzzle. To document both the nature o f the puzzle and the resolution, Figure 5 displays the impulse response o f Pt to eight different contractionary monetary policy shock measures. The top and bottom rows display the effects o f shocks to systems in which St is measured by FFt and NBRt, respectively. Columns 1-4 correspond to policy shock measures in which (i) the current value o f Pt, Yt and current and lagged values o f P C O M t are omitted from g2t, (ii) current and lagged values o f P C O M t are omitted from £2¢, (iii) the current value o f P C O M t is omitted from £2t, and (iv) g2t is given by our benchmark specification, respectively. A number o f interesting results emerge here. First, policy shock measures based on specifications in which current and lagged values o f P C O M are omitted from g2t imply a rise in the price level that lasts several years after a contractionary policy shock. Second, according to the point estimates, the price puzzle is particularly pronounced for the specification in which the current values o f Yt and Pt are also excluded from g2t (column 1). Recall that deleting Pt from (2t had virtually no effect on our results. These findings suggest that current Y and current and past P C O M play a similar role in purging policy shock measures of nonpolicy disturbances. Third, the 95% confidence intervals displayed in Figure 5 indicate that the price puzzle is statistically significant for the Fed Funds based shock measures associated with columns 1 and 2 in Figure 5.41
39 The first paper that documents the "price puzzle" for the USA and several other countries appears to be Sims (1992). 4o There do exist some models that predict a temporary rise in the price level after a contraction. These models stress the role of self fulfilling shocks to expectations in the monetary transmission mechanism. See for example Beaudry and Devereux (1995). Also there exist some limited participation models of the monetary transmission mechanism in which the impact effect of contractionary monetary policy shocks is so strong that prices rise in the impact period of the policy shock. See for example Fuerst (1992) and Christiano et al. (1997a). 4~ We used the artificial data underlying the confidenceintervals reported in Figure 5 to obtain a different test of the price puzzle. In particular, we computed the number of times that the average price response over the first 2, 4 and 6 quarters was positive. For the FF model underlying the results in column 1 the results were 96.4%, 97.2%, and 98.0%, respectively. Thus, at each horizon, the price puzzle is significant at the 5% significance level. For the FF model underlying the second column, the results are 95.6%, 94.6%, and 89.8%, so that there is a marginally significant price puzzle over the first year. Regardless
100 0.70
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Fig. 5. The impulse response of Pt to eight different contractionary monetary policy shock measures. The top and bottom rows display the effects of shocks to systems in which St is measured by F F t and NBRt, respectively. Columns 1-4 correspond to policy shock measures in which (i) the current value of Pt, Yt and current and lagged values o f P C O M t are omitted from g2t, (ii) current and lagged values of P C O M t are omitted from E2t, (iii) the current value of P C O M t is omitted from g2,, and (iv) ~2t is given by our benchmark specification, respectively. Fourth, consistent with results in Eichenbaum (1992), the price p u z z l e is less severe for the N B R based policy shocks. Fifth, little evidence o f a price puzzle exists for the benchmark F F and N B R policy shocks. We conclude this section by noting that, in results not reported here, we found that the dynamic responses o f nonprice variables to monetary policy shocks are robust to deleting current and lagged values o f P C O M from £2t.
4.4.3. Equating the policy instrument, St, with MO, M1 or M 2 There is a long tradition o f identifying monetary policy shocks with statistical innovations to monetary aggregates like the base (M0), M 1 and M2. Indeed this was
of the horizon, the price puzzle was not significant at even the 10% significance level for the other specifications in Figure 5.
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the standard practice in the early literature on the output and interest rate effects of an unanticipated shock to monetary policy. 42 This practice can be thought of as setting St equal to a monetary aggregate like M0, M1 or M2 and using a particular specification of Y2t. We refer the reader to Leeper, Sims and Zha (1996) and Cochrane (1994) for critical reviews of this literature. Here we discuss the plausibility of identification schemes underlying M based policy shock measures by examining the implied response functions to various economic aggregates. Figure 6 reports estimated response functions corresponding to six policy measures. Columns 1 and 2 pertain to policy shock measures in which St is set equal to M0t. Column 1 is generated assuming that g2t consists of 4 lagged values of Yt, Pt, PCOMt, FFt, NBRt and M0t. For column 2, we add the current value of Yt, Pt, and PCOMt to g2t. Columns 3 and 4 are the analogs of columns 1 and 2 except that M0t is replaced by Mlt. Columns 5 and 6 are the analogs of columns 1 and 2 except that MOt is replaced by M2. We begin by discussing the dynamic response functions corresponding to the M0 based policy shock measures. Notice that the responses in column 1 are small and estimated very imprecisely. Indeed, it would be difficult to reject the hypotheses that Y, P, PCOM, and F F are all unaffected by the policy shock. Once we take sampling uncertainty into account, it is hard to argue that these response functions are inconsistent with the benchmark policy shock measure based response functions. In this limited sense, inference is robust. Still, the point estimates of the response functions are quite different from our benchmark results. In particular, they indicate that a contractionary policy shock drives Pt and FFt down. The fall in Pt translates into a modest decline in the rate of inflation. 43 After a delay of one or two periods, Zt rises by a small amount. The delay aside, this response pattern is consistent with a simple neoclassical monetary model of the sort in which there is an inflation tax effect on aggregate output [see for example Cooley and Hansen (1989)]. The response functions in column 2 are quite similar to those in column 1. As before, they are estimated with sufficient imprecision that they can be reconciled with various models. The point estimates themselves are consistent with simple neoclassical monetary models. Compared to column 1, the initial decline in Yt after a contractionary policy shock is eliminated, so that the results are easier to reconcile with a simple neoclassical monetary model. The impulse response fimctions associated with the M1 based policy shocks in columns 3 and 4 are similar to those reported in columns t and 2, especially when sampling uncertainty is taken into account. The point estimates themselves seem harder to reconcile with a simple monetary neoclassical model. For example, according to 42 See for example Barro (1977), Mishkin (1983), S. King (1983) and Reichenstein (1987). For more recent work in this tradition see King (1991) and Cochrane (1994). 43 The fall in P~ translates into an initial .20 percent decline in the annual inflation rate. The maximal decline in the inflation rate is about .25 percent which occurs after 3 periods. The inflation rate returns to its preshock level after two years.
L.J. C h r i s t i a n o et al.
102 Money Model: MO first
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I
Ch. 2:
103
Monetary Policy Shocks." What Have we Learned and to What End? Money Model
Money Model: M2 first
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104
L.J. Christiano et al.
column 3, output falls for over two quarters after a contractionary policy shock. The fact that output eventually rises seems difficult to reconcile with limited participation or sticky wage/price models. This is also true for the results displayed in column 4. Moreover, the results in that column also appear to be difficult to reconcile with the neoclassical monetary model. For example, initially inflation is hardly affected by a monetary contraction, after which it actually rises. Sampling uncertainty aside, we conclude that the M I based policy shock measures are difficult to reconcile with known (at least to us) models of the monetary transmission mechanism. Finally, consider the M2 based policy shock measures. Here a number of interesting results emerge. First, the impulse response functions are estimated more precisely than those associated with the M0 and M1 based policy shock measures. Second, the impulse response functions share many of the qualitative properties of those associated with the benchmark policy shocks measures. In particular, according to both columns 5 and 6, a contractionary monetary policy shock generates a prolonged decline in output and a rise in FFt. Also the price level hardly changes for roughly 3 quarters. This is true even for the policy shock measure underlying column 5 where the price level is free to change in the impact period of the shock. There is one potentially important anomaly associated with the M2 based policy shock measures: after a delay, NBR and M2 move in opposite directions. In sum, the M based policy shock measures provide mixed evidence on the robustness of the findings associated with our benchmark policy shocks. The response functions associated with the M0 and M1 policy shock measures are estimated quite imprecisely. In this sense they do not provide evidence against robustness. The point estimates of the response functions associated with the M1 based policy shock measures are hard to reconcile with existing models of the monetary transmission mechanism. But the point estimates associated with the M0 based policy shock measures are consistent with simple neoclassical monetary models. I f one wants evidence that is not inconsistent with simple neoclassical monetary models, this is where to look. Finally, apart from the anomalous response of NBR, qualitative inference about the effects of a monetary policy shock are robust to whether we work with the M2 based policy shock measure or the benchmark policy shock measures.
4.4.4. Using information from the federal funds futures market An important concern regarding the benchmark policy shock measures is that they may be based on a smaller information set than the one available to the monetary authority or private agents. Rudebusch (1996) notes that one can construct a marketbased measure of the one-month ahead unanticipated component of the federal fimds rate. He does so using data from the federal funds futures market, which has been active since late 1988. 44 He recognizes that a component of the unanticipated move in the 44 See Brunner (1994), Carlson et al. (1995), and Krueger and Kuttner (1996) for further discussion and analysis of the federal funds futures market.
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federal funds rate reflects the Federal Reserve's endogenous response to the economy. To deal with this problem, he measures the exogenous shock to monetary policy as the part o f the unanticipated component of the federal funds rate which is orthogonal to a measure o f news about employment. In Rudebusch's view, the correlation between the resulting measure and our F F benchmark policy shock measure is sufficiently low to cast doubt upon the latter. 45 But policy shock measures can display a low correlation, while not changing inference about the economic effects of monetary policy shocks. We now investigate whether and how inference is affected by incorporating federal funds futures market data into the analysis. To study this question, we repeated the benchmark F F analysis, replacing FFt with F F t - F M t - 1 in the underlying monthly VAR. 46 Here FMt_I denotes the time t - 1 futures rate for the average federal funds rate during time t. 47 We refer to the orthogonalized disturbance in the F F t - FM¢ I equation as the F M policy shock. In addition, because o f data limitations, we redid the analysis for what we refer to as the Rudebusch sample period, 1989:04-1995:03. Because of the short sample period, we limit the number o f lags in the VAR to six. Before considering impulse response functions to the policy shocks, we briefly discuss the shocks themselves. Panel A of Figure 7 displays the F M policy shocks for the period 1989:10-1995:03. In addition, we display F F policy shocks for the same period. These were computed using our benchmark, monthly VAR model, estimated over the whole sample period, using six lags in the VAR. Panel B is the same as Panel A, except that the VAR underlying the benchmark F F policy shocks is estimated using data only over the Rudebusch sample period. A few features o f Figure 7 are worth noting. First, the shock measures in Panel A are o f roughly similar magnitude, with a standard deviation o f the benchmark and F M policy shocks being 0.22 and 0.16, respectively. Consistent with the type o f findings reported by Rudebusch, the correlation between the two shock measures is relatively low, 0.34. 48 Second, when we estimate the VARs underlying the benchmark F F and F M policy shocks over the same sample period, the correlations rise to approximately 0.45. Interestingly, the F F policy shocks now have a smaller standard deviation than the F M policy shocks. 49 We now proceed to consider robustness o f inference regarding the effects o f monetary policy shocks. The dynamic response functions to an F M policy shock, together with 95% confidence intervals, are displayed in column 1 of Figure 8. There
45 See Sims (1996) for a critique of Rudebusch's analysis. 46 Evans and Kuttner (1998) find that small, statistically insignificant deviations from futures market efficiencypartially account for the low correlations between variants of the FF benchmark policy shocks and FF~ - F Mt 1. 47 These data were taken from Krueger and Kuttner (1996). 48 Rudebusch actually reports the R2 in the regression relation between the two shocks. This is the square of the correlation between the two variables. So, our correlation translates into an R2 of 0.12. 49 Given the short sample, it is important to emphasize that the standard deviations have been adjusted for degrees of freedom.
L.d Christiano et al.
106
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are two obvious features to these results. First, the policy shock itself is very small (a little over 10 basis points). Second, with the exception o f F F L - FMt-1, the response o f the other variables is not significantly different from zero at all lags. To compare these results with those based on the benchmark F F policy shocks, we need to control for the difference in sample periods and lag lengths. To this end, we report the impulse response functions and standard errors o f the 6 lag benchmark F F model estimated over the Rudebusch sample period. These are displayed in column 2 o f Figure 8. We see that the same basic message emerges here as in column 1: over the Rudebusch sample period, the shocks are small and the impulse response functions are imprecisely estimated. We conclude that there is no evidence to support
Ch. 2:
Monetary Policy Shocks." What Haue we Learned and to What End?
Short Sample
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!
3~
Fig. 8. The dynamic response functions to an FM policy shock, together with 95% confidence intervals, are displayed in column 1. There are two obvious features to these results. First, the policy shock itself is very small (a little over 10 basis points). Second, with the exception o f F F t -FM~ 4, the response of the other variables is not significantly different from zero at all lags.
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the notion that inference is sensitive to incorporating federal funds market data into the analysis. This conclusion may very well reflect the limited data available for making the comparison. 4.4.5. Sample period sensitioity
Comparing the results in Figure 8 with our full sample, benchmark F F results (see column 1, Figure 2) reveals that the impulse response functions are much smaller in the Rudebusch sample period. A similar phenomenon arises in connection with our benchmark NBR model. Pagan and Robertson (1995) characterize this phenomenon as the "vanishing liquidity effect". Wong (1996) also documents this phenomenon for various schemes based on the recursiveness assumption. These findings help motivate the need to study the robustness o f inference to different sample periods. We now proceed to investigate subsample stability. Our discussion is centered around two general questions. First, what underlies the difference in impulse response functions across subsamples? Here, we distinguish between two possibilities. One possibility is that the difference reflects a change in the size o f the typical monetary policy shock. The other possibility is that it reflects a change in the dynamic response to a shock o f a given magnitude. We will argue that, consistent with the findings in Christiano's (1995) discussion o f the vanishing liquidity effect, the evidence is consistent with the hypothesis that the first consideration dominates. Second, we discuss robustness o f qualitative inference. Not surprisingly in view o f our findings regarding the first question, we find that qualitative inference about the effects o f a monetary policy shock is robust across subsamples. This last finding is consistent with results in Christiano et al. (1996b). In the analysis that follows, we focus primarily on results for the benchmark F F policy shocks. We then briefly show that our conclusions are robust to working with the NBR policy shocks. To begin our analysis o f subsample stability, we test the null hypothesis that there was no change at all in the data generating mechanism for the Rudebusch sample period. To this end, we constructed confidence intervals for the impulse response functions in column 2 o f Figure 8 under the null hypothesis that the true model is the one estimated using data over the full sample. 50 The resulting confidence intervals are reported in column 3. In addition, that column reports for convenience the estimated response functions from column 2. We see that the estimated impact effect o f a one standard deviation policy shock on the federal funds rate (see the
so These confidence intervals were computed using a variant of the standard bootstrap methodology employed in this paper. In particular, we generated 500 artificial time series, each of length equal to that of the full sample, using the six lag, benchmark full sample FF VAR and its fitted disturbances. In each artificial time series we estimated a six lag, benchmark FF VAR model using artificial data over the period corresponding to the Rudebusch sample period. The 95% confidence intervals are based on the impulse response functions corresponding to the VARs estimated from the artificial data.
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fourth row of column 3) lies well below the 95% confidence interval. So, we reject the null hypothesis that there was no change at all in the data generating mechanism in the Rudebusch sample. 51 Next, we modified the null hypothesis to accommodate the notion that the only thing which changed in the Rudebusch sample was the nature of the monetary policy shocks. In all other respects, the data generating mechanism is assumed to remain unchanged. Under this null hypothesis, we generated 95% confidence intervals for the estimated impulse response functions in column 2 of Figure 8. s2 These confidence intervals are reported in column 4 of Figure 8, which also repeats for convenience the point estimates from column 2. Notice that, with one exception, all of the estimated impulse response functions lie within the plotted confidence intervals. 53 The exception is that the impulse response function o f P C O M lies just outside the plotted confidence intervals for roughly the first six periods. Based on these results, we conclude that there is little evidence against the joint hypothesis that (i) the response of the aggregates to a given policy shock is the same in the two sample periods and (ii) the size of the shocks was smaller in the post 1988:10 period. For any particular subsample, we refer to these two conditions as the modified subsample stability hypothesis. We now consider the stability of impulse response functions in other subsamples. Figure 9 reports response functions to monthly benchmark F F policy shocks, estimated over four subsamples: the benchmark sample, and the periods 1965:1-1979:9, 1979:10-1994:12, and 1984:2-1994:12. In each case, the method for computing confidence intervals is analogous to the one underlying the results in column 4 of Figure 8. 54 From Figure 9 we see that the estimated response functions for 51 The procedure we have used to reject the null hypothesis of no change versus the alternative of a change in 1989 implicitly assumes the choice of break date is exogenous with respect to the stochastic properties of the data. There is a large literature (see Christiano (1992) and the other papers in that Journal of Business and Economic Statistics volume) which discusses the pitfalls of inference about break dates when the choice of date is endogenous. In this instance our choice was determined by the opening of the Federal Funds Futures Market. Presumably,this date can be viewed as exogenous for the purpose of our test. 52 With one exception, these confidence intervals were computed using the procedure described in the previous footnote. The exception has to do with the way the shocks were handled. In particular, the artificial data were generated by randomly sampling from the orthogonalized shocks, rather than the estimated VAR disturbances. Residuals other than the policy shocks were drawn, with replacement, from the full sample period set of residuals. The policy shocks were drawn from two sets. Shocks for periods prior to the analog of the Rudebusch sample period were drawn, with replacement, from the pre-Rudebusch sample fitted policy shocks. Shocks for periods during the analog of the Rudebusch sample period were drawn, with replacement, from the Rudebusch sample fitted policy shocks. 53 In this manuscript, we have adopted the extreme assumption that the stochastic properties of the policy shock changed abruptly on particular dates. An alternative is that the changes occur smoothly in the manner captured by an ARCH specification for the policy shocks. Parekh (1997) pursues this interpretation. He modifies our bootstrap procedures to accommodate ARCH behavior in the shocks. 54 That is, they are computed under the assumption that the data generating mechanism is the six lag, full sample estimated VAR with policy shocks drawn only from the relevant subsample.All other shocks are drawn randomly from the full sample of fitted shocks.
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Fed Funds Model 65:01 to 94:12
Fed Funds Model 65:01 to 79:09
Fed Funds Model 79:10 to 94:12
Fed Funds Model 84:02 to 94:12
MP Shock => EM
MP Shock => EM
MP Shock => EM
MP Shock => EM
MP Shock => Price
MP Shock => Price
MP Shock => Price
MP Shock => Pcom
MP Shock => Pcom
MP Shock
ii ................................. J
MP Shock => Pcom
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
::2:t
.....................
MP Shock => FF
MP Shock => FF
e\
/\
=> Pcom
MP Shock => FF
~\
= L , " .........................................r MP Shock => NBR
MP Shock => NBR
MP Shock=> NBR
MP Shock => NBR
MP Shock => TR
MP Shock => TR
MP Shock => TR
MP Shock = > M1
MP Shock => M1
¢ MP Shock => TR
iil
::J
I
..........
g
17
25
MP Shock => M1
.
.
.
1
.
.
.
.
.
5
~
~1
MP Shock => M1
2', .....: T T ? . : . : : , : . : 2 L ~
"
~ 0,
..........................
Fig. 9. Response functions to monthly benchmark F F policy shocks, estimated over four subsamples: the bencl~mark sample, and the periods 1965:1-1979:9, 1979:10-1994:12, and 1984:2-1994:12.
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employment, P, P C O M , and M1 almost always lie within the confidence intervals. For the third and fourth sample periods there is no evidence against the modified subsample stability hypothesis. There is some marginal evidence against the hypothesis in the first subsample. In particular, the P C O M and price level responses lie outside the plotted confidence interval at some horizons. We find these results somewhat troubling, since they may indicate that the benchmark F F policy shocks are contaminated by other shocks to which the Fed responds. Despite this, the overall impression one gets from these results is that the modified subsample stability hypothesis is not rejected for the benchmark F F policy shocks. At the same time, there is strong evidence that the variance of the policy shocks changed over the sample. One interpretation is that the early 1980s were a period in which policy shocks were very large, but that the shocks were of comparable magnitude and substantially smaller size throughout the rest of the post-war period. One bit of evidence in favor of this view is that the estimated policy shocks in the second and fourth sample periods are reasonably similar in size, 20 basis points versus 12 basis points, respectively. We now briefly point out that qualitative inference is robust across subsamples. For each subsample we find evidence consistent with a liquidity effect. Specifically, a policy-induced rise in the federal funds rate is associated with a decline in nonborrowed reserves, total reserves and M1. In addition, the contractionary policy shock is associated with a delayed response of employment and a very small change in the price level. We now consider the results for the benchmark N B R policy shocks, reported in Figure 10. The overall impression conveyed here is similar to what we saw in Figure 9. There is relatively little evidence against the modified subsample sensitivity hypothesis. For the most part, the point estimates all tie within the plotted confidence intervals. Note that the impulse response functions are qualitatively robust across subsamples. We now turn to a complementary way of assessing subsample stability, which focuses on the magnitude of the liquidity effect. Panels A and B of Table 1 report summary statistics on the initial liquidity effect associated with the benchmark F F and N B R identification schemes, respectively. In that table, F F / N B R denotes the average of the first three responses in the federal funds rate, divided by the average of the first three responses in nonborrowed reserves. These responses are taken from the appropriate entries in Figure 9. As a result, F F / N B R denotes the percentage point change in the federal funds rate resulting from a policy-induced one percent change in NBR. F F / M 1 denotes the corresponding statistic with the policy-induced change in M1 in the denominator. Because of the shape of the impulse response function in M1, we chose to calculate this statistic by averaging the first six responses in F F and M1. The statistics are reported for the four sample periods considered in Figure 9. In addition, the 95% confidence intervals are computed using the appropriately modified version of the bootstrap methodology used to compute confidence intervals in Figure 9. Panel B is the exact analog of Panel A, except that the results are based on the N B R policy shocks.
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N B R M o d e l 85:01 to 9 4 : 1 2
N B R M o d e l 85:01 to 7 9 : 0 9
N B R M o d e l 7 9 : 1 0 to 9 4 : 1 2
N B R M o d e l 8 4 : 0 2 to 9 4 : 1 2
MP Shock => EM
MP Shock => EM
MP Shock => EM
MP Shock => EM
MP Shock => Price
MP S h o c k - > Price
MP S h o c k => Price
MP Shock
MP Shock - > Pcom
MP Shock - > Poem
ii!! .
MP Shock => Pcom
.
.
.
~5
~
41
>Pcom
11,t1
MP Shock
>FF
~__~ ~ ~ . ~
MP Shock => FF
MP S h o c k - > FF
MP Shock => g g
MP Shock => NBR
MP Shock - > NBR
MP Shook => NBR
MP Shock - > NBR
MP Shock => TR
MP Shock => TR
MP Shock => TR
2~............................................ MP Shock => M1
~ 23 .............................
i
MP Shock => M1
..... L ~
J MP Shock => M1
i
Fig. 10. Results for the benchmark NBR policy shocks.
MP Shock => M1
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113
Table 1 The liquidity effect, sample period sensitivity Subsample
FF/NBR
FF/M 1
-0.94 (-1.30, -0.73) -0.70 ( 2.64,-0.55) 0.71 (-1.95,-0.64) -0.69 (-5.52, 1.86)
-2.17 (-3.61, -1.36) -1.88 (-8.32,-0.72) -1.13 ( 4.48, 0.82) -0.97 (-13.92, 13.39)
-0.23 ( 0.29, 0.17) -0.07 (-0.36, -0.11) -0.27 ( 0.35, 0.13) -0.13 (-0.45,-0.04)
-1.14 ( 2.10,-0.59) -2.08 (-4.86, -0.14) -0.37 (-3.56,-0.15) -0.47 (-10.12, 5.35)
Panel A: FF Policy Shock
65:01~4:12 65:01-79:09 79:10-94:12 84:02-94:12 Panel B: NBR Policy Shocks
65:01-94:12 65:01-79:09 79:1044:12 84:02-94:12
We begin our discussion by reviewing the results in panel A. The full sample results indicate that a one percent policy-shock induced increase in nonborrowed reserves results in roughly a one percentage point reduction in the federal funds rate. A one percent policy-shock induced increase in M1 results in roughly a two percentage point decline in the federal funds rate. The point estimates do vary across the subsamples. However, the evidence suggests that the differences in estimated responses can be accounted for by sampling uncertainty. In particular, there is little evidence against the null hypothesis that the true responses are the same in the subsamples. This is evident from the fact that the confidence intervals in the subsamples include the point estimates for the full sample. Turning to panel B, we see that, using the N B R identification scheme, we obtain point estimates of the responses that are generally smaller. Again, there is little evidence against subsample stability. We now summarize our findings regarding subsample stability. We have two basic findings. First, there is evidence that the variance of the policy shocks is larger in the early 1980s than in the periods before or after. Second, we cannot reject the view that the response of economic variables to a shock of given magnitude is stable over the different subsamples considered.
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We conclude this section by noting that other papers have also examined the subsample stability question. See, for example Balke and Emery (1994), Bernanke and Mihov (1995) and Strongin (1995). These papers focus on a slightly different question than we do. They investigate whether the Fed adopted different operating procedures in different subperiods, and provide some evidence that different specifications of the policy rule in Equation (2.1) better characterize different subsamples. At the same time, Bernanke and Mihov (1995) and Strongin (1995) do not find that the dynamic response functions to a monetary policy shock are qualitatively different over the different subsample periods that they consider. In this sense, their results are consistent with ours. 4.5. Discriminating between the benchmark identification schemes
In the introduction we sketched a strategy for assessing the plausibility of different identification schemes. The basic idea is to study the dynamic response of a broad range of variables to a monetary policy shock. We dismiss an identification scheme if it implies a set of dynamic response functions that is inconsistent with every model we are willing to consider. The first subsection illustrates our approach by comparing the plausibility of two interpretations of an orthogonalized shock to NBR. These amount to two alternative identification schemes. The first corresponds to the benchmark NBR identification scheme described in Section 4. Under this scheme, an orthogonalized contractionary shock to NBR is interpreted as a negative money supply shock. The second scheme, recently proposed by Coleman, Gilles and Labadie (1996), interprets the same shock as either a positive shock to money demand, or as news about a future monetary expansion. When we use our strategy to assess their identification scheme, we find that we can dismiss it as implausible. 55 The second subsection contrasts our approach to discriminating among identification schemes with one recently proposed in Bernanke and Mihov (1995). We review their methodology and explain why we think our approach is more likely to be fruitful. 4.5.1. The Coleman, Gilles and Labadie identification scheme
According to Coleman, Gilles and Labadie (1996), understanding why an NBR policy shock may not coincide with an exogenous contractionary shock to monetary policy requires understanding the technical details about the way the Fed allocates the different tasks of monetary policy between the discount window and the Federal Open Market Committee. They argue, via two examples that a contractionary NBR shock may correspond to other types of shocks.
55 The discussion presented here summarizes the analysis in Christiano (1996).
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115
In their first example, they argue that a negative NBR shock may actually correspond to a positive shock to the demand for money. The argument goes as follows. Suppose that there was a shock to either the demand for TR, M 1 or M2 that drove up the interest rate. Absent a change in the discount rate, this would lead to an increase in Borrowed Reserves via the discount window. Suppose in addition that the FOMC believes that the managers of the discount window always over accommodate shocks to the demand for money, and respond by pulling nonborrowed reserves out of the system. An attractive feature of this story is that it can potentially account for the fact that the federal funds rate is negatively correlated with nonborrowed reserves and positively correlated with borrowed reserves [see Christiano and Eichenbaum (1992)]. Unfortunately, the story has an important problem: it is hard to see why a positive shock to money demand would lead to a sustained decline in total reserves, M1 or M2. But this is what happens after an NBR policy shock (see Figure 2). In light of this fact, the notion that a negative NBR policy shock really corresponds to a positive money demand shock seems unconvincing. In their second example, Coleman, Gilles and Labadie argue that a negative NBR shock may actually correspond to a positive fi~ture shock to the money supply. The basic idea is that the Fed signals policy shifts in advance of actually implementing them, and that a signal of an imminent increase in total reserves produces an immediate rise in the interest rate. Such a rise would occur in standard neoclassical monetary economies of the type considered by Cooley and Hansen (1989). Suppose that the rise in the interest rate results in an increase in borrowed reserves. If the Fed does not wish the rise in borrowed reserves to generate an immediate rise in total reserves, it would respond by reducing nonborrowed reserves. This interpretation of the rise in the interest rate after an NBR policy shock is particularly interesting because it does not depend on the presence of a liquidity effect. Indeed, this interpretation presumes that the interest rate rises in anticipation of a future increase in the money supply. To the extent that the interpretation is valid, it would constitute an important attack on a key part of the evidence cited by proponents of the view that plausible models of the monetary transmission mechanism ought to embody strong liquidity effects. Again there is an important problem with this interpretation of the evidence: the anticipated rise in the future money supply that the contractionary NBR policy shock is supposed to be proxying for never happens: TR, M 1 and M 2 f a l l for over two years after a contractionary NBR policy shock. In light of this, the notion that a contractionary NBR policy shock is proxying for expansionary future money supply shocks seems very unlikely. 4.5.2. The Bernanke-Mihov critique
The preceding subsection illustrates our methodology for assessing the plausibility of different identification schemes. Bernanke and Mihov (BM) propose an alternative approach. Under the assumption that the policy function is of the form of Equation (2.1), they develop a particular test of the null hypothesis that e7 is a monetary policy shock
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against the alternative that Ei~' is confounded by nonmonetary policy shocks to the market for federal funds. To implement their test, Bernanke and Mihov develop a model o f the federal funds market which is useful for interpreting our benchmark identification schemes. These schemes are all exactly identified, so that each fits the data equally well. To develop a statistical test for discriminating between these schemes, BM impose a particular overidentifying restriction: the amount that banks borrow at the discount window is not influenced by the total amount o f reserves in the banking system. BM interpret a rejection o f a particular overidentified model as a rejection o f the associated NBR, F F or NBR/TR identification scheme. But a more plausible interpretation is that it reflects the implausibility o f their overidentifying restriction. This is because that restriction is not credible in light o f existing theory about the determinants o f discount window borrowing and the empirical evidence presented below.
4.5.2.1. A model o f the federal funds market. reserves is given by rRt = fTR(Y2t) -- aFFt + OdEd,
BM assume that the demand for total
(4.4)
where fTR(g2t) is a linear function o f the elements o f £2,, a, oa > 0, and eta is a unit variance shock to the demand for reserves which is orthogonal to Or. According to Equation (4.4), the demand for total reserves depends on the elements o f g2t and responds negatively to the federal funds rate. The demand for borrowed reserves is:
BRt =fBR(Qt)+ [3FFt - yNBRt + Obebt,
(4.5)
wherefBR(g2t) is a linear function o f the elements o f f2t and ob > 0. The unit variance shock to borrowed reserves, Etb, is assumed to be orthogonal to f2t. BM proceed throughout under the assumption that y = 0. Below, we discuss in detail the rationale for specification (4.5). 56 Finally, they specify the following Fed policy rule for setting NBRt:
NBR~ = fxgR ( g2t) + et,
(4.6)
where
et = Oa a d ( + Ob Obetb + ase 7
(4.7)
Here, e7 is the unit variance exogenous shock to monetary policy. By assumption, etd, e), e7 are mutually orthogonal, both contemporaneously and at all leads and lags.
56 We follow BM in not including the interest rate charged at the discount window (the discount rate) as an argument in Equation (4.5). BM rationalize this decision on the grounds that the discount rate does not change very often.
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The parameters q~d and Cb control the extent to which Fed responds contemporaneously to shocks in the demand for total reserves and borrowed reserves. Using the fact that TR = N B R + BR, and solving Equations (4.4)-(4.7), we obtain TRt ] NBRt I = F(f2t) + ut, ut = Be~, FF, J
(4.8)
where
"fTR(~3 F(g2t)
=
0
1
1
y 1
0
fNBR(~2t)
(4.9)
1
and (Td~ CdaY+OJaflU~--aO's~-~a/3--tZ~b~-l+O~Y-¢bfi+a B =
~ra4)a
~d~d7 ~ a+l ~+a
G 7 1
~s ~
(4.10)
G ~ z' ~ -1+¢b7-~ ~
~'b
e, =
(4.11)
4/ We now turn to the problem of identifying the parameters of the money market model. As in Section 3, we first estimate ut using the fitted disturbances, ~tt, in a linear regression of the money market variables on g2t, and then estimate et from ~, = B-lht using a sample estimate of B. The latter can be obtained by solving 4.5.2.2. Identifying the p a r a m e t e r s o f the model.
(4.12)
V = B B ~,
where V is the Gaussian maximum likelihood estimate of E u t d t which respects the restrictions, if any, implied by Equation (4.12) and the structure of B in Equation (4.10). The estimate, V, is obtained by maximizing T 2 {loglVl+tr(SV-')},
where
1 T S =~Zht~t;,
(4.13)
t=l
subject to conditions (4.10)-(4.12). When the latter restrictions are not binding, the solution to this maximization problem is V = S. s7 57 BM use a slightlydifferentestimation strategy. See the appendix in BM.
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Denote the model's eight structural parameters by ~p = [a, [3, 7, cpd , o b, ad2, 0-2, 0-21.
(4.14)
Let ~0r denote a value of ~0 which implies a B that satisfies condition (4.12). The model is underidentified if there exist other values of ~ that have this property too. The model is exactly identified if ~ r is the only value of ~ with this property. Finally, the model is overidentified if the number of structural parameters is less than six, the number of independent elements in S. Given the symmetry of V, condition (4.12) corresponds to six equations in eight unknown parameters: a, fi, 7, q~d, q~b, 0-2, 0-I, 0-2. To satisfy the order condition discussed in Section 3, at least two more restrictions must be imposed. Recall that the F F , N B R and N B R / T R identification schemes analyzed in the previous section correspond to a particular orthogonality condition on the monetary policy shock. These conditions are satisfied in special cases of the federal funds market model described above. Each special case corresponds to a different set of two restrictions on the elements of ~p. In each case, the estimation procedure described above reduces to first setting V = S and then solving the inverse mapping from V to the free elements of ~p in condition (4.12). The uniqueness of this inverse mapping establishes global identification. When St = NBRt, relations (4.8)-(4.10) imply that the measured policy shock is given by Equation (4.7). So, from the perspective of this framework, our N B R system assumes: q~d = q~b = O.
(4.15)
The free parameters in ~p are uniquely recovered from V as follows: V21 0 -2 = V22, V32, V l l + uV31
(4.16)
a --
fi -
V31+ aV33'
(4.17)
g
1
=
a~2 = ( 1 3 + a ) [ V 3 1 + aV33]'
V21(~ + a ) (/0-2 ,
2 O~i = V 3 3 ( a + / 3 ) 2 _ O - c ~ _ ( 1 _
7) 2 a s2 .
(4.18)
where V O. refers to the (i,j) element of V. When St = FFt then ~ p d ( y _ l ) + l a + e ; -l+q~b(g-1)~+e~, y - 1 fi + a fi + a ~et.s
et -
(4.19)
From the perspective of this framework, the benchmark F F system assumes: q~d _
1 1--•'
q~b = _q)d.
(4.20)
The free parameters in ~p are recovered from V as follows: _ C
~/- 1 /~ q- a '
0-2 _ V322 V33,
V31 Of = -- V3~,
(4.21)
Ch. 2:
Monetary Policy Shocks: What Have we Learned and to What End?
-;=
gll
_ a2C2 0-2 s,
[
y = 1 --
0-2 = ( 1 - y)2 [V22 L
0-2 (1 -
V21-k
]
0-2
7) 2
"
1
(4.22)
acq,2J ' /3 = (y
119
V32
-
-- a . 1) V77ss
(4.23)
'
The NBR/TR system assumes: a = q}b = O.
(4.24)
Under these conditions, it is easy to verify that the error of the regression of NBRt on g2t and TRt is e~. The free parameters of the money model are recovered from V as follows: 0-; = g l l ,
~d _ V21
0"2 = V 2 2 - ( 0 d ) 20.2d,
(4.25)
V~I' cl -
V32 - Odv3~ 0.2 ,
d :/32 [v33- c20-~2 -
c2-
cZos2],
v31 02,
fi= [c2-0~cl] -~ ,
y = /3C1 + 1.
(4.26) (4.27)
Restrictions (4.15), (4.20), (4.24) guarantee that the benchmark NBR, F F and NBR/TR policy shock measures are not polluted by nonmunetary policy shocks, respectively. 4.5.2.3. The Bernanke-Mihov test. Recall that the basic purpose of the money market model discussed above is to help assess whether different monetary policy shock measures are polluted by nonpolicy shocks to the money market. In the case of the NBR policy system this amounts to testing restriction (4.15). For the F F and NBR/TR systems this corresponds to testing restrictions (4.20) and (4.24), respectively. The problem is that, since each of these systems is exactly identified, the restrictions cannot be tested using standard statistical procedures. From this perspective, the money market model is not helpful. As the model stands, to assess the different identification schemes, one must revert to the strategy laid out in the previous section. Namely, one must examine the qualitative properties of the impulse response functions. Instead BM impose an additional maintained assumption on the model. Specifically, they assume y = 0, i.e., the demand for borrowed reserves does not depend on the level of nonborrowed reserves. With this additional restriction, the NBR, F F and NBR/TR models have only five structural parameters, so each is overidentified. Consequently, each can be tested using standard likelihood ratio methods. An important limitation of this approach is that we can always interpret a rejection as evidence against the maintained hypothesis, y = 0, rather than as evidence against the NBR, F F or NBR/TR identification schemes. A rejection would be strong evidence against one of these identification schemes only to the extent that one had overwhelmingly sharp priors
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that y really is zero. In fact, there are no compelling reasons to believe that y is zero. Just the opposite is true. Standard dynamic models of the market for reserves suggest that y is not zero. Consider for example Goodfriend's (1983) model of a bank's demand for borrowed reserves. Goodfriend highlights two factors that affect a bank's decision to borrow funds from the Federal Reserve's discount window. The first factor is the spread between the federal funds rate and the Fed's discount rate (here assumed constant). The higher this spread is, the lower is the cost of borrowing funds from the discount window, relative to the cost of borrowing in the money market. The second factor is the existence of nonprice costs of borrowing at the Federal Reserve discount window. These costs rise for banks that borrow too much or too frequently, or who are perceived to be borrowing simply to take advantage of the spread between the federal funds rate and the discount rate. Goodfriend writes down a bank objective function which captures both of the aforementioned factors and then derives a policy rule for borrowed reserves that is of the following form: O 0, is strongly rejected. This calls into question the usefulness o f their model for interpreting the benchmark F F identification scheme for the sample period as a whole. 62 Finally, note from Table 2b that the N B R / T R model is not strongly rejected by BM's likelihood ratio test and the parameter values are consistent with all o f BM's sign restrictions. In sum, B M have proposed a particular way to test whether the policy shock measures associated with different identification schemes are polluted by nonpolicy shocks. The previous results cast doubt on the effectiveness o f that approach. 4.6. Monetary policy shocks and volatility
Up to now we have focussed on answering the question, what are the dynamic effects o f a monetary policy shock? A related question is: How have monetary policy
62 BM actually argue that this model is most suitable for the pre-1979 period. Here too, their point estimate of a is negative and significantly different from zero.
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shocks contributed to the volatility of various economic aggregates? The answer to this question is of interest for two reasons. First, it sheds light on the issue of whether policy shocks have been an important independent source of impulses to the business cycle. Second, it sheds light on identification strategies which assume that the bulk of variations in monetary aggregates reflect exogenous shocks to policy. For example, this is a maintained assumption in much of the monetized real business cycle literature. 63 Table 3 summarizes the percentage of the variance of the k step ahead forecast errors in P, Y, PCOM, FF, NBR, TR and M 1 that are attributable to quarterly benchmark FF, NBR and NBR/TR policy shocks. Analog results for policy shock measures based on M0, M 1, and M2 are reported in Table 4. We begin by discussing the results based on the benchmark policy measures. First, according to the benchmark F F measure, monetary policy shocks have had an important impact on the volatility of aggregate output, accounting for 21%, 44% and 38% of the variance of the 4, 8 and 12 quarter ahead forecast error variance in output, respectively. However, these effects are smaller when estimated using the NBR/TR policy shock measures and smaller still for the benchmark NBR policy shocks. Indeed, the latter account for only 7%, 10% and 8% of the 4, 8 and 12 quarter ahead forecast error variance of output. Evidently, inference about the importance of monetary policy shocks depends sensitively on which policy shock measure is used. In addition, conditioning on the policy shock measure, there is substantial sampling uncertainty regarding how important policy shocks are in accounting for the variance of the k step forecast error. Second, none of the policy shock measures account for much of the volatility of the price level, even at the three year horizon. In addition, only the F F benchmark policy shock measure accounts for a nontrivial portion of the variability of PCOM. Evidently, monetary policy shocks are not an important source of variability in prices, at least at horizons of time up to three years in length. Third, regardless of whether we identify St with the federal funds rate or NBR, policy shocks account for a large percent of the volatility of St at the two quarter horizon. However, their influence declines substantially over longer horizons. Fourth, according to the benchmark F F and NBR/TR measures, monetary policy shocks play a very minor role in accounting for the variability in TR and M1. Policy shocks play a more important role according to the benchrnark NBR measure. Even here, most of the volatility in TR and M1 arises as a consequence of nonpolicy shocks. Identification strategies which assume that monetary aggregates are dominated by shocks to policy are inconsistent with these results. Finally, policy shocks are more important in explaining the volatility in M2 than for TR or M1. This is true regardless of which benchmark policy measure we consider. Still, the variation in M2 due to policy shocks never exceeds 50%.
63 See Cooley and Hansen (1989), Chaff et al. (1996) and Christiano and Eichenbaum (1995).
Ch. 2."
Monetary Policy Shocks: What Have we Learned and to What End?
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Next we consider the results obtained for policy shock measures based on M0, M1, and M2. The VAR's underlying these results correspond to the ones underlying the results reported in columns 2, 4 and 6 in Figure 6. In each case, St is equated to either M0, M I or M2, and the information set, g2t, includes current and past values of Yt, Pt, PCOMt as well as lagged values o f F F t , TRt and St. A number of results are interesting to note here. First, the M 0 and Ml-based policy shock measures account for only a trivial fraction of the fluctuations in output. In contrast, at horizons greater than a year, M2-based policy measures account for a noticeably larger fraction of output variations. While they account for a smaller fraction of output volatility than do the F F policy shocks, they are similar on this dimension to the NBR/TR policy shock measures. Second, neither the M0 or Ml-based policy shock measures account for more than a trivial part of the volatility of P and PCOM. Policy shock measures based on M2 play a somewhat larger role at horizons of a year or longer. However, there is considerable sampling uncertainty about these effects. Finally, at horizons up to a year, M0, M 1 , and M2-based policy shocks account for sizeable percentages of M0, M1, and M2, respectively. At longer horizons the percentages are lower. Viewed across both sets of identification strategies that we have discussed, there is a great deal of uncertainty about the importance of monetary policy shocks in aggregate fluctuations. The most important role for these shocks emerged with the FF-based measure of policy shocks. The smallest role is associated with the M0 and Ml-based policy shock measures. We conclude this subsection by noting that even if monetary policy shocks have played only a very small role in business fluctuations, it does not follow that the systematic component, f in Equation (2.1), of monetary policy has played a small role. The same point holds for prices. A robust feature of our results is that monetary policy shocks account for a very small part of the variation in prices. This finding does not deny the proposition that systematic changes in monetary policy, captured by f , can play a fundamental role in the evolution of prices at all horizons of time.
5. The effects of monetary policy shocks: abandoning the recursiveness approach In this section we discuss an approach to identifying the effects of monetary policy shocks that does not depend on the recursiveness assumption. Under the recursiveness assumption, the disturbance term, e)~, in the monetary anthority's reaction function [see Equation (2.1)] is orthogonal to the elements of their information set g2t. As discussed above [see Equation (4.1)] this assumption corresponds to the notion that economic variables within the quarter are determined in a block recursive way: first, the variables associated with goods markets (prices, employment, output, etc.) are determined; second, the Fed sets its policy instrument (i.e., NBR in the case of the
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benchmark NBR system, and F F in the case of the benchmark F F system); and third, the remaining variables in the money market are determined. To help compare the recursiveness assumption with alternative identifying assumptions, it is convenient to decompose it into two parts. First, it posits the existence of a set of variables that is predetermined relative to the policy shock. Second, it posits that the Fed only looks at predetermined variables in setting its policy instrument. Together, these assumptions imply that monetary policy shocks can be identified with the residuals in the ordinary least squares regression of the policy instrument on the predetermined variables. The papers discussed in this section abandon different aspects of the recursiveness assumption. All of them drop the assumption that the Fed only looks at variables that are predetermined relative to the monetary policy shock. This implies that ordinary least squares is not valid for isolating the monetary policy shocks. Consequently, all these papers must make further identifying assumptions to proceed. The papers differ in whether they assume the existence of variables which are predetermined relative to the monetary policy shock. Sims and Zha (1998) assume there are no variables with this property. In contrast, papers like Sims (1986), Gordon and Leeper (1994), and Leeper, Sims and Zha (1996) assume that at least a subset of goods market variables are predetermined. Under their assumptions, these variables constitute valid instruments for estimating the parameters of the Fed's policy rule. The section is organized as follows. First, we discuss the identifying assumptions in the paper by Sims and Zha (1998). We then compare their results with those obtained using the benchmark identification schemes. Finally, we briefly consider the analyses in the second group of papers mentioned above.
5.1. A fully simultaneous system
This section is organized as follows. In the first subsection we discuss the specification of the Sims and Zha (1998) (SZ) model and corresponding identification issues. In the second subsection, we compare results obtained with a version of the SZ model to those obtained using the benchmark policy shocks. 5.1.1. Sims-Zha: model specification and identification
We begin our discussion of the SZ model by describing their specification of the money supply equation. It is analogous to our policy function (2.1), with St identified with a short term interest rate, Rt. Sims and Zha (1998) assume that the only contemporaneous variables which the Fed sees when setting St are a producer's price index for crude materials (Pcm) and a monetary aggregate (M). In addition, the Fed is assumed to see a list of lagged variables to be specified below. Note that unlike the benchmark systems, £2t does not contain the contemporaneous values of the aggregate price level
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and output. A s Sims and Zha (1998) point out, this is at best only a reasonable working hypothesis. 64 The reaction function in the SZ model can be summarized as follows: Rt = const. + aiMt + a2Pcmt +.fs(Zt-1 . . . . , Zt-q) + a ~ ~,
(5.1)
where fs(Z~-i . . . . , Zt_q) is a linear function o f past values o f all the variables in the system, q > 0, a > 0, and Ets is a serially uncorrelated monetary policy shock. Sims and Zha (1998) assume that P c m and M are immediately affected by a monetary policy shock. As noted above, this rules out ordinary least squares as a method to estimate Equation (5.1). Instrumental variables would be a possibility i f they made the identifying assumption that there exists a set o f variables predetermined relative to the monetary policy shock. However, they are unwilling to do so. They make other identifying assumptions instead. First, they postulate a money demand function o f the form: Mt - Pt - Yt : const. + blRf +fM(Zt 1, . . . , Zt-q) + aMe M.
(5.2)
Here, f M ( Z t _ l
.... , Zt-q) is a linear function o f past values o f all the variables in the system, aM > 0, and eM is a serially uncorrelated shock to money demand. Recall, Yt and Pt denote aggregate output and the price level. Note that the coefficients on Et and Pt are restricted to unity. Sims and Zha display a model which rationalizes a money demand relationship like Equation (5.2). 65 Second, they assume that Pcmt responds contemporaneously to all shocks in the system. They motivate this assumption from the observation that crude materials prices are set in auction markets. Third, as noted above, they are not willing to impose the assumption that goods market variables like P and Y are predetermined relative to the monetary policy shock. Clearly, they cannot allow P and Y to respond to all shocks in an unconstrained way, since the system would then not be identified. Instead, they limit the channels by which monetary policy and other shocks have a contemporaneous effect on P and Y. To see how they do this, it is convenient to define a vector o f variables denoted by Xt, which includes Pt and Yr. Sims and Zha impose the restriction that Xt does not respond directly to Mt or Rt, but that it does respond to Pcmt. A monetary
64 This is because the Fed does have at its disposal various indicators of price and output during the quarter. For example, the Fed has access to weekly reports on unemployment claims and retail sales. Also, two weeks prior to each FOMC meeting, policymakers have access to the "Beige Book", which is compiled from nationwide surveys of business people. In addition, FOMC members are in constant contact with members of the business community. Moreover, the Fed receives, with a one month lag, various monthly measures of output and prices (e.g. employment, wages and the consumer price level). 65 Their model rationalizes a relationship between the contemporaneous values of Mr, Pt, Yt and St. One can rationalize the lagged terms in the money demand equation if there is a serially correlated shock to the marginal product of money in their model economy. Ireland (1997) and Kim (1998) rationalize similar relationships with Y replaced by consumption.
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policy shock has a contemporaneous impact on the variables in Xt via its impact on Pcmt.
To see this, first let
Xt =
Wt Pimt
,
Zt=
Lrbk' j
R,
"
x,
where P i m denotes the producer price index of intermediate materials, W denotes average hourly earnings of nonagricultural workers, Tbk denotes the number of personal and business bankruptcy filings. The assumptions stated up to now imply the following restrictions on the matrix A0 in representation (3.2) of Zt: - a l l a12 a13 a14 a15 al6 a17 a18 0 0 0 0 a22 a 2 3 - a 2 2 - a 2 2 a33 0 0 0 0 0 a31 a32 A0 =
a41 asa a61 aT1
0 0 0 0
as1 0
0 0 0 0 0
a44 a54 a64 a74 a84
a45 a55 a65 a75 a85
a46 a56 a66 a76 a86
a47 a57 a67 a77 a87
a48 a58 a68 a78 a88
(5.3)
The first row of A0 corresponds to the P c m equation. The second and third rows correspond to the money demand equation (5.2), and to the monetary policy rule (5.1), respectively. The next five rows correspond to Xt. The second and third elements of et in Equation (3.2) correspond to ~ and e7. It is evident from Equation (5.3) that the impact of a monetary policy shock operates on Xt via its influence on Peru. Specifically, this reflects the fact that the (4, 1) to (8, 1) elements of A0 are potentially nonzero. I f we impose that these elements are zero, then, given the other zero restrictions in the second and third columns of A0, the variables in Xt are predetermined relative to a monetary policy shock. We now consider identification o f the SZ model. Notice that the last five rows in A0 have the same restrictions, suggesting that Equation (3.2) is not identified. To see that this is in fact the case, consider the following orthonormal matrix:
,o]
(3×3) (3;5)
m =
0 ' (5×3) (5x5) J
(5.4)
where the dimensions are indicated in parentheses and ~" is an arbitrary orthonormal matrix. Note that if A0 satisfies (i) the restrictions in Equation (5.3) and (ii) the relation
Ch. 2.. Monetary Policy Shocks: What Have we Learned and to What End?
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Ao I (Aol) ' = V, then WAo does too. Here, V denotes the variance covariance matrix of the fitted residuals in the VAR (3.1), for Zt. By the identification arguments in Section 3, representation (3.2) with A0 and with WAo are equivalent from the standpoint o f the data. That is, there is a family o f observationally equivalent representations (3.2), for the data. Each corresponds to a different choice o f A0. We now discuss the implications o f this observational equivalence result for impulse response functions. Recall from Equation (3.6) that, conditional on the Bz's characterizing the VAR o f Zt, the dynamic response functions o f Zt to et are determined by Ao 1. Also, note that (WAo) -1 = Ao I W I. Two important conclusions follow from these observations. First, the impulse response functions o f Zt to the first three elements o f et are invariant to the choice o f A0 belonging to the set o f observational equivalent A0's defined above, i.e., generated using W~s o f the form given by Equation (5.4). Second, the dynamic response functions to the last five elements o f et are not. To the extent that one is only interested in the response functions to the first three elements o f et, the precise choice o f ff~ is irrelevant. Sims and Zha choose to work with the A0 satisfying Equation (5.3) and the additional restriction that the square matrix formed from the bottom right 5 × 5 matrix in A0 is upper triangular. 66 The corresponding dynamic response functions o f Zt to the last five shocks in et simply reflect this normalization. We now make some summary remarks regarding identification of the SZ model. In Section 3 we discussed an order condition which, in conjunction with a particular rank condition, is sufficient for local identification. According to that order condition, we need at least 28 restrictions on A0. The restrictions in Equation (5.3), along with the normalization mentioned in the previous paragraph, represent 31 restrictions on A0. So, we satisfy one o f the sufficient conditions for identification. The rank condition must be assessed at the estimated parameter values. Finally, to help guarantee global identification, Sims and Zha impose the restriction that the diagonal element o f A0 are positive. 5.1.2. Empirical results
We organize our discussion o f the empirical results around three major questions. First, what are the effects of a contractionary monetary policy shock using the SZ identification scheme? Second, how do these effects compare to those obtained using the benchmark identification scheme? Third, what is the impact on Sims and Zha's (1998) results o f their assumption that the variables in Xt respond contemporaneously to a monetary policy shock?
66 The A0 matrix is contained in the set of observationallyequivalentA0's as long as that set is non-empty. To see this, suppose there is some A0 that satisfies (i) Equation (5.3) and (ii) the relation A~1 (Aol)I = V. Let QR denote the QR decomposition of the lower right 5 × 5 part of this matrix ~. The 5 × 5 matrix Q is orthonormal and R is upper triangular. Then, form the orthonormal matrix W as in Equation (5.4), with W = Qt. The matrix WAo satisfies (i) and (ii) with the additional restriction on Equation (5.3) that the lower fight 5 × 5 matrix in A0W is upper triangular. This establishes the result sought.
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To answer these questions, we employ a version o f the SZ model in which Mt corresponds to M 2 growth and Rt corresponds to the 3 month Treasury Bill Rate. 67 The four-lag VAR model was estimated using data over the period 1965Q3-1995Q2.68 Our results are presented in column 1 o f Figure 11. The solid lines correspond to our point estimates o f the dynamic response o f the variables in Zt to a contractionary monetary policy shock. The dotted lines represent 95% confidence intervals about the mean o f the impulses. 69 The main consequences o f a contractionary SZ policy shock can be summarized as follows. First, there is a persistent decline in the growth rate o f M2 and a rise in the interest rate. Second, there is a persistent decline in the GDP deflator and the prices o f intermediate goods and crude materials. Third, after a delay, the shock generates a persistent decline in real GDP. Finally, note that the real wage is basically unaffected by the SZ policy shock. Comparing these results with those in Figure 2, we see that the qualitative response o f the system to an SZ policy shock is quite similar to those in the benchmark F F and N B R systems. It is interesting to note that the estimated SZ policy shocks are somewhat smaller than the estimated benchmark F F policy shocks. For example, the impact effect o f a benchmark F F policy shock on the federal funds rate is about 70 basis points, while the impact o f a SZ policy shock on the three-month Treasury bill rate is about 40 basis points. At the same time, the SZ policy shock measure is roughly o f the same order o f magnitude as an N B R policy shock. In both cases a policy shock is associated with a forty basis point move in the federal funds rate. We now turn to the third question posed above. We show that Sims and Zha's insistence that Xt is not predetermined relative to a monetary policy shock has essentially no impact on their results. To do this, we simply shut down the coefficients in A0 which allow a monetary policy shock to have a contemporaneous impact on Xt and reestimate the system. Column 2 in Figure 11 reports the results. Comparing columns 1 and 2, we see that inference is virtually unaffected. It is interesting to compare the SZ model with the analysis in Leeper et al. (1996). They work with a system that contains more variables. But, the fundamental difference is that they impose the assumption that goods market variables are predetermined
67 The variable, Tbk, is not used in our analysis. Also, SZ measure M as the log level of M2. Comparing the estimated dynamic response functions to a monetary shock in our version of SZ with those in SZ it can be verified that these two perturbations make essentially no difference to the results. 68 The variable, Pcm, was measured as the log of the producer price index for crude materials, SA; Pim is the logged producer price index for intermediate materials, SA; Y is logged GDP in fixed-weight 1987 dollars, SA; P is the logged GDP deflator derived from nominal GDP and GDP in fixed-weight 1987 dollars, SA; R is the three-month Treasury bill rate; and the change in the log of M2, SA. These data series are taken from the Federal Reserve Board's macroeconomic database. Logged average hourly earnings of private nonagricultural production workers are divided by the GDP deflator, SA, and are derived from the Citibase data set. 69 These were computed using the procedure described in Sims and Zha (1995).
Ch. 2:
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Monetary Policy Shocks." What Have we Learned and to What End?
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134
L.J. Christiano et al.
relative to a monetary policy shock. 70 The response to a monetary policy shock of the variables that these analyses have in common is very similar. This is consistent with our finding that the absence of predeterminedness of good market variables in the SZ model is not important. A number of other studies also impose predeterminedness of at least some goods market variables. These include Sims (1986), who assumes predeterminedness of investment, and Gordon and Leeper (1994), who assume all goods market variables and the 10 year Treasury rate are predetermined. Inference about the dynamic response of economic aggregates is very similar across these papers, Sims and Zha (1998), Leeper et al. (1996) and the benchmark systems.
6. Some pitfalls in interpreting estimated monetary policy rules In Sections 4 and 5 we reviewed alternative approaches for identifying the effects of a monetary policy shock. A common feature of these different approaches is that they make enough identifying assumptions to enable the analyst to estimate the parameters of the Federal Reserves's feedback rule. A natural question is: why did we not display or interpret the parameter estimate? The answer is that these parameters are not easily interpretable. In this section we describe three examples which illustrate why the estimated policy rules are difficult to interpret in terms of the behavior of the monetary authority. We emphasize, however, that the considerations raised here need not necessarily pose a problem for the econometrician attempting to isolate monetary policy shocks and their consequences. The central feature of our examples is that the policy maker reacts to data that are different from the data used by the econometrician. In the first example, the decision maker uses error-corrupted data, while the econometrician uses error-free data. In the second and third examples the decision maker reacts to a variable that is not in the econometrician's data set. The policy rule parameters estimated by the econometrician are a convolution of the parameters of the rule implemented in real time by the policy maker and the parameters of the projection of the missing data onto the econometrician's data set. It is the convolution of these two types of parameters which makes it difficult to assign behavioral interpretations to the econometrician's estimated policy rule parameters. Our first example builds on the measurement error example discussed in Section 2. We assume there is measurement error in the data used by real time policy makers, while the econometrician uses final revised data. We suppose xt + vt corresponds to the
70 In their description of the model, monetary policy shocks impact on the analog of X via a limited set of variables. In practice, however,they set the coefficients on these variables equal to zero. So, all their estimated systems have the property that the goods market variables are predetermined relative to the monetarypolicy shock.
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raw data received by the primary data collection agency and that o t reflects classical reporting and transmission errors that are uncorrelated with the true variable, xt, at all leads and lags. In addition, we suppose that the reporting errors are discovered in one period, so that ut in Equation (2.2) is zero. We assume that the data collection agency (or, the staff of the policy maker) reports its best guess, ~?t, of the true data, xt, using its knowledge of the underlying data generating mechanism and the properties of the measurement error process. 71 Finally, suppose that xt evolves according to xt = p l S t - 1 + p2xt-1 + oat,
where rot is uncorrelated with all variables dated t - 1 and earlier. Suppose the data collection authority computes ~?t as the linear projection o f x t on the data available to it. Then, fft = P [ x t [St 1, xt + ut, xt 1] = a 0 S t - I + a l (xt + ut) + a 2 x ~ l ,
(6.1)
where the ai's are functions of pl,/)2, and the variances of oat and or. Now, suppose that the policy authority is only interested in responding to xt, and that it attempts to do so by setting (6.2)
St = a2ct
in real time. Substituting Equation (6.1) into this expression, we see that Equation (6.2) reduces to Equations (2.1) and (2.4) with t o = a a o , [31 = a a l ,
[32 = aa2.
(6.3)
Notice how different the econometrician's estimated policy rule, (2.4) and (6.3), is from the real time policy rule (6.2). The [3's in the estimated policy rule are a convolution of the behavioral parameter, a, the measurement error variance, and the parameters governing the data generating mechanism underlying the variables that interest the policy maker. 72 Also notice that an econometrician who estimates the policy rule using the recursiveness assumption will, in population, correctly identify the monetary policy shock with aalvt. This example shows how variables might enter f , perhaps even with long lags, despite the fact that the policy maker does not care about them p e r se. In the example, the variables St-l and xt-i enter o n l y because they help solve a signal extraction problem. Finally, the example illustrates some of the dangers involved in trying to give
71 For a discussion of the empirical plausibility of this model of the data collection agency, see Mankiw et al. (1984), and Mankiw and Shapiro (1986). 72 See Sargent (1989), for a discussion of how to econometrically unscramble parameters like this in the presence of measurement error.
L.J. Christiano et al.
136
a structural interpretation to the coefficients in f Suppose a0 and a are positive. An analyst might be tempted to interpret the resulting positive value of/3o as reflecting a desire to minimize instrument instability. In this example, such an interpretation would be mistaken. Significantly, even though the estimated policy rule has no clear behavioral interpretation, the econometrician in this example correctly identifies the exogenous monetary policy shock. For our second example, we assume that the policy maker responds only to the current innovation in some variable, for example, output. In particular suppose that, St = aet + cr~.Et, where e~ is the innovation to which the policy maker responds, a is the policy parameter, and E~ is the exogenous policy shock. Suppose that et is related to data in the following way, et = ~i~-o/3ixt-i, so that in Equation (2.1), O Price
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143
8. Conclusion
In this chapter we have reviewed the recent literature that grapples with the question: What happens after a shock to monetary policy? This question is of interest because it lies at the center of the particular approach to model evaluation that we discussed: the Lucas program applied to monetary economics. The basic step in that program involves subjecting monetary models to a particular experiment: a monetary policy shock. Since alternative models react very differently to such a shock, this experiment can, in principle, be the basis of a useful diagnostic test. But to be useful in practice, we need to know how the actual economy responds to the analog experiment. Isolating these data based experiments requires identifying assumptions. We argued that qualitative inference about the effects of a monetary policy shock is robust across many, but not all the sets of identifying assumptions that have been pursued in the literature. A key question remains: How can the results of the literature we reviewed be used to quantitatively assess the performance of a particular model? Much of the empirical literature on monetary policy shocks proceeds under the assumption that monetary policy is highly reactive to the state of the economy. In sharp contrast, analyses of quantitative general equilibrium models often proceed under much simpler assumptions about the nature of the monetary authority's reaction function. This leads to an obvious problem: unless the monetary policy rule has been specified correctly, the nature of the monetary experiment being conducted in the model is not the same as the experiment in the data. One way to deal with the problem is to solve theoretical models using estimated reaction functions taken from the policy shock literature. There are two potential problems associated with this approach. First, and most importantly, it is often the case that models have multiple equilibria when policy is specified as a relationship between endogenous variables. Second, the complexity of estimated reaction functions makes it difficult (at least for us) to gain intuition for the way a monetary policy shock impacts on a model economy. Christiano et al. (1997b) suggest an alternative approach to ensuring the consistency between model and data based experiments. The basic idea is to calculate the dynamic effects of a policy shock in a model economy under the following representation of monetary policy: the growth rate of money depends only on current and past shocks to monetary policy. Formally such a specification represents the growth rate of money as a univariate, exogenous stochastic process. However this representation c a n n o t be developed by examining the univariate time series properties of the growth rate of money, say by regressing the growth rate of money on its own lagged values. Instead the representation must be based on the estimated impulse response function of the growth rate of money to a monetary policy shock. The rationale underlying the proposal by Christiano et al. (1997b) is as follows. To actually implement a particular monetary policy rule, the growth rate of money m u s t (if only implicitly) respond to current and past exogenous shocks in an appropriate way. This is true even when the systematic component of policy is thought of as a
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relationship between endogenous variables, like the interest rate, output and inflation. The literature on monetary policy shocks provides an estimate o f the way the growth rate o f money actually does respond to a particular shock - a monetary policy shock. For concreteness we refer to the estimated impulse response function of the growth rate o f money to a policy shock as "the exogenous monetary policy rule". 79 Suppose that an analyst solves a monetary model under the assumption that policy is given by the exogenous policy rule. In addition, suppose that the model has been specified correctly. In this case, the dynamic responses o f the model variables to a policy shock should be the same as the dynamic response functions o f the corresponding variables to a policy shock in the VAR underlying the estimate o f exogenous policy rule [see Christiano et al. (1997b)]. This is true even if the monetary policy shock was identified in the VAR assuming a policy rule that was highly reactive to the state o f the economy. So, the empirical plausibility of a model can be assessed by comparing the results o f an exogenous policy shock in the model to the results o f a policy shock in a VAR. It is often the case that a model economy will have multiple equilibria when policy is represented as a relationship between endogenous variables. Each may be supported by a different rule for the way the growth rate o f money responds to fundamental economic shocks. Yet, for any given rule relating the growth rate o f money to these shocks, it is often (but not always) the case that there is a unique equilibrium [see Christiano et al. (1997b) for examples]. Under these circumstances the proposal by Christiano et al. (1997b) for evaluating models is particularly useful. The monetary policy shock literature tells us which exogenous policy rule the Fed did adopt and how the economy did respond to a policy shock. These responses can be compared to the unique prediction o f the model for what happens after a shock to monetary policy. However, it is unclear how to proceed under a parameterization of monetary policy in which there are multiple equilibria. We conclude by noting that we have stressed one motivation for isolating the effects o f a monetary policy shock: the desire to isolate experiments in the data whose outcomes can be compared with the results o f analog experiments in models. Authors like Sims and Zha (1998) and Bernanke et al. (1997) have pursued a different motivation. These authors argue that if the analyst has made enough assumptions to isolate another fundamental shock to the economy, then it is possible to understand the consequences o f a change in the systematic way that monetary policy responds to that shock, even in the absence o f a structural model. Their arguments depend in a critical way on ignoring the Lucas critique. This may or may not be reasonable in their particular applications. We are open minded but skeptical. For now we rest our
79 Christiano et al. (1997b) argue that a good representation for the exogenous monetary policy rule relating the growth rate of M1 to current and past policy shocks is a low order M A process with a particular feature: the contemporaneous effect of a monetary policy shock is small while the lagged effects are much larger. In contrast, the dynamic response function of the growth rate of M2 to current and past policy shocks is well approximated by an AR(1) process.
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c a s e for t h e u s e f u l n e s s o f t h e m o n e t a r y p o l i c y s h o c k literature o n the m o t i v a t i o n we h a v e p u r s u e d : t h e desire to b u i l d structural e c o n o m i c m o d e l s that c a n b e u s e d to t h i n k a b o u t s y s t e m a t i c c h a n g e s i n p o l i c y i n s t i t u t i o n s a n d rules.
References Balke, N.S., and K.M. Emery (1994), "The federal funds rate as an indicator of monetary policy: evidence from the 1980's", Economic Review (Federal Reserve Bank of Dallas) First Quarter, 1-16. Ball, L. (1995), "Time-consistent policy and persistent changes in inflation", Journal of Monetary Economics 36(2):329-350. Barro, R.J. (1977), "Unanticipated money growth and nnemployment in the United States", American Economic Review 67(2): 101-115. Bartle, R.G. (1976), The Elements of Real Analysis, 2nd edition (Wiley, New York). Beaudry, R, and M.B. Devereux (1995), "Money and the real exchange rate with sticky prices and increasing returns", Carnegie-Rochester Conference Series on Public Policy 43:55-101. Bernanke, B.S. (1986), "Alternative explanations of the money-income correlation", Carnegie-Rochester Conference Series on Public Policy 25:49 99. Bernanke, B.S., and A.S. Blinder (1992), "The federal funds rate and the channels of monetary transmission", American Economic Review 82(4):901 92I. Bernanke, B.S., and M. Gertler (1995), "Inside the black box: the credit channel of monetary policy transmission", Journal of Economic Perspectives 9(4):27-48. Bernanke, B.S., and I. Mihov (1995), "Measuring monetary policy", Working Paper No. 5145 (NBER). Bernanke, B.S., M. Gertler and M.W. Watson (1997), "Systematic monetary policy and the effects of oil price shocks", Brookings Papers on Economic Activity 1997(1):91 142. Boschen, J.E, and L.O. Mills (1991), "The effects of countercyclical monetary policy on money and interest rates: an evaluation of evidence from FOMC documents", Worldng Paper 91 20 (Federal Reserve Bank of Philadelphia). Brunner, A.D. (1994), "The federal funds rate and the implementation of monetary policy: estimating the federal reserve's reaction function", International Finance Discussion Paper No. 466 (Board of Governors of the Federal Reserve System). Campbell, J. (1997), "Cyclical job creation, job destruction and monetary policy", manuscript (University of Rochester). Carlson, J.B., J.M. McIntire and J.B. Thomson (1995), "Federal funds futures as an indicator of future monetary policy: a primer", Federal Reserve Bank of Cleveland Economic Review 31(1):20-30. Chari, V.V, RJ. Kehoe and E.R. McGrattan (1996), "Sticky price models of the business cycle: the persistence problem", Staff Report 217 (Federal Reserve Bank of Minneapolis). Chari, VV, L.J. Christiano and M. Eichenbaum (1998), "Expectation traps and discretion", Journal of Economic Theory 81(2):462 492. Christiano, L.J. (1991), "Modeling the liquidity effect of a money shock", Federal Reserve Bank of Minneapolis Quarterly Review 15(1):3 34. Christiano, L.J. (1992), "Searching for a break in GNP", Journal of Business and Economic Statistics 10(3):237 250. Christiano, L.J. (1995), "Resolving the liquidity effect: conunentary", Federal Reserve Bank of St. Louis Review 77(3):55 61. Christiano, L.J. (1996), "Identification and the liquidity effect: a case study", Federal Reserve Bank of Chicago Economic Perspectives 20(3):2 13. Christiano, L.J., and M. Eichenbaum (1992), "identification and the liquidity effect of a monetary policy shock", in: A. Cukiern~an, Z. Hercowitz and L. Leiderman, eds., Political economy, growth and business cycles (MIT Press, Cambridge and London), 335 370.
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Christiano, L.J., and M. Eichenbaum (1995), "Liquidity effects, monetary policy and the business cycle", Journal of Money, Credit and Banking 27(4):1113-1136. Christiano, L.J., M. Eichenbaum and C.L. Evans (1996a), "The effects of monetary policy shocks: evidence from the flow of funds", Review of Economics and Statistics 78(1): 16-34. Christiano, L.J., M. Eichenbaum and C.L. Evans (1996b), "Identification and the effects of monetary policy shocks", in: M. Blejer, Z. Eckstein, Z. Hercowitz and L. Leiderman, eds., Financial Factors in Economic Stabilization and Growth (Cambridge University Press, Cambridge) 36-74. Christiano, L.J., M. Eichenbaum and C.L. Evans (1997a), "Sticky price and limited participation models: a comparison", European Economic Review 41(6):1201-1249. Christiano, L.J., M. Eichenbaum and C.L. Evans (1997b), "Modeling money", Working Paper 97-17 (Federal Reserve Bank of Chicago). Clarida, R., and J. Gali (1994), "Sources of real exchange rate fluctuations: how important are nominal shocks?", Carnegie-Rochester Conference Series on Public Policy 41:1-56. Clarida, R., and M. Gertler (1997), "'How the Bundesbank conducts monetary policy", in: C.D. Romer and D.H. Romer, eds., Reducing Inflation: Motivation and Strategy (University of Chicago Press)" 363~406. Clarida, R., J. Gali and M. Gertler (1997), "Monetary policy rules and macroeconomic stability: evidence and some theory", manuscript (New York University). Clarida, R., J. Gali and M. Gertler (1998), "Monetary policy rules in practice: some international evidence", European Economic Review 42(6):1033 1067. Cochrane, J.H. (1994), "Shocks", Carnegie-Rochester Conference Series on Public Policy 41:295 364. Coleman II, W.J., C. Gilles and EA. Labadie (1996), "A model of the federal funds market", Economic Theory 7(2):337-357. Cooley, T.E, and G.D. Hansen (1989), "The inflation tax in a real business cycle model", American Economic Review 79(4):733-748. Cooley, T.E, and G.D. Hansen (1997), "Unanticipated money growth and the business cycle reconsidered", Journal of Money, Credit and Banking 29(4, Part 2):624-648. Cushman, D.O., and T. Zha (1997), "Identifying monetary policy in a small open economy under flexible exchange rates", Journal of Monetary Economics 39(3):433~448. Eichenbaum, M. (1992), "Comment on interpreting the macroeconomic time series facts: the effects of monetary policy", European Economic Review 36(5):1001-1011. Eichenbaum, M., and C.L. Evans (1995), "Some empirical evidence on the effects of shocks to monetary policy on exchange rates", Quarterly Journal of Economics 110(4): 1975-1010. Evans, C.L., and K. Kuttner (1998), "Can VARs describe monetary policy", Research Paper 9812 (Federal Reserve Bank of New York). Faust, J., and E.M. Leeper (1997), "When do long-rtm identifying restrictions give reliable results?", Journal of Business and Economic Statistics 15(3):345-353. Fisher, J. (1997), "Monetary policy and investment", manuscript (Federal Reserve Bank of Chicago). Friedman, M., and A.J. Schwartz (1963), A Monetary History of the United States: 1867-1960 (Princeton University Press, Princeton, NJ). Fuerst, T. (1992), "Liquidity, loanable funds, and real activity", Journal of Monetary Economics 29(1): 3-24. Gall, J. (1992), "How well does the IS-LM model fit post war data?", Quarterly Journal of Economics 107(2):709 738. Gali, J. (1997), "Technology, employment, and the business cycle: do technology shocks explain aggregate fluctuations?", Working Paper No. 5721 (NBER). Gertler, M., and S. Gilchrist (1993), "The role of credit market imperfections in the monetary transmission mechanism: arguments and evidence", Scandinavian Journal of Economics 95(1):43-64. Gertler, M., and S. Gilchrist (1994), "Monetary policy, business cycles and the behavior of small manufacturing firms", Quarterly Journal of Economics 109(2):309-340.
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Geweke, J.E, and D.E. Rankle (1995), "A fine time for monetary policy?", Federal Reserve Bank of Minneapolis Quarterly Review 19(1): 18-31. Goodfriend, M. (1983), "Discount window borrowing, monetary policy, and the post-October 6:1979 Federal Reserve operating procedure", Journal of Monetary Economics 12(3):343056. Goodfriend, M. (1991), "Interest rates and the conduct of monetary policy", Carnegie-Rochester Conference Series on Public Policy 34:7-30. Gordon, D.B., and E.M. Leeper (1994), "The dynamic impacts of monetary policy: an exercise in tentative identification", Journal of Political Economy 102(6):1228-1247. Grilli, V, and N. Roubini (1995), "Liquidity and exchange rates: puzzling evidence from the G-7 Countries", Working Paper No. S/95/31 (New York University Solomon Brothers). Hamilton, J.D. (1994), Time Series Analysis (Princeton University Press, Princeton, NJ). Hamilton, J.D. (1997), "Measuring the liquidity effect", American Economic Review 87(1):80-97. Ireland, EN. (1997), "A small, structural, quarterly model for monetary policy evaluation", CarnegieRochester Conference Series on Public Policy 47:83 108. Kashyap, A.K., J.C. Stein and D.W. Wilcox (1993), "Monetary policy and credit conditions: evidence from the composition of external finance", American Economic Review 83(1):78-98. Killian, L. (1998), "Small-sample confidence intervals for impulse response functions", Review of Economics and Statistics 80(2):218-230. Kim, J. (1998), "Monetary policy in a stochastic equilibrium model with real and nominal rigidities", Finance and Economics Discussion Series, PA8-02 (Board of Governors of the Federal Reserve System). Kim, S., and N. Roubini (1995), "Liquidity and exchange rates, a structural VAR approach", manuscript (New York University). King, R.G. (1991), "Money and business cycles", Proceedings (Federal Reserve Bank of San Francisco). King, S. (1983), "Real interest rates and the interaction of money, output and prices", manuscript (Northwestern University). Krueger, J.T., and K.N. Kutmer (1996), "The fed funds futures rate as a predictor of Federal Reserve policy", Journal of Futures Markets 16(8):865-879. Leeper, E.M. (1997), "Narrative and VAR approaches to monetary policy: common identification problems", Journal of Monetary Economics 40(3):641 657. Leeper, E.M., and D.B. Gordon (1992), "In search of the liquidity effect", Journal of Monetary Economics 29(3):341-369. Leeper, E.M., C.A. Sims and T. Zha (1996), "What does monetary policy do?", Brookings Papers on Economic Activity 1996(2): 1 63. Lucas Jr, R.E. (1980), "Methods and problems in business cycle theory", Journal of Money, Credit and Banking 12(4):696 715. Lucas Jr, R.E. (1988), "Money demand in the United States: a quantitative review", Carnegie-Rochester Conference Series on Public Policy 29:137-167. Lucas Jr, R.E. (1994), "On the welfare cost of inflation", Working Papers in Applied Economic Theory 94-07 (Federal Reserve Bank of San Francisco). Mankiw, N.G., and M.D. Shapiro (1986), "News or noise: an analysis of GNP revisions", Survey of Current Business 66(5):20 25. Mankiw, N.G., D.E. Runkle and M.D. Shapiro (1984), "Are preliminary announcements of the money stock rational forecasts?", Journal of Monetary Economics 14(1): 15-27. McCallum, B.T. (1983), "A Reconsideration of Sims' Evidence Regarding Monetarism", Economic Letters 13(2,3):167-171. Mishkin, ES. (1983), "A rational expectations approach to testing macroeconomics: testing policy ineffectiveness and efficient-markets models" (University of Chicago Press, Chicago, IL). Oliner, S.D., and G.D. Rudebusch (1996), "Is there a broad credit channel for monetary policy?", Federal Reserve Bank of San Francisco Review 1:3-13.
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Pagan, A.R., and J.C. Robertson (1995), "Resolving the Liquidity Effect", Federal Reserve Bank of St. Louis Review 77(3):33-54. Parekh, G. (1997), "Small sample bias, unit roots, and conditional heteroskedasticity in macroeconomic autoregression", PhD dissertation (Northwestern University). Ramey, V.A., and M.D. Shapiro (1998), "Costly capital reallocation and the effects of government spending", Carnegie-Rochester Conference Series on Public Policy 48:145-194. Reichenstein, W. (1987), "The impact of money on short-term interest rates", Economic Inquiry 25(11): 67-82. Romer, C.D., and D.H. Romer (1989), "Does monetary policy matter? A new test in the spirit of Friedman and Schwartz", NBER Macroeconomic Annual 1989 (MIT Press, Cambridge) 121-170. Romer, C.D., and D.H. Romer (1994), "Monetary policy matters", Journal of Monetary Economics 34:75-88. Rotemberg, J.J., and M. Woodford (1992), "Oligopolistic pricing and the effects of aggregate demand on economic activity", Journal of Political Economy 100(6): 1153-1207. Rotemberg, J.J., and M. Woodford (1997), "An optimization-based econometric framework for the evaluation of monetary policy", NBER Macroeconomics Annual, 297-345. Rudebusch, G.D. (1995), "Federal Reserve interest rate targeting, rational expectations and the te~wn structure", JotLr~al of Monetary Economic 35(2):245-274. Rudebusch, G.D. (1996), Do measures of monetary policy in a VAR make sense?, Working Papers in Applied Economic Theory 96-05 (Federal Reserve Bank of San Francisco). Sargent, T.J. (1984), "Autoregressions, Expectations and Advice", American Economic Review 74(2): 408-415. Sargent, T.J. (1987), Macroeconomic Theory, 2nd edition (Academic Press, Boston, MA). Sargent, T.J. (1989), "Two models of measurements and the investment accelerator", Journal of Political Economy 97(2):251-287. Shapiro, M.D. (1994), "Federal Reserve policy: cause mad effect", in: N.G. Mankiw, ed., Monetary Policy (University of Chicago Press, Chigago, IL). Sims, C.A. (1980), "Macroeconomics and reality", Econometrica 48(1): 1-48. Sims, C.A. (1986), "Are forecasting models usable for policy analysis?", Federal Reserve Bank of Minneapolis Quarterly Review 10(1):2-16. Sims, C.A. (1992)i "Interpreting the macroeconomic time series facts: the effects of monetary policy", European Economic Review 36(5):975 1000. Sims, C.A. (1996), "Comments on 'Do measures of monetary policy in a VAR make sense?' by Glen Rudebusch", manuscript (Yale University). Sims, C.A., and T. Zba (1995), Error bands for impulse responses. Working paper number 95-6 (Federal Reserve Bank of Atlanta). Sims, C.A., and T. Zha (1998), "Does monetary policy generate recessions?", Working Paper 98-12 (Federal Reserve Bank of Atlanta). Strang, G. (1976), Linear Algebra and its Applications (Academic Press, New York). Strongin, S. (1995 ), "The identification of monetary policy disturbances: explaining the liquidity puzzle", Journal of Monetary Economics 34(3):463-497. Uhlig, H. (1997), "What are the effects of monetary policy? Results from an agnostic identification procedure", manuscript (Tilburg University). Wong, K.-E (1996), "Variability in the effects of naonetary policy on economic activity", unpublished manuscript (Universaty of Wisconsin) October.
Chapter 3
MONETARY POLICY REGIMES A N D ECONOMIC PERFORMANCE: THE HISTORICAL RECORD MICHAEL D. BORDO
Rutgers University, New Brunswick, and NBER
ANNA J. SCHWARTZ National Bureau of Economic Research, New York Contents
Abstract Keywords 1. Policy regimes, 1 8 8 0 - 1 9 9 5 1.1. Definition of a policy regime 1.2. Types of regimes 1.3. Rules vs. discretion in monetary regimes 2. International m o n e t a r y r e g i m e s 2.1. The gold standard 2.1.1. Gold as a monetary standard 2.1.2. Gold and the international monetary system 2.1.3. Central banks and the rules of the game 2.1.4. Theory of commodity money 2.1.5. The gold standard as a rule 2.1.6. The viability of the gold standard 2.2. Interwar vicissitudes of the gold standard 2.3. Bretton Woods 2.4. The recent managed float and the European Monetary System 3. Episodes in U S central b a n k i n g history 3.1. Origins of US central banking 3.2. Federal Reserve 1914 3.3. Interwar years, 1919-1941 3.3.1. 1919-1929 3.3.2. The Great Depression of 1929-1933 3.3.2.1. Policy continuity? 3.3.2.2. Banking panics 3.3.2.3. Transmission of the monetary collapse to the real economy 3.3.2.4. The October 1929 stock market crash Handbook of Macroeconomics, Volume 1, Edited by J.B. Taylor and M. WoodJbrd © 1999 Elsevier Science B. g All rights' reserved 149
151 151 152 152 152 153 153 153 154 155 156 157 158 160 161 163 167 168 168 172 175 175 178 179 180 181 183
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3.3.2.5. Would stable money have attenuated the depression? 3.3.2.6. gold standard policies in transmitting the Great Depression 3.3.3. 1933-1941 3.4. Bretton Woods, 1946 1971 3.4.1. 1946-1951 3.4.2. Federal Reserve discretionary regime, 1951-1965 3.4.3. Breakdown of Bretton Woods, 1965-1971 3.5. Post-Bretton Woods, 1971-1995 3.5.1. 1971 1980 3.5.2. Shifting the focus of monetary policy, 1980-1995 3.6. Conclusion 3.6.1. Breakdown of the gold standard, 1914-197l 3.6.2. The Great Depression, 192%1933 3.6.3. The Great Inflation, 1965-1980 4. M o n e t a r y r e g i m e s and e c o n o m i c p e r f o r m a n c e : the e v i d e n c e 4.1. Overview 4.2. Theoretical issues 4.3. Measures of macroeconomic performance, by regime 4.4. Inflation and output levels and variability 4.4.1. Inflation 4.4.2. Real per capita income growth 4.5. Stochastic properties of macrovariables 4.6. Inflation persistence, price level predictability, and their effects on financial markets 4.6.1. Inflation persistence 4.6.2. Price level uncertainty 4.6.3. Effects on financial markets 4.7. Temporary and permanent shocks 5. Overall assessment o f m o n e t a r y p o l i c y r e g i m e s Acknowledgments A p p e n d i x A. Data sources A.1. United States of America A.2. United Kingdom A.3. Germany A.4. France A.5. Japan References
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Abstract Monetary policy regimes encompass the constraints or limits imposed by custom, institutions and nature on the ability of the monetary authorities to influence the evolution of macroeconomic aggregates. This chapter surveys the historical experience of both international and domestic (national) aspects of monetary regimes from the nineteenth century to the present. We first survey the experience of four broad international monetary regimes: the classical gold standard 1880-1914; the interwar period in which a short-lived restoration of the gold standard prevailed; the postwar Bretton Woods international monetary system (1946-1971) indirectly linked to gold; the recent managed float period (1971-1995). We then present in some detail the institutional arrangements and policy actions of the Federal Reserve in the United States as an important example of a domestic policy regime. The survey of the Federal Reserve subdivides the demarcated broad international policy regimes into a number of episodes. A salient theme in our survey is that the convertibility rule or principle that dominated both domestic and international aspects of the monetary regime before World War I has since declined in its relevance. At the same time, policymakers within major nations placed more emphasis on stabilizing the real economy. Policy techniques and doctrine that developed under the pre-World War I convertible regime proved to be inadequate to deal with domestic stabilization goals in the interwar period, setting the stage for the Great Depression. In the post-World War II era, the complete abandonment of the convertibility principle, and its replacement by the goal of full employment, combined with the legacy of inadequate policy tools and theory from the interwar period, set the stage for the Great Inflation of the 1970s. The lessons from that experience have convinced monetary authorities to reemphasize the goal of low inflation, as it were, committing themselves to rule-like behavior.
Keywords gold standard, Bretton Woods, managed float, Federal Reserve, domestic policy regime, convertibility rule, stabilization goals, Great Depression, Great Inflation of the 1970s, rules, nominal anchor, exchange rate arrangements, inflation level, inflation variability, output level, output variability, trend stationary process, difference stationary process, inflation persistence, price level uncertainty, permanent shocks, temporary shocks J E L classification: E42, E52
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1. Policy regimes, 1880-1995 1.1. Definition of a policy regime Monetary policy regimes encompass the constraints or limits imposed by custom, institutions and nature on the ability o f the monetary authorities to influence the evolution o f macroeconomic aggregates. We define a monetary regime as a set o f monetary arrangements and institutions accompanied by a set o f expectations - expectations by the public with respect to policymakers' actions and expectations by policymakers about the public's reaction to their actions. By incorporating expectations, a monetary regime differs from the older concept o f a monetary standard, which referred simply to the institutions and arrangements governing the money supply 1.
1.2. Types of regimes Two types o f regimes have prevailed in history: one based on convertibility into a commodity, generally specie, and the other based on fiat. The former prevailed in the U S A in various guises until Richard Nixon closed the gold window in August 1971, thereby terminating the gold convertibility feature o f the Bretton Woods international monetary system. The latter is the norm worldwide today. The two types o f regimes relate closely to the concept o f a nominal anchor to the monetary system. A nominal anchor is a nominal variable that serves as a target for monetary policy. Under specie convertible regimes, the currency price o f specie (gold and/or silver coin) is the nominal anchor. Convertibility at that price ensures that price levels will return to some mean value over long periods o f time 2. Regimes have both a domestic (national) and international aspect. The domestic aspect pertains to the institutional arrangements and policy actions of monetary authorities. The international aspect relates to the monetary arrangements between nations. Two basic types o f international monetary arrangements prevail - fixed and flexible exchange rates, along with a number o f intermediate variants including adjustable pegs and managed floating.
I See Leijonhufvud (1984) and Bordo and Jonung (1996). Eichengreen (1991a, p. 1) defines "a regime as an equilibrium in which a set of rules or procedures governing the formulation of public policy generates stable expectations among market participants". He views a monetary regime "as a set of rules or procedures affecting money's ability to provide one or more of [the] three functions [of money]". 2 A moving nominal anchor is used by central banks today. The monetary authorities pursue an inflation target based on the desired growth rate of a nominal variable, treating the inherited past as bygones. In this regime, although the inflation rate is anchored, the price level rises indefinitely [Flood and Mussa (1994)].
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1.3. Rules vs. discretion in monetary regimes
Alternative monetary regimes can be classified as following rules or discretion. The convertible metallic regimes that prevailed into the twentieth century were based on a rule - adherence to the fixed price of specie. The rule operated in both the domestic and the international aspects of the regime. In the international aspect, maintenance of the fixed price of specie at its par value by its adherents ensured fixed exchange rates. The fixed price of domestic currency in terms of specie provided a nominal anchor to the international monetary system. Fiat or inconvertible regimes can also be based on rules if the authorities devise and credibly commit to them. At the domestic level, setting the growth rates of monetary aggregates or those targeting the price level are exanaples of rules. At the international level, fixed exchange rate regimes such as the European Monetary System (EMS) are based on a set of well-understood intervention principles and the leadership of a country dedicated to maintaining the nominal anchor. This chapter surveys the historical experience of both international and domestic (national) aspects of monetary regimes from the nineteenth century to the present. We first survey the experience of four broad international monetary regimes: the classical gold standard 1880-1914; the interwar period in which a short-lived restoration of the gold standard prevailed; the postwar Bretton Woods international monetary system (1946-1971) indirectly linked to gold; the recent managed float period (1971-1995). We then present in some detail the institutional arrangements and policy actions of the Federal Reserve in the United States as an important example of a domestic policy regime. The survey of the Federal Reserve subdivides the demarcated broad international policy regimes into a number of episodes. A salient theme in our survey is that the convertibility rule or principle that dominated both domestic and international aspects of the monetary regime before World War I has since declined in its relevance. At the same time, policymakers within major nations placed more emphasis on stabilizing the real economy. Policy techniques and doctrine that developed under the pre-World War I convertible regime proved to be inadequate to deal with domestic stabilization goals in the interwar period, setting the stage for the Great Depression. In the post-World War II era, the complete abandonment of the convertibility principle, and its replacement by the goal of full employment, combined with the legacy of inadequate policy tools and theory from the interwar period, set the stage for the Great Inflation of the 1970s. The lessons from that experience have convinced monetary authorities to reemphasize the goal of low inflation, as it were, committing themselves to rule-like behavior.
2. International monetary regimes 2.1. The gold standard
The classical gold standard which ended in 1914 served as the basis of the convertibility principle that prevailed until the third quarter of the twentieth century.
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We discuss five themes that dominate an extensive literature. The themes are: gold as a monetary standard; gold and the international monetary system; central banks and the "rules of the game"; the commodity theory of money; the gold standard as a rule. 2.1.1. G o l d as a m o n e t a r y s t a n d a r d
Under a gold standard the monetary authority defines the weight of gold coins, or alternatively fixes the price of gold in terms of national currency. The fixed price is maintained by the anthority's willingness freely to buy and sell gold at the mint price. There are no restrictions to the ownership or use of gold. The gold standard evolved from earlier commodity money systems. Earlier commodity money systems were bimetallic - gold was used for high-valued transactions, silver or copper coins for low-valued ones. The bimetallic ratio (the ratio of the mint price of gold relative to the mint price of silver) was set close to the market ratio to ensure that both metals circulated. Otherwise, Gresham's Law ensured that the overvalued metal would drive the undervalued metal out of circulation. The world switched from bimetallism to gold monometallism in the 1870s. Debate continues to swirl over the motivation for the shift. Some argue that it was primarily political [Friedman (1990a), Gallarotti (1995), Eichengreen ( 1996)] - nations wished to emulate the example of England, the world's leading commercial and industrial power. When Germany used the Franco-Prussian War indemnity to finance the creation of a gold standard, other prominent European nations also did so 3. Others argue that massive silver discoveries in the 1860s and 1870s as well as technical advances in coinage were the key determinants [Redish (1990)]. Regardless of the cause, recent research suggests that the shift both was unnecessary and undesirable since France, the principal bimetallic nation, had large enough reserves of both metals to continue to maintain the standard [Oppers (1996), Flandreau (1996)]; and because remaining on a bimetallic standard, through the production and substitution effects earlier analyzed by Irving Fisher (1922), would have provided greater price stability than did gold monometallism [Friedman (1990b)]. The simplest variant of the gold standard was a pure gold coin standard. Such a system entails high resource costs and, consequently in most countries, substitutes for gold coin emerged. In the private sector, commercial banks issued notes and deposits convertible into gold coins, which in turn were held as reserves to meet conversion demands. In the public sector, prototypical central banks (banks of issue) were established to help governments finance their ever expanding fiscal needs [Capie, Goodhart and Schnadt (1994)]. These notes were also convertible, backed by gold reserves. In
3 Gallarotti (1995) describes the shift of political power in favor of the gold standard in Germany. See Friedman and Schwartz (1963) and Friedman (1990b) for a discussion of the US switch de facto to gold in 1879.
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wartime, convertibility was suspended, but always on the expectation of renewal upon termination of hostilities. Thus the gold standard evolved into a mixed coin and fiduciary system based on the principle of convertibility. A key problem with the convertible system was the risk of conversion attacks of internal drains when a distrustful public attempted to convert commercial bank liabilities into gold; and external drains when foreign demands on a central bank's gold reserves threatened its ability to maintain convertibility. In the face of this rising tension between substitution of fiduciary money for gold and the stability of the system, central banks learned to become lenders of last resort and to use the tools of monetary policy to protect their gold reserves [Bagehot (1873), Redish (1993), Rockoff (1986)]. The gold standard, both the pure coin variety and the more common mixed standards, were domestic monetary standards which evolved in most countries through market driven processes. By defining its unit of account as a fixed weight of gold or alternatively by fixing the price of gold, each monetary authority also fixed its exchange rate with other gold standard countries and became part of an international gold standard. 2.1.2. Gold and the international monetary system
The international gold standard evolved from domestic standards by the fixing of the price of gold by member nations. Under the classical gold standard fixed exchange rate system, the world's monetary gold stock was distributed according to the member nations' demands for money and use of substitutes for gold. Disturbances to the balance of payments were automatically equilibrated by the Humean price-specie flow mechanism. Under that mechanism, arbitrage in gold kept nations' price levels in line. Gold would flow from countries with balance of payments deficits (caused, for example, by higher price levels) to those with surpluses (caused by lower price levels), in turn keeping their domestic money supplies and price levels in line. Some authors stressed the operation of the law of one price and commodity arbitrage in traded goods prices, others the adjustment of the terms of trade, still others the adjustment of traded relative to nontraded goods prices [Bordo (1984)]. Debate continues on the details of the adjustment mechanism; however, there is consensus that it worked smoothly for the core countries of the world although not necessarily for the periphery [Ford (1962), DeCecco (1974), Fishlow (1985)]. It also facilitated a massive transfer of long-term capital from Europe to the new world in the four decades before World War I on a scale relative to income which has yet to be replicated. Although in theory exchange rates were supposed to be perfectly rigid, in practice the rate of exchange was bounded by upper and lower limits - the gold points - within which the exchange rate floated. The gold points were determined by transactions costs, risk, and other costs of shipping gold. Recent research indicates that although in the classical period exchange rates frequently departed from par, violations of the gold points were rare [Officer (1986, 1996)], as were devaluations [Eichengreen (1985)]. Adjustment to balance of payments disturbances was greatly facilitated by short-term
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capital flows. Capital would quickly flow between countries to iron out interest rate differences. By the end of the nineteenth century the world capital market was so efficient that capital flows largely replaced gold flows in effecting adjustment. 2.1.3. Central banks and the rules o f the game
Central banks also played an important role in the international gold standard. By varying their discount rates and using other tools of monetary policy they were supposed to follow "the rules of the game" and speed up adjustment to balance of payments disequilibria. In fact many central banks violated the rules [Bloomfield (1959), Dutton (1984), Pippenger (1984), Giovannini (1986), Jeanne (1995), Davutyan and Parke (1995)] by not raising their discount rates or by using "gold devices" which artificially altered the price of gold in the face of a payments deficit [Sayers (1957)]. But the violations were never sufficient to threaten convertibility [Schwartz (1984)]. They were in fact tolerated because market participants viewed them as temporary attempts by central banks to smooth interest rates and economic activity while keeping within the overriding constraint of convertibility [Goodfriend (1988)]. An alternative interpretation is that violations of the rules of the game represented the operation of an effective target zone bordered by the gold points. Because of the credibility of commitment to gold convertibility, monetary authorities could alter their discount rates to affect domestic objectives by exploiting the mean reversion properties of exchange rates within the zone [Svensson (1994), Bordo and MacDonald (1997)]. An alternative to the view that the gold standard was managed by central banks in a symmetrical fashion is that it was managed by the Bank of England [Scammell (1965)]. By manipulating its Bank rate, it could attract whatever gold it needed; furthermore, other central banks adjusted their discount rates to hers. They did so because London was the center for the world's principal gold, commodities, and capital markets, outstanding sterling-denominated assets were huge, and sterling served as an international reserve currency (as a substitute for gold). There is considerable evidence supporting this view [Lindert (1969), Giovannini (1986), Eichengreen (1987)]. There is also evidence which suggests that the two other European core countries, France and Germany, had some control over discount rates within their respective economic spheres [Tullio and Wolters (1996)]. Although the gold standard operated smoothly for close to four decades, there were periodic financial crises. In most cases, when faced with both an internal and an external drain, the Bank of England and other European central banks followed Bagehot's rule of lending freely but at a penalty rate. On several occasions (e.g. 1890 and 1907) even the Bank of England's adherence to convertibility was put to the test and, according to Eichengreen (1992), cooperation with the Banque de France and other central banks was required to save it. Whether this was the case is a moot point. The cooperation that did occur was episodic, ad hoc, and not an integral part of the operation of the gold standard. Of greater importance is that, during periods of financial crisis, private capital flows aided the Bank. Such stabilizing capital movements likely
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reflected market participants' belief in the credibility of England's commitment to convertibility. By the eve of World War I, the gold standard had evolved de facto into a gold exchange standard. In addition to substituting fiduciary national monies for gold to economize on scarce gold reserves, many countries also held convertible foreign exchange (mainly deposits in London). Thus the system evolved into a massive pyramid of credit built upon a tiny base of gold. As pointed out by Triffin (1960), the possibility of a confidence crisis, triggering a collapse of the system, increased as the gold reserves of the center diminished. The advent of World War I triggered such a collapse as the belligerents scrambled to convert their outstanding foreign liabilities into gold. 2.1.4. Theory of commodity money
The gold standard contained a self-regulating mechanism that ensured long-run monetary and price level stability, namely, the commodity theory of money. This was most clearly analyzed by Irving Fisher (1922) although well understood by earlier writers. The price level of the world, treated as a closed system, was determined by the interaction of the money market and the commodity or bullion market. The real price (or purchasing power of gold) was determined by the commodity market; and the price level was determined by the demand for and supply of monetary gold. The demand for monetary gold was derived from the demand for money while the monetary gold stock was the residual between the total world gold stock and the nonmonetary demand. Changes in the monetary gold stock reflected gold production and shifts between monetary and nonmonetary uses of gold [Barro (1979)]. Under the self-equilibrating gold standard, once-for-all shocks to the demand for or supply of monetary gold would change the price level. These would be reversed as changes in the price level affected the real price of gold, leading to offsetting changes in gold production and shifts between monetary and nonmonetary uses of gold. This mechanism produced mean reversion in the price level and a tendency towards longrun price stability. In the shorter run, the shocks to the gold market or to real activity created price level volatility. Evidence suggests that the mechanism worked roughly according to the theory [Cagan (1965), Bordo (1981), Rockoff (1984)] but other factors are also important - including government policy towards gold mining and the level of economic activity [Eichengreen and McLean (1994)]. This simple picture is complicated by a number of important considerations. These include technical progress in gold mining; the exhaustion of high quality ores; and depletion of gold as a durable exhaustible reserve. With depletion, in the absence of offsetting technical change, a gold standard must inevitably result in long-run deflation [Bordo and Ellson (1985)]. Although there is evidence that the gold standard was self-regulating, the lags involved were exceedingly long and variable (between 10 and 25 years, according to Bordo (1981), so that many observers have been unwilling to rely on the mechanism as a basis for world price stability, and prominent contemporary
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authorities advocated schemes to improve upon its performance. Others, e.g., Keynes (1930), doubted the operation of the self-regulating mechanism and attributed whatever success the gold standard had before 1914 to purely adventitious acts - timely gold discoveries in Australia and California in the 1850s, invention of the cyanide process in the 1880s, and gold discoveries in South Africa and Alaska in the 1890s. 2.1.5. T h e g o l d s t a n d a r d a s a rule
One of the most important features of the gold standard was that it embodied a monetary rule or commitment mechanism that constrained the actions of the monetary authorities. To the classical economists, forcing monetary authorities to follow rules was viewed as preferable to subjecting monetary policy to the discretion of wellmeaning officials. Today a rule serves to bind policy actions over time. This view of policy rules, in contrast to the earlier tradition that stressed both impersonality and automaticity, stems from the recent literature on the time inconsistency of optimal government policy. In terms of the modern perspective of Kydland and Prescott (1977) and Barro and Gordon (1983), the rule served as a commitment mechanism to prevent governments from setting policies sequentially in a time inconsistent manner. According to this approach, adherence to the fixed price of gold was the commitment that prevented governments from creating surprise fiduciary money issues in order to capture seigniorage revenue, or from defaulting on outstanding debt [Bordo and Kydland (1996), Giovannini (1993)]. On this basis, adherence to the gold standard rule before 1914 enabled many countries to avoid the problems of high inflation and stagflation that troubled the late twentieth century. The gold standard rule in the century before World War I can also be interpreted as a contingent rule, or a rule with escape clauses [Grossman and Van Huyck (1988), DeKock and Grilli (1989), Flood and Isard (1989), Bordo and Kydland (1996)]. The monetary authority maintained the standard - kept the price of the currency in terms of gold fixed - except in the event of a well understood emergency such as a major war. In wartime it might suspend gold convertibility and issue paper money to finance its expenditures, and it could sell debt issues in terms of the nominal value of its currency, on the understanding that the debt would eventually be paid off in gold or in undepreciated paper. The rule was contingent in the sense that the public understood that the suspension would last only for the duration of the wartime emergency plus some period of adjustment, and that afterwards the government would adopt the deflationary policies necessary to resume payments at the original parity. Observing such a rule would allow the government to smooth its revenue from different sources of finance: taxation, borrowing, and seigniorage [Lucas and Stokey (1983), Mankiw (1987)]. That is, in wartime when present taxes on labor effort would reduce output when it was needed most, using future taxes or borrowing would be optimal. At the same time positive collection costs might also make it optimal to use the inflation tax as a substitute for conventional taxes [Bordo and V6gh (1998)].
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A temporary suspension o f convertibility would then allow the government to use the optimal mix o f the three taxes 4. It is crucial that the rule be transparent and simple and that only a limited number o f contingencies be included. Transparency and simplicity avoided the problems o f moral hazard and incomplete information [Canzoneri (1985), Obstfeld (1991)], i.e., prevented the monetary authority from engaging in discretionary policy under the guise o f following the contingent rule. In this respect a second contingency - a temporary suspension in the face o f a financial crisis, which in turn was not the result o f the monetary authority's own actions - might also have been part o f the rule. However, because o f the greater difficulty o f verifying the source o f the contingency than in the case o f war, invoking the contingency under conditions o f financial crisis, or in the case o f a shock to the terms o f trade - a third possible contingency - would be more likely to create suspicion that discretion was the order o f the day. The basic gold standard rule is a domestic rule and it was enforced by the reputation o f the gold standard itself, i.e., by the historical evolution o f gold as money. A n alternative commitment mechanism was to guarantee gold convertibility in the constitution as was done in Sweden before 1914 [Jonung (1984)]. The gold standard contingent rule worked successfully for the "core" countries o f the classical gold standard: Britain, France, and the U S A [Bordo and Schwartz (1996a)]. In all these countries the monetary authorities adhered faithfully to the fixed price o f gold except during major wars. During the Napoleonic War and World War I for England, the Civil War for the U.SA., and the Franco-Prussian War for France, specie payments were suspended and paper money and debt were issued. But in each case, after the wartime emergency had passed, policies leading to resumption at the prewar parity were adopted. Indeed, successful adherence to the pre-World War I rule may have enabled the belligerents to obtain access to debt finance more easily in subsequent wars. In the case o f Germany, the fourth "core" country, no occasions arose for application o f the contingent aspect o f the rule before 1914. Otherwise its record o f adherence to gold convertibility was similar to that o f the other three countries. Unlike the core countries, a number o f peripheral countries had difficulty in following the rule
4 The evidence on revenue smoothing is mixed. According to Mankiw (1987), both the inflation tax and conventional taxes should follow a Martingale process and a regression of the inflation rate on the average tax rate should have a positive and significant coefficient as the former as well as Poterba and Rotemberg (1990) and Trehan and Walsh (1990) found for the post-World War I United States. However, Bordo and White (1993) for the Napoleonic War suspension of convertibility by Britain, Lazaretou (1995) for Greece in periods of inconvertibility in the nineteenth century, and Goff and Toma (1993) for the USA under the classical gold standard reject the hypothesis of revenue smoothing but not that of tax smoothing. As Goff and Toma (1993) argue, seigniorage smoothing would not be expected to prevail under a specie standard where the inflation rate does not exhibit persistence (which was the case during the British and during the Greek inconvertibility episodes). The Bordo and White, and Lazaretou results suggest that, although specie payments were suspended, the commitment to resume prevented the government from acting as it would under the pure fiat regime postulated by the theory.
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and their experience was characterized by frequent suspensions of convertibility and devaluations. One author argues that the commitment to gold convertibility by England and the other core countries was made possible by a favorable conjuncture of political economy factors. The groups who were harmed by the contractionary policies, required in the face of a balance of payments deficit to maintain convertibility, did not have political power before 1914. By contrast, in some peripheral countries, powerful political groups, e.g., Argentine ranchers and American silver miners, benefited from inflation and depreciation [Eichengreen (1992)]. The gold standard rule originally evolved as a domestic commitment mechanism but its enduring fame is as an international rule. As an international standard, the key rule was maintenance of gold convertibility at the established par. Maintenance of a fixed price of gold by its adherents in turn ensured fixed exchange rates. The fixed price of domestic currency in terms of gold provided a nominal anchor to the international monetary system. According to the game theoretic literature, for an international monetary arrangement to be effective both between countries and within them, a time-consistent credible commitment mechanism is required [Canzoneri and Henderson (1991)]. Adherence to the gold convertibility rule provided such a mechanism. Indeed, Giovannini (1993) finds the variation of both exchange rates and short-term interest rates within the limits set by the gold points in the 1899-1909 period consistent with market agents' expectations of a credible commitment by the core countries to the gold standard rule. In addition to the reputation of the domestic gold standard and constitutional provisions which ensured domestic commitment, adherence to the international gold standard rule may have been enforced by other mechanisms [see Bordo and Kydland (1996)]. These include: the operation of the rules of the game; the hegemonic power of England; central bank cooperation; and improved access to international capital markets. Indeed the key enforcement mechanism of the gold standard rule for peripheral countries was access to capital obtainable from the core countries. Adherence to the gold standard was a signal of good behavior, like the "good housekeeping seal of approval"; it explains why countries that always adhered to gold convertibility paid lower interest rates on loans contracted in London than others with less consistent performance [Bordo and Rockoff (1996)]. 2.1.6. The viability of" the goM standard
The classical gold standard collapsed in 1914. It was reinstated as a gold exchange standard between 1925 and 1931, and as the gold dollar standard from 1959 to 1971. The gold standard, while highly successful for a time, lost credibility in its 20th century reincarnations and was formally abandoned in 1971. Among the weaknesses which contributed to its abandonment were the cost of maintaining a full-bodied gold standard. Friedman (1953) estimated the cost for the USA in 1950 as 1.5 percent of real GNR Shocks to the demand for and supply of
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gold that produced drift in the price level also weakened support for the gold standard, leading many economists to advocate schemes for reform [Cagan (1984)]. Finally, in a growing world, the gold standard, based on a durable exhaustible resource, posed the prospect of deflation. The key benefits of the gold standard, in hindsight, were that it provided a relatively stable nominal anchor and a commitment mechanism to ensure that monetary authorities followed time consistent policies. However, the gold standard rule of maintaining a fixed price of gold meant, for a closed economy, that continuous full employment was not a viable policy objective and, for an open economy, that domestic policy considerations would be subordinated to those of maintaining external balance. In the twentieth century few countries have been willing to accept the gold standard's discipline [Schwartz (1986b)]. 2.2. Interwar vicissitudes o f the goM standard
The outbreak of World War I in August 1914 led to a massive worldwide financial crisis as investors across the world scrambled to liquidate sterling and other financial assets in exchange for domestic currency and gold. The response to the crisis and the need by the European belligerents for gold to pay for war material led to the breakdown of the gold standard. After the war the UK and other countries expressed a strong preference to return to gold parity at the original parity following the gold standard contingent rule [see the Cunliffe Report (1918)]. At the Genoa Conference in 1922, the Financial Commission, under British leadership, urged that the world return to the gold standard. However, the system they advocated was a gold exchange standard that encouraged member countries to make their currencies convertible into gold but to use foreign exchange (the currencies of key reserve countries, the UK and the USA) as a substitute for gold. The experts also encouraged members to restrict the use of gold as currency, thus establishing a gold bullion standard, and to cooperate when raising or lowering their discount rates to prevent competition for gold. The motivation to economize on gold was a belief that the world would suffer a severe gold shortage in coming decades. The gold standard was restored worldwide in the period 1924-1927. It only lasted globally until 1931. The key event in its restoration was the return in April 1925 by the UK to convertibility at the prewar parity of $4.86. It is believed to have overvalued sterling between 5 and 15 percent depending on the price index used [Keynes (1925), Redmond (1984)] 5.
5 A vociferousdebate continues between the followers of Keynes who attribute the UK's weak economic performance and high unemploymentin the 1920s to the decision to return to gold at an overvalued parity, and those who attribute the high unemploymentto policies that raised the replacement ratio (the ratio of unemployment benefits to money wages), as well as other supply side factors. See, e.g., Pollard (1970); Thomas (1981); and Benjamin and Kochin (1979, 1982). For a recent discussion of the economics of resumption in 1925, see Bayoumiand Bordo (1998).
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Countries with high inflation, such as France and Italy, returned to gold but at a greatly devalued parity. It took France seven years to stabilize the franc after the war. As described by Eichengreen (1992), the franc depreciated considerably in the early 1920s reflecting a war of attrition between the left and the right over the financing of postwar reconstruction and over new fiscal programs [Alesina and Drazen (1991)]. The weakness of the franc was halted by Poincar6's 1926 stabilization program which restored budget balance, low money growth, and an independent central bank [Sargent (1984), Prati (1991)]. Germany, Austria, and other countries, which had endured hyperinflation, all stabilized their currencies in 1923/1924 and, with the aid of the League of Nations, all returned to gold convertibility at greatly devalued parities 6. The gold standard was restored on the basis of the recommendations of Genoa. Central bank statutes typically required a cover ratio for currencies of between 30 and 40 percent, divided between gold and foreign exchange. Central reserve countries were to hold reserves only in the form of gold. The gold exchange standard suffered from a number of serious flaws compared to the prewar gold standard [Kindleberger (1973), Temin (1989), Eichengreen (1992, 1996)]. The first problem was the adjustment problem. The UK with an overvalued currency ran persistent balance of payments deficits and gold outflows which imparted deflationary pressure, and in the face of sticky prices and wages, low growth, and high unemployment. This also required the Bank of England to adopt tight monetary policies to defend convertibility. At the other extreme, France with an undervalued currency enjoyed payments surpluses and gold inflows. The Banque de France did not allow the gold inflows to expand the money supply and raise the price level. It sterilized the inflows and absorbed monetary gold from the rest of the world 7. At the same time the USA, the world's largest gold holder, also sterilized gold inflows and prevented the adjustment mechanism from operating [Friedman and Schwartz (1963)]. The second problem was the liquidity problem. Gold supplies were believed to be inadequate to finance the growth of world trade and output. This in turn was a legacy of high World War I inflation which reduced the real price of gold. The League of Nations in the First Interim Report of the Gold Delegation [League of Nations (1930)] tried to forecast the world demand for and supply of gold in the next decade. The Report argued that, unless further attempts to economize on gold succeeded, the world was destined to suffer from massive deflation. That happened in the period 1929-1933, not because of a gold shortage but because of the Great Depression [Bordo and Eichengreen (1998)].
6 Accordingto Sargent (1984), because the reform package was credibly believed to signal a change in the monetary regime, the price level stabilized with no adverse real effects. Wicker (1986), by contrast, presents evidence of a substantial increase in unemployment in Austria, Hungary, and Poland, which persisted for several years. 7 According to Eichengreen (1992), a change in the statutes of the Banque de France following the Poincar6 stabilization, prevented the Banque from using open market operations to expand the money supply. Meltzer (1995b, Chapter 5) disputes this interpretation, arguing that the Banque was not required to deflate the world economyby selling foreign exchange for gold.
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In the face of the perceived gold shortage, following the strictures of Genoa, central banks substituted foreign exchange for gold. This in turn created a confidence problem. As outstanding pounds and dollars increased relative to gold reserves in London and New York, the greater the likelihood that some shock would lead to a speculative attack on sterling or the dollar by foreign holders fearful that they would be unable to convert their balances. Indeed this is what happened to sterling in 1931 [Capie, Mills and Wood (1986)] and to the dollar in 1933 [Wigmore (1987)]. The final problem plaguing the gold exchange standard was a lack of credibility. A change in the political complexion of many European countries (the growth of labor unions and left-wing parties) after World War I made it more difficult to defend convertibility if it meant adopting deflationary monetary policy [Eichengreen (1992, 1996), Simmons (1994)]. Speculative attacks made short-term capital flows destabilizing instead of stabilizing, as they were before World War I. The lack of credibility could have been offset, according to Eichengreen (1992), by increased central bank cooperation but it was not forthcoming. The system collapsed in the face of the shocks of the Great Depression 8. 2.3. Bretton Woods
Bretton Woods was the world's last convertible regime. It fits within the context of the gold standard because the USA, the most important commercial power, defined its parity in terms of gold and all other members defined their parities in terms of dollars. The planning that led to Bretton Woods aimed to avoid the chaos of the interwar period [Ikenberry (1993)]. The ills to be avoided were deduced from the historical record: floating exchange rates, condemned as prone to destabilizing speculation in the early 1920s; the subsequent gold exchange standard that enforced the international transmission of deflation in the early 1930s; and devaluations after 1933 that were interpreted as beggar-thy-neighbor actions and declared to be wrong, as was resort to trade restrictions, exchange controls, and bilateralism [Nurkse (1944)]. To avoid these ills, an adjustable peg system was designed that was expected to combine the favorable features of the fixed exchange rate gold standard and flexible exchange rates. Both John Maynard Keynes representing the UK and Harry Dexter White representing the United States planned an adjustable peg system to be coordinated by an
Eichengreen (1990) contrasts two alternative explanations for the collapse of the gold exchange standard: it collapsed after the start of the Great Depression in 1929 because of a scramble by central banks for gold in the face of a loss of confidence in the reserve country currencies; it collapsed as a consequence of inappropriate policies followedby the USA and France in sterilizing gold inflows and thereby creating deflationarypressure on the international monetary system. Cross-countryregressions for 24 countries over the period 1929-1935 explaining the demands for international reserves, gold and foreign exchange, including dummy variables for the USA and France, provide strong support for the latter hypothesis.
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international monetary agency. The Keynes plan gave the International Clearing Union substantially more resources and power than White's United Nations Stabilization Fund, but both institutions were to exert considerable power over the domestic financial policy of the members. The British plan contained more domestic policy autonomy than did the US plan, while the American plan put more emphasis on exchange rate stability. The Articles of Agreement signed at Bretton Woods, New Hampshire, in July 1944 represented a compromise between the American and British plans. It combined the flexibility and freedom for policy makers of a floating rate system which the British team wanted, with the nominal stability of the gold standard rule emphasized by the USA. The system established was a pegged exchange rate, but members could alter their parities in terms of gold and the dollar in the face of a fundamental disequilibrium. Members were encouraged to rely on domestic stabilization policy to offset temporary disturbances to their payments balances and they were protected from speculative attack by capital controls. The International Monetary Fund (IMF) was to provide temporary liquidity assistance and to oversee the operation of the system [Bordo (1993a)]. Although based on the principle of convertibility, with the USA rather than England as the center country, Bretton Woods differed from the classical gold standard in a number of fundamental ways. First, it was an arrangement mandated by an international agreement between governments, whereas the gold standard evolved informally. Second, domestic policy autonomy was encouraged even at the expense of convertibility, in sharp contrast to the gold standard where convertibility was key. Third, capital movements were suppressed by controls [Marston (1993), Obstfeld and Taylor (1998)]. The Bretton Woods system faced a number of problems in getting started, and it took 12 years before the system achieved full operation. Each of the two key problems in the early years - bilateralism and the dollar shortage - was largely solved by developments outside the Bretton Woods arrangements. The dollar shortage was solved by massive US Marshall Plan aid and the devaluation of sterling and other currencies in 1949. Multilateralism was eventually achieved in Western Europe in 1958 following the establishment in 1950 of the European Payments Union [Eichengreen (1995)]. The period 1959-1967 was the heyday of Bretton Woods. The system had become a gold dollar standard whereby the United States pegged the price of gold and the rest of the world pegged their currencies to the dollar. The dollar emerged as the key reserve currency in this period, reflecting both its use as an intervention currency and a growing demand by the private sector for dollars as international money. This growth in dollar demand reflected stable US monetary policy. Also the system evolved a different form of international governance than envisioned at Bretton Woods. The IMF's role as manager was eclipsed by that of the USA in competition with the other members of the G-10. According to Dominguez (1993), although the IMF provided many valuable services, it was not successful in serving as a commitment mechanism.
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The Bretton Woods system, in its convertible phase from 1959 to 1971, was characterized by exceptional macroeconomic performance in the advanced countries (see Section 4 below). It had the lowest and most stable inflation rate and highest and most stable real growth rates of any modern regime. However, it was short-lived. Moreover, it faced smaller demand and supply shocks than under the gold standard. This suggests that the reason for the brevity of its existence was not the external environment but, as with the gold exchange standard, structural flaws in the regime and the lack of a credible commitment mechanism by the center reser,)e country. The three problems of adjustment, liquidity, and confidence dominated academic and policy discussions during this period. The debate surrounding the first focused on how to achieve adjustment in a world with capital controls, fixed exchange rates, and domestic policy autonomy. Various policy measures were proposed to aid adjustment [Obstfeld (1993)]. For the United States, the persistence of balance of payments deficits after 1957 was a source of concern. For some it demonstrated the need for adjustment; for others it served as the means to satisfy the rest of the world's demand for dollars. For monetary authorities the deficit was a problem because of the threat of a convertibility crisis, as outstanding dollar liabilities rose relative to the US monetary gold stock. US policies to restrict capital flows and discourage convertibility did not solve the problem. The main solution advocated for the adjustment problem was increased liquidity. Exchange rate flexibility was strongly opposed. The liquidity problem evolved from a shortfall of monetary gold beginning in the late 1950s. The gap was increasingly made up by dollars, but, because of the confidence problem, dollars were not a permanent solution. New sources of liquidity were required, answered by the creation of Special Drawing Rights (SDRs). However, by the time SDRs were injected into the system, they exacerbated worldwide inflation [Genberg and Swoboda (1993)]. The key problem of the gold-dollar system was how to maintain confidence. If the growth of the monetary gold stock was not sufficient to finance the growth of world real output and to maintain US gold reserves, the system would become dynamically unstable [Triffin (1960), Kenen (1960)]. Indeed the system was subject to growing speculative attacks, in which market agents anticipated the inconsistency between nations' financial policies and maintenance of pegged exchange rates [Garber and Flood (1984), Garber (1993)]. Although capital flows were blocked in most countries, controls were increasingly evaded by various devices including the use of leads and lags - the practice of accelerating payments in domestic currency and delaying foreign currency receipts in the expectation of a devaluation of the domestic currency [Obstfeld and Taylor (1998)]. Thus successful attacks occurred against sterling in 1947, 1949 and 1967 and the franc in 1968 [Bordo and Schwartz (1996b)]. From 1960 to 1967, the United States adopted a number of policies to prevent conversion of dollars into gold. These included the Gold Pool, swaps, Roosa bonds, and moral suasion. The defense of sterling was a first line of defense for the dollar. When none of the measures worked the dollar itself was attacked via a run on the
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London gold market in March 1968 leading to the adoption of the two-tier gold market arrangement. This solution temporarily solved the problem by demonetizing gold at the margin and hence creating a de facto dollar standard. The Bretton Woods system collapsed between 1968 and 1971 in the face of US monetary expansion that exacerbated worldwide inflation. The United States broke the implicit rules of the dollar standard by not maintaining price stability [Darby et al. (1983)]. The rest of the world did not want to absorb dollars and inflate. They were also reluctant to revalue. The Americans were forced by British and French decisions to convert dollars into gold. The impasse was resolved by President Richard Nixon's closing of the gold window, ending convertibility on 15 August 1971. Another important source of strain on the system was the unworkability of the adjustable peg under increasing capital mobility. Speculation against a fixed parity could not be stopped by either traditional policies or international rescue packages. The breakdown of Bretton Woods marked the end of US financial predominance in the international monetary system. The absence of a new center of international management set the stage for a multipolar system. Under the Bretton Woods system, as under the classical gold standard, a set of rules was established, based on the convertibility of domestic currency into gold, although under Bretton Woods only the United States was required to maintain it 9. Also, as under the gold standard, the rule was a contingent one. Under Bretton Woods the contingency, which would allow a change of parity, was a fundamental disequilibrium in the balance of payments, although fundamental disequilibrium was never clearly defined. Unlike the example of Britain under the gold standard, however, the commitment to maintain gold convertibility by the USA, the center country, lost credibility by the mid-1960s. Also the contingency aspect of the rule proved unworkable. With fundamental disequilibrium being ill-defined, devaluations were avoided as an admission of failed policy. In addition, devaluations invited speculative attack even in the presence of capital controls. Once controls were removed, the system was held together only by G-10 cooperation and once inconsistencies developed between the interests of the USA and other members, even cooperation became unworkable. In conclusion, under Bretton Woods gold still served as a nominal anchor. This link to gold likely was important in constraining US monetary policy, at least until the mid-1960s, and therefore that of the rest of the world. This may explain the low inflation rates and the low degree of inflation persistence observed in the 1950s and 1960s [Alogoskoufis and Smith (1991), Bordo (1993b)]. However, credibility was considerably weaker than under the gold standard and it was not as effective a nominal anchor [Giovannini (1993)]. Moreover, when domestic interests clashed with convertibility, the anchor chain was stretched and then discarded [Redish (1993)]. This was evident in the US reduction and then removal of gold reserve requirements in 1965
9 McKinnon (1993) also views Bretton Woods and the gold standard as regimes based on a set of rules.
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and 1968, the closing of the Gold Pool in 1968 and the gold window itself in 1971. The adoption of the Second Amendment to the IMF Articles of Agreement in 1976 marked the absolute termination of a role for gold in the international monetary system. With the closing of the gold window and the breakdown of the Bretton Woods system, the last vestiges of the fixed nominal anchor of the convertibility regime disappeared. The subsequent decade under a fiat money regime and floating exchange rates exhibited higher peacetime inflation in advanced countries than in any other regime. An interesting unanswered question is whether the demise of the fixed nominal anchor and the convertibility principle explains the subsequent inflation or whether a change in the objectives of monetary authorities - full employment rather than convertibility and price stability - explains the jettisoning of the nominal anchor.
2.4. The recent managed float and the European Monetary System As a reaction to the flaws of the Bretton Woods system, the world turned to generalized floating exchange rates in March 1973. Though the early years of the floating exchange rate were often characterized as a dirty float, whereby monetary authorities extensively intervened to affect both the levels of volatility and exchange rates, by the 1990s it evolved into a system where exchange market intervention occurred primarily with the intention of smoothing fluctuations. Again in the 1980s exchange market intervention was used by the Group of Seven countries as part of a strategy of policy coordination. In recent years, floating exchange rates have been assailed from many quarters for excessive volatility in both nominal and real exchange rates, which in turn increase macroeconomic instability and raise the costs of international transactions. Despite these problems, the ability of the flexible regime to accommodate the problems of the massive oil price shocks in the 1970s as well as other shocks in subsequent years without significant disruption, as well as the perception that pegged exchange rate arrangements amongst major countries are doomed to failure, render the prospects for significant reform of the present system at the world level remote. Based upon the Bretton Woods experience, major countries are unwilling to compromise their domestic interests for the sake of the dictates of an external monetary authority or to be subject to the constraints of an international exchange rate arrangement which they cannot control [Bordo (1995)]. This is not the case at the regional level where there is a greater harmony of interests than between major countries. Indeed Europe is moving unsteadily towards creating a monetary union with a common currency. On the road to that end, the EMS established in 1979 was modelled after Bretton Woods (although not based on gold), with more flexibility and better financial resources [Bordo (1993b)]. It was successful for a few years in the late 1980s when member countries followed policies similar to those of Germany, the center country [Giavazzi and Giovannini (1989)]. It broke down in 1992 to 1993 in a manner similar to the collapse of Bretton Woods in 1968-1971. It also collapsed for similar reasons - because pegged exchange rates, capital mobility, and
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policy autonomy do not mix. It collapsed in the face of a massive speculative attack on countries that adopted policies inconsistent with their pegs to the D-mark and also on countries that seemingly observed the rules, but whose ultimate commitment to the peg was doubted. The doubt arose because of rising unemployment in the latter. The lesson from this experience is that the only real alternatives for the European countries are monetary union, perfectly fixed exchange rates and the complete loss of monetary independence, or else floating. Halfway measures such as pegged exchange rate systems do not last. Schemes to reimpose capital controls [Eichengreen, Tobin and Wyplosz (1995)] will be outwitted and will only misallocate resources. The legacy of the gold standard and its variants for EMU is the role of gold as the nominal anchor and of a credible policy rule to maintain it. Cooperation and harmonization of policies under the gold standard was episodic and not by design - in contrast with Bretton Woods, EMS and EMU. For the EMU to succeed, members must have the same credible commitment to their goal as did the advanced nations to the gold standard rule a century ago. That is, they must sacrifice domestic to international stability. The advent of generalized floating in 1973 allowed each country more flexibility to conduct independent monetary policies. In the 1970s inflation accelerated as advanced countries attempted to use monetary policy to maintain full employment. However, monetary policy could be used to target the level of unemployment only at the expense of accelerating inflation [Friedman (1968), Phelps (1968)]. In addition, the USA and other countries used expansionary monetary policy to accommodate oil price shocks in 1973 and 1979. The high inflation rates that ensued led to a determined effort by monetary authorities in the USA and UK and other countries to disinflate. The 1980s witnessed renewed emphasis by central banks on low inflation as their primary (if not sole) objective. Although no formal monetary rule has been established, a number of countries have granted their central banks independence from the fiscal authority and have also instituted mandates for low inflation or price stability. Whether we are witnessing a return to a rule like the convertibility principle and a fixed nominal anchor is too soon to tell. We now turn from the general discussion of domestic and international monetary regimes to survey an important example of a domestic regime - the USA.
3. Episodes in US central banking history 3.1. Origins o f US central banking
Before the passage of the Federal Reserve Act in 1913, the United States did not have a central bank, but it did adhere successfully to a specie standard from 1792 on, except for a brief wartime suspension at the end of the War of 1812 and the 17-year greenback episode from 1862 to 1879. From 1879 to 1914, the United States adhered to the gold standard without a central bank. With the exception of a period in the 1890s, when
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agitation for free coinage of silver led to capital flight and threats of speculative attacks on the dollar [Grilli (1990), Calomiris (1993)], US commitment to gold convertibility was as credible as that of the other core countries [Giovannini (1993)]. Although a formal central bank was not in place before 1914, other institutions performed some of its functions. The Independent Treasury, established in 1840, served as a depository for federal government tax receipts in specie. On a number of occasions, by transferring specie to commercial banks, by judicious timing of its debt management, and by disbursement of the budget surplus, the Treasury mitigated financial stress. It even engaged in primitive open market operations, according to Timberlake (1993, ch. 6). Clearing house associations in various financial centers, beginning with New York in 1857, provided lender of last resort services of a central bank by issuing emergency currency [Timberlake (1993), ch. 14], but often after rates became extremely high - 100 percent in 1907. The Federal Reserve system was established to deal more systematically than had the Treasury and the clearing houses with the perceived problems of the banking system including periodic financial panics and seasonally volatile short-term interest rates. It came into existence at the end of the classical gold standard era, yet it was founded directly upon the precepts of central banking under the gold standard: use of discount rate policy to defend gold convertibility, and the importance of a lender of last resort [Meltzer (1995a), ch. 2]. In addition, the new institution was organized to smooth seasonal movements in short-term interest rates by providing an elastic money supply. By accommodating member bank demand for redisconnts, based on eligible, self-liquidating commercial bills, the reserve banks were designed to promote sufficient liquidity to finance economic activity over the business cycle [Meltzer (1996), ch. 3]. The remaining subsections cover episodes of the eighty-odd years of the Federal Reserve's existence within the broad regimes demarcated in Section 2:1919-1941 ; 1946-1971; 1971-1995 l0 The environment in which the system operated in each of these episodes was vastly different from that envisioned by the founders. Monetary policy changes took place. The changes reflected the influence of three sets of players, who shaped the saga of the system: Congress, by legislation and oversight; the system's officials, by their efforts to fulfill its mission, as they understood it; and the research community, by its interpretation and evaluation of the system's performance. Our discussion comments on these sources of influence on the system. To accompany the discussion, Figures 3.1 a - f present annual series for six important macroeconomic aggregates, 1914-1995: CPI and real per capita income; M2 and the monetary base; the short-term commercial paper rate, and a long-term bond yield. Vertical lines on each plot mark the separate monetary policy episodes that distinguish the Federal Reserve era.
I0 We omit war years, 1915 1918 and 1941 1946. World War II for the USA began later than for the European countries, hence the difference between the dating of the Fed episodes and the broad international regimes in sections 2 and 4.
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1
2
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6
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'3515 1920 1925 1930 1935 1940 1945 1950 1955 198C 1965 1970 1575 1980 1985 1590 1995
Fig. 3.1a. Real per capita income, 1914-1995, USA. Regimes: 1. World War II; 2. 1920's; 3. Great Contraction; 4. the recovery; 5. interest rate peg; 6. Fed. discretionary regime; 7. breakdown of convertibility principle; 8. shifting the focus of monetary policy. Data sources: see Appendix A.
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~915 19~ 1925 1930 1935 1940 1945 1950 1955 1980 1965 1970 1975 1980 1985 1990 1995
Fig. 3.1b. CPI, 1914-1995, USA. See Figure 3.1a for legenda.
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2
3
4
5
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8
1915 1920 1925 1930 1935 1940 1945 1950 1955 1980 1965 1910 1975 1980 1985 1990 1995
Fig. 3.1c. Monetary base, 1914-1995, USA. See Figure 3.1a for legenda.
Ch. 3." Monetary Policy Regimes and Economic Performance." The Historical Record
1
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5
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1915 1920 1925 1930 1935 1940 1945 1950 1955 1950 1365 1970 1975 198D 1985 1990 t995
Fig. 3.1d. M2, 1914-1995, USA. See Figure 3.1a for legenda.
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01915 ................................................................. 197"0 1925 1930 1'9r35 1940 1945 1950 1955 1980 1965 1970 1975 1980 1985 I'~9~' '1'995 Fig. 3.1e. Short-term interest rate, 1914 1995, USA. See Figure 3.1a for legenda.
1
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1915 1920 1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 199'0' 1995
Fig. 3.1£ Long-term interest rate, 1914-1995, USA. See Figure 3.1a for legenda.
171
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3.2. Federal Reserve 1914
In the 30 sections of the Federal Reserve Act that was signed into law on 13 December 1913 Congress sketched the outlines of the system it sought to create. Its structure included a board based in Washington, DC, of five (increased to six, June 1922) appointees of the President, one of whom he would designate as the Governor, plus the Comptroller of the Currency, and the Secretary of the Treasury as ex officio chairman; no fewer than eight and no more than twelve Federal Reserve banks, each located in a principal city, the final number and boundaries of the districts to be determined by a committee of the Secretaries of Treasury and Agriculture, and the Comptroller of the Currency; a Federal Advisory Council of one banker elected by each reserve bank. By this structure Congress intended to create a system of semi-autonomous regional reserve banks, loosely subject to the supervision of the Washington board. Over the next two decades the board and the reserve banks and the reserve banks among themselves would be pitted against one another in a struggle to determine which one was dominant. The principal change the Federal Reserve Act introduced was the provision of an "elastic currency", Federal Reserve notes (or, equivalently, member bank deposits at the reserve banks). "Elastic" meant that the new Federal Reserve money would be subject to substantial change in quantity over short periods, thus requiring some body to control the creation and retirement of the money, some means for creating and retiring the money, and some criteria to determine the amount to be created or retired [Friedman and Schwartz (1963)]. Both the board and the reserve banks, without clear lines of demarcation of their respective powers, were given joint control of the creation and retirement of Federal Reserve money. The means for creating it were gold inflows, rediscounting of "eligible" paper, discounting of foreign trade acceptances, and open market purchases of government securities, bankers' acceptances, and bills of exchange. Retirements involved the converse. The criteria for determining the amount of Federal Reserve money, on the one hand, was a gold standard rule, imposing the requirement of a 40 percent gold reserve against notes and a 35 percent gold reserve against deposits, and convertibility of Federal Reserve money in gold on demand at the Treasury Department or in gold and lawful money at any Federal Reserve bank; and, on the other hand, a real bills doctrine, according to which the amount issued would be linked to "notes, drafts, and bills of exchange arising out of actual commercial transactions" (section 13), offered for discount at rates to be established "with a view of accommodating commerce and business" (section 14d). In addition to gold backing, each dollar of Federal Reserve notes was also to be secured by a 60 percent commercial paper collateral requirement. The two criteria were on the surface contradictory. While the gold standard rule requires the stock of notes and deposits to be whatever is necessary to balance international payments over the long run, in the short run, the stock of gold reserves and international capital market flows can accommodate temporary imbalances.
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However, the gold standard does not determine the division of the stock of money between currency and deposits, although facilitating shifts between the two forms of money was a crucial attribute of the new institution. The real bills criterion, by contrast, which was linked to this division, sets no limit to the quantity of money. A basic monetary problem that the Federal Reserve Act was intended to solve was an attempt by the public to shift from holding deposits to holding currency. Such attempts had led to a series of banking crises before 1914 [Schwartz (1986a)]. The solution was to introduce a form of currency that could be rapidly expanded - the role of the Federal Reserve note - and to enable commercial banks readily to convert their assets into such currency - the role of rediscounting. By limiting the lender of last resort to rediscounting only such paper as arose from "actual commercial transactions" as opposed to paper arising from "speculative transactions" (i.e., loans backed by stock market collateral), the Federal Reserve Act sustained the real bills doctrine but, in so doing, it confused the elasticity of one component of the money stock relative to another and the elasticity of the total. Systemwide open market operations were not contemplated in the Act. Each reserve bank had discretion to choose the amount of government securities to buy and sell and first claim on the earnings of its government securities portfolio. The Federal Reserve Act gave the board and the reserve banks the right to regulate interest rates. As a result, the behavior of short-term interest rates changed. Before the Federal Reserve began operations, nominal interest rates displayed extreme seasonality, which was linked to financial crises [Kemmerer (1910), ch. 2; Macaulay (1938), chart 20; Shiller (1980), pp. 136-137; Clark (1986), Miron (1986), Mankiw, Miron and Weil (1987), Miron (1996)]. Once in operation, it apparently altered the process generating short-term interest rates. According to Barro (1989), the shifts in monetary policy involved changes in the process for monetary-base growth. Federal Reserve policy did not completely eliminate seasonality in nominal interest rates, but substantially reduced its amplitude. Why the policy of smoothing was quickly effective in reducing seasonality and other transitory movements in nominal interest rates has been the subject of debate. Was it the founding of the Federal Reserve, as Miron (1986) and Goodfriend (1991) contend, or the abandonment of the gold standard by many countries in 1914 that led to diminished interest rate seasonality, as Truman Clark (1986) contends, or was there no regime change at all, as Fishe and Wohar (1990) maintain? Whichever interpretation one adopts, if one regards the nominal interest rate as the implicit tax on holding real money balances, smoothing the nominal interest rate over the year is a benefit but only of small consequence in raising welfare. McCallum (1991) suggests, however, that seasonal interest rate smoothing encouraged Federal Reserve smoothing in nonseasonal ways also, which was probably detrimental to monetary policy more generally. Goodfriend (1988) asks how the Federal Reserve was able to combine a commitment to a fixed dollar price of gold, on its founding, with interest rate smoothing. His answer is that, under a gold standard, the Federal Reserve could choose policy rules for both
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money and gold. It varied its stockpile of gold in supporting a fixed price of gold, and used monetary policy to target interest rates. Semi-autonomous reserve banks, according to the Federal Reserve Act, would each establish discount rates in accordance with regional demand for and supply of rediscounts, subject to review and determination of the board (section 13). Discount rates were to vary by types of eligible paper and by different maturities. Where the power rested to initiate discount rate changes would become contentious. The example of the Bank of England in setting its rate above market rates influenced early reserve bank belief that discount rates should be penalty rates. This belief conflicted with the political interest to use the Act to achieve a low level of interest rates [Meltzer (1996), ch. 3]. The Federal Reserve Act also included a fiscal provision (section 7). Member banks own the reserve banks, and are paid a 6 percent cumulative dividend on their capital stock, as if the reserve banks were a public utility and the board were the regulatory body [Timberlake (1993)]. Expenses of both the reserve banks and the board were paid from earnings on assets. Timberlake finds a contradiction between regarding the reserve banks as both the income-earning utility and regulators of the commercial banking system. The net earnings of the reserve banks, according to the law, after payment of dividends were to be divided between the surplus account and the Treasury. However, before they needed to turn over any part of their earnings to the government, the reserve banks could build up their surplus until the accounts equaled (originally 40 percent, changed in March 1919 to 100 percent of) their subscribed capital and, even then, 10 percent of net earnings would continue to be added to the surplus before the remainder was paid to the Treasury as a franchise tax on the note issue. The objective of the Federal Reserve was to serve as a lender of last resort and thus eliminate financial crises, to be achieved by interest rate smoothing, according to the consensus view of writers on its founding. It would issue notes and deposits, based on real bills, and convertible into gold on demand. Toma (1997) regards the foregoing specification of the intent of the Federal Reserve Act as misconceived. Based on a public choice approach, he describes the reserve banks as a network of competitive clearinghouses that were to provide liquidity for retail banks. Assigning money creation powers to the Federal Reserve was a way of funding the general government, which could indeed raise revenue for itself by granting monopoly status to a clearinghouse and taxing its profits. That strategy, however, would reduce the liquidity the clearinghouse were to offer banks. Greater government financing needs meant less liquidity supplied by the reserve industry and greater bank fragility. Hence, for Toma, the founding of the Federal Reserve reflected a tradeoff between government revenue needs and financial stability. Since prospective government seigniorage requirements were low in 1913, financial stability goals dominated. Toma also disputes the role of interest rate smoothing. The solution to the financial crisis problem in his view did not rely on interest rate control. Instead, the Federal
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Reserve rebated earnings to large city banks through an in-kind payment of checkclearing services, and subsidized loans during the fall when discount rates were constant and market interest rates rose. Hence probability of a financial crisis was reduced. Manipulation of market interest rates was not required. Toma's emphasis on government revenue needs as an important element in the thinking of the founders of the Federal Reserve would carry weight if he would cite evidence to this effect during the lengthy debate preceding the law's enactment. As it is, his evidence is that public finance considerations accounted for the creation of the national banking system and 19th century central banks. These examples do not clinch his case. Similarly, Toma's argument that interest rate smoothing was not needed for financial stability because it was achieved by the alternative means he identifies does not challenge the fact that smoothing occurred. 3.3. Interwar years, 1919-1941 3.3.1. 1919-1929
The system's experiences during World War I and the aftermath left the policy guidelines of the Federal Reserve Act of questionable value. The gold criterion had become operative only when inflation rose in 1919-1920, and the system's gold reserve ratio plunged. In the face of that decline, the system had contracted. However, when gold inflows followed, and the gold criterion signaled the need to lower interest rates, the real bills criterion signaled the opposite policy. The real bills criterion had been emasculated by wartime finance considerations, but in 1920 member bank indebtedness to the reserve banks and their large portfolios of government securities signaled a need for higher interest rates. Moreover, the steep discount rates in 1920-1921 were not penalty rates since they were lower than open market rates on commercial paper [Meltzer (1996), ch. 3]. In the deep contraction of 1920-1921 the system had no compass by which to steer to keep to a chosen course. The violent swings of prices that marked the inflation of 1919 and deflation of 1920 was the background to Federal Reserve performance in the years before the Great Depression. No disputes exist about what the Federal Reserve's actions were, but a contentious literature has arisen about the interpretation of those actions. The issues concern the Federal Reserve's commitment to the gold standard criterion and the real bills doctrine, and whether stabilization of the business cycle became its goal. With respect to the gold standard criterion, the problem for the Federal Reserve was that gold standard rules appeared to be inapplicable in a world where only the United States maintained gold payments. The flow of gold to the United States in 1921-1922 threatened monetary stability if the authorities responded with expansionary actions. But gold sterilization was incompatible with using the gold reserve ratio as a guide to Federal Reserve credit. From 1923 on gold movements were largely offset by movements in Federal Reserve credit, so essentially no relation is observed between the gold movements and the
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monetary base [Friedman and Schwartz (1963), pp. 279-284]. The system justified sterilization of gold movements on three grounds: pending the return to gold standards by countries abroad, much of the gold was in this country only temporarily; gold movements could not serve their equilibrating role with most of the world not on the gold standard; sterilization of the inflow was desirable to increase the gold stock in view of increased short-term foreign balances here. Once other countries returned to the gold standard, however, these reasons were no longer valid, although the system still repeated them. Wicker's (1965, pp. 338-339) objection to regarding gold sterilization as a significant indicator of monetary policy is that "Federal Reserve monetary policy may not have been at all times rationally conceived and administered" (p. 338). He sees a conflict between sterilization for domestic considerations and the commitment to fully convertible gold currencies abroad, but he concedes that the Federal Reserve rejected the reserve ratio as a guide, although only until the international gold standard would be fully restored. To replace the gold reserve ratio, the Tenth Annual Report [Federal Reserve Board (1924)] of the Federal Reserve system maintained that credit would not be excessive "if restricted to productive uses". This seems to be a procyclical needs of trade doctrine. The Report distinguishes between "productive" and "speculative" use of credit, the latter referring to speculative accumulation of commodity stocks, not stock market speculation. Wicker argues that the Report emphasized a quantitative as well as a qualitative criterion for the adequacy of bank credit, and that the system was not guilty of the real bills fallacy (1965, pp. 340-341). How the quantitative criterion was to be applied in practice Wicker does not explain. Strong in 1922 in a speech at Harvard showed that he understood that the qualitative criterion was ineffectual, noting that "the definition of eligibility does not affect the slightest control over the use to which the proceeds (of Federal Reserve credit) are put" [Chandler (1958), p. 198; Meltzer (1996), ch. 3]. A third issue that divides commentators on monetary policy during the 1920s is whether the system consciously pursued the goal of stabilizing the business cycle. After its unfortunate experience with the discount rate in 1919-1920 as the instrument to implement monetary policy, in the following years the system experimented with open market operations. They were initially regarded as a means to obtain earnings for the reserve banks. The banks individually bought government securities without apparent concern for the influence of those purchases on the money market, with the result that their uncoordinated operations disturbed the government securities market. The Treasury's dismay led the reserve banks in May 1922 to organize a committee of five governors from eastern reserve banks to execute joint purchases and sales and to avoid conflicts with Treasury plans for new issues or acquisitions for its investment accounts. The committee met for the first time on 16 May at the New York reserve bank and elected Strong as permanent chairman. Although the centralization of open market operations led to a recognition of the bearing of purchases and sales on monetary policy, it did not happen immediately.
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Opposition to open market operations was voiced by Adolph Miller, an economist member of the reserve board. He argued that changes in member bank borrowing offset open market operations and therefore had no effect on credit conditions. In his view the reserve banks should limit provision of credit to rediscounting bills that member banks submitted. Opposition to open market operations was related to a general view of monetary policy that distinguished sharply between discounts and bankers' acceptances, on the one hand, and government securities on the other as sources of credit expansion. Reserve creation by buying bills and discounting bills was regarded as financing a genuine business transaction, while reserve creation by buying government securities had no such direct connection with the needs of trade. Reserve creation in the latter case might filter into loans on Wall Street. These conflicting domestic policy positions were intertwined with international considerations. The system attached great importance to the reestablishment of a worldwide gold standard, but official literature contained no discussion of the policy measures appropriate to achieve the objective. Strong played the leading role in the system's relations with other countries, promoting credit arrangements with countries that returned to the gold standard during the 1920s. From Strong's standpoint, easing measures in 1927 served two purposes: overcoming slack business conditions, despite his concern about speculation in the stock market; and helping to strengthen European exchange rates. Recession in the United States reached a trough in November 1927, and European exchange rates strengthened. Wicker (1965, p. 343) disputes that stabilization through skilful open market operations was Strong's objective. He contends that open market purchases in 1924 and 1927 were intended to reduce US interest rates relative to Britain's to encourage a gold flow to London. According to Wicker, for Strong it was through the restoration of the world gold standard that stabilization of national economies would automatically occur. Wicker concludes, "The error of assigning too much weight to domestic stability as a major determinant of monetary policy has arisen.., out of a faulty and inadequate account of the nature of Benjamin Strong's influence on open market policy and a tendency to exaggerate the extent to which some Federal Reserve officials understood the use of open market policy to counteract domestic instability". Wheelock (1991) models econometrically the Federal Reserve's open market policy from 1924 to 1929 with alternative explanatory variables. His results confirm that the Federal Reserve attempted "to limit fluctuations in economic activity, to control stock market speculation, and to assist Great Britain retain gold" (p. 29). He is unable, however, to discriminate between Wicker's approach and that of Friedman and Schwartz. Toma (1997) disputes the Friedman and Schwartz view that the Federal Reserve in the 1920s discovered how to use open market policy to fine tune the economy and that those years were the high tide of the system. He contends that the system had no such stabilization powers. Open market purchases tend to reduce the volume of discounting and open market sales to increase it - the so-called scissors effect - that
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Adolph Miller had earlier mentioned, for reasons different from Toma's. For Toma the private banking system eliminated any lasting effect these operations might have had on Federal Reserve credit (p. 80), and the relative stability o f the 1920s cannot be attributed to fine tuning by the Federal Reserve (p. 87). In his view the stability is associated with monetary restraint that competitive open market operations o f profitseeking reserve banks induced. The period o f the 1920s, for him, was "one of reserve bank competition interrupted by occasional episodes o f coordination" (p. 73). Toma also contends that the Federal Reserve did not use centralized open market operations to smooth interest rates during the 1920s (p. 80). He reports that seasonal behavior o f Federal credit during 1922-1928 was "driven by the demands of the private banking system (i.e., discount loans and bankers' acceptances) rather than by open market operations". Two fallacies undermine Toma's positions. He treats the scissors effect as if it were a one-to-one offset o f open market operations. The inverse relation between borrowing and open market operations was hardly that close [Meltzer (1995b), ch. 5]. In addition, Toma's insistence that open market operations continued to be decentralized after the OMIC was established is incorrect. His portrayal o f the system in a public choice framework seems far removed from the facts. 3.3.2. The Great Depression o f 1929-1933
No period o f Federal Reserve history has elicited as much discussion as the four years that set off the economic collapse that began in August 1929 and ended in March 1933. Since our subject is monetary regimes, we exclude the view that the contraction can be explained by real business cycle theory [Prescott (1996)] 11. Instead we deal with issues on which opinions are divided among students of the period for whom monetary policy is the central focus. There are six principal issues: (1) Was there a significant change in Federal Reserve conduct o f monetary policy between 1923-1929 and 1929-1933? (2) Were bank failures a significant contributor to the economic collapse? (3) How was the monetary collapse transmitted to the real economy? (4) Did the stock market crash in October 1929 play an important role in initiating the economic decline? (5) Had the Federal
11 Bernanke and Carey (1996) note that "any purely real theory" (p. 880) is unable to give a plausible explanation of the strong inverse relationship they find (across a panel of countries over the period 1931-1936) between output and real wages, and of their finding that countries that adhered to the gold standard typically had low output and high real wages, while countries that left the gold standard early had high output and low real wages. The dominant source of variation between the two sets of countries was differences in money stocks and hence in levels of aggregate demand. Another view attributes the severity of the Great Depression to the collapse of world trade following the passage of the Smoot-Hawley tariffin 1930 [Meltzer (1977), Crucini and Kahn (1996)]. The importance of the tariff act and the retaliation it provoked are minimized as an important cause of the downturn in Eichengreen (1989) and Irwin (1996).
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Reserve not allowed the money stock to decline, would the depression have been attenuated? (6) Did gold standard policies transmit the depression to the rest of the world? 3.3.2.1. Policy continuity?. Friedman and Schwartz (1963) maintain that during the
1920s the Federal Reserve responded effectively to fluctuations in economic activity, but during the depression it did not. They attribute the change to the death of Benjamin Strong in 1928. It removed from the scene the dominant figure in the system who had the best understanding of its capabilities. No one with equal authority replaced Strong. Power within the system shifted from him to a leaderless conference of reserve bank governors and a board that had no stature. Challenges to the foregoing position have been mounted by Wicker (1965), Brunner and Meltzer (1968), Temin (1989), Wbeelock (1991), and Meltzer (1995b). They find no shift in Federal Reserve performance between the Strong years and the depression years. For Wicker, who believes international considerations dominated open market operations in the 1920s, the reason the Federal Reserve saw no need for action in 1929-1931 was that those years posed no threat to the gold standard. When Britain abandoned the gold standard in 193l, however, the system raised discount rates in order to maintain convertibility. It was acting on the consistent principle that domestic stability was subordinate to the gold standard. Temin agrees with Wicker. Brunner and Meltzer (1968, p. 341) do not accept the argument for continuity based on the primacy of international considerations. Rather, they trace the continuity to the Federal Reserve's mistaken monetary policy strategy, which they assert has been an unchanging characteristic of its performance. For the system, a low level of nominal interest rates and of member bank borrowing are indicators of monetary ease, a high level, of monetary tightness. In 1924 and 1927, interest rates and bank borrowing had declined only moderately, hence they indicated relative monetary tightness, justifying open market purchases. During the depression years, since interest rates and member bank borrowing were at exceptionally low levels, they signified to the Federal Reserve that there was monetary ease and that injections of reserves were unneeded. Based on regression estimates of the demand for borrowed reserves for all member banks in the New York reserve district and for weekly reporting member banks in New York City, Wheelock (1991) also finds Federal Reserve behavior largely consistent throughout the 1920s and the depression. Meltzer (1995b, ch. 5) disagrees with the view that, had Strong lived, policies would have differed. He describes Strong as an ardent upholder of market interest rates and borrowing as the main indicators of monetary policy. Since Strong approved of the deflationary policy of 1920-1921, he sees no reason to believe that Strong would have opposed deflation from 1929 to 1931. Meltzer notes that, while the real bills doctrine and member bank borrowing as policy indicator were the prevailing principles of Federal Reserve officials, and some so-called liquidationists supported a more deflationary policy, support for expansionary policy was at best a future possibility, "much of the time" not under consideration
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during the depression years. For Friedman and Schwartz, Strong was not a slavish follower of the prevailing principles of the Federal Reserve, and there is enough evidence in his speeches and Congressional testimony to suggest that he would not have passively observed cataclysmic economic decline without championing policies he knew had succeeded in 1924 and 1927. Hetzel (1985) provides the evidence on Strong's views. The expansionist position taken by the New York reserve bank during 1930 is also persuasive evidence that policy would have been different had Strong then been alive. 3.3.2.2. Banking panics. Friedman and Schwartz (1963 ) identified four banking panic s between October 1930 and March 1933, and found them largely responsible for the steep contraction in the stock of money that took place. A bank failure not only eliminated its deposits from the money stock, but also diminished the public's confidence in other banks, with the result that holding currency became preferable to holding a bank's liabilities. A withdrawal of deposits in the form of currency reduced bank reserves. Given Federal Reserve policy to hold back on the provision of reserves, both the deposit-currency and the deposit-reserve ratios declined, contributing far more to the decline in the money stock than did a bank failure. Recent research on banking panics has centered on whether it is accurate to designate the cluster of bank failures in November 1930-January 1931 as the first banking panic, as Friedman and Schwartz do; the geographical boundaries of each of the panics and whether they had a national impact; whether periods additional to those Friedman and Schwartz designated qualify as bona fide panics; whether panics during the Great Depression differed from pre-1914 examples; causes of bank suspensions. Wicker (1996) is the author of the most substantial empirical work on the microeconomic level of banking panics during the Great Depression. He has combed old newspaper files to learn the names and locations of failed banks, and compiled data on currency inflows and outflows by Federal Reserve districts to track the fallout from a concentration of bank failures in panic subperiods. Controversy over the validity of the assignment by Friedman and Schwartz of special significance to the failure of the Bank of United States in December 1930 dates back to Temin (1976). He asserted that failures were induced by the decline in agricultural income and in the prices of relatively risky long-term securities held by banks, and that the failure of the Bank of United States did not precipitate a liquidity crisis. In his Lionel Robbins lecture [Temin (1989)], he repeated the view that the first banking panic was a minor event. White (1984), who found that balance sheets of failed banks in the 1930s did not differ from those of the 1920s, denied the characterization by Friedman and Schwartz of bank failures in 1930 as cases of illiquidity, unlike pre-1914 cases of insolvency. White overlooks the fact that runs on banks in distress in the 1920s were rare [Schwartz (1988)], but in the 1930s were common. Wicker (1980) called attention to the omission by Friedman and Schwartz of the failure in November 1930 of Caldwell and Company, the largest investment banking house in the South, that led to runs on 120 banks in
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four states. He concludes [Wicker (1996), p. 32] that, on the evidence of Temin, White, and his own research, "the 1930 crisis was a region specific crisis without noticeable national economic effects". He believes the second crisis from April to August 1931 perhaps is also region specific, and without clearly identifiable national effects (p. 18). Wicker also identifies a fifth panic in June 1932 in the city of Chicago, comparable in severity, he says, to the 1930 panic. Measured by the deposit-currency ratio, however, and the money stock, which are national in coverage, their fall unmistakably records the incidence of the first two bank crises. Regional disparities are not incompatible with national effects. As for the absence of noticeable national economic effects, does Wicker suggest that economic activity did not deteriorate between October 1930 and August 1931? Some attention has been given to the question whether banks that fail during panics are in the main illiquid or insolvent. Calomiris and Gorton (1991) find the answer depends on which of two rival theories applies. The random withdrawal theory associates bank suspensions with illiquidity induced by contagion of fear. The asymmetric information theory associates suspensions with insolvency due to malfeasance. Saunders and Wilson (1993) found contagion effects in a sample of national banks 1930-1932, but did not examine separately panic and nonpanic months. Wicker also notes contagion effects in the Caldwell collapse in November 1930. Wicker (1996) highlights a difference between pre-1914 and Great Depression panics. In the former the New York money market was the center of the crisis. In 1930 and 1931, however, the crisis originated in the interior of the country, with minimal central money market involvement. Wicker credits the Federal Reserve with this result: "there were no spikes in the call money rate or other short-term interest rates" (p. 23). However, he faults the Federal Reserve for not attempting to restore depositor confidence through open market purchases.
3.3.2.3. Transmission o f the monetary collapse to the real economy. The literature
on the propagation of the depression takes two different approaches. One stresses real wage and price rigidity as the propagator [on price setting in product markets and wage setting in labor markets, see Gordon (1990)]. The other approach stresses the consequences of price deflation, whether anticipated or unanticipated. The disruption of the process of financial intermediation owing to bank failures has also been studied as a nonmonetary link to output decline. O'Brien (1989) provides empirical evidence on nominal wage rigidity in the late 1920s and thereafter. Manufacturing firms became convinced following the 19201922 steep wage and price decline that maintaining wage rates during a downturn was necessary if precipitous sales declines were to be avoided. They did so collectively and voluntarily. The puzzle is why firms adhered to the policy once the severity of the sales decline in 1929-1931 became evident. It took until the fall of 1931 for many firms to decide that wage cuts would not have adverse consequences for productivity.
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Based on data for 22 countries, 1929-1936, Bernanke and Carey (1996) assess empirically whether slow adjustment of nominal wages was an important factor in the depression. They found a strong inverse relationship between output and real wages. They do not offer their own explanation of the failure of wages and other costs to fall along with prices that thus contributed to the rise in unemployment and the decline in sales. They cite conjectures by other researchers that coordination failures or politicization of wage and price setting as possible explanations 12. The issue whether the price deflation during the Great Depression was anticipated or not is important for choosing between the debt deflation hypothesis or high ex ante real interest rates as the explanation for the severity of the Great Depression. According to the debt deflation hypothesis, unanticipated deflation increases the real burden of nominal debt, curtails expenditures, and makes it more difficult for borrowers to repay bank loans. As a result bank balance sheets deteriorate, and banks ultimately may fail. Financial intermediation is reduced, with negative effects on economic activity. However, if deflation was anticipated, the debt explanation for the severity of the Great Depression turns on a collapse of consumption and investment expenditures driven by high real interest rates. No conclusive evidence can be cited in support of deflation as either unanticipated or anticipated. Research findings diverge. Barsky (1987) and Ceechetti (1992) concluded that simple time series models predicted price changes. An opposite conclusion was reached by Dominguez, Fair and Shapiro (1988), on the basis of forecasts from VAR models using data ending at various dates between September 1929 and June 1930. Hamilton (1987, 1992) links unanticipated deflation to the Federal Reserve's tight monetary policy in 1928, and shows that deflation was not anticipated in selected commodities markets for which he examined the relationship between spot and futures prices. Nelson (1991) found in reviewing the contemporary business press that there was some expectation that prices would decline but not the degree or the duration of the decline. Evans and Wachtel (1993) construct a test using data on inflation and interest rates that suggests that time series forecasts of price change, such as Cecchetti reported, are not accurate representations of what people expected prices would be. The prospect of future policy changes or knowledge of past changes of policy made them highly uncertain about the future behavior of prices. They expected little of the deflation that actually occurred. Evans and Wachtel indicate that, in 1930-1933, with anticipated deflation of no more than 2 percent and nominal interest rates ranging between 5 and
12 Bordo, Erceg and Evans (1997) simulate over the interwar period an equilibrium model of the business cycle with sticky wages embodiedin Fischer (1977) and Taylor(1980) staggeredcontracts. They show that monetary contraction closely replicates the downturn in output until early 1933. Thereafter, their monetary model produces a much faster recovery than actually occurred. Other forces, such as Roosevelt'sNIRA policy [Weinstein (1981)] and technology shocks may be important in accounting for the recovery.
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1 percent, the ex ante real rate of interest was unlikely to have exceeded 7 percent and was probably much smaller. The foregoing studies focus on the United States. Bernanke and James (1991) in an examination of the experience of 24 countries find that the extent of the worldwide deflation was less than fully anticipated in view of two facts: the nominal interest rate floor was not binding in the deflating countries, and nominal returns on safe assets were similar whether countries did or did not remain on the gold standard. The issue whether price deflation during the Great Depression was anticipated or unanticipated is still unresolved. Another nonmonetary channel that served to propagate the depression has also been studied. Bernanke (1983) introduced the decline in financial intermediation as a nonmonetary shock, operating as an independent force in producing real economic decline in the 1930s. The disruption of financial markets as a result of the reduction in banks' ability to lend engendered a fall in the net worth of households and firms holding nominally fixed debt. The ensuing debt crisis became an important propagator of economic contraction, increasing the number of bankruptcies [see also Bernanke (1995), Bernanke and Gertler (1989), Calomiris (1993)]. Brunner and Meltzer (1968, 1993) accept Bernanke's emphasis on the importance of the credit market in the transmission of shocks but not his treatment of it and the debt crisis as a separate and independent exogenous shock. They regard it as an induced response to the monetary authorities' failure to counter deflation.
3.3.2.4. The October 1929 stock market crash. The Dow Jones Industrial Index was between 300 and 320 during the first half of 1929 until the end of June, when it stood at 333. It-climbed during the following months and peaked at 381 on 2 September. By the end of September, the index had fallen to 343. On 23 October stock prices dropped to 305. The crash came on 24 October, "Black Thursday". By 6 November the index was down to 231. A week later the index had fallen to 199. This was the low following the crash [Wigmore (1985), pp. 4-26 and Table A-19]. It is commonly believed that the stock market crash reduced the willingness of consumers to spend. It is said to have caused "a collapse in domestic consumption spending" [Romer (1993), p. 29] because it created uncertainty, decreased wealth and reduced the liquidity of households' balance sheets [Mishkin (1978)]. Temin (1976) specifically rejects an explanation of the fall in consumption as reflecting the effect on wealth of the stock market crash, on the ground that the wealth effect was too small. He regards the fall as autonomous and unexplained. Yet econometric evidence in support of this proposition is far from convincing. In her recent paper Romer bases her regressions on her intuition that stock market variability made people temporarily uncertain about the level of their future income and thus caused them to postpone durable goods purchases and stimulate consumer spending on nondurables. Her model predicts a greater fall in 1930 in durables than actually occurred, does not predict the slight fall in perishables, and overpredicts a rise in semidurables.
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Romer goes on to examine the estimated effect of stock market variability following the October 1987 crash and suggests that uncertainty was both more severe and more persistent in 1929-1930 than in 1987-1988, and that this explains why consumers began spending again in 1988 while they continued to defer purchases of durable goods in 1930. A key difference that Romer does not note is that the stock of money grew 4.9 percent (M1; 5.5 percent M2) in the year following the 1987 crash. A policy issue that has not been addressed in recent research on the 1929 stock market crash is whether the Federal Reserve then should have made itself an "arbiter of security speculation" (in the words of the press statement released by the board on 9 February 1929). The board wrangled with the reserve banks by insisting that moral suasion rather than raising the discount rate would curb speculation. In the end the discount rate was raised. It broke the bull market but also sacrificed stable economic growth. The question of the system's responsibility for stock market valuations applies not only to 1929 but to 1987 and 1997. 3.3.2.5. Would stable money have attenuated the depression?. McCallum (1990) showed that his base rule (with feedback) would have avoided the severe decline in nominal income that occurred between 1929 and 1933. Following McCallum's methodology of using an empirical model of the economy based on interwar data to examine how a counterfactual policy would have performed, Bordo, Choudhri and Schwartz (1995) considered two variants of Milton Friedman's constant money growth rule and estimated separate relations for output and the price level. Basic simulations of both variants yielded results consistent with claims that, had a stable money policy been followed, the depression would have been mitigated and shortened. The view that a k percent rule (constant money growth rule) is suboptimal [Eichenbaum (1992)] compares economic performance under constant money growth with alternative rules or discretion that yield a superior outcome. Focus on the constant money growth policy relative to actual performance during the depression shows that it was clearly preferable. 3.3.2.6. gold standard policies in transmitting the Great Depression. Recent research gives the gold standard a major role in the causation and transmission of the depression, but assigns no special significance to US monetary policy, although Bemanke and James (1991) note that US panics may have contributed to the severity of the world deflation. They stress the close connection between deflation and nations' adherence to the gold standard, but find the case for nominal wage stickiness or real interest rates as transmission mechanisms dubious. They favor financial crises as the mechanism by which deflation can induce depression. Another view [Temin (1989, 1993)] is that gold standard ideology, which accorded external balance more weight than internal balance, produced the transmission, with financial crises constituting another transmission channel. According to Temin (1989, p. 84), dealing only with the United States, it is hard to explain how the initial downturn
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was spread and intensified to produce three or four years o f contraction, much less the international propagation mechanism 13. The operation o f the gold standard in interwar years was impaired by forced contraction in countries losing gold without producing expansion in countries gaining gold [Eichengreen (1992)]. Instead o f gold standard ideology, Meltzer (1995b) emphasizes the hold o f the b e l i e f that there had been a speculative situation between 1921 and 1929; he asks (1995b, ch. 5) why deficit countries chose to deflate rather than suspend convertibility, which happened many times in the 19th century. His answer is that policy makers in many o f these countries believed that deflation was the corrective needed in response to previous speculative excesses. What was paramount in their minds was not so much the gold standard imperative as it was the real bills doctrine. Similarly, with respect to Federal Reserve failure to p u r c h a s e government securities in 1930 and most o f 1931, when the system's reserve ratio was generally twice the required ratio, and subsequently when the "free gold problem" 14 was alleged to prevent such action, the explanation for Meltzer was the real bills doctrine, the belief that deflation was exacted by earlier speculative credit expansion. The board could have suspended reserve requirements in 1932-1933 rather than compel intensified contraction, but did not 15 Meltzer's perspective suggests that it was not an unyielding commitment to the gold standard that enforced deflation on the world. It was the failure o f policy makers to exercise temporary release from the commitment, which was a well-established feature o f the gold standard, in response to an internal or external drain [Bordo and Kydland (1995)]. A n d the failure can be traced to the hold o f the real bills doctrine and unawareness o f the distinction between nominal and real interest rates. A subject that needs to be explored is whether it is true that expansionary monetary policy by the Federal Reserve would have been futile because it would have aroused suspicion that the United States intended to leave the gold standard, and consequently resulted in gold losses. For two reasons this scenario is hard to credit. In the first place,
13 A response to this view was made by Haberler (1976, p. 8): Given the dominant position of the US economy and the monetary arrangements and policy maxims of the time - fixed exchanges trader the new gold standard the depression that came about in the United States was bound to spread to the four corners of the world. This does not mean that there were no other focal points of depression elsewhere in the world, for example in Central Europe; but the American infection clearly was the most virulent and the United States was in the strongest position to stop the slide. 14 Eichengreen (1992) argues that low free gold reserves prevented the system from conducting expansionary policy after 1931. Friedman and Schwartz (1963) and Meltzer (1995b, Ch. 5) regard free gold reserves as a pretext for the system's inaction that is explained by totally different reasons. 15 On 3 March 1933, when the New York reserve bank's reserve percentage fell below its legal limit, the board suspended reserve requirements for thirty days, too late to alter the imminent collapse of the system.
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it does not acknowledge the enormous size of US gold reserves. In February 1933, when there was both an internal and external drain, reflecting lack of confidence in Roosevelt's commitment to gold, the gold loss was $263 million. Gold reserves of $4 billion remained. In the second place, had expansionary monetary policy been in place, it would have stabilized the money supply and propped up the banking system. A quantitative estimate of the gold loss coefficient under these conditions, we conjecture, would reveal it to be modest in size, and would dispose of the argument that the possibility of expansionary monetary policy was illusory.
3.3.3. 1 9 3 3 - 1 9 4 1
The passivity of the Federal Reserve during the depression continued after it ended but under wholly different circumstances. New Deal institutional changes transformed monetary policy. Institutional changes that enhanced the authority of the board at the expense of the reserve banks ironically were the setting in which the Federal Reserve was overshadowed by the Treasury. The Treasury became the active monetary authority, while the Federal Reserve was passive. The main source of growth in the base was gold imports, which surged as foreigners took advantage of the steadily higher price of gold in 1933 that was fixed at $35 by the Gold Reserve Act. When the Treasury bought gold, it paid with a check at a reserve bank, which increased member bank reserves. The Treasury could print a corresponding amount of gold certificates, which it could deposit at the reserve bank to restore its deposits. These transactions accounted for the major movements in the monetary base. However, as a result of the gold sterilization program the Treasury adopted in December 1936, in the first nine months of 1937 the monetary base did not reflect the growth of the gold stock. During that period, the Treasury paid for the gold it bought by borrowing rather than by using the cash balances it could create on the basis of the gold. This was similar to sterilization by the Federal Reserve in the 1920s, when it sold government securities to offset the effect on the monetary base of gold inflows. The difference was that in the 1930s the Treasury rather than the Federal Reserve sold the bonds and took the initiative in sterilizing gold. The Treasury's gold sterilization program became effective at a time when the Federal Reserve undertook its first monetary policy action since the New Deal was in place. The sharp rise in member bank excess reserves beginning after the banking panic of 1933 was seen as raising dangers of future inflation. Sales of securities would have been desirable but for the need for adequate earnings. The system's room for maneuver was further limited by the political context within which it had to operate, since the Treasury could nullify anything it wished to do. The one option the Federal Reserve thought it had was to reduce excess reserves by exercising the power to double reserve requirements that the Banking Act of 1935 gave it. It did so in three steps between
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August 1936 and May 1937. Given the banks' demand for prudential reserves, the action backfired and led to recession. Reserve requirements were not reduced until April 1938 to a level that eliminated one-quarter of the combined effect of earlier rises. A start toward Treasury desterilization was made in September 1937, when the board requested the Treasury to release $300 million from the inactive gold account. The board itself, of course, could have taken the economic equivalent by buying $300 million of government securities. On 19 April 1938 the Treasury discontinued the inactive gold account. Romer (1992) highlights money growth in stimulating real output growth between 1933 and 1942. Three other studies examine Federal Reserve behavior during those years: Eichengreen and Garber (1991), Calomiris and Wheelock (1998), and Toma (1997). Eichengreen and Garber regard monetary policy in 1933-1940 as foreshadowing wartime practices. The Federal Reserve acceded to Treasury requests in 1935 to moderate the rise in interest rates, and it purchased long-term government bonds for the first time in its history. In April 1937 after the second increase in reserve requirements the Federal Reserve again bought government bonds to moderate interest rate rises, acknowledging in 1938 its responsibility for "orderly conditions in the government securities market". The reason it did so, according to Eichengreen and Garber, was that changes in bond prices might endanger financial and economic security. Calomiris and Wheelock attribute the Treasury's dominance to the increase in its resources generated by gold and silver purchase programs which enabled it to alter bank reserve positions and to intervene directly in financial markets. In fact, the Treasury always had these powers. It was the New Deal political environment which was hospitable to their use. That had not been the case in preceding administrations. A shift in the focus of monetary policy away from markets for commercial paper and bankers acceptances and toward the market for government securities seems to Calomiris and Wheelock less a result of economic conditions than of Administration pressure. With the gold standard constraint absent and Federal Reserve independence diminished, monetary policy was free to monetize government debt, Calomiris and Wheelock conclude. Of course, it was the continued growth of the monetary gold stock that freed the Federal Reserve from the gold reserve constraint, not the absence of a legal gold standard constraint. In Toma's (1997) interpretation of the New Deal period, the government's financing requirements took center stage and induced changes in monetary institutions. In his view, New Deal legislation increased the seigniorage capacity of the monetary sector and fundamentally changed the Treasury's monetary authority. The Treasury took possession of the monetary gold stock and with the allowance for change in the dollar price of gold (the weight of the gold dollar at any level between 50 and 60 percent of its prior legal weight, of which the President specified 59.06 percent), a long-run constraint on the government's monetary powers was relaxed. A positive probability of
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future upward revaluation of the official gold price created the opportunity for future Treasury profits. The Treasury had money-creating powers equal to those of the Federal Reserve. Neither the Federal Reserve nor the Treasury had to share with each other revenue from money creation. After 1933 the Federal Reserve could keep all its earnings and make no transfers to the Treasury. And only the Treasury benefited from gold inflows since the gold certificates the Federal Reserve received did not give it legal title to the gold. Toma explains the Federal Reserve constant credit policy as a way of assigning monopoly rights to the Treasury as the money producer. The Treasury happened to be the least cost producer; it could provide the government's seigniorage requirement by the increase in the monetary base that was equal to or less than the value of gold inflows. In effect, the Federal Reserve paid the Treasury for the right to operate by forgoing its role as money producer. The doubling of reserve requirements, on Toma's interpretation, occurred because of an increase in the government's financing needs. The legislative authorization of flexibility in reserve requirements provided not only for the government's needs but also for the Federal Reserve's earnings objective. Had reserve requirements not been increased, the government's seigniorage revenue would have been lower, and income tax rates would have been higher, damaging real economic activity. Higher reserve requirements imposed costs on retail banks, so policy makers established federal deposit insurance as one way to moderate adverse stability implications for the financial system. Toma's version of events does not square with the record. The Federal Reserve was concerned with its own earnings needs, not with maximizing the government's seigniorage revenue. The reserve requirement increases led to government securities sales by member banks that raised interest rates for the Treasury, hardly the optimal principal agent relationship. Toma's linkage of the passage of federal deposit insurance with the reserve requirement increases rewrites the history of that act, which was a response to depression bank failures. 3.4. Bretton Woods, 1946-1971 3.4.1. 1946-1951
As in World War I, Federal Reserve credit outstanding rather than gold accounted for the increase in the monetary base during World War II. The Federal Reserve again became the bond-selling window of the Treasury and used its powers almost entirely for that purpose. After World War II ended, as after World War I, the system continued the wartime policy of providing the reserves demanded at a fixed cost: through supporting the price of government securities at unchanged levels. During the immediate postwar period and for some time thereafter, the Federal Reserve did not question the desirability of supporting the price of government
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obligations. On 10 July 1947, however, the posted 3/8 of 1 percent buying rate on Treasury bills and the repurchase option granted to sellers of bills were terminated. The Treasury, which had been reluctant to see any change in the pattern of rates, was reported to have consented to the rise in interest costs on its short-term debt owing to the offset created by the adoption on 23 April 1947 by the system of a policy of paying into the Treasury approximately 90 percent of the net earnings of the reserve banks. The next step in the program of raising the support rates somewhat was the sharp narrowing of the difference between short and long rates as a result of a rise in rates on bills and certificates. This led to a shift to short-term securities by individual holders and to a reverse shift by the Federal Reserve. The $5 billion of bonds the system bought was offset by a reduction of some $6 billion in its holdings of short-term securities, so there was monetary contraction in 1948. It was not, however, recognized and inflation fears prevailed, when inflationary pressure in fact was waning. Banks were urged to avoid making nonessential loans, discount rates were raised to 1.5 percent in 1948, reserve requirements were raised in September after Congress authorized a temporary increase in the legal maximum, and consumer credit controls were reinstated. The system was slow in reacting to the cyclical decline that began in November 1948. Not until March-April 1949 were credit controls eased. Between May and September, six successive reductions were made in reserve requirements. In June the system announced that it would not seek to prevent bond prices from rising. For the time being, the system regained some control over its credit outstanding. After the final reduction in reserve requirements in September 1949, the system held outstanding credit roughly constant for the balance of the year and early 1950, and hence refrained from offsetting the expansionary influence of the released reserves. The outbreak of the Korean War in June 1950 unleashed a speculative boom. The accompanying rise in interest rates pushed up yields to levels at which the Federal Reserve was committed to support government security prices. Concern grew that the support program would become the engine for an uncontrollable expansion of the money stock. The system's desire to be freed from this commitment was, however, accomplished only after protracted negotiations with the President and the Treasury, which was fearful of losing the advantage of a ready residual buyer of government securities and of low interest rates. In March 1951 an agreement with the Treasury was finally reached, relieving the system of responsibility for supporting the government security market at pegged prices. Eichengreen and Garber (1991) contend that the existing literature lacks a formal analysis of why investors were willing to hold Treasury securities at low interest rates in the 1940s, and why this willingness disappeared at the end of the decade. They build on the explanation by Friedman and Schwartz (1963) that expectations of deflation after the war induced the public to hold higher amounts of liquid assets than they otherwise would, and that expectations of inflation after 1948 induced the public to hold smaller amounts of liquid assets than they otherwise would. In 1946-1948, the implication of the target zone approach that they adopt is that the 1948 increases in reserve
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requirements and the 1949 bond sales by the Federal Reserve can be thought of as keeping the price level below the upper bound. Bank liquidity declined, and inflationary pressure subsided. Eventually the Federal Reserve reduced reserve requirements as if the price level was approaching the lower bound of the implicit price zone, and by the end of 1949 M1 began to rise. Interest rates rose with inflationary expectations and the cap on interest rates became inconsistent with Korean War imperatives. That is why the Accord with the Treasury was negotiated, if the Eichengreen and Garber analysis is accepted. A question Eichengreen and Garber pose and answer is why the Federal Reserve was concerned about price and interest rate stability - referring to an interest rate peg, not a target - in the aftermath of World War II and not in other periods. They say it was not the system's subservience to the Treasury's pursuit of low debt-service costs that is the answer. Instead, it was fear that a rise in interest rates would cause capital losses on commercial bank portfolios and undermine the stability of the banking system. Despite the fact that by 1951 the banks' vulnerability to capital losses had been attenuated, the Federal Reserve was still concerned to minimize them, and the Treasury helped by offering at par nonmarketable bonds with 2.75 percent yields in exchange for 2.5 percent long-term bonds marketed in 1945. Toma (1997) disagrees with Eichengreen and Garber that the Federal Reserve adopted the stable interest rate program for financial stability reasons. He assigns the seigniorage motive as the driving force with financial stability as at best a secondary consideration. According to Toma, coordination between the Treasury and the Federal Reserve as the two money producers substituted for the gold standard in limiting monetary growth. It seems to us quixotic, however, to describe wartime inflationary monetary growth as a substitute for the gold standard. 3.4.2. Federal Reserve discretionary regime, 1951-1965
The Treasury-Federal Reserve Accord overthrew the dominance of Treasury financing needs over monetary policy. In 1951, after more than 20 years of depression and war, the Federal Reserve had to formulate the criteria by which it would operate as an independent central bank. At that date the Bretton Woods system was in a formative stage, but under its aegis the US commitment to the convertibility of the dollar into gold initially seemed impregnable. By the end of the 1950s, however, as the gold stock began to decline, preventing gold outflows became a major objective of the Treasury as well as the Federal Reserve. A more immediate criterion for monetary policy than the convertibility principle was that the Federal Reserve should "lean against the wind", by taking restrictive action during periods of economic expansion, and expansionary action during periods of economic contraction. The countercyclical theme in the period ending 1965 was generally described in terms of avoiding either inflation or deflation, but full employment was also accepted as an equally important goal of monetary policy.
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The specific operating strategy for implementing "leaning against the wind" that the Federal Reserve adopted was unchanged from its practice in the 1920s [Calomiris and Wheelock (1998)]. It used open market operations to affect the level of discount window borrowing and free reserves - excess reserves minus borrowings. The theory of bank borrowing the Federal Reserve developed was that a change in nonborrowed reserves i.e., reserves provided by open market operations, forced banks to adjust the amount they borrowed. A tradition at the Federal Reserve against borrowing acted to restrain borrowing, even if it were profitable for banks to do so. According to the theory, when free or net reserves were high, market interest rates tended to fall, and bank credit and the money supply tended to grow. When free reserves were low or negative, i.e., net borrowed reserves, market rates tended to rise, bank credit and the money supply tended to contract [Brunner and Meltzer (1964)]. Because of this framework, the Federal Reserve has seen itself as exercising a dominant influence on the evolution of short-term market interest rates. In the 19511965 period, it targeted the Federal funds rate indirectly by using the discount rate and borrowed reserves target. This is now known as interest rate smoothing, a procedure that was earlier known as free reserves or net borrowed reserves targeting [Goodfriend (1991)]. The intention of indirect targeting is to avoid fluctuations and minimize surprise changes in interest rates. Removing seasonality in interest rates, however, is not the main aspect of smoothing under consideration here. Goodfriend describes the modus operandi of indirect targeting in the 1950s as follows. The Federal Reserve estimated the banks' demand for reserves during a defined period and provided most of the reserves by open market purchases. The balance had to be obtained from the discount window where borrowing became a privilege not a right. The Federal Reserve thus targeted borrowed reserves. The amount the banks were willing to borrow, however, depended positively on the spread between the Federal funds rate and the discount rate. Accordingly, the Federal Reserve targeted the Federal funds rate indirectly. Because the demand for borrowed reserves was unstable, it could not target borrowing exactly. In the relation between borrowed reserves and a discount rate-Federal funds rate combination, there was no tight linkage between the Federal funds rate and the discount rate. As a result, the market could not readily determine precisely what the indirect Federal funds rate target was, but it could estimate the range in which the funds rate should fall. Goodfriend's explanation for the Federal Reserve's preference for indirect targeting, even if the result was market misinterpretation of its intention, was that the procedure gave it the option to make changes quietly, keeping target changes out of the headlines. As we shall see, in 1994 it reversed the position it had held for decades and began to announce changes in the Federal funds rate, by that time a directly targeted rate, immediately after an FOMC decision. Capturing headlines did not have the adverse effects on monetary policy the Federal Reserve had for so long claimed would occur. For monetarist criticism of interest rate smoothing one must turn to earlier studies [Brunner and Meltzer (1964), Meigs (1962)]. Essentially, the criticism of interest rate smoothing is that, if the Federal Reserve sets the price of bank reserves and lets
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the market determine the quantity demanded, it abdicates control over the quantity. Goodfriend does not pose the normative question whether the procedure is optimal. Poole (1991), the discussant, does. He tries to make the case for the Federal Reserve's implementation of policy through the Federal funds rate rather than through monetary aggregates control, the preferable alternative for him. The smoothing arguments for interest rate control - it smooths the flow of revenue from the inflation tax; it stabilizes unemployment and inflation; it stabilizes rates at all maturities - in Poole's analysis lack substance. The only argument that he finds plausible is the belief that asset prices under the alternative policy of steady money growth could differ significantly from full-employment equilibrium levels and that the Federal Reserve can anchor interest rates at approximately the correct level when the market cannot do as well. Successful central banks, according to Poole, permit short-run fluctuations in monetary growth but adjust money market interest rates as necessary to constrain money aggregates in the long run from growing too fast or too slow. The Federal Reserve's performance since 1992 provides support for Poole's conclusion. Interest rate smoothing by the Federal Reserve during the decade and a half from 1951 did not preclude a low average inflation rate, but it also yielded unstable industrial output, as contemporaries judged it. Whether this outcome could have been avoided had the Federal Reserve's objective been only the price level and not also output is a subject to which we return when we discuss the 1990s. 3.4.3. Breakdown o f Bretton Woods, 1965-1971
Money growth accelerated in the early 1960s and persisted through the 1970s. US inflation began to accelerate in 1964, with a pause in 1966-1967, and was not curbed until 1980. An inflationary monetary policy was inappropriate for the key reserve currency in the Bretton Woods system. US balance of payments deficits from the late 1950s threatened a convertibility crisis as outstanding dollar liabilities rose and the monetary gold stock dwindled. To prevent conversion of dollars into gold, the United States and other central banks formed the London Gold Pool in 1961 to peg the price of gold at $35 an ounce, established a network of currency swaps with the other central banks, and issued bonds denominated in foreign currencies. These measures fell short. If the link with the dollar was unbroken, US inflation condemned the rest of the world to inflate. The only way to restrain US policy was to convert dollars into gold. French and British intentions to do just that prompted US suspension of gold convertibility in August 1971. Generalized floating of exchange rates followed (see Section 2.4 above). 3.5. Post-Bretton Woods, 1971-1995 3.5.1. 1971-1980
As tenuous as the convertibility obligation had become by the mid-1960s, its absence after the early 1970s totally removed the discipline of convertibility from domestic
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monetary policy. The Federal Reserve was freed o f commitment to maintain a stable price level. To cope with inflation that they blamed on supply-side shocks or shifts in demand for money, policy makers turned to incomes policy which soon failed. Peacetime inflationary episodes as a result came to be associated with discretionary monetary policy. The episode from 1965 to 1980 is commonly attributed to the willingness o f the Federal Reserve to fund government expenditures for the Vietnam war and Great Society social programs and to the authority's belief that it could exploit short-run Phillips curve tradeoffs. Raising monetary growth to provide employment was consonant with Federal Reserve discretion. When the inflation rate accelerated, the authority became ensnared in a trap it itself had set. Monetarist doctrine had convinced Federal Reserve officials that reducing monetary growth in order to curb inflation would produce a recession. They could not bring themselves to choose that option, because o f the political costs. So they pernaitted continuance o f high monetary growth rates and everrising inflation until Paul Volcker broke the spell in 1979. Monetary policy in this period, as in earlier ones, was implemented by control over interest rates rather than control over money growth. The dangers of operating with an interest rate instrument became clear when rising interest rates from the mid-1960s on reflected growing fears o f inflation, not restrictive monetary policy. Rising interest rates were accompanied by high money growth. In January 1970, in response to criticism o f its policymaking, the FOMC for the first time adopted a money growth target. In 1975 Congress passed Joint Congressional Resolution 133 requiring the Federal Reserve to adopt and announce 1-year money growth targets and, in October 1979, the reason for the change in Federal Reserve operating procedures was said to be more precise control o f money growth. The Federal Reserve annotmced the target growth range each year on a base equal to the actual level o f the money stock in the fourth quarter o f the previous year. In the late 1970s, above-target money growth in one year was built into the next year's target, and in 1981, below-target money growth was built into the 1982 target. The Federal Reserve thus permitted base drift, contributing to instability o f money growth. These differences between targets and actual money growth were a consequence o f the Federal Reserve's policy o f maintaining a narrow, short-run target range for the Federal funds rate, unchanged from its operating procedures before monetary growth targets were adopted 16. One change in Federal Reserve operational procedure during the period was its gradual shift during the early 1970s from indirect targeting to direct targeting o f the Federal funds rate within a narrow band specified by the FOMC each time it
16 Differences between an aggregate selected for monetary control and a stable relationship with prices and nominal income that existed before the adoption of the targeted aggregate are said to arise because of financial innovations. The breakdown of the relationship has come to be known as Goodhart's Law [Goodhart (1989)]. It is true that financial innovation does occur and affects the definition of any monetary aggregate and the predictability of its velocity. There is no evidence, however, that links monetary targeting to innovation.
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met [Goodfriend (1991)]. The range within which the rate was allowed to move was commonly 25 basis points. The Federal Reserve managed the rate within the band by open market operations, adding reserves to maintain the rate at the upper bound o f the band, subtracting reserves to maintain the rate at the lower bound. A move o f the band up or down signaled a change in the target, which the market readily perceived. The financial press usually reported a change the day after the Federal Reserve implemented it [Cook and Hahn (1989)]. To support the current target, the Federal Reserve had to accommodate changes in money demand. It had to supply the level o f reserves that would keep the Federal funds target within the narrow band the FOMC set for periods between meetings. This is another way o f explaining how it became an engine o f inflation during the second half o f the 1970s, given that it had no nominal anchor and that the current target could be too low. If the Federal Reserve was slow in raising the target and, when it did raise the target, did not raise it enough, as total nominal spending in the economy rose, rapid money growth resulted, and accordingly higher inflation. Furthermore, interest rate smoothing could itself be a determinant o f the inflation generating process. In Goodfriend's (1987) model, he shows that rate smoothing with a price level objective induces a nontrend-stationary process for the money stock and the price level. This contributes to both money stock trend and price level drift. Interest smoothing increases both the price level forecast error variance and the variability o f expected inflation. So interest rate smoothing tends to create macroeconomic instability 17.
3.5.2. Shifting the focus of monetary policy, 1980-1995 In the period following the inflation episode of 1965-1980, operating procedures at the Federal Reserve underwent modifications. The adoption by the FOMC on 6 October 1979 o f targeting on nonborrowed reserves in place o f direct Federal funds rate targeting represented an admission that earlier interest rate smoothing had failed to provide noninflationary monetary growth. The new procedure was designed to supply banks with the average level o f total reserves that would produce the rate o f monetary growth the FOMC desired over the period from a month before a meeting to some future month, without regard for the accompanying possible movement o f the Federal funds rate outside a widened range o f 400 basis points.
17 An empirical study of UK monetary policy, 1976-1985, by Bordo, Choudhri and Schwartz (1990) suggests that rate smoothing by the Bank of England allowed money stock base drift to reduce the predictability of the trend price level. Had the Bank of England followed a trend-stationary money supply rule, it would have reduced the variance of the trend in prices by more than one-half. Ireland (1993) extends this analysis to the US case. He shows that the Friedman rule would have reduced long-run price uncertainty by 82 percent over the 1915-1990 period.
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At each FOMC meeting a decision was made and never kept not only about the desired growth rate of M1 and M2 but also about the average level of borrowed reserves that it was assumed the banks would desire over the intermeeting period. The staff then estimated a weekly total reserves path from which it subtracted the borrowing assumption to arrive at a nonborrowed reserves path on which the Open Market Desk targeted open market purchases. It sought to keep the average level of nonborrowed reserves between FOMC meetings equal to the nonborrowed reserves path. Under this procedure an increase in the demand for reserves was not mechanically accommodated; in the event, to keep total reserves on its path, nonborrowed reserves might be decreased. When total reserves were above the path level, the level of the nonborrowed reserves path or the discount rate was adjusted to reduce deviations of the money aggregates from their desired rate of growth. When the nonborrowed reserves path was lowered, banks were compelled to increase their borrowings, as a result of which the Federal funds rate rose. A 3 percent surcharge on discount window borrowings by banks with deposits of $500 million or more that borrowed frequently that was first imposed by the Federal Reserve on 14 March 1980 was eliminated a few months later, then reimposed at a lower rate, which was subsequently raised, and later again lowered until finally eliminated on 17 November 1981. Despite the official description of the operation of the nonborrowed reserves procedure, movements in the Federal funds rate were far from automatic [Cook (1989), Goodfriend (1993)]. There were judgmental adjustments to the nonborrowed reserve path at FOMC meetings and between FOMC meetings that changed what the reserves banks were expected to borrow at the discount rate, in effect changing the funds rate target. There were also changes in the discount rate and, as just noted, in the surcharge. Goodfriend concludes that the 1979-1982 period was one of aggressive Federal funds rate targeting rather than of nonborrowed reserve targeting. At the 5 October 1982 FOMC meeting, it abandoned nonborrowed reserve targeting. The Federal Reserve interpreted its experience over the preceding three years as demonstrating that short-run control of monetary aggregates was inferior to interest rate smoothing for stabilization. The outcome of the experiment was that, although M1 growth slowed on average, its volatility tripled compared to the period preceding October 1979 [Friedman (1984)], the Federal funds rate became highly volatile [Gilbert (1994)], and both nominal and real GDP displayed exceptionally large fluctuations quarterly [Friedman (1984)1. Goodfriend (1983) attributed the Federal Reserve's difficulty with reserve targeting to the unreliability of the demand function for discount window borrowing on which its operating procedure critically depended. Pierce (1984) found that the flaw in the operating procedure was produced by lagged reserve accounting in effect at the time, under which required reserves were based on deposit liabilities two weeks earlier. Therefore, only free reserves could serve as a target and, hence, borrowing estimates, which were inaccurate, became crucial. The upshot was that open market operations destabilized money growth.
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On 5 October 1982, when the Federal Reserve suspended the nonborrowed reserves procedure, it shifted to targeting borrowed reserves. In line with this change, the FOMC at each meeting stated its instruction to the Open Market Desk for open market operations to achieve either more or less reserve restraint. More restraint was equivalent to a higher level of borrowings, less, to a lower level. If the demand for total reserves increased, the Federal funds rate and borrowings would rise. In order to reduce borrowed reserves to their desired predetermined level, nonborrowed reserves had to increase, with the effect of reducing the Federal funds rate. No change in borrowed reserves or the funds rate would then occur. This amounted to indirect targeting of the Federal funds rate. To keep the total of reserves the banks borrowed near some desired level, the spread between the Federal funds rate and the discount rate had to be such that banks would have an incentive to borrow that level of reserves. An increase in the spread induced banks to increase their borrowings. It could be achieved by changing the discount rate or the Federal funds rate. The target level of borrowings was attained by providing the appropriate amount of nonborrowed reserves. The borrowed reserves target operated with loose control of the funds rate. Sometime about 1992 the Federal Reserve began to target the Federal funds rate directly in a narrow band. Target changes were made in small steps of 25-50 basis points, usually separated by weeks or months, and not soon reversed. The FOMC directive has not, however, specified the target Federal funds rate, but refers to degrees of reserve restraint that would be acceptable. The model of this regime that Rudebusch (1995) sets up and simulates replicates Federal Reserve operations. Nevertheless, since February 1994, the Federal Reserve during FOMC meetings has announced a change in the funds rate if one has been made. A further procedural change was made in mid-December 1996 in Federal Reserve daily money market operations, revealed at a press conference at the New York reserve bank. The system will announce when it enters the market the size of its open market operations, to be conducted from system accounts, rather than from its customer accounts. The objective is to inform the market about the amount of liquidity the open market operations provide to or withdraw from the banking system. So in the 1920s and since the 1950s, the Federal Reserve in one way or another has targeted the Federal funds rate, while simultaneously announcing a money growth target. In the years since 1992 it has apparently taken low inflation as its sole objective and has succeeded in adjusting the target rate. A side effect is that monetary volatility has been low, and the real economy has not been buffeted by monetary shocks, facilitating low unemployment and financial market stability. Only possible inflation of equity market prices seems troubling. The Federal Reserve along with other central banks changed its policy goals during this period. The primary goal became resisting inflationary pressures. It did so aggressively in 1980-1982. Disinflation was largely accomplished by 1983, when the inflation rate declined to 4 percent per annum. Goodfriend (1993) interprets rising long-term rates in 1983 and 1987 as signaling expectations that the Federal Reserve might again allow inflation to increase. The Federal Reserve met the test by raising
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the Federal funds rate long enough to contain the inflation scare. Goodfriend remarks on the fragility of the credibility of the Federal Reserve and on how costly it is to maintain. In 1996-1997 the long rate at 6.5-7 percent was high enough to suggest that Goodfriend's assessment of the Federal Reserve's credibility is accurate. The duration of a 30-year bond at an interest rate of 6.75 percent is 14.8 years. Who would confidently predict that the then current inflation rate of 2.5 percent would not increase over that horizon? So the expectations explanation for the success of monetary policy targeted on the funds rate seems questionable. The basic problem is that there are no institutional underpinnings of the low-inflation policy. There is no guarantee that the successor to the present chairman of the Federal Reserve will also have a strong aversion to inflation. The durability of Federal Reserve commitment to price stability is a question that only the future will determine. Of the 82 years that the Federal Reserve has been in existence, only 18 can be termed years of stable (consumer) prices - 1923-1929 (average per year change of 0.3 percent); 1960-1965 (average per year price change of 1.3 percent); 1992-1995 (average per year price change of 2.8 percent). The most recent episode is too brief to take for granted its staying power. Arguments in favor of a stable price level in preference to a low inflation rate have been advanced by Feldstein (1996, 1997) and Svensson (1996a,b). Svensson compares price level and inflation targeting, when society (the principal) delegates the choice to a central bank (the agent), under the assumption that output and employment are at least moderately persistent. The decision rule the central bank follows under discretion for inflation targeting is a linear feedback rule for inflation on employment. The variance of inflation is proportional to the variance of employment. Under price level targeting, the decision rule is a linear feedback rule for the price level on employment. Inflation, the change in the price level, is a linear function of the change in employment. Based on a very special set of assumptions, Svensson concludes that society will be better off assigning a price level target rather than an inflation target to the central bank because the variance of inflation will be lower, there is no inflation bias, and employment variability will be the same as under inflation targeting. Feldstein bases his argument on the interaction of taxes and inflation that bias the allocation of resources in favor of current consumption and in favor of owner-occupied housing. The higher the inflation rate, the bigger the bias. Reducing the inflation rate by 2 percent would raise the level of real GDP by 2/3 of 1 percent each year in the future as long as the inflation rate remained at the lower level. Feldstein maintains that the arguments against going from low inflation to price stability do not singly or collectively outweigh the tax-inflation case for going to price stability or even to a lower inflation rate. One argument for inflation targeting is that reducing the permanent rate of inflation requires a loss of output. With a target price path, the monetary authority offsets past errors, creating more uncertainty about short-term inflation than with an inflation target [Fischer (1994), pp. 281-284]. Feldstein's response is that the output loss is
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temporary, a shortfall of GDP below what it would otherwise be of 2.5 percent for two years to reduce the inflation rate by 2 percentage points. That is why he compares the one-time loss of reducing the inflation rate with the permanent increase of real GDP from reducing the tax-inflation effect. Another argument for inflation targeting has been made by Akerlof, Dickens and Perry (1996). They contend that a very low level of inflation may lead to higher unemployment than at a higher inflation level because workers are unwilling to accept nominal wage decreases. Feldstein's response is that, by reducing fringe benefits, it is possible to reduce a worker's compensation without reducing his money wage rate. They also assume that workers don't learn that falling prices raise real wages. Whether the price level or inflation is the target, a central bank has to determine the pace at which to try to achieve either one. The question is whether it is optimal to move immediately to the target. One answer is that gradualism is acceptable in the absence of a cost in terms of growth foregone [Dornbusch (1996), p. 102]. The information and transactions costs of moving from the old to the new regime also argue for a gradual return to a noninflationary position. Long-term borrowing and lending contracts and employment contracts arranged under the old regime need to be unwound. Advance announcement of the gradualism policy would give the private sector time to adjust its expectations. The speed of adjustment of monetary policy should respond to the speed with which expectations adjust and the gradualist prescription is that expectations adjust slowly. Feldstein suggests that this view needs to be modified and disinflation should proceed forthwith when political support for the policy permits it to go forward, since political support is indispensable but is not always at hand. A stronger argument for speedy adjustment than Feldstein's is the rational expectations approach that treats expectations as adjusting quickly, and hence finds shock treatment is preferable. Sargent's view (1986, p. 150) is that "gradualism invites speculation about future reversals, or U-turns in policy". A major consideration in the choice between gradualism and shock treatment is the initial position. With moderate inflation of 8-10 percent, as observed in advanced countries, gradualism may be the answer. With very high inflation rates of 1000 percent per year, as recently experienced in Latin America, gradualism is meaningless. Only shock treatment will suffice. Still another view, dubbed "opportunistic disinflation" [Orphanides and Wilcox (1996)], argues that the Federal Reserve should conduct contractionary monetary policy only during business expansions; during recessions, it should abstain, counting on recessionary tendencies themselves to produce further disinflation. McCallum (1996, p. 112] notes a confusion in this view between regime design, with which the paper advocating opportunistic disinflation is concerned, and the issue of managing the transition from one regime with higher inflation to a regime with a lower level of inflation. Opportunistic disinflation is not a contribution to the literature on the timing of disinflation during the transition. If there is a temporary cost in bringing down inflation, how high is that cost? Unfortunately, no quantitative estimates exist of the cost in lost output and employment
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of a disinflation of a given magnitude pursued over a given period. Hypothetical scenarios based on differing models arrive at qualitatively different conclusions. The announcement of a perfectly credible disinflation will either entail no expected output loss [King (1996)] or, perhaps, an increase in cumulative output [Ball (1994)]. The cost depends on the speed of adjustment of anticipations, which in turn depends on the underlying price level performance of the monetary regime. Alan Greenspan at the Tercentenary Symposium of the Bank of England [Greenspan (1994, p. 259)] remarked: "... the pressure towards reserving or rather focusing central bank activity to the equivalent of the gold standard will become increasingly evident". If this is a correct prediction that price stability will be the single goal of the Federal Reserve over the long term, and if it is achieved, price stability may well become a credible surrogate for convertibility. The system will then end up fulfilling a key element of the vision of its founders. 3.6. Conclusion
Three events stand out in our survey of monetary policy episodes and macroeconomic performance. One is the breakdown of the gold standard in stages over the period from 1914 to 1971. The second is the Great Depression of 1929-1933. The third is the Great Inflation of 1965-1980. To escape from the macroeconomic experience that marked the economy in each of these watershed happenings became the driving force for change. The change was intellectual, reflecting what was perceived as the problem and deduced as its solution. It also led to a change in the monetary policy episode that succeeded each of these events. The new episode in turn exhibited unforeseen deficiencies. To conclude the section, we comment on the way the triad of events unfolded. 3.6.1. Breakdown o f the gold standard, 1914-1971
After World War I, the discipline of the gold standard came to be regarded as an impediment to the management of the economy to achieve the objectives of growth and high employment. The deep depressions of the interwar years were the measure by which the economy under a gold standard was judged to be a failure. The loosening of the link to gold after World War I presaged its abandonment 50 years later. Although price stability was generally included among the goals of the post-World War II era, stability of employment took precedence. The instability of the interwar years led to the creation of the Bretton Woods system, which had a good record of price and output stability until the mid-1960s. Nevertheless, the convertibility principle lost favor. Improving the real performance of the economy was given pride of place. To achieve the improvement, the task was assigned to government management of monetary and fiscal policy, not to impersonal market forces. The simple rule for governments to maintain a fixed price of gold was set aside in 1971, but the seeds of the downfall of that rule were sown earlier in the postwar
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years as country after country opted for monetary independence, full employment, and economic growth. Countries rejected the restraints that the operation of a fixed exchange rate imposed on the pursuit of these widely supported national objectives. In the United States, where the share of international trade was a minor factor in aggregate national income, the view prevailed that the domestic economy should not be hostage to the balance of payments. Maintenance of the price of gold was not an objective of the Employment Act of 1946. The growth of government itself has destroyed the viability of a gold standard. A real gold standard was feasible in a world in which government spent 10 percent of national income, as in Britain and the USA pre-World War I. It is not feasible in a world in which governments spend half or more of national income. 3.6.2. The Great Depression, 1929-1933
The Great Depression was sui generis. To explain it, it is necessary to examine policy errors and the weaknesses of the interwar gold standard. It is a consensus view that monetary contraction began in the United States, and was transmitted to the rest of the world by fixed exchange rates. Monetary contraction began in 1928 to curb a boom on the New York Stock Exchange. Although the stock market crashed in October 1929, the policy of contraction was not then halted. Instead, it was pursued relentlessly by the Federal Reserve until the spring of 1932. The Federal Reserve mistakenly believed that monetary policy had been overexpansionary in the 1920s and that deflation was the proper remedy. In fact the system had achieved stable economic growth from 1922 to 1929 with falling wholesale prices. The US gold stock rose during the first two years of the 1929-1933 contraction, but the Federal Reserve did not permit the inflow of gold to expand the US money stock. It not only sterilized the inflow, it went much further. The US quantity of money moved perversely, going down as the gold stock went up, contrary to gold standard rules. Under a fixed exchange rate system, shocks in one country's income, employment, and prices tend to be transmitted to its trading partners' income, employment, and prices. Absent policy changes in the USA, the only recourse for countries on the gold standard was to cut the fixed exchange rate link. The first major country to do so was Britain. After runs on sterling, it abandoned the gold standard in September 1931. The international monetary system split in two, one part following Britain to form the sterling area; the other part, the gold bloc, following the United States. The trough of the depression in Britain and in other countries that accompanied her in leaving gold was reached in the third quarter of 1932. In the two weeks following Britain's departure from gold, central banks and private holders in foreign countries converted substantial amounts of their dollar assets in the New York money market to gold. The US gold stock declined by the end of October 1931 to about its level in 1929. The Federal Reserve, which had not responded to
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an internal drain from December 1930 to September 1931 as a series of runs on banks, bank failures, and shifts from bank deposits to currency by anxious depositors produced downward pressure on the US quantity of money, responded vigorously to the external drain. A sharp rise in discount rates ended the gold drain but intensified bank failures and runs on banks. In October 1931, unlike the situation in 1920, the system's reserve ratio was far above its legal minimum. The system overreacted to the gold outflow and magnified the internal drain. Federal Reserve officials believed that purchases of government securities, which would have relieved monetary contraction, were inconsistent with the real bills doctrine that the Federal Reserve Act enshrined. They resisted engaging in such purchases until March 1932, when they undertook doing so, following which there was widespread revival in the real economy in the summer and fall. The termination of the purchase program during the summer was followed in the six months from October 1932 by mounting banking difficulties. States began to declare banking holidays. By February 1933, fears of a renewed foreign drain added to the general anxiety. For the first time also, the internal drain took the form of a specific demand by depositors for gold coin and gold certificates in place of Federal Reserve notes or other currency. The Federal Reserve reacted as it had in September 1931, raising discount rates in February 1933 in reaction to the external drain but not seeking to counter either the external or internal drain by extensive open market purchases. The drains continued until 4 March, when the Federal Reserve banks and all the leading exchanges did not open for business. A nationwide banking holiday was proclaimed after midnight on 6 March by the incoming administration, which ushered in a new regime.
3.6.3. The Great Inflation, 1965-1980
By the mid-1960s, the convertibility principle no longer dominated central bank policies. The goal of full employment supplanted it in the minds of central bank and govenmaent officials. The Phillips curve presented them with a course of action that promised higher employment at the cost of rising inflation, a cost that was typically dismissed as insignificant. An additional factor that nurtured an acceleration of inflation was central bank reliance on short-term interest rates as the instrument to control monetary growth. Under noninflationary conditions, this practice produced a procyclical movement in monetary growth. Under the gathering inflationary conditions from the mid-1960s, the inflation premium that became imbedded in interest rates made the instrument unreliable as an indicator of restriction or ease. Reliance on it contributed to a rise in the rate of monetary growth. It was not until the 1970s, when ever higher inflation was accompanied by a decline in economic activity and a rise in unemployment that pressure arose to reverse the policies and procedures that led to the Great Inflation. The upshot was a shift to a new regime in 1979, in which disinflation was the guiding principle. The regime since
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the last decade has focused on price stability, reviving the peacetime domestic objective of the classical gold standard.
4. Monetary regimes and economic performance: the evidence 4.1. O v e r v i e w
Having surveyed the history of international monetary regimes and of the institutional arrangements and episodes in Federal Reserve history viewed as a domestic policy regime, we ask the question, under what conditions is one or another type of monetary regime best for economic performance? One based on convertibility into specie (gold and or silver), in which the monetary authority defines its monetary unit in terms of a fixed weight of specie and ensures that paper money claims on the specie monetary unit are always interchangeable for specie? Or one based on government fiat? Alternatively, in the international monetary sphere, which international monetary regime is superior, one based on fixed exchange rates? One based on floating rates? Or some intermediate variant such as the adjustable peg that characterized the Bretton Woods system and the EMS? Or the managed float which prevails in the world today? Evidence on the performance of alternative monetary regimes is crucial in assessing which regime is best for welfare. 4.2. T h e o r e t i c a l issues"
Traditional theory posits that a convertible regime, such as the classical gold standard that prevailed 1880-1914, is characterized by a set of self-regulating market forces that tend to ensure long-run price level stability. These forces operate through the classical commodity theory of money [Bordo (1984)]. According to that theory, substitution between monetary and nonmonetary uses of gold and changes in production will eventually offset any inflationary or deflationary price level movements. The fixed nominal anchor also ensures long-run price predictability and hence protects long-term contracts. It also may foster investment in long-lived projects [Klein (1975), Leijonhufvud (1984), Flood and Mussa (1994)]. Adherence to the fixed nominal anchor by providing credibility to monetary policy contributes to low inflation both by restraining money growth and by enhancing money demand [Ghosh et al. (1996)]. However, while ensuring long-run price stability and predictability, a gold standard provided no immunity to unexpected shocks to the supply of or demand for gold. Such shocks could have significant short-run effects on the price level. In a world with nominal rigidities they would generate volatility in output and employment 18.
18 According to Fischer (1994), in a comparison of price level stability versus low inflation, these volatility costs outweighthe benefits of long-run price level predictability.
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Indeed, because of the problem of wide swings in the price level around a stable mean under the gold standard, Fisher (1920), Marshall (1926), Wicksell (1898), and others advocated reforms such as the compensated dollar and the tabular standard that would preserve the fixed nominal anchor yet avoid swings in the price level [Cagan (1984)]. In an inconvertible fiat money regime, without a nominal anchor, monetary authorities in theory could use open market operations, or other policy tools, to avoid the types of shocks that may jar the price level under a specie standard and hence provide both short-run and long-run price stability. However, in the absence of a fixed nominal anchor, some other type of commitment would be required to prevent the monetary authority from using seigniorage to satisfy the government's fiscal demands, or to maintain full employment. In its international dimension, the convertible regime was one of fixed exchange rates and a stable nominal anchor for the international monetary system. Stability, however, came at the expense of exposure to foreign shocks through the balance of payments. In the presence of wage and price stickiness, these shocks again could produce volatile output and employment. Adherence to the international convertible regime also implied a loss of monetary independence. Under such a regime the monetary authorities' prime commitment was to maintain convertibility of their currencies into the precious metal and not to stabilize the domestic economy. In a fiat (inconvertible) money regime, adhering to a flexible exchange rate provides insulation against foreign shocks 19. However, as in a convertible regime, countries in fiat money regimes can adopt fixed exchange rates with each other. The key advantage is that it avoids the transactions cost of exchange. However, a fixed rate system based on fiat money may not provide the stable nominal anchor of the specie convertibility regime unless all members define their currencies in terms of the currency of one dominant country (e.g., the USA under Bretton Woods or Germany in the EMS). The dominant country in turn must observe the rule of price stability [Giavazzi and Pagano (1988)]. The theoretical debate on the merits of fixed and flexible exchange rates stemming from Nurkse's (1944) classic indictment of flexible rates and Friedman's (1953) classic defense is inconclusive 20. It is difficult to defend an unambiguous ranking of exchange rate arrangements 21. Hence, evidence on the performance of alternative monetary
19 Theoretical developments in recent years have complicatedthe simple distinction between fixed and floating rates. In the presence of capital mobility, currency substitution, policy reactions, and policy interdependence, floating rates no longer necessarily provide insulation from either real or monetary shocks [Bordo and Schwartz (1989)]. Moreover, according to recent real business cycle approaches, no relationship may exist between the international monetary regime and transmission of real shocks [Baxter and Stockman (1989)]. 20 For surveys, see Frenkel and Mussa (1985) and Bordo and Schwartz (1989). Also see McCalltnn (1997) p. 15. 21 See, for example, Helpman and Razin (1979) and Helpman (1981).
204
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r e g i m e s is crucial in assessing the condition u n d e r w h i c h one or another r e g i m e is best for welfare 22.
4.3. Measures o f macroeconomic performance, by regime In Table 4.1 we present annual data on two key m e a s u r e s o f e c o n o m i c p e r f o r m a n c e , the inflation rate ( G N P deflator) and the g r o w t h rate o f real per capita i n c o m e ( G N P ) for the five largest industrial countries across four r e g i m e s over the p e r i o d 1 8 8 1 1995 23. The r e g i m e s covered are: the classical g o l d standard (1881 1913); the interwar p e r i o d ( 1 9 1 9 - 1 9 3 8 ) ; Bretton Woods ( 1 9 4 6 - 1 9 7 0 ) ; the present floating e x c h a n g e rate r e g i m e ( 1 9 7 3 - 1 9 9 5 ) 24. We divide the Bretton W o o d s p e r i o d into two subperiods: the preconvertible phase ( 1 9 4 6 - 1 9 5 8 ) and the convertible phase (1959-1970)25. We divide the recent float into two subperiods: h i g h inflation ( 1 9 7 3 - 1 9 8 2 ) and low inflation (1983-1995). For the U n i t e d States o v e r the p e r i o d 1 8 8 0 - 1 9 2 9 , we show data f r o m two sources: Balke and G o r d o n (1986), and R o m e r (1989). A l l sources for the U S A and other countries are shown in the Data A p p e n d i x . For e a c h variable and e a c h country w e present two s u m m a r y statistics: the m e a n and standard deviation. A s a s u m m a r y statistic for the countries taken as a group, we s h o w the grand m e a n 26. We c o m m e n t o n the statistical results for e a c h variable.
22 Meltzer (1990) argues the need for empirical measures of the excess burdens associated with flexible and fixed exchange rates the costs of increased volatility, on the one hand, compared to the output costs of sticky prices on the other hand. His comparison between EMS and non-EMS countries in the postwar period, however, does not yield clear-cut results. 23 For similar comparisons for the G-7 see Bordo (1993b). For 21 countries including advanced and developing countries see Bordo and Schwartz (1996b). Other studies comparing historical regime performance include: Bordo (1981); Cooper (1982); Meltzer (1986); Schwartz (1986b); Meltzer and Robinson (1989); Eichengreen (1993b); and Mills and Wood (1993). 24 One important caveat is that the historical regimes presented here do not represent clear-cut examples of fixed and floating exchange rate regimes. The interwar period is not an example of either a fixed or floating rate regime. It comprises three regimes: a general floating rate system from 1919 to 1925, the gold exchange standard from 1926 to 1931, and a managed float to 1939. For a detailed comparison of the performances of these three regimes in the interwar period, see Eichengreen (1991b). We include this regime as a comparison to the other three more clear-cut cases. The Bretton Woods regime cannot be characterized as a fixed exchange rate regime throughout its history. The preconvertibility period was close to the adjustable peg envisioned by its architects, and the convertible period was close to a defacto fixed dollar standard. Finally, although the period since 1973 has been characterized as a floating exchange rate regime, at various times it has been subject to varying degrees of management. 25 We also examined the period (1946-1973), which includes the three years of transition from the Bretton Woods adjustable peg to the present floating regime. The results are similar to those of the 1946-1970 period. 26 Bordo (1993b) also presents data on seven other variables: money growth, nominal and real short-term and long-term interest rates and nominal and real exchange rates. Bordo and Schwartz (1996b) show the same data plus the government budget deficit relative to GDP for fourteen additional countries.
Ch. 3." Monetary Policy Regimes and Economic Performance: The Historical Record
205
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Fig, 4.1, Annual inflation rate, 1880 1995, five countries. 4.4. Inflation and output levels and uariability 4.4.1. Inflation
The rate of inflation was lowest during the classical gold standard period (Figure 4.1). This was true for every country except Japan which did not go on the gold standard until 1897. During the interwar period mild deflation prevailed. The rate of inflation during the Bretton Woods period was on average and for every country except Japan lower than during the subsequent floating exchange rate period. During the Bretton Woods convertible period the inflation rate in the USA, the UK and France was higher than in the preceding subperiod; the reverse was true for Germany and Japan but on average there was not much difference between the
Ch. 3: Monetary Policy Regimes and Economic PerJbrmance: The Historical Record
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subperiods. During the floating regime inflation has been lower in the recent subperiod o f low inflation than during the Bretton Woods convertible subperiod except in the U S A and U K 27. The Bretton Woods period had the most stable inflation rate as j u d g e d by the standard deviation. The managed float and the gold standard periods were next. The interwar period was the most unstable. However, when subperiods o f the regimes are distinguished, the recent decade o f low inflation was the most stable, followed by the Bretton Woods convertible regime, then the inflation phase o f the float, and last, the gold standard period. In general, the descriptive evidence o f lower inflation under the gold standard and the Bretton Woods convertible regime than is the case for the other regimes is consistent with the view that convertible regimes provide an effective nominal anchor. The marked low inflation o f the recent decade suggests that the equivalent o f the convertibility principle may be operating. At the same time, evidence that inflation variability on average was higher in the classical gold standard period than in most other regimes is consistent with the commodity theory o f money and the price-specie flow mechanism which posits offsetting changes in the monetary gold stock 28. The evidence on inflation and inflation variability is also consistent with the behavior o f two other nominal variables [Bordo (1993b)]. First, m o n e y growth was generally lowest under the gold standard across all countries, followed by the Bretton Woods convertible regime. It was most stable during the Bretton Woods convertible regime. Second, long-term nominal interest rates were lowest during the classical gold standard period. During Bretton Woods they were lower than in the recent float [see also McKinnon (1988)]. 4. 4.2. Real p e r capita income growth Generally, the Bretton Woods period, especially the convertible period, exhibited the most rapid output.growth o f any monetary regime, and, not surprisingly, the interwar
27 The dispersion of inflation rates between countries was lowest during the classical gold standard and to a lesser extent during the Bretton Woods convertible subperiod compared to the floating rate period and the mixed interwar regime [Bordo (1993b)]. This evidence is consistent with the traditional view of the operation of the classical price-specie-flow mechanism and commodity arbitrage under fixed rates and insulation and greater monetary independence under floating rates. 28 Supporting evidence is provided in a recent study by Ghosh et al. (1996). Classifying the exchange rate systems for 136 countries over the period 1960 to 1990 into pegged, intermediate, and floating, they adopt a methodology similar to that of Table 4.1. They find that the unconditional mean inflation rate for countries on pegged exchange rates was significantly lower than for those that did not peg. This result holds up, controlling for the 1960s during which most countries adhered to Bretton Woods. The only exception was high-income floating countries which had lower than average inflation rates. Their results are unchanged when conditioned on a set of determinants of inflation, and when account is taken of possible endogeneity of the exchange rate regime. With respect to the volatility of inflation, they found it to be highest among floaters, again with the exception of high income countries. For them, it was the lowest.
208
M.D. Bordo and A.J. Schwartz
period the lowest (Figure 4.2). Output variability was also lowest in the convertible subperiod of Bretton Woods, but because o f higher variability in the preconvertibility period, the Bretton Woods system as a whole was more variable than the floating exchange rate period. Both pre-World War II regimes exhibit considerably higher variability than their post-World War II counterparts. The comparison does not apply to the U S A based on the Romer data 29, 30, 31 To link rapid growth in the industrialized countries in the quarter century following World War II to the Bretton Woods international monetary system [Bretton Woods Commission (1994)], seems less compelling than for other aspects of macroeconomic performance. First, there is little conclusive evidence linking exchange rate volatility to either trade flows or the level o f investment [Mussa et al. (1994)], avenues by which a stable exchange rate regime might have affected economic growth. Although Ghosh et al. (1996) find evidence linking real growth to the growth of investment and trade for pegged countries, they also find total factor productivity growth to be an important channel o f growth for floaters. Second, although trade liberalization may have played an important role in the acceleration o f growth rates in the European economies during the Golden Age, most o f the liberalization of trade, before nations declared Article VIII current account convertibility in December 1958, was under the aegis o f institutions developed outside o f the Bretton Woods framework - the Marshall Plan, Organization for European Economic Cooperation (OEEC), European Payments Union (EPU), and European Coal and Steel Community (ECSC) [Eichengreen (1995)]. Finally, the Bretton Woods arrangements might have contributed to postwar growth by being part o f the overall package creating political and economic stability - "the Pax Americana", that was a reaction to the chaos o f the interwar and World War II periods. In this view, rapid postwar growth represented a "catch up" by the European nations and Japan from low levels o f per capita output compared to that o f the leading industrial country, the USA. The "catch up" by these nations was encouraged by the USA. They adopted the leader's best-practice technology and hence grew at a much more rapid rate than before [Abramovitz (1986)] 32.
29 The Bretton Woods regime also exhibited the lowest dispersion of output variability between countries of any regime, with the interwar regime the highest [Bordo (1993b)]. The lower dispersion of output variability under Bretton Woods may reflect conformity between countries' business fluctuations, created by the operation of the fixed-exchange rate regime [Bordo and Schwartz (1989)]. 3o The Hodrick Prescott filter alternative to the first differences used in Table 4.1, yields basically the same rankings of regimes. 3~ In their 1960 1990 sample, Ghosh et al. (1996) find little connection between adherence to a pegged exchange rate and growth, once account is taken of the 1960s experience. High-income floaters generally had more rapid growth than low-income floaters. There was little correlation between output volatility and the regime. 32 in an institutional vein, it has been argued that the Bretton Woods framework(plus GATT) contributed to growth by providing an overall framework of rules. Within them Western European nations solved a hierarchy of coordination problems, allowing them to encourage investment in growth-generating export
Ch. 3:
Monetary Policy Regimes and Economic Performance." The Historical Record 0 ~i
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sectors [Eichengreen (1995)]. Without the Bretton Woods framework it might not have been possible to solve prisoner's dilemma games between labor and capital within each country taken in isolation and, for the OEEC, EPU, Marshall Plan, and ECSC to liberalize trade on comparative advantage lines between the members. Given that the European regional arrangements occurred outside of, and because of, shortcomings in the Bretton Woods arrangements, one wonders if institutional developmentswould have been much different if the European countries were not party to Bretton Woods at all.
M.D. Bordo and A.J. Schwartz
210
O,
and (la), (Ta-d). The maximand in Equation (10) is the number of people multiplied by what each enjoys in present discounted value of utility from consumption c. The k constraint says that capital accumulates from the output left over after total consumption and depreciation. Coefficient 0 parametrizes the intertemporal elasticity of substitution in consumption, while p is the discount rate. We emphasize that we have restricted p to be not just nonnegative but to exceed the sum of the rates of population growth and technical change, p>v+~.
(11)
Equation (10) determines consumption and thus savings and investment to maximize social welfare. Define ~ to be per capita consumption normalized by technology, i.e., = c/A. Appendix A shows that the necessary first order conditions to Equation (10) are: _ f(k) -
C
- (Vf(/~) - {p + (5 + 0~1)
0 -1 ,
(9b)
lim lc(t) e -(p-v-~)t = O.
t-~OO
A balanced-growth equilibrium is a positive time-invariant technology-normalized capital stock k (together with implied ~ = f ( k ) ) such that under Equation (9a)
XN. Durlauf and D.T. Quah
248
.f ([c)[c-1
m,
o
k'
Fig. 2. Solow-Swan growth and convergence. Function f(lcfic 1 is continuous, and tends to infinity and zero as fc tends to zero and infinity respectively. Moreover, it is guaranteed to be monotone strictly decreasing. The vertical distance betweenf(~:),~-1 and (6 + v + ~)'c 1 is v t~/k. Convergence to steady state k* therefore occurs for all initial values k. and under Equation (9b) c=0
where = f ( k ) - (6 + v + ~) lc E (0, f(/})). (Our balanced-growth equilibrium definition implies that we can specialize to timeinvariant k.) Balanced-growth predictions are identical under either accumulation assumptions (9a) and (9b). To see this, note that at balanced-growth equilibrium under assumption (9b) we can find r in (0, 1) such that =f(/c) - (6 + v + ~)X: = (1 - O f ( k ) as both k and ~ are constant through time; Equation (9b) thus reduces to (9a). Two questions arise from this formulation. First, does a balanced-growth equilibrium always exist? And, second, even if both formulations have the same empirical implications in long-run steady state, do transitions to steady state differ? Figure 2 shows that a unique balanced-growth equilibrium exists and that lc satisfying assumption (9a) is dynamically stable everywhere in the region lc > 0 (Appendix A also proves this). Since ~ = f ( k ) , we immediately have that output per effective worker too has a unique, globally stable steady state.
Ch. 4." The New Empirics of Economic Growth
249
The dynamics of this model can be understood further by taking a Taylor series expansion in log/c about steady-state lc*,
k - T (Vf(k)-f(k)k
1)
× ( l o g / c - l o g k*).
For F Cobb-Douglas, F ( K , N A ) = K a ( N A ) l-a,
a C (0, 1)
~ f(Tc) =/c a,
(12)
this first-order series expansion becomes d log /c =a -(1 - a ) ( 6 + v + ~) × (log k - log/c*) = ft. × (log k - log/c*) where we have defined j, de__f--(1 -- a)(6 + V + ~) < 0.
(13)
Solving this differential equation gives log lc(t) - log k* = (log lc(0) - log/c*)e it log y(t) - log ~* = (log ~(0) - log/p*)e xt ~ 0 as t ---+ oo,
(14a)
i.e., log lc and log F converge to their respective steady-state values log/c* and log~* de=f logf(/c*) exponentially at rate [~[. As a increases to 1 this rate of convergence approaches 0: thus, the larger is the Cobb-Douglas coefficient on physical capital, the slower does log ~ converge to its steady-state value. Under the Cobb-Douglas assumption (12), the accumulation equation (9a) and Figure 2 imply the steady-state level
= [(6+ v + ~) lr]a/(l 6)
(15)
Equation (15) gives steady-state income levels as depending positively on the saving rate and negatively on the labor force growth rate. Before discussing in detail the empirical implications of Equation (14a), we turn to how the Solow-Swan and the general equilibrium Cass-Koopmans versions of this
250
S.N. Durlauf and D.Z Quah 42
~
Z, + ~ = p - v - ~+ 0~> 0
X,X2 = V2f(!c~')g" 0 '< 0
Fig. 3. Eigenvalues in the Cass-Koopmans model. Since V2f(k *) ~*0 ' - (V2f(/~*) lc*) [J(/c*)//~* (6 + v + ~)] 0 1, if f(/0 =/~a, with a E (0, 1) then as a increases towards unity the negative eigenvalue ~2 rises towards zero. model differ in their observable predictions. First, rewrite the first two equations in Equation (9b) as d (log~)= dt \ l o g
( ~ - ( 6 + v + ~ ) ) (Vf(k)_[p+6+O~])O-1
(16) "
Define the zero of (~/~:, c/~) by (~:*, ~*). (Appendix A establishes that this is welldefined.) Then the first-order Taylor series expansion of (log/~, log ~) about steady state (log lc*, log ~*) is:
d log )a \ log
1
= ~
V2.f(~g)k0 1
('og log *) 0
x
log ~
log 8" (17)
{ log lc - log k* ) deftM × \ log ~ - log ~* _ " Coefficient matrix M in Equation (17) has determinant Vzf(k)~0 -1 < 0 so its eigenvalues are real and of opposite sign. Moreover, its trace is Vf(k*) - (fife*) - ~*)/k* = ( p + 6 + 0~) - (6 + v + ~)
=p-(v+~)+O~ >O. Denote the eigenvalues of M by )h > 0 > )~2. Figure 3 uses these determinant and trace properties to establish how A1 and A2 vary with the parameters of the model. For
Ch. 4.. The New Empirics of Economic Growth
251
the Cobb-Douglas technologyf(]~c) = fca, eigenvalue Z2 increases towards 0 as c~ rises towards 1. Eigenvalue ~2 determines dynamics local to the steady state as: log lc(t) - log lc* = (log lc(0) - log lc*) e z2t,
(18)
log ~(t) - log ~* = (log ~(0) - log ~*) e x:t, with [log fc(0) - log lc*] and [log ~(0) - log ~*] satisfying a specific proportionality condition described in Appendix A. Then for technology (12), with ~* = (lc*) a, the first equation in (18) gives log ~(t) - log y* = (log y(0) - log ~*) e z2t ~ 0 as t ~ oc.
(14b)
Comparing equations (14a) and (14b) we see that assumptions (9a) and (9b) deliver identical observable implications - not just in steady-state balanced growth, but also locally around steady state. The convergence rates Z and Z2 have different interpretations as they depend on different economic parameters. However, they vary in the same way when the technology parameter a changes. How are these common observable implications useful for understanding patterns of cross-country growth? Parallel to the theoretical development above, we interpret the bulk of the empirical literature as concerned with two sets of implications: first, steady-state balanced-growth predictions and, second, (convergence) predictions local to steady state. Without loss, write the convergence coefficient as Z in both (14a) and (14b). From observed per capita income y = ~HA = )~A we have: log y(t) = log y(t) + log A(t) = log ~* + [log ~(0) - log ~*]e xt + log A(0) + ~t. Moreover, since .p* =f(/c*) a n d f ( k * ) / k * = (6 + v + ~ ) r l, there is some function g such that ~* = g((6 + v + ~) I r). We can therefore write the implied sample path in observable per capita income as log y(t) = log(g((6 + v + ~) 1z-)) + log A(0) + ~t + [log y(0) - (log(g((6 + v + ~) 1T)) + log A(0))]e xt,
(19)
and its time derivative d - log y(t) = ~ + Z × [log y(0) - (log (g((6 + v + ~) 1r)) + log A(0))]e xt dt
(19')
From Equation (19) log y can be viewed as having two components: a convergence component (the term involving e xt) and a levels component (the rest of the right-hand side).
252
~N. DurlaufandD.Z Quah
logy(t) l°gy~(O)
fit + (log~* + logA(O))a
logy3(O) logya(0) /
~
~
~t + (logy*+ logA(0))b
logy4(0) I
Fig. 4. Growth and convergence in the neoclassical model: two different possible steady-state paths corresponding to two possible values for the sum log~* + logA(0) - log(g((6 + v + ~) ~r)) + logA(0). As long as this sum remains unobserved or unrestricted, any pattern of cross-country growth and convergence is consistent with the model. As drawn, the a value applies to economies at yl(0) and y2(0) while the b value to y3(0) and y4(0). Economies 1 and 2 converge towards each other, and similarly economies 3 and 4. At the same time, however, economies 2 and 3, although each obeying the neoclassical growth model, are seen to approach one another, criss-cross, and then to diverge.
Figure 4 displays a graphical representation o f Equation (19) for two possible values o f log(g((6 + v + ~ ) - l r ) ) + log A(0). The figure shows two different possible steady-state paths - corresponding to two possible values for the sum log ~* + log A(0) = log(g((6 + v + ~)-1 r)) + log A(0). Relative to typical claims in the literature, Figure 4 conveys a negative message. As long as log j?* + log A(0) remains unobserved or unrestricted, any pattern o f crosscountry growth and convergence is consistent with the model. As drawn in Figure 4, the a value applies to economies at yl(0) and ye(0) while the b value to y3(0) and y4(0). Economies 1 and 2 converge towards each other, as do economies 3 and 4. A t the same time, however, economies 2 and 3, although each obeying the neoclassical growth model, are seen to approach one another, criss-cross, and then diverge. We can now organize those empirical studies that use the neoclassical growth model for their theoretical underpinnings. Cross-section regression analyses, such as Barro and Sala-i-Martin (1992), Baumol (1986), DeLong (1988), Mankiw, Romer and Weil (1992), and Sachs and Warner (1995) estimate variants o f Equation (19). Mankiw, Romer and Weil (1992), in particular, consider two versions o f Equation (19): first, when the term in e zt is already at its limiting value, then the first component o f the
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253
expression is taken to "explain" the steady-state cross-section distribution of income. 3 Second, when the term in e zt is taken to be central - and the rest o f the right-hand side o f Equation (19) is given (or are nuisance parameters) - the equation is taken to "explain" convergence in income. This second interpretation motivates the convergence analyses o f the other papers mentioned above. 4 In our reading o f the empirical literature, there is some confusion over the goals o f the analysis. On the one hand, a researcher might study Equation (19) to estimate the coefficients o f interest in it. But the only parameters related to the economic reasoning in Equation (19) are those in the function g, i.e., parameters o f the production function. Thus, standard econometric techniques applied to this equation might be useful for recovering such parameters. A researcher might go further and seek, in an ad hoc way, to parameterize A(0) and ~ as functions o f other economic variables. While this might be useful for regression fitting, its results are difficult to interpret in terms o f the original economic analysis. After all, A(0) and ~ played no integral role in the theoretical reasoning and it is unclear that a structural model incorporating these other variables would produce a regression o f the type typically estimated. A second goal o f an empirical analysis o f Equation (19) is to address questions o f cross-country patterns o f growth. We think, however, that all such analyses, even at their most successful, are silent on those questions. From Figure 4, as long as A(0) is unrestricted or omitted from the analysis, no study o f Equation (19) can reveal how cross-country incomes evolve. One interpretation o f the preceding is that the basic model's key implications are both too strong and too weak. I f A(0) were required to be identical across economies, then the growth and convergence predictions in Figure 2 are likely inconsistent with the inequality dynamics in cross-country incomes we described in Section 2. If, on the other hand, a researcher goes to the opposite extreme and allows A(0) to differ arbitrarily across economies, then the theoretical model says little about cross-country patterns of growth. The free parameters A(0) carry the entire burden o f explanation. Finally, should a researcher take a middle path, and restrict A(0) to depend on specific economic variables in an ad hoc manner, then that researcher might well end up fitting the data satisfactorily. However, the results of such a procedure can be difficult to interpret within the Solow-Swan (or Cass-Koopmans) growth model. 5
3 The Mankiw Romer-Weil formulation, of course, includes human capital accunmlation. That feature is ignored for expositional convenience here as it does not affect our basic point. We return to it below. 4 An earlier literature [e.g., Grief and Tullock (1989)] studied similar regression equations with growth on the left-hand side and explanatory variables on the right. We distinguish this from the work described in the text only because that earlier research did not show any preoccupation with convergence. It instead investigated, using exploratory empirical techniques, only the determinants of growth an important question, certainly, but distinct from the simultaneous interest in convergence that characterizes the newer literature. 5 Mankiw, Romer and Weil (1992) is a key exception. Those authors focus on that part of the steadystate path that depends on savings and population growth rates, not on A(0), and suggest that their human capital modification of the Solow-Swan model does fit the data. We discuss that model below.
S.N. Durlaufand D.T. Quah
254
Empirical studies such as Bernard and Durlauf (1995, 1996), Durlauf and Johnson (1995), and Quah (1997) seek to circumvent some of the criticisms we have just described. One strand of this work estimates models that explicitly nest the traditional neoclassical setup. Another strand seeks to identify those features of the long-run behavior of cross-country incomes that are invariant with respect to finely-detailed structural assumptions. Before turning to more detailed empirics, however, we describe models that depart from the basic set of assumptions in the neoclassical growth model. This is easy to do given the structure we have set up. Again, our goal is not to repeat discussion already found elsewhere, but to survey in a unified way the empirical implications of the different classes of models. 4.2. The neoclassical model: multiple capital goods A well-known model due to Mankiw, Romer and Weil (1992) (hereafter MRW) adds human capital to the Solow-Swan model, and develops empirics that potentially better explain the cross-country income data than models that account only for physical capital accumulation following Solow's original work. The MRW model fits in our framework as follows. Again, take production technology (la), and assume (7a-c). In place of Equation (7d), let K have two components, the first called physical capital Kp and the second human capital Kh: K = (Kp, Kh)'.
(7d')
(Distinguish Kh from that concept of human capital that is H - the latter multiplies the labor input N to produce effective labor input N, while the former is an entry in the vector of capital stocks, and thus is better viewed as analogous to physical capital Kp.) Extend the accumulation assumption (9a) to Rp = vpY - 6pKp,
vp, 6p > O,
Kh = vhY - 6hKh,
~h, 6h > 0,
(9a')
Z'p+ Th < 1. Then technology-intensive effective capital stocks k = @,, L ) ' with/Cp = Kp/NA and L = Kh/NA satisfy
~
Y -(a~+ v+~),
L
Y -(ah+ v+~).
A balanced-growth equilibrium is a positive time-invariant triple (~, kp, kh)* such that
Y
Y = 6h+v+~.
Ch. 4: The New Empirics of Economic Growth
255
When F is Cobb-Douglas so that
fffCp, fCh)=(lcp)a~ffch)ah,
etp, ah > O and ap+Cth < l,
(20)
straightforward calculation establishes that a balanced-growth equilibrium has:
-ap -(1 - ap) ( log ((6p + V + ~)'Cpl) )
log k;
\
×
log ((6h + v+~)rh 1)
= (1 - ap - ah) -~
× a~log((ap+v+~) ~r,~)+(~-a~)log((a~+~+~)-~r,,) and log ~* = (1
ap - ah)-' [ap log ((6p + V + ~)-1Tp) + ah log ((6h + v + ~)-l rh)].
0s')
Equation (15 ~) is the MRW counterpart to the Solow-Swan levels prediction (15). It specializes to the latter when ah is set to 0; otherwise, it comprises a geometric average of contributions from physical and human capital. It is easy to show in state space (/Cp,fch) that this system is globally stable and converges to balanced-growth equilibrium. In general, then, all dynamics - including those of y - depend on the bivariate state vector (/~p, fch). This would suggest that, in a growth regression, studying the (one-dimensional) coefficient on initial income alone, with or without auxiliary ad hoc conditioning, gives a misleading picture of dynamics local to steady state. However, with additional restrictions on model parameters, conditioning on the level of~(t) can render the local convergence behavior of~ independent of the state (fop(t), [ch(t)). Mankiw, Romer and Weil (1992) achieve this by setting equal the depreciation rates on human and physical capital, i.e., 6p = Oh. From Equation (20), and taking the firstorder Taylor series expansion in log ~, log lop, and log lch, we have:
~p +ah~ =%
Tp~pp
Y- + ah [r~ L a
(ah + v + ~ ) l ~
= ~ [~a~ + ~ + ~) (Clog ~ - log ~*) - Oog ~,, - log k ; ) ) ] + a~ [(ah + v + ~) (Clog ~ - log ~*) - Clog ~ - log ~;))]
256
S.N. Durlauf and D.T. Quah
so that 6p - 6h = b then gives
33 -
(1 - % - ah)(6
+ V + if) ×
(log 33--
l o g 33*).
(21)
Under this MRW specification the sample path (19) changes so that the levels and convergence components include terms in rh and c% The observable implications remain unchanged: observed per capita income evolves in balanced-growth equilibrium as A(t); away from steady state, observed per capita income converges towards that balanced-growth path. The dynamics are still as given in Figure 4. The MRW model has been used as the basis for numerous empirical studies. To aid our subsequent discussion of those studies, we develop a more explicit representation for the model's predictions. From Equation (21) now let )~ ~f -(1 - % - ah)(6 + v + ~) < 0,
(22)
so that log 33(t) - log 33* = [log 33(0) - log 33*] e zt log 33(t + T) - log 33" = [log 33(t) - log 33*] e zT". Transforming to get observable log y(t), this becomes: log y ( t + T) - [log A(0) + (t + T)~] = (1 - e zr) log 33* + [log y(t)
log A(0) - t~] e zr
log y ( t + T) - log y(t) = (1 - e zT) log 33* + (e zr - 1) log y ( t ) + (1 - eZV) log A(0) + (t + T - eZrt)~ Substituting in Equation (15 1) for steady state log 33* gives log y ( t + T) - log y(t) = (1 - e zr) log A(0) + (t + T - eZrt)~ + (e zT" - 1)log y(t) + ( 1 - e zr) l _ a pap ah log rp
(23)
ah - ah log rh + (1 - e ~v) 1 - ap - (1
-
e ~'v) 1 --a pa+ p a-h
ah l o g ( 6 + v + ~ ) .
In words, growth depends on some (exogenously given) constants, the initial level log y(t), savings rates, technological parameters, and the population growth rate. Since ). < O, the coefficient on the initial level log y(t) should be negative.
Ch. 4: The New Empirics of Economic Growth
257
Comparing MRW's convergence rate (22) with Solow-Swan's (13), the only difference is the addition o f ah in the former. Thus, keeping fixed ap (physical capital's coefficient), 6, v, and ~, MRW's addition o f human capital to the neoclassical model implies )~ closer to zero, or a slower rate of convergence, than in the Solow-Swan model. In both the MRW and traditional neoclassical models the levels o f balanced-growth income time paths can vary with the parameters o f preferences and technology (r, p, 0, and a). However, the rate o f change in those balanced-growth time paths in incomes is always just the exogenously given ~ = ft/A. This is useful to remember when working with representations such as Equation (23) - although the dependent variable in the regression equation is a growth rate, these models do not explain growth rates over long time horizons. It is this that makes it useful to label these models of exogenous growth.
4.3. Endogenous growth: asymptotically linear technology We now consider a range o f models that generate long-run growth from other than exogenous technical change. When possible, we will show how such models can be derived by straightforward perturbations o f the parameterizations we have used to describe the neoclassical model. 6 Assume, as in the standard one-capital neoclassical model, Equations (la) and (7a-d), but instead o f Equation (8), suppose that V2)A > 0 :
lim F ( K , NA) > 0. x~oc K
(24)
For instance, the CES production function
F ( K , N A ) = [~/xK a + ]IN(~(A) a] 1/a is homogeneous o f degree 1, concave, and satisfies Equations (2), (3) and (24) with VNA > 0"
lim F ( K , N A ) _ g~x/~ > O. K--+o~ K
Call a production function satisfying condition (24) asymptotically linear. The motivation for this terminology comes f r o m f ( k ) varying linearly with ~: as the latter gets large. 7
6 Such a strategy is inspired by Solow (1956, Example 3); see also Jones and Manuelli (1990). 7 Of course, even if the limitingf(k)k I were zero rather than positive, we would still have asymptotic linearity (albeit trivially), but we hereafter ignore this possibility when using the phrase. A useful alternative is to say that condition (24) impliesf(k) is O(~:)(or big-oh k), following standard terminology in statistics and elsewhere. Dully and Papageorgiou (1997) find that a CES specification for the aggregate production function fits cross-cotmtry data better than a Cobb-Douglas, and moreover that the elasticity of substitution between capital and labor exceeds one. This evidence implies the possibility for endogenous growth of the kind described in Jones and Manuelli (1990) and this subsection.
S.N. Durlauf and D.T. Quah
258
lim~ ~ o *
f
('~)k-I
> (6 + v + ~) r
i
(6+v+~)r
o
~
Fig. 5. Asymptotically linear (O(k)) growth and convergence. The continuous function J(Tc)fc 1 tends to infinity as lc tends to zero and to limi~oof(Tc)Tc 1 > 0 as k tends to infinity. Moreover, it is
guaranteed to be monotone strictly decreasing for finite lc. The vertical distance between f(k)Ic 1 and ((~ + V d- ~).g--I is T l~/~f. If limk~o~f(fc)Tc-1 < (6 + V + ~)T-1 then convergence occurs as in the Solow-Swan model with some constant finite Ic* describing balanced-growth equilibrium. However, if lim~ ~oof(Tc)Tc-1 > (6 + v + ~)r 1 then ~/tc is always positive, and balanced growth obtains only as /c 7 ec. Every initial ~:(0) is part of an equilibrium tending towards balanced growth. By l'Hopital's rule, condition (24) gives lim ~ ( k ) =
lim f(~)~-I >0 ~
1~
s(k) = 1,
so that, following the reasoning in Section 3, balanced-growth equilibria with positive ) / ~ are now possible. Let capital accumulation follow (9a) as before. Whereas previously Figure 2 established existence of a unique balanced-growth equilibrium with finite (y*, k*) and k / k = 0, Figure 5 now shows a range of possibilities. Taking technology parameters as fixed, define the threshold savings rate T=
-
6+v+ff lim1_~oo f ( k ) ~ 1"
The numerator is the rate at which technology-adjusted physical capital per worker naturally "dissipates", given the rates of discount, population growth, and exogenous technology development. The denominator is physical capital's limiting average product, which equals the limiting marginal product. This expression thus displays a tension between two opposing forces: the more productive physical capital is in the limit, the lower is the threshold savings rate, whereas the faster capital naturally dissipates, the higher is the threshold. If _r is at least 1, then all feasible savings rates r E (0, 1) imply the same behavior as the Solow-Swan outcome: growth in y occurs
Ch. 4." The New Empirics of Economic Growth
259
in the long run at rate ~. However, if_T is less than 1, more intricate long-run dynamics can manifest. When an economy has r less than _r, again, the result is the Solow-Swan outcome. But when economies have sufficiently high savings rates, i.e., T C (_r, 1), then k/k always exceeds a time-invariant positive quantity, and has limiting behavior given by
tli~rn~ ~:(t)
linL f(k) k '
z- - (b + V + ~) > O.
Moreover, such (y, k) paths tend towards balanced-growth equilibrium since Vf(k(t))
To(t)
~(t)
/~(t)
[
k(t)
~-
[1 f(~(t))~(t)_ I It(t)
~
--4
0
as t --, oc.
As long-run growth rates are then Y"9-~+ [(~Em f(fc)Tc
1)r--(6-}-V+~)1
> ~,
they increase in r, meaning that economies saving a higher fraction of their income grow faster in the long run. It is this growth effect that makes the current specification an "endogenous growth" model. Compare this with the standard neoclassical growth model where savings rates affect only the levels of balanced-growth sample paths, not growth rates. This relation between savings and long-run income growth applies only to those economies with savings rates exceeding the threshold value _r. All economies with savings rates below this value cannot influence long-run income growth rates by changing their savings behavior (unless they move savings rates above that threshold). What observable implications follow from this? If savings rates were uniformly distributed across countries, there should be one cluster of economies around the same low per capita income growth rate and a different group with scattered income growth rates increasing in savings rates; see, for instance, Figure 6. As in the standard neoclassical model, this asymptotically linear technology model can be given a general equilibrium interpretation. Recall assumption (9b), and assume the preference parameter 0 satisfies l i m ~ f ( f c ) fc-1 - (p + 5)
> 0 >
lim~__,o~f(k) k-' - (p + 6) > O. p-v
(25)
From Equation (10) the parameter 0 is the inverse of the intertemporal elasticity of substitution. Thus, Equation (25) states that elasticity can be neither too high nor too low - it must respect bounds varying with technology parameters.
S.N. Durlauf and D.Z Quah
260 long-run e l y
Y Fig. 6. Threshold effect of savings on long-run income growth rates in O(~:) model. For economies with savings rates r less than the threshold value _% the long-run income growth rate is ~ independent of r. If T > 2, however, then savings rates positively affect long-run growth.
From p > v, Equation (25) implies that limk~o~f(Tc) fc i > p + 6. For the interval of feasible values for 0 to exist, it suffices that ~ < p - v, which in turn follows from Equation (11). Finally, these relations imply lim f ( k ) ~:
1 > 6 + v +
~,
lc---+oo
which had been used earlier to guarantee r < 1. Thus, Equation (25) is related to but strengthens the assumption underlying Figure 5. In Appendix A, we show that Equation (25) implies that there exists a balancedgrowth equilibrium with a positive growth rate given by tlimoo~ =
f(k)k-l-[p+6+O~]
0 I >0,
and that for every initial k(0) there exists an equilibrium tending towards balanced growth. If, however, 0 is too large, then the unique balanced-growth equilibrium has limt--,o~ )(t)@(t) = 0. The equilibria have exactly the character described above in the discussion surrounding Figure 5, only with 0 -~ replacing r. The models in Rebelo (1991) and Romer (1986) differ from those above in several important ways. Rebelo (1991) uses a linear AK specification in place of the usual convex production technologies. (Linearity, of course, implies asymptotic linearity.) Equilibrium in that model tends towards balanced growth. Romer (1986) distinguishes the productive effects of individual-specific physical capital from economy-wide externalities induced by private accumulation. Romer's
Ch. 4: The New Empirics of Economic Growth
261
model uses the production technology (lb) with the arguments to F identified as the actions of private agents, and lets A depend on K, but with K defined as the social or aggregate outcome. Private agents ignore the effects of their actions on A; there is an externality in private agents' decisions to accumulate physical capital. In Romer's model, as far as private agents are concerned, A still evolves exogenously. In equilibrium, of course, A depends on the purposeful actions of economic agents, and thus is properly viewed as endogenous. Private agents' optimizing decisions on consumption and savings remain identical to those in the standard neoclassical model. At the same time, the equilibrium aggregate outcome can display ongoing, endogenously-determined growth differing from the standard model. Moreover, the model also allows evaluating the efficiency properties of particular decentralized economic equilibria. Some versions of Romer's model imply equilibria tending towards balanced growth; others display ongoing growth but with no tendency towards balanced growth. 8 Essential economic features therefore differ. However, the model of Rebelo (1991) and certain versions of the general model in Romer (1986) resulting in ongoing endogenous growth have, in essence, the same mathematical structure as that described earlier in this section. Their observable implications, therefore, are also the same. One apparently natural conclusion from these models is that the researcher should now calculate regressions across economies of income growth rates on savings rates, tax rates, and so on - variables that in the analyses of Jones and Manuelli (1990), Rebelo (1991), and Romer (1986) potentially affect long-run growth rates. Such regressions would resemble the MRW regression (23) except that there is now no reason for the initial condition log y(t) to appear with a negative coefficient. This line of reasoning suggests that what distinguishes exogenous and endogenous growth models is whether the initial condition log y(t) enters negatively in an equation explaining growth rates. Note, though, that this endogenous growth analysis does not imply that the initial condition log y(t) should never appear in an estimated regression. By contrast, that initial condition is absent only in the balanced-growth limit, i.e., with lc infinite. But in any balanced-growth limit, even the exogenous-growth neoclassical model has the initial condition vanish from the right of relation (19), (19I), or (23). 4.4. Nonconvexities and poverty traps
An alternative class of models has focused on specific nonconvexities in the aggregate production function. 9 This research has analyzed the implications of such nonconvexities for the relation between initial conditions and the steady-state behavior
8 A suitably parameterized model following Example 1 in Romer (1986, p. 1028) yields equilibria tending towards balanced growth. 9 Increasing returns to scale, of the kind studied in Romer (1986), is also a nonconvexity, of course. What we mean instead are those nonconvexitiesassociated specificallywith certain threshold effects we will describe below.
S.N. Durlauf and D.T. Quah
262
of aggregate output. Models with nonconvexities, unlike the neoclassical model, lead to long-run dependence in the time-series properties of aggregate output. Specifically, nonconvex models can display poverty traps, where economies with low initial incomes or capital stocks converge to one steady-state level of per capita output, while economies with high initial incomes or capital stocks converge to a different steadystate level. Examples of such models include those by Durlauf (1993), Galor and Zeira (1993), and Murphy, Shleifer and Vishny (1989). The model due to Azariadis and Drazen (1990) is particularly convenient for illustrating the empirical differences between this framework and the neoclassical approach. The Azariadis-Drazen model works off thresholds in the accumulation of human and physical capital. These thresholds stem from spillovers between individual investments arising when aggregate capital is sufficiently high. In effect, economies with insufficient aggregate capital have different production functions from those with sufficiently high aggregate capital. We present the basic ideas of Azariadis and Drazen (1990) in our framework as follows. Modify the MRW production technology (20) to:
ap(t) = f ~p if lOp(t) > tOp(t) ap
ah(t) =
otherwise;
(26)
{ a h i f L ( t ) > tch(t) c~h otherwise;
where the explicit (t) indicates variables changing through time and the coefficients ap(t), ah(t) vary with the underlying state (lop, lch). The quantities t :3 "
............................. {;;ii;5 :::............ z i / //" / // /
t Fig. 10b. Coincident/3 and a convergence. The figure shows a cross section of economies where/3 and a convergence coincide. All economies converge smoothly in towards the common steady-state growth path. Similarly, the dispersion of the cross-section distribution declines to zero.
For an arbitrary pair of time points tl and t2 with tl < t2, the population cross-section regression of log y(t2) on a constant and log y(tl) is, by definition, the projection P [log y(t2) I 1, log y(ti) ] = Ec log y(t2) + b (log y(tl) - Ec log y(t,)), where b = Varc 1 (log Y(h))" Covc (log y(t2), log y(tl)),
S.N Durlauf and D.T. Quah
274
\ ," i~~; ~ ] ............;;~/......... ....
/...
~ ,,'" ...................... ~,7
constant a
Fig. 10c. (r convergent limit with ongoing intra-distribution churning. The figure shows a cross section of economies at the steady-state distribution limit, but displaying ongoing intra-distribution dynamics. This situation might be viewed as the distributional endpoint of the earlier Figure 10a. Y
}
-
Yj
constant o
t Fig. 10d. a-convergent limit without intra-distribution churning. Figure shows a cross section of economies at the steady-state distribution limit, but unlike in Figure 10c there are no ongoing intradistribution dynamics. All economies simply move in parallel.
the C subscript denotes appear on the left gives
cross-section. Rearranging
the projection so that growth rates
P [log y(t2) - log Y(h) I 1, log y ( q ) ] = [Ec log y(t2) bEc log y(tl)] - (1 - b) log
(30)
y(h).
The sign o f the coefficient on log Y(h) in this regression depends on whether b exceeds 1. The projection coefficient b, in turn, depends on how large the covariance
Ch. 4: The New Empirics of Economic Growth
275
between growth and initial income is relative to the variance of initial income. Suppose that we are in the situation described by Figure 10c, where long-run stationary steady state has been reached and log y(t) has its cross-sectional variances invariant in time. Since t2 > tl, equation (30) is a regression of growth rates on initial conditions. The Cauchy-Schwarz inequality ICovc(log y(t2), log y(ti))l ~< Var~/2(log y(t2))Var~/Z(log Y(h)) (with the inequality strict except in degenerate cases) then implies that (1 b) in Equation (30) is negative. In words, the conditional average - for that is what is represented by a cross-section regression - shows its growth rate negatively related to its initial level. That might, at first, suggest that we should see converging crosssection dynamics like those in Figure 10b, where the poor eventually attain the same income levels as the rich. However, recall that this negative relation between growth rates and initial levels has been constructed precisely when the cross-section dynamics are instead those in Figure 10c, where the gap between poorest and richest is always constant. More elaborate examples are easily constructed. For one, we need not consider situations only at long-run steady state. Since - outside of degenerate cases - the Cauchy-Schwarz inequality is strict, it is easy to find examples where -(1 - b) is negative even when Varc(log y(t2)) is bigger than Varc(log Y(h)), i.e., the cross-section dispersion is increasing even as the regression representation is suggesting dynamics like Figure 10b. Moreover, if one perturbs the regressor so that it is not log y(tl) but instead some other log y(to) then the same argument shows that the regression coefficient on the "initial" level can be positive regardless of whether the cross-section distribution is expanding, diminishing, or unchanged in dispersion. Different interpretations can be given to the effects we have just described - one early manifestation of these is known in the statistics literature as Galton's Fallacy or Galton's Paradox [see, e.g., Friedman (1992), Maddala (1988, 3.12), Stigler (1986, ch. 8), or Quah (1993b)]. 18 We prefer to regard the situation constructed above as one where knowledge of what happens to the conditional average (the regression representation) is uninformative for what happens to the entire cross section. In this interpretation, further fi-convergence regression analysis of the growth equation (23) be it with cross-section data, panel-data, or any other structure; be it conditional or unconditional - cannot reveal whether the poor will catch up with the rich. These considerations suggest instead directly analyzing the dynamics of the cross-section distribution. Doing so goes beyond studying just a-convergence, as the latter studies only one aspect of the distribution at each point in time. Moreover, a-convergence is silent on whether clusters form within the cross section (as in the emerging twin peaks
18 This connection had been impressed on Quah by G.S. Maddala and Marc Nerlove separately in private commtmications.
276
S.N. Durlaufand D.T. Quah
of Figure 1) and on whether transitions occur within the distribution: both Figure lOc and Figure lOd show the same a-convergence dynamics, yet economic behavior across them must differ dramatically. 5.2. Augmented cross-section regression
More recent empirical growth studies have tried to go beyond the original cross-section regressions and, instead, emphasize identifying those factors that explain international differences. Relative to the neoclassical growth model of Section 4, these exercises can be interpreted as parameterizing A. Table 2 surveys those regressors that, in the literature, have been used in crosscountry regressions 19. In addition to the four variables suggested by the augmented Solow-Swan model (initial income and the rates of human capital investment, physical capital investment, and population growth), the table includes 36 different categories of variables and 87 specific examples. Recall that the sample to which nearly all these additional control variables have been applied has only about 100 observations (the size of the subsample typically used from the Heston-Summers dataset). While these augmented cross-section regression studies have suggested some insightful extensions of the neoclassical growth model, we find problematic the lessons drawn from some of the empirical findings. First, many studies fail to make clear whether the regressions they consider can be interpreted within some economic model. It is certainly always possible to let A be a linear function of arbitrary control variables. But exploiting that hypothesized linear function need not be a useful way of studying the control in question. For example, the threshold externality in the Azariadis-Drazen model can be viewed as a latent variable indexing the aggregate production function. Such an interpretation is plausible for factors ranging from international market access to political regime - the ability of a society to innovate and to exploit readily available opportunities is influenced by political culture, with well documented historical examples going as far back as Athens and Sparta. However, we conclude from the model that these factors induce nonlinearities in the growth relation. Linear regressions are, not surprisingly, unable to get at the features of interest. Moreover, it is unclear what exercise a researcher conducts by adding a particular control variable, even when the variable is motivated by a particular economic theory. The basic Solow-Swan model admits an immense range of extensions through factors such as inequality, political regime, or trade openness. These are often highly correlated with one another, and are neither mutually exclusive nor prioritized as possible explanations of growth. Hence, it is difficult to assign much import to the statistical
19 Temple (1996) provides an excellent literataxreoverview discussing some of these studies in greater detail.
Ch. 4."
277
The New Empirics of Economic Growth
Table 2 Growth regression compilation a Explanatory variable
Reference
Finding b
Change in labor force participation rate Blomstrom, Lipsey and Zejan (1996) Corruption
+*
Mauro (1995)
Capitalism (level)
Sala-i-Martin (1997)
+*
Democracy, some
Barro (1996, 1997)
+*
more
Barro (1996, 1997)
-*
overall
Alesina, Ozler, Roubini and Swagel (1996)
?
Domestic credit, growth rate
Levine and Renelt (1992)
+f
volatility of growth rate
Levine and Renelt (1992)
+f
Barro and Lee (1994)
-
Barro and Lee (1994) Barro (1996) Barro (1997) Caselli, Esquivel and Lefort (1996) Forbes (1997)
-* -*
Education, college level female
+* -*
female growth
Barro and Lee (1994)
-*
male
Barro and Lee (1994) Caselli, Esquivel and Lefort (1996) Forbes (1997)
+* -* +*
male growth
Barro and Lee (1994)
+*
overall
Barro (1991) Knowles and Owen (1995) Levine and Renelt (1992) Mankiw, Romer and Weil (1992)
+* + +f +*
primary
Barro (1997)
-
Barro (1996) Barro and Lee (1994) Easterly (1993) Harrison (1995) Levine and Renelt (1992) Sala-i-Martin (1997)
-* -*
Easterly (1993) Harrison (1995) Sala-i-Martin (1997)
-
Barro (1996, 1997) Barro and Lee (1994) Caselli, Esquivel and Lefort (1996) Easterly, Kremer, Pritchett and Summers (1993)
+* + +* +*
External debt (dummy)
Easterly, Kremer, Pritchett and Summers (1993)
-
Fertility
Barro (1991, 1996, 1997) Barro and Lee (1994)
-* -*
Exchange rates (real), black market premium
distortions
terms of trade improvement
-* -f -*
-*
continued on next page
278
S.N. Durlauf and D.Z Quah
Table 2, continued Explanatory variable
Reference
Finding b
Financial repression
Easterly (1993)
-*
Financial sophistication
King and Levine (1993)
+*
Fraction college students, engineering
Murphy, Shleifer and Vishny (1991)
+*
Murphy, Shleifer and Vishny (1991)
-*
Barro (1991, 1996, 1997) Barro and Lee (1994) Caselli, Esquivel and Lefort (1996)
-* -* +*
growth in consumption
Kormendi and Meguire (1985)
+
deficits
Levine and Renelt (1992)
_t"
investment
Barro (1991)
+
Alesina, Ozler, Roubini and Swagel (1996)
+*
Alesina, Ozler, Roubini and Swagel (1996) Easterly, Kremer, Pritchett and Summers (1993)
+ +
Health (various proxies)
Barro (1997) Barro and Lee (1994) Caselli, Esquivel and Lefort (1996) Knowles and Owen (1995)
+* +* -* +*
Inequality, democracies
Persson and Tabellini (1994)
-*
non-democracies
Persson and Tabellini (1994)
+*
overall
Alesina and Rodrik (1994) Forbes (1997)
-* +*
law Government, consumption
Growth rates, G7 G7 lagged
Inflation, change
Kormendi and Meguire (1985)
level (above 15%)
Barro (1997)
level
Levine and Renelt (1992)
variability
Barro (1997) Levine and Renelt (1992)
+
Barro (1991, 1997) Barro and Lee (1994) Barro and Sala-i-Martin (1992) Ben-David (1996) Caselli, Esquivel and Lefort (1996) Cho (1996) Kormendi and Meguire (1985) Levine and Renelt (1992) Mankiw, Romer and Weil (1992) Romer (1993)
-* -* -* -* -* +* -* _r * -*
Barro (1997)
-*
Initial income
(interacted with male schooling)
-* f f
continued on next page
Ch. 4."
279
The N e w Empirics o f Economic Growth
Table 2, continued ExplanatOry variable
Reference
Finding b
Investment ratio
Barro ( 1991) Barro (1996, 1997) Barro and Lee (1994) Caselli, Esquivel and Lefort (1996) Levine and Renelt (1992) Mankiw, Romer and Weil (1992)
+* + +* +* +r +*
Investment, equipment, fixed capital
Blomstrom, Lipsey and Zejan (1996) DeLong and Summers (1993) Sala-i-Martin (1997)
+* +*
Sala-i-Martin (1997)
+*
Sala-i-Martin (1997)
+*
Mining (fraction of GDP)
Sala-i-Martin (1997)
+*
Money growth
Kormendi and Meguire (1985)
+
Politics, civil liberties
Barro and Lee (1994) Kormendi and Meguire (1985) Levine and Renelt (1992) Sala-i-Martin (1997)
* + ?f +*
instability
Alesina, Ozler, Roubini and Swagel (1996) Barro (1991) Barro and Lee (1994) Caselli, Esquivel and Lefort (1996) Levine and Renelt (1992) Sala-i-Martin (1997)
-* -* -* -* -f *
political rights
Barro and Lee (1994) Sala-i-Martin (1997)
+* +*
Barro and Lee (1994) Kormendi and Meguire (1985) Levine and Renelt (1992) Mankiw, Romer and Weil (1992)
+ -* -f *
< 15 years
Barro and Lee (1994)
-*
_> 65 years
Barro and Lee (1994)
?
Easterly (1993) Harrison (1995)
+ -*
Barro (1991) Easterly (1993)
* -*
non-equipment Latitude (absolute)
Population growth
Price distortion, consumption investment Price levels, consumption investment
Easterly (1993)
+
Easterly (1993)
-* continued on next page
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280
Table 2, continued Explanatory variable
Reference
Finding b
Regions, latitude (absolute)
Sala-i-Martin (1997)
+*
East Asia
Barro (1997) Barro and Lee (1994)
+ +
former Spanish colony
Sala-i-Martin (1997)
Latin America
Barro (1991) Barro (1997) Barro and Lee (1994) Sala-i-Martin (1997)
-*
Barro (1991) Barro (1997) Barro and Lee (1994) Sala-i-Martin (1997)
-*
Religion, Buddhist
Sala-i-Martin (1997)
+*
Catholic
Sala-i-Martin (1997)
-*
sub-Saharan Africa
*
-* -*
-* -*
Confucian
Sala-i-Martin (1997)
+*
Muslim
Sala-i-Martin (1997)
+*
Protestant
Sala-i-Martin (1997)
-*
Rule of law
Barro (1996, 1997) Sala-i-Martin (1997)
+* +*
Scale effects, total area
Sala-i-Martin (1997)
?
total labor force
Sala-i-Martin (1997)
?
Frankel and Romer (1996) Frankel, Romer and Cyrus (1996) Harrison (1995) Levine and Renelt (1992)
+* +*
primary products in total exports (fraction)
Sala-i-Martin (1997)
-*
export-GDP ratio (change)
Kormendi and Meguire (1985)
+*
FDI relative to GDP
Blomstrom, Lipsey and Zejan (1996)
-
Trade, export/import/total trade as fraction of GDP
machinery and equipment imports Romer (1993) Trade policy, import penetration
+f
+*
Levine and Renelt (1992)
?f
Learner index
Levine and Renelt (1992)
-f
openness (change)
Harrison (1995)
+*
openness (level)
Harrison (1995) Levine and Renelt (1992)
+* ?f continued on next page
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Table 2, continued Explanatory variable
Reference
Finding b
Trade policy (cont'd), outward orientation tariffs years open, 1950-1990 Variability, growth innovations
Levine and Renelt (1992)
?f
money War, casualties per capita duration occurrence
Barro and Lee (1994) Sala-i-Martin (1997) Kormendi and Meguire (1985) Ramey and Ramey (1995) Kormendi and Meguire (1985) Easterly, Kremer, Pritchett and Summers (1993) Barro and Lee (1994) Barro and Lee (1994) Sala-i-Martin (1997)
+* -* -* * + + -*
a In this table we can give no more than a flavor of the findings extant. Detailed variable definitions can be found in the individual references. b Symbols: * denotes a claim of significance (authors' significance levels differ across studies, and are not always explicitly reported); ? denotes that the author(s) did not report the result; and f and r indicate fragility and robustness in the sense used by Levine and Renelt (1992).
significance o f an arbitrarily chosen subset o f possible controls. We therefore find unpersuasive claims that these regressions are able to identify economic structure. The problem o f open-ended alternative models also extends to various attempts in the literature to find instruments for the various baseline and augmented S o l o w Swan regressors, which are o f course typically endogenous themselves. Frankel and Romer (1996) use geographic variables to instrument their measure o f trade openness. However, that these variables are exogenous with respect to trade openness does not make them legitimate instruments. For example, from the perspective o f European and Asian history it is wholly plausible that land mass correlates with military expenditures and military strength, which themselves correlate with tax rates and political regime two alternative augmentations o f the Solow model which have been proposed. Because growth explanations are so broad, it is especially easy to construct plausible reasons why "exogenous" instruments are less useful than they might first appear. The failure o f the growth model to naturally generate useful instruments contrasts with rational expectations models whose structure produces such instruments automatically from the orthogonality o f forecast errors and available information. This reasoning has led to a reexamination o f the empirical conclusions from this line o f work. The issue has been addressed in two ways. First, Levine and Renelt (1992) have challenged many o f the findings in cross-country growth regressions. They emphasized that findings o f statistical significance may be fragile due to dependence on additional controls whose presence or absence is not strongly motivated by any theory.
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By applying Leamer's [Leamer (1978)] extreme bounds analysis (thereby identifying the range of coefficient estimates for a given regressor generated by alternative choices of additional regressors) they found that only the physical capital investment rate and, to a weaker degree, initial income are robustly related to cross-country growth rate differentials. Levine and Renelt (1992) have identified a serious problem with the empirical growth literature. However, their procedure for dealing with the problem is itself problematic. The difficulty may be most easily seen in the following example. Suppose that one is interested in the coefficient b0 relating variables X and Y, where the true data generating process is given by
~
= X j b 0 + ~j,
with X deterministic and c normally distributed N(0, a2). Suppose the researcher considers a set of controls {Zl : integer l}, each ZI being separately entered in the regression: Yj = X j b + Ztjct + ej.
(31)
Assume that the Zl's are nonstochastic and that, in sample, have zero cross-product with X. Denote the sample second moments of X and ZI by IlXll2 and 1121112 respectively. Then OLS on Equation (31) produces ,5 estimates that are draws from the normal distribution N(b0, (112112+ IlZlll2) 10"2). Since the Zl are deterministic, as researchers increase the number of separate ZI used in different regression analyses, so does the probability increase that some draw on b will have sign opposite to that on the true b. 2o The problem is that the b distribution has support that can become unbounded due to sampling variation induced by the arbitrarily chosen regressors. Without a theory on how to control this problem, it is difficult to draw strong conclusions about the fragility of regression coefficients. Hence, while we find the Levine and Renelt analysis suggestive, the import of the challenge is unclear. Said-i-Martin (1997) has attempted to deal with this limitation by calling "robust" only those variables found statistically significant in 95% of a group of regressions in a wide range of possible combinations of controls. This work finds that many more variables appear to be robust. These variables fall into 9 categories: (1) region (dummy variables for Sub-Saharan Africa and Latin America), (2) political structure (measures of rule of law, civil liberties and political instability), (3) religion, (4) market distortions (measured with reference to official and black market exchange rates), (5) equipment investment, (6) natural resource production, (7) trade openness, (8) degree of capitalism, and (9) former Spanish colonies.
20 The basic argumentclearlystill applieswhenthe Zt are stochastic, eventhoughthen the b distributions are not typicallynormal.
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However, it is again unclear how to interpret such results. Suppose that one were to take a given regression relationship and begin to include alternative sets o f right hand side variables which were in each case orthogonal to the original regressors. The presence or absence o f these regressors would have (by assumption) no effect on estimated coefficient size or estimated standard errors. Hence, one could always generate an arbitrarily large number o f regressions with the same significant coefficient but with no implications as to whether the coefficient estimate is or is not robust. Hence, it is impossible to know whether Sala-i-Martin's exercise actually reveals something about robustness, or merely something about the covariance structure of the controls which he studies. Further, the exercise assumes that robustness is interesting outside of the context of which variables are under study. The fact that the presence o f one variable in a growth regression renders another insignificant is not vitiated by the fact that others do not do so, when the first is o f economic interest, and the others are not. The problem with both these approaches to robustness o f control variables in growth regressions is that they attempt to use mechanical statistical criteria in identifying factors whose interest and plausibility is motivated by economic (or social science) theory. The dimensions along which one wants estimates to be robust are determined by the goals o f the researcher, which cannot be reduced to algorithms o f the kind that have been employed. 5.3. Panel-data analysis
To permit unobservable country-specific heterogeneity in growth regressions, Benhabib and Spiegel (1997), Canova and Marcet (1995), Casetli, Esquivel and Lefort (1996), Evans (1998), Islam (1995), Lee, Pesaran and Smith (1997), and Nerlove (1996) have used panel-data methods to study the cross-country income data. Following traditional motivation in panel-data econometrics [e.g., Chamberlain (1984)], many such studies seek to eliminate, in the notation o f Section 4, unobservable country-level heterogeneity in A(0). Those heterogeneities, denoted individual effects in the language o f panel-data econometrics, constitute nuisance parameters that within the conventional framework the researcher attempts to remove. 21 Panel-data studies proceed from the neoclassical (MRW) model (23) as follows. Assume that depreciation 6 and technology growth ~ are constant across economies.
21 Canova and Marcet (1995) and Evans (1998) are exceptions to this. Canova and Marcet analyze a Bayesian-motivatedparameterization of the individual effects, and conclude that those effects do, indeed, differ across economies. Evans, using a different statistical technique, concludes the same. Evans follows Levin and Lin (1992) and Quah (1994) in taking an underlying probability model where both time and cross-section dimensions in the panel dataset are large. This contrasts with standard panel-data studies where the time dimension is taken to be relatively small. The large N, large T framework then allows inference as if the individual effects are consistently estimated, and permits testing for whether they differ across countries. See also Ira, Pesaran and Shin (1997).
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284
Fix horizon T, append a residual on the right, and redefine coefficients to give, across economies j, the regression equation log yj(t + T) - log yi(t) = bo + b~ log yj(t) + b2 log Tpj + b3 log Th,i+ b4 log(b + Y/+ ~) + (-j,t (32) with def
b0 = (1 - e;W)log A(0) + (t + T - e;Wt)~, def" e~.T bl = -
1,
b2 def = (1 _ eZr)_l
ap - a p - ah
b3 def (1 -- e zr) 1
ah --
def
b4 = - ( 1
C/p
, --
ah
eZZ) % + ah -
1 - %-ah"
Let T = 1 and assume that b0 is a random variable with unobservable additive components varying i n j and t: log yj(t + 1) - log yj(t) = I~j + tft + bl log yj(t) + b2 log "gpj -t- b3 log rhj + b4 log(b + vj + ~) + (~j,t. (33) This formulation differs from the original MRW specification in two ways. First, the law of motion for output is taken in one-period adjustments. This is inessential, however, and the researcher is free to recast Equation (32) with T set to whatever the researcher deems appropriate. Second, the (originally) constant b0 is decomposed into economy-specific and time-specific effects: b0 =/~j + tot.
(34)
Panel-data methods, applied to the model above, have produced a wide range of empirical results. While Barro and Sala-i-Martin (1991, 1992) defend a 2% annual rate of convergence from cross-section regressions, estimates from panel-data analyses have been more varied. Lee, Pesaran and Smith (1997, 1998) conclude annual convergence rates are approximately 30% when one allows heterogeneity in all the parameters. Islam (1995) permits heterogeneity only in the intercept terms, and finds annual convergence rates between 3.8% and 9.1%, depending on the subsample under study. Caselli, Esquivel and Lefort (1996) suggest a convergence rate of 10%, after conditioning out individual heterogeneities and instrumenting for dynamic endogeneity. Nerlove (1996), by contrast, finds estimates of convergence rates that are even lower than those generated by cross-section regression. He explains this difference as being due to finite
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sample biases in the estimators employed in the other studies using the neoclassical growth model. The disparate results across panel-data studies can sometimes, but not always, be attributed to the different datasets that different researchers have employed. The use of a panel-data structure has advantages and disadvantages. One significant advance comes from clarifying the difficulties in interpreting the standard cross-section regression. In particular, the dynamic panel (33) typically displays correlation between lagged dependent variables and the unobserved residual. The resulting regression bias depends on the number of observations in time and only disappears when that number becomes infinite. Moreover, the bias does not disappear with time averaging. Thus, if the dynamic panel were the underlying structure, standard cross-section regressions will not consistently uncover the true structural parameters. But beyond simply pointing out difficulties with the cross-section OLS formulation, the panel-data structure has been argued, on its own merits, to be more appropriate for analyzing growth dynamics. For instance, Islam (1995) shows how time- and country-specific effects can arise when per capita output is the dependent variable instead of output per effective worker (Islam argues this substitution to be appropriate). Alternatively, one might view the error structure as a consequence of omitted variables in the growth equation, whereupon the separate time and country effects in Equation (34) have alternative natural interpretations. These instances of the greater flexibility (and, thus, reduced possibilities for misspecification) allowed by panel-data analyses - unavailable to cross-section regression studies - account for their broader econometric use more generally, not just in studies of economic growth. However, the putatively greater appeal of panel-data studies should not go unchallenged. To see the potential disadvantages, consider again the decomposition in Equation (34). For researchers used to the conventions in panel-data econometric analysis, this generalization from a constant unique b0 is natural. But for others, it might appear to be a proliferation of free parameters not directly motivated by economic theory. Freeing b0 so that it can vary across countries and over time can only help a theoretical model fit the data better. Restricting b0 to be identical across countries and over time - when, in reality, b0 should differ - can result in a model that is misspecified, thereby lowering confidence that the researcher has correctly identified and estimated the parameters of interest. This advantage of a panel-data approach applies generally, and is not specific to growth and convergence. But for convergence studies, the flexibility from decomposing b0 into economy-specific and time-specific components can instead be problematic, giving rise to misleading conclusions. We describe two scenarios where we think this might naturally occur. First, note that Equation (32) implies that A(0) (and thus b0 through/~j) forms part of the longrun path towards which the given economy converges (see again Figures 10a-10d). Ignore Galton's Fallacy to sharpen the point here. If the researcher insists that A(0) be identical across economies, then that researcher concludes convergence to an underlying steady-state path precisely when catching up between poor and rich takes place. Thus, the implication from a convergence finding is transparent: it translates
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directly into a statement about catching up (again, abstracting away from Galton's Fallacy). By contrast, when the researcher allows A(0) to differ across economies, finding convergence to an underlying steady-state path says nothing about whether catching up occurs between poor and rich: Figures 10a-10d show different possibilities. This is not just the distinction between conditional and unconditional convergence. In panel-data analysis, it is considered a virtue that the individual heterogeneities A(0) are unobservable, and not explicitly modelled as functions of observable right-hand side explanatory variables. By leaving free those individual heterogeneities, the researcher gives up hope of examining whether poor economies are catching up with rich ones. The use of panel-data methods therefore compounds the difficulties in interpreting convergence regression findings in terms o f catchup from poor to rich. For the second scenario, recall the problem the panel-data regression Equation (33) traditionally confronts is the possibility that the #j's, the individual-specific effects, are correlated with some of the right-hand side variables. If not for this, OLS on Equation (33) would allow both consistent estimation and (with appropriately corrected standard errors) consistent inference. 22 One class of solutions to the inconsistency problem derives from transforming Equation (33) to annihilate the /~j. For instance, in the so-called "fixed-effects" or within estimator, one takes deviations from timeaveraged sample means in Equation (33), and then applies OLS to the transformed equation to provide consistent estimates for the regression coefficients. But note that in applying such an individual-effects annihilating transformation, the researcher winds up analyzing a left-hand side variable purged of its long-run (timeaveraged) variation across countries. Such a method, therefore, leaves unexplained exactly the long-run cross-conntry growth variation originally motivating this empirical research. The resulting estimates are, instead, pertinent only for higher-frequency variation in the left-hand side variable: this might be o f greater interest for business cycles research than it is for understanding patterns o f long-run economic growth across countries. 23 Our point is general: it applies not just to the fixed-effects estimator, but also to the first-difference estimator, and indeed to any panel-data technique that conditions out the individual effects as "nuisance parameters". In dealing with the correlation between individual effects and right-hand side variables - a properly-justified problem in microeconometric studies [again see, e.g., Chamberlain (1984)] - the solution
22 OLS might not be efficient, of course, and GLS might be preferred where one takes into account the covariance structure of the #j's. 23 This statement clearly differs from saying that fixed-effects estimators are inconsistent in dynamic models without strict exogeneity of the regressors [e.g., Chamberlain (1984)]. The absence of strict exogeneity characterizes Equation (33), and thus is an additional problem with fixed-effects estimators. This shortcoming has motivated studies such as Caselli, Esquivel and Lefort (1996) that use techniques appropriate for such correlation possibilities. However, those techniques do nothing for the shortrun/long-run issue we raise.
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offered by panel-data tecbafiques ends up profoundly limiting our ability to explain patterns of cross-country growth and convergence. 24 Interestingly, that conditioning out country-specific effects leaves only highfrequency income movements to be explained creates not only the problem just described, but also its dual. Over what time horizon is a growth model supposed to apply? Many economists (or Solow and Swan themselves in the original papers for that matter) regard growth analyses as relevant over long time spans. Averaging over the longest time horizon possible - as in cross-section regression work - comes with the belief that such averaging eliminates business cycle effects that likely dominate per capita income fluctuations at higher frequencies. By contrast, Islam (1995, p. 1137) has argued that since Equation (23) is "based on an approximation around the steady state ... it is, therefore, valid over shorter periods of time". However, we think this irrelevant. Different time scales for analyzing the model are mutually appropriate only if the degree of misspecification in the model is independent of time scale. In growth work, one can plausibly argue that misspecification is greater at higher frequencies. Taking Islam's argument seriously, one might attempt using the neoclassical growth model to explain even weekly or daily income fluctuations in addition to decadal movements. 5.4. Time series: unit roots and cointegration
An alternative approach to long-run output dynamics and convergence based on timeseries ideas has been developed in Bernard and Durlauf (1995, 1996), Durlauf (1989), and Quah (1992). Convergence here is identified not as a property of the relation between initial income and growth over a fixed sample period, but instead of the relationship between long-run forecasts of per capita output, taking as given initial conditions. Bernard and Durlauf (1996) define time-series forecast convergence as the equality of long-term forecasts taken at a given fixed date. Thus, given ~t the information at date t, economies j a n d f show time-series forecast convergence at t when: lim E(yj(t + T ) - yj,(t + T) ] ~t) = O,
T~oo
i.e., the long-term forecasts of per capita output are equal given information available at t. It is easy to show that time-series forecast convergence implies/3-convergence when growth rates are measured between t and t + T for some fixed finite horizon T. The critical distinction between time-series forecast convergence and/3-convergence is that an expected reduction in contemporary differences (/3-convergence) is not the same as the expectation of their eventual disappearance.
24 Quah (1996c, p. 1367) has also argued this.
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This dynamic definition has the added feature that it distinguishes between convergence between pairs of economies and convergence for all economies simultaneously. Of course, if convergence holds between all pairs then convergence holds for all. Some of the theoretical models we have described - in particular, those with multiple steady states - show that convergence need not be an all or nothing proposition. Subgroups of economies might converge, even when not all economies do. To operationalize this notion of convergence, a researcher examines whether the difference between per capita incomes in selected pairs of economies can be characterized as a zero-mean stationary stochastic process. Hence, forecast convergence can be tested using standard unit root and cointegration procedures. Under the definition, deterministic (nonzero) time trends in the cross-pair differences is as much a rejection of convergence as is the presence of a unit root. In the literature applying these ideas, two main strands can be distinguished. The first, typified by Bernard and Durlauf (1995, 1996), restricts analysis to particular subgroups of economies, for instance the OECD. This allows the researcher to use long time series data, such as those constructed by Maddison (1989). Multivariate unit root and cointegration tests reject the null hypothesis that there is a single unit-root process driving output across the OECD economies - thus, across all the economies in the OECD grouping, time-series forecast convergence can be rejected. At the same time, however, individual country pairs - for instance, Belgium and the Netherlands do display such convergence. In a second strand, Quah (1992) studies the presence of common stochastic trends in a large cross section of aggregate economies. He does this by subtracting US per capita output from the per capita output of every economy under study, and then examines if unit roots remain in the resulting series. Because the number of time-series observations is the same order of magnitude as the number of countries, random-field asymptotics are used to compute significance levels. Quah's results confirm those of Bernard and Durlauf described above. He rejects the null hypothesis of no unit roots in the per capita output difference series; in other words, he finds evidence against convergence (in the sense given by the forecasting definition). Time series approaches to convergence are subject to an important caveat. The statistical analysis under which convergence is tested maintains that the data under consideration can be described by a time-invariant data generating process. However, if economies are in transition towards steady state, their associated per capita output series will not satisfy this property. Indeed, as argued by Bernard and Durlauf (1996), the time series approach to convergence, by requiring that output differences be zeromean and stationary, requires a condition inconsistent with that implied in crosssection regressions, namely that the difference between a rich and poor economy have a nonzero mean. Time-series and cross-section approaches to convergence rely on different interpretations of the data under consideration. Hence they can provide conflicting evidence; in practice, the two approaches commonly do.
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5.5. Clustering and classification
Following Azariadis and Drazen's (1990) theoretical insights, Durlauf and Johnson (1995) study Equation (27), and find evidence for multiple regimes in cross-country growth dynamics. They do this in the dataset originally used by MRW [Mankiw, Romer and Weil (1992)] by identifying sample splits so that within any given subsample all economies obey a common linear cross-section regression equation. Durlauf and Johnson allow economies with different 1960 per capita incomes and literacy rates (LR) to be endowed with different aggregate production functions. Using a regression-tree procedure 25 to identify threshold levels endogenously, Durlauf and Johnson find the MRW dataset display four distinct regimes determined by initial conditions: (1) yj(1960) < $800; (2) $800 ~~ 1 :
O,+s = (M')'Ot.
(36)
Since M is a transition probability matrix, its largest eigenvalue is 1, and the left eigenvector corresponding to that eigenvalue can be chosen to have all entries
292
s.)~ Durlauf and D.T. Quak
nonnegative summing to 1. Generically, that largest eigenvalue is unique, so that M s converges to a rank-one transition probability matrix. But then all its rows must be equal, and moreover equal to that probability vector satisfying:
The vector q~o~ is the E r g o d i c row vector; it corresponds to the limit o f relation (36) as s --~ oc. In words, q ~ is the long-rtm limit o f the distribution o f incomes across economies. 26 Table 3 shows that limiting distribution to be twin-peaked. Although in the observed sample, economies are almost uniformly distributed across cells - if anything, there is a peak in the middle-income classes - as time evolves, the distribution is predicted to thin out in the middle and cluster at rich and poor extremes. This polarization behavior is simply a formalization o f the tendencies suggested in Figure 1. Such analysis leads to further questions. How robust are these findings? The discretization to construct the transition probability matrix is crude and ad hoc. Moving from a continuous income state space - Figure 1 - to a discrete one comprising cells - Table 3 - aliases much o f the fine details on the dynamics. Does changing the discretization alter the conclusions? To address these issues, we get rid o f the discretization. In Appendix A we describe the mathematical reasoning needed to do this. The end result is a s t o c h a s t i c k e r n e l the appropriate generalization o f a transition probability matrix - which can be used in place o f matrix M in the analysis. Quah (1996b, 1997) estimates such kernels. Figures 1 la and 1 lb show the kernel for the transition dynamics across 105 countries over 1961 through 1988, where the transition horizon has been taken to be 15 years. The twin-peaked nature o f the distribution dynamics is apparent now, without the aliasing effects due to discretization. Bianchi (1997) and Jones (1997) eschew dealing with the stochastic kernel by considering the cross-section distribution Ft for each t in isolation. This ignores information on transition dynamics, but is still useful for getting information on the shape dynamics in F. Each Ft is estimated nonparametrically. Bianchi (1997) goes further and applies to each Ft a bootstrap test for multimodality (twin-peakedness, after all, is just bimodality). Bianchi finds that in the early part of the sample (the early 1960s) the data show unimodality. However, by the end o f the sample (the late 1980s) the data reject unimodality in favor o f bimodality. Since Bianchi imposes less structure in his analysis - nowhere does he consider intradistribution dynamics, or in the language of Appendix A, the structure o f T* - one guesses that his findings
26 Potential inconsistency across M matrices estimated over single- and multiple-period transitions is a well-known problem from the labor and sociology literature [e.g., Singer and Spilerman (1976)]. Quah (1993a) shows that, in the Heston-Summers cross-country application, the long-run properties of interest are, approximately, invariant to the transition period used in estimation.
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15.year.Horizon
.% %.q
0.•
~.~
o.~
?
%
0.2 0r0
Fig. 1la. Relative income dynamics across 105 countries, 1961 1988. For clarity, this stochastic kernel is one taken over a fifteen-year transition horizon. The kernel can be viewed as a continuum version of a transition probability matrix. Thus, high values along the diagonal indicate a tendency to remain. A line projected from a fixed value on the Period t axis traces out a probability density over the kernel, describing relative likelihoods of transiting to particular income values in Period t + 15. The emerging twin-peaks feature is evident here, now without the aliasing possibilities in discrete transition probability matrices. are more robust to possible misspecification. Here again, however, twin-peakedness manifests. We have taken care, in building up the theoretical discussion from the previous sections, to emphasize that those models give, among other things, ways to interpret these distribution dynamics. A n observed pattern in the distribution dynamics o f crosscountry growth and convergence can be viewed as a reduced form - and one can ask i f it matches the theoretical predictions o f particular classes o f models. We view in exactly this way the connection between the empirics just discussed and the distribution dynamics o f models such as Lucas's (1993) described in Section 4 above. The work just described, while formalizing certain facts about the patterns o f crosscountry growth, does not yet provide an explanation for those patterns. Putting this differently, we need to ask what it is that explains these reduced forms in distribution dynamics. In light o f our discussion above on the restrictions implied by cross-
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15.year.Horizorl
2 o q)
Period
2 ~+15
3
Contour plot levels at 0.2, 0.35, 0.5 Fig. 1 lb. Relative income dynamics across 105 countries, 1961-1988, contour plot. This figure is just the view from above of Figure 1 la, where contours have been drawn at the indicated levels and then projected onto the base of the graph.
country interactions, we conjecture that this "explaining distribution dynamics" needs to go beyond representative-economy analysis. Quah (1997) has addressed exactly this issue: in the spirit of our discussion above on theoretical models with crosscountry interaction, Quah asks for the patterns of those interactions that can explain these reduced-form stochastic kernels. He finds that the twin-peaks dynamics can be explained by spatial spillovers and patterns of cross-country trade - who trades with whom, not just how open or closed an economy is.
6. Conclusion
We have provided an overview of recent empirical work on patterns of cross-country growth. We think the profession has learned a great deal about how to match those empirical patterns to theoretical models. But as researchers have learnt more, the
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criteria for a successful confluence of theory and empirical reality have also continued to sharpen. In Section 2 we described some of the new stylized facts on growth - they differ from Kaldor's original set. It is this difference, together with the shift in priorities, that accounts for wishing to go beyond the original neoclassical growth model. Neither the newer empirical nor theoretical research has focused on preserving the stability of the "great ratios" or of particular factor prices. Instead, attention has shifted to a more basic set of questions: why do some countries grow faster than others? What makes some countries prosper while others languish? Sections 3 and 4 described a number of well-known theoretical growth models and presented their empirical implications. Although a considerable fraction of the empirical work extant has studied growth and convergence equations - whether in cross-section or panel data - we have tried to highlight first, that those equations might be problematic and second, that in any case they need not be the most striking and useful implications of the theory. Distribution-dynamics models make this particularly clear. Appropriate empirical analysis for all the different possibilities we have outlined above is an area that remains under study. Section 5 described a spectrum of empirical methods and findings related to studying patterns of cross-country growth. The range is extensive and, in our view, continues to grow as researchers understand more about both the facts surrounding growth across countries and the novel difficulties in carrying out empirical analyses in this research area.
At the same time, we feel that the new empirical growth literature remains in its infancy. While the literature has shown that the Solow model has substantial statistical power in explaining cross-country growth variation, sufficiently many problems exist with this work that the causal significance of the model is still far from clear. Further, the new stylized facts of growth, as embodied in nonlinearities and distributional dynamics have yet to be integrated into full structural econometric analysis. While we find the new empirics of economic growth to be exciting, we also see that much remains to be done. Appendix A. Proofs and additional discussions This appendix collects together proofs and additional discussion omitted from the main presentation. It is intended to make this chapter self-contained, but without straying from the empirical focus in the principal sections. A. 1. Single capital good, exogenous technical progress
The classical Cass-Koopmans [Cass (1965), Koopmans (1965)] analysis produces dynamics (9b) from the optimization program (10). To see this, notice that given assumptions (la) and (7a-c), K ( t ) = Y(t) - c ( t ) N ( t ) - 6 K ( t )
296
S.N. Durlauf and D.T. Quah
can be rewritten as k = y - c - (6 + v)k,
The original problem (10) can then be analyzed as max
U(c(t))e-(p-~)t dt
subject to k = F ( k , A ) - c - ( 6 + v)k.
{c(t),k(t)),~o
The first-order conditions for this are: ~U" = (p + 6 - OF(k, A)/Ok) U', k = F ( k , A ) - c - ( 6 + v)k,
lim k ( t ) e (p v ) t = O.
t~oo
Rewrite these in growth rates and then in technology-normalized form; use the parameterized preferences U from program (10); and recall that F homogeneous degree 1 means its first partials are all homogeneous degree 0. This yields the dynamics (9b). Turn now to convergence. In order to understand Figure 2 note that if we define g(~) derf(~)/c ~, then on k > 0 function g is continuous and strictly decreasing: Vg(k) = Vf(k)k -1 - f ( k ) k -2 =
[~:Vf(k)-/(k)]
k -2 < 0
by concavity and lim1~ 0 f ( k ) /> 0 from Equation (2). Moreover, limk~ 0 g(k) --, oc (directly if limk~ 0 f ( k ) > 0; by l'Hospital's Rule and Equation (3) otherwise) and limk~ ~ g(k) = 0 from Equation (8). These endpoints straddle (6 + 7 + ~)r l, and therefore the intersection k* exists, and k satisfying T-'~/lc = g(lc) - (6 + v + ~)v -1 is dynamically stable everywhere on/} > 0. To see that (k*, ~*), the zero of Equation (16), is well-defined, let lc* solve
vf(/) = p + 6 + 0~ (k* > 0) and notice that then
~. de__f[f_(k*) k W - - ( 6 + v + ~ ) j l k* >o since
f(k*) - k*
~> V f ( / ~ * ) = p + 0 + 0 ~
from the assumption p > v + ~.
> 6+v+~
297
Ch. 4: The New Empirics of Economic Growth
To see how Equation (18) follows from Equation (17), notice that since M's eigenvalues are distinct and different from zero, we can write its eigenvalue-eigenvector decomposition: M =
1,
with VM full rank and having columns equal to M's right eigenvectors, and
Then the unique stable solution of Equation (17) is
log ~(t)- log ~*
\ log ~(0)- log ~* eX2t'
with log fc(0) - log fc* ) log ~(0) log ~* having 0 as its first entry.
VM~ ×
(This proportionality property can always be satisfied since ~(0) is free to be determined while k(0) is given as an initial condition.) This timepath constitutes a solution to the differential Equation (17) for it implies d dt
Vml ~
log ~(t)-log 0*
=)~2× \ l o g ~ ( t ) _ l o g ~ .
log ~(t) - log ~*
=
~2 × VM1 ~ log 0(t) - log 0*
= ~ \ log ~(t) -- log ~*
=M
)~2 VMI \ log 0(t) -- log ~* log ~(t) - log ~*
This solution is clearly stable. Since any other solution contains an exponential in ,~i, this solution is also the unique stable one. A.2. Endogenous growth: asymptotically linear technology
We need to verify that Equations (24) and (25) imply the existence of a balancedgrowth equilibrium with positive limt~o~ ~/7c and limt~o~ ~(t)/lc(t). Along the
S.N. Durlaufand D.T. Quah
298
optimal path ~ is a function of ~:. Consider conditions (9b) as k ~ oo (since we are interested in equilibria with ~(t)/k(t) bounded from below by a positive quantity). Then lim =~ = l i m f ( k ) k l _ ( 6 + v + ~ ) _ ~ l i m ~,~k
f(k)k-l-[p+6+O~] using limk~ ~ ~Tf(k) = l i m ~ J ' ( k ) k
)
0 1,
I. For these to be equal,
lim ~= = lim f(k)lc -1 - ( 6 + v ) - [ l f l ~ J ( k ) k ' k--+~ k ~ - ~ > limf(k) k 1_(6+v)_(p v) k--+~ -
lim f(k)
- (p+6)]O'
~: I _ ( p + 6 ) > 0.
The long-run growth rate is
(k~na f (k) k -I - [p + 6 + O~]) 0 -1, which is positive from Equation (25) 0 < lim~__+~f(k)k l _ ( p + 6) o0
p-v
~p-
v - ~ > [tim f(k)lc -1 -(p+6+O~)l 0-'.
If 0 is too large [exceeding the upper bound in Equation (25)] then this model collapses to the traditional neoclassical model where balanced-growth equilibrium has finite (fi*, k*), and neither preference nor technology parameters (apart from ~) influences the long-run growth rate.
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The New Empirics of Economic Growth
299
A.3. Distribution dynamics
Rigorous expositions o f the mathematics underlying a formulation like Equation (35) can be found in Chung (1960), Doob (1953), Futia (1982), and Stokey and Lucas (1989) (with Prescott) 27. Since we are concerned here with real-valued incomes, the underlying state space is the pair (IR, 91), i.e., the real line R together with the collection 91 of its Borel sets. Let B(1R, 91) denote the Banach space of bounded finitely-additive set functions on the measurable space (R, 91) endowed with total variation norm:
in B(R, 91) "
Iqol =sup ./
where the supremum in this definition is taken over all { A / : J - 1, 2 . . . . , n} finite measurable partitions o f R. Empirical distributions on R can be identified with probability measures on (R, 9l); those are, in turn, just countably-additive elements in B(R, 91) assigning value 1 to the entire space IR. Let ~3 denote the Borel a-algebra generated by the open subsets (relative to total variation norm topology) of B(IR, 91). Then (B, ~3) is another measurable space. Note that B includes more than just probability measures: an arbitrary element q) in B could be negative; qo(lR) need not be 1; and q0 need not be countably-additive. On the other hand, a collection o f probability measures is never a linear space: that collection does not include a zero element; if ¢1 and ¢2 are probability measures, then ¢1 - ¢2 and ¢1 + ~2 are not; neither is x¢1 a probability measure for x E R except at x = 1. By contrast, the set o f bounded finitely-additive set functions certainly is a linear space, and as described above, is easily given a norm and then made Banach. Why embed probability measures in a Banach space as we have done here? A first reason is so that distances can be defined between probability measures; it then makes sense to talk about two measures - and their associated distributions - getting closer to one another. A small step from there is to define open sets o f probability measures, and thereby induce (Borel) a-algebras on probability measures. Such a-algebras then allow modelling random elements drawn from collections of probability measures, and thus from collections o f distributions. The data o f interest when modelling the dynamics o f distributions are precisely random elements taking values that are probability measures.
27 Economic applications of these tools have also appeared in stochastic growth models [e.g., the examples in Stokey and Lucas (1989, ch. 16] (with Prescott), income distribution dynamics [e.g., Loury (1981)], and elsewhere. Using these ideas for studying distribution dynamics rather than analyzing a time-series stochastic process, say, exploits a duality in the mathematics. This is made explicit in Quah (1996a), a study dealing not with cross-country growth but business cycles instead.
S.N. Durlaufand D.Z Quah
300
In this scheme then, each Ot associated with the observed cross-sectional income distribution Kt is a measure in (B, ~3). If (g2, 5, Pr) is the underlying probability space, then q~t is the value of an ~/~3-measurable map ~ : (£2, 5) ~ (B, ~3). The sequence {qst " t ~> 0} is then a B-valued stochastic process. To understand the structure of operators like Z*U t it helps to use the following: Definition" Stochastic Kernel Definition. Let cp and ~p be elements of B that are probability measures on (R, 91). A stoehastie kernel relating q) and ~p is a mapping M(~0,v) : (1R, 91) --+ [0, 1] satisfying: (i) Vy in N, the restriction M(~,e)(y, .) is a probability measure; (ii) VA in 91, the restriction M(cp,~l,)(',A) is 91-measurable; (iii) VA in 91, we have cp(A) = fM(~,v)(y, A) d~p(y). To see why this is useful, first consider (iii). At an initial point in time, for given y, there is some fraction d~p(y) of economies with incomes close to y. Count up all economies in that group who turn out to have their incomes subsequently fall in a given 91-measurable subset A C_ IR. When normalized to be a fraction of the total number of economies, this count is precisely M(y, A) (where the (q), ~p) subscript can now be deleted without loss of clarity). Fix A, weight the count M(y, A) by d~p(y), and sum over all possible y, i.e., evaluate the integral f M(y, A) d~p(y). This gives the fraction of economies that end up in state A regardless o f their initial income levels. If this equals q)(A) for all measurable subsets A, then q) must be the measure associated with the subsequent income distribution. In other words, the stochastic kernel M is a complete description of transitions from state y to any other portion of the underlying state space R. Conditions (i) and (ii) simply guarantee that the interpretation of (iii) is valid. By (ii), the right hand side of (iii) is well-defined as a Lebesgue integral. By (i), the right hand side of (iii) is a weighted average of probability measures M(y, .), and thus is itself a probability measure. How does this relate to the structure of T~ ? Let b(R, 91) be the Banach space under sup norm of bounded measurable functions on (1R, 91). Fix a stochastic kernel M and define the operator T mapping b(lR, 91) to itself by V f in b(R, 91), Vy in R •
(Tf)(y) =
f f(x)M(y,dx).
Since M(y, .) is a probability measure, the image Tf can be interpreted as a forward conditional expectation. For example, if all economies in the cross section begin with incomes y, and we take f to be the identity map, then (i~f)(y) = f x M ( y , dx) is next period's average income in the cross section, conditional on all economies having income y in the current period. Clearly, T is a bounded linear operator. Denote the adjoint of T by T*. By Riesz Representation Theorem, the dual space of b(IR, 91) is just B(IR, 9l) (our original collection of bounded finitely additive set functions on 91); thus T* is a bounded linear
Ch. 4: The New Empirics of Economic Growth
301
operator mapping B(R, 9l) to itself. It turns out that T* is also exactly the mapping in (iii) of the Stochastic Kernel Definition, i.e., V V probability measures in B, VA in ~ "
(T*
V)(A) f M(y, A) dV(y).
(This is immediate from writing the left-hand side as
(T*V)(A)= f IA d(T* V)(Y)= f (TIA)(y)dV(y)
/[flA(x)M(y,dx)]
dV(y)
(adjoint) (definition of T)
. f M(y, A)dV(y),
(calculation)
with 1A the indicator function for A.)
Appendix B. Data
The data used in Section 2 are from version V6 of Summers and Heston (1991). Income is taken to be real GDP per capita in constant dollars using Chain Index (at 1985 international prices) (series RGDPCH). Economies not having data in 1960 and 1989 were excluded. The remaining sample comprised 122 economies (integers immediately before country names are the indexes in the Summers-Heston database): 1
(1) Algeria
2
(2) Angola
3
(3) Benin
4
(4) Botswana
5
(5) Burkina Faso
6
(6) Burundi
7
(7) Cameroon
8
(8) Cape Verde Islands
9
(9) Central African Republic
10
(10) Chad
11
(11) Comoros
12
(12) Congo
13
(14) Egypt
14
(16) Gabon
15
(17) Gambia
16
(18) Ghana
17
(19) Guinea
18
(20) Guinea Bissau
19
(21) Ivory Coast
20
(22) Kenya
21
(23) Lesotho
22
(25) Madagascar
23
(26) Malawi
24
(27) Mali
25
(28) Mauritania
26
(29) Mauritius
S.N. Durlauf and D.T. Quah
302 27
(30) Morocco
28
(31) Mozambique
29
(32) Namibia
30
(33) Niger
31
(34) Nigeria
32
(35) Reunion
33
(36) Rwanda
34
35
36
37
(38) Seychelles (40) Somalia
(37) Senegal (39) Sierra Leone
38
(41) South Africa
39
(43) Swaziland
4O
(44) Tanzania
41
(45) Togo
42
(46) Tunisia
43
(47) Uganda
44
(48) Zaire
45
(49) Zambia
46
(50) Zimbabwe
47
(52) Barbados
48
49
5O
51
(55) Costa Rica (58) E1 Salvador
(54) Canada (57) Dominican Republic
52
(6o) Guatemala
53
(61) Haiti
54
(62) Honduras
55
56
(64) Mexico
57
(63) Jamaica (65) Nicaragua
58
59
(67) Puerto Rico
6O
(66) Panama (71) Trinidad and Tobago
61
(72) USA
62
(73) Argentina
63
(74) Bolivia
64
65
66
67
(76) Chile (78) Ecuador
(75) Brazil (77) Colombia
68
(79) Guyana
69
(80) Paraguay
7O
(81) Peru
71
(82) Suriname (84) Venezuela
72
(83) Uruguay
74
(86) Bangladesh
(88) China (90) India
76
(89) Hong Kong
78
(91) Indonesia
(92) Iran (95) Japan
8O
(94) Israel
82
(96) Jordan
73 75 77 79 81
(97) Korean Republic 85 (102) Myanmar 87 (106) Philippines
84 (100) Malaysia
89 (109) Singapore 91 ( l l l ) Syria
9O (110) Sri Lanka 92 (112) Taiwan
83
86 (105) Pakistan 88 (108) Saudi Arabia
Ch. 4:
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303
93 (113) Thailand 94 (116) Austria 95 (117) Belgium 96 (119) Cyprus 97 (120) Czechoslovakia 98 (121) Denmark 99 (122) Finland 100 (123) France 101 (125) Germany, West 102 (126) Greece 103 (128) Iceland 104 (129) Ireland 105 (130) Italy 106 (131) Luxembourg 107 (132) Malta 108 (133) Netherlands 109 (134) Norway 110 (136) Portugal 111 (137) Romania 112 (138) Spain 113 (139) Sweden 114 (140) Switzerland 115 (141) Turkey 116 (142) UK 117 (143) USSR 118 (144) Yugoslavia 119 (145) Australia 120 (146) Fiji 121 (147) New Zealand 122 (148) Papua New Guinea The clustering-classification results described in Section 5 derive from the following subsample split [taken from Durlauf and Johnson (1995), Table IV]: (1) yj(1960) < $800: Burkina Faso, Burundi, Ethiopia, Malawi, Mali, Mauritania, Niger, Rwanda, Sierra Leone, Tanzania, Togo, Uganda; (2) $800 ~< yj(1960) ~< $4850 and LRj(1960) < 46%: Algeria, Angola, Benin, Cameroon, Central African Republic, Chad, Congo (People's Republic), Egypt, Ghana, Ivory Coast, Kenya, Liberia, Morocco, Mozambique, Nigeria, Senegal, Somalia, Sudan, Tunisia, Zambia, Zimbabwe, Bangladesh, India, Jordan, Nepal, Pakistan, Syria, Turkey, Guatemala, Haiti, Honduras, Bolivia, Indonesia, Papua New Guinea; (3) $800 ~< yj(1960) ~< $4850 and 46% ~< LRj(1960): Madagascar, South Africa, Hong Kong, Israel, Japan, Korea, Malaysia, Philippines, Singapore, Sri Lanka, Thailand, Greece, Ireland, Portugal, Spain, Costa Rica, Dominican Republic, E1 Salvador, Jamaica, Mexico, Nicaragua, Panama, Brazil, Colombia, Ecuador, Paraguay, Peru; (4) $4850 < yj(1960): Austria, Belgium, Denmark, Finland, France, Germany (Federal Republic), Italy, Netherlands, Norway, Sweden, Switzerland, UK, Canada, Trinidad and Tobago, USA, Argentina, Chile, Uruguay, Venezuela, Australia, New Zealand. References
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Islam, N. (1995), "Growth empirics: a panel data approach", Quarterly Journal of Economics 110(443): 1127-1170. Jones, C.I. ( 1995a), "R&D-based models of economic growth", Journal of Political Economy 103 (3):759784. Jones, C.I. (1995b), "Time series tests of endogenous growth models", Quarterly Journal of Economics 110:495-525. Jones, C.I. (1997), "On the evolution of the world income distribution", Journal of Economic Perspectives 11 (3): 19-36, Summer. Jones, L.E., and R.E. Manuelli (1990), "A convex model of equilibrium growth: theory and implications", Journal of Political Economy, 98(5, part 1): 1008-1038. Kaldor, N. (1963), "Capital accumulation and economic growth", in: EA. Lutz and D.C. Hague, eds., Proceedings of a Conference Held by the International Economics Association (Macmillan, London). Kelly, M. (1992), "On endogenous growth with productivity shocks", Journal of Monetary Economics 30(1 ):47-56. King, R.G., and R. Levine (1993), "Finance and growth: Schumpeter might be right", Quarterly Jounlal of Economics 108(3):717-737. Knowles, S., and RD. Owen (1995), "Health capital and cross-country variation in income per capita in the Mankiw-Romer-Weil model", Economics Letters 48(1):99-106. Kocherlakota, N.R., and K.-M. Yi (1995), "Can convergence regressions distinguish between exogenous and endogenous growth models?", Economics Letters 49:211~15. Koopmans, T.C. (1965), "On the concept of optimal economic growth", in: The Econometric Approach to Development Planning (North-Holland, Amsterdam). Kormendi, R.C., and R Meguire (1985), "Macroeconomic determinants of growth: cross-country evidence", Journal of Monetary Economics 16(2): 141-163. Lamo, A.R. (1996), "Cross-section distribution dynamics", Ph.D. Thesis (London School of Economics, February). Learner, E.E. (1978), Specification Searches: Ad Hoc Inference from Non-Experimental Data (Wiley, New York). Lee, K., M.H. Pesaran and R.P. Smith (1997), "Growth and convergence in a multi-country empirical stochastic Solow model", Journal of Applied Econometrics 12(4):357-392. Lee, K., M.H. Pesaran and R.R Smith (1998), "Growth empirics: a panel data approach - a comment", Quarterly Journal of Economics 113(452):319-323. Leung, C., and D. Quah (1996), "Convergence, endogenous growth, and productivity disturbances", Journal of Monetary Economics 38(3):535-547. Levin, A., and C. Lin (1992), "Unit root tests in panel data: asymptotic and finite-sample properties", Working paper, Economics Department, UCSD, San Diego. Levine, R., and D. Renelt (1992), "A sensitivity analysis of cross-country growth regressions", American Economic Review 82(4):942-963. Loury, G.C. (1981), "lntergenerational transfers and the distribution of earnings", Econometrica 49(4): 843 867. Lucas Jr, R.E. (1988), "On the mechanics of economic development", Journal of Monetary Economics 22(3):3-42. Lucas Jr, R.E. (1993), "Making a miracle", Econometrica 61(2):251-271. Maddala, G.S. (1988), Introduction to Econometrics (Macmillan, New York). Maddison, A. (1989), The World Economy in the 20th Century (Development Centre of the OECD, Paris). Mankiw, N.G., D. Romer and D.N. Weil (1992), "A contribution to the empirics of economic growth", Quarterly Journal of Economics 107(2):407-437. Mauro, E (1995), "Corruption and growth", Quarterly Journal of Economics 110(3):681-713. Murphy, K.M., A. Shleifer and R.W. Vishny (1989), "industrialization and the big push", Journal of Political Economy 97(4):1003-1026.
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Chapter 5
N U M E R I C A L S O L U T I O N OF D Y N A M I C E C O N O M I C M O D E L S * MANUEL S. SANTOS Department of Economics, University of Minnesota
Contents Abstract Keywords 1. Introduction 2. T h e m o d e l and p r e l i m i n a r y considerations 3. B e l l m a n ' s e q u a t i o n and differentiability o f the value function 3.1. Bellman's equation and the contraction property of the dynamic programming algorithm 3.2. Differentiability of the value function
4. A numerical dynamic programming algorithm 4.1. 4.2. 4.3. 4.4. 4.5.
Formulation of the numerical algorithm Existence of numerical solutions and derivation of error bounds Stability of the numerical algorithm Numerical maximization Numerical integration
5. Extensions of the basic algorithm 5.1. 5.2. 5.3. 5.4.
Multigrid methods Policy iteration Modified policy iteration Polynomial interpolation and spline functions 5.4.1. Polynomial interpolation 5.4.2. Spline functions
6. Numerical approximations of the Euler equation 6.1. Numerical methods for approximating the Euler equation 6.2. Accuracy based upon the Euler equation residuals
7. Some numerical experiments 7.1. A one-sector deterministic growth model with leisure 7.2. A one-sector chaotic growth model 7.3. A one-sector stochastic growth model with leisure
312 312 313 314 319 320 321
324 324 326 328 329 332
334 334 336 338 340 340 344
345 347 352
355 355 362 364
* The author is grateful to Jerry Bona, Antonio Ladron de Guevara, Ken Judd, John Rust, John Taylor and Jesus Vigo for helpful discussions on this topic. Special thanks are due to Adrian Peralta-Alva for his devoted computational assistance. Handbook of Macroeconomies, Volume 1, Edited by JB. Taylor and M. WoodJbrd © 1999 Elsevier Science B. V All rights reserved 311
312 8. Quadratic approximations 9. Testing economic theories 10. A practical approach to computation References
M.S. Santos
368 375 379 382
Abstract This chapter is concerned with numerical simulation of dynamic economic models. We focus on some basic algorithms and assess their accuracy and stability properties. This analysis is useful for an optimal implementation and testing of these procedures, as well as to evaluate their performance. Several examples are provided in order to illustrate the functioning and efficiency of these algorithms.
Keywords dynamic economic model, value function, policy function, Euler equation, numerical algorithm, numerical solution, approximation error JEL classification: C61, C63, C68
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1. Introduction
This chapter offers an overview of some important methods for simulating solutions of dynamic economic models with the aid of high-performance computers. The recent surge of research in this area has been impelled by current developments in computer processing, algorithm design, software and data storage. This progress has fostered the numerical analysis of a wide range of problems to limits beyond what one could possibly foresee a few years ago. Since advances in our computational capabilities are likely to continue, it is expected that numerical simulation of economic models will be an attractive and expanding research field. A basic concern in science is understanding model predictions. Classical mathematical methods can help us derive basic qualitative properties of solutions such as existence, uniqueness and differentiability. But these methods usually fail to afford us with the specific information necessary to test a model. There are some well-known economic examples in which optimal decisions have an analytical representation or closed-form solution (e.g., models with linear decision rules, or with constant elasticities for consumption and saving). In these cases, optimal policies are generally derived from algebraic manipulations or analytical techniques, and computational methods are usually not needed. Such a state of affairs, however, is not the most common situation. Most dynamic economic models feature essential nonlinearities stemming from intra and intertemporal substitutions over non-constant margins. (These nonlinearities become more pronounced when uncertainty is present in the decision problem.) Digital computers are then the most plausible way to understand the behavior of a given model with a view toward its eventual testing. And one should expect that computational techniques will help to bridge the traditional gap between theoretical developments and empirical economic analysis. Over the past decades, economic thinking has achieved levels of rigor and argumentation comparable to any other scientific discipline. The principles of axiomatization and mathematical logic are well rooted in economic theory. Also, empirical work has endorsed the underlying postulates of statistical analysis. If our main objective is to collect the fruits of this scientific endeavor, the same accepted practices should prevail for solving economic models. A framework for carrying out and reporting numerical experiments is presented in Bona and Santos (1997). Our purpose here is to focus on the accuracy and stability properties of some algorithms currently used by economists, and evaluate their performance in the context of some growth models. Accuracy seems to be a minimal requirement for judging a numerical simulation. And once we have a theory of the error involved in a numerical approximation, we are in a better position to devise more efficient algorithms, and to test and debug the computer code. Stability is concerned with possible variations that numerical errors and misspecifications of parameter values may inflict on the computed solution. Unstable algorithms may lead to odd outcomes, and may considerably lessen the power of a numerical simulation in testing a particular theory. Our study of accuracy and stability properties will be
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complemented with some numerical experiments where we discuss further aspects of the implementation and performance of these algorithms. Computational tools have been applied to a wide variety of problems in economics and finance. Rather than providing a thorough review of these applications, the present work focuses on the analysis o f some fundamental algorithms as applied to some simple growth models. Once the functioning of these algorithms is understood in this basic context, these same techniques should be of potential interest for solving other model economies, even though the assumptions of strong concavity and differentiability are fundamental to some of our results. There are several survey papers on this topic, which to a certain extent may be complementary to the present one. Kehoe (1991) reviews the literature on static general equilibrium models, along with certain numerical methods for dynamic economies. Our paper is in the spirit of Taylor and Uhlig (1990), who describe several computational methods and evaluate their performance. In our case, we shall concentrate on fewer methods, place more emphasis on their accuracy and stability properties, and carry out alternative numerical tests. Marcet (1994) reexamines the literature on the so called parameterized expectations algorithm and presents a sample of its applications. This method computes the optimal law of motion from a direct approximation of the Euler equation. A variety of other methods that approximate the Euler equation are laid out in Judd (1992, 1996), who has advocated for the use of polynomial approximations with certain desired orthogonality properties. Variants of both Marcet's and Judd's procedures along with some other related algorithms are reevaluated in Christiano and Fisher (1994). Finally, Rust (1996) considers several numerical methods for solving dynamic programming programs, and analyzes their complexity properties. Complexity theory presents an integrated framework for assessing the efficiency of algorithms, although some of these asymptotic results may not be binding in simple applications.
2. The model and preliminary considerations
We begin our analysis with a stochastic, reduced-form model of economic growth in which the solution to the optimal planning problem may be interpreted as the equilibrium law of motion of a decentralized economy. Our framework is encompassed in the class of economies set out in Stokey and Lucas (1989). The reader is referred to this monograph for some basic definitions and technical points raised in the course of our discussion. The usefulness of this relatively abstract setting for carrying out numerical computations will be illustrated below with some simple examples. Let (K, K) and (Z, Z ) be measurable spaces, and let (S,,9) = (K x Z, K; x Z) be the product space. The set K contains all possible values for the endogenous state variable, Z is the set of possible values for the exogenous shock, and S is the set of state values for the system. The evolution of the random component {zt}t~o is governed by a stochastic law defined by a function q) : Z × Z ~ Z
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and an i.i.d, process {st}¢>~l where zt = qo(zt-l,et). It follows that the mapping q0 induces a time-invariant transition function Q on (z, Z). Moreover, for each z0 in Z one can define a probability measure /d (z0, ") on every t-fold product space (Z t, Z t) = (Z × Z x ... × Z, Z × Z x ... x Z ) comprising all partial histories of the form z t = (Zl, . . . , zt). The physical constraints of the economy are summarized by a given jeasible technology set, f2 C K x K × Z, which is the graph o f a continuous correspondence, F : K × Z ---, K. The intertemporal objective is characterized by a one-period return function u on g2 and a discount factor, 0 0~
P
[(3r, Zt+l). DZvz,(k, kt+l) • (zt, Jrt+l)] pt(z0,dz t)
subject to Zt+l = q~(z~,et+l),
(3.6)
t = 0, 1,2 . . . . .
Here the maximization proceeds over all measurable functions {~t}~0, ~t : Z t-1 ~ R/, for t ~> 1 with ~c0 = x0 and z0 fixed, the one-period objective O2vz,(kt, kt+l) is the Hessian matrix of the mapping v(., ",Zt) for given zt, and {kt,zt}~_ 0 is the optimal contingency plan to problem (2.1) for the initial value (ko,zo). From this characterization, one readily proves that the optimal plan {~}~-0 to maximization problem (3.6) determines the derivative of the policy function with respect to/co. That is, ~t = Dlgt(ko,zo) • ;to for t ~> 1, where Dlgt(ko,zo) denotes the derivative of the function g ( g ( . . , g(ko, zo). • .), zt 2), zt-1) with respect to k0 for every possible realization (Zl,Z2. . . . . zt 1). Given that (x0, 0, 0, 0 .... ) is a feasible solution to maximization problem (3.6), we obtain that
IID~lW(ko,zo)ll 0.) Such conditions are nevertheless pathological (i.e., not robust to small perturbations o f the model), although further sources ofnon-differentiability may arise in multivariate models with complex behavior, or in stochastic frameworks. In conclusion, there exists a reasonably general theory for first- and second-order derivatives o f the value function, and such derivatives may be bounded from primitive elements o f the model. Relatively little is known about existence and characterization o f higher-order derivatives 4.
4. A numerical dynamic programming algorithm Our purpose now is to formulate a numerical algorithm for computing functions W and g, and to study its accuracy properties. In the actual implementation o f the algorithm, there are additional errors stemming from the numerical maximization and integration o f these functions, and these errors may unfold over the iterative process. We shall present here a stability analysis o f these approximations, along with a discussion o f available methods for numerical maximization and integration. In subsequent sections we shall be concerned with possible extensions o f this basic algorithm to speed up computations, and with its numerical implementation.
4.1. Formulation of the numerical algorithm Our numerical algorithm is based on a discretized version of the method of successive approximations. The basic idea underlying this analysis is to restrict the set o f functions ~/V to a finite-dimensional domain so that the algorithm can be coded in a finite number o f computer instructions. Our functions will be defined on the whole domain K × Z via piecewise affine interpolation o f their values over a finite set o f regular points. In the following section, we shall consider alternative interpolation schemes. In the sequel we assume that the state space S = K x Z is a polyhedron. This does not entail much loss of generality for most economic applications. Let {SJ } be a finite
4 A thoughtful method to compute high-order derivatives of the value and policy functions is laid out in Judd (1996) and Gaspar and Judd (1997). This method is only operative for steady-state solutions, assuming the existence of such derivatives.
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family of simplices 5 such that UjS j = S and int(S i) Nint(SJ) = 0 for every pair S', SJ. Define the grid level or mesh size as h = max diam (k S J ~,/ . J Let (kJ,z j) be a generic vertex of the triangulation. Consider then the space o f piecewise affine functions
t/Vh = { V h " S ---+R
V h is bounded, continuous and DV h is constant in int (S j) for each S j
f
It follows that every function V h in I'Vh is determined by the nodal values Vh(k j, Z J), for all vertex points (k J, z J). These nodal values uniquely define a piecewise affine function over the whole domain S, compatible with the given triangulation {S J}. Also, 142h is a closed subspace o f I/V, equipped with the norm =
max
IIV ll (k,z cK×z
[Vh(k,z)[
for
V h E W h.
For a given triangulation {S J} with mesh size h, we now consider the following algorithm for value function iteration: (i) Initial step: Select an accuracy level TOLW and an initial guess W0h. (ii) Value function evaluation: Let /,
" ~J ,z0) J = max v(kj, W£+l(k kl
+/3 /
Jz
j dz1) m nh(kl ,z1) Q ( z O,
(4.1)
subject to (kj, kl,z j) E f2 for each vertex point (kJ,zJ).
Ilw2+,- w211
(iii) End of iteration: If TOL , stop; else, increment n by 1, and return to step (ii). For present purposes, it should be understood that the maximization and integration operations in algorithm (4.1) are performed exactly. As a matter of fact, in the actual implementation o f the algorithm, one solves for the integral on the right-hand side o f Equation (4.1), and then maximizes over kl. These functional evaluations define a mapping T h : I/V ---+ I/Vh such that Wn/+l = Th(W2). The mapping T h is a discretized version o f the dynamic programming algorithm.
5 A simplex S j in Rt is the set of all convex combinations of 1+ 1 given points. [cf. Rockafellar (1970)]. Thus, a simplex in R l is an interval, a simplex in R2 is a triangle, and a simplex in R3 is a tetrahedron.
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Accuracy level TOLW is selected so as to provide a good approximation W2+l to the fixed point W h o f the following functional equation
j j = max W h(k6,z0) kl
(kd, k,,zd)+/3[Wh(kl,zl)e(zX,
dzl)
,JZ
(4.2)
subject to (k~,kl,z j) c f2 for each vertex point (k~,zo).J j This is the corresponding discretized version o f Bellman's Equation (3.1), which is required to hold at the finite set o f vertex points. Using the above iterative scheme, we now establish existence o f the fixed point W h and provide estimates for the approximation error involved in this discretization.
4.2. Existence of numerical solutions and derivation of error bounds A basic topic in numerical analysis is existence and accuracy properties of numerical solutions. Relatively little is known, however, about the existence and accuracy properties o f most numerical algorithms used by economists. Li (1993) and Marcet and Marshall (1994) are notable exceptions to this trend. Li provides a derivation of error botmds for a simple monetary model. Marcet and Marshall prove certain asymptotic properties of a parameterized expectations algorithm. Although Marcet and Marshall's results are comforting, these results cannot be generally invoked in numerical computations since they simply assert that the approximation scheme converges to the exact solution as the computational cost goes to infinity. We now review some accuracy properties o f our numerical model 6. L e m m a 4.1. Under Assumptions (1)-(4), Equation (4.2) has a unique solution W h
in I/Vh. Proof: The proof is the standard one. One immediately sees that T h is a contraction mapping with modulus 0 < / 3 < 1. By a well known fixed-point theorem, equation (4.2) has a unique fixed point W h in 142h. [] It is worth noticing that the contraction property o f the operator T h implies that the sequence {W~}n~>0 generated by Equation (4.1) converges linearly to the fixed point W h. L e m m a 4.2. Let W be the value fimction in Equation (3.1). Let 7 = under Assumptions (1)-(4) it must hold that IIzw- ZhWII.< ~h 2
IIo2wll Then,
6 The present analysis is taken from Santos and Vigo (1998). One can find related discretization procedures and derivation of error bounds for the computation of the value function [e.g. Bertsekas (1975), Chow and Tsitsiklis (1991), Falcone (1987), Fox (1973), Kitanidis and Foufoula-Georgiou(1987), and Whitt (1978, 1979)]. None of these latter papers, however, derives a quadratic order of convergence for the computed value function in an infinite-horizon optimization setting (as h goes to zero).
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Observe that at each nodal value Th(W)(kJ,z j) = W(kJ,zJ); moreover, ThW is piecewise affine, and function W is C 2. Then, the result follows from a standard application of Taylor's theorem [see Santos and Vigo (1998) for further details]. Theorem 4.3. Let W be the fixed point of Equation (3.1) and W h be the fixed point of Equation (4.2). Then, under Assumptions (1)-(4) it must hold that
IIv/- v/hll ~ ~
h2,
Proof: Let T and T h be as defined previously from Equations (3.2) and (4.1), respectively. We must then have,
[Iv/- whl[ = IITV/- T~V/~II ~< IITV/- T~V/II + [[Thv/- Thv/~[[ ~< I[Tv/- r~v/]l +/3 Ilv/- v/~[[, where use is made in these computations of the triangle inequality and of Lemma 4.1. Therefore,
IIW- whll ~ @
I[Tw- Thv/II
Theorem 4.3 is now a direct consequence of Lemma 4.2. []
v/ll
Inequalities (3.7)-(3.10) provide upper estimates for parameter y = lID2 These estimates can be useful to bound the observed error in specific applications. It should be noted that constant 7/2(1 -/3) becomes unbounded for/3 = 1. This singularity seems to be related to the fact that if the value of the approximation error in a single period may be up to (7/2)h 2 (Lemma 4.2), then the cumulative error over the entire infinite horizon may extend up to [7/2(1 -/3)] h 2. An immediate consequence of Theorem 4.2 is the following useful result: Corollary 4.4. Let g(k j, z j) be the optimal policy Jbr the original t)aIue function W at a vertex point (kJ, zJ), and let gh(kJ,zJ) be the optimal policy for the approximate value function W h at vertex point (k j, zJ). Then, /
Ilg(kJ,zJ)-g,h(kJ,zJ)[[ ~ ',
27
\ I/2
/[n(i-/3)l h forall(kJ, zJ).
It follows then from this analysis that for the computed value function the order of convergence is quadratic on h, whereas for the computed policy function the order of convergence is linear. As it is to be expected, a key assumption for the linear convergence of the optimal policy is a positive lower estimate t / o n the concavity of the return function as postulated in Assumption 2.
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4.3. Stability of the numerical algorithm Our results indicate that the numerical model has some desired accuracy properties. In particular applications, however, these computations are subject to round-off errors and inaccuracies due to numerical maximizations and integrations. These numerical errors may get propagated in unexpected ways over the iteration process. A numerical algorithm is said to be unstable if small computational errors produce large variations in the solution. Unstable algorithms should be avoided. We now show that our algorithm has some desired stability properties. For a given triangulation {S -i} with mesh size h, define a functional operator T~'" }d2 ---+}42h such that [Th(V)(kJ,z j) - Th(V)(kJ,zJ)[ 0. The interpretation is that under the operator T~ the computational error is not greater than e for all functions in the space W. If such distance is preserved for all nodal values V(kJ, z J), it follows that the constructed piecewise linear interpolations of {Th~(V)(k/,z/)} and {Th(V)(kJ,zJ)} are also within an e-distance over the whole domain S. In accordance with this interpretation, we postulate the following regularity conditions on functional operator T): (i) Monotonicity: T[zV ' ~ T~h V for V' /> V; (ii) Discounting: T~(V + a) 0 is the possible computational error in each iteration, then e/(1 - f i ) should be an upper bound for the cumulative error over the entire infinite horizon. Indeed, this estimate can be obtained from the recursion [(cumulative error)t [ ~< [(current error)t [ + fi [(cumulative error)t_l[. The bounds established in Theorem 4.5 can be useful for an efficient design of the computer code. Thus, it would not be optimal to operate with a very fine grid
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of points in cases where the approximation errors from maximization and integration are relatively large. An efficient implementation of the algorithm requires that these errors should be balanced. That is, h2/e ~ 2/y. This benchmark value may be adjusted depending upon the computational cost of reducing each of these errors, and on further operational features of the algorithm discussed in subsequent sections. Of course, the computational error e stems from the maximization and integration approximations, and these individual errors should also be balanced. Routines for maximization and integration usually provide good estimates for these approximations. Moreover, it should be realized that if the maximization is carried out over a discrete set of grid points {k j} with mesh size h, then the additional error involved in this approximation is of order h 2, since the first-order derivative at a maximizer is equal to zero. On the other hand, if the integration is performed over a discretized space, as an approximation for an underlying continuous-valued random variable, then the additional error will depend on the integration scheme and the differentiability properties of the value fimction. [Observe that in general variable z has a more pronounced effect on the derivatives of the value function, cf. Equations (3.4) and (3.10).] Thus, one should make reasonable choices concerning discretizations of state spaces K and Z so that the involved approximation errors are of the same magnitude. It seems that commonly found computations which restrict the uncertainty space Z to very few states as compared to the space of capitals K [e.g., Christiano (1990) and Danthine, Donaldson and Mehra (1989)] may obtain more accurate approximations for the same computational cost by considering more balanced grids over the whole space K x Z. 4.4. Numerical maximization Methods for numerical optimization are covered in Gill, Murray and Wright (1981), Kahaner, Moler and Nash (1989), and Press et al. (1992). In addition, there are several professionally designed subroutines for the solution of various specialized problems, as well as the two all-purpose libraries NAG and IMSL. Here, we shall offer a brief introduction to these methods along with a discussion of some specific issues concerning the analysis of error and implementation of these numerical procedures. As in classical mathematical analysis, a distinction is made in numerical maximization between smooth and non-smooth problems, global and local optimization, constrained and unconstrained solutions, and maximization in one and several dimensions. These properties not only dictate the nature of the techniques employed to tackle the optimization, but they also bear on practical computational considerations. Thus, a method for numerical maximization of non-smooth functions is generally inefficient for smooth problems. Likewise, a numerical method for maximization in several variables will not generally be suitable for one-dimensional problems. Algorithms for numerical maximization of smooth functions usually search over the whole domain of feasible solutions, as opposed to restricting the search to a grid of prespecified points. The software usually indicates the tolerance level or interval of
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uncertainty. Since at an interior maximum point the derivative of the function is equal to zero, it should be understood that if the computed maximum is at an e-distance, then the approximation error in the functional value is of order e 2. If the search for the maximum is restricted to a grid of prespecified points with mesh size h [e.g., Christiano (1990)], then the additional error incurred in this approximation would be of order h 2. Although the search for a maximum over a grid of points is a very simple strategy associated with a relatively small approximation error, methods based on functional evaluations are not generally computationally efficient for smooth problems. There are faster, more powerful algorithms that take advantage of the information provided by the derivatives of the functions. In our case, our mappings are piecewise linear, and hence the gradient is defined at almost every point. Moreover, the curvature of these mappings can be bounded in a certain sense, since the first-order derivatives are determined by the envelope theorem [cf. equality (3.3)], and an upper estimate of the rate of change of these derivatives is the maximum value of the secondorder derivative of the return function [cf. inequality (3.7) above, or Montrucchio (1987) for more general arguments]. Hence, our functions possess some smoothness properties, and have bounded curvature. Then, numerical maximization methods based on simple functional evaluations would generally be inefficient. Of course, smoothness can be obtained under higher-order interpolations; and if these interpolations preserve concavity, the optimization problem may be more tractable. Another important consideration is the dimensionality of the optimization problem. Numerical methods for maximization in several dimensions are generally based on one-line maximizations. The choice of these directions is key for defining the search method. This choice, however, is trivial for unidimensional problems. For univariate maximization, a typical initial step is to bracket the maximum. That is, in some very simple way one selects a triplet of points a < b < c such that f ( b ) is greater than both f ( a ) and f(c). (This choice guarantees the existence of a maximum inside the chosen interval; moreover, if the objective is concave such solution is the desired global maximum.) Once the maximum has been bracketed, then the searching process should exploit regularity properties of the function, using either smooth approximations or functional evaluations. There are also hybrid procedures that combine both types of information. This is the strategy followed by Brent's method [cf. Press et al. (1992), Sect. 10.2], and it seems suitable to our case where the functions have kinks at the vertex points, but at the same time preserve certain smoothness properties. The method proceeds along the following steps: (a) Smooth approximation. The routine selects three given function values, and constructs a parabolic approximation. Then it quickly determines the maximum of the parabola. If this maximum point falls within certain limits (i.e., the maximum is cooperative), then this value is added in the next iteration for a subsequent parabolic approximation, until a desired level of accuracy is achieved. Convergence to the true maximum is of order 1.324.
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(b) Golden-section search. I f the parabolic approximation is not a reasonable one, then the routine switches to a more reliable but slower method called goldensection search. This procedure is analogous to the familiar method of bisection for finding the zeroes of a univariate function. Given at each stage a bracketing triplet of points, golden-section search tries a point that is a fraction 0.38197 into the largest of the two intervals from the central point of the triplet. With the four points now available, the procednre then selects a new bracketing triplet. Following this iterative process, the interval of search is eventually reduced at each stage by 1 - 0.38197 = 0.61803, which corresponds to the rate of convergence of this method. Brent's method falls into the class of so-called safeguarded procedures, which combine fast algorithms with slower, more reliable ones. Press et al. (1992, Sect. 10.3) discuss another method of this nature which seems appropriate for univariate concave optimization. The method proceeds as follows. Given a bracketing triplet of points a < b < c, one determines the direction of the derivative at the intermediate point, b. This information then defines the next interval of search, which would be either [a, b] or [b, c]. The value of the derivatives of the two chosen points can then be used to produce another intermediate point by some root finding procedure such as the secant method. If this method yields values beyond certain limits, then one bisects the interval under consideration. Of course, in order to implement this latter safeguarded procedure, concavity and smoothness properties of the univariate problem are essential. In the unidimensional case, concavity is always preserved by piecewise linear interpolations. As for differentiability, one could compute for instance one-side derivatives, or else resort to higher-order interpolations preserving first-order differentiability. Regarding multivariate optimization, there are also a host of algorithms, the usefulness of which will depend on the dimensionality, smoothness and concavity properties of the optimization problem. In recent years, there has been a considerable amount of attention devoted to numerical procedures on non-smooth optimization for both concave and non-concave objectives [e.g., see Bazaraa et al. (1993, Ch. 8), HiriartUrruti and Lemarechal (1993) and Shor (1985)]. A simple algorithm in this class of non-differentiable problems is the downhill simplex method (also called the polytope algorithm). For smooth optimization, two popular procedures are quasi-Newton and conjugate-gradient methods. These latter two methods can also be applied to nonsmooth problems, making use of firfite-difference approximations for the first-order derivatives. In an n-dimensional space, the downhill simplex method considers n + 1 functional evaluations, say at points xo,xL,... ,x,,. These points can be visualized as the vertices of a simplex or polytope. Then, in the next step a new simplex is constructed by producing a new vertex that will replace the point with the worst functional evaluation. Depending on the new value, the polytope may further expand or contract in that direction. A new iteration then starts by replacing the worst point, and this iterative process goes on until a desired solution is attained. Under this procedure, the search for a maximum is not
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usually guaranteed, but this seems to be a convenient way to find a maximum in cases where one cannot approximate the derivatives o f the function. Quasi-Newton methods derive estimates o f the curvature of the function, without explicitly computing the second-order derivatives. Thus, each iteration starts at a point xk with a matrix Bk which reflects second-order information, and which is supposed to be an approximation o f the true Hessian if the function is sufficiently smooth. (At the initial stage one usually starts with B0 equal to the identity matrix, and in such case the algorithm reduces to the steepest descent method.) Then, the search direction, Pk, is the solution to Bk •Pk = -gk, where gk is the gradient vector. Subsequently, maximization is carried out in this direction, that is, on the line xk + apk. A choice o f a number ~ defines a new point xk+l = xk + aPk, and completes the iteration. The Hessian estimate Bk+l is then updated following some standard methods [cf. Press et al. (1992), p. 420]. The whole process stops when the gradient gk is sufficiently small. Conjugate-gradient methods construct a sequence o f searching directions which satisfy certain orthogonality and conjugacy conditions so as to improve at each step the search for a maximum. Conjugate-gradient methods do not require estimates or knowledge o f the Hessian matrix. Hence, their applicability extends more naturally to large-scale problems.
4.5. Numerical integration The integral operation in Equation (4.1) can be effected by standard numerical procedures. In general, numerical integration can be easily carried out in one dimension, and it becomes costly in several dimensions. Professionally designed software usually provides an estimate o f the approximation error or tolerance, which in most cases can be adjusted. An n-point quadrature formula, ~ 7 _ I w i f ( x i ) , is an estimate o f a given integral,
I = f~'f(x) dx, where the wi and xi are called weights and nodes. These values depend on a, b and n, but not on f . The difference Rn = f~a'f(x)dx - ~7:~ wif(xi) is called the remainder or error. There are several well known quadrature rules, which under certain regularity conditions yield bounds for the approximation error. The mid-point rule takes w = b - a, and x = ½(b + a). The trapezoidal rule has weights wl = w2 = ½(b - a), and nodes Xl = a, x2 = b. Each o f these rules can be compounded. For example, let us divide the interval [a, b] into N equally-sized panels, and let SN be the integral value by applying the trapezoidal rule to each o f these panels. Then, computations for S:v can be useful to calculate S2N. The three-point compounded Simpson's rule can be defined a s S u : ~4 S 2 x -- ~1S N , and for sufficiently smooth functions the approximation
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error under this latter rule is "fourth order" [cf. Kahaner et al. (1989) and Press et al. (1992)]. Compounding allows to attain higher accuracy using previous functional evaluations. Compounding is also very convenient to track numerically the approximation error. Theoretical error bounds are usually too conservative. Further, a quadrature rule is not useful unless there is some way to estimate the remainder R~ [Kahaner et al. (1989), p. 150]. Another basic family of integration rules is Gaussian quadrature. Here, the weights and the nodes are freely selected so that certain integrands can be more effectively approximated. An n-point Gaussian quadrature rule can integrate exactly every polynomial up to degree 2n - 1, integration of polynomials of higher degree would generally entail an approximation error. Hence, Gaussian quadrature is very efficient for the integration of smooth, polynomial-like functions, but may not have a good performance for other types of integrands. More generally, Gaussian quadratures are constructed so as to integrate exactly polynomials times some weighting function, p(x); that is, weights w i and nodes x i are chosen to satisfy f]°p(x)p(x)dx ~ 7 = i wip(xi), where p(x) is a polynomial. For the particular choice p(x) = l/x/1 - x 2, the rule is termed Gauss-Chebyshev integration, and for p(x) = 1 the rule is termed Gauss-
Legendre integration. Compounding is not possible for Gaussian rules, since the nodes of an n-point rule are distinct from those of an m-point rule. (Only, ifn and m are odd, the rules will have the mid-point in common.) There are, however, ways to estimate the error for Gaussian quadratures. Let G~ = ~ 7 _ I w f ( x i ) be an n-quadrature of polynomial degree 2n - 1. Then, define n+l
i-1
j-1
Here, Kzn+l has n + 1 additional nodes, and different coefficients for all ai, hi. These values can be specified so that Kzn+l is of polynomial degree 3n + 1. The two rules (Gn, K2n+l) form a Gauss-Kondrod pair. The difference IGn -K2~+I I is generally a fairly pessimistic error bound for the integral estimate K2n+l. Gauss-Kondrod quadrature rules are usually regarded as very efficient methods for calculating integrals of smooth functions. For double integrals, f~ fJf(x,y)dy dx, an obvious procedure is to solve iteratively one-dimensional integrals. This is called a product rule. Here, the approximation error can be bounded from the estimates of the one-dimensional quadratures. In some cases, especially for integrals over many dimensions, it is optimal to acknowledge the multidimensional nature of the approximation problem, and resort to a more direct integration rule. This would be called a non-product rule. Sometimes, suitable transformations can be made to facilitate computations, but numerical integration of multiple integrals may become rather costly or infeasible. [Davis and Rabinowitz
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(1984) and Stroud (1972) are still useful references.] An alternative route is MonteCarlo integration, which is much easier to implement. Monte-Carlo methods approximate the value of the integral from an average of functional evaluations over random points. Now, the error is actually a stochastic variable, and hence one can make some probabilistic inferences. Thus, if N is the number of sampled points, then the expected error goes to zero at a rate N -u2. This result holds under certain mild conditions on the integrand, without reference to the dimensionality. Over the past three decades, there has been active research to improve these estimates of Monte-Carlo integration, using "quasi-random" methods or "low discrepancy points". The general idea of these deterministic methods (with some randomized extensions) is to sample more carefully the regions of integration, so that the error may exhibit on a worst-case basis a convergence rate close to N 1 [e.g., see Geweke (1996), Niederreiter (1992) and Press et al. (1992) for an introductory account to this theory, and Papageorgiou and Traub (1996), Paskov (1996), and Tan and Boyle (1997) for some numerical evaluations]. These latter results usually require some sort of Lipschitz continuity, and the constants involved in the orders of convergence may depend on the dimensionality of the domain.
5. Extensions of the basic algorithm In this section we introduce some variations of the preceding algorithm which may accelerate the computation of the value and policy functions. There are two natural ways to modify the original method of successive approximations: (a) Introducing an alternative iteration process, or (b) appending a different interpolation procedure, under the same iteration structure. The accuracy properties of algorithms in class (a) remain unchanged, even though one may obtain considerable savings in computational time. The idea is that the method of successive approximations is slow, and there are other possible ways to compute the fixed point. The accuracy properties of algorithms in class (b) may change, although it should be recalled that higher-order approximants do not always yield higher accuracy, and these approximations may be more costly to implement. A common feature of these extensions is that they have not been extensively used and tested in the economic literature; but some of them seem to be of potential interest, and may be quite useful in particular applications. 5.1. Multigrid methods Multigrid methods have been widely applied for solving partial differential equations [cf. Stoer and Bulirsch (1993)]. In a natural way, these methods have been proposed by several authors for the solution of dynamic programs. Chow and Tsitsiklis (1991) have argued that the complexity of a multigrid algorithm is, in a certain sense, optimal.
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Also, Santos and Vigo (1995) implement this algorithm for computing solutions of economic growth problems, and report substantial speedups with respect to the single grid method, especially for cases with fine grids or with high discount factors. To motivate this approach, assume that in computing the value function the desired precision parameter h has been fixed, and there is not a good initial guess /47oto start the process. If W h is the fixed point of T h, and W is the fixed point of T, then it follows from the contraction property of these operators that in the first iteration the approximation error is bounded by /3[IWh-Wo[I+Mh
2.
(5.1)
Likewise, in the nth iteration the approximation error is bounded by
IIwh- w01[ +Mh 2
(5.2)
In these calculations, the distance between W0 and the value function W has been decomposed in the following way. The term /3n ][Wh - W01] bounds the distance between the fixed point W h and the nth iteration of W0 under T h for n ~> 1, whereas M h 2 bounds the distance IIW - Wh]] for M = y/[2(1 -/3)]. This second component is fixed on h, whereas the first one decreases by the factor 0 < /3 < 1. Hence, if our original guess is not close to W, initial reductions in the approximation error are relatively large (in absolute value) as compared to the second term M h 2. Since these gains are basically related to the distance ][W h - W0H and the contraction property of the operator T h, a coarser grid may lead to similar results, even though it entails less computational cost. Thus, it may be beneficial to consider coarser grids in the early stages of the computational process. Formally, the multigrid method proceeds as follows. Let ({S hi })i be a sequence of triangulations, for i = 0, 1 , . . . , n. Let hi be the mesh size of triangulation {S hi }. Suppose that h0 > h~ > ..- > hi > .." > hn. Then, take an arbitrary initial guess W0 and implement the above iterated method (4.1) under the coarsest partition {S h' }, to obtain a fixed point W h°. Next, choose Who as the initial condition for computing WhI . And follow the same procedure for subsequent finer grids: Pick W h~-~ as the initial choice for computing W h', for i = 1, 2, 3 , . . . , n. For the same level of accuracy, this method may reduce the computational burden, since early computations are performed over coarser grids with less computational cost. Likewise, one can generally obtain reasonable initial guesses of the value and policy functions at a very low cost from numerical computations over coarse grids. Our previous error analysis becomes useful for an optimal implementation of this method. As discussed in the preceding sections, inaccuracies of maximization and integration operations must be of order of magnitude h~, where hi is the grid level of triangulation {S h~}. Hence, in early stages of the iteration process these subroutines can be set up at a lower tolerance level to be effected more quickly. Furthermore, from early maximizations one can derive good guesses, so that the search for the maximum
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can be restricted to smaller regions while proceeding with finer partitions. Another crucial issue is the timing of the grid change. A coarser grid is more cheaply effected, but the gain in the approximation error is also smaller; i.e., the right-hand term of expression (5.2) is larger, and the contraction property of the operator does not apply to that term. For an optimal grid change, it seems that an appropriate benchmark value is the gain achieved per work expended. Let us then estimate the approximation errors in expression (5.2), and then proceed with the corresponding calculations for a specific illustration. A good bound for the first term of expression (5.2) can be obtained from the contraction property of our operators; indeed,
:h- will
II
- v: ll
1-/3
As for the term Mh 2, we need an estimate of the second-order derivative of the value function. This estimate may be calculated from our theoretical analysis of Section 3, although more operational bounds are usually obtained from computational experiments (cf. Section 7). To discuss the issue under concern in a more precise manner, assume then that there is but one state variable, and the grid sizes (hi)7-0 follow the relation hi = ho/2 i, i = 1,2 . . . . , n. Then, moving from mesh size hi to mesh size hi+l will roughly double the computational cost in each iteration, since there are twice as many vertex points. On the other hand, the additional gain in the approximation error is determined by the second term of expression (5.2), and for a grid change this term is estimated to 3 2 be ~Mh i , since convergence is quadratic. Therefore, to benefit from the grid change, the gain in the approximation error must double, and this happens when the ratio Ilw"3 For multidimensional problems, the optimal ratio would be smaller, as the computational cost increases exponentially.
v: llmh2
5.2. Policy iteration 7 The method of policy iteration is credited to Bellman (1955, 1957) and Howard (1960). [see Puterman and Brumelle (1979) for some early history.] In economics, the method has been successfully applied to some optimization problems involving large discrete state spaces [e.g., Rust (1987)]. There are, however, certain unresolved issues concerning the implementation of this procedure. From a theoretical point of view, the iterative scheme exhibits locally a quadratic convergence rate, and it has not been established an optimal radius or region of convergence. On the other hand, further computational work is needed to assess its performance in a wide range of economic applications. In the context of our previous discretization procedure, for a given triangulation {S J) with mesh size h, we now consider the following iterative scheme for policy
iteration: 7 This presentation draws upon unpublished work of the author with John Rust.
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(i) Initial step." Select an accuracy level TOLW and an initial guess W0h. (ii) Policy improvement step." Find gnh(k~, J Zo) J that solves
B(W;,h)(k dj :o)j =_W~,(k~,zj)+mkax v(kg,k,,zJo)+[3fz W2(kl'z')Q(z/'dz') (5.3) for each vertex point (k~,z0).J J (iii) Policy evaluation step." Find W,'+ h l(kd, J z 0) J satisfying I"
h j O/ -_ v(kd,g~;(k;,zo), J h j j z j )+: :3L Z/;+~(g.(k;, h h j z~),z')Q(z j,dz,) wL~(kd,z (5.4) for each vertex point (k j, z~). (iv) End of iteration." I f ][W2+,- W2[[ ~< TOLW, stop; else, increment n by 1, and return to step (ii). It should be understood that all functions derived from these iterations are piecewise linear, and compatible with the given triangulation {S J}. Thus, B maps the space of continuous fimctions, 142, into the space of piecewise linear functions, W h. Of course, the fixed point of Equations (5.3)-(5.4) corresponds to the fixed point of Equation (4.2), and the existence of such a unique solution W h has been established in L e m m a 4.1. Therefore, the error bounds derived in Theorem 4.3 and Corollary 4.4 apply here. Observe that step (ii) corresponds to a single iteration of the method of successive approximations. The innovation of policy iteration is in step (iii). For continuous random variables this step is generally non-trivial, since it involves the computation of a function W2, which appears on both sides, and under an integral sign. In such situations, in order to facilitate calculations one may resort to a complete discretization of the space Z, with a further loss of accuracy s. To be more specific about step (iii), let us consider a return function v(kt, kt+l), with kt, kt+l in R+. Assume that {ki}~, are the grid points. Then, for a piecewise linear function g h, Equation (5.4) can be written as follows: . ~::
= [I
-
# P g::] v/Zh+,,
(5.5)
where vg/; is an N-dimensional vector with elements v(ki,ghn(ki)), i = 1.... ,N; on the right-hand side, I is the N x N identity matrix, and Pg) is an N × N matrix generated by policy g~ in the following way: I f kl = g~(ki), for i = 1. . . . . N, and kl = ).U ~ + (1 - Z)kJ for some j, then the ith row Pgl; would be all zeroes, except for the (j - 1)th a n d j t h entries, which are equal to Z and (1 - Z), respectively. As a result
8 Technically, this is a Fredholm equation of the second kind. A natural approach for solving this problem is to use numerical integration and collocation [cf. Dahlquist and Bjorck (1974), pp. 396-397], and limit the search for the fixed point to a finite system of linear equations.
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of this construction, each row of matrix Pg) is made up of non-negative elements that add up to unity. Hence, the matrix [I -/3Pg~I] is always invertible, for 0 1, and /~ = 1 if j = 0. Observe that all non-linearities in system (6.7) appear on the right-hand side. This seemingly simple form may facilitate computation of a fixed point a*, and such a simple structure stems from both the method of approximating the Euler equations and the orthogonality properties of the polynomials. Let Oy(a) = I~ ~ n
Tj(xi) log ma(~(xi)),
j
O, 1 . . . . . n - 1.
i-1
Then, step 4 involves the computation of a fixed point a* = q~"(a*); i.e., solving the equation system (6.6) or (6.7). This computation may be most efficiently effected via standard solution methods for non-linear equations. Christiano and Fisher (1994) resort to a Newton-Raphson algorithm, taking as initial guess the solution of an associated quadratic optimization problem. For more complex problems, homotopy methods may be more effective, or successive refinements of the algorithm, using as initial guess the solution of a coarser discretization (cf. Example 7.2, below). Alternatively, one could iterate on the map a~+l = q~n(at), and verify if the sequence converges to a fixed point, but unless we are sure of the stability properties of mapping ~ , convergence here could be slower or even more problematic. As illustrated by Christiano and Fisher (1994), the algorithm has been fast and accurate in some simple test cases. However, this good performance may not be observed in more complex applications, since computation of a fixed point a* in Equation (6.7) is a rather delicate problem, and the indiscriminate use of polynomial interpolation may result in a poor approximation. As a matter of fact, there is relatively little theoretical work on the performance and asymptotic properties of this class of algorithms, and it seems that the following issues need further investigation:
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Existence: The non-linear system of equations characterizing a particular algo-
rithm may not have a solution a*, and hence we may be unable to produce a reasonable approximation. (ii) Multiplicity: There may be multiple solutions. (iii) Computability: Even if there is a unique solution, the system of non-linear equations may not be amenable to computation using standard numerical techniques. (iv) Accuracy: There is no theory on error bounds or accuracy properties for this family of algorithms. A related discussion on the existence of a fixed point a* is contained in Marcet and Marshall (1994). These authors suggest the use of the Brouwer fixed-point theorem. To the best of our knowledge, at present there are neither rigorous proofs, nor well known counterexamples, regarding the existence of a fixed point a*; moreover, it is hard to document if existence has been a serious operational issue. Under our assumptions in Section 2, the previously discussed algorithm may generate multiple solutions. The multiplicity of solutions may signal the presence of singularities, which limit the range of methods for solving systems such as (6.7); however, multiplicity should not raise further logical concerns. If there are multiple solutions, one would be advised to select the one with smallest Euler equation residuals (cf. Theorem 6.1, below); else, if all solutions exhibit small residuals, then the corresponding policy functions cannot be far apart form each other. Application of Newton-type methods for solving systems of non-linear equations involves inversions of matrices of derivatives, and these matrices may be singular (or nearly singular). Although Newton-type methods can attain quadratic orders of convergence to a fixed point a*, collinearities may result in inaccurate solutions or in rather small regions of convergence. Such difficulties may especially arise in complex stochastic models, with costly numerical integrations, and where a given level of accuracy may require fine grids or polynomials of high degree. Regarding accuracy, there is no formal derivation of error bounds for the class of algorithms studied in this section. In this analysis two types of accuracy results are pertinent: (a) Accuracy of a given numerical scheme approximating the Euler equation. (b) Derivation of error bounds for the policy and value functions using the Euler equation residuals. Concerning point (a), in the absence of a good theory for bounding or estimating highorder derivatives of the policy function, it seems a difficult task to obtain tight error estimates for polynomial approximations, especially in stochastic models with several state variables. Likewise, stability properties [cf. Section 4.4] for algorithms using Newton's method will depend primarily on the conditioning number of the Jacobian matrix, and this has to be directed specifically to each particular application. On the other hand, understanding how the size of the Euler equation residuals translates into approximation errors for the value and policy functions is not only essential to derive theoretical error bounds for the computed value and policy functions
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M.S. S a n t o s
under this family of algorithms, but it is also a key step for further implementational issues. For instance, evaluation of the residuals could allow us to assess the accuracy of competing numerical solutions, or the accuracy of a given numerical solution regardless of our confidence on the algorithm.
6.2. Accuracy based upon the Euler equation residuals" In practical applications, accuracy can be checked in simple test cases against analytical solutions, or against more reliable numerical methods. These indirect procedures may nevertheless be awkward or infeasible in some situations. That is, computing models with closed-form solutions is only illustrative of the performance of the algorithm in real applications, and the use of reliable numerical methods to test an algorithm involves further computational cost. We shall present here a recent result that allows us to bound the approximation error for any arbitrary solution. This analysis is based upon the computation of the Euler equation residuals. Computation of the residuals is a relatively easy task which involves functional evaluations, and hence it can be effected for arbitrarily large samples of points at a reasonable computational cost. In order to proceed more formally we need the following terminology. Let ~ be a measurable selection of the technology correspondence, F. Define W~ as (DO
v/
(ko, zo ) = Z
fit ~z' v(g t ( k°' zo ), ~t+l (/co,zo), zt ) ~tt(zo, dzt ).
t=0
As before, g/(ko,zo) = ~(~(...~(ko,zo)...),zt 2),zt-i) for every possible realization (z l, z2 . . . . . zt-1). The interpretation is that ~ is the computed policy function, and W~ is the resulting value function under the plan generated by ~. The following result applies to every numerical solution ~ independently of the algorithm under which may have been secured. Theorem 6.1 ]Santos (1999)]. Let e > O. Assume that
Dzv(ko,~(ko,zo),zo)+[3 fzDlv(~(ko,zo),~2(ko,zo),zl)Q(zo, dZl ) `)(1 - aft)
l=
>`(1 - a) + (1 - >,)(1 - a/3)
We fix parameter values, fi = 0.95, >` = ½, A = 10, a = 0.34, 6 1, p = 0.90. Also, we restrict the feasible domain so that k E [0.1, 10], e E [-0.032,0.032], and z is such that logz c [-0.32, 0.32]. The random process e comes from a normal distribution, where the density has been rescaled in order to get a cumulative mass equal to unity. As in Prescott (1986) we assume a standard deviation ~re = 0.008. Observe then that the end-points o f the domain o f variable e are four standard deviations away from the mean. One can again check that under these restrictions Assumptions (1)-(4) are satisfied. Indeed, in this simple case one can show that the model has a globally stable invariant distribution. As the random shock has a small variance, all paths eventually fluctuate around the point (k*,~) = (1.9696, 1), where k* is the state value o f the deterministic model and 5 is roughly the unconditional mean of the random process. Consequently, as in Section 7.1, to estimate the value M o f Theorem 4.3 it is reasonable to restrict ourselves to a certain domain containing the ergodic set, such as P = {(x,z)l ½ ~< k ~< 10, e -°32 ~< z ~< e°32}. This set is large enough to encompass most plausible economic applications using this framework, and here Mh 2 = 64.9374h 2. As in the preceding example, to this estimate we should add the other component of the observed error concerning the fact that the iteration process is stopped in finite time. Over the feasible domain o f state variables we set out a uniform grid o f vertex points (k j, 2 j) with mesh size h. Our numerical procedure then follows the iterative process specified in Equation (4.2) with an initial value W0 --= 0. As in Section 7.1, the algorithm is written so that only unidimensional maximizations need to be considered. Thus, each iteration proceeds as follows h W~+a (/co,z0) _- max >`log + (1 - >`) log l0 ,o \ (1 - >`)(1 = 7 0 ~ J
+[3~W:((zoAk~(1-1o)-a[(1
lo)
M°;1->`a)]),z,)Q(zo, dz~).
(7.4) All the integrations have been carried out under the subroutines qsimp and qtrap, as specified in Press et al. (1992, Sect. 2.4), with TOLI= 10-9. These subroutines follow
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Table 16 Example 7.3. Computational method: dynamic programming algorithm with linear interpolation a Vertex points
Mesh size
Iterations
CPU time
Max. error in g
Max. error in W
43 × 3
0.3872
10
1.44
1.12x10 ~
2.61
143 × 9
10 t
57
87.49
3.63×10 2
2.41x10 1
500 × 33
0.0282
108
2198.95
1.06× 10 2
6.03 × 10 2
a Parameter values:/3 = 0.95, )t = 7, A - 10, a = 0.34, 6 = 1, p = 0.9 and a s. = 0.008. 1
m
Table 17 Example 7.3. Computational method: dynamic programming algorithm with linear interpolation a Vertex points
Mesh size
Iterations
CPU time
Max. error in g
Max. error in W
43 x 3
0.3872
25
3.98
1.27×10 1
10.57
143 × 9
10 1
286
445.89
3.68x10 2
1.11
500 × 33
0.0282
550
9839.58
1.06x 10 2
1.56× 10 1
a Parameter values:~/3= 0.99, 2~= ½, A = 10, a = 0.34, 6 = 1, p = 0.9 and oe = 0.008. an N - s t a g e refinement o f an e x t e n d e d trapezoidal rule, and as argued in this treatise, they are fairly efficient for the integration o f relatively simple problems. Again, for univariate n u m e r i c a l m a x i m i z a t i o n s we have e m p l o y e d Brent's a l g o r i t h m w i t h tolerance level TOLM = 10 -8. The iteration process in E q u a t i o n (7.4) stops w h e n II w~+l - w~H ~< r o L w
- h2
It should be n o t e d that for the i m p l e m e n t a t i o n o f the algorithm (but not for the final calculation o f the a p p r o x i m a t i o n errors) all interpolations n e e d only be unidimensional. That is, although W~ in E q u a t i o n (7.4) is defined over a grid in a t w o - d i m e n s i o n a l space, a u n i d i m e n s i o n a l interpolation over z allows us to c o m p u t e the integral in E q u a t i o n (7.4) for g i v e n kJ. Then, the integral values are interpolated over k to define a continuous objective for the univariate m a x i m i z a t i o n . Table 16 presents i n f o r m a t i o n on our n u m e r i c a l e x p e r i m e n t for several values o f h. It can be o b s e r v e d that the error t e r m is always b o u n d e d by 24h 2. Hence, the constant s t e m m i n g f r o m our computations is b o u n d e d above by 24, whereas our estimate o f the o b s e r v e d error,
eh(k,z) = I Wh(k,z) - W2(k,z)[ k*) and underestimates output fluctuations for small values o f k. These countervailing effects
18 In all the figures, a number mEn means mxl0".
M.X Santos
372
6.0000E+O0 5.6000E+00 C
o
5.2000E+00
t,.}
4.8000E+00 >,
0 D.
of 4.4000E+00 4.0000E+O0 3.6000E+00
O O -ILLI O O O
O O + ILl CO ~" C~
O O 4LIJ CO C,l CO
CO
~
co
co
~-
~
e~
O O 4IJJ ¢O O ~"
O O 4LU O CO 131
O O + LU I~ Lib tg)
,-u5
O O 4LIJ '~" CO ~
CO ~
O O + LLI *'I~
u~ co
,--
Capital Domain Fig. 2. E x a m p l e 7.3. Policy f u n c t i o n s for the d y n a m i c p r o g r a m m i n g a l g o r i t h m (DP) a n d q u a d r a t i c a p p r o x i m a t i o n ( L Q ) for z = 1. P a r a m e t e r v a l u e s : / 3 = 0.99, 3. = 5, l A = 17, a = 0.34, 6 = 1, p = 0.9 a n d a~. = 0.008. In the deterministic case, (i.e., for a t = 0), this p a r a m e t e r i z a t i o n yields a s t e a d y - s t a t e value k* = 4.669.
m cO
1.6000E-O1
"
1.2000E-01
"6
8.0000E-02
e-,
4.0000E-02 O.O000E+O0 e=
~
-4.0000E-02 O O + LLI O O O L~
O O 4LLI CO '~" C,l ~
O O 4LLI t.D C',l CO CO
O O 4LU CO O ~" LO
O O + LLI O CO O') v--
O O 4ILl I"-LO LO CO•
O O + LLI ~ ¢0 ~ If)
O O 4LU
e,i
e5
co
**
u5
u~
~o
r~
T--
Capital d o m a i n Fig. 3. E x a m p l e 7.3. D i f f e r e n c e b e t w e e n the c o m p u t e d p o l i c y functions, gLQ _ gDP, for z = 1. P a r a m e t e r v a l u e s : / 3 = 0.99, 2~ = ½, A = 17, a = 0.34, 6 = 1, p = 0.9 a n d ae = 0.008. In the d e t e r m i n i s t i c case (i.e., f o r cre = 0), this p a r a m e t e r i z a t i o n yields a s t e a d y - s t a t e value k* = 4.669.
Ch. 5:
373
Numerical Solution o f Dynamic Economic Models
Table 23 Example 7.3. Standard deviations, a(i), and correlation coefficients, corr(i,j), for i,j = k, c, u,y a Method
d(y)
~(i)
a(c)
a(u)
corr(w, u)
corr(c,y)
PEA-collocation
0.0283
0.0170
0.0142
0.0034
0.5436
0.8867
Dynamic programming
0.0287
0.0174
0.0142
0.0035
0.5498
0.8838
Quadratic approximation
0.0280
0.0167
0.0141
0.0033
0.5507
0.8917
a Parameter values:/3 - 0.99, ;t = ½, A = 1.1, a = 0.34, 6 = 0.05, p = 0.9 and at = 0.008.
b a l a n c e e a c h o t h e r a n d lead to a s i m i l a r e s t i m a t e for o ( y ) . S i n c e t h e a p p r o x i m a t i o n e r r o r is m o r e p r o n o u n c e d for s m a l l v a l u e s o f k, q u a d r a t i c a p p r o x i m a t i o n s w o u l d t h e n h a v e a t e n d e n c y to u n d e r e s t i m a t e o ( y ) . Finally, Table 23 a n d F i g u r e s 4 a n d 5 r e p l i c a t e t h e a b o v e c o m p u t a t i o n s for a m o r e realistic p a r a m e t e r i z a t i o n w i t h d e p r e c i a t i o n f a c t o r (5 = 0.05. T h i s c a l i b r a t i o n o f the m o d e l does n o t p o s s e s s a c l o s e - f o r m solution, a n d c o n s e q u e n t l y w e c a n o n l y r e p o r t c o m p u t a t i o n s c o r r e s p o n d i n g to o u r n u m e r i c a l m e t h o d s - q u a d r a t i c a p p r o x i m a t i o n , P E A - c o l l o c a t i o n a n d t h e d y n a m i c p r o g r a m m i n g a l g o r i t h m . C o n c e r n i n g the a c c u r a c y 1.0000E-02 ~-I-
__..-.--I- ~ - I -
2.0000E-03
/+
"0
=,?.
-6.0000E-03
~+
e-
/
0 .m ,,-I
-1.4000E-02
14,1
/
-2.2000E-02
-I-
+
-3.0000E-02 O O -ILU O O O
O O -ILU 03 03 03
O O + ILl LO (,O I~
O O -ILU CO %-
O O "t" LU O 03 LO
O O + LLI 03 ~ 133
O O "t" ILl LO 03 O,I
O O + LU CO I'~ LO
Capital domain Fig. 4. Example 7.3. Euler equation residuals for PEA and quadratic approximation (LQ) policy functions for z = 1. Parameter values:/3 = 0.99, 2~= ½, A = 1.1, a = 0.34, 6 = 0.05, p = 0.9 and o e = 0.008. In the deterministic case, this parameterization yields a steady-state value k* = 5.0294.
374
M.S. Santos
4.0000E-02 tO
3.2000E-02
U
>,
.2 "6 D,. e-
2.4000E-02
1.6000E-02
,,,.,,
8.0000E-03 U e-
=_"
O.O000E+O0
rl
-8.0000E-03 0 + W
0 + W
o + W
0 + W
~ + W
0 + W
0 + W
o
~
~
~
~
~
~
0
~
~
~
~
~
~
0 + W
Capital domain
Fig. 5. Example 7.3. Difference between the computed policy fimctions, gLQ _ gD?, for z = 1. Parameter values: fi = 0.99, ;t = ½, A = 1.1, a = 0.34, 6 = 0.05, p = 0.9 and at. = 0.008. In the deterministic case (i.e., for a e = 0), this parameterization yields a steady-state value k* = 5.0294. o f quadratic approximations, there are no substantial variations with respect to the preceding experiment. As before, this approximation yields good estimates for the secondorder moments, and the most significant differences are downward biases for the standard deviation of both output and investment, which are nevertheless fairly small. Also, Euler equation residuals and approximation errors for the linear policy function are of order 10 3 in a significant region. This approximation becomes increasingly less accurate as we deviate from the steady-state solution, especially for small values o f k. In this experiment, we should stress the good performance of PEA-collocation, since an 8 x 5 vertex-point grid achieves Euler equation residuals o f order 10.6 for all points in the domain (cf. Figure 4). Obtaining residuals o f this order o f magnitude under the dynamic programming algorithm requires considerably more grid points, and a computational effort o f the order o f a thousand times higher. One should bear in mind that in all these computations, the standard deviation o f the innovation is very small, i.e., a~. = 0.008. Peralta-Alva and Santos (1998) explore in detail the sensitivity o f quadratic approximations to variations in the parameters fi, 6, p and a~. The most sizeable deviations regarding second-order moments are observed for variations in parameters p and ae, although substantial changes in other parameter values may also have non-negligible effects.
Ch. 5:
375
Numerical Solution of Dynamic Economic Models" Observations Selected for Calibration .•°
Economic Model
Exact Solutions
I I
I AccuracyTests [ Numerical Solutions
Numerical Model
Observations Selected for Testing
Fig. 6. Testing economic theories.
9. Testing economic theories In this section, we focus on certain basic issues concerning the process of validation of economic models, where computer simulations must play a prominent role. Our aim is to provide a useful framework for the analysis and evaluation of numerical procedures. As a result of some conjecture or proposed theory, we consider that a mathematical model has been constructed, with the ultimate goal of rationalizing an economic situation, phenomenon or activity. As is well understood, in general exact solutions are not readily available. Hence, for the purpose of testing an economic model one has to follow the indirect route of computing suitable approximations. To gain confidence in these computations, one needs to weigh a wide range of sources of error that go from the construction of the model to the formal testing of the solutions. A framework for this discussion is outlined in Figure 6, which highlights some major steps in the process of scientific inquiry. In order to perform quantitative experiments, a model must be calibrated. That is, specific functional forms and corresponding parameter values must be stipulated so as to derive definite predictions. Considering that models are artificial constructs, it does not seem plausible to assign these values independently of the underlying postulates. Thus, a parameter such as the elasticity of intertemporal substitution for aggregate consumption can be difficult to identify in the data. But even if it is identified, such parameter may proxy inherent simplifications in the model such as the absence of a government sector, the absence of leisure, or of home production, or of some other observable or non-observable (i.e., not easy to measure) components not explicitly modelled. Besides, from a conceptual perspective this parameter may be attached a different meaning in envirolunents with bounded rationality or with alternative mechanisms for forming expectations.
376
M.S. Santos
Evidence from panel data and microeconomic studies may help to pin down parameter values, but this evidence is not conclusive on most occasions. Thus, Browning, Hansen and Heckman (ch. 8, this volume) observe that estimates from microeconomic studies are not readily transportable to macroeconomic models. Likewise, estimating a subset of parameter values from some independent observations may lead to biases and inconsistencies in the ulterior calibration of the remaining ones [cf. Gregory and Smith (1993) and Canova and Ortega (1996)]. In view of all these complexities, a more operative strategy for calibrating a model is simply to select a set of observations where all model parameters may be jointly estimated. For instance, a common practice nowadays is to calibrate a business cycle model from some facts related to growth theory [e.g., Christiano and Eichenbaum (1992) and Cooley and Prescott (1995)] without focussing so much attention on microeconomic studies. Of course, alternative sets of observations selected for calibration may lead to different parameter values, and the ability of a certain range of parameter values to account for a wide group of well established observations may enhance our confidence in the model chosen. All these calibrations may be associated with standard errors stemming from uncertainties present in the data [cf. Christiano and Eichenbaum (1992)]. The approach just described presents a subtle difference with respect to what would be called a naive interpretation of the positivist (or falsificationist) view, often associated with the writings of Friedman (1953) and Popper (1965). Under these latter methodological programs, a model could in principle be tested without a previous confrontation to an independent or unrelated set of data. In such circumstances, functional forms and parameter values could be chosen so as to insure a best fit for the data or sets of data to be accounted for, without further regard to other seemingly unrelated events that the theory was not initially purported to explain. After an economic model has been specified, one can proceed to its further analysis. Some basic properties such as existence of solutions, existence of stationary paths or invariant distributions, monotonicity, differentiability, uniqueness and stability, may be explored by classical mathematical methods. But most often these tools fail to provide us with the quantitative information necessary to test a model. In such circumstances, numerical approximations may be needed to deepen our understanding of the model's predictions. The numerical model is not usually aimed to rationalize an economic situation. This model is generally an algorithmic device, to be implemented in a computer with a view toward the simulation of the true behavior of the original economic model. Therefore, all errors involved in these approximations should be made sufficiently small so that our inferences based upon the numerical simulations remain basically true for the exact solution. (Accuracy tests are designed to bound the error of the approximate solution.) It should be emphasized that there is no universal criterion or hard and fast rule for the approximation error that can be valid for all applications. The appropriate size of the error will depend on further considerations such as the purpose of the investigation, the sensitivity of the solution to initial conditions and parameter values,
Ch. 5.. Numerical Solution o f Dynamic Economic Models
377
and the conditions under which a model is tested. In other words, our ability to make good inferences from approximate solutions is not unambiguously determined by the size of the approximation error. Since few mathematical models yield closed-form solutions, one may be tempted to specify from the start a numerical model (i.e., a collection of rules which could be coded as a finite set of computer instructions), avoiding formulation of an abstract economic model. This point of view may seem very appealing 19, but some practical considerations work in favor of formulating an abstract model. First of all, there is a well developed mathematical theory concerning regular spaces of functions over continuum quantities, which has no counterpart for discrete variables. Indeed, our most powerful mathematical tools apply to locally linear domains, with continuous or differentiable functions. Second, computations introduce approximations and round-off errors, and it becomes tedious to verify certain properties such as existence, stability, monotonicity of solutions or the constancy of a given elasticity. Validation of these results via numerical computations would require a full sampling of the state and parameter spaces, and this is not usually the most effective way to establish a given property of the solution. Thus, in most situations it is not plausible to support our entire analysis on a numerical model, and a more abstract framework is necessary. Computer simulations, though, may help us make further theoretical progress, since these computations may give way to reasonable conjectures that can be subsequently examined by mathematical analysis or by further numerical experiments. Finally, predictions and properties of solutions of the economic model should be contrasted with the underlying real-world situation. In some simple cases, these critical comparisons may be effected without resort to a formal analysis (e.g., a model may be dismissed if its predictions about the rates of interest are very distant from the observed data). In more subtle situations, statistical techniques are usually needed. Some recent work has focussed attention on testing the output of computer simulations [cf. Canova and Ortega (1996), Christiano and Eicheubaum (1992), Gregory and Smith (1993), Kim and Pagan (1995), and references therein.] For setting up an appropriate framework for model testing, one should realize that data sets and computer simulations are subjected to sources of uncertainty of a different nature. Data sets are characterized by measurement and sampling errors, whereas simulations involve approximations of the original model along with further numerical errors. While measurement errors may resemble round-off errors and machine failures, sampling errors may be avoided in model simulations. Indeed, the moments of an invariant distribution of a model can be accurately reproduced by resorting to arbitrarily large sample paths or by theoretical analysis, although these model statistics may not be univocally defined in the presence of multiple equilibria or multiple
19 Afterall, one is ultimately interested in quantitative assessments; moreover,the numerical model may often offer a more faithful approximationof the real world situation, where some features and quantities are also discrete.
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M.S. Santos
invariant distributions, and are subject to the uncertainty stemming from parameter calibrations 20. Consequently, for evaluating the performance of a model system we require a detailed study of both the economic and numerical models, together with an analysis of the available data. Each element in this chain has its own specificities, which may give rise to different types of errors that must be accounted for in the process of testing a particular theory or conjecture. Let us briefly discuss some of the major components: (a) Sensitivity of solutions to initial conditions: Small changes in the state variables may lead to fundamentally different predictions, and such pathological behavior may appear in both theoretical and numerical models, especially in environments with chaotic dynamics, multiple steady states or invariant distributions, or indeterminate equilibria. These are instances in which there are no definite predictions, and one should investigate how these instabilities unravel in the numerical model and in the data analysis. But even if solutions are reasonably well-behaved, a distinction should be made between steady-state behavior and transitional dynamics. Often, it is useful to compute the speed of convergence to a given steady state or invariant distribution. (b) Uncertainty of the calibrated parameters: A model may have definite predictions regarding steady-state behavior and transitional dynamics, but such predictions are subjected to the uncertainty stemming from the parameter space. For a given set of parameter values, arbitrarily good estimates of unconditional moments of the model's invariant distributions may generally be obtained by large sets of simulations. These moments depend on parameter values, and correspondingly follow the probability law induced by the parameter space. (c) Rounding and chopping errors: Computer arithmetic is not exact, and hence calculations are subject to errors. For rounding errors, it is reasonable to presume that they are normally distributed. Furthermore, their order of magnitude is small, in most cases around 10 15. But care should be exercised, since in the presence of instabilities these errors may grow over the iterative scheme in many unexpected ways. Their cumulative effect over the computed solution is usually more sizeable. (d) Approximation errors: These are the errors involved in the discretization of the theoretical model. Again, their cumulative effect over the computed solution may be considerably larger than that of the functions being approximated (cf. Lemma 4.2 and Theorem 4.3). Approximation errors are often impaired by systematic components, and hence their statistical modelization may become problematic. For instance, the numerical algorithm presented in Section 4 always underestimates the value function. Systematic biases may also affect the curvature of the interpolants.
20 A further issue in model testing is to determine if the data selected correspond to a steady-state situation, or if transitional dynamics are playing an important role in the analysis.
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(e) Sampling errors: As we only observe a limited number o f realizations of a random
variable, it is well understood that our inferences are subject to sampling error. Statistical theory can help us gauge this error for both small and large data samples. (f) Measurement errors: This is also a classical topic o f statistical theory. It is well known that imperfect measurement may bias the sample moments of our estimates. Several recent attempts at analyzing computer simulations [see the recent surveys by Canova and Ortega (1996), and Kim and Pagan (1995)] contemplate some o f these sources o f error. For instance, Christiano and Eichenbaum (1992) advance an statistical framework for testing second-order moments assuming that uncertainty stems from calibrated parameter values and sample observations. As other alternative approaches, the implicit postulate for the purpose at hand is that some other errors are so small that can be safely ignored. But, in cases where sensitivity o f solutions to initial conditions and approximation errors become relevant considerations, a formal analysis of these influences is needed before embedding them into a statistical framework. For the same significance level, approximation errors may change or widen the critical region, or lessen the power o f a test. If errors are o f different orders o f magnitude, an obvious step is to lessen the influence o f the most critical ones. Approximation or numerical errors can generally be reduced at the expense o f more computational effort, and the gains from reducing these errors should be evaluated against the incurred cost. O f course, these considerations apply for all other errors, but often their costs are prohibitive. Summarizing, this section started with the basic idea that numerical analysis is a useful tool in the study o f economic models, and our purpose has been to highlight several sources o f error to be accounted for in testing an economic theory. Numerical analysis yields approximate solutions. Thus, to make proper inferences about an economic model, the approximation error must be sufficiently small. This error must be evaluated in conjunction with further properties of both the theoretical and numerical models, and the statistical properties o f the data. It does not seem plausible to derive a universal measure o f error - or any other purely econometric statistic - that can yield definite answers in all situations. And the benefits derived from a better approximation have to be balanced against the additional computational effort.
10. A practical approach to computation This chapter has reviewed several numerical techniques for solving economic models. Our analysis has been restricted to a family o f standard growth models in which optimal solutions may be decentralized as competitive allocations. This equivalence between optimal solutions and competitive allocations will break down in the presence of externalities, incompleteness o f financial markets, taxation, public expenditure, money, and other frictions or governmental interventions. Additionally, there are
380
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alternative frameworks that have not been considered here, such as overlapping generations economies or models based on game theoretical assumptions. The array of methods presented in this paper should nevertheless be of potential interest for solving these other modelizations. Indeed, one should expect that in the near future a good share of research efforts will leave the frictionless economic framework considered in this paper and aim at a rigorous simulation of more complex economies. For a good start to some of these topics, the reader is referred to the recent monograph edited by Cooley (1995). Chapters 1-4, 7 and 12 of that volume focus especially on computational issues. For further related work and extensions, the following is a very partial list of theoretical and applied papers on computation: (a) Economics with heterogeneous agents and borrowing constraints: Castafieda, Diaz-Gimenez and Rios-Rull (1997), Huggett and Ventura (1997), Krieger (1996), Krusell and Smith (1995), and RiosRull (1997); (b) Economies with taxes: Bizer and Judd (1989), Chari and Kehoe (ch. 26, this volume), Coleman (1991), and Jones, Manuelli and Rossi (1993); (c) Suboptimal equilibria: Baxter (1991), and Greenwood and Huffman (1995); (d) Monetary economies: Cooley and Hansen (1989), Giovannini and Labadie (1991), and Lucas and Stokey (1987); (e) Finance models: Duffle (1996, ch. 11), Boyle, Broadie and Glasserman (1997), and Heaton and Lucas (1996); (f) Overlapping generations economies: Auerbach and Kotlikoff (1987), Kehoe and Levine (1985), and Kehoe, Levine, Mas-Colell and Woodford (1991); (g) Asymmetric information: Prescott (1997), and Phelan and Townsend (1991); (h) Game theory: McKelvey and McLennan (1996), and Dutta and Sundaram (1993). Given the ample variety of economic models and computational techniques, a researcher will most often be faced with the choice of an appropriate numerical method. Our purpose now is to outline a set of preliminary steps which may prove useful in the selection and implementation of a numerical procedure. The first basic principle to bear in mind is that there is no numerical method that will perform better in all situations. Hence, the choice will generally be complex, and the numerical procedure must be suited to the analytic nature of the problem under consideration. At this stage, a theoretical study of the model is most helpful. Qualitative properties of optimal solutions, such as existence or differentiability, should shed light on the error stemming from different approximations. The existence and stability properties of steady states or invariant distributions will help determine the mesh size or order of the approximant, as well as the most suitable restriction of the domain. In addition, a theoretical analysis of the model should provide valuable clues in the computation process, such as the choice of an initial guess for the solution, the efficient manipulation of state and control variables to simplify the model, or the appropriate subroutines for integration, maximization, and related operations. Also, it is useful to undertake a theoretical analysis of the numerical model, and examine differences between the dynamic behavior of these solutions and their continuum analogues. A second point to be stressed is that for smooth, concave models involving one or two state variables there are generally reliable algorithms that can compute the solution
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Numerical Solution o f Dynamic Economic Models
381
in reasonable time at a desired level of accuracy. As already stressed, subroutines for integration and maximization in one dimension are fairly efficient. Moreover, technological developments will facilitate the application of reliable methods in a near future. It seems then that the use of less rigorous or less reliable approximation procedures becomes more attractive for more difficult, time consuming computational problems. Although quadratic approximations and methods approximating the Euler equation performed remarkably well for all models considered in this paper, it should be realized that in most cases these models can be solved via the discretized dynamic programming algorithm combined with spline interpolation. And an efficient use of the multigrid algorithm (and to a lesser extent, with policy and modified policy iteration) may help minimize the computing time. There are, however, computational problems in which the use of reliable methods becomes awkward or infeasible. These models may simply lack concavity, interiority of solutions, or smoothness, or involve several state variables. Techniques for the computation of non-smooth or discrete problems are generally less powerful, and sometimes model-specific; hence, at this general level of discussion, it seems difficult to offer specific guidelines. On the other hand, for the computation of large-scale models, we suggest the following sequential procedure: (i) Quadratic approximations; (ii) Globally convergent numerical methods in which accuracy can be controlled; (iii) Faster computational procedures, which may lack global convergence or a formal derivation of error bounds. If the model contains a globally stable steady state or invariant distribution, then it seems natural to start with a quadratic approximation. To assess the accuracy of this approach it may be helpful to calculate the (exact) derivatives of the value and policy functions, or simply check the Euler equation residuals. From these residuals, one may be able to estimate approximation errors for the value and policy functions (cf. Section 6.2). The quadratic model may also provide a good initial guess of the solution for the remaining, more sophisticated computational methods. Reliable algorithms, which are amenable to a progressive control of the approximation error, can also be useful in the computation of large-scale dynamic problems. In these cases, one may resort to coarse grids with higher-order approximants. (For instance, Johnson et al. (1993) solve a four-dimensional dynamic model with a discretized version of the dynamic programming algorithm with spline interpolation.) Comparisons of outcomes with progressively finer grids - or against simple test cases with closed-form solutions - may serve to appraise numerically the constants involved in the approximation errors. Likewise, a close examination of the Euler equation residuals may allow us to evaluate the accuracy of these methods. Reliable algorithms should also provide reasonable initial guesses for the computation of a model under faster numerical methods. Sometimes, the only feasible route for solving a model is via faster algorithms, which may lack global convergence or are not easily amenable to a formal error analysis. The efficient design of such computational procedures often involves
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a combined application of standard techniques from numerical analysis in ways suggested by a previous analysis of error for reliable algorithms. There are also some subtle issues concerning implementation o f these procedures. As suggested in Section 6, one should check for existence and uniqueness of solutions; moreover, application of Newton-type methods to find zeroes o f non-linear equations requires that the system be locally well-conditioned. For the practical operation of the algorithm, it may be helpful to start with an initial candidate from (i) or (ii), and then compute successively finer approximants, taking as initial guess in each step the solution obtained from the previous approximation. Comparisons of the outcomes and coefficients obtained from successive approximants may shed light on the stability and accuracy of the numerical method. Moreover, an evaluation of the Euler equation residuals may be the most effective way to estimate numerically the approximation error [cf. Santos (1999)].
References Amman, H.M. (1996), "Numericalmethods for linear-quadraticmodels", in: H.M. Amman,D.A. Kendrick and J. Rust, eds., Handbook of ComputationalEconomies (Elsevier, Amsterdam) 588-618. Anderson, E.W., L.P. Hansen, E.R. McGrattan and T.J. Sargent (1996), "Mechanics of forming and estimating dynamic linear economies", in: H.M. Amman, D.A. Kendrick and J. Rust, eds., Handbook of Computational Economics (Elsevier, Amsterdam) 173-252. Araujo, A. (1991), "The once but not twice differentiability of the policy function", Econometrica 59:1383 1391. Auerbach, A.J., and L.J. Kotlikoff (1987), Dynamic Fiscal Policy (Cambridge University Press, Cambridge). Baxter, M. (1991), "Approximatingsuboptimaldynamic equilibria: an Euler equation approach", Journal of Monetary Economics 27:173~00. Bazaraa, M.S., D.H. Sherali and C.M. Shetty (1993), Nonlinear Programming (Wiley,New York). Becker, G.S. (1993), "Nobel lecture: The economic way of looking at behavior", Journal of Political Economy 101:385-409. Becker, R., and I. Zilcha (1997), "StationaryRamsey equilibriumunder uncertainty",Journal of Economic Theory 75(1):122-140. Bellman, R. (1955), "Functional equations in the theory of dynamic programming V Positivity and quasilinearity", Proceedings of the National Academy of Sciences, USA 41:743 746. Bellman, R. (1957), Dynamic Programming (Princeton University Press, Princeton, NJ). Bellman, R., R. Kalaba and B. Kotkin (1963), "Polynomial approximation - a new computational technique in dynamicprogramming:Allocationprocesses", Mathematicsof Computation17:155-161. Benveniste,L.M., and J.A. Scheinkman(1979), "On the differentiabilityof the value function in dynamic models of economics", Econometrica47:727-732. Bertsekas, D.P. (1975), "Convergence of discretization procedures in dynamic programing", IEEE Transactions on Automatic Control 20:415-419. Bizet, D., and K. Judd (1989), "Taxation and uncertainty", American Economic Review 79:331-336. Blackwell, D. (1965), "Discounteddynamicprogramming",Annals of Mathematical Statics 36:226-235. Blume, L.E., D. Easley and M. O'Hara (1982), "Characterizationof optimal plans for stochastic dynamic programs", Journal of Economic Theory 28:221~34. Boldrin, M., and L. Montrucchio (1986), "On the indeterminacy of capital accumulationpaths", Journal of Economic Theory 40:26-39.
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Chapter 6
I N D E T E R M I N A C Y A N D SUNSPOTS IN M A C R O E C O N O M I C S JESS BENHABIB New York University
ROGER E.A. FARMER UCLA
Contents
Abstract Keywords 1. Introduction 2. W h y should we care? 2.1. Technical aspects of linear models 2.2. Indeterminacy and propagation mechanisms in real models of business cycles 2.3. Indeterminacy and propagation mechanisms in monetary models of business cycles 3. I n d e t e r m i n a c y in real m o d e l s 3.1. A framework for comparing different models 3.2. The one-sector model with increasing returns 3.3. The two-sector model with increasing returns 3.4. The two-sector model with constant marginal returns 3.5. Fixed costs and the role of profits 3.6. Models with variable markups 4. I n d e t e r m i n a c y in m o n e t a r y m o d e l s 4.1. Monetary models with one state variable and fixed labor supply 4.2. Money in the utility function and the production function 4.3. Monetary models with one state variable and a variable labor supply 4.4. Monetary models with several state variables 5. I n d e t e r m i n a c y and p o l i c y f e e d b a c k 5.1. Fiscal policy feedback 5.2. Monetary policy feedback 5.3. Interest rate rules and indeterminacy 5.4. Monetary models and sticky prices due to frictions in the trading process 6. I n d e t e r m i n a c y and m o d e l s o f endogenous growth 7. S o m e related w o r k Handbook of Macroeconomics, Volume 1, Edited by J.B. Taylor and M. WoodJbrd © 1999 Elsevier Science B.V. All rights reserved 387
388 388 389 390 391 393 395 398 398 400 401 403 405 405 407 408 410 411 413 416 416 417 419 422 423 426
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8. Empirical aspects of models with indeterminacy 8.1. Real models and propagation dynamics 8.1.1. One-sectormodels 8.1.2. Two-sectormodels 8.1.3. Multi-sectormodels 8.2. Monetary models and the monetary transmission mechanism 9. Some criticisms of the use of models with indeterminate equilibria to describe data 9.1. Equilibrium selection 9.2. Equilibrium forecast functions 9.3. Does indeterminacyhave observableimplications? 10. Conclusion Acknowledgements References
427 427 428 433 435 437 438 438 440 442 442 443 443
Abstract
This chapter gives an overview of the recent literature on indeterminacy and sunspots in macroeconomics. It discusses of some of the conceptual and the technical aspects of this literature, and provides a simple framework for illustrating the mechanisms of various dynamic equilibrium models that give rise to indeterminate equilibria. The role of external effects, monopolistic competition, and increasing returns in generating indeterminacy is explored for one-sector and multi-sector models of real business cycles and of economic growth. Indeterminacy is also studied in monetary models, as well as in models where monetary and fiscal policy are endogenous and determined by feedback rules. Particular attention is paid to the empirical plausibility of these models and their parametrizations in generating indeterminate equilibria. An overview of calibrated macroeconomic models with sunspot equilibria is given, and their successes and shortcomings in matching properties of data are assessed. Finally some issues regarding the selection of equilibria, the observable implications, and difficulties of forecasting that arise in such models are briefly addressed.
Keywords indeterminacy, multiple equilibria, sunspots J E L classification: E00, E3, 040
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1. Introduction M o d e m macroeconomics is based on dynamic general equilibrium theory and for some time it has been known that, unlike static general equilibrium theory, in dynamic general equilibrium economies equilibria may be indeterminate 1. Indeterminacy means that there may be an infinite number of equilibria, all very close to each other, and the existence of indeterminacy in a dynamic model has, in the past, been considered to be a weakness of a theory that should be avoided by careful modeling assumptions. In contrast, a recent literature has grown up in macroeconomics that exploits the existence of an indeterminate set of equilibria as a means of understanding macroeconomic data. This chapter surveys this literature and places it in the context of other recent developments in macroeconomics. The literature on quantitative aspects of indeterminacy can be organized around three strands. First, there is work that uses models with indeterminate equilibria to explain the propagation mechanism o f the business cycle. Second, there is a group of papers that uses indeterminacy to explain the monetary transmission mechanism, specifically the fact that prices are sticky, and third, there is work in growth theory that uses indeterminacy to understand why the per capita incomes of countries that are similar in their fundamentals nevertheless save and grow at different rates. In this survey we explain the ideas that underlie each of these strands; we discuss the mechanisms that lead to indeterminacy and we report on the current state of quantitative models of the business cycle. We pay particular attention to areas in which models with indeterminate equilibria might offer a significant improvement over a more conventional approach. A closely related concept to that of indeterminacy is the idea of a sunspot equilibrium, an idea developed by Cass and Shell [Shell (1977), Cass and Shell (1983)], to refer to equilibrium allocations influenced by purely extrinsic belief shocks in general equilibrium models 2. A sunspot equilibrium is one in which agents receive different allocations across states with identical fundamentals; that is, preferences,
1 Gale (1974) first demonstratedthat indeterminacyoccurs in Samuelson's "consumption-loans"model and Calvo (1978) was one of the first to discuss the issue in this context. Kehoe and Levine (1985) have an excellent discussion of the conditions under which indeterminacy can and cannot occur in infinite horizon general equilibrium economies. 2 Azariadis (1981) was the first published paper to show that sunspots may be responsible for business cycles although he uses the term self-fulfilling prophecies, originally coined by Robert K. Merton (1948). Woodford (1986, 1988) further demonstrated how sunspots could be relevant to understanding macroeconomic fluctuations. Howitt and McAfee (1992) use the term 'animal spirits' (popularized by Keynes in the General Theory) to refer to the same concept. It is perhaps unfortunate that these terms are now closely connected. Jevons, for example, who worked on sunspots in the 19th century, did not intend that 'sunspots' should refer to extrinsic uncertainty; instead he believed that there was a real link between the sunspot cycle, the weather and the agricultural sector of the US economy. Similarly,Keynes did not use animal spirits to mean self-fulfilling beliefs; instead his view of uncertainty was closer to Frank Knight's concept of an event for which there is too little information to make a frequentist statement about probabilities.
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endowments and technology are the same but consumption and or production differs. Sunspot equilibria can often be constructed by randomizing over multiple equilibria of a general equilibrium model, and models with indeterminacy are excellent candidates for the existence of sunspot equilibria since there are many equilibria over which to randomize. Sunspots cannot occur in finite general equilibrium models with complete markets since their existence would violate the first welfare theorem; risk averse agents will generally prefer an allocation that does not fluctuate to one that does. Examples of departures from the Arrow-Debreu structure that permit the existence of sunspots include (1) incomplete participation in insurance markets as in the overlapping generations model, (2) incomplete markets due to transactions costs or asymmetric information, (3) increasing returns to scale in the technology, (4) market imperfections associated with fixed costs, entry costs or external effects, and (5) the use of money as a medium of exchange. We have drawn attention to three strands of literature; business cycles, monetary transmission and economic growth. The literature that uses indeterminacy and sunspots to understand business cycles is more fully developed than the work on growth theory, and models have been developed of both business cycles and the monetary transmission mechanism that provide quantitative explanations of economic data. These models exploit two ideas; first that indeterminacy may provide a rich source of propagation dynamics to an equilibrium model and second that sunspots may provide an alternative impulse to technology or taste shocks. In monetary models the dynamics of indeterminate equilibria have been exploited to explain how a purely nominal shock may have real effects in the short run without invoking artificial barriers to price adjustment. In addition to their contribution to the theory of economic fluctuations, models of indeterminacy have been used in the literature on economic growth to explain different and sometimes divergent growth rates of countries and regions that start out with similar endowments and wealth levels. Despite the explosion of research in modern growth theory, many important questions remain unsettled and results are frequently not robust to alternative empirical specifications [see Levine and Renelt (1992)]. The growth literature on indeterminacy highlights the possibility that economic fundamentals alone will not pin down the savings rates for different countries since countries with identical endowments and wealth levels may coordinate on different equilibrium savings rates that may be determined by cultural, social or historical considerations.
2. Why should we care?
The initial work on indeterminacy in general equilibrium models was often abstract and far removed from issues of economic policy. Part of our goal in this survey is to dispel the misconception that indeterminacy is an esoteric area that is unconnected with the core of macroeconomics. We will show that, if one accepts dynamic general
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equilibrium theory as an organizing principle, the possibility o f indeterminacy is part of the package. Furthermore, indeterminate equilibria can illuminate a number o f issues that are otherwise puzzles. Two issues that we will discuss in this section are (1) the role of beliefs in business fluctuations, and (2) the monetary transmission mechanism. In our concluding comments at the end o f the chapter we will draw attention to some unanswered questions associated with the research agenda. These include the question o f co-ordination o f beliefs on a specific equilibrium and the way that an equilibrium is maintained. 2.1. Technical aspects o f linear models
In deterministic models o f dynamic economies, indeterminacy implies the existence o f many equilibrium paths for quantities and prices that can be indexed by specifying initial conditions for prices. In stationary stochastic contexts, the effect of initial conditions on the evolution o f the economic variables fade away. Nevertheless the indeterminacy of equilibrium in these environments allows the introduction o f exogenous shocks that are not based on fundamentals. Such shocks would be inconsistent with equilibrium if the rational expectations equilibrium were unique. However, this is no longer the case in the presence o f indeterminacy. As long as the sunspot shocks follow a stochastic process that is consistent with the expectations of agents, equilibrium conditions can be satisfied, and sunspots will affect the evolution of real economic variables. Since the stochastic process for sunspots can typically be chosen from a wide class, there are many possible stationary rational expectations equilibria. The particular equilibrium that prevails depends upon the beliefs that agents use to forecast future values o f prices and it is in this sense that sunspots "select" a stochastic equilibrium. We begin by discussing some technical aspects o f linear stochastic models in order to illustrate the content of indeterminacy for the applied econometrician. Our discussion centers on solution methods for linear models and illustrates the implications o f indeterminacy for the methods that are used to formulate, simulate and estimate these models 3. Later in the chapter we will discuss the class o f behavioral models that give rise to approximate linear models. These behavioral models are typically derived from an infinite-horizon maximizing problem solved by a representative agent, although there is no reason to maintain the representative agent assumption and similar linear models follow from a much larger class o f dynamic general equilibrium models 4.
3 For a more complete discussion of solntion methods in linear models with and without indeterminacies the reader is referred to Farmer (1993), King, Plosser and Rebelo (1987) or Blanchard and Kahn (1980). For an alternative, excellent, treatent of sunspots in a variety of models the reader is referred to the survey by Chiappori, Geoffard and Guesnerie (1992). 4 Kehoe and Levine (1985) show that infinite horizon models with a finite number of agents behave very much like the finite commodity model. The key distinction [originally pointed out by Shell (1971)] is between competitive models with a finite number of infinite lived agents, in which there is generically
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We start with the assumption that we have already solved for the equilibrium of a dynamic model and that the non-stochastic version of this model contains a balanced growth path. Linearizing around this balanced growth path leads to a system of equations of the form Yt = A y t 1 + B E t
(2.1)
[Yt+l] + C x t + ut,
(2.2)
Xt = D x t 1 + vt,
where y is a vector of endogenous variables, x is a vector of policy variables, u and v are conformable vectors of stochastic shocks, and A, B, C, and D are matrices of parameters that are found by taking first-order Taylor series approximations to the functions that describe a non-stochastic version of the model around its balanced growth path. These equations consist of market clearing conditions, Euler equations and static first-order conditions and a set of transversality conditions that impose boundedness conditions on the elements of Yr. We assume that policy is stationary, that is, the roots of D are all within the unit circle. To find the rational expectations solution to this model, one can rewrite Equations (2.1) and (2.2) as follows: 74zt = Bzt+l +
Ce,+l,
(2.3)
where
zt ~
Yt-1
,
=-
~4 ~
LXt_l
I 0
0
(2.4)
-B
et+l ~
vt Yt+I - E t
,
[Yt+l]
C
[i° 00J 0 -I
.
(2.5)
Premultiplying by j - 1 using the notation q~ = ~-1~, F ~ A 1~, leads to zt = ~ z t + l + Fet+l.
(2.6)
Generally, one can invert Equation (2.6) and write zt+l as a function ofzt, but since some of the roots of the matrix ~ lie outside the unit circle, this procedure does not typically allow one to construct the stochastic process for zt that constitutes the rational expectations equilibrium. The problem is that arbitrary solutions to Equation (2.6) fail
a finite odd number of equilibria, and those with an overlapping generations structure in which there is a double infinity of goods and agents.
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to remain bounded and they violate the transversality conditions o f one or more o f the agents in the underlying equilibrium model. In other words, the system (2.7)
Zt+l = C[9-lzt - (/)-l/'et+l,
is explosive. In determinate rational expectations models one can eliminate the effect of explosive roots o f q5 I by placing restrictions on zt. Suppose we partition the vector zt into two disjoint sets z~, of dimension nl, and z 2, o f dimension n2, where z) contains those variables that are predetermined at date t, and z 2 contains those variables that are free to be chosen by the equilibrium conditions of the model. Let ,~ be the roots o f qs, and partition ;t into two sets )t I , o f dimension ml, and )t 2, of dimension m2, where/~1 consists o f the roots o f q~ that are outside the unit circle and ~2 o f those that are inside the unit circle 5. The condition for a determinate solution is that one has exactly as many non-predetermined variables as non-explosive roots o f q~, in other words, n2 = m2. The solution to the determinate model is found by diagonalizing the matrix q~ and writing Equation (2.7) as a system o f scalar equations in the (possibly complex) variables ~t, where the ~ are linear combinations of the zt formed from the rows of the inverse matrix o f eigenvectors of q). The elements o f ~ associated with stable roots of q~ are set to zero; in the case when m2 n2 these elements provide exactly enough linear restrictions to exactly determine the behavior o f the zt. In the case when there are fewer stable roots of q5 than non-predetermined initial conditions there is the possibility o f multiple indeterminate equilibria. We illustrate the importance o f this issue for economics with two examples. =
2.2. Indeterminacy and propagation mechanisms in real models o f business cycles The idea behind models o f indeterminacy as a source o f propagation of shocks can be best understood within the context of the Cass-Koopmans stochastic growth model. The equilibrium o f this model can be described as the solution to a system of difference equations. I f we allow for a productivity shock that enters the model as a disturbance to the production function this system has three state variables, consumption, capital and productivity:
[ct ~st
= q) [ kt+, k ~t+l
+F
.
(2.8)
[ ut+l
The variables c and k are consumption and capital, and s is productivity. The tilde's denote deviations from the balanced growth path: fi is the innovation to the productivity shock and ~, defined as et+l = ~'t+l -- Et [Ct+l],
(2.9)
5 There is an important knife-edge special case in which one or more roots are exactly on the unit circle which we shall not explore in detail.
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is the one-step-ahead forecast error of consumption. In the linearized model, there are two other endogenous variables, output and labor, that are described as linear functions of the "state variables" c, k and s. Using the definitions from the previous section, k and s are predetermined and c is non-predetermined. When the model is derived from a maximizing model with a constant returns-to-scale technology one can show that the matrix q~ has two unstable roots (outside the unit circle) and one stable root (inside the unit circle). Since there is one non-predetermined variable, c, and one stable root of q~ one has a unique equilibrium; this equilibrium is found by eliminating the influence of the stable root of q~. Since the roots of q~ are the inverses of the roots of q~ I this procedure eliminates the influence of the unstable root of q~ l in the equation
kt+l L s~+i
= ~ 1 ~7t ~
-q0 1/7 LUt+l 1.] ,
(2.10)
by making ~ a function of k and 2. In the special case when there are no shocks to productivity, ~ is identically zero and the steady state of the bivariate system in ~ and is a saddle point. In this case the unique solution makes ~ a function of k that places the system on the stable branch of the saddle. In the stochastic case ~ depends not only on fc but also on ~. In the stochastic model, the expectational error Ot+l is a function only of the innovation to the productivity shock fit+l and there is thus no independent role for errors in beliefs to influence outcomes. This was the original point of the rational expectations revolution; it is possible to show that, if there is a unique rational expectations equilibrium, expectations must be a unique function of fundamentals. When there is a unique equilibrium, it can be described as the solution to a secondorder stochastic difference equation of the form
St+l with a set of side conditions,
E'I E1 It
~t
(2.12)
'
that determine the values of the other variables of the system. The matrices ~ and are functions of the elements of q~ and F found by replacing ~ with a function of and ~. ~ and 7 represent output and employment, and the elements of C are found from the linearized first-order conditions of the original equilibrium model. The main idea of the recent indeterminacy literature is that small departures from the assumptions of the real business cycle model lead to big departures from its
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implications. Farmer and Guo (1994, 1995) take a variant of the Cass Koopmans stochastic growth model, originally studied by Benhabib and Farmer (1994), in which the technology displays increasing returns 6. Their model has a representation of the same form as equation (2.8) but, in contrast to the standard version of this model, in the Farmer-Guo version all the roots of the matrix q~ lie outside of the unit circle. It follows that all roots of q~ 1 are inside the unit circle and hence Equation (2.10) describes a stationary process for an arbitrary series of iid shocks 0t. In the standard model the forecast errors 0t are functions of the fundamental errors fit. In contrast, in the increasing returns model, the forecast errors enter as independent shocks to the business cycle. There are two important implications of models of indeterminacy for business cycle dynamics. The first is that animal spirits can represent an independent impulse to the business cycle even in a model in which all agents have rational expectations. The second is that models with indeterminacy display much richer propagation dynamics than models with a unique determinate equilibrium. In the model with increasing returns, for example, the matrix q~ may generate dynamics that lead to hump shaped responses to shocks, wherever these shocks originate. In the standard model, the dynamics of consumption, capital and GDP are dictated mainly by the assumed form of the dynamics of the shocks 7. 2.3. Indeterminacy and propagation mechanisms in monetary models of business cycles We have described how models with indeterminate equilibria can potentially be used to understand business cycles. A second area in which indeterminacy may prove important is in models of the monetary transmission mechanism. Once again, it is the propagation mechanism inherent in models with indeterminacy that sets these models apart. Early papers on the issue of indeterminacy and monetary propagation were set in the context of the two-period overlapping generations model; papers in this literature include Geanakoplos and Polemarchakis (1986), Azariadis and Cooper (1985), Farmer and Woodford (1997), Farmer (1991, 1992) and Chiappori and Guesnerie (1994). Later work has switched focus and more recently there has been an attempt to include money in infinite-horizon models of money. Papers that exploit the existence of indeterminate monetary equilibria in an infinite-horizon framework include Woodford (1986, 1988), Beaudry and Devereux (1993), Bennett (1997), Lee (1993), Matheny (1992, 1998), Matsuyama (1991b), Benhabib and Farmer (1996b), and Farmer (1997). One of the key ideas in this literature is that indeterminacy can be used to understand the monetary transmission mechanism. 6 The model used by Farmer and Guo is the one explored by Benhabib and Farmer (1994), although similar results would follow from the models of Gali (1994) or Rotemberg and Woodford (1992). 7 See, for example, the paper by Cogley and Nason (1995) who point out the discrepancies between the dynamic predictions of the RBC model and the much richer dynamics apparent in US data.
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The equilibria of a simple monetary economy can often be characterized by a fimctional equation of the form m t = E t [ G (mr+l,/~t+l, ut)] ,
(2.13)
where mt represents the real value of monetary balances,/~t+l is the ratio of the money supply at date t + 1 to the money supply at date t, ut is a fundamental shock with known probability distribution, E t is the expectations operator and G is a known function. There exist examples of more complicated monetary rational expectations models that include additional state variables and allow for endogenous capital formation. These more c o m p l i c a t e d models rely on the same key insight as simpler models with a single state variable: in models with indeterminacy, prices may be predetermined one period in advance and yet all markets may clear at all dates and agents may have rational expectations. Assuming that Equation (2.13) has a steady state we may linearize around it to generate an equation that must be (approximately) satisfied by any sequence of real balances in a rational expectations equilibrium:
Fnt = aEt [~,+~] +/3#~+~ + yu,.
(2.14)
The variables ~ and/t represent deviations from the non-stochastic steady state. In a standard monetary model the parameter a is between zero and one in absolute value. In standard models one solves equation (2.14) by iterating forwards to find the current value of real balances as a function of the rule governing the evolution of/~t. In models with indeterminacy on the other hand, a may be greater than one and in this case there exist many solutions to this model of the form mt+l = - m r a
~t+] -
ut + et+t,
(2.15)
where et÷l is an arbitrary iid sunspot sequence. Farmer and Woodford (1997) showed that one of these equilibria has the property that the price at date t + 1 is known at date t; in this sense prices are "sticky" even though there is no artificial barrier to prevent them from adjusting each period. The existence of a predetermined price equilibrium is significant because it offers the possibility of using equilibrium theory to understand one of the most difficult puzzles in monetary economics; the characteristics of the monetary transmission mechanism. In his classic essay "Of Money", published in the eighteenth century, David Hume described the empirical facts that at that time were known to characterize the aftermath of what today we would call an "unanticipated monetary injection". Following an addition of money to an economy we typically observe (1) a short-run increase in real economic activity, (2) a fall in short rates of interest, and (3) an increase in the real value of money balances. In the short run, the price level does not respond. Over a longer period of time the interest rate increases to its initial level, real economic
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activity contracts and the full impact o f the monetary increase is felt in prices. David Hume's observations were based on the effect of the importation o f gold to Europe in the aftermath o f the discovery o f the "New World" but his observations have proven remarkably consistent with m o d e m econometric evidence based on the analysis o f vector autoregressions that allow us to construct estimates o f impulse response functions 8. The sequence o f events described in the previous paragraph constitutes a description o f what we believe to be a consensus view o f the facts concerning the "monetary transmission mechanism". But although these facts have been known for two hundred years we still have to reach a consensus theory that can account for them. The current leading contenders as explanations o f the monetary transmission mechanism are some version o f the "menu costs" model due to Akerlof and Yellen (1985) and Mankiw (1985), the contract approach o f Taylor (1980), the staggered price setting model of Calvo (1983) or the closely related cost-of-adjustment models o f Rotemberg (1982, 1996). Each o f these approaches has its merits. However, using standard approaches, it is often difficult to generate impulse response fimctions that resemble the data. Chari, Kehoe and McGrattan (1996) point out some problems of the staggered price setting approach in capturing monetary dynamics. The cost-of-adjustment model of Rotemberg (1996) with quadratic adjustment costs does a better job empirically, and new research in this area is likely to achieve further improvements in explaining propagation dynamics. In part this progress may arise from recognition that monetary models, with or without menu costs, staggered price setting, or informational problems that rationalize labor contracts, can contain a continuum o f rational expectations equilibria (see for example Section 5.4 below). By allowing the data to pick an equilibrium in which prices are slow to respond to new information, one may make considerable progress in reconciling equilibrium theory with the facts. This is the main message o f the recent literature that we will review in Sections 4 and 5.2. As with the work on indeterminacy and business cycles, the main criticisms of monetary models with indeterminacy have been leveled at the plausibility o f the mechanisms that cause indeterminacy to arise. For example, one can show that for certain classes o f fiscal policy, monetary overlapping generations models possess two steady states; one is determinate and one is indeterminate. Since these example are difficult to match with time series data they are open to the criticism that they have little relevance to real world economies. This criticism led to a set of papers based on the infinite-horizon model in which money enters because o f a cash-in-advance constraint, or as a result o f money in the utility function or in the production function. Models in this class have proved harder to dismiss than two-period overlapping
8 There is a huge array of work that studies the empirical characteristics of monetary impulse response functions in the USA [e.g. Sims (1980, 1989)]. One of the constant features of this data is the fact that the price level is slow to respond to purely nominal disturbances; that is, prices are "sticky". Nor does the characteristic of a slow price response seem to be peculiar to the USA as demonstrated by Sims (1992) who compares the USA, the UK, France, Germany and Japan.
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generations models although they leave open a number of issues to which we will return in Section 4. The main unanswered issues are: (1) how do agents co-ordinate on a particular equilibrium, (2) are models with indeterminacy possible for plausible parameter values and (3) what are the implications of these models for a theory of optimal monetary policy.
3. Indeterminacy in real models In Section 3 we turn our attention the specific mechanisms that give rise to indeterminacy. Our goal is to provide a common framework to assess the mechanisms that have been discussed in the recent literature. In the class of models that we will focus on, the source of indeterminacy may be viewed as arising from a coordination problem and in this sense our survey provides an extension of the survey of static coordination problems, by Cooper and John (1988), to models in which there is an essential dynamic element to the equilibrium concept. We begin by providing a very simple example that contains the essential elements of the models discussed in recent literature, and in the remaining parts of Section 3 we elaborate on the elements that are specific to each of a number of variants of this basic mechanism. Consider a specific equilibrium path for prices and rates of return; we are going to illustrate how, beginning from one particular equilibrium path, it may be possible to construct another. Suppose that agents collectively change their expectations and they come to believe that the rate of return on an asset will increase. As a consequence of this belief, they begin to accumulate this asset at a faster rate and its price increases. Suppose the return on the asset indeed tends to increase with higher stocks, maybe because of the presence of increasing returns, or a mechanism that mimics increasing returns. Since the overall rate of return on assets must remain equal to an intertemporal rate of discount, maintenance of the new belief as an equilibrium path requires an expected depreciation in the price of the asset, or a capital loss to offset the initial increase in its rate of return. If this price decline is sufficient to contain the explosive accumulation of the asset, then the resulting new path is also an equilibrium. We can now repeat this argument starting with the new equilibrium path, and construct yet another equilibrium. Since there are infinitely many such paths, the original equilibrium is indeterminate. Some of the models that we will discuss work through a mechanism in which increasing returns to scale is an essential part of the argument. However, as a number of the models described in this survey will demonstrate, indeterminacy does not necessarily require increasing returns to scale. 3.1. A framework for comparing different models Recent interest in models with increasing returns was inspired by the literature on endogenous growth initiated by Lucas (1988) and Romer (1990). Early work by these authors showed how to make increasing returns consistent with an equilibrium growth
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model by building externalities or monopolistic competition into an otherwise standard dynamic general equilibrium model. The work of Hall (1988, 1990) and of Caballero and Lyons (1992) provided a further impetus to the increasing returns agenda by suggesting that externalities might be important not only in generating growth but also as a business cycle propagation mechanism. Their work suggested that the degree of increasing returns, exhibited via external effects or markups, was significant in many sectors of the economy. Subsequently a number of authors, notably Basu and Fernald (1995, 1997), Burnside, Eichenbaum and Rebelo (1995), and Burnside (1996), have scaled down the early estimates of Hall, bringing them closer to constant returns, and in certain cases even finding decreasing returns in some industries. Earlier theoretical models of indeterminacy, for example the model of Benhabib and Farmer (1994), had relied on large increasing returns in line with Hall's original estimates. Subsequent theoretical work however has substantially reduced the degree of increasing returns needed to generate indeterminacy [as in Benhabib and Farmer (1996a)], and it now has become clear that in an economy with some small market imperfections, even a technology with constant marginal returns can generate indeterminacy 9. To illustrate the critical elements that generate indeterminacy, we begin with a simple equilibrium structure that can, with slight modifications, accommodate a number of different models. The standard model with infinitely-lived identical agents maximizing the discounted sum of instantaneous utilities given by U(c)+ V(1 - L ) , where c is consumption and L is labor, gives rise to the following system of equations:
U'(c) =p,
(3.1)
U (c)wo(k,L; k,L) = V'(1 - L ) ,
(3.2)
k = y(k, L(p, k); L L(p, k)) - gk - c,
(3.3)
r= p-+ (w~(k,p;LL(p,k))-g)
(3.4)
I
--
-
P
Here w0 is the marginal product of labor, at this point taken to be equal to the real wage, and w~ is the rental rate on capital, equal to the marginal product of capital: the marginal products are with respect to private inputs, keeping k and L fixed. The depreciation rate is g, the discount rate is r, the shadow price of capital is p, and the production function is given by y. In Equations (3.3) and (3.4), it is assumed that L(k,p) has been obtained by solving Equations (3.1) and (3.2). Equation (3.2) represents the labor market equilibrium, Equation (3.3) equates net investment to capital accumulation, and Equation (3.4) is the standard "Euler" equation requiring
9 A separate and related branch of the literature that we do not have space to cover in this survey has shown that search externalities also give rise to indeterminacy. Examples of papers that illustrate this possibility are those by Howitt and MeAfee (1988) and Boldrin, Kiyotaki and Wright (1993). See also Matsuyama (199la) for indeterminacy in a model of industrialization.
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the equality of the return on the asset (its net marginal product plus its shadow price appreciation), to the rate of discount. The only non-standard feature of the model above is the inclusion of external effects, generated by the aggregate inputs k and L in the production function. Of course some deviation from the standard framework in the form of a market imperfection must be introduced to obtain indeterminacy, since the standard representative agent model has locally unique equilibria. 3.2. The one-sector m o d e l with increasing returns
We start by investigating the one-sector model of Benhabib and Farmer (1994), which demonstrates how indeterminacy can arise in a representative agent model with increasing returns 10. Increasing returns in this model is reconciled with private optimization by introducing either external effects in production, or a monopolistically competitive market structure where firms face downward sloping demand curves. Benhabib and Farmer show that the simple one-sector model with external effects is identical, in the sense of giving rise to the same reduced form, to a model with monopolistic competition and constant markups. To see how indeterminacy comes about in a model with external effects and increasing returns, consider starting from an equilibrium path, but let the agents believe that there is an equilibrium in which the shadow price of investment p is higher than its current value, and that future returns justify a higher level of investment. If agents act on this belief, the higher current price of investment reduces consumption and induces agents to divert GDP from consumption to investment. If there were no externalities, investment would increase, and the marginal product of capital would begin to decline as additional capital is accumulated. This decline would have to be offset by capital gains in the form of increases in the shadow price of capital in order to validate the belief of agents that higher rates of investment will yield the appropriate return. Trajectories of this sort for investment and prices may be maintained for a number of periods, but the resulting over-accumulation of capital and the exploding prices will violate transversality conditions: an agent will never get to consume enough in the future to justify the sacrifice of a higher rate of investment. In other words, an agent conjecturing such a path of prices and returns would be better off consuming, rather than accumulating additional capital at such a pace. Consider now an alternative parametrization of this model in which externalities are sufficiently big to permit indeterminate equilibria. Once again, let agents conjecture that there exists an alternative equilibrium path, starting with a price of investment p, which is higher than the one in the current steady-state equilibrium. The higher price will cause agents to divert GDP from consumption to investment but, if externalities are strong enough they will simultaneously increase their consumption of leisure. The
10 See also Benhabib and Rustichini (1994), Boldrin and Rustichini (1994) and Chamley (1993).
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increase in leisure will cause GDP to decline, and investment will eventually fall as well. Benhabib and Farmer show that this dynamic argument has a representation in terms of labor demand and supply curves, with strong increasing returns to the labor input, and where the labor demand curve slopes up more steeply than the labor supply curve. In this framework shifts of the curves generated by conjectured changes in the shadow price of capital can lead to the contractionary employment effects mentioned above. As the marginal product of capital falls with the decline of labor, the shadow price of investment must appreciate to produce a capital gain, because in equilibrium the overall return on capital must be equal to the rate of discount. This reinforces the original impulse of a higher relative price for the capital good. The contraction of labor however causes GDP and investment to decline, and the capital stock starts to fall. The decline in the stock of capital reverses the process, because it shifts the labor demand curve down. Since the labor demand curve slopes up more steeply than the labor supply curve, a downward shift in labor demand tends to increase employment. This is the critical element that gives rise to indeterminacy in the model. Higher employment and the low level of the capital stock both cause the marginal product of capital to increase, and intertemporal equilibrium now requires a depreciation, rather than appreciation, in the price of capital to equate the overall return to the discount rate. As the price of capital falls, the economy returns to its original steady state along this new equilibrium path. The key to indeterminacy in this model then lies in the non-standard slopes of the labor demand and supply curves, which induce a perverse labor market response. This feature, which requires a high enough externality to induce increasing returns in the labor input by itself, is what makes the model empirically implausible. 3.3. The two-sector model with increasing returns
A more satisfactory specification leading to indeterminacy is given by Benhabib and Farmer (1996a). They start with a two-sector model, but with identical production functions in the consumption and investment sectors. One might think that by making the production functions of the two sectors identical, the model would collapse to a one-sector economy. However, the two-sector structure is preserved by the distinct external effects in each sector, each arising from their own sectoral outputs, rather than from the aggregate output. The model yields a linear production possibilities surface (ppf) from the private perspective, but one that is convex (to the origin) from the social perspective. To develop a parallel exercise conducted for the one-sector case above, consider again starting at a steady state and increasing the (shadow) price of capital p. Since we have a two-sector model, Equation (3.1) has to be modified to reflect the relative price of capital in terms of the consumption good. If we denote this price by q, we get qU' (c) =p.
(3.5)
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A higher q now raises consumption since, given the convexity of the ppf, an increase in consumption relative to investment will be associated with an increase in the relative price of capital. The impact effect on p is ambiguous because Ul(c) declines as well, but if the curvature of U(c) is not too severe, p and q will change in the same direction. Benhabib and Farmer (1996a) use logarithmic utility of consumption in their model. Nevertheless, the reader may want to think of U~(c) as constant in order to get a clearer picture of the logic of the argument. When consumption increases, the supply curve of labor shifts to the left, and since the demand curve for labor slopes down in the two-sector model, the result is a contraction in labor, and also in investment. As in the previous case, the decline in labor decreases the marginal product of capital. Maintaining intertemporal equilibrium now requires an appreciation in the (shadow) price of capital to keep the overall return equal to the discount rate, reinforcing the initial rise in q, and therefore in p. However, the decline of investment and of the capital stock must eventually raise the marginal product of capital, and reverse the appreciation of p: intertemporal equilibrium eventually requires a depreciation of the shadow price of capital. The process therefore is reversed, and the economy moves back towards the steady state, giving rise to an alternative equilibrium trajectory. It is clear in this case that the elements responsible for the "stability" of the steady state, and therefore for "indeterminacy," are no longer the perverse slopes of demand and supply in the labor market, but the convexity of the ppf. Benhabib and Farmer (1996a) find that even a small output externality, resulting in increasing returns as low as 1.07, coupled with standard values for the other parameters, is sufficient to generate indeterminacy. Furthermore their calibration analysis incorporating iid sunspot shocks does quite well by the standards of the recent real business cycle analysis. In spite of an apparent improvement over the one-sector model, one may question the implication of a convex social ppf which implies that the sectoral aggregate supply curves slope down. This issue has been empirically investigated by Shea (1993). He studies 26 manufacturing industries and finds that in 16 of them, the supply curves slope up. The question therefore arises as to whether indeterminacy requires increasing returns in all sectors. Recent results of Harrison (1996) on the two-sector model show that indeterminacy obtains for roughly the same parametrizations as in Benhabib and Farmer (1996a), with the exception that only the investment sector is assumed to have increasing returns. Furthermore, her estimates of increasing returns and externalities, obtained by allowing them to vary across sectors, indicate that they may be sufficiently high in the investment sectors to support indeterminate equilibria 11 Perli (1994) obtains similar results by introducing a home production sector as in Benhabib, Rogerson and Wright (1991), with sector-specific external effects only in the
11 Basu and Fernald (1997) do indeed find that there is heterogeneity of returns to scale across industries.
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market sector, and he obtains indeterminacy with small increasing returns. Furthermore the times series generated by his calibrated model are comparable to US postwar series. In a recent paper Weder (1996) also shows that in a two-sector model indeterminacy can arise with mild increasing returns in the investment sector alone. His specification relies on imperfectly competitive product markets rather than external effects. 3.4. The two-sector model with constant marginal returns
So far it would seem that some increasing returns may be necessary to generate indeterminacy. In a recent paper Benhabib and Nishimura (1998) demonstrate that social constant returns to scale, coupled with some small external effects that imply very mild decreasing returns at the private level, can also generate indeterminacy. In their framework there are private decreasing returns to scale and small external effects that give rise to constant returns to scale at the level o f the aggregate economy. In the two models we discussed in Sections 3.2 and 3.3, a rise in the shadow price of capital eventually induces the capital stock to fall and its marginal product to increase. Consequently, intertemporal equilibrium requires the price o f capital to fall, reversing its original increase. This mechanism can be duplicated in a two-sector model without upward sloping labor demand curves or a convex ppf. The reason is that in models with more than one sector, the marginal product of capital depends not only on factor inputs, but also on the composition o f output and on the relative factor intensities o f the underlying technology. We can express the technology of the two-sector economy in per capita variables as a transformation surface given by c = T ( y , k ) . Such a technology implies that T1 = - p and T2 - wl, where the subscripts o f T indicate derivatives with respect to the appropriate argument. T1 then is the slope o f the ppf, while T2 corresponds to the marginal product o f capital in the production o f the capital good 12. Consider first a simple two-sector model without externalities, and production functions that differ across the consumption and investment sectors. For simplicity also assume that total labor is fixed, and that the utility function is linear in consumption, so that in terms of Equation (3.5) we have q = p. I f the production possibility frontier is strictly concave we can invert the relation T1 (y, k) = - p to obtain the output of the investment good as y = y ( k , p ) . Equation (3.3) must now be modified as follows: (3.6)
k = y ( k , p ) - gk.
Equations (3.4) and (3.6) now fully describe the dynamics of the system in (k,p). With external effects suppressed and a fixed labor supply, the local dynamics around the steady state will depend on the Jacobian matrix J : j=
N-g 0
N -~+(r+g)
"
12 That Ta - w is not immediate, but follows from efficiency conditions and envelope theorems.
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Note that the lower left submatrix, ./21 - [-Owl~Ok], is identically zero. This is because under constant (social) returns to scale relative factor prices uniquely determine prices as well as input coefficients. To demonstrate that equilibrium is determinate we must show that the roots of J have opposite sign; but, since J21 is zero, the roots of J are equal to the elements on the main diagonal, Jll and J22. Determinacy is the assertion that these elements have opposite sign. Benhabib and Nishimura demonstrate that the signs of J~l and J22 are related to two familiar theorems in the international trade literature, the Stolper Samuelson theorem and the Rybczinski theorem. We deal here only with the case in which the investment good is labor intensive although the argument that we will present can be easily extended to the case when the investment good is capital intensive, by reversing the signs of the two inequalities that we will present. Consider first the Stolper-Samuelson theorem which asserts (in the labor intensive case) that a rise in p will decrease the rental price of capital, ~ol. In symbols this asserts that
?;;l Since, at the steady state, o)1 = p ( r + g), the Stolper-Samuelson theorem implies that the element J22 is positive. Now consider the Rybczinski theorem which asserts that if more capital is used in the investment goods sector, output of investment goods will rise less than proportionately. In symbols this is represented by the inequality
Since, at the steady state, y = gk, the Rybczinski theorem implies that the element J11 is negative. It follows,.for the case when J21 iN zero, that the Stolper-Samuelson and Rybczinski theorems can be used to establish that equilibrium is determinate. More generally, in multi-sector models J11 and J22 will be matrices rather than scalars, and with linear utility J2i will still be a zero matrix, so that the roots of J will be given by the roots of Jll and J22. However, as shown in Benhabib and Nishimura (1998), the duality between the Stolper-Samuelson theorem and the Rybczinski theorem will imply that at least half of the roots of J will be positive, implying that the equilibrium is determinate 13 What happens when there are external effects? Benhabib and Nishimura (1998) establish that in this case one can break the reciprocal relation between the Rybczinski and Stolper-Samuelson theorems. Output effects on investment are still governed by the logic of the Rybczinski theorem, but the Stolper-Samuelson theorem requires that
13 In some cases, more than half of the roots of J can have positive real parts. This situation however is associated with the presence of optimal cycles and not indeterminacy. See Benhabib and Nishimura (1979).
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costs of production equal prices. With external effects, markets are distorted and the relevant factor intensities can be reversed. This reversal may break the saddle-point property (even when external effects are 'small') and generate a situation in which more than half of the roots of J are negative and there are multiple indeterminate equilibria. Benhabib and Nishimura (1998) calibrate and simulate a discrete-time version of their model, with one consumption and two investment goods, which incorporates iid sunspot shocks and logarithmic utility. Their calibration can match the moments of GDP, consumption, aggregate investment and hours in US data as well as any other standard RBC model, and can generate impulse responses to technology shocks that resemble the hump-shaped impulse responses generated with vectorautoregressions on US data. (See Section 8.1.2 below.) 3.5. Fixed costs and the role o f profits
The results sketched above suggest that introducing even small market imperfections into the standard infinitely-lived representative agent model can produce empirically plausible indeterminacy even under constant social returns. Increasing returns to scale at the level of the aggregate social production function are not necessary for indeterminacy. In order to generate this result, however, there must be decreasing returns at the level of private firms and an implication of this is that firms will earn positive profits. In the parametrized examples given by Benhabib and Nishimura (1998) these profits are quite small because the size of external effects, and therefore the degree of decreasing returns needed for indeterminacy are minimal. Nevertheless positive profits would invite entry, and unless the number of firms is fixed, a fixed cost of entry must be assumed to determine the number of firms along the equilibrium path. Such a market structure would exhibit increasing private marginal costs but constant social marginal costs, which is in line with current empirical work on this subject. It seems therefore that models of indeterminacy based on market imperfections which drive a wedge between private and social returns must have some form of increasing returns, no matter how small, either in variable costs, or through a type of fixed cost that prevents entry in the face of positive profits 14. The point is that while some small wedge between private and social returns is necessary for indeterminacy, this in no way requires decreasing marginal costs, or increasing marginal returns in production. 3.6. Models with variable markups
The use of variety in intermediate or consumption goods, coupled with a monopolistically competitive market structure, has been incorporated by a number of
14 For a more detailed statement of this argument see the papers by Gali (1994) and Gali and Zilibotti (1995).
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authors into the standard optimal growth model. Woodford (1987) first demonstrated that a monopolistically competitive market structure with flee entry, coupled with variable markups, can lead to indeterminacy and to self-fulfilling sunspot equilibria. He constructs a model in which aggregate output, investment, as well as employment, are driven by expectations of aggregate demand. Furthermore, kinks in the demand curves faced by firms give rise to variable markups, allowing the quantity of labor supplied to adjust to the quantity demanded through changes of aggregate demand in the goods market. As noted by Woodford, this labor market structure with variable markups relaxes the rigid relation between wages and the marginal product of labor that is a feature of competitive market models, and avoids the implication that wages must move countercyclically in the absence of technology shocks 15. Two related approaches, using variable markups, are those of Gali (1994, 1996) and Rotemberg and Woodford (1992). Gali develops a monopolistically competitive market structure in which firms markup price over marginal cost. He shows that changes in the composition of aggregate demand between investment and consumption may cause the markup to vary systematically over the business cycle and he uses the countercyclical markup to demonstrate the possibility that there may be many indeterminate equilibria. Rotemberg and Woodford develop a model of the business cycles based on implicit collusion among firms that strategically vary markups depending on the state of aggregate demand. Their model also gives rise to indeterminacy, and although the exact mechanism is somewhat different from that in Gali or Benhabib and Farmer, the implications for business cycle data are similar as demonstrated in the recent paper by Schmitt-Groh6 (1997). We will return to this paper in Section 8.1.1 in which we discuss empirical aspects of indeterminacy. We can illustrate the mechanism that gives rise to indeterminacy in the models of Gali, and of Rotemberg and Woodford, in the context of our simplified model consisting of Equations (3.1), (3.3) and (3.4). For simplicity we assume that labor is fixed, although variable labor would allow a smaller and less variable markup to generate indeterminacy. First we note that the return on (or user cost of) capital in Equation (3.4) will have to be modified: it is not the marginal product but the marginal product divided by the markup:
r = ) + ( w'(k'P' ~ Z(P'k)) - g)
(3.7)
Here, we simply assume that the markup # is related to p, although in the works cited above this dependence of the variability of the markup is derived from more basic structural assumptions. The influence of a variable markup works in the following way. Starting at a steady-state equilibrium, an increase in p raises the share of investment
15 This issue is also related to the comovementsof consumption and output, and is discussed further below in Section 8.1.2.
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and leads to faster capital accumulation. The higher investment share also lowers the markup and raises wl/t~, even though wl declines. With r fixed, ~b/p then must decline, reversing the process and driving the economy back towards the steady state. This then is another equilibrium, implying that the initial value of p, and therefore the equilibrium trajectory, is indeterminate. The same process would work if the markup depended directly on the capital stock, so that wl/l~ increased with a rise in the capital stock. Gali (1994, 1996) produces variable markups by introducing different demand elasticities for investment and consumption goods. The average markup depends on the relative shares of consumption and investment in aggregate output. Gali assumes the elasticities are such that the markup is negatively related to investment share, and he presents evidence from US data to support his contention that this assumption is a reasonable first approximation to the facts. In the model of Rotemberg and Woodford (1992) the variability of/~ results from a collusion arrangement between firms that share the market and earn a stream of profits in an implicit collusion arrangement. The equilibrium conditions imply that the markup /~ depends on the ratio of the values of the firms to aggregate output. Implicit collusion among firms requires countercyclical markups to maintain discipline, and to prevent deviation from the collusive arrangement. One of the important implications of models with variable markups, as shown in the paper by Schmitt-Groh~ (1997), is that indeterminacy in these models can occur with a lower degree of increasing returns to scale than in the case of constant markups. We return to this idea in Section 8.1.1.
4. Indeterminacy in monetary models In this section we discuss indeterminacy in models that include the real value of money balances as an argument of the utility function or in the production function, following the work of Patinkin (1956). Brock (1974) is the first to formally discuss the possibility of indeterminacy associated with self-fulfilling hyperinflationary and deflationary equilibria, as well as conditions that rule out such equilibria, in a model of an infinitely lived representative agent that has real balances in the utility function. Since Brock's paper, much of the literature on indeterminacy in monetary models has focused on self-fulfilling inflations and deflations, rather than the possibility of a continuum of equilibria converging to a steady state, possibly because the former can occur under relatively weak conditions. Hyperinflationary equilibria can be ruled out if it is assumed that it is prohibitively costly for the economy to operate at low levels of real balances, or close to a barter economy. For certain classes of monetary policies, speculative deflations can also be ruled out by restrictions on preferences. Obstfeld and Rogoff (1983) provide conditions under which such hyperinflationary and deflationary equilibria can be ruled out. An extensive overview of indeterminacy in a cash-inadvance or a cash good-credit good model is given by Woodford (1994). He analyzes the conditions for the existence of hyperinflationary and deflationary equilibria as well
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as equilibria described by a continuum o f paths that converge to a steady state and which can give rise to stationary sunspot equilibria 16 One o f the first to note the possibility o f indeterminacy in the form o f a continuum o f equilibria converging to a steady state was Calvo (1979), and there have been many related papers since. Wilson (1979) gives some o f the early examples o f indeterminate equilibria in the context o f a cash-in-advance model. Taylor (1977) has one o f the first discussions o f indeterminacy in a monetary model where he proposes a selection principle that picks the equilibrium solution exhibiting the m i n i m u m variance. McCallum (1983) provides a survey o f early literature on monetary indeterminacy and also proposes a 'minimal state variable solution' to select an equilibrium. Rather than try to survey additions to this vast body o f work we concentrate instead on recent papers that have exploited indeterminacy to address and study empirical features o f the business cycle. Our point o f departure then is the recent interest in calibrated versions o f indeterminate models and in particular the idea that indeterminacy can be a feature that allows us additional freedom to explain properties o f economic fluctuations that are otherwise difficult to understand.
4.1. M o n e t a r y models with one state variable and f i x e d labor supply
We begin our discussion with a class o f models in which money plays the role o f facilitating exchange and in which all other aspects o f the model are stripped down to a bare minimum. These are models in which real money balances is the sole state variable. By focusing on the simplest possible monetary model we will highlight an idea that holds in more general examples; indeterminacy is most easily obtained in monetary models when changes in the stock o f real balances have large effects on output. These effects can come from including m o n e y in the utility function as in Calvo (1979), money in the production function as in Benhabib and Farmer (1996b), or from a cash-in-advance constraint as in the calibrated monetary models o f Cooley and Hansen (1989, 1991). To relate monetary models to our discussion o f indeterminacy in real economies, suppose that output depends on real balances: y = y ( m ) , with y increasing and strictly concave 17. We assume that money is injected into the economy in equal lump-sum
16 See also the symposium on "Determinacy of equlibrittm under alternative policy regimes" in Economic Theory, volume 4, no. 3, 1994. There is also an extensive emipircal literature initiated by
Flood and Garber (1980) that tests for the existence of speculative bubbles. 17 One way to derive a model of money in the production function is by assuming that finns face transaction costs, T(q, m), where q is gross output and m is real balances. In this case net output is given by y(m) = q - T(q, m). If one specifies T as a Leontief function then the model reduces to a cash in advance specification. One could also replace q with consumption c in T to derive a variant of a model with money in the utility function.
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transfers to all agents, and that nominal balances g r o w at the rate o. T h e n the net rate o f return on h o l d i n g m o n e y is
y'(m)- or=y'(m)- (o-
~) ,
w h e r e ~ is inflation, and or _= ( o - / n / m ) in equilibrium. I f we a s s u m e that m o n e y is the only asset, then b = J ( m ) / n . N o w using equations (3.1) and (3.4), we obtain /n
+
u"(c) J
=
+ o-
y'(m))
.
(4.1)
I f we define the elasticities
eL.-
U"(c) c U'(c) '
era-
y'(m) m y(m) '
(4.2)
then E q u a t i o n (4.1) b e c o m e s /n =
m (r + o -y'(m))
1 -ecem
(4.3)
E q u a t i o n (4.3) is a differential equation that must be obeyed by paths for real balances that are consistent w i t h rational expectations equilibrium. The m o d e l has an e q u i l i b r i u m in w h i c h real balances are constant and equal to m*. This steady-state e q u i l i b r i u m is defined as the solution to the equation:
r + o = y'(m*). Since equilibria can be described as functions o f a single state variable, there will exist a set o f indeterminate equilibria i f the steady state in equation (4.3) is locally stable: this requires the right-hand side o f the equation to be decreasing in m at m* 18. Since m (r + ~ r - J ( m ) ) is increasing in m at m*, a sufficient condition for i n d e t e r m i n a c y is that m o n e y is sufficiently productive, or alternatively put, that em is large e n o u g h so that emec > 1 19 The intuition for this result is straightforward. W h e n m o n e y is not sufficiently productive, an increase in n o m i n a l balances or alternatively an initial low price level
18 It is easy to show that over the range of m for which J(m) > 0, higher values of initial m lead to higher welfare because m and output will remain higher at all points in time. 19 Note that if the optimal quantity of money rule is implemented so that a - -r, at the steady state we will have j(m) = 0 and em = 0. Once a level of m is attained at which J(m) - O, it is natural to think that this will continue to hold for higher levels of m because even if money balances cannot cut transaction costs any further, they cannot hurt either. In such a case we will have a continuum of steady state equilibria with real balances satisfying y(m) = 0, as discussed in Woodford (1987).
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that corresponds to real balances higher than m*, generates an excess supply o f money that spills onto the goods market: prices have to jump so that output, y(m*), and real balances, m*, remain fixed at their steady-state values. If prices did not jump to restore the steady-state values, real balances would rise, the return on money y1(m) would decline, and the higher money balances would be held only if some further deflation were expected. If such deflationary expectations were confirmed, real balances would eventually explode, and unless certain assumptions are made to rule out such hyperinflationary equilibria, we would still have indeterminacy, even though the steady-state level of real balances is unstable. On the other hand if money were sufficiently productive, an increase in real balances at m* would increase output and create a net excess d e m a n d for money, rather than an excess supply. More formally, the higher real balances would raise consumption and reduce the marginal utility o f current consumption so much that the agents would want to hold higher levels o f the monetary asset 2°. Therefore an increase in nominal balances, or a low initial price that places real balances above m*, would generate a net excess demand for money which would have to be offset by expected inflation in order to restore equilibrium. Inflation would then drive real balances back down to their steady-state value without requiring an instantaneous jump in prices. Thus the steady state is stable, with a continuum o f trajectories o f real balances converging to it. 4.2. M o n e y in the utility function and the production function The model that we discussed in Section 4.1 included money as an argument in the production function. In this section we will show how models of this kind can be extended to include money as an argument of production and utility functions. This modification will be useful in our discussion o f policy rules in Section 5.3 where money in the utility function allows us to demonstrate that indeterminacy can arise in broader range of cases than in simple production function model o f Section 4.1. We model the utility function with the specification U(c, m) where U is increasing and concave in c and m. When money enters both the production and utility functions, Equation (4.3) becomes m (r+ cr-y'(m)-
U,,,(c,m)~ Uc(c,m) I ,
In =
(4.4)
1 - ecem - F,cm where U(c, m) = U ( y ( m ) , m) and the cross partial term ecru -
- m Ucm
Uc
(4.5)
2(1 Note here that this effect is the result not only of the high marginal productivity of money, reflected in era, but also of the rate at which marginal utility declines, reflected in e,,. This is clear from the condition of indeterminacy which requires the product emec to be large.
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plays an important role in the way the model behaves. This term measures the effect of holding extra real balances on the marginal utility of consumption. The term Um/U~., given by the expression
is also important in determining whether equilibrium is determinate since if Um/U~. is decreasing in m the steady state will be locally stable, and therefore indeterminate whenever 1 - e~e~ - ec,n < 0. It seems reasonable to assume that neither consumption nor money are inferior goods,
(Vmm
-go
cmj < 0.
But this is not enough to determine whether Um/U~. is increasing or decreasing in m. It might seem that the discussion in this section is of little empirical relevance. Perhaps utility functions that allow for peculiar cross partials can be easily ruled out by data and we should restrict attention to logarithmic functions or at least to utility functions that are separable in consumption and real balances. Unfortunately, this argument negates the very reason for including money in the utility fimction in the first place since it is precisely changes in the marginal utility of transacting that one might expect to characterize a monetary economy. In Section 4.4 we discuss a calibrated model used by Farmer (1997) in which he demonstrates that models with indeterminacy can be used to mimic impulse response functions in the data in addition to capturing the more salient features of velocity and the rate of interest in the US data. Farmer includes money in the utility function and he chooses in a sum of weighted CES utility functions that allows the term d(Um/Uc)/dm to have either sign. It is precisely this flexibility that allows the model to capture the empirical features that we will describe in Section 4.4. 4.3. Monetary models with one state variable and a variable labor supply
In this section we will show how the extension of monetary models to allow for a second factor of production, labor, can increase the ability of these models to describe data by generating indeterminacy for a more plausible range of the parameter space. Models of this kind still have a single state variable since one can show that, in equilibrium, hours worked are a function of real balances. As in the case of models with a fixed supply of labor, indeterminacy is most likely to occur when money has a big effect on output. There is a growing literature using cash-in-advance or money in the utility function approaches to calibrated models of money that finds a unique rational expectations equilibrium; examples include Cooley and Hansen (1989, 1991) and related literature.
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One reason why calibrated models of money may appear to have a unique determinate equilibrium is that these models often use simple functional forms that allow for a single parameter to capture the magnitude of the importance of money. Recent work by Benhabib and Farmer (1996b) and Farmer (1997) demonstrates that indeterminacy may occur in a monetary model f o r realistically calibrated parameter values by modeling the role of money with more flexible functional forms that nest the cash-in-advance model as a special case. For example, in Benhabib and Farmer (1996b), output is produced using labor, and the service of money: (4.6)
y - y(m, I).
I f one makes the assumption that the technology is Cobb-Douglas, there is a single parameter that captures the effect of money; the elasticity of output with respect to real balances. This parameter can be directly measured in the same way that one measures the elasticity of output with respect to labor, through the share of resources used by the firm in transacting. This leads to the calibrated measure Xm = roy'(m, l) _ im y(m, 1) y(m, l)'
(4.7)
where i is the opportunity cost of holding money and the left-hand side of Equation (4.7) is the elasticity of real balances. Since the opportunity cost of holding money cannot be much more than 2% and the velocity of circulation, y/m, is around 5 in post-war data, the elasticity of money in production must be small, less than half of one percent. This kind of magnitude is not enough to generate big effects. Suppose, however, that money is highly complementary with other factors. In this case Benhabib and Farmer (1996b) show that indeterminacy may hold in a monetary model with an otherwise standard constant returns to scale technology. They use a technology of the form Y = (ame + le )
1/e,
which collapses to a Leontief technology as e approaches - o c and to a Cobb-Douglas technology as e approaches 0. The Leontief (or cash-in-advance) technology is rejected by the data since it can be shown to imply that the interest elasticity of the demandfor-money should be zero. The Cobb-Douglas function is also rejected since it would imply an elasticity of substitution of minus one. In the data, recent studies 21 find that the interest elasticity of the demand for money is close to -0.5 and one can use this fact to calibrate the value of e. Models that are calibrated to capture both money's share of output and the elasticity of money demand can be calibrated to display indeterminate equilibria. The reason
21 See, for example, Hoffman, Rasche and Tieslau (1995).
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why indeterminacy is more easily obtained in this case is that, in equilibrium, there is a relationship between real balances and labor demand that is found by solving firstorder conditions in the labor market. I f one solves for labor demand as a function of real balances [call this function l(m)] using this condition and substitutes the result back into the production function one arrives at the equation y=y(m,l(m)).
Calibrations of the production function using Equation (4.7) lead to the conclusion that the elasticity of y with respect to its first argument is small. However, although the direct effect of money on output may be small, the indirect effect through the fact that labor and real balances are increased together, the elasticity o f y with respect to its second argument may be large. Benhabib and Farmer (1996b) exploit the fact that their parametrization leads to indeterminacy to match a number features of the monetary propagation mechanism. 4.4. Monetary models with several state variables
The Benhabib and Farmer explanation of monetary dynamics works by picking an equilibrium in which the price is predetermined one period in advance and hence an increase in the nominal quantity of money causes an increase in real balances and employment. Beaudry and Devereux (1993) and Farmer (1997) build on this idea by building money into versions of a real business cycle economy. The paper by Beaudry and Devereux adds money to a structure in which there is already a real indeterminacy because of increasing returns to scale. The work by Farmer adds money into the utility function and has a production sector that is identical to the standard real business cycle model. Both sets of authors calibrate their economies to fit the broad features of the US economy (both real and monetary) and both models perform as well or better than a standard RBC model at replicating the second moments of US time series on consumption, investment, capital, GDP and employment. The following discussion is based on Farmer (1997) who allows for a fairly general specification of utility of the form U = U(C,m,l),
where C is consumption, m is real balances and I is labor supply. In the spirit of the real business cycle models of King, Plosser and Rebelo, Farmer argues that one should restrict attention to utility functions that allow for growth to be balanced and he shows that this implies that utility must be homogenous of degree p (a real number less than one) in m and C. The class of functions used in the paper is of the form U- X(C'm)l p 1-p
W(C,m)I-PV(1),
(4.8)
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J Benhabib and R.E.A. Farmer
where X and W are CES aggregators and V is an increasing convex function that measures the disutility of working. The following discussion is based on the special case of this utility function: cl-p
U-
-1-p
ml-PV(l),
p>
1.
The production side of the model is a standard RBC economy in which output is produced with the technology Y = F ( K , l) S,
where F is Cobb-Douglas and S is an autocorrelated productivity shock. Farmer considers two kinds of monetary policies. Policies in which there is an interest rate rule of the kind
and money growth rules of the kind Mt = g M t 1,
where i is the nominal rate of interest, M is the nominal quantity of money and/~ is the money growth factor. In the case when the monetary authority fixes the money growth rate in advance, the model can be described by a four-variable difference equation of the form 22 lull
~t+ 1
Ct+l Kt+l /'/~t+ 1
=A
G Kt mt
! U2+l L
where u 1 and //2 are fundamental shocks and e I and e 2 are sunspot shocks. Unlike the sunspot models that we have discussed so far, Farmer allows for multiple shocks to both sunspots and fundamentals and he calibrates the magnitude of the shocks by estimating the variance co-variance matrix of the residuals from a four variable vector autoregression on US data. Indeterminacy in this model can be understood by appealing to the Benhabib and Farmer (1994) results on the real model with increasing returns. Consider the case in which the Central Bank pegs the nominal interest rate. It is well known that this policy
22 The variables in Farmer (1997) are divided by a growing productivity term to deal with nonstationarities in the data. For brevity we omit this refinement in our discusion.
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rule leads to price level indeterminacy. What Farmer shows is that for utility functions in the class described by Equation (4.8) there may also be a real indeterminacy. Optimal decisions in Farmer's model are characterized by three Euler equations one for capital, one for money and one for bonds - and one static first-order condition describing the labor market. One may combine the Euler equations for money and bonds to yield a second static first-order condition: = ~' +
I+X P
7- 17, P
(4.9)
where the variables ~, rh, and 7 are the logarithms of consumption, real balances and labor supply. This equation plays the role of the "demand for money" in this model. The labor market equations can also be broken down into demand and supply of labor equations as in the real model discussed in Section 3.2. These demand and supply equations are: (1 - p ) ~ + p ~ +X1 = 6),
(4.10)
(1 - a ) k + ( a - 1)7 = 6),
(4.11)
where 6) is the log of the real wage and k is the log of capital. Equation (4.11) is a "labor demand" equation. I f we were to graph the real wage against labor demanded and supplied it would be represented by a downward sloping line, shifted by changes in the capital stock. Equation (4.10) is a "labor supply equation". On the same graph, it would be represented by an upward sloping line that was shifted by changes in consumption or changes in real balances. The key to understanding indeterminacy in the monetary model is to notice that one can replace ~ in Equation (4.10) by a fimction of ~, 7 and ], from the money market equilibrium condition, Equation (4.9). This leads to the hybrid equation ~-
-P]+ P
-1
7 =6).
(4.12)
P
In the real model in continuous time Benhabib and Farmer show that indeterminacy occurs when the labor demand and supply curves cross with the wrong slopes. This occurs in their model as a result of increasing returns to scale in production that causes the labor demand curve to slope up. I f one eliminates real balances from the labor market equations, using Equation (4.10), the resulting model has exactly the same structure as the real model of Benhabib and Farmer, with the additional twist that the interest rate enters as an exogenous variable. Equation (4.12) is a compound equation that combines the labor supply curve and the money demand equation; this plays the same role as the standard labor demand equation in the real model. Notice that this hybrid "labor supply curve" slopes down whenever (1 + X ) / P is less than 1. Using the Benhabib-Farmer indeterminacy condition, it follows that the monetary model has a
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real indeterminacy whenever the "labor supply" curve slopes down more steeply than the labor demand curve; for reasonable calibrations this occurs when Z is small (elastic labor supply), a equals 2/3 (labor's share of national income) and p is bigger than 1.5.
5. Indeterminacy and policy feedback So far we have discussed models in which indeterminacy arises in the context of models with a government sector; but we have allowed only government policies that are determined by simple rules such as fixed money growth rates or fixed government debt. In this section we will examine indeterminacy that may arise as a consequence of more complicated government policies that allow for feedback from the private sector to future values of fiscal or monetary policy variables.
5.1. Fiscal policy feedback We begin with a class of models in which there are "fiscal increasing returns", first discussed and elaborated on by Blanchard and Summers (1987). In the simplest formulation of such a model an increase in the capital stock can increase the posttax return on capital, because it expands the tax base and reduces the tax rate. If G is the constant real government expenditures and G = "cf(kt), where T is the tax on capital a n d f ( k ) is income, we can obtain the analogue of Equation (3.4) as
L) ) If the after-tax return f ' ( k ) ( 1 - G/f(k)) is increasing in k, a shift in p will raise investment and the capital stock, as well as the return to capital, so that Equation (5.1) will be satisfied only if p/p falls. This reverses the original rise in p and moves the system back toward the steady state, generating another equilibrium path. In fact, as shown in Velasco (1996), such a system has two steady-state values for k, corresponding to a high and a low tax rate, with the low-tax steady state representing a saddlepoint. Note that the term (1 - G/f(k)) is analogous to the reciprocal of a markup that varies inversely with the stock of capital k. Two related papers are those of Guo and Lansing (1998) and Schmitt-Groh6 and Uribe (1997a). Guo and Lansing explicitly compare the welfare properties of alternative fiscal policies in a model with increasing returns in the production sector. Their focus is on the ability to Pareto rank alternative equilibria with an eye to asking if models of indeterminacy might eventually be used to conduct welfare analysis, and to design optimal fiscal policies to select the best equilibrium. The model of SchmittGroh6 and Uribe includes labor and capital taxes, and generates two steady states by fixing government revenues and requiring the tax rate to be determined endogenously. Their model does not rely on explicit increasing returns to generate indeterminacy,
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although the labor market effects are similar to those o f Benhabib and Farmer (1994) with upward sloping labor demand curves. The mechanism that operates in their paper works through increases in employment that decrease equilibrium tax rates, and raise the after-tax return on labor. The tax rates at the indeterminate steady state are below those that maximize the revenue on the Laffer curve. Schmitt-Groh6 and Uribe provide a calibration o f their model to fit the US data and show that a successful calibration requires an elastic labor supply and a labor tax rate above the share of capital in aggregate income. They introduce a non-taxed home production sector, which allows indeterminacy under realistic tax rates and labor supply elasticities. 5.2. Monetary policy f e e d b a c k
The monetary models discussed so far assume no feedback from the private economy to government behavior. In practice, however, central banks typically react to the private sector, and the existence of central bank reaction functions has led to the development of a literature in which it is the central bank that is itself responsible for indeterminacy. Many o f the early monetary models simply assumed that the path of the money supply is an exogenous process determined by the central bank. In practice, central banks do not control a monetary aggregate directly. For example, in the USA the Federal Reserve system manipulates non-borrowed reserves on a day to day basis in an attempt to peg an interest rate (the Federal Funds rate) at a level that is revised periodically in light o f economic conditions. Why does much o f the literature assume that the central bank controls the money supply when in practice interest rate control is more common? One reason is that, as pointed out by Sargent and Wallace (1975), interest rate rules lead to price level indeterminacy and until recently, most authors have avoided building models with indeterminate equilibria because it was not known how to match up models o f this kind with data. Recently there has been more interest in the design o f central bank operating rules and this has led to a revival of interest in indeterminacy and its implications in calibrated monetary models 2s. One o f the first to derive an indeterminate equilibrium from a central bank reaction function is Black (1974). He assumes that the central bank responds, at time t, to the inflation rate between times t - 1 and t, decreasing (increasing) real money balances if this inflation rate is positive (negative). In the absence o f a central bank reaction o f this kind, higher inflation would be required to sustain equilibrium in response to an initial (upward) departure o f the initial price from its unique equilibrium level. When the central bank follows a contractionary reactive policy, inflation is no longer necessary to
23 TO do justice to the literature on central bank policy rules would require a separate survey and the reader is merely referred to two recent papers in the area: Taylor (1996) and Svensson (1996) and the literature cited therein. Also see the conference issue of the Journal of Monetary Economics (1997) no. 5, that collects together a number of related papers on the issue of 'Rules and Discretion in Monetary Policy.'
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sustain equilibrium. If the monetary policy response is sufficiently strong, prices must decline to offset the expected further contraction of nominal balances, reversing the inflation, and returning the system to its steady-state level of real balances. Therefore deviations of real balances from steady-state levels are reversed, and the initial price level is indeterminate. More recently Leeper (1991) and Schmitt-Groh~ and Uribe (1997b) have studied similar models where the monetary policy rule ties the nominal rate of interest to past inflation. One way to interpret the policy rule in models of this class is to assume that current inflation is forecast by past inflation. I f one assumes that marginal utility and endowments are constant, and that the utility function is separable in real balances and consumption, Leeper's model can be characterized by a discrete Euler equation of the form
Pt+l Pt
-/3it+l.
(5.2)
In this equation/3 is the discount factor, it+l is the nominal interest rate, representing the payout at time t + 1 to an investment in t, and Pt is the price level at time t. I f we assume a simplified feedback rule, with the constants a and y,
it+l
=
P, - apt l
+ ~,
(5.3)
and combine this with Equation (5.2) to obtain a first-order difference equation in the inflation rate, one can show that there exists an indeterminate set of equilibria ira/3 < 1. If on the other hand the nominal rate responds to the contemporaneous inflation, or in a stochastic model to expected inflation, so that
( Pt+l ~ it+, = aE \ ~ - t J + g' real indeterminacy disappears and the inflation rate is pinned down. The price level however is indeterminate because the interest rate rule can now accommodate any level of nominal balances consistent with price expectations, as noted by Sargent and Wallace (1975) for policies that peg the nominal interest rate. Woodford (1991, 1995, 1996) has argued that the price level can nevertheless be pinned down to eliminate nominal indeterminacy if we introduce departures from "Ricardian equivalence," that is if the government does not follow policies that are solvent under all possible equilibria. Woodford's distinction between Ricardian and non-Ricardian regimes is similar to Leeper's (1991) distinction between active and passive monetary policies. In the case of a non-Ricardian policy it is the requirement that private agents have confidence in the solvency of the government budget rule that 'selects' an equilibrium. In this case Woodford argues that the price level is determined by fiscal, rather than monetary, policy 24. [See also Sims (1997).] 24 See also footnotes 27 and 30 below.
Ch. 6: Indeterminacy and Sunspots in Macroeconomics
419
The discussion of Leeper's analysis makes clear that the presence and nature of indeterminacy is influenced by the existence of lags in the implementation of policy. In particular, with separability of consumption and real balances in the utility function, real indeterminacy would be ruled out in a continuous-time formulation, unless delays were explicitly introduced into the policy rule. The continuous-time formulation of monetary dynamics under interest rate policy rules given in the next subsection also illustrates this point. On the other hand, even with separability in the utility function and interest rules where the nominal rate responds to contemporaneous inflation, the slightest price stickiness can convert price level indeterminacy into real indeterminacy, as shown later in the next subsection, in the discussion of the sticky price model of Calvo (1983). 5.3. Interest rate rules and indeterminacy
In general, policy rules that tie the interest rate to current inflation, or in a stochastic model to expected inflation, imply growth rates for nominal balances that are not constant. By backing out the implied growth rates of nominal balances, we can study equilibria of monetary models of the type studied in Section 4. To illustrate how such monetary feedback policies can generate indeterminacies, we will use a simple continuous-time model, based on work by Benhabib, Schmitt-Groh6 and Uribe (1998). We begin by describing the structure of the private economy. For simplicity we assume that there is no production, and that a representative agent receives a constant, nonstorable endowment e at each moment in time. The agent carries wealth from one period to the next by holding bonds, B, and money real balances, M. We define the real holdings of these two assets as m - M/P and b - B/P, where P is the price level and we let a refer to total real assets so that at each instant a = b + m. Money balances do not pay interest but bonds pay a nominal interest which we denote by i. The flow budget constraint of an agent is given by i~=a(i
¢c)-im+e+ T-c,
where ¢~ = P / P is the inflation rate and T are lump-sum transfers or taxes. Now we turn to the structure of preferences. We denote the utility function of the agent by U(c, m), which we assume to be increasing and concave in consumption and real balances and we assume that the agent maximizes the discounted sum of utilities over an infinite horizon, with discount rate p. The first-order conditions for this problem yield the equation Uc(c, m) = p
[this is the monetary analog of Equation (3.1) given in Section 3.1], the portfolio condition which equates the return on bonds to the marginal benefit of holding money:
J Benhabib and R.E.A. Farmer
420
and the following Euler equation which is the analog of Equation (3.4):
/, _ p + ~ _ p
urn.
(5.5)
u~
Since endowments are constant, market clearing in goods requires that c = e, so that b = 0. Totally differentiating Uc(c, m) = p, and noting that ~ is zero, we have Ucm t~l = fo. If we substitute this into Equation (5.5), and use the money market identity =
(:) cr-
,
(5.6)
where o is the growth rate of nominal money balances, Equation (5.5) becomes P
p
Ecm~ =
+
m
UcJ
,
(5.7)
Ecm = --Ucrn m/Uc. To discuss policy we use a continuous-time version of the same rule given by Equation (5.3). Notice that unlike the case when the central bank picks ~, we must find an expression for a as a function of the interest rate by solving for ~ from the policy rule. To accomplish this task we write the following representation of the monetary policy rule: where
i= R + a(ar-(R-
p)),
where R and a are constants. If we use the definition of inflation (5.6) and the firstorder condition (5.4),
i =~=R+a(ar-(R-p))=R+a
6---m
(R-p)
,
(5.8)
we can find an expression for the money growth rate o:
g = -- +
-a-lR+R-p.
(5.9)
m Finally, by substituting this expression into Equation (5.7) we obtain the following differential equation that characterizes time paths for real balances that are consistent with equilibrium in the economy with interest rate feedback 25' 26: m = (ecru) -~
m ~- -R
( a - 1).
(5.10)
Once again it seems reasonable to suppose that money is a normal good. This implies that Um/Uc will be decreasing in m, which implies that the nominal rate i and the 25 Note that as in the discussionof Leeper's (1991) discrete-timemodel, if the utility function is separable in consumption and money so that Eem = 0, there is no real indeterminacy when the nominal interest rate is set as a function of current exected inflation: Equation (5.10) determines only the level of m. 26 The model easily generalizes to the case where money is productive as well, if we replace e with y(m), with/(m) >~O. In that case Equation 5.4 becomes i - (Um/Ue)+/(m). Details of the analysis are straightforward and are left to the reader.
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421
demand for real balances m are inversely related. This model has a unique steady state, defined by the level o f real balances, rh, for which (Um/Uc)[rh] = R. Further, the differential equation (5.10) will be stable if ~c,, and ( a - 1) are o f the same sign. Since a measures the sensitivity of the central bank to the inflation rate in setting the nominal interest rate, it follows that, depending on the sign o f Ec,7, there may be multiple equilibria with interest rate rules that are either sensitive (a > 1) or insensitive (a < 1) to the inflation rate 27. The mechanism at work here depends on the feedback rule: for example a rise in real balances causes the nominal rate, which in equilibrium must be equal to Um/Uc to fall. This induces a tighter monetary policy that reigns inflation in. Therefore, even if Um/Uc declines, the net return to holding money may either increase or decrease, depending on the strength o f the central bank response to the nominal rate. The other channel through which the demand for money is affected is through the effect o f money on the marginal utility o f goods, as discussed in Section 4.1: depending on the sign o f e~m, the demand for money may increase or decrease with a rise in real balances. Therefore both a and Ecm play a role in determining the nature o f the dynamics of m and the stability o f the steady state. The results in this section also cover a cash-in-advance economy as a special case. A cash-in-advance model is equivalent to having money in the utility function where consumption and money are combined with a CES aggregator, which in the limit becomes a Leontief production function as the elasticity o f substitution goes to zero. Since in such a case e~m < 0, indeterminacy with the interest rate rule used above and a cash-in-advance constraint is only possible if a < 128. The results obtained in this section depend on the linearity o f the feedback rule given by Equation (5.5). In effect from the perspective of global analysis the situation is more complicated. The nominal interest rates must be bounded below since the central bank cannot enforce negative rates. Benhabib, Schmitt-Groh6 and Uribe (2000) then show that if the feedback rule used by the central bank, i(¢c), is non-decreasing, and there exists a steady state at 7c* where a = i~(7~*) > 1, that is where monetary policy is active, then there must also exist another steady-state value o f Jr at which a < 1, that is where monetary policy is passive. [This can easily be seen simply by graphing both sides of the steady-state relationship, or the Fisher equation, p + Jr = i(7c)]. In such cases global indeterminacy holds, even though local analysis around one o f the steady states may indicate local determinacy.
27 Requiring the discounted value of government debt to remain asymptotically finite, as in Woodford (t996), eliminates price level indeterminacy but not the real indeterminacies discussed in this section. Benhabib, Schmitt-Groh6 and Uribe (1998) show that under indeterminacy, any one of the equilibrium trajectories of the real variables will have the discounted value of government debt remain asymptotically finite for an appropriate choice of the initial price level. See however footnote 30 for the case with sticky prices. 2s See Benhabib, Schmitt-Groh~ and Uribe (1998).
422
J. Benhabib and R.E.A. Farmer
5.4. Monetary models and sticky prices due to frictions in the trading process In this section we will discuss the role of interest rate rules in generating indeterminacy under "sticky" prices. Recently Woodford (1996), Chari, Kehoe and McGrattan (1996), Clarida, Gali and Gertler (1997), and Kiley (1998), among others, have studied models with sticky prices, based on variants of a staggered price setting model originally due to Calvo (1983) and Taylor (1980). These papers study monetary policies that target the nominal interest rate as a function o f past or current inflations, and each of them has noted the possibility of indeterminacy in models in which staggered price setting is assumed to be part of the environment rather than part of the equilibrium concept. One approach to modelling sticky prices due to Calvo (1983) is to specify that firms can change their prices at random intervals, but with a continuum of firms, a fixed fraction of them can do so at each instant. The firms set their prices fully expecting that their price will remain fixed over a random interval while some of the other firms will change their price, and aggregate demand will also vary. This structure may be interpreted as one of monopolistic competition with firms facing downward sloping demand curves which depend on aggregate demand and on the prices of other firms. The following example is based on the Calvo model. We assume that money enters the utility function and we write the Euler equation for consumption as ec
U~
r-Jr
(5.11)
where the portfolio condition again implies that i = Um/Uc. Substituting the policy rule for the nominal interest rate given by the first equality in Equation (5.8) into Equation (5.11), we can rewrite the Euler equation as C
b = -- ((a - 1) (Jr - (R - r))). ec
(5.12)
Under sticky prices the inflation rate Jr is governed by the dynamics of staggered prices which leads to the following equation describing the rate of change o f inflation: ~c = b (q - c).
(5.13)
Here q and b are constants, with q representing a capacity level associated with full employment: (q - c) may be interpreted as excess aggregate demand 29. Equations (5.13) and (5.12) constitute a system of differential equations in (c, jr), where neither c nor Jr are predetermined variables and the local dynamics of these equations depend on the Jacobian of the system evaluated at the steady state. If a < 1,
29 For a discussion of the relation between this equation and the standard Phillips curve see Calvo (1983).
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423
the steady state is indeterminate since the Jacobian o f the linearized dynamics around the steady state has one negative root. I f a > 1, the relevant roots are imaginary with zero real part, and the stability properties o f the steady state depend on higher-order terms o f the Taylor expansion in the linearization 3°. The novelty o f the class o f models with staggered price setting is that indeterminacy may arise for reasons that are i n d e p e n d e n t o f other mechanisms, in the sense that real indeterminacy may disappear if one removes staggered price setting. In our earlier formulation with flexible prices and a nominal interest rate feedback rule, real indeterminacy was only possible i f money entered the utility function in a nonseparable manner. But with Calvo-style price setters, real indeterminacy may occur even with a separable utility function. It follows that real indeterminacy in this case is attributable directly to the monopolistically competitive price setting mechanism that we introduced to model sticky prices 31. One way to interpret these results is to note that price level indeterminacy that occurs under interest rate rules and flexible prices with separable preferences, turns into real indeterminacy as soon as we introduce some degree o f price stickiness. This is in contrast to our earlier discussion o f indeterminacy in monetary models in which sticky prices implement one o f the possible set o f equilibria. In contrast, in the staggered price setting literature it is the sticky prices that c a u s e indeterminacy rather than the other way around.
6. Indeterminacy and models of endogenous growth Our discussion so far has centered on models o f business cycles. Another important area in which indeterminacy plays role is economic growth. Recently, Levine and Renelt (1992) demonstrated the lack o f robustness o f many o f the empirical results explaining the differences in the growth rate o f countries b y institutional and policy differences, and by differences in their rates o f factor accumulation, initial wealth and income distribution. The presence o f indeterminacies offers an additional and
30 Benhabib, Schmitt-Groh6 and Uribe (1998) also discuss a sticky price model based on Rotemberg (1996), where agents optimally choose how much to adjust their prices at each instant. They show that indeterminacy obtains for a < 1 just as in Calvo's model, but that it also obtains for a > 1 under some conditions. In the latter case the steady state can has two stable roots rather than one, so the stable manifold is of dimension two. Benhabib, Schmitt-Groh~ and Uribe (1998) show that requiring the discounted value of government debt to remain asymptotically finite, as in Woodford (1996), restricts initial conditions so the dimension of the restricted stable manifold is reduced to one: this however still implies real indeterminacy since neither c nor Jr are predetermined variables. Fm'thermore they also show that when a > 1, the steady state may be totally unstable with two positive roots, in which case indeterminacy takes the form of the stability of a limit cycle rather than of the steady state. See also Benhabib, Schmitt-Groh~ and Uribe (2000). 31 As noted by Kiley (1998), sticky prices have the effect of increasing the responsiveness of output to monetary shocks, and in this sense they are "productive."
424
J. Benhabib and R.E.A. Farmer
complementary explanation of why countries that have very similar endowments and fundamentals, nevertheless save and grow at different rates 32. The recent literature on endogenous growth, initiated by Lucas (1988) and Romer (1990), contains elements of market imperfections that can be shown to generate indeterminacies under reasonable parametrizations. In contrast to the business cycle literature however, in models of endogenous growth it is the balanced growth path that is indeterminate, rather than the steady-state level of GDE The distinctive feature of endogenous growth models is their production technology which allows proportional growth in some accumulated assets like human or physical capital, or the stock of research and development. The fact that the technology allows for linear growth implies that there must exist increasing returns at the social level to overcome diminishing returns at the private level. It is a small step from here to generate indeterminacy through complementarities between the factors of production. An interesting feature of endogenous models is their ability to generate multiple balanced growth paths in conjunction with indeterminacy. We can illustrate how multiple balanced growth paths and indeterminacy can arise in such models with small modifications to the simple structure of Equations (3.1), (3.2), (3.3) and (3.4). We will rely on a simple illustrative structure of production that is linear in an accumulated asset, and with sufficiently strong external effects from the labor input. Our endogenous growth model will have a balanced growth path, along which the ratio of the asset to consumption will be constant. If we denote the asset by k, we will have k = s c , where c is consumption and s is a constant. For simplicity lets assume that the utility of consumption is logarithmic, and that the production function is of the Cobb-Douglas form, y = k a k O-a) L ~, where lc represents an external effect. Consider the endogenous growth version of Equation (3.4), where we have replacedp by c using Equation (3.1):
C
-- ( w I ( L ) -- ( r q- g ) ) .
Note that since y is linear in k, Wl only depends on L, and is given by wl = can also write Equation (3.3) for the goods market equilibrium as
k k - a(L) - s - g,
(6.1)
a L ft.
We
(6.2)
32 For a study of the empiricalrelevanceof indeterminacyin explainingeconomicgrowth,see Benhabib and Gali (1994).
Ch. 6: Indeterminacy and Sunspots in Macroeconomics
425
where a(L) is the average product of capital and only a function of L because y is linear in k: a(L) = L ~. Since s is a constant along the balanced growth path, the difference between the right-hand sides o f Equations (6.1) and (6.2) must be zero 33: a ( L ) - S - Wl(L) + r = s - r + a(L)(1 - a) = O.
(6.3)
The second equality follows because the marginal and average products o f capital, a(L) and yI(L), are proportional, and in our Cobb-Douglas example their difference
is a(L)(1 - a). We can also express s as a function o f L by using the labor market equilibrium condition given by Equation (3.2): s-
c
k
- -
m(L)
-
v'(1 -L)
- v(L).
(6.4)
Here m(L) is the marginal product of labor divided by k. Substituting this expression into Equation (6.3) we have a(L) (1 - a) - o(L) = r.
(6.5)
Equation (6.5) can have one, two or no solutions corresponding to the balanced growth paths, depending on the parameters of the model. The right-hand side o f Equation (6.5) is monotonic in L if v(L) is decreasing, but if v(L) is increasing, there may be two balanced growth paths. A n increasing v(L) however is only possible if the marginal product of labor is increasing, and this requires a significant labor externality. This is precisely what happens in the endogenous growth version o f the model in Benhabib and Farmer (1994), when the labor externalities are high enough. One of the balanced growth paths is determinate, while the other is indeterminate. A more extensive analysis o f a related mechanism in the Lucas (1988) model, with small external effects confined only to the research sector, is given in Benhabib and Perli (1994) [see also Xie (1994)]. They show that multiple balanced growth paths and indeterminacy can appear for reasonable parametrizations o f the model 34. A similar mechanism produces indeterminacy and multiple balanced growth paths in Romer's (1990) model, as analyzed by Benhabib, Perli and Xie (1994) and by Evans,
33 This approach of equating two growth rates to (graphically) characterize nmltiple steady states is also taken by Evans, Honkapohja and Romer (1996). To generate sufficiently high returns that justify the higher investment and growth rates, the authors rely on production complementarities giving rise to external effects from the introduction of new intermediate goods, rather than from postulating increasing returns to the labor input. 34 The conditions for indeterminacy in the Lucas model are less stringent than those presented above because it is a two-sector model. In fact the two-sector structure allows for indeterminacy with a fixed labor supply just like the two sector model discussed in Section 3.1, but requires the utility of consumption not to exhibit too much curvature. This same feature arises in the two-sector model above, and it is the reason for introducing a third sector in Benhabib and Nishimura (1998).
426
J. Benhabib and R.E.A. Farmer
Honkapohja and Romer (1996). Evans, Honkapohja and Romer (1996) also study a modification of the Romer model by introducing adjustment costs that generate a nonlinear production possibility curve between the consumption and investment sectors. Their model has three balanced growth paths, two of which are stable under a learning mechanism. Introducing sunspots induces jumps across the two stable (indeterminate) balanced growth paths, and generates fluctuations in the growth rate. Such regime switching equilibria giving rise to sunspot fluctuations in the growth rate are also studied, both theoretically and empirically, in Christiano and Harrison (1996). As in Benhabib and Perli (1994), they observe that indeterminacy can arise even if the balanced growth paths are locally determinate, because rates of investment can be chosen to place the economy on either one of them. Another specification generating indeterminacy is given in the endogenous growth model of Gali and Zilibotti (1995). They use a model with monopolistic competition, coupled with fixed costs and entry. Markups are inversely related to entry and to the capital stock, so that raw returns can increase in k. This model is an endogenous growth version of the variable markup model of Gali (1994). It gives rise to two balanced growth paths, one with zero growth representing a corner solution, and the other one with a positive growth rate. Furthermore there is a range of initial conditions for capital in which the equilibrium trajectory is indeterminate, and may converge to either of the balanced growth paths depending on the initial choice of consumption.
7. S o m e related w o r k
So far the framework presented by Equations (3.1)-(3.4) assumed that the preferences were standard, and in particular that the discount rate was constant. We may however allow the discount rate to be affected by some social norm, proxied for example by the value of aggregate consumption. If preferences and the discount rate are subject to such external effects, it is clear from Equation (3.4) that they can substitute for external effects and increasing returns in technology. A higher price of investment may lead to a higher capital stock, and may well decrease the marginal returns to capital. If the discount rate declines as well however, increasing price appreciations in the price of the capital good may be unnecessary to sustain equilibrium. The price of the investment good may well decline, and move back towards its stationary equilibrium value, generating indeterminacy. Such a mechanism is explored in detail in a recent paper by Drugeon (1996). In general, endogenous preferences coupled with some market imperfections are likely to provide a basis for multiple equilibria and indeterminacy. An alternative route to indeterminacy may be through increasing returns not in the production function, but in the utility function. In such a setup there must be sufficient discounting of the future to assure that utilities remain finite in equilibrium. In a recent paper Cazzavilan (1996) studies indeterminacy in such a model, where public goods financed by taxes enter the (constant returns to scale) production function. Since the
Ch. 6: Indeterminacy and Sunspots in Maeroeconomics
427
public goods are productive, they create externalities because agents take the tax rate as given. The result is an endogenous growth structure with an indeterminate balanced growth path. Indeterminacy can also arise from variations in capacity utilization if utilization rates co-move with labor, as would be the case if intensified utilization accelerates capital depreciation. This possibility has recently been shown by Wen (1998). In his model a shift in production towards investment will raise the capital stock, but an associated increase in labor will cause the marginal product of capital to increase rather than decrease, very much like the model of Benhabib and Farmer (1994). The reason for the expansion in labor however is not an upward sloping demand curve for labor due to external effects, but a rightward shift in the labor demand curve due to increased capacity utilization. Wen calibrates his model to US data and finds that indeterminacy can provide a remarkably good match to the data with mild increasing returns in the order of 0.1. Guo and Sturzenegger (1994) study the application of indeterminacy to the study of international consumption data. The RBC model has trouble with the fact that consumption across countries is predicted to be perfectly correlated under simple variants of the international RBC model with complete markets. But in practice the correlation between consumption across countries is quite low. The Guo and Sturzenegger explanation drives business cycles with sunspots as in the single-country model of Farmer and Guo (1994), but they assume that agents are unable to perfectly insure across countries. Their calibrated model does a fairly good job of explaining the cross-country data and is one of the first applications of empirical models of indeterminacy to international data sets. We should note that we have not touched upon the literature that deals with indeterminacy in overlapping generations models or in finite markets with incomplete participation or market imperfections. Some recent overviews of these topics can be found in Balasko, Cass and Shell (1995) or Bisin (1997), among others.
8. Empirical aspects of models with indeterminacy In Section 2 we mentioned two areas in which models with indeterminate equilibria might potentially improve upon existing models of the business cycle. The first is that of propagation dynamics and the second is related to monetary features of business cycles. In this section we elaborate on the claim that indeterminacy might be a fruitful research direction by surveying known results in which some progress has been made on each of these issues. 8.1. Real models and propagation dynamics
The real business cycle literature represented a major departure from the Keynesian models that preceded it. On the theoretical front RBC theorists argued that the correct
428
J. Benhabib and R.E.A. Farmer
way forward for macroeconomics is some version of dynamic general equilibrium theory. On the empirical front they argued that the standards for what should be considered a successful description of the data should be considerably relaxed from the requirements imposed by time-series econometricians. Following the approach initiated by Kydland and Prescott (1990), much of the RBC literature dispenses with attempts to study the low-frequency components of time series by passing data (both actual and simulated) through a filter that leaves only high-frequency components 35. If simulated data from an artificial model can replicate a few of the moments of the data from an actual economy then RBC economists argue that the model is a successful description of the real world. There is much to disagree with in the RBC methodology. It has nevertheless had the effect of providing a unified framework for comparing and evaluating alternative economic theories. In this section of the survey we will turn our attention to calibrated models of indeterminacy that have used the RBC methodology to provide competing explanations of business cycle phenomena. These models all build on some simple variant of a representative-agent economy and the variables they describe includes consumption, investment, GDP and employment as a subset. It is therefore possible to ask how their predictions compare with those of the benchmark model 36. 8.1.1. One-sector models
In Section 2.2 we pointed out that the one-sector real business cycle model, driven by productivity shocks, has a representation as a difference equation in three state variables. We reproduce this equation here:
~t-I-1 k ~t+l
: ~-1
~g ~t
4~-lF Lu,+l ]j'
(8.1)
The variables ~, /c and ~ represent deviations of consumption, capital and the productivity shock from their balanced growth paths; ~ is a belief shock and ~t is an innovation to the productivity shock. The variables kt+l and st+l are determined at date t but ct+l is free to be determined at date t + 1 by the equilibrium conditions of the model. If the matrix q~ 1 has three roots inside the unit circle then it is possible to construct equilibria in which the business cycle is driven purely by iid sunspot errors (the variable et+l) and the artificial data constructed in this way can be compared with actual data in the same way that one matches RBC models by comparing moments. This idea was exploited by Farmer and Guo (1994) who pointed out that there are
35 Hodrick and Prescott (1980) advocated this approach in their widely circulated discussion paper. Although the HP filter is widely used in the literature it has also been widely criticized since the filter itself can alter the covarianceproperties of the filtered data in ways that may introduce spurious cycles. 36 For all interesting perspective on this issue, see Kamihigashi (1996).
Ch. 6: Indeterminacy and Sunspots in Macroeconomics
429
some dimensions in which the sunspot model can perform better than models driven by fundamentals. We return to this idea shortly. To get a better idea of how an array of sunspot models compare with each other, with the RBC model and with the data, Schmitt-Groh6 (1997) analyses four different models, all of which are calibrated in a similar way, and all of which have a representation of the kind illustrated in Equation (8.1). The models that she studies are (1) a model similar to that of Gali (1994) in which changes in the Composition of Aggregate Demand (the CAD model) allow the markup to be countercyclical; (2) a model based on Rotemberg and Woodford (1992) in which markups may again be countercyclical but in this case the variability of the markup follows from Implicit Collusion (the IC model); (3) a model with increasing returns and decreasing marginal costs (the IR model); and finally (4) a model with externalities (the EXT model) based on the work of Farmer and Guo (1994). The main question addressed by her work is "For what values of the parameters can the matrix q~-i in Equation (8.1) have three roots all inside the unit circle?" This is an interesting question in light of the results of Farmer and Guo (1994) since, when all of the roots of q~-I are inside the unit circle, one can generate artificial time series for consumption, investment, hours and GDP by simulating sequences of variables using the equation
I
~t+ 1
(8.2)
where @1~ and [~P-IF],, are the 2 × 2 upper left blocks of @ ' and [~P ' F I. The formulation of the model in Equation (8.2) is one in which equilibrium business cycles in which all agents are fully rational are driven purely by sunspots. Schmitt-Groh6 (1997) simulates series of artificial data for all four types of onesector model. In each case she calibrates the baseline parameters as in a standard RBC model, laid out in work by King, Plosser and Rebelo (1987), and she sets the increasing returns, externality and markup elasticity parameters in a way that minimizes the degree of aggregate increasing returns but still allows equilibria to be indeterminate. Table 1 reproduces sections a, b and c of Table 6, page 136 in Schmitt-Groh6 (1997). The first two data columns of this table are reproduced from King, Plosser and Rebelo (1987) and they illustrate the dimensions on which the RBC model is often evaluated. The column labeled "RBC model" contains statistics generated by simulations of a 'standard' RBC model in which the source of business cycle dynamics is a highly persistent productivity shock. Column 1, for comparison, gives the US data. Columns 3 through 6 are statistics generated by Schmitt-Groh6 in which each of the four sunspot models are used to simulate data but, in contrast to the RBC model, each of these columns simulates data generated by a pure sunspot shock. The main point of this table is to illustrate that by the standards of the calibration literature, the models driven purely by sunspots perform about as well. This in itself is an interesting observation because although early work on sunspot models had demonstrated that sunspots could exist, much of this literature had little or no connection with data.
430
J. Benhabib and R.E.A. Farmer
Table 1 Results of different models US data
RBC model
CAD model
IC model
IR model
EXT model
a. Relative standard deviation: std(x)/std(output) Output
1.00
1.00
1.00
1.00
1.00
1.00
Consumption
0.69
0.64
0.35
0.39
0.82
0.91
Investment
1.35
2.31
3.36
3.41
2.32
1.82
Hours
0.52
0.48
0.71
0.70
0.43
0.32
Real Wage
1.14
0.69
0.42
0.44
0.83
0.91
0.60
0.81
b. Autocorrelation coefficient AR(1)
Output
0.96
0.93
0.89
0.71
Consumption
0.98
0.99
0.98
0.98
1.00
1.00
Investment
0.93
0.88
0.88
0.66
-0.08
0.16
Hours
0.52
0.86
0.88
0.66
-0.24
-0.12
Real Wage
0.97
0.98
0.94
0.88
0.97
0.99
0.58
0.84
0.92
c. Contemporaneous correlation with output
Consumption
0.85
0.82
0.65
Investment
0.60
0.92
0.97
0.96
0.82
0.82
Hours
0.07
0.79
0.85
0.86
0.56
0.42
Real Wage
0.76
0.90
0.91
0.86
0.90
0.95
Earlier in this survey we drew attention to two aspects in which sunspot models with indeterminate equilibria are different from standard models with a unique equilibrium. The first was that models with sunspots can generate an alternative source of the impulse to the business cycle, and it is this claim, that sunspots may be a primary impulse, that is evaluated in Table 1. A second, and perhaps more interesting feature of models with indeterminacy, is that they offer an alternative explanation of propagation dynamics. To evaluate this claim, Farmer and Guo (1994) generate a set o f impulse response fimctions from three different models and they compare these impulse response functions with those from US data. The impulse response functions to innovations in output for US data are derived from a vector autoregression of output, employment, consumption and investment with a linear time trend and five lags, over the period 1954.1 to 1991.3. The three models are a standard RBC economy (the same calibration as the RBC economy in Table 1) and two different models with externalities. One of these models is calibrated as in work by Baxter and King (1991) who introduce externalities but calibrate these externalities in a way that is n o t large
Ch. 6: Indeterminacy and Sunspots in Macroeconomics
431
enough to generate indeterminacy. The second is a calibration w i t h indeterminacy in line with the EXT model discussed by Schmitt-Groh& Figure 1 compares the impulse responses in each of these three models with the impulse response to a set of shocks in the US data. Notice, in particular, the dynamic pattern of investment in the data and compare it with models 1 and 2 in Figure 1. The impulse responses for US data shows clear evidence of a cyclical response pattern whereas models 1 and 2 (the RBC model and the externality model w i t h o u t indeterminacy) both show monotonic convergence patterns. Farmer and Guo point out that monotonic convergence in the RBC economy follows from the fact that, although the dynamics in k and s are two-dimensional, there is no feedback in the dynamics of the system from the productivity shock s to the capital stock k. The system
[
~+1
that characterizes the RBC dynamics has monotonic impulse response functions because the matrix A is upper triangular and it necessarily has two real roots. The equation
t+l] [};+et+lJ,
I kt+l
on the other hand, that characterizes the dynamics of the sunspot models, incorporates feedback both from ct to kt and vice versa, hence the matrix A that determines the properties of the impulse response functions in this case c a n have a complex roots. It is this feature that Farmer and Guo exploit to generate the features of the dynamic responses illustrated in Figure 1. Although the one-sector models discussed above do a relatively good job of describing data they rely on large markups and imperfections to generate indeterminacy, which may not be empirically plausible. Schmitt-Groh6 (1997) concludes that while "... the relative volatility, autocorrelation, and contemporaneous correlation properties of macroeconomic aggregates predicted by each of the endogenous business cycle models are broadly consistent with those actually observed in the US data ... the degree of market power or returns to scale required for the existence of expectation driven business cycles lies in the upper range of available empirical estimates.". The more recent models of Perli (1994), Schmitt-Groh6 and Uribe (1997a) and Wen (1998), which modify the one-sector model by introducing home production, taxes, and variable capacity utilization, are also successful in their calibration analysis but they avoid the high degree of increasing returns to scale required by Benhabib and Farmer (1994) to generate indeterminacy. In the next subsection we discuss the empirical performance of multi-sector models which do not rely on large market distortions or external effects to generate indeterminacy.
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J. Benhabib and R.E.A. Farmer
~J
E
~
E
1 then there is a set of equilibria of the form Pt+l
1 =
~st
a
-
~ P t + et+l,
(9.6)
where et+x is an arbitrary lid sunspot shock with zero conditional mean. In the determinate case, agents can forecast using the equilibrium function (9.5). Plugging this function into Equation (9.2) leads to the expectation El [Pt+l] = E t [ St+l ] = 0 k a a
(9.7)
which implies that demand is given by the function x D = apt.
(9.8)
As the current price varies, market demand varies with current price according to Equation (9.8). A Walrasian auctioneer, calling out prices, would find a unique price, P t = st~a, at which demand equals supply.
Ch. 6:
441
Indeterminacy and Sunspots in Macroeconomics
In the indeterminate case it is not so obvious how an equilibrium could be maintained. Suppose that agents forecast the future using the equilibrium pricing rule, Equation (9.6). Substituting this rule back into Equation (9.2) leads to the demand function X? = S t
which is identical to supply for all possible values o f p t . Equation (9.6) cannot be used to forecast the future price since if agents were to use the equilibrium price function demand equals supply for any possible price. But although Equation (9.6) cannot be used to forecast, there is a rule that can. Suppose that agents use only lagged information to forecast the future price. In particular, suppose that they use the rule 1 a a a2 Pt+l = ~st + et+l - -~ffst 1 - ~et + bfPt-l,
(9.9)
which is obtained by lagging Equation (9.6) by one period. We will refer to this rule as a forecast function. Using Equation (9.9) they can compute an expectation of the price in period t + 1: 1
a
E[pt+l] = ~ s t - ~ S t
a2
a
1 -- ~ e t + ~ P t
1-
(9.10)
Plugging this expectation back into Equation (9.2) leads to the demand function a
a2
x~ = apt + s t - ~st 1 - a e t + ~ P t 1
(9.11)
that shows how current demand varies with price if agents use Equation (9.9) to forecast. Equating demand to supply, it follows that the current price will be determined by the equation 1 a pt = ~st-1 + et - ~Pt-1,
(9.12)
which is the equilibrium pricing rule that we introduced in Equation (9.6). Let us recapitulate what we have said. We have shown that if agents forecast the future price using the forecast function (9.9) then the actual price will be described by the stochastic difference equation in Equation (9.12). Since the forecast function was obtained by lagging the actual pricing rule, the forecast function is rational. To verify this, one can substitute the equilibrium price rule (9.12) into the forecast function. Furthermore, the sequence of error terms et is arbitrary. We have shown that there are arbitrary forecast functions each of which can support a different rational expectations equilibrium 41.
41 For a generalization of this argument to a higher-dimensional linear model see Matheny (1996).
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442
9.3. Does indeterminacy have observable implications? Some authors have been concerned that models with indeterminate equilibria may not be useful models since, it might be thought, anything can happen. This argument is false. In fact, models with indeterminate equilibria place relatively strong restrictions on the moments of data once one closes these models by specifying a process that determines the formation of beliefs. For example, consider the Farmer-Guo version of the RBC model with increasing returns. We showed earlier that this model is described by a set of equations of the form
~+,
:q'~
St+l
kt St
q, 1F
~t+l .
(9.13)
Ut+l
It is true that if one allows the sequence of forecast errors ~t to be arbitrary that this model allows additional freedom to describe the data 42. But once one specifies a stationary stochastic process for the joint determination of sunspots and fundamentals, this model places strong restrictions on the joint process determining the evolution of the state variables. Indeed, Aiyagari (1995) has argued that these restrictions are falsified in data, and this criticism of the Benhabib-Farmer (1994) model is in part responsible for the research agenda on two-sector models that we described above. Although models of indeterminacy do place restrictions on data, these restrictions are often less severe than the standard real business cycle model. Indeed, it is the fact that some of the restrictions of the standard model are often rejected by the data that is one of the prime motivations for considering a wider class of economies.
10. Conclusion
The central theme of this chapter is that the standard infinite-horizon model, modified to incorporate some mild market imperfection, often supports an indeterminate set of equilibria. When the non-stochastic version of a model has an indeterminate set of equilibria, variants of the model that explicitly incorporate uncertainty will typically support a continuum of stationary rational expectations equilibria, some of which may be driven by sunspots. In this sense the property that the equilibria of finite ArrowDebreu economies are determinate is fragile. An implication of this argument is that minor perturbations of the (Hamiltonian) structure of a representative agent model allows self-fulfilling expectations to have a significant influence on the dynamics of prices and output. Furthermore, the economic
42 On the other hand, even with arbitrary forecast errors for sunspot shocks, without technology shocks it would not be possible to match the procyclicalityof consumption in the data for the reasons cited in Section 8.1.2.
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mechanisms which give rise to such perturbations are varied, and the investigation of these mechanisms is a fruitful one, since it can potentially account for features of the time series data that are otherwise difficult to understand. The models that we have discussed in this survey may lead to the development o f a rich theory of economic policy. In some situations, as in models involving monetary policies with feedback rules, sunspots may exist under some policy regimes but not under others. In other instances, as in models where coordinating on higher investment rates leads to Pareto-superior outcomes, the kind o f policies needed to achieve such coordination may be quite complex, and even difficult to implement. The important consideration however is not so much to find policies that eliminate the possibility o f multiple or sunspot equilibria, but to design policies that will select and implement the best possible equilibrium. Even if it is not possible to design policies that will select the best equilibrium, or to completely eliminate sunspot equilibria, the models that we have described in this survey may enable us to design Pareto-improving policy rules. The argument that equilibria are indeterminate may be wrong; but interventionist policy arguments couched in this language are at least capable o f comparison with their noninterventionists counterparts. I f a dialogue is to be developed between those who favor active intervention and those who do not, it is important that the two groups speak the same language. Dynamic general equilibrium theory, allowing for indeterminacies, is exactly the kind o f vehicle that is required to further communication in this debate.
Acknowledgements We wish to thank Roland Benabou, Jordi Gali, Stephanie Schmitt-Groh6, Jang Ting Guo, Sharon Harrison, Takashi Kamihigashi, Roberto Perli, Martin Uribe and Michael Woodford for very useful discussions and comments. Technical support from the C.V. Starr Center for Applied Economics at New York University and from the Program for Dynamic Economics at U C L A is gratefully acknowledged. Farmer's research was supported by the National Science Foundation, grant #952912.
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Chapter 7
LEARNING DYNAMICS GEORGE W. EVANS University of Oregon SEPPO HONKAPOHJA
University of Helsinki Contents
Abstract Keywords 1. Introduction 1.1. Expectations and the role of learning 1.1.1. Background 1.1.2. Role of learning in macroeconomics 1.1.3. Alternative reduced forms 1.2. Some economic examples 1.2.1. The Muth model 1.2.2. A linear model with multiple REE 1.2.3. The overlapping generations model with money 1.3. Approaches to learning 1.3.1. Rational learning 1.3.2. Eductive approaches 1.3.3. Adaptive approaches 1.4. Examples of statistical learning rules 1.4.1. Least squares learning in the Muth model 1.4.2. Least squares learning in a linear model with multiple REE 1.4.3. Learning a steady state 1.4.4. The seignorage model of inflation 1.5. Adaptive learning and the E-stability principle 1.6. Discussion of the literature 2. General m e t h o d o l o g y : recursive stochastic algorithms 2.1. General setup and assumptions 2.1.1. Notes on the technical literature 2.2. Assumptions on the algorithm 2.3. Convergence: the basic results 2.3.1. ODE approximation Handbook of Macroeconomics, Volume 1, Edited by JB. Taylor and M. WoodJbrd © 1999 Elsevier Science B. V All rights reserved 449
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2.3.2. Asymptotic analysis 2.4. Convergence: further discussion 2.4.1. Immediate consequences 2.4.2. Algorithms with a projection facility 2.5. Instability results 2.6. Further remarks 2.7. Two examples 2.7.1. Learning noisy steady states 2.7.2. A model with a unique REE 2.8. Global convergence 3. Linear e c o n o m i c m o d e l s 3.1. Characterization of equilibria 3.2. Learning and E-stability in univariate models 3.2.1. A leading example 3.2.1.1. A characterization of the solutions 3.2.1.2. E-stability of the solutions 3.2.1.3. Strong E-stability 3.2.1.4. E-stability and indeterminacy 3.2.2. The leading example: adaptive learning 3.2.2.1. Adaptive and statistical learning of MSV solution 3.2.2.2. Learning non-MSV solutions 3.2.2.2.1. Recursive least squares learning: the AR(1) case 3.2.2.2.2. Learning sunspot solutions 3.2.3. Lagged endogenous variables 3.2.3.1. A characterization of the solutions 3.2.3.2. Stability under learning of the AR(1) MSV solutions 3.2.3.3. Discussion of examples 3.3. Univariate models further extensions and examples 3.3.1. Models with t dating of expectations 3.3.1.1. Alternative dating 3.3.2. Bubbles 3.3.3. A monetary model with mixed datings 3.3.4. A linear model with two forward leads 3.4. Multivariate models 3.4.1. MSV solutions and learning 3.4.2. Multivariate models with time t dating 3.4.3. Irregular models 4. L e a r n i n g in nonlinear m o d e l s 4.1. Introduction 4.2. Steady states and cycles in models with intrinsic noise 4.2.1. Some economic examples 4.2.2. Noisy steady states and cycles 4.2.3. Adaptive learning algorithms
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4.2.4. E-stability and convergence 4.2.4.1. Weak and strong E-stability 4.2.4.2. Convergence 4.2.4.3. The case of small noise 4.2.5. Economic models with steady states and cycles 4.2.5.1. Economic examples continued 4.2.5.2. Other economic models 4.3. Learning sunspot equilibria 4.3.1. Existence of sunspot equilibria 4.3.2. Analysis of learning 4.3.2.1. Fornmlation of the learning rule 4.3.2.2. Analysis of convergence 4.3.3. Stability of SSEs near deterministic solutions 4.3.4. Applying the results to OG and other models 5. E x t e n s i o n s a n d r e c e n t d e v e l o p m e n t s 5.1. Genetic algorithms, classifier systems and neural networks 5.1.1. Genetic algorithms 5.1.2. Classifier systems 5.1.3. Neural networks 5.1.4. Recent applications of genetic algorithms 5.2. Heterogeneity in learning behavior 5.3. Learning in misspecified models 5.4. Experimental evidence 5.5. Further topics 6. C o n c l u s i o n s References
511 512 513 513 513 513 514 515 516 517 517 518 520 520 521 521 521 523 524 525 527 528 530 531 533 533
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Abstract
This chapter provides a survey of the recent work on learning in the context of macroeconomics. Learning has several roles. First, it provides a boundedly rational model of how rational expectations can be achieved. Secondly, learning acts as a selection device in models with multiple REE (rational expectations equilibria). Third, the learning dynamics themselves may be of interest. While there are various approaches to learning in macroeconomics, the emphasis here is on adaptive learning schemes in which agents use statistical or econometric techniques in self-referential stochastic systems. Careful attention is given to learning in models with multiple equilibria. The methodological tool is to set up the economic system under learning as a SRA (stochastic recursive algorithm) and to analyze convergence by the method of stochastic approximation based on an associated differential equation. Global stability, local stability and instability results for SRAs are presented. For a wide range of solutions to economic models the stability conditions for REE under statistical learning rules are given by the expectational stability principle, which is treated as a unifying principle for the results presented. Both linear and nonlinear economic models are considered and in the univariate linear case the full set of solutions is discussed. Applications include the Muth cobweb model, the Cagan model of inflation, asset pricing with risk neutrality, the overlapping generations model, the seignorage model of inflation, models with increasing social returns, IS-LM-Phillips curve models, the overlapping contract model, and the Real Business Cycle model. Particular attention is given to the local stability conditions for convergence when there are indeterminacies, bubbles, multiple steady states, cycles or sunspot solutions. The survey also discusses alternative approaches and recent developments, including Bayesian learning, eductive approaches, genetic algorithms, heterogeneity, misspecifled models and experimental evidence.
Keywords expectations, learning, adaptive learning, least squares learning, eductive learning, multiple equilibria, expectational stability, stochastic recursive algorithms, sunspot equilibria, cycles, multivariate models, MSV solutions, stability, instability, ODE aproximation, stochastic approximation, computational intelligence, dynamic expectations models J E L classification: E32, D83, D84, C62
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1. Introduction
1.1. Expectations and the role of learning 1.1.1. Background In modern macroeconomic models the role o f expectations is central. In a typical reduced form model a vector o f endogenous variables yt depends on lagged values yt-1, on expectations o f the next period's values, Yt+l, and perhaps on a vector o f exogenous shocks ut, e.g. taking the form Yt = F ( y t 1,Yt+l, ut), where for the moment assume F to be linear. O f course, in some models the dependence on Yt-l or ut m a y be absent. The information set available when Yt+l is formed typically includes {Yt-i, ut-i, i = 1,2, 3 . . . . } and may or may not also include the contemporaneous values Yt and ut. A useful notation, i f y t , ut are in the information set, is E{yt+l and we write the reduced form as
Yt - F ( y t l,EtYt+l, ut).
(1)
I f y t and ut are not included in the information set then we write yet+l as ET_yt+l. In the economic models we consider in this survey, these expectations are those held by the private agents in the economy, i.e. o f the households or the firms. Models in which policy makers, as well as private agents, must form expectations raise additional strategic issues which we do not have space to explore l . Following the literature, we restrict attention to models with a large number o f agents in which the actions o f an individual agent have negligible effect on the values yr. Closing the model requires a theory o f how expectations are formed. In the 1950s and 1960s the standard approach was to assume adaptive expectations, in which expectations were adjusted in the direction o f the most recent forecast error, e.g. in the scalar case, and assuming Yt is in the information set, ETyt+l = ET_lyt + Y(yt - ET_lyt) for some value o f 0 < ~/ ~< 1. Though simple and often well-behaved, a well known disadvantage o f adaptive expectations is that in certain environments it will lead to systematic forecast errors, which appears inconsistent with the assumption o f rational agents. The rational expectations revolution o f the 1970s has led to the now standard alternative assumption that expectations are equal to the true conditional expectations in the statistical sense. Rational expectations (in this standard interpretation used in macroeconomics) is a strong assumption in various ways: it assumes that agents know the true economic model generating the data and implicitly assumes coordination o f expectations by the agents 2. It is, however, a natural benchmark assumption and is widely in use.
1 See Sargent (1999) for some models with learning by policy makers. 2 A rational expectations equilibrium can be interpreted as a Nash equilibrium, a point made in Townsend (1978) and Evans (1983). It is thus not rational for an individual to hold "rational expectations" unless all other agents are assumed to hold rational expectations. See the discussion in Frydman and Phelps (1983).
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More recently a literature has developed in which the RE (rational expectations) assumption has been replaced by the assumption that expectations follow a learning rule, either a stylized or a real-time learning rule, which has the potential to converge to RE. An example of a learning rule is one in which agents use a linear regression model to forecast the variables of interest and estimate the required parameters by least squares, updating the parameter estimates each period to incorporate new data. Modeling expectations in this fashion puts the agents in the model in a symmetric position with the economic analyst, since, when studying real economies, economists use econometrics and statistical inference. In contrast, under RE the agents in the model economy have much more information than the outside observer. It is worth emphasizing that most of the literature on learning reviewed in this paper has followed standard practice in macroeconomics and postulates the assumption of a representative agent as a simplification. This implies that the expectations and the learning rules of different agents are assumed to be identical. Some recent papers allow for heterogeneity in learning and this work is discussed below. 1.1.2. Role o f learning in macroeconomics
Introducing learning into dynamic expectations models has several motivations. First, learning has been used to address the issue of the plausibility of the RE assumption in a particular model: could boundedly rational agents arrive at RE through a learning rule? This issue is of interest as it provides a justification for the RE hypothesis. The early work by DeCanio (1979), Bray (1982) and Evans (1983) focused on this, and some further papers are Bray and Savin (1986), Fourgeaud, Gourieroux and Pradel (1986), Marcet and Sargent (1989b), and Guesnerie (1992). This view is forcefully expressed by Lucas (1986), though he views the adjustment as very quick. Secondly, there is the possibility of models with multiple REE (rational expectations equilibria). If some REE are locally stable under a learning rule, while others are locally unstable, then learning acts as a selection device for choosing the REE which we can expect to observe in practice. This point was made in Evans (1985) and Grandmont (1985) and developed, for example, in Guesnerie and Woodford (1991) and Evans and Honkapohja (1992, 1994b, 1995a). Extensive recent work has been devoted to obtaining stability conditions for convergence of learning to particular REE and this work is discussed in detail in the later sections of this paper. A particular issue of interest is the conditions under which there can be convergence to exotic solutions, such as sunspot equilibria. This was established by Woodford (1990). Thirdly, it may be of interest to take seriously the learning dynamics itself, e.g. during the transition to RE. Dynamics with learning can be qualitatively different from, say, fully rational adjustment after a structural change. This has been the focus of some policy oriented papers, e.g. Taylor (1975), Frydman and Phelps (1983), Currie, Garratt and Hall (1993) and Fuhrer and Hooker (1993). It has also been the focus of some recent work on asset pricing, see Timmermann (1993, 1996) and Bossaerts (1995). Brian Arthur [see e.g. papers reprinted in Arthur (1994)] has emphasized path-
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dependence of adaptive learning dynamics in the presence of multiple equilibria. If the model is misspecified by the agents, then this can effectively lead to persistent learning dynamics as in Evans and Honkapohja (1993a), Marcet and Nicolini (1998) and Timmermann (1995). Even if the model is not misspecified, particular learning dynamics may not fully converge to an REE and the learning dynamics may be of intrinsic interest. This arises, for example, in Arifovic (1996), Evans and Ramey (1995), Brock and Hommes (1996, 1997), and Moore and Schaller (1996, 1997) 3. The theoretical results on learning in macroeconomics have begun to receive some support in experimental work [e.g. Marimon and Sunder (1993, 1994) and Marimon, Spear and Sunder (1993)] though experimental work in macroeconomic set-ups has so far been less than fully studied. We review this work in Section 5.4. The implications of these results have led also to one further set of issues: the effects of policy, and appropriate policy design, in models with multiple REE. For example, if there are multiple REE which are stable under learning, then policy may play a role in which equilibrium is selected, and policy changes may also exhibit hysteresis and threshold effects. The appropriate choice of policy parameters can eliminate or render unstable inefficient steady states, cycles or sunspot equilibria. For examples, see Evans and Honkapohja (1993a,b, 1995b). Howitt (1992) provides examples in which the stability under learning of the REE is affected by the form of the particular monetary policy 4. A further application of learning algorithms is that they can also be used as a computational tool to solve a model for its REE. This point has been noted by Sargent (1993). An advantage of such algorithms is that they find only "learnable" REE. A well-known paper illustrating a computational technique is Marimon, McGrattan and Sargent (1989). A related approach is the method of parameterized expectations, see Marcet (1994) and Marcet and Marshall (1992).
1.1.3. Alternative reduced forms The models we will consider have various reduced forms, and some preliminary comments are useful before turning to some economic examples. The form (1) assumed that contemporaneous information is available when expectations are formed. If alternatively the information set is {Yt-i, ut-i, i = 1,2, 3,...} then Yt may also (or
3 These lines of research in macroeconomicscorrespond to parallel developments in game theory. For a survey of learning in economics which gives a greater role to learning in games, see Marimon (1997). See also Fudenberg and Levine (1998). 4 Herepolicy is modeled as a rule which atomistic private agents take as part of the economicstructure. Modeling policy as a game is a different approach, see e.g. Cho and Sargent (1996a) and Sargent (1999) for the latter in the context of learning.
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instead) depend on E[_lyt, the expectation o f y t formed at t - 1, so that the reduced form is Yt = F ( y t - l , E t * _ l y t , E t lYt+l,/At) o r Yt = F ( y t 1,E t lYt, ut).
The Muth model, below, is the special case Yt = F(E[_,yt, ut). In nonlinear stochastic models a point requiring some care is the precise quantity about which expectations are formed. Even assuming that lagged values of Yt-i are not present, the required reduced form might be Yt = H ( E [ G ( y t + l , Hi+l), Ut).
In many of the early examples the model is nonstochastic. The reduced form then becomes Yt = H ( E t G ( y t + I ) ) . If H is invertible then by changing variables to Y~ = I-I-~ (yt) the model can be transformed to ~t = E [ f ( ~ t + l ) ,
(2)
w h e r e f ( ~ ) = G(H(.~)). The form (2) is convenient when one considers the possibility of stochastic equilibria for models with no intrinsic randomness, see Section 4.3. In nonstochastic models, if agents have point expectations, these transformations are unnecessary and the model can again simply be analyzed in the form yt - f ( E t Y~+I),
(3)
where f ( y ) = H ( G ( y ) ) . This is standard, for example, in the study of learning in Overlapping Generations models. Finally, mixed datings of expectations appear in some models. For example, the seignorage model of inflation often considers a formulation in which
1.2. Some economic examples
It will be helpful at this stage to give several economic examples which we will use to illustrate the role of learning. 1.2.1. The Muth model
The "cobweb" model of a competitive market in which the demand for a perishable good depends on its price and the supply, due to a production lag, depends on its
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expected price, was originally solved under rational expectations by Muth (1961). Consider the structural model qt = m l - m2Pt + Olt, q¢ = r l E ~ lpt + r ~ w t - i + vzt,
where m2, rl > 0, Vlt and vzt are unobserved white noise shocks and wt 1 is a vector o f exogenous shocks, also assumed white noise for convenience, qt is output, Pt is price, and the first equation represents demand while the second is supply. The reduced form for this model is (4)
Pt = ~ + a E t lPt + Y l w t 1 + ~lt,
where ~ = m l / m 2 , ] / = - r 2 / m 2 , and a = - r l / m 2 There is a unique REE in this model given by
~t = (vlt - 02~)/m2. Note that a < 0.
pt = a + blwt-1 + ~/t, where a=(1-a) ~/~, b = ( 1 a ) - l ) ,. Under RE, E t _ l P t = ~l + [Jwt_ 1. L u c a s a g g r e g a t e s u p p l y m o d e l . A n identical reduced form arises from the following simple macroeconomic model in the spirit o f Lucas (1973). Aggregate output is given
by qt = 7t+ O(pt - E t _ l P t ) + ~t,
while aggregate demand is given by the quantity theory equation mt + vt = P t + qt,
and the money supply follows the policy rule mt = ff~ +[9out + p l Wt 1.
Here 0 > 0 and the shocks ~t, vt, ut and wt are assumed for simplicity to be white noise. Solving for Pt in terms o f E ~ l P t , wt-1 and the white noise shocks yields the reduced form (4). For this model 0 < a = 0(1 + 0) -1 < 1. 1.2.2. A l i n e a r m o d e l w i t h m u l t i p l e R E E
Reduced form models o f the form Yt = a + [3Ei* yt+l + 6 y t - , + x w t + vt
(5)
arise from various economic models. Here Yt is a scalar, vt is a scalar white noise shock and wt is an exogenous vector o f observables which we will assume follows a stationary first-order VAR (vector auto-regression) wt = p w t 1 + et.
(6)
Variables dated t are assumed in the time t information set. An example is the linearquadratic market model described in Sections XIV.4 and XIV.6 o f Sargent (1987).
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The standard procedure is to obtain solutions of the form Yt = ?t + byl-I + Uwt + [tvt,
(7)
where D satisfies the quadratic 82 - / 3 18 +/3-1,~ = 0. For many parameter values there will be a unique stationary solution with 181 < 1. However, if externalities or taxes are introduced into the model as in Section XIV.8 of Sargent (1987), then for appropriate parameter values both roots of the quadratic are real and have absolute value less than unity, so that there are two stationary solutions of the form (7). (In this case there also exist solutions that depend on sunspots.) 1.2.3. The overlapping generations model with m o n e y
The standard Overlapping Generations model with money provides an example of a model with REE cycles. 5 Assume a constant population of two-period lived agents. There are equal numbers of young and old agents, and at the end of each period the old agents die and are replaced in the following period by young agents. In the simple version with production, the utility function of a representative agent born at the beginning of period t is U ( C t + l ) - W(nt), where Ct+l is consumption when old and nt is labor supplied when young. U is assumed increasing and concave and W is assumed increasing and convex. We assume that output of the single perishable good qt for the representative agent is given by qt = nt, and that there is a fixed stock of money M. The representative agent produces output in t, trades the goods for money, and then uses the money to buy output for consumption in t + 1. The agent thus chooses n t , M t and ct+~ subject to the budget constraints ptnt = Mt = Pt+lCt+l, where Pt is the price of goods in year t. In equilibrium ct = nt, because the good is perishable, and N(t = M. The first-order condition for the household is ~ ' ( n 3 = Et. (p,+~ pt g'(ct+l)). Using the market clearing condition ct+l = nt+l and the relation Pt/Pt+l = nt+l/nt, which follows from the market clearing condition p t n t = M , we obtain the univariate equation n t W ' ( n t ) = E t (nt+l U'(nt+O). Since w(n) =- n W ' ( n ) is an increasing function we can invert it to write the reduced form as nt = H ( E t G(nt+l)),
(8)
where H(-) - w 1(.) and G(n) = nU'(n). I f one is focusing on nonstochastic solutions, the form (8) expresses the model in terms of labor supply (or equivalently 5 Overlapping generations models are surveyed, for example, in Geanakoplos and Polemarchakis (1991).
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nt
rl t
~) t+l) I
I
n~
n t+l
n2
(a)
I
;,
n 3
at+ 1
(b)
rl t
f(nt+l)
nt
f(nt+< g)
i nL
n H
I
nL nu
nt+l
(c)
I
nH
~-
nt+~
(d) Fig. 1.
real balances) and assuming point expectations one has E[G(nL+I) = G(E[nt+l) and one can write nt = f ( E [ n t + l ) f o r f = H o G. The model can also be expressed in terms of other economically interpretable variables, such as the inflation rate :vt = p t / P t - l . One takes the budget constraint and the household's first-order condition which, under appropriate assumptions, yield savings (real balances) as a function mt ==-M/pt = S(E[Yf;t+I). Then the identity mt~5t = mt t yields ~t = S(E[ l~t)/S(E[Jvt+~). Depending on the utility functions U and W, the reduced form function f can have a wide range of shapes. If the substitution effect dominates everywhere then f will be increasing and there will be a single interior steady state (see Figure 1a), but if the income effect dominates over part of the range t h e n f can be hump-shaped. (An
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autarkic steady state state can also exist for the model.) In consequence the OG model can have perfect foresight cycles as well as a steady state (see Figure lb). Grandmont (1985) showed that for some choices o f preferences there coexist perfect foresight cycles of every order 6. Whenever there are cycles in the OG model, there are multiple equilibria, so that the role o f learning as a selection criterion becomes important. Various extensions o f the OG model can give rise to multiple (interior) steady states. Extension 1 (seignorage model). We briefly outline here two extensions o f the OG model which lead to the possibility o f multiple steady states. In the first extension we introduce government purchases financed by seignorage. Assuming that there is a fixed level o f real government purchases g financed entirely by printing money then g = (Mr - M t i)/pt. The first-order condition for the household is the same as in the basic model. Using the market clearing conditions p t n t = Mr, pt+lct+~ = Mt and Ct+l = nt+l - g we have pt/pt+l = (nt+l - g ) / n t which yields nt = H(E[((nt+l - g)U'(nt+l - g))) or nt = f(E[(nt+l - g)) for nonstochastic equilibria 7. In the case where the substitution effect dominates and where f is an increasing concave function which goes through the origin, this model has two interior steady states provided g > 0 is not too large. See Figure lc. It can be verified that the steady state n = nH corresponds to higher employment and lower inflation relative to n = nL. Extension 2 (increasing social returns). Assume again that there is no government spending and that the money supply is constant. However, replace the simple production function qt = nt by the function qt - 5C(nt, Art), where Aft denotes aggregate labor effort and represents a positive production externality. We assume U1 > 0, U2 > 0 and 5ell < 0. Here Art = Knt where K is the total number of agents in the economy. The first-order condition is now W'(nt) = E? Pt .T'l(nt,Knt) Pt+l
Ul(Ct+l).
Using pt~ot+l = qt+l/qt and Ct+l = qt+l we have W' (nt ).T'(nt, Knt ) .Ul(nt,Knt)
= Et.F(nt+~,Knt+l) U'(.U(nt+l,Knt+l)).
Letting #(nt) denote the left-hand-side function, it can be verified that #(nt) is a strictly increasing function o f nt. Solving for nt and assuming point expectations
6 Conditions for the existence of k-cycles are discussed in Grandmont (1985) and Guesnerie and Woodford (1992). 7 Alternatively, the model can be expressed in terms of the inflation rate in the form S(E; l:r,) S(E?:rt+l) - g
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yields nt = f~(E[nt+l) for a suitable j~. For appropriate specifications of the utility functions and the production functions it is possible to obtain reduced-form functionsj7 which yield three interior steady states, as in Figure ld. Examples are given in Evans and Honkapohja (1995b). Employment levels nL < n v < nH correspond to low, medium and high output levels, and the steady states nL and n v can be interpreted as coordination failures. 1.3. Approaches to learning
Several distinct approaches have been taken to learning in dynamic expectations models. We will broadly classify them into (i) Rational learning, (ii) Eductive approaches and (iii) Adaptive learning. The focus of this chapter is on adaptive learning, but we will provide an overview of the different approaches. 1.3.1. Rational learning
A model of rational learning, based on Bayesian updating, was developed by Townsend (1978) in the context of the cobweb model. In the simplest case, it is supposed that agents know the structure of the model up to one unknown parameter, ml, the demand intercept. There are a continuum of firms and each firm has a prior distribution for m I. The prior distributions of each firm are common knowledge. Townsend shows that there exist Nash equilibrium decision rules, in which the supply decision of each firm depends linearly on its own mean belief about ml and the mean beliefs of others. Together with the exogenous shocks, this determines aggregate supply qt and the price level Pt in period t, and finns use time t data to update their priors. It also follows that for each agent the mean belief about ml converges to ml as t --+ oc, and that the limiting equilibrium is the REE. Townsend extends this approach to consider versions in which the means of the prior beliefs of other agents are unknown, so that agents have distributions on the mean beliefs of others, as well as distributions on the mean of the markets distributions on the mean beliefs of others, etc. Under appropriate assumptions, Townsend is able to show that there exist Nash equilibrium decision rules based on these beliefs and that they converge over time to the REE. This approach is explored further in Townsend (1983). Although this general approach does exhibit a process of learning which converges to the REE, it sidesteps the issues raised above in our discussion of the role of learning. In particular, just as it was asked whether the REE could be reached by a boundedly rational learning rule, so it could be asked whether the Nash equilibrium strategies could be reached by a learning process. In fact the question of how agents could ever coordinate on these Nash equilibrium decision rules is even more acute, since they are based on ever more elaborate information sets. The work by Evans and Ramey (1992) on expectation calculation can also be regarded as a kind of rational learning, though in their case there is not full convergence to the REE (unless calculation costs are 0). Here agents are endowed with calculation
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algorithms, based on a correct structural model, which agents can use to compute improved forecasts. Agents balance the benefits o f improved forecasts against the time and resource costs o f calculation and are assumed to do so optimally. Formally, since their decisions are interdependent, they are assumed to follow Nash equilibrium decision rules in the number o f calculations to make at each time. Because o f the costs of expectation calculation, the calculation equilibrium exhibits gradual and incomplete adjustment to the REE. In a "Lucas supply-curve" or "natural rate" macroeconomic model, with a reduced form close to that o f the "cobweb" model, they show how monetary nonneutrality, hysteresis and amplification effects can arise. As with Townsend's models, the question can be raised as to how agents learn the equilibrium calculation decision rules 8. 1.3.2. E d u c t i v e a p p r o a c h e s
Some discussions o f learning are "eductive" in spirit, i.e. they investigate whether the coordination o f expectations on an REE can be attained by a mental process o f reasoning 9. Some o f the early discussions o f expectational stability, based on iterations o f expectation functions, had an eductive flavor, in accordance with the following argument. Consider the reduced form model (4) and suppose that initially all agents contemplate using some (nonrational) forecast rule E°t_lPt = a ° + b°'wt 1.
(9)
Inserting these expectations into Equation (4) we obtain the actual law o f motion which would be followed under this forecast rule: Pt = (t~ + a a °) + ( a b ° + 7)'wt x + tit,
and the true conditional expectation under this law o f motion: E t - l p t = (~ +ota °) + (orb ° + y ) ' w t 1.
Thus if agents conjecture that other agents form expectations according to Equation (9) then it would instead be rational to form expectations according to E11pt = a l
+bltwt_l,
where a 1 = ~ + a a ° and b 1 = y + ab °.
8 Evans and Ramey (1998) develop expectation calculation models in which the Nash equilibrium calculation decision rules are replaced by adaptive decision rules based on diagnostic calculations. This framework is then more like the adaptive learning category described below, but goes beyond statistical learning in two ways: (i) agents balance the costs and benefits of improved calculations, and (ii) agents employ a structural model which allows them to incorporate anticipated structural change. 9 The term "eductive" is due to Binmore (1987).
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Continuing in this way, if agents conjecture that all other agents form expectations according to the rule EN1pt = a N + bNtwt_l, then it would be rational to instead form expectations according to
EN~lpt = (t~ + aaN) + (]I + (ybN)twt 1. Letting 0 N! = (a N, bNt), the relationship between Nth-order expectations and (N + 1)thorder expectations is given by
~)N+a = T(oN),
N = 1,2,3,...,
(10)
where
T(O)' : (Ta(a, b), Tb(a, b) I) = (l~ + aa, g' + ab').
(11)
One might then say that the REE is "expectationally stable" if limk~o~ q~N=~= (fi, ~)~)/. The interpretation is that if this stability condition is satisfied, then agents can be expected to coordinate, through a process o f reasoning, on the REE 10. Clearly for the problem at hand the stability condition is lal < 1, and if this condition is met then there is convergence globally from any initial ~b°. For the Lucas supply model example above, this condition is always satisfied. For the cobweb model, satisfaction of the stability condition depends on the relative slopes o f the supply and demand curves. In fact we shall reserve the term "expectational stability" for a related concept based on the corresponding differential equation. The differential equation version gives the appropriate condition for convergence to an REE under the adaptive learning rules. To distinguish the concepts clearly we will thus refer to stability under the iterations (10) as iterative expectational stability or iterative E-stability. The concept can be and has been applied to more general models. Let q~ denote a vector which parameterizes the expectation fimction and suppose that T(¢) gives the parameters o f the true conditional expectation when all other agents follow the expectation function with parameters q~. An REE will be a fixed point ~} of T (and in general there may be multiple REE o f this form). The REE is said to be iteratively E-stable if q~N __+ ~} for all q~0 in a neighborhood o f ~}.
10 Interpreting convergence of iterations of (10) as a process of learning the REE was introduced in DeCanio (1979) and was one of the learning rules considered in Bray (1982). [Section 6 of Lucas (1978) also considered convergence of such iterations.] DeCanio (1979) and Bray (1982) give an interpretation based on real time adaptive learning in which agents estimate the parameters of the forecast rule, but only alter the parameters used to make forecasts after estimates converge in probability. The eductive argument presented here is based on Evans (1983), where the term "expectational stability" was introduced. Evans (1985, 1986) used the iterative E-stability principle as a selection device in models with multiple REE. Related papers include Champsaur (1983) and Gottfries (1985).
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An apparent weakness of the argument just given is that it assumes homogeneous expectations of the agents. In fact, the eductive argument based on iterative E-stability is closely related to the concept of rationalizability used in game theory, which allows for heterogeneity of the expectations of agents. The issue of rationalizability in the cobweb model was investigated by Guesnerie (1992). In Guesnerie's terminology the REE is said to be strongly rational if for each agent the set of rationalizable strategies is unique and corresponds to the REE, Guesnerie showed that if lal < 1 in Equation (4) then the REE is strongly rationalizable, so that in this case the eductive arguments are indeed compelling. Guesnerie (1992) shows that the strong rationality argument can be extended to allow also for heterogeneity in the economic structure, e.g. a different supply curve for each agent, due to different cost functions. The argument can also be extended to cases with multiple REE by making the argument local. In Evans and Guesnerie (1993) the argument is extended to a multivariate setting and the relationship between strong rationality and iterative E-stability is further examined. I f the model is homogeneous in structure, then (even allowing for heterogeneity in beliefs) an REE is strongly rational if and only if it meets the iterative E-stability condition. However, if heterogeneity in the structure is permitted, then iterative E-stability is a necessary but not sufficient condition for strong rationality of the REE. For an investigation of strong rationality in univariate models with expectations of future variables, see Guesnerie (1993). Guesnerie (1996) develops an application to Keynesian coordination problems. 1.3.3. Adaptive approaches
We come now to adaptive approaches to learning, which have been extensively investigated over the last 15 years. In principle, there is a very wide range of adaptive formulations which are possible. As Sargent (1993) has emphasized, in replacing agents who are fully "rational" (i.e. have "rational expectations") with agents who possess bounded rationality, there are many ways to implement such a concept 11. One possibility is to extend the adaptive expectations idea by considering generalized expectation functions, mapping past observations of a variable into forecasts of future values of that variable, where the expectation function is required to satisfy certain reasonable axioms (including bounded memory in the sense of a fixed number of past observations). This approach was taken, in the context of nonstochastic models, in the early work by Fuchs (1979) and Fuchs and Laroque (1976), and the work was extended by Grandmont (1985) and Grandmont and Laroque (1986). Under appropriate assumptions it can be shown that the resulting dynamic systems can converge to perfect
It Sargent (1993) provides a wide-ranging overviewof adaptive learning. See Honkapohja(1996) for a discussion of Sargent's book. Adaptive learning is also reviewed in Evans and Honkapohja (1995a) and Marimon (1997). Marcet and Sargent (1988) and Honkapohja (1993) provide concise introductions to the subject.
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foresight steady states or cycles. Using a generalization of adaptive expectations, the conditions under which learning could converge to perfect foresight cycles were also investigated by Guesnerie and Woodford (1991). A second approach is to regard agents as statisticians or econometricians who estimate forecasting models using standard statistical procedures and who employ these techniques to form expectations of the required variables. This line of research has naturally focussed on stochastic models, though it can also be applied to nonstochastic models. Perhaps the greatest concentration of research on learning in macroeconomics has been in this area, and this literature includes, for example, Bray (1982), Bray and Savin (1986), Fourgeaud, Gourieroux and Pradel (1986), Marcet and Sargent (1989c), and Evans and Honkapohja (1994b,c, 1995c). A third possibility is to draw on the computational intelligence 12 literature. Agents are modeled as artificial systems which respond to inputs and which adapt and learn over time. Particular models include classifier systems, neural networks and genetic algorithms. An example of such an approach is Arifovic (1994). Cho and Sargent (1996b) review the use of neural networks, and the range of possibilities is surveyed in Sargent (1993) 13. We discuss these approaches in the final section of this paper. Finally, we remark that not all approaches fall neatly into one of the classes we have delineated. For example, Nyarko (1997) provides a framework which is both eductive and adaptive. Agents have hierarchies of beliefs and actions are consistent with Bayesian updating. For a class of models which includes the cobweb model, conditions are given for convergence to the Nash equilibrium of the true model. The focus of this survey is on adaptive learning and the main emphasis is on statistical or econometric learning rules for stochastic models. We now illustrate this approach in the context of the economic examples above. 1.4. Examples o f statistical learning rules 1.4.1. Least squares learning in the Muth model
Least squares learning in the context of the Muth (or cobweb) model was first analyzed by Bray and Savin (1986) and Fourgeaud, Gourieroux and Pradel (1986). They ask whether the REE in that model is learnable in the following sense. Suppose that firms believe prices follow the process Pt = a + b'wt 1 + rh,
(12)
corresponding to the unique REE, but that a and b are unknown to them. Suppose that firms act like econometricians and estimate a and b by running least squares
12 This term is now more common than the equivalent term "artificial intelligence". 13 Spear (1989) takes yet another viewpoint and looks at bounded rationality and learning in terms of computational constraints.
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regressions ofpt on wt 1 and an intercept using data {Pi, wi}i=o. t l Letting (at l, bt-1) denote their estimates at t - 1, their forecasts are given by
E[_]pt = at i + bl_~wt-i.
(13)
The values for (at 1,bt-1) are given by the standard least-squares formula / ~-1
\ 1
( aZt z- 1i l)z=; l[) b , \i=1
t-1
"~
(~Zi-lPi,~ /
\i=1
where
z, = (1 w , ) .
(14)
/
Equations (4), (13) and (14) form a fully specified dynamic system, and we can ask: Will (at, bt) I ---+ (gl, t~l)I as t --+ oo.9 The above papers showed that if a < 1 then convergence occurs with probability 1. It is notable that this stability condition is weaker than the condition lal < 1 obtained under the eductive arguments. Since ct < 0 always holds in the Muth model (provided only that supply and demand curves have their usual slopes), it follows that least squares learning always converges with probability one in the Muth model. The stability condition can be readily interpreted using the expectational stability condition, formulated as follows. As earlier, we consider the mapping (11) from the perceived law of motion (PLM), parameterized by q~t = (a, bt), to the implied actual law of motion (ALM) which would be followed by the price process if agents held those fixed perceptions and used them to form expectations. Consider the differential equation dO aT - T(q~) - ~b, where z- denotes "notional" or "artificial" time. We say that the REE ~ = (fi, b~)~ is expectationally stable or E-stable if ~} is locally asymptotically stable under this differential equation. Intuitively, E-stability determines stability under a stylized learning rule in which the PLM parameters (a, b) are slowly adjusted in the direction of the implied A L M parameters. It is easily verified that for the Muth model the E-stability condition is simply a < 1, the same as the condition for stability under least-squares learning. The formal explanation for the reason why E-stability provides the correct stability condition is based upon the theory of stochastic approximation and will be given in later sections. It should be emphasized that in their role as econometricians the agents treat the parameters of Equation (12) as constant over time. This is correct asymptotically, provided the system converges. However, during the transition the parameters of the A L M vary over time because of the self-referential feature of the model. Bray and Savin (1986) consider whether an econometrician would be able to detect the transitional misspecification and find that in some cases it is unlikely to be spotted 14. 14 Bullard (1992) considers some recursive learning schemes with time-varying parameters. However, the specification does not allow the variation to die out asymptotically.
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Consider now the model (5) and suppose that agents have a PLM (perceived law of motion) of the form (15)
Yt = a + byt 1 + c'wt + ~k,
and that they estimate the parameters a, b, and c by a least squares regression ofyt on Yt-I, wt and an intercept. Letting = (a,,b,c;),
z; = ( 1 , y ,
1,w;),
the estimated coefficients are given by
(~t=~i~oZiZi)
(16)
(i~oZiYi)
and expectations are given by Etyt+l - at + byt + e;pwt,
where for convenience we are assuming that p is known. For simplicity, estimates q~t are based only on data through t - 1. Substituting this expression for E;yt+l into Equation (5) and solving for Yt yields the ALM (actual law of motion) followed by Yt under least squares learning. This can written in terms of the T-map from the PLM to the ALM: Yt = T(~)t)'zt + W((~t) vt, where
T(0 ) = r
W
= [ (1-[3b)-'6 ] , and \ (1 -/3b)-i (/f + [3cp) /
= (1-/3b) i.
(17) (18)
(19)
Note that fixed points ~ = (fi, b, U) ~ of T(~b) correspond to REE. The analysis in this and more general models involving least squares learning is facilitated by recasting Equation (16) in recursive form. It is well known, and can easily be verified by substitution, that the least squares formula (16) satisfies the recursion g)t = ~)t-1 + ~tR/Igt-l(Yt-I - ~ 1zt l), Rt = Rt I + ~'t(zt lZ; I -Rt-~),
(20)
for Yt = 1/t and suitable initial conditions 15. Using this RLS (Recursive Least Squares) set-up also allows us to consider more general "gain sequences" Yr. The dynamic system to be studied under RLS learning is thus defined by Equations (17)-(20).
15
Rt is an estimate of the moment matrix for zt. For suitable initial conditionsR t = t 1 ~ti=l 0 ZiZ~"
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Marcet and Sargent (1989c) showed that such dynamic systems fit into the framework o f stochastic recursive algorithms which could be analyzed using the stochastic approximation approach. This technique, which is described in the next section, associates with the system an ordinary differential equation (ODE) which controls the motion o f the system. In particular, only asymptotically stable zeros ~} o f the differential equation are possible limit points o f the stochastic dynamic system, such that Ct --+ ~}. In the case at hand the ODE is dO
dT R-IMz(O)(T(())-O)'
dz " - Mz(0)-R'
where M~(O) = lim E [zt(O)zt(O)'] t-~oo
for zt(O)' = (1,yt l(0),w~) and Y,(0) = T(~)'zt(O)+ W(O)vt. Here T(0 ) is given by Equation (18). Furthermore, as Marcet and Sargent (1989c) point out, local stability o f the ODE is governed by d 0 / d r = T(0) - 0. It thus follows that E-stability governs convergence of RLS learning to an REE o f the form (7). For the model at hand it can be verified that if there are two stationary REE o f the form (7), then only one o f them is E-stable, so that only one o f them is a possible limit point o f RLS learning. This is an example of how RLS learning can operate as a selection criterion when there are multiple REE 16
1.4.3. Learning a steady state We now consider adaptive learning of a steady state in nonstochastic nonlinear models of the form (3):
Yt =f(E[yt+l). The basic OG model and the extensions mentioned above fit this framework. One natural adaptive learning rule is to forecast Yt+l as the average o f past observed values, * = (t - - 1)-l ~ i = t-I oYi for t = 1,2,3 . . . . . Since the model is nonstochastic, i.e. Etyt+l the traditional adaptive expectations formula E[yt+x = Et*_yt + Y(Yt 1 - E[_lYt) for fixed 0 < ~/~< 1 also has the potential to converge to a perfect foresight steady state.
16 Marcet and Sargent (1989c) focused on set-ups with a unique REE. Evans and Honkapohja (1994b) showed how to use this framework in linear models with multiple REE and established the connection between RLS learning and E-stability in such models. In these papers convergence with probability 1 is shown when a "Projection Facility" is employed. Positive convergence results when learning does not incorporate a projection facility are given in Evans and Honkapohja (1998b), which also gives details for this example. See also Section 2.4.2 for discussion.
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Both of these cases are covered by the following recursive formulation, in which for convenience we use q~t to denote the forecast at time t ofyt+l: E[yt+l = q~t, where Ot = ()t-i + Yt(yt 1 -q)t-l), and where the gain sequence 7t satisfies
0 2 / y - 1 (these possibilities correspond to the cases yf'(~) < 1 and y f f ( ~ ) > 1). Under decreasing gain the condition is again simplyf1(~) < 1 (which again is also the small-gain limit for the constant-gain case) 18. S u m m a r y under small gain. Thus for the small-gain case, i.e. assuming either decreasing gain or a sufficiently small constant gain, the condition for local stability under adaptive learning is not affected by the timing assumption and is simply f t ( ~ ) < 1. Returning to our various examples, it follows that the steady states in Figures la and lb are stable under adaptive learning. In Figure lc, the high-output, low-inflation steady state is locally stable, while the low-output, high-inflation steady state is locally unstable. Finally, in Figure ld the high- and low-output steady states nL and nH are locally stable, while nu is locally unstable. As is clear from the above discussion, in the case of sufficiently large constant gains the stability condition is more complex and can depend sensitively on timing assumptions. [Lettau and Van Zandt (1995) analyze the possibilities in detail for some frameworks.] Our treatment concentrates on the decreasing-gain case in large part because in stochastic models, such as the linear models discussed above, decreasing gain is required to have the possibility of convergence to an REE. This also holds if intrinsic noise is introduced into the nonlinear models of this section, e.g. changing model (2) to Yt = E t f ( Y t + l ) + vt. Even if Ut is iid with arbitrarily small support, the above learning rules with constant gain cannot converge to an REE while with decreasing gain and appropriate assumptions we still obtain (local) convergence to the (noisy) steady state if the stability condition is met 19. Finally, we remark that the key stability condition, f f ( ~ ) < 1 for stability of a steady state under adaptive learning with small gain, corresponds to the E-stability condition. In this case the PLM is taken to be simply Yt = ~ for an arbitrary ~b. Under this PLM the appropriate forecast Ei*f(Yt+l) is f(~b) and the implied A L M is Yt = f(q~). The
18 The formal results for the decreasing-gain case can be established using the results of Section 2. Alternatively, for a direct argument see Evans and Honkapohja (1995b). 19 However, we do think the study of constant-gain learning is important also for stochastic models. For example, Evans and Honkapohja (1993a) explore its possible value if either (i) the model is misspecified or (ii) other agents use constant-gain learning.
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T-map from the PLM to the A L M is just f ( 0 ) , so the E-stability differential equation is dqVdT = f ( q ~ ) - q~, giving the stability conditionf'(~}) < 1. Stability under adaptive learning and E-stability for cycles and sunspots in nonlinear models are reviewed in Section 4. 1.4.4. The seignorage model o f inflation
The preceding analysis of learning a steady state is now illustrated by a summary discussion of the implications of adaptive learning for the seignorage model of inflation. This model is chosen because of its prominence in the macroeconomic literature. The seignorage model is discussed, for example, in Bruno (1989) and in Blanchard and Fischer (1989), pp. 195-201. In this model there is a fixed level of government expenditures g financed by seignorage, i.e. g = (Mt - Mt O/pt or g = Mt/pt - (Pt 1/Pt)(Mt-l/Pt-1). The demand for real balances is given by Mt/pt = S(E[pt+I/pt) and it is assumed that S t < 0. The model can be solved for inflation :rt = pt~ot-i as a function of Et*:rt+l ~ E[pt+l/pt and El_ 1srt oi" equivalently (in the nonstochastic case) can be written in terms o f M S p t and Et(Mt+l~Ot+l ). For the Overlapping Generations version of this model, given as Extension 1 of Section 1.2.3, nt = Mt/pt and the model was written as nt = f ( E [ n t + l ) . In order to apply directly the previous section's analysis of adaptive learning, we initially adopt this formulation. The central economic point, illustrated in Figure lc, is that for many specifications there can be two steady states: a high real balances (low-inflation) steady state nt = nil, err = :q satisfying 0 < f ( n H ) < 1, and a low real balances (high-inflation) steady state nt = hE, Jrt = Sr2 satisfying f~(nL) > 1. In the economic literature [e.g. Bruno (1989)] the possibility has been raised that convergence to nL provides an explanation o f hyperinflation. The analysis of the previous section shows that this is unlikely unless the gain parameter is large (and the alternative timing assumption used). In the smallgain case, the low-inflation/high real balances steady state nH is locally stable under learning and the high-inflation/low real balances steady state nL is locally unstable, in line with E-stability. Suppose instead that the model is formulated in terms of inflation rates. In this case the reduced form is :rt -
S(Et l,Tgt) S(E?ev~+I) - g"
In the usual cases considered in the literature we now have that h(~) -= S ( ~ ) / ( S ( s r ) - g ) is increasing and convex (for Jr not too large), and of course for small deficits g we have the two steady-state inflation rates as fixed points of h(~). The low-inflation (high real balances) steady state ~1 then satisfies 0 < h~(~l) < 1 and the high-inflation (low real balances) steady state ~2 satisfies h~(~2) > 1. E-stability for the PLM ~t = q) is determined by d 0 / d r = h(q~) - q~, so that again the low-inflation steady state is E-stable and the high-inflation steady state is not.
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Under adaptive learning of the inflation rate we assume that E[~t+l = 0t with either the standard timing Ot = Or-1 + gt(~t 1 - q~t-1) or the alternative timing assumption Ot = Ot-1 + gt(J~t - Ct-1). This set-up has been examined in detail by Lettau and Van Zandt (1995). They find that in the constant-gain case, for some values of y, the high-inflation steady state can be stable under learning under the alternative timing assumption. However, their analysis confirms that with small constant gain or decreasing gain the low-inflation steady state is always locally stable under adaptive learning and the high-inflation steady state is always locally unstable under adaptive learning. To conclude the discussion we make two further points. First, in some papers the learning is formulated in terms of price levels, rather than real balances or inflation rates, using least squares regressions of prices on lagged prices. Such a formulation can be problematic since under systematic inflation the price level is a nonstationary variable 2o. Second, the seignorage model has been subject to experimental studies, see Marimon and Sunder (1993) and Arifovic (1995). Their results suggest convergence to the low-inflation, high-output steady state. Such results accord with the predictions of decreasing or small-constant-gain learning. 1.5. Adaptive learning and the E-stability principle
We have seen that when agents use statistical or econometric learning rules (with decreasing gain), convergence is governed by the corresponding E-stability conditions. This principle, which we will treat as a unifying principle throughout this paper, can be stated more generally. Consider any economic model and consider its REE solution. Suppose that a particular solution can be described as a stochastic process with a particular parameter vector } (e.g. the parameters of an autoregressive process or the mean values over a k-cycle). Under adaptive learning our agents do not know ~} but estimate it from data using a statistical procedure such as least squares. This leads to estimates ~bt at time t and the question is whether q~t ---+~ as t ---, c~. For a wide range of economic examples and learning rules we will find that convergence is governed by the corresponding E-stability condition, i.e. by local asymptotic stability of ~}under the differential equation
dO dr
T(~0) - q},
(22)
where T is the mapping from the PLM q~ to the implied ALM T(q~). The definition of E-stability based on the differential equation (22) is the formulation used in Evans (1989) and Evans and Honkapohja (1992, 1995a). This requirement of E-stability is less strict than the requirement of iterative E-stability based on
20 See also the discussion in Section 5.1.4.
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Equation (10)21. As became evident from the results o f Marcet and Sargent (1989c), it is the differential equation formulation (22) which governs convergence of econometric learning algorithms. This form o f E-stability has been systematically employed as a selection rule with multiple REE in linear models by Evans and Honkapohja (1992, 1994b) and Duffy (1994), and in nonlinear models by Evans (1989), Marcet and Sargent (1989a), and Evans and Honkapohja (1994c, 1995b,c). O f course, there may be alternative ways to parameterize a solution and this may affect stability under learning. In particular, agents may use perceived laws of motion that have more parameters than the REE o f interest, i.e. overparameterization o f the REE may arise. This leads to a distinction between weak vs. strong E-stability. An REE is said to be weakly E-stable if it is E-stable as above, with the perceived law o f motion taking the same form as the REE. Correspondingly, we say that an REE is strongly E-stable if it is also locally E-stable for a specified class o f overparameterized perceived laws o f motion. (The additional parameters then converge to zero.) 22 We remark that, since it may be possible to overparameterize solutions in different ways, strong E-stability must always be defined relative to a specified class o f PLMs 23. Finally, as a caveat it should be pointed out that, although the bulk of work suggests the validity o f the E-stability principle, there is no fully general result which underpins our assertion. It is clear from the preceding section that the validity o f the principle may require restricting attention to the "small-gain" case (gain decreasing to zero or, if no intrinsic noise is present, a sufficiently small constant gain). Another assumption that will surely be needed is that the information variables, on which the estimators are based, remain bounded. To date only a small set o f statistical estimators has been examined. We believe that obtaining precise general conditions under which the E-stability principle holds is a key subject for future work.
1.6. Discussion of the literature In the early literature the market model o f Muth (1961), the overlapping generations model and some linear models were the most frequently used frameworks to analyze learning dynamics. Thorough treatments o f learning dynamics in the Muth model were given by Bray and Savin (1986) and Fourgeaud, Gourieroux and Pradel (1986). Interestingly, without mentioning rational expectations, Carlson (1968) proposed that price expectations be formed as the mean o f observed past prices in study the linear
21 There is a simple connection between E-stability based on Equation (22) and the stricter requirement of iterative E-stability. An REE ~ is E-stable if and only if all eigenvalues of the derivative map DT(~) have real parts less than one. For iterative E-stability the requirement is that all eigenvalues of DT(~) lie inside the unit circle. 22 Early applications of the distinction between weak and strong stability, introduced for iterative E-stability in Evans (1985), include Evans and Honkapohja (1992), Evans (1989) and Woodford(1990). 23 In an analogous way, E-stability can also be used to analyze non-REE solutions which are tmderparameterized. See Section 5.3 below.
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non-stochastic cobweb (or Muth) model. Auster (1971) extends the convergence result for the corresponding nonlinear setup. Lucas (1986) is an early analysis of the stability of steady states in an OG model. Grandmont (1985) considers the existence of deterministic cycles for the basic OG model. He also examines learning using the generalizations of adaptive expectations to finite-memory nonlinear forecast functions. Guesnerie and Woodford (1991) propose a generalization to adaptive expectations allowing possible convergence to deterministic cycles. Convergence of learning to sunspot equilibria in the basic OG model was first discovered by Woodford (1990). Linear models more general than the Muth model were considered under learning in the early literature. Marcet and Sargent (1989c) proposed a general stochastic framework and technique for the analysis of adaptive learning. This technique, studied e.g. in Ljung (1977), is known as recursive stochastic algorithms or stochastic approximation. (Section 2 discusses this methodology.) Their paper includes several applications to well-known models. Margaritis (1987) applied Ljung's method to the model of Bray (1982). Grandmont and Laroque (1991) examined learning in a deterministic linear model with a lagged endogenous variable for classes of finite memory rules. Evans and Honkapohja (1994b) considered extensions of adaptive learning to stochastic linear models with multiple equilibria. Other early studies of learning include Taylor (1975) who examines learning and monetary policy in a natural rate model, the analysis of learning in a model of the asset market by Bray (1982), and the study Blume and Easley (1982) of convergence of learning in dynamic exchange economies. Bray, Blume and Easley (1982) provide a detailed discussion of the early literature. The collection Frydman and Phelps (1983) contains several other early papers on learning. Since the focus of this survey is on adaptive learning in stochastic models we will not comment here on the more recent work in this approach. The comments below provide references to approaches and literature that will not be covered in detail in later sections. For Bayesian learning the first papers include Turnovsky (1969), Townsend (1978, 1983), and McLennan (1984). Bray and Kreps (1987) discuss rational learning and compare it to adaptive approaches. Nyarko (1991) shows in a monopoly model that Bayesian learning may fail to converge if the true parameters are outside the set of possible prior beliefs. Recent papers studying the implications of Bayesian learning include Feldman (1987a,b), Vives (1993), Jun and Vives (1996), Bertocchi and Yong (1996) and the earlier mentioned paper by Nyarko (1997). A related approach is the notion of rational beliefs introduced by Kurz (1989, 1994a,b). The collection Kurz (1997) contains many central papers in this last topic. The study of finite-memory learning rules in nonstochastic models was initiated in Fuchs (1977, 1979), Fuchs and Laroque (1976), and Tillmann (1983) and it was extended in Grandmont (1985) and Grandmont and Laroque (1986). These models can be viewed as a generalization of adaptive expectations. A disadvantage is that the finite-memory learning rules cannot converge to an REE in stochastic models, cf.
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e.g. Evans and Honkapohja (1995c). Further references of expectation formation and learning in nonstochastic models are Grandmont and Laroque (1990, 1991), Guesnerie and Woodford (1991), Moore (1993), B6hm and Wenzelburger (1995), and Chatterji and Chattopadhyay (1997). Learning in games has been subject to extensive work in recent years. A small sample of papers is Milgrom and Roberts (1990, 1991), Friedman (1991), Fudenberg and Kreps (1993, 1995), Kandori, Mailath and Rob (1993), and Crawford (1995). Recent surveys are given in Marimon and McGrattan (1995), Marimon (1997), and Fudenberg and Levine (1998). Kirman (1995) reviews the closely related literature on learning in oligopoly models. Another related recent topic is social learning, see e.g. Ellison and Fudenberg (1995) and Gale (1996).
2. General methodology: recursive stochastic algorithms 2.1. General setup and assumptions In the first papers on adaptive learning, convergence was proved directly and the martingale convergence theorem was the basic toot, see e.g. Bray (1982), Bray and Savin (1986), and Fourgeand, Gourieroux and Pradel (1986). Soon it was realized that it is necessary to have a general technique to analyze adaptive learning in more complex models. Marcet and Sargent (1989b,c) and Woodford (1990) introduced a method, known as stochastic approximation or recursive stochastic algorithms, to analyze the convergence of learning behavior in a variety of macroeconomic models. A general form of recursive algorithms can be described as follows. To make economic decisions the agents in the economy need to forecast the current and/or future values of some relevant variables. The motions of these variables depend on parameters whose true values are unknown, so that for forecasting the agents need to estimate these parameters on the basis of available information and past data. Formally, let Ot E N ~ be a vector of parameters and let Ot = Ot-I q- ytQ(t, Ot-l,Xt)
(23)
be an algorithm describing how agents try to learn the true value of 0. It is written in a recursive form since learning evolves over time. Here Yt is a sequence of"gains", often something like Yt = t 1. Xt C tRk is a vector of state variables. Note that in general the learning rule depends on the vector of state variables. This vector is taken to be observable, and we will postulate that it follows the conditionally linear dynamics Y t = A ( O t - l ) Y t l + B(Ot
l)mt,
(24)
where Wt is a random disturbance term. The detailed assumptions on this interrelated system will be made below 24. 24 Note that somewhat different timing conventions are used in the literature. For example, in some expositions Wt 1 may be used in place of Wt in Equation (24). The results are unaffected as long as W~ is an iid exogenousprocess.
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Note that the least squares learning systems in Section 1.4 can be written in the form (23) and (24). For example, consider the system given by Equations (17) and (20). Substituting Equation(17) into Equation (20) and setting St-i = Rt yields an equation of the form (23), with O[ = (at, bt, c;, vec(St)') and X/ = (1,yt 1, w~,yt-2, w;_ 1, oi-0, and it can be checked that Xt follows a process of the form (24) 2s.
2.1.1. Notes on the technical literature The classical theory of stochastic approximation, see Robbins and Monro (1951) and Kiefer and Wolfowitz (1952), was developed for models without full state variable dynamics and feedback from parameter estimates. Recent expositions of stochastic approximation are given e.g. in Benveniste, Metivier and Priouret (1990), Ljung, Pflug and Walk (1992), and Kushner and Yin (1997). A widely cited basic paper is Ljung (1977), which extended stochastic approximation to setups with dynamics and feedback. Ljung's results are extensively discussed in the book by Ljung and S6derstr6m (1983). A further generalization of Ljung's techniques is presented in Benveniste, Metivier and Priouret (1990). A somewhat different approach, based on Kushner and Clark (1978), is developed in Kuan and White (1994). An extension of the algorithms to infinite-dimensional spaces is given in Chen and White (1998). Stochastic approximation techniques were used by Arthur, Ermoliev and Kaniovski (1983, 1994) to study generalized urn schemes. Evans and Honkapohj a (1998a,b) and the forthcoming book Evans and Honkapohj a (1999a) provide a synthesis suitable for economic theory and applications. The exposition here is based primarily on these last-mentioned sources. Other useful general formulations are Ljung (1977), Marcet and Sargent (1989c), the appendix of Woodford (1990), and Kuan and White (1994).
2.2. Assumptions on the algorithm Let Ot c IRd be a vector of parameters and Xt E IRk be a vector of state variables. At this stage it is convenient to adopt a somewhat specialized form of Equation (23), so that the evolution of 0t is assumed to be described by the difference equation 0 t = Or_ 1 q- ]/tJ-~(Ot 1,Xt) + ]/2tPt(Ot-l,~t ).
(25)
Here ~(.) and Pt(') are two functions describing how the vector 0 is updated (the second-order term Pt(') is often not present). Note that in Equation (25) the function Q(t, Ot-l,Xt) appearing in Equation (23) has been specialized into first- and secondorder terms in the gain parameter Yr.
25 Here vec denotes the matrix operatorwhich forms a column vector from the matrix by stacking in order the columns of the matrix.
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Next we come to the dynamics for the vector of state variables. In most economic models the state dynamics are assumed to be conditionally linear, and we postulate here that Xt follows Equation (24). Without going into details we note here that it is possible to consider more general situations, where Xt follows a Markov process dependent on 0t-l. This is needed in some applications, and the modifications to the analysis are presented in detail in Evans and Honkapohja (1998a). For local convergence analysis one fixes an open set D C N J around the equilibrium point of interest. The next step is to formulate the assumptions on the learning rule (25) and the state dynamics (24). We start with the former and postulate the following: (A.1). 7t is a positive, nonstochastic, nonincreasing sequence satisfying O(3
OOforallgE~, 8~8", (a) U(8*)=O, 0~8", (b) U'(8) h ( 8 ) < O f o r a l l g E ~ , (c) U(8) --~ oc if 8 ---+ 0 ~ or 181 -+ 0 0 for the contour sets of the Lyapunov function. Also let P~ ..... be the probability distribution of (X~, 8t)t>~n with X~ = x, On = a. The following theorem is the basic convergence result for recursive stochastic algorithms (25) and (24): T h e o r e m 1. Let 8" be an asymptotically stable equilibrium point of the ODE d 8 / d r = h(0(r)). Suppose assumptions (A) and (B) are satisfied on D = int(K(c)) for some c > O. Suppose that for 0 < cl < c2 we have K(c2) C D. Then (i) Va E K(Cl),n >/O,x one has
Pn, x,a{Ot leaves K(c2) infinite time or Ot ---+8*} = 1, and (ii) .for any compact Q c D there exist constants' B2 and s such that Va E Q, n >~ 0, x"
P~ .... (8, --+ 8*} ~ x -O2 (a + fxlS)j(n), where J(n) is a positive decreasing sequence with limn~o~ J(n) = O.
Remark:J(n) isinfactgivenbyJ(n)=
(
1+
~ t=n+l
Z2
~ t=n+l
)
7~ •
To interpret the results one first fixes the contour sets K(cl) C K(c2). The theorem states two things. First, the algorithm either converges to 0* or diverges outside K(c2). Second, the probability o f converging to 8" is bounded from below by a sequence o f
26 For converses of the Lyaptmov stability theorems see Hahn (1963, 1967). 27 0~ denotes the boundary of ~.
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numbers which tends to unity as n ~ oc. In other words, if at some large value of t the algorithm has not gone outside K(c2) then it will converge to 0* with high probability. 2.4. Convergence: fi~rther discussion 2.4.1. Immediate consequences The following two results are special cases for obtaining statements about convergence when starting at time 0. The first result is an immediate consequence of the second part of Theorem 1: Corollary 2. Suppose Yt = ~g[, where ~ satisfies (A.1). Let the initial value o f 0 belong to some compact Q c D. Then V6 > 0 : 3~* such that V0 < ~ < ~* and aEQ:
Po, x,a{ot ~ o*} >1 1 - 6 . This is the case of slow adaption. For slow enough adaption the probability of convergence can be made "very close" to one. For general adaption speeds and with additional assumptions it is possible to obtain convergence with positive probability: Corollary 3. Assume that O* is locally asymptotically stable for the ODE. Assume that each component of Wt is' either a random variable with positive continuous density or else is constant. Fix a compact set Q c D, such that O* E int(Q), and a compact set J c II{k. Suppose that for every Oo E Qo and Xo E Jo in some sets Qo and Jo, and for every n > O, there exists a sequence Wo. . . . . WT, with T ) n, such that Or E int(Q) and X r E int(J). Then
P0..... { 0 ~ 0 " } > 0 for all a C Qo and x c Jo. It must be emphasized that it is not in general possible to obtain bounds close to unity even for the most favorable initial conditions at this level of generality. The reason is that for small values of t the ODE does not approximate well the algorithm. For early time periods sufficiently large shocks may displace Ot outside the domain of attraction of the ODE. 2.4.2. Algorithms with a projection facility In the earlier literature [e.g., Marcet and Sargent (1989b,c), Evans and Honkapohja (1994b,c, 1995c)] this problem was usually avoided by an additional assumption, which is called the Projection Facility (PF). It is defined as follows: For some 0 < Cl < c2, with K(c2) c D, the algorithm is followed provided Ot c int(K(c2)).
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Otherwise, it is projected to some point in K(ci). An alternative to PF, see e.g. Ljung (1977), is to introduce the direct boundedness assumption that the algorithm visits a small neighborhood o f the equilibrium point infinitely often. This condition is often impossible to verify. The hypothesis of a PF has been criticized as being inappropriate for decentralized markets [see Grandmont (1998), Grandmont and Laroque (1991) and Moreno and Walker (1994)]. The basic results above do not invoke the projection facility which has in fact a further strong implication. With a PF the probability for convergence to a stable equilibrium point can be made equal to unity: Corollary 4. Consider the general algorithm augmented by a projection facility. Then
Vx, a : Po, x,a{Ot --+ 0"} = 1.
We omit the proof, which is a straightforward consequence of the main theorems, see Evans and Honkapohja (1998a). Finally, we note here that almost sure local convergence can be obtained in some special models, provided that the support o f the random shock is sufficiently small, see Evans and Honkapohja (1995c). Also for nonstochastic models there is no need to have a PF when one is interested in local stability. However, for some nonstochastic models problems with continuity o f the functions in the learning algorithm may arise 28.
2.5. Instability results We will now consider the instability results which will, broadly speaking, state the following: (i) The algorithm cannot converge to a point which is not an equilibrium point o f the associated ODE, and (ii) the algorithm will not converge to an unstable equilibrium point o f the ODE. We will have to adopt a new set o f conditions 29. Let again Ot C IRa be a vector o f parameters and adopt the general form (23) for the algorithm, i.e. Ot = Ot 1 + ~/tQ(t, Ot-I ,~t). Below we will impose assumptions directly on Q(.). Again, Xt E 1Rk is the vector of observable state variables with the conditionally linear dynamics (24), i.e. Xt = A(Ot_i)Xt_l + B(Ot_l)Wt. Select now a domain D* C IRd such that all the eigenvalues o f A(O) are strictly inside the unit circle V0 c D*. The final domain o f interest will be an open and connected set
28 For example, the moment matrix in recursive least squares can become singular asymptotically. See Grandmont and Laroque (1991) and Grandmont (1998) for a discussion. Evans and Honkapohja (1998b) and Honkapohja (1994) discuss the differences between stochastic and nonstuchastic models. 29 The main source for the instability results is Ljung (1977). (We will adopt his assumptions A.) A slightly different version of Ljung's results is given in the appendix of Woodford (1990). For an instability result with decreasing gain in a nonstochastic setup see Evans and Honkapohja (1999b).
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D C D* and the conditions below will be postulated for D. We introduce the following assumptions: (C.1). Wt & a sequence o f independent random variables with IWtl < C with probability one f o r all t. (C.2). Q(t, O,x) is C 1 in (O,x) f o r 0 E D. For fixed (O,x) the derivatives are bounded in t. (C.3). The matrices A(O) and B(O) are Lipschitz on D. (C.4). limt~oo EQ(t, O,f(t(O))= h(O) exists f o r 0 E D, where Xt(O) = A(O)Xt_l(O) + B(O)Wt. (C.5). Yt is a decreasing sequence with the properties ~ - ~ Yt = oo, ~-2~ ~t < oc Jbr
It 1] < oo.
some p, and lim suPt~o ~ ~ - ~
With these assumptions the following theorem holds [see Ljung (1977) for a proof]: T h e o r e m 5. Consider the algorithm with assumptions (C). Suppose at some point O* E D we also have the validity o f the conditions (i) Q(t, 0*,Xt(0*)) has a couariance matrix that is bounded below by a positive definite matrix, and (ii) EQ(t, O, Xt(O)) is C 1 in 0 in a neighborhood o f O* and the derivatives converge uniformly in t. Then if h(O*) ~ 0 or if Oh(O*)/O0 ~ has an eigenualue with positive real part, Pr(O~ --~ 0 " ) = 0. In other words, the possible rest points of the recursive algorithm consist of the locally stable equilibrium points of the associated ODE 30. It is worth mentioning the role of condition (i) in the theorem. It ensures that at even large values of t some random fluctuations remain, and the system cannot stop at an unstable point or nonequilibrium point. For example, if there were no randomness at all, then with an initial value precisely at an trustable equilibrium the algorithm would not move off that point. If the system is nonstochastic the usual concept of instability, which requires divergence from nearby starting points, is utilized instead. 2.6. Further remarks" The earlier stability and these instability results are the main theorems from the theory of recursive algorithms that are used in the analysis o f adaptive learning in economics. 30 This assumes that the equilibrium points are isolated. There are more general statements of the result.
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We note here that there exist some extensions yielding convergence to more general invariant sets of the ODE under further conditions. I f the invariant set consists of isolated fixed points and the dynamics can be shown to remain in a compact domain, then it is possible to prove a global result that learning dynamics converges to the set of locally stable fixed points 31. Another global convergence result for a unique equilibrium under rather strong conditions will be discussed in a moment. As already mentioned in Section 1.5 in the Introduction a simpler way of obtaining the appropriate convergence condition for adaptive learning is the concept of expectational stability. The method for establishing the connection between E-stability and convergence of real-time learning rules is obviously dependent on the type of the PLM that agents are presumed to use. For nonlinear models one usually has to be content with specific types of REE, whereas for linear models the entire set of REE can be given an explicit characterization and one can be more systematic. We remark that the parameterization of the REE and the specification of who is learning and what (i.e. perceived law of motion) can in principle affect the stability conditions. This situation is no different from other economic models of adjustment outside a full equilibrium. However, it is evident that the local stability condition that the eigenvalues of T(.) have real parts less than one is invariant to 1-1 transformations q~ --+ /3 = f ( ~ ) , where f and f-1 are both C 1 . Recall, however, that if agents overparameterize the solution this may affect the stability condition, which is captured by the distinction between weak and strong E-stability.
2.7. Two examples 2.7.1. Learning noisy steady states We consider univariate nonlinear models of the form
Yt = H(E[ G(yt+,, or+l), or),
(26)
where ut is an iid shock. Here E[G(yt+t, vt+l) denotes subjective expectations of a (nonlinear) function of the next period's value of Yt+l and the shock ut+l. Under REE E[ G(yt+ I, Vt+l) = Et G(y~+l, Vt+l), the true conditional expectation. As mentioned previously, various overlapping generations models provide standard examples that fit these frameworks.
31 Woodford(1990) and Evans, Honkapohjaand Marimon (1998a) are examples of the use of this kind of result.
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A noisy steady state for Equation (26) is given by 0 = E G ( y ( o ) , v), such that O = EG(H((), v), o),
Yt = H(O, v~).
Note that Yt is an iid process. For learning a steady state the updating rule is O, = Ot-i +t l [G(yt, v t ) - O t - l ] ,
(27)
which is equivalent to taking sample means. Contemporaneous observations are omitted for simplicity, so that we set E'[G(yt+l,vt+l) = 0t 132. Thus yt = H(Ot l,Vt) which is substituted into Equation (27) to obtain a stochastic approximation algorithm o f the form (24), (25). The convergence condition for such a "noisy" steady state is
~--~E(G(H(O, v), v) < 1.
This condition can also be obtained from the E-stability equation, since the T-map is in fact T(O) = E ( G ( H ( O , v), v). The extension to the E-stability of cycles is discussed in Evans and Honkapohja (1995c) and in Section 4 below.
2. 7.2. A model with a unique R E E The market model of Muth (196 l) was introduced in Section 1.2.1 above. We consider briefly its generalization to simultaneous equations, e.g. to multiple markets, discussed earlier in Evans and Honkapohja (1995a, 1998b): Yt = ~ + AEt_lYt + Cwt,
wt = Bwt 1 + ot.
Here Yt is an n x 1 vector o f endogenous variables, wt is an observed p x 1 vector o f stationary exogenous variables, and vt is a p x 1 vector of white noise shocks with finite moments. The eigenvalues of Bp×p are assumed to lie inside the unit circle. For simplicity, the matrix B is assumed to be known. E[_jyt denotes the expectations o f agents held at time t - 1 based on their perceived law o f motion. Assume also that I - A is invertible.
32 This avoids a simultaneity betweeny~ and 0t, see Section 1.4.3 for further discussion and references.
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This model has a unique RISE Yt = ?t + [~wt 1 + ~h,
where fi = ( I - A ) 1~, ~ = ( / _ A ) - I C B and ~Tt = Cut. Is this REE expectationally stable? Consider perceived laws of motion of the form Yt = a + bwt-1 + ~lt
for arbitrary n × 1 vectors a and n x p matrices b. The corresponding expectation function is E ; l Y t = a + bwt 1, and one obtains the actual law of motion Yt = (t~ + A a ) + ( A b + CB)wt_l + ~t,
where ~t = Cot. The T mapping is thus T ( a , b) = ( # + A a , A b + CB).
E-stability is determined by the differential equation da
db
dv - Iz+(A-I)a,
dv - CB+(A-I)b.
This system is locally asymptotically stable if and only if all eigenvalues of A negative real parts, i.e. if the roots of A have real parts less than one. In real-time learning the perceived law of motion is time-dependent:
I have
yt = at-i + bt-l Wt-1 + ~]t,
where the parameters at and bt are updated running recursive least squares (RLS). Letting ¢ = (a, b), z; = (1, w;), et = Y t - (9t-tzt-1, RLS can be written 1
-1
t
Rt = Rt 1 + t l(zt_lz~_l - Rt-I).
This learning rule is complemented by the short-run determination of the value for Yt which is Yt = T(¢t 1)zt 1 + Cot,
where T(¢) = T ( a , b) as given above. In order to convert the system into standard form (25) we make a timing change in the system governing Rt. Thus we set SL-1 Rt, so that =
st =sL 1 +t-'(ztz;-S,
1 ) + t -2
(')
(ztz;-st_l)
The last term is then of the usual form with pt(St 1, zt) = -TZT(ztzt t - St-l). The model is of the form (25) with Ot = vec(q);, St) and X / = (1, w;, w; I, vt). The dynamics for
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the state variable are driven by the exogenous processes and one can verify that the basic assumptions for the convergence analysis are met. The associated ODE can be obtained as follows. Substituting in for et and Yt one obtains for Ot
O; = O ; l + t ls,~z,-~ [T(0, 1)z, 1 + C v , - 0 t
~z~_~]'
= ~)~-1 -}- t-latllzt-lZ~l [Z(~t-l) - ~t-1]' -t- t-lStllzt lVffC'. Taking expectations and limits one obtains the ODE as d q ~ ' - R 1Mz[T(q))-q)]', dT
dR-Mz-R, dr
where Mz denotes the positive definite matrix Mz = Eztz~. The second equation is independent of q~ and it is clearly globally asymptotically stable. Moreover, since R -+ Mz, the stability of the first equation is governed by the E-stability equation
d~ - ~r(0) - 0. dr Its local stability condition is that the eigenvalues of A have real parts less than one, see above. Thus the E-stability condition is the convergence condition for the RLS learning algorithm in this model. In the next section we establish a global result that is applicable for this model.
2.8. Global convergence In this section we provide a stronger set of conditions than conditions (A) and (B) of Section 2.2, which guarantees global convergence of the recursive algorithm
0, : 0t ~ + z t ~ ( 0 , - 1 , x , ) + yTp,(O,-1,x,). The new assumptions are: (D.1). The functions 7-[(0, x) and pt( O,x) satisfy for all O, O' E iRd and all x, x' c iRk: (i) [7-{(O,x 1 ) - 7-[(O,x2)l > po for some a, po > O. Then the sequence On converges Po ..... almost surely to 0".
A proof is outlined in Evans and Honkapohja (1998b). In that paper it is also shown how this theorem can be used to establish global convergence in the multivariate linear model of Section 2.7.2. 3. Linear economic models
3.1. Characterization o f equilibria Many linear rational expectations models have multiple solutions, and this is one o f the reasons why the study o f learning in such models is o f considerable interest, as previously noted. Consider the following specification: g k Yt = a + ~-~OiYt i+ Z [ 3 i E t i-I
lYt+i+Ut,
(28)
i-O
in which a scalar endogeneous variable Yt depends on its lagged values, on expectations o f its current and future values, and on a white noise shock yr. Here Et-lYt+i denotes the expectation o f yt+i based on the time t - 1 information set. For this model it is possible to give a complete characterization o f the solutions, using the results o f Evans and Honkapohja (1986) and Broze, Gourieroux and Szafarz (1985). The technique is based on the method o f undetermined coefficients, but rather than guessing a solution of a particular kind it is applied systematically to find a representation for all possible solutions. Every solution can be written in the form
Yt
~ - 1' ~[Jk i 1--[30y t k ~e 6i -Z..~ffYt [3~nlOt- ¢....~ [jk Yt i + [,4k i= l lak i=1 k k -
k i+Vt
(29)
q-~CiOt-i+Edict-i, i-1 i-1
where et is an arbitrary martingale difference sequence, i.e. a stochastic process satisfying Et Iet = 0, and where c i , . . . , ck, dl . . . . . di¢ are arbitrary 33. Various particular 33 There is an extensive literature on solution techniques to linear RE models and different possible representations of the solutions. Some central references are Gourieroux, Laffont and Monfort (1982), Broze, Gourieroux and Szafarz (1990), Whiteman (1983), McCallum (1983), Pesaran (1981), d'Autume (1990), and Taylor (1986).
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solutions can be constructed from Equation (29) by choosing the values for the ci and di and the et process appropriately.
In the literature attention is most often focused on so-called minimal state variable (MSV) solutions to Equation (28) 34. These solutions are of the form g Yt = a + Z
biYt-i + Or.
i-I
Many macroeconomic models have expectations for which the information set includes the current values o f the variables. A characterization similar to Equation (29) is available for such models. Some models in the literature have mixed datings of expectations and/or incorporate exogenous processes other than white noise. Although there exists a general characterization for the set o f solutions in Broze, Gourieroux and Szafarz (1985), it is often easier to be creative and derive the representation by the principles outlined above. The references in the footnote above provide detailed discussions of the methods in particular frameworks. 3.2. L e a r n i n g and E-stability in unit)ariate models
In this section we give a comprehensive analysis o f adaptive learning dynamics for some specific linear setups. Although these models appear to be relatively simple, they cover a large number o f standard macroeconomic models that have been developed in the literature. Another advantage o f focusing initially on simple models is that we can study the learning dynamics for the full set o f solutions and obtain complete analytic results. It is possible to generalize the analysis o f learning to more general setups (including various multivariate models) and derive, for example, the conditions for stability of specific solutions under learning. However, these conditions become easily abstract, so that analytic results are limited and it becomes necessary to resort to numerical methods. 3.2.1. A leading example
Consider the univariate model Yt = a + [3oEt* ly t + [31E[~lYt+l + t~t,
(30)
where vt is assumed to be an exogenous process satisfying Et_l c~t = O.
34 The terminology is due to McCallum (1983), but our usage differs from his in that we only use his primary solution principle to define MSV solutions. McCallum also introduces a subsidiary principle, and he defines MSV solutions as those that satisfy both principles. McCallum (1997) argues that his MSV criterion provides a classification scheme for delineating the bubble-free solution.
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E x a m p l e 3.1. Sargent a n d Wallace (1975) "ad-hoc" model: qt = al + a2(Pt - Et_lPt) + ult, qt = bl + b2(rt - ( E t lPt+l
where a2 > 0;
Et_lPt)) + u2t,
m = co + P t + c l q t + c 2 r t + u 3 t ,
where b2 < 0;
where Cl > 0, C2 < 0.
Here q,p, and m are the logarithms o f output, the price level and the money stock, respectively, and the money stock is assumed constant, r is the nominal interest rate. This fits the reduced form (30) with yt = p t , and/31 > 0 and/3o +/31 < 1. E x a m p l e 3.2. R e a l balance m o d e l [Taylor (1977)]: qt = al + a2(m - P t ) + ult,
where a2 > 0;
qt = bl + b 2 ( r t - ( E ? _ l p t + l - E [ _ l p t ) ) + b 3 ( m - p t ) + u 2 t , m = co +Pt + qt + c2rt + c 3 ( m - p t ) + u3t,
where b2 < 0, b3 > 0;
where c2 < 0, 0 < c3 < 1.
The reduced form is Equation (30) with Yt = Pt and/31 = -rio, where /30 = b2(b3 + b2(1
a2 - c3)c~ ~ - a2) -l.
For appropriate choice o f structural parameters, any value/30 ~ 0 is possible. 3.2.1.1. A characterization o f the solutions. The set o f stochastic processes Yt = -/311a + fijl (1 - /3o) yt 1 +or + C l Ut-I +diet-1
(31)
characterizes the possible REE. cl and dl are free, and et is an arbitrary process satisfying Et let = 0. et is often referred to as a "sunspot", since it can be taken to be extrinsic to the model. We will refer to Equation (31) as the ARMA(1,1) set of solutions. These solutions can either be stochastically (asymptotically) stationary or explosive, depending on the parameter values. The ARMA(1,1) solutions are stationary if 1/311(1 --/30)] < 1. Choosing d l = 0 and cl = -/311(1 -/3o) gives an ARMA(1,1) process with a common factor for the autoregressive and moving average lag polynomials. When cancelled this yields the MSV solution 35 a Y t -- 1 -- /30 -- [31
+ yr.
(32)
The MSV solution is, o f course, often the solution chosen in applied work, and it is the unique non-explosive solution if I/3~-1(1 -/30)[ > 1. Various terminologies are in use
35 See Evans and Honkapohja (1986, 1994b) for details of this technique in general setups.
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for this situation: the model is equivalently said to be "saddle-point stable" or "regular" and the MSV solution is said to be "locally determinate". If instead l[311(1 -/30)l < 1 then the model is said to be "irregular" and the MSV solution is described as "locally indeterminate". It is precisely in this case that the A R M A solutions are stationary. We will now consider the E-stability of the various solutions, taking no account of whether the ARMA(1,1) solutions are stationary or explosive (an issue to which we will return). Obviously for this model the MSV solution is always stationary. 3.2.1.2. E-stability o f the solutions. Posit a PLM (Perceived Law of Motion) of the same form as the MSV solution:
(33)
yt = a + yr.
Under this PLM we obtain y t = a + ([3o + [31)a + vt
as the Actual Law of Motion implied by the PLM (33). For E-stability one examines the differential equation da d r - a + ([3o +[3,) a - a
(34)
with unique equilibrium ~ = a/(1 - [ 3 0 - [31). The E-stability condition is
[30 +[31 < 1.
(35)
Next consider PLMs of the ARMA(1,1) form Yt = a + b y t - l +cvt l +det-1 +vt,
(36)
where et is an arbitrary process satisfying Et l gt = O, assumed observable at t. The implied A L M is Yt = a + [3oa +/31a(1 + b) + ([3ob + [3lb2)yt-1 + (flOC + [31be)or 1 + ([3od + [31bd)et-t + yr.
(37)
The mapping from PLM to A L M thus takes the form T ( a , b , c , d ) = (a+[30a+[31a(1 +b),[3ob+[31b2,[[3oc+[31bc,[3od+[31bd),
(38)
and we therefore consider the differential equation d d~(a, b, c, d) = T(a, b, c, d) - (a, b, c, d).
(39)
Note first that (a, b) form an independent subsystem, d ( a , b) = Tab(a, b) - (a, b). Evaluating the roots of DTab - I at the ARMA(1,1) solution values a = -/311a, b = [311(1 -[30), it follows that E-stability for the ARMA(1,1) solutions requires [30 > 1,
/31 < 0.
(40)
It is then further possible to show that if (a, b) converge to the ARMA(1,1) solution values, then under Equation (39) (c, d) also converge to some value [see Evans and
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Honkapohja (1992) for details]. Hence Equations (40) are the conditions for the ARMA(1,1) solution set to be E-stable. 3.2.1.3. StrongE-stability. Reconsider the MSV solution. Suppose agents allow for the possibility that Yt might depend on Yt l, vt j and et 1 as well as an intercept and yr. Is the MSV solution locally stable under the dynamics (39)? Evaluating D T - I at ( a , b , c , d ) = ( a / ( 1 - / 3 o - / 3 1 ) , 0 , 0 , 0 ) one obtains for the MSV solution the strong E-stability conditions:
/30+/31 < 1,
/3o < 1.
(41)
These conditions are stronger than the weak E-stability condition (35). For the ARMA(1,1) solution class one obtains that they are never strongly E-stable, if one allows for PLM o f the form Yt = a + blyt-i + b2Yt 2 + c o t 1 + det 1 +yr.
(42)
The argument here is more difficult (since the linearization Of the differential equation subsystem in (b~, b2) has a zero eigenvalue), and is given in Evans and Honkapohja (1999a). See Evans and Honkapohja (1992, 1994b) for related arguments. (In fact the lack o f strong E-stability is also delicate, since the differential system based on Equation (42) exhibits one-sided stability/instability)36. 3.2.1.4. E-stability and indeterminacy. The overall situation for the model (30) is
shown in Figure 2 37. In terms o f E-stability, there are 4 regions of the parameter space. Iffl0 +/31 > 1 and fil > 0 then none of the REE are E-stable. I f Equation (41) holds then the MSV solution is strongly E-stable, while the ARMA(1,1) solution is E-unstable. If fi0 + fil < 1 and/3o > 1 then the MSV solution is weakly but not strongly E-stable, and the ARMA(1,1) solutions are also weakly E-stable. Finally if/3o +/31 > 1 and /31 < 0 then the ARMA(1,1) solution set is weakly E-stable, while the MSV solution is E-unstable. In Figure 2 the region o f indeterminacy (in which there are multiple stationary solutions) is marked by the shaded cones extending up and down from the point (1, 0). Outside this region, the MSV solution is the unique stationary solution, while inside the indeterminacy region, the A R M A solutions as well as the MSV solution are stationary. For this framework the connection between indeterminacy and E-stability can be summarized as follows. In the special case/3o = 0, indeterminacy arises iff
36 Under Equation (42) the strong E-stability condition for the MSV solution remains (4i). 37 We comment briefly on the relationship of the results given here and those in Evans (1985) and Evans and Honkapobja (1995a). The results in Evans (1985) are based on iterative E-stability, which is a stronger stability requirement. In addition both Evans (1985) and Evans and Honkapohja (1995a) used a stronger definition of weak E-stability for the MSV solution, using PLMs with yt_1 included.
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MSV solution strongly E - sta )le, ARMA solution explosive and [ i - unstable
All solutions E - unstable, ARMA solutions stationary .
All solutions E - unstable, ARMA solutions explosive
.13o MSV solution E - unstable, A R M A solutions explosive and weakly E - stable "[I ~' solution I solution '1 I strongly weakly but E - stable, not strongly 1 ARMA solutions E - stable, stationary and A R M A solutions E - unstable stationary and weakly E - stable
Fig. 2. I /31 l> 1, but the A R M A solutions are never E-stable. However if/30 > 1, cases of (weakly) E-stable A R M A solutions arise in the right-hand half o f the lower cone of indeterminacy. Thus in general there is no simple connection between weak E-stability and determinacy 3s. Applying these results to examples 3.1 and 3.2, we have the following. In the Sargent-Wallace "ad hoc" model, the MSV solution is uniquely stationary and it is strongly E-stable, while the other solutions are E-unstable. In the Taylor real-balance model we have/31 = -/3o. There are three cases: (i) if/3o < ½ then the MSV solution is uniquely stationary and is strongly E-stable, while the other solutions are E-unstable; (ii) if ~l 1 then the MSV solution is stationary and weakly (but not strongly) E-stable and the ARMA(1,1) solutions are also stationary and weakly (but not strongly) E-stable. 3.2.2. The leading example: adaptive learning 3.2.2.1. Adaptive and statistical learning o f M S V solution. Since the MSV solution is an lid process, the natural statistical estimate is the sample mean, t at = t l ~ - ~ Y t - i , i-1
which is, in recursive form, at = at-i
+ t-l(yt
-
(43)
at-l).
Inserting Yt = ol + ([30 q-/31) at 1 + ot into the recursive equation we obtain the dynamic equation at = a t - 1 - t - t
l(a-]-(/30-}-/31)
at 1--at 1 + Ut).
(44)
Thus the model (30) with PLM (33) and learning rule (43) leads to the stochastic recursive algorithm (44), which can be analyzed using the tools of Section 2. The associated ODE is just the E-stability equation (34). It follows that if the E-stability condition/30 +/31 < 1 is met, then there will be convergence of at to fi = a/(1 -/3o -/31 ) and hence of the process followed by yt to the MSV solution. Indeed, for this set-up there is a tmique zero of the ODE and under the E-stability condition it is globally stable. Thus if/3o +/31 < 1 then at ~ fi with probability 1 globally, i.e. for any initial conditions. 3.2.2.2. Learning non-MSV solutions. Consider next whether suitable statistical learning rules are capable of learning the non-MSV solutions. Since there are technical complications which arise from the continua of solutions, we start with a particular solution from the set (31) in which cl = d l = 0:
Yt = --/311a +/311( 1 --/30)Yt-1 +Vt.
(45)
We also restrict attention to the "irregular" case 1/311(1 -/30)[ < 1, so that we are considering an asymptotically stationary solution. We remark that if the model (30) is regarded as defined for t /> 1, then the solution set (45) has an arbitrary initial condition Yo, the influence of which dies out asymptotically (in the irregular case). In nonstochastic models, vt - 0 and the MSV solution is the steady state Yt = a/(1 -/30 -/31). The solutions (45) then constitute a set of paths, indexed by
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the initial Y0, converging to the steady state, and, as mentioned above, the steady state is then said to be "indeterminate". Thus the question we are now considering is the stochastic analogue to whether an adaptive learning rule can converge to an REE in the "indeterminate" case. 3.2.2.2.1. Recursiue least squares learning: the AR(1) case. We thus assume that agents have a PLM o f the AR(1) form yt = a + byt
l + Ut .
Agents estimate a and b statistically and at time t - 1 forecast Yt and Yt+l using the PLM Yt = at 1 + bt-lYt-~ + Or, where at-l, bt 1 are the estimates o f a and b at time t - 1. Inserting the corresponding forecasts into the model (30) it follows that Yt is given by Yt = Ta(at-l,bt 1)+ Tb(at l,bt-1)Yt
l+Ut,
(46)
where Ta(a,b) = a +/3oa+/31a(1 +b),
Tb(a,b) = fiob+[31b 2.
We assume that (at, bt) are estimated by ordinary least squares. Letting ¢~ = (at, bt),
zt' 1 : (1,yt 1),
least squares can be written in recursive form as Ot = Ot-i + t IRtlz t l(yt - z ~ Rt = Rt l + t
1
10t-I),
t
(47)
(zt 1Zt_l - Rt-l).
Equations (47) and (46) define a stochastic recursive algorithm, and the tools of Section 2 can be applied. In particular, for regions o f the parameter space in which Ibl < 1 we obtain the associated differential equation de d r - R - ' M z ( ¢ ) ( T ( O ) - ¢)'
dR dT - Mz(¢) - R ,
(48)
where M~(¢) = E[zt(¢)zt(¢)'] and zt(¢) is defined as the process for zt under Equation (46) with fixed Ct = ¢. Here ¢' = (a, b) and T(¢) = (Ta(a,b), Tb(a, b)). Provided that /30 +/31 < 1 and /30 > 1, the A R ( I ) solution (45) is stationary and weakly E-stable and it can be shown that the ODE (48) is locally stable at (a, b) = (-/311 a,/311 (1 -/30))39. It follows that under these conditions the solution (45) is locally stable under least squares learning. 3.2.2.2.2. Learning sunspot solutions. Consider now the full class o f ARMA(1,1) solutions (31). Assuming that ut and the sunspot variable el are observable at t, we
39 That stability of ODEs of the form (48) is governed by the stability of the differential equation de -- T(¢) - ¢ is shown in Marcet and Sargent (1989c). ~7
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can consider least squares learning which allows for this more general dependence. We now set (9~ = (at, bt, ct, dr),
z~l = (1,yt-1, vt-1, et 1),
(49)
and we continue to assume that agents use recursive least squares to update their coefficient estimates, ~bt. Thus under least squares learning the dynamic system is given by Equations (47), (49) and the equation Yt = T(Ot)'zt 1 + ut, where T(q)) is given by Equation (38). This again defines a stochastic recursive algorithm and for q~ = ( a , b , c , d ) with [b I < 1 the ODE is again o f the form (48). It is again straightforward to show that local stability of the ODE is governed by the differential equation dqVdr = T(~b) - q~ defining E-stability. There is a technical problem in applying the stochastic approximation tools, however: the assumptions are not satisfied at the ARMA(1,1) set of solutions since they include an unbounded continuum. Although this prevents a formal proof of convergence to the ARMA(1,1) set under least-squares learning, simulations appear to show that there is indeed convergence to an ARMA(1,1) solution if the E-stability condition is met 4°. See Evans and Honkapohja (1994b) for an illustration of convergence to sunspot solutions in a related model. 3.2.3. L a g g e d e n d o g e n o u s variables
In many economic models, the economy depends on lagged values as well as on expectations o f the future. We therefore extend our model (30) to allow direct feedback from Yt-~ and consider models of the form Yt = a + 6yt-1 + [3oE?_lyt + [31E[_lyt+l + or.
(50)
This reduced form is analyzed in Evans and Honkapohja (1992). E x a m p l e 3.3. Taylor (1980) o v e r l a p p i n g c o n t r a c t model: i 1 * Xt = ~ X t 1 + " ~ E t - l X t + l
+
i * * ~Y(Et-lqt + E~-lqt+~) + Ul~,
w, = ½(xt +xt 1), qt = k + mt - wt + uet, mt = Fn + (1 - ~ ) wt,
where xt is the (log) contract wage at time t, wt is the (log) average wage level, qt is aggregate output, and mt is money supply. 0 < ~ < 1, and 1 - cp is a measure of accommodation of price shocks. The reduced form is: xt = a + ½(1 - ½cPY)xt 1 1 ~cP~Et_~xt * + ½(1 1 ~q~y)E t* lXt+l + yr.
4o Recently Heinemaun (1997b) has looked at the stability under learning of the solutions to this model when agents use a stochastic gradient algorithm.
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E x a m p l e 3.4. Real balance model with p o l i c y feedback: Augment Example 3.2 with a policy feedback mt = m + dpt 1 + u4t. In this example /31 = --/30 and any value of fi0 ~ 0 can arise. 3.2.3.1. A characterization o f the solutions. The MSV solutions are o f the AR(1) form.
Guess a solution of the form (51)
Yt = ~P+PYt 1 +vt.
A solution o f this form must satisfy /31p 2 +([30 - 1)/9+ b = 0,
a(1 -[30 -/31(1 + p ) ) - I = ~fl.
If ( / 3 0 - 1) 2 -4/316 > 0 there are two solutions Pl and P2 o f the form (51). One also has the ARMA(2,1) class Yt = -/31~ a + /311( 1 - /3o)Yt 1 - (~/311yt-2 +
Ut + C l O t
1 + d, et-1,
(52)
where ~t is an arbitrary sunspot and where cl and dl are arbitrary. 3.2.3.2. Stability under learning o f the AR(1) M S V solutions. Assume that agents have a PLM of the AR(1) form yt = a + blyt 1 + or. The (weak) E-stability conditions are /3o+fil-l+/31bl
O,
where mt is the log o f the money supply at time t, and Pt is the log of the price level at time t. This can be solved for the above form with yt =--pt, wt =- mr, [3 = 7/(1 + y), and )~ = 1/(1 + 7).
43 Adaptive learning methods can be extended to show convergenceto explosive REE such as this one see Evans and Honkapohja (1994a). For further analysis of learning in nonstationary setups, see Zenner (1996).
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E x a m p l e 3.6. P P P model o f the exchange rate: In a small open economy with flexible exchange rates, perfect capital mobility, exogenous output and purchasing power parity (PPP) we have m t - P t - a - c i t + ot, c > O it = tt + E t el+l - et Pt = fix + ex. Here it is the nominal interest rate, it is the foreign nominal interest rate, et is the log of the exchange rate, mt is the log of the money snpply, pt is the log of the price level, and/Sx is the log of the foreign price level, rot, Or, 19t and ix are treated as exogenous. The
first equation represents monetary equilibrium, the second is the open parity condition and the third is the PPP equation. Solving for et we arrive at the form (54), with Yt =- et, wt a linear combination of the exogenous variables, and/3 = c/(1 + c).
Example 3.7.
Asset p r i c i n g with risk neutrality: Under risk neutrality and appropriate assumptions, all assets earn expected rate of return 1 + r, where r > 0 is the real net interest rate, assumed constant. If an asset pays dividend de at the end of period t then its price pt at t is given by Pt = (1 + r ) - I ( E t p t + l +dr).
We again have the form (54), with yt =--pt, wt =- dt and [3 = (1 + r ) 1. There is a unique M S V solution to the model given by Yt = a + bwt + vt,
f i - (1 - / 3 ) - l a / 3 b
where and
b = (1 -/3~p)-1/t.
In the context of the model (54) and particularly in the case of the asset pricing application, the MSV solution is often referred to as the f u n d a m e n t a l solution 44. Is the fundamental solution stable under least squares learning? We first obtain the (weak) E-stability conditions. The T-map is T,(a, b) =/3a + ot/3b,
Tb(a, b) -/3~flb + )~,
and it is easily verified that the fundamental solution is E-stable if [3 < 1 and/3~p < 1. We have assumed [~Pl < 1 and in all three of our economic examples above 0 0 such that for all 0 < c ~ < c there exists an e'-SSE relative to (21,Y2). (ii) I f f ' ( 2 1 ) ~ 1 andf'(j32) ~ 1 at a pair o f distinct steady states (21,;v2), there is an e such that for all 0 < e' < e there exists an e'-SSE relative to (21,22) 66 (iii) There is an e > 0 such that for all 0 < e ~ < e there exists an e'-SSE relative to a single steady state 33 if and only if [f'(33)[ > 1. The overlapping generations models sketched above provide simple examples o f SSEs and e-SSEs since these models can exhibit multiple steady states, steady states with [f~(33)[ > 1 and cycles. To conclude the discussion on the existence of SSEs we remark here that for fully specified models it is sometimes possible to utilize arguments based on global analysis (such as the index theorem o f Poincar6 and Hopf) to prove the existence o f SSEs, see the surveys cited previously.
4.3.2. Analysis of learning 4.3.2.1. Formulation of the learning rule. For learning sunspot equilibria the agents must have a perceived law o f motion that in principle can enable them to learn such 66 This kind of SSE may be called an "animal spirits" cycle in accordance with Howitt and McAfee (1992).
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an REE: If agents believe that the economy is in an SSE, a natural estimator for the value o f y t in the two different sunspot states is, for each state of the sunspot process in the past, the computation o f the average o f the observations ofyt which have arisen in that state of the sunspot st. This is a form o f state-contingent averaging. Thus let q}t = (~blt, O2t) be the estimates o f the values that Yt takes in states 1 and 2 o f the sunspot. Let also ~ t = 1 if& = j and ~Pjt = 0 otherwise be the indicator function for state j o f the sunspot. The learning rules based on state-contingent averaging can be written in the form Ojt =Oj, t-I + t ll])/,t-lq/lt_l(Yt-l--Oj, t l-t-et 1),
qjt = qj, t l + t
l(~#/,t-1--qj, t-1),
(77)
Yt = lPlt[Ygllf(Olt) + (1 -- ~11)f(02t)] + /P2t [(1 -- a't'22)f(01t) + a'g22f(02t)]
for j = 1,2. We note here that in the learning rules agents are assumed to use observations only through period t - 1. This is to avoid a simultaneity between Yt and expectations E t f ( y,+l ). Equations (77) are interpreted as follows, tqj, t_ 1 is the number o f times state j has occurred up to time t - 1. The recursion for the fraction o f observations o f state j is the second of Equations (77). The first equation is then a recursive form for the state averages, with one modification to be discussed shortly. Finally, the third of Equations (77) gives the temporary equilibrium for the model, since the right-hand side is the expectation o f the value o f f ( y t + O given the forecasts ~jt. We make a small modification in the learning rule by including a random disturbance et to the algorithm. This can be interpreted as a measurement or observation error, and it is assumed to be iid with mean 0 and bounded support (tetl < C, C > 0, with probability 1) 67.
4.3.2.2. Analysis o f convergence. We now show that, under a stability condition, the learning rule (77) above converges locally to an SSE. For this we utilize the local convergence results reviewed in Section 2. First introduce the variables
O[=(01t, e)2t,qlt,q2t),
X / = (~Pl,t 1, ~P2,t-l, et-l)
and the functions
~J'(0t l , ~ t ) = q~j,t-lq/,lt-l(Yt 1-Oj, t 1 +Et 1), ~2+i(0t 1,Xt) = *Pi,t 1 - q i , t-1,
j=
1,2,
i = 1,2.
For state dynamics we note simply that Xt is a Markov process independent of 0t. The system is then in a standard form for recursive algorithms 6s. 67 The observation error is needed only for the instability result. 68 The formal analysis requires an extension of the basic framework of Section 2 to non-iid shocks or alternatively to Markovian state dynamics as summarized in the appendix of Evans and Honkapohja (1998b) and treated in detail in Evans and Honkapohja (1998a). The formal details for the former approach are given in Woodford (1990) and Evans and Honkapohja (1994c).
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The associated differential equation governing local convergence is d 0 / d r = h(O), where
hi(0) = Yglql[~ll/(~l)+( 1 - ~ 1 1 ) / ( ~ 2 ) - ~ 1 ] , h2(0) =~2q2[(1 13g22)f(01)w,7g22]c(~2)-~2], h3(0) = ~ l - q l , h4(0) = f g 2 - q 2 .
Here (~l, £c2) is the limiting distribution of the states of the Markov chain. Clearly, at the equilibrium point ql = Yrl, q2 = ~2 and (~1, q~2)is an SSE. In the ODE d 0 / d r = h(O) the subsystem consisting of the last two components of h(O) is independent of (Ol, q52) and one has global stability for it in the domain qi E (0, 1), i = 1, 2. It follows that the entire ODE is locally stable provided DT(e)I, ~2) has all eigenvalues with real parts less than unity, where
T(qJI' q~2) =
( T I ( ~ I , ~2) ~ = ( ; r l l f ( q ) l ) + (1 -- Jrl 1)f(~2) "~ T2(~bl,q~2)J (1-~22)f(Ol)+~22f(O2)J"
(78)
Note that the function T(q~I, 02) = [T1 (¢1, ~2), T2(~l, ¢2)] defines the mapping from the perceived law of motion [Yt+x = ~ if st+l = 1, yt+l = ¢2 if st+l = 2] to the actual law of motion [Yt+i = ¢~ ifst+l = 1, Yt+l = 0~ if St+l = 2], where (¢~,~b~) = T(¢~,¢2). The condition on the eigenvalues can thus be used to define the concept of E-stability for sunspot equilibria. We have obtained the following result:
Proposition 11. The learning rule (77) converges locally to an SSE (y~,y~) provided it is weakly E-stable, i.e. the eigenvalues of DT(y~,y~) have real parts less than one.
Remark: The notion of convergence is as in Theorem 1 in Section 2. If the algorithm is augmented with a projection facility, almost sure convergence is obtained. It is also possible to derive an instability result along the lines of Evans and Honkapohja (1994c) for SSEs which are not weakly E-stable:
Proposition 12. Suppose that an SSE (y~,y~) is weakly E-unstable, so that DT(y~,y~) has an eigenvalue with real part greater than unity. Then the learning dynamics (77) converge to (Yl ,Y2) with probability zero. The stability result can also be developed for the general model (26) or (69). In this framework sunspot equilibria are noisy, because the equilibrium is influenced by both the sunspot variable as well as the exogenous preference or technology shock. This is discussed in Evans and Honkapohja (1998a).
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The preceding result shows that local convergence to SSEs can be studied using E-stability based on Equation (78). Computing D T we have
DT(y) =
a-gllf'(yl) (1 - ~ll)f'(Y2) ) (1 - ~22)f'(Yl) a-g22f'(y2) "
The analysis of E-stability of SSEs near deterministic solutions (e-SSEs) is based on two observations. First, D T ( y ) can be computed for the deterministic solutions, which are limiting cases for e-SSEs. Second, under a regularity condition, the fact that eigenvalues are continuous functions o f the matrix elements provides the E-stability conditions for e-SSEs in a neighborhood o f the deterministic solution. This approach yields the following results: (i) Given a 2-cycle ~ = (YI,.~2) w i t h f ' ( ~ l ) f ' ( ~ 2 ) -~ 0, there is an e > 0 such that for all 0 < c' < e all e'-SSEs relative to ~ are weakly E-stable if and only if P is weakly E-stable, i.e., it s a t i s f i e s f ' ( ~ 0 f ' ( ~ 2 ) < 1. (ii) Given two distinct steady states Yl ¢ Y2 there is an e > 0 such that for all 0 < e ~ < e all e~-SSEs relative to ~ = (Yl,Y2) are weakly E-stable if and only if both steady states are weakly E-stable, i . e . , f ' ( ~ l ) < 1 a n d f ' ( ~ 2 ) < 1. Analogous results are available when a 2-cycle is strongly E-stable or a pair of distinct steady states are strongly E-stable [see Evans and Honkapohja (1994c) for the definition and details]. For the case o f a single steady state the situation is more complex, but the following partial result holds: Let ~ be a weakly E-unstable steady state, i.e. S ( . v ) > 1. Then there exists an e > 0 such that for all 0 < e' < e all c'-SSEs relative to .~ are weakly E-unstable. One may recall from Proposition 3 that SSEs near a single steady state ~ also exist when f ' ( ~ ) < -1. For this case it appears that both E-stable and E-trustable e-SSEs relative to 33 may exist. However, it can be shown that there is a neighborhood of such that SSEs in the neighborhood are E-unstable in a strong sense. 4.3.4. Applying the results to OG and other models The hyperinflation model, Example 4.4 in Section 4.2.1, has often been used as an economic example for sunspot equilibria. This construction relies on the two distinct steady states o f the model. The application o f the results above shows that such equilibria near a pair o f steady states are unstable under learning. In order to construct a robust example o f such "animal spirits" sunspot solutions it is necessary to have a pair o f steady states that are both stable when agents try to learn them. Since under certain regularity conditions two stable steady states are separated by an unstable one, the construction o f a robust example o f sunspot equilibria, which is based on distinct steady states, normally requires the existence o f three steady states at a minimum.
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The model of increasing social returns, Example 4.3 in Section 4.2.1, is a simple OG model with this property. Evans and Honkapohja (1993b) develop this extension and provide simulations illustrating convergence to such an SSE. Other similar robust examples of these endogenous fluctuations are the "animal spirits" equilibria in Howitt and McAfee (1992) in a model of search externalities, and equilibrium growth cycles in Evans, Honkapohja and Romer (1998b) in a model of endogenous growth with complementary capital goods. Alternatively, stable sunspot solutions can be obtained when the model exhibits a k-cycle which is by itself stable under learning. If such a k-cycle is found, then normally there also exist stable sunspot solutions nearby, provided agents allow for the possibility of sunspots in their learning behavior. OG models with downward-sloping offer curves provide simple examples of sunspot equilibria near deterministic cycles. In addition to these local results, the original analysis of Woodford (1990) showed how to use index theorem results to obtain global stability results for SSEs in the OG model.
5. Extensions and recent developments
In this section we take up several further topics that have been analyzed in the area of learning dynamics and macroeconomics. These include some alternative learning algorithms, heterogeneity of learning rules, transitions and speed of convergence results, and learning in misspecified models. 5.1.
Genetic algorithms, classifier systems and neural networks
Some of the models for learning behavior have their origins in computational intelligence. Genetic algorithms and classifier systems have found some applications in economics. 5.1.1.
Genetic algorithms
Genetic algorithms (GA) were initially designed for finding optima in non-smooth landscapes. We describe the main features of GAs using the Muth market model which is one of the very first applications of GAs to economics. The exposition follows Arifovic (1994). We thus consider a market with n firms with quadratic cost functions Cit = x q i t + 1 2 ~ynqit, where qit is the production by firm i, and x and y are parameters. Given 1 2 price expectations Pte the expected profit of firm i is l i f t = P te qit - xqit - ~ynqit, and one obtains the supply function for firm i as qit = ( y n ) - l ( P ~ - x ) . The demand function is taken to be pt - A - B ~ - 1 qit, and the RE solution Pt = P~ yields qit = qt = (A - x ) / [ n ( B +y)].
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Arifovic (1994) considers some alternative GA's. We outline here her "singlepopulation" algorithm. Formally, there is a population A t of 'chromosomes' Ait which are strings o f length ~ o f binary characters 0, 1 : a I
g
Air = ( it, . . . ~ ait)~
k
where
ait
= 0 or 1.
To each chromosome Air one associates a production decision by firm i by the formula g
xit qit = ~ ,
where
= V" ak 2 k 1 xit ~ it • k=l
Here k is a norming factor 69. Short-run profits/~t = Hit = Ptqit - Cit provide a measure of 'fitness' for alternative chromosomes (production decisions). Here P t is the shortrun equilibrium price, given a configuration o f n chromosomes. The basic idea in a genetic algorithm is to apply certain genetic operators to different chromosomes in order to produce new chromosomes. In these operators the fitness measure provides a criterion o f success, so that chromosomes with higher fitness have a better chance o f producing offsprings to the population. The following operators are used by Arifovic (1994): (1) R e p r o d u c t i o n : Each chromosome Air produces copies with a probability which depends on its fitness. The probability o f a copy Cit is given by P ( c i t ) = ~ A i t / ( ~ ; = 1 [Ait)" The resulting n copies constitute a 'mating pool'. (2) C r o s s o v e r : Two strings are selected randomly from the pool. Next, one selects a random cutoff point, and the tails o f the selected chromosomes are interchanged to obtain new chromosome strings. Example. If there are two strings [ 110101111 ] and [001010010], and tails o f length 4 are interchanged, then the new strings are [110100010] and [001011111]. Altogether n / 2 pairs are selected (assume that n is even, for simplicity). (3) M u t a t i o n : For each string created in step 2, in each position 0 and 1 is changed to the alternative value with a small probability. These are standard genetic operations. In her analysis Arifovic (1994) adds another operator which is not present in standard G A s 70. (4) E l e c t i o n : The new 'offsprings' created by the preceding three operators are tested against their 'parents' using the profit measured at the previous price as the fitness criterion. The rules for replacement are if one offspring is better than both parents, replace the less-fit parent, -
69 Note that for large g the expressions xit can approximate any real number over the range of interest. 70 The market model does not converge when this operator is absent. Since mutation is always occurring, unless it is made to die off asymptotically, something like the election operator must be utilized to get convergence.
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if both offsprings are better, replace both parents, - if parents are better than offsprings, they stay in the population. These four operations determine a new population of size n and, given this configuration, a new short-run equilibrium price is determined by the equality of demand and output. After this the genetic operators are applied again using the new market price and profits as the fitness measure. Arifovic (t994) shows by simulations that this algorithm converges to the RE solution irrespective of the model parameter values 71. This result is remarkable, since it happens in spite of the myopia in the fitness criterion. (The system, however, has no stochastic shocks.) For some specifications it also turns out that the time paths of the GA corresponds reasonably well with certain experimental results for the market model. These genetic operations can be given broad interpretations in terms of economic behavior. First, reproduction corresponds to imitation of those who have done well. Second, crossover and mutation are like testing new ideas and making experiments. Finally, election means that only promising ideas are in fact utilized. To conclude this discussion we remark that as a model of learning the genetic algorithm is probably best interpreted as a framework of social rather than individual learning, cf. Sargent (1993). Indeed, individual firms are like individual chromosomes who are replaced by new ones according to the rules of the algorithm. -
5.1.2. Classifier systems Classifier Systems provide a different variety of learning algorithms which can be made more akin to thought processes of individuals than a GA. This allows a direct behavioral interpretation with individual economic agents doing the learning. A classifier system consists of an evolving collection of 'condition-action statements' (i.e. decision rules) which compete with each other in certain specified ways. The winners become the active decisions in the different stages. The strengths (or utility and costs) of the possible classifiers are a central part of the system and accounts are kept of these strengths. When a 'message' indicating current conditions arrives, one or more classifiers are activated as the possible decisions given the signal. Next, the competition stage starts to select the active classifier. The strengths are updated according to the performance of the active classifier. (The updating rules in fact mimic the updating of parameter estimates in stochastic approximation.) Typically, there are also ways for introducing new classifiers 72. A well-known economic application of classifier systems is Marimon, McGrattan and Sargent (1989). They introduce classifier system learning into the model of money
71 Thisfinding is consistent with the E-stabilitycondition and corresponds to the Least Squares learning results, see Sections 1.4.1 and 2.7.2: downward sloping demand and upward sloping supply is sufficient for global convergence. 72 Sargent (1993), pp. 77-81, and Dawid (1996), pp. 13-171 provide somewhat more detailed descriptions of classifier systems.
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and matching due to Kiyotaki and Wright (1989). Using simulations Marimon et al. show that learning converges to a stationary Nash equilibrium in the Kiyotaki-Wright model, and that, when there are multiple equilibria, learning selects the fundamental low-cost solution. Another recent application is Lettau and Uhlig (1999). They utilize a classifier system as a rule-of-thumb decision procedure in the usual dynamic programming setup for consumption-saving decisions. The system does not fully converge to the dynamic programming solution, and Lettau and Uhlig suggest that this behavior can account for the 'excess' sensitivity of consumption to current income. 5.1.3. Neural networks
Another very recent approach to learning models based on computational intelligence has been the use o f neural networks 73. The basic idea in neural networks is to represent an unknown functional relationship between inputs and outputs in terms o f a network structure. In general the networks can consist o f several layers o f nodes, called neurons, and connections between these neurons. The simplest example o f a network is the perceptron which is a single neuron receiving several input signals and sending out a scalar output. Infeedforward networks information flows only forward from one layer o f neurons to a subsequent one. Such a network usually has several layers o f neurons, organized so that neurons at the same layer are not connected to each other, and neurons in later layers do not feed information back to earlier layers in the structure. In network structures signals are passed along specified connections between the different neurons in the network. In each neuron input signals are weighted by some weights and the aggregate is processed through an activation function of that neuron. The processed signal is the output from that neuron, and it is sent to further neurons connected to it or if at the terminal layer as a component o f the output o f the whole network. A n important property o f these networks is that they can provide good approximations o f the unknown functional relation between the inputs and the outputs. To achieve this the networks must be 'trained': the weights for inputs at each neuron must be determined so that, given the training data, the network approximates well the functional relation present in the input and output data. This training is often based on numerical techniques such as the gradient method, and in fact many training schemes can be represented as stochastic approximation algorithms. The training can be done with a fixed data set, so that it is then an 'off-line' algorithm, or it may been done 'on-line' as a recursive scheme. In the latter case the basic setup corresponds closely to adaptive learning.
73 The use of neural networks in economics is discussed e.g. in Beltratti, Margarita and Terna (1996), Cho and Sargent (1996b), and Sargent (1993). White (1992) is an advanced treatise discussing the relationship of neural networks to statistics and econometrics.
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In economic theory, neural networks have very recently been utilized as representations of approximate functional forms, as computational devices and as an approach to bounded rationality and learning. One use of neural networks has been the computation of (approximate) solutions to economic models, see e.g. Beltratfi, Margarita and Terna (1996) for various illustrations from economics and finance. Another use of neural networks has been in modelling bounded rationality and learning. Cho (1995) uses perceptrons in the repeated prisoner's dilemma game, so that the perceptrons classify the past data and through a threshold this leads to a decision in accordance with the output of the perceptron. Such strategies are quite simple, and thus the modeled behavior is very much boundedly rational. Nevertheless, the efficient outcomes of the game can be recovered by use of these simple strategies. Cho and Sargent (1996a) apply this approach to study reputation issues in monetary policy. Other papers using neural networks as a learning device in macroeconomic models include Barucci and Landi (1995), Salmon (1995), Packal~n (1997) and Heinemann (1997a). The last two studies look at connections to E-stability in the Muth model. 5.1.4. Recent applications o f genetic algorithms
The paper by Arifovic (1994) demonstrated the potential of GAs to converge to the REE, and a natural question is whether such convergence occurs in other models, and whether, when there are multiple equilibria, there is a one-to-one correspondence between solutions which are stable under statistical or econometric learning rules and solutions which are stable under GAs. The expectational stability principle, which states that there is a close connection between stability under adaptive learning rules and expectational stability, would argue for a tight correspondence between stability under econometric learning and under GAs. One setup in which this question can be investigated is the OG model with seignorage, in which a fixed real deficit is financed by printing money. Recall that, provided the level of the deficit is not too large, there are two REE monetary steady states. E-stability and stability under adaptive learning was discussed in Sections 1.4.3 and 1.4.4. Under small-gain adaptive learning of the inflation rate, the lowinflation steady state is locally stable while the high-inflation steady state is locally unstable, consistent with the E-stability results. Learning in this model was actually first investigated under least-squares learning by Marcet and Sargent (1989a). They assumed that agents forecast inflation according to the perceived law of motion Pt+l = [3tPt, where fit is given by the least squares regression (without intercept) of prices on lagged prices. They showed that there could be convergence only to the lowinflation steady state, never to the high-inflation steady state. In addition, in simulations they found some cases with unstable paths leading to expected inflation rates at which there was no temporary equilibrium (i.e., at which it was impossible to finance the deficit through money creation). Arifovic (1995) sets up the GA so that the chromosome level represents the first period consumption of the young. Using GA simulations (with an election operator),
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she also finds convergence to the low-inflation steady state and never to the highinflation steady state. There are some differences in detail from Least Squares learning. From some starting points which lead to unstable paths under (Marcet-Sargent) leastsquares learning there was convergence under GA learning. It is possible that some of these apparent discrepancies arise from the particular least-squares learning scheme followed. Since the price level in either steady state is a trended series, whereas the inflation rate is not, it would be more natural to an econometrician to estimate the inflation rate by its sample mean rather than by a regression of prices on past prices. In any case, there does appear to be a close connection in this model between the local stability properties of statistical and GA learning, and the key features of learning dynamics are revealed by E-stability. In Bullard and Duffy (1998a), GAs are used to look at the issue of convergence to cycles in the standard deterministic OG endowment model with money. Recall that Grandmont (1985) showed that for appropriate utility functions it is straightforward to construct models in which there are regular perfect foresight cycles. Recall also that Guesnerie and Woodford (1991) and Evans and Honkapohja (1995c) provide local stability conditions for the convergence of adaptive and statistical learning rules to particular RE k-cycles. For "decreasing-gain" rules these are the E-stability conditions which are given in the above section on nonlinear models. It is therefore of interest to know whether GAs exhibit the same stability conditions. In Bullard and Duffy (1998a) agent i uses the following simple rule for forecasting next period's price: F [ [ P ( t + 1)] = P ( t - ki - 1). Different values of ki are consistent with different perfect foresight cycles. (Note that every value of ki is consistent with learning steady states). The value of ki used by agent i is coded as a bit string of length 8, so that the learning rule is in principle capable of learning cycles up to order 39. Given their price forecast, each agent chooses its optimal level of saving when young and total saving determines the price level. A GA is used to determine the values o f k i used in each generation. Note that in this setup [in contrast to the approach in Arifovic (1994, 1995)] the GA operates on a forecast rule used by the agent, rather than directly on its decision variable 74. The question they ask is: starting from a random assignment of bit strings, will the GA converge to cycles? To answer this question they conduct GA simulations for a grid of values of the parameter specifying the relative risk aversion parameter of the old. Their central finding is that, with only a handful of exceptions, there is convergence either to steady states or 2-cycles, but not to higher-order cycles. This finding raises the possibility that GAs may have somewhat different stability properties than other learning rules. However, the results are based on simulations using a GA
74 This makes GA learning closer in spirit to least squares and other adaptive learning of forecast rules. Using GAs to determine forecast rules was introduced in Bullard and Duffy (1994). Bullard and Duffy (1998b) show how to use GAs to directly determine consumption plans in n-period OG endowment economies.
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with a particular specification of the initial conditions and the forecast rule. Thus many issues concerning stability under GAs remain to be resolved 75. We close this section with a brief description of two other recent papers which use GAs in macroeconomic learning models. Arifovic (1996) considers an OG model with two currencies. This model possesses a continuum of stationary perfect foresight solutions indexed by the exchange rate. In the GA set-up each agent has a bit string which determines the consumption level and the portfolio fractions devoted to the two currencies. Fitness of string i used by a member of generation t - 1 is measured by its ex-post utility and is used to determine the proportion of bit strings in use in t + 1 according to genetic operator updating rules. The central finding is that the GA does not settle down to a nonstochastic stationary perfect foresight equilibrium, but instead exhibits persistent fluctuations in the exchange rate driven by fluctuations in portfolio fractions. Arifovic, Bullard and Duffy (1997) incorporate GA learning in a model of economic development based on Azariadis and Drazen (1990). This model, which emphasizes the roles of human capital and threshold externalities, has two perfect foresight steady states: a low-income zero-growth steady state and a highincome positive-growth steady state. In the GA set-up the bit strings encode the fraction of their time young agents spend in training and the proportion of their income they save 76. The central finding, based on simulations, is that, starting from the low-income steady state, economies eventually make a transition to the high-income steady state after a long, but unpredictable length of time. These examples illustrate that GAs can be readily adapted to investigate a wide range of macroeconomic models. An advantage of GAs in economics is that they automatically allow for heterogeneity. A disadvantage is that there are no formal convergence results. Although in some cases there are supporting theoretical arguments, the findings in economics to date rely primarily on simulations. This literature is growing fast. Dawid (1996) provides an overview of GAs and discusses their applications to both economic models and evolutionary games. Lettau (1997) considers the effects of learning via genetic algorithms in a model of portfolio choice. 5.2. Heterogeneity in learning behavior
In most of the literature on statistical and econometric learning it is assumed that the learning rules of economic agents are identical. This is a counterpart and an addition to the assumption of the existence of a representative agent. Some studies have considered models in which agents have different learning rules. An early example is Bray and Savin (1986), who allow for agents to have heterogeneous priors in the context of the Muth model. Howitt (1992) incorporates different learning rules in his analysis of the instability of interest rate pegging. Evans, Honkapohja and Marimon (1998a)
75 GA learning of 2-cycles has also recentlybeen investigatedin Arifovic (1998). 76 In this model all of the standard genetic operators are used except the election operator.
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extend the deficit financing inflation model to include a continuum of agents with identical savings functions but different learning rules. Marcet and Sargent (1989b) consider a model in which two classes of agents with different information form different expectations. Soerensen (1996) looks at adaptive learning with heterogeneous expectations in a nonstochastic OG model. In this literature there are two techniques for setting up and analyzing models with heterogenous learning. First, as pointed out by Marcet and Sargent (1989c), when setting up the problem as a recursive algorithm it is straightforward to allow for a finite range of possibly heterogeneous expectations by expanding the state vector accordingly. This is easily done when there are a finite number of different agent types. Second, in some models it may be possible to aggregate the different learning rules and obtain for mean expectations a rule that is amenable to standard techniques. Evans, Honkapohja and Marimon (1998a) is an example of this latter methodology. The stability conditions for learning are in general affected by behavioral heterogeneity. However, many models with heterogeneous agents make the assumption that the dynamics of endogenous variables in the reduced form depend only on average expectations 77. It turns out that, when the basic framework is linear, the stability condition for convergence of learning with heterogeneous expectations is identical to the corresponding condition when homogeneous expectations are imposed, see Evans and Honkapohja (1997). Finally, we remark that the models based on GAs and classifier systems discussed above can incorporate heterogeneity in learning behavior, as can the approach developed in Brock and Hommes (1997). Using the latter approach, Brock and de Fontnouvelle (1996) obtain analytical results on expectational diversity.
5.3. L e a r n i n g
in m i s s p e c i f i e d m o d e l s
In most of the literature it has been assumed that agents learn based on a PLM (perceived law of motion.) that is well specified, i.e. nests an REE of interest. However, economic agents, like econometricians, may fail to correctly specify the actual law of motion, even asymptotically. It may still be possible to analyze the resulting learning dynamics. An early example of this idea, in the context of a duopoly model, is Kirman (1983). Maussner (1997) is a recent paper focusing on monopolistic competition. As an illustration, consider the Muth model of Sections 1.2.1 and 1.4.1 with reduced form (4). Agents were assumed to have a PLM of the form P t = a + btwt_l + l"lt, corresponding to the REE. Suppose that instead their PLM is Pt = a + rh, so that
77 Frydman (1982) and some papers in the volume Frydman and Phelps (1983) have stressed the importance of average opinions.
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agents do not recognize the dependence o f price on wt-1, and that they estimate a by least squares. Then at = at-i + t l ( p t - a t - l ) , and the PLM at time t - 1 is Pt = at-1 + ~/t with corresponding forecasts E~_lpt = at-l. Thus the A L M is Pt = t.t + aat-i + yl wt 1 + 17t and the corresponding stochastic recursive algorithm is at = at-1 + t 1(~+ (a - 1)at-1 + g'wt-1 + ~Tt). The associated ODE is d a / d z = ~t + (a - 1)a, and thus from Section 2 it follows that at --+ fi = (1 - a ) - l g almost surely. (We remark that the ODE da/d'c can also be interpreted as the E-stability equation for the underparameterized class o f PLMs here considered). In this case we have convergence, but it is not to the unique REE which is Pt = (1 - a) 1# + (1 - a)-ly~wt I + rh. Agents make systematic forecast errors since their forecast errors are correlated with wt-i and they would do better to condition their forecasts on this variable. However, we have ruled this out by assumption: we have restricted PLMs to those which do not depend on wt-1. Within the restricted class o f PLMs we consider, agents in fact converge to one which is rational given this restriction. The resulting solution when the forecasts are Ei*lpt = fi is pt = (1 - a ) - l ~ + Y'w,-1 + ~,. We might describe this as a restricted perceptions equilibrium since it is generated by expectations which are optimal within a limited class o f PLMs. The basic idea o f a restricted perceptions equilibrium is that we permit agents to fall short o f rationality specifically in failing to recognize certain patterns or correlations in the data. Clearly, for this concept to be "reasonable" in a particular application, the pattern or correlation should not be obvious. In a recent paper, Hommes and Sorger (1998) have proposed the related, but in general more stringent, concept o f consistent expectations equilibria. This requires that agents correctly perceive all autocorrelations o f the process. The restricted perceptions equilibrium concept is closely related to the notion o f reduced order limited information R E E introduced in Sargent (1991). Sargent considers the Townsend (1983) model in which two classes o f agents have different information sets and each class forms expectations based on a PLM which is a fixed-order vector A R M A process, e.g. a first-order A R process. This gives a mapping from the PLM to the A L M and a fixed point o f this map is a limited information REE, which was studied under learning in Marcet and Sargent (1989b). Sargent shows that this solution has reduced order, i.e. agents could make better forecasts using a higher-order
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ARMA process. In Sargent (1991), agents use an ARMA process, which is shown to yield full-order equilibrium 78. Some recent literature has explored learning dynamics in economies which are subject to recurrent structural shifts. As pointed out in Evans and Honkapohja (1993a), there are in principle two approaches if agents understand that these shifts will recur. One approach is for them to construct a hypermodel which allows for the structural shifts. If the agents misspecify such a model, they may converge to a restricted perceptions equilibrium, as above. An alternative approach is to allow for the structural shifts using a constant- or nondecreasing-gain learning algorithm which can potentially track the structural change. The constant-gain procedure was followed in Evans and Honkapohja (1993a). The choice of gain parameter involves a trade-off between its tracking ability and forecast variance, and an equilibrium in this class of learning rules was obtained numerically. In this kind of framework, policy can exhibit hysteresis effects if the model has multiple steady states. The recent analysis of Sargent (1999) also employs a constant-gain algorithm. In two recent papers the agents use algorithms in which the gain parameter is reset as a result of structural change. Timmermann (1995) looks at an asset pricing model with decreasing gain between structural breaks. It is assumed that agents know when a structural change has occurred and reset their gain parameters accordingly. This leads to persistent learning dynamics with greater asset price volatility 79. Marcet and Nicolini (1998) consider the inflation experience in some Latin American countries. Using an open economy version of the seignorage model in which the level of seignorage is exogenous and random, they assume that agents use decreasing gain unless recent forecast errors are high, in which case they revert to a higher fixed gain. They show that under this set-up the learning rule satisfies certain reasonable properties. Under their framework, recurrent bouts of hyperinflation are possible, and are better explained than under rational expectations. 5.4. Experimental evidence
Since adaptive learning can have strong implications for economic dynamics, experimental evidence in dynamic expectations models is of considerable interest. However, to date only a relatively small number of experiments have been undertaken. The limited evidence available seems to show that, when convergent, time paths from experimental data converge towards steady states which are stable under smallgain adaptive learning. Perhaps the clearest results are from experiments based on
78 Evans, Honkapohjaand Sargent (1993) consider an equilibrium in which a proportion of agents have perfect foresight and the rest, econometricians,have the optimal model from a restricted class of PLMs. Mitra (1997) considers a model with these two types of agents in which the econometricianschoose an optimal memory length. 79 In Timmermann(1993, 1996) excess asset price volatility is shown during the learning transition in a model with no structural breaks.
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the hyperinflation (seignorage) OG model. Recall that in this model the high real balance/low-inflation steady state is E-stable, and thus stable under adaptive learning, whereas the low real balance/high-inflation steady state is unstable 8o. This theoretical result is strongly supported by the experiments described in Marimon and Sunder (1993) [related experiments are reported in Arifovic (1995)]: convergence is always to the high real balance steady state and never to the low real balance steady state. Marimon, Spear and Sunder (1993) consider endogenous fluctuations (2-cycles and sunspot equilibria) in the basic OG model. Their results are mixed: persistent, beliefdriven cycles can emerge, but only after the pattern has been induced by corresponding fundamental shocks. These papers also consider some aspects of transitional learning dynamics. One aspect that clearly emerges is that heterogeneity of expectations is important: individual data show considerable variability. Arifovic (1996) conducts experiments in the 2-currency OG model in which there is a continuum of equilibrium exchange rates. These experiments exhibit persistent exchange rate fluctuations, which are consistent with GA learning. For the same model, using a Newton method for learning decision rules, simulations by Sargent (1993), pp. 107-112, suggest path-dependent convergence to a nonstochastic REE. These results raise several issues. First, it would be useful to simulate learning rules like the Newton method with heterogeneous agents and alternative gain sequences. Second, given the existence of sunspot equilibria in models of this type one should also investigate whether such solutions are stable under adaptive learning. Finally, Marimon and Sunder (1994) and Evans, Honkapohja and Marimon (1998a) introduce policy changes into experimental OG economies with seignorage. The former paper considers the effects of prealmounced policy changes. The results are difficult to reconcile with rational expectations but the data are more consistent with an adaptive learning process. The latter paper introduces a constitutional constraint on seignorage which can lead to three steady states, two of which are stable under learning. The experiments appear to confirm that these are the attractors. The learning rules in this paper incorporate heterogeneity with random gain sequences, inertia and experimentation. This generates considerable diversity and variability during the learning transition which has the potential to match many aspects of experimental data. 5.5. Further topics
The speed of convergence for learning algorithms is evidently an important issue for the study of learning behavior. The self-referential nature of many learning models invalidates the direct application of the corresponding results from classical statistics. At present very few studies exist on this subject. An analytic result on asymptotic speed of convergence for stochastic approximation algorithms is provided in Benveniste,
80 At least providedthe gain is sufficientlysmall. See Sections 1.4.3 and 1.4.4.
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Metivier and Priouret (1990), on pp. 110 and 332. In particular, suppose that the gain sequence is 7t = C/t. Then, provided the real parts of all eigenvalues of the derivative of the associated ODE are less than -0.5, asymptotic convergence occurs at rate v't. (No analytic results are available in this case if the eigenvalue condition fails.) Marcet and Sargent (1995) have applied this result to adaptive learning in a version of the Cagan inflation model. They also carried out Monte Carlo simulations. The numerical results appear to accord with the analytics if the model satisfies the eigenvalue condition. However, the speed of convergence can be very slow when the eigenvalue condition fails 81. In the discussion of statistical learning procedures it is a standard assumption that the PLM can be specified parametrically. However, just as an econometrician may not know the appropriate functional form it may be reasonable to assume that agents face the same difficulty. In this case a natural procedure is to use nonparametric techniques. This is discussed in Chen and White (1998). As an illustration consider learning a noisy steady state in a nonlinear model (26) in Section 2.7.1 which we repeat here for convenience: Yt = H(E[G(yt+I, Vt+l), vt). Previously, the shock was assumed to be iid and in this case a noisy steady state y(vt) could be described in terms of a scalar parameter 0* = EG(y(v), v) (here the expectation is taken with respect to the distribution of v). Chen and White (1998) instead consider the case where vt is an exogenous, stationary and possibly nonlinear AR(1) process. A natural PLM is now of the form Ei*G(yt+l, vt+~) = O(vt), and under appropriate assumptions there exists an REE O(vt) in this class. Agents are assumed to update their PLM using recursive kernel methods of the form
Or(o)
=
Ot 1(0) + t -1 [G(yt, v,) - Ot-i (v)] 91((vt - vt 1)/ht)/ht,
where 9l(.) is a kernel function (i.e. a density which is symmetric around zero) and {ht} is a sequence of bandwidths (i.e. a sequence of positive numbers decreasing to zero). Chen and White establish that under a number of technical assumptions and an E-stability-like condition the learning mechanism converges to O(vt) almost surely, provided a version of the projection facility is employed. Another new approach employs models in which agents choose a predictor from some class of expectation functions. Brock and Hommes (1997) suggest the notion of an adaptively rational expectations equilibrium in which agents make a choice among finitely many expectations functions on the basis of past performance. This choice is coupled with the dynamics of endogenous variables, and the resulting dynamics can sometimes lead to complicated global dynamics. A related paper is Hommes and Sorger (1998). The approach is similar in spirit to models of choice of forecasting functions in the presence of nonlinear dynamics or structural shifts, cf. Evans and Honkapohja (1993a), Marcet and Nicolini (1998), and Mitra (1997).
gl Vives (1993) has establisheda similar asymptoticspeed of convergenceresult for Bayesianlearning.
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6. Conclusions
Increasingly, macroeconomists are investigating models in which multiple rational expectations equilibria can arise. Traditionally, this was considered theoretically awkward: which solution would the economy follow? Examining adaptive learning in such circumstances is particularly fruitful. Requiring stability of equilibria under adaptive learning can greatly reduce the degree of multiplicity. In some models there is a unique equilibrium which is (locally) stable under learning, while other models can have more than one stable equilibrium. Even in the latter case, incorporating learning dynamics provides a resolution of the indeterminacy issue, since models with multiple stable equilibria are converted into models with path dependence. The dynamics of such an economy are determined by its initial conditions (including expectations) and by the equations of motion which include the learning rules as well as the usual structural equations of the model. In particular, the ultimate equilibrium can in part be determined by the sequence of random shocks during the transition. As was indicated above, there is some experimental evidence supporting the important role played by adaptive learning in models with multiplicity. A number of important policy issues can arise in such models, and learning dynamics need to be taken into account in formulating economic policies. In some cases policy rules can lead to unstable economic systems even though the equilibria themselves may seem satisfactory. In cases with multiple stable equilibria, the path dependence exhibited in models with adaptive learning can lead to hysteresis effects with changes in policy. In addition, temporarily inefficient policies may be necessary to guide the economy to a superior equilibrium. Finally, even in cases with a unique equilibrium, learning dynamics can be important in characterizing data in situations where there are sudden changes in policy regimes. The dynamics with learning can be very different from fully rational adjustments after such a change. Although our discussion has focused most heavily on asymptotic convergence to REE, some of these other issues, which have been less studied, are likely to receive more attention in the future. Learning dynamics is a new area of research where many issues are still open and new avenues no doubt remain to be discovered. We look forward to future work with excitement.
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Chapter 8 MICRO
DATA
AND
GENERAL
EQUILIBRIUM
MODELS*
MARTIN BROWNING
Institute of Economics, Copenhagen University, Copenhagen, Denmark email:
[email protected] LARS PETER HANSEN
University of Chicago, Chicago, IL, USA email:
[email protected] JAMES J. HECKNIAN
University of Chicago, Chicago, IL, USA email:
[email protected] Contents Abstract Keywords Introduction 1. Stochastic g r o w t h m o d e l 1.1. Single consumer model 1.1.1. Parameterizations 1.1.2. Steady states 1.1.3. Micro evidence 1.2. Multiple agents 1.2.1. Preferences 1.2.2. Labor income 1.2.3. Market structure 1.2.4. Preference homogeneity 1.2.5. Risk aversion or intertemporal substitution? 1.2.6. Preference heterogeneity 1.2.7. Questionnaire evidence on the scale and distribution of risk aversion 1.3. Incomplete markets 1.3.1. Microeconomic uncertainty 1.3.1.1. Estimated processes for wages and earnings
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* We thank Marco Cagetti, John Heaton, Jose Scheinkman, John Taylor, Edward Vytlacil and Noah Williams for comments. Hansen and Heckman gratefully acknowledge funding support by the National Science Foundation.
Handbook of Macroeconomics, Volume 1, Edited by J.B. Taylor and M. Woodford © 1999 Elsevier Science B.V. All rights reserved 543
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1.3.1.2. Missing risks 1.3.1.3. Statistical decompositions 1.3.2. Limited commitment and private information 1.3.2.1. Limited commitment 1.3.2.2. Private information 2. Overlapping generations model 2.1. Motivation 2.2. Economic models of earnings and human capital investment 2.2.1. The Ben-Porath framework 2.2.2. The HLT model of earnings, schooling and on-the-job training 2.3. Structure of the model 2.3.1. Equilibrium conditions under perfect foresight 2.3.2. Linking the earnings function to prices and market aggregates 2.4. Determining the parameters of OLG models 3. Micro evidence 3.1. Introduction 3.2. Defining elasticities 3.2.1. Frisch demands 3.2.2. Other demand functions 3.2.3. An example 3.2.4. The life-cycle participation decision 3.3. Consumption estimates 3.4. Labor supply 3.4.1. Labor supply estimates 3.5. Heterogeneity in the marginal rate of substitution between goods and leisure Summary and conclusion References
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Abstract Dynamic general equilibrium models are required to evaluate policies applied at the national level. To use these models to make quantitative forecasts requires knowledge o f an extensive array o f parameter values for the economy at large. This essay describes the parameters required for different economic models, assesses the discordance between the macromodels used in policy evaluation and the microeconomic models used to generate the empirical evidence. For concreteness, we focus on two general equilibrium models: the stochastic growth model extended to include some forms o f heterogeneity and the overlapping generations model enriched to accommodate human capital formation. Keywords general equilibrium models, microeconomic evidence, stochastic growth model, overlapping generations model, calibration
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Introduction An extensive literature in macroeconomics and public finance uses dynamic stochastic general equilibrium models to study consumption savings, capital accumulation, and asset pricing and to analyze alternative policies. Except for a few special cases, the economies studied cannot be analyzed using "paper and pencil" style analysis. It is often difficult to produce general theorems that are true for all parameter values of dynamic general equilibrium models. This is a general feature of non-linear dynamic models in economics as well as in the physical sciences. For such models, knowing which parameters govern behavior is essential for understanding their empirical content and for providing quantitative answers to policy questions. For the numerical output of a dynamic equilibrium model to be interesting, the inputs need to be justified as empirically relevant. There are two sources of information that are commonly used in rationalizing parameter values. One is the behavior of time series averages of levels or ratios of key variables. These time series averages are often matched to the steadystate implications of versions of the models that abstract from uncertainty. The other input is from microeconomic evidence. In this essay we discuss the use of evidence from both sources, concentrating mostly on microeconomic evidence. See King and Rebelo (1999) and Taylor (1999) for extensive discussions of calibrating real-business cycle and staggered contract models, respectively. It was once believed to be a simple task to extract the parameters needed in general equilibrium theory from a large warehouse of stable micro empirical regularities. Indeed, Prescott (1986) argued that: The key parameters of growth models are the intertemporal and intratemporal elasticities of substitution. As Lucas (1980) emphasizes,"On those parameters, we have a wealth of inexpensive available data from census and cohort information, from panel data describing market conditions and so forth". While this Lucas-Prescott vision of calibration offers an appealing defense for building models with microeconomic foundations, implementing it in practice exposes major discrepancies between the micro evidence and the assumptions on which the stylized dynamic models are based. The microeconomic evidence is often incompatible with the macroeconomic model being calibrated. For example, a major finding of modem microeconomic data analysis is that preferences are heterogeneous. For reasons of computational tractability, dynamic general equilibrium model-builders often abstract from this feature or confront it in only a limited way. This chapter explores the discordance between micro evidence and macro use of it and suggests ways in which it can be diminished. Our chapter raises warning flags about the current use of micro evidence in dynamic equilibrium models and indicates the dangers in, and limitations of, many current practices. It also exposes the weak micro empirical foundations of many widely used general equilibrium modeling schemes. The decision to incorporate micro evidence in an internally consistent manner may alter the structure and hence the time series implications of the model. While
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steady-state approximations may be useful for some purposes, compositional changes in labor supply or in market demand alter the microeconomic elasticities that are relevant for macroeconomics. Like several of the other contributions to this Handbook, ours is more of a guide for future research than a summary of a mature literature. Because the micro empirical literature and the macro general equilibrium literature have often moved in different directions, it is not surprising that they are currently so detached. The main goal of this essay is to foster the process of reattachment. Macro general equilibrium models provide a framework within which micro empirical research can be fruitfully conducted. At the same time, dynamic general equilibrium theory will be greatly enriched if it incorporates the insights of the micro empirical literature. The micro fotmdations of macroeconomics are more fruitfully built on models restructured to incorporate microeconomic evidence. This essay explores three challenges for closing the gap between empirical microeconomics and dynamic macroeconomic theory: • Heterogeneity: Any careful reading of the empirical microeconomics literature on consumption saving and labor supply reveals quantitatively important heterogeneity in agent preferences, in constraints, in dimensions of labor supply and skill, and in human capital accumulation processes. Accounting for heterogeneity is required to calibrate dynamic models to microeconomic evidence. • U n c e r t a i n t y : Modern macroeconomics is based on models of uncertainty. Aggregating earnings across members in a household and across income types may create a disconnect between uncertainty as measured by innovations in time series processes of earnings and income equations and actual innovations in information. Government or interfamily transfers provide insurance that should be accounted for. Alternative risk components such as risks from changes in health, risks from unemployment and job termination, and risks from changes in family structure, have different degrees of predictability and are difficult to quantify. Measuring the true components of both micro and macro uncertainty and distinguishing them from measurement error and model misspecification remains an empirical challenge that is just beginning to be confronted. • Synthesis: Synthesizing evidence across micro studies is not a straightforward task. Different microeconomic studies make different assumptions, often implicit, about the economic environments in which agents make their decisions. They condition on different variables and produce parameters with different economic interpretations. A parameter that is valid for a model in one economic environment cannot be uncritically applied to a model embedded in a different economic environment. Different general equilibrium models make different assumptions and require different parameters, many of which have never been estimated in the micro literature. In order to be both specific and constructive, in this essay we limit ourselves to two prototypical general equilibrium models: (a) a stochastic growth model and (b) a perfect foresight overlapping generations model. The first model is sufficiently rich to enable us to explore implications of uncertainty, market structure and some
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forms of heterogeneity in the preferences and opportunities of microeconomic agents. The second model introduces explicit life-cycle heterogeneity and demographic structures in appealing and tractable ways. We consider a recent version of the second model that introduces human capital formation, heterogeneity in skills, and comparative advantage in the labor market. These attributes are introduced to provide a framework for analyzing labor market policies, to account for a major source of wealth formation in modern economies, and to account for the phenomenon of rising wage inequality observed in many countries. The plan of this chapter is as follows. We first present two basic theoretical models analyzed in this chapter and the parameters required to implement them. We summarize the main lessons from the micro literature that pertain to each model and their consequences for the models we consider. The models are presented in Sections 1 and 2, respectively, with some accompanying discussion of the relevant micro literature. Section 3 presents further discussion of the micro evidence on intertemporal substitution elasticities.
1. Stochastic growth model This part of the chapter presents alternative variants of a Brock-Mirman (1972) stochastic growth model and discusses the parameters needed to calibrate them. We explicitly consider the consequences of heterogeneity for the predictions of this model and for the practice of research synthesis. It is often not the median or "representative" preferences that govern behavior asymptotically; rather it is the extreme. The agents with the smallest rates of time preference or smallest risk aversion may dominate the wealth accumulation process, but not the supply of labor. Understanding the source and magnitude of the heterogeneity is required before microeconomic estimates can be "plugged" into macroeconomic models. We also explore the measurement of microeconomic uncertainty needed to quantify the importance of precaution in decision-making and to calibrate equilibrium models with heterogeneous agents. We use the Brock-Mirman (1972) stochastic growth model as a starting point for our discussion because of its analytical tractability. Moreover, it is the theoretical framework for the real-business cycle models of Kydland and Prescott (1982) and Hansen (1985) and for subsequent multiple consumer extensions of it by Aiyagari (1994), Krusell and Smith (1998) and others. Our use of the stochastic growth model is not meant as an endorsement of its empirical plausibility. Much is known about its deficiencies as a model of fluctuations [e.g., see Christiano (1988), Watson (1993), and Cogley and Nason (1995)] or as a model of security market prices implied by a Lucas-Prescott (1971) type of decentralization [e.g., see Hansen and Singleton (1982, 1983), Mehra and Prescott (1985), Weil (1989), Hansen and Jagannathan (1991), and Heaton and Lucas (1996)]. Nevertheless, the Brock-Mirman model and its extensions provide a convenient and widely used starting point for investigating the difficulties
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in obtaining plausible parameter configurations from microeconomic data and from aggregate time series data. 1.1. Single consumer model
Suppose that there is a single infinitely-lived consumer. This consumer supplies labor and consumes in each period, evaluating consumption flows using a v o n N e u m a n n Morgenstern discounted utility function: oo
F~~ /3t U(c,), t=O
where ct is consumption at date t, U is an increasing concave function and 0 0. Given prior knowledge of the constant interest rate r, we may use this formula and formula (1.18) to deduce the variance of the Act:
Var(Act)=Ik=~o(l~r)kak]'[j=~o(l~r)JaJJ =
Cov(Ae,2k,Ac;). k=0
From these calculations then we may again infer how the true income process responds to a linear combination of the t/t shock vector 26. I f the {~/t} process is scalar, then we have full identification of the information structure confronting the consumer that is pertinent for the evolution of income. Thus the use of consumption data, even if measured with error, can help to identify the true income uncertainty confronting economic agents. We conclude this subsection with a brief discussion of the sources of market incompleteness.
1.3.2. Limited commitment and private inJbrmation In our incomplete markets model, we made no attempt to justify the form of market incompleteness. Two common justifications include problems of enforceability and observability. Measurements of microeconomic uncertainty are critical ingredients to economies that explicitly account for limited commitment and private information.
1.3.2.1. Limited commitment. Kehoe and Levine (1993) and Kocheflakota (1996) propose the following alternative to the incomplete markets models we have considered thus far. Suppose consumers are permitted to walk away from obligations, but when they do so they are excluded from future participation in markets. Instead they are restricted to use their own backyard storage technologies or simply consume only their 26 Hansen, Roberds and Sargent (1991) use a closelyrelated argumentto show that present-valuebudget balance has testable implication when consumption is a martingale. They do not, however, explore the ramifications of measurement error in consumption and income.
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labor income. As a consequence, in equilibrium, consumers are guaranteed a lower bound on their discounted utilities at each date and state. Consumption allocations are obtained by solving Pareto problems subject to utility lower bounds implied by the utility threat points: the points at which consumers are indifferent between honoring their obligations and defanlting. While there is no default in equilibrium, allocations are altered by the presence of the utility threat points. Alvarez and Jennann (1999) consider an economy with limited commitment in which consumers have no access to backyard storage technologies and hence are punished by constraining future consumption to equal future income, period by period. They show that the resulting allocations may be decentralized by introducing personspecific solvency or borrowing constraints 27. Euler equations are replaced by Euler inequalities as in Luttmer's (1996) work on asset pricing when investors face solvency constraints. Alvarez and Jermann (1999) are able to imitate asset pricing features of an economy with solvency constraints, but with different predictions about when agents will be up against financial market constraints. Individual Euler equations linking consumption to asset prices continue to characterize the behavior of some individuals in each time period. When and which consumers are constrained in their ability to borrow can be ascertained by computing the utility threat points, which in turn depend on the microeconomic uncertainty they would be forced to confront in the absence of risk sharing. Thus the same problem of measuring uncertainty discussed for economies with incomplete markets applies to economies in which limited commitment is the only source of financial market frictions. 1.3.2.2. Private information. In the limited commitment economies considered by Kehoe and Levine (1993) and by Alvarez and Jermann (1999), idiosyncratic endowment shocks are publicly observed. Suppose instead that they are only known to the individual agents and not to the public. Again, the options and information available to agents matter. For instance, suppose that a capital accumulation technology is only available to the society as a whole and not to individuals. In this case, individual consumption contracts can be enforced because the agents are unable to privately transfer consumption from one period to the next. There is a substantial body of work on optimal resource allocation subject to incentive constraints that relies on the enforceability of consumption contracts [see Green (1987), Phelan and Townsend (1991), Atkeson and Lucas (1992) among others]. In general, the efficient allocations do not have decentralizations that look like the incomplete market structure previously described. Economies with a simple security market structure are not Pareto efficient even after accounting for the incentive constraints [e.g., see Lucas (1992)]. Of course, security markets could be supplemented by other institutions designed to reduce or
27 The person-specific nature of the solvency constraint stretches a bit the notion of a decentralized economy.As an alternative, we may view these limited commitmenteconomies as (constrained) efficient benchmarks to which we might compare alternative institutional arrangements.
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eliminate the efficiency wedge. In contrast, when a capital accumulation technology is privately available, individual agents can hide their consumption from the public. Thus individual consumption contracts are no longer enforceable. For some special versions of these environments, Allen (1985) and Cole and Kocherlakota (1997) show that the incomplete security market economy we described previously fully decentralizes the Pareto efficient allocations subject to incentive constraints. Even when there is an efficiency wedge, the specification ofmicroeconomic uncertainty is a critical ingredient in both the decentralized economy and in the Pareto efficient economy subject to incentive constraints. The problems of measuring microeconomic uncertainty arise in private information economies as well.
2. Overlapping generations model 2.1. Motivation
The overlapping generations model (OLG), now widely used in macroeconomics and public finance, captures heterogeneity among cohorts, something not captured by the models considered in Section 1. Thus it provides valuable information on cohortspecific consequences of economic policies and of macro disturbances. Influential versions of the OLG model by Auerbach and Kotlikoff (1987) and Fullerton and Rogers (1993) are widely used to evaluate a variety of policy reforms, including tax and social security policies. The versions of this model that are widely used are based on perfect foresight. A major computational advantage of this form of the OLG model is that it is relatively straightforward to calculate transitional paths for high dimensional specifications, as demonstrated by Greenwood and Yorukoglu (1997). This is in contrast with the stochastic growth model with incomplete markets, where computation with high dimensional state vectors is still a very difficult problem. However, perfect foresight is a strong assumption. Huggett (1996) considers a version of an OLG model with idiosyncratic uncertainty, but he only analyzes steady states. Most empirical implementations of the OLG model ignore decisions to accumulate human capital by taking skills to be exogenous endowments. On the other hand, human capital is a more substantial component of total wealth than physical capital. For this reason, we present a generalization of the Auerbach-Kotlikoff overlapping generations model due to Heckman, Lochner and Taber (1998) (HLT) that incorporates human capital, in addition to the physical capital that is the centerpiece of the AuerbachKotlikoff analysis. We consider the evidence required to empirically implement this and other versions of the OLG model using either micro or macro data. We use our exposition of this extended version of the OLG model to consider the benefits of introducing richer forms of heterogeneity in skills and human capital production technologies and to consider choices at both the intensive and extensive margins. Schooling decisions are made at the extensive margin. Moreover, investment
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in schooling and on-the-job training are distinct activities and produce distinct skills, and output is produced by a skill mix supplied by different agents. The following facts motivate our choice o f models: • Empirical evidence in labor economics demonstrates that comparative advantage in factor markets is the rule and that different persons specialize in different skills [see Sattinger (1993)]. Widely-used efficiency units corrections that collapse skill to a single dimension do not explain either cross-section wage distributions or the evolution o f wage distributions over time. • There are substantial cohort effects in earnings functions. • Heterogeneity in saving behavior and wealth accumulation helps in fitting OLG models to data. Life-cycle models with preference homogeneity in each cohort do not generate enough dispersion in savings to explain the observed capital holdings. To accommodate these facts, we follow HLT by embedding an extension o f the model o f Ben Porath (1967) of individual human capital production into a general equilibrium setting. In Ben-Porath's model, in the stochastic growth model discussed in Section 1, and in the Auerbach-Kotlikoff model or the Fullerton-Rogers model, wage inequality can only be generated by differences in amounts o f a common skill (the so-called "efficiency units" model). All skill commands the same price. In the model considered by HLT and in this part o f our chapter, different levels o f schooling enable individuals to invest in different skills through on-the-job training in the post-schooling period. In the aggregate, the skills o f different schooling groups are not perfect substitutes 28. Within schooling groups, however, persons with different amounts o f skill are perfect substitutes 29. For our purposes, this model provides us with a useful platform for integrating empirical results from labor economics into a general equilibrium, macroeconomic model. Before turning to the details o f the HLT model, it is fruitful to place their work in the context o f current models o f earnings determination widely used in the literature. 2.2. Economic models o f earnings and human capital investment
Three different frameworks for interpreting wage and earnings functions coexist in the literature on empirical labor economics. Their coexistence in the literature is a constant source o f confusion. The only agreement among users o f these alternative frameworks is their agreement to call the coefficient on schooling in a log wage equation a "rate o f return", although the conditions required to do so are known to be restrictive 3°.
28 This specification is consistent with evidence that the large increase in the supply of educated labor consequent from the baby boom depressed the returns to education. [See Freeman (1976), Autor, Katz and Krueger (1997) and Katz and Murphy (1992).] 29 This specification accords with the empirical evidence summarized in Hamermesh (1993, p. 123) that persons of different ages but with the same education levels are highly substitutable for each other. 3o In order to interpret the schooling coefficient as an internal rate of return, it is necessary to assume an environment of perfect certainty and to further assume that earnings are multiplicatively separable
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The first two frameworks are scalar and vector attribute pricing equations, respectively. The third framework accounts for personal investment and the estimated coefficients on attributes of fitted earnings functions are not prices although they are often interpreted to be so. The most widely used empirical framework for wages in aggregative analysis writes the wage w of person i (wi) as the product of the price R of labor services denominated in "efficiency units" with the quantity of efficiency units Qi embodied in the agent: Wi =
RQi,
(2.1)
where Qi is a function of characteristics of individuals such as education and work experience. In this framework, R = FQ(Q, K) where F is the aggregate production function and k is the aggregate capital stock and Q = ~/~_ 1 Qi where N is the number of agents working in the economy. Equation (2.1) underlies the calibrated real-business cycle models of Kydland (1984, 1995) and a variety of microeconomic studies. [See the one-sector model in Heckman and Sedlacek (1990) or the aggregative models of Stokey and Rebelo (1995), Lucas (1988), Uzawa (1965) and Caballe and Santos (1993).] In logs, Equation (2.1) has the strong implication that aggregate shocks operate only through intercepts of log wage equations. This framework cannot explain the well-documented rise in inequality in means among skill classes or increases in variances of log wage equations that are central features of many modern economies [see, e.g. Katz and Murphy (1992)] unless quality changes within skill groups. Since the mean of log wages and the variance of log wages increase greatly over short periods of time for groups with stable skill characteristics, such an explanation of rising wage inequality is implausible. It is tested and rejected in the study of Katz and Murphy (1992). Real wages respond to aggregate unemployment (see Table 2.1). However, the response to local unemployment rates varies by gender, education and age (see Table 2.2). The wages of unskilled persons are much more cyclically sensitive to local unemployment rates. This evidence does not support a one-dimensional efficiency units model, In addition, an efficiency units model is inconsistent with the well-documented evidence on comparative advantage in the labor market. [See Heckman and Sedlacek (1985), or Sattinger (1993).] A model in which persons choose sectors based on their comparative advantage in them explains the distribution of US wage data whereas a model with efficiency units does not [Heckman and Sedlacek (1990)]. Finally, the evidence reported in Topel (1986) and Heckman, Layne-Farrar and Todd (1996) demonstrates that the wages of different skill groups respond differently to local labor market shocks. This is inconsistent with efficiency units wage specification (2.1). in schooling and experience and that all persons of all education levels have the same post-school experienceprofiles. See Heckman and Klenow (1997) and Heckman, Lochner and Taber (1999) for a derivation.
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Table 2.1 Elasticity of real wages with respect to aggregate unemployment: first-differenced panel data Author
Data Set
Elasticity
(S.E.)
Bils (1985)
NLS young men (whites), 1966 1980 10 changes
-0.089
(0.019)
Rayack (1987)
PSID males (whites), 1968-1980 12 changes
-0.081
(0.016)
Blank (1990)
PS1D males (whites), 1969-1982 13 changes, pairwise balanced
-0.081a
(0.043)
Solon, Barsky and Parker (1994)
PSID males, 1967 1987 20 changes
-0.085
(0.022)
a Blank (1990) regresses change in real wages on percentage change in real GNP. Her estimate is transformed to unemployment elasticity using an estimated "Okun" coefficient of 0.30.
An alternative specification of the wage equation that meets some of these objections to the efficiency units wage model is the Gorman-Lancaster model of earnings. [Welch (1969), applies this model.] In this framework, the wage of person i is a function of characteristics embodied in the person times their price. Letting Ai be a J x 1-dimensional vector o f attributes for person i and RA a 1 x J-dimensional vector o f their economy wide prices,
(2.2)
Wi = R A A i .
The prices are determined by the aggregate production function using the aggregates N
j = 1,...,J
and
Rj =
f j ( A 1. . . . . A j , / ~ ) ,
i-I
where Rj is the jth entry of RA. In principle, this model can explain rising wage inequality in log earnings if aggregate shocks, or local labor market shocks, do not operate uniformly on the arguments of the production function, Aj, j = 1. . . . . J, and persons possess more than one type of skill. Equation (2.2) is inconsistent with the evidence on comparative advantage in the labor market, however. Heckman and Scheinkman (1987) demonstrate that for the US economy skill prices RA are not uniform across different sectors for a variety of different types o f sectors. Since prices are not uniform, a model of sectorat self-selection and comparative advantage in the labor market is required. Heckman and Sedlacek (1985, 1990) find that such a model better explains both the cross section and time series of the US wage distribution than do efficiency units models. The usual least squares estimates of wage equations reported in the literature do not estimate skill prices unless analysts account for self selection across sectors where skill prices are different.
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Table 2.2 Elasticities of wages, hours, and earnings with respect to state unemployment rates a Category
Hourly wage Actual (1) Adjusted (2)
1. All
Annual hours Actual (3) Adjusted (4)
Annual earnings Actual (5) Adjusted (6)
-0.07 (0.02)
-0.08 (0.02)
-0.11 (0.01)
-0.12 (0.01)
-0.18 (0.02)
-0.20 (0.02)
a. Women
-0.06 (0.02)
-0.06 (0.02)
-0.08 (0.02)
-0.09 (0.02)
-0.14 (0.03)
-0.16 (0.03)
b. Men
-0.08 (0.02)
-0.09 (0.02)
0.13 (0.01)
-0.15 (0.01)
-0.21 (0.02)
-0.24 (0.02)
a. 0. Moreover, (~wA O'AA
( Crww(YAZ-- ~wZOwA )
al
~
iOi
)
Some o f these covariances can be signed. From concavity o f preferences, aAX < 0 (diminishing marginal utility o f wealth); assuming that goods and leisure are normal in all periods, awX ~< 0 and in a cross section, it usually happens that awA > 0 (higher wage people have higher wealth). Thus, in general, the bias o f &i for al is ambiguous. Observe, however, that there is no bias if income effects are small. If they are negligible, awx = aAX = 0 and the cross-section wage elasticity in a leisure demand equation recovers the Frisch labor supply parameter. Moreover, in an environment o f perfect certainty, the weaker the correlation over time in wages, the weaker is C~w~,the more likely is the bias o f &l for al to be downward and the more likely is &2 to be negative. If At is omitted from the model, as was the common practice in many of the crosssection studies surveyed in Table 3.4, the OLS estimate o f al when )~t is omitted from the regression (&l) has: plim&~=a'(
l+°w~)'aww/
(3.27)
which is definitely downward biased since C~wZ< 0. From a l , w e can produce a lower bound on al 54 The preceding arguments remains valid if there are cohort effects or if we introduce age effects by relaxing the restriction that/3(1 + r) = 1 back into the model, provided that the assumed properties o f the covariances also apply to age-adjusted versions o f
54 A version of this bounding argmnent was first presented in Heckman (1971). The argument is more general and does not require contemporaneous additive separability.
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them. In the case of secular exponential growth in wealth across cohorts, the coefficient on age includes a growth rate of wealth term. Provided that aXw < 0, the argument also extends to an uncertain environment with less than full insurance. In Table 3.4 we present cross-section estimates from 15 studies. The estimates correspond to &l or &a above. The first nine studies in the table produce estimates of O O,nh in w _ ( 2 ~ ) ~O,l n l where T - h = 1. Thus, the upper bounds for Frisch leisure demands are lower bounds for Frisch labor supply elasticities The first nine studies are the traditional least squares estimates which are plagued by bad data on asset income. All but one of the numbers reported in the first nine rows of this table are below the estimates of the Frisch labor supply elasticity reported in Table 3.3 vindicating our analysis and suggesting that the cross-section labor supply estimates constitute lower bounds for the Frisch labor supply parameter. Studies (10) and (11) of Table 3.4 report that cross-sectional annual-hours-of-work labor supply elasticities for married women are as negative as they are for married men. Studies (12) and (13) reveal that labor supply elasticities are more positive when participation is the dependent variable. (But recall from our earlier discussion that these elasticities are not the eis or the Hicks-Slutsky elasticity.) This point is emphasized in the studies of Heckman (1978, 1993) and Blundell and MaCurdy (1999). Study (12) reveals that the curvature of labor supply is more elastic the lower the wage. Study (14) attempts to replicate the influential cross-sectional estimates of Hausman (1981), which are sometimes used by macroeconomists as measures of the Frisch parameter. His estimates are much higher than the other estimates reported in the table. In a careful analysis, the authors of study (14) are unable to replicate Hausman's reported estimates using his own data. Instead their unrestricted estimates of labor supply elasticities are negative and inconsistent with utility maximization. When the restrictions of utility maximization are imposed, the model exhibits a zero labor supply elasticity for males. It is the estimates in row (14), and not Hausman's (1981), that fall in line with the other estimates reported in the tables and should be used for evidence on uncompensated static labor supply. Study (15) summarizes the literature that establishes that the traditional labor supply model does not satisfy Slutsky symmetry or integrability conditions. However, Heckman (1971) establishes that when the cross-section model is embedded in a lifecycle setting, the evidence is much stronger for the traditional household model of labor supply. 3.5. Heterogeneity in the marginal rate o f substitution between goods and leisure
Beginning with the work of Heckman (1974), and continuing on in the work of Burtless and Hausman (1978), Hausman (1981), MaCurdy (1983) and MaCurdy, Green and Paarsch (1990), labor economists and econometricians have estimated the extent of variability in the marginal rate of substitution between goods and leisure. Heckman (1974) builds a life-cycle model of labor supply, asset accumulation, and the demand for child care for an environment of perfect certainty. In his model the
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marginal rate of substitution between goods and leisure is explicitly parameterized and allowed to depend on both observed and unobserved factors. In later work, MaCurdy (1983) applies and extends this framework to an environment of uncertainty using Euler equation methods. Other researchers have used static one-period models of labor supply to document heterogeneity in the preference for leisure. For brevity we only summarize the static labor supply evidence reported by Heckman (1974), who presents the clearest evidence on preference heterogeneity. The marginal rate of substitution function (or slope of the indifference curve) at a given level of prework income Y is
m = m(Y, h),
(3.28)
where we ignore variations in the prices of other goods and where h is hours of work 55. A consumer possesses a family of indifference curves indexed by level indicator Y, the no-work level of income or consumption We know that Om/OY > 0 if leisure is a normal good. From diminishing marginal rate of substitution between goods and time, we know that Ore/Oh > O. If a consumer faces a parametric wage w, at initial income position Y, she works if w > m(Y, 0). I f this inequality applies, the consumer's decision of how much to work is characterized by
w = m(Y*, h*),
(3.29)
where Y* is a level index appropriate to the indifference curve and is the amount of income (or consumption) that would make the consumer indifferent to a choice between working h* hours at wage rate w to gain total resources wh* + Y, or not working and receiving income Y*. Without knowledge of Y*, one cannot deduce the relationship between w and h* predicted by consumer optimization. However, given Y*, we know that optimality also requires that h* satisfy h*
wh* + Y = f m(Y*, h) dh + Y*.
(3.30)
0
From Equation (3.29), if m is monotonic in Y* we may solve for Y* as a function of w and h: Y* = g(w, h). Using this value in Equation (3.30), we implicitly define the labor supply function by h
wh + Y = f m [g(w, s), s] ds + g(w, h). 0
55 In the life-cycle version of the model, Y is determined by a two-stage budgeting argument.
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Ch. 8:
623
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A wide variety of fimctional forms may be used to specify m. Heckman's (1974) preferred specification is (3.31)
lnm = a o + a ~ Y + a z h + a 3 Z + u ,
where Y is the prework level of income, h is hours of work, and Z is a vector of variables to be discussed more fully below. A random variable designated "u" with zero mean and variance ~r~2, reflects variation in preferences for work among individuals. The previous analysis leads us to the prediction that al > 0 (normality of leisure) and a2 > 0 (diminishing marginal rate of substitution between goods and leisure). The resulting static labor supply function is implicitly defined by u = lnw-oto-ot2h-ot3Z-al
wh+ Y-
~22(1 - e a2h) .
(3.32)
This labor supply curve can become backward bending beyond a certain value of hours worked. Estimates for this specification are presented in Table 3.5 where the price of child care is also introduced as a determinant of the marginal rate of substitution. This table reveals that the estimated marginal rate of substitution function for married women has considerable heterogeneity depending both on observed variables (education, the number of children and the wealth level of the family) and on unobservables. Children, education, and asset income all raise the value of leisure. The standard deviation in the unobservables is also large. This heterogeneity arises from variation over people and not sampling variation. Figure 3.1 plots the median, third quartile and first quartile standard deviation in the population marginal rate of substitution for a group with median non-labor income and other characteristics, excluding education from preferences. Population variation in tastes is a central empirical regularity in the micro literature. Moreover, as documented in Heckman and Willis (1977) and Heckman (1982), these tastes are stable over time giving rise to persistence in employment status over time.
Summary and conclusion This chapter has documented the empirical evidence supporting heterogeneity in preferences, constraints and skills and some of their consequences for modern macroeconomics. It has also examined the problems of measuring uncertainty from time series processes of earnings and wage functions. Finally, it has discussed the problem of extracting the parameter values required in dynamic general equilibrium theory from the "large shelf" of micro estimates. The gulfs between the theoretical environments presumed by the dynamic economic models and estimation environments used by empirical microeconomists leaves the shelf of directly usable numbers virtually empty. What is there requires careful interpretation and qualification.
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While dynamic general equilibrium models may suggest new directions for empirical macroeconomic research, it is essential to build the dynamic economic models so that the formal incorporation of microeconomic evidence is more than an afterthought. Macroeconomic theory will be enriched by learning from many of the lessons from modern empirical research in microeconomics. At the same time, microeconomics will be enriched by conducting research within the paradigm of modern dynamic general equilibrium theory, which provides a framework for interpretation and synthesis of the micro evidence across studies.
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in the United States and Canada, with special attention to the impact of income taxes", Econometrica 49(2):451-490. Negishi, T. (1960), "Welfare economics and existence of an equilibrium for a competitive economy", Metroeconomica 12:92 97. Pencavel, J. (1986), "Labor supply of men: a survey", in: O. Ashenfelter and R. Layard, eds., Handbook of Labor Economics, vol. 1 (North-Holland, Amsterdam) ch. 1, 3-102. Phelan, C., and R.M. Townsend (1991), "Computing multi-period, information-constrained optima", Review of Economic Studies 58(5):853-881. Prescott, E.C. (1986), "Theory ahead of business cycle measurement", Federal Reserve Bank Minneapolis Quarterly Review 10(4):9-22. Rayack, W (1987), "Sources and centers of cyclical movement in real wages: evidence from panel data", Journal of Post-Keynesian Economic 10(1):3~1. Rogerson, R. (1988), "Indivisible labor, lotteries and equilibrium", Journal of Monetary Economics 21(1):3 16. Rosen, S. (1976), "A theory of life earnings", Journal of Political Economy 84(Suppl.):345-382. Rubinstein, M. (1974), "An aggregation theorem for securities markets", Journal of Financial Economies 1:225-244. Ryder, H., E Stafford and E Stephen (1976), "Labor, leisure and training over the life cycle", International Economic Review 17:651-674. Samwick, A. (1997), "Discount rate heterogeneity and social security reform", mimeograph (Dartmouth College). Santos, M.S. (1999), "Numerical solution of dynamic economic models", ch. 5, this Handbook. Sattinger, M. (1993), "Assignment models of the distribution of earnings", Journal of Economic Literature 31(2):831-80. Scheinkman, J.A., and L. Weiss (1986), "Borrowing constraints and aggregate economic activity", Econometrica 54(1):23-45. Shaw, K. (1989), "Life cycle labor supply with human capital accumulation", International Economic Review 30(2):431-456. Shea, J. (1995), "Union contracts and the life-cycle permanent income hypothesis", American Economic Review 95(1):18(~200. Solon, G., R. Barsky and J.A. Parker (1994), "Measuring the cyclicality of real wages: how important is composition bias?", Quarterly Journal of Economics 109(1): 1-25. Stockman, A.C., and L. Tesar (1995), "Tastes and technology in a two country model of the business cycle: explaining international co-movements", American Economic Review 85(1):168-185. Stokey, N.L., and S.T. Rebelo (1995), "Growth effects of flat-rate taxes", Journal of Political Economy 103(3):519-550. Taylor, J.B. (1999), "Staggered wage setting in macroeconomics", ch. 15, this Handbook. Topel, R. (1986), "Local labor markets", Journal of Political Economy 94(3):111-43. US Bureau of the Census (1960), 1960 Census Public Use Sample (United States Government Printing Office, Washington, DC). Uzawa, H. (1965), "Optimum technical change in an aggregative model of economic growth", International Economic Review 6(1): 18-31. Watson, M.W (1993), "Measures of fit for calibrated models", Journal of Political Economy 101(6): 1011 1041. Well, E (1989), "The equity premium puzzle and the risk-free rate puzzle", Journal of Monetary Economics 24(3):4-421. Weiss, Y. (1986), "The determination of life cycle earnings: a survey", in: O. Ashenfelter and R. Layard, eds., Handbook of Labor Economics, vol. 1 (Elsevier Science, New York) part 5, 603-640. Welch, E (1969), "Linear synthesis of skill distribution", Journal of Human Resources 4(3):311 325. Wilson, R. (1968), "The theory of syndicates", Econometriea 36(1):119-132.
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Zeldes, S.P. (1989), "Consumption and liquidity constraints: an empirical investigation", Journal of Political Economy 97(2):305-46.
Chapter 9
NEOCLASSICAL GROWTH THEORY ROBERT M. SOLOW
Massachusetts Institute of Technology, Department of Economics, E52-383B, Cambridge, MA 02139, USA
Contents
Abstract Keywords 1. Introduction 2. The Harrod-Domar model 3. The basic one-sector model 4. Completing the model 5. The behaviorist tradition 6. The optimizing tradition 7. Comparing the models 8. The Ramsey problem 9. Exogenous technological progress 10. The role of labor-augmentation 11. Increasing returns to scale 12. Human capital 13. Natural resources 14. Endogenous population growth and endogenous technological progress in the neoclassical framework 15. Convergence 16. Overlapping generations 17. Open questions References
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Abstract
This chapter is an exposition, rather than a survey, of the one-sector neoclassical growth model. It describes how the model is constructed as a simplified description of the real side of a growing capitalist economy that happens to be free of fluctuations in aggregate demand. Once that is done, the emphasis is on the versatility of the model, in the sense that it can easily be adapted, without much complication, to allow for the analysis of important issues that are excluded from the basic model. Among the issues treated are: increasing returns to scale (but not to capital alone), human capital, renewable and non-renewable natural resources, endogenous population growth and technological progress. In each case, the purpose is to show how the model can be minimally extended to allow incorporation of something new, without making the analysis excessively complex. Toward the end, there is a brief exposition of the standard overlapping-generations model, to show how it admits qualitative behavior generally absent from the original model. The chapter concludes with brief mention of some continuing research questions within the framework of the simple model.
Keywords growth, technological progress, neoclassical model JEL classification: 04, E1
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1. Introduction
As part of macroeconomics, growth theory functions as the study of the undisturbed evolution of potential (or normal capacity) output. The force of "undisturbed" in this context is the maintained assumption that the goods and labor markets clear, i.e., that labor and capital are always fully or normally utilized (or, at the very minimum, that the degree of utilization does not vary). The scope of specifically "neoclassical" growth theory is harder to state, because it is a matter of judgment or convention how much more of the neoclassical general equilibrium apparatus to incorporate in a model of undisturbed growth. As in most of macroeconomics, modeling strategy in growth theory tends to be weighted away from generality and toward simplicity, because the usual intention is to compare model with data at an early stage. Simplicity does not mean rigidity. On the contrary, it will emerge from this review that the neoclassical growth model is extraordinarily versatile. Like one of those handy rotary power tools that can do any of a dozen jobs if only the right attachment is snapped on, the simple neoclassical model can be extended to encompass increasing and decreasing returns to scale, natural resources, human capital, endogenous population growth and endogenous technological change all without major alteration in the character of the model. In this survey, the completely aggregated one-sector model will be the main focus of attention. Models with several sectors (agriculture and industry, consumption goods and capital goods) have attracted attention from time to time, but they tend to raise different issues. To discuss them would break continuity. The main loss from this limitation is that the important literature on open-economy aspects of growth theory has to be ignored. [The main reference is Grossman and Helpman (1991) and the work they stimulated.] Apart from the underlying restriction to "equilibrium growth" (meaning, in practice, the full utilization already mentioned), the most important neoclassical attribute is the assumption of diminishing returns to capital and labor. Here "capital" means (the services of) the stock of accumulated real output in the strictest one-good case, or the complex of stocks of all accumulatable factors of production, including human capital and produced knowledge, when they are explicitly present. The further assumption of constant returns to scale is typically neoclassical, no doubt, but it is not needed in some unmistakably neoclassical approaches to growth theory. The text will be neutral as between the ultra-strong neoclassical assumption that the economy traces out the intertemporal utility-maximizing program for a single immortal representative consumer (or a number of identical such consumers) and the weaker assumption that saving and investment are merely common-sense functions of observables like income and factor returns. The long-run implications tend to be rather similar anyway. Much of growth theory, neoclassical or otherwise, is about the structural characteristics of steady states and about their asymptotic stability (i.e., whether equilibrium paths from arbitrary initial conditions tend to a steady state). The precise definition of a steady state may differ from model to model. Most often it is an evolution
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along which output and the stock of capital grow at the same constant rate. It would be possible to pay much more attention to non-steady-state behavior, by computer simulation if necessary. The importance of steady states in growth theory has both theoretical and empirical roots. Most growth models have at least one stable steady state; it is a natural object of attention. Moreover, ever since Kaldor's catalogue of "stylized facts" [Kaldor (1961)], it has generally, if casually, been accepted that advanced industrial economies are close to their steady-state configurations, at least in the absence of major exogenous shocks. The current vogue for large international cross-section regressions, with national rates of growth as dependent variables, was stimulated by the availability of the immensely valuable Summers-Heston (1991) collection of real national-accounts data for many countries over a fairly long interval of time. The results of all those regressions are neither impressively robust nor clearly causally interpretable. Some of them do suggest, however, that the advanced industrial (OECD) economies may be converging to appropriate steady states. There is nothing in growth theory to require that the steady-state configuration be given once and for all. The usefulness of the theory only requires that large changes in the determinants of steady states occur infrequently enough that the model can do meaningful work in the meanwhile. Then the steady state will shift from time to time whenever there are major technological revolutions, demographic changes, or variations in the willingness to save and invest. These determinants of behavior have an endogenous side, no doubt, but even when established relationships are taken into account there will remain shocks that are too deep or too unpredictable to be endogenized. No economy is a close approximation to Laplace's clockwork universe, in which knowledge of initial positions and velocities is supposed to determine the whole future.
2. The H a r r o d - D o m a r model
This survey is not intended as a history of thought, It is worth saying, however, that neoclassical growth theory arose as a reaction to the Harrod-Domar models of the 1940s and 1950s [Harrod (1939), Domar (1946)]. (Although their names are always linked, the two versions have significant differences. Harrod is much more concerned with sometimes unclear thoughts about entrepreneurial investment decisions in a growing economy. Domar's more straightforward treatment links up more naturally with recent ideas.) Suppose that efficient production with the aggregate technology requires a constant ratio of capital to (net) output, say Y = vK. Suppose also that net saving and investment are always a fixed fraction of net output, say I = d K / d t = s Y . Then, in order for the utilization rate of capital to stay constant, capital and output have to grow at the proportional rate st;. Since labor input is proportional to output, employment would then grow at the same rate. If labor productivity were increasing at the rate m, the growth rate of employment would be sv - m. Let the growth of the labor force, governed
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mainly by demography, be n. Then the persistence of any sort of equilibrium requires that s v = m + n; if so > m + n there would be intensifying labor shortage, limiting the use of capital, while if so < m + n there would be increasing unemployment. But the equilibrium condition so = m + n is a relation among four parameters that are treated, within the Harrod-Domar model, as essentially independently determined constants: s characterizes the economy's propensity to save and invest, v its technology, n its demography, and m its tendency to innovate. There is no reason for them to satisfy any particular equation. An economy evolving according to Harrod-Domar rules would be expected to alternate long periods of intensifying labor shortage and long periods of increasing unemployment. But this is an unsatisfactory picture of 20th century capitalism. The Harrod-Domar model also tempted many - though not its authors to the mechanical belief that a doubling of the saving-investment quota (s) would double the long-term growth rate of a developing or developed economy. Experience has suggested that this is far too optimistic. There was a theoretical gap to be filled. The natural step is to turn at least one of the four basic parameters into an equilibrating variable. Every one of those four parameters has its obvious endogenous side, and models of economic growth have been built that endogenize them. Within the neoclassical framework, most attention has been paid to treating the capital intensity of production and the rate of saving and investment as variables determined by normal economic processes. The tradition of relating population growth to economic development goes back a lot further; and labor-force participation clearly has both economic and sociological determinants. The case of technological progress is interesting. Most neoclassical growth theory has treated it as exogenous. Some authors, e.g., Fellner (1961) and von Weizsficker (1966), have discussed the possibility that internal economic factors might influence the factor-saving bias of innovations. It is then no distance at all to the hypothesis that the voltune of innovation should be sensitive to economic incentives. This was certainly widely understood. But there was little or no formal theorizing about the rate of endogenous technological progress until the question was taken up in the 1980s. The original references are Lucas (1988) and Romer (1986), but there is now a vast literature.
3. The basic one-sector model
The model economy has a single produced good ("output") whose production per unit time is Y ( t ) . The available technology allows output to be produced from current inputs of labor, L ( t ) , and the services of a stock of"capital" that consists of previously accumulated and partially depreciated quantities of the good itself, according to the production function Y = F ( K , L). (Time indexes will be suppressed when not needed.) The production function exhibits (strictly) diminishing returns to capital and labor separately, and constant returns to scale. (More will be said about this later.)
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Constant returns to scale allows the reduction Y : F ( K , L) : L F ( K / L , 1) : L F ( k , 1) = L f ( k ) ,
and thus finally that y =f(k), where y is output per unit of labor input, i.e., productivity, and k is the ratio of capital to labor input, i.e., capital intensity. From diminishing returns, f ( k ) is increasing and strictly concave. The usual further assumptions (Inada conditions) about f ( k ) eliminate uninteresting possibilities: f ( 0 ) ~>0, f ' ( 0 ) = cx~, f ' ( c ~ ) = 0. These are overly strong: the idea is that the marginal product of capital should be large at very low capital intensity and small at very large capital intensity. [No more than continuity and piecewise differentiability is required of f(.), but nothing is lost by assuming it to be at least twice continuously differentiable, so strict diminishing returns means that f l ' ( k ) < 0.] The universal assumption in growth theory is that each instant's depreciation is just proportional to that instant's stock of capital, say D = d K . This is known to be empirically inaccurate, but it is the only assumption that makes depreciation independent of the details of the history of past gross investment. The convenience is too great to give up. Since this point is usually glossed over, it is worth a moment here. A much more general description is that there is a non-increasing survivorship function j ( a ) , with j(0) = 1 and j(A)=0. (A may be infinite.) The interpretation is that j ( a ) is the fraction of any investment that survives to age a. Then if I ( t ) is gross investment, K ( t ) = f A I ( t -- a ) j ( a ) da. Now differentiation with respect to time and one integration by parts with respect to a leads to K' : I(t) -
~0'AI ( t
- a) d(a) da,
(3.1)
where d ( a ) = - j ' ( a ) is the rate of depreciation at age a. So net investment at time t depends on the whole stream of gross investments over an interval equal to the maximum possible lifetime of capital. It can be checked that only exponential survivorship, j ( a ) = e -da, simplifies to K ' = I ( t ) - d K ( t ) . This assumption will be maintained for analytical convenience. The more complicated formula could easily be adapted to computer simulation. 4. Completing the model At each instant, current output has to be allocated to current consumption or gross investment: Y= C + I . It follows that K ' = Y - d K - C = F ( K , L ) - d K - C.
(4.1)
If the labor force is exogenous and fully employed, L(t) is a given function of time. (In this completely aggregated context, the clearing of the markets for goods and
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labor amounts to the equality of saving and investment at full employment.) Then any systematic relationship that determines C(t) as a function of K ( t ) and t converts Equation (4.1) into an ordinary differential equation that can be integrated to determine the future path of the economy, given L(t) and the initial value of K. Suppose L(t) = e nt. Then simple transformations convert Equation (4.1) into autonomous per capita terms: k/-f(k)
(d+n) k
c,
(4.2)
where, of course, c = C/L. The last component to be filled in is a rule that determines consumption per capita. Here there are two distinct strategies plus some intermediate cases. The simplest possibility, as mentioned earlier, is just to introduce a plausible consumption function with some empirical support. This was the earliest device [Solow (1956), Swan (1956)]. The other extreme, now more common, is to imagine the economy to be populated by a single immortal representative household that optimizes its consumption plans over infinite time in the sort of institutional environment that will translate its wishes into actual resource allocation at every instant. The origins are in Ramsey (1928), Cass (1965) and Koopmans (1965), but there is a large contemporary literature on this basis. For excellent surveys with further references, see Barro and Sala-i-Martin (1995), Blanchard and Fischer (1989, Ch. 2), and D. Romer (1996, Chs. 1, 2).
5. The behaviorist tradition
The two simplest examples in the "behaviorist" tradition are (a) saving-investment is a given fraction of income-output, and (b) saving-investment is a given fraction (which may be unity) of non-wage income, however the distribution of income between wages and profit or interest is determined in the society at hand. The case where different fractions of wage and non-wage income are saved amounts to a mixture of (a) and (b) and does not need to be examined separately. [Complications arise if the correct distinction is between "workers" and "capitalists" instead of wages and non-wages, because workers who save must obviously become partial capitalists. See Samuelson and Modigliani (1966), and also Bertola (t994).] In all this, an important role is played by the maintained assumption that investment always equals saving at full utilization. Under the first of these hypotheses, (4.2) becomes k' = f ( k ) - (d + n) k - (1 - s ) f ( k ) = s f ( k ) - (d + n)k,
(5.1)
where of course s is the fraction of output saved and invested. The conditions imposed o n f ( k ) imply that the right-hand side (RHS) of Equation (5.1) is positive for small k
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(a+d+n)k///
Fig. 1. because f'(O) > (d + n)/s, first increasing and then decreasing because f " ( k ) < 0, and eventually becomes and remains negative because i f ( k ) becomes and remains very small. It follows that there is a unique k* > 0 such that U ( t ) > 0 when k(t)k*. Thus k* is the globally asymptotically stable rest point for k (leaving aside the origin which may be an unstable rest point [if f ( 0 ) = 0 ] . The phase diagram, Figure 1, drawn for the case f ( 0 ) = 0 , makes this clear.) The properties of k* will be discussed later. For now it is enough to note that, starting from any initial capital intensity, the model moves monotonically to a predetermined capital intensity defined from Equation (5.1) by s f ( k * ) - ( d + n ) k *= O. [Note that k* precisely validates the Harrod-Domar condition because f ( k ) / k corresponds precisely to v, now a variable. The depreciation rate appears only because Equation (5.1) makes gross saving proportional to gross output instead of net saving proportional to net output.] When the economy has reached the stationary capital intensity k*, the stock of capital is growing at the same rate as the labor force - n - and, by constant returns to scale, so is output. The only sustainable growth rate is the exogenously given n, and productivity is constant. A reasonable model of growth must obviously go beyond this. The second hypothesis mentioned earlier, that saving-investment is proportional to non-wage income, requires a theory of the distribution of income between wages and profits. The usual presumption is the perfectly competitive one: profit (per capita) is k S ( k ) because f t ( k ) is the marginal product of capital. Some further generality is almost costlessly available: if the economy is characterized by a constant degree of monopoly in the goods market and monopsony in the labor market, then profit per capita will be proportional to kf~(k) with a factor of proportionality greater than one. If sk is the fraction of profits saved and invested (or the product of that fraction and the monopoly-monopsony factor) Equation (4.2) can be replaced by
U = skkf'(k) - (d + n) k.
(5.2)
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d+n ( k)
k ~-
Fig. 2. The analysis is not very different from that of Equation (5.1). Indeed if F(K,L) is Cobb-Douglas with elasticities b and 1 - b , so that kfl(k)=by, Equations (5.1) and (5.2) coincide, with s replaced by skb. More generally, the conditions imposed on f(k) do not quite pin down the behavior of kfr(k), though they help. For instance, kf'(k) < f ( k ) as long as the marginal product of labor is positive; so the fact that in Figure 2 the graph of sf(k) eventually falls below the ray (n + d)k, irrespective of s, implies that the RHS of Equation (5.2) becomes and remains negative for large k. The other Inada condition is more complicated. Obviously f ( 0 ) = 0 implies that kH(k) goes to zero at the origin. Now the derivative ofkf'(k) i s f ( k ) + kf"(k) 1/b). This makes at most a trivial difference in the qualitative behavior
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of the solution of Equation (5.2). For some parameter choices the origin is the only steady state; for the rest there is one and only one non-zero steady state, and it is an attractor. So nothing special happens. In discrete time, however, the qualitative possibilities are diverse and complex. The discrete analogue of Equation (5.2) can easily exhibit periodic or chaotic dynamics (and even more so if there is saving from wages). It is not clear how much "practical" macroeconomic significance one should attach to this possibility; but it is surely worth study. For an excellent treatment, see B6hm and Kaas (1997). Since k f ' ( k ) / f ( k ) = e ( k ) , the elasticity of f ( . ) with respect to k, the RHS of Equation (5.2) could be translated as s k e ( k ) f ( k ) - ( n + d ) k . As this suggests, a wide variety of assumptions about (market-clearing) saving and investment can be incorporated in the model if Equation (5.1) is generalized to
k' (t) = s ( k ) f (k) - (n + d) k.
(5.3)
For example, suppose s(k) is zero for an initial interval of low values of k and y, and thereafter rises fairly steeply toward the standard value s. This pattern might correspond to a subsistence level of per capita income, below which no saving takes place. The modified phase diagram now has two non-zero steady-state values of k, the larger of which is as before. The smaller steady state is now unstable, in the sense that a small upward perturbation will launch a trajectory toward the stable steady state, while a small downward perturbation will begin a path leading to k = y = 0. This is a sort of low-equilibrium trap; similar variations can be arranged by making n a function of, say, the wage rate, and thus of k. The details are straightforward.
6. The optimizing tradition These formulations all allocate current output between consumption and investment according to a more or less mechanical rule. The rule usually has an economic interpretation, and possibly some robust empirical validity, but it lacks "microfoundations". The current fashion is to derive the consumption-investment decision from the decentralized behavior of intertemporal-utility-maximizing households and perfectly competitive profit-maximizing firms. This is not without cost. The economy has to be populated by a fixed number of identical immortal households, each endowed with perfect foresight over the infinite future. No market imperfections can be allowed on the side of firms. The firms have access to a perfect rental market for capital goods; thus they can afford to maximize profits instant by instant. For expository purposes, nothing is lost by assuming there to be just one household and one firm, both pricetakers in the markets for labor, goods, loans and the renting of capital. The firm's behavior is easy to characterize because it can afford to be myopic under these assumptions. To unclutter the notation, take d = 0. The market provides a real wage w(t) (in terms of the produced commodity) and a real (own) rate of interest i(t).
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It is always profit-maximizing for the firm to hire labor and rent capital up to the point where i(t) = S ( k ( t ) ) ,
(6.1a)
w ( t ) = f (k(t)) - k f ' (k(t)),
(6. lb)
the two RHS expressions being the marginal products of capital and labor. [To allow for depreciation, just subtract d from the RHS of Equation (6.1a).] As before, the size of the household grows like e nt, and each member supplies one unit of labor per unit time, without disutility. (This last simplifying assumption is dispensable.) The household's preferences are expressed by an instantaneous utility function u(c(t)), where c(t) is the flow of consumption per person, and a discount rate for utility, denoted by r. The household's objective at time 0 is the maximization of (6.2)
fo °~ e r t u(c(t)) e nt dt = U.
(The term e ~t can be omitted, defining a slightly different but basically similar optimization problem for the household or clan.) The maximizing c(t) must, of course, satisfy a lifetime budget constraint that needs to be spelled out. Let J ( t ) = Jot i(t) dt so that e -J(t) is the appropriate factor for discounting output from time t back to time zero. The household's intertemporal budget constraint requires that the present value (at t = 0) of its infinite-horizon consumption program should not exceed the sum of its initial wealth and the present value of its future wage earnings. In per-capita terms this says
ff
e J(O c(t) e nt dt = ko +
e J(t) w ( t ) e nt dt,
(6.3)
where non-satiation is taken for granted, so the budget constraint holds with equality, and k0 is real wealth at t = 0. Maximization of Equation (6.2) subject to condition (6.3) is standard after introdnction of a Lagrange multiplier, and leads to tile classical (Ramsey) first-order condition
IC~Ff~C~ u'(c)
1 c'(t) = i(t) - r.
c
(6.4)
The first fraction is the (absolute) elasticity of the marginal utility of consumption. So the optimizing household has increasing, stationary, or decreasing consumption according as the current real interest rate (real return on saving) exceeds, equals, or falls short of the utility discount rate. For a given discrepancy, say a positive one, consumption per head will grow faster the less elastic the marginal utility of consumption.
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R.M. Solow
((k):n
C~
CO I I
k*
ko
k
Fig. 3. In the ubiquitous special case of constant elasticity, i.e., if u ( c ) = (c 1 - h _ 1)/(1 - h), Equation (6.3) becomes l_c,(t) _ i ( t ) - r _ f ' ( k ( t ) )
c
h
- r
h
(6.3a)
by Equation (6.1a). Under these rules of the game, the trajectory of the economy is determined by Equation (6.4) or, for concreteness, Equations (6.3a) and (4.2), reproduced here with d = O as k ' ( t ) = f ( k ( t ) ) - n k ( t ) - c(t).
(6.5)
The phase diagram in c and k is as shown in Figure 3. c ' ( t ) = 0 along the vertical line defined b y f t ( k * ) = r , with e increasing to the left and decreasing to the right. U = 0 along the locus defined by c = f ( k ) - n k , with k decreasing above the curve and decreasing below it. Under the normal assumption that r > n [otherwise Equation (6.2) is unbounded along feasible paths] the intersection of the two loci defines a unique steady state:
k=k*
f'(k*)
= r,
c* = f ( k * ) - nk*.
(6.4a,b)
7. Comparing the models
This steady state is exactly like the steady state defined by a "behaviorist" model: capital per head and output per head are both constant, so capital and output grow
Ch. 9: Neoclassical Growth Theory
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at the same rate as the labor force, namely n. In the steady state, the ratio of saving and investment to output is a constant, nk*/f(k*). The steady-state investment rate is higher the higher k* turns out to be, and thus, from Equation (6.4a) the lower is r. So far as steady-state behavior is concerned, choosing a value for s is just like choosing a value for r, a higher s corresponding to a lower r. Out-of-steady-state behavior differs in the two schemes. In the usual way, it is shown that the steady state or singular point (c*, k*) is a saddle-point for the differential equations (4.2) and (6.3a). History provides only one initial condition, namely k0. If the initial value for c is chosen anywhere but on the saddle path, the resulting trajectory is easily shown to be non-optimal for the household (or else ultimately infeasible). The appropriate path for this economy is thus defined by the saddle path, which leads asymptotically to the steady state already discussed. Of course the saving-investment rate is not constant along that path, although it converges to the appropriate constant value (from above if k0 0. The difference is that in this steady state aggregate capital, output and consumption are all proportional to e (a+n)t SO that capital, output and consumption per person in natural units are all growing at the exponential rate a, to be thought of as the growth rate of productivity. This growth rate is obviously independent of s. The effect of a sustained step increase in s, starting from a steady state, is a temporary increase in the aggregate and productivity growth rates that starts to taper off immediately. Eventually the new path approaches its own steady state, growing at the same old rate, but proportionally higher than the old one. [There is a possibility of over investment, if f ~ ( k * ) < a + n + d, in which case higher s increases output but decreases consumption. This will be elaborated later, in connection with the Diamond overlapping-generations model.] The situation is slightly more complicated in the optimizing version of the model because the argument of c(t) must continue to be current consumption per person in natural units, i.e. consumption per effective unit of labor multiplied by e nt. This does not change the structure of the model in any important way. The details can be found in D. Romer (1996, Ch. 2). It goes without saying that the introduction of exogenous technical progress achieves a steady state with increasing productivity, but does not in any way explain it. Recent attempts to model explicitly the generation of A(t) fall under the heading of "endogenous growth models" discussed in the original papers by Lucas (1988) and EM. Romer (1990), and in the textbooks of Barro and Sala-i-Marfin (1995) and D. Romer (1996). A few remarks about endogenizing aspects of technical progress within the neoclassical framework are deferred until later.
10. The role of labor-augmentation
The question remains: what is the role of the assumption that exogenous technical progress is purely labor-augmenting? It is clear that either version of the model, and especially easily with the behaviorist version, can be solved numerically without any such assumption. It is just a matter of integrating the differential equation k I = s f ( k ; t) - (n + d) k. The deeper role of labor-augmentation has to do with the importance - in theory and in practice - attached to steady states. It can be shown that purely labor-augmenting technical progress is the only kind that is compatible with the existence of a steady-state trajectory for the model. This observation was due originally to Uzawa (1961). Since the proof is not easily accessible, a compact version is given here. To begin with, it is worth noting that labor-augmenting technical progress is often described as "Harrod-neutral" because Roy Harrod first observed its particular
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significance for steady states. We have defined a steady state as a growth path characterized by a constant ratio of capital to output. For a well-behaved f ( k ; t) it is clear that constancy of the average product of capital is equivalent to constancy of the marginal product of capital. So a steady state might just as well be characterized as a path with constant marginal product o f capital. In the same way, because y/k is monotone in k, one can express k as a function of k/y and t and therefore y = f ( k ; t) =f(k(k/y; t); t) = g(z; t), where z stands for k/y. Now a straightforward calculation leads to dy/dk = gz/(g +zgz). The requirement that dy/dk be independent o f time for given k/y says that the RHS of this equation is independent of t and therefore equal to a function of z alone, say c(z). Thus one can write gz(z; t)=c(z)[g(z;t)+zgz(z; t)], and finally that gz(z; t)/g(z; t ) = c ( z ) / ( 1 - z c ( z ) ) . The RItS depends only on z, and thus also the LHS, which is d(lng)/dz. Integrating this last, one sees that lng(z; t) must be the sum of a function of t and a function of z, so that g(z; t) = y = A(t)h(z). Finally z = k/y = h -1(y/A), whence k/A = (y/A) h -1 (y/A) =j(y/A) and y/A =j-I (k/A). This is exactly the purely-labor-augmenting form: Y = F(K, AL) means Y = ALF(K/AL, 1) or
y/A = f (k/A). The assumption that technical progress is purely labor-augmenting is thus just as arbitrary as the desire that a steady-state trajectory should be admissible. That property brings along the further simplifications.
11. Increasing returns to scale There is an almost exactly analogous, and less well understood, way of dealing with increasing returns to scale. Leaving the model in extensive variables, one sees that the equation K'(t) = sF[K(t),A(t)L(t)] can be integrated numerically for any sort of scale economies. Trouble arises only when one looks for steady-state trajectories, as a simple example shows. Suppose F is homogeneous of degree h in K and AL. If K and AL are growing at the same exponential rate g, Y =F(K,AL) must be growing at the rate gh. Unless h = 1, steady state trajectories are ruled out. There is a simple way to restore that possibility. Let h be a positive number not equal to 1 and suppose the production function F [ K , A L h] is homogeneous of degree 1 in K and AL h. Production exhibits increasing returns to scale in K and L if h > 1: doubling K and L will more than double AL h and thus more than double output, though F is generally not homogeneous of any degree in K and L. (Obviously, if F is C o b b Douglas with exponents adding to more than 1, it can always be written in this special form.) But now, if A grows at the exponential rate a and L at the rate n, it is clearly possible to have a steady state with Y and K growing at rate g = a + nh. (The same goes for h < 1, but the case of increasing returns to scale is what attracts attention.) It is an interesting property of such a steady state that productivity, i.e., output per unit of labor in natural units, Y/L, grows at the rate g - n = a + ( h - 1)n. Thus the model with increasing returns to scale predicts that countries with faster growth of the labor force
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will have faster growth rates of productivity, other things equal. This seems empirically doubtful. This discussion speaks only to the existence of a steady state with non-constant returns to scale. More is true. Within the behaviorist model, the steady state just described is a global attractor (apart from the trivial trap at the origin). To see this it is only necessary to redefine y as Y/AL h and k similarly. The standard calculation then shows that k' = s f ( k ) - (a + hn + d) k, with a unique stable steady state at k*, defined as the unique non-zero root of the equation sf(k*) = (a + hn + d) k*. Note that, with h > 1, a higher n goes along with a smaller k* but a higher productivity growth rate. The appropriate conclusion is that the neoclassical model can easily accommodate increasing returns to scale, as long as there are diminishing returns to capital and augmented labor separately. Exactly as in the case of exogenous technical progress, a special functional form is needed only to guarantee the possibility of steady-state growth. The optimizing version of the model requires more revision, because competition is no longer a viable market form under increasing returns to scale; but this difficulty is not special to growth theory.
12. Human capital Ever since the path-breaking work of T.W. Schultz (1961) and Gary Becker (1975) it has been understood that improvement in the quality of labor through education, training, better health, etc., could be an important factor in economic growth, and, more specifically, could be analogized as a stock o f " h u m a n capital". Empirical growth accounting has tried to give effect to this insight in various ways, despite the obvious measurement difficulties. (For lack of data it is often necessary to use a current flow of schooling as a surrogate for the appropriate stock.) See for just a few of many examples, Denison (1985) and Collins and Bosworth (1996). The U.S. Bureau of Labor Statistics, in its own growth-accounting exercises, weights hours worked with relative wage rates, and other techniques have been tried. These considerations began to play a central role in theory with the advent of endogenous growth theory following Romer and Lucas, for which references have already been given. Here there is need only for a sketch of the way human capital fits into the basic neo-classical model. Corresponding empirical calibration can be found in Mankiw, Romer and Weil (1992) and Islam (1995). Let H ( t ) be a scalar index of the stock of human capital, however defined, and assume as usual that the flow of services is simply proportional to the stock. Then the extensive production function can be written as Y = F ( K , H, L). If there is exogenous technical progress, L can be replaced by AL as before. Assume that F exhibits constant returns to scale in its three arguments. (If desired, increasing returns to scale can be accommodated via the device described in the preceding section.) Then the intensive productivity function is y = F(k, h, 1) = f ( k , h). In the endogenous-growth literature, it
R.M. Solow
654 k t= 0
I~"
/Y "-~ 'h:0
k
Fig. 4.
is more usual to start with the assumption that Y = F(K, HL), so that HL is interpreted as quality-adjusted labor input. The really important difference is that it is then assumed that F is homogeneous of degree 1 in the two arguments K and HL. Obviously this implies that there are constant returns to K and H, the two accumulatable inputs, taken by themselves. This is a very powerful assumption, not innocent at all. Within the neo-classical framework, the next step is a specification of the rules according to which K and H are accumulated. Simple symmetry suggests the assumption that fractions sx and sH of output are invested (gross) in physical and human capital. (This is undoubtedly too crude; a few qualifications will be considered later.) Under these simple assumptions, the model is described by two equations:
k' = s ~ f ( k , h ) - ( a + n + d x ) k ,
h' = s H f ( k , h ) - ( a + n + d H ) h .
(12.1)
As usual, a + n is the rate of growth of the (raw) labor supply in efficiency units and dx and dH are the rates of depreciation of physical and human capital. Under assumptions on f(.,.) analogous to those usually made on f(.), there is just one non-trivial steady state, at the intersection in the (h, k) plane of the curves defined by setting the LHS of Equation (12.1) equal to zero. In the Cobb-Douglas case If(k, h) = kbh c, b + c < 1] the phase diagram is easily calculated to look like the accompanying Figure 4. With more effort it can be shown, quite generally, that the locus of stationary k intersects the locus of stationary h from below; since both curves emanate from the origin, the qualitative picture must be as in Figure 4. Thus the steady state at (h*,k*) is stable. [It is obvious from Equation (12.1) that k*/h* =sx/sH if the depreciation rates are equal; otherwise the formula is only slightly more complicated.] Thus, starting from any initial conditions, K, H and Y eventually grow at the same rate, a + n. This model with human capital is exactly analogous to the model without it. But this model is unsatisfactory in at least two ways. For one thing, the production of human capital is probably not fruitfully thought of, even at this level of abstraction, as a simple diversion of part of aggregate output. It is not clear how to model the production
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of human capital. The standard line taken in endogenous-growth theory has problems of its own. (It simply assumes, entirely gratuitously, that the rate of growth of human capital depends on the level of effort devoted to it.) Nothing further will be said here about this issue. The second deficiency is that, if investment in physical capital and in human capital are alternative uses of aggregate output, the choice between them deserves to be modeled in some less mechanical way than fixed shares. One alternative is to treat human capital exactly as physical capital is treated in the optimizing-competitive version of the neo-classical model. Two common-sense considerations speak against that option. The market for human capital is surely as far from competitive as any other; and reverting to infinite-horizon intertemporal optimization on the part of identical individuals is not very attractive either. It is possible to find alternatives that give some economic structure to the allocation of investment resources without going all the way to full intertemporal optimization. For example, if in fact one unit of output can be transformed into either one unit of physical or one unit of human capital, market forces might be expected to keep the rates of return on the two types of investment close to one another as long as both are occurring. This implies, given equal depreciation rates, that f l ( k , h)=.f2(k, h) at every instant. The condition f12 > 0 is sufficient (but by no means necessary) for the implicit function theorem to give k as a function of h. I f F(K, H, L) is Cobb-Douglas, k is proportional to h; the same is true for a wider class of production functions including all CES functions. The simplest expedient is to combine this with something like k ~= s f ( k , h ) (a + n + d) k with h replaced by h(k). Then physical investment is a fraction of output, and human-capital investment is determined by the equal-rate-of-return condition. In the Cobb-Douglas case, this amounts to the one-capital-good model with a CobbDouglas exponent equal to the sum of the original exponents for k and h. It happens that this set-up reproduces exactly the empirical results of Mankiw, Romer and Weil (1992), with the original exponents for k and h each estimated to be about 0.3. A more symmetric but more complicated version is to postulate that aggregate investment is a fraction of output, with the total allocated between physical and human capital so as to maintain equal rates of return. With depreciation rates put equal for simplicity, this reduces to the equation k ~+ H = sf(k, h) - (a + n + d)(k + h), together with h = h(k). The Cobb-Douglas case is, as usual, especially easy. But the main purpose of these examples is only to show that the neoclassical model can accommodate a role for human capital, with frameworks ranging from rules of thumb to full optimization.
13. N a t u r a l r e s o u r c e s
There is a large literature on the economics of renewable and nonrenewable resources, some of it dealing with the implications of resource scarcity for economic growth. [An early treatise is Dasgupta and Heal (1979). See also the Handbook of Natural
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Resource and Energy Economics, edited by Kneese and Sweeney (1989) for a more recent survey.] This is too large and distant a topic to be discussed fully here, but there is room for a sketch of the way natural resources fit into the neoclassical growththeoretic framework. The case of renewable natural resources is simplest. Some renewable resources, like sunlight or wind, can be thought of as providing a technology for converting capital and labor (and a small amount of materials) into usable energy. They require no conceptual change in the aggregate production function. More interesting are those renewable resources - like fish stocks and forests - that can be exploited indefinitely, but whose maximal sustainable yield is bounded. Suppose the production function is Y = F ( K , R , e(g+n)t), with constant returns to scale, where R is the input of a renewable natural resource (assumed constant at a sustainable level) and the third input is labor in efficiency units. If a constant fraction of gross output is saved and invested, the full-utilization dynamics are K ~ = s F ( K , R, e (g+n)t) - dK, where R is remembered to be constant. For simplicity, take F to be Cobb-Douglas with elasticities a, b and 1 - a - b for K, R and L respectively. The model then looks very much like the standard neoclassical case with decreasing returns to scale. It is straightforward to calculate that the only possible exponential path for K and Y has them both growing at the rate h = (1 - a - b)(g + n)/(1 - a). If intensive variables are defined by y = Ye ht and k K e -ht, the usual calculations show that this steady state is stable. In it, output per person in natural units is growing at the rate h - n = [(1 - a - b ) g - bn]/(1 - a ) . For this to be positive, g must exceed b n / ( 1 - a - b). This inequality is obviously easier to satisfy the less important an input R is, in the sense of having a smaller Cobb-Douglas elasticity, i.e., a smaller competitive share. If the resource in question is nonrenewable, the situation is quite different. In the notation above, R/> 0 stands for the rate of depletion of a fixed initial stock So given at t = 0. Thus the stock remaining at any time t > 0 is S(t) and S(t) = ft ~ R(u) du, assuming eventual exhaustion, so that R ( t ) = - S ~ ( t ) . Along any non-strange trajectory for this economy, R(t) must tend to zero. Even if F ( K , O , A L ) = 0, it is possible in principle for enough capital formation and technological progress to sustain growth. But this has not been felt to be an interesting question to pursue. It depends so much on the magic of technological progress that both plausibility and intellectual interest suffer. The literature has focused on two other questions. First, taking L to be constant, and without technological progress, when is a constant level of consumption per person sustainable indefinitely, through capital accumulation alone? The answer is: if the asymptotic elasticity of substitution between K and R exceeds 1, or equals 1 and the elasticity of output with respect to capital exceeds that with respect to R. For representative references, see Solow (1974), Dasgupta and Heal (1979), and Hartwick (1977). Second, and more interesting, how might such an economy evolve if there is a "backstop" technology in which dependence on nonrenewable resources is replaced by dependence on renewable resources available at constant cost (which may decrease =
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through time as technology improves). In pursuing these trails, capital investment can be governed either by intertemporal optimization or by rule of thumb. The depletion of nonrenewable resources is usually governed by "Hotelling's rule" that stocks of a resource will rationally be held only if they appreciate in value at a rate equal to the return on reproducible capital; in the notation above, this provides one differential equation: dFR/dt=FRFK. The other comes from any model of capital investment.
14. Endogenous population growth and endogenous technological progress in the neoclassical framework Making population growth and technological progress endogenous is one of the hallmarks of the "New" growth theory [see Barro and Sata-i-Martin (1995) for references]. Needless to say, one way of endogenizing population growth goes back to Malthus and other classical authors, and has been adapted to the neoclassical-growth framework from the very beginning. There was also a small literature on endogenous technical progress in the general neoclassical framework, e.g., Fellner (1961) and von Weizs~icker (1966), but it was concerned with the likely incidence of teclmical change, i.e., its labor-saving or capital-saving character, and not with its pace. However the same simple device used in the case of population can also be used in the case of technology. It is outlined briefly here for completeness. The Malthusian model can be simplified to say just that the rate of population (labor-force) growth is an increasing function of the real wage; and there is at any time a subsistence wage - perhaps slowly changing - at which the population is stationary. In the neoclassical model, the real wage is itself an increasing function of the capital intensity (k), so the subsistence wage translates into a value k0 that separates falling population from growing population. There is no change in the derivation of the standard differential equation, except that the rate of growth of employment is now n(k), an increasing function of k vanishing at k0. One might wish to entertain the further hypothesis that there is a higher real wage, occurring at a higher capital intensity kl such that n(k) is decreasing for k > kl, and may fall to zero or even beyond. Technical progress can be handled in the same way. Imagine that some unspecified decision process makes the rate of (labor-augmenting) technological progress depend on the price configuration in the economy, and therefore on k. (The plausibility of this sort of assumption will be discussed briefly below.) In effect we add the equation Al(t) = a(k)A(t) to the model. The remaining calculations are as before, and they lead to the equation k' = s f ( k ) - ( d + n ( k ) + a ( k ) ) k ,
(14.1)
where it is to be remembered that k = K / A L stands for capital per unit of labor in efficiency units.
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(d+a(k)+h(k))k s~f(k)
~ kl~
~
k2
k3
Fig. 5.
The big change is that the last term in Equation (14.1) is no longer a ray from the origin, and may not behave simply at all. It will start at the origin. For small k, it will no doubt be dominated by the Malthusian decline in population and will therefore be negative. One does not expect rapid technological progress in poor economies. For larger values of k, n(k) is positive and so, presumably, is a(k). Thus the last term of Equation (14.1) rises into positive values; one expects it to intersect sf(k) from below. Eventually - the "demographic transition" - n(k) diminishes back to zero or even becomes negative. We have no such confident intuition about a(k). On the whole, the most advanced economies seem to have faster growth of total factor productivity, but within limits. Figure 5 shows one possible phase diagram, without allowing for any bizarre patterns. The steady state at the origin is unstable. The next one to the right is at least locally stable, and might be regarded as a "poverty trap". The third steady state is again unstable; in the diagram it is followed by yet another stable steady state with a finite basin of attraction. Depending on the behavior of a(k), there might be further intersections. For a story rather like this one, see Azariadis and Drazen (1990). There are many other ideas that lead to a multiplicity of steady states. The interesting aspect of this version of the model is that k is output per worker in efficiency units. At any steady state k*, output per worker in natural units is growing at the rate a(k*). It is clear from the diagram that a change in s, for instance, will shift k* and thus the steady-state growth rate of productivity. It will also shift n(k*) and this is a second way in which the aggregate growth rate is affected. So this is a neoclassical model whose growth rate is endogenous. The question is whether the relation A ~ ~ a(k)A has any plausibility. The Malthusian analogue L / = n(k)L has a claim to verisimilitude. Birth and death rates are likely to
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659
depend on income per head; more to the point, births and deaths might be expected to be proportional to the numbers at risk, and therefore to the size of the population. One has no such confidence when it comes to technical change. Within the general spirit of the neoclassical model, something like a(k) seems reasonable; k is the natural state variable, determining the relevant prices. But the competitive market form seems an inappropriate vehicle for studying the incentive to innovate. And why should increments to productive knowledge be proportional to the stock of existing knowledge? No answer will be given here, and there may be no good answer. The relevant conclusion is that population growth and technological progress can in principle be endogenized within the framework of the neoclassical growth model; the hard problem is to find an intuitively and empirically satisfying story about the growth of productive technology.
15. Convergence The simplest form of the neoclassical growth model has a single, globally stable steady state; if the model economy is far from its steady state, it will move rapidly toward it, slowing down as it gets closer. Given the availability of the Summers-Heston crosscountry data set of comparable time series for basic national-product aggregates, it is tempting to use this generalization as a test of the neoclassical growth model: over any common interval, poorer countries should grow faster than rich ones (inper capita terms). This thought has given rise to a vast empirical literature. Useful surveys are Barro and Sala-i-Martin (1995) and Sala-i-Martin (1996), both of which give many further references. The empirical findings are too varied to be usefully discussed here, but see chapter 4 by Durlauf and Quah in this volume, and also chapter 10. Sala-i-Martin distinguishes between/j-convergence and a-convergence. The first is essentially the statement given above; it occurs when poor countries tend to grow faster than rich ones. On the other hand, a-convergence occurs within a group of countries when the variance of their p e r capita GDP levels tends to get smaller as time goes on. Clearly/j-convergence is a necessary condition for a-convergence; it is not quite sufficient, however, though one would normally expect/J-convergence to lead eventually to a-convergence. Something can be said about the speed of convergence if the neoclassical model holds. Let gt, r stand for the economy's per capita growth rate over the interval from t to t + T , meaning that gt, r = T-11og[y(t+T)/y(t)]. Then linearizing the neoclassical model near its steady state yields an equation of the form gt, r = const. - T-l(1 - e -/~r) logy(t).
(15.1)
Obviously gt,0 =/Jlogy. Moreover, in the Cobb-Douglas case with f ( k ) = k b, it tma~s out that/3 = (1 - b)(d + n + a). Another way to put this is that the solution to the basic differential equation, near the steady state at k*, is approximately k(t) - k* ~ e-b(a+n+")t(k0 - k*).
(15.2)
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Since b is conventionally thought to be near 0.3, this relation can be used to make /3-convergence into a tighter test of the neoclassical model. [It usually turns out that b must be considerably larger than that to make the model fit; this has led to the thought that human capital should be included in k, in which case the magnitudes become quite plausible. On this see Mankiw, Romer and Weil (1992).] One difficulty with all this is that different countries do not have a common steady state. In the simplest model, the steady-state configuration depends at least on the population growth rate (n) and the saving-investment rate (s) or the utility parameters that govern s in the optimizing version of the model. One might even be permitted to wonder if countries at different levels of development really have effective access to a common world technology and its rate of progress; "backwardness" may not be quite the same thing as "low income". In that case, an adequate treatment of convergence across countries depends on the ability to control for all the determinants of the steadystate configuration. The empirical literature consists largely of attempts to deal with this complex problem. On this, see again chapter 10 in this Handbook. The natural interim conclusion is that the simple neoclassical model accounts moderately well for the data on conditional convergence, at least once one allows for the likelihood that there are complex differences in the determination of steady states in economies at different stages of development. The main discrepancy has to do with the speed of convergence. This is perhaps not surprising: actual investment paths will follow neither optimizing rules nor simple ratios to real output. Outside the simplest neoclassical growth model, there may even be multiple steady states, and this clearly renders the question of/3-convergence even more complicated. This possibility leads naturally to the notion of club-convergence: subsets of "similar" countries may exhibit/3-convergence within such subsets but not between them. Thus the states of the United States may exhibit convergence, and also the member countries of OECD, but not larger groupings. This is discussed in Galor (1996). See also Azariadis and Drazen (1990) for a model with this property.
16. Overlapping generations Literal microfoundations for the optimizing version of the standard neoclassical model usually call for identical, immortal households who plan to infinity. An equivalent and equally limiting - assumption involves a family dynasty of successive generations with finite lives, each of which fully internalizes the preferences of all succeeding generations. An alternative model avoids some restrictiveness by populating the economy with short-lived overlapping generations, each of which cares only about its own consumption and leaves no bequests. The simplest, and still standard, version involves two-period lives, so that two generations - young and old - coexist in each period. As the previous sentence suggests, overlapping-generations models are written in discrete time, although this is not absolutely necessary [Blanchard and Fischer (1989),
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p. 115]. There is a large literature beginning with Samuelson (1958) [anticipated in part by Allais (1947)]. An excellent exposition is to be found in D. Romer (1996, Ch. 2), and a full treatment in Azariadis (1993), where further references can be fotmd. The OG model has uses in macroeconomic theory generally [for instance Grandmont (1985), Hahn and Solow (1996)], but here attention is restricted to its use in growth theory, beginning with Diamond (1965). There is only one produced good, with the production function Yt = F ( K t , A t N t ) as usual. In the standard notation, we can write Yt = f ( k t ) , where y and k are output and capital per unit of labor in efficiency traits. In each period, then, the competitively determined interest rate is rt =f~(kt) and the wage in terms of the single good is Atwt = A t ( f ( k t ) - ktf~(kt)). Note that wt is the wage per efficiency unit of labor; a person working in period t earns Atwt. Nt families are born at the beginning of period t and die at the end of period t + 1. Set Nt = (1 + n) t so the total population in period t is (1 + n) t 1 + (1 + n) t. Each family supplies one unit of labor inelastically when it is young, earns the going (real) wage Atwt, chooses how much to spend on the single good for current consumption etl, earns the going rate of return rt+l on its savings ( A t w t - c t l ) , and spends all o f its wealth on consumption when old, so that ct2 = (1 + rt+l)(Atwt - Ctl). Note that savings in period t are invested in period t + 1. As with other versions of growth theory, it is usual to give each household the same time-additive utility function u ( c t l ) + ( 1 + i ) lu(ct2). It is then straightforward to write down the first-order condition for choice of Ctl and c~2. It is U'(Ct2)/U'(Ctl) = (1 +i)/(1 +rt+l); together with the family's intertemporal budget constraint it determines etl and ct2, and therefore the family's savings in period t as a function of rt+l and Atwt. In the ever-popular special case that u(x) = (1 - m)-lx I m (SO that m is the absolute elasticity of the marginal utility function), it follows directly that the young family saves a fraction s(r) of its wage income, where
(1 + r) (1-m)/m s(r) = (1 + r)( 1-m)/m + (1 + i) 1/m
(16.1)
and the obvious time subscripts are omitted. The formulas for young and old consumption follow immediately. Now assume that At -- (1 + a) t as usual. Since the savings of the young finance capital investment in the next period, we have Kt+l = s(rt+l)AtwtNt. Remembering that k is K l A N , we find that kt+l = (1 + n) l(1 + a)-ls(rt+l)wt.
(16.2)
Substitution of rt+l =f~(kt+l) and wt = f ( k t ) - k t f t ( k t ) leaves a first-order difference equation for kt. In simple special cases, the difference equation is very well behaved. For instance (see D. Romer 1996 for details and exposition), if f ( - ) is Cobb-Douglas and u(.) is
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kt+l
J
k0
kI
k2
k3
k~
kt
Fig. 6.
logarithmic, the difference equation takes the form kt+l = const.ktb and the situation is as in Figure 6. (Note that logarithmic utility implies that the young save a constant fraction o f their earnings, so this case is exactly like the standard neoclassical model.) There is one and only one stationary state for k, and it is an attractor for any initial conditions k0 > 0. Exactly as in the standard model, k* decreases when n or a increases and also when i increases (in which case s naturally decreases). Since k =K/AN, the steady-state rate of growth of output is a + n and the growth rate of labor productivity is a, both independent of i (or s). There are, however, other possibilities, and they can arise under apparently "normal" assumptions about utility and production. Some o f these possibilities allow for a multiplicity of steady states, alternately stable and unstable. This kind of configuration can arise just as easily in the standard model when the saving rate is a function of k. They are no surprise. The novel possibility is illustrated in Figure 7. The curve defined by Equation (16.2) in the (kt, k t + l ) plane may bend back, so that in some intervals - in the diagram, when kt is between km and kM - kt is compatible with several values of kt+t. The difference equation can take more than one path from such a kt. This is the situation that gives rise to so-called "sunspot" paths. See Cass and Shell (1983), Woodford (1991), and an extensive treatment in Farmer (1993). The mechanism is roughly this. Suppose sl(r) < 0. Then a young household at time t that expects a low value of rt+l will save a lot and help to bring about a low value of r. If it had expected a high value of r next period, it would have saved only a little and helped to bring about a high value of r. The possibility exists that the household may condition its behavior on some totally extraneous phenomenon (the "sunspot" cycle) in such a way that its behavior validates the implicit prediction and thus confirms the significance of the fundamentally irrelevant signal. In this particular model, the sunspot phenomenon seems to require that saving be highly sensitive to the interest
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kt+l
km
kM
kt
Fig. 7.
rate, and in the "wrong" direction at that. This goes against empirical findings, so that indeterminacy of this kind may not be central to growth theory, even if it is significant for short-run macroeconomic fluctuations.
17. Open
questions
This survey has stayed close to basics and has not attempted anything like a complete catalogue of results in growth theory within the neoclassical framework. In that spirit, it seems appropriate to end with a short list of research directions that are currently being pursued, or seem worth pursuing. The role of human capital needs clarification, in both theoretical and empirical terms. Human capital is widely agreed to be an important factor in economic growth. Maybe more to the point, it seems to offer a way to reconcile the apparent facts of convergence with the model. One difficulty is that the measurement of human capital is insecure. See Judson (1996) and Klenow and Rodriguez-Clare (1998). School enrollment data are fairly widely available, but they clearly represent a flow, not a stock. Direct measurement of the stock runs into deep uncertainty about depreciation and obsolescence, and about the equivalence of schooling and investment in human capital. Mention has already been made of the use of relative wages as indicators of relative human capital; the well-known Mincer regressions can also be used, as in Klenow and Rodriguez-Clare (1998). (Better measurement might throw some light on the way human capital should enter the production function: as a labor-augmentation factor, as a separate factor of production, or in some other way. On this, as on several
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other matters, the distinction between "neoclassical" and "endogenous" growth theory seems to be artificial.) It was suggested in the text above that there is no mechanical obstacle to the endogenization of technical change within the neoclassical model. But the analytical devices mentioned by example were all too mechanical. The modeling of technological progress should be rethought, and made more empirical using whatever insights come from micro-studies of the research-and-development process. It seems pretty clear that the endogenous-growth literature has been excessively generous in simply assuming a connection between the level of innovative effort and the rate of growth of the index of technology. It is impossible to know what further empirical work and theoretical modeling will suggest about the nature of that connection. But it is a central task to find out. One of the earliest stories about endogenous technical change was Arrow's model of "learning by doing" (Arrow 1962), which is well within the neoclassical framework. It, too, was rather mechanical, with automatic productivity increase as simple fall-out from gross investment. Many economists have found the basic idea to be plausible; it is a source of technical change that is entirely independent of R&D. But very little econometric work has taken off from learning-by-doing, and there seems to have been no attempt to test it. Recently there have been renewed efforts to elaborate and improve the underlying idea [Young (1993), Solow (1997)]. The next step should probably be empirical. Very similarly, the notion that technical change has to be "embodied" in new investment in order to be effective seems instantly plausible [Solow (1960), Jorgenson (1966)]. For many years, however, it proved to be impossible to verify its importance in macroeconomic time series data. Just recently there has been a revival of interest in this question. Lau (1992) may have isolated a significant embodiment-effect in a combined time-series analysis involving several advanced and newly-industrialized countries. And Wolff (1996) claims to have found in the embodiment-effect an explanation of the productivity slowdown that occurred almost worldwide in the early 1970s. Hulten (1992), on the other hand, came to different conclusions using different methods and data. The interesting possibility of using changes in the relative prices of investment goods and consumption goods to isolate the embodiment effect has opened up new vistas. Greenwood, Hercowitz and Krusell (1997) is a pioneering reference. See also the survey by Hercowitz (1998) and an (as yet) unpublished paper by Greenwood and Jovanovic (1997). Some new theory might help develop this work further. A powerful embodiment effect (and the same could be said about learning by doing) will strengthen the connection between short-run macroeconomic fluctuations and long-run growth. As things stand now, the only effect of business cycles on the growth path comes through the "initial" value of the stock of capital. These more sophisticated mechanisms would also link growth to cycle through the level of achieved technology. There are no doubt other ways in which better integration of growth theory and business-cycle theory would improve both of them. A last issue that needs exploring is the matter of increasing returns to scale. It was shown earlier that the neoclassical model can easily accommodate increasing (or,
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for that matter, decreasing) returns to scale, just as a matter o f modeling production. The important question lies elsewhere. The ubiquity o f increasing returns to scale implies the ubiquity o f imperfect competition as a market form. There is plenty o f microeconomic theory to link imperfect competition with investment and perhaps with innovative activity. The systematic relation - if any - between imperfect competition and growth has not been fully explored. What there is has come mostly through the endogenous-growth literature [Aghion and Howitt (1992, 1998), Romer (1990)), and there it has been an appendage to specialized models o f the R & D process. Imperfect competition is finding its way slowly into general macroeconomics. Growth theory should not be far behind.
References Aghion, P., and P. Howitt (1992), "A model of growth through creative destruction", Econometrica 60:323 351. Aghion, P., and P. Howitt (1998), Endogenous Growth Theory (MIT Press, Cambridge, MA.). Allais, M. (1947), Economic et InterSt (Imprimerie Nationale, Paris). Arrow, K. (1962), "The economic implications of learning by doing", Review of Economic Studies 26:155-173. Azariadis, C. (1993), Intertemporal Macroeconomics (Blackwell, Oxford). Azariadis, C., and A. Drazen (1990), "Threshold externalities in economic development", Quarterly Journal of Economics 105:501-526. Barro, R.J., and X. Sala-i-Martin (1995), Economic Growth (McGraw-Hill, New York). Becker, G. (1975), Human Capital, 2nd edition (National Bureau of Economic Research/Columbia University Press, New York). Bertola, G. (1994), "Wages, profits and theories of growth", in: L. Pasinetti and R. Solow, eds., Economic Growth and the Structure of Long-Term Development (St. Martin's Press, New York) 90-108. Blanchard, O.J., and S. Fischer (1989), Lectures in Macroeconomics (MIT Press, Cambridge, MA). B6hm, V., and L. Kaas (1997), "Differential savings, factor shares, and endogenous growth cycles", Working Paper (Department of Economics, University of Bielefeld). Cass, D. (1965), "Optimum growth in an aggregative model of capital accumulation", Review of Economic Studies 32:233~40. Cass, D., and K. Shell (1983), "Do sunspots matter?", Journal of Political Economy 91:193~27. Collins, S., and B. Bosworth (1996), "Economic growth in East Asia: accumulation vs. assimilation", Brookings Papers on Economic Activity 1996(2): 135-191. Dasgupta, P., and G. Heal (1979), Economic Theory and Exhaustible Resources (Cambridge University Press, Cambridge). Denison, E.E (1985), Trends in American Economic Growth, 1929-1982 (The Brookings Institution, Washington, DC). Diamond, EA. (1965), "National debt in a neoclassical growth model", American Economic Review 55:1126-1150. Domar, E. (1946), "Capital expansion, rate of growth and employment", Econometrica 14:137 147. Farmer, R. (1993), The Macroeconomics of Self-fulfilling Prophecies (MIT Press, Cambridge). Fellner, W (1961), "Two propositions in the theory of induced innovations", Economic Journal 71: 305-308. Galor, O. (1996), "Convergence? Inferences from theoretical models", Working Paper No. 1350 (Center for Economic Policy Research, London). Grandmont, J.-M. (1985), "On endogenous competitive business cycles", Eeonometriea 53:995-1046.
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Greenwood, J., and B. Jovanovic (1997), "Accounting for growth", unpublished. Greenwood, J., Z. Hercowitz and R Kxusell (1997), "Long-run implications of investment specific technological change", American Economic Review 87:342-362. Grossman, G.M., and E. Helpman (1991), Innovation and Growth in the Global Economy (MIT Press, Cambridge, MA). Hahn, E, and R.M. Solow (1996), A Critical Essay on Modem Macroeconomic Theory (MIT Press, Cambridge, MA). Harrod, R. (1939), "An essay in dynamic theory", Economic Journal 49:14-33. Hartwick, J. (1977), "Intergenerational equity and the investing of rents from exhaustible resources", American Economic Review 66:972 974. Hercowitz, Z. (1998), "The 'embodiment' controversy: a review essay", Journal &Monetary Economics 41:217-224. Hulten, C. (1992), "Growth accounting when technical change is embodied in capital", American Economic Review 82:964-980. Islam, N. (1995), "Growth empirics: a panel data approach", Quarterly Journal of Economics 110: 1127-1170.
Jorgenson, D. (1966), "The embodiment hypothesis", Journal of Political Economy 74:1 17. Judson, R. (1996), "Measuring human capital like physical capital: what does it tell us?", Working Paper (Federal Reserve Board, Washington, DC). Kaldor, N. (1961), "Capital accumulation and economic growth", in: EA. Lutz and D.C. Hague, eds., The Theory of Capital (St. Martin's Press, New York). King, R.G., and S.T. Rebelo (1993), Transitional dynamics and economic growth in the neoclassical model, American Economic Review 83:908-931. Klenow, RJ., and A. Rodriguez-Clare (1998), "The neoclassical revival in growth economics: has it gone too far?", in: O.J. Blanchard, ed., Macroeconomics Annual 1997 (MIT Press, Cambridge, MA). Kneese, A., and J. Sweeney, eds (1989), Handbook of Natural Resources and Energy Economics, 3 volumes (Elsevier Science, Amsterdam). Koopmans, T.C. (1965), "On the concept of optimal economic growth", Scientific Papers of Tjalling C. Koopmans (Springer, New York). Lau, L. (1992), "The importance of embodied technical progress: some empirical evidence from the group-of-five countries", Publication No. 296, mimeograph (Center for Economic Policy Research, Stanford University, Stanford, CA). Lucas, R. (1988), "On the mechanics of economic development", Journal of Monetary Economics 22:3-42. Mankiw, N.G., D. Romer and D.N. Weil (1992), "A contribution to the empirics of economic growth", Quarterly Journal of Economics 107:407-448. Ramsey, E (1928), "A mathematical theory of saving", Economic Journal 88:543-559. Romer, D. (1996), Advanced Macroeconomics (McGraw-Hill, New York). Romer, P.M. (1986), "Increasing returns and long-run growth", Journal of Political Economy 94:1002-1037. Romer, P.M. (1990), "Endogenous technological change", Journal of Political Economy 98: $71 S 102. Sala-i-Martin, X. (1996), The classical approach to convergence analysis, Economic Journal 106:1019-1036. Samuelson, EA. (1958), "An exact consumption-loan model of interest, with or without the social contrivance of money", Journal of Political Economy 66:467-482. Samuelson, P.A., and E Modigliani (1966), "The Pasinetti paradox in neoclassical and more general models", Review of Economic Studies 33:269-301. Schultz, T.W ( 1961), Investment in human capital, American Economic Review 51:1-17. Solow, R.M. (1956), "A contribution to the theory of economic growth", Quarterly Journal of Economics 70:65-94.
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Solow, R.M. (1960), "Investment and technical progress", in: K. Arrow, S. Karlin and R Suppes, eds., Mathematical Methods in the Social Sciences (Stanford University Press, Palo Alto, CA) 89-104. Solow, R.M. (1974), "Intergenerational equity and exhaustible resources", Review of Economic Studies 41:29-45. Solow, R.M. (1997), Learning from 'Learning by Doing' (Stanford University Press, Palo Alto, CA). Summers, R., and A. Heston (1991), "The Penn world trade (Mark 5): an expanded set of international comparisons 1950-1988", Quarterly Journal of Economics 106:329 368. Swan, T.W. (1956), "Economic growth and capital accumulation", Economic Record 32:334-361. Uzawa, H. (1961), "Neutral inventions and the stability of growth equilibrium", Review of Economic Studies 28:117-124. von Weizs/icker, C. (1965), "Existence of optimal programs of accumulation for an infinite time horizon", Review of Economic Studies 32:85 104. yon Weizs/icker, C. (1966), "Tentative notes on a two-sector model with induced technical progress", Review of Economic Studies 33:245~52. Wolff, E. (1996), "The productivity slowdown, the culprit at last? Follow-up on Hulten and Wolff", American Economic Review 86:1239-1252. Woodford, M. (1991), "Self-fulfilling expectations and fluctuations in aggregate demand", in: N.G. Mankiw and D. Romer, eds., New Keynesian Economics, vol. 2 (MIT Press, Cambridge, MA) 2:77-110. Young, A. (1993), "Invention and bounded learning by doing", Journal of Political Economy 101: 443-472.
Chapter 10
EXPLAINING CROSS-COUNTRY INCOME DIFFERENCES ELLEN R. McGRATTANand JAMES A. SCHMITZ, Jr. Federal Reserve Bank of Minneapolis
Contents 670 670 671 674 678 678 687 688 695 695 695 702 707 709 709 715 720 724 730 733 734 734
Abstract Keywords 1. Introduction 2. Some basic facts 3. Accounting 3.1. Levels accounting 3.2. Growth accounting 4. Growth regressions 5. Quantitative theory 5.1. Effects of policy on disparity 5.1.1. Policies distorting investment 5.1.2. Policies affecting trade 5.1.3. Other policies 5.2. Effects of policy on growth 5.2.1. Policies in a two-sector AK model 5.2.2. Policies in an R&D model 6. Two growth models and all of the basic facts 6.1. An exogenous growth model 6.2. An endogenous growth model 7. Concluding remarks Acknowledgements References
Handbook of Macroeconomics, Volume 1, Edited by J.B. Taylor and ~ © 1999 Elsevier Science B.V. All rights reserved 669
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Abstract This chapter reviews the literature that tries to explain the disparity and variation of GDP per worker and GDP per capita across countries and across time. There are many potential explanations for the different patterns of development across countries, including differences in luck, raw materials, geography, preferences, and economic policies. We focus on differences in economic policies and ask to what extent can differences in policies across countries account for the observed variability in income levels and their growth rates. We review estimates for a wide range of policy variables. In many cases, the magnitude of the estimates is under debate. Estimates found by running cross-sectional growth regressions are sensitive to which variables are included as explanatory variables. Estimates found using quantitative theory depend in critical ways on values of parameters and measures of factor inputs for which there is little consensus. In this chapter, we review the ongoing debates of the literature and the progress that has been made thus far.
Keywords cross-country income differences, growth accounting, growth regressions, endogenous growth theory J E L classification: E62, E65, O11, O 4 1 , 0 4 7
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1. Introduction
Gross domestic product (GDP) per worker of rich countries like the USA is about 30 times that of poor countries like Ethiopia. The fastest growing countries now grow at 9 percent per year, whereas 100 years ago the highest rates of growth were around 2 percent. Over the postwar period, there is virtually no correlation between income levels and subsequent growth rates, and growth rates show very little persistence. This chapter reviews the literature that tries to explain these and other facts about the crosscountry income distribution. There are many potential explanations for the different patterns of development across countries, including differences in luck, raw materials, geography, preferences, and economic policies. As in most of the literature, we focus on economic policy and ask to what extent can differences in policies across countries account for the variability in levels of income and their growth rates. Are policies responsible for only a few percent of the income differences or for most of the variation? If they are responsible for most of the variation, which policies are particularly helpful or harmful? We show that while some progress has been made in answering these questions, it has been fairly limited. There are estimates of the effects of policy on income and growth for a wide range of policy variables. However, in most cases, their magnitudes are under debate. Moreover, there is little consensus concerning methodology. We review two approaches used to obtain estimates of the effects of policy on income and growth. The most widely used approach is to run cross-sectional regressions of growth rates on initial levels of income, investment rates, and economic policy or political variables. [See, for example, Kormendi and Meguire (1985), Barro (1991), and Barro and Sala-i-Martin (1995).] Policy variables found to have a significant effect on growth in these regressions include measures of market distortions such as the average government share in GDP or the black market premium, measures of political rights or stability, and measures of financial development. For example, Barro and Lee (1994) show that as a result of differences in the ratio of government consumption to GDP and in the black market premium between a group of East Asian countries and a group of sub-Saharan African countries, the East Asian countries were predicted to grow 3.5 percent per year faster. The actual difference in growth rates was 8.1 percent per year. Thus, these differences in the two variables account for a large fraction of the difference in growth rates. In this literature, the estimated coefficients on variables designated as policy variables have been shown to be sensitive to which variables are included in the regression. Levine and Renelt (1992) find that a large number of policy variables are not robustly correlated with growth. Hence, estimates of the impact of economic policy on growth are under debate. Another approach to calculating the effects of economic policy, which we call quantitative theory, is to specify explicit models of economic development, parameterize them, and derive their quantitative implications. In our review of quantitative theory, we start with studies that explore the extent to which differences in economic
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policies account for differences in levels of income. We consider the effects of fiscal policies, trade policies, policies affecting labor markets, and policies impeding efficient production. [Examples of such studies include Chari et al. (1997) on investment distortions, Romer (1994) on tariffs, Hopenhayn and Rogerson (1993) on labor market restrictions, Parente and Prescott (1994, 1997) on barriers to technology adoption, and Schmitz (1997) on inefficient government production.] To illustrate the quantitative effects of some of these policies, we derive explicit formulas for cross-country income differences due to inefficient government production, taxes on investment, and tariffs. These formulas show that measured differences in policies can potentially explain a significant fraction of observed income disparity. However, there is also debate in this literature about the magnitude of the impact of policy on income. Much of the debate centers around the choice of model parameters. For example, measured differences in investment distortions can account for a significant fraction of observed income disparity if shares on accumulable factors are on the order of ~ or larger. Shares on the order of 1 imply very little disparity in incomes. Measured differences in tariff rates imply significant differences in incomes if the number of imports are assumed to vary significantly with the tariff rate. Otherwise, the effects of tariffs are very small. We also review studies in quantitative theory that explore the extent to which differences in economic policies account for differences in growth rates of income. We review two standard endogenous growth models: a two-sector "AK" model and a model of research and development (R&D). For the AK model, we consider the effects of changes in tax rates on long-run growth rates as in King and Rebelo (1990), Lucas (1990), Kim (1992), Jones et al. (1993), and Stokey and Rebelo (1995). To illustrate the quantitative effects of these tax policies, we derive explicit formulas for the steady-state growth rate in terms of tax rates and parameters of the model. Here, too, the estimated impact of tax changes on growth varies dramatically in the literature. For example, the predicted decline in the growth rate after an increase in the income tax rate from 0 percent to 20 percent ranges from 7/10ths of a percent to 4 percentage points. Using the explicit formulas, we show how the estimates of tax effects on growth are sensitive to certain model parameters. Unlike the AK model, there has been little work to date assessing the effects of policy changes on growth rates in the R&D models. [See, for example, Romer (1990), Grossman and Helpman (1991a, b), and Aghion and Howitt (1992).] This is likely due to the fact that the main quantitative concern for these models has been their predicted scale effects. That is, most of these models predict that the growth rate increases with the number of people working in R&D. We describe a discrete-time version of the model in Romer (1990) and Jones' (1995a) version of the model which eliminates scale effects. [See also Young (1998).] We also discuss the possible growth effects of policies such as the subsidization of R&D and show that these effects depend critically on certain model assumptions. Both approaches to estimating the effects of policy, then, the growth regression approach and the quantitative theory approach, have provided estimates of the impact
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of policy on income and growth. But, as the examples above indicate, within each approach, the magnitude of the impact of policy is under some debate. But in comparing the two approaches, we need to compare more than the precision of their estimates of policy's effect on incomes and growth. For example, the growth regression literature has come under considerable criticism because of econometric problems. [See, for example, Mankiw (1995), Kocherlakota (1996), Sims (1996), and Klenow and Rodriguez-Clare (1997a).] One serious problem is the endogeneity of right-handside variables in these regressions. The quantitative theory approach is not subject to such econometric criticisms. Hence, while the growth regression approach is the most widely used approach, we think the quantitative theory approach will ultimately be the predominant one. Thus, we place more emphasis on it in our review. The rest of our review proceeds as follows. Section 2 presents some basic facts about the cross-country income distribution using data on GDP per worker for 19601990 compiled by Summers and Heston (t991) and on GDP per capita for 18201989 compiled by Maddison (1991, 1994). In Section 3, we review the accounting literature which has been a source of data on factor inputs and total factor productivity. Studies in the accounting literature attempt to apportion differences in country income levels or growth rates to technological progress and factor accumulation. [See, for example, Krueger (t968), Christensen et al. (1980), Elias (1992), Mankiw et al. (1992), Young (1995), Hsieh (1997), Klenow and Rodriguez-Clare (1997a), and Hall and Jones (1998).] These studies do not directly address why factor inputs differ across countries, but they do provide measures of labor and capital inputs, estimates of the shares of these inputs, and thus an estimate of either the level or the growth rate of total factor productivity (TFP). We show that, as yet, there is still no consensus on the level or growth of human capital and TFP or on the size of factor shares. The remainder of the chapter is concerned with estimating the effects of policy on income and growth. In Section 4, we review the empirical growth literature. In Section 5 we review studies applying the quantitative theory approach - considering first those concerned with differences in income levels and then those concerned with growth. The two literatures within quantitative theory, that examining disparity and that examining growth, have developed in large part separately from each other. There have been few attempts to account for more than one key regularity in the data and few attempts to compare the implications of competing theories for data. We conclude the chapter by considering the implications of two standard growth models, the neoclassical exogenous growth model and the AK model, for some of the basic features of the data from Maddison (1991, 1994) and Summers and Heston (1991). To make a direct comparison, we use the same tax processes as inputs in both models. We show that these models do fairly well in accounting for the large range in relative incomes, the lack of correlation in incomes and subsequent growth rates, and the lack of persistence in growth rates. However, both models have trouble replicating the large increase in maximal growth rates observed over the past 120 years.
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301 1989 Relative GDP Per Capita (World Average = 1)
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2. Some basic facts
In this section, we review some basic facts about the distribution of country incomes and their growth rates. We have historical data for various years over the period 18201989 from Maddison (1994) for 21 countries. For the period 1870-1989, data are available from Maddison (1991) in all years for 16 countries. More recent data are from the Penn World Table (version 5.6) of Summers and Heston (1991) and cover as many as 152 countries over the period 1950-1992 I. These data show that disparity in incomes is large and has grown over time, that there is no correlation between income levels and subsequent growth rates, that growth rate differences are large across countries and across time, and that the highest growth rates are now much higher than those 100 years ago. These basic features of the data are summarized in Figures 1-4. [See Parente and Prescott (1993) for a related discussion.] In Figure 1, we provide two perspectives on the disparity of per capita GDP across countries. First, we plot per capita GDP in 1985 US dollars for 21 countries for various years between 1820 and 1989. These data are taken from Maddison (1994). Each country-year observation is represented by a square. Second, for 1989, we display the distribution of relative GDP per capita using the 137 countries with available data in the Summers and Heston data set (variable RGDPCH). To construct the relative GDP,
1 All of the data used in this chapter are available at our web site.
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we divide a country's per capita GDP by the geometric average for all 137 countries. A value o f 8 implies that the country's per capita GDP is 8 times the world average, and a value o f ½ implies that the country's per capita GDP is g1 o f the world average. One noteworthy feature o f Figure 1 is the increase in disparity in GDP per capita over the last 170 years in Maddison's (1994) 21-country sample 2. The ratio o f the highest per capita GDP to the lowest in 1820 is 3.0, whereas the ratio in 1989 is 16.7. Hence, the range o f GDPs per capita in this sample increased by a factor of 5.6 (16.7 + 3). If we consider the Summers and Heston (1991) sample of 137 countries in 1989 (shown in the insert o f Figure 1), we find that the average GDP per capita for the top 5 percent o f countries is nearly 34 times that of the bottom 5 percent. Another notable aspect o f the 1989 distribution is its near uniformity in the range ½ to 8. Thus, it is not true that being very rich (having a per capita GDP from 4 to 8 times the world average) or being very poor (having a per capita GDP from g1 to 1 of the world average) is uncommon. Furthermore, over the period 1960-1990, the ratio o f the relative incomes o f rich to poor has been roughly constant; 1989 is not an unusual year 3. The data that we plot in Figure 1 are GDP per capita since we do not have data on the number o f workers prior to 1950. However, much o f our analysis in later sections will deal with GDP per worker. If we instead use GDP per worker to obtain an estimate o f disparity in 1989, we get a similar estimate to that found with GDP per capita. In 1989 the average GDP per worker for the most productive 5 percent of the countries is about 32 times that o f the least productive 5 percent. Next consider Figure 2, which has in part motivated the cross-sectional growth literature. Figure 2 presents average annual growth rates in GDP per worker over the 1960-1985 period versus the relative GDP per worker in 1960. For this period, data are available from Summers and Heston (1991) for 125 countries. There are two key features to note. First, there is no correlation between 1960 productivity levels and subsequent growth rates. The correlation is 0.01. Second, the range in average annual growth rates is large. Even over a 25-year period, some countries had average growth rates o f over 5 percent per year while some countries had average annual growth rates that were negative. These features o f the data are also found for GDP per capita and for the subset o f the Summers and Heston countries that have data available through 1990. [For example, see Barro and Sala-i-Martin (1995), who use GDP per capita.] Figure 3 presents average annual growth rates o f GDP per worker for a country over 1973-1985 versus average annual growth rates over 1961-1972 for the same sample o f countries used in Figure 2. As Easterly et al. (1993) note, the correlation between growth rates in the two subperiods is low. The correlation in this case is 0.16. A striking feature of the growth rates is the magnitudes across subperiods. For example, Saudi
2 Prescott (1998) calculates the disparity between western and eastern countries and finds a significant increase in disparity over the past 200 years. 3 The same is true of the Maddison 21-cotmtry sample. The ratio of the highest to lowest per capita GDP was 19.0, 19.6, and 16.7 in 1950, 1973, and 1989, respectively.
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Y8'80 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 n Year Fig. 4. Maximum GDP per capita growth, 1870-1990. Arabia grew at a rate of 8.2 percent in the first half of the sample and then at a rate o f - 1 . 8 percent in the second half. Guinea's growth rate in the first half of the sample was about 0 and jumped to 4.2 in the second half. Figure 4 plots growth rates of the fastest growing countries over time. Starting in 1870, for each country for which data are available, we calculate the average annual growth rate within each decade between 1870 and 1990. For each decade, we select the country that achieved the maximum growth rate and plot this growth rate along with the country names in Figure 4. For example, the USA achieved the maximum average annual growth over the 1870-1880 decade, about 2.5 percent. The sample of countries in Figure 4 are from two sources. From 1870 to 1950, the data are GDP per capita from Maddison (1991). Over this period, there are only 16 countries 4. From 1950 to 1990, the data are GDP per capita from Summers and Heston (1991). We included all countries with available data. The pattern in Figure 4 is striking. The maximum average annual growth rates over a decade have increased dramatically through time, from the 2-3 percent range in the late 1800s to the 8-9 percent range that we currently observe. An obvious concern is that the pattern in Figure 4 is driven by the fact that the sample of countries increased dramatically after 1950. The countries in Maddison (1991) are the ones that are the
4 Unlike the 21-country sample used for Figure 1, the data from Maddison (1991) are primarily rich countries.
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E.R. McGrattan and J.A. Schmitz, Jr.
most productive today; they are the most productive today because they had the greatest growth in productivity from 1870 to 1950. There may have been episodes during this period in which some of the poorer countries had miraculous growth rates. But, to our knowledge, no such episodes have been identified. Thus, we suspect that if data for all countries were available back to 1870 and we again drew Figure 4, the picture would not change very much 5. Before reviewing the progress that has been made in estimating the effects of policy on income and growth, we review the levels and growth accounting literatures. The objective of these literatures is to estimate the contributions of physical capital, labor, educational attainment, and technological progress to differences in levels or growth rates of output. While they do not directly address why factor inputs differ across countries, the accounting exercises are nonetheless important steps to explaining cross-country income differences. For example, to estimate the effects of policy in quantitative theories, reliable estimates for certain parameters, like the capital shares, are needed. The accounting exercises provide careful measures of labor and capital inputs, estimates of the shares of these inputs, and an estimate of TFP or its growth rate.
3. Accounting We start this section with some results of levels accounting. We show that the estimates of TFP are sensitive to the measurement of human capital and the shares of income to physical and human capital. As yet, there is little consensus on the size of the stock of human capital or on the magnitude of the factor shares. Thus, when we calculate the fraction of income differences explained by differences in observed factor inputs, we find a wide range of estimates. We then discuss some recent work in growth accounting estimating the growth in TFP for the East Asian newly industrialized countries. Here, there is less disagreement about whether good economic performances were due in large part to factor accumulation or to total factor productivity.
3.1. Levels accounting
The objective in levels accounting studies is to apportion differences in income levels to differences in levels of total factor productivity and factor inputs. Typically, the
5 In fact, if we use the Maddison (1991) data, which are availableuntil 1980, to construct growth rates between 1950 and 1980, the pattern is the same for all years except 1970-1980.
Ch. 10: ExplainingCross-CountryIncome Differences
679
starting point is an aggregate production function F - assumed to be the same across countries - o f the form
Y=F(K,H,L,A),
(3.1)
where Y is output, K is the stock o f physical capital, H is the stock o f hmnan capital, L is the labor input, A is an index o f the technology level, and income is defined to be output per worker (Y/L). These studies construct measures o f K, H , and L and treat A as a residual in Equation (3.1). Many levels accounting studies assume that the production function has a C o b b Douglas form given by
y =KakHa,,(AL)I ak a,~,
(3.2)
where ak and ah are capital shares for physical and human capital, respectively, and ak + ah < 1. Equation (3.2) is then rearranged to get
ctk
ah
y =A
,
(3.3)
where y = Y/L. With measures o f K/Y and H/Y, these studies ask, To what extent do cross-country variations in these capital intensities account for the large variation in y?6 There is substantial disagreement on the answer to this question. For example, Mankiw et al. (1992) argue that differences in K/Y and H/Y can account for a large fraction o f the disparity in y whereas Klenow and Rodriguez-Clare (1997b) and Hall and Jones (1998) argue that it accounts for much less. In this section, we ask the following question: To what extent can differences in capital intensities account for the income disparity between the richest and poorest countries? To be precise, we calculate the ratio
ak Yrich
Nr
Ypoor
1 (Ki)~ N,~ E i 6 poor ~
~k
ah ~,,
,
(3.4)
(Hi)l~ah
where the "rich" are the Nr most productive countries and the "poor" are the Np least productive countries. Note that Equation (3.4) assumes no differences in technology A
6 A notable exception is Krueger (1968), who does not have measures of physical capital. She estimates income levels that could be attained in 28 countries if each country had the same physical capital per worker and natural resources as the USA, but each country had its own human resources. Krueger finds that there would still be large per capita GDP differences between the USA and many of these countries even if they had the physical capital and natural resources of the USA. Using logged differences in incomes, her findings imply that the fraction of the income disparity explained by differences in human capital is in the range of 20 to 40 percent.
E.R. McGrattan and JA. Schmitz, Jr.
680
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F i g 5 Physical capital-output ratio versus income, 1985
across countries. Thus, if we use observations of K/Y and H / Y in Equation (3.4), the ratio is a prediction of the disparity in income levels due only to variations in capital intensities. In our calculations, we use the measures of capital intensities in Mankiw et al. (1992), Klenow and Rodriguez-Clare (1997b), and Hall and Jones (1998). The measure of K/Y is very similar across the studies of Mankiw et al. (1992), Klenow and Rodriguez-Clare (1997b), and Hall and Jones (1998). Therefore, we use the same K/Y for all of the calculations that we do. We construct estimates of the capital stock for each country using the perpetual inventory method. With data on investment, an initial capital stock, and a depreciation rate, we construct a sequence of capital stocks using the following law of motion for/(~ 7: Kt+~ = (1 - 6)Kt + Xkt,
(3.5)
where 6 is the rate of depreciation. We choose a depreciation rate of 6 percent. For the initial capital stock, we assume that the capital-output ratio in 1960 is equal to the capital-output ratio in 1985 in order to get our estimate 8. In Figure 5, we plot the physical capital-output ratio, K/Y, for 1985 versus the relative GDP per worker in 1985 for all countries that have complete data on GDP 7 We use I x RGDPCH x POP from the Penn World Table of Summers and He ston (199 l) for investment. 8 This way of estimating the final capital-output ratio leads to a good approximation if the economy is roughly on a balanced growth path. As a check, we tried other initial capital stocks and found that the final capital-output ratio was not sensitive to our choices.
Ch. 10:
Explaining Cross-Country Income Differences
681
per worker and investment [variables R G D P W and I in Summers and Heston (1991)] over the sample period 1960-1985. There are 125 countries in the sample. The figure shows that capital-output ratios for the most productive countries are on the order o f 3, whereas capital-output ratios for the least productive countries are around 1 or below. The correlation between the capital-output ratio and the logarithm o f relative GDP per worker is 0.67. We next consider measures of H/Y which vary a lot across the three studies. We start with the measure used by Mankiw et al. (1992). Motivated by the work o f Solow (1956), Mankiw et al. (1992) assume that g
Y
sh
(3.6)
n+g+6'
where sh is the fraction o f income invested in human capital, g is the growth rate o f world-wide technology, n is the growth rate o f the country's labor force, and c5 is the rate at which both physical and human capital depreciate. The expression in Equation (3.6) is a steady-state condition o f Solow's (1956) model augmented to include human capital as well as physical capital. Mankiw et al. (1992) use the following measure for sh:
sh = secondary school enrollment rate x
1 5 - 19 population] ~ - - - ~ population] '
(3.7)
which approximates the percentage o f the working-age population that is in secondary school. To construct this measure, we use Equation (3.7) with secondary school enrollment rates from Barro and Lee (1993) [variables Sxx, xx=60, 6 5 , . . . , 85] and population data from the United Nations (1994). We construct sh for each o f the six years (1960, 1965 . . . . , 1985) in which data are available and take an average 9. This investment rate is divided by n + g + 6 with g = 0.02 and 6 = 0.03 as in Mankiw et al. (1992) and n given by the growth rate o f the country's labor force constructed from the Summers and Heston data set 10. In Figure 6, we plot average secondary school enrollment rates [the average o f variables Sxx, xx=60, 65 . . . . . 85 from Barro and Lee (1993)] versus the relative GDP per worker in 1985. Figure 6 has two noteworthy features. First, there is a very strong correlation between the secondary enrollment rate and the logarithm o f output per worker across countries. The correlation is 0.83. Second, there is a large range in secondary enrollment rates. There are many countries with secondary enrollment rates under 10 percent, and as many with rates over 60 percent. Weighting the enrollment
9 Data are unavailable in all years for Namibia, Reunion, Seychelles, Puerto Rico, Czechoslovakia, Romania, and the USSR. 10 Mankiw et al. (1992) use working-age population, while we construct growth rates of the labor force using Summers and Heston's (1991) RGDPCHxPOP/RGDPW. The results are quantitatively similar.
682
E.R. McGrattan and J.A. Schmitz, Jr.
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Fig. 6. Secondary enrollment versus income, 196~1985. rates by the population [as in Equation (3.7)] and deflating them by n + g + 6 [as in Equation (3.6)] does little to change the pattern displayed in Figure 6. Hence, there are large differences between the human capital-output ratios for the most productive and least productive countries, with the correlation between H / Y and the logarithm o f G D P per worker equal to 0.79. With this measure o f H/Y for Mankiw et al. (1992), the K/Y series described above, and values for the capital shares ak and ah, we can calculate the ratio o f incomes in Equation (3.4). In this calculation, N~ and Np in Equation (3.4) are the richest 5 percent o f countries and the poorest 5 percent o f countries, respectively. In Table 1, we report our results. In the first row o f the table, we assume that ak = 0.31 and ah = 0.28 as estimated by Mankiw et al. (1992). In this case, the predicted income disparity assuming only differences in capital stocks - between the richest and poorest countries in 1985 is 12.8. The actual ratio is 31.4. The numbers in the last column o f Table 1 are the ratios o f predicted to actual income disparity - both in logarithms. This is a measure o f the gap in productivities attributable to variation in human and physical capital. For M a n k i w et al.'s (1992) human capital measure and parameter values, we find that 74 percent [that is, log(12.8)/log(31.4)] o f the gap in actual incomes can be explained by differences in capital intensities 11.
H Mankiw et al. (1992) run a regression of the logarithm of output per worker on their measures of K/Y and H/Y. They find an R2 statistic of 0.78 and parameter estimates &k = 0.31 and &h = 0.28. They
Ch. 10: Explaining Cross-Country Income Differences
683
Table 1 Income disparity due to different factor intensities, 1985 Human capital measure based on:
Mankiw et al. (1992) Variation on Mankiw et al.
Physical capital Human capital share share
Predicted income disparity a
Percentage difference explained b
0.31
0.28
12.8
74
1/3
1/3
33.7
102
0.31
0.28
5.4
49
1/3
1/3
9.9
66
1/3
0.43
30.1
99
Klenow and Rodriguez-Clare (1997b) c
0.30
NA
3.4
36
Hall and Jones (1998) c
1/3
NA
4.0
40
a Income disparity is defined to be the ratio of the average income of the richest 5 percent of countries to the average income of the poorest 5 percent (where income is output per worker). b The percentage difference explained is defined to be the logarithm of the predicted income disparity divided by log(31.4), which is the logarithm of the actual income disparity. c NA means not applicable. No value of a h is reported because the production function used in this study can be written as Y = Kak(ALg(s)) i c,k, where the function g does not depend on either capital share. Thus, the income disparity does not depend on %.
We also find that the correlation b e t w e e n the predicted and actual logarithms o f G D P per worker is 0.84. We should note, however, that the results are v e r y sensitive to the choice o f capital shares. For example, suppose that we use slightly h i g h e r values for capital shares; 1 The results o f this case are r e p o r t e d in the second row o f Table 1. say, al~ = ah = 5" In this case, the prediction for the ratio o f productivities o f the top 5 percent to the b o t t o m 5 percent is 33.7 - m o r e than t w i c e what it was in the case with ak = 0.31 and ah = 0.28, and a l m o s t exactly in line with the data. K l e n o w and R o d r i g u e z - C l a r e (1997b) argue that M a n k i w et al.'s (1992) m e a s u r e o f h u m a n capital overstates the true variation in educational attainment across the world b e c a u s e it e x c l u d e s p r i m a r y school enrollment, w h i c h varies m u c h less than does secondary. Figure 7 plots the p r i m a r y e n r o l l m e n t rates [the average o f variables Pxx, xx=60, 65 . . . . . 85 f r o m Barro and L e e (1993)] versus G D P per worker. Again, there is a strong positive correlation. But note that there is m u c h less variation in p r i m a r y e n r o l l m e n t rates than in s e c o n d a r y e n r o l l m e n t rates, w h i c h are displayed in Figure 6. O n l y ten countries have a rate b e l o w 0.40. Suppose that we redo our calculation o f the ratio Yrich/Ypoor u s i n g a measure o f sh in E q u a t i o n (3.6) that includes primary, secondary, and p o s t - s e c o n d a r y e n r o l l m e n t rates.
view the high R2 statistic and the reasonable estimate for physical capital's share as strong evidence that variation in factor inputs can account for most of the variation in output per worker.
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In particular, suppose that we use the fraction of 5- through 64-year-olds who are enrolled in school averaged over the period 1960-1985; this is a weighted average of the three enrollment rates in Barro and Lee (1993). In Table 1, in the row marked "Variation on Mankiw et al.," we report the predicted disparity, which is only 5.4. This ratio implies that roughly half (49 percent) of the observed disparity in incomes is explained by differences in capital--output ratios. Thus, adding primary and tertiary enrollment rates to the measure of sh significantly reduces the contribution of human capital to income differences. (Compare the first and third rows of Table 1.) Although the predicted disparity is smaller, we still find a strong positive correlation between yi and (Ki/Yi) ak/(1 ak-ah)(Hi/Yi) ak/(1-ak-ah). The correlation in this case is 0.79. As before, the magnitude of this disparity depends critically on the capital shares. Making a slight change from ak = 0.31 and ah = 0.28 to ak = 51 and ah = ½ leads to an increase in the percentage explained from 49 percent to 66 percent. I f we choose ak = ½ and ah = 0.43, then almost all of the income disparity can be explained by differences in capital stocks across countries. As Mankiw (1997) notes, we have little information about the true factor shares - especially for human capital. Klenow and Rodriguez-Clare (1997b) and Hall and Jones (1998) argue that a more standard way of measuring human capital is to use estimates of the return to schooling from wage regressions of log wages on years of schooling and experience. [See Mincer
Ch. 10: Explaining Cross-Country Income Differences
685
(1974).] For example, Klenow and Rodriguez-Clare (1997b) report estimates o f the human capital-output ratio constructed as follows: 1-c¢ k
~- -
e ~'Is
o)i e ~'2expi+y3exp~
AL y'
(3.8)
where s is the average years of schooling in the total population over age 25 taken from Barro and Lee (1993) [variable HUMAN85], expi is a measure of experience for a worker in age group i and is equal to ( a g e / - s - 6), and mi is the fraction o f the population in the ith age group. The age groups are {25-29, 30-34 . . . . , 6 0 - 6 4 } and age/ E { 2 7 , 3 2 , . . . , 62}. The coefficients on schooling and experience in Equation (3.8) are given by Yl = 0.095, Y2 = 0.0495, and Y3 = -0.0007, which are averaged estimates from regressions of log wages on schooling and experience. The measure o f the human capital-output ratio used by Hall and Jones (1998) does not depend on experience and is given by
H _ (eO(,~) ~ y
AL__
(3.9)
y'
where s is the average years of schooling in the total population over age 25 taken from Barro and Lee (1993) [variable HUMAN85] and q~(.) is a continuous, piecewise linear function constructed to match rates o f return on education reported in Psacharopoulos (1994) 12. For schooling years between 0 and 4, the return to schooling q~/(s) is assumed to be 13.4 percent, which is an average for sub-Saharan Africa. For schooling years between 4 and 8, the return to schooling is assumed to be 10.1 percent, which is the world average. With 8 or more years, the return is assumed to be 6.8 percent, which is the average for the OECD countries. It turns out that the measures o f H / Y constructed by Klenow and Rodrlguez-Clare (1997b) and Hall and Jones (1998) are very similar. I f we set Y2 and Y3 equal to 0 in Equation (3.8) and ignore experience, then we get roughly the same capital intensities as those constructed by Klenow and Rodrlguez-Clare (1997b). Similarly, if we set qi(s) = 0.095s in Equation (3.9) and assume the same rate o f return on education across countries, then we get roughly the same capital intensities as those constructed by Hall and Jones. As a result, the residuals, A, constructed in these two studies are very similar. The correlation between the two residual series is 0.88 if we use the countries that appear in both data sets. We now see how the measures o f human capital defined by Klenow and RodrlguezClare (1997b) and Hall and Jones (1998) affect the ratio Y~ch/Ypoor in Equation (3.4).
cxk
12 Substituting H/Y in Equation (3.9) into Equation (3.3) and simplifying gives y = A ( K / Y ) ~ e¢(~'), which is the form of the GDP per worker used in Hall and Jones (1998). We have written their implied H/Y so as to compare it to that of Mankiw et al. (1992).
E.R. McGrattan and JA. Schmitz, Jr.
686 1.4 • o --
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Fig. 8. Human capital-output ratio versus income, 1985. For our calculations, we assume, as they do, that ak is the same across all countries. In Table 1, we report our predictions o f income disparity and the gap in productivities attributable to differences in capital intensities. For Klenow and Rodriguez-Clare (1997b), we find that only 36 percent o f the gap in productivities is attributable to d i f f e r e n c e s in capital stocks. For Hall and Jones, 40 percent o f the difference in productivities is explained by differences in capital stocks 13. To see why the results reported in Table 1 are so different across studies, consider the data in Figure 8. We plot the human capital-output ratios o f Mankiw et al. (1992) and Klenow and Rodriguez-Clare (1997b). Due to data availability, only 117 o f the original 125 countries in our sample are included. Both measures o f human capital to output are plotted against the relative GDP per worker in 1985. For both series, we fit an exponential curve. As is clear from the figure, there is much larger variation in the human capital-output ratio o f Mankiw et al. (1992) than in that o f Klenow and Rodriguez-Clare (1997b). For Klenow and RodHguez-Clare (1997b), the correlation between H / Y and GDP per worker is close to zero. In fact, i f we had used ak = ½ and ah = 1 when constructing H/Y for Klenow and Rodrlguez-Clare (1997b), we would
13 Klenow and Rod6guez-Clare (1997b) and Hall and Jones (1998) find that the average contribution of A to differences in y is about ½ (that is, the mean of the Ai's relative to A for the USA is approximately ½).
Ch. 10: ExplainingCross-Country Income Differences
687
have found a negative correlation between the human capital-output ratio and GDP per worker.
3.2. Growth accounting The objective in growth accounting is to estimate the contributions of technological progress and factor accumulation to differences in growth rates of output. As we saw in Figure 4, the growth rates of the fastest growing countries were on the order of 8 or 9 percent in the post-World War II period. These growth rates far exceed those of the fastest growing countries a century before. During three of the four decades between 1950 and 1990, East Asian countries led the pack. During the 1950s and 1960s, Japan had the highest growth rate. During the 1980s, South Korea had the highest growth rate. Among countries growing at 6 percent or better over the 19501990 period are three other East Asian countries, namely, Hong Kong, Singapore, and Taiwan. A question which has interested many people, then, is, Is factor accumulation or TFP growth responsible for growth rates of 8 or 9 percent? The studies of Young (1995) and Hsieh (1997) focus on the newly industrialized countries in East Asia, namely, Hong Kong, Singapore, South Korea, and Taiwan. Young (1995) finds that the extraordinary growth performances of the Asian countries are due in large part to factor accumulation. The output growth rates over the period 1966-1990 for Hong Kong, Singapore, South Korea, and Taiwan are 7.3, 8.7, 10.3, and 9.4, respectively 14. The estimates of average TFP growth over the same period for Hong Kong, Singapore, South Korea, and Taiwan are 2.3, 0.2, 1.7, and 2.6, respectively. Hsieh (1997) estimates TFP growth for the East Asian countries using both the primal approach as in Young (1995) and the dual approach. The primal approach is to use the growth rates of quantities of capital and labor to back out measures of TFP growth whereas the dual approach is to use the growth rates of prices of these factors. Hsieh finds TFP growth rates for Hong Kong, Singapore, South Korea, and Taiwan of 2.7, 2.6, 1.5, and 3.7, respectively IS. As these estimates suggest, there is much more agreement in these two studies than in the levels accounting studies reviewed above. Both agree that factor accumulation was much more important than TFP growth. There is some disagreement on the estimate for Singapore. Young (1995) finds that factor accumulation, especially of capital, is the whole story behind Singapore's high growth rates, whereas Hsieh (1997) finds that a significant fraction is due to TFP growth. Hsieh argues that while capital increased significantly, the real return to capital did not fall. The higher is the growth rate in the real return of capital, the higher would be Hsieh's estimate of TFP growth.
14 The data on output for South Korea and Taiwan do not include agriculture, and the period for the Hong Kong data is 1966-1991. 15 The period for Singapore used in Hsieh (1997) is 1971-1990. Using the primal approach yields an estimate of -0.7 for this shorter sample.
688
E.R. McGrattan and J.A. Schmitz, Jr.
While growth rates of TFP on the order of 2 percent are high, they are not extraordinarily high. In the USA, for example, Christensen et al. (1980) find growth rates of TFP of 1.4 percent over the period 1947-1973 when growth rates in output were around 4 percent. The growth rates in output for the East Asian countries over the period 1966-1990 were significantly higher since the growth in capital and labor was extraordinarily high. In the remainder of the chapter, we turn to the literatures which directly estimate the impact of policy on income and growth.
4. Growth regressions In this section, we review a literature - the cross-sectional growth literature - that quantifies the effects of observed policies on country growth rates. We begin with a brief overview of the literature. We discuss the motivation for the studies in this literature and the typical growth regression that is run. We then describe the results of Barro and Lee (1994) and their estimates of the effects of policies on growth. Finally, we discuss some critiques of the methodology used in the literature. As we noted in Section 2, average growth rates vary widely across countries and are uncorrelated with initial income levels. (See Figure 2.) The fact that income levels and subsequent growth rates are uncorrelated was at one time thought to be a puzzle for standard growth theory which predicted that poor countries should grow faster than rich countries in per capita terms. Such a prediction would imply a negative correlation between income levels and subsequent growth rates. This result depends, of course, on countries having the same steady-state income levels. If countries do not converge to the same steady-state income levels, the pattern predicted by theory is potentially consistent with Figure 216. Analyses in the growth regression literature attempt to uncover the relationship between initial incomes and subsequent growth rates, holding constant variables that determine countries' long-run steady-state income levels. The typical exercise involves regressing the growth rate of per capita GDP on the initial level of GDP, initial factor endowments such as educational levels, and control variables which are assumed to be determinants of the steady-state level of per capita output. Without the initial factor endowments and control variables, the coefficient on initial GDP is positive (as suggested by Figure 2). With these variables included, the coefficient on initial GDP is negative. The set of control variables typically includes the ratio of investment to GDP, measures of market distortions such as the ratio of government consumption to GDP and the black market premium on foreign exchange, measures of political instability, and measures of financial development. Again, the purpose of these variables is to sort countries into more homogeneous groups, that is, groups that have similar steady
16 In Section 5, we provide a different explanation for this fact.
Ch. 10: Explaining Cross-Country Income Differences
689
0.35 p = -0.62 i
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0.3
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-
-
0.25
0 © -=~ 0.2 •
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0~
"~ 0.15 0.1-
0
t O,
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where Tt is the value of these transfers at date t. In this case, the profit-maximization problem solved by the investment-goods firm is given by max P X rKx-wLx, subject t o X =,~ gYr I-r (5.7) x~,Lx l + Z x and the problem of the consumption-goods firm is the same as before. The specification in Equation (5.7) implies that bribes are proportional to the scale of the investment. We now derive an explicit formula for differences in income due to differences in investment distortions that, like bribes, are simply transfers from one agent to another. For now we assume that a = y. In this case, the relative price of investment to consumptionp is proportional to the distortion 1 + rx in equilibrium. For the model we assume that the world prices of consumption and investment goods are one (so that the world price of investment equals the price of a country with a distortion of zero). If we assume that all investment is measured in national income accounts, then GDP per worker in the model is given by C/L + X/L. Assuming that the only difference across countries is the level of rx that they face, we find that the ratio of productivities of countries i and j is given by yi _ ( l + rxi'] ~'
y1
\
j
(5.8)
in the steady state. If we assume, as Chari et al. (1997) do, that half of the capital stock is organizational capital and is therefore not measured in national income accounts, then GDP per worker in the model is given by C/L + ½X/L. [See Prescott and Visscher (1980) for a discussion of the concept of organization capital.] In this case, the ratio of productivities of countries i a n d j is Yi
Yj
_
a(1
+ Txi) aal -
b(1
+
Txi) al-1
a(1 + rxj)~-~r - b(1 + rxj)~l~ '
where a and b are positive constants that depend on/3, 6, and a (and growth rates of population and world-wide technology which we have abstracted from here). For the
Ch. 10: Explaining Cross-Country Income Differences
701
parameters used in Chaff et al. (1997), a is about 5 times larger than b. Therefore the ratio of measured incomes is approximately equal to the expression in Equation (5.8). Consider again the data in Figure 14. Is the range in relative prices large enough to account for the 30-fold difference in relative incomes? It is, if one views K as a broad measure of capital that includes not only physical capital but also stocks of human capital and organizational capital. For example, if we assume a capital share on the order of 2, then differences in relative prices (and hence, differences in the ratio (1 + Txi)/(1 + rxj)) on the order of 5 or 6 imply a factor of 30 difference in incomes since we square relative prices. In Figure 14, we see that four of the poor countries have relative prices exceeding 4. If we compare these countries with the richest countries who have relative prices that fall below 1, we can get relative productivities on the order of 30. There is a potential bias in the measure of distortions that we plot in Figure 14. If consumption goods are largely nontraded tabor-intensive services, we would expect that they are systematically cheaper in capital-poor countries. In this case, the relative price overstates the real distortion. To demonstrate this, we can use the steady-state conditions of the model to derive an expression for the relative price of investment to consumption in terms of the distortions T~. The expression is given by i-a
p = B ( 1 + rx)v~, where B depends on parameters assumed to be the same across countries. Above we assumed a = 7 and therefore had p = B(1 + Tx). However, if production of investment goods is more capital intensive than production of consumption goods (~ > a), then the ratio of prices of two countries is larger than the ratio of their true distortions; that is, Pi/Pj > (1 + "gxi)/(1 + T~j) where rxi > rxj. Chari et al. (1997) find for Mexico and the USA that, if anything, the relative prices understate the true distortion since estimates of capital shares imply a > 7 for both countries. The estimates derived in this section illustrate that the effects of certain policies distorting investment are potentially large. Inefficient government production can explain 22 to 33 percent of the productivity gap between countries like Egypt and the USA. For more comprehensive measures of distortions like the relative price of investment to consumption, the implied differences in incomes across countries are large if we assume that the distortions affect not only physical capital but also human and organizational capital. However, the estimates of the impact of policy on income found above are sensitive to choices of the capital share and to magnitudes of measured versus unmeasured capital. For example, if we assume a capital share of ½, then differences in relative prices on the order of 5 imply differences in incomes on the order of the square root of 5, which is significantly smaller than differences on the order of the square of 5. Before closing this section, we have three general comments concerning this literature. First, a number of quantitative studies have extended the basic neoclassical model explored above. One aim of these studies is to ask whether observed policy
E.R. McGrattan and J.A. Schmitz, Jr.
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differences have a larger impact on measured incomes in the extended models as compared to the standard model. Jovanovic and Rob (1998) extend the basic model to include vintage capital. The extended vintage capital model yields predictions for income disparity that are similar to those of the standard model. Parente et al. (1997) introduce home production into the standard model. Policies that influence capital accumulation now also have an impact on the mix o f market and non-market activity. Their model can imply (for a given difference in policies) a much larger difference in income disparity across countries than does the standard model. Second, the analysis above illustrates that theories o f the kind described here cannot rely on variations in TFP (that is, variations in A~.) alone to explain income differences. For example, the following is true in the steady state with a = 7: a
Y
l+Txx
1
1 + 6,
(5.9)
where Y = AcKaL l-a. This condition shows why variation across countries in the residual Ac is not enough. There are large differences in K/Y across countries. With rx constant, this model predicts that K/Y is constant. Thus, we need variation in some intertemporal distortion (for example, ~'x) in order to generate differences in the capital-output ratio 20. Third, to simplify matters, we assumed no cross-country variation in TFP in deriving our predictions o f income differences. However, when there is unmeasured capital, it is hard to distinguish between an economy with a small capital share and variations in both Ac and z'~ (where Ac and r~ are correlated) and an economy with a larger capital share and only variations in T~. For example, if Klenow and Rodriguez-Clare (1997b) or Hall and Jones (1998) were to construct measures o f TFP simulated from a stochastic version of the model above with half o f the capital stock unmeasured, they would conclude that TFP accounts for much o f the variation in output per worker even if it accounted for none o f the variation in output per worker. Thus, one must be cautious when interpreting the results o f Klenow and Rodriguez-Clare (1997b) and Hall and Jones (1998). And, as Prescott (1998) points out, a theory o f TFP differences is still needed. We turn next to the trade literature and again derive formulas relating policies to differences in income levels. 5.1.2. Policies affecting trade The earliest research using models to measure the impact of policies on country income and welfare focused on trade policies 21. The trade literature is large. In this section, we 2o Even if we do not abstract from growth, the theory with only variations in A~. will do poorly. The capital-output ratio is highly correlated with income, but growth rates of the poor and rich are not very different. 21 In fact, Johnson (1960) discusses the work of Barone (1913) who attempts to measure the impact of tariffs on income and welfare.
Ch. 10: Explaining Cross-Country Income Differences
703
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Fig. 15. Tariff rates versus income, 1980. provide a broad historical outline o f this literature. We first discuss several measures o f trade restrictions and their relationship to country productivity. We then discuss work relating trade restrictions to differences in income levels. Figure 15 presents measures of tariff rates on capital goods and intermediate goods constructed by Lee (1993) versus the relative GDP per worker for 91 countries in 1980. When plotting the data, we dropped the observation for India, where income in 1980 was approximately ½ o f the world average and the tariffrate was 132 percent. This point was dropped so we could more easily view the other data points. Not surprisingly, there is a negative relationship between tariff rates and incomes. The correlation between tariff rates and incomes is -0.38. As Easterly and Rebelo (1993) point out, taxes on international trade fall as a share o f government revenue as income rises, while the share o f income taxes rises. For many o f the low- and middle-income countries, the tariffrates are in the range o f 25 to 50 percent. But rates among the rich are, in general, quite low. In Figure 16, we present additional evidence on trade restrictions. We plot Sachs and Warner's (1995) measure o f a country's "openness" for the period 1950-1994 against relative GDP per worker in 1985. A country is open if (i) nontariff barriers cover less than 40 percent o f its trade; (ii) its average tariff rates are less than 40 percent; (iii) any black market premium in it was less than 20 percent during the 1970s and 1980s; (iv) the country is not socialist under Kornai's (1992) classification; and (v) the country's government does not monopolize major exports. Sachs and Warner construct
704
E.R. McGrattan and J.A. Schmitz, Jr. 1
•
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Fig. 16. Fraction of years open versus income, 1950 1994. an index that measures the fraction o f years in the period that a country has been "open." As we see from Figure 16, the correlation between the Sachs and Warner index and GDP per worker is strongly positive; economies with policies that promote trade are those with high productivities. In an early paper, Johnson (1960) reviews and extends prior studies that measure the cost o f protection. His measure of the cost o f protection is defined to be "the goods that could be extracted from the economy in the free-trade situation without making the country worse off than it was under protection - some variation o f the Hicksian compensating variation" (p. 329). In the two-good version o f his general equilibrium model, the cost o f protection in percentage o f national income is Cost = ~
t/V,
where r is the tariff on imports, t/is the compensated elasticity o f demand for imports, and V is the ratio of imports at market prices to domestic expenditure. Johnson argues that the cost is small given it is an elasticity multiplied by three fractions, each o f which is small. The example he gives is a tariff o f 331 percent and an import share of 25 percent. To obtain a cost o f 4 percent o f national income, the compensated demand elasticity has to be slightly above 5 - a value he dismisses as implausibly high. When Johnson extends the analysis to many goods, he cannot conclude as easily that the cost of protection is small. However, when he analyzes data from two studies
Ch. 10: ExplainingCross-Country Income Differences
705
on Australia's and Canada's commercial policies, he concludes that the cost is small in both countries. During the 1960s, a number of studies continued the work reviewed by Johnson. A good reference is Balassa (1971). The findings of Balassa (1971) are similar to those of Johnson (1960). The cost of protection is on the order of a few percent of GDP, with the highest cost being 9.5 percent of income in Brazil. A further development in the quantitative study of tariffs was the computational general equilibrium (CGE) literature. There are a number of good surveys of this literature, such as Shoven and Whalley (1984). Some notable contributions to this literature are Cox and Harris (1985), Whalley (1985), and the papers in Srinivasan and Whalley (1986). Most of the CGE literature found - as did the earlier literature that reductions in observed tariffs would lead to small increases in welfare and income, typically on the order of 1 percent of GDR Since the mid-1980s, there have been attempts to extend the models in this literature under the presumption that larger gains in income follow tariff reductions. One avenue has been to develop dynamic models in which the capital stock adjusts to the reductions in tariffs. The CGE literature typically studied static models, so, for example, the models did not consider the response of capital stocks to changes in tariffs. A recent paper that looks at such responses of capital stocks is Crucini and Kahn (1996). This study examines the increase in tariffs that followed passage of the Smoot-Hartley tariff during the Great Depression. They find that if "tariffs had remained permanently at levels prevailing in the early 1930s [due to the Smoot-Hartley tariff], steady-state output [in the USA] would have declined by as much as 5 percent" as a result of the higher tariffs (p. 428). At least for this episode then, considering changes in the capital stock does not significantly change the conclusion that the effects of tariffs on income are small. Another recent paper is that by Stokey (1996) who examines dynamic gains from trade as capital stocks adjust. Stokey finds larger gains from capital adjustment than do Crucini and Kahn. Another avenue that has been pursued is to allow for changes in the set of goods available in the economy as tariffs change. One example is Romer (1994) who argues that tariffs may have a large impact on productivity. He constructs an example of a small open economy which imports specialized capital inputs to use in a love-forvariety production function. Foreign entrepreneurs that sell the capital inputs face fixed costs of exporting to the small open economy. In the model, increases in tariffs result in a narrowing of goods imported and a fall in productivity. Romer's back-of-theenvelope calculations show that the effects on productivity may be large. Here, we review his calculations and discuss Klenow and Rodriguez-Clare's (1997c) study of this mechanism for Costa Rica. Romer (1994) considers a small open economy that produces a single good according to the production function y = L 1 a fo N x~ di,
E.R. McGrattan and J.A. Schmitz, Jr.
706
where L is the labor input and xi is the input of the ith specialized capital good, i c [0, N]. The capital goods are imported from abroad. The number of types of goods imported N is not a priori fixed; in equilibrium, it will depend on the tariff rate. Each specialized capital good is supplied by a foreign monopolist. The foreign monopolist faces a constant marginal cost of producing each unit equal to c and a fixed cost to export equal to co(i) = l~i, where /2 is a positive constant. The small open economy charges a tariff of T percent on all purchases of the specialized capital goods. Let the timing of events be as follows. The small open economy announces a tariff r. Given this v, foreign entrepreneurs decide whether or not to export to the country. Because of the symmetry of the capital goods in final production, all foreign entrepreneurs that export face the same demand curve and earn the same revenue. Profits differ, of course, since fixed costs differ. Marginal entrepreneurs are those whose profit just covers their fixed cost. The product of the marginal entrepreneur is N. The problem facing the foreign entrepreneur i if he enters is max (1 - T ) p ( x i ) x i - cxi, xi
where the inverse demand function p ( x i ) = o~(L/xi) a-1 is derived from the marginal productivity condition for capital. It is easy to show that the profit-maximizing price is a simple markup over marginal cost and that the profit-maximizing quantity is
which depends on the level of the tariff r. Since the tariffis the same for all producers, we have dropped the index i on x. Setting gross profit equal to fixed costs, we can solve for the marginal product N as a function o f x ( r ) ; that is, N(T)
-
(1
-
a)Cx(r)"
With these expressions for x(T) and N(T), we can write GDP in equilibrium as y = L I - a N ( T ) [ x ( T ) ] a.
What is the impact of tariffs on GDP? One way to measure the impact is to compare the GDP of a country with no tariffs to one with tariff rate r; that is, y(v = o) y(r - - > O)
_ N(0) Cx(0))a =(1_ r)g+l. ~
t x(T))
(5.10)
This expression assumes that the labor input is the same in the two countries. Before making some back-of-the-envelope calculations with this ratio, let us present another formula. Romer (1994) argues that the effects of tariffs on GDP can be
Ch. 10: Explaining Cross-Country Income Differences
707
large and, in particular, much larger than traditional analyses have suggested. In the traditional calculations, the implicit assumption is that the set of products does not change with tariffs. In the context of the above model, the traditional analysis assumes a different timing of events. The timing in the traditional analysis assumes that entrepreneurs decide to export or not, assuming there is a zero tariff. After this decision is made, the small open economy posts an unanticipated tariff of r. Because the fixed costs are sunk, entrepreneurs continue to export (as long as net profits are positive). What is the impact of tariffs in this case? In this case, the relevant ratio is y(T=0) y(v > 0)
N
= =N(0)
(x(0)~a =(1-v)~. ~ X('g)/]
(5.11)
Note that the key difference between Equations (5.10) and (5.11) is that N(0) replaces N(r). In essence, the key difference between these formulas is the exponent on (1 - r). In the latter case, where the number of imports varies with the tariff rate, the exponent is larger. 1 Suppose that T is To do some rough calculations, Romer (1994) assumes that a = g. 25 percent. Using the formula in Equation (5.10), we find that GDP is 2.4 times higher without tariffs than with tariffs. Using the formula in Equation (5.11), we find that GDP is only 1.3 times higher without tariffs than with tariffs. Thus, we significantly underestimate the effect on GDP if we do not allow the number of goods to vary with the tariff rate. Furthermore, the result is nonlinear. I f we assume that r is 50 percent, then the first formula yields a ratio of 8, while the second yields a ratio of 2. Two conclusions can be drawn from these simple calculations. First, the effects of tariffs on productivity may be much larger when we consider that the set of products changes with tariffs. Second, the effects of tariffs on GDP are potentially large. The rough calculations that we did above use rates in the range observed for the low- and middle-income countries. (See Figure 15.) Romer's (1994) estimates led Klenow and Rodriguez-Clare (1997c) to consider the effects of tariffs in Costa Rica. Klenow and Rodrlguez-Clare (1997c) find that considering changes in the set of goods imported can significantly change the traditional cost-of-tariff calculation. For example, they find that their cost-of-tariff calculation leads to a loss from trade protection that is up to 4 times greater than the traditional calculation. In the particular case they study, the Costa Rican tariff reform in the late 1980s, the traditional calculation leads to rather small gains from tariff reduction. Hence, Klenow and Rodriguez-Clare's (1997c) estimates of the gain are also rather small - just a few percent of GDE Still, it may be that in other countries or time periods, their formula may imply gains from tariff reductions that are a large fraction of GDE 5.1.3. Other policies
There are many other studies that have examined the quantitative impact of particular policies on income. In labor economics, there are studies of the effects of labor
708
E.R. McGrattan and JA. Schmitz, Jr.
market restrictions such as impediments to hiring and firing workers on productivity and income. In industrial organization, there are studies assessing the quantitative effects of policies toward monopoly. In public finance, there are studies concerned with the quantitative effects of tax policies on income. In this section, we discuss some examples. In many countries (developed and less developed), there are legal restrictions on the actions of employers. These laws range from requiring paying termination costs when firing employees to prohibiting firms from closing plants. Putting such legal restrictions on the actions of employers obviously influences their decision to hire employees. The laws, then, have implications for the equilibrium level of employment. A number of studies have tried to quantify the effects of such laws on aggregate employment and income. For example, Hopenhayn and Rogerson (1993) study the costs of imposing firing costs on firms. They construct a general equilibrium and use it to study the consequences of a law that imposes a tax equal to one year's wages if a firm fires an employee. They find that such a policy reduces employment by about 2.5 percent and reduces average productivity by 2 percent 22. An old issue is the relationship between monopoly and economic progress. In much of the R&D literature discussed later, there is an emphasis on the idea, attributed to Schumpeter, that entrepreneurs need to capture rents in order to innovate and introduce new products. Hence, this idea suggests that monopoly leads to economic progress. There is, of course, some truth to this idea. But for developing countries in which the issue is primarily one of technology adoption and not creation, the idea may be of little quantitative importance. Developing countries need to worry less about the incentives to invent new products than do developed countries. Hence, if monopolies have costs as well, monopolies may be more costly in developing countries. But the cost of monopoly is low in most models. The cost of monopoly is usually due to a restriction on output. The costs of such output restrictions are usually estimated to be a small share of GDR Bigger costs would emerge if monopoly were tied to restrictions on technology adoption. Parente and Prescott (1997) present a new model that argues that monopoly does restrict technology adoption. They study the consequences of giving a group the right to use a particular technology. If the group is given such a right, then it may try to block the adoption of new technologies that would reduce the gain from the monopoly right. Moreover, the group may use existing technologies inefficiently. There is also a branch of the CGE literature that studies public finance issues. Among the policies that have been quantitatively explored in this literature are the abolition of government taxes, indexation of tax systems to inflation, and replacement of income taxes with consumption taxes. A good survey of some of this literature is contained in Shoven and Whalley (1984).
22 Other work in this area includes Bertola (1994) and Loayza (1996) who study the effects of certain labor market restrictions on growth.
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709
5.2. Effects of policy on growth Up to now, we have focused on disparity in the levels o f income across countries. However, much o f the recent literature has focused instead on income growth. O f particular interest is the significant increase in the standard o f living of the richest countries over the past 200 years and the recent growth miracles in East Asia. (See Figures 1 and 4.) An objective in this literature - typically referred to as the endogenous growth literature - is to develop models in which growth rates are endogenously determined. One of the main questions o f this literature has been, What are the determinants o f the long-run growth rate? To illustrate the kinds o f quantitative predictions that have been found, we analyze two prototype endogenous growth models. The first is a two-sector model with growth driven by factor accumulation. The second model assumes that growth is driven by research and development. For both models, we derive steady-state growth rates and show how they depend on economic policies. Under certain assumptions, measured differences in policies imply significant differences in growth rates.
5.2.1. Policies in a two-sector AK model In this section, we analyze the balanced growth predictions o f a prototype two-sector endogenous growth model 23. There are three main differences between this model and the exogenous growth model discussed in Section 5.1.1. First, here we assume that there are constant returns to scale in accumulable factors. Second, we introduce elastic labor supply. Adding elastic labor supply does not change the results o f Chari et al. (1997) significantly, but does have a large effect on the predictions of the endogenous growth models. Third, we add taxes on factor incomes as in Stokey and Rebelo (1995). We assume that there is a representative household which maximizes
Z [3tU(c,, gt) Nt,
(5.12)
t-0
where c is consumption per household member, g is the fraction o f time devoted to work, and Nt is the total number of household members. Since we are using a representative household in our analysis, we refer to the units o f c as per capita and to Art as the total population at date t. As before, we assume here that the growth rate o f the population is constant and equal to n. For our calculations below, we assume that U(c,g) = {c(1 - g)~P}l-~/(1 - a ) with tO > 0.
23 For more discussion of this model, see Rebelo (1991) and Jones and Manuelli (1997). In Section 5, we analyze simulations of the model using as inputs the process for investment tax rates estimated in Chari et al. (1997).
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E.R. McGrattan and JA. Schmitz, Jr.
There are two sectors of production in the economy. Firms in sector 1 produce goods which can be used for consumption or as new physical capital. The production technology in this sector is given by (5.13)
c + x k = y = A(kv)~k(hgu) ah,
where xk is per capita investment in physical capital; A is the index of the technology level; v and u are the fractions of physical capital and labor, respectively, allocated to sector 1; and k and h are the per capita stocks of physical and human capital, respectively. In this case, we assume constant returns to the accumulable factors; that is, ak + ah = 1. The human capital investment good is produced in sector 2 with a different production technology, namely, Xh = B (k(1 - v)) °z' (hg(1
u)) °h ,
(5.14)
where xh is per capita investment in human capital and B is the index of the technology level. Again, we assume constant returns in accumulable factors so that 0h + Oh = 1. As do Uzawa (1965) and Lucas (1988), we allow for the possibility that the production of human capital is relatively intensive in human capital (that is, Oh > ah). Note that if ak = Ok and A = B, then this model is equivalent to a one-sector endogenous growth model. The laws of motion for the per capita capital stocks k and h are given by (1 + n ) k t + l = (1 - - 6 k ) k t + x k t , (1 + n)ht+l = (1 - 6h)ht +xht,
(5.15) (5.16)
where the term (1 + n) appears because we have written everything in per capita terms. Households supply labor and capital to the firms in the two sectors. Their income and investment spending are taxed. A typical household's budget constraint is given by ct + (1 + 'cxkt)x~t + (1 + "cxh3qtxht
~< (1 - "cklt)rltktvt + (1 - "ck2t)r2tkt(1 - Or) + (1
-
"Chit)Wltgthtut
(5.17)
+ (1 -'chet) w2te-tht(1 - ut) + It,
where q is the relative price of goods produced in the two sectors, "cxk is a tax on physical capital investment, 'cxh is a tax on human capital investment, rj is the rental rate on physical capital in sector j, wj is the wage rate in sector j, "ckj is a tax on income from physical capital used in sector j, rhj is a tax on income from human capital used in sector j, and T is per capita transfers. We assume that households maximize Equation (5.12) subject to Equations (5.15), (5.16), and (5.17), the processes for the tax rates "cxk, "cxh, "ckj, "chj, j = 1,2, and
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Table 3A Parameter values for tax experiments in the two-sector endogenous growth model Parameters
Model King and Rebelo (1990)
Lucas (1990)
Kim (1992)
Jones et al. (1993)
0.33 0.33
0.24 0.0
0.34 0.34
0.36 0.17
Physical capital (6k)
0.1
0.0
0.05
0.1
Human capital (610
0.1
0.0
0.01
0.1
0.988 0.0 1.0 0.0 0.126
0.98 0.5 2.0 0.014 0.078
0.99 0.0 1.94 0.0 0.048
0.98 5.0 1.5 0.0 0.407
Capital shares
Sector 1 (ak) Sector 2 (0~) Depreciation rates
Preferences
Discount factor (/3) Share on leisure (~0) Risk aversion (o) Growth in population (n) Technology level (B)
given factor prices. Assuming competitive markets, one finds that factor prices in equilibrium are marginal products derived using the technologies in Equation (5.13) and Equation (5.14). We turn now to some calculations. Following Stokey and Rebelo (1995), we parameterize the model to mimic different studies in the literature. In Table 3A, we display four such parameterizations corresponding to the studies o f King and Rebelo (1990), Lucas (1990), K i m (1992), and Jones et al. (1993). For all four models and all o f the numerical experiments we run, we normalize the scale o f technology in sector 1 with A = 1 and adjust B so as to achieve a particular growth rate in our baseline cases. Although there are slight differences between the model described above and those we are comparing it to, when we run the same numerical experiments as these studies, we find comparable results. Here we run the same numerical experiment for all four models. The experiment is motivated by the data on income tax revenues and growth rates for the U S A reported in Stokey and Rebelo (1995). Stokey and Rebelo note that in the USA, there was a large increase in the income tax rate during World War II. Despite this, there was little or no change in the long-run US growth rate. Stokey and Rebelo argue that this evidence
712
E.R. McGrattan and J.A. Schmitz, Jr
Table 3B Steady-state growth for a 0 percent and 20 percent income tax in the two-sector endogenous growth model Model King and Rebelo (1990)
Lucas (1990)
Kim (1992)
Jones et al. (1993)
2.00 -0.62
2.00 1.17
2.00 1.31
2.00 -1.99
2.18 182
1.28 5.12
1.23 3.89
3.31 2924
Steady-state growth rate
Tax rate = 0 Tax rate = 0.2 Ratio of incomes
After 30 years After 200 years
suggests that the models in the literature predict implausibly large growth effects o f fiscal policies 24. Suppose that we parameterize our model using the values given in Table 3A. The parameter B is set so as to achieve a steady-state growth rate o f 2 percent when all tax rates are 0. N o w consider an increase in the tax rates rk~, rk2, rhb and Th2 from 0 percent to 20 percent. In Table 3B, we display the after-tax steady-state growth rates for all four parameterizations. The new growth rates range from a value o f - 1 . 9 9 for Jones et al.'s (1993) parameters to 1.31 for Kim's (1992) parameters. To get some sense o f the magnitudes, imagine two countries that start out with the same output per worker but one follows a 0 percent tax policy and the other a 20 percent tax policy. After 30 years, one would predict that their incomes differ by a factor o f 1.23 using Kim's (1992) parameters and 3.31 using Jones et al.'s (1993) parameters. After 200 years, the factors would be 3.89 versus 2,924. Thus, there is a large difference between the predictions o f Lucas (1990) or K i m (1992) and King and Rebelo (1990) or Jones et al. (1993) i f growth rates are compounded over many years. Table 3B shows clearly that the estimated impact o f policy on growth varies dramatically in the literature. Here, too, there is still much debate about the magnitude o f the estimates o f policy effects. To get some sense o f why the results are so different, we consider two special cases o f the model and derive explicit formulas for the growth rate o f productivity in the steady state. Suppose first that incomes from capital and labor used in sector j are
24 The small change in growth could also be due to the fact that there were other policy changes such as lower tariffs or increased public spending on education as in Glomm and Ravikumar (t998) that had offsetting effects on the growth rate.
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taxed at the same rates. That is, let rj- = rkj = Thj. Suppose that tax rates on physical and human capital investment are equal; that is, z'x = rxk = rxh. Suppose also that the capital shares are equal in the two sectors, with a = ak = Ok. Finally, assume that the depreciation rates are equal for physical and human capital, and let 6 = 6k = 6h. In this case, the steady-state growth rate for output per worker is given by 1
g
/3 1-b+[Aoc(1-rl)]a[B(1-a)(1-T2)] I-ag(v) -~+~ ~] ]j -1,
(5.18)
where z- = (rx, rl, v2) is the vector o f tax rates and g(v) denotes the fraction o f time spent working in the steady state, which is a function o f the tax rates. From the expression in Equation (5. t 8), we see that predicted effects o f a tax increase depend on the discount factor, the depreciation rate, the capital share, and the elasticity o f labor. The parameters o f King and Rebelo (1990) fit the special case in Equation (5.18). But they further assume that labor is supplied inelastically, and therefore, g(T) = 1. Consider two variations on King and Rebelo's (1990) parameter values given in Table 3A. First, suppose they had assumed 6 = 0 rather than 6 = 0.1. Using the formula in Equation (5.18) with a = 0.33, 6 = 0, /3 = 0.988, a = 1, A - 1, and B = 0.0154, we find that the pre-tax growth rate is 2 percent and the after-tax growth rate is 1.36 percent, which is significantly higher than -0.62. (See Table 3B.) Now consider increasing o. If we set 6 = 0, a = 2, and B = 0.032 so as to get a pre-tax growth rate o f 2 percent, then the after-tax growth rate is 1.48 percent, which is even higher than the estimate found with Kim's (1992) parameter values. We now consider a second special case. Suppose that the sector for producing human capital uses no physical capital. In this case, the steady-state growth rate for output per worker is given by 1
1 + vxh JJ
(5.19)
where r = (rx, Tkl, rk2, Tha, Th2) is the vector o f tax rates and g(T) is the time spent working in the steady-state equilibrium. The parameters o f Lucas (1990) fit this special case. In this case, no physical capital is allocated to sector 2, and therefore, changes in Vkz have no effect at all. Furthermore, changes in tax rates in sector 1 only affect growth if they affect the supply of labor. If labor is inelastically supplied, the taxes levied on factors in sector 1 have no growth effects at all. Lucas (1990) chooses a near-inelastic labor supply elasticity (~p = 0.5). Suppose, for his case, we use ~p = 5, implying an elastic labor supply as in Jones et al. (1993), and set B = 0.219 to hit the baseline growth rate. With these changes, the steady-state labor supply g is 0.209 when the tax rates are 20 percent and 0.283 when the tax rates are 0 percent. Using the formula in Equation (5.19), we find that the pre-tax growth
E.R. McGrattan and J.A. Schmitz, Jr.
714
rate is 2 percent and that the after-tax growth rate is 0.79 percent. Thus, the growth effects are sensitive to the choice of labor elasticity. The formulas in Equation (5.18) and Equation (5.19) illustrate how sensitive the quantitative predictions are to certain parameter assumptions. In particular, the predictions are sensitive to choices of the labor elasticity, depreciation rates, and the intertemporal elasticity of substitution. Stokey and Rebelo (1995) attribute the wide range of estimates of the potential growth effects of tax increases cited in the literature to different assumptions for these parameters. The conclusion that Stokey and Rebelo (1995) draw from the US time series evidence is that tax reform would have little or no effect on growth rates in the USA. They do not dispute that the two-sector endogenous growth model yields a good description of the data if it is parameterized as in Lucas (1990) or Kim (1992). Jones (1995b), however, uses the US time series as evidence that the model is not a good description of the data. He notes that after World War II, we saw large increases in the investment-output ratio in France, Germany, Great Britain, Japan, and the USA. But growth rates in these countries changed little. If the data were well described by a one-sector AK growth model, then Jones (1995b) argues that we should have seen larger increases in the growth rate accompanying the increases in the investmentoutput ratio. The model Jones (1995b) works with is a one-sector version of the model above in which labor is supplied inelastically and the total population is constant. Suppose that A = B, a = ak = 0k, 6 = 6~ = 6h, ~P = 0, and n = 0. In this case, the ratio of human to physical capital is given by the ratio of their relative shares (1 - a ) / a . Here, as in the A K model, total output can be written as a linear function of k, namely, as A k ~ h 1 a = A[(1 - a ) / a ] 1 ~k. Thus, the growth rate in output is equal to the growth rate in capital. From Equation (5.15), we can derive the steady-state growth rate in capital, which we denote by g, by dividing both sides of the equation by kt and subtracting 1. The growth rate in this case is 2¢k
(5.20)
C +X k +Xh
where we have used the steady-state relation between capital and total output c + xk + xh. Jones (1995b) points out that while investment-output ratios have risen over the postwar period, growth rates have stayed roughly constant or have fallen. The formula in Equation (5.20) implies the opposite: increases in investment-output ratios should be accompanied by increases in growth rates. There are several caveats to be noted with Jones' (1995b) argument. First, in countries such as the USA, the changes in the investment-output ratio are not that large, and by Equation (5.20) we would not expect a large change in the growth rate. 1 Suppose a = 51 and A is set equal to ~1 to get a capital-output ratio of roughly 2g. Suppose also that the depreciation rate is 5 percent. These values would imply that an increase in the investment-output ratio from 16.5 percent to 18.1 percent, as reported
Ch. 10: Explaining Cross-Country Income Differences
715
by Jones (1995b) for the USA over the period 1950-1988, should lead to a change in the growth rate from 1.55 percent to 2.18 percent. Given the size of growth rate variations in the data, it is hard to detect such a small change in the long-run growth rate over such a short period of time. Second, the relationship between growth rates and the investment-output ratio is not given by Equation (5.20) as we relax many of the assumptions imposed by Jones (1995b). For example, if labor is elastically supplied or the two sectors of the model have different capital shares, then Equation (5.20) does not hold. In such cases, we have to be explicit about what is changing investmentoutput ratios in order to make quantitative predictions about the growth rates. If, for example, we use Lucas' (1990) model to investigate the effects of income tax changes, we find a small effect on growth rates but a big effect on investment-output ratios. In this section, we discussed the effects of changes in tax rates on growth. The AK model has also been used to study the effects of monetary policy on growth. For example, Chari et al. (1995) consider an AK model with several specifications of the role for money. In all cases, they find that changes in the growth rate of the money supply has a quantitatively trivial effect on the growth rate of out-put. As we saw above, large growth effects require large effects on the real rate of return. Changes in tax rates can have a potentially large effect on the real rate of return, but changes in inflation rates do not. On the other hand, Chari et al. (1995) find that monetary policies that affect financial regulations such as reserve requirements on banks can have nontrivial effects (on the order of a 0.2 percentage point fall in the growth rate with a rise in inflation from 10 to 20 percent) if the fraction of money held as reserves by banks is high (on the order of 0.8). These effects are small, however, relative to the effects of fiscal policy that have been found. 5.2.2. Policies in an R & D model
A large literature has developed theoretical models of endogenous growth based on devoting resources to R&D. This literature includes new product development models [such as in Romer (1990)] and quality-ladder models [such as in Grossman and Helpman (1991a, b) and Aghion and Howitt (1992)]. As compared to the theoretical literature that explores the quantitative link between policies and disparity (as in Section 5.1) and the two-sector endogenous growth literature that explores the quantitative link between policies and growth (as in Section 5.2.1), this R&D literature has far fewer studies exploring the quantitative link between policies and growth. This is likely due to the fact that the main quantitative concern for these models has been their predicted scale effects. Though there has been little explicit analysis of the effect of policy in these models, we think that it is important to review this important literature. We begin by describing a discrete-time version of the model in Romer (1990). Recall that in Section 5. t.2 we considered the problem of a small open economy importing intermediate goods that had already been developed in the rest of the world. Here we focus on the R&D activity. Technological innovation - new blueprints for intermediate
716
E.R. McGrattan and J.A. Schmitz, Jr.
inputs - is the driving force behind growth in this model. We show that the model implies a scale effect: the growth rate increases with the number of people working in R&D. This implied scale effect has been criticized by Jones (1995a) who offers a possible solution without significantly changing the model. [See also Young's (1998) model of quality ladders.] We review Jones' (1995a) model in which there is no scale effect. We lastly turn to the evidence on this point. The discrete-time version of the economy in Romer (1990) that we consider has three production sectors. In the research sector, firms use existing blueprints and human capital to produce new blueprints. In the intermediate goods sector, firms use existing blueprints and capital to produce intermediate capital goods. In the final goods sector, firms use intermediate capital goods, labor, and human capital to produce a final good that can be consumed or used to produce new capital. In addition, there is a household sector. Households buy consumption and investment goods with wages, rental earnings, and profits. Consider first the problem of the final goods producers. Their production function is given by y = H~L r
~0N ~
a-r di,
where Hy is human capital devoted to final goods production, L is labor, N is the total number of intermediate goods currently in existence, and xi is the quantity of the ith intermediate good. Final goods producers choose inputs to maximize their profits and, therefore, solve max Y ].Iy,L, {xi}
wHHy--WLL--fO N pixidi,
(5.21)
where WH is the price of a unit of human capital, WL is the wage rate for labor, Pi is the price of intermediate good i, and the final good is the numeraire. Profit maximization implies that Pi = (1 - a - 7)H~.Lrx7 a-r
(5,22)
and that
N WH = a H ~ - I L r fo
x~-a y di.
Consider next the problem of intermediate goods producers. We start by assuming that the blueprint for intermediate good i has been purchased. The technology available to the producer of intermediate good i is linear and is given by xi = l ki,
(5.23)
Ch. 10: Explaining Cross-Country Income Differences
717
where ki is the capital input. Conditional on having purchased blueprint i, the producer
of intermediate good i maximizes profits ~i: ori
m a x p(xi) xi - rki
(5.24)
Xi
subject to Equation (5.23), where p(-) is the demand function given by Equation (5.22) and r is the rental rate for capital. The decision to purchase a blueprint is based on a comparison of the cost of the blueprint versus the benefit of a discounted stream of profits from using the blueprint. Free entry into intermediate good production implies that
PN, = ,_._, j=t s=t
where PN~ is the price of blueprint N at date t and ~j are profits at date j. Next we consider the problem of research firms who produce new blueprints and sell them to intermediate goods producers. Given an input of human capital,/:/, a firm can produce 6I-IN new blueprints, where 6 is a productivity parameter and N is the total stock of blueprints in the economy. Let Hut denote the aggregate human capital input in R&D; then the stock of blueprints evolves according to Nt+l = Nt + 6HNtNt.
(5.25)
In equilibrium, it must be true that WH = PN 6N.
Lastly, consumers maximize expected utility subject to their budget constraint. Preferences for the representative household over consumption streams are given by oo
C; - ° -
1
t=0
where Ct are units of consumption at date t. Denoting the interest rate by rt, one finds that the maximization of utility subject to the household's budget constraint implies that Ut(Ct) = l.3Ut(Ct+l)(1 + rt+i).
(5.26)
We now compute a steady-state equilibrium growth rate for output. Assume that the total stock of human capital H = HN + Hy and the supply of labor L are both fixed. Romer (1990) shows that a symmetric equilibrium exists in which output Y, consumption C, and the number of blueprints N all grow at the same rate. Denote
E.R. McGrattan and J.A. Schmitz, Jr.
718
this growth rate by g, and denote the quantities, prices, and profits in the intermediate good sector by x, p, and £v. From Equation (5.25), we know that g = (SHN.Thus, to compute the growth rate of output, we need to derive the stock of human capital devoted to R&D in equilibrium. The returns to human capital in the research sector and in the final goods sector must be equal in equilibrium; therefore,
PNON = aH~ ILYN21-a-Y.
(5.27)
Using Equations (5.22), (5.23), and the first-order condition from Equation (5.24), we have that = (gg + y ) ( 1
- a -
y)H~LY~c l-a-y.
Equating the price of blueprints to the discounted value of the profits from use of the blueprints implies that PN
=
1
1
--f'E : Y
--{(a F
q-
~/)(1 - a -
y)H~LY21-a-Y}.
(5.28)
Substituting Equation (5.28) in Equation (5.27) and simplifying yields the following expression for human capital in production: aP
Hv = 6(1 - a - y)(a + ]/)' Therefore, the growth rate is
ar g : 6 ( H-6(l_a_-~,)(a+ y) )=6H-Ar,
(5.29)
where A = a / [ ( l - a)(1 - a - V)]. From the household's first-order condition in Equation (5.26) we have that g = [fi(1 + r ) ] } - 1.
(5.30)
Thus, in Equations (5.29) and (5.30), we have two equations from which we can determine the growth rate g and the interest rate r on a balanced growth path. Notice that g depends positively on the stock of human capital H . Thus, there is a scale effect, as we noted above. As Jones (1995a) points out, one need not even proceed past the specification of growth in the number of blueprints and the description of technologies to know that there is a scale effect. The main assumption of the model is that a doubling of the number of people working on R&D implies a doubling o f the growth rate by Equation (5.25). However, in many countries, particularly the OECD countries, there has been a dramatic increase in the number of scientists and engineers
Ch. 10: Explaining Cross-Country lncome Differences
719
and a dramatic increase in the resources devoted to R&D with little or no increase in growth rates over a sustained period. Within the context of Romer's (1990) model that we just described, Jones (1995a) offers a possible solution to the problem of the existence of a scale effect. In particular, he assumes that the evolution of blueprints is given by
Nt+l = Nt + 6HZNtNtO,
(5.31)
with 0 < / l ~< 1 and q~ < 1, rather than by Equation (5.25). Jones (1995a) also assumes that g = 0 and that the growth rate of H is given by n, where H is now interpreted to be the total labor force. On a balanced growth path, HXNt must grow at the same rate as Nt 1-0. [See Equation (5.31).] Thus, it follows that /in
g = 1 - 0'
(5.32)
where g is the growth rate of blueprints and output per worker. Note that g now depends on the growth rate of the labor force rather than the total number of researchers. Thus the scale effect is removed. Is the relationship in Equation (5.32) consistent with the data? The answer to this question depends a lot on how the model is interpreted. For example, if we interpret this model as one of a typical country, then the answer is no. The correlation between growth rates of GDP per worker and growth rates of the labor force over the period 1960-1985 is -0.12 (based on all countries with GDP per worker available). The relationship in Equation (5.32) implies a positive correlation. I f we interpret this model as one relevant only for countries in which there is a lot of activity in R&D, we still find that the correlation between the growth rates of GDP per worker and the labor force are around zero or slightly negative. Suppose, finally, that we view the model as one relevant for the world economy. Using the data of Bairoch (1981), we see that the growth rate in real GDP per capita for the world and the growth rate of the world population followed the same trend over the period 1750-1990. Hence, under this interpretation, the model is roughly consistent with the data assuming that the growth rate of the world population is a good proxy for the growth rate of the number of researchers world-wide. With growth in output per worker determined by a variable assumed to be exogenous, namely the growth rate in the labor force, policies intended to encourage innovation such as subsidies to R&D or capital accumulation have no growth effects in Jones' (1995a) model. However, we do know that there are many countries that have a large fraction of R&D spending financed by their government. Examples include France, Germany, Japan, and the USA. In fact, Eaton and Kortum (1996) attribute more than 50 percent of the growth in each of the OECD countries to innovation in Germany, Japan, and the USA. But the effects of policies encouraging innovation are still being debated. [See, for example, Aghion and Howitt (1998).]
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E.R. McGrattan and J.A. Schmitz, Jr.
6. Two growth models and all of the basic facts In Section 5.1, we reviewed the literature that studies the implied disparity in incomes in various parameterized models, while in Section 5.2, we reviewed the literature that studies the implied growth rates in various parameterized models. These studies typically focus on one or the other dimension of the income distribution - that is, disparity or growth. As Lucas (1988) argues, "the study of development will need to involve working out the implications of competing theories for data other than those they were constructed to fit, and testing these implications against observation" (p. 5). We turn to that task in this section. We look at implications of two of the above parameterized models for numerous dimensions of the income distribution over the last 100 or so years (in particular, the features in Figures 1-4). As we just mentioned, few studies actually perform this exercise. A big difficulty in performing such an exercise is coming up with reasonable measures of factor inputs such as human capital and economic policies for such a long period of time. We do not solve that problem here. What we do is take the measure of distortions on capital investment that Chari et al. (1997) use for the post-World War II period and suppose the process applies to the last 200 years or so 25. Our purpose in this section is not to argue that investment distortions were the only factor determining variations in incomes but only to show what can be learned by conducting the exercise that Lucas (1988) suggests. For example, we learn that the parameterized models, with the distortions from Chari et al. (1997), do a reasonable job in explaining some figures, but not all. The models that we analyze here are a standard exogenous growth model and a standard AK endogenous growth model. For both models, we generate panel data sets and compare them to the data compiled by Maddison (1991, 1994) and Summers and Heston (1991). This is done by producing analogues of Figures 1-4 for the two models. The exogenous growth model can be written succinctly as the following maximization problem: OQ
max Z / 3 / ( c t ( 1 - g t ) ~ ) > ° / ( 1 - o) {ct,~,,xkt,xh~} t=0 subject to
ct + (l + rxt)(x,~ + xht) 0, fi =/3(1 + g ) l - ° ( 1 + n ) , and ak + ah < 1. Original variables have been converted to per capita terms and, i f necessary, divided by the level o f technology in order to make them stationary (for example, gt = L t ~ t , ct = C,/(AtNt), kt = K t / ( A t N t ) , and so on, where Nt = (1 + ny is the total population and At = A(1 + g)t is the level o f technology). A bar over the variable denotes the economy-wide level 26. The endogenous growth model can be written succinctly as the following maximization problem
max Z/3t(c'(1 {c,,g,,xk,,xh,,v~,u,} t=0 subject to
- gt)~P)l-a/(1 - or)
ct + (1 + rxt)(xkt + qtxht)
