Rules, Reputation and Macroeconomic Policy Coordination

In this book David Currie and Paul Levine address a broad range of issues concerning the design and conduct of macroeconomic policy in open economies. Adopting neo-Keynesian models for which monetary and fiscal policy have short-term real effects, they analyse active stabilisation policies in both a single- and multicountry context. Questions addressed include: the merits of simple policy rules, policy design in the face of uncertainty, and international policy coordination. A central feature of the book is the treatment of credibility and the effect of a policy-maker's reputation for sticking to announced policies. These considerations are integrated with coordination issues to produce a unique synthesis. The volume develops optimal control methods and dynamic game theory to handle relationships between governments and a conscious rational private sector, and produces a unified, coherent approach to the subject. This book will be of interest to students and teachers of open economy macroeconomics, and to professional economists interested in using macroeconomic models to design policy.

Rules, reputation and macroeconomic policy coordination

Rules, reputation and macroeconomic policy coordination

DAVID CURRIE and PAUL LEVINE

j CAMBRIDGE UNIVERSITY PRESS

Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 1RP 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia © Cambridge University Press 1993 First published 1993 A catalogue record for this book is available from the British Library Library of Congress cataloguing in publication data

Currie, David A. Rules, reputation and macroeconomic coordination / David Currie, Paul Levine. p. cm. ISBN 0 521 44196 X 1. Economic policy. 2. Economic policy-International cooperation. 3. Macroeconomics. I. Levine, Paul, 1944— II. Title. HD87.C87 1993 339.5-dc20 92-30613 CIP ISBN0 521 44196 X hardback Transferred to digital printing 2004

To James, Tim, Ben and Sam

CE

Contents

Acknowledgements

page ix

Introduction

1

Part I

General issues

1

Macroeconomic policy design and control theory - a failed partnership?

21

2

International policy coordination - a survey

43

3

The European road to monetary union

77

Part II Theory and methodology 4

The design of feedback rules in linear stochastic rational expectations models

95

5

Credibility and time consistency in a stochastic world

122

6

Should rules be simple?

148

7

Macroeconomic policy design using large econometric rational expectations models 176 with Nicos Christodoulakis and Jessica Gaines

Part III Fiscal and monetary policy in interdependent economies 8

Macroeconomic policy design in an interdependent world

213

9

Does international macroeconomic policy coordination pay and is it sustainable?: a two-country analysis 247 vii

viii

Contents

10

International cooperation and reputation in an empirical two-bloc model 288 with Nic Vidalis

11

Fiscal policy coordination, inflation and reputation in a natural rate world 321 with Ullrich Hoffmeyer

12

The use of simple rules for international policy coordination with Jessica Gaines

13

347

Evaluating the extended target zone proposal for the G3 with Simon Wren-Lewis

385

Bibliography

407

Index

420

Acknowledgements

We are deeply indebted to Patrick McCartan of Cambridge University Press for his encouragement and support throughout the preparation of this volume. Both authors are fortunate to have been able to collaborate with a number of talented colleagues. Five appear as co-authors in this volume: Nicos Christodoulakis, Jessica Gaines, Nic Vidalis, Ullrich Hoffmeyer and Simon Wren-Lewis. Thanks are also due to Durmus Ozdemir for carefully reading and correcting the entire text. Finally we are grateful to the following copyright holders for giving permission to reprint published papers: The Royal Economic Society, the Money Study Group, European Research Press, Elsevier Science Publishers, Springer-Verlag, Kluwer Academic Publishers, Oxford University Press, Cambridge University Press, Macmillan Press, Academic Press and the Greek Economic Review.

IX

Introduction

Overview

This book brings together thirteen contributions written by the authors and collaborators in the general area of macroeconomic policy for open economies. The chapters appear essentially in their original published form. A number of typographical errors and some more substantive errors, discovered on rereading, have been corrected. In addition, in a few places material has been removed or added to avoid repetition and aid coherence. The volume covers a broad area ranging from single open economy issues to games between national policy-makers. Nevertheless, a number of common features and themes link these selected contributions. First the models employed are 'ad hoc' and for the most part neo-Keynesian, displaying a short-run trade-off between output and inflation even where policy rules are anticipated. In all but one of the chapters we assume that exchange-rate behaviour obeys the uncovered interest-rate parity condition and that expectations in the foreign exchange market are rational in the sense that they are consistent with the predictions of the model. Thus the models we deploy can be broadly described as being in the Mundell-Flemming-Dornbusch tradition. Apart from exchange-rate behaviour, the private-sector relationships postulated in most of our ad hoc models are clearly susceptible to the Lucas Critique. In using backward-looking equations with 'constant' parameters it may be argued that we ignore the crucial links between a change of regime and private-sector behaviour via private-sector expectations of future policy. Despite the success of rigorous models based on micro-foundations, it is by no means obvious that all areas of economic behaviour should be modelled as forward-looking and rational. And, even if we were convinced that all economic agents behaved in this way, backward-looking relationships can be regarded as empirically acceptable

2

Introduction

approximations if the influence of the past on current decisions greatly outweighs the influence of future (rational) expectations. The stability of many macroeconomic relationships in the face of many changes in regime indicates that, for whatever reason, the Lucas Critique may not be all pervasive (see Hendry, 1988). A common theme in the volume is the idea of formulating fiscal and monetary policy in the form of a stabilisation rule. Since the term 'rule' is sometimes used in different ways in the policy literation (for instance to imply precommitment) we should immediately make our terminology clear. We use rule in the sense of a state-contingent or feedback rule in which fiscal and monetary policy instruments are changed period-byperiod in response to changes in target variables such as output and inflation. Feedback rules bring us to a second theme of the book which is that there are both advantages and disadvantages in formulating such rules to be simple. The meaning and significance of simplicity in this context is best understood by considering optimal rules which are arrived at by optimising a social welfare function under the constraint of a model. Even when the problem is simplified to the case of a linear model, a quadratic welfare function and an infinite time horizon, it still turns out that the optimal rule is extremely complex involving a time-invariant feedback on a potentially large number of variables (depending upon the size of the model). Our simple rules retain the time invariance of the optimal rule but assign each instrument to a limited number of target variables. The precise form simple rules should take is the 'assignment problem' first considered by Mundell (1962, 1964) and about which much of the discussion of monetary and fiscal policy in open economies has since centred. The problem as originally posed by Mundell is the assignment of one target to one instrument in such a way that the target variables will eventually converge to their desired values. The sub-optimality of simple rules of this type was not of real concern in this early literature and consequently there was no attempt to justify simplicity per se. More recently, Vines, Maciejowski and Meade (1983), justify one-to-one targeting of instruments or 'decoupled-rules' to target variables on the basis that it permits the decentralisation of decisionmaking. Other arguments advanced for simple rules (not necessarily decoupled) is that they are more easily understood and can be made intuitively convincing. This is clearly an important consideration in arguing for their adoption within the political sphere. A further reason for adopting rules that are simple is related to a third theme of these contributions which is the credibility issue. Notions of

Introduction

3

'credibility' and 'reputation', which relate to private-sector beliefs regarding future government policy, are now commonplace in the macroeconomic literature. At a more informal level, these ideas frequently enter into discussions of policy effectiveness, especially relating to the disinflationary fiscal and monetary stances pursued by OECD countries in the early 1980s. At the theoretical level there has been much recent progress in developing and clarifying the issues (see Persson, 1988 and Levine, 1990, for recent surveys). A credibility problem may arise whenever policy-makers try to precommit themselves to future policies either in the form of planned paths for fiscal and monetary instruments or in the form of a feedback rule. For a single economy with one policy-maker credibility becomes an issue if policies are arrived at by maximising some welfare function given that at least some private-sector expectations are rational. In an international context, even without rational expectations, a credibility problem arises when governments try to enforce a cooperative agreement. The manner in which rational expectations can fundamentally alter the nature of optimal policies was first highlighted by Kydland and Prescott (1977). Using a very general model in which the private sector's current decisions depend upon expectations of future government policies, they showed that optimal, welfare-maximising policies become sub-optimal simply with the passage of time. In other words policies which are optimal ex ante become sub-optimal ex post. The term time inconsistency is used to describe this property. If governments were able to make binding commitments to their ex ante optimal policy, then time inconsistency would not be a serious problem. However, in the absence of some institutional arrangements which effectively deter any change in the previously announced rule, an irresistible incentive to renege on time-inconsistent policy may arise. If the rational expectations assumption is extended to assuming that the private sector has full information on the government's optimisation problem, then it follows that such reneging is anticipated. Time-inconsistent policies must therefore lack credibility. A number of quite distinct responses to the time-inconsistency problem are found in the literature. One approach argues that the only credible policies are those which are constrained to the time consistent. These can be found in minimising the welfare loss in each period in the knowledge (shared by both the policy-maker and the private sector) that a similar procedure will be used one period later (see chapters 4, 5 and 6 for details of this procedure). A major problem with time-consistent policies however is that they can be severely sub-optimal. In the normative spirit of the chapters in this volume we are therefore interested in alternatives.

4

Introduction

One alternative is to relax the assumption of complete information. (By complete information in this context we mean that the private sector knows both the true structure of the model and the policy-maker's welfare criteria.) In the large literature which adopts this approach (for example, Backus and Driffill, 1985a, 1985b; Barro, 1986a; Vickers, 1985; Cukierman and Meltzer, 1986; and others) the main result that emerges is that uncertainty on the part of the private sector regarding the 'type' of government may be sufficient to sustain the ex ante optimal policy for much of the time. Another strand of literature appeals to partial precommitment as a means by which governments can bind their successors. This idea was first applied to the Ramsey optimal taxation problem by Lucas and Stokey (1983) who show that debt precommitment, appropriately arranged, can achieve an optimal taxation policy (given government expenditure) which is time consistent. A third alternative to accepting the time-consistency constraint is to consider reputational equilibria in which the optimal policy - or at least policies superior to the time-consistent policy - can be supported by a private-sector trigger mechanism. Following Barro and Gordon (1983b) this is assumed to take the form of a loss of credibility or reputation following reneging. Then the policy-maker is forced to switch from the optimal time-inconsistent policy to the sub-optimal time-consistent policy for the period in which the 'punishment' lasts. The prospect of this loss of reputation may, in some circumstances, sustain the ex ante optimal policy and resolve the time-inconsistency problem. The approach to the time-inconsistency problem adopted in this volume is the last of these alternatives. In both a single-country and a multicountry context we show that time-consistent policies can be very inefficient and that punishment mechanisms of the type discussed can enforce the ex ante optimal policy. For the most part credibility is examined for welfare-maximising policies which are not constrained to be simple. Chapter 12 however provides a treatment of reputational equilibria for the case of simple rules. The credibility issue provides a further motivation for studying simple rules. Indeed one of the conclusions of Kydland and Prescott is that optimal rules should be eschewed and that 'economic theory may be used to evaluate policy rules and that one with good operating characteristics be selected. It is preferable that selected rules be simple and easily understood so that it is obvious when a policy-maker deviates from the policy.' The 'credibility-requires-simplicity' line of reasoning is often used as a rationale for a non-contingent percentage money supply rule. The

Introduction

5

problem with an 'open-loop' rule however is that it provides no flexibility for the government to respond to unforeseen shocks to the economy. The type of feedback rules we have discussed can provide this flexibility and, in addition, constitute a form of precommitment. The concern with monitoring expressed in the quotation from Kydland and Prescott suggests an incomplete information game between the government and private sector. Monitoring is irrelevant to the credibility question in a complete information game with the private sector, in particular, knowing the full nature of the government's calculations. In an incomplete information setting a further reason for adopting simple rules emerges. The private sector may not believe in policy announcements and will only change its priors by observing and learning about the implemented rules. Evidently the learning process is speeded up if the rules are relatively simple. This argument for simplicity associated with credibility and the need for the private sector to be able to easily monitor policy is formalised in chapter 6. We have mentioned the benefits of feedback rules in responding to unforeseen shocks. This brings us to the final theme which is designing rules in the face of uncertainty. Many of the general arguments against stabilisation rules focus on the existence of uncertainty. To take a famous example, Friedman (1968) rules out feedback control from the money supply to prices because of an inability to predict just what effect a particular monetary action will have on the price level. Monetary policy, according to Friedman, has effects which are 'long, variable and uncertain' and in these circumstances 'attempting to control the price level is therefore likely to make monetary policy itself a source of economic disturbance'. Hence he argues for an open-loop monetary policy in the form of a A: per cent growth rule. Subsequent analysis, drawing on developments in stochastic control theory, show that Friedman's intuitive insights are correct only in the limited sense that, when lags are both long and variable in an uncertain way, then the gains from the use of a stabilisation policy are correspondingly smaller (Fischer and Cooper, 1973). However it is also wellestablished that Friedman is wrong in the general advocacy of open-loop control as a means of facing model uncertainty. Under the form of uncertainty considered by Friedman, feedback control may need to be more cautious (see Turnovsky (1977, chapter 14)), but some carefully designed feedback policy is bound to be stabilising relative to open-loop control. The concern with long, variable and uncertain lags focused on one form of uncertainty, namely uncertainty about model parameters. The design of robust policy rules in the face of this form of uncertainty is a difficult

6

Introduction

and on-going branch of research (see Petkovski, 1989 and Astrom, 1983, for recent surveys and Christodoulakis, Kemball-Cook and Levine, 1990, for a macroeconomics application). The chapters in this volume address the question of uncertainty when it occurs as unanticipated exogenous shocks to the economy. A recurrent feature of the chapters is the search for simple rules which are robust with respect to this particular form of uncertainty. In the neo-Keynesian models of this volume the role for an active stabilisation policy goes hand-in-hand with policy spillovers and the role for policy coordination. In the final section we address the need to avoid inefficient non-cooperative outcomes and the sustainability of cooperative policy rules. The issues of credibility, simplicity and robustness of policy design are now developed in a multicountry context. Taken together with the earlier chapters on the single economy, the volume then provides a unified coherent approach to the design of macroeconomic stabilisation rules for the open economy. Part I

General issues

The three chapters presented in this section reflect two major trends in macroeconomics research over the last two decades. The first trend is the development of a more systematic approach to macroeconomic policy design drawing upon formal control methods. The second is the treatment of policy design in an international context with the focus on interdependencies between national economies and on coordination issues. Coordination questions addressed in this section range from those in a Group of Seven (G7) context to European coordination and the issue of monetary union. Chapter 1 assesses the impact that control theory has had on both the economics profession and - more crucially - on policy-makers themselves. It traces the history of ideas beginning with Phillips in the 1950s, continuing with the Ball Committee (1978) and through to contemporary debates. Chapter 2 surveys the academic literature on macroeconomic policy coordination, an area that has grown dramatically over the past tqn to fifteen years. Attempts at international policy coordination - ranging from the ambitious package of macroeconomic and trade agreements at the Bonn economic summit of 1978 to the more limited efforts to manage exchange rates within the G7 countries and the EMS - have given rise to a number of questions of both a theoretical and empirical nature. How large are coordination gains and, indeed, can coordination ever be counterproductive? Assuming there are gains from coordination, can

Introduction

7

cooperative agreements be sustained or are they vulnerable to reneging by countries participating in the agreement? This question raises the problem of credibility both as it relates to the promises that countries make to each other and those made individually or collectively to the private sector. Various forms of uncertainty raise particular problems for policy coordination and here the question arises as to whether uncertainty undermines or strengthens the case for coordination. A final issue concerns the scope for limited forms of cooperation such as agreements focusing on the exchange rate or on simple rules of the type discussed above. We return to many of these themes in the papers of part III. The Exchange Rate Mechanism (ERM) within the European Monetary System provides a notable example of sustained international policy coordination based on a rule-based arrangement. The transition from the ERM to Economic and Monetary Union provides the theme for chapter 3. Three main issues which concern any form of monetary union are addressed in this chapter. The first concerns the gains to be reaped from the elimination of transaction costs of a single currency. The second is the issue which arouses the most passion - the question of sovereignty. Finally we consider credibility in both an ERM and EMU context. The benefits and dangers of EMU may be assessed by considering a spectrum of exchange-rate regimes which in fact correspond to the present evolution of exchange-rate management with the European Community. At the ends of the spectrum are the floating exchange rates as existed pre-ERM and the EMU regimes. Within these two extremes the ERM has progressed from a 'soft-ERM' with frequent currency realignments and wider bands for some, to a 'hard-ERM' with less frequent alignments and narrow bands for all. The question posed by chapter 3 is whether a prolonged period with a 'hard-ERM' provides a viable alternative to a swift movement to full monetary union as envisaged in the Delors Report. Comparing the four regimes only full EMU will eliminate transaction costs, but exchange-rate uncertainty is progressively lowered as one moves away from floating rates. Regarding monetary sovereignty there are two aspects to be considered. First, a 'semi-small' open economy, trading in less than fully competitive output markets, can influence its real exchange rate by means of control over the nominal exchange rate. Second, in a neo-Keynesian world monetary sovereignty allows a country to choose - at least in the short term - its own preferred point on the inflation/output trade-off in its use of a stabilisation policy. The 1992 programme of market unification will, if successful, erode this first aspect of sovereignty. In the meantime the sovereign manipulation of real exchange rates has a strong beggar-thy-neighbour character and the

8

Introduction

recent literature on monetary policy coordination, surveyed in the previous chapter, has highlighted this form of the prisoners' dilemma. The remainder of chapter 3 focuses on the credibility problem involved in the sovereign choice of the output/inflation combination. A generally accepted characterisation of a hard-ERM is that it allows the Bundesbank the freedom to set German monetary policy for its domestic anti-inflation concerns; and then other member countries fix their exchange rate vis-a-vis the Deutsch Mark thereby buying into Bundesbank credibility in resisting inflation. Thus the Bundesbank has acted as a leader forcing a tight and credible monetary policy on all members of the ERM. We can relate this phenomenon of German leadership to the earlier discussion of time inconsistency. The Bundesbank clearly has a reputation for being tough on inflation. But it is also plausible to suggest that the Bundesbank has a reputation for precommitment to any well-defined policy rule which may, in some circumstances, involve more accommodating stances towards inflation. If an authority enjoys reputation in this sense then it can exercise the greatest leverage on the economy with the best results. A central bank with a reputation can exploit the relationship between expectations of future policy and the current state of the economy to smooth the effects of demand and supply shocks, i.e., to engage in a stabilisation policy. Similarly in the foreign exchange market reputational monetary policy can give the central bank leverage over nominal exchange rates. The great fear regarding EMU however is that the new ECB might lack the reputation now enjoyed by the Bundesbank. Pooling of sovereignty also means pooling of credibility. The result may well be a ECB which is not in a position to make credible policy commitments which are believed by the private sector. The loss of reputation in the labour market (regarding inflation) and in the foreign exchange market (regarding interest rates) will then result in a long-run inflationary bias and a reduction in the effectiveness of a stabilisation policy. This scenario is not evitable; a careful choice of constitution for the ECB may circumvent this problem. Or, alternatively, the ECB may acquire reputation over time. Nonetheless chapter 3 highlights the risk that EMU will deliver higher average inflation and a less effective stabilisation policy than a hard-ERM. Part II

Theory and methodology

The four chapters in this section develop the general methodology for the chapters that follow and provide applications to macroeconomic policy

Introduction

9

design for the single open economy. Chapter 4 lays the foundation for the methodology by considering the optimal control problem for dynamic stochastic models with rational expectations. Throughout the volume (with the exception of chapter 13) we work within a linear, quadratic framework with an infinite time horizon; that is, the models are linear (or linearised) and the policy-maker's objective function is quadratic and defined over an infinite time horizon. In the absence of rational expectations it is a standard result in control theory that the optimal policy may be expressed as a time-invariant feedback rule which has the following properties. First, providing the future is not discounted too heavily, the optimal rule stabilises the endogenous variables. (Strictly, there is a further condition of complete controllability for this to be satisfied. However this condition does not turn out to be stringent and all the linear models used in this volume are completely controllable.) The second property we have mentioned is the discussion of simple rules. The optimal rule is linear in the entire state vector. For large models this implies that the feedback rule involves a feedback on a large number of endogenous variables and is therefore far from simple. A third property of the optimal rule is that it is independent of the initial state of the system. For a deterministic model optimal policy may also be expressed in terms of trajectories for instruments and this 'open-loop' form of policy is of course dependent upon the initial state. But the feedback rule itself does not have this dependency. A fourth property concerns the stochastic problem when white noise is added to the dynamic relationships and possibly to the instruments. For this form of additive uncertainty the optimal rule is the same as for the deterministic problem and is independent of the covariance matrix of the white noise disturbances. This is the property of certainty equivalence. The third and fourth properties describe two invariances for the optimal rule. An immediate consequence is the fifth property - the optimal feedback rule is time consistent. The optimal rule described at time t > 0 is simply the continuation of the rule chosen at time / = 0 and in neither case does the rule depend upon the current state of the system. Suppose now that the policy-maker is looking for rules with a particular assignment of instruments to targets. 'Quasi-optimal' rules within this class - simple rules - do not have these last three properties. In other words the best simple rule depends upon both the initial state and the covariance matrix of disturbances. Simple rules do not have either the certainty equivalence nor the time-consistency property. The policy implication of these results is that there are costs of simplicity to be weighed against the advantages in the choice of rules.

10

Introduction

All these conclusions apply to the standard control problem where there are no rational expectations. For models with rational expectations we have seen in the earlier discussion that the optimal rule (in addition to any simple rule) is time inconsistent. However it is possible to design a rule which is optimal within the constraint of being time consistent. Thus we have three types of rules: the optimal but time-inconsistent rule and two sub-optimal rules which are the best available within their class - the simple rule (of a particular form) and the time-consistent rule. The rest of chapter 4 sets out the solution procedures for evaluating these three types of rule and the associated welfare loss under control for continuous time models. Illustrative examples are provided with numerical solutions. For the most part the dynamic models in this volume do not allow for analytical solutions - indeed this seems to be a feature of even the simplest dynamic model. Our research strategy has therefore been to resort to simulation methods using especially written software for the task (see Gaines, al-Nowaihi and Levine (1987)). Chapter 5 returns to the time-inconsistency problem. Following Barro and Gordon (1983b) we examine whether reputational considerations can restore credibility for policy-makers and allow them to improve upon the sub-optimal time-consistent rule. Barro and Gordon consider an economy described by a Lucas-Sargent-Wallace supply curve in which the policy-maker is able to control inflation. They analyse three possible policy regimes. If the policy-maker has a reputation for precommitment then the 'ideal' policy of zero inflation is optimal. Reputation is necessary to sustain this policy because it is time inconsistent: there exists an incentive or 'temptation' to renege since a bout of surprise inflation can lower real wages and increase output, at least in the short run. If the authorities have no reputation then the optimal policy is one of non-zero inflation. This policy, which is a Nash non-cooperative equilibrium of the game between the government and the private sector, is time consistent but inferior to the ideal policy. Finally, there is a third regime, which consists of announcing a policy of zero inflation but implementing non-zero inflation. This 'perfect cheating' policy is superior to the 'ideal' policy, but is obviously implausible because it ignores the private sector's response to reneging. The question is therefore whether one can build into the model a mechanism which can, in some circumstances, ensure reputation for precommitment. Barro and Gordon achieve this by postulating a threat or punishment by the private sector. Following an act of reneging by the government the private sector will, for some punishment interval of P periods, believe only in the time-consistent or Nash non-cooperative

Introduction

11

inflation rate, from which there is no incentive to renege. After the punishment interval it believes in the ideal policy once again. Since the ideal policy is characterised as cooperation in the Barro-Gordon game, this amounts to saying that the threat by the private sector is to withdraw cooperation for some interval P. We describe this absence of cooperation as a loss of reputation. With the punishment mechanism operating, the government must weigh any temptation to renege against the costs associated with a loss of reputation. If these costs exceed the temptation then no incentive to renege exists. The ideal policy then becomes credible and 'sustainable'. Chapter 5 extends the Barro-Gordon game in two ways. First, their analysis applies only to essentially static models. We generalise the analysis to models with structural dynamics. Second, we examine how continuing stochastic disturbances affect the sustainability of the optimal or 'ideal' rule. Our main result is that, provided the degree of discounting is not too great, then the optimal rule will be sustainable in a sufficiently stochastic environment. The intuition behind this result is that the inferior stabilisation properties of the time-consistent policy becomes more serious when the policymaker must respond to continuous rather than one-off shocks. Then the temptation to respond to a particular configuration of the economy by reneging must be weighed against the reduced ability to respond to future disturbances. As the policy-maker becomes more far-sighted, the latter penalty effect dominates and sustainability is established. These results are illustrated using a continuous time dynamic model of an open economy. The demand side consists of a sluggishly adjusting IS curve with wealth effects. On the supply side prices are sticky but in the long run the labour market clears and output depends only on the real exchange rate. Exchange-rate dynamics are given by the uncovered interest-rate parity condition with rational expectations. Finally the policy-maker uses both fiscal and monetary instruments. Chapter 6 develops the 'ease-of-monitoring' argument for simple rules. The concern with monitoring suggests an incomplete information game between the private sector and the government. Following the Barro and Gordon policy game, which assumes complete information, the credibility literature has switched its focus to games where the private sector knows the general nature of government behaviour but may be missing some information such as the weight on output in the objective function. This chapter assumes a rather less well-informed public which does not attempt to infer the policy-makers objectives but instead estimates the rule directly by observing the data and applying standard econometric techniques.

12

Introduction

In an overlapping contract model with rational expectations it is shown that learning about the optimal rule, by a recursive OLS procedure, proves to be slow as compared with learning about a simpler rule of a lower dynamic order. In this incomplete information setting simple rules perform significantly better than the (originally) optimal counterparts. This result, then, provides some formal reasoning to support the less formal argument for simplicity, proposed above. The analysis of chapter 6 is conducted in discrete time. An appendix provides the discrete time analogue of the continuous time solution procedures presented in chapter 4. Taken together these chapters provide a comprehensive set of procedures for deriving fully optimal, time-consistent and simple rules for linear, stochastic, dynamic models in both continuous and discrete time. Up to this point the models used have been of a rather simple stylised character and have been 'calibrated' rather than estimated. Chapter 7 develops the methodology to handle large econometric models with an application to the London Business School (LBS) forecasting model. The design methodology proceeds through four stages. Since our techniques apply only to linear models we first obtain a linear representation of the original non-linear model. This linear representation is expressed in deviation form from a baseline forecast. The second stage is to evaluate a central path for instruments which reflect the policy-maker's preferences in relation to the baseline (e.g., more output and less inflation). Optimal control methods developed in earlier chapters then result in open-loop trajectories for fiscal and monetary instruments which, in this exercise, are chosen to be real government spending and short-term interest rates. In addition a once-and-for-all change in the direct tax rate is allowed at the beginning of the exercise. Since the LBS model assumes rational expectations in the financial sector, the optimal policy is time inconsistent and its sustainability must be examined using the approach of chapter 5. The third stage concerns the feedback component of policy. The openloop trajectories for instruments and target variables constitute expected paths which will be adhered to in the absence of unexpected shocks or modelling errors. The feedback component of policy is formulated in deviation form about these central, expected paths and specifies exactly how policy is to be adjusted in the face of either unexpected exogenous shocks or other divergences arising from the inadequacies of the model used. The optimal policy can provide both these components of policy this is the certainty equivalence property - provided that the only source of uncertainty lies in additive white noise disturbances. But as we have argued feedback rules should be simple. The third stage of the design

Introduction

13

procedure is then to evaluate simple rules that provide this adjustment in the face of uncertainty. The precise form of the simple rule involves both proportional and integral control. A variant of Tinbergen's rule is that integral control with n instruments can drive n target variables towards the central trajectories in the long run. The two target variables corresponding to our two instruments are chosen to be output and inflation. A further feature of the rule is the introduction of a four quarter lag in the fiscal component of the rule. This enables the policy-maker to announce adjustments relative to the central path over the next four quarters based on observations of output and inflation over the previous four quarters. Fiscal policy then involves precommitment to levels of expenditure over the coming year plus a central path and a simple feedback adjustment thereafter. This form of rule allows the policy-maker to make absolute commitments to departments and local authorities for the financial year but, at the same time, respond in a flexible way to unexpected changes to the macroeconomy. The final stage of the procedure is to validate the open-loop plus feedback components of policy by simulations on the original LBS model. These four stages constitute a comprehensive approach to policy design which address the issues of credibility and uncertainty raised earlier. Probably the most significant empirical result is that for the LBS model a form of simple rule can be found which is robust in the sense that it is effective at stabilising the UK economy in the face of a range of important external shocks. Part III

Fiscal and monetary policy in interdependent economies

In part II the studies considered policy design for the small open economy facing an external environment which is regarded as exogenous. Part III turns to the examination of policy rules for large economies behaving strategically with respect to each other. Chapter 8 sets up a dynamic stochastic rational expectations model of an open economy with a wage/price spiral, private-sector wealth accumulation and uncovered interest-rate parity in the foreign exchange market. Solution procedures developed in chapter 4 are used to compare a variety of simple rules for the conduct of monetary and fiscal policy. These include the use of the money supply, the exchange rate, nominal income and the price level as indicators for the adjustment of interest rates. They also include a number of decoupled rules for monetary and fiscal policy. Of these rules, the price rule involving a feedback from the price level to the short-term nominal interest rate stands out. It performs well in the

14

Introduction

face of a variety of unexpected exogenous disturbances - in this sense it is robust. It is also robust with respect to wide variations in crucial parameters. These conclusions apply strictly to the single open economy. In a world consisting of two identical interdependent economies with the same structure and empirical properties as the single economy, the ranking of the various rules changes dramatically. In particular the price rule designed for the single open economy results in global destabilisation if implemented generally. Because the rule relies on the link between the exchange rate and the price level, it triggers off an excessive interest-rate response which turns out to be beggar-thy-neighbour in character. The existence of externalities in the use of monetary policy is, of course, not a new result - Hamada (1976) stressed this feature in his seminal paper on policy interdependence. The main novelty of chapter 8 lies in the formulation of policy in terms of simple, assignment rules and in the generalisation of the earlier analysis of Hamada to models with complex, structural dynamics. Chapter 9 makes the game-theoretic framework of part III more explicit. Interdependencies in a two-bloc world are characterised in terms of a three-person game whose players consist of the two governments and a homogeneous forward-looking private sector. There are then two sets of bilateral relationships to consider. Considering first those between the governments and the private sector, the former may have the possibility of pursuing the reputational, optimal but time-inconsistent policy. In the absence of reputation or some trigger mechanism (considered later), governments must pursue the time-consistent non-reputational policy. Regarding government/government relationships the two countries can cooperate and jointly select their policies to minimise an agreed global welfare loss function. In the absence of cooperation the governments must act independently resulting in a non-cooperative equilibrium. We assume the governments either jointly enjoy or do not enjoy reputation. This gives rise to four possible policy outcomes or 'regimes': the cooperative reputational policy (CR); the cooperative nonreputational policy (CNR); the non-cooperative, reputational policy (NCR); and the non-cooperative non-reputational policy (NCNR). The details of these four regimes and the alternative non-cooperative equilibria for continuous time and discrete time models are given in chapters 9 and 12 respectively. Chapter 9 reports numerical results for a stylised, calibrated continuous time model similar to that in chapter 8. Chapter 10 reports a similar exercise on 'Minilink', a reduced linear two-bloc (USA and the rest of the OECD) representation of the OECD Interlink model. On both the calibrated model in chapter 9 and the more empirically

Introduction

15

based Minilink in chapter 10 we find rather larger gains from cooperation than some other studies. In their pioneering study Oudiz and Sachs (1984) estimated that the gains from cooperation among the G3 countries in the mid 1970s would have been worth no more than 0.5 per cent of GNP for each country compared with the best non-cooperative outcomes. Our results depend critically on the persistence of external shocks and on whether governments can sustain reputational, time-inconsistent policies. With reputation, cooperation (under regime CR) gives rise to significant benefits in the face of permanent shocks. In the Minilink exercise of chapter 10 the reported average welfare gains for the two blocs can be as much as an equivalent 3 per cent increase in GNP. One source of these gains is straightforward. The short-run trade-off between output and inflation coupled with the department from purchasing-power parity which characterises Minilink leads to a strong role for stabilisation policy. When exercised independently this has the familiar beggar-thy-neighbour properties. Forward-looking rational exchange-rate behaviour exacerbates this inefficiency and leads to greater gains from coordination provided that the governments can pursue reputational policies. This contrasts with some earlier studies which ignored reputational effects. In the Minilink studies the gains from cooperation without reputation (regime CNR) are small and sometimes negative. This confirms the result of Rogoff (1985), based on an analytical study, that cooperation may be counterproductive. Our studies suggest a further result. Without cooperation, reputational policies perform badly and may even destabilise the economies. Thus to benefit from reputation requires cooperation and to benefit from cooperation requires reputation. Chapters 9 and 10 also examine the possibility that cooperation with reputation may be sustained as a reputational equilibria supported by a trigger mechanism. There are two aspects of this sustainability problem. The first concerns the incentive for governments jointly to renege on the cooperative reputational policy. This aspect is precisely the time-inconsistency problem for a closed economy as examined by Barro and Gordon. The second concerns the incentive for either government to renege on the cooperative agreement with each other. There are now two trigger mechanisms to consider - that of the private sector and that of each government. As for the single-country case the appropriate trigger mechanism for the private sector is to believe in reputational policies, if they are adhered to, and to believe in nonreputational (time-consistent) policies if reneging occurs. For the governments cooperation continues if no reneging by either country occurs; otherwise a switch to a non-cooperative policy is made.

16

Introduction

In chapters 9 and 10, only infinite punishment periods are considered. A sufficient condition for sustainability is that cooperation with reputation (CR) is superior to the outcome after a permanent switch to any of the three alternative regimes (CNR, NCR and NCNR) everywhere along the trajectory of CR and for both blocs. This may be too strong a condition because it can be argued that if governments renege on each other then their reputation for precommitment with respect to the private sector is also lost. Only switches to the two non-reputational regimes (CNR and NCNR) are then relevant for examining sustainability. In fact for Minilink we find that a bargaining parameter that weights the joint welfare loss of the two blocs can be found which slightly favours the rest of the OECD and which ensures that no incentive to renege occurs for both blocs in the face of permanent demand and supply shocks. This, it should be noted, is a purely deterministic exercise. In a stochastic world sustainability will be further enhanced as the arguments of chapter 4 demonstrate. A number of authors including Rogoff (1987), Levine (1988) as well as Barro and Gordon themselves have stressed the fact there is a multiplicity of efficient solutions which may be supported depending on the length of the punishment period. We only consider infinite punishment periods, but it is of interest to consider how punishment periods can be made endogenous to the model. If we assume that private agents are atomistic there is clearly no prospect for an endogenous, strategic choice of the punishment period, P. The determination of P becomes a postulate about privatesector behaviour which may or may not be regarded as empirically sound. However, large players, such as the two policy-makers, can be assumed to make a strategic choice of P sufficient in length to support the optimal cooperative policy. The assumption about private-sector behaviour is then that they choose the same punishment period allowing the strategic players to take the lead. The use of trigger mechanisms to support reputational policies appears then to be less problematic in the context of internationally coordinated policies, though the manner in which players may choose punishment lengths strategically, weighing costs against benefits, would benefit from being modelled in a game-theoretic way (see al-Nowaihi and Levine (1992) for a general approach to this problem). In the third of this group of chapters, chapter 11, the same analysis is applied to a two-block natural rate model with perfectly flexible prices. Unlike any of the other chapters in the book, this model is sufficiently simple to permit an analytical treatment. Apart from tractability there are two further reasons for adopting this natural rate model. First it is interesting to examine whether and how problems of coordination may arise even in a natural rate world. Second the analysis throws some light

Introduction

17

on the long-run results reported in the Minilink study which is a nonmarket clearing in the short-run but is a natural rate model in the long run. The final two chapters of the book return to the theme of simple assignment rules. Chapter 12 reexamines the issues covered in chapters 9 and 10 regarding the gains from coordination and the sustainability of cooperative agreements but now under the constraint that rules should be simple. In fact the rules examined in this chapter are extremely simple with both fiscal and monetary policy being assigned to inflation and output using only proportional feedback. First, the significant welfare gains from policy coordination reported earlier for optimal rules are confirmed for simple rules of this form. ^Second, we show that there is considerable scope for using agreements in the form of simple rules as a surrogate for more far-reaching agreements on international policy coordination. Finally, we show that simple rules are sustainable; indeed simplicity in policy design may actually improve the prospects of sustainability. The final chapter of the volume examines the 'Extended Target Zone' proposal of Williamson and Miller (1987) for the conduct of monetary and fiscal policy using the National Institute/London Business School Global Econometric Model (GEM). The scheme takes the following form. First, countries should determine a consistent set of real exchange-rate targets, chosen so as to ensure medium- to longer-run current-account equilibrium. Second, countries should choose targets for the growth of nominal demand. These targets should take account of the need to reduce inflation towards zero, of the need to expand demand in the face of a fall in capacity utilisation, and of the need to adjust current-account balances towards equilibrium. Third, differences in interest rates between countries should be varied to limit the deviation of currencies from their target levels, aiming to keep exchange rates within a given band around the target. Fourth, the average level of world interest rates should be varied to stabilise the aggregate growth of nominal demand around the sum of national targets for nominal demand growth. Fifth, national fiscal policy should be varied with a view to achieving national targets for nominal demand growth. Proportional/integral feedback rules consistent with this assignment are applied to the G3 countries (the US, Japan and Germany) and each country is assumed to operate the same feedback rule. Optimal parameter values are computed over the period 1975 to 1986 with the aim of minimising a welfare loss that penalises elevations of capacity utilisation, inflation, government spending and the real exchange rate from desired levels. Fundamental Equilibrium Exchange Rates (FEERs) calculated using GEM are chosen as the desired real exchange-rate paths.

18

Introduction

The results suggest that policy cooperation using this assignment rule leads to substantial Pareto welfare improvements relative to history and, at the same time, reduces real exchange-rate variability. The results therefore lend support to the Target Zone proposal of Williamson and Miller. They also support our conclusion in the survey article (chapter 2) that an increased emphasis on assignment rules for the conduct of fiscal and monetary policy may well offer both an effective and a practical way forward in the evolution of international macroeconomic policy coordination.

I

General issues

1

Macroeconomic policy design and control theory - a failed partnership?

The subject of this chapter is the applicability of control theory to the pressing questions of macroeconomic policy design that face us, both domestically and in the international arena. I approach the issue with some trepidation, mindful of the dangers of claiming too much in the field of policy design. Those were nicely stated in a broader context by that eminent biologist, Sir Peter Medawar (1984), when he wrote: It is not their wrongness so much as their pretensions to Tightness that have brought economic predictions and the theory that underlies them into welldeserved contempt. The dogmatic self-assurance and the asseverative confidence of economists are additional causes of grievance - self-defeating traits among people eager to pass for scientists.

The control theory that has mostly been applied in economics has been taken over from engineering. This theory tells us how best to control a physical system (a space-ship, for example) in order to achieve certain objectives (to reach and return from the moon) by appropriate adjustment of the instruments of control (the rockets and stabilisers of the space ship), subject to the physical laws constraining the behaviour of the system. These control techniques have been applied with considerable success in a wide range of physical and engineering applications - the control of chemical process plants, electronic systems, rockets, to name just three. The problem of control in economics is often posed in similar terms. Thus, in macroeconomics, for example, policy-makers may wish to ensure that the economy progresses, without undue fluctuations in output, unemployment, prices and other key variables, along a course that yields the most prosperous outcome for society, by suitable The Economic Journal, 95 (June, 1985), 285-306. This is a slightly revised version of David Currie's inaugural lecture presented at Queen Mary College on 22 May 1984 and draws on many formal and informal discussions with economists too numerous to acknowledge. 21

22

General issues

adjustment of instruments - interest rates, taxes, government spending and so on - subject to the constraints imposed by the structure of the macroeconomy, as expressed in a suitable macroeconometric model. The essential problem - the design of control rules for the instruments that best meet the preferences of the controller given the structure of the system to be controlled - appears to be common with that already solved in the field of engineering. With techniques available from the engineering literature it is not surprising that economists took these over and applied them to the design of macroeconomic policy. Control methods entered economics quite early on, most notably through the work of Bill Phillips (1954, 1957) at the London School of Economics. But I think it is fair to say that their influence was only fully felt in the 1970s, when computational techniques caught up and the theory of control entered the tool-kit, albeit the specialised one, of economists. In the UK, the QMC based Programme of Research into Econometric Methods was important in hastening this process. By 1976, matters had developed so far that an official committee was established under the chairmanship of Professor Jim Ball, to invesigate the usefulness of optimisation techniques for policy formulation. The report of that Committee, Ball (1978), gave a cautious welcome to control methods as an additional weapon in the armoury of policy formulation. It gave little support to those - and I regret there were some - who naively conveyed the impression that control techniques were the answer to policy-makers' dreams, that all one needed to do was to take the Treasury model, apply control methods, and, after a process in which the preferences of policy-makers would be established, the budget could be read straight from the computer printout. Now there are a variety of reasons why the use of control theory in economics is likely to be less straightforward than in engineering applications, and some of these I shall touch on later. But there is one in particular on which I wish to focus. It is distinctive to economics, and social systems more generally. Essentially it arises from the fact that the analogy between engineering and economic systems which inspired much of the early work on control theory in economics is, quite simply, misleading. Economic systems, like all systems of social interaction, are intelligent in a way that engineering systems are not. In economics, we are concerned with the control of intelligent systems that think about what policy-makers do to them and act accordingly. This distinction is so obviously important once it is stated that it is surprising how little it figures in the report of the Ball Committee. That reflects not on the quality of the report, but rather on the state of thinking about these issues within the profession at the time. It is this feature of

Macroeconomic policy design and control theory

23

economic systems - that the components of the economic system that we wish to control reflect upon and react to economic policy in an intelligent, non-mechanical manner - that has made us rethink our approach to macropolicy-making. In the process, the subject has become much richer, greatly exciting, and, I believe, more useful. I want to try to give a flavour of the subject, of the issues with which we have been grappling and of our attempts to resolve them. Unfortunately it can only be a flavour, for it is a subject that can properly be expressed only in rather forbidding mathematics. Let me give two simple examples of how the control problem is complicated by intelligent private-sector behaviour. My examples concern the two main instruments of policy - fiscal and monetary. My first is the use of tax changes - fiscal policy - to control the economy. This was the standard fare of the annual budget, at least until a recent fetish with the Public Sector Borrowing Requirement overthrew all that. The standard view was that, to raise the level of demand and output and reduce unemployment, the Chancellor should cut taxes. Conversely to reduce overheating - inflationary pressures - he should raise taxes. This policy will indeed work, at least in the short to medium run, in all the current main macromodels, and it operates by regulating the level of aggregate consumption in the economy. Now at the same time, our theories of consumption emphasise the wish of consumers to spread out unevenness in their flow of income so as to achieve a rather smoother pattern of consumption. If we take this permanent income or life-cycle view of consumption as it has come to be known, and I think the evidence supports it, we are forced to think with greater care about the consequences of tax changes. In particular, we must think about whether the tax change is permanent or temporary. A cut in income tax that is perceived to be permanent will increase the future stream of income, and lead to a rise in consumption. But a temporary cut has very limited effects on life-time earnings, and consumers are likely to spread out its benefits over a considerable time. Hence a transitory cut in income tax may do very little to affect consumption. But, if the object of the tax cut is to offset a temporary fall in aggregate demand, this argument suggests that a cut in income tax, necessarily temporary, will not be effective. To engineer a temporary boost to consumption, the Chancellor needs an alternative: thus a cut in indirect taxes, which lowers prices temporarily, will make it advantageous to buy now rather than later. This example illustrates the point that the effects of a tax cut depend very much on whether consumers see it as temporary or permanent, and this depends in turn on what policy is perceived to lie behind the tax cut. My second example concerns monetary policy and the exchange rate.

24

General issues

Suppose one is asked what effect an increase in the money supply has on the exchange rate - a simple straightforward question to which one might think a simple answer should be possible, if we have our wits about us. But I have at least two contradictory views to offer. On one view, a larger money supply implies higher domestic prices, so that a depreciation of the exchange rate will occur, maintaining the relationship between domestic and foreign prices when expressed in a common currency.1 Indeed the overshooting hypothesis suggests that, if prices adjust slowly to the money supply change, the exchange-rate depreciation may be exaggerated, overshooting its final equilibrium and then appreciating back.2 But on an alternative view, the money supply increase will cause an exchangerate appreciation. This is because the foreign exchange market expects the Bank of England to raise interest rates to dampen monetary growth and bring the money supply back onto target. 3 This is the sort of contradiction that gives economists a bad press. But it is apparent, not real, for the arguments apply to quite different circumstances. The first argument rests on the assumption that changes in the money supply tend not to be reversed - the money supply, for example, follows a random walk. In the second, monetary changes are reversed, and the authorities pursue a policy of monetary targets. Two distinct policies underlie the two cases. What my argument illustrates is not that economists can never agree, but rather that to ask how the foreign exchange market reacts to a money supply change is not a well-defined question. The answer to the question depends crucially on how the market expects the authorities to react. Once again the behaviour of the system depends on the perceived policy or control rule of the authorities. In designing policy, we must take account of this dependence. This is a very general proposition that applies in almost every area of macroeconomics. Its consequences are pervasive. It blurs the usual distinction between technical advice on 'how the economy works', on the one hand, and policy advice, on the other. One cannot discuss how the economy works without first specifying a background of what policy is in force; and it is then very natural to question how changes in that policy influence the behaviour of the system. I hope by now that I have persuaded you that there is an extra complication, a new dimension, to the control problem in economics. As in engineering applications, the design of the control rule depends on the structure of the system to be controlled; but, unlike engineering, the structure of the system depends on what control rule is perceived to be in force. To get a grip on this problem, it is clear that we must model the way in which people form their expectations about the future. We therefore need

Macroeconomic policy design and control theory

25

a theory of the economy in which people's expectations about the future enter as explanatory variables; and we need a theory (or theories) of how people think about economic events and form expectations of the future. The natural way to progress is, of course, to use direct observations of expectations drawn from surveys. Unfortunately such data are rather sparse, available only for relatively short time periods, and usually relate only to the very short-term future, not the one, two orfiveyears or longer that are all too often relevant for economic decisions. Cross-sectional panel data on expectations would be an invaluable aid to research in this area. It is therefore regrettable that we are not actively seeking new sources of expectations data - instead Rayner's axing of the general database available to economists offers little encouragement in this. In the absence of satisfactory expectations data, economists have proceeded in one of two ways. The first is to assume that expectations are formed by some rule of extrapolation, more or less complicated, based on the past. Thus inflation may be forecast from observation of past inflation using some type of adaptive or Box-Jenkins forecasting procedure. Conventional methods of this kind may serve quite well for forecasting a variable such as inflation or output over a one- or two-year period provided present developments are not very different from the past. But it serves rather badly if circumstances change markedly - for example, if the price of oil alters dramatically as in 1973 and 1978, or if policy shifts abruptly as in 1979/80. Moreover, it is most unlikely to perform well in financial or foreign exchange markets, where conventional rules of thumb cannot be expected to produce profits. The second approach is what has come to be known as 'rational or consistent expectations'.4 This cuts through the problem of having two distinct models - one for how the economy works and the other for how people form expectations - by assuming that people form their expectations on the basis of knowledge of the true structure of the economy. Thus, when testing an economic theory, we also assume that people's expectations are formed consistently with that same theory, and the quantitative estimates of the model used in forming expectations coincide with those of the model itself. If the theory is correct, this amounts to assuming that expectations are optimal predictors, in the sense of being unbiassed and having minimum variance. Now the assumption that people form their expectations rationally is rather breathtaking in its range, and may remind you of Sir Peter Medawar's dictum that I cited earlier. After all, the experiments of psychologists reveal that most people are ignorant of even the most elementary principle of statistical inference. We find it particularly difficult, so the work of Kahnemann, Tversky and other experimental psychologists

26

General issues

indicates, to make rational decisions where the problem is too openended.5 In so far as they admit the term 'rational', psychologists typically characterise our mode of reasoning as taking the form of bounded rationality. Whether consciously or unconsciously, people limit the range of options to be considered at any particular time. We all, no doubt, can give instances of such behaviour, of course in others, never ourselves. Inevitably, this form of decision-making can be severely sub-optimal, particularly when having to cope with new circumstances that need novel decisions. Despite this, I hope you will not dismiss the assumption of national forecasts too easily. One way in which each of us can come to terms with our lack of even elementary skills in forecasting is to draw on outside advisers. A number of groups forecast key macroeconomic variables regularly, and these are available for only the price of a newspaper. Of course, such forecasts have a bad name - I recall, in particular, Sir John Mason's comparison, in his British Association Presidential address last year, between economic forecasting and weather forecasting.6 Sir John spoke of weather forecasting from the informed position of a former Director of the Met Office. But he overstated, I think, the inaccuracy of macroeconomic forecasts, which have indeed a reasonable success rate.7 Provided that their models are well estimated, and provided the models are allowed to influence the forecasts (both of which conditions I emphasise for they are by no means always met), those who use these forecasts may well behave as though they are making careful forecasts, even though they have no knowledge of economics and statistics. Of course, the extent to which people draw on macroforecasts depends not only on their accuracy, but also on the costs of being wrong, and one might well expect these to be greater in financial markets than in other markets. It is no surprise to find, therefore, that the assumption of rational expectations seems to work much better infinancialand foreign exchange markets than elsewhere. My second defence of the assumption of rationality in forecasting, and decision-making more generally, is that it may be a reasonable aggregate approximation. Even if individuals are subject to bounded rationality, divergences from full rationality at the individual level may cancel out in the aggregate, so that aggregate behaviour is such that it is as if individuals can forecast rationally. This idea has been developed formally in the work of John Cross on adaptive strategies of behaviour.8 He demonstrates that, in a stable environment, adaptive behaviour, based on repeating more frequently those successful strategies and shifting away from those unsuccessful ones, leads to a distribution of individuals around the rational outcome in

Macroeconomic policy design and control theory

27

a wide class of problems. Sub-optimal strategies persist, though, because, in a large population, some ill-conceived strategies will pay-off and some well-conceived strategies will fail in any particular experiment, so that non-rational strategies will be able to survive. This view is helpful since it means that we do not have to accept that each and every one of us forms our expectations rationally in order to apply the theory in the aggregate. It also emphasises the need to understand processes of learning, and the importance of the flourishing literature that is currently integrating the analysis of learning and expectations, as well as relaxing the extreme informational assumptions that underlay the early analysis of behaviour under rational expectations.9 There is, I think, an even more compelling reason for adopting the assumption of rational expectations in policy appraisal, though in this context the term consistent expectations is probably better. In assessing policies under consistent expectations, one is testing them under conditions where their effects are understood. I submit that a good performance under these conditions is a necessary condition for a satisfactory policy. For if a policy performs badly under these circumstances, but well under different ones, it can only be because it works through systematic forecasting errors by the private sector. But since there will be an incentive for the private sector (or its forecasting agents) to alter its forecasting method if it generates systematic error, this is a rather weak and vulnerable basis for policy. A policy that performs badly when its effects are understood must be unsatisfactory. To be sure, we must be sensitive to our uncertainty about model structure, but this can be handled within the framework of consistent expectations by methods of robustness testing to check that our policy advice is not too model dependent. I hope that I have persuaded you that the notion of rational or consistent expectations is useful in the control setting. There is no doubt that it has revolutionised our way of thinking about macroeconomics. This is because, notwithstanding the severe technical problems it poses for analytical and empirical macroeconomics, it provides the simplest possible way of incorporating intelligent behaviour on the part of the private sector, and this is essential for most policy problems in macroeconomics. But initially rational expectations were thought to be destructive for the use of control theory in economics. This was for two quite separate reasons. The first has a long pedigree stretching back to Adam Smith's invisible hand - the notion that perfect markets and intelligent individuals will lead to an optimal outcome, without any need for government intervention. Essentially the private sector can act as controller, making government control superfluous. This appeared in the early work on rational expectations in the form of a class of models in which govern-

28

General issues

ment could influence the real economy only by fooling the private sector. 10 Since rational expectations imply that such fooling cannot be systematic, government could only add extra unsystematic noise to the system, not assist in stabilisation. To be sure, one might object that government may be better informed than the private sector - the Civil Service would hardly be doing its job otherwise - but then the answer is to make public that better information. Stabilisation policy becomes the province of the Press Office, not the Treasury. But this argument will not wash. It requires us to believe that the problems of coordination and adjustment within the private sector are negligible, so that each of us can enter into flexible contracts for wages, financial transactions and other dealings, with clauses contingent upon each possible macroeconomic state of the world.11 Just imagine how long the negotiations between Ian McGregor and Arthur Scargill would last in that sort of world. Consider the earnings that this would give to lawyers in drawing up complicated contracts, and I am sure you will agree that such a world would be severely sub-optimal. We all resent government for some reason or other, just as we curse traffic lights, particularly when they go wrong. But how many of us would really welcome the abolition of traffic lights as the resolution of our traffic problems? But although consistent expectations do not dispose of the control problem they greatly complicate and enrich it. This is because in order to assess the performance of control rules we must take into account their effect on the way the private sector forecasts and behaves.12 There is, moreover, an extra dimension to the control problem. For different control rules alter the informational content of variables such as interest rates and the exchange rates. 13 Intelligent agents make inferences about disturbances to the system from observations of such variables, and their behaviour will depend on these inferences. Policy may operate through these informational channels, as well as through more conventional control mechanisms. For those of a theoretical bent of mind, there is also the possible need for government to intervene to avoid forms of structural instability and chaotic motion that may arise in rational expectations models. 14 The second objection to control with rational expectations was equally erroneous, but highlighted an important problem in the control of intelligent systems which has come to be known as time inconsistency.15 It is a quite general phenomenon in social decision-making, and will be familiar to those who have decisions imposed upon them, even if decision-makers themselves tend to have a blind spot for it. I shall refer to government in what follows, but you may equally well think of management, head of department, examiner, dean, governor - whatever you will.

Macroeconomic policy design and control theory

29

Consider how government should respond to an inflationary disturbance, such as a surge in wages. Before the event, a government concerned with inflation should give the impression that it will not accommodate the wage increase by maintaining aggregate demand and output. For by so threatening that wage rises will result in loss of jobs and unemployment, it may persuade those bargaining over wages to limit their demands. This notion of a threat effect undoubtedly underlies current policy, and is also a plank, though only one of several, in the New-Keynesian policies advocated by James Meade. 16 Before the wage push, therefore, government should be hawkish. After the event, however, the threat effect has served its purpose. Bygones being bygones, the government should consider how best to respond to the wage surge. Under quite general conditions, the optimal response is partially to accommodate the wage surge, so that the effects on unemployment are mitigated, though at the expense of a rise in the price level. The optimal response, it seems then, is to renege on the initial hawkish stance. It is paradoxical that this problem was first analysed in a two-period setting, where, correctly analysed, the problem disappears. For the private sector, if it is bright enough, can figure out that the government will renege, for government has no reason not to do so. The private sector will, therefore, entirely discount the government's hawkish stance. You may be aware of similar chains of reasoning that prove a rational government could not use the UK nuclear deterrent. The only way the government can retain any credible threat in this situation is by appearing irrational in its decision-making. With nuclear war, the world as we know it may end if the threat is called; but fortunately the effects of a wage surge are less drastic. It is therefore much more natural to consider the problem as a continuing game, not in the restricted two-period setting. This simplifies matters. For it is clear that government cannot go on reneging on its commitments and continue to retain its credibility. Yet it is the credibility of government announcements that gives them their usefulness for control purposes. There is, therefore, a choice between two types of policy. The first type, called time-consistent policies, is where the government has no credibility. The private sector assumes that the government will succumb to short-run temptation, and calculates its expectations of the future on that basis. Government then formulates policy subject to this lack of credibility. Thus in my example the private sector calculates that the government will accommodate wage surges, and ignores all government pronouncements to the contrary. In the stylised history of the post-war period, it is suggested that such was the pre-Thatcher state - before the

30

General issues

Flood. Given that no one believes what it says, the best that the government can then do is, indeed, to accommodate. The second type of policy is time-inconsistent. Government has credibility, because it does not succumb to any short-run incentive to renege. Because of this, government policy does influence private-sector behaviour through the private sector's expectations of the future, as well as through more conventional control mechanisms. In our example, a policy of not accommodating wage surges, despite the short run temptation so to do, is time inconsistent. The view is sometimes expressed that time-inconsistent policies are not feasible, because government cannot credibly commit itself to succumb to temptation. I find this hard to accept. It is clear that a government with credibility has a wider range of policy options available to it than if it lacked credibility; and it can therefore expect a better policy performance. 17 The cost of reneging on a well-chosen time-inconsistent policy is that you end up without credibility with a time-consistent policy yielding worse results. Some of the QMC PRISM (Programme of Research into Small Macromodels) Group's results suggest that this cost can be very high. 18 A government that is concerned with more than the very short term has a real incentive to adhere to time-inconsistent policies, and is therefore credible in so doing. Those who feel that the chain of reasoning has now become tortuously theoretical, a feature of much of the game theoretic reasoning of this kind, may be reassured by a more down-to-earth argument. Mrs Thatcher has demonstrated the feasibility of sticking to your guns, with only rather minor wavering in the face of strong temptations to renege. The oftenrepeated slogan TINA - there is no alternative - was recognised, of course, even (or perhaps especially) within government, to have no intellectual basis. But it served to raise the political costs of a U-turn, and hence to make politically credible the time-inconsistent policy that has been pursued. In this sense Mrs Thatcher has done much to restore the credibility of the notion that government can stick to policies through thick and thin. Unfortunately, time inconsistency is not a sufficient condition for a good policy, and Mrs Thatcher has restored credibility to the policy domain by pursuing what I shall suggest later has been a severely sub-optimal time-inconsistent policy, so sub-optimal that there must be time-consistent policies that would have dominated it. But, if such costly strategies can be made credible, a better designed policy should be all the more credible. Now if you expect me at this point to reveal the nature of that better designed policy, I must disappoint you. I have untried prototypes, and many ideas on what not to do, but I have no streamlined policy to wheel

Macroeconomic policy design and control theory

31

on shining and gleaming, fully tested, with knobs and whistles. But what we do now have are control methods available to us much better suited to the problems of policy design in economics. We can devise optimal time-inconsistent or time-consistent policies in rational expectations models, whether in a deterministic or a stochastic setting.19 We can alter the information assumptions in such analysis, permitting us to analyse properly for the first time questions of indicator and intermediate target regimes.20 We can analyse the bargaining, game theoretic aspects of policy that I discuss later. And we have developed experience of the use of these methods that suggests that they are powerful tools for policy appraisal. 21 There is, of course, the worry that our control technology is in danger of out-stripping the capabilities of the models available to us, and that is a concern which I share. After all, in engineering applications of control theory, the laws of motion of the system are usually well known; and, if they are not, some experimentation can usually reveal the salient features for control purposes. In macroeconomics, our understanding is much poorer, and I say that despite being firmly of the view that the area of common ground between macroeconomists is much larger than is generally appreciated. Moreover, experimentation is not an attractive option. To make matters worse, the large macroeconometric models (with hundreds or sometimes even thousands of equations) to which control methods have usually been applied do not necessarily incorporate the latest state of understanding. This is not surprising given their sheer size relative to the staff resources available for their support. A frequently recurring analogy in this area, particularly with those like myself who favour smaller models, is with painting the Forth Bridge. But in view of the large number of equations in these models that are just inadequate whether because they fit very badly, have features that fly in the face of commonsense, have not been updated, or have been imposed in violence to the data - this image does not go far enough. Instead I invite you to think of a Forth Bridge made up in sections of the latest engineering design, while in other places we have sections made of wood, or even, in the case of the monetary sector so crucial to the present government's strategy, submerged stepping stones. Worse still, some would argue, all of this is built across a narrow stretch of river, so that a simpler, smaller design would suffice. Not surprisingly, the heavy technology of control theory reveals all the weaknesses of model construction. Moreover, these models appear more vulnerable in the light of rational expectations, which sheds doubt on the stability of their structure and on the nature of the dynamics that they embody. However, this is far from being a necessary state of affairs - on the

32

General issues

contrary, it is changing fast. Applied econometrics has advanced enormously in the past decade, and good econometric technique is now the norm, rather than the exception. In the past few years, we have had studies which systematically compare the sectors of different models, and which have been most helpful in identifying best, and worst, practice. 22 Increasingly expectations are being modelled seriously, with corresponding gains in understanding - to cite just two examples, work at the National Institute has highlighted the importance of forward-looking expectations in understanding the stock-led recession of 1980/1, and productivity movements in the recent past. 23 1 have no doubt that the next generation of models will incorporate the modelling of expectations quite generally. An important extra element is the creation of the new MacroModelling Bureau at the University of Warwick, funded by the Economic and Social Research Council, which is making the large forecasting models available to the academic community. If the Bureau takes the process of model comparison further and more systematically, as it should, in a few years' time we shall, I think, have better models and considerably greater convergence of model properties. My own guess is that in the process models will become smaller, sleeker and fitter. But policy design cannot, and should not, await these developments. We will never have the perfect model, the state of knowledge in macroeconomics being what it is, and if we did we probably would not agree on it. What we need is an approach to policy design that is ever alert to these uncertainties and differences. This has been by no means so in the past. For example, the control methods that we have applied have often treated the macromodel as a rather precise representation of the economy, instead of being vague and uncertain. In consequence, the resulting control rules may lean far too heavily on quirky and unsatisfactory features of the model. Techniques that avoid these pitfalls are, of course, familiar, but they are computationally expensive, particularly when applied to the very large forecasting models. 24 Moreover, they require knowledge of the system properties of the estimates of the model; and this is simply not available for the large models which are generally estimated separately equation by equation, or at best block by block, rather than as a system. This is yet another reason why the future for policy appraisal may well lie in a new generation of small-scale macromodels, to which systematic robustness tests can be applied. Perhaps a more serious problem has been that control techniques have all too often been used to show what policy performs best in each model separately. This is despite the fact that these so-called best policies may perform rather badly in other models. Consider the problem from the policy-maker's point of view. He has available to him a variety of views of

Macroeconomic policy design and control theory

33

the world represented by each of the models and supported by, usually, highly intelligent and persuasive arguments from each of our very articulate and numerate modelling teams. It is not very helpful for him to know that a certain policy performs well in one model, when it performs badly in another. What instead would be of considerable interest would be to know that a certain policy performs tolerably well in all the models. Such policies might be agreed upon by the different modelling groups despite their important differences in outlook. What the racehorse owners at the Treasury and the Bank of England should be looking for is 'a horse for all courses', not a well-refined pure-bred capable of performing well only in special circumstances.25 At the moment, the sheer diversity of models means that robust policies of this kind are probably not available. No policy could satisfy simultaneously the Merseyside Monetarism of Patrick Minford and the Cambridge Keynesianism of Wynne Godley. But we can reasonably expect to see a narrowing of these differences under the joint imperatives of the ESRC and Warwick as part of the process of model evaluation and selection that I alluded to before. It would not surprise me then if a robust policy, a horse for all courses, could be found. This is part of a more general argument that our advocacy of particular policies should be sensitive to the important gaps in our knowledge. This may seem evident enough, but it is not taken seriously in practice. A standard question for all who advocate particular policies should be: what if the world is different, in plausible respects, from that which you assume? And this approach needs to be formally assimilated into our control theory. Control engineers have led the way in this, motivated by similar considerations, and there is a whole body of design theory drawing upon general, rather than specific, features of structure, that could well guide us in this task. There are some remaining technical barriers here - the complications introduced by rational expectations make it hard to take this body of knowledge over wholesale - but there is no reason to suppose that these problems will not be resolved. If we can systematically and routinely examine the robustness of our conclusions to those features of the control problem about which we are most uncertain, I have no doubt that the quality and usefulness of our policy advice will be greatly enhanced. So far my discussion of the control problem incorporates an important asymmetry. Government is aware that its actions modify private-sector behaviour, and takes this interdependence into account in determining its best course of action; but the private sector does not take into account any dependence of government strategy on its, the private sector's, actions. Were it to do so, so that the asymmetry disappears, we would be in a world of bilateral bargaining, in which the private sector thinks strategically.

34

General issues

The usual justification for this asymmetry is that the private sector is made up of many separate agents who cannot act together. For many applications I suppose that assumption will do; though sceptics can point to the coordinating role of the business lunch, the conformity to fashions of opinion in the City that so strikes the outsider, as well as to the more well-worn example of trade union behaviour. But the sphere in which strategic thinking becomes all important is in the international arena, when questions of international policy coordination between governments come under scrutiny. For much of the post-war era, up to the late 1960s, macroeconomic policy was conducted within the international constraints of the system established by Keynes and White at Bretton Woods. But that system showed increasing signs of strain in the late 1960s, as the adjustable peg mechanism for the exchange rate, entailing discrete adjustments to the exchange-rate parity, became incompatible with the increasing mobility offinancialcapital internationally. In the consequent move to generalised floating in the early 1970s, the rules of the game were swept away. Each country has pursued its own individual macroeconomic objectives with little regard for its international consequences. Although we have seen within Europe the framework for coordination under the auspices of the European Monetary System, the actual degree of coordination has been limited, in part because of the acute pressures from uncoordinated policy in the rest of the world. Now there is a view that these matters are best left to individual governments. That view is well expressed in the work of Milton Friedman, who emphasises the role of the exchange rate in insulating domestic from foreign developments and conversely, and control of the money supply to stabilise domestic developments.26 Such results, it is true, can be derived from a rather primitive international monetarist model of the early 1970s vintage. But once one takes account of the full interdependencies between countries - through prices, real demands, asset prices and the flow of funds - it becomes very clear that such independence does not hold. We cannot sidestep the question of how best to formulate policy in an interdependent world. These issues are currently high on the agenda of debate and the PRISM group at QMC is much concerned with them as part of a broader programme of research under the auspices of the newly formed Centre for Economic Policy Research.27 How then should we address this problem? One approach - the hardnosed realistic one - would be to accept that we are in a world where cooperation is limited, and design the best type of policy that one can expect for the United Kingdom. In other words, we design our policy treating the behaviour of other countries as given. This, it should be

Macroeconomic policy design and control theory

35

noted, is the standard approach to policy design, since almost all our models are of the single open economy, treating the rest of the world as exogenous. If all countries do the same, we arrive (perhaps iteratively) at a non-cooperative solution to the policy problem. The second approach the soft-headed idealist one - is to assume a world of cooperation, and design policy internationally in such a way as to secure the best overall performance, subject to the cooperating members sharing in the resulting benefits. To do this, of course, requires a model of many economies, specifying the interdependencies between them. The difficulty is that the non-cooperative solution, hard-nosed though it is, may yield pretty disastrous outcomes. Let me give an example of this. Consider a government that wishes to stabilise the trend of prices in the economy (it could equally well be nominal income), in the face of aggregate demand or aggregate supply disturbances. It turns out that in a wide class of models it is not very difficult to do that using monetary policy. What one makes use of is the fairly strong and well-established linkage from the exchange rate through to prices, using the fact that import prices figure directly in consumer prices because of the import component of the price index, as well as indirectly through domestic costs and wages. Essentially government responds to rising inflationary pressures by raising interest rates and inducing an exchange-rate appreciation, which acts to dampen the rise in prices. Pursued vigorously enough, such a policy can stabilise variations in prices and other nominal magnitudes fairly effectively. Moreover, some recent PRISM work suggests that one can devise a rule of this kind for targeting the price level that is robust in the multiple sense that it stabilises the system well in the face of a variety of disturbances, wide parameter variation, and model variation.28 From the standard perspective of single-country optimisation, it looks a good buy. But, if all countries wish to reduce inflation and all seek to do so by appreciating their exchange rate, they must fail, simply because of the elementary fact that one country's appreciation is another's depreciation. The aggregate consequence of such policies is an interest-rate war, of the kind that we have seen in the last five years internationally.29 Thus single-country policy design is prone to generate beggar-myneighbour policies. This points to the need to examine policy in the interdependent, global setting, searching for policies that perform satisfactorily in the aggregate as well as being able to cope with countryspecific disturbances that generate divergences between one country and the rest of the world. We have some elementary analysis of such problems, and one or two blueprints for a possible international policy design, notably that of James Meade.30 But we have, as yet, no systematic appraisal of policy in this area. Given its importance in the international

36

General issues

debate, it is perhaps fortunate that this issue is now the object of intense research and I expect the next year or two to see theflowof important and usable research results. But the hard-nosed realist will respond by asking what is the point of devising elaborate policy designs that would work well if all countries adhered to them, but will not be implemented because all countries have an incentive to renege, to carry out a different, free-riding policy. The problem is, after all, like the so-called prisoners' dilemma in game theory: there is a cooperative outcome that is best all round, but each player has an incentive to choose a different strategy, since that seems to improve his payoff irrespective of the strategy chosen by the other players. Yet if everyone acts in this way, all are left worse off. Thus the problem is one of coordination, given the private incentive to renege on cooperative behaviour. Had Keynes and White taken this sceptical view, and with the background of the 1930s there was every reason for them to do so, we would never have seen Bretton Woods and the post-war settlement. Fortunately the problem is not as intractable as it first appears, for we are concerned with a repeated prisoners' dilemma game, not an isolated one. In this context, the question is whether one can devise a set of threats of penalties to be imposed on those players or countries who renege. That sounds an impossibly complicated task. But recent advances in the theory of noncooperative game theory suggest that it may be much more amenable than it appears.31 Rather than attempt to describe and motivate the complicated theorems that have been thrown up in the past few years, let me instead describe a revealing experiment that was conducted several years ago.32 Game theorists were invited to submit computer programs to play a game of the repeated prisoners' dilemma kind that I described earlier, in a league play-off against all other entries. The winner was very simple and familiar - tit-for-tat - you are nice to people until they are nasty to you, and then you are nasty to them until they are nice to you again. On the second round, competitors were invited to resubmit, having the benefits of a comprehensive analysis of the results of the first round. Once more tit-for-tat came out ahead of the other strategies, some of them enormously complex, devious and Machiavellian in design. I find this result, which has been confirmed by subsequent analysis suggested by the experiment as well as other work, most reassuring. The notion that simple, nice strategies of the type that we all commonly use are robust, effective strategies suggests that the cooperation on which all aspects of our social life depend is not as fragile as analysis sometimes suggests. Tit-for-tat's strength lies in its capacity to elicit and reward cooperation in other players - the key to success in games of the repeated

Macroeconomic policy design and control theory

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prisoners' dilemma type. It does this by offering cooperation, retaliating speedily to non-cooperation, but forgiving equally rapidly; and in all this being transparent, clear and predictable throughout. Results of this kind being generated by non-cooperative game theory have important implications throughout social science, in particular in the theories of social institutions and social conventions, as well as in other areas such as biology.33 In the field of international economic cooperation, it suggests that the design of a threat system to sustain a cooperative policy design need not be as hard as at first it seems, and that the crucial question that faces us is rather whether we can devise a satisfactory cooperative policy that copes with the problems of interdependence. The design of a new international order is a formidable task, particularly in the changed balance of forces that now prevails in the world, but it is a challenge to which economists as a profession should, and I believe can, rise. On the basis of the review that I have given of developments in the field of economics and control, you will perceive that I am optimistic about the future. Unfortunately, and this takes me to my final theme, I am not optimistic that these developments will be translated into a more effective economic policy, at least in the foreseeable future. I base this pessimism on the experience of the past few years, when we have seen a policy implemented that flies in the face of what we know, and knew then, and that seems almost wilfully designed to provide an object lesson in how not to manage our macroeconomic affairs. The points that I wish to argue are not new - they have been argued repeatedly, forcefully and at great length both inside and outside Whitehall. But there has been an absence of interest in taking these arguments seriously, to engage in rigorous policy appraisal in their light. Lest I be misunderstood, let me emphasise that it is not part of my case to argue that the government was wrong to embark upon a policy of reducing inflation by sole reliance on monetary and fiscal retrenchment. That view can be argued, and has been argued many times, and I sympathise with it. But it was the government's prerogative to take that decision and make it stick; and it is the electorate's prerogative to judge whether it was right. What I wish to argue instead is this: that had it opened its ears to the technical advice available to it, this government might well have managed its basic strategy much more effectively, with less cost in terms of output foregone, investment and employment. Since 1979, policy has been conducted in accordance with the principles of the Medium Term Financial Strategy, MTFS for short, formally unveiled in the budget of 1980. This laid down target ranges for the growth of the money supply, initially £M3, and a supporting stance for fiscal policy in the form of targets for the Public Sector Borrowing

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General issues

Requirement. At the heart of the strategy was the notion (drawn from Friedman, and developed by Laidler and in the United Kingdom context most notably by Ball, Burns and their colleagues at the London Business School) that the best way to reduce inflation was to do so gradually by means of a phased reduction in the money supply.34 Once a low inflation rate is established, continued targeting of the money supply at a low constant rate was hoped to prevent any resurgence of inflation. Now the conception behind the MTFS - that if one is to embark on a disinflationary programme, it should be gradualist - is a good one. Because of the sluggishness of wage and price inflation in adjusting downwards, for which there is overwhelming evidence, deflation is best administered slowly to smooth the effects on output. This intuition is formally confirmed by the optimal control-based analysis of a number of researchers.35 But while the conception may have been right, the delivery went sadly wrong. For while a phased reduction in the money supply may deliver gradualism in thefinanciallyclosed economy of Friedman's mind, it has a very different effect in the open economy that is the United Kingdom. The error was to neglect the consequences of intelligent behaviour in the foreign exchange market. It was clear that the strict implementation of the MTFS gave the prospect of higher UK interest rates for a run of years, which together with lower domestic inflation made sterling a more attractive investment. Funds flowed into sterling and the exchange rate therefore showed a marked appreciation, resulting in an historically unprecedented loss of international competitiveness. Socalled gradualism inflicted a sudden sustained contractionary blow to the traded goods sector, notably manufacturing, with consequences for jobs and output that we observe today. Matters were made still worse by an inauspicious rise in indirect taxes, proclaimed in the Alice in Wonderland economics of the time as non-inflationary. This pushed up prices, and exacerbated the upward pressure on interest rates and the fall in output. This suggests that the MTFS was a poorly designed, high-risk strategy. In saying this, I am not speaking with the advantage of hindsight. The essence of the analysis was set out in the work of Rudiger Dornbusch (1976) on exchange-rate overshooting. And there were papers circulating widely in both academic and official circles in 1979 and 1980 that spelt out these points most clearly - notably in the symposium organised by Oxford Economic Papers?6 Moreover, the most obvious resolution of these problems - to engage in the popular pastime of rate-capping, but in the context of the foreign exchange market to stem the rise in sterling, or at the very least to relax monetary stringency in the face of a rising exchange rate - these policies had a highly respectable pedigree, with the Swiss Central Bank following such a course only the year before.37

Macroeconomic policy design and control theory

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Equally damaging has been the straitjacket in which the MTFS has placed fiscal policy. A long-standing feature of the fiscal system has been the operation of automatic fiscal stabilisers. By this is meant the tendency of the fiscal deficit to fall as demand rises, and conversely to rise as demand falls, because of the link of revenues and expenditures - income tax and unemployment benefit, for example - to demand. By allowing this effect to operate, fluctuations in demand are dampened, helping to stabilise output movements. But over the period in question, the government has been pursuing an absolute target for the deficit, expressed in terms of the Public Sector Borrowing Requirement. Fiscal stabilisers have therefore been switched off, and in consequence the ride has been bumpier the unusual severity of the stock cycle through which we have just been owes much to this feature of policy. Now it is true that there are problems in operating fiscal stabilisers alongside short-run monetary targets. It is a feature of a great many models, sometimes not always appreciated by their originators, that this policy combination generates stochastic instability.38 This is because the need to finance by bond sales residual budget imbalances arising from demand fluctuations generates a rising level of volatility in interest rates, demand and output. The Treasury model, amongst others, shares this feature. But if that is the reason for this bizarre policy of PSBR targeting, it does not stand up. It would not be hard to devise low-frequency fiscal adjustments that permit automatic fiscal stabilisers to operate while avoiding such instabilities. One might equally look at the many alternatives to short-run monetary targeting that provide assurances about longer-run inflation without emasculating fiscal policy. The consequences of this fiscal conservatism have been severe. With the economy in deep recession, with unemployment well above estimates of the level at which inflation is non-accelerating, the United Kingdom has been running a substantially contractionary fiscal policy. With other EEC countries following the United Kingdom example, it is no surprise that eminent economists should be calling for some fiscal relaxation to help the European economy out of the trap of depressed demand and rising unemployment. 39 But our policy-makers seem set to disregard the strong case that has been advanced for a supply-side friendly policy of demand expansion. But perhaps the most bizarre aspect of the whole strategy is its centrepiece - the targeting of the money supply. It may seem heretical to challenge such ideas - clearly I must be of unsound mind - but let me remind you that we managed without monetary targeting for most of our country's history; and that indeed there has been almost no period in which the money supply has not been free to vary within reasonable limits

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General issues

in response to movements over the business cycle. We are, after all, not interested in the money supply as such, but rather with prices and output. The attention given to the money supply is usually justified in terms of its being an intermediate target or indicator, containing useful information about the future course of prices and output. But no one has shown that the money supply, however one defines it, is the sole variable that should be used in that way, or that it is very satisfactory in that role: supposed relationships involving money break down as quickly as they are discovered. I am not asserting that the money supply, suitably defined, is unimportant - even though we do not have compelling evidence in its favour, we would be most unwise to disregard money supply movements. But my point is that there are many other variables that are important as well. And one of the damaging features of overconcern with the money supply has been the consequent neglect of international considerations, notably the exchange rate. United Kingdom policies have, as a result, been of a severely beggar-my-neighbour character. But questions of this kind are, I am afraid, of little interest to our current masters. Most recently, we observed the Chancellor of the Exchequer dismissing David Hendry's demolition of Friedman's empirical work for the United Kingdom with the lofty words 'I am not interested really in the arcane quibbles of econometricians', and then saying in the next moment 'the basic concept is very clear'. 40 There is no empirical regularity involving money that has stood up for more than a brief time that can justify the current naive approach to policy-making. The latest in a string of fads is the targeting of MO, cash plus bank reserves. The recent study by Barry Johnston - a fine piece of econometrics incidentally, but widely misinterpreted - has been said to be the econometric basis of the reformulated MTFS. 41 Yet we can expect in advance that it will not stand up for long, that it will collapse as the latest victim of Goodhart's Law. The study itself made clear just how much financial innovation - the spread of branches credit cards and cash tills - has altered the demand for MO over the past decade, and this in a period when no one was concerned with MO. How much greater and faster can we expect innovation to be when government is creating an incentive to economise by tracking the cash available to the public and to the banks and with the spread of electronic cash management. If the banks are not busy considering strategies for expanding their business without MO, they are failing as bankers. Can we really expect to control the activities of the financial institutions, amongst the most powerful, innovative and creative of our industries, by devices of this kind? All of this should be a matter of deep concern to us. For these policies, wished on us by those for whom 'the basic concept is very clear', are laid

Macroeconomic policy design and control theory

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at the door of the economics profession when they fail. And those advances in our understanding which I mentioned earlier will, I think, go untapped so long as this intellectual climate prevails. We must insist that those who would experiment with the British economy and our fellow citizens should first subject their favoured schemes to a process of rigorous scrutiny and appraisal against available alternatives. But I am not hopeful that we will see this happen. My title refers to a failed partnership, but left matters open with a question mark. We must, I fear, take down that question mark: the partnership, rich, rewarding and exciting though its potential may be, is foundering on a failure to listen and to learn. Notes 1 This is emphasised by the monetary approach. See, for example, Frenkel and Johnson (1976). 2 Dornbusch (1976), Buiter and Miller (1981, 1982). 3 Currie (1984). 4 See the papers collected in Lucas and Sargent (1981). For excellent surveys, see Begg (1982) and Sheffrin (1983). 5 Kahnemann et al. (1982). On the notion of bounded rationality in economics, see particularly Simon (1982, 1983). 6 Mason (1983). 7 See, for example, the National Institute (1983, 1984), Kmenta and Ramsey (1981). 8 Cross (1983). 9 For models of learning, see Bray (1982), Bray and Savin (1986), and Frydman and Phelps (1983). For analysis of partial information, see Frydman and Phelps (1983), Minford and Peel (1983), Pearlman et al. (1986) and Townsend (1983). See also Barro and Gordon (1983a, b), Kreps and Wilson (1982) and Backus and Driffill (1985a, b) for consideration of credibility and reputation. 10 See, for example, Lucas (1972), Sargent and Wallace (1976), Barro (1976). 11 For the consequences of less flexible contracts, see, for example, Fischer (1977), Phelps and Taylor (1977) and Taylor (1980). 12 Lucas (1976). 13 See Pearlman et al (1986). 14 See Currie (1985a), Begg (1984), van der Ploeg (1986). 15 See Kydland and Prescott (1977), Prescott (1977), Calvo (1978), Driffill (1980), Buiter (1981), Miller and Salmon (1985a). 16 Meade (1982a), Vines et al. (1983). 17 Buiter (1981). 18 Levine and Currie (1987b). What we here refer to as the time-inconsistent policy may be rendered time-consistent if reputations are explicitly analysed (see Backus and Driffill 1985a, b) or if a threat strategy is followed (see Oudiz and Sachs, 1984).

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19 Buiter and Dunn (1982), Chow (1981), Driffill (1980, 1982), Currie and Levine (1983), Levine and Currie (1987b). 20 Pearlman et al (1986), Currie and Levine (1985a). 21 Miller and Salmon (1985a, 1985b); Currie and Levine (1985a, 1985b). 22 See the set of papers from the National Institute dealing with comparisons of models of exports, bank lending, wages and employment; a useful summary is given in Hendry (1983). 23 See Hall et al (1983), Wren-Lewis (1984). 24 Astrom (1970), Holly et al (1979). 25 Chow (1980), Currie and Levine (1983). 26 Friedman (1953, 1968). 27 See Currie and Levine (1985a, b). 28 Levine and Currie (1987b), Currie and Levine (1985a). 29 Currie and Levine (1985a, b). 30 Meade (1982b, 1984). 31 Basar and Olsder (1982). 32 Axelrod (1980a,b, 1981, 1984), Axelrod and Hamilton (1981), for a convenient summary, see Hofstadter (1983). 33 Schotter (1981), Axelrod and Hamilton (1981), Maynard Smith (1982), Dawkins (1982). 34 Friedman (1975), Ball et al (1977, 1979), Laidler (1982). For a recent assessment of the work of Ball et al from a sympathetic background, see Budd and Longbottom (1984). 35 Driffill (1982), Currie and Levine (1983). 36 Eltis and Sinclair (1981). See particularly Buiter and Miller (1981), Artis and Currie (1981), and the suggestively titled paper by Scott (1981). 37 Artis and Currie (1981), Schiltknecht (1981). 38 Blinder and Solow (1973), Currie (1976, 1978), Christ (1979), Currie and Gazioglu (1983), Whittaker and Wren-Lewis (1983), Blackburn and Currie (1984). 39 Buiter and Miller (1984), Buiter (1984a), Layard et al (1984). 40 Hendry and Ericsson (1983), Brown (1983), Treasury and Civil Service Committee (1984). 41 Johnston (1984), Treasury and Civil Service Committee (1984).

2

1

International policy coordination a survey

Introduction

International economic policy coordination is the process whereby 'countries modify their economic policies in what is intended to be a mutually beneficial manner, taking account of international economic linkages' (Group of Thirty, 1988). This is a broad definition, encompassing a spectrum of forms of coordination ranging from the rather limited to the ambitious. At the ambitious end is the Bonn economic summit of 1978, where the Group of Seven (G7) countries agreed to a full-blown package deal on macroeconomic and trade policies (see Putnam and Bayne, 1987). At the more limited end is the multilateral surveillance process carried out by the International Monetary Fund under the Bretton Woods fixed exchange-rate system. Recent years have seen a resurgence of interest in international policy coordination, particularly in the monetary sphere which is the concern of this survey. The first half of the 1980s saw a period when governments were primarily concerned to 'put their own house in order', combating inflation by means of tight monetary policy. In this period, which saw a large and sustained appreciation of the dollar, international policy coordination was out of favour. But, by 1985, concern over the substantial misalignment of the dollar led to renewed interest in monetary coordination, particularly on the part of the USA. The Plaza Agreement of September 1985 was to coordinate monetary policy actions to manage the steady decline of the dollar from its February 1988 peak. A series of G7 summits since then have reaffirmed cooperation over monetary policy. The most significant of these was the Louvre Accord, which agreed to 'cooperate closely to foster stability of exchange rates around current levels' (for a review of this period, see Funabashi, 1988). This set in place a Surveys in Monetary Economics, C. J. Green and D. T. Llewellyn (eds.), Basil Blackwell Ltd.

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General issues

loose arrangement of unannounced exchange-rate zones that has since then influenced policy, particularly amongst the G3 countries (the USA, Germany and Japan), despite particular episodes when these informal zones have been threatened by foreign exchange market pressure. This period of coordination may provide the basis for a move towards a more formalised system of monetary coordination based on exchange-rate targeting (see Group of Thirty, 1988), although many obstacles may impede that development (Currie, Holtham and Hughes Hallett, 1989). These developments have given rise to a number of questions concerning policy coordination. Is coordination desirable in principle? Can cooperative agreements be sustained or are they vulnerable to reneging by one or more of the countries participating in the arrangement. How large are the coordination gains in practice? Does model uncertainty undermine the case for coordination? What is the scope for limited forms of cooperation such as agreements focusing on the exchange rate or on simple rules which assign monetary policy to stabilising specified target macroeconomic variables? In the following, we survey the existing literature on policy coordination, and consider what answers, possibly partial, can be provided to these questions. In section 2, we set out the basic theoretical framework, deriving from Hamada, for analysing the potential inefficiency of noncoordinated policy-making, using an illustrative model that we also draw on in later sections. In section 3, we extent the analysis using the model to take account of issues of reputation and credibility: the recent literature has demonstrated that in an interdependent world the benefits of reputation and coordination are interlinked. Section 4 considers the vital issue of whether coordinated policies are sustainable, and which types of policies are vulnerable to reneging. Section 5 reviews the empirical literature on measuring the gains from policy coordination. Section 6 examines whether these benefits of coordination are lost if there is considerable uncertainty about the way in which the international economy operates. Section 7 considers the benefits from adopting simple rules for the conduct of monetary, and more general macroeconomic, policies. Sections 5 and 7 are based on material presented in Currie, Holtham and Hughes Hallett (1989). One area that we neglect wholly is the developing literature on NorthSouth interactions and policy coordination (see Currie and Vines, 1988). This is an important area, but one that is less crucial for our topic of international monetary cooperation. The field that we have covered is a fast-moving one, where many issues remain the subject of active research. Inevitably, therefore, this survey is a snap-shot, and intermediate in character. The conclusions that we draw in

International policy coordination

45

the concluding section 8 are therefore preliminary, and may be overturned by subsequent research. Reflecting this activity, we seek in the conclusion to point to research issues that would merit greater attention in future work. 2 The inefficiency of non-cooperative policies

This section begins the survey of the analytical literature dealing with international monetary policy issues. In general terms the case for policy coordination arises because of policy spillovers between countries which result in the potential for Pareto-inefficient non-cooperative outcomes. In a series of seminal articles Hamada demonstrates the gains from both full cooperation and from partial cooperation in the form of internationally agreed 'rules of the game' (Hamada, 1974, 1976, 1979, 1985). We examine Hamada's contribution using a simple two-country model which will also serve to demonstrate the more recent contributions to the literature in subsequent sections. Hamada adopts a stylised game-theoretical framework which others have subsequently followed. Each country or bloc is regarded as one entity or 'player' in an international macroeconomic policy game and each has a number of macroeconomic policy objectives such as targets for GDP, inflation and the current-account balance. To achieve these objectives the government of each country has a small number of instruments, say onefiscalinstrument and one monetary instrument. In order to assess policies each policy-maker adopts a welfare measure (or welfare loss function) which penalises deviations of target variables and instruments about desired values. In a two-country world, Hamada then examines how successful the countries would be if they cooperated in pursuit of their objectives. This requires countries to agree to coordinate their policies in an appropriate manner, to minimise a joint welfare loss function. This joint welfare function is a weighted average of the two individual welfare functions, with the weight being determined by relative bargaining power. (It should be noted that this does not require countries to share common objectives: indeed there is nothing to prevent the individual objectives of countries being totally at odds with one another.) At the polar extreme of this framework is independent non-cooperative decision-making. Here Hamada considers two non-cooperative alternatives. In the first, countries act independently taking the actions of the other as given. This gives a Nash equilibrium outcome. For the second alternative, countries still act independently but one country (the leader) anticipates how the other country reacts to its policy. This leads to a Stackelberg equilibrium.

46

General issues

The general approach can be illustrated with the following two-country model. The countries have identical economies and pursue symmetrical objectives. On the demand side the model is given by yt = aiet-a2rt yf = ~ axet-

+ a3y*

(2.1)

a2rf + a3yt

(2.2)

where yt denotes output at time /, et is the real exchange rate measured so that a rise represents a depreciation and rt is the expected real interest rate. Asterisks denote the 'second' country, all variables except the interest rate are in logarithms and all are measured in deviation form about an equilibrium in which output is at its natural rate. The supply side of the model is given by y t = -bxet-

b2rt + b3(nt -

TT?,,)

b2r* + b3(n*-n*d

(2.3) (2.4)

where ifut denotes expectations of inflation nt based on information available at the beginning of period t. Hereafter n%t. Equations (2.3) and (2.4) are Lucas supply curves augmented with real exchange-rate and real interest-rate effects. The former arises in an open economy because a real exchange-rate appreciation (a reduction in et) drives a wedge between the producer and consumer real wage (see, for example, Artis and Currie, 1981). An increase in the expected real interest rate reduces output because it depresses the desired level of capital stock. The model is completed with the uncovered interest-rate parity condition which, in terms of the real exchange rate and the expected real interest rate, implies that = r* —

(2.5)

Q

where e t+Xj denotes expectations of et+\ formed at time t. For the first policy-maker we assume an intertemporal welfare loss function at time t of the form

Wt = i 2 Xi(yt+i - yf + anhil

(2.6)

i= 0

with a similar expression with variables marked with asterisks for the second country. A, where 0 < X < 1, is a discount factor assumed to be the same for both countries. The quadratic function penalises output deviations around a target y and non-zero inflation. The monetary instrument is taken to be the rate of inflation nt. Following Aoki (1981) it is analytically convenient to consider separately the 'aggregate system' and the 'divergence system'. Let yf = \

International policy coordination

47

(jt + yf) and yf = \(yt — yf) with similar definitions for other variables. Then the aggregate demand and supply-side functions are given by y?=-(l-a3yla2r?

(2.7)

and yf—~

(2.8)

b2rf H

whilst the corresponding divergence system is given by (2.9)

yf = (1 + a3)- 1 (flie ( -a 2 '-?) yf=-bxet-

- 6 2 r? + ft3(^? - nt)

(2.10)

eUu

(2.11)

Equating supply and demand gives the reduced form of the model as yf = y a (t? —: n?)

(2.12)

yf = yd(^? - itf*)-&+u e /f=-a 2 - 1 (i -aJyXxf-Ti? )

(2.13)

rf = - (2ax +a2)-1 yd((l + a3)(nf - nf) ^

(2.14) (2.15)

where ya =

(a2-b2(\-a3)yla2b3

yd = (2ax +a2 + (2bx - b2)(\ + a3))~l (2ax + a2)b3 ^ = (1 + a3y\ax

- ydb3\ax

+ ^(1 + a3)))

and et is given by (2.11). Equations (2.11) and (2.12) to (2.15) then express the real exchange rate and the aggregate and divergence values of output and expected real interest rates in terms of the monetary instruments nt and nf and the expectations of inflation and the future real exchange rate. The subsequent analysis will be conducted using these reduced form equations. In order to demonstrate Hamada's analysis we first consider a nonrational expectations version of the model treating expectational variables as exogenous. For convenience we put nQt = nfe = eet+Xj = 0. Then adding (2.12) and (2.13) we obtain yt = yf + yf = W

+ f)nt + (ya - yd)nf)

(2.16)

which highlights the nature of the policy spillovers for this model. We assume that a2 > b2{\ - a3) so that ya > 0 and surprise inflation in (2.12)

48

General issues

has the appropriate positive effect on output. We further assume that 2b\ > b2. Then ya > b3 but yd < b3. Hence ya > yd and inflation in country two has a positive effect on output in country one. There are two channels through which monetary policy in country two affects output in country one. The first is that monetary expansion lowers the real exchange rate of country two and the real exchange rate depreciates (i.e., et rises). This represents an appreciation for country one which tends to increase its output. At the same time the real interest-rate differential for country two decreases which may imply an increase in the real interest rate of country one (although the average rate falls). This tends to decrease the output of country one; but provided that parameter values are as stated this effect will be dominated by the exchange-rate effect. Now consider the regimes examined by Hamada. Under cooperation the symmetry of the problem implies that the governments would minimise a simple average \(Wt + Wf) = W? + Wf where Wa is as in (2.6) with yt and nt replaced by ya and na respectively and Wf is similarly obtained but with y replaced by half the difference between output objectives, i.e., equal to zero. Thus the global welfare loss can be expressed as an 'aggregate loss' and a 'divergence loss'. But again by symmetry Wf = 0. The model is also static so that minimising the intertemporal loss function reduces to a single-period minimisation of y)2 + anf)

(2.17)

which by symmetry gives the cooperative inflation rate as

say. For the Nash non-cooperative solution each country minimises its welfare loss given the inflation rate of the other country. There is no agreement that inflation rates should be equal, though in fact this turns out to be the outcome. Country one then (from (2.16)) minimises its single-period welfare loss zt = i((0int + 92nf -yf

+ an])

(2.19)

given nf where 9X = (ya + yd)/2 and 02 = (ya - y d )/2. This gives country one's reaction function nt = {6\ + a)-'ex(y-62n?)

(2.20)

By symmetry the reaction function of country two is n* = (d2l + ayl0l(y-927zt)

(2.21)

International policy coordination

49

nt*

Figure 2.1 The gains from cooperation and leadership

Thus, solving these two equations, the Nash non-cooperative solution is given by nt = n? =-

a

+ f)y

2a

,NC

- = 71

(2.22)

say (substituting back for 9X and 02). Comparing (2.22) with the cooperative solution (2.18), it is straightforward to show that inflation rates under cooperation are higher than those under non-cooperation provided that ya > yd. Thus non-cooperation leads to the familiar contractionary bias; because some of the benefits of each country's monetary expansion are exported to the other country both are inhibited in using their monetary instrument to the level that they would if the benefits were internalised. These results are illustrated in diagrammatic form in figure 2.1. The loss function (2.19) for country one and its counterpart for country two are families of ellipses in the (n,n*) plane. The bliss point, B l9 for country one which results in zero welfare loss is at nt = 0, n? = 02ly with a symmetric result for the bliss point, B 2, of country two. The reaction

50

General issues

function R of country one is obtained by treating 7r* as parametric and is the locus of points where the welfare curve is tangential to the vertical lines nf = constant. Reaction function R* follows similarly and the Nash solution N lies at the intersection of the two reaction functions. Pareto efficient combinations of policies lie along the contract curve which is the locus of tangencies between the welfare curves. The cooperative solution C is the symmetric point where nt = nf on the contract curve. The diagram can also demonstrate a possible Stackelberg solution to the game. Suppose country one is the leader. Then it chooses a point on country two's reaction function which minimises its welfare loss. The outcome is shown as point S in the figure. Compared with the Nash point N the inflation rate of the leader is less under this regime at the expense of a higher inflation rate in country two. The foregoing analysis serves to illustrate the general approach of Hamada's work. Much of his analysis takes place within a regime of fixed exchange rates in which the objectives of the countries are inflation rates, output and the balance of payments. We return to the question of fixed exchange rates as rules of the game in a later section. Canzoneri and Gray (1985) and Turnovsky and d'Orey (1986) adopt a similar framework to explore the games that may be played by monetary authorities in two structurally identical economies following a common external shock such as an oil price increase. More recently the literature on coordination issues has grown considerably. Three main developments stand out. First whereas Hamada analysed interdependence in terms of a single-shot game, subsequent work considers repeated games or the international policy 'supergame'. We return to this scheme in section 4 where we deal with the sustainabihty of the cooperative outcome. The second development tackles the issues raised by assuming that the private sector forms rational expectations. This in turn raises important questions of reputation which are surveyed in the next section. For the most part the analytical literature is confined to static models. Repeated games introduce strategic time dependence. Of more general interest, however, are models with structural time dependence where the welfare loss in any one period depends on past actions as well as on actions taken in that period. The games now become dynamic which introduces greater complexity and, in the main, precludes neat analytical results. The principal benefit of proceeding from repeated to dynamic games is that the theory can be applied to empirical models which generally have high-order dynamics. This work is surveyed in section 5.

International policy coordination 3

51

Reputation

The model employed in the previous section followed that of Hamada in that expectations (explicitly in our model and implicitly in Hamada's model) are non-rational. Introducing rational expectations means that the 'reputation' or 'credibility' problem needs to be addressed. Notions of 'reputation' which relate to private-sector beliefs regarding future government policy are now commonplace in the macroeconomic literature. At a more informal level these ideas frequently enter into discussions of policy effectiveness especially relating to the disinflationary fiscal and monetary stances pursued by OECD countries in the early 1980s. At the theoretical level there has been much recent progress in developing and clarifying the issues (see the survey by Persson (1988) which emphasises reputation in macroeconomic public finance and by Levine and Holly (1989) which focuses on reputation in structurally dynamic models). What might be called the reputation problem was first highlighted by Kydland and Prescott (1977). They examined models where private agents are forward-looking so that future government policies, if believed, can affect the present. The problem is that with the passage of time-optimal policies formulated by minimising some welfare loss function become sub-optimal. The term time inconsistency is used to describe this property. If governments are able to make binding commitments to their ex ante optimal policy, time inconsistency would not be a serious problem. In the absence of some institutional arrangements which, in the words of Kydland and Prescott, makes it a 'difficult and time-consuming process to change policy rules', an incentive to renege on the time-inconsistent policy occurs. This creates the reputation problem - the private sector with information of the government's optimisation problem can anticipate future reneging so that time-inconsistent policies lack credibility. The only credible policies, or so it would appear, are those which are time consistent. These unfortunately can be severely sub-optimal. The main question posed by the literature is whether the ex ante optimal or 'ideal' policy can be made self-enforcing and therefore be sustained in the absence of binding commitments. In the Barro and Gordon (1983b) policy game with complete information, it is shown that the policy-maker's concern for his or her reputation for precommitment can, in some circumstances, sustain the ex ante optimal policy or at least policies far superior to those which are time consistent. The literature cited is concerned with the game between a single government and the private sector. International policy questions intro-

52

General issues

duce a new set of strategic relationships; those between many policymakers representing countries of blocs. To explore how reputation transforms the nature of the international policy game let us consider the rational expectations variant of our model which in reduced form is given by equations (2.12) to (2.15). First consider cooperation with governments agreeing to minimise an average of their welfare losses jointly. As before the global single-period welfare loss function is given by if)

(2.17')

and aggregate output is given by a Lucas supply curve yf = ya(nf - nr)

(2.12')

The solution to this cooperative regime is identical with the closed economy problem considered by Barro and Gordon (1983b). The best outcome or the 'ideal rule' is zero inflation. Then a rational expectations solutions is nf = nfe = yf = 0 which by symmetry also holds for the inflation rates and outputs of each individual country. However given expectations of zero inflation there exists a temptation on the part of the policy-maker to choose a non-zero inflation obtained by minimising (2.17') treating private-sector expectations nfe as parametric. This results in (ya + a)

^;

'

yt>0

(3.1)

If 7iae = 0 this results in the cooperative inflation rate (2.18) obtained for the non-rational expectations model. However, the private sector is assumed (in a rational expectations setting) to know the nature of the policy-makers' calculations. It will therefore form expectations in accordance with (3.1). This leads to

Thus output remains at the natural rate but inflation is higher. As the problem has been posed, the ideal policy nf = 0 lacks credibility because the private sector can anticipate that there exists an incentive to renege to n? given by (3.1) (with nfe = 0). Credibility may be achieved however if the policy-makers enjoy a reputation for precommitment. This may be achieved through one of two mechanisms. The first is through some binding institutional constraint which forces policy-makers to precommit. The second is the result of a private-sector expectational mechanism whereby they only believe in the higher 'discretionary'

International policy coordination

53

Table 2.1. Four symmetric regimes Relations between government Cooperation (C) Relations between governments and private sector

Non-cooperation (NC)

Reputation (R)

CR

NCR

Non-reputational (NR)

CNR

NCNR

inflation rate for some punishment period. We return to this question in the next section. Whether credibility is achieved by postulating a legal constraint or by assuming an appropriate private-sector response to reneging we shall refer to the resulting policy as 'reputational'. Where neither mechanism exists or is ineffective the only policies open to the policy-makers are then 'non-reputational'. In analysing cooperation between governments in a two-country world, we can therefore distinguish between reputational and non-reputational policies. Allowing either government to pursue reputational or non-reputational policies gives rise to eight possible 'regimes'. Of these, four are symmetric and the other half are asymmetric in that only one country enjoys reputation. The four symmetric regimes are shown in table 2.1. Of these four regimes, the cooperative reputational (CR) policy cannot be improved upon. We have seen that nt = n* = 0

(3.3)

for regime CR whilst

say, for cooperation without reputation (CNR). Consider next the two non-cooperative regimes, NCR and NCNR. Non-cooperation with reputation (NCR) can be dealt with in a straightforward fashion. If governments pursue reputational policies they do not spring inflation surprises. But since anticipated monetary policy is neutral in our natural-rate model, there is no other mechanism by which it can have real effects. Thus, the chosen inflation rate must be zero. In other words, we have that for policy NCR, both countries' inflation rates are zero. Finally consider non-cooperation without reputation (NCNR). Putting

54

General issues

yt = yf + yf in the single-period welfare loss of country one and substituting yf and yf given by (2.12) and (2.13) gives

zt = Winf

- nf) + y\yf - *f) - &t+ i., - yf + an?)

(3.5)

Country one acting independently and without reputation minimises zt given (3.5) with respect to nt taking nf and all expectational variables as given. This leads to an inflation choice u

nt= -

2a

y)

(y? + yf-&+i,,-y)

(3.6)

(noting that nf = \(nt + nf) and nf = \(nt - nf)). For country two a similar result holds with the sign on eQt+ 1>r reversed. In a rational expectations equilibrium n* = nt and nfe = nf. Thus yt = yf = yf = yf = 0 and adding the two equations for the two countries gives (ya + / ) * + **

y

Thus by symmetry

x>+nf =JL 9= NCNR

i^~ *

^ ^

(3 8)

-

say. The important feature of this result is found by comparing 7rCNR given by (3.4) with ;r NCNR given by (3.8) we have TT NCNR < TTCNR a

d

(3.9)

provided that y > y which holds given our assumptions about parameter values in the original structural model. Thus we have shown that cooperation without reputation may be counterproductive, a result first shown by Rogoff (1985) and confirmed by subsequent work (for dynamic models) by Miller and Salmon (1985b), Levine and Currie (1987a) and (for a static model) by Canzoneri and Henderson (1988). The intuition behind this result should be clear from our model. For ya > yd part of the output gains from surprise inflation are exported through the depreciation of the exchange rate. This reduces the incentive for a country, acting individually, to engage in inflation surprises. With cooperation all gains are internalised and the incentive to spring inflation surprises increases, leading to a higher level of inflation in the rational expectations equilibrium. Analogous to the Rogoff result that cooperation without reputation may not pay is the possibility that reputation without cooperation may also be counterproductive. In other words, regime NCR may be inferior

International policy coordination

55

to regime NCNR. This result has been shown by Oudiz and Sachs (1985) and Levine and Currie (1987a). Both 'paradoxes' are discussed in Canzoneri and Henderson (1988). In the essentially static model of this section regime NCR in fact yields zero inflation for both countries as opposed to non-zero inflation for NCNR. Thus reputation without cooperation does pay in this case. However, in the models of Oudiz and Sachs and of Levine and Currie wage/price sluggishness is featured. The resulting dynamic models are then very prone to instability under regime NCR because, with reputation, governments are more encouraged to manipulate the exchange rate to combat inflation using fiscal policy to avoid excessive output loss. This results in self-defeating competitive appreciations of the exchange rate which for some combinations of model parameters and welfare criteria can actually destabilise the economies. To summarise this section, we have discussed four 'policy regimes' cooperation with and without reputation (CR and CNR) and noncooperation with and without reputation (NCR and NCNR). For our illustrative model we have shown that the optimal choice of monetary policy gives the following inflation rates for each country 0 = nCK

= TTNCR < TT NCNR < TTCNR

(3.10)

In other words, reputation without cooperation does pay (and in fact renders cooperation unnecessary) but cooperation without reputation is counterproductive. Other work employing dynamic model has produced examples where reputation without cooperation may also be counterproductive. The outstanding issue that remains is how either cooperative or reputational policies may be sustained given the incentives to renege. This is the subject of the next section. 4

Sustainability

There are two aspects of the sustainability problem to be considered. The first is whether the two countries will honour the cooperative agreement with each other. The second is the credibility problem familiar in the reputation literature: will governments cheat against the private sector? The sustainability of the 'ideal policy', cooperation with reputation, must address both forms of reneging. If supranational bodies exist which can legally enforce precommitment by governments then the ideal policy will certainly be credible and sustainable. The outside agency can then prevent governments from reneging on each other or on the private sector. This approach is identified by Canzoneri and Henderson (1988) as a 'loss of sovereignty' and is

56

General issues

contrasted with a form of sovereign policy-making whereby countries 'coordinate' rather than 'cooperate' on an agreed outcome (such as zero inflation in our example above) and employ trigger mechanisms to enforce that outcome. The distinction between coordination between sovereign states and cooperation enforced by supranational institutions is a useful one. However both outcomes require agreement on some jointly chosen policies which in turn requires agreement on the appropriate global welfare measure. The process by which countries bargain to arrive at a cooperative of coordinated agreement can be modelled in a number of ways (Hughes Hallett, 1986b). For identical economies, welfare criteria and shocks, the outcome of the bargain must be symmetrical as in our example. In what follows we shall continue to identify cooperation with mutually agreed policies (with or without some outside monitoring agency) and non-cooperation with independent policy-making. Trigger mechanisms which may enforce efficient outcomes in the absence of legal constraints were introduced into oligopoly theory by Friedman (1971, 1977) and employed in a macroeconomic context by Barro and Gordon (1983b). To illustrate the concept first consider the Hamada analysis using the non-rational expectations model. The cooperative inflation rate nc = yay/(ya2 + a) (see (2.18)) is greater than the non-cooperative, Nash inflation rate nNC = (ya + yd) y(ya(ya + yd) + 2tf) (see (2.22)). But precisely because nt = nf = nc is not a Nash equilibrium there exists an incentive to renege on this cooperative agreement. Given that country two is sticking to inflation nf = nf, country one can improve its welfare by reneging to an inflation rate on its reaction function (2.20) given by nt = (0? + a)-l0x(y

- 62nc) = TTREN c

(4.1)

say. A little algebra shows that TT < n so that given a cooperative agreement to pursue inflation rates nt = nf = nc each country will have an incentive to lower its inflation rate and free-ride on the higher inflation of the other country. In a one-period game both countries will switch to inflation rates along their own reaction function and the equilibrium outcome will be as in the familiar prisoner's dilemma, the Pareto-inefficient non-cooperative inflation rate 7CNC. How then can the cooperative inflation rate be enforced? The answer is to consider repetitions of the single-period game or in other words to consider a repeated game with an intertemporal welfare loss given by (2.6). Suppose that country one employs the following trigger mechanism strategy. REN

International policy coordination nt+i

= nc

if

^+.

= 7rNc

if

n* = nc U t 7

tnc

C

= 7l

57

i=l,2,...oo

i=i92,...P

(4.2)

I = P + 1, i> + 2, . . .oo

What this trigger mechanism says is that country one abides by the cooperative agreement if country two does likewise. If country two reneges then in the next period country one switches to the Nash noncooperative policy for P periods and subsequently cooperation is restored. An identical strategy is employed by country two. Since in a Nash equilibrium both countries are on their reaction functions and hence are acting in an optimal way the threat is credible. The equilibrium in which both countries pursue strategies described by (4.2) is then said to be subgame perfect. But will the outcome be the cooperative inflation rate 7rc? To answer this question we need to consider the one-period gains from reneging or the temptation and compare this with the costs of reneging or the enforcement which arises from the trigger strategy. Consider the single-period welfare loss for country one given by zt which we write as zt=f(nt9n*)

(4.3)

The temptation for country one is then given by temptation = z c - z REN

(4.4)

where zc=f(nc,nc) and z REN =/(7r REN ,7r c ) with rcREN given by (4.1). The temptation in other words is the welfare gain for country one from switching from nt = nc to nt = 7iREN assuming country two still honours its commitment to n* = nc. This gain lasts for only one period after which the trigger mechanism (4.2) comes into operation. Country two now switches to nf = n™0 in which case the optimal inflation for country one is nt = 7rNC too. The costs of reneging are then the increased welfare loss discounted for P periods or enforcement = (z NC - z°)(X + I2 + . . . I*)

(4.5)

where z NC =/(7r NC ,7r NC ) and X is the discount factor. The cooperative inflation rate is sustainable if enforcement exceeds temptation in which case there is no incentive to renege. From (4.4) and (4.5) this requires (z NC

_ Z C } * 0 ~ ^) 1 A

>ZC_

using X + X2 + . . . + kp = 1(1 -

Z

58

General issues

Two extreme cases can now be distinguished. The first (assumed by Friedman) is to assume an infinite punishment period (P = oo). Then the sustainability condition (4.5) becomes X

ZC — z

since ^ - ^ 0 as />-> oo (assuming X < 1). From (4.7) we arrive at Friedman's balanced temptation theorem: there exists a discount factor X sufficiently close to unity for which enforcement exceeds temptation and nt = nf = nc is sustainable. This follows because the left-hand side of (4.7) can be made as large as is necessary by making X approach unity. The other extreme is P = 1 in which case (4.6) becomes (4-8) Since X < 1 (4.8) requires that the single-period gains from reneging z c - z R E N will not exceed the single-period gains from cooperation z N C - z c . There is no reason why this condition should hold in general so there may be particular combinations of models and welfare criteria for which (4.8) can never hold. This does not mean that all is lost, however. Following Barro and Gordon (1983b) for this case we can always find an intermediate inflation rate n* between nc and 7rNC for which the temptation to renege is just exceeded by the enforcement effect. One feature of the trigger mechanism approach to sustainability should now be apparent. A number of authors including Rogoff (1987), Canzoneri and Henderson (1988) and Levine (1988) as well as Barro and Gordon themselves have stressed the fact that there is a multiplicity of efficient solutions which can be supported depending on the length of the punishment period. This may not be a serious problem for countries acting strategically because one can envisage trigger strategies being chosen (possibly by a supranational agency) to be sufficiently damaging (i.e., with P sufficiently large) to support the cooperative equilibrium. However if we turn to the rational expectations model and consider cooperation with reputation then the choice of punishment period by the private sector becomes more problematic. Now consider the rational expectations case. The question to examine is whether cooperation with reputation (policy CR) can be supported by appropriate trigger mechanisms by both the governments and the private sector. Suppose that the governments operate a trigger strategy as before except that the non-cooperative equilibrium may now be with or without reputation (NCR and NCNR). Following Barro and Gordon the private

International policy coordination

59

sector adopts a trigger mechanism with respect to its beliefs about future policy. In particular for country one ft*t+ i,t = ft _e _ _NCNR

=

i f ftt *f

' + ' ' ' = TTCR

ft , 7T CR

'

/ = 1, 2 , . . . oo 1 = 1 2 P

( 4 9^

/ = p Y l , P + 2,...oo

where 7r?+/,, denotes expectations of inflation in period t + i formed on the basis of information available at time /. Clearly, in our symmetrical model an identical mechanism holds for country two. It is now clear that the appropriate choice of non-cooperative equilibrium for the government's trigger strategy is 7i NCNR . Reputation and cooperation both require a commitment to policy CR. If reneging occurs both reputation and the cooperative agreement between governments break down, when the two trigger mechanisms operate simultaneously. The sustainability condition (analogous to (4.6)) now becomes (z NCNR - z CR ) A ( 1 " k*> > z CR - z REN 1 — A

(4.10)

For this case we shall pursue the matter further. Recall that nCK = 0 and ^NCNR _ ^ a _j_ y^yj2a. The one-period reneging inflation rate assumes that nf - nfe = nf = 0 and can therefore be obtained from the reaction function for the non-rational expectations model (4.1). Thus, if we put ^REN

=

01 J?

putting nc = 0 in (4.1). The corresponding single-period losses are given by

z C R = iF2

(4.12)

Z NCNR = 1(\

and Z REN

^"y

< ZCR

Hence on substituting in (4.10) the sustainability condition becomes ^(1"^>

^

(4.15)

It emerges that factors which help sustainability are: a discount factor close to unity, a long punishment interval and a high value of ya 4- yd relative to a. The intuition behind the first two effects is obvious but the

60

General issues

last point is less so. The expression (ya + yd) captures the output benefits of surprise inflation for country one taking all expectations and country two's inflation as given. As ya + yd increases the benefits of surprise inflation increase but so does the non-reputational inflation rate TT NCNR . Thus both temptation and enforcement increase with ya + yd. For this model and choice of welfare loss function the latter dominates so that high values of ya + yd relative to a helps to achieve sustainability. Suppose we assume that a 1 per cent increase in output is associated with a 2 per cent inflation surprise (on an annual basis). Then ya + yd = 0.5 and if we put a = 1 (4.15) becomes >0.94

(4.16)

For the least effective trigger strategy, P = 1, this becomes k > 0.94 or the discount rate must be less than around 6 per cent a year which is not implausible. At the other extreme P = o° (4.16) gives k > 0.48 which is a very lax restriction. The prospects for the trigger strategy supporting the cooperative policy with reputation, zero inflation, seems good at least for this model and choice of welfare criteria. As we have mentioned, an important shortcoming of the trigger strategy approach is that the choice of punishment period is indeterminate. This is particularly serious for the private-sector trigger mechanism if we assume private agents to be atomistic. In this case there is no prospect for a strategic choice of punishment period, P. The determination of P becomes a postulate about private-sector behaviour which may or may not be regarded as empirically sound. However, large players such as the two policy-makers can be assumed to make a strategic choice of P sufficient in length to support the optimal cooperative policy. The assumption about private-sector behaviour is then that they choose the same punishment period allowing the strategic players to take the lead. The use of trigger mechanisms to support reputational policies then appears to be less problematic in the context of internationally coordinated policies, though the manner in which players may choose punishment lengths strategically, weighing costs against benefits, would benefit from being modelled in a game-theoretic way. This in turn may require models with more explicit microfoundations as are found in Kehoe (1987) and van derPloeg (1988). We conclude this section by briefly mentioning two developments in the use of trigger mechanisms to establish sustainability. The first is the generalisation of the preceding analysis to structurally dynamic models with price/wage sluggishness, lagged responses of demand to competitiveness and so on. The main changes lie in the added complexity of

International policy coordination

61

reputational and non-reputational optimal policies and that the sustainability condition becomes time-varying and must be examined along all possible trajectories of the regime to be sustained (see Levine (1988) for a single-country treatment and Currie, Levine and Vidalis (1987) for an application to international policy coordination between two blocs). The second development is to assume that stochastic shocks hit the economy of either a demand- or supply-side character. The principal conceptual change is that temptation becomes stochastic in character so that, for unbounded shocks, it is impossible to achieve sustainability for all realisations. However in a dynamic context enforcement, which now becomes the expected costs arising from the trigger strategy, also increases so that on average sustainability is improved but for some exceptional shock it breaks down. If the probability of this occurring is small it is then possible to assert that the ideal policy is an 'approximate' equilibrium in the sense used by Radner (1981). Canzoneri and Henderson (1988, 1992) for repeated games and Levine, Currie and Gaines (1989) for dynamic games apply this analysis to the policy coordination problem in a stochastic environment. 5

Policy coordination: does coordination pay?

In this section, we turn from theoretical analysis to consider the measurement of the benefits (or costs) to be derived from the coordination of monetary policy. In so doing, we broaden the areas of interest to consider also the benefits of monetary and fiscal policy coordination, not monetary policy alone. This is for the practical reason that most empirical studies of international policy coordination consider the benefits of coordinating both monetary and fiscal policy. The approach to assessing the empirical gains from international policy coordination is to use the available international macroeconometric models. These models have the advantage of incorporating many important features of macroeconomics which the simplified models in the theoretical literature ignore such as high-order structural dynamics and asymmetries in national economies. The empirical importance of asymmetries in the economic structures of the USA and Europe is highlighted in Karakitsos (1988). By comparing the outcome under cooperative decision-making with that under a non-cooperative benchmark, the benefits of coordination can be assessed. Since models are imperfect representations of the real world, the issue of model uncertainty is clearly crucial, and we address that in the following section. Empirical studies which evaluate the potential gains from policy coordination have generally found the benefits to be significant but not large. In

62

General issues

their pioneering study, Oudiz and Sachs (1984) estimated that the gains from cooperation among the G3 countries in the mid 1970s would be worth no more than 0.5 per cent of GNP to each country, compared with the best non-cooperative outcomes. Later studies have suggested that the gains from coordination among the OECD economies may be somewhat larger. In a more general analysis which allowed for dynamic decision-making, Hughes Hallett (1986a, b, 1987a) found that the gains became larger - between 0.5 per cent and 1.5 per cent of GNP for the USA, EEC and Japan. More recently Canzoneri and Minford (1986), Minford and Canzoneri (1987) and Currie, Levine and Vidalis (1987) have also suggested relatively small gains in the absence of major shocks, based on calculations from versions of the Liverpool and OECD models for the USA and the EEC or OECD respectively. However, those results turn out to vary significantly with the size and persistence of external shocks and the perceived reputations of the governments concerned. Persistent shocks and the existence of 'reputation' appreciably increases the relative value of coordination (Currie, Levine and Vidalis (1987)): governments with reputation may derive very large benefits for cooperation in the face of permanent shocks. There has been little work on the likely distribution of cooperation gains between countries. Oudiz and Sachs (1984) found gains distributed roughly 2:1 in favour of Germany relative to the US for two different econometric models. Hughes Hallett's (1986b) study of the USA and EEC in the mid 1970s using a wide range of bargaining models suggests gains distributed 2:1 in favour of the EEC, corroborating this finding. Later work showed this result to be somewhat sensitive to alternative types of exogenous shocks (Hughes Hallett, 1987b), although in no case was the position of the EEC as main gainer overturned. Hughes Hallett, Holtham and Hutson (1989) examine surrogate cooperation in the form of agreed exchange-rate paths. They find that the gains to be asymmetrically distributed among the G5 countries in the late 1980s and that it is extremely difficult to find ways of improving the lot of those countries which benefit least under coordination. These are important but awkward results because they suggest that, whatever the overall gains, (i) it will be hard to secure and maintain a coordination agreement in the face of significant uncertainties, and (ii) if those who make the gains and those who shoulder the burden of adjustment are different sets of people, there are going to be political difficulties in securing any agreement in the first place. However, it may well be that these distributional problems can be reduced by a hierarchical approach to coordination, focusing first on coordination amongst the G3 countries and then subsequently within regional groupings.

International policy coordination

63

These results have been drawn from empirical macromodels. They are not inconsistent with a separate strand of the literature which has sought to examine these findings from a theoretical vantage point, using small demonstration models in which each economy is represented by just a few simplified equations. Typically these equations restrict us to a world of two identically symmetric economies with either no dynamics or steadystate dynamics, and in which the policy responses are known with certainty and there are no information innovations. These limitations reduce the interest of the resulting research findings. Nevertheless these simplified models also suggest that the gains from coordination are likely to be fairly small, but not insignificant; see, for example, the highly stylised models of Oudiz and Sachs (1985), Currie and Levine (1985b), Levine and Currie (1987a) and Miller and Salmon (1985a) and the simple, but estimated models of Taylor (1985), Carlozzi and Taylor (1985) and Sachs and McKibbin (1985). The gains of coordination relative to non-coordination may well be substantially smaller than those of efficient non-cooperative policies over strategies which ignore predictable policy changes abroad. Thus, coordination in the sense of information exchanges, rather than detailed coordination across all variables, may supply part of the improvements available from policy coordination. If this is so, an important function of international fora for policy discussion is the exchange of information between policy-makers concerning policies and the state of their economies. Interestingly, this conclusion can hold even when the information exchanged was found to contain prediction errors (Hughes Hallett, 1987b). It has also been confirmed in a series of experiments carried out by Minford and Canzoneri (1987), using a very different model of the G7 countries over three different episodes in the 1980s. Thus information exchanges do appear to be a key part of the coordination process, irrespective of the model or time period. One might suppose that the wider the range of policies reviewed, the greater are the benefits of information exchange to the decisions subsequently taken (Bryant, 1987). Certainly prior consultation would alert policy-makers to potential and self-defeating conflicts, such as incompatible exchange-rate or trade balance targets which would lead to competitive appreciations or depreciations, or inconsistent fiscal and monetary programmes. Prior consultation could also help policy-makers avoid any losses due to conditioning their own decisions on erroneous information about other policy-makers' intentions (e.g., what priorities they have, what target paths they aim at, what model they use for policy selection, etc.). Whatever the difficulty of predicting the true state of the world, errors due to mistakes made about the information base being used by other decision-

64

General issues

makers should be avoidable and there is no point in adding unavoidable (or genuinely random) errors. Information exchanges which take place through regular consultation may be routine, but we should not be surprised if they make significant contributions to improving policy choice. None of the studies of policy coordination reviewed so far considered exchange rates to be a target of policy, either in their own right or as a surrogate target. Yet much of the recent policy debate has been concerned with exchange-rate management, with the aim of either stabilising exchange rates or of making controlled realignments. This may serve as a means of improving relative coordination between countries. The Hughes Hallett (1987c) study points out that, although exchange rates will then be included among the targets during policy selection, they can either be included in the associated objective function evaluations (in which case exchange-rate stability is a target in its own right) or excluded from those evaluations (in which case exchange rates are just an intermediate target, instrumental in securing improvements elsewhere). If the former holds, the gains from coordination appear larger than before, about 3-6 per cent of GNP as estimated across seven multicountry models. If exchange rates are treated merely as intermediate targets the gains are significantly smaller, much the same as in the earlier literature cited above. Later work which looked at the exchange-rate targeting issue in more detail for the G5 countries (Hughes Hallett et ah, 1989) generated the same findings: gains at 0.7 per cent to 1.7 per cent of GNP, inclusive of exchange-rate stability, were larger than others had estimated, but those figures were reduced by 0.3 per cent for each country when exchange rates were just taken as intermediate targets. The point here is that an exchange rate is a shared variable the domestic impact of which is the same whether exchange-rate changes originate at home or abroad. In contrast, most other variables have impacts which are significantly smaller internationally than domestically. The need to limit exchange-rate spillovers is therefore greater than for other linkage variables, and the potential gains for coordination appear correspondingly larger when that is done. In fact, coordination itself appears to generate extra stability in the target variables and more continuity in the policy interventions - a characteristic which had already been noted in nonexchange-rate exercises (Hughes Hallett, 1986a). The explanation, according to Cooper (1969), is that ignoring independence within or between economies leads to oscillation and overshooting because the implied assignments (country by country) ignore the international side effects of decisions taken in the domestic interest until after they have appeared. Corrections have then to be applied. These are the costs of

International policy coordination

65

imposing policy assignments, in this case one to one by countries rather than for specific variables on a dynamic system. Target stability and policy continuity are therefore particularly sensitive to exchange-rate management. Success with shared targets, whether treated as intermediate or not, requires a measure of coordination of how they are controlled, and also some consensus about the target path which they should pursue. If that is missing, countries will inevitably waste policy power by pushing against each other in a vain attempt to achieve the impossible. This follows from the n - 1 problem: there are one fewer independent exchange rates (or current accounts) than independent policy-makers. However, it does not follow that any agreed target path is better than none. Hughes Hallett et al. (1989) find that, among the G5 countries, jointly specifying exchangerate target paths independently of other objectives is better than moderate disagreement about what that target path should be as long as the suggested targets are not inconsistent with other objectives. Hence the problem appears to be one of choosing an appropriate set of target paths, not of securing precise agreement on some path(s). Whether this conclusion is true of other shared variables, such as the current account, is an open question. 6

The consequences of model uncertainty

The empirical results discussed up to now depend on the particular model that is used in the experiment. Important questions follow: how robust are the estimates of cooperation gains to the choice of model? Suppose that policy-makers design cooperative policies using one particular world model. How robust are the resulting policies if the 'correct' model turns out to be different? The extent to which models offer quite different guidelines for policymakers can be seen from the work of Frankel and Rockett (1988). Table 2.2 shows the estimates of twelve world models of the effects of a change in US monetary policy keeping fiscal policy fixed. The disagreement is clearly substantial particularly in the effect on the rest of the world. The effect in the second year on US output of a 4 per cent increase in the US money supply is estimated variously to range from 3.0 per cent to 0.1 per cent. The effect on GDP in the rest of the world varies from - 0.7 per cent to 0.4 per cent. It is clear that there is not even agreement on the sign, let alone the magnitude, of relevant monetary spillovers. Frankel and Rockett's next step was to compare the effects of cooperative and non-cooperative policies under the assumption that policymakers believe different models. Concentrating on six of the models and

66

General issues

Table 2.2. Money multipliers in international models: effect of a 4% increase in US money supply % difference after 2 years According to MCM EEC EPA

LINK Liverpool MSG

MINIMOD VAR

OECD Taylor Wharton DRI

USA GNP

+ 1.5 + 1.0 + 1.2 + 1.0 + 0.1 + 0.3 + 1.0 + 3.0 + 1.6 + 0.6 + 0.7 + 1.8

ROECD

The

GNP

dollar

-0.7 + 0.2 -0.4 -0.1 -0.0 + 0.4 -0.2 + 0.4 + 0.3 -0.2 + 0.4 -0.6

-6.0 -4.0 -6.4 -2.3 -3.9 -2.0 -5.7 -22.9 -2.6 -4.9 -1.0 -14.6

Notes: The models MCM EEC EPA

the US Federal Reserve Board's Multicountry Model, the COMPACT model of the EEC staff, the World Model of the Economic Planning Agency of the Japanese Government, the model of Project LINK, LINK Liverpool The Liverpool University Model, the McKibbin-Sachs Global Model, MSG MINIMOD the model of Haas and Masson at the IMF, the vector autoregressive model of Sims and Litterman, VAR OECD the OECD staffs INTERLINK model, TAYLOR the model of John Taylor, Stanford University, WHARTON the Wharton World Econometric Model, DRI the DATA Resources Inc. multicountry model. Source: Frankel and Rockett (1988). considering cooperation between the USA and the rest of the OECD bloc (ROECD), each of the two policy-makers has six choices of model on which to base policy decisions, giving 6 x 6 = 36 possible pairs of beliefs. Frankel and Rockett computed cooperative and non-cooperative policies for these thirty-six cases and constructed a payoff matrix which showed the gain from cooperation for each policy pair if each model in turn turned out to be correct.

67

International policy coordination Welfare of USA Y (a = 0.65)

X

Z (a = 0.45) Non-cooperative outcome

Cooperative agreements

Welfare of ROECD

Figure 2.2 Sustainable cooperative agreements There are now 6 x 6 x 6 = 216 outcomes (cooperative versus noncooperative) to examine. Frankel and Rockett showed that cooperation results in a welfare improvement in 62 per cent of these cases for both the USA and ROECD. On the basis of this evidence the probability is that cooperation will benefit the world, but the risks of cooperation being counterproductive are substantial. The pessimistic conclusion of the effects of model uncertainty on the gains from cooperation has been questioned by more recent work. Holtham and Hughes Hallett (1987) have pointed out an oddity in many of the failure of cooperation cases in the Frankel-Rockett work: in these cases, one or other party would be worse off as a result of the agreement. They refer to these as 'weak bargains'. Even in the absence of altruism, cooperating countries are likely to avoid weak bargains for one very good reason. In a weak bargain one party perceives that there is an incentive for the other to renege. In other words the agreement is not perceived to be sustainable.

As we have seen sustainability is a crucial concept in considering agreements of any type. The ability to sustain a particular agreement depends on the nature of the bargain which is struck over the distribution of the gains. This involves the choice of weights a and 1 - a say, used to average the individual welfare measures in arriving at the global/welfare criterion. Figure 2.2 shows a typical distribution of the welfare measure for each country under a cooperative agreement. The curve through points Y and Z plots the set of feasible cooperative agreements as the parameter a

68

General issues

varies. The shaded region is where the welfare for both blocs is higher than under the non-cooperative outcome (point X). In the hypothetical case shown all values of a between 0.45 and 0.65 result in cooperative agreement which benefits both blocs. Only these agreements are sustainable. The problem with many of the outcomes considered by Frankel and Rockett was that one or other party perceived the agreement as falling outside the sustainable region. In 'strong bargains' both blocs believe that neither party is worse off under cooperation than under non-cooperation. Holtham and Hughes Hallett went on to reexamine the Frankel and Rockett payoff matrix, restricting possible agreements to 'strong bargains'. They found that within the smaller number of possible outcomes the success rate of cooperation increased significantly. One interpretation of these studies is that they highlight the advantage of cooperative policy-making being preceded by an agreement on the appropriate model on which to base the policies. But even if policymakers can agree to use, say, the MCM model or some composite, compromise model incorporating average behaviour, it would still be important that agreed policies are reasonably robust with respect to the choice of model. Holtham and Hughes Hallett examined the case where policy-makers agree to use one of eight possible world models (including a composite model) to design a cooperative set of policies. They then calculated a payoff matrix in which one of these eight views of the world turns out to be correct. (This study overlaps with that of Frankel and Rockett. The main difference is that the latter only considered static optimisation whereas Holtham and Hughes Hallett considered optimal policies over a six-year period.) The payoff matrix is shown in table 2.3. The entries (USA, ROECD) show the percentage welfare gains from cooperation as compared to non-cooperation for the USA and ROECD respectively. Negative entries indicate that non-cooperation is best if the two blocs design policy on the basis of one model when the truth is captured by another model. There are seventy-nine out of 128 negative entries. This proportion of failures is in fact higher than the Frankel and Rockett result and indicates that model uncertainty is as serious an obstacle to policy coordination as model disagreements. The MCM is the only model for which policies give significant gains to cooperation whichever model turns out to be true. It would seem then that there is a strong argument for implementing a cooperative policy based on MCM. However this conclusion requires qualification. If policymakers are extremely risk averse, an alternative criterion is to examine the

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69

worst-case scenario for each policy from among all possible outcomes. A 'minimax' strategy then chooses the policy which gives the best worst-case outcome. This turns out to be the non-cooperative policy based on the LINK model. The argument for cooperative policy thus depends on the policy-makers' attitude to risk. Do these studies across models undermine the case for policy coordination? There are a number of reasons for rejecting a pessimistic conclusion. Firstly, the Holtham and Hughes Hallett study included only two models which incorporate forward-looking expectations, the Taylor and MSG models. Only in these two cases can one consider the reputational effects discussed earlier. The consequence of this is that these studies do not fully capture the importance of combining reputation with cooperation highlighted in the Currie-Levine papers. Secondly, Ghosh and Masson (1988) show that if policy-makers learn about the model (through observations of macroeconomic variables and Bayesian learning) then the performance of coordinated relative to noncooperated policies substantially improves. A third reason for not necessarily drawing an anti-cooperation conclusion from the multimodel studies is more technical. The policies designed by Holtham and Hughes Hallett are 'open-loop' in character, i.e., fiscal and monetary policies take the form of paths over the six-year period. Precommitment to these transition paths (for government spending and monetary growth) do not allow policy-makers to revise policy in the light of modelling errors which are discovered in the course of using one particular model. A more flexible form of precommitment would take the form of feedback rules which allow adjustments to be made as outturns differ from forecasts. It is a standard result from control theory that feedback rules improve robustness in the face of modelling uncertainties (and, incidentally, provide protection against exogenous shocks). In the next section we consider some recent work which approaches international policy coordination in the form of agreed feedback rules. 7

Simple rules

A difficulty with the empirical literature surveyed in sections 5 and 6 is that although policies can generally be formulated in terms of feedback rules, these turn out to involve extremely complex adjustments (an exception is the study by Hughes Hallett et al. (1989) which formulates cooperative agreements in the form of agreed exchange-rate paths). It is often argued that policy needs to follow rules that are easily implemented, easily monitored and whose advantages are intuitively apparent if they are acceptable to policy-makers. The monitoring argument for simplicity

Table 2.3. Pay-off matrix: percentage gains to cooperation (Maintained model) REALITY

LINK

OECD

MCM

EEC

TAYLOR

MSG

MINIMOD

AVERAGE

33.6 37.3

-57.6 -33.6

-0.2 -18.2

88.9 88.6

87.7 85.5

75.9 77.3

66.1 64.9

5.2 -42.2

-35.6 -1193.2

12.4 57.6

- 133.7 -1107.2

93.5 64.0

94.3 - 17.2

91.6 61.0

62.1 - 13.9

-0.7 - 1115.1

EEC

-86.4 - 185.9

- 855.0 10.0

20.4 35.8

78.5 86.1

65.5 58.0

7.6 50.5

1.4 -6.8

-85.0 27.9

MCM

-3218.4 - 1289.5

- 1488.3 - 366.2

- 1698.2 -3831.3

52.3 66.5

- 15.9 51.3

7.4 - 186.0

- 844.6 - 272.9

- 2894.7 - 3353.4

TAYLOR

- 1377.2 - 209.4

- 603.4 - 160.0

- 1035.2 - 145.4

40.7 81.3

93.1 87.5

- 135.5 55.1

-2152.0 - 152.3

- 2983.9 - 133.2

MSG

- 593.4 - 1010.6

-73.3 - 433.7

-31.9 -1192.2

34.2 33.4

83.3 -0.9

81.9 75.3

-819.6 -81.7

-531.5 - 1400.8

MINIMOD

- 152.6 - 197.1

- 133.2 -44.6

- 509.6 - 785.6

52.0 68.8

83.5 74.3

81.3 60.3

70.7 55.5

-318.1 - 532.9

AVERAGE

35.2 - 632.4

-92.7 -44.6

-54.0 -9.7

90.1 90.1

95.1 84.0

93.3 85.2

61.8 76.0

76.2 77.0

MINIMAX

-3218.4 - 1289.5

- 1488.3 - 433.7

- 1698.2 -3831.3

34.2 33.4

- 15.9 -17.2

- 135.5 - 186.0

-2152.0 - 272.9

- 2983.9 - 3353.4

LINK OECD

Note: The difference between the welfare loss under non-cooperation and cooperation as a percentage of the former: note ' + implies a gain from cooperation and ' - ' implies a loss. The first line in each entry refers to the USA, the second to ROECD. Source: Holtham and Hughes Hallett (1987).

72

General issues

carries particular force in the international sphere where the need to monitor policy commitments applies to both the private sector and to the countries entering into agreements. It may well be that well-designed simple rules will increase the credibility of policy commitments in the eyes of the private sector and foreign governments. In that case, how far can simple policy rules be designed to replicate the gains from full coordination? This raises the issue of what is meant by simplicity in this context. One aspect of simplicity is the need to ensure that policy rules have a simple dynamic structure. A second aspect is to restrict the range of variables or information to which policy instruments respond. If this goes together with a specialisation whereby different instruments respond to distinct subsets of variables then one arrives at assignment rules, which represent a specific form of simplification. One system of simplified rules that has been proposed is the Williamson-Miller (1987) extended target zone proposal, which blends each of these elements of simplicity. They propose that fiscal policy should be used to manage internal nominal demand growth, while monetary policy is aimed at maintaining external balance by holding exchange rates within wide bands around equilibrium or fundamental equilibrium exchange-rate (FEER) levels. In addition, the absolute level of world interest rates is used to steer world nominal income growth. Objections to this policy are that monetary policy has rather limited effect on the current account, so that the scheme may be rather poor at dampening current-account imbalances, and that fiscal policy is too inflexible for successful management of internal demand. An alternative scheme, proposed by Boughton (1989), suggests that fiscal policy should instead be assigned to achieving external current-account balance in the medium term, while monetary policy steers internal demand growth. Empirical testing of these simple policy rules is still fairly sparse. Currie and Wren-Lewis (1989a, b, c, 1990), using the GEM model, find that the extended target zone scheme could well have improved on historical performance over the past decade, and that this conclusion is fairly robust with respect to changes in the objective function (including the implied flexibility of fiscal policy). Moreover, the extended target zone proposal is found to outperform the alternative scheme proposed by Boughton. These findings are for the G3 countries alone, supporting an hierarchical structure of coordination: it is not clear whether they would generalise to a larger group of countries adopting the target zone proposal in a non-hierarchical way. However, it should be said that these comparisons are with historical outcomes. Further work is required to show whether or not these schemes genuinely out-perform non-cooperative decision-

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73

making and therefore help to contain the potential inefficiencies of noncooperative outcomes. Taylor (1989) uses his own model of the G7 countries to examine a number of simple monetary rules. A fixed exchange-rate regime is compared with a flexible exchange rate in which monetary policy is assigned to stabilising the price level or nominal income or both price and output with different elasticities. The general conclusion is that the latter mixed rule is likely to have superior stabilisation properties in the face of exogenous shocks to the economy and that an agreement to fix nominal interest rates between the USA, Japan and Germany achieves little with respect to internal or external stability. Frenkel et al. (1989) evaluate some simple rule for monetary and fiscal policy using MULTIMOD which include the Williamson/Miller target zone proposal and Boughton's reverse assignment. As with the Taylor exercise they considered stochastic exogenous shocks. Simulations of individual shocks emphasise a point in the theoretical literature going back to Poole (1970) that the performance of simple policy rules varies with the nature of the shocks facing the economy. Rules that perform best for some shocks may perform less well for others. The authors are cautious in drawing strong conclusions from the full stochastic simulations which attempt to meet this problem. However there appears to be some indication that the Miller/Williamson proposal does better than the reverse assignment providing some support for the findings of Currie and Wren-Lewis. The studies so far examine the efficacy of coordinated simple rules by comparing one rule with another or with history. Levine, Currie and Gaines (1989) compare cooperative with non-cooperative regimes on the assumption that policy is conducted in terms of particular simple rules. The incentive-compatibility problem we have surveyed for optimal rules is extended to simple rules. Their main findings, using a reduced two-bloc version of the OECD Interlink model, are that, first, policy coordination can be effective under the constraint that rules must be simple. This confirms the results of other studies that we have discussed. Second, there is considerable scope for using agreements in the form of simple rules as a surrogate for more far-reaching agreement on international policy coordination. Levine, Currie and Gaines find that if the USA and the rest of the world agree only on the assignment of fiscal and monetary policy instruments to targets (inflation and output in this study) then even if the decisions over the feedback parameters are made non-cooperatively, the outcome is far superior to non-cooperative decision-making with no agreement whatsoever. The recent interest in simple feedback rules derives from their intuitive

74

General issues

appeal, from the need for simplicity and credibility and from their relative ease of implementation. We also know that feedback rules are less sensitive than open-loop policies to modelling errors. It has yet to be established that robust rules can be found that retain simplicity but perform reasonably well across a range of different views of the world economy. This remains a major challenge for economists working in this area. 8 (i)

Conclusions What have we learned about monetary policy coordination?

The principal contribution of the theoretical literature is methodological; that is to say it provides a framework for analysing macroeconomic policy in interdependent economies in terms of a game in which the players are the policy-makers in different blocs or countries and an atomised private sector. By specifying policy-makers' objectives in terms of a welfare function the gains from coordination can be assessed by comparing the values of the welfare criteria under cooperative and non-cooperative equilibria. Using this framework, Hamada demonstrated the potential for inefficient non-cooperative outcomes between countries as a result of policy spillovers. His seminal studies were limited to simple symmetrical twocountry models which lacked dynamics and implicitly assumed backward-looking expectations on the part of the private sector. By introducing strategic dynamics and rational expectations, subsequent literature has addressed issues to do with reputation and the incentivecompatibility of cooperative agreements. In the light of these developments, is coordination desirable in principle and can cooperative agreements be sustained? The unequivocal superiority of cooperation demonstrated by Hamada now needs careful qualification. Cooperation in a context where policy-makers lack reputation for precommitment may be counterproductive. Cooperation is only unambiguously superior to non-cooperative outcomes if policy-makers enjoy reputation. Cooperative agreements can in principle be sustained in the face of incentives to renege if both governments and the private sector adopt suitable trigger mechanisms. These specify switches in privatesector expectations and switches in government policies if governments jointly or individually renege. Turning to the empirical literature, how large are the gains in practice? The general consensus arising from a large number of studies is that the gains from the coordination of monetary and fiscal policy among the

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75

OECD countries may be quite small, though significant. However, this conclusion needs careful qualification. The outcome of these exercises is model dependent and depends on the choice of welfare function, whether (in rational expectations models) the governments enjoy reputation for precommitment or not. In particular if shocks are persistent, if policymakers have conflicting objectives, particularly incompatible exchangerate targets, and if reputational policies can be pursued then the gains from coordination increase appreciably. A serious limitation of much of the literature is that it assumes that policy-makers know the global model. We have reviewed studies which alert us to the dangers of policy coordination where the model is unknown and different policy-makers may use different models. However if policy-makers can agree on which model to use then the literature suggests that a robust policy exists which gives cooperation gains irrespective of which model turns out to be correct. The effects of model uncertainty can be further ameliorated if learning is allowed and policy is formulated in terms of feedback rules. Another limitation of the literature is that they assume full cooperation involving both monetary and fiscal policy and that the implied optimal rules are extremely complex. The theoretical literature on the EMS and fixed exchange-rate regimes in general show that limited agreements which tie the hands of policy-makers by making them choose policies constrained to maintaining fixed nominal or real exchange rates can be useful forms of surrogate cooperation. Empirical work on agreed exchange-rate paths as a form of cooperation is less encouraging. Cooperation gains are small and asymmetrically distributed between participating countries. A number of studies look at simple rules such as the extended target zone proposals of Williamson and Miller (1987) and the reverse assignment of Boughton (1989). Thefindingsare favourable when outcomes are compared with history; but it has not been established whether these schemes outperform non-cooperative decision-making, possibly based on similar rules. (ii) Future research The literature that we have surveyed in this chapter is, of course, in a continuing state of development, so that this survey represents an intermediate report. It may therefore be helpful to conclude by pointing to issues that would merit greater attention in future work. One of the considerable advances in this field has been the application of rigorous game theoretic techniques to questions of policy coordination, following the innovative contribution of Hamada. However,

76

General issues

much remains to be done to clarify and strengthen the game theoretic foundations of this literature. In particular, the introduction of general trigger strategies greatly multiplies the number of admissible equilibria, and the multiplicity in turn generates an indeterminacy into the analysis of coordination. Recent advances in game theory have been in the direction of narrowing down the range of admissible equilibria by imposing stricter equilibrium conditions (see Abreu, 1988 and Harsanyi and Selton, 1988). The introduction of this work into the analysis of coordination should be a fruitful line of research. It seems likely that advances will be made in incorporating learning into models with forward-looking behaviour, relaxing the extreme assumptions of rational expectations. This will have important implications for the literature on policy coordination. In particular, it should allow a more developed analysis of the consequences of model uncertainty for policy coordination than that surveyed in section 6 of this chapter. It should also allow rigorous analysis of the benefits of information exchanges and perhaps substantiate the suggestion in our survey that the benefits of information exchanges are at least as important as those of coordination per se. More narrowly, we would expect the literature to continue the investigation of effective, sustainable simple rules for policy coordination. This line of enquiry is important, because practical rules are likely to be simple in design. Yet advantages of simplicity should not be at the expense of sustainability, credibility or robustness. The systematic investigation of rules along these lines is likely to be a continuing area of productive research. A notable advance in this area in the 1980s has been the increasing flow of empirically based results, making the theoretical research much more relevant to policy-makers. It is possible that the analysis is now outstripping the capabilities of our empirical macroeconometric models. There are two respects, in particular, where this may be so. First the analysis of alternative regimes requires careful attention to issues concerning the Lucas critique which have been insufficiently addressed in the context of the international macroeonometric models used in this literature. Second in assessing regimes and policy rules, notably the EMS and European monetary union, long-run relationships are of appreciable importance. The current state of empirical international macromodels is probably not sufficiently advanced to incorporate these adequately. The advance of this area of research, in particular in measuring the benefits or otherwise of coordination and in coming forward with practical policy advice, depends in a major way on improvements in empirical model.

3

The European road to monetary union

What exactly does Monetary Union involve? A monetary union is an arrangement between participating states in which internal exchange rates are permanently fixed and with no institutional barriers to the free movement of capital or to the circulation of currencies. This definition of monetary union still leaves open the question of whether monetary union is compatible with the existence of national currencies or whether it implies the adoption of a common currency. The Delors Committee took the latter view - Europe under EMU would use a European currency which would evolve out of the European Currency Unit (ECU). In what follows we take this as our paradigm of a monetary union. Before going on to discuss European monetary union in detail, let me set out the main issues which need to be addressed in considering any form of monetary union. These may be summarised under the following headings Gains from the elimination of transaction costs and exchange-rate uncertainty The consequences of losing or 'pooling' sovereignty The credibility of the monetary authority or authorities Transaction costs and exchange-rate uncertainty

As foreign travellers we are all familiar with the transaction costs involved in changing currencies. These charges made by a bank reflect the European Research, Vol. 1, part 6, November, 14-22. This chapter is a revised version of Paul Levine's professorial lecture given on 27th February 1990. Thanks are due to Nigel Healey for extremely helpful comments on an early draft and to Kathleen Levine for providing an intelligent layperson's view of the penultimate draft. 77

78

General issues

deployment of resources - personnel and equipment - as well as the opportunity costs of holding stocks of foreign exchange (i.e., the foregone interest payments). The non-trivial nature of these charges is demonstrated by the story of the Europhile who leaves London with £1,000 to visit all the other eleven members of the EC changing his currency at each border. Taking into account unfavourable exchange rates tourist transactions may cost as much as 10 per cent. This would mean our traveller would return to England having spent more than two-thirds of the £1,000 simply on exchanging currencies. Can we put a figure on the size of these transactions costs for the EC as a whole? Eltis (1989) in a NEDO paper quotes charges of 0.5-1 per cent for commercial transactions and 2.5-3 per cent for tourist transactions. Taking both trade and capital transactions together rough calculations suggest a savings of between 2-3 per cent of GDP if we move to a single currency in the EC (Artis, 1989). Turning to exchange-rate uncertainty, either a single currency or permanently fixed exchange rates will bring about the reduction of the uncertainty which exchange-rate fluctuations bring to trade and financial transactions. It is true that increasingly sophisticated financial institutions provide a form of insurance against exchange-rate uncertainty. However, these 'hedging' facilities are not free and their cost reflects the potential savings for a community with a single currency or fixed exchange rates. Moreover for long time horizons forward facilities are not universally available. Again one could in principle put a figure on the savings involved in doing away with the need for hedging facilities. But these calculations, which quantify the gains from the elimination of transactions costs and exchange-rate uncertainty, focus on what is really a secondary issue. The primary benefits will arise from the increase in trade, capital flows and labour mobility within the nations of Europe which are a consequence of reduced costs and uncertainty. The ultimate gains will take the form of a more efficient allocation of resources within the community. These are difficult to quantify. But it is widely agreed that the EC is still fragmented into a series of national and regional markets with various obstacles to trade. The 1992 Programme is aimed at eliminating barriers to trade arising from customs procedures, technical regulation and public procurement practices. Monetary union will remove a further obstacle to trade and in this respect is a natural complement to the Programme to 'complete the internal market'. All this is relatively uncontroversial. What really divides economists in their assessment of EMU are the 'strategic' issues of sovereignty and credibility.

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The aspect of EMU (and indeed EMS) which arouses the most passion is the question of sovereignty. Keen European federalists support EMU partly because it will help the political integration of the EC. Of course there are those who instinctively oppose both EMU and EMS for precisely the same reason. My intention is to avoid these broad political considerations and focus on the merits or otherwise of'losing sovereignty' over monetary policy (as opponents of EMU would phrase it) or 'pooling sovereignty' and 'cooperating over monetary policy' (as supporters would put it). Before considering the particular policy choices facing the EC, let me first pose a very general question; should nations cooperate or 'compete' in the setting of macroeconomic policy? Policy cooperation: the general case In general terms the case for policy cooperation arises for two reasons. The first is the simple fact that policy objectives of different governments do not coincide. National governments are concerned primarily, if not exclusively, with their own country's GDP, inflation, unemployment, etc. If in the EC, say, the authorities all pursued the same community-wide objectives then the need for cooperation would not arise. The second reason why cooperation may be beneficial is the existence of significant spillovers in the effect of macroeconomic policy between countries. These arise from the interdependence of economies through flows of trade, capital and multinational production. This is particularly important for the EC because as their economies become more integrated policy spillovers will become more pronounced. Figure 3.1 gives an indication of the growing integration of the UK with the rest of the EC through trade links. Similarly, for the EC as a whole, the share of intra-EC trade as a percentage of total EC trade has risen from around 35 per cent to 60 per cent over the period 1958-87. These figures partly reflect the enlargement of the EC during that period; but they also reflect the higher growth of intra-EC trade. As a consequence a relatively autonomous European economic zone of interdependent economies is emerging. Interdependence implies policy spillovers. As a consequence there exists the potential for inefficient 'competing' or non-cooperative outcomes. For example, consider a situation in which all governments face a drop in domestic and overseas demand and the prospects of a recession. Each country could unilaterally counteract this drop in demand by fiscal relaxation (e.g., increasing government spending). But if a country expands demand in isolation, thereby worsening its trade balance, pressure on the balance of payments causes the exchange rate to fall, thus fuelling domestic inflation. On the other hand if each country envisages a

80

General issues

43.5 Imports Exports 35.2

26.9

18.6

1966

I

1972

i

I

I

I

I

I

1978

I

I

1

L_

1984

1987

Figure 3.1 Trade with EEC as percentage of total relaxation of policy by other governments, there is an incentive to freeride, i.e., rely on overseas demand to provide export-led growth, and engage in only a limited change of domestic policy. The problem is that part of the increase in demand is exported as a positive externality. Only by cooperating and internalising that externality can the socially desirable level of expansion be effected. Another example of the inefficiency of non-cooperative policies is in the response to inflationary shocks, such as an increase in the price of oil. One way for an individual country to tackle inflationary pressures is to respond with tight monetary policy (that is high interest rates). This would produce a rise in the nominal exchange rate which, by reducing import prices, would be counter-inflationary. The problem with each country pursuing a tighter monetary policy is that it is clearly beggar-thy-neighbour in character. One country's exchange-rate appreciation is a depreciation for the rest of the world. In effect each country is trying to export inflation. The global outcome would take the form of an interest-rate war resulting in an excessively tight world monetary stance. Under a cooperative policy stance, however, this negative externality would be internalised and this time a socially desirable level of contraction would be achieved. In these examples, both of which have been observed in the world economy, the countries are acting quite rationally given the policy actions of others. Nonetheless the global outcome is still one which all parties would prefer to avoid. In pursuing policies independently and in the

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81

Table 3.1. Payoff matrix for the prisoners' dilemma Prisoner 2

Prisoner 1

Does not confess

Confesses

Does not confess

(3,3)

(20,1)

Confesses

(1,20)

(10,10)

absence of coordination they have been trapped by the prisoners'

dilemma. The prisoners' dilemma is a well-known parable cited by game theorists (see Levine (1985) for a detailed account of game theory in economics). Two partners in a serious crime are caught and held in separate cells. Each prisoner is made the same offer. If both confess they will go to jail for ten years. If only one confesses he will only receive a one-year sentence whilst his partner gets the maximum of twenty years. If neither confesses they will be charged with a less serious offence and both get three years. The eventual outcome of this 'game' can be seen by examining the 'payoff to each prisoner from the four possible outcomes. In table 3.1 which represents the 'payoff matrix', thefirstnumber in the brackets is the jail sentence for prisoner 1 and the second refers to prisoner 2. For prisoner 1, examination of the matrix reveals that it is always better to confess whatever prisoner 2 does. In game theoretic terminology, the strategy to confess 'dominates'. The same holds for prisoner 2, so that the outcome of the game is that both prisoners confess. This is true despite the fact that if both were to remain silent, they would both be better off. The prisoners' dilemma demonstrates one of the fundamental general results of economics. Where there are interdependencies between economic agents and spillovers (or 'externalities') resulting from agents' actions, rational self-interested action taken in an uncoordinated manner can lead to outcomes that nobody wants. Only by cooperation can these externalities be internalised and the most efficient outcome reached. Floating exchange rates, the ERM and EMU

So far we have considered the benefits for monetary policy coordination at a very general level. Our discussion has perhaps been more directly

82

General issues

relevant to global coordination between the G7 countries (i.e., North America, Japan and Western Europe) rather than Europe. In Europe monetary policy coordination has taken a very particular form, namely the Exchange Rate Mechanism (ERM) within the European Monetary System. The ERM was introduced in 1979. It was always seen as a transition stage towards full monetary union. In that respect the ERM was an attempt to resurrect the ambitions of an earlier attempt to move towards EMU in the late sixties. The main impetus for the ERM was the general dissatisfaction with the floating exchange-rate system which came to replace the fixed rates of the Bretton Woods system in the early seventies. There was general concern with the high level of exchange-rate volatility which, it was felt, prevented European business from reaping the full benefits of the Common Market. Full EMU is a irrevocable commitment to fix bilateral exchange rates against each other. This would represent a fundamental change from the ERM which allows currencies to fluctuate by 2.25 per cent above or below a central rate. In addition the ERM allows for mutually agreed realignments. The ERM is thus a flexible system of exchange-rate cooperation. Whether flexibility is a good thing or a bad thing is something I shall return to presently. The ERM can evolve towards full monetary union if realignments become less frequent (as has been the case in the late eighties) and if bands become narrower. But even a 'hard-ERM' with narrow bands and infrequent or even no realignments still differs fundamentally from full EMU. In order to develop this point I would like to consider a spectrum of possible exchange-rate regimes. At one end of the spectrum is the floating exchange-rate regime. Within such a system countries are unconstrained by exchange-rate commitments in their pursuit of monetary (or fiscal) policies. At the other end of the spectrum is EMU which for the purposes of the argument we take to mean the adoption of a common currency and with a European Central Bank setting interest rates for the entire Community. Within these two extremes consider a 'soft-EMS' which has wide bands and allows frequent realignments and a 'hard-EMS' with narrow bands and with very infrequent or even no realignments. These four regimes are summarised in table 3.2. Now consider transaction costs in the contest of these regimes. Clearly only EMU eliminates the costs arising from changing currencies and the remaining three regimes are equivalent on this criterion. Regarding the elimination of exchange-rate uncertainty it is clear that there exists a progressive improvement as one moves from a floating exchange rate to EMU.

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83

Table 3.2. Exchange-rate regimes Floating exchange rate

Soft-EMS

Hard-EMS

EMU

No constraint

Wide bands Frequent currency realignments

Narrow bands No realignments

One currency

Using the elimination of transaction costs and exchange-rate uncertainty as the criteria, EMU then clearly comes out best. Let us next turn to the question of monetary sovereignty. Sovereignty over monetary policy, by which we mean the choice of the money supply and short-term interest rates by the central bank, gives a nation the ability to influence the exchange rate and domestic inflation. Consider a regime which allows exchange-rate sovereignty (i.e., floating or soft-EMS regime). We have discussed ways in which a government might respond to a supply shock such as an increase in the price of oil by raising interest rates and causing the exchange rate to appreciate. But the nominal exchange rate is not what is important in this respect, it is the real exchange rate or the relative price of domestically produced goods to imports. To what extent can real bilateral exchange rates be changed with the EC? If the EC consisted of perfectly integrated and competitive markets and if transport costs are negligible compared with other costs of production then the 'law of one price' would hold. In other words there would only be one price (in any one currency) at which any particular commodity is sold. Now consider table 3.3. This shows the relative prices in different EC countries of three goods and one service. It is striking that the three goods - cars, Pharmaceuticals and domestic appliances - are extensively traded and yet at the same time exhibit a substantial departure from the 'law of one price'. These figures indicate that the EC is still far from achieving the degree of integration that the 1992 programme sets as its ultimate objective. In the EC as it now exists there is then scope for real exchange-rate sovereignty between members. Governments by raising or lowering interest rates can cause changes in the nominal exchange rate. Because markets are not perfectly integrated these do not immediately feed into price changes which would leave the real exchange rate unchanged. Suppose, however, that rapid progress was made towards market unification. In such a world a change of your exchange rate rapidly feeds through into prices and the price of exports relative to trading partners

84

General issues

Table 3.3. Prices in European markets

Belgium France West Germany Italy Netherlands UK

German cars

Pharmaceuticals

Life insurance

Domestic appliances

100 115 127 129 n.a. 142

100 78 174 80 164 114

100 75 59 102 51 39

100 130 117 110 105 93

Source: Nicolaides and Baden Fuller, 1987.

remains unchanged. Sovereignty in the sense of being able to control competitiveness (relative prices) is then an illusion. With flexible exchange rates, sovereignty of monetary policy still enables you to raise (or lower) inflation rates. But is this worthwhile? In traditional Keynesian models popular some ten years ago there existed a trade-off between inflation and output. You could raise output and lower unemployment but at the cost of higher inflation. The main mechanism by which this trade-off worked was through the persistent underforecasting of inflation by workers, either individually or collectively, as inflation rose. This meant that as prices rose the real wage fell and firms then found it profitable to employ more workers. If this forecasting error persisted indefinitely then a long-term trade-off between output and inflation existed. Milton Friedman pointed out that any sensible forecasting rule should eliminate systematic forecasting errors in the long run, thereby also eliminating the long-term unemployment/inflation trade-off. The neo-classical Rational Expectations school further argued that a stable forecasting rule cannot result in systematic forecast errors even in the short term. This effectively ruled out even a temporary trade-off unless there were other rigidities in the labour market such as workers being locked into long-term contracts that prevented them responding to their correct perceptions of higher inflation. In this 'natural rate world' output and employment is invariant to monetary policy. Sovereignty over monetary policy may still exist. But this is only nominal sovereignty. But in a world of integrated product markets and flexible labour markets with rational expectations, its only effect is to change inflation. Note that this result does not require the integration of labour markets across national boundaries.

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More recently Keynesian models of the labour market have made something of a come-back. Responding to the neo-classical criticism of 'ad hocness', an alternative view of wage and price rigidities has been developed based on rigorous theoretical foundations. We now have a better undertanding of how imperfections in product markets can spill over into the labour market. The nature of contracts between employers and workers is also better understood. A further development in labour-market economics is the concept of 'hysterisis'. According to hysterisis theories, the equilibrium or natural rate of unemployment depends on the actual history of unemployment due to the deterioration in skills and morale of the unemployed. Unemployment is then difficult to reverse and may not ever be completely reversible. These developments support the view that monetary policy can effect output certainly in the short term and possibly in the long term although not in the manner envisaged by the old Keynesian models. To summarise, with a regime that allows nominal exchange-rate changes, sovereignty in the conduct of monetary policy allows sovereign control of competitiveness (the real exchange rate) only in a world where product markets are not integrated. The 'completion of the internal market' by 1992 aims at removing remaining fiscal, technical and physical barriers to the integration of the economies of the EC. There is some controversy over whether '1992' will happen and if so over its contribution to freer trade. Nevertheless there is no doubt the EC is becoming more integrated and we are therefore moving closer to a Euorpe where sovereignty over monetary policy amounts merely to a sovereign choice of inflation rates. In a Keynesian world with adjustable exchange rates this still leaves countries able to independently choose their own desired inflation/output combination according to their own priorities. In a natural-rate world the sovereign choice of your inflation rate does not affect output. Whether this limited form of sovereignty is a good thing depends crucially on a government's reputation, a question we return to presently. Even in the case where sovereignty in the conduct of monetary policy gives a nation influence over the real exchange rate as well as over the unemployment/inflation trade-off it may still be undesirable to exercise that sovereignty. Because one country's devaluation is another country's revaluation, monetary policy in open economies has a beggar-thyneighbour character and leads to the prisoners' dilemma. In a general sense cooperation should lead to a more efficient outcome, but this conclusion needs careful qualification when we come to consider credibility or 'time inconsistency'.

86

General issues

The time-inconsistency issue

One feature of an economic system which distinguishes it from an engineering system is that current decisions of individuals in areas such as consumption, investment, wage-setting and foreign exchange operations are dependent upon expectations of future events. If these expectations are in turn based on full knowledge of the process generating the economic variation in question (i.e., are 'rational') then this leads to the 'time-inconsistency' problem. As parents many of us are familiar with this problem. Children's expectations of future events, in the form of appropriate rewards and punishments, affect their current behaviour. The rewards and punishments are in turn designed with this relationship in mind. But as time passes there emerges an incentive not to carry out the promises or threats. After all bygones are bygones; the announced regime of rewards and punishments has had its effect on the children's behaviour. To put it more formally the nature of your optimisation problems changes and you have an incentive to renege on the original announcement. Children not being stupid of course realise and anticipate this from the outset. Your original promises and threats therefore lack credibility and your ability to control your children's behaviour is correspondingly diminished. The important thing is to be able to precommit yourself to a particular course of action and to acquire a reputation for sticking to that precommitment. Governments in settingfiscaland monetary policy face a similar problem. If they enjoy a reputation for precommitment to a particular set of policies then they can exert the most leverage in the economy with the best results. If they lack that credibility or reputation their ability to regulate the economy may be greatly diminished. Credibility and EMU

What bearing do these considerations have on EMU? So far we have ignored a vitally important asymmetry in the way the ERM works which reflects an important difference between the monetary authorities in the European Community. The ERM was designed to encourage a symmetrical response to bilateral exchange-rate changes on the part of the central bank whose currency appreciated towards the top of its band and the central bank whose currency depreciated. However this is not the way the system has operated. In fact the burden of adjustment has fallen on weak currency countries and the Germans have generally been unwilling to lower interest rates when the Deutsch Mark has been strong.

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87

The Bundesbank has generally pursued a monetary policy aimed at keeping inflation very low. When they have raised interest rates in pursuit of this objective the remaining countries in the ERM have been forced to follow suit in order to prevent their currencies sinking to the bottom of the band. When the other countries might have wished to reduce their interest rates, in order for instance to reduce unemployment, due to the large German trade surplus the Bundesbank has not felt it necessary to prevent a Deutsch Mark appreciation by easing their monetary policy. Thus the Bundesbank has acted as a leader forcing a tight monetary policy on all members of the ERM. If we move to a hard-EMS then Germany would in effect set interest rates for the whole of the EC. In this respect the ERM is similar to previous fixed exchange-rate regimes - the gold standard and Bretton Woods. All three have involved a leader or 'hegemon'; the UK in the case of the gold standard and the US under Bretton Woods. Although this feature was not intended by the instigators of the ERM it has turned out to be perhaps its most important advantage. Without German leadership the system would not have a disinflationary bias and would not have achieved the main success pointed out by its proponents - which has been to bring down inflation rates in inflation-prone countries. We can relate this phenomenon of German leadership to our earlier discussion of time inconsistency. The Bundesbank clearly has a reputation for being tough on inflation. But it is also plausible to suggest that the Bundesbank has a reputation for precommitment to any well-defined policy rule which may in some circumstances involve a more accommodating stance towards inflation. If an authority enjoys reputation in this sense, then it can exercise the greatest leverage on the economy with the best results. In a Keynesian world with a trade-off between inflation and output, at least in the short term, a central bank with reputation can exploit the relationship between expectations of future policy and the current state of the economy to smooth the effects of demand and supply shocks, i.e., to engage in stabilisation policy. Similarly in the foreign exchange market reputational monetary policy can give the central bank leverage over nominal exchange rates. We have seen that this does not necessarily imply control over real exchange rates within the EC, but a European Central Bank would be in a position to change the ECU/$ or the ECU/Yen real exchange rate because the degree of integration in product markets between the EC and the outside world is far less than that within the EC. The great fear regarding EMU however is that the new ECB might lack the reputation now enjoyed by the Bundesbank. Pooling of sovereignty also means pooling of credibility. The result may well be a ECB which is

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General issues

Table 3.4. Ranking of regimes Regime

Transaction costs Exchange-rate uncertainty Exchange-rate sovereignty Non-integrated product markets Integrated product markets Inflation sovereignty Keynesian world 'Natural-rate' world Prisoners' dilemma Credibility

Floating

Soft-EMS

Hard-EMS

EMU

2=

2=

2=

4

3

2

1 1

1

2

3

4

1=

1=

1=

1=

1

2

3

4

1=

1=

1=

1=

4 3

3 2

2 1

1 4

not in a position to make credible policy commitments which are believed by the private sector. The loss of reputation in the labour market (regarding inflation) and in the foreign exchange (ECU) market (regarding interest rates) will then result in a long-run inflationary bias and a reduction in the effectiveness of stabilisation policy. Is there then an alternative to the speedy transition to EMU as set out in the Delors Report? In a recent paper with Currie, Levine and Pearlman (1992) we compare EMU on the assumption that the ECB lacks reputation with hard-EMS. The latter we have seen involves German leadership with the Bundesbank effectively setting interest rates for the entire EC, but doing so with reputation. Now consider table 3.4 which ranks the four exchange-rate regimes of table 3.2 according to all the criteria discussed up to this point. The main conclusion that emerges is that hard-EMS has most of the benefits of EMU without the drawbacks on the credibility issue. One drawback of hard-EMS however is that it involves German leadership. Whilst this may be beneficial in eliminating the underlying inflationary bias present in the other regimes, this feature raises problems when it comes to stabilisation policy. Hard-EMS is not a cooperative regime but one where the Bundesbank is concerned with German inflation and possibly other domestic objectives. Stabilisation policy involves monetary policy being adjusted to shocks hitting the economies of the EC according

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to a well-defined rule. If shocks are symmetrical German leadership may not lead to any problems. But if shocks are asymmetrical this is no longer the case. Suppose for instance that household savings fall unexpectedly in the EC outside Germany. If this change in behaviour is temporary then the long-term trend for inflation, growth and monetary policy remains unchanged; but in a Keynesian world stabilisation policy is called for to smooth the transition of the economy back to the long-run path. Under hard-EMS, because Germany is relatively unaffected and concerned only with the domestic economy, the appropriate community response will not be forthcoming. On the other hand with EMU, even without reputation the outcome may still be better for the non-German members of the EMS, because the ECB would take a global view of the EC economies. This argument suggests that hard-EMS may be unsustainable in the face of asymmetrical shocks because it would be unacceptable to the non-German members. Rapid progress to EMU may then be unavoidable despite the loss of reputation involved. However Germans who have the most to lose from a premature EMU have every incentive to compromise in their exercising of leadership. Compromise in this sense means taking into account the needs of the whole EC in the setting of monetary policy. Formally we may construct a welfare criterion which penalises deviations of specified targets (such as GDP, inflation and unemployment) about some desired levels. Policies are then chosen to minimise the resulting welfare loss subject to a view of how the economy works and in particular how targets and monetary policy are related. If we construct two welfare losses - W° for Germany and WREMSforthe rest of the EMS say - then German compromise entails minimising some weighted combination

where the weight a set at zero involves no compromise, but in general a lies between zero and one. In the chapter quoted simulations of hard-EMS and EMU are undertaken for a model which has only partly integrated product markets, has Keynesian features in the short run but has a natural rate of output in the long run. The results are two-fold: On the assumption that the ECB lacks reputation for precommitment, in the long run and on average hard-EMS is unambiguously superior to EMU for both Germany and the REMS even where the Bundesbank fails to compromise (i.e., a = 0). If we consider short-run responses to temporary asymmetric shocks

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General issues

(stabilisation policy in other words) then hard-EMS out-performs EMU only if the Bundesbank compromises to the extent of setting a to be at least 0.4. This is shown in figure 3.2 which shows the welfare losses (W^EMS, W°) under hard-EMS for various degrees of German compromise for a typical asymmetric demand or supply shock. The shaded area gives outcomes which are superior to EMU for both the REMS and Germany. Thus it turns out (for this model) that a must be at least 0.4 for hard-EMS to be superior to EMU for both Germany and the REMS. This study suggests that there does exist an effective alternative to EMU in the form of a hard-EMS exchange-rate regime; but its sustainability in the face of shocks which are not common to Germany and the REMS may require considerable German compromise. If however we are to move quickly to EMU, which given the bold spirit of the New Europe in the 1990s seems possible, then the results of the study highlight the need to consider very carefully the constitution of the ECB. For it is the modus operandi of the new bank which may decide the success or failure of the whole venture. The constitution of the ECB

The debate over the constitution of the ECB focuses on two principal issues. The first regards the assigned objectives of the bank. Should it be concerned only with controlling inflation and set monetary policy for the EC towards that single aim? Or should the central bank be concerned with the general economic performance of member states? The second issue is accountability. Should the bank be fully accountable to the European Parliament or should it be ensured a high degree of independence? As the Economist (10-16 February 1990) put it in favouring the latter, 'the only good central bank is one that can say no to politicians'. At one extreme in the spectrum of accountability is the model offered by the Bundesbank. A law of 1957 ensures its independence from the federal government in the conduct of monetary policy. The influence of government is confined to the appointment of the bank's council for eight-year terms and to the right to challenge and delay decisions; but these latter powers are rarely used. At the other extreme is the Bank of England. Its relationship to the government and the Treasury in particular is best summed up by Nigel Lawson's statement to a House of Commons select committee in 1987 - 'I make the decisions and the Bank carries them out'.

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91

a= 1 (German altruism)

EMU

0.4 a = 0 (No German compromise)

l/l/REMS

Figure 3.2 Incentive-compatible choice of a

The temptation will be to design a ECB to resemble a Bundesbank writ large; indeed its president Mr Pohl may insist on these arrangements as a condition for German support. However the creation of a new social institution provides a unique opportunity to take a fresh, deeper look at what is required. The fundamental question is whether the monetary authority acquires a reputation for precommitment. With reputation the central bank can influence today's decisions of individuals and firms by promises of tomorrow's monetary stance. This may involve a policy that gives absolute priority to inflation control; but equally it is quite consistent with a monetary rule that also concerns itself with output growth and unemployment. Democratic accountability is not incompatible with a central bank that commands reputation. The constitution of the ECB must however ensure that monetary rules are revised infrequently. How could this work in practice? The announced rule could consist of two components: a central expected path for the money supply together with central forecasts for inflation, interest rates and output growth. Low monetary growth rates and model-based forecasts of low inflation rates would anchor inflationary expectations. The second component of the rule would indicate exactly how monetary policy would be adjusted in the event of unexpectedly higher or lower inflation and growth rates. For instance if growth rates fell unexpectedly this would require a lowering of interest rates below the forecast. If inflation rose unexpectedly the rule might specify a raising of interst rates. There would have to be upper and lower bounds

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General issues

on interest rates. Rule revisions within the period of commitment could be allowed if the circumstances were both clearly specified and due to events beyond the control of the authority. Rules of this type areflexiblebut consistent with precommitment. They can be subjected to the full process of democratic accountability; but once decided the rule should remain unchanged for as long a period as possible (say eight years, the period for which the Bundesbank's president serves). If rules are simple in their design then the public can monitor their operation and the central bank can acquire reputation simply by being seen to adhere to its promise. In this way a ECB may emerge which combines the reputation of the Bundesbank with the ability to engage in effective stabilisation policy. Conclusions

I have discussed the benefits and dangers associated with EMU. A possible half-way house in the form of a hard-EMS has been explored as have ways in which the ECB may acquire a reputation for precommitment, the essential ingredient for the success of EMU. A theme running through the chapter is that the interdepencence of EC economies requires cooperative solutions to common problems. We have focused on monetary union - in effect monetary policy cooperation. But the need for cooperative solutions to global, social and economic problems clearly extends beyond the monetary policy in the EC. The same principles apply to problems of income imbalances in the community, to the challenge posed by new social and economic order in Eastern Europe, to global environmental imperatives and to the most glaring and persistent imbalance in the world between North and South. Compared with these issues EMU may seem a rather parochial and unimportant concern hardly deserving a chapter of this length. But the ability of the EC to be an effective and progressive player on the world stage depends on its cohesion and collective purpose. EMU can play an important part in bringing this about.

II

Theory and methodology

4

1

The design of feedback rules in linear stochastic rational expectations models

Introduction

This chapter is concerned with the optimal control problem for a linear stochastic continuous time model where expectations are rational, in the sense that they are consistent with the model, and where the model includes forward-looking 'jump' variables. The chapter draws upon work for deterministic models by Calvo (1978), Driffill (1982), Miller and Salmon (1985a) and Buiter (1984b) and extends their results to the stochastic case. For the most part, we confine ourselves to solutions with the familiar saddle-path property, but feedback rules which completely stabilise the system (i.e., have no unstable roots) are considered in a later section. For stochastic models, the closed-loop or feedback representation of policy is particularly useful in policy design. For the standard control problem (without rational expectations), with a quadratic loss function and an infinite time horizon, the optimal policy may be represented as a linear time-invariant feedback rule on the state vector. This is no longer the case when we examine models with rational expectations and forwardlooking jump variables. We show that, in such models, optimal feedback control can be expressed and implemented as a form of integral control. The added complexity of the full optimal closed-loop rule strengthens the case for considering simple sub-optimal rules of the type which have recently been advocated, for example, by Vines et al. (1983), Currie and Levine (1985a) and Taylor (1985). The plan of the chapter is as follows. Section 2 sets out the results of standard control theory in order to provide a framework for the rest of the chapter and to highlight the differences between optimal control in models with and without rational expectations. Journal of Economic Dynamics and Control, 11 (1987), 1-28, Elsevier Science Publishers B.V. (North-Holland).

95

96

Theory and methodology

In section 3 we consider the deterministic control problem for models with rational expectations (RE). We first set out the initially optimal, but time-inconsistent, rule when the government is able to precommit itself. We show that such rules are not linear in the state vector. This leads us to consider simple 'quasi-optimal' rules w = Dy where w is a vector of instruments, y is the state vector and D is a matrix of constants with appropriate restrictions which give the rules their simple character. Finally in section 3 we drop the assumption that governments can precommit themselves so that optimal policies must be found within the constraint of being time consistent. Section 4 addresses the full stochastic problem. In this section we show that the principle of certainty equivalence applies to the stochastic problem with rational expectations. We also consider certainty equivalence in relation to the simple quasi-optimal rules of section 3. Section 5 addresses the implementation of the initially optimal policy. We derive another feedback representation of the optimal rule which appears to be in the class w = Dy. However, if private agents only know the original model, this rule 'over-stabilises' the system and we show that over-stable rules of this form are always inferior to optimal rules with the saddle-path property. The chapter concludes by demonstrating the main results using two illustrative examples. 2

The standard control problem

Consider a dynamic system described by the stochastic differential equation dy = Ay At + Bwdt + dv

(2.1)

where y is an n x 1 observable state vector, w a n r x l control vector and dv is an n x 1 vector of white noise disturbances independently distributed with dv ~ N(0, Vdt). Matrices A, B and Fhave time-invariant coefficients and, in addition, V is symmetric and non-negative definite. All variables are measured as deviations about their long-run equilibria. In this section, y is a vector of predetermined variables with y(0) given. Equation (2.1) could represent a macroeconomic model where the economy is being subjected to continuous shocks and, more importantly, where there are no forward-looking expectations. Examples are text-book macroeconomic models prior to the 'Rational Expectations Revolution' such as continuous time versions of those in Turnovsky (1977). Throughout the chapter, we shall adopt an expected quadratic loss function with an infinite time horizon and non-zero discount factor, which we write as E0(W0) where

Linear stochastic rational expectations models

^0 = I I

^Qy

+ 2yTUw + wTRw]dt

97

(2.2)

Jo

and Eo denotes expectations, formed at time t = 0. In (2.2), p > 0 is the discount rate, matrices Q, U, R have time-invariant coefficients and, in addition, R is symmetric and positive definite and Q is symmetric and non-negative definite. The optimal control problem is to minimise Eo( Wo) subject to (2.1). It is useful to decompose this problem into two parts. Thefirstis a deterministic problem that uses the deterministic analogue of (2.1), namely y = Ay + Bw

(2.3)

The second is a purely stochastic problem that uses Ay = Aydt + Bwdt + dv

(2.4)

where y — y — y and w = w - w. Substituting y = y + y and w = w + w and noting that y is linear in dv and E0(dv) = 0 (by the definition of white noise), we obtain (2.5) where roo

tfo = i

Jo

z~pt\yTQy + 2yTUw + wr/?w]df

(2.6)

and

Wo = \ \ Q~pt[yTQy + 2fUw + wTRw]dt (2.7) Jo The two parts of the control problem are then the minimisation of Wo subject to (2.3) and the minimisation of E0(W0) subject to (2.4). Consider first the deterministic problem. We shall confine ourselves to a system with (A,B) completely controllable. A necessary and sufficient condition for this is that We are interested in linear feedback rules of the form w = Dy where D is time-invariant. Provided that (A,B) is completely controllable, it is a standard result that at least one value for D can be found that stabilises (2.3), i.e., one D exists such that A + BD has eigenvalues all with negative real parts. It is also standard that the optimal rule with loss function (2.6) is linear in y, stabilises the system and is found to be w = - R~\UT + BTS)y

(2.9)

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Theory and methodology

where S is the unique non-negative definite symmetric solution to the Riccati equation S(A - \pl) + (AT-\pI)S+Q-

(SB + U)R-\UT + BTS) = 0 (2.10)

An important point to note is that the feedback rule (2.9) is independent of y(0), the initial value of the state vector. Turning to the stochastic problem, an important concept in stochastic control is that of certainty equivalence. According to the principle of certainty equivalence, when optimising under uncertainty, one simply adopts the policy appropriate for the deterministic problem. This would mean that w= - R~\UT+BTS)y

(2.11)

would minimise Eo( Jf'o) subject to (2.4). This result is a consequence of the linearity of the dynamic system (2.1) and the quadratic form of the loss function. It follows from (2.9) and (2.11) that w= -R-\UT

+ BTS)y

(2.12)

is the optimal rule for the full stochastic problem (with a deterministic component). Certainty equivalence depends crucially on the result that the optimal feedback rule in the deterministic problem is invariant with respect to the initial state of the system. The same rule applied to the stochastic problem which, in turn, is independent of the covariance matrix of the disturbance vector. These results are extremely useful to a policy-maker because they imply that a feedback rule appropriate to all initial states of the system and to all types of disturbances is available. An immediate consequence of these two 'invariances' is that the optimal feedback rule is time consistent in the sense that Kydland and Prescott (1977) first used the term, i.e., an optimal rule decided at time t > 0 is simply the continuation of the rule chosen at time t = 0. In neither case does the rule depend on the current value of the state vector. In both cases, w = - R~\UT + BTS)y is optimal. However, suppose that the controller is looking for optimal rules of a restricted character which impose restrictions on the coefficients of/) for the feedback rule w = Dy. For instance, a decoupled rule with each instrument feeding back on a particular combination of state variables may be of interest (see Vines, Maciejowski and Meade (1983) for a case for marrying particular instruments with particular 'targets' in macroeconomic policy design). This type of restricted rule is 'quasi-optimal' in

Linear stochastic rational expectations models

99

the sense that it is sub-optimal in the general class of linear feedback rules but optimal within its own class. The most striking feature of these 'simple' feedback rules is that certainty equivalence does not apply. The reason for this is that quasi-optimal rules are not invariant with respect to the initial state of the economy, nor with respect to the covariance matrix of the disturbances. (See Poole (1970) and, in a rational expectations setting, Levine and Currie (1985a).) Quasi-optimal rules are also, in a sense, time inconsistent. The timeinvariant optimal rule computed at time t > 0 will be dependent upon y(i) and will therefore differ from the rule decided at t = 0. The source of time inconsistency lies in the two constraints imposed on the optimisation problem. These are first, the time invariance of the coefficients of D and second the restrictions on the coefficients which make them simple rules. If we lift either of these restrictions, in the absence of forward-looking relationships in the model, optimisation will result in policies which are time consistent. To summarise this section, in the standard control problem where the controller minimises a quadratic loss function with infinite time horizon Eo(Wo) where Wis given by (2.2) and the constraint is (2.1), the optimal feedback rule: (i) stabilises the system, (ii) is linear in the state vector y, (iii) is the same as for the deterministic problem (the principle of certainty equivalence), (iv) is independent of both the initial displacement of the state vector and the covariance matrix of the disturbances, (v) is time consistent. Quasi-optimal linear rules, by contrast, do not have properties (iii), (iv) and (v). All these conclusions apply to the standard control problem, a 'game' between man and nature, where there are no forward-looking expectations. Turning to rational expectations (RE), it is now wellestablished that in models with this feature, policy rules - whether optimal or not - generally do not satisfy time consistency. A controller who wishes to adopt such time-inconsistent rules needs to be able to convince the private sector that he or she will not renege and reoptimise at a future date. In other words, the credibility of announcements of time-inconsistent rules requires precommitment on the part of the controller. However, failing precommitment there does exist a class of sub-optimal rules which are time consistent (Holly and Zarrop, 1983; Buiter, 1984c; Cohen and Michel, 1988; Miller and Salmon, 1985a; Levine and Currie, 1985). In the remaining sections we consider all these five aspects of the

100

Theory and methodology

control problem in models with RE for both fully optimal and simple 'quasi-optimal' time-invariant feedback rules. 3

The deterministic control problem with rational expectations

We now generalise (2.1) to include forward-looking relationships. We write the dynamic system as dv

(3.1)

where z is an (n — m) x 1 vector of predetermined variables with z(0) given, x is an m x 1 vector of non-predetermined or free variables, and dxe = xe(t + dt, t) where xe(t, T) denotes the rational expectation of x(t) formed at time x. The deterministic analogue of (3.1) is (2.3) where y is now defined by [zrjc7]7'. 3.1

The optimal policy with precommitment

The following solution procedure, employing Pontryagin's maximum principle, draws upon work by Calvo (1978), Driffill (1982), Miller and Salmon (1985a) and Buiter (1984b). Consider for the time being a finite time horizon with loss function \ z~pt{yTQy + 2yTUw + wTRw)dt (3.2) -< Jo where yT = [z r x 7 ]. To ease the notational burden, we shall drop the bars in y and w (denoting the deterministic component of the trajectory). The optimisation at time t = 0 is then to minimise Wo subject to y = Ay + Bw. Define the Hamiltonian H = \t~pt(yTQy

+ 2yTUw + wTRw) + k(Ay + Bw)

(3.3)

where k{t) is a 1 x n row vector of costate variables. By the maximum principle, we minimise the Lagrangian L=W+

[k{Ay + Bw-y)dt= \{H- ky)dt (3.4) Jo Jo with respect to k{t\ w{t) and y(i). Consider first arbitrary variations on k. Then SL = 0 iff dH/dk = y

(3.5)

which from (3.3) simply gives the dynamics of the model, i.e., y = Ay + Bw.

Linear stochastic rational expectations models

101

Now consider arbitrary variations in w and y. Integrating (3.4) by parts, we have !' 0. Removing the two sources of time inconsistency - the constraints on Dx and the leadership role of the controller in exploiting TV = N(DX) - leaves us with a rule, equation (3.38), which is time consistent (see also Miller and Salmon (1985a) and Cohen and Michel (1988) for alternative derivations of this rule). 4

The stochastic control problem with RE

Having solved the deterministic control problem, the full stochastic problem (with a deterministic component) may be obtained, as in the standard problem, by a decomposition into deterministic and stochastic parts. Then (2.2)-(2.7) still apply with y = I v I and dy interpreted as J ~e x In this section we show that the principle of certainty equivalence applies to the stochastic control problem with rational expectations, i.e., the optimal feedback rule (3.19) for the deterministic component is also optimal for the stochastic component and hence for the full stochastic problem. We also consider certainty equivalence in relation to the simple quasi-optimal rules of section 3.2. We proceed by means of the following lemma which establishes a general relationship between the deterministic and purely stochastic components of the control problem.

108

Theory and methodology

Lemma. Consider a dynamic system given by (3.1). Let w = Dxz and w = Dxz be time-invariant feedback rules for the deterministic and stochastic parts of the control problem respectively. Assume that A + Bl D and A + BlD have the saddle-path property. Assume further that p > 0. Then if the welfare loss at time t = 0 for the deterministic component is written as Wo = (/(Z(0), Dx) where Z(0) = z(0)z r (0), the corresponding welfare loss for the stochastic problem takes the form E o (^o) =f(P~l Vu, Dx) where cov(dv1) = Vndt (havingpartitioneddv = [dv 2 ] conformably with [ J ] ) . Proof Consider the dynamics of the stochastic components with w = Dxz. Then corresponding to (3.28) the unique saddle-path is given by x = — Nz where N is as for the deterministic problem (see, for example, Currie and Levine, 1982). Substituting for w and x the control problem reduces to minimising

= i f %-''tr(/E(£i7))df,

(4.1)

Jo

where

NTQX2 -Q21N+ NTQ22N +DfU+ UTDl + DfRD

P=Qn-

and U is defined as before. The dynamic constraint is dz = £zdt + dv\

(4.2)

where K = An - Al2N + BxDx. Note that P and K are simply P and K defined in (3.30) and (3.32) with Dx replaced by Dx. It is a standard result (see, for example, Kwakernaak and Sivan (1972, p. 101)) that Z(t) = E0(z(t)zT(t)) with z(i) given by (4.2) satisfies the dynamic Lyapunov equation Z=KZ + ZKT+VU

(4.3)

pt

Define f = fS>e~ Z(t)dt by analogy with Tin (3.33). Then integrating by parts and putting lim,_»ooe~^Z(f) = 0 (assuming the solution to be stochastically stable) and Z(0) = 0 (since z(0) = z(0) is given) we arrive at the result yi) + p-lVu=0

(4.4)

Thus we have obtained the welfare loss E o ^ o ) = itr(Pf) with Tgiven by (4.4). Comparing (3.30) and (4.4) we can see that the only differences between the deterministic and stochastic parts of the control problem are that Dx and Z(0) in the former are replaced by Dx and p~l Vxx in the latter. Thus the theorem is proved. We can now use the lemma to obtain the following three results on certainty equivalence.

Linear stochastic rational expectations models

109

Theorem 2. Certainty equivalence applies to models with or without rational expectations if at least one of the following holds: (i) Vu = kZ(0) for some constant k, or (ii) the optimal value of Dx (that minimises Wo =/(Z(0), 50) is independent of Z(0). Proof

Both (i) and (ii) follow immediately from the lemma.

Theorem 3. Certainty equivalence applies to the full optimalfeedback rule for which the minimal welfare loss for the full stochastic problem is given by o) = ~ itr(7Vn(Z(0) + p~l Vu)) Proof Consider the following stochastic generalisation of the costate vector of p given by (3.16): dp = dp + Ldv

(4.5)

where p2(0) = p2(0) = 0 and L is some matrix to be determined. Partition L = [L2] conformably with [^] and augment the dynamic system (3.1) by (4.5). Then we have dz dp2 dpi

dxe

=

An G2l Gu A2\

0 #22

H2l 0

0 Hl2 Hn 0

An

z

G22

Pi

G\2

P\

A22

d/ +

X

B 0 wdt + 0 B2

dv1

L2dv L'dv 2 dv

(4.6) A rigorous proof that the stochastic system under control takes the form (4.6) can be found in Bismut's stochastic generalisation of Pontryagin's principle (Bismut, 1978, and see also Markose, 1986). To establish this theorem we first note that eq. (4.6) is in the general form described by (3.1) with z, p predetermined variables with

Uo)i k o b2(O)J [ o

and x} px non-determined variables. Denote L2dv by dv and cov[ dv J = Vdt. Also since p2(0) = 0 we have that 0

0

= Z(0)

110

Theory and methodology

say. From the theorem above we may now write E0(W0) =W0 + E0(W0) =/(Z(0), D) +f(p~l

V, D)

(4.7)

for feedback rules and

w = D\ „

From the deterministic result in section 3 above we know that the first expression in (4.7) is minimised at D given by (3.19) and that D is independent ofZ(0). It follows that the second term is minimised at D = D or in other words that certainty equivalence holds. The welfare loss for the stochastic problem from (3.24) and the lemma is then given by 1

'"

'"*)

(4.8)

To complete the solution to the stochastic problem we require L2. (Note that Lldv, as for dv2, plays no role in the solution.) Write (4.8) as (4.9, From section 5 we shall establish that Nu>09 N22 < 0 (both in the positive definite or negative definite sense) and NX2 = - Mi- It follows on partitioning v that (4.9) becomes Eo(^o) = ^ " tr[7Vn Vu + N22 V22]

(4.10)

The optimal value of dv = L2dv is then simply to set L2 = 0 in which case

Eo(Wo)= --p~ltr[NuVn]=

-?p-ltr(NuVn)-

The welfare loss for

the combinea deterministic and stochastic problem is then as given in the theorem. Theorem 4. Certainty equivalence applies to the optimal time-consistent feedback rule for which the minimal welfare loss for the full stochastic problem is given by

Proof Certainty equivalence follows immediately from the lemma and the result that DY as given by (3.38) is independent of Z(0). The welfare loss also follows immediately from the lemma. Theorems 2, 3 and 4 now enable us to see when certainty equivalence stands or falls. From theorem 2(i) if we consider displacements or

Linear stochastic rational expectations models

111

stochastic shocks to only one variable in the state vector, certainty equivalence clearly holds. Otherwise (i) is not satisfied in which case certainty equivalence rests on condition (ii). As we have seen in theorems 3 and 4, theorem 2(ii) does apply to the full optimal rule and to the optimal time-consistent rule. But it does not apply in general to simple rules. This will be demonstrated in the illustrative examples of section 6. An alternative way of representing and implementing the different feedback rules considered here is in the form of an open-loop trajectory plus a disturbance-contingent component. For example the initially optimal rule (3.19), which we have shown applies to the full stochastic problem, may be written as

where J= J\\ — J\2Nand we have used (3.22). A similar remark applies to the simple rules and the time-consistent rule. Formulating policy in the form of (4.11) will be particularly useful if the disturbance vector dv1 takes a simple (i.e., sparse) form. Otherwise the policy will appear at least as complex as the associated feedback rule (3.26) and must be ruled out if simplicity is required in policy design. We conclude this section by briefly discussing the case of no discounting, i.e., p = 0. If there is a stochastic component the welfare loss function (2.2) is inappropriate as the integral diverges. An alternative objective in this case would be to minimise the asymptotic loss function defined as asy E(W) = iasy E(yTQy + 2yTUw + wTRw)

(4.12)

Indeed, it has been argued by Taylor (1985) that the appropriate focus for macroeconomic policy is this stochastic equilibrium. The problem then becomes purely stochastic and the time inconsistency associated with the deterministic component disappears. This solution to time inconsistency presupposes that the 'rules of the game' would constrain governments to adopt a welfare loss function (4.12), i.e., having a zero discount factor. Moreover unless the fully optimal feedback rule is adopted any simple policy design (also advocated by Taylor (1985)) has to confront the problem that certainty equivalence does not apply in this case. Then the policy-maker needs to know the covariance matrix of the disturbances to arrive at the quasi-optimal rule. This issue is pursued further in our illustrative example 2 in section 6.

112 5

Theory and methodology An 'overstable' feedback rule

We have seen from section 3.2 that the optimal rule may be expressed as a feedback on the current value of z plus an integral feedback term on a discounted linear combination of past values of z. We now turn to ways of expressing the optimal rule as a feedback on [ J ] alone. However, we shall show that implementation of the policy in this form in fact results in a sub-optimal outcome. Equation (3.18) holds for the stochastic case, so we may write for both the deterministic and stochastic problem:

Pi = -Nuz-Nl2p2,

(5.1)

x = -N2lz-N22p2

(5.2)

1

Thus, providing Nn and N22 exist, we have z = - Nu\P\ + NX2p2) and p2= - N22(x + N2Xz). We can now eliminate z in (3.19) and obtain a feedback on [^] or we may eliminate p2 and obtain a feedback on [ J j . The latter is particularly interesting because this would appear to be a feedback rule in the simple class examined in section 3.2. It takes the form (5.3) Substituting for D from (3.20), this becomes

w=- J R- 1 [C/ r +5 r Jr]^]

(5.4)

Xn=-Nn

(5.5)

where

X2l=

+ Nl2NiJN2l9

-N22'N2U

Xl2 = Nl2N22\ l

X22 = - N22

(5.6) (5.7) (5.8)

On comparing (5.3) with (2.9), we can see that matrix X turns out to be the unique non-negative definite solution to the Riccati equation (2.10). In what follows we shall assume that X is positive definite. Suppose that the authorities announce their optimal rule in the form (5.4). This rule only has the saddle-path property required for a unique rational expectations solution in the context of the model (3.1) augmented with the dynamic relationships for p, i.e., for a 2n x In dynamic system. If, as we have assumed, private agents are rational in the sense that they perceive (or act as if they perceive) of the model (3.1) alone, then rule (5.4)

Linear stochastic rational expectations models

113

completely stabilises the system (i.e., has n stable eigenvalues). Then for our RE model, we have a case of too many stable eigenvalues. Rational expectations models with too many stable roots have an infinite number of stable paths leading to the equilibrium. A number of recent studies have addressed this problem (Taylor, 1977; Minford, Matthews and Marwaha, 1979; Buiter, 1984b; Turnovsky and Scarth, 1983). The problem in essence is to find n' missing boundary conditions where ri is the number of 'unwanted' stable roots. In our example nr = m and the problem is to choose x(0) in some way. There is no general agreement on the appropriate choice of x(0) in the literature cited. However, the following theorem establishes the superiority of the saddlepath solution (the rule announced as (3.19)) over the overstable rule (5.4) whatever the choice ofx(0). Theorem 5. The feedback rule (5.4), perceived by the private sector in association with the model (3.1), will have too many stable roots for a saddle-path solution. Whatever the chosen missing boundary conditions x(0) for this rule, the optimal rule (3.19) or (3.26) will be at least as good. Proof The welfare loss for the overstable rule, by theorem 3 or 4, is given by Eo(^o) = itr(AXr(0) + p~l V)) T

T

T

(5.9) T

where 7(0) = y(0)y (0) and y (0) = [z (0)x (0)]. Note that the stochastic component dv2 now affects the solution. We are missing m boundary conditions to define x(0). However, we can find the minimum value of Eo(^o) with respect to x(0). Considering the deterministic component first we have min Wo = min\(z T (0)X u z(0) + 2z r (0)^ 12 x(0) + xT(0)X22x(0)) (5.10) Differentiating, the first-order condition is dWo/dx(0) = 2X{2z(0) + 2X22 x(0) = 0

(5.11)

and differentiating again d2W0/dx(0)2

= 2X2r2

(5.12)

Since X22 > 0 it follows from (5.11) and (5.12) that Wo is minimised at x(0) given by x(0)=-X22lX2lz(0)

(5.13)

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Theory and methodology

(using Xj2 - X2i). Substituting (5.13) into (5.10) the minimal value of Wo is given by Wtn = hT{<S){Xu - Xl2X22'X2l)z(0)

(5.14))

Substituting for Xlu Xl2, X2 and X22 from (5.5)-(5.8), (5.14) becomes tn = hT(0)NuzT(0) = hv(NuZ(0))

(5.15)

which turns out to be the welfare loss for the correctly implemented optimal policy (3.19). For the full stochastic problem we can similarly minimise E0(W0) with respect to x(0) and dv2 to give p~l Vn))

(5.16)

where x(0) is given by (5.13) and dv2 = - X22X2Xdvx. Thus we have shown the overstable rule perceived of by the private sector alongside the model (3.1) leads to an outcome which depends on initial conditions x(0) and the stochastic component dv2. Only for the best possible outcome where x(0) = - X22 X2X z(0) and dv2 = - X22 X2Xdvx will the performance of this rule be as good as the optimal policy implemented as (3.19). This completes the proof. Of the various proposed solutions to the non-uniqueness perhaps the most straightforward is that of Minford, Matthews and Marwaha (1979). They argue that forces outside the model would impose a 'most stable path' in the same that they would impose the saddle-path solution in the case of n - m stable roots. From the private sector's point of view (as opposed to the controller's), a most stable path would imply that their decision variables, the non-predetermined variables x, jump immediately to their equilibria, i.e., JC(O) = 0. This is not the best possible outcome from the controller's viewpoint (this would be given by (5.13)) so that, given this solution to the non-uniqueness problem, the overstable rule will lead to an inferior performance as compared with the correctly implemented optimal rule. 6 6.1

Illustrative examples Model 1

Our first model consists of two dynamic relationships: dm = n(axe - a2r - m)dt + dvi, e

de = rdt,

(6.1) (6.2)

Linear stochastic rational expectations models

115

where the following notation is used: m e r dvi

= nominal money supply; = exchange rate (home currency unit/foreign currency unit); = domestic rate of interest; = white noise disturbance.

All variables are measured in terms of deviations of their logarithm from equilibrium except for the interest rate which is measured as deviations of proportions. All parameters are defined to be positive. Equation (6.1) is a sluggish adjustment process for money demand with mean lag ju~l and a long-run demand for money function m = ax e + a2r. Equation (6.2) models the exchange rate as asset market determined under conditions of perfect capital mobility. The expected rate of depreciation of the exchange rate in an interval dt (denoted by dee = ee(t + d/, /) - e(t) where ee(r, i) is the expected exchange rate at time r, formed at time i) exactly offsets the interest-rate differential in favour of the home currency. Unlike the money stock which adjusts slowly and is a predetermined variable, the exchange rate is non-predetermined and can make discrete jumps in response to the changes in exogenous variables or policy rules. The white noise disturbance vector dvx indicates the presence of a shock to the demand for money. Equations (6.1) and (6.2) can be written in the form (3.1), with w = r,

[x] = [T] and

o o J'

B

-[i J

(6 3)

*

Let us consider the first optimal policy with precommitment for the deterministic policy. For the welfare loss function we assume no time discounting (p = 0) and we put

O - [ ; "]. U-0. R-l

(6.4)

We choose parameter values a = b = 2, ax = 1, a2 = 2 and ju = 0.4. Then from the solution procedure set out in section 3.1 we have

J=

0 -a 0

0 0 -1

IMi\

—b

nua-,

o0

and by numerical computation we obtain, for matrix N,

(6.5)

116

Theory and methodology

- 1-28

-0.58

(6.6)

from which the optimal rule is obtained as r = R~1[-

BxNlu B2 - BiN12]\m

IP2]

= 1.02m - 0.54/?2

(6.7)

where p2 = [0.65m - 0.79/?2] If we set m(0) = 1, the welfare loss is - itr(JV n Z(0)) = 0.64 from (6.6). The trajectory for the money supply is from (3.22) given by

where - 1.45 0.65

0.54 - 0.79

for chosen parameter values. The trajectory for the exchange rate e, from (3.18), is then given by

e=-[N2l

AT22][jJ = [-0.58 0.28] [ j j

(6.9)

Consider next a feedback rule of the form r = Z>[T] which, from (3.29), may be written r = /?m, i.e., as a feedback only on the predetermined variable m. From the eigenvalues and eigenvectors of A + BD, with D = W 0], it is straightforward to show that the unique saddle-path for this example is given by e=

fiax

m=-Nz

(6.10)

where the positive eigenvalue 1 is given by (6.11) Thus N =f(fi) where it should be noted f(fi) is a non-linear function in p. Consider now the 'quasi-optimal' rule within the class of rules r = /3m for the deterministic problem. Then P = a + bN2 + p2 [in (3.30)], K= - ju(l + a2p + axf{P)) [in (3.32)], and 7 = - m\Q)/2K\\n (3.33)]. It follows that

Linear stochastic rational expectations models

117

Table 4.1. Comparison of the optimal rule with precommitment, the quasi-optimal rule with precommitment and the time-consistent rule for model 1

Welfare loss

Optimal rule with precommitment

Quasi-optimal rule with precommitment

Time-consistent rule

0.64

0.66

0.67

Trajectory for initial disturbance m(0) = 1 Time t (in years)

m(t)

0 1 2 3 4 5

1.00 0.29 0.13 0.07 0.04 0.03

*t)

1.03 0.18 0.03 0.01 0.00 0.00

m{t) 1.00 0.27 0.07 0.02 0.01 0.00

r{i) = Pm{i) £ = 0.83

m{t)

0.83 0.22 0.06 0.02 0.00 0.00

1.00 0.21 0.03 0.03 0.00 0.00

r{i) = Pm(t) P= 1.07

1.07 0.22 0.03 0.03 0.00 0.00

which can be minimised with respect to ft by a standard numerical technique. For our selected parameter values and m(0) = 1 the quasioptimal rule turns out to be r = 0.83m with minimal welfare loss 0.66. The trajectory for m is given by m{t) = m(0)eKt

(6.13)

where K— - 1.31. Then the trajectory for e follows from (6.10). We now turn to the third type of rule discussed in section 3, the optimal time-consistent rule. Following the general procedure we find that the solution is r = fim where ft = jua2Sand S is the positive root of the Riccati equation - 2//(l + axN)S - (a + bN2) + fi2a22S2 = 0

(6.14)

Solving (6.14) we obtain /? = 1.07 and the minimal welfare loss works out to be 0.67. The trajectories for m and e follow as for the simple rule preceding but with K= - 1.57. The results for the three types of rules are tabulated in table 4.1. Two points of note emerge from this table. First there are welfare losses associated with both an insistence on a simple feedback rule on m(t) and with time consistency (about 5 per cent in both cases). Second, the

118

Theory and methodology

relationship between r{t) and m(t) is not linear and time-invariant for the optimal rule with precommitment. We finally turn to a feedback rule of the form discussed in theorem 5. From (5.5)—(5.8) (or direct solution of the Riccati equation) we obtain Z=

[2.49 2.08 [ 2 . 0 8 3.57

The feedback rule associated with Xis, from (5.4), r= -R-xBTx\™\=

-0.09m-1.9 h?

(6.16)

This is certainly an odd-looking rule but, nevertheless, it is one of the ways in which the optimal rule (6.7) can be expressed. If, however, the rule is announced in this form and seen by the private sector in the context only of the model (6.1) and (6.2), then the resulting system will have two stable eigenvalues. A rational expectations solution will then exist if an additional initial boundary condition for e(0) can be specified. The 'most stable path' solution of Minford, Matthews and Marwaha (1979) suggests that this extra condition is e(0) = 0. For the deterministic problem with ra(0) - 1 this gives a welfare loss of 1.25 confirming the inferior performance of the overstable rule for this choice of e(Q). 62

Model 2

In our second model, we retain (6.2) but replace (6.1) with a more developed output/price/money interaction. The model consists of Ay = Vfaie-p)

- a2(r -pe) - y]dt + dv1?

(6.17)

dp = [fix y + p2(e - p)] dt + dv 2 ,

(6.18)

dm = Y2[yxy - y2r + p - m]df,

(6.19)

e

de = rdt,

(6.2')

where the following additional notation is used: y = real output; p = general price index; pe = expected inflation rate. Equation (6.17) represents the IS curve with output adjusting sluggishly to competitiveness and the real interest rate with mean lag W^1. Equation (6.18) is a Phillips curve with prices adjusting sluggishly to changes in output and import costs. Equation (6.19) is a sluggishly adjusting LM curve. We consider dv = (dv!dv 2 ) r to be a vector made up an aggregate demand disturbance and a supply disturbance, respectively.

Linear stochastic rational expectations models

119

Table 4.2. Comparison of the initially optimal rule with precommitment, quasi-optimal rules with precommitment and the time-consistent rule for model 2 Actual disturbance (unit variance)

Expected disturbance All

dv, dv2

Policy rule

dv,

dv2

Certainty equivalent rules Optimal with precommitment Time-consistent rule (r = 0.1>> + 0.01/?) Overstable rule (r = 0.1\y + 0.02/? + O.OOOle)

0.85 0.98 1.05

1.16 2.09 1.90

Quasi-optimal rules with precommitment General (r = 0.06j> - 0.1/? + 0.5m) Monetary (r = 0.61m) General (r = - 1.307 - 0.02/? + 0.21m) Monetary (r = 1.72)

0.86 0.86 1.80 0.88

1.49 1.40 1.16 1.35

By the assumption of RE, pe, the expected inflation rate, is given by (6.18) as pe = /?i y + /?2(e - p). Substituting into (6.17), we may write the model in the form (3.1) as

Pi V2 0

0 0 ^2

0

0 0 (6.20)

B=

120

Theory and methodology

Unlike model 1, we consider only the stochastic problem (i.e., Wo = 0) with no time discounting (p = 0). We put a

0 b

Q= 0

, «-.,

0

(6.21)

0

and parameter values chosen are a = b = 2, ax = 0.3, a2 = 0.1, yx = y2= 1.0, px = 0.15, #> = 0.25 and WX = W2 = 0.5. Two forms of feedback rules of the type w = Dz are considered. First, a general linear rule of the form r = dxy + d2p + d3m, and second, a monetary indicator rule r = /?ra. For this model we shall consider the purely stochastic problem discussed at the end of section 4 corresponding to a stochastic world with p = 0. Consider the disturbances one at a time. Then for each of these two rules a quasi-optimum form may be found using the general technique of section 3. Their properties may be demonstrated in a 'payoff matrix in which the ijth element is the welfare loss when disturbance dv, is planned for, but dv7 occurs. The results are displayed in table 4.2 and compared with those for the optimal, the time-consistent and the overstable rules. The latter are certainty equivalent; but the performance of the time-inconsistent quasioptimal rules are dependent on the disturbance for which the rule is designed and on the actual outcome. For instance, the general rule when designed for a supply shock (dv2) matches the full optimal rule if a supply shock actually occurs. If a demand shock (dv^ occurs, the same rule performs very badly. The poor performance of both the time-consistent and the overstable rule is very striking. As strategies, they are dominated by three out of four of the restricted rules and are considerably inferior to the full optimal rule. It is interesting to note that for this model they are also very similar. The main point which the results for model 2 demonstrate is how the non-applicability of certainty equivalence greatly complicates the task of policy design if we choose to confine ourselves to simple feedback rules. Even with only two disturbances to consider, only one of the four simple rules (r = 0.06y - 0.1/7 + 0.5m) is dominated. (How a policy-maker could deal with the problems this raises is discussed further in Levine and Currie, 1985). However, it is possible that simple rules do exist which perform reasonably well for all disturbances. The search for simple rules which are 'robust' in this sense is the subject of current research by the authors.

Linear stochastic rational expectations models

121

Notes 1 A weaker, necessary condition for a minimum is, from (3.6), X(T)y(x) = 0. 2 The transversality condition for the infinite time horizon is

or, if the condition in footnote 1 is adopted,

Model stability ensures both conditions are satisfied and is the adopted assumption throughout the chapter.

Credibility and time consistency in a stochastic world

1

Introduction

It is a familiar feature of optimal policies in models with forward-looking rational expectations that they are time inconsistent (see, for example, Kydland and Prescott, 1977; Calvo, 1978; Driffill, 1982). This is because, if optimal policy is formulated by Pontryagin's maximum principle, the passage of time leads to an incentive to renege on the initial optimal plan and to adopt a new one. Since the private sector can anticipate this, the initial policy lacks credibility. Appreciation of this problem has motivated the examination of policies that are optimal within the constraint of being time consistent (see, for example, Buiter, 1983; Cohen and Michel, 1988; Miller and Salmon, 1985a; Oudiz and Sachs, 1985; Levine and Currie, 1985). This analysis has been conducted in the context of developments of the standard control state space approach. An alternative game theoretic approach to the issue has examined the possibility that reputation effects may sustain the full optimal or ideal rule. The existing literature considers this in the context of very simple model structures. Thus Barro and Gordon (1983b) show how reputation effects may move the best sustainable rule towards the ideal rule. Backus and Driffill (1985a, 1985b) demonstrate how the existence of uncertainty over the objective function of government may be sufficient to sustain the ideal rule in a simple model structure. But the generalisation of these results to models with more complicated dynamic structures is by no means straightforward. In this chapter, we seek to bring together the game theoretic and state space approaches to this problem, and examine how continuing stochastic Journal of Economics, 47(3) (1987), 225-52. Springer- Verlag. This chapter was written while David Currie was an Houblon-Norman Research Fellow at the Bank of England; this financial support, together with that of the Rockefeller Foundation, is gratefully acknowledged.

122

Credibility and time consistency

123

noise affects the sustainability of the ideal rule. Our main result is that, provided the degree of discounting is not too great, the ideal or full optimal rule will be sustainable in a sufficiently stochastic environment. The essential idea is a simple one. Government may renege on the ideal rule, but thereafter their commitment to pursue the ideal rule will not be credible. After reneging, therefore, government will be constrained to pursue the time-consistent rule. In considering whether to renege, government will weigh the advantages of reneging on commitments made with respect to past disturbances against the costs of being constrained thereafter to pursue the time-consistent rule with respect to future shocks. Since the performance of the time-consistent rule is, in general, inferior to that of the ideal rule, the cost of reneging is an inferior policy performance in handling all future disturbances to the system. Clearly if the rate of discount is not too high, this future cost will outweigh the gains from reneging. If this is the case, government has an incentive to adhere to the ideal rule, rather than reneging. Because the private sector can figure this out, the ideal rule is credible and sustainable. The significance of stochastic noise in this context is that it transforms the nature of the policy problem from the one-shot deterministic policy game into a repeated stochastic policy game. The repeated character of the game gives an incentive for government to invest in its reputation to secure long-term policy gains, rather than obtain a short-run advantage from reneging. This provides a rationale for the doubts expressed by Taylor (in commenting on Barro and Gordon, 1983b) about the usefulness of the time-inconsistency model as a positive theory of macroeconomic policy. Before deriving the main result of the chapter and illustrating it in the context of a specific dynamic model, some preparatory analysis is required. In section 2, we derive the full optimal or ideal rule. In section 3, we demonstrate the problem of time inconsistency, while in section 4 we show how the time-consistent rule may be derived. Section 5 then proves our central result, that with continuing stochastic noise the ideal rule may be sustainable provided that the discount rate is not too high. Section 6 then sets up a specific dynamic model, and section 7 reports results derived from this model illustrating the propositions of section 5. 2 2.1

Optimal policy with precommitment The optimisation problem

The economic systems considered in this chapter have, as a general form, the following linear stochastic differential equation

124

Theory and methodology

f + ifwdf+dv

(2.1)

where z is an (n - m) x 1 vector of predetermined variables, x is an m x 1 vector of non-predetermined variables which can freely jump in response to 'news' and w is an r x 1 vector of control instruments. The stochastic environment is modelled by v, a b x 1 vector of disturbances with E(dv) = 0, cov(dv) = Vdt and dxe denotes the rational expectations of dx formed at time t on the basis of the information set I(t) = {z(s), x(s): s^ t} and knowledge of the model (2.1). The matrices A, B and V all have time-invariant coefficients and, in addition, V is symmetric and nonnegative definite. All variables are measured as deviations from some long-run trend deterministic equilibrium. Exogenous variables that follow ARIMA processes may be incorporated into (2.1) by a suitable extension of the vector z. The initial conditions of the system at time 0 are given by z(0). Turning to the optimisation problem, the controller is concerned about outputs s = Fy + Ow

(2.2)

where s is an n° x 1 vector, y = [*] and F and 0 are timeinvariant matrices. The welfare loss to be minimised at time £ = 0 is assumed to be of the form Eo(Wo(z(0)) where Eo denotes expectations formed at time t = 0 and Wo(z(0)) = \ \ t-pt[sTQxs + wTQ2w]dt

(2.3)

where p ^ 0 is a discount rate, Qx and Q2 have time-invariant coefficients and, in addition, Q2 is symmetric and positive definite and Q\ is symmetric and non-negative definite.1 In the absence of discounting, the objective is to minimise the asymptotic loss function defined by The control problem is then to minimise (2.3) with respect to w subject to (2.1) and (2.2). We proceed by decomposing this problem into separate deterministic and stochastic components (see Chow, 1975). To do this let y = [ | ] and w = w denote the solution to the deterministic problem with dynamics given by y = Ay + Bw

(2.4)

where z(0) = z(0) are the initial conditions and we have used the fact that xe = x in this deterministic case. Now let y — y — y and w — w — w. Then subtracting (2.4) from (2.1) we obtain

Credibility and time consistency e

= Ay At + BwAt + dv

125

(2.5)

Substituting y = y + y and w = w + w into (2.3) and noting that y is of order dv and E(dv) = 0 we arrive at = Wo(z(0)) + Eo(Wo(z(0)))

(2.6)

where Wo(z(0)) and Wo(z(0)) are given by (2.3) and (2.2) with (y,w) replaced by (y, vi>) and (y, w) respectively. The two parts of the control problem are then the deterministic problem given by the minimisation of Wo(z(0)) subject to (2.4) and the stochastic problem given by the minimisation of Eo(W(z(0)) subject to (2.5) with £(0) = 0. 2.2

The deterministic optimisation problem

The deterministic problem has been the subject of a number of recent papers. (See Calvo, 1978; Driffill, 1982; Buiter, 1984c; Miller and Salmon, 1985a; Levine and Currie, 1987b.) The solution, employing Pontryagin's maximum principle, may be outlined as follows. Let us define the Hamiltonian H=h~pt[sTQiS~+

wTQ2w] + k(Ay + By)

(2.7)

where s = Fy + 0w and X(t) is a 1 x n row vector of costate variables. The first-order conditions for a minimum are dH/dX = p, dH/dw = 0 and X = - dH/dy. Differentiating (2.7), these lead to the following dynamic system under optimal control

y ]\A-BR-*UT p \~[UR-XUT-Q

-BR~'BT ]\y pI-AT+UR-lBT\l

where U = rTQx 0 then the result will be a rule of the form of (3.4), but with the integral evaluated over [t0, t], not [0, /]. It is this feature of the rule which changes with the passage of time, so that the rule may be timeinconsistent (see Kydland and Prescott, 1977; Cohen and Michel, 1988). This may also be seen by noting that the optimal policy sets/?2(0) = 0, but that, from (2.7), p2(t) is generally non-zero for t > 0. Reoptimisation at

128

Theory and methodology

time t0 > 0 leads the controller to setting p(t0) = 0, thereby reneging on the policy formulated at time t = 0. If precommitment of future policy (whether that of the controller or his successors) is ruled out, a critical question is whether this rule will carry credibility with the private sector, or whether they will expect the controller to renege on the rule in the future. If expectations are based on the prospect of reneging, and not consistently with the rule itself, then the rule is, of course, not optimal, for its optimality properties are based on the assumption that expectations are formed consistently with it. The extent to which policy (3.1) announced at t = 0 with p2(0) = 0 becomes sub-optimal at time t > 0 can be seen from the welfare loss Et(Wt) where E, denotes expectations formed at time t and

Wt =f(z(tlP2(t)) = \ f % - « * - V & s + wrg2vv)dT

(3.5)

computed at time /. (In the notation of (3.5), Wo(z(0)) of (2.3) now becomes /(z(0), 0) since p2(0) = 0.) As in section 2 we first consider the deterministic problem and then apply certainty equivalence. By Pontryagin's principle at time t we have

Then writing p = Sy where S is the Riccati equation and integrating we obtain

Wt = yTSy = yTy

(3.7)

Using (2.10) and the results N{2 = — N2i and iVYi = Nn (see appendix) we may substitute for/?! and x in (3.7) to obtain

N22p2(t)pI(t))

(3.8)

T

where 2{i) = z(i)z {i). As we see from the appendix, N22 < 0 so that the original policy (3.1) announced at time t = 0 becomes sub-optimal at time t > 0 in the sense that the welfare loss may be reduced by an amount - ^tT(N22p2p2) by reoptimising and puttingp2{t) = 0. This also holds for the stochastic problem since (3.8) then becomes E , W ) = - itr(7V n (Z(/) + p~x Vu) + N22p2{i)pZ(i))

(3.9)

so that the potential gains from reoptimisation are - \tr(N22p2p2) which is now a stochastic expression whose value will depend on the pattern of disturbances up to time t.

Credibility and time consistency

4

129

Time-consistent policy

The time inconsistency of optimal policy arises because it is not derived from dynamic programming which assumes Bellman's 'Principle of Optimality'. The following sets out a solution procedure for obtaining an optimal policy which satisfies Bellman's principle and is therefore time consistent. It generalises the solution of Cohen and Michel (1986) which only considers the case of one predetermined variable, one non-predetermined variable and one instrument; but neglects the questions discussed by these authors relating to the micro-foundations of systems in the form (2.1). Our solution also provides a continuous-time analogue of the solution provided by Oudiz and Sachs (1985) and leads, by a different route, to the time-consistent result proposed by Miller and Salmon (1985a). From the discussion in the previous section we have seen that a necessary condition for a time-consistent rule is that it must not depend on past values of the state vector (contrasting with the optimal timeinconsistent rule (3.4)). We therefore seek a solution w = Dxz for the time-consistent feedback rule (i.e., a rule without memory). Substituting into (2.1) we then have

]]^]d? + dv

(4.1)

which has a solution x = - N(Dx)z where N = M22 M2U M = \_M\\ M*2\ partitioned conformably with [*] being the matrix of left-eigenvectors of the dynamic matrix in (4.1) arranged so that the last m rows are associated with the unstable eigenvalues. (We assume that Dx is such that A + B[DX 0] in (4.1) has the saddle-point property.) Substituting x = - Nzwe may write the expected welfare loss at time t as Et(Wt) where Wt =/(z(0) = 1 f t~p{x~t\zTQz

+ 2zTUw + wTRw)dt

(4.2)

Jt

where Q = Qn - NTQl2 - Q2XN + NTQ22N, U = Ul - NTU2, U= [££] and Q = [Q2[ Q222\ both partitioned conformably with [^]. Along the saddle-path, substituting x= - Nz into (2.1) we have dz = (Au - Al2N)zdt + Bl wdt + dv1

(4.3)

The optimisation problem at t = 0 is then to minimise /(z(0)) given by (4.2) subject to (4.3) within the class of rules w = Dxz. It is important to stress that in (4.2) and (4.3) N = N(DX). This relationship can be

130

Theory and methodology

considered to be the reaction function of the forward-looking private agents capturing their response, x = - N(Dx)z, to a policy announcement. If the controller were to exploit its knowledge of this reaction function, then it would be acting as a Stackelberg leader (by analogy with standard oligopoly theory). The non-linear optimisation problem that emerges can be solved (see Levine and Currie, 1985); but the rule obtained in this way will not be time consistent. A time-consistent solution must satisfy Bellman's 'Principle of Optimality' which states that 'An optimal policy has the property that whatever the initial state and decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision' (Bellman, 1957). It follows from this that the optimal value of Wt at time /, /*(z(/)) say, must satisfy the following 'fundamental recurrence relation' /•(z(/)) = min[/(z(0, w(t))At +/*(z(/ + At))]

(4.4)

w

where It = \[zTQz + 2zTUw + wTRw\ We can now envisage Dx converging to a stationary time-consistent value by the following iterative procedure. Without loss of generality we imagine the last optimisation taking place at some time T over a time horizon [T, oo] leading to a rule w = D^z. Then we may let T-^oo, This optimisation will then have a relationship to the optimisation at time T- At an instant earlier given by (4.4). The important point to note is that the saddle-path x = - N{Dx)z will be determined by the last optimisation, i.e., N = NiD^). Then the penultimate optimisation leading to w = Dxil)z must take N as given. The minimisation of (4.2) subject to (4.3) with N given is a standard optimisation problem. By Bellman's equation derived from (4.4), the deterministic solution is found by minimising

J(z(0, w(t)) + \^J(Az + Bl w)

(4.5)

with respect to w where A = Au - AUN and Af is held fixed at N(D{°\ Alternatively Pontryagin's maximum principle can be employed. By either route the following standard solution is obtained (4.6) where &l) satisfies the Riccati equation 0 (see, for example, Levine and Currie, 1987b).

(4.7)

Credibility and time consistency

131

Repeating the iteration with N = N(D{1)) etc. we thus have an infinite sequence D{°\ D{1\ . . . . We have not been able to derive any convergence properties for this sequence analytically. However, for the model chosen in this chapter (and for a number of other models) we have found the sequence to converge to a stationary value, Dx say, provided a suitable initial rule M0) is chosen. The rule w = Dxz will then satisfy Bellman's Principle of Optimality and therefore be time consistent. Corresponding to (3.9) the welfare loss for the stochastic case at time t is given by E,W) = itr(5(Z(r) + p~l Vn))

(4.8) 0

(2)

where S is the stationary value of the sequence S* , S . . . 5

The sustainability of the optimal policy

We begin by summarising the results of sections 3 and 4. Denote the welfare loss at time / for the initially optimal but time-inconsistent policy by Et(W?p) and for the time-consistent policy by E^WJ0). Then we have E£W?P) = - ±tr(Nu(Z(t) + / T 1 Vn) + N22p2{t)p^i))

(5.1)

EiWf0) = 2tr(S(Z(f) + p~l Vn))

(5.2)

and where N is given by (2.10), p2 by (3.3) and S is obtained iteratively from the Riccati equation (4.7). We now examine the possibility of the government sustaining a better rule than the time-consistent policy given in section 4.3 We assume for the argument that there exists a single government with an infinite life. As we have argued the optimal policy lacks credibility because there are potential gains from reoptimisation at time /, by putting p2(t) = 0, after the initial policy announcement. We stress potential gains because a government that wished to pursue a time-inconsistent policy will need to establish a reputation for being a government that does not renege on earlier commitments. An act of reneging, by announcing a new reoptimised policy, will destroy that reputation. However, a switch to a timeconsistent policy is a possible policy option, because after losing its reputation of a government that resists the temptation to reoptimise, the time-consistent policy becomes optimal. In what follows we assume the reputation a government requires to implement a time-inconsistent policy can never be re-established after reneging.4 (This is an extreme assumption and further work in progress will consider the more general but difficult case of a finite 'punishment period'.) In this case an incentive to renege at any time t can be measured

132

Theory and methodology

by the welfare loss difference B^W?*) - E^WJ0). If this quantity becomes positive for some value t > 0, then the optimal time-inconsistent policy cannot be sustained if the private sector is aware of the incentive that now exists to renege. In what follows, we assume that the private sector knows the objective function of the government so that it can work out the welfare loss difference. We therefore need to focus on the condition Et(W?p) < Et(Wj°) for all t, which from (5.1) and (5.2) may be written as Nn)(Z(t) + p-l Vn)) > " tr(N22p2(t)p^(t)) Vt

(5.3)

Since the welfare loss for the deterministic problem under the timeconsistent rule must be greater than under the optimal rule, we have S > - Nu. Moreover, from the appendix, N22 < 0. Hence both sides of (5.3) are positive. In principle, then, condition (5.3) could be violated. In fact if dv has an undoubted distribution since p2 and Z(i) are stochastic variables which can take any values with some finite distribution there will always be some non-zero probability that (5.3) is violated. We will therefore consider in turn the cases of bounded and unbounded variables. Case (i): Bounded variables (discrete time only) If dv is bounded, it follows that all variables of the system are bounded, including costate variables. However, it is a standard result of stochastic processes that any continuous martingale with the property E(dv) = 0, Cov(dv) = Vdt is Gaussian, and therefore unbounded. In discrete time we may consider an analogue of (2.1) with disturbances ut which are bounded and have mean zero and constant variances. In the limit as the unit of time becomes small the results derived in previous sections will hold approximately. We then have the following proposition. In a stochastic context, if the rate of discount is sufficiently low, then there is no incentive to switch from the optimal policy to the time-consistent policy. To prove this proposition let us define5 fc, = kY(p) = max( - ix{N22p2pl + (S + N and k2 = h2{p) = tr((S + Nx0 Vx0 > 0

(5.4)

Then condition (5.3) is satisfied if p~xk2(p) > kx(p). For small p we may expand kx(p) and k2(p) by Taylor's theorem and approximate Mp) s fci(0) +fcl(0)/>and k2(p) = k2(0) + k\($)p where fc}(0) and k\ denote first derivatives at p = 0. Then the condition is satisfied approximately if

Credibility and time consistency

133

(5.5) l

Suppose first that k 2(0) > &i(0). Then the right-hand side of (5.5) is negative and the condition is satisfied for all p. Ifk2(0) < ki(O) then (5.5) is satisfied if

which proves the proposition. Case (ii) dv ~ N(0, Vdt) The proposition above must now be amended to the following. In a stochastic context, an incentive to switch from the optimal to the timeconsistent policy exists but with probability 6 per unit of time, where 6 can be made to be as close to zero as we like by choosing the rate of discount to be sufficiently low. To prove this proposition, we first redefine kx in (5.4) to be kx = max( - tr(N22p2(t)p2r + (0 + (S + Nn)Z(t)))

(5.7)

which is ki measured along the expected trajectory at t = 0. Then provided 0 < p < fc2(0)/(£i(0) - kl2(0)) the condition Et(Wjc)>Et(W?p) holds along the deterministic (expected) trajectory. Now consider stochastic variations about the expected trajectory. Write z = z + z and/?2 = Pi + p2- Then the condition to be satisfied, (5.3), becomes (tr((5 + Nu)(Z(t) + p-lVu) + N22p2pl) > - (tr((S + Nn)(zzT + 2zf) + N22(p2p2T + 2p2p2T)

(5.8)

The right-hand side of (5.8) is stochastic, but we may find a bounded confidence interval for it. The left-hand side clearly tends to oo as p tends to zero. From this, it is clear that, for arbitrarily small probability 9, we may find an upper bound for the discount rate below which the righthand side of (5.8) exceeds the left-hand side with probability smaller than 6. From this, the result follows. For subsequent sections, we require an explicit confidence interval for the right-hand side of (5.8). In what follows, we derive a very conservative confidence interval. This also serves to formalise the argument of the previous paragraph. To do this, we note that the right-hand side, Q, of (5.8) may be rewritten as

134

Theory and methodology

which is a linear plus a quadratic term in [zTp7]. We also have that

M=

E]

(5.10)

where the covariance matrix E can be obtained by standard analysis (see chapter 4, section 4). It follows that

Turning to the quadratic term in (5.9), using the result obtained in the appendix we may write M

ii x I ^yz

A

"ZIFI — 'I '*•!

where rji2 ~ %2(l), A is the matrix of eigenvalues of -LTN22L T

(5.12)

and

1

£22 = LL partitioning E = [j^ , j ^ J . Hence the right-hand side of (5.8), Q, such that

"> + # „ 0 0 N22 y\S+NU

[ 0

at time t with probability 1 - a where za and %L a r e critical values for the standard normal distribution and x2 distribution respectively. (Note that since {A,} are the eigenvalues of a positive definite matrix, Xi > 0 for all i.) Since the left-hand side of (5.8) tends to oo as p tends to 0 whilst the right-hand side remains finite, a can be made as close to 0 as we like by choosing p to be sufficiently low. This completes the proof. 6

An application to a specific dynamic model

To illustrate the issues examined in the previous sections, we consider the formulation of optimal policy in the context of the following four equation model: dy = y/(ai(e — p) — a2(r — pc) + a 3 v — a4s — y)dt + vx

dv = [ - §Qs - <j)ij + §2(e -p)]dt

- dp

(6.1)

(6.2)

Credibility and time consistency

135 (6.3)

e

de = rdt

(6.4)

where all variables are measured in terms of deviations from a given long-run trend path and where the following notation is used: y e p r v s dvi,dv 2

logarithm of real output; logarithm of nominal exchange rate (measured as the price of foreign exchange); logarithm of domestic prices; nominal interest rate (measured as a proportion); logarithm of the real net financial wealth of the private sector; autonomous taxes; demand and supply shocks respectively.

Equation (6.1) represents the IS curve, with output adjusting sluggishly in responses to competitiveness, the expected real interest rate, real wealth and autonomous taxes. Equation (6.2) is derived from a log-linearisation of the equation for private-sector net nominal financial wealth accumulation, depending on autonomous taxes, income and real competitiveness as the determinants of the sum of the government budget deficit and the current account of the balance of payments. Real financial wealth accumulation is then obtained by deducting the rate of inflation, p (revaluation effects being neglected). Equation (6.3) represents a Phillips curve relationship, with inflation depending on output and real competitiveness (all variables relative to trend). Finally, equation (6.4) imposes perfect capital mobility and perfect substitutability between domestic and foreign bonds. We could augment the model with a money demand function, but since we treat autonomous taxes, s, and nominal interest rates, r, as the instruments of policy, this adds nothing to the results. We may write the model in the form of (2.1) as: - y/{\ - Pi a2 dv dp dee

a2p2)

p2

0 0

0 y/a2 0 0 1

0 0

i^(a, + a2p2^

4>2 - p2 Pi 0

F

- dv 2 dv 2 0

We assume a loss function of the form Eo( Wo) where

dt

(6.5)

136

Theory and methodology o=

\ e-^( Jo

(6.6)

and parameter values y/ = 0.5, ax = 0.3, a2 = 0.1, a 3 = 1 . 0 , a4 = 0.4, $o = 1.0, cf>i = 1.3, 0.

where

e~N(0,I).

Then

E = LL T , Q > 0 and A is xTQx = eTWe

Diagonalising W we have W=CACT

where

and hence

xTQx = nTAn where rj = CTe~ N(0,I) (since CTC = I) and 1 7 ? - / ( I ) (i.e., a Chi-square distribution).

Notes 1 Note that the welfare loss WQ reduces to (2.2) in chapter 4 with Q = U = rrQl 0 and R = Q2 + OTQl 0. 2 If private agents know the model (2.1) then the policy announcement (2.13) together with the process for p2 (the bottom m rows of (2.11)) will give a perceived saddle-path stable model under control mathematically identical to (2.8). Thus the optimal policy can be implemented as (2.13) (see Levine, 1984). 3 Backus and Driffill (1986), drawing on an earlier version of our paper and working in discrete time, have derived our first proposition below as well as a number of other results similar to ours. 4 Following Barro and Gordon (1983b) one can think of the private sector as having a 'threat' to believe only in the time-consistent policy from time t onwards for the duration of the punishment period, if reneging occurs prior to t. Note that this does not require agents to be coordinated and to act strategically. It only assumes that, if reoptimisation takes place, private agents will believe that it will recur each period over the punishment interval. Furthermore each private agent knows that all other private agents know this. There is one feature of the Barro-Gordon game which is not reproduced here. We assume that either new policies are announced on implementation or that private agents can instantly observe a change in regime. Thus there is no period of 'cheating' where the private sector believes the optimal rule is still in force when in fact a new rule is being pursued. As a result we are exaggerating

Credibility and time consistency

147

the overall costs of reneging; but as it turns out, with stochastic disturbances and an infinite punishment period, these costs are so large that the introduction of a period of cheating would make no qualitative difference to the results. 5 The expressions on the right-hand side are bounded because we assume a bounded rational expectations solution [ J J to the deterministic trajectory and that the disturbances are bounded.

6

1

Should rules be simple?

Introduction

This chapter examines one aspect of the 'rules versus discretion' (or, more accurately, the 'precommitment versus discretion') debate over the conduct of macroeconomic policy. Given that precommitment yields better outcomes than discretion what form should it take? Much of the literature proposes that precommitment should be in the form of a non-contingent or 'open-loop' k percentage money supply rule. The problem with open-loop policies however is that they provide no flexibility for the government to respond to unforeseen shocks to the economy. A feedback rule can provide this flexibility and, in addition, constitutes a form of precommitment. The use of control theory provides feedback rules which perform this twin function: flexibility and precommitment. One difficulty with contingent rules however is they may prove difficult for a sceptical private sector to monitor. How are contingencies to be distinguished from reneging on the commitment, in which case how can contingent rules be credible? Monitoring is irrelevant to the credibility question where the relationship between the private sector and the government is modelled as a game of complete information, with the private sector in particular knowing the full nature of the government's calculations. In this case the private sector can evaluate for itself the precommitment rule and deduce the relevant policies for different contingencies. The incentive compatibility of the rule then requires a trigger mechanism on the part of the private sector which 'punishes' the government who reneges by only believing in discretionary policy for some 'punishment period'.1 Economics of Planning, 25 (1992), 113-138. Kluwer Academic Publishers. The support of the ESRC (grant number BO 1250012) and the Leverhulme Trust is gratefully acknowledged.

148

Should rules be simple?

149

The concern with monitoring suggests instead an incomplete information game. Following the Barro and Gordon (1983b) policy game with complete information, the credibility literature has switched its focus to games with less than full information as a means of eliminating the inefficiencies of discretionary policy (Backus and Driffill, 1985a,b; Barro, 1986a; Vickers, 1985; Canzoneri, 1985; Cukierman and Meltzer, 1986 surveyed in Levine, 1990). However these incomplete information games constitute only a rather limited departure from complete information. Throughout this literature it is assumed that the private sector knows the general nature of government behaviour but may be missing some information regarding parameters such as the weight on output in the welfare function. The private sector then observes policy instruments (for example, the money supply) and infers the values of the unknown parameters. An alternative approach to learning is to assume a less well-informed public which estimates the rule directly by observing the relevant data and applying standard econometric techniques. Bray and Savin (1986) exemplify this approach in a paper which examines private-sector learning about a cob-web model of price-output behaviour. In section 4, a learning model of this kind is presented in which the public observes the policy instrument and learns about the rule by a recursive OLS (or discounted least squares) procedure.1 This brings us back to the question posed earlier: what form should precommitment rules take? In an overlapping contract model with rational expectations, the optimal rule with precommitment is shown in section 3 to be rather complicated, taking the form of an error correction mechanism. It is shown that learning about a rule of this type proves to be slow as compared with learning about a simpler rule of lower order. Although the simple rule is sub-optimal in a complete information setting, when information is withdrawn and learning is introduced its performance becomes significantly better than its originally optimal counterpart. The rest of the chapter is set out as follows. Section 2 describes the (illustrative) model used to pursue these issues. Essentially the model is a dynamic generalisation of the Lucas supply curve adopted by Barro and Gordon with wage-price sluggishness and an inflation-output trade-off in the short run. Section 3 sets out the three policy rules implicit in the preceding discussion: the optimal rule with precommitment (in a complete information setting), the discretionary rule and a simple precommitment rule. Section 4 considers the outcome where the public has to learn about either the optimal or simple rules with priors consisting of some weighted

150

Theory and methodology

average of the implemented and discretionary rules. Section 5 concludes the chapter.

The model We employ a model which has a natural level of output in the long run but exhibits inflation inertia and some output-inflation trade-off in the short run. It is in spirit of overlapping contract models found in Calvo (1983), Taylor (1980) and Buiter and Miller (1983). We set out below the model, notation and the policy-makers welfare loss function: nt=y/yt + ct

(2.1)

ct = (1 - //)(?,_! + mt-2

+ n2qt-, + . . .)

qt = (1 - ju)(net+ut + junet+2,t + li2nUxt

+ •• •)

(2.2) (2.3)

where nt - inflation rate ct = core inflation yt = aggregate demand (equals output) qt = contract inflation. All variables are measured in deviation form about a given long run with output at its natural rate and inflation determined by the long-run values of fiscal and monetary instruments. It is convenient to consider this given long-run inflation rate, about which inflation under different rules is measured, as zero. Welfare loss at time t is Wt where Wt = \ 2 X(<m}+i + b(yt+i - yt)2)

(2.4a)

yt is desired output and X is the discount factor. y t + x=pyt,

0

partitioned so that Nu

is 2 x 2 , are functions of the

model's parameters. Even for this simple model I have not been able to obtain analytical expressions for those matrices. Instead, numerical computations are provided at the end of this section for all the policies using 'ACES', a software package for policy exercises on linear rational expectations models (Gaines, al-Nowaihi and Levine, 1989). 3.2 The discretionary policy: a time-consistent dynamic programming solution The time inconsistency of the previous policy is most clearly seen from (3.4). At time / = 0 a condition for Wo to be minimal isp20 = 0. But this is only an ex ante optimal policy. At t > 0, p2t ^ 0 in general and from (3.4) there is an incentive to reoptimise by setting p2t = 0. If in the unlikely event the policy-maker's precommitment to the new policy is believed by the private sector (or in other words reputation is maintained) then the gains from reoptimisations are \N22p2t which is always positive because, as shown in Levine (1988), N22 < 0. To obtain a discretionary time-consistent policy the cost-to-go Wt is written as Wt = \{an] + b(yt - yt)2) + XWt+,

(3.5)

Then a time-consistent policy is obtained by dynamic programming; Wt is minimised given that a similar procedure will be used to minimise Wt+xdX time t + 1. The structure of the solution is of the general form (3.1) to (3.4) above except the costate variable p2t is absent. The policy rule and the system under the rule are then of lower order than for the optimal policy. For this particular model it is possible to obtain an analytical form of the discretionary rule. Contract inflation qt depends only upon expectations of future inflation. It follows from (2.2) that core inflation ct also depends only upon inflation expectations both past and present. In evaluating discretionary policy, expectations must be treated as parametric. Hence core inflation may also be treated as parametric. Minimisation of Wt given by (2.4a) with respect to {y/+/} is then equivalent to minimising

154

Theory and methodology

W, = i 2 A'(ai?+, + b(W-\tt+i - ct+l) -yf)

2

(3.6)

i= 0

with respect to {nt+} treating {ct+} as parametric. Optimisation may now be carried out period-by-period to yield the inflation rule at time / b(ct + y/y) V - ) ay/ + 6 3 7

which is equivalent to the aggregate demand rule (3 8)

-

3.3

An optimal 'simple' rule with precommitment

The feedback form of the optimal policy with precommitment is given by (3.1) where form (3.2) Pit+1 = C3lyt + C32ct + C33p2u

[

P20 = 0

(3.9)

c c c 1 C2\ C22 C23 . The variable p2t is not observable directly. It is

a costate variable introduced by the policy-maker in order to undertake the minimisation problem. However the rule can be expressed purely in terms of the observed contract inflation ct by solving (3.9) to give Pit = i ) (C33y-l(C3lyt-T

+ C32c,_T)

(3.10)

T = l

for t > 0. Hence the rule (3.1) becomes yt = Dxyt + D2ct + D3 £ (C33y-l(C3lyt-T

+ C32ct.r)

(3.11)

T = l

for t > 0 with >>o = D\y + ^>2^o and where D = [D{ D2 D3\ The rule consists of proportional elements feeding back on current values of yt and ct plus feedback on lagged contract inflation with geometrically declining weights. Thus even for the very simple model considered here, the feedback rule is rather complicated. For more general dynamic models of higher order, rules corresponding to (3.11) will become exceedingly complicated. The simple rule we examine has the form (3.11) with D3 = 0. The rule in other words is a proportional rule of the same form as the discretionary time-consistent rule. However it should be stressed that it is a reputational rule which requires credible precommitment.

Should rules be simple?

155

Substituting yt = Dxyt + D2ct into (2.4) the policy-maker's choice problem at time t = 0 is to minimise with respect to D{ and D2. (3.12) where P depends on [DUD2] and parameters a, b and y/. The solution procedure for finding the optimal value of [Df D\] is given in the appendix. An important feature of the optimal rule, within the class of simple proportional rules, is that [Df Z>J] are not independent of [yo Co] the initial values of the state variables. This feature, which is not found for the optimal discretionary policy, means simple rules must be time inconsistent because the optimal values [Df DJ], as functions of [yo Co], will in general differ from those calculated at time t as functions of \yt Ctl 3.4

Simulations

Table 6.2 summarises the properties of the three policy rules. Parameter values chosen are ju = 0.75, ^ = 0.5, X = 0.988, a = 0.0032, ft = 0.001, p = 1 and % — 5. We assume a quarterly model so that // = 0.75 corresponds to a contract length of one year (see (2.5)) and X = 0.988 to a 5 per cent discount rate per year. Weights on a and b give annual inflation and output priority of 1:5 in the welfare function. The weights are also scaled to make the welfare loss lie between 0 and 10. p = 1 gives a constant bliss point for output set at 5 per cent above the natural rate. Finally y/ = 0.5 means that (from (2.1)) a 1 per cent increase in output is associated with a 2 per cent increase in annual inflation in the short term. Two initial displacements are considered. For the first the economy is in a zero inflation equilibrium (c0 = 0). For the second the policy-maker faces an extreme initial inflationary state with c0 = 5 (or 20 per cent per year). The inflationary bias of the discretionary policy - familiar from the Barro-Gordon paper - is apparent from table 6.2 and figures 6.1 and 6.2. For both the zero inflation and high inflation initial states, the long-run inflation rate under the optimal policy is zero compared with 3 per cent under discretion. There are some short-run output gains from higher inflation but overall the welfare loss under the discretionary policy is over twice that under the optimal, precommitment policy. As stressed in the previous sub-section, the optimal simple rule depends upon the initial state. For c0 = 0, the simple rule involves very strong feedback on contract inflation compared with the other rules as table 6.1 shows. This has the effect of quickly stabilising inflation at a rate above

156

Theory and methodology

Table 6.1. The three policy rules Optimal rule:

yt = 0.56yt - 0.89c, + 10.06/?2,

Dynamics:

\ct+i i _ r

Discretionary rule:

yt = 0.56y, - 0.89c,

Dynamics:

C+I-[0.11 0.83]

Simple rule 1:

yt = 0.5Syt - 20c,

Simple rule 2:

yt=-

Dynamics:

c,+ I =[0.22

0.04 0.80 o.oi l r -1.86 - 1 . 3 3 0.81 1 Ct

0.65c, 0.4,,

[5]

the long run of the optimal rule but well below that of the discretionary rule. Output is initially above that of the other rules but is quickly stabilised at the natural rate after three years. For c0 = 5, the commitment to a substantial lowering of output brings contract inflation down to only 2 per cent immediately for both the fully optimal and simple rule. For this initial displacement the latter involves only a mild feedback on inflation but the initial output loss is much stronger. Overall, using the welfare measure, the simple rules compare very well with the discretionary rule and is almost as good as the optimal rule. To summarise: the ranking of the three rules, using the welfare loss criterion, is given by Optimal rule > Simple rule > Discretionary rule

(3.13)

The good performance of the two simple rules very much depends on the correct matching of the appropriate rule to the initial state. If the simple rule designed for a zero inflation initial state is deployed when in fact initial inflation is high, then the outcome is far inferior to that under both the fully optimal and discretionary rules. Thus this rule is unsuitable for handling unanticipated inflation shocks. However a simple rule designed for the high inflation initial state and which responds only to inflation changes, performs reasonably well for initial low inflation.4 These results can be seen from the 'payoff matrix' - table 6.2c - which calculates the welfare losses under the various rules for initial dis-

Should rules be simple?

157

Table 6.2a. Simulations of the three policies: no initial inflation Variable/time (years)

0

1

2

3

4

5

Core inflation ct Optimal Discretionary Simple rule 1

0 0 0

0.46 1.68 0.14

0.37 2.45 0.15

0.21 2.81 0.15

0.10 2.98 0.15

0.04 3.06 0.15

0.00 3.13 0.15

Contract inflation qt Optimal Discretionary Simple rule 1

0.89 2.19 0.34

0.46 2.69 0.15

0.20 2.92 0.15

0.07 3.03 0.15

0.02 3.08 0.15

0.00 3.10 0.15

0.00 3.13 0.15

Inflation nt Optimal Discretionary Simple rule 1

1.39 1.39 1.46

0.83 2.32 0.18

0.40 2.75 0.15

0.17 2.95 0.15

0.06 3.04 0.15

0.01 3.09 0.15

0.00 3.13 0.15

Output yt Optimal Discretionary Simple rule 1

2.78 2.78 2.92

0.73 1.29 0.09

0.07 0.60 0.00

-0.08 0.28 0.00

-0.04 0.13 0.00

-0.02 0.06 0.00

0.00 0.00 0.00

Cost-to-go Wt Optimal Discretionary Simple rule 1

1.05 2.20 1.05

1.07 2.27 1.06

1.07 2.30 1.07

1.07 2.34 1.07

1.07 2.34 1.07

1.07 2.34 1.07

1.07 2.35 1.07

00

placement c0 = 0 and c0 = 5. A 'no feedback' control rule (i.e., an openloop policy) is also included in order to assess the gains from feedback control. Three important features emerge from table 6.2c. The first is that the gains from using any rule are at best small, starting from zero inflation; however the stabilisation gains are substantial following an inflationary shock. The second feature is that the simple rule designed for the inflationary initial state is relatively robust in that its performance is satisfactory if a benign supply shock occurs and inflation turns out to be rather less. The third feature is that 'loss of reputation' is far more serious in the welfare sense than the constraint of simplicity (if the correct rule is chosen). The credibility of the 'reputationaP (i.e., the time-inconsistent rules which, it should be noted, include the no feedback, open-loop rule) requires either the introduction of some trigger strategy on the part of the private sector, or the relaxation of the complete information assumption

158

Theory and methodology

Table 6.2b. Simulations of the three policies: with initial inflation Variable/time (years)

0

1

2

3

4

5

Core inflation ct Optimal Discretionary Simple rule 2

5.00 5.00 5.00

2.42 3.99 2.60

1.10 3.53 1.35

0.35 3.31 0.70

0.09 3.21 0.36

0.01 3.17 0.19

0.00 3.13 0.00

Contract inflation qt Optimal 1.95 Discretionary 3.69 Simple rule 2 1.98

0.62 3.39 1.03

0.13 3.25 0.53

0.00 3.18 0.28

0.00 3.15 0.14

0.00 3.14 0.07

0.00 3.13 0.00

Inflation nt Optimal Discretionary Simple rule 2

4.17 4.17 3.39

1.53 3.61 1.76

0.44 3.35 0.91

0.07 3.23 0.47

0.00 3.17 0.25

0.00 3.15 0.13

0.00 3.13 0.00

Output yt Optimal Discretionary Simple rule 2

-1.67 -1.67 -3.23

-1.79 -0.77 - 1.68

- 1.13 -0.36 -0.97

-0.57 -0.17 -0.45

-0.25 -0.08 -0.23

-0.09 -0.04 -0.12

0.00 0.00 0.00

Cost-to-go Wt Optimal Discretionary Simple rule 2

1.22 2.45 1.22

1.10 2.39 1.12

1.05 2.37 1.07

1.03 2.35 1.06

1.02 2.35 1.05

1.05 2.35 1.05

1.02 2.34 1.04

00

and the introduction of learning. The latter is the route chosen for the rest of the chapter. 4

Learning under incomplete information

4.1 OLS learning In the previous section the good performance of the two precommitment rules, the optimal and simple rules, requires credible precommitment. In this section we examine the outcome where policy announcements of time-inconsistent rules are not fully believed and the private sector lacks information of the policy-maker's welfare function. Instead it observes and learns about the rule in operation. Learning about rules in a deterministic setting would be uninteresting

Should rules be simple?

159

Table 6.2c. Payoff matrix: welfare loss relative to minimum Rule

actual displacement

Optimal Discretionary Simple rule 1 Simple rule 2 No feedback control

c0 = 0

c0 = 5

1.00 2.15 1.01 1.02 1.02

1.00 2.02 10.90 1.01 1.49

(and unrealistic). We therefore introduce noise into the model which is confined to instrument noise only. Thus aggregate demand yt is related to planned aggregate demand y? by yt = yf + ut9

M,~NID(0,i(l - C33) + D3C3l], etc.

y,-\],

160

Theory and methodology

3 -

^ ^

^ discretionary rule

# c o

2-

V

CO

c

optimal rule

1 -

simple rulei

(a)

i

—-^ •—v

2

_

3

Time (years)

discretionary rule 3

^

o

\

c

simple rule 2 Opti mal rule

\ \ v

(b) 0

1

1

1

2

3

4

5

Time (years)

Figure 6.1 (a) Inflation trajectories: no initial inflation, (b) Inflation trajectories: initial inflation (c0 = 5)

We shall assume that the private-sector priors /?0, say, are a linear combination of the discretionary rule (4.1) and the announced optimal rule (4.3) which we write as yf = /Fz,, i.e., (4.5)

Should rules be simple?

V -

^

2

o

1 -

discretionary rule

V^s.

optimal rule simple rule 1 * ^ 1

o-

(a)

161

— I

2

^ ^ ^ I 3

I

Time (years)

2

3

Time (years)

Figure 6.2 (a) Output trajectories: no initial inflation, (b) Output trajectories: initial inflation (c0 = 5)

The parameter /?E[0,1] is a measure of the initial reputation of the government. In the extreme case of 'no reputation', policy announcements have no impact, p = 0 and the priors consist of the discretionary rule. For/? = 1 announcements are believed and the policymaker may enjoy the benefits of the full optimal rule. The OLS updating rule is

162

Theory and methodology

% = $t-X + Ptzt(yt-znt-x)

(4.6)

l

where Pt = Sr and St = Ej = {Zjzf, i.e., S, = S,_ i + xtzj and /?o = 0 f ^2 0 0] (see Harvey, 1981, chapter 7, for example). The updating formula (4.6) begins after the regime change for 2 periods for the simple rule (at least two observations are needed to estimate two coefficients) and after four periods for the optimal rule. In each period forward-looking contract inflation is determined by solving the rational expectations model under the latest estimate of the rule. The saddle-path which corresponds to (3.3) is then qt=-N0t)zt

(4.7)

Writing the model in the general form

the dynamics of the predetermined block zt is given by zt+l = [An - Al2N$t)]z,

+ B2(yf + ut)

(4.9)

partitioning A = [A2\ A^i a n d B = [B2] conformably with \zj qt\ A problem arises where the latest estimate /?, does not give rise to a system under control with the saddle-path property. The system under the expected rule may be unstable (with too many unstable eigenvalues) or may have a multiplicity of solutions (too few unstable eigenvalues). The possibility of this occurring increases as the variance of the noise a2 increases. An estimate of the rule which violates the saddle-path property cannot be used by private agents to calculate the saddle-path (4.7). To deal with this problem the updating procedure is modified as follows. The latest estimate for N is used in (4.7) only if the saddle-path property holds. Otherwise the matrix N and hence the saddle-path is calculated using the most recent estimate of ft that leads to a unique rational expectations solution. It should be noted that for the case where the latest estimate of the rule $t implies instability, then this assumption is a logical consequence of the standard hypothesis in the rational expectations literature that private agents 'do not readily participate in any economy which is following an explosive path' (Begg, 1982, p. 38). For the case where /?, implies too few unstable eigenvalues then the assumption provides a plausible criterion for pinning down a unique rational expectations solution. It should also be stressed that when the saddle-path property breaks down agents do continue to update their estimate of the rule and learning is not suspended.

Should rules be simple?

163

An alternative learning criteria is a discounted least squares updating rule (DLS) which is given by (4.6) where Pt = Srl but now St is given by St = dSt-x + ztzJ

(4.10)

where 8 E (0,1). If 8 = 1, /?, is the OLS estimator as before. The effect of introducing 8 less than one is to speed up the process by which the private sector changes its priors. In this way a breakdown in the saddle-path property is rather less serious because the private sector continues to forget about its (false) priors at a faster rate. 4.2

Simulations

Tables 6.3 to 6.5 summarise results obtained by numerical stochastic simulations using a NAG random number generator and averaging over ten simulations in each case. The standard deviation of instrument noise a takes values 0.1 (0.1) 0.5. All welfare losses are measured relative to the minimal welfare loss under the fully optimal rule with no instrument noise. In the first two rows of tables 6.3a and 6.3b the policy-maker implements the optimal and discretionary rule respectively with the private sector adopting correct priors. In the third row, the private sector initially believes in the discretionary rule but subsequently learns about the optimal rule being implemented (i.e., initial reputation p is zero in (4.5)). In the fourth row initial reputation is again zero but this time a simple rule is adopted by the policy-maker. The learning process is OLS updating as described by (4.6). In table 6.3a there is no initial displacement (c0 = 0) and the appropriate simple rule is rule 1 in table 6.1. In table 6.3b c0 = 5 and the corresponding simple rule is rule 2. From row one we see there is a significant deterioration in the performance of the optimal rule as instrument noise increases even where privatesector priors are correct. This is particularly true in the inflationary initial state. When initial reputation is zero (row three) this deterioration increases markedly; indeed for the inflationary initial state the optimal rule under learning can be outperformed by the less complex discretionary rule. There is no such problem for the simple rule. Learning about simple rules is swift and as a result their performance does not deteriorate greatly as instrument noise increases (see row four). Simple rule 1 performs almost as well as the optimal rule at the lower end of the standard deviation range, 8 = 0.1; as a increases, it performs better and its relative performance becomes progressively better as well. In the case of initial inflation, table 6.3b shows that the relative superiority of simple rule 2 over the optimal rules is even more pronounced.

Table 6.3a. Learning about rules: stochastic simulation results with no initial inflation (c 0 = 0) Welfare loss over 200 periods operation

expectations

(7 = 0.1

(7 = 0.2

(7 = 0.3

(7 = 0.4

(7 = 0.5

Optimal Discretionary Optimal

Optimal Discretionary Priors: discretionary OLS learning Priors: discretionary OLS learning

1.02 2.10 1.10

1.02 2.10 1.26

1.10 2.03 1.35

1.10 2.08 1.51

1.18 2.09 1.57

1.16

1.17

1.18

1.20

1.21

Simple

Table 6.3b. Learning about rules: stochastic simulation results with initial inflation (c 0 = 5) Welfare loss over 200 periods operation

expectations

(7 = 0.1

(7 = 0.2

(7 = 0.3

(7 = 0.4

(7 = 0.5

Optimal Discretionary Optimal

Optimal Discretionary Priors: discretionary OLS learning Priors: discretionary OLS learning

1.01 1.75 1.12

1.17 1.76 1.17

1.26 1.77 1.90

1.28 1.79 1.51

1.33 1.80 1.66

1.04

1.04

1.05

1.06

1.08

Simple

166

Theory and methodology

Table 6.4a. Detailed breakdown of switch from discretionary to optimal rules with OLS learning: a = 0.2, no initial inflation (c0 = 0) Actual rule: [ft ft ft ft] = [0 - 0.889 0.648 0.914]

ft

ft

No times saddle-path property breaks down

0.83 0.31 0.49 0.96 0.65 0.44 0.77 1.29 0.48 0.94

0.84 0.74 0.78 0.84 0.83 0.81 0.86 0.94 0.81 0.93

17 5 5 5 5 9 19 5 27 6

Last estimated rule Iteration

ft

1 2 3 4 5 6 7 8 9 10

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

ft - 1.02 -0.64 -0.76 -1.17 -0.86 -0.67 -0.98 - 1.42 -0.72 - 1.08

^Welfare ] OSS 1.30 1.11 1.17 1.11 1.11 L27 L58 1.14 1.69 1.19

Tables 6.4a and 6.4b give a detailed breakdown of the simulations for the case where the standard deviation o = 0.2. For c0 = 0, table 6.4a shows the problem faced by the optimal rule under learning in the form of a possible breakdown of the saddle-path property which impedes the learning process. By contrast simulation results for both simple rules (not reported) reveal that the saddle-path condition never breaks down. The high values of the welfare loss are clearly associated with a high number of breakdowns over the 200 periods of the simulation. For the case c0 = 5 the number of times the saddle-path property breaks down is considerably less. The reason for this is straightforward; higher variation in the inflation rate reduces the variances of the OLS estimates and reduces the number of times and estimated rule departs significantly from the actual rule. The reported final estimates in table 6.4b confirm this explanation. Even in this second case the simple rule still performs markedly better than the optimal rule. Indeed its relative superiority is, if anything, greater. This suggests that there is an additional reason for the better performance of simple rules which is not reliant on the possible breakdown of the saddle-path condition under the optimal rule. The high initial inflation simple rule 2 appears to be more robust than the optimal rule in the sense that the latter is more sensitive to misperceptions of the private sector even where these involve only small deviations from the true rule.

Should rules be simple?

167

Table 6.4b. Detailed breakdown of switch from discretionary to optimal rules with OLS learning: o = 0.2, no initial inflation (c0 = 5) Actual rule: [ft ft ft ft] = [0 - 0.889 0.648 0.914] Last estimated rule Iteration

ft

1 2 3 4 5 6 7 8 9 10

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

ft -0.74 -0.88 -0.90 -0.83 -0.90 -0.90 -0.95 -0.78 -0.83 -0.89

ft

ft

No times saddle-path property breaks down

0.41 0.58 0.66 0.55 0.68 0.69 0.78 0.53 0.56 0.71

0.69 0.77 0.82 0.76 0.82 0.83 0.90 0.78 0.76 0.87

1 3 1 0 4 1 0 1 1 2

^Welfare ] OSS 1.13 1.17 1.14 1.15 1.16 1.20 1.13 1.16 L.16 L30

The fact that optimal rules may lack robustness in this sense is a wellknown result in control theory (see Rosenbrock and McMorran, 1971). The final set of simulations reported in tables 6.5a and 6.5b generalises the previous results in two directions. First, the relative performance of the different rules is examined under a range of priors from no initial reputation (p = 0) to full initial reputation (p = 1). Second, an alternative learning mechanism - discounted least squares learning - is examined. Different values of the discount factor, 8, were tried. The familiar problem of instability when 8 was too low was encountered and results are reported for 8 = 0.95. In table 6.5a the performance of the optimal rule with no initial inflation improves substantially as priors become more sympathetic to the government. However the same holds for the simple rule so that its relative superiority remains unless priors become totally sympathetic (p = 1). Discounted least squares learning improves the outcome under the optimal rule somewhat, for the reasons discussed in the previous section, but does not alter the main conclusions. With initial inflation, table 6.5b, the picture is rather different. There is no longer a progressive improvement under the optimal rule as prior p approach unity. The reason for this is the extreme sensitivity of the performance of the optimal rule to small variations in perception which in turn depend upon particular stochastic realisations. This result is not

168

Theory and methodology

Table 6.5a. The effect of priors on OLS learning: o = 0.2; no initial inflation (c0 = 0) Priors Rules in operation

Learning mechanism

Optimal Simple Optimal

OLS OLS DLS (S = 0.95) DLS (S = 0.95)

Simple

p=0

p = 0.25

p = 0.5

p = 0.75

p= 1.0

1.27 1.17 1.18

1.17 1.04 1.16

1.14 1.02 1.12

1.08 1.02 1.06

1.02 1.01 1.02

1.17

1.03

1.02

1.01

1.01

Table 6.5b. The effect ofpriors on OLS learning: a = 0.2; with initial inflation (c0 = 5) Priors Rules in operation

Learning mechanism

Optimal Simple Optimal

OLS OLS DLS (S = 0.95) DLS (S = 0.95)

Simple

p=0

p = 0.25

p = 0.5

p = 0.75

p= 1.0

1.17 1.04 1.17

1.14 1.03 1.14

1.15 1.01 1.14

1.10 1.01 1.09

1.17 1.00 1.13

1.04

1.02

1.01

1.01

1.00

significantly affected by the choice of OLS or DLS learning. Overall the main conclusion is that the superiority of the simple rules in a learning context remains intact (indeed enforced) under different assumptions regarding priors and the learning mechanism. 5

Conclusions

The argument of the chapter is that in a situation where the private sector lacks information of government objectives and has to learn about the policy rule by direct observation and estimation, simple rules may outperform the more complicated 'optimal rule'. This provides some formal reasoning to support the more informal arguments for simplicity associ-

Should rules be simple?

169

ated with the need for the private sector to be able to monitor policy (see Barro, 1986b for instance). The question of whether simple rules of the type considered here can be made incentive compatible given that they are sub-optimal and hence time inconsistent is not considered. This appears to be a difficult problem. The Barro-Gordon trigger mechanism approach, which offers the most promising route, requires a plausible punishment where the government reneges on its time-invariant feedback rule. Since in our setting the public is estimating a fixed rule with time-invariant parameters, reneging will introduce misspecification into their estimated equations. The appropriate response may be to start estimating a time-varying feedback rule. But then, as Bray and Savin (1986) put it, 'econometricians recognise that there are no precise rules for how to go about constructing such a model. Thus once agents embark upon model revision they are more or less on unchartered seas.' Given the uncertainties involves here it is by no means apparent that if rule revisions occur then a rational expectation equilibrium, in which public perceptions correspond on average to the actual rule, does exist. Informally speaking the punishment for reneging is the great uncertainty which policy-makers are well advised to avoid. We have advanced one reason for conducting macroeconomic policy in the form of a simple rather than the more complex optimal rule. There are of course other, possibly more compelling, reasons for simple rules such as their intuitive appeal or their ease of implementation. A further reason for simplicity is that rules within this category may be found which have good robustness features in the face of model uncertainty. Ultimately however these issues are primarily empirical and can only be resolved on empirically based models. Appendix The three policy rules The model in the chapter in deterministic form can be generalised to the following

1 H

A*

(Al)

where zt is an (n — m) x 1 vector of predetermined variables including non-stationary processes, z0 is given, wt is a vector of policy variables, xt is an m x 1 vector of non-predetermined variables and x?+i,, denotes rational (model consistent) expectations of xt+\ formed at time t. Then *?+1,/ = xi+1 and letting yj = \zfxj\9 (Al) becomes yt+1 = Ayt + Bwt

(A2)

170

Theory and methodology

Define target variables st by st = Myt + Hwt

(A3)

and the policy-maker's welfare loss at time t by Wt = \JjX\sJ+iQxst+i+

wT+iQ2wt+l]

(A4)

i=0

which we rewrite as wt = \ f ) liyT+iQyi+iQyi+i + 2yJ+iUwt+i

+ wT+iRwt+l]

(A5)

i=0

where Q = MTQXM, U=MTQlH, R = Q2 + HTQlH, Qx and Q2 are symmetric and non-negative definite R is required to be positive definite and X E (0,1) is a discount factor. The procedures for evaluating the three policy rules are outlined in the rest of this appendix (or Levine (1988) for a more detailed treatment). (i)

The optimal policy with precommitment

Consider the policy-maker's ex-ante optimal policy at / = 0. This is found by minimising Wo given by (A5) subject to (A2) and (A3) and given z0. We proceed by defining the Hamiltonian Ht(yt,yt+ujut+l)

= WiyJQyt + 2yjUwt + wfRwt) + //,+ x(Ayt + Bwt - yt+ {)

(A6)

where jut is a row vector of costate variables. By standard Lagrange multiplier theory we minimise Lo(yo, yu . •., w0, wl9 ..., //i, ju2, ...) = 2 Ht

(A7)

t=0

with respect to the arguments of L o (except z0 which is given). Then at the optimum, Lo = Wo. Redefining a new costate vector pt = A~'//f, the first-order conditions lead to (A8) T

XA pt+ ! - / > , = - (Qyt + Uwt)

(A9)

Substituting (A8) into (A2) we arrive at the following system under control XBR~lBT T

l 7

0 ^A 'UR- B )\[pt+l\

]\yt+l]\A-BR-lUT

[-{Q-UR-XUT)

l\[ (A10)

Should rules be simple?

171

To complete the solution we require In boundary conditions for (A 10). Specifying z0 gives us n — m of these conditions. The remaining condition is the 'transversality condition' lim fij = lim llpt = 0

(All)

and the initial condition />2o = 0

(A12)

where pi = [pft p2t] is partitioned so that p u is of dimension (n- m)x 1. Equations (A3), (A8), (A 10) together with the 2n boundary conditions constitute the system under optimal control. Solving the system under control leads to the following rule

- - i - k -UfeJ t::Hk lH-k -UfeJ •311

^ 1 2 ^ 2 2 * > 2 1

_ C - 1 C

— O22 *J21

^ 1 2 ^ 2 2

_ 7 V l l

C - l

—

*>22

J

AT

L^21

y

y

.

AT

K^lD)

^22J

xt= ~[N2lN22] r*

(A16)

[P2t\

where F = - (R + BTSB)~1 (BTSA + U7),G = A-BF and

s=

(Ai7)

[k k]

partitioned so that Sn is (n- m)x (n- m) and 522 is m x m is the solution to the steady-state Ricatti equation S=Q-UF-FTUT

+ FTRF 4- X{A - BF)TS(A - BF)

(A18)

The cost-to-go for the optimal policy (OP) at time t is W?p = - 2-(tr(7VnZ,) + tr(N22p2tpl))

(A19)

where Zt = ztzl. To achieve optimality the policy-maker sets p20 = 0 at time t = 0. At time t > 0 there exists a gain from reneging by resetting p2t = 0 which is given by the last term in (2.12). Since N221

(A21)

The welfare loss at time t for the ex-ante optimal policy was from (A 13) found to be a quadratic function of xt and p2t. We have seen that the inclusion of p2t was the source of the time inconsistency in that case. We therefore seek a lower-order controller wt — - Fzt with the cost-to-go quadratic in zt only. We then write Wt+ x = \zj+ x St+1 zt+1 in (A20). This leads to the following iterative process for Ft wt = - Ftzt

(A22)

where Ft = (Rt + XBjSt+ i$tyl

(UT + XBTSt+, At) 27

(A22b)

+ B2)

(A22c)

Rt = R + K[Q22Kt + t/ "^ + KJU

Kt=

- (A22 + Nt+iAnyl(Nt+iBl

(A22a)

2

l

Bt = B + Ai2Kt

(A22d)

Ut=Ul + Qi2Kt + JJU2 + JTQ22Jt

(A22e)

Jt = — (^22 + N f + i ^ ^ ) " ^ ^ ! ^ ! ! + v412)

(A22Q

y|, = An + ^12^/

(A22g)

1

_l_ If A

D J7\T o

/ -7

D r'\

(A22h)

Q, = Qu+ JTQzi + QnJ, + ./r&s./,

(A22i)

N,=

(A22j)

-J, + K,F,

where B = [ # ] , U = [(72], A = [A2\ A ^ an< l Q similarly are partitioned

Should rules be simple?

173

conformably with the predetermined and non-predetermined components of the state vector. Equation (A22) describes an iterative process for Ft, Nt and St starting with some initial values for Nt and St. If the process converges to stationary values, F, Nand S say, then the time-consistent feedback rule is Wt= — Fzt with welfare loss at time / (A23) (Hi) Optimal simple rules We now consider simple sub-optimal rules of the form (A24) where D is constrained to be sparse in some specified way. Rule (A24) can be quite general. By augmenting the state vector in an appropriate way it can represent a PID (proportional-integral-derivative) controller (though the paper is restricted to a simple proportional controller only). Substituting (A24) into (A5) gives Wt = ^ryT+iPt+iyt+i

(A25)

where P = Q + UD + DTUT + DTRD. The system under control (Al), with wt given by (A24), has a rational expectations solution with xt = — Nzt where N = N(D). Hence yJPyt = zjTzt

(A26)

where T=PnNTP2l - Pl2N+ NTP22N, P is partitioned as for S in (A 17) onwards and zt+x = (Gn-GX2N)zt

(A27)

where G = A + BD is partitioned as for P. Solving (A27) we have z, = (G n -G 12 AO'z 0

(A28)

Hence from (A25), (A26) and (A28) we may write at time t Wt = hrVzt = \tv(VZt)

(A29)

where Zt = ztzj and V satisfies the Lyapunov equation V=T+HTVH

(A30)

174

Theory and methodology

where H= Gn — GX2N. At time t = 0 the optimal simple rule is then found by minimising Wo given by (A29) with respect to the non-zero elements of D given z0 using a standard numerical technique. An important feature of the result is that unlike the previous solution the optimal value of D, D* say, is not independent of z0. That is to say D* = D*(z0)

(iv)

The stochastic case

Consider the stochastic generalisation of (Al), and where

er = real exchange rate pe = price of equity pg = price of gilts g = real government spending

182

Theory and methodology

r = short-term interest rates (per cent) x = direct taxation rate (income tax plus employees' National Insurance contributions per cent) y* = real OECD industrial production po = dollar price of oil r* = world short-term interest rates (per cent) p* = world price level wr = average wage rate y = real GDP nt = consumer price annual inflation (per cent) psbr = PSBR as a proportion of money GDP (per cent) bal = balance of payments (current account) as a proportion of money GDP (per cent) Variables er, pe, pg, gt, y*, p°, p*, wr and y are measured as proportional per cent changes about the February 88 forecast. The remaining 'rate' variables r, x, r*9 n and bal are measured as deviations about the baseline forecast. The vector of predetermined variables in (3.1), xt with x0 = 0, is a composite vector of the predetermined variables in the original model. Finally pet+1 denotes expectations of pt+1 formed at time t. Let y, ft, psbr, bal be the policy-maker's desired values for the target variables and g, f, i be the corresponding desired values for instruments. Then we may write down a general quadratic welfare loss function for the policy-maker at time t = 0 as t=T

+ d{bal, - bal,)2 + e(g, - g,f + / ( r , - f,f + h(r, - i,)2} (3-3) where X EL (0,1) is a discount factor, Tis the time horizon and a,b,c. . . are weights reflecting the policy-maker's priorities for the achievement of 'bliss' values yt = yt, etc. If Wo, given by (3.3), is minimised subject to a linear rational expectations model, we obtain an optimal open-loop policy (i.e., trajectories) for instruments gt, rt and xt for t = 0,1, . . ., T. If we put T= oo then in addition the optimal policy may be represented as a constant coefficient feedback rule which, in the face of unexpected exogenous disturbances or modelling errors, will stabilise the economy about the open-loop trajectories (see Levine (1988) for details of all the computations discussed in this section). Feedback rules are, in effect, correction mechanisms which prescribe how instruments are to be revised when the actual out-turns of the economy depart from its deterministic forecast.

Macroeconomic policy design

183

3.2 The time-inconsistency problem The policy described above is only optimal ex ante. Ex post it ceases to be optimal and there exists an incentive to renege. This is the problem of time inconsistency. Kydland and Prescott regard the time-inconsistency property as a fundamental problem which leads them to argue against the use of optimal control methods for macroeconomic policy design. However it is possible to design an optimal time-consistency policy which, though inferior to the ex ante optimal policy, has good stabilisation properties. In Barro and Gordon (1983b) a different approach to the time-inconsistency problem is offered which has been widely pursued since. They examine whether reputational considerations can restore credibility for policy-makers and hence avoid the inferior outcome of the time-consistency constraint. It is assumed that policy-makers suffer a loss of reputation if they renege on earlier commitments. With this 'punishment' mechanism in place, Barro and Gordon show that policies superior to the time-consistency policy exist which can be credible and sustainable. The model in the Barro-Gordon paper has a natural rate of output and is static. In Levine (1988) a generalisation of the analysis to any structural dynamic linear rational expectations model is provided. The particular form a 'loss of reputation' takes in this dynamic setting is that the policy-maker must pursue a time-consistent (or 'non-reputational') policy for the duration of the punishment period. The policy-maker's decision can be schematically presented as: Optimal but Pursue timetime-inconsistent consistent Return policy (OPT). -* policy (TCT) ^ to OPT forP periods for) -> continue with OPT

The gains from reneging ('temptation') must now be weighed against the costs arising from a loss of reputation for P periods ('enforcement'). Let the welfare loss after reneging be Wf and the corresponding welfare loss for the optimal, time-inconsistent policy be W?PT. Then no incentive to renege exists if W?FT < Wf

for all t along policy OPT

(3.4)

This is necessary and sufficient condition for the optimal policy to be sustainable. Condition (3.4) is henceforth referred to as the sustainability condition.

The analysis outline rests on the acceptability of the 'trigger mechanism' that takes the form that the private sector only believes in the discretionary

184

Theory and methodology

or time-consistent policies if the policy-maker reneges. This is rational in the sense that the time-consistent policy is the outcome of repeated reoptimisation which is anticipated by the private sector. The ad hoc element of the trigger mechanism lies in the length of punishment length P. We assume a long punishment length which is implemented as P = oo. This admittedly biases the outcome in favour of sustainability. However, we ignore any stochastic element in the disturbances. Currie and Levine (1987) and Levine (1988) show that introducing stochastic disturbances adds to enforcement because policy-makers without reputation are prevented from using the best reputational rule to deal with repeated stochastic disturbances during the interval of punishment. Thus in a stochastic world sustainability is enhanced. For each choice of punishment length P there exists a rational expectations equilibrium superior to the time-consistent solution which can be supported by this trigger strategy. If Pis sufficiently large then the optimal policy itself may be supportable. The existence of this multiplicity of solutions is a well-known problem with repeated games. As the analysis stands P is 'chosen' by the private sector; the longer the punishment length the less forgiving are the public. The choice of P = oo might be valid if indeed the public are unforgiving towards governments which break their promises. Another approach to the choice of P is to regard the punishment length as chosen by governments strategically in order to enforce the best outcome. An atomised private sector cannot act strategically but governments can. Suppose that we regard democracy as a game between successive alternating party-based governments. Each government has identical preferences and chooses a strategy which consists of honouring the precommitment of its predecessor to the optimal (but time-inconsistent) policy; but if its predecessor reneges then the incumbent government switches to the discretionary time-consistent policy. This remains in force for P periods of office and applies to both itself and the alternative government. If both prospective governments pursue this trigger strategy then the equilibrium it supports is sub-game perfect. As before there is a multiplicity of equilibria. But now the actual punishment period P can be chosen so that the gains from cheating are outweighed by the losses incurred by the pursuit of a socially inferior policy.3 As we shall see even P = oo is the only rational choice of punishment length. 33

The complexity of the optimal feedback rule

The optimal rule is exceedingly complex involving a feedback on the entire state vector of the linear model. Moreover, for the linearised model, the

Macroeconomic policy design

185

Table 7.1. The use of instruments

Instruments

Frequency of change

Frequency of policy revisions

Short-term interest rates rt Government spending gt Tax rate T,

Quarterly Quarterly Once at t = 0

Quarterly Annual No revision

state vector turns out to be a composite vector of variables with no economic interpretation. We now consider the design of simple feedback rules. Simplicity means, among other things, that feedback must be on a limited number of economic variables or 'indicators'. A simple rule might restrict the instruments to stabilising a subset of the target variables yu nu psbrt and balt in Wo. Other considerations in specifying rules are that some instruments can be changed more frequently than others and for some, changes must be announced in advance. In particular for the UK the practice is that two instruments, interest rates, rt, and government spending, gt, can change each quarter (at least) while tax rates, xu have been varied less often. In the stylised exercise below we allow the tax rate to be changed only once at the start. Plans for government spending in the UK are usually revised annually. The form of the feedback rule we adopt allows for annual revisions of previously announced open-loop plans for government spending and quarterly revisions of interest rates. These institutional features of the use of instruments are summarised in table 7.1. The form of simple rule considered here involves both proportional and integral control. Integral action on a particular target variable will drive it towards a given trajectory in the long run. With two instruments integral action on at most two target variables is then permissible. These are chosen to be the primary targets in (3.3), output, yt and inflation, nt. In addition proportional control on the remaining targets is feasible. Previous experience of simulations on the LBS model has established that the balance of payments, balt, can become implausibly large without apparently destabilising inflation or the PSBR. For this reason we restrict the additional proportional control to balt. The precise choice of simple rule is then a proportional integral (PI) controller on yt and nt plus proportional control on balt. Assuming these economic variables can be observed (or estimated) with a lag of one quarter, a controller with the frequency characteristics of table 7.1 takes the form

186

Theory and methodology

t

-yt-\) t-l - yt-X) + 8X2(yt-X 02\A(nt-x - fct-j) + 022(nt-x - nt-X) 03lA(balt-l-balt-l)

(3.5)

~ gt) = 04XA(yt-4 - yt-4) + 042(yt-4 - yt-*) + 051 A(nt-4 - Sf__4_) + 052(nt-4 - nt-4) + 06lA(balt-4-balt-4)

(3.6)

where A denotes a first difference and f,, yt9 gt and nt denote open-loop trajectories. By introducing a lag of four quarters in (3.6), at time t the policy-maker can plan and announce revisions of previous plans for gt over the next four quarters based on observations of yt and nt over the previous four quarters. The structure of the rules means that instruments deviate from their announced course (£t, rt) only when forecasts of targets are not met. The rule (3.5) and (3.6) is fairly general and includes a number of simple rules, which appear in the literature, as special cases. For instance with On = #2i = #3i= 042 = 05i = 06i = O, and 0n = 022, 94l = 052 the rule becomes O-r, = 0n(y,_i +pct-i -yt-\

-pCt-i)

gt- gt = 04i(yt-4+pct-4-yt-4-pct-4)

(3.7) (3.8)

wherepc is the consumer price index (in deviation form). The rule is now a proportional nominal income rule for both instruments. With 0 n = 012 = 021 = 03! = e42 = 051 = 052 = 06i = 0 the rule is rt ~ ft = 622{pct-x -\-pct-i)

(3.9)

gt-gt

(3.10)

= 04X{yt-4 + yt-4)

which is the 'decoupled' proportional price and output rule found by Currie and Levine (1985) to perform well in a small, semi-empirical model of the UK economy. 3.4

Summary

To summarise this section, a general welfare loss function is specified with targets for GDP, inflation, the public-sector borrowing requirement and the current-account surplus. Instruments are the short-term interest rate, government spending on goods and services, and a composite direct tax rate. Using a linearised LBS model, an optimal policy is computed which consists of open-loop trajectories for instruments plus a feedback rule which stabilises the economy about its open-loop path in the face of exogenous or modelling uncertainties.

Macroeconomic policy design

187

The policy thus obtained is time inconsistent. Its credibility requires that there is no incentive to renege at any time during its implementation. This feature may be achieved if it is assumed that reneging leads to 'loss of reputation' and an inability to pursue a time-inconsistent policy for some interval of time. This leads to a sustainability condition which must be examined empirically. Policy design is simplified thus: the tax rate is fixed at the outset of the new policy and no further revisions occur. The interest rate can be revised quarterly about its originally announced open-loop path but government spending can only be revised annually. In addition the feedback rule for these two instruments is restricted to a proportional-integral controller on output and inflation and a proportional controller on the balance of payments. 4 4.1

Results The optimal open-loop policies

The control exercise is carried out assuming two types of policy-maker with contrasting welfare loss functions. Both policy-makers have the same targets y and n for higher output and lower inflation. These, in deviation form, are taken to be y = 5 per cent (achieved by higher capacity utilisation of a higher level of capital stock) and n = - 3 per cent which would reduce inflation to near zero in the long run. The priority of policy-maker type 1 is to raise output towards its 'bliss' value whilst policy-maker 2 aims to reduce inflation. Thus the weights a and b in the welfare loss function attached to output and inflation respectively are set such that a is much greater than b for policy-maker 1 and b is much greater than a for policy-maker type 2. Regarding taxation, policy-maker 1 is satisfied with the 1987-8 standard rate of income tax of 27p in the £. The baseline forecast has incorporated the reduction to 25p from 1988(2) which is roughly equivalent to a 1 per cent reduction in the composite direct tax rate x. Thus we may set f = 1 for policy-maker 1 from that data. For policy-maker 2, f = 0 and the remaining targets bdl, psbr, f and g are set at zero indicating that the baseline forecasts are satisfactory.4 The assumptions regarding the two policy-makers are summarised in table 7.2. The next step is to simulate an interactive process between the civil servant designing the details of policy and the policy-maker herself to reveal the appropriate values of the weights a,b,c, . . . in the welfare loss function (3.3). The achievement of bliss points y = y and n = n is not possible (on the LBS model) and both the policy-makers know this.

188

Theory and methodology

Table 7.2. The policy-makers' preferences Policy-maker 1

Policy-maker 2

Output targets Inflation targets Taxation targets

y= 5 fc= - 3 f =1

y= 5 h= - 3

Weights in welfare loss function

a>b

b>a

T= 0

Notes: a = output weight. b = inflation weight.

Policy-maker 1 will be satisfied with an outcome that has output rising by as much as possible, subject to an upper limit on inflation, the PSBR and the current-account deficit. From past experience of simulations on the LBS model we know that the inflation and current-account constraints are the crucial ones. For inflation an upper limit of 2.5 per cent per year (above the baseline forecast), over the nine years of the simulation, is set. For the current-account deficit an upper limit of 1.5 (as per cent of nominal GDP) above the base-run is chosen. Experiments with numerous combinations of weights on the linear model narrowed the choice for policy-maker 1 to the following: a = 10, b = 2, c = 0, 10 < d < 30, e = / = 1

(4.1)

with the tax rate rfixedat the outset at % = z = 1. For both policy-makers the (quarterly) discount factor is set at I = 0.988 implying a 5 per cent discount rate a year. Similarly for policy-maker 2 the objective, to squeeze inflation out of the economy, is constrained by a concern that output does not fall too much. A maximum fall of 2 per cent in output is permitted. By experimentation, the following choice of weights for policy-maker 2 was made: a = 2, 10 < b < 20, c = d = 0, e =f= 1

(4.2)

Table 7.3 gives the long-run equilibria for both the optimal, time-inconsistent and the time-consistent open-loop policies. Two welfare loss functions have been selected for each policy-maker at the extreme ends of the ranges in (7.2) and (7.3). For policy-maker 1 who is prepared to see the balance of payment deficit (current account) rise by as much as 1.35 per cent, as a per cent of money GDP (d= 10), government spending under the optimal policy

Table 7.3. Long-run equilibria for optimal open-loop policies in deviation form (%): results from linear model Policy-maker 1 Economic variable (see section 3) y n bal Unemployment rate er g r T

Welfare loss

d = 10

Policy-maker 2 d = 30

b = 10

b = 20

OPT

TCT

OPT

TCT

OPT

TCT

OPT

TCT

2.54 2.64 -1.35 - 1.83 10.57 5.28 -2.32 1 6037

2.35 2.40 -1.19 - 1.67 9.93 5.15 -1.77 1 6020

1.76 1.48 -0.94 - 1.32 6.23 6.48 -1.47 1 8166

1.51 1.20 -0.74 - 1.12 5.39 6.29 -0.80 1 8170

-1.12 - 1.32 0.82 0.88 -4.05 - 1.24 2.54 0 4630

-0.88 - 1.32 0.65 0.69 -3.11 -0.90 2.06 0 4693

-1.70 - 1.96 1.21 1.32 -6.26 -2.07 3.59 0 5331

-1.48 - 1.72 1.07 1.16 -5.40 - 1.71 3.26 0 5402

190

Theory and methodology

(OPT) rises by over 5 per cent in real terms and the interest rate is lowered by over 2 per cent in the long run. Output rises by 2.54 per cent, inflation by 2.64 per cent per year and the real exchange rate falls by over 10 per cent in the long run. The time-consistent policy (TCT) is not substantially different, the main difference being the less rigorous use of monetary policy and with this change a smaller output increase, less inflation and less real exchange depreciation. The alternative case with d = 30 represents the more cautious policymaker 1 who is concerned about the current-account deterioration of the previous simulation. Now there is a shift from monetary to fiscal policy with a much smaller decrease in nominal interest rates. The outcome is a smaller rise in output but less inflation and less deterioration of the current account. Turning to policy-maker 2, inflation is squeezed out of the economy by reducing government spending and increasing interest rates. For b = 10 an annual drop in inflation of about 1.32 per cent requires a drop of 1.12 per cent in output from the optimal policy. When a greater priority is placed on inflation (b = 20), a 2 per cent drop in annual inflation costs a 1.7 per cent fall in output. The difference between the optimal and time-consistent policies is more significant for policy-maker 2, but not dramatic. Table 7.4 summarises the gains from optimal control for the two policy-makers. The control exercise considered up to now has been the use of open-loop policies in pursuit of output and inflation targets (y and h respectively). The first row of the table compares the welfare loss under control under optimal but time-inconsistent and time-consistent policies (W$PT and WoCT respectively) with the welfare loss under 'no control' which is the baseline forecast (Woc). Under control the welfare loss is reduced to 52-64 per cent of that under no control for policy-maker 1 and to 57-82 per cent for policy-maker 2. A striking result is that, using the welfare loss as a measure, there is almost no time inconsistency involved for either policy-maker in these open-loop policies. Suppose now that the economy is subjected to exogenous shocks. The five shocks considered in the chapter are permanent changes to OECD industrial production (y*), the dollar price of oil (p°), the world short-term interest rate (r*), the world price level (p*) and the average wage rate (wr). If these shocks are unanticipated the policy-maker must implement the optimal or time-consistent policy in feedback form. As mentioned in the previous section this involves a very complex form of rule. But putting this problem to one side for the moment, let us consider the welfare loss measured about the open-loop trajectories of the controlled economy with and without this added feedback rule. The ratios given in the bottom part of the table show the gains from feedback control in the face of the five

Macroeconomic policy design

191

Table 7.4. Gains from optimal control: results from linearised model Type of policy-maker Policy-maker's objectives

Policy-maker 1

Pursuit of targets y, n Stabilisation in response to shock: wr y* p° r* P*

d=30

d=K ) W°0PT

Policy-maker 2

wiCT

6 = 1C)

rpoPT

TXTCT

M/NC

TJ/NC

M/OPT

6 = 20 U/TCT

W°0PT

wiCT

JXTMC

0.64

0.64

0.52

0.52

0.81

0.82

0.57

0.58

0.24 0.64 0.90 0.70 0.25

0.24 0.65 0.92 0.72 0.25

0.18 0.55 0.71 0.52 0.20

0.18 0.55 0.71 0.54 0.20

0.26 0.42 0.59 0.39 0.29

0.26 0.48 0.60 0.49 0.32

0.16 0.30 0.40 0.28 0.19

0.16 0.36 0.41 0.38 0.22

Notes:

WoPT: Welfare loss at time t = 0 under optimal policy. Welfare loss at time / = 0 under time-consistent policy. *oc\ Welfare loss at time t = 0 under 'no control'. wr: Average earnings. y*: OECD industrial production. p°: Dollar price of oil. r*: World interest rate. /?*: World price index.

permanent shocks. These gains are considerable especially for wr and p* shocks. For policy-maker 1 there is again practically no time inconsistency in the feedback rule of any of the shocks. For policy-maker 2 there is significant time inconsistency for three of the shocks, y*, r* and /?*. Furthermore the values of the ratio IVQCT/W^PT, a measure of time inconsistency, increases for policy-maker 2 as the weight b on inflation increases. It would appear then that the extent to which time inconsistency is a problem depends on the concern for inflation on the part of the policymaker. This is an interesting result which contrasts with that in Barro and Gordon (1983b) where a high weight on inflation reduced the degree of time inconsistency. The reason for this difference is that the source of time

192

Theory and methodology

inconsistency is different for the two models involved. Barro and Gordon's model uses a static natural rate model in which time inconsistency arises from the benefits of surprise inflation only. In the LBS model surprise inflation effects in the labour market are not very significant and expectations in this sector are assumed to be formed adaptively anyway. Time inconsistency arises from the forward-looking rational character of the financial sector and mainly from the ability to manipulate the forward-looking exchange rate. This feature is more pronounced for the second policy-maker for whom real exchange-rate appreciation is the main mechanism for reducing inflation. Thus, with the latter policymaker, for a given change in output the real exchange rate changes by far more. This reflects the different monetary/fiscal policy mix which is more 'monetarist' for policy-maker 2. 4.2 Time inconsistency and sustainability We now turn to the issue of sustaining the optimal policy given that it is time inconsistent with a possible incentive to renege. As we have seen, time inconsistency is only a problem of policy-maker 2 and so we focus on this case. Consider both the optimal open-loop trajectory and the stabilisation responding to the five permanent shocks using the optimal policy in feedback form. In this chapter we only examine the case where reputation is lost forever if the policy-maker reneges and the one period of'cheating' is ignored. The results are displayed in table 7.5. The columns marked 'before' on the table show the welfare loss, Wt, from time t onwards ('cost-to-go') along the optimal trajectory. The columns marked 'after' are the welfare loss following reoptimisation and a permanent loss of reputation. If this value dips below the cost-to-go in the preceding column then the sustainability condition (7.5) is not satisfied. Table 7.5 shows that the sustainability condition is satisfied along the open-loop trajectory and following permanent shocks to wr, po and wr. By contrast there is an incentive to switch to a non-reputational policy in response to shocks to y* and r*. The underlined numbers indicate where the welfare loss, after reneging, falls below that under the optimal policy. We have not succeeded, then, in establishing sustainability for policies in response to a permanent change to shocks y* and r*. However, the incentive to renege (as measured by the difference between the 'before' and 'after') are very small compared with the welfare gains from control. Moreover the addition of stochastic disturbances can completely alter the picture in favour of sustainability.5

Table 7.5. The sustainability condition (policy-maker 2): results from linear model y*

wr

0 4 8 12 16 20 24 28 32 36

r*

P°

p*

Before

After

Before

After

Before

After

Before

After

Before

After

Before

After

5474 5459 5479 5487 5472 5443 5413 5387 5365 5349

5534 5515 5539 5550 5535 5506 5475 5449 5426 5437

22.43 22.15 22.59 22.93 23.00 22.97 22.94 22.91 22.86 22.80

22.96 22.59 23.08 23.44 23.49 23.44 23.40 23.37 23.31 23.25

0.24 0.12 0.09 0.09 0.09 0.10 0.10 0.10 0.10 0.10

0.29 0.11 0.09 0.09 0.09 0.10 0.10 0.10 0.10 0.10

0.091 0.073 0.074 0.075 0.074 0.073 0.073 0.072 0.072 0.072

0.093 0.075 0.075 0.076 0.076 0.075 0.074 0.074 0.073 0.073

0.42 0.12 0.06 0.04 0.04 0.03 0.03 0.03 0.03 0.03

0.56 0.09 0.04 0.04 0.04 0.04 0.03 0.03 0.03 0.03

0.14 0.11 0.10 0.10 0.11 0.11 0.11 0.11 0.11 0.11

0.16 0.11 0.10 0.11 0.11 0.11 0.11 0.11 0.11 0.11

Notes: 'Before' denotes welfare loss before reneging. 'After' denotes welfare loss after reneging.

194

4.3

Theory and methodology

Stabilisation using a simple rule

We now turn to the design of simple feedback rules for fiscal and monetary policy which can get as close as possible to emulating the performance of the optimal feedback rule. In principle each policy-maker could adopt a different feedback rule reflecting the contrasting priorities for stabilisation of output and inflation. We shall consider, however, a 'neutral' policymaker who places equal weights on output and inflation. The weights chosen are a

= b = 5;d= \0;e=f=

1

(4.3)

The rule that emerges from this neutral policy-maker could be thought of as a 'technocrat's rule' which is a constitutionally binding form of precommitment as recommended by Kydland and Prescott in their seminal paper on time inconsistency. An important feature of simple rules is that they do not have the 'certainty-equivalence' property, i.e., a rule designed for one particular shock is sub-optimal when a different shock occurs (see Levine and Currie, 1987b). One approach to the non-certainty-equivalence problem is to minimise an expected payoff calculated on the basis of a weighted average of the shocks, the weights being chosen on the basis of the frequency and variance of the shocks. An alternative approach, adopted here, is to seek a robust rule which performs at least as well as no control irrespective of which shock hits the economy. Table 7.6b shows the 'stabilisation gains' measured as the ratio of the welfare loss under the rule to the welfare loss under no feedback control. For each of the five permanent shocks an optimal simple rule of the form Art = axAyt-i + a2yt-\ + c ^ - i + a4Abalt-i Agt = px Ayt-A + p2yt-4 + j837c/_4 + p4Abalt-4

(4.4)

has been computed following the optimisation procedure described in Levine (1988). All variables are now measured as deviations about a given open-loop trajectory such as the optimal path or the base-run. These rules just consist of proportional plus integral control on output and proportional control only on the price level and the current account. Rules (i) to (v) in table 7.6b have been designed in turn for the five permanent shocks. The diagonal entries in the table show the gains from control if the shocks are correctly anticipated so that the actual shock is that for which the rule is designed. The off-diagonal entries indicate the problem arising from the failure of certainty-equivalence for simple rules. Of the five rules only two of them - those designed for shocks j * and p° are better than no feedback control for all shocks.

Macroeconomic policy design

195

Table 7.6a. The feedback coefficients for rules (i)-(vi) (see equation (4.4)) Feedback coefficients

Rule

ax

(i) (ii) (iii) (iv) (v)

1.8 0.50 0.70 5.76 1.72

a2 -0.02 0.03 0.03 0.04 0.001

a3 0.40 0.51 0.40 0.17 0.88

a4

px

fi2

ft

ft

-0.43 -0.52 -0.42 -0.02 -0.63

-0.9 -0.23 -0.35 -1.62 -0.82

0.01 -0.04 -0.05 -0.02 -0.001

-0.30 -0.35 -0.27 -0.08 -0.66

0.87 0.19 0.25 -0.09 0.52

Table 7.6b. The welfare loss compared with 'no feedback control'for simple rules designed for specific shocks: results from linearised model Actual shock Expected shock

wr

y*

P°

r*

P*

wr Rule (i)

0.26

0.94

1.01

0.81

0.34

y* Rule (ii)

0.26

0.78

0.92

0.87

0.34

P° Rule (iii)

0.26

0.79

0.89

0.85

0.34

r* Rule (iv)

0.27

1.35

1.35

0.76

0.37

P* Rule (v)

0.28

0.93

1.10

0.85

0.33

Optimal policy

0.27

0.57

0.85

0.69

0.30

Table 7.6a gives the actual values of coefficients for the five rules. The 'specialised' character of rule (iv) in particular is shown by the extreme values of coefficients ax and ft. The rule which appears to perform best irrespective of the shock (i.e., is robust) is rule (iii) designed specifically for thtp° disturbance. This is the rule selected as the optimal, simple rule of the form (4.4).

196

Theory and methodology

4.4 Validation All the results up to this point have been based on optimal control analysis and simulations on the linearised model. The latter, as we can see in appendix 1, is a good approximation to the full LBS model. We expect then the designed policies to perform well when applied to the original model and this we now examine. Table 7.6 and figure 7.1 show the open-loop trajectories of selected variables under the optimal policies for the cases d = 30 for policy-maker 1 and b = 10 for policy-maker 2. For these simulations and those which follow the terminal conditions, imposed in 1996(4), consisted of a constant growth rate for pe (price of equity) and a constant level for er (real exchange rate) anApg (price of gilts). The open-loop trajectories closely follow those for the linear model for the first five years or so but due to the different terminal conditions (imposed at infinity on the linear model) diverge somewhat thereafter. Thus the terminal values in table 7.7 differ significantly for some variables from the long-run equilibria in table 7.3. However, the important feature of the results for the full model is that they reflect the two policy-makers' objectives in a satisfactory way. Policy-maker 1 raises output by 1-2 per cent and reduces unemployment by 1-1.3 per cent (representing up to 500,000 off the register) at an annual inflation cost of about 1 per cent and at a cost of 1 per cent to the current account after eight years. Policymaker 2 reduces inflation by almost 1 per cent at an output cost of 0.5-1.4 per cent over the period. The main reason for the asymmetry between the two policy-makers is that policy-maker 1 is prepared to raise taxation equivalent to about 2p on the basic rate of income tax. This helps to contain both inflation and the deterioration in the current-account deficit. Table 7.8 shows the gains from using the optimal simple rule on the full LBS model to stabilise the economy about some desired path in the face of the five permanent shocks considered previously. The ratios shown are the welfare loss for the neutral policy-maker under feedback control relative to that under no feedback control for three different time horizons. Clearly the simple rule performs extremely well for four out of five of the shocks and is still better than no feedback control for the worst shock, y*. Figure 7.2 compares trajectories for GDP, the consumer price level (pc) and bal the current account as a per cent of nominal GDP (as deviations about the forecast) under feedback control with those under no feedback control.6 Following a permanent 1 per cent shock to nominal earnings, figure 7.2 shows a steep rise in the price level to a maximum of 1.5 per cent, a rise in GDP to a maximum of 0.4 per cent and a deterioration in bal of 0.5 per cent after eight years. The feedback rule brings about a tightening

Table 7.7. Open-loop optimal trajectories: results from LBS model

Economic variable GDP

Consumer price index Unemployment rate Current-account

Policy-maker 1(^=30)

Policy-maker 2(b=

Time

Time

10)

1

2

5

8

1

2

5

8

quarter

years

years

years

quarter

years

years

years

1.6 0.0

1.9 2.2

1.9 5.0

1.5 7.4

-0.3 0.0

- 1.4 -2.0

- 1.3 -4.4

-0.5 -5.7

-0.9

- 1.2

- 1.3

- 1.2

0.1

0.6

0.9

0.7

- 1.1

-0.4

-0.8

-0.9

0.6

0.3

0.7

0.7

9.8

5.5

4.8

3.0

-7.4

-4.8

-2.4

0.6

6.0

5.9

6.2

6.4

-0.0

-0.7

- 1.1

- 1.3

- 1.3

-0.8

- 1.2

- 1.4

1.0

1.0

1.0

1.0

0.9 0

1.5 0.0

2.3 0

2.6 0

BOP

Real exchange rate Government spending Interest rate Tax rate

Open-loop trajectories: policy-maker 1 G

6.0 Change

6.0 c

4.0

CD

u >5

2.0-

BAL s^* -2.0 ~\ 1 1 1 1 I I I I Q3 Q3 Q3 Q3 Q3 Q3 Q3 Q3 Q3 1988 1989 1990 1991 1992 1993 1994 1995 1996

Q3 Q3 Q3 Q3 Q3 Q3 Q3 Q3 Q3 1988 1989 1990 1991 1992 1993 1994 1995 1996

1.0 7

8.0 -

R ^ ^- - "

2.0 -

O) CD

----*

3.0 -

c

GDP

20 y

Open-loop trajectories* policy-maker 2

CD

4.0

0.0

0.0 -2.0

Open-loop trajectories: policy-maker 1

8.0

8.0

CD D) C CD .C

(J

SI

(J

0.0 ^ I

^ I

I

^ I

"

"

•

"

•

"

"

•

—

—

—

«.

Open-loop trajectories: policy-maker 2 BAL

4.0 2.0 0.0 -

G I

"

6.0

r - ' " "

1.0

"

I

I

.1

1

Q3 Q3 Q3 Q3 Q3 Q3 Q3 Q3 Q3 1988 1989 1990 1991 1992 1993 1994 1995 1996

> ^

^

PC

2.0 ~| 1 1 I 1 i i i """" r Q3 Q3 Q3 Q3 Q3 Q3 Q3 Q3 Q3 1988 1989 1990 1991 1992 1993 1994 1995 1996

Figure 7.1 Open-loop trajectories

Macroeconomic policy design

199

Table 7.8. The welfare loss compared with 'no feedback control for simple rule (4.4), coefficients as in rule (in), table 7.6a: results from LBS model Time horizon Shock

8 quarters

20 quarters

32 quarters

wr y* p° r* P*

0.62 0.99 0.49 0.55 0.61

0.49 0.97 0.47 0.61 0.61

0.48 0.97 0.53 0.71 0.62

of fiscal and monetary policy to counteract these spending falls by about 0.3 per cent after eight years and interest ratesriseby 0.4 per cent. This has the effect of bringing both GDP and the price level down to their baseline values by the terminal date. To assess what 'might have been' if feedback control in the form of our simple rule had been in operation in the recent past, consider the oil price shock. Assuming approximate linearity we can deduce the following from the 10 per cent permanent shock to the dollar price of oil (figure 7.3). Without feedback control the effect of a doubling of the oil prices experienced in 1979-80 is that the real output falls by about 5 per cent after one year and recovers slowly thereafter. The price level falls by about 25 per cent over eight years representing about 3 per cent off inflation a year. With the feedback rule output falls by 3 per cent but quickly recovers to 1 per cent below its baseline after five years. The cost of considerable output gains over the period is that the price level falls by only 10 per cent over eight years so there are less gains for inflation control. To achieve these feedback gains requires a loosening of fiscal and monetary policy according to the rule. Thus from figure 7.3 government spending would have to rise by 4 per cent and interest rates to fall by 5 per cent over eight years. These calculations assume a permanent doubling of the price of oil and so exaggerate the necessary adjustment to the actual oil price behaviour of 1979-80 which was not, of course, permanent. Nonetheless the simulations do illustrate the potential gains from properly designed fiscal and monetary policies in feedback form.

Average wage shock {wr) positive constant adjustment of 1%

Average wage shock (wr) positive constant adjustment of 1%

0.60

R(C)

0.40 , which is assumed to be known by economic agents along with the model and the current state vector. The solution yields the asymptotic variances and covariances of all endogenous variables, and hence permits us to evaluate the loss function. We can then implement an iterative search procedure in which the unconstrained elements of D are varied so as to minimise the loss function. This procedure yields optimal simple linear time-invariant feedback rules. That policy should be expressible in this form is restrictive for two reasons. First, as we indicate in the following, the full optimal policy cannot be implemented in the form of a linear time-invariant feedback rule of the form of (2.2), so that to insist on policy formulated in this form is restrictive. (This is an important difference between control of models with, and those without, free variables. However, as indicated in chapter 4, the optimal rule can be implemented by means of linear feedback on the z vector, together with integral control terms involving z.) Second, restrictions on the coefficients of the D matrix, such as the zero restrictions that arise naturally from indicator regimes or decoupled control, limit policy design further. It is helpful to be able to assess the costs of these restrictions on policy design. For, if the costs are not high, the benefits of design simplicity discussed in the introduction to this chapter may make simple policy desirable. This requires some benchmark against which the performance of simple rules can be judged. The obvious benchmark is that of the performance of the full optimal rule. The full optimal rule is particularly useful as a benchmark since it has the important property of certainty equivalence. This means that the

218

Fiscal and monetary policy in interdependent economies

closed-loop feedback solution to the deterministic and stochastic control problems are of the same form, so that the solution to the stochastic problem is independent of the disturbance covariance matrix, E. Use of the full optimal rule means that policy-makers need not assess the combination of shocks likely to perturb the system. In this sense, the full optimal rule provides 'a horse for all courses'. By contrast, linear time-invariant feedback rules of the form (2.2) do not satisfy certainty equivalence. This is because the optimal choice of the parameters of/) depends on the disturbance covariance matrix, E. This is a major disadvantage of simplicity in policy design, for policy-makers may have no reasonable estimate of E. It is not helpful to policy-makers to have a 'horse for each course' if the actual course is not known. However, the lack of certainty equivalence does not rule out the existence of simple rules that are robust in the sense of performing reasonably well whatever the disturbance covariance matrix, E. (We here judge performance relative to that of the full optimal rule.) In our design evaluation of section 4 we pay particular attention to whether robust rules of this kind exist. Of course, robustness with respect to changes in the parameters of the disturbance covariance matrix is merely one aspect of the broader question of robustness with respect to other model parameter and specification changes, which we also consider. Before turning to the detailed derivation of our control methods, we must briefly consider the issue of time inconsistency. Both the full optimal rule and the linear time-invariant rules considered here are time inconsistent, because they offer to policy-makers a short-term incentive to renege on the rule which the private sector have assumed in formulating their plans. However, reneging imposes longer-run costs by undermining faith in the ability of governments to keep to commitments. If such faith is undermined altogether, the result is likely to be a closed-loop Nash equilibrium. This may be arrived at by means of a type of Cournot adjustment process, with each side determining its decision sequentially on the assumption of a given feedback rule on the part of the other.1 Alternatively, it may come about by the private sector assuming that the government will renege if there is any short-run incentive to do so, calculating the government's optimal action accordingly, and then determining its optimal plan in the light of this; the government then determines its optimal plan subject to this procedure for private-sector decision-making. This equilibrium is time consistent, and is necessarily inferior to the time-inconsistent full optimal policy considered in this chapter (see Buiter, 1983). If the closed-loop Nash equilibrium is markedly inferior to the equilibrium under time-inconsistent optimal policy, whether full or simple,

Macroeconomic policy design in an interdependent world

219

this provides a strong incentive for a far-sighted government not to succumb to the temptation of reneging. Our earlier analysis (Currie and Levine, 1987 and chapter 5 in this volume) suggests this to be the case, at least for certain simple models. It also shows that well-designed simple rules may perform significantly better than the closed-loop Nash equilibrium, so that a similar point applies to this class of rules. This does not, of course, mean that a more myopic government will not renege, but it does suggest that the problems of time-inconsistent policies need not be over-stressed.2 The solution procedures described above apply to cooperative policymaking. We now turn to non-cooperative equilibria. Equation (2.1) is sufficiently general to represent an ^-country model of interdependent economies. In particular for the case of two countries, denote as unstarred variables those referring to country 1 (the 'home' country) and starred variables those referring to country 2 (the 'overseas' sector). Then we may write a two-country model in the form z 'dz ' dz* z* =A X dx dx*

[B,B*]

dt

X*

dv{ dvf dv2 dv*

(2.4)

where the dimensions of z, x*, x and x* are ( « - m ) x l , ( « * - m*) x 1, m x 1 and m* x 1 respectively. Consider first the case of the two countries pursuing simple rules of the form (2.2). Suppose country 2 adopts a policy given by (2.5)

Then substituting (2.5) into (2.4), country 1 faces the dynamic constraint dz dz* dx dx*

z B*D*]

+ Bwdt + dv

(2.6)

where we denote dv r = [dvf dv^ In a Nash game each 'player' chooses its own move taking the current observed 'moves' of other players as given. In a closed-loop Nash game these moves of other players are observed in feedback form on the state vector. For the two-country game presented here this means that country 1 chooses a feedback rule

220

Fiscal and monetary policy in interdependent economies z w= D

(2.7)

x

taking D* as given. Suppose country 1 has a loss function asy E(W) where W=[zTz*TxTx*T]Q

+ wTRw

(2.8)

with a similar loss function asy E(W*) for country 2, Q and R being replaced with Q* and R* respectively. Then the optimisation problem of country 1 - to minimise asy E(W) subject to (2.6) - is in the form of the problem discussed above and leads to an optimal value for D in the form of a reaction function D =/(/)*). Similarly country 2 has a reaction function D* = g(D) and the Nash equilibrium is at the fixed-point of/g for D and gfTor D*. A plausible adjustment process by which countries move towards a closed-loop Nash equilibrium is the following Cournot-type sequence of moves. Given an initial feedback rule for country 2 with D* denoted by Z>*(0), country 1 chooses D = D(l) =/(/)*(0)). Then country 2 revises its choice of D* to Z)*(l) = g(Z)(l)). Country 1 then reoptimises and so on. This process may or may not converge to an equilibrium depending on the initial value of D*. Indeed a Nash equilibrium may not exist and if it does it may not be unique. Given the complexity of models with rational expectations, it seems a formidable task to obtain analytical conditions for the existence, uniqueness and convergence of a Nash equilibrium and we do not attempt to do so here. Consider next the full optimal policy for the two-country model. Suppose country 2 adopts an optimal rule from which chapter 5, equation 3.1 can be implemented in the form w* = D*

(2.9)

dp* = p*

(2.10)

where

pf has dimensions (m + m*) x 1 and P* depends on A, B, B* and country 2's loss function. (Country 2 could arrive at (2.9) by assuming country 1

Macroeconomic policy design in an interdependent world

221

was pursuing a simple rule of the type considered above.) In a closed-loop Nash game country 1 takes (2.9) as given and optimises subject to the dynamic constraint •dz

'z

•

dz*

"dvi"

= A Pi

dpi

dx dx*

dvf

>

T*

dt +

X

0 wdt + B2

.x*.

0 dv2 dvf_

(2.11)

where A=

Ft

B*2Dt

A2\ and

we

have

P*

An 0 A22

(2.12)

partitioned

is A so that An B so Bl that is (n + w* - ra - ra*) x r, Z>* = [Z)f, /)*] with D?r x (w + w* - ra - ra*) and P* similarly. The optimal feedback rule of country 1 is then of the form

(n + n* — m — ra*) x (n + n* — raw*),

w=D

Pi

(2.13)

P2

where

(2.14)

=P P2

and/? 2 has dimensions (m + m*) x 1. Comparing (2.9) and (2.13) we see that matrices D and D* defining the feedback rules are not comparable as they have different dimensions. The reaction function analysis adopted from simple rules is now not applicable. If we envisaged a Cournot-type adjustment process, then with each iteration the model defining the dynamic constraint for the optimising country increases its dimension by the number of non-predetermined variables m + m*. The game may well converge in the sense that the coefficients of D and D* relating to the additional p2 and/?f terms tend to zero and the welfare loss tends to a finite quantity. 3 If we confine ourselves to simple (and, in general, time-inconsistent) rules of the form (2.5) and (2.7) this complex convergence problem does not arise. Nor does it arise if

222

Fiscal and monetary policy in interdependent economies

one focuses on time-consistent (but sub-optimal) policies for government or on Nash open-loop games between countries (see Miller and Salmon, 1985 and chapter 9). All these options form interesting directions for research; but in this chapter we shall analyse non-cooperative behaviour only in terms of the first, namely simple policy rules. 3

The model

Throughout the rest of this chapter we use variants of the following eleven equations continuous-time stochastic model: dy — ii/\\a\q — a2(r — pe) + a3 v — a4s + a5y* — y]dt + du\ (3.1) dm = y/2[y\y — y2r + p + y3v — m]dt + du2 At - dp = dw' = y/ y(r)dT + pf c - w\ w \ dt + du3 V3\P\ y(r)dr 31 px | L J-oo J

(3.2) (3.3) (3.4)

p = 0pd + (l- 9)(w* + e)

(3.6)

dp=y/4(p-p)dt

(3.7)

c

de = (r-r*)dt q

= w* + e -

(3.8) (3.9)

w

dw*= - / / 1 w * d / + dw4

(3.10)

d r * = -/u2r*dt

(3.11)

+ du5

where the following notation is used: e y q r s v m p pd w w* r* dut

nominal exchange rate (defined as the price of foreign exchange) real output competitiveness domestic nominal rate of interest autonomous taxation real net financial wealth of the private sector nominal money supply general price index price index of domestic output nominal wages nominal wages overseas foreign nominal rate of interest white noise disturbance

Macroeconomic policy design in an interdependent world

223

Because we are concerned in this chapter only with stabilisation of the system around an exogenously given long-run equilibrium (which may incorporate trends), all variables are measured in terms of deviations of their logarithm from equilibrium, except for interest rates which are measured as deviations of proportions. All parameters are defined to be positive. An 'e' superscript denotes an expectation formed at time / on the basis of information available up to time t\ while a '*' superscript denotes the foreign counterpart to the variable in question. A bar denotes a partial equilibrium value. Equation (3.1) represents the IS curve with output adjusting sluggishly with a mean lag of y/rl to competitiveness, the real interest rate, real financial wealth, autonomous taxes and foreign demand. Equation (3.2) represents the LM curve. The money supply is assumed to be demand determined for any given level of interest rates, and money demand adjusts sluggishly to output, interest rates and real financial wealth with a mean lag of ^i" 1 . Equation (3.3) determines the change in real wealth from the determinants of the sum of the government budget deficit and the current account of the balance of payments. Neglecting interest payments and approximately this relationship loglinearly, this makes the change in real wealth depend positively on competitiveness and foreign output, and negatively on domestic output, autonomous taxes and inflation. Equation (3.4) determines the level of nominal wages. Taking the derivative of its deterministic part, we have that long-run wage inflation is determined by an expectations-augmented Phillips curve, but with actual wage inflation adjusting sluggishly towards this long-run relationship. The sluggishness of wave adjustment generates fluctuations in real output in the face of demand disturbances, even under the rational expectations (see, for example, Buiter, 1980). Equation (3.5) is a partial equilibrium relationship giving the price index of domestic output in equilibrium as a weighted average of domestic and foreign wages (the influence of the latter variable working partly through a mark-up on costs and partly through competitive pricing effects). The corresponding general price index in partial equilibrium is given from (3.6) as a weighted average of domestic prices and foreign wages. Actual prices adjust quickly but not instantly according to (3.7) where y/4 is large.4 Equation (3.8) models the exchange rate as asset market determined under conditions of perfect capital mobility.5 The expected rate of depreciation of the exchange rate in an interval dt (denoted by dee = ec(t + dt, t) — e(t) where ee(r, i) is the expected exchange rate at time T, formed at time i) exactly offsets the interest-rate differential in favour of the home currency. (Note that r = 0 in equilibrium corresponds to the domestic and foreign interest rates being equal.) Unlike other variables, which adjust slowly and are predetermined variables, the exchange rate is

224

Fiscal and monetary policy in interdependent economies

non-predetermined and can make discrete jumps in response to changes in exogenous variables or policy rules. Equation (3.9) defines competitiveness in terms of relative costs. Equations (3.10) and (3.11) specify exogenous first-order autoregressive processes for foreign wages and foreign interest rates respectively. Some persistence in these disturbances is required if they are to have any impact on domestic variables, and this process involves the minimum additional complication. Equations (3.1) and (3.11) specify our model of the small open economy. For the two-country analysis of interdependent economies, we assume an identical structure for the overseas economy. Our model of the overseas sector is therefore given by equations (3.1)—(3.9), with unstarred variables being replaced by starred variables and vice versa.6 In the appendix we set out the single-country model and the two-country model in the form of equation (2.1). For our subsequent policy design analysis, we need to define a suitable loss function. Our loss function for the single country is assumed to take the form W=ay* + bp2 + cr2 + s2

(3.12)

while for the two-country case we assume an aggregate loss function of the form W* = a(ya)2 + b(p*)2 + c(ra)2 + (saf

(3.13)

where the 'a' superscript denotes the sum of the relevant variables over the two countries (e.g. ya = y + y*). We may also consider a divergence loss function given by: Wd = a(ydf + b(pdf + c{rdf + (sd)2

(3.14)

where the 'd' superscript denotes the divergence of the relevant variable between the two countries (e.g. yd = y - y*). As shown in the appendix, the assumption of two identical countries permits us to decompose the two-country model into two orthogonal parts, the aggregate model (given by (A.4)) and the divergence model (given by (A.5)). As explained in section 6, we may therefore choose aggregate rules to minimise the aggregate loss function, given the aggregate model; and, quite separately, choose divergence rules to minimise the divergence loss function, given the divergence model. Since Wa= Wd = 2{W+ W*) where W* is the value of (3.12) evaluated over foreign variables, this amounts to minimising a loss function consisting of the sum of the individual countries' loss functions. Our assumed parameter values are set out in table 8.1, together with

Macroeconomic policy design in an interdependent world

225

Table 8.1. Parameter values Parameter y/x y/2 y/3 y/4 ax a2 a3 a4

Low

0.1

7, 72 y3 A 4>o 4>i

ck 3 c, 0 /i, & a b c

Central 0.5 0.5 0.5 10.0 0.3 0.1 1.0 0.4

High

0.5 2.0

1.0 0.1

0.5 0.5

1.0 1.0 0.3 1.0 1.3

0.1 0.5 0.7 0.7 0.5 0.5 2.0 2.0 1.0

2.0

5.0

variants of a number of parameter values to test for robustness of policies with respect to parameter change. The parameter a5 is set equal to the degree of openness, 1 - 0\C\. For the parameter of the objective function, we assume c = 1, penalising equally variations in r and s. We penalise price fluctuations twice as much (b = 2) relative to the instruments, and for our central assumptions penalise output and pricefluctuationsequally (a = 2). We also consider a Keynesian variant in which output fluctuations are penalised much more heavily (a = 5). Only results for central parameter values are reported in tables 8.2-14. 4 The design of rules for monetary and fiscal policy in a small open economy

In this section we review results considered in more detail in Currie and Levine (1985a) for the small open economy. We first consider monetary policy alone, thus holding the fiscal instrument, autonomous taxes,

226

Fiscal and monetary policy in interdependent economies

constant. Since tax receipts fluctuate with the level of economic activity, this amounts to allowing automatic fiscal stabilisers to operate unimpeded. Our simple rules for monetary policy alone take the form _

Pm pe Piy + P) Pp

monetary rule exchange-rate rule nominal income rule price level rule

(.

..

where p is chosen optimally by the procedure outlined in section 2. Each of the rules provides a long-run anchor for expected nominal variables provided that disturbances to the system follow stationary processes.7 In the case of the exchange-rate target, this anchor depends on foreign prices following a stationary process, but trends in foreign prices may be offset by a suitable trend in the exchange rate along the long-run equilibrium path. We next consider the use of fiscal policy (represented in the model by autonomous tax changes) in conjunction with monetary policy. We consider three forms of decoupled control rules given by

DCRI

J \s = p2 r 1 DCRII II \ r J * ^ , \s = p(y+p)

(4.2)

The first, DCR I, represents a Meade-type assignment of using fiscal policy to keep nominal income on track and using monetary policy to stabilise the real exchange rate. DCR II is similar, except monetary policy now tracks the nominal, rather than the real, exchange rate. DCR III represents an extension of the price rule in (4.1) whereby monetary policy is assigned to the price level while fiscal policy reacts to real output fluctuations. This rule implies that a rise in output, for example, generates a contraction of fiscal policy, with a subsequent tightening of monetary policy as and when there is an effect of higher output on inflation. All these rules are of a simple proportioned form: as noted below, elements of integral control may also be desirable. The solution procedure described in section 2 can now be used to find values of p for rule (4.1), and of px and /?2 for (4.2), that minimise asy (JV) with W given by (3.12).8 As noted in that section certainty equivalence does not apply for simple rules so that the design of each category of rules

Macroeconomic policy design in an interdependent world

227

Table 8.2. Best policy rules and welfare losses for the single open economy Disturbances

Policy rule

dw,

du2

du3

dw4

du5

Minimal control Optimal monetary Optimal monetary + fiscal

2.44 2.28 1.50

0.00 0.00 0.00

1.22 0.39 0.22

0.96 0.09 0.09

3.83 0.39 0.31

r = 0.56 (y+p)

3.08 3.82 2.31

0.00 0.48 0.00

0.82 0.68 0.47

0.16 0.27 0.16

1.04 1.09 0.46

2.04

0.00

0.38

0.19

0.75

2.49

0.00

0.33

0.15

0.60

r = 0.77 m r = 4.02/7 r=\0p

1

r = 2.87 pi s = 1.24 y]

depends on the covariance matrix of the disturbances. Our approach to this problem is to consider one disturbance at a time and choose the optimal value of the parameter (or parameters) for each rule. We then evaluate the welfare loss for the chosen optimal rule when each of the other shocks in turn hits the system. For each form of simple rule then there exist up to five optimal rules corresponding to anticipated disturbances uh i= 1,5. In table 8.2 we report only those with superior performance for central parameter values and loss function asy E(2y2 + 2p2 + r2 + s2). For monetary policy alone the exchange-rate rule, because of the segmentation of the model under this regime, gives no improvement on minimal control for all disturbances except u5 and even for that disturbance it performs relatively badly.9 There is little to choose between the best nominal income and monetary rules reported; the former performs better for U\ and u2 shocks and the latter for a u3 shock. But the rule that completely dominates in category (4.1) is the price rule, i.e., irrespective of the nature of the disturbance the best price rule is superior to the best monetary rule, nominal income or exchange-rate rules. In addition, comparison with the performance of the full optimal rule shows that its performance compares quite well with this benchmark, particularly for domestic shocks. Thus the costs of simplicity do not appear to be enormous for this rule. The other simple rules, by contrast, perform rather badly overall.

228

Fiscal and monetary policy in interdependent economies

Table 8.3. The consequences of monetary targeting. Policy rules r = fim, with increasing values of ft asy var(ra)

fi dii,

5 10 15 20

Optimal = minimal Minimal control du2

5

10 15 20

Optimal = minimal Minimal control dw3

5 10 15 20

Optimal (fi = 0.77) Minimal control dw4

5 10 15 20

Optimal (fi > 40) Minimal control du5

5

10 15 20 Optimal (fi > 40) Minimal control

asy var(r)

asy var(/?)

iisy ^ /ar(»

Welfare loss

0.07 0.02 0.01 0.01 1.00 1.00

1.72 2.07 2.20 2.27 0.00 0.00

1.07 1.17 1.20 1.22 0.03 0.03

1.20 1.16 1.15 1.14 1.19 1.19

6.25 6.73 6.90 6.99 2.44 2.44

0.16 0.09 0.06 0.05 1.00 1.00

4.11 9.01 9.01 18.95 0.00 0.00

0.09 0.06 0.06 0.03 0.00 0.00

().00 ().01 ().01 ().01 ().00 (3.00

4.29 9.13 14.06 19.03 0.00 0.00

0.01 0.00 0.00 0.00 0.12 0.62

0.22 0.25 0.26 0.27 0.07 0.00

0.13 0.13 0.13 0.13 0.12 0.37

(3.15 (3.14 (114 (3.14 (3.19 (3.24

0.77 0.79 0.79 0.68 0.68 1.22

0.00 0.00 0.00 0.00 0.00 0.29

0.03 0.03 0.03 0.03 0.003 0.00

0.07 0.07 0.06 0.06 0.06 0.46

(3.01 (3.01 (3.01 (3.01 (3.01 (3.02

0.18 0.17 0.17 0.16 0.16 0.96

0.01 0.00 0.00 0.00 0.00 1.14

0.12 0.13 0.13 0.13 0.13 0.00

0.28 0.26 0.25 0.24 0.24 1.82

(3.02 (3.02 (3.02 (3.02 (3.02 (3.10

0.72 0.68 0.66 0.65 0.64 3.83

It is a familiar result from the government budget constraint literature (see, for example, Blinder and Solow, 1973; Christ, 1979) that monetary targeting may be unstable if the wealth effects on money demand are large relative to those on expenditure. To check whether this was at the root of the poor performance of the monetary targets, we examined the consequences

Macroeconomic policy design in an interdependent world

229

of increasing p so that the money supply is kept strictly on a fixed track. The results reported in table 8.3 indicate that there is no tendency for instability of the model under strict monetary targeting, suggesting that monetary targets perform badly because of their failure to dampen volatility rather than because of inherent instability. For the central parameter set we also find the price rule is rather robust in its performance. A choice of /? somewhere in the range between 2 and 10 yields a similar performance that is insensitive to disturbance uncertainty. To examine the consequences of parameter variation, we subjected the model to a variety of parameter changes (see table 8.1). These include a more Keynesian objective function, an increased degree of openness, a large impact of demand on inflation, an increased effect of wealth on money demand, an increased direct influence of monetary policy on demand, and reduced wealth effects on aggregate demand and money demand. While the details of these results vary, the superiority and robustness of the price-level rule remains intact; while the performance of the other simple rules remains poor. The only significant difference is if the influence of demand on inflation is stepped up, when the performance of the price-level rule relative to the full optimal rule falls significantly in dealing with domestic shocks. Turning to rule (4.2) we find that DCR I and DCR II perform badly and are ruled out as plausible policy rules. DCR III in the form of two variants performs well but in neither case is there a clear-cut improvement on the best-price rule using monetary policy alone. The first variant of DCR III improves the performance in the face of ux and u3 shocks at the expense of a deterioration in the face of u4 and u5. The second variant provides still more improvement in the face of w3 shocks at the expense of a deterioration for ux and u5 shocks. The top of table 8.2 presents comparisons with full optimal policies, both for monetary policy alone and forfiscaland monetary policy together. The additional use offiscalpolicy gives a significant gain in handling domestic disturbances. In consequence, optimal policy significantly outperforms our best simple rules with respect to all disturbances. Moreover, additional simulations showed that the performance of the optimal rule is robust with respect to parameter changes. Whenfiscalpolicy is considered, therefore, the costs of simplicity are high. However, the price-level rule continues to perform well relative to other forms of simple rule. The poorer performance of the price rule relative to optimal policy when fiscal policy is considered may reflect the absence of any elements of integral control in the design of our simple rules. It remains to be investigated whether better rules can be devised that incorporate integral as well as proportional feedback, whilst remaining simple in design.

230

5

Fiscal and monetary policy in interdependent economies

The cooperative two-country control problem

Hitherto in this chapter, we have considered policy design in the small open economy, treating the rest of the world as exogenous. In this section, we examine the consequences of assuming two interdependent economies, each identical to the model described in section 3 and assumed throughout our previous analysis. Our assumed loss function for the aggregate problem is described by (3.13), and penalises deviations in aggregate variables summed over the two countries. An orthogonal divergence problem is that defined by minimising a loss function (3.14) defined over deviations of the differences in variables between the two countries (see appendix). The joint solution to the aggregate and divergence problem is equivalent to minimising a loss function which is the sum of the individual country loss functions defined by (3.12). We focus first on the aggregate problem, and consider the performance of simple rules in this context. Since the exchange rate does not enter the aggregate problem, we ignore the exchange-rate rule given in 4.1, as well as DCRI and DCRII. We also consider three new variants of rule defined by NIR

fc:

[s = /fely + P) (5.1)

NIFR

s

= p2(

NIR represents a nominal income rule where both monetary and fiscal policy are used jointly to track nominal income. DCR IV represents a form of decoupled control, where interest rates are used to track the money supply andfiscalpolicy tracks nominal income. NIFR represents a nominal income target pursued byfiscalpolicy alone. Table 8.4 presents results for the aggregate problem for the rules using monetary policy alone (to be compared with the results of section 4). Because the scope for monetary policy is much less for the aggregate problem, with the channel of influence via the exchange rate ruled out, the gain from control is not large. This may be seen by comparing the results reported for minimal and optimal control respectively. Of the simple rules, the price rule gives no gain whatever relative to minimal control. Both nominal income and monetary rules give some gain, the nominal income rule coping better with demand {ux) shocks and the monetary rule performing better for supply shocks (w3).

Macroeconomic policy design in an interdependent world

231

Table 8.4. The aggregate two-country problem: monetary policy alone

Expected disturbance (unit variance)

Actual disturbance (unit variance) Policy rule

dwf

dw|

all

Minimal control Optimal

(r = 0.00/7)

19.30 14.79

17.05 14.50

dwf

Nominal income Monetary Price

(r=1.01(y+/?)) (r = 0.66 m) (r = 0.006/?)

15.45 16.18 19.26

17.84 15.71 17.05

dw|

Nominal income Monetary Price

(r = 0.38 (>;+/?)) (r = 0.50 m) 0 = 0.02/?)

16.70 16.27 19.35

16.40 15.61 17.05

Open economy best rules: r = 0.56(y+/?) r = 0.17 m r = 4.02/? Optimal

16.53 16.03 15.86 16.23 unstable 23.02 16.90

The bottom part of table 8.4 reports the performance of the best simple rules designed for the single economy reported in section 4 when applied in both countries simultaneously. The nominal income and monetary rules are not dissimilar to those devised for the aggregate country. The price rule, by contrast, represents a rather active feedback on prices, and for the aggregate economy it is totally destabilising. Thus the best rule designed for the single open economy is disastrous in its performance when applied generally. Similar problems apply to the full optimal rule designed for the single economy, when applied generally. Although it is stable, table 8.4 shows its performance to be very poor. This highlights the dangers of the usual approach to policy design which focuses on the single open economy. The results in table 8.5 for the two-country problem with joint fiscal and monetary policy indicate the important role for fiscal policy in the aggregate problem. The optimal policy gives very considerable gain when fiscal policy is used. Of the simple rules examined in table 8.5, DCR III gives the best performance. However, the parameter in the interest rate part of this rule indicates that monetary policy is playing a minimal part

232

Fiscal and monetary policy in interdependent economies

Table 8.5. The aggregate two-country problem: fiscal and monetary policy Actual disturbance (unit variance)

Expected disturbance (unit variance)

Policy rule

all

optimal

dwf

DCR III NIR

DCR IV NIFR dw|

DCR III NIR

DCR IV

3.30

8.31

4.48

13.66

5.28

38.84

s = 1.29 (y+p) s= 1.30 (y+p)

5.32 5.34

37.33 37.36

r = 0.0005/? 5 = 0.34;; r = 0.210+/?) s = 0.14 (y+p)

6.54

10.52

11.59

14.86

r = 0.29 m

12.02

14.83

s = 0.12 (y+p) s = 0.\5(y+p)

11.98

15.00

7.36

18.12

5.35

35.70

r = 0.0001/? s = \.\6y r = 0.25 (y+p) s = 1.29 (y+p) r = 0.09 m

NIFR Open economy best rules: r= 10/? s = 0.46 y r = 2.87 p

s = \.24y

optimal

dw|

dwf

unstable

in this rule, and the full burden of control falls onfiscalpolicy. This rule reduces to simple Keynesian policy of controlling output by fiscal feedback on output. The performance of the other rules is broadly similar, because of the limited influence of monetary policy, and amounts to fiscal feedback on nominal income. This performs poorly relative to optimal policy or DCR III. The performance of the best rules for monetary and fiscal policy reported in section 4 for the single open economy are reported at the bottom of table 8.5. In contrast to the use of monetary policy alone, the simple rules do not totally destabilise the system. However, their performance is poor, whilst the full optimal rule designed for the single open economy is totally destabilising when applied in the aggregate.

Macroeconomic policy design in an interdependent world

233

Table 8.6. The divergence component of the two-country problem Monetary policy alone (s = 0) Expected disturbance all dux

du3

Actual disturba

Policy rule optimal minimal control exchange rate nominal income monetary price exchange rate nominal income monetary price

(r = 0.0001 m)

(r = 0.0001 (y+p)) (r = 0.0001 m) (r = 0.008 p)

(r = 0.0001 (y+p)) (r = 0.0001 m) (r = 0.005/?)

Monetary plus fiscal policy

dwj

dw3

1.39 1.40

0.10 0.10

1.40 1.40 1.40 1.40

0.10 0.10 0.10 0.10

1.40 1.40 1.40 1.40

0.10 0.10 0.10 0.10

Actual disturba

Expected disturbance

Policy rule

d#!

dw3

all

optimal

1.15

0.07

d«!

DCR

r = 0.01/7 s = 0.29 y

1.34

0.09

du3

DCR III

r = 0.0002/7 s = 0.7001 y

1.47

0.08

We have not carried out any thorough testing of the robustness of the results of this section. However, we have examined the effects of giving monetary policy a greater channel of influence in the aggregate problem by choosing the higher variant for a2 in table 8.1 Although this gave monetary policy a greater role it did not alter the broad results. In particular, the results that policies devised for the single open economy perform badly in the aggregate continues strikingly to stand out. We have derived these results for the two-country case, but they have a more general interpretation. This is because the aggregate problem can be derived from the aggregation of the ^-country generalisation of the two-country analysis presented in the appendix. The larger the number of

234

Fiscal and monetary policy in interdependent economies

countries, the greater will be the difficulty in sustaining the optimal rules for the aggregate stabilisation problem presented in tables 8.4 and 8.5, and in preventing single countries adopting the optimal rules reported in section 4. Our results focus attention directly on the incentive to renege on cooperative forms of international behaviour, despite the rather serious consequences for performance if all countries do, indeed, renege. The results for the cooperative solution to the divergence problem are reported in table 8.6. For monetary policy alone (reported in the upper part of the table), there is no gain whatever to control, whether simple or full optimal. This contrasts markedly with the solution to the singlecountry problem. The lower part of the table shows that there is some scope for fiscal action of a mild kind. However, these benefits of control show up only for the optimal rule, and the simplefiscalrules show little or no benefit relative to minimal control. These results once more highlight the marked differences between the cooperative two-country solution to the control problem and that thrown up by single-country optimisation. They suggest that there is a serious free-rider problem in the design of international policy. To examine this further, in the next section we consider policy design in an explicit game theoretic framework. 6

Two-country non-cooperative games

In this section we consider non-cooperative behaviour for two identical countries in the form of a Cournot-type adjustment process leading to a closed-loop Nash equilibrium. We assume that each country pursues the same form of simple rule and we consider monetary policy only. To simplify matters still further we consider shocks (uuuf), and (w3,w*) in pairs so that each country is experiencing either an aggregate demand shock or a supply shock in common. We put var(dw/) = var(dwf) = 0.5 dt for i = 1 and 3 and assume that disturbances in different countries are independent.10 Then var(dwf) = var(dwf) = At and the welfare loss may be compared with that for cooperative policies already considered. Suppose a cooperative policy is agreed in the form of the simple rules examined in table 8.4. We can investigate the incentive to renege (i.e., the short-term gain) and the long-term consequence by following a Cournottype sequence of decisions. The results are displayed in tables 8.7-14 below for central parameter values with loss functions asy E(2y2 + 2p2 + r2) and asy E(2j*2 + 2/?*2 + r*2), for a monetary rule (r = fim and r* = /?ra*), a nominal income rule (r = /3(y + p) and r* = (}(y* + p*)) and a price level rule (r = ftp and r* = /?/?*). An exchangerate rule (r = /te and r* = - $e) is not considered because in this case an

Table 8.7. The Cournot adjustment process for policy rule r = Expected disturbance dwf var(d«!) = var(dwf) = 0.5 d/ cov(dw,, dwf) = 0

Expected disturbance dwf var(dw3) = var(dw?) = 0.5 dt cov(dw3, dwf) = 0 Country 2

Country 1

Country 1

Country 2

Iteration number

fi

Welfare loss

fi

Welfare loss

P

Welfare loss

0 1 2 3 4 5

0.66 0.88 0.88 0.98 0.98 0.98

4.73 4.70 4.92 4.92 4.94 4.94

0.66 0.66 0.95 0.95 0.98 0.98

4.73 4.92 4.88 4.97 4.94 4.94

0.50 3.13 3.13 3.31 3.31 3.32

3.95 3.16 5.10 5.10 5.10 5.10

Note: Parameters: central loss function: asy E(2j^2 + 2p2 + r2).

Welfare loss 0.50 0.50 3.34 3.34 3.34 3.34

3.95 6.17 5.03 5.09 5.09 5.10

236

Fiscal and monetary policy in interdependent economies

Table 8.8. Welfare loss for Nash equilibrium for policy rule: r = Actual disturbance (unit variance)

Expected disturbance (unit variance)

Policy rule

dw!

dw|

dw? dwf

r = 0.98 m r = 3.33 m

4.94 6.44

4.10 5.10

optimal /? is indeterminate ruling out the existence of unique reaction functions. Consider first the monetary rule. For a demand disturbance in view of the slight benefits of divergence control the aggregate policy rd = 0.71, md of table 8.4 implies that r = 0.71 m and r* = 0.71 m* for the two countries. From table 8.7 it can be seen from iteration 1 that the benefits to country 1 from reneging and pursuing an optimal policy given country 2's policy r* = 0.71 m* are very small and the eventual Nash equilibrium results in a rather greater (but still small) welfare loss. By contrast, for a supply disturbance, both the short-term gains and the long-term losses are quite considerable and at the Nash equilibrium the monetary rule is far stronger (r = 3.32m) than at the cooperative policy (r = 0.53m). These results suggest that if a joint policy is agreed between two countries in the form of a monetary rule, a supply shock is far more likely to undermine that agreement than a demand shock if the countries indulge in short-sighted behaviour. For a nominal income rule from table 8.9 both types of disturbances result in similar incentives to renege with only a slight long-term loss for a demand shock as against a significant long-term loss for a supply shock. This suggests that even a far-sighted country would be tempted to renege when faced with a demand disturbance. An interesting point to note about both monetary and nominal income rules is that if countries are engaged in a Nash closed-loop game it may actually pay if they are wrong about the nature of the shock hitting the two countries. Thus from table 8.8 a monetary rule designed with u\ in mind results in a better outcome than a rule designed from u% even when u\ actually occurs. It follows that countries pursuing a monetary rule benefit if they wrongly rule out the possibility of a u% shock. For a nominal income rule table 8.10 indicates that countries will benefit if they are wrong on all occasions. If a u\ shock occurs it is better if countries expect a u% shock and if a u% shock occurs it is better if they expect a u\ shock!

Table 8.9. The Cournot adjustment process for policy rule r = /?(y + p) Expected disturbance dwf var(dw!) = var(dwf) = 0.5 dt cov(dw1? duf) = 0 Country 1

Expected disturbance dwf var(dw3) = var(dwf) = 0.5 dt cov(dw3, duf) = 0 Country 1

Country 2

Country 2

Iteration number

fi

Welfare loss

fi

Welfare loss

fi

Welfare loss

fi

Welfare loss

0 1 2 3 4 5

1.01 0.49 0.49 0.42 0.42 0.42

4.47 4.20 4.56 4.55 4.56 4.56

1.01 1.10 0.43 0.43 0.42 0.42

4.47 4.89 4.51 4.56 4.56 4.56

0.38 0.98 0.98 1.07 1.07 1.08

4.13 3.64 4.62 4.61 4.59 4.59

0.38 0.38 1.10 1.10 1.08 1.08

4.13 5.10 4.47 4.59 4.59 4.59

238

Fiscal and monetary policy in interdependent economies

Table 8.10. Welfare loss for Nash equilibrium for policy rule: r = fl(y + p)

Expected disturbance (unit variance) dwf dw|

Actual disturbance (unit variance) Policy rule

dwf

dw|

= 0.42 (y+p) r= 1.08 (y+p)

4.56 4.49

4.14 4.59

r

For both monetary and nominal income rules the closed-loop Nash equilibrium is symmetrical. For our final regime, a price level rule, this is no longer the case. Starting at the best cooperative policy which in this case is minimal control, the Cournot process converges to an asymmetrical outcome with the country that moves first benefiting considerably at the expense of the other, in both the short term and long term. (Note that we have imposed an upper limit of P = 10 for all the rules.) For both demand and supply shocks the country that reneges moves immediately to the strongest possible feedback rule r = 10/? leaving the second country's best rule as minimal control (see tables 8.11 and 8.12). Tables 8.13 and 8.14 report results starting at r = p rather than minimal control. For disturbance dwf the country that moves first loses out and the second country benefits whereas for a u\ disturbance, the final outcome is a symmetrical Nash equilibrium at r = 0.76/?. As yet we have not been able to find a starting point which yields a symmetrical equilibrium for a u\ shock. These results suggest that the incentive to renege on the cooperative monetary or nominal income rule, while adhering to the overall constraint of such a simple rule, is not large. By contrast, the price rule offers a very considerable incentive to renege, particularly since the country that reneges first secures long-run, not just short-run, benefit. It seems that the price rule, and policies like it, are inimical to international cooperation. 7

Conclusions

In this chapter, we have been concerned with the design of macroeconomic policy in a stochastic interdependent world. This analysis was conducted in terms of a model that is rather more complex in its interactions and dynamics than is usual in the analytical literature on policy interdependence, allowing for wage/price dynamics, asset accumulation and exchange-rate dynamics.

Table 8.11. The Cournot adjustment process for policy rule r = (}p Expected disturbance dwf var(dw3) = var(dw3*) = 0.5 dt cov(dw3, du*) = 0

Expected disturbance dwf var(dwj) = 1var(dwf) = 0.5 dt cov(dw,, duT) = 0

Iteration number 0 1 2 3 4 5

Country 1

Country 2

Country 1

P

Welfare loss

0.006 10.00 10.00 10.00 10.00 10.00

5.18 4.74 4.74 4.73 4.73 4.73

P

Welfare loss

0.006 0.006 0.0002 0.0002 0.0002 0.0002

5.18 8.53 8.53 8.53 8.53 8.53

Country 2

P

Welfare loss

P

Welfare loss

0.02 10.00 10.00 10.00 10.00 10.00

4.23 2.63 2.55 2.55 2.55 2.55

0.02 0.02 0.0002 0.0002 0.0002 0.0002

4.32 10.68 10.64 10.64 10.64 10.64

240

Fiscal and monetary policy in interdependent economies

Table 8.12. Welfare loss for Nash equilibrium for policy rule: r = ftp Expected disturbance (unit variance)

Actual disturbance (unit variance) Policy rule

dwf

dw|

dwf

r = 10/? (r = 0.0002/?)*

4.73 (8.53)

2.55 (10.64)

dw|

r= 10/? (r = 0.0002/7)*

4.73 (8.53)

2.55 (10.64)

Note: * Country 2 in brackets (where different). Section 2 set out techniques for deriving optimal control rules in stochastic rational expectations models with complex dynamics, and also showed how restricted or simple optimal rules may be derived. This latter aspect of policy design assumes significance in view of the importance attached to restrictions on policy design (e.g., monetary or nominal income targeting) in current policy debates. But simple or restricted design carries with it the cost that certainty equivalence no longer holds, so that the design of policy is no longer independent of the nature of the shocks perturbing the system. Methods of handling these complications are set out in sections 2 and 4. In addition, section 2 sets out the model for deriving the Nash solution to a two-country policy game as the outcome of a Nash game between the countries pursuing simple rules. The Nash game between the countries pursuing the optimal rule is shown to lead to an ever-expanding state vector. In our subsequent analysis, we applied these methods to an examination of the effectiveness of the full optimal policy and a variety of simple rules in stabilisation. What our results bring out clearly is the divergence between policy design in the single open economy and in the global economy. This divergence arises from externalities in policy design in an interdependent world. It leads to the possibility of free-riding behaviour, as countries renege on cooperative policy design. This raises the question as to how best to contain such free-riding behaviour within a system of international policy coordination. Advances in the theory of non-cooperative game theory suggest that this may not be as intractable a problem as is usually assumed. 11 This is because forms of tit-for-tat strategy can be shown to be rather robust strategies in dealing with repeated games of the prisoners' dilemma type so frequently

Table 8.13. The Cournot adjustment process for policy rule r = (Ip Expected disturbance dwf = var(dwf) = 0.5 dt cov(dw,, duf) = 0

Expected disturbance dw| var(dw3) = var(dw*) = 0.5 dt cov(dw3, duf) = 0

va^dt/j)

Country 1 Iteration number 0 1 2 3 4 5 20

Country 2

P

Welfare loss

1.00 0.17 0.17 0.0002 0.0002 0.0002 0.0002

7.69 6.99 9.30 8.53 8.53 8.53 8.53

Country 2

Country 1

P

Welfare loss

1.00 1.00 10.00 10.00 10.00 10.00 10.00

7.69 5.62 5.30 4.73 4.73 4.73 4.73

P

Welfare loss

P

Welfare loss

1.00 0.61 0.61 0.67 0.66 0.70 0.76

7.23 7.03 7.67 6.66 6.46 6.46 6.16

1.00 1.00 0.89 0.89 0.84 0.84 0.76

7.23 5.56 5.55 5.80 6.65 6.65 6.16

242

Fiscal and monetary policy in interdependent economies

Table 8.14. Welfare loss for Nash equilibrium for policy rule: r = ftp Expected disturbance (unit variance)

dwf

Actual disturbance (unit variance) Policy rule

dwf

d,|

r= 10/7 (r = 0.0002/7)*

4.73 (8.53)

2.55 (10.64)

r = 0.76/7

6.82

6.16

Note: * Country 2 in brackets (where different).

encountered in problems of international policy coordination. However, our results also show that not all aspects of the international policy game are of the prisoner's dilemma type. This is illustrated by the Nash game under the price rule, where one country can secure a lasting gain at the expense of the other. This arises because, in this game, tit-for-tat amounts to the threat to destabilise the system totally, and may therefore not be credible. Similar results may well apply more generally to strategies that rely on manipulating the exchange rate under the regime of floating rates to secure domestic objectives at the expense of global aims. It may be that other forms of macropolicy threats may be credible in containing this form of behaviour, and this is an issue that needs more detailed consideration. Alternatively, it may be that a wider class of threats, involving factors outside the field of international policy, is required to sustain international cooperation. If this is so, our analysis offers a possible additional argument for simple rules, particularly concerning the targeting of the money supply or nominal income: that by formulating policy in these terms, one may rule out the non-cooperative forms of behaviour implied by the price level and by full optimal behaviour. Appendix

In this appendix, we set out the models analysed in this chapter in the form of equation (2.1); that is, = AZX \dt + Bwdt + dv

(A.I)

Macroeconomic policy design in an interdependent world

(i)

243

The single-country model

We let n = I

yds, so that An = ydt. We also treat7* as given exogenou-

sly, and therefore incorporated into the disturbance terms in (3.1) and (3.3). Then we may write (3.1)—(3—9) in the form of (A.2), where x-e and:

- ¥\

0

¥27\

- y/2

0 0 0 0 0 0 0

-4).

o 0 l 0 0 0

y/\ay ¥273 0 0 0 0 0 0 0

y/i(y/4a2dl - a l) 0

- y/i^42)

¥4

- y/}

¥3

- ¥4 0 0 0 0

¥4&\ 0 0 0 0

1 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0

¥\{d\ + ¥4( i2d2) 0

¥3P\ 0 0 0 0 0

0

4>2 -

Vl&l

^4^2

0 0 0

0 0 0 0 0 0 0 ~ M2 - 1

¥\(a 0 4)2" 0 y/2S2

0 0 0 0

o" 0 0 0 0 0 0

1

-1 2

(A.2)

dw 3 "4 -

5

_

•

and where 2) if

y/4S 0 0

¥4 ¥3

0 0

0 0 0 0 0 0 0 0 0 ¥3^1 0 0 0

¥\\a\

+ ¥4°-2O2)

0

4>2-f(O)=-JVnz(O) * fixed

from (3.12). Hence integrating

(3.19)

Does international macroeconomic policy coordination pay?

W(z(P),w*) = hT(0)Nuz(0)

+ W(0,w*)

259 (3.20)

From (3.20) we can see that W cannot be obtained directly from the costate vector due to the presence of the second term W(0, w*) > 0. This term arises from spillover effects from the policy of country 2 into country 1. Rather than attempting to calculate W(0, w*) it is more straightforward to compute the co variance matrix for (3.11) directly and obtain the full welfare loss W(z(Q), w*) from this. Proceeding in this way writing D = [DUD2, D3] in (3.18) we may express the welfare loss function for country 1 for the full stochastic problem as (3.21) where 'Qu + DfRDi Q=

0 0 Q2l

and d = [zTp2rp2*TplpfTxT]T

D{RD2 D{RD3 0 0 Q 12 Ri D2 D{RD3 0 0 0 Ri D2 D3RD3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 222

(3.22)

is the state vector in (3.11). Then (3.23)

p

7

where D = Jo° e" 'E(dJ )d/ can be shown to be obtained from

where Z is an (n + m) x (n + m) matrix satisfying (J -hpr)Z

+ Z(JT - \pl) + Z0(0) + p~'27O = 0

(3.25)

with r r „ , „ [z(0)z [z(0)z (0)( 01 , „ Zo(0) and =[

[ o

oj

\En

01

^ = [ 0 oj

(see Levine and Currie, 1987b for details). The welfare loss for country 2 follows similarly.

Fiscal and monetary policy in interdependent economies

260 4

Other non-cooperative equilibria

(i)

Closed-loop Nash equilibrium with reputation

Suppose that the Nash policies of the previous section were announced in the form (3.15) and (3.16). If both countries perceived the other country's policy in this form then the game would be of a closed-loop character. We consider now a Cournot-type adjustment process which we present not so much as a feasible form of adjustment behaviour, but rather as a means of obtaining the closed-loop Nash equilibria. Country 1 announces its openloop Nash policy in feedback form z w(,0)

=

z =

D(0)

P^

(4.1)

_jp*(0)_

r pto) "1 where

^

is given by bottom 2m rows of (3.13).

The constraint facing country 2 then consists of dt + B2w*dt

(4.2)

The dynamic system (4.2) has a state vector [z1p^p%m x] which is of order n + 2m. The optimal policy for country 2 will then take the form (4.3) /,2*(0)

where pi(1) is formed as for pf> and pf(0). After iterating n times (for n even) we arrive at

§.-0

and

(4.4)

Does international macroeconomic policy coordination pay?

/*«»

p*

261

(4.5)

(n-\)

for the two countries. Whatever the order of dynamics of the rule for one country, the response of the other is a higher-order rule. The existence of a closed-loop Nash equilibrium with reputation then rests on the convergence of the process described, i.e., on the conditions lim D^p+ft- i)/?f("~!) = 0

(4.6) (4.7)

for all t. Conditions (4.6) and (4.7) are necessary for w{n)(t) and w*(n)(t) to tend to finite values as n -> oo. The closed-loop Nash equilibrium with reputation, if it exists, is an extremely complicated rule and must be approximated by rules of the form (4.4) and (4.5) for some (possibly large) n. In practical terms the possibility of countries actually engaging in games involving policy rules of such complexity seems remote. We therefore do not pursue this form of non-cooperative behaviour further in this chapter. If the closed-loop Nash equilibrium with reputation is to be ruled out on the grounds that either it does not exist or it is too complex, the same must apply with greater force for a CCVE. As we have seen in the discussion in section 2(iii), it is even more complex than the closed-loop Nash equilibrium and there is again no guarantee it even exists. Again, as for the closed-loop Nash equilibrium, we do not consider the CCVE (with reputation) to be a plausible form of non-cooperative solution. (ii)

The non-reputational closed-loop Nash equilibrium

If the government lacks the credibility to precommit itself to preannounced policies, optimal policies must be found which are time consistent without reputation. The following is an iterative solution for a time-consistent non-reputational closed-loop Nash equilibrium between two countries (see also Miller and Salmon, 1985b) and the discrete time analogue of Oudiz and Sachs, 1985). It is now well-established in the literature that time-consistent nonreputational rules must take the form of proportional feedback on the

262

Fiscal and monetary policy in interdependent economies

predetermined variables, i.e., the rules must be of the form w = Dz, w* = D*z. (This is because feedback on additional costate variables of the optimisation decision introduces potential of time inconsistency: see, in addition to work cited above, Cohen and Michel (1988) and Currie and Levine (1987).) Suppose the two countries begin with w = Doz and w* = D$z. Substituting into (3.1) the system under control will then be = [A + BXDO + B2D$]\

\dt + dv

(4.8)

The rational expectations solution to (4.8) has a saddle-path given by x = - Noz

(4.9)

where No = N0(D0,D0) = M22 M2U with M = IM21 M*2\ being the matrix of left-eigenvectors of the transition matrix in (4.8) arranged so that the last m rows are associated with the unstable eigenvalues and partitioned conformably with [^J. Country 1 now takes 7V0 as given (the time-consistency constraint) and Z)*o as given (the closed-loop Nash assumption between countries). Substituting for x from (4.9) it then minimises E0(W0) where e=pt(zTQz + wTRw)dt

(4.10) Jo Jo where Q = Qn ~ N%Ql2- Q2lN0 + NZQ22N0 and Q = [ | ; | g ^ ] partitioned conformably with [^]. The dynamic constraint is dz = [ i u - Al2N0]z dt + Bxwdt + dv1

(4.11) 2

where A = A + B2[D$,0] and A, [/)*>0] and dv = [dv ] are all partitioned conformably with [^]. The minimisation of E o (^ o ) with Wo given by (4.10) and subject to (4.11) is a standard control problem with solution w-Dxz where Dx = - R~xBJSX and Sx satisfies the Riccati equation SX(AXX - AX2N0 - \pT) + ( i n - i127V0 + Q-SxBxR-lBxTSx

\pI)TSx =0

(4.12)

There are a number of ways in which the iteration may now proceed. The following we found to be numerically successful for the model given in section 6. Keeping x = - Noz with A^o = No(Do,Do) we iterate between countries to obtain a convergent sequence Du D2, D 3 , . . . for country 1 and Df, Z>f, D$, . . . for country 2, with stationary values D and Z>* respectively. Then TVQ is up-dated based on the latest D and D* and the process is repeated. This leads to a time-consistent closed-loop Nash equilibrium.

Does international macroeconomic policy coordination pay?

263

Let the stationary values for the two coupled Riccati equations associated with countries 1 and 2 be S and S* respectively. Then corresponding to (3.23) we have that the expected welfare loss is given by Eo(^o) = itr(S(Z(0) + p~lSn))

(4.13)

r

where Z(0) = z(0)z (0), for country 1 with a similar expression (with S replaced by S* and p by /?*) for country 2. 5

The cooperative equilibrium

For the cooperative equilibrium the two countries combine to choose [v^*] to minimise a joint welfare loss function which is a linear combination of the individual country's loss functions, i.e., E0(W0) where o

~pt[sT(aQ + (1 - a)Q*)s + awTRw + (1 - a)w*TR* w*]dt (5.1)

and 0 ^ a ^ 1. The choice of a is discussed in part (iii) of this section. The discount factor p is assumed to be the same for each country acting independently (i.e., p = /?*). (i)

The optimal policy with precommitment

The control problem for the two countries acting together is then to minimise E0(W0) with Wo given by (5.1) subject to (3.1). Proceeding as before by Pontryagin's principle, we define a Hamiltonian.

^ ] ) (5.2) where

X is a costate

row vector,

Qc = aQ + (1 - a)Q* and

Rc - [°0 (l - a)/?*]- Then the first-order conditions give, for the deterministic problem **> = p=(pI-A7)-Qcs pt T

(5.3) (5.4)

where p = e l and the In boundary conditions are given by z(0), p2(0) = 0 and lim,.^ e~ptp(i) = 0. By certainty equivalence the stochastic system under optimal control is then

264

Fiscal and monetary policy in interdependent economies

where dse = \&c*\ F° = - BRBT and G =

pI-AT.

The saddle-path relationship is now

where Nc is defined analogously in (3.12) and the solution is as before with the feedback rule becoming r _i (5.7)

T

c

r wc

NC i x * has been partitioned

where B '= K? = [Kf,K$] and N = ^ conformably with [zTpUT. The welfare loss calculation is now far more straightforward. For the deterministic case, Wo = Wo(z(0)) so that corresponding to (3.19) we have _Nciz(0)

(58)

Hence integrating = -itr(JVfiZ(O))

(5.9)

r

where Z(0) = z(0)z (0) and we have used W0(0) = 0. For the full stochastic problem (5.9) is replaced with £0(^0)= - itr(JVfi(Z(O) + p-xZu)) where Lu = cov(dv1) (Levine and Currie, 1987). E0(W0) is the joint or global welfare loss. To obtain the individual welfare losses for the countries separately the method used for the noncooperative open-loop Nash equilibrium (equations (3.21)—(3.25)) must be employed. (ii)

Non-reputational optimal

The optimal non-reputational cooperative policy can be calculated as for the closed-loop Nash time-consistent non-cooperative policy of section 4(ii). Of course, for the cooperative case the iterations only precede between the collusive governments and the private sector.

Does international macroeconomic policy coordination pay?

(Hi)

265

The choice of a

The parameter a should be chosen in accordance with the 'bargaining power' or relative gains from cooperation for the two countries. In standard bargaining theory the outcome is usually taken to be the Nash arbitration point which maximises the product of the cooperative gains. Let E(JPCP) and E0(W^CP) represent the expected welfare loss for our chosen cooperative and non-cooperative solutions respectively for country 1 and let E o ( ^ c p ) , E 0 ( ^ N C P ) be the corresponding values for country 2. Then the Nash point is given by the value of a that maximises the product (E0WP) - EoWcp))(EoWCP) - E0(^NCP))

(5.10)

(see Hughes Hallett (1985) for a discussion of the Nash point and alternatives). 6

The model

We apply the methods developed in the previous sections of the chapter to a two-country model. The model consists of the following equations Ay = y/\(diC + a2y* — a3r — a4s — y)dt + dv!

(6.1)

&p = (Pxy + p2c)At + dv2

(6.2)

e

dc = ( r - r * ) d / c* = - c

(6.3) (6.4)

with two further relationships analogous to (6.1)—(6.2) with starred variables and parameters replacing unstarred variables and vice versa, and where the following notation is used: y = real output p = price level r = short-term expected real rate of interest s = autonomous component of real taxes c = real exchange rate = e + p* — p where e is the nominal exchange rate dvi = white noise demand disturbance dv2 = white noise supply disturbance All variables are measured as logarithms with the exception of the real rate of interest which is a proportion and all variables are measured as a deviation about a long-run trend. All parameters are specified to be non-negative.

266

Fiscal and monetary policy in interdependent economies

Equation (6.1) represents the IS curve, with output adjusting sluggishly to competitiveness, the real rate of interest, real taxes and overseas real output. Equation (6.2) represents a Phillips curve relationship with inflation depending on output deviations and on real competitiveness, this latter effect arising from the effects of shifts in the terms of trade on the real consumption wage and hence on potential supply. The model is then completed by an uncovered real interest arbitrage condition (equation (6.3)) which assumes that domestic and foreign bonds are perfect substitutes, so that the real rates of return, adjusted for expected real exchange-rate changes, are equalised. We treat autonomous taxes and short-term interest rates as the policy instruments.10 Certain features of the model deserve further comment. The money market is not specified explicitly, but its inclusion would add nothing to our analysis. This is because, with interest rates acting as the instrument of policy, the money supply adjusts passively to clear the money market at prevailing interest rates. Our model also neglects the effects of changes in wealth resulting from current-account imbalances orfiscaldeficits. It also neglects expectations in the inflation equation, and aspects of forwardlooking behaviour that might arise in the determination of expenditures. There would be interest in working with a more developed model, preferably empirically based, that rectifies these omissions, and this will be the aim of future work by the authors. Nonetheless, our simple model incorporates key macroeconomic relationships of interest, and provides a useful testbed for an initial exploration of questions of international cooperation. Our two-country model may be set up in the form Ay dy* dp dp* dce

~¥\ Pi 0 0

y/la2

- v* 0

fit 0

0 0 0 0 0

0 0 0 0 0

Via, f}2

-f% 0

•

0 0 0 0

•

M k

o

-y/faf dt

0 0 -1

dt

C m

a3

•

y y* p p*

- ft a*

•

0 0 0 1

•

m

o • y/faf 0 0 0

"dv.

dvf

[s*\ dt + dv2

dvf 0

(6.5) which is comparable to (3.1) with z = [yy*pp*]T, x = c, w = [rs]T and _ [r*s*]T. w,* =

Does international macroeconomic policy coordination pay?

267

The welfare loss function adopted for country 1 is E(W) where W= i [ Q~pt(ay2 + bp2 + r2 + s2)dt Jo so that in (3.3)

(6.6)

0" 0

(6.7) 0

0

and R = I. E(W*) is given similarly for country 2 with starred variables and parameters and 0

0 a

(6.8)

0 0

b

The analysis and associated software is quite general and can be used for asymmetrical economies. However in this chapter we confine ourselves to identical countries. The starred and unstarred parameters are then equal. In the results that follow the following 'central' values are chosen: y/x = y/f = a2 = af = 0.5, ax = af = 0.3, a3 = a? = 0.1, a4 = a * = 0.4, ^ = # = 0.15, ft = j8? = 0.25, a = a* = b = b* = 2.0, p = p* = 0.05, a = 0.5. Where alternative parameter values are used, this is indicated. 7

Results

In sections 3 to 5 we have described four possible non-cooperative equilibria and two cooperative equilibria. Of the four non-cooperative solutions we have ruled out the closed-loop Nash with reputation (i.e., when governments can implement the full optimal policy) and all types of CCVE on the grounds of their complexity. This leaves two forms of non-cooperative behaviour. The equilibria concepts are summarised in table 9.1 below with the abbreviations used subsequently. (i)

The trajectories of CR, CNR, NCR and NCNR

Tables 9.2-9.9 show the deterministic trajectories and the corresponding welfare losses for the four equilibria for an initial negative displacement

268

Fiscal and monetary policy in interdependent economies

Table 9.1. Taxonomy of equilibria Government/government relationships Government/privatesector relationships

Cooperative

Non-cooperative Nash

Optimal policy with reputation Optimal non-reputational policy

CR CNR

NCR (open-loop) NCNR (closed-loop)

of real output, y(0) = - 1, and a positive initial price-level displacement, p(0) = 1, in country 1. For the two time-reputational policies, CR and NCR, the importance of potential time inconsistency can be assessed by comparing the welfare loss before and after reoptimisation (or 'reneging') along the trajectories. These values are shown in the last two columns. (The first number in the brackets refers to country 1, the second to country 2 and the number above is the total.) For the initial output displacement y(0) = - 1 in country 1 all four policies involve a relaxation offiscaland monetary policies. Both the real interest rates (r,r*) and real taxes (s,s*) fall immediately (and more in country 1), and then move gradually back to their steady-state values. The real exchange rate, c, rises immediately (representing a depreciation) and then falls gradually to its long-run equilibrium. Output rises, but, with it still below its natural rate, prices fall, returning eventually to their equilibria. For the initial price displacement p(0) = 1 the picture is broadly reversed. Fiscal and monetary policies are tightened (and more so in country 1). The real exchange rate undershoots and then rises back to its long-run value. The price level falls but with output losses experienced by both countries. For cooperative policy the time-inconsistent and time-consistent policies, CR and CNR respectively, differ principally in the use of the exchange rate. When governments can precommit themselves they can exercise a leadership role with respect to the forward-looking private sector. Thus with credible announcements, policy CR enables the governments jointly to utilise the exchange-rate mechanism more and we see this in the greater degree of overshooting for the case of an output displacement and a greater degree of undershooting for a price displacement, especially the latter. The consequence of this greater use of the exchange rate is that for CR there is a far more equal distribution of the welfare loss between the

Table 9.2. Expected trajectories and welfare loss for CR: initial displacement y(0) — - 1 Welfare loss Time t (years)

y(t)

0

-1.00

1

y*(t) 0.00

p{t)

p*(t)

c(t)

r(t)

s(t)

r*(t)

s*(t)

0.00

0.00

0.277

-0.148

-0.591

-0.091

-0.366

- 0 . 5 1 3 -0.113

-0.052

-0.068

0.174

-0.152

-0.404

-0.029

-0.319

2

- 0 . 2 6 3 -0.109

-0.079

-0.115

0.066

-0.113

-0.294

-0.026

-0.262

3

-0.130 -0.068

-0.099

-0.136

0.005

-0.073

-0.224

-0.035

-0.210

4

-0.056 -0.027

-0.115

-0.141

-0.016

-0.047

-0.176

-0.039

-0.169

5

-0.012

-0.124

-0.138

-0.016

-0.032

-0.142

-0.037

-0.137

0.005

Before reneging

After reneging

1.48 (1.12,0.35) 0.77 (0.47, 0.30) 0.52 (0.27, 0.25) 0.41 (0.20, 0.21) 0.35 (0.17,0.18) 0.30 (0.15,0.15)

1.48 (1.12,0.35) 0.77 (0.48, 0.29) 0.52 (0.28, 0.24) 0.41 (0.20, 0.21) 0.35 (0.17,0.18) 0.30 (0.15,0.15)

Table 9.3. Expected trajectories and welfare loss for CR: initial displacement p(0) = 1 Welfare loss Time t (years)

y(i)

y*(t)

p(t)

p*(t)

0.00

1.00

0.00

c(t)

r(t)

s(t)

r*(t)

s*(t)

-1.095

0.091

0.366

0.148

0.591

0

0.00

1

-0.191

0.014

0.734

0.251

-0.830

0.317

0.379

-0.104

0.473

2

-0.254

-0.031

0.546

0.405

-0.410

0.276

0.317

-0.098

0.396

3

-0.250

-0.095

0.445

0.458

-0.119

0.179

0.279

-0.028

0.325

4

-0.228

-0.148

0.399

0.449

0.015

0.101

0.250

0.029

0.270

5

-0.207

-0.178

0.376

0.415

0.048

0.058

0.223

0.055

0.228

Before reneging

After reneging

4.77 (2.94, 1.83) 3.94 (2.18, 1.76) 3.34 (1.71, 1.63) 2.86 (1.41 1.44) 2.44 (1.20, 1.24) 2.08 (1.03, 1.05)

4.77 (2.94, 3.90 (2.25, 3.31 (1.77, 2.85 (1.45, 2.44 (1.21, 2.08 (1.03,

1.83) 1.65) 1.54) 1.40) 1.23) 1.05)

Table 9.4. Expected trajectories and welfare loss for CNR: initial displacement y(0) = - 1 Time t (years)

yit)

y*(t)

Pit)

P*(t)

c(t)

r(t)

s(t)

r*(t)

s*(t)

Welfare loss 1.51 (1.25,0.26) 0.80 (0.59, 0.21) 0.54 (0.37,0.17) 0.42 (0.28,0.14) 0.36 (0.24,0.12) 0.31 (0.20,0.11)

0

- 1.00

0.00

0.00

0.00

0.101

-0.151

- 0.605

- 0.088

-0.351

1

- 0.528

- 0.099

- 0.092

- 0.028

0.174

-0.152

- 0.404

- 0.029

-0.319

2

- 0.276

- 0.095

-0.139

- 0.055

0.036

- 0.077

- 0.036

- 0.062

- 0.250

3

-0.136

- 0.063

-0.161

- 0.075

0.026

- 0.058

- 0.232

-0.051

- 0.202

4

- 0.054

- 0.028

-0.169

- 0.087

0.020

- 0.045

-0.181

- 0.041

-0.164

5

- 0.007

- 0.000

-0.169

- 0.094

0.017

- 0.036

-0.145

- 0.033

-0.133

Table 9.5. Expected trajectories and welfare loss for CNR: initial displacement p(0) = 1 Time t (years)

At)

y*(t)

Pit)

P*(t)

c(t)

r(t)

s(t)

r*(r)

s*(t)

Welfare loss 6.10 (5.51,0.60) 5.20 (4.66, 0.53) 4.41 (3.93, 0.48) 3.73 (3.29, 0.44) 3.14 (2.74, 0.40) 2.63 (2.27, 0.36)

0

-0.00

0.00

1.00

0.00

-0.213

0.145

0.582

0.113

0.454

1

-0.117

- 0.059

0.941

0.045

-0.185

0.119

0.476

0.094

0.376

2

-0.181

-0.104

0.874

0.076

-0.163

0.099

0.398

0.079

0.315

3

-0.213

-0.133

- 0.806

- 0.096

-0.144

0.084

0.337

0.067

0.267

4

- 0.226

- 0.050

- 0.740

0.109

-0.127

0.073

0.290

0.057

0.229

5

- 0.227

-0.158

0.676

0.115

-0.113

0.063

0.253

0.050

0.198

Table 9.6. Expected trajectories and welfare loss for NCR: initial displacement y(0) = - 1 Welfare loss Time / (years) y{i) 0

-1.00

1

y*(t)

p(t)

p*(t)

c(t)

r{t)

s{t)

r*(/)

0.00

0.00

0.00

0.242

-0.098

-0.392

-0.030

-0.544

-0.141

-0.062

-0.064

0.157

-0.052

-0.208

2

-0.295

-0.140

-0.094

-0.115

0.076

-0.023

3

-0.147

-0.088

-0.115

-0.144

0.022

4

-0.050

-0.025

-0.127

-0.154

5

0.020

0.032

-0.132

-0.151

s*(t)

Before reneging

-0.120

After reneging

-0.026

1.53 (1.14,0.39) - 0 . 1 0 6 0.96

1.53 (1.14,0.39) 1.03

-0.092

-0.014

-0.055

-0.003

-0.011

0.001

-0.005

0.013

0.050

0.014

-0.013

0.024

0.097

0.025

(0.56,0.40) 0.77 (0.39, 0.38) 0.003 0.72 (0.35, 0.37) 0.057 0.70 (0.34, 0.36) 0.101 0.69 (0.34, 0.35)

(0.61,0.42) 0.82 (0.43, 0.39) 0.68 (0.34, 0.34) 0.70 (0.28, 0.29) 0.44 (0.22, 0.22)

Table 9.7. Expected trajectories and welfare loss for NCR: initial displacement p(0) = 1 Welfare loss Time t (years)

At)

p(t)

p*(t)

c(t)

r{t)

s(t)

r*(t)

s*(t)

0

0.00

0.00

1.00

0.00

-0.911

0.011

0.044

0.029

0.116

1

-0.119

0.050

0.776

0.219

-0.754

0.019

-0.077

0.010

0.042

2

-0.192

-0.006

0.600

0.376

-0.460

-0.042

-0.169

-0.016

-0.065

4

-0.281

-0.186

0.417

0.462

-0.050

-0.077

-0.309

-0.068

-0.274

6

-0.281

-0.344

0.328

0.374

0.0424

-0.102

-0.407

-0.102

-0.406

8

-0.410

-0.417

0.229

0.240

0.023

-0.112

-0.447

-0.113

-0.447

10

-0.412

-0.416

0.110

0.108

0.003

-0.105

-0.422

-0.106

-0.420

Before reneging

After reneging

6.52 (3.90, 2.62) 6.02 (3.28, 2.74) 5.70 (2.93, 2.77) 5.13 (2.53, 2.60) 4.35 (2.15,2.20) 3.38 (1.68, 1.70) 2.42 (1.21, 1.21)

6.52 (3.90, 5.96 (2.80, 5.26 (2.80, 5.26 (1.78, 1.93 (0.95, 0.87 (0.43, 0.54 (0.27,

2.62) 2.46) 2.46) 1.79) 0.98) 0.44) 0.27)

Table 9.8. Expected trajectories and welfare loss for NCNR: initial displacement y(0) = - 1 Time / (years)

y{t)

0

- 1.00

0.00

0.00

1

- 0.544

-0.123

2

- 0.276

3

p*(t)

s(t)

s*(t)

Welfare loss

- 0.048

-0.190

-0.252

- 0.045

-0.181

-0.077

-0.306

- 0.062

- 0.250

0.028

-0.048

-0.193

- 0.039

-0.156

-0.112

0.021

-0.039

-0.155

- 0.033

-0.133

-0.123

0.017

-0.032

-0.128

- 0.028

-0.113

1.56 (1.27,0.29) 0.92 (0.43, 0.25) 0.54 (0.37,0.17) 0.58 (0.35, 0.21) 0.51 (0.30, 0.21) 0.49 (0.28, 0.20)

c(t)

r(t)

0.00

0.123

-0.127

-0.507

-0.089

-0.034

0.067

-0.063

-0.095

-0.139

-0.055

0.036

-0.165

-0.099

-0.163

-0.093

4

- 0.085

-0.063

-0.175

5

- 0.035

-0.031

-0.179

P(t)

Table 9.9. Expected trajectories and welfare loss for NCNR: initial displacement p(0) = 1 Time t (years)

y(0

y*(t)

pit)

P*it)

c{t)

r(t)

s(t)

r*(t)

s*(t)

Welfare loss 5.91 (5.17,0.74) 5.14 (4.39, 0.75) 4.49 (3.73, 0.76) 3.93 (3.17,0.76) 3.45 (2.70, 0.75) 3.03 (2.30, 0.73)

0

-0.00

0.00

1.00

0.00

- 0.296

0.111

0.444

0.050

0.199

1

-0.100

-0.013

0.925

0.066

- 0.243

0.093

0.371

0.047

0.188

2

-0.149

- 0.038

0.850

0.118

- 0.203

0.080

0.318

0.044

0.174

3

-0.172

- 0.062

- 0.779

0.170

-0.170

0.070

0.278

0.040

0.161

4

-0.181

-0.081

0.714

0.185

-0.144

0.062

0.247

0.037

0.149

5

-0.182

- 0.094

0.654

0.205

-0.121

0.055

0.055

0.221

0.139

Does international macroeconomic policy coordination pay?

277

country experiencing the displacement of the welfare loss between the country experiencing the displacement and the second country. For instance in the case of/?(0) = 1, for CR output falls to - 0.21 in country 1, - 0.18 in country 2 and the price level falls to 0.38 in country 1 and rises to 0.42 in country 2 by year 5. (Since the variables are in log form these represent per cent changes in the original variables.) For CNR by contrast output falls to - 0.23 in country 1, - 0.16 in country 2 and the price level falls to 0.68 in country 1 but only rises to 0.12 in country 2 by year 5. Turning to the non-cooperative policies, again the reputational policy NCR involves a much greater use of the exchange rate than the nonreputational policy NCNR. This is particularly true for an initial price displacement. In this case country 1 'exports' its price-level disturbance and its output loss to country 2. But whereas with CR the real interest rate in country 1 is kept below that in country 2 (and is even negative for a period), with NCR the Nash response of country 2 to an immediate tightening of monetary and fiscal policy in country 1 is to tighten its policy even more. The Nash equilibrium is a rather unusual response to a price displacement consisting of a relaxation of fiscal and monetary policy by both countries with eventually, after year 6, the interest rate in country 1 being above that in country 2. (The persistence of the price displacement means a ten-year time horizon is necessary to examine NCR fully.) Non-cooperative Nash equilibria can of course give outcomes which no player wants (recall the prisoner's dilemma) and NCR is a case in point. Indeed we have found for high values of the discount rate in the region p > 0.12 (with the remaining parameters at their central values) NCR is actually unstable (i.e., has too many unstable roots for the saddle-path condition). For p at its central value 0.05 the model is stable at NCR but the performance is very bad for an initial price displacement. (For the initial output displacement there is not a great deal of difference in the performance across all four equilibria.) An interesting feature of NCR is that it exhibits potentially extreme time inconsistencies in the sense that very large gains can be obtained if both countries reoptimise especially in later years. However, if reoptimisation takes place after year 1, country 1 actually loses out, all the benefits from reneging going to country 2. Again given the Nash assumption that countries optimise given the actions of the other this is a quite possible unintended effect of 'reneging'. (ii)

The gains from cooperation

Consider first the deterministic case with displacements y(0) = - 1 , = y*(p) = _ i ? p(0) = 1, and p(0)=p*(0) = 1. Table 9.10 shows the initial welfare loss for the two countries separately (in brackets) and the

y(0)

278

Fiscal and monetary policy in interdependent economies

Table 9.10. The gains from cooperation (a = 0.5): deterministic case Equilibrium solution CR CNR NCR NCNR

y(0) = - 1

y(0) = y*(0) = - 1 p(0) = 1

p(0) = p*(0) = 1

1.48 (1.12, 0.35) 1.51 (1.24,0.16) 1.53 (1.14, 0.39) 1.57 (1.27,0.30)

4.78 (2.39, 2.39) 4.78 (2.39,2.39) 5.04 (2.52, 2.52) 5.04 (2.52,2.52)

17.20 (8.60, 8.60) 17.20 (8.60,8.60) 24.34 (12.17, 12.17) 18.04 (9.02,9.02)

4.77 (2.94, 1.83) 6.10 (5.51,0.60) 6.52 (3.90, 2.62) 5.91 (5.17,0.74)

total (above) for these four displacements for the equilibria CR, CNR, NCR and NCNR. It should be emphasised that the welfare loss for the two cooperative solutions CR and CNR are for the case of equal weights (a = 0.5) attached to the two countries collective welfare loss function in (5.1). In this case, comparing CR with either NCR or NCNR, it can be seen that the country not faced with an output or price displacement loses out by cooperating. However for symmetrical displacements (y(0) = }>*(0) = 1 and/?(0) = /?*(0) = 1), for which a = 0.5 is appropriate, CR is superior to both non-cooperative solutions for both countries although the gains for this model and parameter values are not large unless NCR is chosen as the threat point. A further interesting point is revealed by table 9.10. It regards the gains from cooperation when the governments cannot precommit themselves and are thus constrained to pursue time-consistent policies. Then cooperation can actually increase the total welfare loss of the two countries and thus be counterproductive. Comparing CNR with NCNR we can see this is true for a price level (p(0) = 1) but not for the output displacement (y(0) = - 1). This result confirms similar findings by Rogoff (1985) and Miller and Salmon (1985). It should be noted that for symmetrical displacements y(0) = j*(0) = - 1 and for p(0) = /?*(0) = 1 the optimal policy is always time consistent since symmetry implies no exchange-rate movement is possible. Policies CR and CNR then coincide since the exchange rate is the only forward-looking variable that can give rise to time inconsistency, and these cooperative policies must always be superior to any non-cooperative equilibrium. The conclusion that must be drawn from table 9.10 is that in a

Does international macroeconomic policy coordination pay? =0.25

1.5

c ID

279

1.4

orC

O

1.3 .«=0.4iNCNR > a = 0.42 (Nash point w.r.t. NCNR) = 0.45 n NCR =0.5

CO CO

Wei

_o 1.2 1.1 1.0

0.10 0.15

0.20

0.25

0.30

0.35

0.40

Welfare loss for Country 2

Figure 9.2 Welfare losses under CR for 0.25 < a < 0.5. Initial displacement y(0) = - 1

deterministic setting the cooperative solution CR is possible on the basis of equal weights a = 0.5 only for symmetrical displacements. For asymmetrical displacements a must be changed in favour of the country not facing any initial displacement. The results are displayed infigures9.2 and 9.3. The shaded regions are those pairs of welfare losses for the two countries which are Pareto-superior to the two non-cooperative equilibria NCNR and NCR. Thus for an initial displacement y(0) = - 1 (figure 9.2) choices of a between about 0.38 to 0.45 give a CR equilibrium Paretosuperior to NCNR. The ringed point, a = 0.42, is the Nash bargaining point with respect to a threat point NCNR. However a must lie between about 0.47 to 0.58 for CR to be Pareto-superior to the alternative non-cooperative outcome NCR. The important point to note of course is that these intervals do not intersect, i.e., there is no choice of a that will sustain CR against both non-cooperative equilibria NCNR and NCR. From figure 9.3 we see that is also true for an initial price displacement ^(0) = 1. We conclude that if NCNR and NCR are regarded as possible noncooperative alternatives or threats following a breakdown in the cooperative agreement, CR, there is no choice of a that will make CR sustainable in a deterministic context. This leaves the possibility that CR is sustainable in a stochastic setting. We start by assuming a particular covariance matrix En where cov(dv!) = EnAt and dvT = (dv^vf dv2dvf). We put

280

Fiscal and monetary policy in interdependent economies

= 0.2

5.5-

NCNR a = 0 .25 0.27 (Nash point w.r.t. NCNR)

5.0-

a =0.3

I 4.5

o

4.0

%

15 MPR

-

I l

% Q)

3.5 -

i%

/

3 . 0 _-

2 5

1

0.50

1

0.75

1

1

1.00

1.25

1 1.50

1.75

I l l I l I

a = 0 .5

1 % 2.00

1

2.25

2.50

1

I

2.75

1

Welfare loss for Country 2

Figure 9.3 Welfare losses under CR for 0.2 < a < 0.5. Initial displacement p(0) = 1

(7.1)

2;,,= 0

1

so that shocks are uncorrelated and have the same unit variances.11 In a stochastic world these disturbances are in addition to the initial displacement of the system considered for the deterministic case. Rather than choose an arbitrary relationship between initial displacements and the disturbance covariance matrix we scale the former with respect to the latter as follows. Imagine the system in stochastic equilibrium under policy CR. Given Xu we can then derive the asymptotic variance matrix, Z, of the predetermined variables z = (y, y*9 p and /?*). (This is found by solving a Lyapunov equation analogous to (3.25).) Then for each zt (i.e., each predetermined variable) in turn, we can calculate a 99 per cent confidence region. We then choose two initial displacements by putting y and/? in turn on the extreme boundary of this 99 per cent confidence interval. Thus for Eu given by (7.1) this gives (table 9.11) z(0) = 3.01 and /?(0) = 4.86 for displacements 1 and 2 respectively. The other components of the initial displacements are calculated from the standard regression formula

Does international macroeconomic policy coordination pay?

281

Table 9.11. The gains from cooperation (a = 0.5): stochastic case

Equilibria CR CNR NCNR NCR

Displacement 1

Displacement 2

3.0l" y*(0) = 1.59 -2.65 P(O) p*(0) -2.89

- 1.62" y(0) y*(0) = - 1 . 4 1 4.86 P(O) p*(0) 3.98

322 (165, 157) 376 (193, 183) 372 (191, 181) 420 (214, 206)

528 (269, 259) 584 (309, 275) 588 (310, 278) 719 (365, 354)

•-ZijZ^zjO)

(7.2)

The consequence of this procedure is that initial displacements are chosen to be the extreme 1 per cent tail of the distribution of y, y*,p, and/?* that would be observed in stochastic equilibrium under CR. In this sense the initial displacement is chosen to be large in relation to stochastic disturbances. This is done because large initial displacements relative to stochastic noise reduce the sustainability of CR with respect to both the two governments and the governments and the private sector. Thus if anything we are biasing our results against the possibility of sustaining CR. The results for the initial welfare losses are shown for these two extreme initial displacements in table 9.11. In contrast to the deterministic case for a = 0.5 we now see that CR is easily Pareto-superior to both non-cooperative outcomes NCNR and NCR for both displacements. However all the results of this sub-section are for initial welfare losses at the time at which policies are calculated and announced. To establish the sustainability of CR we need to show that CR is Pareto-superior to NCNR and NCR at any point in time. The credibility of CR also requires that CR is Pareto-superior to CNR at all times. These matters are examined in the final sub-section. (Hi) The supergame strategy We now turn to the supergame strategy discussed in section 2. We wish to establish whether the cooperative, initially optimal solution CR can be

282

Fiscal and monetary policy in interdependent economies

sustained at all time t ^ 0. Thefirstconsideration is whether there exists an incentive for the two countries to renege jointly on the private sector. Because of time inconsistency, CR is only initially optimal (at t = 0) and for t > 0 it is sub-optimal, the optimal policy then being (5.7) with p2(t) reset to zero. Reoptimising in this way, however, is not possible as timeinconsistent policy announcements would then lack credibility with the private sector. But a switch to the time-consistent policy is always a possible policy option because this is precisely the policy the private sector expects when the government loses its reputation for precommitment (see Currie and Levine (1987) and chapter 5 for a full discussion of this point). The incentive for the countries jointly to renege on the private sector can then be assessed by comparing the cooperative optimal policy CR with the cooperative time-consistent policy CNR. This is done in tables 9.12 and 9.13 for the deterministic and stochastic cases respectively. For the deterministic case with a = 0.5 we can see that for an initial price displacement p(0) = 1 the welfare loss under CNR, although considerably greater than under CR at t = 0, becomes less at times t = 2 and 3 years. It is then optimal for jointly optimising countries to renege to CNR at any time between two and three years. Whether the countries would agree to do this is another matter because in each case the benefits from reneging in this way accrue entirely to the second country and the first country loses out. The most likely outcome of this incentive to switch to a time-consistent policy would be the non-cooperative time-consistent equilibrium NCNR. The second condition for the sustainability of CR is that there is no incentive for either country to switch to a non-cooperative policy. As we have seen in the previous section with equal weights attached to the joint welfare junction (a = 0.5) CR is not Pareto-superior to either NCNR or NCR unless the shocks are symmetrical. Furthermore CR and NCR are so far apart that there is no value of a that makes CR Pareto-superior to both NCNR and NCR. Cooperation cannot even get off the ground in a deterministic world. In a stochastic world we have seen that there are considerable initial gains from cooperation for both countries. In table 9.13 we consider the passage of time along the average (expected) trajectory of CR. Now we see that CR is Pareto-superior by a large margin to both CNR and the two non-cooperative outcomes NCNR and NCR at all times. Thus the existence of stochastic disturbances renders the best form of cooperative behaviour CR sustainable. This proves to be true, for our model and parameter values, for both a possible joint reneging of the governments on the private sector and a reneging of one country on the other.

Table 9.12. Welfare loss along trajectory ofCR: deterministic case a = 0.5

Time (years)

Displacement y(0) = - 1

Displacement p(0) = 1

CR

CR - After switch to CNR

After switch toNCNR

After switch to NCR

Before reneging

After reneging

1.48 (1.12, 0.35) 0.77 (0.47, 0.30) 0.52 (0.27, 0.25) 0.41 (0.20, 0.21) 0.35 (0.17, 0.18) 0.30 (0.15, 0.15)

1.51 1.57 1.53 1.48 (1.12, 0.35)(1.25, 0.26)(1.27, 0.30) (1.14, 0.77 0.78 0.82 0.83 (0.48, 0.29) (0.26, 0.51)(0.53, 0.29) (0.53, 0.52 0.65 0.52 0.56 (0.27, 0.24) (0.25, 0.27) (0.20, 0.23)(0.27, 0.41 0.55 0.41 0.42 (0.20, 0.21) (0.22, 0.19)(0.22, 0.23) (0.27, 0.35 0.48 0.35 0.37 (0.17, 0.18)(0.19, 0.16)(0.17, 0.20) (0.24, 0.30 0.43 0.30 0.32 (0.15, 0.15) (0.16, 0.14)(0.15, 0.17)(0.21,

0.39) 0.35) 0.28) 0.28) 0.24) 0.22)

Before reneging 4.77 (2.94, 3.93 (2.18, 3.34 (1.17, 0.41 (1.41, 2.44 (1.20, 2.08 (1.03,

After reneging

4.77 1.83) (2.94, 3.89 1.76)(2.25, 3.31 1.63) (1.77, 0.41 1.41)(1.45, 2.44 1.24) (1.21, 2.08 1.24) (1.03,

- After switch to CNR

Displacement (y(0) = /(O) = - 1 Displacement p(O) = p*(O) = 1

After switch toNCNR

Switch to NCR

6.10 6.52 5.91 1.83)(5.51, 0.60) (5.17, 0.74) (3.90, 4.16 4.25 5.49 1.64)(3.17, 0.99)(3.11, 1.14) (3.09, 3.32 4.70 3.45 1.54)(1.95, 1.37)(1.46, 1.99) (2.48, 0.41 0.42 0.55 1.40) (1.39, 1.46) (1.45, 1.50) (2.05, 2.53 3.46 2.45 1.23) (1.13, 1.32)(1.35, 1.18)(1.72, 2.95 2.08 2.16 1.05) (0.97, 1.11)0.02, 1.14) (1.46,

2.62) 2.04) 2.22) 2.00) 1.74) 1.49)

CR = CNR 4.78 (2.39, 2.91 (1.45, 2.05 (1.03, 1.63 (0.81, 1.38 (0.69, 1.20 (0.60,

After switch toNCNR

5.04 2.39) (2.52, 2.52) 3.08 1.45) (1.54, 1.54) 2.18 1.03) (1.09, 1.09) 1.72 0.81) (0.86, 0.86) 1.46 0.69) (0.73, 0.73) 1.26 0.60) (0.63, 0.63)

After switch to NCR

CR = NCR

After After switch switch toNCNR to NCR

5.04 17.20 18.04 24.34 (2.52, 2.52) (8.60, 8.60) (9.02, 9.02) (12.17, 12.17) 3.36 15.06 5.72 21.42 (1.68, 1.68) (7.53, 7.53) (7.86, 7.86) (10.71, 10.71) 2.60 13.13 13.66 18.70 (1.30, 1.30) (6.57, 6.57) (6.83, 6.83) (9.35, 9.35) 2.20 11.36 11.78 16.16 (1.10, 1.10) (5.68, 5.68) (5.89, 5.89) (8.08, 8.08) 1.92 9.75 10.10 13.86 (0.96 0.96) (4.88, 4.88) (5.05, 5.05) (6.93, 6.93) 1.70 8.32 8.60 11.80 (0.85, 0.85) (4.16, 4.16) (4.30, 4.30) (5.90, 5.90)

Table 9.13. Welfare loss along trajectory ofCR: stochastic case a = 0.5 Displacement 1 (see table 9.11) CR Time (years)

Before reneging

After reneging

After r\l LCI switch to CNR

0

322 (165, 305 (154, 293 (147, 284 (142, 277 (136, 272 (136,

322 (165, 305 (154, 293 (147, 284 (142, 277 (136, 272 (136,

376 (193, 183) 359 (182, 178) 348 (175, 173) 339 (170, 169) 332 (163, 163) 327 (163, 163)

1 2 3 4 5

157) 151) 146) 142) 136 136)

157) 151) 146) 142) 136) 136)

Displacement 2 (see table 9.11)

After After •TY1LCI r\LLCI switch switch to NCNR to NCR 372 (191, 181) 354 (179, 175) 342 (172, 170) 333 (167, 166) 327 (164, 163) 322 (161, 161)

420 (214, 206) 398 (201, 197) 382 (192, 190) 370 (185, 185) 360 (180, 180) 354 (177, 177)

CR Before reneging

After reneging

After switch to CNR

/A.1LC1

584 (309, 545 (280, 510 (257, 424 424 479 (212, 212) (212, 212) (239, 452 397 397 (199, 199) (199, 199) (225, 374 374 429 (187, 187) (187, 187) (214,

528 (269, 259) 490 (247, 243) 455

528 (269, 259) 490 (247, 242) 455

After After r\L LCI switch switch to NCNR to NCR

r\L LCI

588 (310,278) 548 264) (281,267) 512 253) (258, 254) 478 240) (239, 239) 451 227) (225, 226) 427 215) (213,214) 275)

719 (365, 663 (334, 614 (308, 570 (285, 532 (266, 498 (249,

354) 329) 306) 285) 266) 249)

Does international macroeconomic policy coordination pay?

8

285

Conclusions

In this chapter, we have been concerned with examining whether international cooperation between governments in macroeconomic policymaking is sustainable in a two-country world. We use for this purpose a fairly standard illustrative model, which incorporates the key international spillover effects, though neglecting asset accumulation. Future work will be directed towards extending this analysis to empirically based models. The difficulty with sustaining cooperative policies is that there may emerge, with the passage of time, a position at which it pays one or other of the governments to renege on the cooperative agreement. (In calculating whether this is so, account must be taken of the consequences of breaking associated commitments with the private sector.) If this is so, cooperative agreements will lack credibility, and will therefore not be entered into by rational governments. In the analysis, questions of sustainability of cooperative policy intersect with questions of the credibility and reputation of government vis-a-vis the private sector. Without government reputation vis-a-vis the private sector, it turns out that cooperation between governments may not pay, and when it does the benefits are minimal (see table 9.11). These results confirm in the context of our model the results of Rogoff (1985). They arise because non-reputational policies may be thought of as noncooperation between the government and private sector, and with noncooperation between certain players in the policy game cooperation between governments may not be advantageous. Our results also suggest a converse: that without cooperation between governments, government reputations may not be advantageous. Thus non-cooperative reputational policy delivers a much worse policy pay-off than do the non-reputational policies, whether cooperative or not, and is actually unstable if the discount rate of policy-makers is too high. (All are necessarily inferior to the cooperative reputational policy.) These results suggest that, without government reputation vis-a-vis the private sector, cooperation will not be advantageous and therefore, a fortiori, not sustainable. It remains therefore to ask whether, with reputations, cooperation is sustainable. Our results suggest that cooperation with reputation is difficult to sustain in a deterministic world, where the policy problem is one of setting policy so as to deal with an undesirable initial position of the system (e.g., reducing high inflation, recovering from recession). Only if the initial displacement is nearly symmetrical between the two countries will cooperation be sustainable. By contrast, cooperation with reputation is more easily sustained in a stochastic world,

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where repeated disturbances (e.g., inflationary shocks, demand fluctuations) continually move the system away from its desired position. In this case, when considering reneging on cooperative agreements, governments will need to weigh the advantages of reneging with respect to past shocks against the costs, in terms of poorer policy performance, of dealing with future shocks in a non-cooperative manner. Provided that the discount rate of policy-makers is not too high, cooperation with reputation will prove advantageous and therefore sustainable. These benefits are measured relative to the non-reputational alternatives. Measured relative to the alternative of non-cooperation with reputation, the gains are still larger. The results suggest an important message. Rogoff's finding is that a move towards international cooperation is not necessarily advantageous in a world where governments lack credibility with their private sectors. Our results suggest that government credibility with their private sectors may be most disadvantageous if unaccompanied by cooperation between governments. Further work will establish whether this result carries over to empirically based models of the international economy. Notes 1 Levine and Currie (1985) earlier derived this optimal time-consistent rule as the limit of a Cournot-adjustment process in policy rules. 2 This assumes an infinite punishment period. It would be of interest to examine the consequences of a finite punishment period (as in Barro and Gordon (1983b) for a very simple model), but this is much more difficult to handle for the class of general models analysed in this chapter. 3 Note, however, that if the only alternative policy is the optimal nonreputational time-consistent policy such an incentive need not necessarily emerge. In the example examined by Currie and Levine (1987), it depends on the nature of the initial displacement of the system. 4 This result is most obvious for the case of no-discounting, where future shocks necessarily assume an infinite weight relative to the current state of the system. Taylor (1985) suggests that the assumption of the case of no discounting is most appropriate for macroeconomic policy design. These findings are analogous to those obtained in the literature dealing explicitly with reputations (see, for example, Barro and Gordon (1983b), Backus and Driffill (1985a, 1985b)). Note, however, that this analysis may be applied in a quite general class of models. 5 The critical rate of discount is found to be so high that the argument could well survive the introduction of finite lived governments because of elections. Formally this may be modelled by assuming a given probability density of the government falling from power in any time interval.

Does international macroeconomic policy coordination pay?

287

6 Feedback control and closed-loop control are synonymous in our linearquadratic framework in which 'certainty-equivalence' applies (see Bar-Shalom and Tse, 1976). 7 This is true only of reputational closed-loop Nash strategies, which involve an element of strategic behaviour of each government vis-a-vis the other. It is not the case for non-reputational closed-loop Nash equilibrium, where this strategic element is absent. 8 Hughes-Hallett reports a general iterative procedure that may fail to converge and typically yields multiple solutions. A modified directed search was then implemented in which at least one country is better off and neither is worse off at each step. Since a first-order CCVE is essentially a Nash-type game with perceptions of both levels and derivatives the justification for this is not altogether clear. 9 Another interesting possibility, which is not considered in this chapter, is that the responsibility for the breakdown in the cooperative agreement can be directed at one of the countries only. Then this country is constrained to pursue time-consistent policies but the other country can still pursue optimal time-inconsistent policies. 10 Since the expected rate of inflation moves in a continuous way in this set-up, use of the nominal short-term interest rate as the instrument is equivalent to use of the short-term expected real rate of interest, r. We therefore treat r as the instrument, and associate instrument costs with movements in real, not nominal, interest rates. 11 Symmetry is then maintained between countries in that they both experience uncorrelated disturbance of equal variances. But this is a minimal amount of symmetry which still leaves the possibility of considerable asymmetry in the displacements at any time.

10 International cooperation and reputation in an empirical two-bloc model

1

Introduction

This chapter is concerned to examine three themes. First, we wish to investigate whether, and in what circumstances, it is advantageous for governments to cooperate internationally in the conduct of macroeconomic policy. Second, we wish to investigate whether it is advantageous in terms of economic performance for governments to establish reputations vis-a-vis the private sector. Third, we wish to examine the sustainability of cooperative and reputational policies. We address these questions in the context of an empirical two-bloc model derived from the OECD Interlink model. In previous empirical work (and in almost all the theoretical work), these three questions have been addressed separately. Here we address them jointly, and with striking conclusions. Our main results may be summarised as follows. To realise the gains from international cooperation requires reputation; and to realise the gains from reputation requires cooperation. Moreover, cooperative policies with reputation are found to be sustainable; and the joint gains from cooperation and reputation are considerable. The broad outlines of our analysis may be summarised in terms of the matrix of policies on page 289. Horizontally we depict the choice between cooperation and non-cooperation between governments. Vertically we depict the choice between reputational policies and non-reputational policies. The matrix yields four distinct policy sets.1 Within each cell, we assume that governments determine policy by an explicit optimisation procedure, maximising a specified welfare function. (This has the disadvantage, of which we are well aware, that it generates rather complex policy rules.) Reputational policies thus correspond to the full optimal Global Macroeconomics: Policy Conflict and Cooperation, R.C. Bryant and R. Portes (eds.), Macmillan (1987). With Nic Vidalis. The financial support of the Rockefeller Foundation, the Ford Foundation and the Alfred P. Sloan Foundation is gratefully acknowledged. The research benefited from the assistance of Jessica Gaines and Ali Al-Nowaihi.

288

International cooperation and reputation

289

Relations between governments Cooperation Non-cooperation (C) (NC) Relations between Reputation (R) governments and xT ^ . . x Non-reputation x private sectors

CR

NCR

CNR

NCNR

rule (or, in the terms of Barro and Gordon, 1983b, the ideal rule). Such policies are frequently regarded as time inconsistent in that an incentive to renege emerges with the mere elapse of time; we take up this issue later. Non-reputational policies correspond to the backward dynamic programming optimisation solution (see Cohen and Michel, 1988; Oudiz and Sachs, 1985), which avoids any potential problem of time inconsistency. Under cooperation, the two governments jointly adopt a Pareto-efficient policy, the choice of policy being determined by bargaining strengths. We model this by assuming that the two governments maximise a weighted average of their individual objective functions, the weights being determined by relative bargaining strengths. Non-cooperative policies are where each government maximises its own objective function, taking as given the policy rule of the other government. The derivation of these four rules for a general two-country dynamic rational expectations model is set out in the appendix to chapter 12.2 In an earlier paper (Levine and Currie, 1987a) we investigated these four policies in the context of a simple two-country analytical dynamic rational expectations model. There we found that the ranking of the two non-reputational policies was unclear, depending on the nature of the shock. This provided partial support for Rogoff's (1985) analytical argument that cooperation need not be advantageous. Thus the outcome under the non-cooperative non-reputational (NCNR) policy may be better for both countries relative to that under the cooperative nonreputational (CNR) policy. However, we also found that the differences in policy performance between the two policies were minor. By contrast, the cooperative reputational (CR) policy was the best, while the noncooperative reputational (NCR) policy was the worst, of the four policies. Indeed the NCR policy risked total destablisation of the system. These results suggested that the gains from cooperation were most significant for reputational policies, and they highlighted the dangers of reputational policies without international cooperation. The object of this chapter is to see whether these conclusions also arise in the context of empirical models.

290

Fiscal and monetary policy in interdependent economies

However, it is not sufficient to establish the potential superiority of the CR policy over the others. It is also necessary to show that it is sustainable. The question of sustainability has two aspects. First there is the issue of whether governments, either jointly or individually, would find it advantageous to renege on the private sector, losing their reputation and switching to a non-reputational policy (that is, a shift downwards in terms of the above matrix). This is the issue of time inconsistency. Second, there is the question of whether either government would find it advantageous to renege on the other, switching to a non-cooperative policy (that is, a shift to the right in terms of the matrix). Levine and Currie (1987a) showed that, in their analytical model, stochastic noise sustained both reputational policies and cooperation under plausible circumstances. We seek to address these questions also, in the context of our empirical model. The plan of the chapter is as follows. The next section describes the provenance and characteristics of Minilink, the reduced version of the OECD Interlink that we have used. In section 3, we go on to examine the performance of the four policies, and assess the gains from cooperation. Section 4 examines the long-run features of each of the four policies under consideration. Section 5 considers the sustainability of reputational and cooperative policies, addressing the issues of time inconsistency and the incentive to renege. Afinalsection summarises our main conclusions. The technical details of our optimisation and solution procedures can be found in the appendix to chapter 12.2 2

Description of Minilink

Our subsequent analysis of policy rules for monetary and fiscal policy is based on the two-bloc version of Minilink. This is a reduced version of the OECD Interlink model as it was in 1983. This reduced version was constructed byfittingregression relations to a set of partial simulations of the Interlink model; this procedure is described in Masson and Richardson (1984). A listing and description of the model is given in Masson, Blundell-Wignall and Richardson (1984). The following provides a brief description of the model. Minilink disaggregates the world into the US and the rest of the world. (A version is also available which disaggregates the rest of the world further into six separate countries and a residual rest of world.) The equations of Minilink relating to the US are as follows: yt = cxpt + c2gt + 2c 3 ra,_i + c4rjt + c5yf i + cX2(rlt - 2nt)

(2.1) (2.2)

International cooperation and reputation

291

ril = cl3r*!+Ut + (l-cl3)rt

(2.3)

rjt = c4i xt-1 + c42rit-2 + c43(et + pt- pf)

(2.4)

Pt-Pt-i

(2.5)

= d3nt + (l -d3)(pf-pf-i

nt = nt-i=

-et + et-x) + d2yt

dx{pt -pt-x-nt-1)

(2.6)

mt=Pt +/o ~f\rt +fiy4 +/ 4 (w / _i -/?/-i) fl/ - 0*-1 = bx (, + b2yf + 63^/ C, = bisCt-i + *i4C/-2 +

(2.7) (2.8)

bn(et+pt-p?)

+ bl2(et-l+pt-l-Pr-l) e, = e? + M + ai(r,-r*)

(2.9) (2.10)

where y is the logarithm of real GDP relative to potential, p is a distributed lag of the real long-term rate of interest, g is government expenditure (measured relative to potential output), a is the stock of net foreign assets, rj and { are distinct distributed lags of the real exchange rate, rl is the long-term nominal rate of interest, n is the expected rate of inflation, e is the nominal exchange rate, p is the price level, r is the short-term nominal rate of interest, r is the mean level of r, and m is the nominal money stock. Starred variables denote the equivalent variables for the rest of the world. The model is a semi-annual one, and rates of interest are measured on an annual basis. The rest of world equations are given by equations similar to equations (2.1)—(2.7), but with different parameter values, and with e, a and x* entering with opposite sign. Parameter values are given in table 10.1, where starred coefficients denote coefficients for the rest of the world. Equation (2.1) specifies output to be a function of government spending, net interest income (fa), foreign output and distributed lags (defined by (2.2) and (2.4)) of the expected long-term real rate of interest and of competitiveness; (2.3) specifies the arbitrage condition between short- and long-term bonds, giving the long-term rate of interest as a forward convolution of short rates; (2.5) specifies a Phillips curve relationship, with a component for direct imported inflation. Inflation expectations are formed adaptively, as given in (2.6); (2.7) defines a conventional money demand function; (2.8) determines the change in the stock of net foreign assets from the current account, which is in turn determined by domestic and foreign output, and a distributed lag of competitiveness given by (2.9). (This is specified so as to incorporate a short-run /-curve effect on the balance of payments.) (2.10) is the uncovered interest arbitrage condition. Our original intention was to make no modification of any kind to

292

Fiscal and monetary policy in interdependent economies

Table 10.1. Parameter values for Minilink US ax bx b2 b3 bn bi2 bn bl4 Cx

c2 C3

c4 c5 Cn C\2

Cn c4i

c42 C43

d, d2 d3 /1

fi ft

Rest of world 0.5 - 0.0412 0.1125 -0.1863 - 1.577 1.936 0.856 -0.215 - 0.573 0.285 0.633 -0.127 0.158 0.754 0.246 0.952 1.413 - 0.657 0.245 0.5 0.0496 0.945 0.774 0.493 0.587

ct ct ct ct ct Cfi C\2

Cn

cl

c*n C4*3

dt dt dt

ft

/?

ft

- 0.347 0.285 0.383 0.124 0.140 0.721 0.279 0.952 1.194 - 0.447 0.253 0.5 0.0496 0.907 0.814 0.316 0.633

Minilink. However, one troublesome feature of the model soon became apparent. The model incorporates a small effect via interest payments of the cumulated current-account balance on aggregate demand (see equation (2.1)), reflecting the accumulation of wealth via current-account surpluses. These effects give rise to very long drawn out dynamics resulting from asset accumulation. Thus we calculated the presence of an eigenvalue just inside the unit circle (about 0.998), which for a semiannual model implies a mean adjustment lag of about 250 years. If this were believed, the effects of the US Federal deficit of the 1980s would roll on for the next millennium! This provides a useful reminder of the fact that negligibly small coefficients sometimes have large effects on model properties. Were the interest-rate effects zero (say, because f=0), the

International cooperation and reputation

293

model would contain a unit root corresponding to net foreign assets, a, but this instability would be confined to this variable and would not spill over to important variables of the model, such as prices and output. But with a small positive value for f this root becomes marginally less than unity, and the corresponding dynamics are transmitted to all variables of the model. These long-lived dynamics seemed implausible and in need of correction. It did not show up in the simulations reported by Masson, Blundell-Wignall and Richardson (1984) and Masson and Richardson (1984), possibly because of the imposition of a terminal condition that suppressed it. It is presumably also a feature of Interlink if run with full rational expectations. The parent model, Interlink, contains a small risk premium, related to the cumulated current account, in its equation for the exchange rate, i.e., a term in a enters equation (2.10). Some versions of Minilink also include this effect. We incorporated this term in our version of the model to assess its effect, but since its coefficient is small it reduced only marginally the troublesome root close to the unit circle. Since long drawn out dynamics of this kind seemed to us to be both implausible and undesirable, the need rose to make changes to Minilink to eliminate this root close to the unit circle. Three main possibilities were considered. The first was to increase the coefficients on the stock of net financial assets (a) so as to reduce the root well below unity. This had the disadvantage that it either implied an implausibly low degree of capital mobility or required large wealth effects on aggregate demand, neither of which are features of either Interlink or Minilink. The second possibility, with which we experimented at some length, was to make the risk premium a forward-looking variable. Instead of the backward cumulation of the current account (as in equation (2.8)), we made the risk premium a forward-discounted summation of future expected currentaccount deficits, together with the initial net asset position. This moved the troublesome root onto the unit circle, counting as an unstable root, thereby ensuring that the saddle-path property was just satisfied. Since the unstable roots do not influence the dynamics of the system in response to unanticipated disturbances, this modification disposes of the very long drawn out dynamics, without greatly distorting the rest of the system. The forward-looking risk premium then acted, in effect, as a way of imposing the intertemporal solvency condition, jumping initially to ensure external solvency. The third possibility was to suppress altogether the dynamic wealth effects arising from the current account (by setting f = 0). This has the potential danger that external solvency is not necessarily ensured, but has the advantage of simplicity. The dynamic behaviour of the model for these three cases was different

294

Fiscal and monetary policy in interdependent economies

in the face of exogenous open-loop changes in fiscal or monetary policy, but only in the long run. However, we found that the differences under control were not significant, largely because the control rule acted effectively to avoid the long drawn out dynamics associated with the troublesome long-lived root. Since the results were all rather similar, in this chapter we report only the results for the third case discussed above, where no asset effects are allowed for. However, it should be borne in mind that results for the long run, discussed particularly in section 4, may be sensitive to this modification. We examined policy design in the face of four possible disturbances: a reduction in aggregate supply (i.e., a shift term in (2.5)), occurring either in the US or in the rest of the world; and a rise in aggregate demand (i.e., a shift in (2.1)), occurring either in the US or the rest of the world. We allowed these shocks to be temporary (one-period) disturbances, or permanent. (We also examined longer-lived transitory shocks, but do not report the results here.) The instruments of policy are assumed to be government spending gt, and the short-term real rate of interest, rt - nt. (Since n is predetermined, this is similar to using the short-term nominal rate as the instrument.) For the purposes of calculating optimal policies, we assumed an objective function for each country of the form Wo=^

X\aAp] + by] + c(rt - nt)2 + dgj]

(2.11)

with weights a = b = 0.01, c = d= 0.005 and A = 0.97. It should be noted that this objective function implicitly takes the baseline trajectory to be the desired path. We discuss the consequences of this in section 4. For the joint welfare loss function, we initially weighted the two welfare functions equally (i.e., we assumed that the a of the appendix to chapter 12 was one half). Subsequently, we varied a to calculate the appropriate Nash bargaining point. 3

The gains from cooperation

The first question that we address is whether or not it is advantageous for governments to cooperate in the setting of their respective monetary and fiscal policies. Both Rogoff (1985) and Miller and Salmon (1985b) report results which suggest that cooperation may not be advantageous. These results may initially appear paradoxical, for it might appear that the outcome under cooperation can be no worse than without cooperation. Since governments could, at worst, agree to carry out the non-cooperative policies, cooperation can surely only improve policy performance.

International cooperation and reputation

295

However, this logic is valid only if governments are the sole players in the policy game. Models with rational expectations contain a further player or players, in the form of private-sector agents who form expectations about future government policy and determine their actions accordingly. In a multi-player game, cooperation between a subset of the players may make matters worse if relationships with the remaining players are non-cooperative. In our context, cooperation between governments may not pay if the relationship between governments and the private sector is non-cooperative. Non-reputational policies may be interpreted as a noncooperative strategy between government and private sector; while reputational policies may be thought of as cooperative, though with government assuming a leadership role. 3 Thus the cooperative reputational (CR) policy is the case where full cooperation occurs between the two governments and between them and the private sector. This case necessarily delivers the best outcome, as confirmed by the results reported in tables 10.2 and 10.3. (The cooperative results in these tables are calculated setting the bargaining parameter, a, to 0.5. We examine the full bargaining solution for the cooperative case in section 5 when examining the questions of the sustainability of the cooperative reputational policy.) The three remaining policies involve some degree of non-cooperation between certain of the players. Thus the cooperative non-reputational (CNR) policy involves cooperation between the two governments, but not with the private sector. The non-cooperative reputational (NCR) policy involves cooperation between the private sector and the two governments, but not between the governments. The non-cooperative non-reputational (NCNR) policy involves non-cooperation, both between the two governments and with the private sector. While the cooperative reputational policy necessarily dominates the other three, the relative ranking of these other three cannot be established a priori, and it may depend on the model structure and the nature of the disturbances to which policy is responding. Rogoff (1985) developed an analytical model in which the non-cooperative non-reputational policy of letting exchange rates float dominated the cooperative non-reputational policy of pegging exchange rates. This can arise because floating exchange rates diminish the incentive for governments to spring monetary surprises on the private sector with a view to generating greater output, since the tendency of the exchange rate to depreciate in the absence of cooperation reduces the attraction of such surprises. Because of this, equilibrium inflation in the non-cooperative case is lower. Levine and Currie (1987a) were the first to examine all four policies. Using a stylised analytical model to examine the efficacy of the alternative policies to cope with

Table 10.2. Welfare losses for permanent disturbances (bargaining parameter a = 0.5) Regime Cooperative

Disturbance US supply shock ROW supply shock US demand shock ROW demand shock

Non-cooperative

Reputational CR

Non-reputational CNR

Reputational NCR

Non-reputational NCNR

5.226 (9.114, 1.338) 4.713 (1.449,7.978) 4.494 (4.856,4.132) 4.497 (4.065, 4.928)

6.635 (9.533, 6.138 (3.598, 8.728 (8.486, 8.548 (8.078,

6.690 (10.796, 2.584) 6.144 (2.660, 9.628) 8.360 (8.561,8.159) 7.707 (7.823, 7.992)

6.224 (8.979, 3.510) 5.771 (3.365,8.176) 7.655 (7.242, 8.068) 7.526 (7.176,7.876)

3.738) 8.677) 8.971) 9.017)

Note: Tables 10.2 and 10.3 report average welfare losses for the two countries. Bracketed values refer to individual welfare losses, the first being that for the US, the second for the ROW.

Table 10.3. Welfare losses for temporary (1-period) disturbances (bargaining parameter a = 0.5) Regime Cooperative

Disturbance US supply shock ROW supply shock US demand shock ROW demand shock

Note: as for table 10.2.

Non-cooperative

Reputational CR

Non-reputational CNR

Reputational NCR

Non-reputational NCNR

0.0041 (0.0076, 0.0006) 0.0039 (0.0009, 0.0068) 0.2206 (0.4314,0.0098) 0.2214 (0.0117,0.4310)

0.0044 (0.0082, 0.0043 (0.0007, 0.2219 (0.4352, 0.2229 (0.0103,

0.0050 (0.0089, 0.0051 (0.0017, 0.2235 (0.4365, 0.2248 (0.0122,

0.0044 (0.0083, 0.0043 (0.0004, 0.2227 (0.4384, 0.2235 (0.0088,

0.0006) 0.0079) 0.0087) 0.4355)

0.0012) 0.0086) 0.0096) 0.4374)

0.0004) 0.0083) 0.0070) 0.4383)

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Fiscal and monetary policy in interdependent economies

different exogenous disturbances, they found that the relative ranking of the non-reputational policies depended on the nature of the disturbances, and that the welfare differences between these two policies were not marked. The Rogoff result that, without reputation, cooperation does not pay was obtained for aggregate supply disturbances, but not for aggregate demand disturbances. Levine and Currie also found that the worst policy was the non-cooperative reputational policy which, at best, verged on the edge of instability and delivered a significantly worse welfare outcome. This suggested a complementary result to that of Rogoff, namely that, without cooperation between governments, reputational policies may be damaging. These findings suggested that to realise gains from cooperation requires reputation; and that to realise the gains from reputation requires cooperation. One aim of this chapter is to examine the validity of these results in an empirical model. Turning to consider the welfare calculations in tables 10.2 and 10.3, we find that they are, indeed, supportive of the earlier analytical results obtained by Levine and Currie. (The cooperative policies reported in these tables are calculated on the assumption that the US and ROW objective functions are weighted equally; that is, a = 0.5. The consequences of variations in a to reflect bargaining strength are examined in section 10.5.) Consider, first, the non-reputational policies (CNR and NCNR), reported in the second and fourth columns of tables 10.2 and 10.3. In the face of temporary disturbances (table 10.3), the non-cooperative non-reputational policy delivers a worse aggregate outcome than the cooperative non-reputational policy in the face of all four disturbances, contrary to Rogoff's result, though in the case of supply disturbances the difference is very small. (See also the fifth column of table 10.4, which reports the ratios of welfare losses.) However, the performances of these two policies are not markedly different. In the face of permanent shocks (table 10.2), a rather different picture emerges. For permanent aggregate demand or aggregate supply disturbances, the Rogoff finding emerges that the non-cooperative outcome is superior. These results suggest that, in the absence of reputation, cooperation will be advantageous in the face of temporary demand disturbances, though the gains will be rather trivial, and will be disadvantageous in the face of long-lived or permanent disturbances. The difference in results between temporary and permanent disturbances is quite striking and requires comment. Our objective function is defined to penalise deviations in output, inflation and the instruments around the initial steady-state equilibrium. Temporary disturbances perturb the system away from this desired equilibrium, but do not render the desired equilibrium infeasible. The control problem is, therefore,

Table 10.4. Ratios of aggregate welfare losses

Disturbances

(1) CNR/CR

(2) NCR/CR

(3) NCNR/CR

(4) NCR/CNR

(5) NCNR/CNR

(6) NCNR/NCR

US supply shock Permanent Temporary

1.270 1.070

1.280 1.288

1.195 1.065

1.008 1.147

0.941 0.995

0.933 0.868

ROW supply shock Permanent Temporary

1.302 1.116

1.304 1.330

1.224 1.122

1.001 1.192

0.940 1.006

0.939 0.844

US demand shock Permanent Temporary

1.942 1.006

1.860 1.013

1.703 1.010

0.958 1.007

0.877 1.004

0.916 0.996

ROW demand shock Permanent Temporary

1.901 1.006

1.714 1.015

1.674 1.009

0.902 1.009

0.881 1.003

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300

Fiscal and monetary policy in interdependent economies USAINF

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i i i i • i i i i i • i i i i i i ii i i i

3 4 5 6 7 8 9 10 11 12 13 14 15 Years USAGOV

—

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—

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•+

Figure 10.1 (a) Cooperative reputational (CR) regime: US supply disturbance: trajectories of output, inflation and competitiveness: (b) Cooperative reputational (CR) regime: US supply disturbance: trajectories of real interest rates and government expenditure

merely that of returning the system to the desired position in the most efficient way, and there is no long-run policy conflict. By contrast, permanent disturbances render the initial, desired equilibrium infeasible. (In the case of demand shocks, this is because of the presence of instrument costs.) There is, in consequence, a long-run policy conflict, in that the government would like to attain a level of output, inflation and instrument settings that is infeasible. As the work of Barro and Gordon (1983b) demonstrates, in the context of a non-dynamic model, such a long-run policy conflict gives rise to differing long-run equilibria under reputational and non-reputational policies. It will also give rise to differing

International cooperation and reputation

301

xiO 1 48 I ( a ) ROWINF 40 ^ COMPET 32 24 ROWOUT 16 USAINF 8 i i i i i i i i i i i i i i i i i 01 j f i 2 3 4 5 6 7 8 9 10 11 12 13 14 15 -8 Years -16' -24 USAOUT -32 -40 -48 x10 1 48 E (b) 40 32 24 E16 8> ^i I i I

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Figure 10.2 (a) Cooperative non-reputational (CNR) regime: US supply disturbance: trajectories of output, inflation and competitiveness: (b) Cooperative non-reputational (CNR) regime: US supply disturbance: trajectories of real interest rates and government expenditure

equilibria as between cooperative and non-cooperative policies. We examine this question in the next section. But the fact that the major difference between the four regimes occurs in the face of permanent disturbances suggests that it is this long-run policy conflict, rather than the short-run stabilisation issue, that is at the heart of the difference between our alternative regimes. (Another important way of introducing such a long-run policy conflict without assuming that targets are infeasible is by introducing ongoing stochastic noise: see Currie and Levine, 1987.) The results also show that the non-cooperative reputational (NCR)

Fiscal and monetary policy in interdependent economies

302

1-12

3 4 5"~6-7-8-9-10-11-12J3J4J5 ROWOUT Years

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USAGOV i I i I i I iT

11 12 13 14 15 Years

-24 -32 r -40

Figure 10.3 (a) Non-cooperative reputational (NCR) regime: US supply disturbance: trajectories of output, inflation and competitiveness: (b) Non-cooperative reputational (NCR) regime: US supply disturbance: trajectories of real interest rates and government expenditure

policy does not perform well, though in contrast to the analytical results of Levine and Currie (1987a) it is not always inferior to the other regimes. For permanent disturbances, it is always inferior to the CR and NCNR regimes for all disturbances and to CNR for supply shocks. But for permanent demand disturbances, it is superior to the CNR regime. However, further examination suggests that the NCR regime is not robust. We found that decreasing even marginally the discount factor, X, led to instability of the NCR. Thus this regime was stable only for values of X in excess of 0.965, corresponding to an annual discount rate of about 7 per cent. By contrast, the other regimes were stable for all values of X

303

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Figure 10.4 (a) Non-cooperative non-reputational (NCNR) regime: US supply disturbance: trajectories of output, inflation and competitiveness: (b) Noncooperative reputational (NCNR) regime: US supply disturbance: trajectories of real interest rates and government expenditure

down to 0.7 (which is the lowest value that we tried). This confirms the earlier analytical finding of Levine and Currie (1987a) that the NCR regime is prone to instability, and the results for this regime are not robust. The source of this tendency towards instability may be discerned from figures 10.1-10.8, which depict the trajectories of key variables under the four regimes in the face of permanent US aggregate supply and aggregate demand disturbances.4 (Because Minilink is fairly symmetrical in its main features, the trajectories for ROW supply and demand shocks look similar, with the countries interchanged, and are therefore not reported.)

304

Fiscal and monetary policy in interdependent economies

(a) x10 1 7i 60 50 - \ U S A O U T 40 3i 20 0 -10. -20 -30 -40 -50 -60 -70

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Figure 10.5 (a) Cooperative reputational (CR) regime: US demand disturbance: trajectories of output, inflation and competitiveness: (b) Cooperative reputational (CR) regime: US demand disturbance: trajectories of real interest rates and government expenditure

In contrast to the other regimes, under the NCR policy, rather little of the burden of adjustment falls on fiscal policy, except in the first few time periods; indeed, as we discuss in the next section, the longer-run response is perverse (see figures 10.3b and 10.7b). Accordingly, a much greater burden of adjustment is thrown on interest rates. This is because, without cooperation, there is the temptation for governments to use monetary policy excessively to operate on the exchange rate, with adverse spillovers. Moreover, with reputation, monetary policy becomes more powerful, since announcements of future policy action are credible and influence the exchange rate immediately. This allows governments to delay their

International cooperation and reputation

305

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70 " (b) 60 50 40 30 20 ROWINT USAINT 10, 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 -10 Years -20 -30 USAGOV -4! -50 ROWGOV

Figure 10.6 (a) Cooperative non-reputational (CNR) regime: US demand disturbance: trajectories of output, inflation and competitiveness: (b) Cooperative non-reputational (CNR) regime: US demand disturbance: trajectories of real interest rates and government expenditure

monetary policy actions without losing policy effectiveness. With discounting, this is advantageous in order to reduce instrument costs. The result is that, under the NCR policy, governments rely heavily on monetary policy action but delay its implementation. Since each government does the same, the effects are partially offsetting, and each government is forced to increase its monetary policy action. The consequence is an interest-rate war. The greater the degree of discounting, the more likely this is to trigger an unstable upward spiral in interest rates. The other three policies, where either cooperation obtains or reputation is absent, avoid this problem.

306

Fiscal and monetary policy in interdependent economies

x10-1 70ft (a) 60 .. 50P\USAOUT

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Figure 10.7 (a) Non-cooperative reputational (NCR) regime: US demand disturbance: trajectories of output, inflation and competitiveness: (b) Non-cooperative reputational (NCR) regime: US demand disturbance: trajectories of real interest rates and government expenditure

Earlier studies (for example, Oudiz and Sachs, 1985; Miller and Salmon, 1985b) have found the gains from international coordination of policy to be relatively small. Moreover, they have also tended tofindquite small differences, in terms of welfare gain, between reputational and non-reputational policies. A striking feature of our results is that they find quite large differences between reputational and non-reputational policies and major gains from coordination of reputational policies. As column one of table 10.4 shows, in the face of permanent disturbances cooperative reputational policies deliver a welfare loss around 25-30 per cent lower than the non-reputational policy in the face of permanent

307

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Figure 10.8 (a) Non-cooperative non-reputational (NCNR) regime: US demand disturbance: trajectories of output, inflation and competitiveness: (b) Non-cooperative non-reputational (NCNR) regime: US demand disturbance: trajectories of real interest rates and government expenditure supply disturbances and nearly one-half lower in the face of permanent demand disturbances. These estimates of the gain from reputational policies are rather greater than previous results. The gains in the face of temporary disturbances of whatever kind are not, however, large, and are almost negligible for temporary demand shocks. However, as we have already noted, for non-cooperative policies, the reputational policy differs from the non-reputational policy in exhibiting a marked tendency towards instability. In particular, it risks destabilising the aggregate economy, as governments play interacting reputational games with the private sector. In the absence of cooperation, reputations may well be damaging.

308

Fiscal and monetary policy in interdependent economies

This is also the basis of our evidence for significant gains from coordination, contrary to the earlier literature. If governments do not coordinate their policies, it may well be better that they adopt non-reputational policies since the reputational ones typically yield a worse, possibly destabilising, outcome. But with coordination, reputational policies become highly advantageous, particularly in the face of permanent disturbances. In order to tap the gains from reputation, cooperation is required. Cooperative reputational policies then yield a markedly better performance than the best non-cooperative alternative. However, this argument for cooperation is open to two possible objections. First, reputational policies may not be credible and sustainable, because of the problem of time inconsistency. Second, cooperation between governments may break down because an incentive emerges for one government to renege on the cooperative policy rule. Both these objections suggest, for different reasons, that the cooperative reputational policy is not feasible. Section 5 examines these points. But first we look at the long-run consequences of the different regimes. 4

The long-run effects of alternative regimes

As we have discussed in the previous section, the long-run equilibrium in the face of permanent disturbances differs between our four regimes. This is because, in the face of such disturbances, the desired long-run equilibrium is no longer feasible; and the resulting long-run policy conflict can generate undesirable long-run outcomes. Thus, for example, the work of Barro and Gordon (1983b) shows how, in the absence of reputation, the tendency for government to expand the economy unduly to obtain shortrun output benefits generates a long-run inflationary bias. In the absence of cooperation, there is also a tendency for governments to resolve the dilemma of stagflation by tightening monetary policy with a view to appreciating the exchange rate and lowering inflation, while expanding fiscal policy to offset the output consequences (see Oudiz and Sachs, 1984; Currie and Levine, 1985a). Adopted by all governments, the result can be a contractionary bias in the aggregate world economy. To investigate these questions, we computed the long-run equilibrium towards which the system tends under our four regimes in the face of permanent supply and demand disturbances. These are reported in table 10.5. Comparing the outcomes under non-reputational policies with the outcome under the equivalent reputational policy (i.e., CNR with CR, NCNR and NCR) gives striking confirmation of the tendency of nonreputational policies to deliver higher inflation in the face of a longrun policy conflict.5 However, the comparison of cooperative and

Table 10.5. Effects of permanent disturbances in long-run equilibrium Real interest

Inflation

Output

Competitiveness

Government expenditures

Welfare losses

US

US

ROW

26.95 107.30 34.27 214.10

4.51 51.53 14.74 104.44

6.35 61.99 22.79 116.93

20.78 96.08 31.19 201.00

5.66 71.24 24.96 139.1

4.68 64.42 16.33 134.25

5.99 75.33 26.35 147.06

4.93 68.12 17.25 142.01

US

ROW

US/ROW

13.82 33.21 -2.85 33.70

4.46 22.92 -3.77 23.90

7.87 0.94 12.02 3.17

39.11 45.08 40.18 43.10

-1.72 - 12.96 7.11 - 9.36

6.12 25.28 -7.51 25.28

10.74 31.21 1.32 32.58

8.23 0.98 12.09 3.30

-56.19 - 48.94 - 58.00 - 50.90

-3.13 - 14.46 3.82 - 10.67

US aggregate demand shock 0 CR 0 0 CNR 0 0 0 NCR 0 0 NCNR

4.63 27.04 -7.86 27.58

3.29 25.59 -3.97 27.10

9.29 1.11 12.65 3.60

30.76 38.24 30.45 36.10

-2.60 - 15.69 4.00 -11.64

ROW aggregate CR CNR NCR NCNR

4.76 27.80 -8.07 28.35

3.38 26.31 -4.08 27.87

9.55 1.14 13.00 3.70

- 48.75 -41.06 - 49.08 - 43.24

-2.67 -16.14 4.11 -11.97

ROW

US US inflation CR CNR NCR NCNR

shock - 10.0 -10.0 - 10.0 - 10.0

ROW inflation shock CR 0 CNR 0 0 NCR NCNR 0

0 0 0 0 -10.0 - 10.0 - 10.0 -10.0

demand shock 0 0 0 0 0 0 0 0

ROW

-2.58 •13.62

2.00 •10.04

-0.54 •12.54

4.96 -8.85 -2.12 •15.34

2.11 •1 1 . 3 9

-2.18 •15.78

2.17 11.71

310

Fiscal and monetary policy in interdependent economies

non-cooperative outcomes is less clear. With reputation, the tendency of the non-cooperation policy towards a contractionary bias is very marked. Indeed, the NCR policy delivers lower inflation in the long run (though not in the short run) in the face of inflationary permanent demand and supply shocks: this represents the end result of a prolonged interest-rate war. However, without reputation, the non-cooperative policy exhibits higher, not lower, inflation relative to the cooperative alternative. This may reflect the fact that, without reputation, the influence of monetary policy on the exchange rate and hence on inflation is rather weaker because policy announcements are not credible, and this in turn reduces the incentive of governments to resort to competitive tightening of monetary policy. An interesting feature of the results is the differing monetary/fiscal mix between the four regimes. For all four disturbances, the ranking between the four regimes is the same. Thus the NCR regime has the tightest stance of monetary policy and the laxestfiscalpolicy (withfiscalpolicy becoming more expansionary as a result of an inflationary shock). Relative to this, the CR regime comes next, with slightly lower real interest rates and a more contractionary fiscal stance, followed by NCNR and finally CNR. These results suggest that reputation results not just in a tighter stance of monetary policy, but also a tighter monetary/fiscal mix. The results also suggest that real interest rates are an important influence on long-run inflation in Minilink. The last two columns of table 10.5 report the welfare losses for the different regimes, and these are compared with the initial welfare losses in table 10.6. For all regimes, the long-run welfare loss is greater than the initial one. This shows the consequences of discounting, for with no discounting the two would be the same. But the differences vary markedly between the regimes. For the CR regime, the differences between the long-run and initial welfare losses are the smallest, increasing by a factor of about three for supply shocks and only by about 15-20 per cent for demand shocks. For the CNR regime, the long-run welfare loss is about twelve times the initial welfare loss for supply shocks and about eight times for demand shocks. For the NCNR regime, the differences are still more marked: an increase by a factor of around twenty-five for supply shocks, and a factor of around eighteen for demand shocks. Interestingly, the NCR regime shows a less marked deterioration than either NCNR or CNR: the welfare loss increases by a factor of about four for supply shocks and around two-and-a-half for demand shocks. These are dramatic changes in welfare and demonstrate the dangers of choosing between regimes solely on the basis of initial welfare calculations. This is despite the fact that the initial welfare represents a

International cooperation and reputation

311

Table 10.6. Comparison of initial and long-run welfare losses for permanent disturbances CR

CNR

NCR

NCNR

US supply shock initial Aggregate long-run US initial long-run ROW initial long-run

5.226 15.729 9.114 26.949 1.338 4.510

6.635 79.419 9.533 107.304 3.738 51.534

6.690 24.505 10.796 34.268 2.584 14.743

6.224 159.273 8.979 214.103 3.510 104.443

ROW supply shock Aggregate initial long-run US initial long-run ROW initial long-run

4.713 13.568 1.449 6.355 7.978 20.781

6.138 79.036 3.598 61.990 8.677 96.083

6.144 26.988 2.660 22.790 9.628 31.186

5.771 158.964 3.365 116.932 8.176 200.997

US aggregate demand shock Aggregate initial long-run US initial long-run ROW initial long-run

4.494 5.173 4.856 5.662 4.132 4.684

8.728 67.833 8.486 71.242 8.971 64.424

8.360 20.646 8.561 24.962 8.159 16.330

7.655 136.688 7.242 139.127 8.068 134.250

ROW aggregate demand shock initial Aggregate long-run initial US long-run initial ROW long-run

4.497 5.470 4.065 5.987 4.928 4.953

8.548 71.728 8.078 75.334 9.017 68.123

7.707 21.802 7.823 26.351 7.992 17.254

7.526 144.538 7.176 147.063 7.876 142.014

discounted sum over the whole future trajectory of the system. Discounting the future has different long-run consequences in the different regimes, and in some, the non-reputational regimes, the consequences can be very serious. Based on long-run considerations, the ranking of the alternative regimes is clear: CR is best, followed by NCR, then by CNR, with NCNR worst. (We must, however, recall the potential instability of the NCR regime, which means that, for a greater degree of discounting, its position in the ranking would slip, and become last with a discount rate

312

Fiscal and monetary policy in interdependent economies

greater than 7 per cent.) On the basis of these long-run results, cooperation is always beneficial, contrary to Rogoff's (1985) analysis, in that CR clearly dominates NCR and CNR clearly dominates NCNR. Moreover, the gains from cooperation are considerable. Ther results also confirm our earlier conclusions that the benefits from cooperation and reputation accrue jointly. The greatest benefits from reputation accrue when used in a coordinated fashion; and the greatest gains from cooperation accrue when governments have reputation. The results in table 10.6 suggest that in the long run the CR policy yields reductions in welfare loss relative to the next-best regime, NCR, of about one-third to one-half for supply shocks and about 75 per cent for demand shocks. Moreover, comparisons with the NCR regime are hazardous because of its marked tendency towards instability. Excluding the NCR regime for this reason, the comparisons with the next-best regime, CNR, give a picture still more advantageous to the CR regime, with reductions in long-run welfare loss of more than 80 per cent for supply shocks and more than 92 per cent for demand shocks. This result underlines the superiority of the cooperative reputational (CR) policy. However, they are of little interest if this policy is not sustainable. It is to the question of the viability of the CR policy that we now turn. 5

The sustainability of the cooperative reputational policy

A common objection, springing from the work of Kydland and Prescott (1977), to the optimal reputational policy is that it will lack credibility because it is time inconsistent: the mere passage of time may create an incentive for government to renege on, and reformulate, its initial policy. But if this is anticipated by the private sector, being sufficiently farsighted, then the policy will lack credibility. Since credibility is essential for the policy to be effective in acting on private expectations, lack of credibility renders the policy non-optimal. Some may wish to argue that the private sector lacks the astuteness or information to anticipate the future reneging by the government. But, even if this is the case, a policy which generates incentives to renege does not provide a satisfactory basis for international cooperation in policy. If governments do renege, it seems plausible to suggest that their reputation will be impaired, and that the choice of policy will be confined to non-reputational policies.6 We may therefore judge the incentive for governments jointly to renege on the cooperative reputational policy by comparison with the welfare losses under the non-reputational alternative.

International cooperation and reputation

313

>a=0.1 12

11

1

•a=0.2

CO CO

_o 2

10

co

9

NCR

NCNR

a =0.6 a = 0.7 I 2

a =0.8 3

_L 4

ROW welfare loss

Figure 10.9 Permanent US supply disturbance: effects of variation in bargaining parameter a on welfare losses

An equally important question of credibility arises concerning the incentive of either government to renege on the cooperative agreement with the other. We know that, with an appropriate initial bargain, the cooperative reputational policy dominates the non-cooperative alternatives. Our results of the previous section also suggest that this holds, with even greater strength, in the long run for permanent disturbances. However, this does not rule out the possibility that a non-cooperative alternative is superior at some point along the transition path. If this were so, the cooperative reputational policy would lack credibility at the outset, since one or other of the governments would anticipate reneging by the other and would be unwilling to commit itself to the cooperative policy. To consider this possibility, we need to examine the trajectory for welfare losses along the transition path. For the case of permanent disturbances, we first consider the initial bargain over the gains from cooperation. Inspection of table 10.2 reveals that, for the US experiencing a permanent domestic supply disturbance, the CR policy is dominated by the NCNR policy. This reflects an inappropriate choice of a, the bargaining parameter, which in previous sections we have set arbitrarily to one-half. We now consider how a might be determined. Figures 10.9 and 10.10 plot variations in welfare losses by

314

Fiscal and monetary policy in interdependent economies 9

NCR

8 7 a) _o £ 0

'"[(y, + / " $?

+

dx(pt+i-pt+i-x)2

+ d2(gt+i - g)2]

(2.8)

i= 0

with a similar expression with '*' variables for the second country. This quadratic function penalises output deviations around some target, y, inflation, and deviations of government spending around some target, g. The target for government spending is assumed to arise from efficiency and/or social considerations influencing the desired level of public spending. Similar considerations would arise were we to choose taxes as the fiscal instrument. No instrument costs are attached to the use of the money supply, since neither social nor efficiency arguments are obviously applicable. Throughout the following, we assume that government's output objectives are ambitious, in the sense that y is in excess of feasible output levels. As in Barro and Gordon (1983b), this may impart an inflationary bias to certain of the regimes. 3

Equilibrium

The determination of the equilibrium of the real economy is illustrated in figures 11.1 and 11.2. Let yf = \(yt + yf) and yf = \(yt - yf), with similar definitions for other variables. Thus the superscript 'a' denotes the aggregate system, while 'd' denotes the divergence system. Then the aggregate

Fiscal policy coordination, inflation and reputation

325

ra

Ya Figure 11.1

ya

Equilibrium of the aggregate real economy

equilibrium for the two countries taken together is given by adding the individual demand and supply functions y? = a5(a0-a2r?

+ a3gf)

(3.1)

yt = b0 - b2r? + b3(p? -

ffi)

(3.2)

where a5 — (1 - a4)~\ Equation (3.1) represents the aggregate demand relationship, depicted by Z)a; while (3.2) represents the aggregate supply relationship, depicted by S a in figure 11.1. We assume throughout the following that Da is more interest elastic than Sa (for reasons of intraperiod stability, i.e., a2a5 > b2), so that an increase in government spending raises real interest rates. Furthermore, we shall assume that bx > \b2. This assumption is not crucial since its converse does not alter any conclusions of this chapter. It simply facilitates comparisons among different regimes. Figure 11.2 represents the equilibrium of the divergence system, obtained by subtracting country 2's demand and supply schedules from country l's. yf = a6(a{et - a2rf + a3gf)

(3.3)

yt=-blet-b2r?

(3.4)

+ b3(p«-p?t)

et= - 2 r f + c? +lt ,= - 2 ^

+

u

(3-5)

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Fiscal and monetary policy in interdependent economies

e

Figure 11.2 Equilibrium of the divergence real economy with a6 = (1 + a4)~l. In equilibrium (3.5) implies that rd = 0. (This condition will of course also hold in the face of permanent shocks). Thus we can depict equilibrium of the difference system as in figure 11.2. Equation (3.3) gives an upward sloping demand schedule, Z>d, between et and yf; while (3.4) gives downward sloping supply schedule, Sd. We first consider the general effects of fiscal policy and monetary policy on output in both countries. A permanent shock to aggregate demand, arising from an increase in government spending in country 1, will shift both demand schedules, Z>a in figure 11.1 and Dd in figure 11.2 to the right. From figure 11.1, this raises aggregate interest rates and lowers world output y? = \(yt + yf). From figure 11.2, this appreciates the real exchange rate of country 1, permitting an increase in output in country 1 to country 2, leading to an increase in yf = \(yt - yf). Output in country 2 therefore falls unambiguously: the depreciation of its real exchange rate and induced rise in real interest rate causes a fall in aggregate supply. However, output in country 1 may rise, if the exchange-rate elasticities are sufficiently large that the rise in output in the divergence system outweighs the fall in the aggregate system. A key feature of the model, arising from its natural rate features, is that the real equilibrium of the system is independent of monetary policy. Thus monetary policy influences the real economy only via price surprises. Such price surprises administer shocks to aggregate supply that are necessarily temporary, lasting only one period. From (3.5), the real exchange rate, eu and the difference in real interest rates, rd, move

Fiscal policy coordination, inflation and reputation

327

inversely on a one-for-one basis. Eliminating rf from (3.3) and (3.4) using this relationship yields an upward-sloping demand schedule, Z>d, as in figure 11.2. This is more elastic with respect to the exchange rate than for permanent shocks. The slope of the supply schedule, S*, is ambiguous, being downward sloping as in figure 11.2 for bx >\b2, but positively sloped for bx < \b2. A price surprise in country 1 therefore shifts both £* and S* to the right, raising aggregate output, lowering average real interest rates and depreciating the exchange rate, regardless of the slope of S*. Output in country 1 rises unambiguously. Whether output rises in country 2 depends on whether the favourable effects of the appreciating exchange rate (assuming that a2 > b2) outweigh the unfavourable effect of possible higher interest rates. 4

Monetary policy

The closed economy case Having considered the influence on the real equilibrium of the system, we nowfixgovernment spending in both countries and focus on the determination of monetary policy and prices. First, consider this problem in the closed-economy context considered by Barro and Gordon (1983b). Solving out for real interest rates from the demand function (2.1) and substituting into the supply function (2.3), the reduced form equation for output becomes: yt = y + B3(pt-fit) l

(4.1) l

where B3 = b3(l — b2a2 )~ and y is equilibrium output. The problem for the government is then to minimise (2.8) subject to (4.1). If we assume that the government has no reputation, this optimisation problem is solved by treating private-sector expectations as given or parametric. This is because a government without reputation perceives that it can have no influence on future expectations of policy actions either because it has zero credibility or it has a negligible probability of survival. The first-order condition for a minimum is given by h\y + hiPt ~ Pit) ~y] + dx{pi - pt- 0 = 0 so that pt = (dj + SI)'l [(y - y + B3plt)B3 + dlPt. i]

(4.2)

However, the private sector can calculate that the government will choose pt as in (4.2), and will therefore form its expectations in accordance with (4.2). Substituting in for plt yields:

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Fiscal and monetary policy in interdependent economies

n

Figure 11.3 Equilibrium inflation without reputation nYR = di lB3(y-y)

(4.3)

where nt = pt -pt-\ is the inflation rate. The inflation rate is therefore related to the gap between target output (y) and equilibrium output (y). This is depicted in figure 11.3. Government indifference curves describe ellipses around the point (yt,nt) = (y,0). The zero inflation equilibrium is at (y,0), at a welfare loss of Wo. However, without reputation this point is not sustainable. This is because, with expectations treated as parametric, the trade-off between inflation and output appears positively sloped, so that the government is tempted to spring a monetary surprise to take the economy to yx. Such a temptation exists for all rates of expected inflation below TT^R and at this rate of inflation the government has no temptation unexpectedly to expand the money supply at a faster rate. Clearly the welfare loss at 7i?R exceeds that at nt = 0. With reputation the government no longer treats expectations as parametric, but may instead seek to influence expectations by announcing its future policy actions. If it does so, and carries out its stated intentions, the best policy is to go for the zero inflation equilibrium with yt = y, yielding a welfare loss of Wo: =0

(4.4)

However, there is still the question as to whether this position is sustainable, or whether the government has an incentive to renege on this policy

Fiscal policy coordination, inflation and reputation

329

by springing a monetary surprise on the private sector. Much here depends on the private sector's reaction when reneging takes place. It seems reasonable to suppose that the private sector will suspend belief in the government's reputation of a period of time (say, n periods), so that after the period of reneging the system reverts to the non-reputational equilibrium for n periods. Then the government must weigh the gains from reneging against the subsequent costs arising from loss of reputation. In terms of the welfare losses of figure 11.3, the welfare gains from the reneging strategy, W, is given by W=WO-W.X-

t=\

= Wo-W-x- h - ^ l / ^ N R - Wo)- 0.042 n .r. n .r. n .r.

0.750 < 0.750 < 0.750 < 0.750 < 0.750
0.1063 in order to guarantee stability. Working backwards for a feasible range of coefficients is somewhat more tedious. We concentrate on the demand side of the model for various rates of p. Table 11.2 gives the values. It shows that as long as the degree of discounting is not too high, stability is guaranteed for virtually every positive parameter of the model, except for a2 (measuring the marginal contribution of the interest rate to the determination of output), which is sensitive for the lower range of/^-values. 7 A closer look at the NCR regime It is of interest to analyse a situation where only country 1 pursues some active policies to achieve growth of output as given by an (ambitious) output target. This means that we now drop the assumption of symmetric behaviour of the two countries. Instead, we look at the case where country 1 and country 2 do not cooperate and where, furthermore, country 1 assumes (correctly) the behaviour of country 2 as being given and being static. We analyse this constellation only for the NCR regime, since it is the resulting dynamics hereby which is of interest. As in section 4 country 1 is now using government expenditure and the exchange rate as

Fiscal policy coordination, inflation and reputation

343

instrument variables (taking expectations about the future exchange rate by the private sector into account). The optimal setting of both policies is determined by the minimisation of the well-known welfare loss function. Consequently, we cannot use the equivalent reaction functions (A.21) and (A.22) for the second country. Instead we analyse the following dynamical system (obtained by inserting (A.21) into (A.22) for kt and A,_i, inserting (A.22) in (A. 19) and substituting for gt from the above): gt-\

et-\ et

pd*iO

t$33* 6

0 pd* eOco — a> 0 0

¥

0 0

gf

(A.30)

gf-i xt

with S =,

j

2

0 =[p(de-4d2)]-1 co = a3a6(a2a6- b2)~l V = 26{pcid(cl - dy) -{p-

d)[d(ca0 - y) + 2d2g]}

To simplify the analysis we use the parametric values from section 5. In addition, we set gf = gf- x = 0 (i.e., government expenditure in country 2 remains on a constant level) and xu the dummy variable, is equal to one. (A.30) then reads gt eet+i,t

0.148 0 0.592

0.797 0 3.189

-3.630 1 25.360

gt-l

_et

0.744

+ 0 0

or, for short

where z, + 1 = [gt et]'; zt = [g r _i ^ _ i ] ' ; xt+Ut

= [^+i,J f and x, = [et]'.

The matrix yl has eigenvalues kx = 0.107; k2 = 0.000 and A3 = 25.481, two of which lie inside the unit circle whereas one lies outside the unit

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Fiscal and monetary policy in interdependent economies

Table 11.3. Trajectories for NCR regime (Allfigures in %) t

X

S?

0 1 2 3 4 5 6 7 8 9 10

-3.25 -0.20 + 3.04 + 6.48 + 10.14 + 14.03 + 18.16 + 22.56 + 27.23 + 32.19 + 37.47

+ 2.17 + 0.13 -2.03 -4.32 -6.76 -9.35 - 12.11 - 15.04 - 18.15 -21.46 - 24.98

r?

*,

-4.33 -0.27 + 4.05 + 8.64 + 13.52 + 18.71 + 24.21 + 30.08 + 36.31 + 42.92 + 49.96

0 0 0 0 0 0 0 0 0 0 0

circle. Thus the condition for stability as introduced in Blanchard and Kahn (1980) is satisfied, which we already know from section 5. We use the solution procedure for linear, rational expectation models, formulated in discrete time, as set out by Currie and Levine (1982). The general solution for xt and zt+1 is then given by xt = - M22lM2lzt zt+l = (Au -

- M22lA2l[0

Al2M22'M2l)zt

| M22]ut AnM22lA2lM22]ut

-

(A.32) (A.33)

where A is the matrix given in (A.31), A is the matrix of eigenvalues, / i s the identity matrix and Mis the matrix of the following (row) eigenvectors to A: 0.183 M = 0.100 0.023

0.983 0.995 0.125

-0.013 -0.025 0.992

MA = AM

(A.34)

Matrices are partitioned conformably as denoted by the subscripts. These submatrices are furthermore assumed to be non-singular. Inserting the parameter values gives as solution gt = 0.232 £,_, + 1.254 et-x + 0.744

(A.35)

et = 0.202 gt-x-

(A.36)

0.126 £?,_!

We may construct an example of instability for the NCR case by choosing p = 0.1. Recall that for the numerical example in section 5 the

Fiscal policy coordination, inflation and reputation

345

governments must discount future welfare losses at a rate greater than 0.1063 in order to guarantee stability. The other parameters are assumed to remain the same. The stability condition (A.27) clearly does not hold, in fact B/C = 1.063 > 1. Instead of obtaining the solution for g? given in (A.26) we have (e.g., (A.25)): g? = 1.063 g?_! + 0.0325

(A.37)

Table 11.3 gives the trajectory for time periods 0 to 10 if it is assumed that the government incorrectly perceives it has reputation. Thus it assumes acting according to the NCNR regime, setting gf initially at - 3.25 per cent (the optimal value), whereas it is the NCR regime which prevails. Since the system is unstable the welfare losses tend to infinity. 8 Reneging In section 5 we have defined a reneging strategy W as follows: W= wK-Wc-

|

= WK - Wc - ( ~ y )

( ^ N R " WK)

(A.38)

which we require to be negative for the reputational policy to be sustainable. WR is the welfare loss under the reputational regime (either with cooperation or without cooperation). Wc is the welfare loss if the government cheats, which we assume it can do for just one period. After that period the private sector will form its expectations in accordance with a non-reputational government. We calculate Wc by assuming that the government is successful in springing a monetary surprise which leads to a higher output than that which would have prevailed under a non-surprise reputational regime. In effect, we calculate Wc by substituting j ? N R into WK. Since ,y?'NR, y?'R the difference between WK and Wc is positive. This difference in welfare losses can be referred to as the gains from cheating. The loss resulting from this policy is the discounted sum of the welfare loss differences between WNR which now prevails for n periods (because the private sector mistrusts any future commitment of the government) and WK, the (lower) welfare loss of what could have been achieved if cheating would not have taken place. We thus require these losses to be greater than the gains if reputation is sustainable. We shall calculate Wfor both the cooperative and the non-cooperative regime, but only for d[l) = 0.5. Substituting the respective values from table 11.1 into (A.38) we get for the cooperative regime 0.9" < 0.905

(A.39)

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Fiscal and monetary policy in interdependent economies

Table 11.4. Minimum value of p for sustainability

n= 1 n= 2 n= 3 n = oo

Pc

/?CN

0.85 0.54 0.49 0.46

0.47 0.35 0.33 0.32

whereas for the non-cooperative regime (A.38) reduces to 0.9" < 0.948

(A.40)

Our numerical example speaks very much in favour of reputation. For every n (n being an integer) greater than zero, reputation is sustainable. Only for n = 0 (note that in this case (A.38) reduces to WK-WC> 0) should the government cheat. This latter result makes sense since the government should indeed try to renege if it faces a public which is either completely ignorant or extremely forgiving with regard to broken commitments of the government. It is of course of interest to have a more general relationship between /?, the discount factor, and n, the punishment period. For the cooperative and the non-cooperative regimes these conditions can be given as w>ln(1.846,-0.846)^ (A4])

^(1.465,7-0.465)^

(A.42)

In p Table 11.4 states the minimum value for p which guarantees sustainability of the reputational policy for various punishment periods n.

Note 1 See, for example, certain of the papers in Buiter and Marston (1985), such as Currie and Levine (1985b), Miller and Salmon (1985b) and Oudiz and Sachs (1985). For recent papers that address both issues together, see Currie and Levine (1987) and Currie, Levine and Vidalis (1987).

12 The use of simple rules for international policy coordination

1

Introduction

In recent years, appreciable advances have been made in the application of dynamic game theory to the design of macroeconomic policies for open, interdependent economies (see, for example, the papers in Buiter and Marston, 1985 and Bryant and Portes, 1987). Whereas the earlier literature (for example, Hamada, 1979; Cooper, 1985) analysed issues of international policy cooperation in the context of games between governments, the more recent literature has recognised the central role played by private agents making rational forward expectations about future policy moves. This literature therefore places at centre stage issues of reputation and credibility in policy-making, and it has integrated the analysis of reputation in a single economy context (see, for example, Barro and Gordon, 1983b) with the literature on international cooperation (Levine and Currie, 1987a). This analysis has permitted a more convincing analysis of the sustainability of cooperation and reputation in policy design. In previous work by the authors, these developments have been applied to an empirically based international model to analyse the gains from cooperation and reputation in international macropolicy. Using Minilink, a reduced two-bloc version of the OECD Interlink model, Currie, Levine and Vidalis (1987) (Chapter 10) examined four separate policy regimes: cooperation with reputation (CR), cooperation without reputation (CNR), non-cooperation with reputation (NCR) and non-cooperation without reputation (NCNR). They found that the benefits from cooperation and reputation were largely joint, in that reputation without cooperation was counterproductive, as was cooperation without reputation. These benefits were appreciable, particularly in the face of permanent disturbances to Blueprints for Exchange Rate Management, M. Miller, B. Eichengreen and R. Portes (eds.), Academic Press (1989). With Jessica Gaines. Financial support from the ESRC (grant number BO 1250012) is gratefully acknowledged. 347

348

Fiscal and monetary policy in interdependent economies

the system. Moreover, the analysis suggested that cooperative reputational policies were sustainable in a fairly wide range of circumstances. A difficulty with this previous work is that the policies considered are all exceedingly complex, involving high-order dynamic feedback rules for the policy instruments. This feature is unfortunate as it can be convincingly argued that policy rules which might appeal to policy-makers and be made credible in the eyes of the private sector should, in fact, be simple. The argument for simplicity carries particular force in the international sphere where the need to monitor policy commitments applies to both the private sector and the countries or blocs entering into agreements. These arguments led the authors in earlier work to examine the design of simple policy rules (see Currie and Levine, 1985a). However, the earlier work examined solely the efficacy of simply policy designs, and did not address questions of their sustainability. It shared that deficiency with other work on simple policy rules. The purpose of this chapter is to develop a methodology for analysing the sustainability of simple rules and to apply the methodology empirically to Minilink. In this, the chapter parallels for simple rules the analysis carried out by Currie, Levine and Vidalis (1987) for complex, optimal rules. The methodology has general applicability, and future work will be concerned to apply it to a systematic appraisal of blueprints for international policy coordination, such as the target zone proposal of Williamson and Miller (1987). The plan of the chapter is as follows. Section 2 sets out the details of Minilink, the model to which our methodology is applied. This empirically based model was derived as a reduced representation of earlier writings of the OECD Interlink model, and therefore suffers from not including recent improvements in the specification of the parent model, as well as from approximation errors. Future work will seek to apply our methodology to more completely specified and more soundly based empirical models. Section 3, together with the appendix, sets out the analytics of the design of simple rules. Section 4 describes techniques for analysing the sustainability of simple rules. Section 5 reports the results of applying these techniques to Minilink, and section 6 briefly concludes. A number of themes run through the chapter. First, we address a number of objections to the use of flexible feedback rules for purposes of macroeconomic stabilisation. Thus we address explicitly the timeinconsistency problem, the scope for beggar-my-neighbour policies in the international sphere, and the objection that effective stabilisation rules are necessarily too complex to be realistically implemented. In future work, we will address the issues raised by model uncertainty. Secondly, we examine the scope for using agreements in the form of simple rules as a

Simple rules for international policy coordination

349

surrogate for more far-reaching agreement on international policy coordination. Finally, we highlight the possibility that simplicity in policy design may improve the sustainability and credibility of the associated macroeconomics policies. 2

Minilink in a stochastic environment

The two-bloc (US and rest of the world) stochastic version of Minilink is described by the following equations. Unstarred variables refer to the US and starred variables to the rest of the world (ROW). (See Masson, Blundell-Wignall and Richardson, 1984, for a full description of the model and the method used to construct the reduced representation from the full Interlink model.) yt = CiPt + c2gt + 2c3na,_i + c4rjt + c5yf + adt

(2.1)

yf = cfp* + cfgf - 2c?r