Economic Growth in the Regions of Europe: Theory and Empirical Evidence from a Spatial Growth Model

Contributions to Economics

For further volumes: http://www.springer.com/series/1262

Sascha Sardadvar

Economic Growth in the Regions of Europe Theory and Empirical Evidence from a Spatial Growth Model

Asst. Prof. Sascha Sardadvar Vienna University of Economics and Business Institute for Economic Geography and GIScience Department of Socioeconomics Nordbergstraße 15 1090 Vienna, Austria [email protected]

ISSN 1431-1933 ISBN 978-3-7908-2636-4 e-ISBN 978-3-7908-2637-1 DOI 10.1007/978-3-7908-2637-1 Springer Heidelberg Dordrecht London New York © Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPI Publisher Services Printed on acid-free paper Physica-Verlag is a brand of Springer-Verlag Berlin Heidelberg Springer-Verlag is part of Springer Science+Business Media (www.springer.com)

Preface

This book is based on the doctoral thesis of the same title, which was completed in April 2009. The study was inspired by the emphasis the European Union puts on its various regions in general, and on the issues of regional development and convergence in particular. Despite growing interest regarding regional economies, theoretical approaches to modelling regional growth appeared unsatisfactory and have motivated the approach adopted in the study. The work could not have been done without encouragement and support from first advisor Manfred M. Fischer, as well as constructive suggestions and valuable advice from second advisor Ingrid Kubin, to both of whom the author owes a debt of gratitude. Technical assistance throughout the time of developing was provided by Monika Bartkowska, Judith Kast-Aigner, Aleksandra Riedl, Petra Staufer-Steinnocher and Anita Wolfartsberger, whose help is greatly appreciated by the author. Special thanks for valuable comments go out to Roger Bivand, Gerlinde Fellner, Wolfgang Fellner, Alexia Fürnkranz-Prskawetz, James LeSage, Thomas Scherngell and Achim Zeileis. Furthermore, the author would like to thank Meaghan Burke for proofreading, and Barbara Feß for editing.

v

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part I

1

Theory of Economic Growth

2 Neoclassical Growth Theory and Standard Models . . . 2.1 From Classical to Neoclassical Growth Theory . . . 2.2 The Basic Outline of Neoclassical Growth Models . 2.3 The Solution with Technological Progress and a Cobb-Douglas Production Function . . . . . . 2.4 Human Capital as an Additional Factor of Production

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9 9 12

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15 18

3 Growth Models with Spatial Externalities . . . . . . . . . . . . . . 3.1 A Neoclassical Growth Model with Identical Steady States . . . 3.2 A Neoclassical Growth Model with Different Steady States . . .

23 23 25

4 Convergence: Theory and Evidence . . . . . . . . . . . . 4.1 Convergence and Disparities of National Economies 4.2 The Formal Derivation of a Convergence Equation . 4.3 Empirical Tests of Convergence, Conclusions and Unsolved Questions . . . . . . . . . . . . . . .

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29 30 34

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38

Part II A Model of Regional Growth 5 Remarks on Regional Growth . . . . . . . . . . . . . . . . . . . 5.1 Agglomeration Effects, Increasing Returns and Polarisation 5.2 Reconsidering Saving and Gross Investment . . . . . . . . . 5.3 Foreign Capital, Its Mobility and the Role of Human Capital

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45 45 48 50

6 Structure of the Model . . . . . . . . 6.1 Definitions and Assumptions . . 6.2 The Production Function . . . . 6.3 Changes of Inputs to Production

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53 53 55 59

7 Evolution of Factors and Output . . . . . . . . . . . . . . . . . . . 7.1 Steady States . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

7.2 7.3

Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . Formal Derivations . . . . . . . . . . . . . . . . . . . . . . . .

8 Implications for Output Growth 8.1 Taylor Approximation . . . 8.2 Solution for Output Growth 8.3 Summary and Outlook . . .

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79 79 82 85

9 Regions, Variables and Data . . . . . . . . . . . . . . . . . . . . . 9.1 Observation Units, Observation Period and Distance Measures 9.2 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 An Illustration of Recent Developments . . . . . . . . . . . .

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91 91 94 98

Part III Empirics

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Spatial Econometric Specification and Estimation 10.1 Spatial Econometric Models . . . . . . . . . 10.2 The Model Specification . . . . . . . . . . . 10.3 Maximum Likelihood Estimation . . . . . .

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105 105 107 109

11

Testing the Theoretical Model . . . . . . . . . . . . . . . . . . . . . 11.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Interpretation and Concluding Remarks . . . . . . . . . . . . .

111 111 114

12

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Appendix 1: Regions of the Observation Area . . . . . . . . . . . . .

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Appendix 2: Supplementary Econometric Results . . . . . . . . . . .

125

Appendix 3: Intra-state Developments 1995 to 2004 . . . . . . . . .

127

Appendix 4: List of Symbols . . . . . . . . . . . . . . . . . . . . . .

131

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

135

List of Figures

2.1 2.2 2.3 2.4 4.1 4.2

Capital stock per labour unit in equilibrium . . . . . . . Steady state per capita growth of output . . . . . . . . . An increase in the saving rate . . . . . . . . . . . . . . The dynamic stability of the Mankiw-Romer-Weil model GDP per capita at PPS from 1820 to 2000 of six areas . . Share of world output of seven economic blocs and associated countries in 1970 and 2005 . . . . . . . . . . 4.3 Relative output per capita of seven economic areas and blocs, 1970–2005 . . . . . . . . . . . . . . . . . . . . . 7.1 Long run development of output . . . . . . . . . . . . . 7.2 Medium run development of regions with low initial output levels . . . . . . . . . . . . . . . . . . . . . . . 7.3 Medium run development of regions with high initial output levels . . . . . . . . . . . . . . . . . . . . . . . 7.4 Medium run development compared to closed economies 7.5 Medium run development in some special cases . . . . . 9.1 Relative GRP per capita, 1995 . . . . . . . . . . . . . . 9.2 Relative GRP per capita, 2004 . . . . . . . . . . . . . . 9.3 Average annual GRP growth per capita, 1995–2004 . . . 9.4 GRP-position changes in the ranking of 255 regions in Europe, 1995–2004 . . . . . . . . . . . . . . . . . . . . 11.1 Regional disparities within countries, 2004 . . . . . . . .

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List of Tables

9.1 9.2 9.3 9.4 11.1 11.2

Thresholds for NUTS regions and examples . . . . . . . Summary of data 1995 . . . . . . . . . . . . . . . . . . Summary of data 2004 . . . . . . . . . . . . . . . . . . Output distribution 1995–2004 . . . . . . . . . . . . . . Estimation results with gross regional product per capita Estimation results with gross value added per gainfully active person . . . . . . . . . . . . . . . . . . . . . . .

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93 99 99 103 112

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Chapter 1

Introduction

Some decrease, others persist or even widen: regional disparities continue to constitute one of the major challenges for European economic policy. The accession of twelve countries to the European Union (EU) on May 1, 2004 and January 1, 2007 has led to two major statistical effects, namely a decrease in the gross domestic product (GDP) per capita of the European Union, and an increase in the gap between the most and the least developed regions. The relative importance of structural policy is reflected in the financial allocation in the current financial framework for 2007–2013: of a total budget of 864 billion euros (price-level of 2004), 308 billion euros are set aside for cohesion policy (European Council 2006, Article 19). Of these, 251 billion euros (European Council 2006, Article 19) are provided to “promote growth-enhancing conditions and factors leading to real Convergence for the least-developed Member States and regions” (European Commission 2006, p. 2, upper cases in the original), of which 153 billion euros are destined for the twelve new member states and their regions (European Commission 2006, p. 3). The origins of assistance for less developed regions lie in the Treaty of Rome (Treaty Establishing the European Economic Community, EEC Treaty), based on which the European Investment Bank was established in order to “facilitate the financing of (. . .) projects for developing less developed regions” (European Economic Community 1957, Article 130). Back in those days, however, the European Community (EC) of then six member states was relatively evenly developed, with the exception of the less productive Southern Italian regions. The situation changed when the United Kingdom and the Republic of Ireland joined the EC in 1973, which led to a significant increase in regional disparities. Nevertheless, despite the Treaty of Rome’s declaration, it was not until 1975 that the European Regional Development Fund was created, since an effective policy on regional structures was seen as “an essential prerequisite to the realization of economic and monetary union” (European Council 1975). The fund’s assistance was set to be allocated according to the relative severity of regional imbalances. Regional disparities in the European Community increased again with the accession of Greece in 1981 and that of Portugal and Spain in 1986. These countries, together with the Republic of Ireland forming the so-called cohesion countries, henceforth benefited from substantial financial support.

S. Sardadvar, Economic Growth in the Regions of Europe, Contributions to Economics, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-7908-2637-1_1, 

1

2

1

Introduction

As a consequence of these steps toward horizontal integration, the EC as of 1986 consisted of 12 member states, twice as many as when it was founded. A major step toward vertical integration followed the same year with the Single European Act, which defines the internal market (nowadays referred to as European Single Market) as “an area without internal frontiers in which the free movement of goods, persons, services and capital is ensured in accordance with the provisions of this Treaty” (European Communities 1987, Article 13, upper case in the original), which had to be established by December 31, 1992. The same treaty declares convergence as an objective of the Community, by adding the title “Economic and Social Cohesion” to the EEC Treaty (European Communities 1987, Article 23): In order to promote its overall harmonious development, the Community shall develop and pursue its actions leading to the strengthening of its economic and social cohesion. In particular, the Community shall aim at reducing disparities between the levels of development of the various regions and the backwardness of the least favoured regions or islands, including rural areas.

Although not made explicit in the Single European Act, it is commonly understood that the simultaneous strengthening of cohesion policy was intended to counterbalance negative effects that the completion of the internal market could have on some countries and regions. Such risks of “aggravated imbalances in the course of market liberalisation” were seen to be best alleviated by “adequate accompanying measures to speed up adjustments in structurally weak regions and countries” (European Commission 2008, p. 9). The respective added article of the EEC treaty reads (European Communities 1987, Article 23): The European Regional Development Fund is intended to help redress the principal regional imbalances in the Community through participating in the development and structural adjustment of regions whose development is lagging behind and in the conversion of declining industrial regions.

In 1989, the European Regional Development Fund was brought together with the European Social Fund and the Guidance Section of the European Agricultural Guidance and Guarantee Fund to form the Structural Funds. The period 1989–1993 saw almost a doubling of cohesion policy’s relative share of the EU budget, from 16% in 1988 to over 30% in 1993, which means that the financial allocations to the Structural Funds reached 20.5 billion ECU in 1993 (price-level of 1993) (European Commission 2008). Since then, regional policy of the European Community has still grown in volume. Despite its sizeable funding, the main principle of the policies of the European Union still aims at allowing economic mechanisms to function as reflected in the Treaty on European Union (Maastricht Treaty), which allows for free factor movement. Considering primarily international disparities, the importance of understanding economic growth was poignantly formulated by Robert E. Lucas Jr. (1988, p. 5), when he commented that “the consequences for human welfare (. . .) are simply staggering: once one starts to think about them, it is hard to think about anything else”. Although economic growth has been investigated by economists for generations, there is still disagreement about its underlying causes (Armstrong and Taylor

1

Introduction

3

2000, p. 65). Neoclassical growth theory has been proven to be able to explain a number of phenomena of national economies’ growth, but problems arise when one tries to translate these to regional economies. The usual assumption of closed economies and abstraction from interaction have so far placed serious limits on the explanation power with respect to regional growth. At least since the introduction of cohesion policy, convergence between the European Union’s economic entities has been observed on the whole. Yet within the EU’s member states, disparities tend to persist or even increase. In recent years, new economic geography has provided formal analyses of how regional disparities may emerge as an equilibrium outcome, for instance as a consequence of lowered trade barriers. Models in the spirit of Paul Krugman’s (1991a and 1991b) core-periphery model help expand the understanding of mechanisms of interrelated regions’ development, but are less successful regarding prospects of growth and long term development. Fingleton (2003, p. 25) notes that “in essence the theory is primarily an exercise in formal, deductive modelling, in which mathematical tricks are employed to allow a neat, general equilibrium solution rather than because they are necessarily realistic.” Probably due to the aforementioned limitations, conventional neoclassical theory so far provides only limited explanatory power for issues of regional growth; in particular, it fails to explain how empirically both divergence and convergence processes are observed at the same time, depending on the choice of spatial aggregation or observation area. This study acknowledges the role of space that is emphasised in new economic geography by developing a strictly neoclassical model of regional growth, where the development of one region is dependent on the development of others, and vice versa. Regions are defined for purposes of this study as economically open, politically interdependent and spatially connected economies that jointly form a superordinate economic system. The research objective of the study is thus to develop a neoclassical model of regional growth with spatial dependence, to transform the theoretical model into a spatial econometric specification and to test the model empirically for European regions. Mankiw, Romer and Weil (1992) have improved the explanation of variation of output of various economies by enhancing Robert Solow’s (1956, 1957) contribution to growth theory with the inclusion of human capital. This study builds on the Mankiw-Romer-Weil model by enhancing it for the possibility of factor movement, in particular of choice of location for gross fixed capital formations. These investment decisions are assumed to be based on expected rates of return, but the investor’s information is limited to neighbouring regions, and hence his operating range is spatially bounded. Expected rates of return depend on current marginal productivity of physical capital, which in turn is a function of the current endowment with production factors. Output growth of one region is dependent on the evolution of its endowment with production factors, which is in turn influenced by its own as well as its neighbours’ factor endowments. These interdependencies result in a system of N connected economies, where the development of the superordinate economic system is of interest in its own right. In equilibrium, all regions are at their identical steady states of factors and output, but during transition periods both convergence and divergence processes may occur.

4

1

Introduction

The model’s growth equation is derived via a Taylor approximation, which results in a system of N differential equations. These equations are solved for output growth between two points in time and directly transformed into a spatial econometric model specification which is tested using 255 European regions for the observation period 1995–2004. The results are in line with the predictions of the model, where human capital constitutes a critical determinant of regional growth: disposability of human capital has a positive impact on productivity and hence attracts investments. It is, however, a mixed blessing, as human capital is to the benefit of a region’s growth if it is found within that region, but is disadvantageous to a region’s prospects on growth if found in neighbouring regions. This negative influence of human capital on neighbouring regions is finally found to serve as a potential answer for the simultaneous observation of regional convergence on a pan-European scale, and divergence between specific groups of regions. The book is structured in three parts: after the Introduction (Chap. 1), Part I explores neoclassical growth theory in detail, by examining its standard models (Chap. 2), growth models that have acknowledged the role of space (Chap. 3), and the related convergence debate (Chap. 4). Part II sets the focus on characteristics of regional growth as opposed to national economies (Chap. 5), the resulting assumptions and structure of the model (Chap. 6), its implications for growth (Chap. 7), and the transformation to an econometrically testable specification (Chap. 8). Finally, Part III discusses availability of data in relation to the model’s assumptions (Chap. 9), explores the spatial econometric model specification and estimation (Chap. 10), and interprets and concludes based on the results (Chap. 11). The study’s three main parts are followed by the Summary (Chap. 12), the References, and four Appendices. In more detail, the study’s three parts are structured as follows: • Part I starts with a brief history of growth theory (Sect. 2.1) before a detailed discussion of the models of neoclassical growth theory on which this study’s model is based on. These include the Solow model in its basic outline (Sect. 2.2), its solution with the particular case of a Cobb-Douglas production function (Sect. 2.3), and its more recent augmentation with human capital (2.4), which has become known as the Mankiw-Romer-Weil model. The relative location in space has so far been largely ignored in neoclassical growth models, with the exception of two recent approaches, one of which considers regions with identical steady states (Sect. 3.1), and one of which features steady states dependent on the relative location in space (Sect. 3.2). The question of whether various economies tend to converge over time arises intuitively from thoughts on economic growth and will be discussed for national economies (Sect. 4.1). This is followed by an exploration of the formal derivation of a convergence equation (Sect. 4.2) and some conclusions, as well as still unanswered questions (Sect. 4.3). • Before the model is laid out in detail, the first section of Part II is devoted to non-neoclassical regional economics, in particular the issues of agglomeration effects, increasing returns and polarisation, and includes a note on new economic geography (Sect. 5.1). The discussion results in a reconsideration of standard assumptions on saving and investment (Sect. 5.2) and the mobility

1

Introduction

5

of investments in physical capital (Sect. 5.3). The model’s definitions and assumptions are laid out in detail (Sect. 6.1), followed by a specification with a Cobb-Douglas production function (Sect. 6.2). Changes in output levels are determined by the evolution of production factors, which serve as the model’s foundation of dynamics (Sect. 6.3). After the derivation of the model’s steady states (Sect. 7.1), its development over time is studied in detail, both by simulation (Sect. 7.2) and formally (Sect. 7.3). A Taylor approximation of the model gives a differential equation of economic growth (Sect. 8.1), whose solution provides a testable spatial econometric specification (Sect. 8.2). Finally, conclusions regarding the model’s implications and an outlook on possible future research are discussed (Sect. 8.3). • The first question to be tackled in view of the empirical study in Part III concerns the model’s concepts and assumptions and how they are reflected in connection with Europe’s ongoing vertical as well as horizontal economic integration (Sect. 9.1), completed by a discussion on the availability, quality, and appropriateness of data (Sect. 9.2). Based on these considerations, a summary of the data in connection with an exploratory analysis is given (Sect. 9.3). Spatial econometrics and the related issue of spatial weight matrices are introduced (Sect. 10.1), to be followed by a detailed description of the spatial econometric specification derived in Part II (Sect. 10.2), which is estimated by maximum likelihood (Sect. 10.3). Results from empirical tests of the model are presented for regions of the European Union and European Free Trade Association on the NUTS 2 and equivalent level (Sect. 11.1), and finally discussed and interpreted (Sect. 11.2).

Part I

Theory of Economic Growth

Robert Solow’s articles from 1956 and 1957, in which he outlined the basics of what is now referred to as neoclassical growth theory, are widely accepted as marking the starting point of modern growth theory. In the 50 years that followed, this theory has been continually further developed, while remaining controversial. Like any model, a neoclassical growth model is a simplified conception of the world. Interestingly, this particular issue has been pointed out and discussed already in Solow’s own contributions but has nevertheless been the primary object of criticism. “Neoclassical growth theory in some sense represents an ‘ideal theory’” (Krelle 1988, p. 86, quotation marks in the original)1 : by building on a small number of reasonable assumptions, it manages to explain a large number of observable phenomena and simultaneously rules out inconsistent developments. It seems inevitable that its attempt at simplifying highly complex social relationships simultaneously constitutes its major strength and weakness. So far, economic growth is understood in a broader sense and as such in most cases implicitly or explicitly refers to national economies. The issue of appropriateness of neoclassical theory is taken up again in Chap. 5 in connection with questions of regional development, whereas in this part of the study the framework of neoclassical growth theory will be explored in detail, after a brief introduction to the history of growth theory. Therefore Chap. 2 sets the focus on the development and application of a neoclassical production function, in particular by focussing on its conclusions for the long run. Chapter 3 examines the few extensions of the original Solow model that consider relative location in space as a determinant of growth. The derivation of an econometrically testable convergence equation and some conclusions drawn from empirical tests can be found in Chap. 4.

1 “In

gewisser Weise ist die neoklassische Wachstumstheorie eine ,ideale Theorie’.”

Chapter 2

Neoclassical Growth Theory and Standard Models

Thoughts and theories on economic growth can be traced back to the classical economists of the eighteenth and nineteenth century, whose works are briefly reviewed alongside the transition to neoclassical growth theory in Sect. 2.1. The basic outline of neoclassical growth models as first developed by Solow (1956) and Swan (1956) is presented in Sect. 2.2. The familiar but nonetheless special case of a Cobb-Douglas production function is examined in Sect. 2.3 in connection with the derivation of steady state levels of factors of production and output. Finally, Sect. 2.4 examines the inclusion of human capital as an additional factor of production and provides a note on endogenous growth theory.

2.1 From Classical to Neoclassical Growth Theory Adam Smith’s “An Inquiry into the Nature and Causes of the Wealth of Nations” includes some considerations on what is now referred to as economic growth. Although Smith (1776) does not develop a long run growth theory as such, conclusions on growth may be deduced, as he refers to the importance and effects of increasing labour productivity as well as saving. The stationary state is defined as a condition where capital accumulation and population size have reached their ceilings, and as a consequence the economy may not progress any more (Smith 1776, p. 82). In contrast to this rather pessimistic view, Smith also refers to technical progress, which raises aggregate output (Smith 1776, p. 75), but considers division of labour as an even more important potential for improving labour productivity (Smith 1776, p. 207).1 However, division of labour may not be improved perpetually: whether long run growth of the aggregate economy is possible in Smith’s model is open to interpretation. The crucial point in Smith’s theorising is population growth – either it would grow to its maximum possible level, or it could be controlled. It follows implicitly that if the latter case were to be achieved, an increase of output per capita in the long run would be possible.

1 Smith

famously exemplifies this with the production of pins.

S. Sardadvar, Economic Growth in the Regions of Europe, Contributions to Economics, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-7908-2637-1_2, 

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2 Neoclassical Growth Theory and Standard Models

When David Ricardo published his now best-known work “On the Principles of Political Economy and Taxation” in 1817, industrialisation was progressing so quickly that he already felt the need to rewrite those parts dealing with technology for the third edition in 1821. Like other economists of the industrial era, when the machine began to take precedence over the worker, it was impossible for him to foresee long run consequences. Ricardo was at first confident that productivityaugmentation due to machinery was to the benefit of all social classes, but revised his conclusions four years later: a scenario where profits rise while wages decline was possible (König 1997). The conclusion of Ricardo’s system concerning general economic growth is that technological progress may postpone, but not prevent the incidence of a stationary state (Pasinetti 1960). Ricardo assumes a two-sector economy that exhibits constant returns to scale in the manufacturing sector, but diminishing returns in the agricultural sector: as more land is cultivated, marginal quality of land worsens (Ricardo 1821). He considers capital owners to be the “productive class” of society, as they devote their profits to capital accumulation. This process of accumulation cannot go on infinitely, however, due to population growth. As a result, less fertile land will be cultivated, thus leading to aforementioned diminishing returns in agriculture (Ricardo 1821). Via feedback effects on employment growth and productivity, the profit rate will decrease until it has fallen to (almost) zero. This will prevent the capitalist class from accumulating, and the economy will have reached its stationary state, where all surplus is taken by the landlords. A couple of decades later, economic development and growth are key themes in the studies of Karl Marx. He considers production to be interwoven with reproduction, distinguishes saving from consumption and accounts for depreciation and technological progress to develop a model of physical capital accumulation (Marx 1872). In this model, one part of the surplus value created in one period is consumed, while the other part becomes next period’s capital; thus concluding that after each turn, “capital has produced capital” (Marx 1872, p. 538).2 Growth manifests itself in the ever-growing share of physical capital relative to labour and increasing labour productivity as a consequence of technological progress. This process is self-reinforcing: the working class, by producing physical capital, induces its own relative redundancy. Marx (1872) concludes that labour-demand growth is too low to compensate for decreases in employment following technological progress. In the long run, unemployment will rise until capitalism is abolished. A further aspect of Marx’ studies is the interdependence of technical progress and the falling tendency of the rate of profit, a topic already featured prominently in Ricardo’s work as discussed above. Sustained (technological) progress requires an increasing ratio of fixed capital to output, which causes crises in the capitalist system and which could lead to a stationary state, although Marx (1894) considers several scenarios and countervailing forces as well. Apart from his visions of the demise of capitalism, Marx (1885) also developed a theory of medium run development

2 “Das

Kapital hat Kapital produziert.”

2.1

From Classical to Neoclassical Growth Theory

11

whereby the (capitalistic) economy grows at a constant rate (Krelle and Gabisch 1972). In this model of medium run growth, there exist a production goods sector and a consumption goods sector. Krelle (1988) later formalises this model for the two sectors and shows that both grow at identical rates, depending positively on the saving rate and the rate of surplus value. This bears the interesting result that wages may increase in absolute terms even in the case of a rising rate of surplus value (Krelle 1988, p. 55).3 Although Marx’ model of medium run growth leaves some questions unanswered, it may be seen as an important forerunner of modern growth theory: it addresses to the same questions asked by economists today, and it anticipates the characteristic feature of neoclassical growth theory, namely the steady state. Until the turn of the twentieth century and long after, economists remained remarkably silent on the issue of growth, with just a few exceptions. Joseph A. Schumpeter greatly stresses the role of innovations and therefore technical and technological progress, but less so the role of capital accumulation. He consciously distinguishes between economic growth and development, the latter being caused by endogenous factors that lead to groundbreaking innovation, thereby changing technique and productive organisation (Schumpeter 1926). The entrepreneur is greatly acknowledged for his pioneering in the field of new technologies, and therefore embodying the driving force of economic development (Schumpeter 1926). It is a crucial characteristic of Schumpeter’s work that development does not happen gradually, but rather cyclical in the wake of innovations. He also used (and popularised) the term creative destruction to describe the process of transformation that accompanies innovations (e.g. Schumpeter 1942, p. 138). In the wake of new conceptions and theories of economic growth, the concept of creative destruction has been taken up and strongly emphasised again in the 1990s by endogenous growth theorists (e.g. Aghion and Howitt 1992). One of the first economists to focus on the rate of growth was Roy F. Harrod (1939). Building on the works of John Maynard Keynes, he develops a theory that sets the base for conditions required for long run equilibrium growth. Keynes himself did not develop a growth theory as such, as he was primarily interested in short term developments, especially in the interplay of aggregate income and investments. Shortly afterwards, Evsey D. Domar (1946) stresses the importance of a dynamic view in order to gain insights into long run growth. Although Harrod and Domar developed their theories independently from each other, they display a number of similarities, which is why the resulting growth model is usually referred to as the Harrod-Domar model. Due to “implausible assumptions” and “undesirable outcomes” (Barro and Sala-i-Martin 1995, p. 49) as a result of building on Keynes’ 3 It is also interesting to note that political arguments in defence of declining wage shares (i.e. rising rates of surplus value) nowadays often go like this: “The wage share may decline, but workers are still better off because wages and salaries continue to grow at real terms” (see, for instance, Der Standard, July 21, 2007, p. 32: “. . .die Verdienste der unselbständig Beschäftigten [schrumpfen] – statistisch gesehen – nicht. Ihre Einkommen wachsen nur langsamer als Kapitalund Unternehmensgewinne.”).

12

2 Neoclassical Growth Theory and Standard Models

short term analysis to study long run problems, the Harrod-Domar model is now of primarily historical interest, but nevertheless remains appreciated as an intermediate step between classical and neoclassical theory.

2.2 The Basic Outline of Neoclassical Growth Models In the introduction to his paper that forms the foundation of neoclassical growth theory, Robert Solow (1956) criticises the Harrod-Domar model by identifying its assumption of fixed proportions of labour and capital as the cause of an equilibrium growth that in fact balances on a knife’s edge (Solow 1956, p. 65). As a tendency toward instability is particularly dissatisfying for any approach dealing with long run problems, Solow (1956) and Swan (1956) turn to neoclassical production functions with varying shares of labour and capital inputs. These two approaches provide the first neoclassical model of long run economic growth and mark the starting point for most studies on economic growth up to the present day, including the model presented in Part II. The Solow model focuses on a closed economy where output Q is produced by the factors labour, L, and capital, K. The production function takes the form Qt = f (Kt , Lt )

(2.1)

where t denotes time. The critical assumption of the production function is that it shows constant returns to scale; Solow (1956) departs here from the classical assumption of scarce land or any non-augmentable resources. Romer (1996) interprets the assumption of constant returns to imply that the economy under consideration is big enough that the gains from specialisation have been exhausted.4 Technically speaking, the neoclassical production function is homogenous of degree one and implies that both factors must be available, or else output would equal zero (i.e. the economy would not exist). The function allows for an unlimited substitutability between capital and labour, which means that to produce any given output, any amount of capital can be efficiently used with the appropriate amount of labour. As a consequence of this assumption, the capital-output ratio can take on any nonnegative value. Furthermore, Eq. (2.1) exhibits positive and diminishing marginal products (quasi-concave) with f (0) = 0, f  (·) > 0, f  (·) < 0. The factors of production grow at constant rates: L˙ t = nLt

(2.2)

K˙ t = sK f (Kt , Lt )

(2.3)

where a dot over a variable denotes a derivative with respect to time, and the labour force growth rate n as well as the saving rate sK are exogenous parameters. 4 The issue of what is actually “big enough” becomes relevant again when dealing with regions, and will be explored in Sect. 6.1.

2.2

The Basic Outline of Neoclassical Growth Models

13

Equation (2.2) implies that Lt = L0 ent , and can be looked at as a supply curve of labour, with the labour force growing at an exponential and completely inelastic rate (Solow 1957, p. 67). Since Lt denotes both labour supply in Eq. (2.2) as well as total employment in Eq. (2.3), the model implies that full employment is perpetually maintained. Under the conditions of full employment and inelastic supply of both labour and capital at any point in time, both factors earn their marginal product, where the real wage v and the real interest rate r are given by vt =

∂Qt ∂Lt

(2.4)

rt =

∂Qt ∂Kt

(2.5)

and where the price level is conveniently assumed to be constant. These two assumptions, of market-clearing conditions at any point and of constant price levels, are seen by some as weak points of neoclassical theory and should be briefly commented upon at this point of the study, as both are seldom found in the real world: as for market-clearing conditions, Krelle (1988, p. 89) notes that a constant degree of monopolisation of markets shifts both distribution of income and the propensity to save, thus changing numerical results, but has no effect on the main results. As for price levels, the underlying assumption may be understood as being not important enough to matter for long run growth or, alternatively, its fluctuations are being brought under control. The latter argument also applies to fiscal and monetary policies. As long as closed economies are considered, output equals income by definition, therefore, it follows from Eqs. (2.1), (2.4) and (2.5) that Qt = vt Lt + rt Kt

(2.6)

which is the sum of wages and profits. In the terminology of macroeconomic accounting, Eqs. (2.1) and (2.6) thus equal gross value added (GVA) at t. Since indirect taxes are assumed to equal zero, gross value added is equal to gross domestic product (GDP). Furthermore, if there is no physical capital migration, gross domestic product is equal to gross national income (GNI).5 The capital-labour ratio kt is defined as kt =

Kt Lt

(2.7)

The derivative of Eq. (2.7) (with no depreciation) with respect to time is K˙ t Lt − Kt L˙ t K˙ t k˙ t = = − nkt Lt Lt2 5 Gross

national income is conceptually identical to gross national product.

(2.8)

14

2 Neoclassical Growth Theory and Standard Models

Since the production function is homogenous of degree one, output per labour unit qt can be expressed as qt = f (kt )

(2.9)

where f (kt ) ≥ 0 for kt ≥ 0. Like Eq. (2.1), (2.9) also exhibits positive and diminishing marginal products with f (0) = 0, f  (·) > 0, f  (·) < 0. From Eqs. (2.3), (2.8) and (2.9), it follows that physical capital equipment per labour unit grows at sK f (Kt , Lt ) − nkt = sK f (kt ) − nkt k˙ t = Lt

(2.10)

investment

The differential equation (2.10) is the fundamental equation of the Solow model: it states that the rate of change of the capital stock per labour unit is the difference between two terms. The first term, sK f (kt ), displays the increment of capital and represents actual investment per labour unit; the second term, nkt , accounts for the increase of labour and as such represents the break-even investment necessary to keep k at its existing level. When k˙ t = 0, the capital-labour ratio is constant, and consequently (with no depreciation) the aggregate capital stock K must be expanding at the same rate as the labour force n. If this is the case, the system is in equilibrium and is defined as being in a state of balanced growth (Solow 1956, p. 70). The stability of the balanced growth path is shown in Fig. 2.1: if the current capital-labour ratio kt is smaller than the equilibrium capital-labour ratio k∗ , then actual investment per labour unit is greater than break-even investment and the capital-labour ratio rises with k˙ t > 0. If kt > k∗ , then k˙ t < 0, and hence k will decrease toward k∗ . Whatever the starting point of the system,6 it converges to the

nk t sf (kt )

k*

kt

Fig. 2.1 Capital stock per labour unit in equilibrium

6 Except for the possibility that there is no physical capital at all, which corresponds to the second intersection of the two lines at the point of origin in Fig. 2.1. As this mathematically possible equilibrium represents a non-existing economy, it is of no further interest.

2.3

The Solution with Technological Progress and a Cobb-Douglas Production Function

15

equilibrium value of kt = k∗ , with k˙ t = 0. Once this point has been reached, represented by the intersection of the two lines in Fig. 2.1 at k = k∗ , the system stays there – thus the equilibrium is stable with k˙ t = 0.7 If for some reason the economy moves temporarily away from equilibrium, it will be forced to return to the balanced growth path. In general, a situation in which the various quantities grow at constant rates is defined as a steady state.8

2.3 The Solution with Technological Progress and a Cobb-Douglas Production Function Both Solow (1956) and Swan (1956) were primarily interested in economic growth as a consequence of capital accumulation, and studied the case of technological progress only briefly in their original papers. In a successive paper, Solow (1957) estimates technological progress for a 1909–1949 time series as a residual of capital and labour for explaining output growth of the US-American economy.9 Here the main conclusion is that technological progress appears to be neutral when it comes to scale effects: shifts in the production function therefore have no effect on marginal rates of substitution at given capital-labour ratios (Solow 1957, p. 316). In general, a production function that takes the form as given in Eq. (2.1), where technological progress enters as labour-augmenting, is defined as Harrod-neutral: Qt = f (Kt , At Lt )

(2.11)

where At represents the level of technology at t. If technological progress is capitalaugmenting of the form Qt = f (At Kt , Lt )

(2.12)

it is defined as Solow-neutral. Finally, if technological progress simply multiplies the production function by an increasing scale factor as Qt = At f (Kt , Lt )

(2.13)

it is defined as Hicks-neutral.10 Barro and Sala-i-Martin (1995) show that technological progress can always be expressed as labour-augmenting. In this context 7A

formal proof for the stability of the equilibrium can be found in Krelle (1988).

8 The term steady state reminds one of the stationary state as discussed in Sect. 2.1. The difference

is that at steady state, an economy continues to grow at constant rate along the steady state growth path. Note that growth at constant rates also includes the possibilities of zero or even negative growth. 9 This residual has become known as the Solow residual. 10 Definitions and discussions of the various ways of implementing technological progress in a neoclassical production function can be found in Beckmann and Künzi (1984).

16

2 Neoclassical Growth Theory and Standard Models

they prove mathematically that only labour-augmenting technological change is consistent with the existence of a steady state. A Cobb-Douglas production function with Harrod-neutral technological progress therefore takes the form Qt = Kta (At Lt )1−a , 0 < a < 1

(2.14)

where the exponents a and 1 − a denote the output elasticities of capital and labour, respectively. Marginal product of each factor is very large when its amount is sufficiently small, and becomes very small when the amount becomes large. This satisfies the Inada-conditions (Barro and Sala-i-Martin 1995, following Inada 1963) of the production function, in particular that the limit of the derivative towards zero is positive infinity, and that the limit of the derivative towards positive infinity is zero. These conditions are fulfilled by a Cobb-Douglas production function, whose intensive form is found by dividing Eq. (2.14) by technology-augmented labour A t Lt :  qˆ t =

Kt At Lt

a

= kˆ ta

(2.15)

where output per unit of effective labour qˆ t is a function of capital per unit of effective labour kˆ t . It follows from Eq. (2.14) that output per labour unit at t equals  a Kt = A1−a kta (2.16) qt = t Lt which corresponds to GVA per worker at t. If capital stock depreciation is considered in the equation that determines the evolution of the capital stock, Eq. (2.3) takes the form K˙ t = sK Qt − dKt

(2.17)

where d is the rate of depreciation. Therefore, the aggregate capital stock grows as long as the left-hand term on the right side of Eq. (2.17) is greater than the right-hand term. In steady state, the amount of capital per unit of effective labour is constant; therefore break-even investment has to take technological progress into account: ˙ kˆ t = sK kˆ ta − (n + g + d)kˆ t

(2.18)

where g is the rate of technological progress: A˙ t = gAt

(2.19)

Thus it can be seen that if steady state growth is defined as an equilibrium where the growth rate of output per unit of effective labour qˆ is nil, while output per

2.3

The Solution with Technological Progress and a Cobb-Douglas Production Function

17

unit of labour (i.e. per worker = per capita) q grows with technological progress g, then aggregate output Q grows with population and technological progress. In steady state, the ratio of capital to effective labour is constant, therefore growth of the physical capital stock per effective labour in steady state also equals zero. In other words, Eq. (2.15) expresses output per labour unit corrected for technological progress, which serves two illustrative purposes: firstly, growth per unit of effective labour in the steady state is a straight line, as can be seen from Fig. 2.2; secondly, if the question of interest is growth per capita that is not due to technological progress, correcting for technological progress makes comparisons easier. The steady state level of kˆ is calculated from the right-hand side of Eq. (2.18) by setting both terms as equal: 

ˆ∗

k =

sK n+g+d



1 1−a

(2.20)

From Eqs. (2.15) and (2.20) it follows that the steady state output per unit of effective labour equals 

∗

qˆ =

sK n+g+d



a 1−a

(2.21)

With A and L growing at constant rates, the model at steady state can be solved at any t for output per labour unit q∗t

∗



= At qˆ = A0 e

gt

sK n+g+d



a 1−a

(2.22)

and aggregate output in steady state at any t equals ∗



= At Lt qˆ = A0 L0 e

(g+n)t

output

Q∗t

sK n+g+d



a 1−a

(2.23)

ln q

ln qˆ t

Fig. 2.2 Steady state per capita growth of output

output

18

2 Neoclassical Growth Theory and Standard Models ln q

ln qˆ

t

Fig. 2.3 An increase in the saving rate

From Eqs. (2.21), (2.22) and (2.23) it follows that an increase in the saving rate sK raises output and therefore income in the long run: as can be seen from Fig. 2.3, output growth will be higher until the economy has reached its new steady state growth path. In other words, a change in the saving rate has a level effect in the long run: if the saving rate increases, growth will temporarily be higher. Long run growth, however, remains unaffected: growth rates of output per labour unit (= per capita), of the capital stock per labour unit and of the wage rate will equal the rate of technological progress as soon as the economy returns to its balanced growth path.

2.4 Human Capital as an Additional Factor of Production One of the most controversial conclusions drawn from the Solow model and its successors is the implication of immense incentives to invest in economies where the marginal product of capital is highest, that is if rates of return across economies differ according to the model. Generally speaking, according to the Solow model as presented above, one would expect capital flows from wealthy countries (those with currently high stocks of physical capital, which for this reason should exhibit a relatively low marginal productivity of physical capital) to less wealthy countries (where conditions are the other way round). In addition, within the model these capital flows are supposed to happen incidentally, that is unhindered by time or space. Capital of course does flow from wealthy countries to less wealthy countries; it is a process which is commonly understood to be one of the crucial characteristics of today’s world economy. In fact, these flows have increased significantly since 1990 (The World Bank 2004a), although large discrepancies exist (The World Bank 2004b): the vast majority of investments takes place within the capital’s origin country, and the vast majority of foreign direct investments (FDI) takes place within the developed world (The World Bank 2008). This phenomenon is often referred to as the Lucas paradox, as discussed by Lucas (1990). He identifies differing human

2.4

Human Capital as an Additional Factor of Production

19

capital endowments and capital market imperfections as candidate answers for the question of why standard neoclassical predictions on capital flows fail at least in part when confronted with reality. Mankiw, Romer and Weil (1992) explore this issue by augmenting the original Solow model to include the accumulation of human capital in addition to physical capital. The Mankiw-Romer-Weil model focuses on economies that converge to their steady states as long as the levels of both physical and human capital per worker rise, where human capital is supposed to be embodied in skilled workers. In analogy to Sects. 2.2 and 2.3, the rates of saving, population growth and technological progress are taken as exogenously given. With a Harrod-neutral Cobb-Douglas production function assumed, production at any point in time t is given by Qt = Kta Htb (At Lt )1−a−b , a > 0, b > 0, a + b < 1

(2.24)

where K captures exclusively the stock of physical capital, H is the stock of human capital, b denotes the output elasticity of human capital, and the other variables are defined as in the Solow model. As can be seen from Eq. (2.24), there is a clear distinction between human capital H, and abstract knowledge A.11 Human capital is defined as consisting of the abilities, skills and knowledge of particular workers, and is thus rival and excludable (Romer 1996, p. 126). Furthermore, although human capital is embodied in workers and hence represents in fact a specific kind of labour, it is treated as a second type of capital in analogy to physical capital. This definition of human capital has two major effects within the modelling framework: firstly, introducing human capital implies that the sum of shares of output paid to capital of both kinds is raised. Secondly, devoting more resources to the accumulation of either type of capital increases the amount of output that can be produced in the future.12 Dividing the production function of Eq. (2.24) by technology-augmented labour At Lt yields output per unit of effective labour qˆ t = kˆ ta hˆ bt

(2.25)

where output per unit of effective labour is now a function of physical capital per unit of effective labour and human capital per unit of effective labour. The evolution of the economy is determined by the key equations ˙ kˆ t = sK qˆ t − (n + g + d)kˆ t 11 The

(2.26)

terms “abstract knowledge”, “state of technology” and, if appropriate, “effectiveness of labour” are usually used as synonyms here as well as in the bulk of growth literature. 12 The term human capital (“Humankapital”) has been voted Ghastly Neologism of the Year 2004 (“Unwort des Jahres”) in Germany, which provoked some controversy. It should be stressed that the jury was appalled by non-technical applications of the expression (Spiegel Online, 18th January 2005 (http://www.spiegel.de/kultur/gesellschaft/0,1518,337259,00.html, queried on 30-July-2007)).

20

2 Neoclassical Growth Theory and Standard Models

and ˙ hˆ t = sH qˆ t − (n + g + d)hˆ t

(2.27)

where sH is the fraction of output invested in human capital, so that sH Qt represents the part of current aggregate output devoted to education expenditures. Note that human capital underlies the same assumptions as physical capital, in particular that human capital depreciates at the same rate, d, as physical capital. With the righthand sides of Eqs. (2.26) and (2.27) set to zero, this system of two equations is solved for the steady state values of physical capital:  kˆ ∗ =

b s1−b K sH n+g+d



1 1−a−b

(2.28)

. . . and the steady state of human capital:  hˆ ∗ =

saK s1−a H n+g+d



1 1−a−b

(2.29)

. . . and, as it follows from Eq. (2.25), steady state output equals: qˆ = kˆ ∗a hˆ ∗b = ∗



s K sH (n + g + d)a+b



1 1−a−b

(2.30)

˙ ˙ The economy is on a balanced growth path if kˆ = 0 and hˆ = 0, in which case output per effective labour unit is constant. The long run growth rate of output per worker thus equals the rate of technological progress g; it applies to the growth rates of both production factors and is identical to the original Solow model. Accordingly, changes in either sK or sH will have a shift-effect on output (equivalent to those sketched in Fig. 2.3), but will leave long run growth unchanged. The phase-diagram (for the method see Chiang and Wainwright 2005) in Fig. 2.4 displays the stability of the two differential equations given in (2.26) and (2.27): ˆ the system converges to the intersection point whatever the initial values of kˆ and h, ˙ˆ ˙ˆ of the two curves k = 0 and h = 0.13 Once the intersection point is reached, the economy stays there. Although the system becomes more complex with the inclusion of a third factor, it remains stable: if the economy has moved away from equilibrium, it will return to the intersection point and hence to the balanced growth path.14 13 In

analogy to Sect. 2.2, the possibility of initial values of either type of capital equalling zero is ignored. 14 A formal proof of the stability of the Mankiw-Romer-Weil model can be found in Gandolfo (1997).

2.4

Human Capital as an Additional Factor of Production

hˆ

21 kˆ = 0

h=0

kˆ

Fig. 2.4 The dynamic stability of the Mankiw-Romer-Weil model

While the similarity of the Mankiw-Romer-Weil model’s qualitative conclusions to those of the Solow model follows from the model’s structure, the introduction of human capital has a considerable impact on quantitative analysis. In particular, raising either saving rate, or both of them, will have shift-effects on long-term output levels. Depending on b, output elasticities with respect to sK , sH , g and n will differ significantly from those of the Solow model. The influence of b can be observed most easily by taking the natural logarithm of steady state output per unit of effective ∗ ∗ ∗ labour, qˆ = kˆ a hˆ b : ∗

ln qˆ =

a b a+b ln sK + ln sH − ln(n + g + d) 1−a−b 1−a−b 1−a−b

(2.31)

Note that Eq. (2.31) is identical to the natural logarithm of Eq. (2.21) if b is set to zero. In an empirical test, Mankiw, Romer and Weil (1992) show that their model is able to explain considerable differences of cross-country income levels, and also serves to explain why it may be not that attractive to invest somewhere just because the current level of physical capital is low. In this sense, the Mankiw-Romer-Weil model gives one of Lucas’ (1990) candidate answers in detail.15 Furthermore, this model provides a seamless expansion of the Solow model, with long run growth still viewed as identical to technological progress – an aspect now seen as a characteristic feature of neoclassical models of (long run) growth. The 1980s marked not only a deepening interest in the role and importance of human capital, but also in the origins of technological progress. Models of long 15 Economic history is somewhat reflected in these two most important neoclassical growth models:

the Solow model adjusts the focus on physical capital accumulation, and provides an explanation of worldwide economic growth after the Second World War; the Mankiw-Romer-Weil model adds a focus on human capital accumulation to the former, and provides an explanation of worldwide economic growth since the decline of industrial employment in Western Europe and North America. Seen from a meta-level, these two models perfectly mirror their respective ages of economic development.

22

2 Neoclassical Growth Theory and Standard Models

run growth were developed that explained technological progress within the model (sometimes called “endogenised”) by relating it to other economic forces. From these models endogenous growth theory (new growth theory) has evolved, which tries to model technological progress and growth – that is, it postulates that long run growth rate can in fact be influenced by economic factors. In this spirit, P. M. Romer (1986) develops a model in which the creation of new knowledge by one firm is assumed to have a positive external effect on the production possibilities of other firms. This non-rivalry of knowledge is further developed by Lucas (1988), who assumes that human capital releases spillovers whereby each producer in an economy benefits from the average level of human capital in the economy. Endogenous growth theory has spawned further noteworthy approaches, such as those by P. M. Romer (1990), Grossman and Helpman (1991), and Aghion and Howitt (1992), whose models focus mainly on the interdependence and feedback effects of various producers and sectors of the economy. These models are able to explain interdependence and developments of factors within one economy, but become highly complex when mutual influence of various economies is considered; thus they are problematic if unambiguous results are needed for empirical tests.

Chapter 3

Growth Models with Spatial Externalities

Although it is widely acknowledged that interaction between economies has a decisive impact on economic development, there have so far been only a handful of attempts to implement space into theoretical models of economic growth, two of which are discussed in what follows: Sect. 3.1 presents a model that examines the incidence of knowledge spillovers from neighbouring regions within the framework of the Solow model. In Sect. 3.2, a spatial enhancement of the Solow model for N economies is reviewed, along with a brief discussion of approaches that refer to the Verdoorn Law.

3.1 A Neoclassical Growth Model with Identical Steady States The assumption of knowledge spillovers between economies has only recently been implemented into long run growth models. Vayá et al. (2004) consider an economy in which the current level of technology A depends on the economy’s current average capital stock per worker, k. In their model, labour productivity in economy i at point in time t is a function of reproducible factors per worker, synthesised into one variable. Although a different terminology is used, the function of labour productivity uses the same standard notation as above and perfectly resembles a Cobb-Douglas production function with Hicks-neutral technological change. Understanding that what the authors refer to as “labour productivity” is technically equivalent to output per worker qi,t , we have: a qi,t = Ai,t ki,t

(3.1)

The model contains two key assumptions: firstly, it allows for externalities from physical capital (referred to by the authors as “reproducible factors”) within the economy, thereby following the reasoning of P.M. Romer (1986) and Lucas (1988), whereby the aggregate level of technology Ai,t is a function of the aggregate level of ki,t . Secondly, externalities related to the aggregate level of neighbours’ technology linked to their respective stocks of physical capital can flow across economies, so that: γ

τ kθ,t Ai,t = ki,t

S. Sardadvar, Economic Growth in the Regions of Europe, Contributions to Economics, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-7908-2637-1_3, 

(3.2) 23

24

3 Growth Models with Spatial Externalities

where  is an exogenous component of technology, τ is a measure of the degree of external effects within economy i, γ is a measure of the degree of external effects originating in neighbouring economies, and θ denotes the (unspecified) set of neighbours of i. Substituting Eq. (3.2) into Eq. (3.1) yields: γ

a+τ kθ,t qi,t = ki,t

(3.3)

Together with the growth of physical capital, we have: k˙ i,t = sK,i qi,t − (n + d)ki,t

(n + d) < 1

(3.4)

In the steady state expressions, the exogenously given growth rate of technology, g, found in the previous sections is missing since its evolution is now considered endogenous. Assuming that the neighbours’ steady states of k are the same as those of economy i, by inserting Eq. (3.3) to Eq. (3.4) and using the same technique as with Eq. (2.18), the steady state of output per worker equals q∗i,t

=

1 1−a−τ −γ



sK n+d



a+τ +γ 1−a−τ −γ

(3.5)

Apart from technological and preference parameters as in Eq. (2.21), the steady state in Eq. (3.5) also depends on the size of externalities τ within the economy as well as external effects γ originating in neighbouring economies. Output per worker in steady state therefore depends positively on spillovers received from other economies, and hence integration that deepens economic interaction is expected to have a positive effect on the steady states. The model can be extended to allow for different types of capital, namely physical capital and human capital as in the Mankiw-Romer-Weil model, without changing its main implications (see LópezBazo et al., 2004). Nevertheless, upon deeper examination of Eq. (3.5), two further and rather unsatisfying conclusions are also to be drawn from this model. Firstly, although there is a positive effect from outside, focus is set on one economy while the responsible neighbours remain unspecified, and thus there is no interdependence. While the model allows for different degrees of spillovers received from neighbours (as captured by γ ) and hence a scenario where a cohort of economies converge to each other and simultaneously possibly diverge from others, it derives the same steady state levels for all economies within a cohort. The model hence results ultimately in a scenario where either all economies are part of one accompanying superordinate economy, or where various accompanying systems’ individual economic units do not interact with those of other systems. Secondly, the model leads to the result that all growth processes eventually come to a halt: in contrast to Eq. (2.22), the rate of technological progress g is no longer found in the steady state level per worker in Eq. (3.5).

3.2

A Neoclassical Growth Model with Different Steady States

25

3.2 A Neoclassical Growth Model with Different Steady States In a recent approach by Ertur and Koch (2007), the above model’s shortcomings are corrected for. They argue that a growth model needs to include global interdependence phenomena in order to explain development and growth: they assume that the international diffusion of technology is geographically localised, in the sense that technological externalities created somewhere in the system decline with geographical distance between economies. Without referring to Vayá et al. (2004) but in a similar vein, they consider a Hicks-neutral aggregate production function of the form: a Qi,t = Ai,t Ki,t Li,t1−a

(3.6)

where dividing by Li,t yields the same level of output per worker qi,t as given in Eq. (3.1). Not unlike in Eq. (3.2), the current level of aggregate technology of a particular economy, Ai,t , is partly endogenised by considering some amount of technology that is exogenously determined and identical to all economies, t , plus externalities from technology embodied in domestic as well as foreign stocks of physical capital per worker. The technology function takes the form Ai,t =

1/(1−γ ) τ t ki,t

N 

τ

∞

kj,t

ν=1 γ

ν (wν )

ij

(3.7)

j=i

where N is the total number of economies, ν is an integer, and (wν )ij is the element of the i th row, j th column of a matrix W, multiplied by itself ν times. This N × N matrix W consists of the connectivity terms wij that capture distance between economies i and j. Equation (3.7) is the key equation for the Ertur-Koch model and deserves a detailed interpretation. The reasoning behind this function relates to the assumption that knowledge is embodied in physical capital, which creates externalities, which in turn enhance the economy’s level of technology: the parameter τ measures the degree of these domestically created externalities. Assuming that these externalities are not constrained by the barriers of one economy but rather extend across borders with diminishing intensity, the level of technology in any economy i depends on its own level of physical capital per worker and on the level of physical capital per worker in its neighbourhood. The degree of international technological interdependence generated by the level of spatial externalities is described by the parameter γ , which is assumed to be identical for each economy. The net effect of these spatial externalities on productivity in economy i depends, however, on the relative spatial connectivity between this economy and its neighbours. Technological interdependence between an economy i and all economies belonging to its neighbourhood is represented by the connectivity terms wij ; for j = 1, 2, ..., N, these terms are assumed to be: 0 < wij < 1

non-negative, non-stochastic and finite

26

3 Growth Models with Spatial Externalities

N j=i

wij = 0

if i = j

wij = 1

for i = 1, ..., N

The more a specific economy i is connected to its neighbours, the higher wij is, and the more economy i benefits from spatial externalities. Equation (3.7) implies that the technology in an economy i depends on the exogenously given level of technology, on the level of its own physical capital stock per worker, and on the level of physical capital per worker in its neighbourhood. Replacing Ai,t in Eq. (3.6) with Eq. (3.7) and dividing by Li,t yields the production function per worker in economy i at t as: qi,t =

N  ∞  τ ∞ γ ν (wν )ij 1/(1−γ ) α+τ [1+ ν=1 γ ν (wν )ij ] t ki,t kj,t ν=1 j=i

(3.8)

The dynamic equation of the evolution of physical capital per worker ki,t is given by k˙ i,t = sK,i qi,t − (n + d)ki,t

with sK,i < 1 (n + d) < 1

(3.9)

The steady state value of physical capital per worker is ∗ ki,t



1  ν ν (1−γ ) 1−α−τ 1+ ∞ ν=1 γ (w )ij



= t

N 

sK,i ni + ς + d



1  ν ν 1−α−τ 1+ ∞ ν=1 γ (w )ij

∞ γ ν (wν )ij 

ν=1  ν ν 1−α−τ 1+ ∞ ν=1 γ (w )ij

×

τ

∗ kj,t

(3.10)

j=i

where ς is the balanced growth rate of physical capital per worker, defined as ς = k˙ i,t /ki,t . Finally, output per worker at steady state of economy i at t equals: q∗i,t =



sK,i ni + ς + d



a+τ 1−a−τ

1 1−a−τ

t

N   j=i

sK,j nj + ς + d



aγ wij 1−a−τ

q∗j,t

(1−a)γ wij 1−a−τ

(3.11)

As can be seen from Eq. (3.11) and contrary to the model of Vayá et al. (2004), the model of Ertur and Koch allows for different steady states of economies that are influenced by each other. Furthermore, these steady states depend on the relative location in space, that is we expect the development of economies to be interrelated with that of neighbouring economies. Depending on the concept that determines the connectivity terms wij , a sustaining core-periphery divide within the model is also possible. Since different saving rates and different labour force growth rates are allowed for, we find the development of one economy to depend on its own as well as on other economies rates of saving and fertility. If each economy is influenced by

3.2

A Neoclassical Growth Model with Different Steady States

27

others and we draw no border between two or more blocks, so that each economy is at least indirectly connected to each other economy, we find that the model reflects Waldo Tobler’s first law of geography: “Everything is related to everything else, but near things are more related than distant things” (cited in Longley et al., 2001, p. 61). Another feature of the model is that each economy exhibits constant returns. This is one of the critical assumptions of the Solow model for one economy under consideration: the economy is sufficiently large that if capital and labour double, the new inputs are used in essentially the same way as the existing inputs, and thus the output also doubles. Reintroducing labour into Eq. (3.8) and then adding up output of all economies, we find that the assumption of constant returns also holds for the entirety of N economies. What distinguishes the approach by Ertur and Koch from previous efforts is its introduction of interdependence into a neoclassical framework that leads to an equilibrium. Equilibrium growth of the physical capital stock per worker in steady state is no longer only a function of an exogenously determined level of technological progress. Rather, the model allows for external effects from physical capital within one economy, and external effects received from other economies. Ertur and Koch thus refer to a concept of knowledge that is abstract in the spirit of the Solow model, but by spilling over within and beyond each economy, it is understood to capture interdependence in a broader context. Apart from its explanation power regarding interdependent development, some of the model’s features leave some questions unanswered: firstly, since the individual aggregate levels of technology are increasing functions in physical capital stocks within and beyond one economy’s borders as displayed in Eq. (3.7), it follows that actual technological progress depends positively on the physical capital stock in any economy, whose evolution in turn depends on other economies’ physical capital stocks. Arguably, the plausibility of a huge and ever-growing physical capital stock that continues to create itself via perpetual reaction-coupling effects is primarily a matter of taste; nevertheless this assumption implies that the entirety of N economies will grow infinitely faster as more physical capital is accumulated. Interestingly, this result reverses the classical view of the capital-output ratio; whether it is more realistic remains open to interpretation. Furthermore, we find that the mere existence of foreign physical capital stocks raises marginal products of labour and physical capital, thus increasing the equilibrium levels of wages and rates of return. Higher investments abroad hence raise the rate of return in one particular economy,1 which also holds the other way round and vice versa, with lower investments abroad reducing the domestic rate of return. However, since domestic saving rates are exogenously given, the propensity to invest remains unchanged, leading to different rates of return in different economies. According to Eq. (3.8), stocks of physical capital in neighbouring economies affect domestic production considerably, allowing sizeable levels of output even in the case

1 This can be seen by taking the first derivative of the production function of economy i in Eq. (3.8) with respect to the stock of physical capital per worker of any region j = i, which will always be positive under the given conditions (i.e. positive values of all variables considered).

28

3 Growth Models with Spatial Externalities

of an almost complete absence of domestic physical capital. This lets one economy grow continuously even if the domestic saving rate is close to zero. It follows that any region that is allowed to choose its saving rate may as well opt to be a free-rider, where economy i is able to produce a considerable amount by relying almost solely on physical capital located beyond its own borders. Furthermore, even though the model recognises interdependence of economies as development of one economy depends on that of the others, factor movement is not allowed for – rather, each economy remains a closed economy in the sense that neither factor movement nor trade takes place. While foreign capital enters the production function directly, it is not allowed as a factor of production as such, but instead operates as a multiplier of domestic production. Hence, foreign capital has an influence on the evolution of the domestic capital stock only indirectly, via its contribution to current production. Foreign capital is never allowed to take advantage of differing rates on return, as it may only be invested at home. One individual economy has control only over the evolution of its product via the saving rate sK,i and the labour force growth rate (i.e. population growth rate) ni , which thus remain the crucial variables of development, just as in the original Solow model. A final note should be made on approaches that try to overcome the difficulties of modelling capital accumulation of N connected economies, namely by considering no capital at all, but rather by relying on knowledge as the sole determinant of growth. A theoretical concept which has gained interest in economic geography and regional science is referred to as the Verdoorn Law (Verdoorn-Kaldor Law), whose rationale refers to an assumption of cumulative causation: increased investment occurs in faster growing regions, thereby reinforcing their higher growth. Theoretical reasoning is sparse and frequently confined to some notes made by Nicholas Kaldor (1970) and Malecki and Varaiya (1986). In the context of regional growth, the Verdoorn Law has recently served as a basis for catching-up processes, to justify a straightforward (possibly spatial) econometric model specification where growth in one regional economy essentially depends on output levels or output growth in others.2 Caniëls (2000) derives a model for N interdependent regional economies, where knowledge is assumed both to be the only factor of production, and to diffuse over space and time without wearing out. Her model’s strength lies in its simplicity – in essence it consists of just one equation – nonetheless for N regions it becomes too complex to be analysed mathematically (Caniëls 2000, p. 56), and the omission of factors of production naturally provides no explanatory power on the issue of factor accumulation.

2 See for instance Pfaffermayr (2007, p. 10), who assumes that “regional spillovers rest on a region’s ability to learn the productivity improvements of its neighbors.”

Chapter 4

Convergence: Theory and Evidence

Considerations on growth theory inevitably raise the question of whether different economies converge to each other in terms of output, income, or related measures. Interest in cross-country disparities of wealth and their causes is omnipresent in international politics. In fact, the issue has become so important that it is now officially a major objective of the European Union, as discussed in the Introduction. Although there exist various definitions of what constitutes convergence, this study will refer to just one definition of convergence, which combines two concepts: The European Commission defines convergence as a process whereby the lesser developed regions approach the mean gross domestic product of the European Union (European Commission 2006, p. 2). As the achievement of this objective would eventually lead to the same level of output in each region, it is technically identical to the definition of convergence commonly used in scientific literature, that “each country eventually becomes as rich as all the others – the cross-section dispersion diminishes over time” (Quah 1993, p. 428).1 One concept of convergence is σ -convergence, which is observed if the variance of income declines over time (Sala-i-Martin 1996b). The complementary concept of β-convergence relies on the speed of convergence and is related to σ-convergence, as it constitutes a necessary but not sufficient condition for σ-convergence to occur. The difference between these two concepts is that β-convergence allows for overtaking, which may lead to an increasing variance. The aim of this chapter is to review and discuss different approaches to the issue of convergence. Section 4.1 identifies the roots of initial divergence and provides a discussion of the (non-)incidence of convergence of the world economy. In Sect. 4.2, the now fashionable convergence equation is formally derived from the Solow model. Finally, Sect. 4.3 briefly reviews some empirical studies on convergence processes and discusses their implications. 1 There is an inconsistency found in these two quotes, as the first refers to output, while the second refers to wealth. These concepts are obviously different from each other despite their frequent treatment as identical concepts. In the context of this chapter, convergence refers to any process where income – either produced or earned – converges to the same level. The next chapter will then take up the topic of why output and income within one economy are in fact two different things, and why they should be distinguished, especially in a regional context.

S. Sardadvar, Economic Growth in the Regions of Europe, Contributions to Economics, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-7908-2637-1_4, 

29

30

4

Convergence: Theory and Evidence

4.1 Convergence and Disparities of National Economies

(a)

(b)

output

The industrial revolution led to dramatic increases in production capacity and, besides influencing almost every aspect of society and daily life, ignited the process of economic growth as it is understood today. Those countries that were first to industrialise their production consequently were also the first to increase their output levels: economic growth in a modern sense and its consequent global income disparities have their roots here. The question of convergence is hence whether the divergence processes that have led to today’s observed disparities show any current signs of reversal. Shortly after the Second World War, Gunnar Myrdal published his analysis of growth and disparities, where he reasons that “it is on the whole the industrialised countries which are industrialising further.” He points to the fact that the major part of investments takes place in high-income countries, these investments being higher in absolute terms as well as relative to the rest of the non-Soviet world (Myrdal 1957, p. 4). Almost 30 years later, Baumol (1986, p. 1073) points to the “remarkable convergence of output per labour hour among industrialized nations” since the Second World War. He also notes, however, that the non-western industrialised countries and even more so a selection of developing countries did not converge to the highest developed economies, but instead have converged to each other, hence displaying club-convergence. Arguably the most prominent example of convergence (within specific countries) was the catching up of European economies first in the wake of industrialisation, and then again after the Second World War. Figure 4.1a plots the natural logarithms of GDP per capita since 1820 (in purchasing power standards of 1990) for the largest four European economies (within today’s political borders). It can be seen that France, Germany and Italy have grown faster than Great Britain, closing the gap or even surpassing. In terms of productivity, all four countries have also moved closer to each other. Interpreting this decreasing gap as evidence for convergence is problematic, however, as it happens after the fact: these four countries may

10

10

9

9

8

8

7

7

6 1820 1840 1860 1880 1900 1920 1940 1960 1980 2000

6 1820 1840 1860 1880 1900 1920 1940 1960 1980 2000

France

Germany

Italy

United Kingdom

Greece

Iran

Italy

Fig. 4.1 (a) and (b) GDP per capita at PPS from 1820 to 2000 of six areas. Note: Missing data has been interpolated; Data source: Maddison (2003)

4.1

Convergence and Disparities of National Economies

31

constitute the largest European economies as of today, but that has not necessarily always been the case. For this reason, retroactive explanations of convergence are not convincing if the economies in the sample happen to be quite similar with respect to most recent data, and in particular when they represent today’s most highly developed economies. If convergence of the world economy takes place, wouldn’t it be more compelling to carry out an ex ante analysis by taking, say, the successor states of great empires of ancient times and seeing how they performed in industrial ages? Figure 4.1b displays economic growth over 180 years as in Fig. 4.1a, but this time for Greece, Iran and Italy within today’s political borders. If anything, one may observe a more or less parallel development, but by no means a sustained trend of convergence. If one then considers that today’s Germany, Greece and Italy did not constitute confined economies in 1820, one may question what has actually been tested in both cases given. These two examples show how empirical studies on convergence are usually restricted by data availability, firstly by a short time span that does not represent long run economic growth, and secondly by a narrowly constricted selection of economies. As the problems with the analyses of Fig. 4.1a, b exemplify, a decision must be made as to which economies to test. This decision should be made a priori and be based solely on theoretical considerations. One may then either test for (see Barro and Sala-i-Martin 1995) • absolute convergence, which involves testing the hypothesis that poor economies tend to grow faster per capita than rich ones without conditioning on any characteristics of economies, or • conditional convergence, by allowing for heterogeneity across economies and testing the hypothesis that an economy grows faster the further it is from its own steady state. An additional issue that arises when testing predictions of neoclassical theory is the fact that steady states are a theoretical concept whose actual levels are unknown in reality, but one must nonetheless make considerations on these. Therefore, when testing for conditional convergence, one may either hold variables constant that distinguish countries from each other, or group sets of countries that can be reasonably compared. In the case that a convergence study is restricted “to sets of economies for which the assumption of similar steady states is not unrealistic,” for instance regions within countries “could be considered as similar ex-ante” (Sala-i-Martin 1996a, pp. 1027–1029, emphasis in the original), one tests for conditional convergence – although initial output is the only explanatory variable, which makes it technically a test for absolute convergence. Moreover, if one expects or observes specific economies to converge within a group, one is referring to what Myrdal (1957) implicitly and Baumol (1986, p. 1080) explicitly describe as the existence of club-convergence. Differing concepts of convergence may be confusing, but they also point out the crucial role played by reference groups of economies. With

32

4

Convergence: Theory and Evidence

differing selections of economies and/or differing sizes of economic entities within one selection, results will be different. Before turning to the formal derivation of a convergence equation and the discussion of the resulting empirical studies in the next two sections, this section ends with the attempt to gain inference from a brief explorative analysis of the world economy by testing for absolute convergence of national economies. While data dating back to the nineteenth century such as those used above are naturally based primarily on estimation, we can now look back on almost 40 years of fairly accurate data on gross domestic product, which are available from the United Nations.2 A sensible approach could be to identify those countries or groups of countries that represent the major economic areas or blocs of the world economy and hence are responsible for the vast majority of world-product. These areas or blocs should be represented by groups consisting of countries that firstly are geographically close to each other, secondly are of roughly comparable size, and thirdly were politically similar to each other for at least most of the past 40 years. Sticking to these criteria, one may identify the following seven major economic areas of the world for the time span 1970–2005: The United States of America (USA), the European Union within the borders from 1995 to 2004 (EU-15),3 Japan, the Union of Soviet Socialist Republics (USSR) and its successor states,4 the Southern Common Market (Mercosur) including its full members and associated members as of 2007,5 the People’s Republic of China (China), and India. As can be seen from Fig. 4.2a, b, together these seven areas made up 83% of the world product in 1970, and 79% Rest 17.5%

(a) India 1.9%

(b)

Rest 20.6%

USA 31.7%

China 2.8%

USA 28.2%

India 1.8%

Mercosur 3.4%

China 4.5% Mercosur 3.4%

USSR 12.5%

USSR 2.3%

Japan 6.2%

EU-15 24.0%

Japan 10.3%

EU-15 28.8%

Fig. 4.2 (a) and (b) Share of world output of seven economic blocs and associated countries in 1970 (left) and 2005 (right). Source: United Nations

2 Available

from: http://unstats.un.org/unsd/default.htm, queried on 05-July-2007. Belgium, Denmark, Finland, France, Germany, Greece, Republic of Ireland, Italy, Luxembourg, the Netherlands, Portugal, Spain, Sweden, United Kingdom; note that both German republics are included for the data from 1970 to 1990. 4 Armenia, Azerbaijan, Belarus, Estonia, Georgia, Kazakhstan, Kyrgyzstan, Latvia, Lithuania, Moldova, Russia, Tajikistan, Turkmenistan, Ukraine, Uzbekistan. 5 Argentina, Bolivia, Brazil, Chile, Colombia, Ecuador, Paraguay, Peru, Uruguay, Venezuela. 3 Austria,

4.1

Convergence and Disparities of National Economies

33

relative output

in 2005. Considering that the share of world population declined for Europe and Japan, who nevertheless managed to expand their shares of worldwide production, there is no evidence of a catching-up process. It shall be further noted that the share of the USSR and successor states shrank to less than a fifth of its 1970 value, and that India’s share declined despite its relative population growth and its increased economic growth in recent years. Sticking to large countries as well as blocs or groups of similar countries makes results easy to survey and compare. From Fig. 4.3, we get a picture in analogy to Fig. 4.1, this time covering around 80% of the world economy from 1970 to 2005 by plotting nominal GDP per capita relative to weighted world product per capita for the same seven areas and blocs as in Fig. 4.2. There is still no evidence of convergence (as defined in the preface to this chapter); South America and the Soviet Union, representing those areas that were technologically and economically closest to the dominating blocs of Western Europe and the USA in 1970, have since fallen further behind instead of catching up. The economic upturn of China receives a lot of press coverage these days, but the figures reveal that it is still very far away from the global mean, not to mention upper values. Half a century after Gunnar Myrdal (1957, p. 4) noted that “in the highly developed countries all indices point steadily upwards,” the impression one gets is a similar one. A report by the United Nations is in line with this view, noting that “there is no systematic tendency for poor countries to grow faster than rich ones” (United Nations 2000, p. 158). Despite occasional catching up of specific countries or economic blocs, in fact nothing much has changed: if anything, the gap between the rich and the poor, as proxied by the more and the less productive, still widens. For these reasons

1000%

100%

10%

1% 1970

USA

1975

EU-15

1980

Japan

1985

USSR

1990

1995

Mercosur

2000

China

2005

India

Fig. 4.3 Relative output per capita of seven economic areas and blocs, 1970–2005. Note: Numbers are displayed on logarithmic scale; Data source: United Nations

34

4

Convergence: Theory and Evidence

we may conclude that the hypothesis of absolute convergence does not hold when confronted with the real world.

4.2 The Formal Derivation of a Convergence Equation Returning to theory, the original Solow model as well as its successors provide at least three reasons to expect convergence (Romer 1996). Firstly, the models predict convergence to a balanced growth path, where the rate of growth is determined by the current state of the economy relative to such a balanced growth path. It follows that if two economies at any point in time are found to be at different points relative to their respective balanced growth paths, then the economy currently lying below the other one is expected to catch up. Secondly, a neoclassical production function assumes a positive marginal product of physical capital which declines as physical capital per unit of labour rises. This implies that ceteris paribus the rate of return on physical capital is higher in economies with lower physical capital stocks, and, consequently, it is expected that capital will flow to these countries. Since output is an increasing function of physical capital, this leads to convergence. Thirdly, as output is also a function of technology, economies with better access to technology will ceteris paribus also perform better – put differently, if the diffusion of knowledge changes in favour of those currently exhibiting relatively low levels of output, these economies are expected to grow faster. Despite these expectations, standard neoclassical models do “not make the loose, and possibly wrong, statement that poor countries tend to grow faster than rich countries” (Gandolfo 1997, p. 187). Indeed, almost none of the models reviewed and discussed above assume identical steady states – the only exception is the model reviewed in Sect. 3.1, which explicitly considers identical regional steady states. What standard neoclassical models do is to make the “precise (and certainly correct) comparative dynamic statement that the growth rate of an economy is directly related to the distance of that economy from its steady state. Hence the poorer an economy is, the faster this same economy tends to grow” (Gandolfo 1997, p. 187, parentheses and emphases in the original). It was not before 1990 that the convergence process of the Solow model was derived formally, as such setting the basis for the debate on the existence of convergence of the 1990s and 2000s.6 This began with a paper by Barro and Sala-i-Martin (1990), where the authors start by reproducing simulation results of the Solow model 6 It

is interesting to note that what nowadays seems to be a common and easily understood term in economics and politics referred to a different thing 20 years ago, as can be seen from contemporary encyclopaedias. For instance, Leipold (1987, p. 1068) describes convergence as a theory that forecasts similar solutions for comparable challenges with which differently organised industrial societies will be confronted (“. . . in der Konvergenztheorie [wird] die Auffassung vertreten, daß unterschiedlich organisierte Industriegesellschaften mit vergleichbaren Herausforderungen konfrontiert sind und dementsprechend auch ähnliche Lösungen anwenden müssen.”).

4.2

The Formal Derivation of a Convergence Equation

35

made by King and Rebelo (1989) via a log-linearised approximation to Eq. (2.15) around the steady state. This method allows for a direct transformation of the Solow model to an empirically testable convergence equation. By taking the natural logarithm of Eq. (2.15) and differentiating this with respect to time, one gets: ˙

 kˆ t d ln qˆ t = a = a sK kˆ ta−1 − n − g − d dt kˆ t

(4.1)

A first-order Taylor approximation of Eq. (4.1) has the form: ∂(d ln qˆ t /dt) ˆ d ln qˆ t (ln kt − ln kˆ ∗ ) ≈ dt ∂ ln kˆ t

(4.2)

where the derivative is evaluated at the balanced growth path value ln kˆ = ln kˆ ∗ . Using Eq. (4.1) to find the first derivative as displayed in Eq. (4.2), we have: ∂(d ln qˆ t /dt) ∂(d ln qˆ t /dt) = kˆ t = (a − 1)asK kˆ ta−1 ∂ ln kˆ t ∂ kˆ t

(4.3)

Inserting the steady state value of physical capital per efficiency unit of labour of Eq. (2.20) into Eq. (4.3) yields: ∂(d ln qˆ t /dt) ≈ a(a − 1)(n + g + d) = −aβ˜ ∂ ln kˆ t

(4.4)

β˜ = (1 − a)(n + g + d)

(4.5)

where

Note that β˜ > 0 because 0 < a < 1, and (n + g + d) > 0. Equation (4.4) can now be inserted into Eq. (4.2): d ln qˆ t ˜ kˆ t − ln kˆ ∗ ) = β(ln ˜ qˆ ∗ − ln qˆ t ) ≈ −aβ(ln dt

(4.6)

Equation (4.6) is the result of the Taylor approximation of Eq. (4.2). As a stand-alone equation, it represents a first-order nonhomogeneous linear differential equation over continuous time, and has the solution:

 ˜ ˜ ln qˆ T = ln qˆ 0 e−βT + ln qˆ ∗ 1 − e−βT

(4.7)

where T refers to the length of the respective observation period (i.e. number of periods). The positive parameter β˜ “governs the speed of adjustment to the steady state” (Barro and Sala-i-Martin 1990, p. 7) and is now commonly referred to as the speed of convergence.

36

4

Convergence: Theory and Evidence

Expanding both sides of Eq. (4.7) by ln qˆ 0 and rearranging yields an approximation of output growth per unit of effective labour between points in time 0 and T: 

˜ (4.8) ln qˆ T − ln qˆ 0 = 1 − e−βT (ln qˆ ∗ − ln qˆ 0 ) In analogy to Eq. (2.22), it can be seen that qT = qˆ T A0 egT

(4.9)

and it follows that ln qˆ T = ln qT − ln A0 − gT and ln qˆ 0 = ln q0 − ln A0 . Inserting these expressions into Eq. (4.8), dividing by number of periods T and rearranging finally yields the famous equation of β-convergence:    ∗ ˜ q 1 − e−βT qT 1 =g+ ln ln 0 T q0 T q0

(4.10)

Equation (4.10) sets the basis for all empirical studies that refer to β-convergence. ˜ which captures the responsiveThe crucial variable is the speed of convergence β, ness of the average growth rate to the gap between ln qˆ ∗ and ln qˆ 0 . It follows that ˜ the greater the responsiveness (Barro and Sala-i-Martin the higher the value of β, 1992). How quickly the derivation of a convergence equation became fashionable can be seen from Mankiw, Romer and Weil (1992), whose derivation is of the same form as Eq. (4.8), with the difference that the speed of convergence equals β˜ = (1 − a − b)(n + g + d)

(4.11)

Since the production function requires that 0 < a < 1 and 0 < b < 1, the speed of convergence implied by Eq. (4.11) is slower than that of Eq. (4.5). The decisive effect of implementing human capital to the Solow model can be illustrated more clearly by inserting the steady state values of Eqs. (2.28), (2.29) and (2.30) into Eq. (4.8): 

a b −(1−a−b)(n+g+d)T ln sK + ln sH ln qˆ T − ln qˆ 0 = 1 − e 1−a−b 1−a−b  a+b − ln(n + g + d) − ln qˆ 0 1−a−b (4.12) Note that in the case of b = 0, Eq. (4.12) also holds for the original Solow model. It can be seen from Eq. (4.12) that the approximated growth of output per unit of effective labour equals ln qˆ T − ln qˆ 0 , and depends positively on both saving rates, sK and sH : as the derivatives of Eq. (4.12) with respect to either saving rate are unambiguously positive, growth will be higher if one of the saving rates is raised, even though the speed of convergence remains unaffected. The inclusion of

4.2

The Formal Derivation of a Convergence Equation

37

human capital makes the derivation of the convergence equation for the MankiwRomer-Weil model a tedious work of cancellations, but leads to a neat result. Even though the derivation of convergence equations has become a standard requisite for research work dealing with neoclassical growth models, allowing for interdependencies between economies under consideration so far causes significant complications when approaching a convergence equation. This can be seen by looking at the models of Sects. 3.1 and 3.2 more closely. Vayá et al. (2004) assume growth of physical capital in economy i to depend on i’s own current stock of physical capital as well as the current stock of physical capital of neighbouring regions, as can be seen from Eqs. (3.3) and (3.4). This serves well enough to gain an overview of the economy’s evolution, but a serious difficulty arises when trying to capture the interdependence of regions: by conception, if economy i is influenced by its neighbours, then these neighbours are influenced by economy i as well – otherwise it would be impossible to derive the steady state level q∗ as in Eq. (3.5). But if such interdependence exists, then it has to be accounted for in the convergence equation as well: clearly, if output is a function of both domestic and foreign physical capital as in Eq. (3.3), then a derivation of output in i with respect to time in analogy to Eq. (4.2) would also have to take into account a function of the evolution of kθ,t . Since kθ,t in turn has to include domestic physical capital ki,t , and because i’s neighbours remain an unspecified bulk of regions as denoted by θ , Vayá et al. (2004) consider an approximation of the growth of ki,t of the form: ˜

ln ki,T − ln ki,0 = (1 − e−βT )(ln k∗ − ln ki,0 )

(4.13)

β˜ = (1 − τ )(n + d)

(4.14)

where

and by taking the natural logarithms of the steady state levels k∗ and y∗ , they come to an approximation of output growth as

˜ −βT

ln qi,T − ln qi,0 = 1 − e



1−γ τ ln  + ln sK 1−τ +γ 1−τ −γ   ˜ ln(n + d) + 1 − e−βT (γ ln kθ,T − ln qi,0 )

τ 1−τ −γ + γ (ln kθ,T − ln kθ,0 ) −

(4.15) In Eq. (4.15), the speed of convergence is a function of τ , n and d, and it is hence not affected by externalities γ which originate from neighbouring economies. However, growth will be increased in presence of such spillovers. It follows that if γ > 0, not only the steady state of output will be positively influenced, but growth as well. In the model of Ertur and Koch (2007) as discussed in Sect. 3.2, economic interdependencies are also accounted for in the authors’ derivation of a convergence equation. Inserting the production function of Eq. (3.8) into Eq. (3.9) yields the

38

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Convergence: Theory and Evidence

evolution of the stock of physical capital per worker: N ∞   k˙ i,t τ ∞ γ ν (wν )ij 1/(1−γ ) −{1−α−τ [1+ ν=1 γ ν (wν )ij ]} = sK,i t ki,t kj,t ν=1 − (ni + d) (4.16) ki,t j=i

A log-linearisation of Eq. (4.16) around the steady state leads to this approximation of the derivation of ln ki,t with respect to time:    ∞  d ln ki,t ∗ (ni + ς + d)(ln ki,t − ln ki,t γ ν (wν )ij ) ≈− 1−α−τ 1+ dt ν=1

+

N  ∞ 

∗ τ γ ν (wν )ij (ni + ς + d)(ln kj,t − ln kj,t )

j=i ν=1

(4.17) as a system of differential linear equations “whose resolution is too complicated to obtain clear predictions” (Ertur and Koch 2007, p. 1040). They replace Eq. (4.17) by another approximation, which finally yields an approximation for ln qi,t that is six lines long. This approximation shall not be reproduced here, as the main implications of the model have already been discussed in Sect. 3.2.

4.3 Empirical Tests of Convergence, Conclusions and Unsolved Questions Barro and Sala-i-Martin (1990) apply Eq. (4.10) empirically in the form of:   1 qi,T = α − β ln qi,0 + ui ln T qi,0 where the intercept is α =g+

  ˜ 1 − e−βT /T ln q∗0

(4.18)

(4.19)

and the regression coefficient equals ˜

β=

1 − e−βT T

(4.20)

and ui is a random variable. An application like Eq. (4.18) tests for absolute convergence as discussed in Sect. 4.1 – that is, all economies under consideration approach the same steady state levels of output per worker (or related measures such as income per inhabitant). Barro and Sala-i-Martin (1995, p. 384) argue that this assumption is more reasonable for regional data sets than for international data sets, noting that “it is more plausible that different regions within a country are

4.3

Empirical Tests of Convergence, Conclusions and Unsolved Questions

39

more similar than different countries with respect to technology and preferences.” Barro and Sala-i-Martin (1990) apply Eq. (4.18) to the states of the USA from 1880 to 1988 on personal income as well as output, and generally find the convergence coefficient β˜ to be greater than zero.7 For the analogous measure of GDP, gross regional product (GRP),8 from 1963 to 1986, they find an estimate βˆ˜ = 0.0180. Barro and Sala-i-Martin (1992) apply Eq. (4.18) to 98 countries with market economies from 1960 and 1985. When testing for absolute convergence of real GDP per capita, they find the convergence coefficient estimate βˆ˜ = −0.0037; the negative value indicates a small tendency for highly developed countries to grow faster than those with lower initial levels of GDP in the observation period. These findings are in accordance with the discussion in Sect. 4.1, as there is no hint of absolute convergence. Dissimilarities between countries are accounted for in another test, in which specific variables considered to have an influence on the steady state levels, such as education levels, are held constant. Now the picture changes significantly, and the convergence coefficient estimate βˆ˜ = 0.0184.9 Barro and Sala-i-Martin (1992) thus come to the conclusion that their empirical results document “the existence of convergence in the sense that economies tend to grow faster in per capita terms when they are further below the steady state position” (Barro and Sala-i-Martin 1992, p. 245). When testing for regional data sets, in addition to the results of Barro and Sala-i-Martin (1990), Sala-i-Martin (1996b) finds evidence for β-convergence for the regions of Canada (observation period 1961–1991), Germany (1950–1990), Great Britain (1950–1990), France (1950–1990), Italy (1950–1990) and Spain (1950–1987) as well as for 90 regions of Western Europe. He notes that “speeds of β-convergence are extraordinarily similar across countries: about 2% per year” (Sala-i-Martin 1996b, p. 1339), and suggests using “a mnemonic rule: economies converge at a speed of about 2% per year” (Sala-i-Martin 1996b, p. 1326).10 Mankiw, Romer and Weil (1992) test the Solow model and compare it to their own model by applying the same data set as Barro and Sala-i-Martin (1992). They consider human capital (measured as the percentage of working-age population with secondary education) as an explanatory variable, and find a convergence coefficient estimate βˆ˜ = 0.0137 for 98 countries. When considering 22 OECD memberstates, they find a speed of convergence of 0.0203. This general result of a 2% convergence-speed is again confirmed by Barro and Sala-i-Martin (2004) for a set 7 Despite these considerations on similarities of regions, Barro and Sala-i-Martin (1990) included three regional dummy variables (south, midwest and west), which made their analysis actually a test on conditional convergence. 8 Gross regional product is called gross state product (GSP) in the USA. 9 The variables considered in the study are measures for primary and secondary school enrolment rates, government consumption, “political stability” and “market distortions.” 10 This idea of a 2% rule has been questioned by Quah (1996), who compares such interpretations of regression results to Galton’s fallacy; a regression towards the mean, it is argued, is uninformative for a distribution’s dynamics.

40

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Convergence: Theory and Evidence

of 87 countries for 1965–1975, 97 countries for 1975–1985, and 83 countries for 1985–1995. They run a number of regressions testing for conditional convergence, considering various control variables of which some are political indicators.11 They find estimated coefficients of the regression term β of –0.0222 for the first period, –0.0231 for the second period, and –0.0338 for the third period, which are equivalent to positive convergence speeds as can be seen from Eq. (4.20).12 Ertur and Koch (2007) test their (spatially augmented) model for 91 countries over the period 1960– 1995 and find evidence on both conditional β-convergence and physical capital externalities. Tests on regional club-convergence within Europe have been carried out by Fischer and Stirböck (2006), who use a spatial econometric specification and test for β-convergence for 256 European Regions from 1995 to 2000 and find evidence for the existence of two convergence-clubs. In summary, since the groundbreaking works by Barro and Sala-i-Martin, the issue of convergence has been extensively econometrically tested, with the conclusion that findings of conditional β-convergence tend to be robust (Islam 2003). Recent contributions to the study of convergence processes may take externalities and spatial dependencies into account, but there remain at least two reasons for lingering discontent with findings of evidence for convergence: • With some studies finding either mixed results or evidence of divergence – especially when focussing on economic units smaller than countries (e.g. Cheshire and Magrini 2000) – the question has been raised whether this concept should ultimately be seen as a statistical artefact, with the following argument: “If conditional convergence is suggested as an approach to describe the behaviour of national states then it should also be valid at a lower level of spatial aggregation. Otherwise, the concept should be considered as somewhat arbitrary. It seems difficult to defend the concept of conditional convergence if it depends on the level of spatial aggregation and thus on arbitrarily drawn borders” (Keilbach 2000, p. 25). • While the argument above is, in principle, reasonable, it tacitly assumes that each economy faces the same conditions for growth. In a meta-analysis of β-convergence, Abreu, de Groot and Florax (2005) find that the more homogenous the spatial units under study are, the higher the convergence rate tends to be. However, robust results are often generated from testing without theorising and hence criticised, “containing a host of variables potentially affecting economic growth” without a clear link to theory (Abreu et al., 2005, p. 395). Moreover, the fixation on β-convergence in empirical studies with or without links to theory may blur the complete picture. Shortly after the emergence of the 11 The

results that follow considered “male upper-level schooling,” life expectancy, fertility rate, government consumption, “rule of law,” “democracy,” changes in terms of trade, investments ratio, and inflation. 12 The regressor, β, tends to β˜ as t tends to 0, and tends to 0 as t approaches infinity, which is why it may also be interpreted on its own.

4.3

Empirical Tests of Convergence, Conclusions and Unsolved Questions

41

concept, Quah (1993) commented on two issues: firstly, “the term convergence can mean a number of different things” (Quah 1993, p. 428). Secondly, convergence of one type does not necessarily occur simultaneously with convergence of another type. For instance, if there is evidence on β-convergence, the rich-poor income disparity may not decrease at the same time (Quah 1996, p. 1364). Another problem that obviously arises from growth regressions that stem from standard models as discussed in Chap. 2 relates to their nature of describing closed economies. These economies are implicitly treated as isolated from each other, an assumption which becomes problematic when regional economies are studied. The model developed in the next part of the study tries to overcome these difficulties, both in terms of theoretical assumptions as well as its econometric specification. The first step is to lay out a precise clarification of the concept of a region. After a comprehensive discussion of the model, at the end of Part II it is shown that the fixation on convergence should be shelved: by acknowledging the role that space plays, a neoclassical growth model is able to explain both convergence and divergence processes.

Part II

A Model of Regional Growth

In Part I, the neoclassical growth model and various extensions including human capital and technological externalities were reviewed. It has been demonstrated how the latter improved on the explanation power of the original Solow model, by accounting for knowledge either as embodied in the workers (human capital), or as diffusing within and across economies (externalities and spillovers). The models thus provide better insights into the functioning of integrated economies. What is missing so far is a link to capital movement, an aspect that appears to be relevant especially in the context of regional economies. The model presented in this part of the study partly builds on the models in the previous part, extending these approaches by considering openness of economies in the sense that foreign direct investments (FDI) are possible. It should be noted that in what follows, the frequently used terms “domestic” and “foreign” in a regional context are not meant to be taken literally, but rather serve as a convenient distinction of what is going on inside and outside a specific regional economy under consideration. The aim of this part is to identify the determinants of growth and development of a set of regions that jointly form a superordinate economic system. The European Commission’s official objective of eventually equal output levels in all its regions is taken as a starting point; López-Bazo (2003) points out that regions within a country, or in a set of integrated economies such as the European Union, are supposed to be more homogenous than a set of worldwide economies. The technology of production, consumer preferences, institutions, and so on may be thought to be largely similar across regions. In addition, the diffusion of ideas and technology is easier across regions than when heterogeneous societies are involved. A certain degree of similarity is also given if some decisions (such as education expenditures) are made by a superior authority for all regions under consideration. This study’s model shows that even with identical levels of output in the long run, divergence processes may occur: during the transition period, the interplay of physical capital, human capital and relative location in space leads to an infinite number of possible intermediate outcomes. Chapter 5 discusses some relevant aspects of regional growth and development, before turning to the outline of the model in Chap. 6. While its production function is similar to the Mankiw-Romer-Weil model, the function of change in the physical

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Part II A Model of Regional Growth

capital stock allows for investments abroad – with decisive effects on growth in the short and medium run. In Chaps. 7 and 8, the model is solved for developments in the medium run as well as in the long run. These latter chapters are primarily dedicated to the transition period, both by simulation results and by formal derivations, before finally some conclusions and possible extensions of the models are discussed.

Chapter 5

Remarks on Regional Growth

A model’s purpose is to provide insights about particular features of the world, which requires making simplifying assumptions. The usual procedure is to start with a very simplified model and then make it increasingly realistic; in the process, one comes to a more sophisticated understanding of the actual system (Krugman 1994, p. 53). Hence before starting, one must check what particular feature of the world is actually of interest if the context slightly changes. Section 5.1 reviews some approaches to regional development, including new economic geography. In Sects. 5.2 and 5.3, some particular features of standard neoclassical growth models are reconsidered in view of setting up a model of regional growth.

5.1 Agglomeration Effects, Increasing Returns and Polarisation Since the heyday of industrialisation, spatial concentration of economic activities has constituted a familiar phenomenon. Alfred Marshall (1890) points to economies of localisation, which occur because firms in the same industry find it advantageous to cluster in the same geographical area for several reasons: firstly, such clusters allow firms to specialise more deeply than when they were widespread. Secondly, clustering strongly facilitates research and innovation in an industry – physical proximity enhances the exchange of information, ideas and knowledge and as such generates technical spillovers. Thirdly, risks for both employers and employees are reduced, as both will find it easier either to have access to workers or to find future prospects of labour (see also Armstrong and Taylor 2000). Related to economies of localisation are agglomeration economies, which arise from geographical concentration of economic activities of different industries: in general, economic activities will tend to concentrate in space as long as positive agglomeration effects exist. Spatial concentration occurs because production is cheaper due to the larger amount of nearby economic activity in agglomerations. The idea that learning takes place within a region has also been previously discussed by Marshall (1890), but it has received new attention in the wake of the tendency of high-technology industries to cluster in space. The most famous example here is Silicon Valley, where firms benefit from spatially concentrating their S. Sardadvar, Economic Growth in the Regions of Europe, Contributions to Economics, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-7908-2637-1_5, 

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Remarks on Regional Growth

activities, which in turn attracts firms to locate in such learning regions. In such a collective learning situation, firms within the innovative milieu will take advantage of the local knowledge base and thus be more successful in producing product and process innovations as would be the case if they were located somewhere else (Acs 2000). In the age of globalisation and the growing importance of knowledge as a factor of production, vertical and horizontal relations of firms are again receiving attention in connexion with spatial systems of innovation (Fischer et al., 2001): economies of scale as caused by agglomeration effects play a crucial role in explaining the emergence of clusters of production and innovation and hence economic development. From a microeconomic perspective, companies innovate products or processes in the search for (temporary) monopoly rents. In order to do so, they require talented personnel, who in return receive scarcity rents. Hence regions where human capital is concentrated above a critical mass will be able to create technological externalities, which in turn make the occurrence of increasing returns possible (Azariadis and Drazen 1990). As a consequence, the region’s terms of trade will improve (Freeman and Soete 1994). Up to now, thoughts on regional peculiarities leading to clusters of production do not form a coherent theory; rather they reveal that regions do not necessarily adhere to the same determinants of growth as national economies. Trends of concentration such as those mentioned above in one way or another represent persisting disparities: they explicitly or implicitly suggest that some regions will perform better than others. In contrast to theories that predict (conditional) convergence, studies on regional polarisation focus on imbalances causing cumulative development processes, which may even increase regional inequality and therefore consolidate already existing polarisation. In general, they stress (see Schätzl 2001) • interregional differences in internal determinants of growth, such as endowment with factors of production, economic structure or consumption functions; • partial immobility of growth determinants; • interregional dependence of regional growth processes, for instance the dependency of peripheral regions to central regions caused by interregional interaction; and • oligopolistic and monopolistic market structures. Gunnar Myrdal is a great critic of theories of stable equilibria that are deduced from the idea that social processes would follow a direction towards selfstabilisation. His study of regional polarisation, which addresses the causes of economic development and underdevelopment, has a spatial aspect, as it pertains to cumulative processes of growth and development both within a country as well as internationally, where a spatial differentiation between growth poles and less developed regions persists.1 Myrdal (1957, p. 26) argues that “the play of forces 1 Hypotheses of polarisation of economic sectors originate in the works of Schumpeter (see Sect. 2.1) and have also been developed by François Perroux, who stresses different growth rates between economic sectors (see Richardson 1973).

5.1

Agglomeration Effects, Increasing Returns and Polarisation

47

in the market normally tends to increase, rather than to decrease, the inequalities between regions.” He develops a model with cumulative causation, where the limited advantages of backward regions (such as cheap labour) are insufficient to offset agglomeration advantages of the centres. The main influence on the rate of growth of lagging regions is the induced effect of growth in advanced regions, which can be divided into centrifugal spread effects and centripetal backwash effects. The former include markets for the products of the lagging regions and diffusion of innovation, while the latter involve flows of labour, capital, goods and services from advanced to developing regions. Significant to the outcome is whether spread or backwash effects are stronger: Myrdal (1957) notes that spread effects are normally outweighed by backwash effects. Thus, the free trade of an interregional system operates to the disadvantage of less developed regions and therefore has a sustaining effect on existing disparities. In a similar vein to Myrdal, Hirschman (1958) notes that an economy will first develop within itself one or several regional centres of economic strength. This process leads to the emergence of growth poles, which means that “international and interregional inequality of growth is an inevitable concomitant and condition of growth itself ” (Hirschman 1958, p. 140). He develops a model of two regions where he labels “polarisation effects” and “trickling-down” what corresponds essentially to Myrdal’s backwash and spread effects. Further studies and refinements of Myrdal’s results have been developed by Kaldor (1970), who explains cumulative causation by the existence of increasing returns. Paul Krugman (1991a, p. 486) refers to Myrdal’s circular causation and develops a formal model to explain how and why industrial production may cluster.2 To this end, Krugman (1991a) develops a variant of a model by Dixit and Stiglitz (1977) that shows how a country can endogenously become differentiated into an industrialised core and an agricultural periphery. Noting that, while industry as well as population obviously tend to cluster, the study of economic geography has gained little attention, he asks a critical question: “Why and when does manufacturing become concentrated in a few regions, leaving others relatively undeveloped?” (Krugman 1991a, p. 484). His model may be seen as a formal interpretation of Hirschman’s and Myrdal’s contributions, as Krugman (1991b, p. 105) identifies two centripetal forces and one centrifugal force at work in his model: firms desire to locate close to the larger market, and workers desire to have access to the goods produced by other workers. At the same time, firms want to move out to serve the peripheral agricultural market. It follows that firms which start production in the peripheral region would need to pay higher wages to attract workers from the industrialised region. The outcome of the model is an equilibrium of either clustering or dispersion, depending on parameters of trade costs (transportation costs), consumer preferences and economies of scale. If trade costs are sufficiently low, the outcome

2 Paul Krugman won the Prize in Economic Sciences in Memory of Alfred Nobel “for his analysis of trade patterns and location of economic activity” in 2008 (http://nobelprize.org/ nobel_prizes/economics/, queried on 17-October-2008).

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Remarks on Regional Growth

will be a stable equilibrium with all industrial production concentrated in just one region, such that it is not profitable for any single firm to leave the region. On the one hand, models in the spirit of Krugman’s exercise expand the understanding of mechanisms of interrelated regions’ development, but are less successful regarding prospects of growth and long-term development. On the other hand, models of neoclassical growth have so far not been very successful in explaining regional disparities, especially because their predictions concerning investment flows are not observed in reality.3 One source of misunderstanding is perhaps the vagueness of what is meant by a region in a specific context. Studies of regional economies usually focus on phenomena such as agglomeration economies in location and urbanisation, increasing effects within clusters etc. If, however, a region is meant to refer to an economy that is large enough to have its own core and periphery, and at the same time that interacts with the outside world, then the assumption of constant returns in the spirit of Solow is appropriate, i.e. a region is big enough that possible gains from specialisation have been exhausted. Kaldor (1970, p. 338) points to the difficulty and the importance of defining a region; an accurate definition for the purpose of this study will be given in Sect. 6.1. Before that, two further sources of misinterpretation of neoclassical growth theory in a regional context will be discussed in the following two sections.

5.2 Reconsidering Saving and Gross Investment The first feature of the neoclassical growth model to be reconsidered relates to the equalisation of income and output. As shown in Sect. 2.2, for a closed economy this assumption holds by definition. In contrast to this, in a world with open economies, investments may take place abroad, which even in the case of perfectly balanced trade-accounts will result in different levels of gross domestic product (GDP) and gross national income (GNI).4 Letting the saving rate take on individual values, as is the case in the traditional Solow model as well as in the other models discussed in Part I, then raises the question of what is actually being theoretically modelled and occasionally empirically measured. Usually empirical work ignores this distinction, by taking GDP as a measure of income. For some countries where these values are approximately the same, this equivalent treatment has no severe effects – for instance in Austria, where total GNI equalled 98.8% of total GDP as of 2004 (Statistik Austria 2005). For other countries, these values may differ significantly – for instance, total GNI of the Republic of Ireland equalled only 85.3% of total GDP

3 An early attempt at a neoclassical model of regional growth with free factor mobility at zero transportation cost is developed by Borts and Stein (1964), where lower wages in one region will attract investments and therefore lead to a higher growth rate of the physical capital stock. This in turn will raise the growth rate of wages, and therefore cause convergence between regions. 4 If just one single moment of disequilibrium is allowed for, to be precise.

5.2

Reconsidering Saving and Gross Investment

49

as of 2004.5 It follows that the equalisation of saving and gross investment is a related issue. Foreign direct investments are meant to be paid back by the labour of domestic workers. While these investments may indeed lead to higher productivity and possibly higher wages, a significant income generated by these investments will stay in the host’s land. The conclusion is that if open economies with investments abroad are considered, and if we assume the domestic saving rate to be a determinant of growth, the dependent variable would be income. In a regional context, however, relying on a regional equivalent of a national saving rate is impractical. Equivalent treatment of national and regional savings would imply that nations and regions are essentially the same thing. By definition, however, regions are part of a superordinate economy, and as such differ decisively from nations regarding their policies, the consequences of their own decisions, or their inhabitants’ behaviour. It follows that if regions are considered, specific economic aspects play different roles in comparison to national economies. In particular, a region is naturally open and as such experiences comparatively more inflows and outflows of goods, residents and investments. For instance, since 1991, 1.7 million people have emigrated from the German state Berlin, while at the same time 1.8 million have immigrated, literally exchanging half of the population of 3.4 million (not counting natural exchange caused by births and deaths).6 Are these people supposed to worry about the (relatively high) regional debt of Berlin as a federal state (i.e. region in the sense of the study) the same way as the national debt of Germany? Even if they do, they are allowed to emigrate at any point in time, while in contrast a nationality may not be changed that easily. The second concern in relation to the equalisation of saving and gross investment is that of public investment, or the issue of autonomous decision-making: even in cases where saving equals gross investment fairly exact on the national scale, this is not necessarily true on the regional scale. The reason for this lies in the fact that governments make consumption and investment decisions following specific desired ends. National governments frequently either transfer funds to lesser developed regions to enhance consumption and investments or, as matter of subsidy, consume and invest in these regions themselves by using taxes at least partly collected in wealthier regions. Although in principle imaginable, in practice government consumption and investment decisions are by no means intended to be exactly proportional to the revenue of taxes in each administrative sub-unit. Rather, they often reflect consciously decided subsidies for specific regions, inevitably leading to an outcome where those citizens (taxpayers) who involuntary invest more (by paying more taxes) face lower returns. The crucial point is that while the “saving equals gross investment” – assumption still holds for the national economy, it does not necessarily hold for its regions: even in the hypothetical case of a negative saving rate

5 Central Statistics Office Ireland, data available from: http://www.cso.ie/statistics/nationalingp. htm, queried on 17-April-2007 6 See Der Spiegel 12/2007.

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Remarks on Regional Growth

in one particular region, it is imaginable that this very region could receive a positive balance of gross investments without ever facing the duty of paying back, thus raising expected future income from saving without having saved itself in the past. Again, one example makes clear why this issue is relevant: between German unification in 1991 and the year 2002, approximately 850 billion euros were transferred by means of policy from the Western German provinces to those of the former German Democratic Republic (Sinn 2004, p. 229). Put differently, in 2002 the sum of total gross regional product (GRP) of the latter provinces stood at 253 billion euros, while aggregate spending of these provinces’ households, firms and governments stood at 366 billion euros. Only one quarter of the difference of 113 billion euros was made up of borrowed money plus foreign direct investments, while of these 113 billion euros, no less than 85 billion euros were made up of transfer payments from the remaining German provinces (Sinn 2004, p. 230), considerably reducing the otherwise enormous supposed deficit on current account of 44.7%. Given that these 85 billion euros have been either used for public investment, or as subsidies for existing or newly founded firms, or, in the case of transfers to private households, have served at least as an opportunity to increase domestic saving, it is apparent that any approach to equalise saving and gross investment on a regional scale will not hold. For purpose of thoroughness, it should be mentioned that such a situation is also imaginable for countries, but it is not applicable to the area of the world that this study deals with. In other words, industrialised countries do not receive considerable gifts from the rest of the world, with one notable exception, which is European cohesion policy as discussed in the Introduction. If anything, this makes a stronger than necessary argument for considering the European Union to be the superordinate economy of the Regions of Europe. However, the main conclusion from the discussion above is that, for regions, the distinction between output and income regarding the issue of growth mechanisms is obsolete. Output is the only reasonable conceptional dependent variable for theoretical approaches to the mechanisms of regional growth, even if it may serve well as an indicator for wealth in empirical studies (GRP per capita is indeed officially and legislatively used as the measure of wealth of regions by the European Union).

5.3 Foreign Capital, Its Mobility and the Role of Human Capital Having found that growth of the physical capital stock in one particular region is not solely dependent on foregone saving in this very region, and before defining whose saving then actually matters, the second feature of the standard neoclassical growth model to be reconsidered deals with the evolution of the physical capital stock itself. One of the more unpleasant conclusions to be drawn from the Solow model and its successors is the notion that these models imply immense incentives to invest in economies where the marginal product of capital is highest – as discussed in Sect. 2.4, the existence of human capital serves as one plausible answer. The second

5.3

Foreign Capital, Its Mobility and the Role of Human Capital

51

likely reason, according to Lucas (1990), is capital market imperfections. Besides sheer forbiddance of capital mobility, there are two causes in particular that prevent capital from flowing from capital-rich to capital-poor countries, namely political risk for the lending country, as it is evident that there must be an effective mechanism for enforcing international borrowing mechanisms, and mistrust of foreigners or, to be more precise, the unknown and supposed possibly unpleasant long run consequences of foreign capital inflows. At least in the context of the European Union these issues may be neglected, as free movement of capital constitutes one of the internal market’s “four freedoms.” This second explanation, however, remains somewhat puzzling, as one would expect foreign capital that actually flows into an economy to be uniformly distributed among the lenders, i.e. ceteris paribus the same level of attractiveness of a given destination of foreign capital investment should apply to all candidate lenders the same way: if economy i attracts investments from economy j, all other economies should be attracted alike. By taking a closer look at international capital flows, indifference with respect to investment targets cannot be observed. For instance, Germany’s share in the stock of foreign direct investment in Austria at the end of 2004 was 12 times higher than France’s share (Oesterreichische Nationalbank 2006, p. 27);7 at the same time, Austria’s share in the stock of foreign direct investment in Hungary was 14 times higher than Great Britain’s (Magyar Nemzeti Bank 2007, p. 37).8 This phenomenon is explained by the Hungarian central bank as a consequence of geographical proximity and historical traditions, by which Austrian enterprises have been traditionally important economic partners to Hungarian companies (Magyar Nemzeti Bank 2007). It follows that, although data on intrastate capital movements cannot be tracked, the mobility of capital and the emergence of foreign direct investments are expected to be spatially bounded for at least three reasons: • cultural connectivity and historical traditions will facilitate the lender’s process of decision making and thus increase capital flows; • increasing trade relations induce capital flows as exports first encourage the establishment of foreign trade offices, to be possibly followed by production facilities on location; • lower transportation costs (i.e. decreasing economic distance) will increase the vertical integration of production across regions and as such increase capital flows. Assuming that Lucas’ (1990) reasoning and Mankiw, Romer and Weil’s (1992) modelling also provide some explanation power for regions, it is expected that income disparities will depend on the relative levels of human capital across regions.

7

The stock of German directly invested capital stood at 17.3 billion euros, and the stock of French directly invested capital at 1.4 billion euros. 8 The stock of Austrian directly invested capital stood at 4.6 billion euros, and the stock of British directly invested capital at 0.3 billion euros.

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Remarks on Regional Growth

Human capital is by definition embodied in labour, hence in equilibrium its marginal product will be earned by the corresponding skilled workers. As a consequence, we would find higher levels of human capital to be accompanied by higher average wages in an economy.9 In turn, human capital is expected to positively influence productivity, and hence to attract investments in physical capital.

9 These developments indeed reveal divergence mechanisms by taking a closer look at the development of wages. In a study of the growing divide of labour incomes in the USA since the 1960s, Galbraith (2000) clusters industries in relation to wage disparities. He comes to the conclusion that the US-American industry is divided into a “knowledge sector” and a “consumption goods sector.” Illustratively, the list of industries with the highest rise in total employee annual earnings per production worker hour by industry is almost identical to definition of high-technology industries according to Standard International Trade Classification (SITC) (for details regarding the classification see Hatzichronoglou 1997).

Chapter 6

Structure of the Model

To include the relative location in space in a growth model means that an economy’s performance no longer lies solely within its range and conditions, but rather also depends on developments and decisions made beyond its borders. This chapter lays out the study’s model, which can be described as a growth model of N regional economies with a neoclassical production function that includes human capital, and where physical capital mobility between regional economies takes place. The configuration of the model, that incorporates both N regional economies and their accompanying superordinate economy is presented in Sect. 6.1. Section 6.2 examines specific characteristics of the neoclassical production. Finally, the key equations of the evolution of factors of production are discussed in Sect. 6.3.

6.1 Definitions and Assumptions Let’s assume a superordinate economy that is defined as being considerably more different and disjointed from the rest of the world than its parts are different and disjointed from each other. The superordinate economy’s parts are referred to as regions, which are similar to each other in terms of technology of production, consumer preferences and institutions. Connectivity between its economic entities in particular refers to factor movement. Converted to the modelling assumptions, the superordinate economy’s individual economic entities are • economically open, in the sense of allowing for migration of factors of production; • politically interdependent, in the sense that parts of decision-making pertaining to economic policy take place collectively, and • spatially connected, in the sense that the relative location in space influences individual investment and migration decisions The individual economies may initially be associated with provinces, and the superordinate economy may initially be associated with a country, but the model’s concept is more comprehensive. In particular, the theoretical model applies to any S. Sardadvar, Economic Growth in the Regions of Europe, Contributions to Economics, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-7908-2637-1_6, 

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54

6 Structure of the Model

group of individual economies that form a superordinate economy, including systems whose parts are countries: the study’s point of interest is the development of economic entities that are at least partly dependent on the development of a superordinate economy. In other words, it takes up the question of how economic entities that are reasonably similar to each other will develop, as specific determinants of their individual economic development are dependent on the system. The assumption that these economies are not totally autonomous makes them regional in the sense of the study, and for this reason the individual economic entities of interest will be hereafter referred to as “regions.” Regions are understood to be spatially confined areas whose borders are selected by specific criteria and appropriate methods. To meet these criteria, a region shall be defined as (i) an administrative entity, (ii) which forms an economic sub-unit below a superordinate economy, (iii) that reflects decision-making pertaining to economic policy at the most crucial level, (iv) and mirrors those social processes and individual behaviour-patterns that ultimately determine economic development. The regions are assumed to be big enough so that the critical assumption of constant returns applies: each economy is sufficiently large that by doubling the amounts of factors of production, the new inputs are used in essentially the same way as the existing inputs, and output thus doubles.1 On the contrary, in a very small economy, there are probably enough possibilities for further specialisation, that doubling the amounts of capital and labour more than doubles output, as discussed in Sect. 5.1. Considering this, the theoretical model applies to any number of economic entities for which the definition above applies, but it does not apply to arbitrarily selected groups of regions that in one way or another belong together, nor is it applicable to any order of administrative fragmentation for regions (i.e. if the assumption of constant returns is not valid). Interdependence, similarity with respect to technological preferences, and free factor movement are consistent with the assumption of an aggregate level of technology that applies to all regions within the superordinate economy. Most importantly and in accordance with Mankiw, Romer and Weil (1992), there is a clear distinction between human capital H and abstract knowledge A: human capital consists of the abilities, skills, and knowledge of particular workers. It is therefore available only once at a time, and as it depreciates over time, it continually requires fresh investments. While human capital is rival and excludable, the current level of technology is available to all regions within the respective superordinate economy and as such takes on the characteristics of a public good. Full availability of technology implies that there are no gains from knowledge spillovers. The model deviates here from the ones presented in Chap. 3, where spillovers are crucial for convergence and hence growth. The model is hence closer to the assumption of similarity of regions as discussed by Barro and Sala-i-Martin (1995) and López-Bazo (2003).

1 Technically,

the production function is a homogenous function of degree one.

6.2

The Production Function

55

Finally, this study assumes that within the superordinate economy, the relative location in space is a determinant of medium run growth. In particular, this study hypothesises that there are certain ties between economies that have an influence on the propensity to invest, and that this propensity is dependent on distance. As noted by Krugman (1991b, p. 53), the concreteness of forces that, in principle, one could examine directly, places constraints on what we can assume, whereas knowledge flows are invisible and “leave no paper trail by which they may be measured or tracked.” This of course does not mean that knowledge spillovers do not exist; the approach here is rather to set the primary focus on tangible developments. The assumption may then be extended to include knowledge spillovers as well, as will be discussed in Sect. 8.3. For the model as presented in the next section, the mobility of capital and the emergence of investment flows are expected to be spatially bounded as discussed in Sect. 5.3.

6.2 The Production Function Let’s consider a set of economically open regions, i = 1, 2, ..., N, over continuous time t. The regions jointly form a superordinate economy, which is assumed to be closed. Aggregate output Qi,t in each region i at time t is produced with the inputs physical capital Ki,t , human capital Hi,t , and labour Li,t , through a Harrod-neutral Cobb-Douglas production function a b Hi,t (At Li,t )1−a−b 0 < a < 0.5 0 < b < 0.5 Qi,t = Ki,t

(6.1)

where At represents the current level of technology available to all regions under consideration. The assumption of constant returns is established by the condition underlying the output elasticities of the production factors, whose sum equals one. Note that regional output Qi,t is equal to income generated within the region’s borders, but due to the openness it is not necessarily equal to income earned within the region’s borders. To see this, let’s distinguish physical capital that is available within a region’s border by its ownership:

Ki,t = Ci,t +

N  j=i

Cji,t −

N 

Cij,t

(6.2)

j=i

Equation (6.2) displays physical capital available for production in region i at t as physical capital that is owned by its inhabitants Ci,t , plus physical capital that is  found within region i and owned by foreign inhabitants N j=i Cji,t , minus physical  capital that is found outside region i but owned by its citizens N j=i Cij,t . The stock of FDI owned by residents within the region (active FDI) is not expected to be

56

6 Structure of the Model

identical to the stock of FDI within the regions which is held by residents of foreign regions (passive FDI), therefore in general we have N  j=i

Cji,t =

N 

Cij,t

(6.3)

j=i

Setting the focus back to the production function, notation and terminology are kept identical to Part I: the assumption of constant returns allows one to express the factors of production as units of effective labour of i at t as Ki,t kˆ i,t = At Li,t

(6.4)

Hi,t hˆ i,t = At Li,t

(6.5)

where kˆ i,t is physical capital per unit of effective labour in i at t, and hˆ i,t is human capital per unit of effective labour. It follows from Eqs. (6.1), (6.4) and (6.5) that output per unit of effective labour qˆ i,t equals a ˆb qˆ i,t = kˆ i,t hi,t

(6.6)

which is identical to the Mankiw-Romer-Weil model, as displayed in (Eq. 2.25). Expressing output per unit of effective labour corrects output for labour force size (which is assumed to equal population size) and technological progress. This makes the analysis more convenient when studying and comparing the relative development of the various regions. Moreover, it should be noted that for any i at any t the following relation between output per worker (equivalent to output per capita) qi,t , and output per unit of effective labour qˆ i,t , holds: qi,t = At qˆ i,t

(6.7)

Under full employment, perfect competition and profit maximising behaviour of companies, compensation at marginal product applies. This has an interesting implication for equilibrium wages: marginal product of unskilled labour equals     Ki,t a Hi,t b ∂Qi,t = (1 − a − b) At = (1 − a − b)At qˆ i,t ∂Li,t At Li,t At Li,t

(6.8)

From the assumption of compensation of marginal products it follows that one unit of human capital is compensated as       Ki,t a Hi,t b Hi,t −1 qˆ i,t ∂Qi,t =b =b ∂Hi,t At Li,t At Li,t At Li,t hˆ i,t

(6.9)

6.2

The Production Function

57

Although human capital represents a specific type of capital, it is embodied in the workers and as such paid as wages. The double role of human capital leads to the conclusion that the total equilibrium wage in Vi,t is captured by the sum Vi,t = Li,t

∂Qi,t ∂Qi,t + Hi,t ∂Li,t ∂Hi,t

(6.10)

From the production function of Eq. (6.1), it follows that Vi,t equals Vi,t = (1 − a − b)Qi,t + bQi,t = (1 − a)Qi,t

(6.11)

Dividing Vi,t by Li,t thus yields average equilibrium wage vi,t : a ˆb vi,t = (1 − a)At kˆ i,t hi,t

(6.12)

It can be seen from Eq. (6.12) that increases in physical capital as well as human capital will always have a positive effect on the average wage. Thus average wage will always rise if the amount of any type of capital in the region rises, and it will decline if any type of capital migrates. Furthermore, the double role of human capital leads to the important conclusion that if human capital is allowed to migrate, the propensity to migrate will not be determined by human capital’s marginal product, but rather by expected wage. Also note that if human capital is assumed to be evenly distributed among one region’s workers, the expected wage for any worker equals vi,t . Finally, we can also express compensation for aggregate physical capital within i at t, Gi,t , as ∂Qi,t = aKi,t Gi,t = Ki,t ∂Ki,t



Ki,t At Li,t

a 

Hi,t At Li,t

b 

Ki,t At Li,t

−1

= aQi,t

(6.13)

Note that the openness of regions implies that Ki,t = Ci,t , hence there are income transfers from capital. In particular, we may split total income from capital with respect to its ownership, so we have: Gi,t = a(Ci,t +

N 

Cji,t −

j=i

= aQi,t

Ci,t −

N

N 

b Cij,t )a Hi,t (At Li,t )1−a−b

j=i

j=i Cij,t

Ki,t

(6.14)

N + aQi,t

j=i Cji,t

Ki,t

From Eq. (6.14), we can derive total income from capital generated that is owned by the inhabitants of i at t, Ji,t , as Ci,t Ji,t = aQi,t − aQi,t Ki,t

N

j=i Cij,t

Ki,t

+a

N  j=i

Qj,t

Cij,t Kj,t

(6.15)

58

6 Structure of the Model

which equals the sum of income generated by physical capital owned by inhabitants of i. From Eqs. (6.13) and (6.14) it can be seen that a considerable amount of total income generated within one region may be earned abroad: if the second term of Eq. (6.15) takes on a larger number than the third term, the region experiences net outflows of profits (translated into the terminology of national accounts, GDP would be lower than GNI). Let’s now consider the superordinate economy, whose aggregate output is given by N 

Qi,t = A1−a−b t

i=1

N 

a b 1−a−b Ki,t Hi,t Li,t

(6.16)

i=1

which is simply the sum of output of all regions under consideration. Nevertheless, a number of key conclusions can be drawn: firstly, the superordinate economy exhibits constant returns as well; secondly, both space and time enter the production function only indirectly through K, H, A and L. These four variables will change over time, and the respective evolution may be influenced by the relative location in space. Thus, the development of the superordinate economy is of interest in its own right. In other words, the evolution of factors of production is determined by time and by the interdependence of the regions, which is why the spatial distribution of factors of production within the superordinate economy has an influence on output growth of the superordinate economy. Note that since the superordinate economy is assumed to be closed, the assumption that output equals income is fulfilled at any time for the superordinate economy. This can be seen by splitting physical capital in Eq. (6.16) in terms of ownership, by inserting Eq. (6.2): N 

Qi,t = A1−a−b t

i=1

N 

(Ci,t +

N 

Cji,t −

j=i

i=1

N 

b 1−a−b Cij,t )a Hi,t Li,t

(6.17)

j=i

From Eqs. (6.11) and (6.15), we know that the sum of incomes earned equals N  i=1

Vi,t +

N  i=1

Ji,t = (1 − a)

N 

Qi,t + a

i=1

N 

⎛ ⎝Qi,t

Ci,t −

N

j=i Cij,t

Ki,t

i=1

+

N  j=i

⎞ Qj,t

Cij,t ⎠ Kj,t (6.18)

 N Since Ki,t = Ci,t + N j=i Cji,t − j=i Cij,t ∀ i, after some cancelling we can express Eq. (6.18) simply as N  i=1

Vi,t +

N  i=1

Ji,t =

N  i=1

Qi,t

(6.19)

6.3

Changes of Inputs to Production

59

which shows that income equals output for the superordinate economy at any t, while we know from inequation (6.3) that for each individual region Vi,t + Ji,t = Qi,t

(6.20)

Equation (6.19) and inequation (6.20) emphasise the distinction between income and output. With the underlying assumptions, a considerable mobility of physical capital investments is expected. As discussed in Sect. 5.2, the identification of ownership on a regional level is both difficult and of no great relevance in practice. Nevertheless, some conclusions concerning regional income distribution can be derived; this issue will be taken up in Sect. 7.2. Before that, the focus is set entirely on distribution of production.

6.3 Changes of Inputs to Production As is the case in both the original Solow model as well as the Mankiw-Romer-Weil model, the initial levels of labour and technology are taken as given and grow at constant rates, therefore the labour force growth rate equals L˙ i,t = ni Li,t

(6.21)

and the level of technology grows at A˙ t = gAt

(6.22)

where a dot over a variable denotes a derivative with respect to time, and ni and g are exogenous parameters. Parallel to the condition that At is the current level of technology available to all regions throughout the superordinate economy, the growth rate of technology g applies to all. In addition, the initial labour force Li is set to be the same for all regions, as is the growth rate ni of region i’s labour force, so that Li,0 = Lj,0 for all i = j, with ni = nj = n, and consequently at any t Li,t = Lj,t

(6.23)

From Eqs. (6.1), (6.21), (6.22) and (6.23), it follows that aggregate output in i at t can be expressed as a b Hi,t (A0 Li,0 )1−a−b Qi,t = e(1−a−b)(g+n)t Ki,t

(6.24)

where A0 and Li,0 are the respective values of A and L. It follows that output per worker (output per capita) in i at t equals: qi,t = A0 egt qˆ i,t

(6.25)

60

6 Structure of the Model

From the assumption of a closed superordinate economy, it follows that the sum of aggregate gross investment in regions i = 1, 2, ..., N equals the sum of aggregate saving taking place in all regions, so that saving equals investment within the superordinate economy.2 Investments may take place in any region inside the superordinate economy. The propensity to invest is assumed to depend on the rate of return, which is determined by the marginal productivity of physical capital. The model hence incorporates what may be the most important interpretation of neoclassical growth theory, namely the issue of gross fixed capital relocations. The assumption of open economies allows to determine investment flows endogenously. The higher physical capital’s marginal productivity within one region, the more attractive this region is supposed to be to potential investors; consequently, net inflows of FDI are expected. Disequilibrium regarding physical capital’s marginal product, however, is not corrected immediately: in any region i a share of output is reinvested, whether by means of domestic saving, borrowing or government investment. In addition, it is assumed that the rate of investment is also affected by the attractiveness of other regions: if marginal productivity of physical capital at t in region j is higher than in region i, the investment-output ratio in region j is ceteris paribus higher than in region i. Furthermore, the net flow of investments from region i to j is assumed to depend on proximity, that is, the closer two specific regions are, the higher the flows. Based on these assumptions, the function of aggregate physical capital growth for region i at point in time t is the key equation of the model and takes the following form: K˙ i,t = sK μt Qi,t

 N   ∂Qi,t /∂Ki,t λwij j=i

∂Qj,t /∂Kj,t

− dKi,t

(6.26)

sK is the saving rate of all regions, i.e. the saving rate of the superordinate economy. d is the rate of depreciation, λ is a measure of the degree of integration and μt is a variable identical to all regions and defined in Eq. (6.29) below. The connectivity term wij measures interdependence of two regions i and j. A region j belongs to the neighbourhood of i if wij > 0, which means that the current combination of factors in production of region j relative to region i affects the capital-accumulation equation of i if it is considered to be a neighbour, and vice versa. The concept of neighbourhood is defined as occurring in some space, therefore related regions will be referred to as “neighbours.” This does not necessarily refer to map-based contiguity relations only, but rather a more general sense of relatedness. In such a broader sense of relatedness, which may include cultural connectivity or trade relations among others, the determination of the values of the connectivity terms wij need not be a simple function of geographical distance. Furthermore, the connectivity terms are defined by the

2 Equivalently, it may also be assumed that the superordinate economy always runs a balanced current account: this would leave the analysis unchanged, but Ci would become unidentifiable.

6.3

Changes of Inputs to Production

61

same attributes as in the model presented in Sect. 3.2, or formally for j = 1, 2, ..., N, the terms wij are assumed to be: 0 < wij < 1

N j=i

non-negative, non-stochastic and finite

wij = 0

if i = j

wij = 1

for i = 1, ..., N

Note that though the sum of connectivities equals one for each economy, in cases where two economies j and j are both neighbours to i, it is not required that wij = wij . Given the assumption of a closed superordinate economy, the superordinate economy’s net investment at t must equal the sum of all net investments of all regions at t: N 

K˙ i,t = sK μt

i=1

N 

⎛ ⎝Qi,t

 N   ∂Qi,t /∂Ki,t λwij j=i

i=1

∂Qj,t /∂Kj,t

⎞ ⎠−d

N 

Ki,t

(6.27)

i=1

To ensure that the sum of net inflows of FDI of all regions equals zero, the following equation has to hold at any t:

sK

N  i=1

Qi,t − d

N 

Ki,t = sK μt

i=1

N 

⎛ ⎝Qi,t

 N   ∂Qi,t /∂Ki,t λwij j=i

i=1

∂Qj,t /∂Kj,t

⎞ ⎠−d

N 

Ki,t

i=1

(6.28) Therefore, μt is dependent on the current spatial distribution of factors and equals: N 

μt =

N  i=1

 Qi,t

Qi,t

i=1

N  j=i



∂Qi,t /∂Ki,t ∂Qj,t /∂Kj,t

λwij  > 0

(6.29)

From Eq. (6.29) it can be seen that μt will always be greater than zero, since both the nominator and the denominator must be positive. It therefore has an influence on the magnitude of investments that occur at each point in time, but does not change the analysis qualitatively. Let’s now have a closer look at the relation between expected rates of return and foreign direct investments between regions, where the former should have a positive influence on the latter, depending on connectivity: different definitions of connectivity will result in different results concerning the number of and the relative distance to neighbours. In the Cobb-Douglas case, Eq. (6.26) equals

62

6 Structure of the Model

K˙ i,t = sK μt Qi,t

 N   Qi,t Kj,t λwij j=i

Qj,t Ki,t

− dKi,t

(6.30)

˙ Since we have qˆ i,t = Qi,t /At Li,t and kˆ i,t = Ki,t /At Li,t , kˆ i,t equals  λwij N  qˆ i,t kˆ j,t K˙ i,t At Li,t − Ki,t A˙ t Li,t − Ki,t At L˙ i,t ˙ˆ ki,t = = sK μt qˆ i,t − (n + g + d)kˆ i,t 2 ˆ i,t A2t Li,t q ˆ k j,t j=i (6.31) Finally, after some rearranging by implementing the production function, change in the physical capital stock per unit of effective labour can be expressed as  −(1−a) λwij N  kˆ i,t hˆ bi,t ˙ˆ a ˆb ˆ ki,t = sK μt ki,t hi,t − (n + g + d)kˆ i,t −(1−a) ˆ b kˆ h j=i

j,t

(6.32)

j,t

It can be seen that even though the production function of Eq. (6.1) does not differentiate between foreign and domestic capital, it determines the evolution of the region: if marginal product of physical capital in region i is higher than in any connected region j = i, it will receive FDI from this region, and vice versa. From Eq. (6.32) it becomes clear that marginal product of physical capital is not a function of the stock of physical capital alone, but also of the stock of human capital. Put differently, region i will receive a positive balance of gross investments from region j = i in the case of: a−1 ˆ b a−1 ˆ b hi,t > kˆ j,t hj,t kˆ i,t

(6.33)

Hence we see that if region i’s stock of human capital is higher than the stock of human capital in region j = i, region i attracts FDI even if its stock of physical capital is higher, as long as this difference remains small enough. By dividing a−1 inequation (6.33) by hˆ bj,t and kˆ i,t , this condition can be expressed formally as 

hˆ i,t hˆ j,t

b

 >

kˆ i,t kˆ j,t

1−a (6.34)

In addition, Eq. (6.32) may also be expressed as a function of the current levels of output and human capital, as such illustrating the influence of two ratios on the balance of gross investments: gross investments will be higher if the endowment

6.3

Changes of Inputs to Production

63

with human capital in region i is higher, or if the level of output is lower than in a neighbouring region j, and vice versa:  λwij b   1−a N a  qˆ j,t λwij a hˆ i,t ˙ˆ ki,t = sK μt qˆ i,t − (n + g + d)kˆ i,t ˆ j,t qˆ i,t h j=i

(6.35)

Equation (6.35) therefore exemplifies how the amount of gross investments in i is influenced by the interplay of human capital and output in all regions of the superordinate economy.3 In contrast to physical capital, human capital is considered to be immobile, and its change in region i over time is given by the following equation: ˙ i,t = sH Qi,t − dHi,t H

(6.36)

where the evolution of human capital is determined by the fraction of output sH invested in education, which is considered to be fixed by the superordinate economy and to be identical for all regions. As is the case in the Mankiw-Romer-Weil model, it is assumed that human capital depreciates at the same rate d as physical capital. From Eq. (6.36) it can be seen that although human capital represents a specific kind of labour, it is treated fairly similarly to physical capital. Therefore, devoting more resources to the accumulation to either type of capital increases the amount of output that can be produced in the future. Moreover, Eq. (6.36) may also be expressed as the change of human capital per unit of effective labour: ˙ hˆ i,t = sH qˆ i,t − (n + g + d)hˆ i,t

(6.37)

Equation (6.37) adheres to the same critical assumptions as those found in the evolution of capital stocks in the Solow model and in the Mankiw-Romer-Weil model. In particular it can be seen that ceteris paribus the growth rate of human capital will be higher, the lower the current stock is. Break-even investment of human capital per unit of effective labour is determined by population growth n, technological progress g, and depreciation d.

3 The capital accumulation equation as given in Eq. (6.35) will serve as the basis for the major part of what follows in this study. In particular, relying on output and human capital as explanatory variables makes empirical interpretations of the model easier, as data on physical capital stocks are most often unavailable (and when available, they should be handled with care), while human capital may be proxied by educational attainment. Arguably, by conception physical capital is less abstract than its human counterpart, if we take into account that the expressions “skills of workers” and “depreciation of human capital” are common in the literature but nevertheless a bit clumsy, and impossible to measure directly.

Chapter 7

Evolution of Factors and Output

The purpose of this chapter is to discuss the dynamic developments of the model presented in the previous chapter. In Sect. 7.1, the model’s long run solutions are derived, and it is shown that under the given assumptions, all regions converge to the same stable output levels in the long run. An eventual outcome, possibly beyond our lifetimes, may be of limited interest; for this reason, a number of medium run scenarios of regional growth are discussed comprehensively in Sect. 7.2. The model makes clear predictions concerning mutual interdependence, initial output levels and stocks of capital, which are derived formally in Sect. 7.3.

7.1 Steady States From the outline of the model, we find that all regions of the superordinate economy face identical conditions except for their initial endowments with physical capital, human capital, and individual connectivity terms. In steady state, the difference between the two terms on the right-hand side of Eq. (6.37) equals zero, therefore in steady state for any region i the following condition must hold: ˆ∗ sH kˆ i∗a hˆ ∗b i = (n + g + d)hi

(7.1)

where kˆ i∗ and hˆ ∗i denote the steady state levels of physical capital per effective labour and human capital per effective labour, respectively. Solving Eq. (7.1) for hˆ ∗i implies that any region i will eventually arrive at the steady state level of human capital per unit of effective labour:  hˆ ∗i

=

sH kˆ i∗a n+g+d



1 1−b

(7.2)

From Eq. (6.32) we see that, as long as marginal productivities of physical capital in any two regions i and j are not identical, we will observe net flows of FDI between these two regions. This process will go on until marginal productivity of physical capital is identical in each region: in equilibrium, different rates of return in different S. Sardadvar, Economic Growth in the Regions of Europe, Contributions to Economics, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-7908-2637-1_7, 

65

66

7 Evolution of Factors and Output

regions have been balanced out, and consequently there is no net outflow or inflow of FDI in any region. Therefore, in equilibrium Eq. (6.29) reduces to: N Qi,t =1 μt = i=1 N i=1 Qi,t

(7.3)

From Eq. (6.32) we find that, if marginal productivity of physical capital is identical in every region, then every ratio of marginal product equals one, and hence in equilibrium the steady state condition equals: ˆ∗ sK kˆ i∗a hˆ ∗b i = (n + g + d)ki

(7.4)

By inserting Eq. (7.2) into Eq. (7.4) and solving for the steady state level of physical capital per unit of effective labour, we get:  kˆ ∗ =

b s1−b K sH n+g+d



1 1−a−b

(7.5)

which is identical for all regions. It is now straightforward to calculate steady state human capital per unit of effective labour:  hˆ ∗ =

saK s1−a H n+g+d



1 1−a−b

(7.6)

and steady state output per unit of effective labour:  ∗

qˆ =

saK sbH (n + g + d)a+b



1 1−a−b

(7.7)

Note that Eqs. (7.5), (7.6) and (7.7) are identical to the corresponding Eqs. (2.28), (2.29) and (2.30) of the Mankiw-Romer-Weil model. From Eqs. (6.25) and (7.7) it follows that if all regions are at their steady state levels, output per worker of region i at t equals  q∗t

= A0 e

gt

saK sbH (n + g + d)a+b



1 1−a−b

(7.8)

Due to the fact that no interaction between regions takes place in the respective accumulation equation, the steady state level of human capital per effective labour for any region i remains unaffected by any region j. The result of identical steady state levels of physical capital for all regions can be demonstrated by considering the

7.1

Steady States

67

evolution of physical capital as a system of equations. In equilibrium, the following condition must hold for any region i: a ˆb sK μt kˆ i,t hi,t

 −(1−a) λwij N  hˆ bi,t kˆ i,t j=i

−(1−a) ˆ b kˆ j,t hj,t

= (n + g + d)kˆ i,t

(7.9)

After inserting Eq. (7.2) into Eq. (7.9), rearranging and taking the natural logarithm of both sides we get: (1 + λ) ln kˆ i,t − λ

N 

wij ln kˆ j,t = ξt

(7.10)

j=i

where ξt may be interpreted as a condition for steady state, equalling ξt =

1 [b ln sH + (1 − b) ln sK − ln(n + g + d)] − μt 1−a−b

(7.11)

Equation (7.10) must hold for all regions if all regions are in steady state, and can be transformed into matrix form for N regions at t as: ∗ ∗ ∗ kˆ + [diag(Wλ)]kˆ − Wλkˆ = ξt ι

(7.12)

* where λ and ξt are scalars, kˆ is an N × 1 vector of the natural logarithms of the steady state levels of physical capital per effective labour, and W is an N × N matrix whose elements are the connectivity measures wij . diag(Wλ) is the corresponding main diagonal from matrix Wλ, and ι is an N × 1 vector of ones. Therefore we have

⎞ ln kˆ 1∗ ⎜ ln kˆ ∗ ⎟ ∗ ⎜ 2⎟ kˆ = ⎜ . ⎟ . ⎝ . ⎠ ⎛

ln kˆ N∗

and ⎛

w11 w12 ⎜ w21 w22 ⎜ W=⎜ . .. ⎝ .. . wN1 wN2

⎞ . . . w1 N . . . w2 N ⎟ ⎟ . ⎟ .. . .. ⎠ . . . wNN

Note that wii = 0 by conception, which is why the diagonal of W solely consists of zeroes. When all regions are at their steady state levels, all levels of physical capital

68

7 Evolution of Factors and Output

are identical. To see this, let’s express Eq. (7.12) as a system of N equations with N unknowns, which must hold if all regions are at their steady states at any t: [1 + +λ(w11 + w12 + w13 + ... + w1 N )] ln kˆ 1∗ − λw11 ln kˆ 1∗ − λw12 ln kˆ 2∗ − ... − λw1 N ln kˆ N∗ = ξt [1 + λ(w21 + w22 + w23 + ... + w2 N )] ln kˆ 2∗ − λw21 ln kˆ 1∗ − λw22 ln kˆ 2∗ − ... − λw2 N ln kˆ N∗ = ξt .. . [1 + λ(wN1 + wN2 + wN3 + ... + wNN )] ln kˆ ∗ − λwN1 ln kˆ ∗ − λwN2 ln kˆ ∗ − ... − λwNN ln kˆ ∗ = ξt N

1

2

N

It can be readily checked by cancelling the terms λwij ln kˆ j∗ that for any N, the solution of this system of equations for any ln kˆ i∗ = ξt . It follows from Eqs. (7.3) and (7.11) that Eq. (7.12) holds if and only if all regions are in steady state. Therefore, the steady state levels of physical capital per effective labour are identical for all regions under consideration, as given in Eq. (7.5). The implications  for growth of the superordinate economy are as follows: during transition period, N i=1 Qi,t will be higher, the higher the integration parameter λ is, leading to increased output for the entire economic area  under consideration. Since the superordinate economy is assumed to be closed, N i=1 Qi,t equals its aggregate income, which is hence also higher. The integration process accelerated by nonzero values of λ will eventually come  to ∗a halt, when the superordinate economy has reached its steady state, with N i=1 Qi,t , so that we get a steady state aggregate income at t of: N 

 Q∗i,t

=e

(g+n)t

i=1

saK sbH (n + g + d)a+b



1 1−a−b

N 

Li,0 A0

(7.13)

i=1

7.2 Simulation Results In the previous section, it has been shown that all regions within the superordinate economy converge to their steady state levels of factors and hence output, which are identical for all regions. This result implies that, given that the regions under consideration have different initial factor-endowments, they will converge to each other with respect to factors and output. The solution of steady states makes no prediction on the length of this transition period, however: long run development regarding steady states and convergence may refer to a very long time, possibly beyond a lifespan.1 It follows that, even though identical steady states are assumed, of primary interest should be the development in the medium run (i.e. during the transition period). Any region i will develop individually as long as kˆ i,t and hˆ i,t are different to their steady state levels, and even in the case when they equal their steady 1 To give just one example, a convergence speed of 2% annually, regularly found in empirical studies, means that it takes about 35 years to reach half of the target value.

7.2

Simulation Results

69

state levels but those of the other regions do not. In a discrete-time case, growth from one period to the next depends on the region’s own current values of both types of capital, and those of the other regions in connection with the integration parameter λ and the connectivity terms wij . If any wij = 0, the corresponding region is influenced indirectly in the next or subsequent periods, since all regions are assumed to be indirectly connected to each other (otherwise the system would consist of more than one superordinate economy). For the following simulation studies, let’s consider a superordinate economy that consists of twelve regions with arbitrarily defined borders and roughly realistic initial values. In particular, let a = 0.3, b = 0.2, n = 0.01, g = 0.02, d = 0.05, λ = 1.2, sK = 0.3, sH = 0.1, A0 = 1, and L0 = 100. These values result in steady state levels of kˆ ∗ = 9.062, hˆ ∗ = 3.021 and qˆ ∗ = 2.416, thus ln kˆ ∗ = 2.204, ln hˆ ∗ = 1.105 and ln qˆ ∗ = 0.882. Consider the vector of natural logarithms of initial physical capital stocks per unit of effective labour at the beginning of the observation period, kˆ 0 , to equal   ’ kˆ 0 = 1.10 1.95 1.61 2.20 2.21 1.61 1.87 1.79 2.08 0.10 0.10 0.10 and consider the vector of natural logarithms of initial physical capital stocks per unit of effective labour at the beginning of the observation period, hˆ 0 , to equal   ’ hˆ 0 = 0.00 −0.69 0.92 1.10 1.13 0.83 0.79 0.96 1.10 0.00 0.00 0.00 From these values it follows that μ0 = 1.28. Furthermore, note that all values are greater than zero, but the vectors’ elements are the respective natural logarithms and hence contain zeroes in the case where hˆ i,0 = 1 and a negative number because hˆ 2,0 = 0.5. Furthermore, consider the matrix of connectivities to equal ⎛

0 ⎜ 0 ⎜ ⎜ 0.2 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0.2 W=⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎝ 0.5 0

0 0 0 0 0.25 0.2 0.33 0 0 0 0 0

0.33 0 0 0.33 0 0.2 0 0 1 1 0 0

0 0 0.2 0 0.25 0.2 0 0 0 0 0 0

0 0.33 0 0.33 0 0.2 0.33 0 0 0 0 0

0.33 0 0.33 0.33 0.2 0 0.33 0 0.25 0.25 0 0 0 0 0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0.33 0 0 0 0 0

0 0 0.2 0 0 0 0 0 0 0 0 0

0 0.33 0 0 0.2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

⎞ 0 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0.5 ⎠ 0

In accordance to the conditions regarding the elements wij as given in Sect. 6.3, W represents a row-standardised matrix. If two regions are neighbours, the element of

70

7 Evolution of Factors and Output

the respective row and column is non-zero; if they are not neighbours, it is zero. The information given in these two vectors and the matrix reads as follows: Region 3 (henceforth “R3”) has an initial level of physical capital per effective labour unit kˆ R3,0 = 5 (or ln kˆ R3,0 = 1.61), and of human capital per effective labour unit hˆ R3,0 = 2.5 (or ln qˆ R3,0 = 0.92), which result in an initial level of output per effective labour unit qˆ R3,0 = 1.95 (or ln qˆ R3,0 = 0.67); R3’s neighbours are R1, R4, R6, R9 and R10. Figure 7.1 plots the natural logarithms of output per unit of effective labour, of the twelve regions as well as the superordinate economy in discrete time for the first 100 periods. The superordinate economy’s output at t is the mean of the output  q ˆ , and is drawn as the bold line; output levels of the of the twelve regions, 12 i=1 i,t twelve regions are drawn in thin lines. The figure reveals two characteristics of the model: firstly, in the long run, all regions converge to the same steady state level of output. Secondly, in the medium run, some regions’ output levels diverge from each other. Taking a closer look at the figure, one can spot that before eventual long run convergence appears, some regions that start at identical levels diverge from each other, some regions get passed over by others, and one region grows before it starts to shrink. These processes can be examined more deeply by taking close-ups of transition periods. Figure 7.2 plots ln qˆ i,t of regions R1, R3, R10, R11 and R12, as well as the superordinate economy (SE) for the first 20 periods. R10, R11 and R12 all start at identically low levels of both types of capital and hence output. Both R10 and R11 have neighbour regions with higher levels of output, but R10’s neighbour, R3, is more developed than R11’s neighbour, R1. Both R1 and R3 are connected to each other as well; R12 is a neighbour to R11 and hence indirectly connected to the developed regions. It can be seen that R10 grows faster than R11, and R11 grows faster than R12, with the accompanying levels of output staying different for a considerably long time. In particular, from Fig. 7.2 it can be seen that (i) regions with low initial levels of output will grow faster the higher their neighbours’ initial levels of output (R10 and R11), (ii) a region with a low initial level of output will grow more

ln qˆ

0

Fig. 7.1 Long run development of output

t

7.2

Simulation Results

71

ln qˆ

t

0 R1

R3

R10

R11

R12

SE

Fig. 7.2 Medium run development of regions with low initial output levels

slowly than its neighbours with low initial levels of output if the latter (R10 and R11) are neighbours to regions with high initial levels of output, but the former (R12) is not, (iii) the development of a region with a medium initial level of output whose neighbours have both low and high initial levels of output is ambiguous, as it experiences both outflows of investments to less developed regions as well as inflows from more developed regions, and (iv) all three regions with lower initial output levels will grow faster than the regions with higher levels of output, so one observes convergence on a global view as well as between R1 and R3, but divergence within the regions R10, R11 and R12. Figure 7.3 does the same as Fig. 7.2, but this time ln qˆ i,t of regions R3, R4, R5, R6 and R9 is displayed. R5 is the only region whose initial output level is (slightly) above steady state, hence it is represented by the uppermost line. R4 has the second highest initial output level, which is slightly below steady state. Also R3, R6 and R9 have initial levels of output above the mean, but below steady state. It can be seen that R4 and R5, which are (amongst others) neighbours to each other, both experience negative growth rates, but with different developments: R4 first shrinks before starting to grow again, while R5 grows, then shrinks, then grows again. Note that R5 is a neighbour to R2 (not plotted in Fig. 7.3), which has a medium initial level of output, but a very low initial level of human capital. Meanwhile, both R3 and R6 grow positively, but R3 gets passed over by R6. Figure 7.3 thus visualises a number of interesting scenarios, as it can be seen that (i) a region may experience a positive growth rate even in the case if it is above its steady state as a matter of its attractiveness to investments, in particular if its endowment with human capital is relatively high so that inequation (6.34) holds (R5 versus R2), (ii) such a region however must eventually experience negative growth rates per unit of effective labour, as all

72

7 Evolution of Factors and Output

ln qˆ

t

0 R3

R4

R5

R6

R9

SE

Fig. 7.3 Medium run development of regions with high initial output levels

regions converge to identical steady state levels (R5), (iii) a region that starts below steady state may experience negative growth rates too, if its initial output level is relatively high (R4), (iv) regions may experience relatively low output growth rates if their neighbours are very attractive, and for that reason even get passed over by other, relatively high developed regions (R3 and R6), (v) evidence of convergence or divergence within these regions depends heavily on the respective observation period. Figure 7.4 plots ln qˆ i,t for R8, R10 and SE in comparison to additional economies that remain closed, labelled I8, I10 and IM. These additional economies start out at identical levels of factors as their counterparts, so we have kˆ I8,0 = kˆ R8,0 , hˆ I8,0 = hˆ R8,0 and so on. The closed economies therefore represent a pure Mankiw-RomerWeil scenario, and it can be seen how all six converge to the same steady state levels of factors and output, but with different courses. In particular, those economies that are part of the superordinate economy grow faster. For instance, note that R8 is a neighbour to R7, from which it receives net FDI inflows during the first 20 periods. Furthermore, it can be seen that the superordinate economy’s counterpart, IM, starts out with a higher level of output, but soon gets passed over. Therefore, from Fig. 7.4 it can be concluded that (i) entry to a more highly developed superordinate economy raises output of less developed regions as a matter of FDI (R10), (ii) highly developed regions gain from openness as well, but regarding output they only gain if they are attractive to investments (R8), (iii) disequilibrium with respect to marginal productivity of physical capital results in inefficient production within the superordinate economy, hence output is lower than in a counterpart with the same initial values (SE and IM), but (iv) due to the superordinate economy’s regions’ convergence processes to their respective steady states and

7.2

Simulation Results

73

ln qˆ

t

0 R8

I8

R10

I10

SE

IM

Fig. 7.4 Medium run development compared to closed economies

simultaneous adjustment of inefficient allocation of physical capital, after some time the superordinate economy passes over its counterpart. In Fig. 7.5, the vectors kˆ 0 and hˆ 0 as well as the matrix W have been slightly changed. In particular, now ln kˆ R10,0 = −0.11, ln kˆ R11,0 = 0.26, ln hˆ R10,0 = 0.30, ln hˆ R11,0 = −0.24, and we have R1 and R10, R3 and R11, R10 and R11, as well as R10 and R12 as neighbours. Expressed verbally, R10 and R11 have the same initial output levels as before, but now R10 has a higher initial endowment with human capital, and a lower endowment with physical capital than R11. Furthermore, R1 and R3 now both neighbour R10 and R11, and R12 is a neighbour to R10 and R11, and vice versa in each case. Figure 7.5 plots ln qˆ i,t of R3, R4, R9, R10 and R11 as well as SE for the modified data. It can be seen that R10 grows much faster than R11 as a result of its higher endowment with human capital. R11’s output growth is still higher than in the scenario above, as it is now a direct neighbour to more highly developed regions and thus attracts more FDI. While R9 continues to grow from its initial high level thanks to its rich endowment with human capital, the significance of interdependent processes of R3 and R4 is as follows: R3 is attractive to investments from R9, but at the same time experiences capital outflows to R10 and R11. These outflows in turn attract inflows from R4, which grows only slowly at the beginning of the observation period, before starting to shrink. Note that R4 is neither a neighbour to R10, R11 or R12, while R3 neighbours all of the plotted regions. From Fig. 7.5 it can be concluded that (i) the initial endowment with human capital plays a decisive role in regional growth in the medium run, especially if the accompanying output level is low, (ii) regions may experience fluctuating and temporarily negative growth rates as a result of the interdependence of regions, (iii) ceteris paribus the relative location in space determines the flow of investments a region receives, and (iv) peripheral regions with low initial levels of output become

74

7 Evolution of Factors and Output

ln qˆ

t

0 R3

R4

R9

R10

R11

SE

Fig. 7.5 Medium run development in some special cases

more attractive for investments as the level of output in their neighbouring regions grows. In conclusion, it may be stated that the model’s predictions on growth and convergence depend on the initial conditions of one region’s stock of human capital and physical capital, and its location in space: the more a region is connected to more developed regions, the more it will gain, especially if its stock of human capital is high. It is a crucial characteristic of the model that it predicts faster growth in the medium run for the superordinate economy as well, as physical capital is allowed to be invested where expected returns are higher. This, however, implies that those regions with initially relatively high levels of physical capital may observe a lower rate of new investments than would otherwise be the case (i.e. with a lesser degree of integration or no integration at all). Hence at first sight it seems that continuing integration is not to the benefit of those with initially high levels of physical capital, as continued net investment would imply declining yet positive returns. This does not mean that income is assumed to be lower than would otherwise be the case: indeed, people who own physical capital are expected to get richer due to the enhanced prospects of investments. Since these investments will necessarily take place outside the region, its output will nevertheless be lower as would be the case if there was no integration. The simulation results above also serve to illustrate a scenario where some regions enter an already existing superordinate economy. Due to the more efficient allocation of physical capital, the aggregate superordinate economy benefits from integration. Therefore, the effect on the share of income that compensates labour (wage share) varies from region to region. Regional income may deviate significantly from regional output as discussed in Sect. 5.2 and shown in Sect. 6.2:

7.3

Formal Derivations

75

Eqs. (6.11) and (6.12) display the parts of output of a region that compensate labour, but they are not equations related to the regional wage share, as this is defined to include any income from physical capital that is owned by the region’s residents. Put differently, if ownership of physical capital is not evenly distributed among the residents of a region, the income gap within the region may widen in the case of regional integration. In cases where regions experience negative growth rates, it may even be the case that the average wage shrinks while income from physical capital rises. A region’s wage share is expected to decline if it experiences continuing net outflows of FDI, as the growth rate of wages is expected to be slower than the growth rate of income generated by physical capital.2 The case of income distribution will not be studied in further detail in this work, but it should be stressed that a cross-regional harmonisation of wages and a cross-regional harmonisation of incomes do not necessarily coincide.

7.3 Formal Derivations The previous section’s simulations focussed on possible occurrences of simultaneous convergence and divergence tendencies between regions during transition periods. The discussion emphasised overall development as such or details thereof. From the perspective of a particular region, however, it is more revealing to study influences from outside: how is region i affected by region j? Since μt does not change the analysis qualitatively, it does not need to be examined in detail and is set to μt = 1 for any t in what follows.3 Taking the natural logarithm of the production function per unit of effective labour of Eq. (6.6) yields: ln qˆ i,t = a ln kˆ i,t + b ln hˆ i,t

(7.14)

By differentiating Eq. (7.14) with respect to time, we get the differential equation ˙ ˙ d ln qˆ i,t hˆ i,t kˆ i,t +b =a dt kˆ i,t hˆ i,t

(7.15)

2 For the reasons outlined in Sect. 5.2, there is no data on regional wage shares, but the mechanism is identical if countries within the EU are considered as regions in the sense of this study; the decline of the wage share of Austria as a consequence of FDI outflows to Central and Eastern European countries has recently been explored by Breuss (2007). 3 μ is always greater than zero, but under specific circumstances it may be smaller than one. In t that case, some regions might experience gross investments that are lower than would be the case if they were closed economies, even though their net inflow of FDI is positive.

76

7 Evolution of Factors and Output

Inserting Eqs. (6.35) and (6.37) into (7.15) and rearranging yields   bλwij   (1−a)λwij N a −(1−a) b  a qˆ j,t hˆ i,t qˆ i,t d ln qˆ i,t + bsH −(a+b)(n+g+d) = asK qˆ i,t a hˆ i,ta ˆ dt qˆ i,t hˆ i,t j=i hj,t (7.16)  where we can make use of the condition that N j=i wij = 1 ∀ i to separate the influˆ ˆ ence of qˆ i,t and hi,t as well as qˆ j,t and hj,t within the term on the left-hand side, and thus express Eq. (7.16) as d ln qˆ i,t = asK qˆ i,t dt

−(1−a)(1+λ) a

b(1+λ) a

hˆ i,t

 1−a  λwaij N  qˆ j,t j=i

hˆ bj,t

+ bsH

qˆ i,t − (a + b)(n + g + d) hˆ i,t (7.17)

Equation (7.17) provides insights into the dynamics of the model for any particular region; to study the influence of a particular region j on the development of region i, we can rearrange Eq. (7.17): d ln qˆ i,t = asK qˆ i,t dt

−(1−a)(1+λ) a

+

b(1+λ) a



hˆ i,t

qˆ 1−a j,t hˆ bj,t

 λwaij

N 



j  =i j  =j

 qˆ 1−a j  ,t hˆ bj ,t

λwij  a

(7.18)

bsH qˆ i,t − (a + b)(n + g + d) hˆ i,t

where the influence of foreign regions is now split into one particular neighbouring region j and the remaining N − 2 regions, indicated by j  . Differentiating Eq. (7.18) with respect to output of a neighbouring region j at t yields ∂(d ln qˆ i,t /dt) = ∂ qˆ j,t −(1−a)(1+λ) a

(1 − a) λw s qˆ   ! ij K i,t

b(1+λ) a

hˆ i,t

(1−a)λwij −a a

qˆ j,t

>0





>0

−bλwij a

hˆ j,t

N  j  =i j  =j



qˆ 1−a j  ,t



λwij  a

>0

hˆ bj ,t

(7.19)

!

The right-hand side of Eq. (7.19) is unambiguously positive, since all variables are defined as being greater than zero. It follows that raising the output level of a neighbouring region j will lead to higher output growth in region i. The magnitude of this

7.3

Formal Derivations

77

positive influence is dependent on the relative and absolute values of the variables. For instance, it can be seen from Eq. (7.19) that the connectivity to a third region j  affects the influence of the output level of region j on i. This specific influence is determined by the ratios of levels of human capital and output of both i and j  . Differentiating Eq. (7.18) with respect to the level of human capital of a neighbouring region j at t yields ∂(d ln qˆ i,t /dt) = ∂ hˆ j,t −(1−a)(1+λ) a

−b! λwij sK qˆ i,t 

b(1+λ) a

hˆ i,t

(1−a)λwij a

qˆ j,t

0

−bλwij −a a

hˆ j,t

N  j  =i j  =j



qˆ 1−a j  ,t



λwij  a

0   !   !   ! 0 >0   !   ! >0

(8.7)

>0

which is unambiguously positive. Note that the right-hand side of Eq. (8.4) is equal to −aβ˜1 , and the right-hand side of Eq. (8.5) is equal to −bβ˜1 + bλ(n + g + d). Introducing β˜1 to Eq. (8.6) and rearranging yields d ln qˆ i,t ≈ −β˜1 (a ln kˆ i,t +b ln hˆ i,t )−β˜1 (a ln kˆ ∗ +b ln hˆ ∗ )+λ(n+g+d)(b ln hˆ i,t −b ln hˆ ∗ ) dt (8.8) Equations (8.6) and (8.8) are identical to the estimation of the convergence process in Mankiw-Romer-Weil model if the degree of integration, λ, equals zero – that is, if no interaction takes place. Besides the initial level of output, the actual pattern of growth in relation to other regions is also dependent on the connectivity to other regions and the initial level of human capital. Since the endowment with production factors at t enters Eqs. (8.6) and (8.8) as well, a reference to a speed of convergence would be misleading in the case of interdependence. From Eq. (7.14) we know that Eq. (8.8) is equal to d ln qˆ i,t ≈ −β˜1 (ln qˆ i,t − ln qˆ ∗ ) + bλ(n + g + d)(ln hˆ i,t − ln hˆ ∗ )  !   ! dt 0

(8.9)

8.1

Taylor Approximation

81

and by defining β˜2 = bλ(n + g + d) > 0

(8.10)

d ln qˆ i,t ≈ −β˜1 (ln qˆ i,t − ln qˆ ∗ ) + β˜2 (ln hˆ i,t − ln hˆ ∗ ) dt

(8.11)

we may express Eq. (8.9) as

which represents a system of first-order linear differential equations with constant coefficients and variable terms. Since the steady states of both factors of production and output are the same for all regions, we have two relations ln qˆ ∗i,t = ln qˆ ∗j,t = ln qˆ ∗

(8.12)

ln hˆ ∗i,t = ln hˆ ∗j,t = ln hˆ ∗

(8.13)

which must hold in steady state. At any t, none, some, or all regions are not in steady q state: Define zi,t = (ˆqi,t − qˆ ∗ )/ˆq∗ and zhi,t = (hˆ i,t − hˆ ∗ )/hˆ ∗ as region i’s percental deviation at t from its steady state levels. Then, if at any t we have qˆ i,t = qˆ ∗ , the following relationships must hold: qˆ i,t ˆ∗ q =q 1 + zi,t

(8.14)

hˆ i,t = hˆ ∗ 1 + zhi,t

(8.15)

q

It follows from Eqs. (8.14) and (8.15) that zi,t and zhi,t are negative if the region in question lies below steady state, and positive if it lies above steady state, and zero if it is exactly at steady state. From Eqs. (8.12) and (8.14) it follows that at any t we may express the steady state levels of output as ln qˆ ∗ = ln qˆ ∗i,t =

N 

wij ln qˆ j,t −

j=i

N 

q

wij ln(1 + zj,t )

(8.16)

j=i

and from Eqs. (8.13) and (8.15), the steady state levels of human capital follow as ln hˆ ∗ = ln hˆ ∗i,t =

N  j=i

wij ln hˆ j,t −

N  j=i

wij ln(1 + zhj,t )

(8.17)

82

8 Implications for Output Growth

Equations (8.16) and (8.17) hold at any t for any i. Inserting both into Eq. (8.11) we have now   N N   d ln qˆ i,t q ˜ ≈ −β1 ln qˆ i,t − wij ln qˆ j,t + wij ln(1 + zj,t ) dt j=i j=i (8.18)   N N   + β˜2 ln hˆ i,t − wij ln hˆ j,t + wij ln(1 + zhj,t ) j=i

j=i

From the algebraic signs of Eq. (8.18) it follows immediately that we expect for region i’s output growth • • • •

a negative influence by its own output level, a positive influence by its neighbours’ output levels, a positive influence by its own human capital endowments, a negative influence by its neighbours’ human capital endowments. q

Note that the terms ln(1 + zj,t ) and ln(1 + zhj,t ) are positive if region j is above the respective steady state, negative if it is below steady state, and zero if it is exactly at steady state. Let’s define q zj,t = −β˜1 ln(1 + zj,t ) + β˜2 ln(1 + zhj,t )

(8.19)

Equation (8.18) may then be expressed in matrix form yˆ = − β˜1 qˆ + β˜1 Wq+ ˆ β˜2 hˆ − β˜2 Whˆ + Wz

(8.20)

where β˜1 and β˜2 are defined as in Eqs. (8.7) and (8.10) (and hence both positive), yˆ is an N × 1 vector on growth approximations with elements yˆ i,t =

d ln qˆ i,t dt

(8.21)

z is an N×1 vector of deviations from steady state values with the unknown elements zj,t , as defined in Eq. (8.19), qˆ is an N × 1 vector consisting of the elements ln qˆ i,t , hˆ is an N × 1 vector consisting of the elements ln hˆ i,t , and W is a an N × N matrix that consists of the elements wij .

8.2 Solution for Output Growth Let’s now hold all variables constant except for ln qˆ i,t . From Eq. (8.18) we get a nonhomogeneous first-order linear differential equation with a constant coefficient and a constant term: d ln qˆ i,t + β˜1 ln qˆ i,t =  dt

(8.22)

8.2

Solution for Output Growth

83

with  = β˜1

N 

wij ln qˆ j,0 + β˜2 ln hˆ i,0 − β˜2

j=i

N 

wij ln hˆ j,0 +

N 

j=i

wij ln zj,0

(8.23)

j=i ˜

By multiplying both sides of Eq. (8.23) with e−β1 t , we get  d ˜ ˜ ln qˆ i,t e−β1 t = e−β1 t dt

(8.24)

from which it follows that growth between two points in time 0 and T ˜

ln qˆ i,T e−β1 T − ln qˆ i,0 =

  −β˜1 T e −1 β˜1

(8.25)

Rearranging Eq. (8.25) yields ln qˆ i,T

    ˜ e−β1 T = + ln qˆ i,0 − ˜ ˜ β1 β1

(8.26)

By subtracting ln qˆ i,0 from both sides of the equation and rearranging, we get output growth in region i between two points in time 0 and T:  ln qˆ i,T − ln qˆ i,0 =

   ˜ − ln qˆ i,0 1 − e−β1 T β˜1

(8.27)

Reintroducing Eq. (8.23) into (8.27), we have  

N ˜ ˜ ln qˆ i,T − ln qˆ i,0 = − 1 − e−β1 T ln qˆ i,0 + 1 − e−β1 T wij ln qˆ j,0 j=i

 β˜  β˜ 

N 2 2 ˜ ˜ + 1 − e −β 1 T ln hˆ i,0 − 1 − e−β1 T wij ln hˆ j,0 β˜1 β˜1 j=i

(8.28)

 1  N ˜ + 1 − e −β 1 T wij ln zj,0 β˜1 j=i We may now rewrite Eq. (8.28) as 



ln qˆ i,T − ln qˆ i,0 = β 1 ln qˆ i,0 + β 2 

+ β4

N  j=i

N  j=i



wij ln qˆ j,0 + β 3 ln hˆ i,0 

wij ln hˆ j,0 + β 5

N  j=i

(8.29) wij ln zj,0

84

8 Implications for Output Growth

where 

β 1 = −1 + exp (− β˜1 T)! < 0  

δi∗ (ν)

1 In principle, there is also the “bishop” case, where vertices count as borders, but common edges do not. This third case, however, is almost never applied in empirical research.

10.2

The Model Specification

107

and where δi∗ (ν) is a defined critical cut-off distance for each region i, so that each region is assigned the same fixed number of closest-lying regions.2 In other words, δ ∗ is the νth order smallest distance between two regions i and j so that each region has exactly ν neighbours. This results in an asymmetric spatial weight matrix where every row has the same number of non-zero elements. The spatial weight matrix should therefore reflect the underlying assumptions of theoretical conceptualisation, and in most studies it is based on either distance or physical contiguity (i.e. common borders). As specified in Sect. 9.1, this study will take travel times by car as a measure of distance relationships.

10.2 The Model Specification Let’s now express the model specification of Eq. (8.37) as 









y = α ι + β 1 q0 + β 2 Wq0 + β 3 h0 + β 4 Wh0 +ρWε

(10.4)

where the vector of deviations from steady state z0 is replaced by a vector of error terms ε, to reflect the fact that the steady states and hence deviations therefrom are  in fact unknown. Furthermore, β 5 is replaced by ρ to adopt the common notation in spatial econometric literature. Rewrite Eq. (10.4) as an econometric model of the form y = Xβ + ε

(10.5)

ε = ρWε + u

(10.6)

  X = ι q0 Wq0 h0 Wh0

(10.7)

where

and where

and 









β = ( α β 1 β 2 β 3 β 4 )

(10.8)

where y is an N × 1 vector of observations on the dependent variable for N regions, W is an N ×N row-standardised spatial weight matrix. X is an N ×(2p+1) matrix of p explanatory variables, which are also lagged in space, and ι as an N × 1 vector of ones. Hence β is a (2p + 1) × 1 vector of regression coefficients, and ε is the vector  w = 1 ∀i, the spatially of error terms. Since W is row-standardised so that N ij j=i

2 This

concept is usually referred to as concept of k-nearest neighbours.

108

10

Spatial Econometric Specification and Estimation

lagged variables for each i equal the arithmetic means of i’s neighbours’ values. In addition, we have ρ as the spatial autocorrelation coefficient of error terms and u is a vector of i.i.d. errors with variance σ 2 . Note that in the specification of this study, we have N = 255 and p = 2. Equations (10.5) and (10.6) can be solved for ε ε = (I − ρW)−1 u

(10.9)

with I being an N × N identity matrix. The variance-covariance matrix for the random vector ε follows as E(εε ) = (I − ρW)−1 E(uu )(I − ρW )−1

(10.10)

After rearranging according to the standard assumption of i.i.d. errors u, with E(uu’) = σ 2 I, the expression simplifies to E(εε ) = σ 2 [(I − ρW) (I − ρW)]−1

(10.11)

Note that in the case where W is not symmetric, we have (I − ρW)−1 = (I − ρW )−1 . With |ρ| < 1, a Leontief expansion of the matrix (I − ρW)−1 yields I + ρW + ρ 2 W2 + ρ 3 W3 + .... The complete structure of the variance-covariance matrix then follows as a sum of terms containing matrix powers and products of W, scaled by the powers of ρ, or more specifically (Anselin 2003): (I − ρW)−1 (I − ρW )−1 = I + ρ(W + W ) + ρ 2 (W2 + WW + W 2 ) + ... (10.12) After adding the products of W and W , the resulting spatial covariance induces global spatial autocorrelation (Anselin 2006) where all locations in the system are related to each other. Moreover, when the number of neighbours is not constant across locations, the diagonal elements in Eq. (10.12) are no longer constant, inducing heteroskedasticity. We may now express the model more compactly from Eqs. (10.5) and (10.9) as y = Xβ + (I − ρW)−1 u

(10.13)

LeSage and Pace (2009) show how, in models that contain spatial lags of dependent and explanatory variables as is the case with the spatial Durbin model (SDM) of Eq. (10.2), interpretation of the parameter estimates becomes more complicated, as the partial derivatives of the dependent variables are not identical to regression parameters. They continue to show how changing a particular explanatory variable for a single observation or for multiple observations has direct as well as indirect effects, which in the case of an SDM are not equal to the regression parameters (LeSage and Pace 2009). A specification which includes a spatial lag of the explanatory variables, as well as spatially dependent disturbances such as that given in Eq. (10.13), is referred to by LeSage and Pace (2009, p. 41) as a spatial Durbin error model (SDEM). They note that “relative to the more general SDM, it [the SDEM]

10.3

Maximum Likelihood Estimation

109

simplifies interpretation of the impacts,” since the direct and indirect impacts are equal to the regression parameters. In the case of this study’s specification in partic  β 1 and β 3 , and indirect impacts ular, we thus have direct impacts as captured by   as captured by β 2 and β 4 . In addition, it should be noted that “this also allows us to use measures of dispersion such as the standard deviation or t-statistic for the regression parameters as a basis for inference regarding significance of the direct and indirect impacts” (LeSage and Pace 2009, p. 42). Estimation of the specification as given in Eq. (10.13) with ordinary least squares (OLS) remains unbiased, but is no longer efficient, and the classical estimators for standard errors will be biased (Anselin 2001). The two main approaches to estimation of spatial models are based on the maximum likelihood principle and the generalised method of moments (Anselin 2006). This study follows the maximum likelihood approach as discussed in the next section.

10.3 Maximum Likelihood Estimation The point of departure for maximum likelihood estimation is an assumption of normality of the error terms (Anselin 2001). The likelihood function  is therefore based on the assumption that u ∼ N(0, σ 2 I), and we thus have (see Anselin 1988)  = (2π)

−N 2

(σ 2 )

−N 2

  1 (|I − ρW|) exp − 2 (y − Xβ) (I − ρW) (I − ρW)(y − Xβ) 2σ (10.14)

It follows that the log-likelihood function ln  is obtained from Eq. (10.14) as (Anselin 2001): N N ln(2π ) − ln(σ 2 ) + ln(|I − ρW|) 2 2 1 − 2 (y − Xβ) (I − ρW) (I − ρW)(y − Xβ) 2σ

ln  = −

(10.15)

A maximisation of this log-likelihood function is equivalent to the minimisation of the sum of squared residuals. The first-order conditions for the estimators βˆ are obtained by taking the partial derivatives of Eq. (10.15) with respect to the parameter vector (for a detailed discussion of the process see Anselin 1988): 1 ∂ ln  = − 2 [−2(X − ρWX) (y − ρWy) + 2(X − ρWX) (X − ρWX)β] (10.16) ∂β 2σ Setting the right hand side of Eq. (10.16) equal to zero then yields for the estimators: βˆ = [(X − ρWX) (X − ρWX)]−1 (X − ρWX) (y − ρWy)

(10.17)

110

10

Spatial Econometric Specification and Estimation

The first order conditions for the variance σ 2 are similarly derived from Eq. (10.15) 1 N ∂ ln  = [(y − ρWy) − (X − ρWX)β] [(y − ρWy) − (X − ρWX)β] − 2 4 ∂σ 2σ 2σ 2 (10.18) and hence we get the estimator for the variance: σˆ 2 =

1 ˆ [(y − ρWy) − (X − ρWX)βˆ ] [(y − ρWy) − (X − ρWX)β] N

(10.19)

A consistent estimator for ρ must be obtained from an explicit maximisation of a concentrated likelihood function (Anselin 2001). By inserting Eqs. (10.17) and (10.19) into (10.15) and ignoring the constant, we get: ln  = −

  1 N ˆ + ln(|I − ρW|) ˆ  (y − Xβ) ln (y − Xβ) 2 N

(10.20)

For purposes of computational feasibility, the Jacobian matrix can be decomposed into in terms of the eigenvalues of W by (see Anselin 2006) det(I − ρW) =

N 

(1−ρυi )

(10.21)

i=1

where det(I − ρW) is the determinant of the matrix (I − ρW), and the υi are the eigenvalues of W. Equation (10.20) thus becomes    N N 1  ˆ ˆ ln(I − ρυi ) ln  = − ln (y − Xβ) (y − Xβ) + 2 N

(10.22)

i=1

which then has to be maximised. It follows from Eq. (10.22) that the condition (1 − ρυi ) > 0 must hold, so we have 1/υmin < γ < 1/υmax , where υmin and υmax are the smallest and largest eigenvalues of W, respectively. In the case of a row-standardised weight matrix, we have υmax = 1, and υmin as typically, but not necessarily, > −1. Computation is facilitated since the eigenvalues only need to be calculated once, and iterating over values of ρ is straightforward (Anselin 2006).

Chapter 11

Testing the Theoretical Model

The final chapter is devoted to tests of the theoretical model using European data for the observation period 1995–2004. They show how a fixation on the incidence of convergence processes may in fact distract from persistent and widening disparities: after presenting the results from tests of the theoretical model, it is demonstrated how these are related to divergence tendencies within subgroups of regions. Section 11.1 presents the basic results, while Sect. 11.2 provides an interpretation and concluding remarks.

11.1 Results The econometric estimations correspond to Eqs. (10.5) and (10.6), so we have (i) one constant, representing technological progress and including price increases during the observation period, (ii) two explanatory variables and their corresponding spatial lags, namely initial output, initial output of neighbouring regions (labelled “W initial output”), human capital, and human capital in neighbouring regions (labelled “W human capital”), and (iii) a non-spherical disturbance term and a vector of remaining i.i.d. errors. Table 11.1 displays the results by maximum likelihood estimation for the dependent variable as the difference between the natural logarithms of GRP per capita in 1995 and 2004 for 255 NUTS 2 and equivalent regions. The explanatory variables in this case are the natural logarithms of GRP per capita in 1995, and the natural logarithms of the percentages of inhabitants with educational levels 5 and 6 according to ISCED97. Table 11.2 displays the results where the dependent variable is the difference between the natural logarithms of GVA per gainfully active person in 1995 and 2004. Equivalently, the explanatory variables are the natural logarithms of GVA per gainfully active person in 1995, and the natural logarithms of the numbers of inhabitants with educational levels 5 and 6 according to ISCED97 as percentage of the numbers of gainfully active persons. There are five results in each table: the results in the most left-hand column stem from physical contiguity, where in the case of islands the respective region that

S. Sardadvar, Economic Growth in the Regions of Europe, Contributions to Economics, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-7908-2637-1_11, 

111

112

11

Testing the Theoretical Model

Table 11.1 Estimation results with gross regional product per capita

Constant Initial output W initial output Human capital W human capital Spatial autocorr. Residual variance LR test LIK AIC BP Wald

Physical contiguity

3 nearest neighbours

4 nearest neighbours

5 nearest neighbours

6 nearest neighbours

2.681 (0.000) –0.183 (0.000) –0.063 (0.084) 0.111 (0.000) –0.070 (0.063) 0.768 (0.000) 0.010 19.348 (0.000) 206.60 –399.21 3.672 (0.452) 405.64 (0.000)

2.819 (0.000) –0.213 (0.000) –0.052 (0.110) 0.158 (0.000) –0.099 (0.023) 0.741 (0.000) 0.010 16.939 (0.000) 201.65 –389.29 2.354 (0.671) 405.09 (0.000)

2.739 (0.000) –0.223 (0.000) –0.036 (0.330) 0.170 (0.000) –0.098 (0.056) 0.785 (0.000) 0.010 10.780 (0.000) 209.12 –404.23 3.416 (0.491) 478.03 (0.000)

2.774 (0.000) –0.233 (0.000) –0.027 (0.502) 0.168 (0.000) –0.107 (0.058) 0.821 (0.000) 0.009 11.557 (0.000) 217.53 –421.06 4.220 (0.377) 570.40 (0.000)

2.882 (0.000) –0.228 (0.000) –0.045 (0.302) 0.164 (0.000) –0.093 (0.093) 0.833 (0.000) 0.009 11.544 (0.003) 215.17 –416.34 5.938 (0.204) 576.49 (0.000)

Notes: Calculations have been carried out with R using the spdep package 0.3–31 by Roger Bivand. p-values are in parentheses. LR test is the likelihood ratio test on restrictions with the p-value for two degrees of freedom given in parentheses. LIK and AIC refer to the values of the maximised log-likelihood and the Akaike information criterion, respectively. BP is the spatially adjusted version of the (non-studentised) Breusch-Pagan test for heteroskedasticity, using the squares of explanatory variables. Wald is the square of the asymptotic standard error.

lies closest in terms of car travel time (which includes time spent on ferries) has been assigned as a neighbour. The results in the other columns stem from weight matrices based on car travel times, and are given for 3, 4, 5 and 6 nearest neighbours. In addition, results have been tested for the restrictions implied by the theoretical     model, namely β 1 = − β 2 and β 3 = − β 4 . Tests on these restrictions are carried out by a likelihood ratio test comparing the corresponding results. The results differ only slightly from each other in the set of estimations based on GRP per capita, as can be seen from Table 11.1: in most cases, the coefficients are significant and have the expected signs. In particular, we have a positive and highly significant constant in all cases. Furthermore, initial output is negative and highly significant, and human capital is positive and highly significant. Neighbouring regions’ output is insignificant in all cases but one, while neighbouring regions’ human capital is negative and significant in all cases. The spatial autocorrelation coefficient is positive and highly significant in all cases. The spatial Breusch-Pagan test (Anselin 1988) does not reject homoskedasticity, and the Wald test as the square

11.1

Results

113

Table 11.2 Estimation results with gross value added per gainfully active person

Constant Initial output W initial output Human capital W human capital Spatial autocorr. Residual variance LR test LIK AIC BP Wald

Physical contiguity

3 nearest neighbours

4 nearest neighbours

5 nearest neighbours

6 nearest neighbours

3.044 (0.000) –0.280 (0.000) –0.015 (0.644) 0.063 (0.000) –0.014 (0.658) 0.767 (0.000) 0.0077 84.619 (0.000) 247.22 –480.44 21.908 (0.000) 406.90 (0.000)

3.359 (0.000) –0.279 (0.000) –0.028 (0.341) 0.093 (0.000) –0.068 (0.090) 0.729 (0.000) 0.008 54.000 (0.000) 236.99 –459.98 14.223 (0.007) 405.09 (0.000)

3.441 (0.000) –0.290 (0.000) –0.007 (0.831) 0.097 (0.000) –0.097 (0.034) 0.766 (0.000) 0.007 56.847 (0.000) 244.40 –474.79 17.111 (0.002) 399.53 (0.000)

3.540 (0.000) –0.299 (0.000) –0.005 (0.884) 0.098 (0.000) –0.115 (0.020) 0.799 (0.000) 0.007 62.486 (0.000) 251.62 –489.24 20.849 (0.000) 452.06 (0.000)

3.398 (0.000) –0.298 (0.000) 0.002 (0.952) 0.095 (0.000) –0.090 (0.103) 0.810 (0.000) 0.007 63.340 (0.000) 249.15 –484.29 23.381 (0.000) 445.17 (0.000)

Notes: See Table 11.1.

of the asymptotic standard error test rejects the absence of spatial dependence in all cases. The measures of the log-likelihood and related Akaike information criterion (Anselin 1988) both indicate that the matrix based on 5 nearest neighbours provides the best fit with the model. The likelihood ratio test (Florax et al. 2003) is a test on the restrictions implied by the theoretical model whereby each regression parameter of a non-lagged variable equals the negative of the corresponding regression parameter of the spatially lagged variable. The implied restrictions are rejected in all cases. The set of estimations based on GVA per gainfully active person (Table 11.2) displays a similar picture: the constant, initial output and human capital have the expected signs and are highly significant in each case. Initial output in neighbouring regions is not significant, while a negative and significant coefficient for human capital in neighbouring regions is found in all cases but two. As before, the spatial autocorrelation coefficient is positive and highly significant in all cases, but now the Breusch-Pagan rejects homoskedasticity. Measures of the log-likelihood and Akaike information criterion again indicate again that the matrix based on 5 nearest neighbours provides the best fit with the model. As before, in all cases the Wald test rejects the absence of spatial dependence, and the likelihood ratio test rejects the restrictions.

114

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Testing the Theoretical Model

11.2 Interpretation and Concluding Remarks In summary, we get a positive value of the constant and domestic human capital as well as a negative value of initial domestic output, with highly significant results in all cases. The observation of a negative influence of the initial level of domestic output follows from the theoretical model, and is also in line with frequently found results in the growth literature. The negative sign may recall tests on conditional convergence, although this study’s interpretation is different. The positive influence of domestic human capital endowment was expected from the theoretical model, as it reflects a higher attractiveness to investment decisions due to higher productivity. The positive sign is consistent with previous studies in growth, where human capital as measured by educational attainment is incorporated. Probably the most striking result is the negative coefficient of human capital in neighbouring regions. Interpretation according to the theoretical model relates to the positive effect of human capital on productivity: ceteris paribus, higher endowments with human capital lead to higher productivity, which will attract investments in physical capital. Since one unit of gross investments may take place only once, this happens at the expense of other regions. It follows that human capital is to the benefit of a region if it is found within its borders, but not so if it is found in its neighbouring regions. The results as presented in Tables 11.1 and 11.2 are robust for distinct but similar concepts of neighbourhood. Upon deeper investigation, it can be found that the statistical significance of the negative influence of human capital endowments in neighbouring regions appears to depend on the number of neighbours: if the number of neighbours remains relatively small, a negative impact of human capital of neighbouring regions is evident, but this impression becomes blurred the more nearest neighbours are included. Similarily, a spatial weight matrix based on physical contiguity of the second or higher order provides no significant results for the spatially lagged variables. These results can be explained by the fact that for each observation, the spatially lagged variables equal the average level of the variable in neighbouring regions: the more regions are considered as neighbours for each observation, the smaller the impact of each individual region becomes. As discussed in Part II, during transition periods, individual growth rates depend on initial output levels and human capital, both within regional borders and in neighbouring regions; the interplay of these four influences determines which outweigh the others. The theoretical model has shown how convergence of output levels on a global scale and divergence within subgroups of regions may coincide, exemplified, for instance, by the incidence of surpassing between regions as visualised in Fig. 9.4. The final question to tackle is where such divergence processes have taken place during the observation period. Figure 11.1 plots GDP per capita for each country as well as lowest and highest respective GRP levels of its NUTS 2 or equivalent regions for the year 2004.

GDP/GRP per capita

11.2

Interpretation and Concluding Remarks

115

80,000 70,000 60,000 50,000 40,000 30,000 20,000 10,000

LU

CH

NO

IE

DK

SE

FI

NL

AT

FR

BE

IT

GR

ES

SI

PT

DE

UK

GRP max

CZ

EE

HU

PL

GDP

SK

LV GRP min

LT

0

Fig. 11.1 Regional disparities within countries, 2004. Notes: The middle row displays EU’s and EFTA’s member states’ GDP per capita at market prices, as well as their least (front row) and highest (back row) NUTS 2 and equivalent regions’ GRP levels. Country codes refer to the official codes as applied by the European Union; main source of data (for details on the sources see Sect. 9.2): Eurostat

Disparities range almost up to fourfold,1 and in many countries they have even widened since 1995: in 13 out of those 19 countries with at least two NUTS 2 or equivalent regions, the gap between the most and least productive region within their respective country has increased.2 Of these 13 cases, in 11 countries the region with the highest initial output value also displayed the highest endowment with human capital.3 Intra-state disparities have displayed a particular increase in former centrally planned economies that accessed the European Union (including Bulgaria and Romania, which are not considered in the study), as exemplified by Slovakia, whose highest to lowest ratio reached threefold level at the end of the observation period.4 1 In Great Britain GRP per capita is 3.83 times higher in Inner London than in Cornwall and Isles of

Scilly (1995: 3.66); in Belgium, Brussels is 3.04 times as high as Hainault (Prov. Hainaut) (1995: 2.97). 2 These are Belgium, the Czech Republic, Finland, Greece, Hungary, Ireland, the Netherlands, Poland, Portugal, Slovakia, Sweden, Switzerland and the United Kingdom. 3 The only exceptions are Belgium and Finland. 4 The ratio of Bratislava (Bratislavský kraj) versus Eastern Slovakia (Východné Slovensko) has risen from 2.83 in 1995 to 3.05 in 2004.

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Testing the Theoretical Model

Common measures of distribution are in line with this rough observation: out of the 18 countries that consist of four or more NUTS 2 and equivalent regions, during the observation period (i) unweighted variance of GRP per capita rose in 12 countries, (ii) weighted variance of GRP per capita rose in 14 countries, (iii) the share of national GDP of the region that is or that contains the country’s capital town increased in 14 countries, and (iv) the Gini coefficient between regions rose in 14 countries. There are only 2 cases where all 4 measures indicate intra-state convergence,5 while in no less than 11 out of 18 countries all 4 measures of disparities were higher in 2004 than in 1995.6 The trend toward increasing disparities is observed in all new member states of the EU (including Bulgaria and Romania), in some cases taking on dramatic shapes. For instance, the weighted standard deviation of the Czech Republic rose from 0.21 to 0.29, and in Hungary from 0.28 to 0.37 (graphs of each measure for each country over the complete observation period are found in Appendix 3). This simultaneous occurrence of convergence and divergence processes goes to show that what we see is a matter of the scale we choose. A booming urban centre boosts the measured national output level, and if it is found in a mainly underdeveloped country it may even lower measured aggregate inequality in an enhanced economic area. At the same time, those who live in the country’s hinterland may experience rising disparities in regional distribution of income. It has been shown how human capital constitutes the critical determinant of regional growth, and how it can be a cause for rising disparities between regions. Investment opportunities abroad have an influence on the wage share and may thus also increase the gap between those who own physical capital and those who do not within a region. Furthermore, there is also a tendency toward a growing income gap within the working class: Joseph Stiglitz (2006, p. 337), among others, expects the income of workers who don’t happen to be identified as human capital to continue to decline, a phenomenon which underlines the relevance of processes of economic growth and development to both science and people’s everyday lives.

5 These

are Austria and Italy. are Belgium, the Czech Republic, Finland, Greece, Hungary, the Netherlands, Norway, Poland, Sweden, Slovakia and the United Kingdom. 6 These

Chapter 12

Summary

The purpose of this study was to contribute to regional growth theory by acknowledging an economy’s relative location in space. At heart of this study is a neoclassical growth model, whose N regional economies are assumed to be big enough to exhibit constant returns, and where each region produces output from the factors physical capital, human capital, and technology-augmented labour. Together, these regions form one superordinate economy. While output in one particular region can only be produced from factors found within the very same region, all regions are interconnected and interdependent with respect to the evolution of output levels. In particular, the choice of location of gross fixed capital formations is viewed as a function of expected rates of return, which depend on physical capital’s current marginal productivity. It is assumed that investment decisions take place either within the originating region, or in its neighbouring regions, from which it follows that the development of a particular region is dependent on its relative location in space. In the theoretical part of the study, it is shown how output growth of one region is a function of the region’s own as well as its neighbours’ current endowments with production factors. In equilibrium, all regions have reached identical steady state levels of factors and output, but their output levels may either converge to, or diverge from each other during transition periods. By introducing the differential equations of factor evolution to the production function, the model’s implications for growth are derived through both simulation and formal derivations. The model’s theoretical results can be summarised as follows: • Investments in physical capital flow to regions where expected rates of return are higher, but since • investment flows are spatially bounded, disequilibrium with respect to marginal productivity of physical capital is not corrected for immediately, and at the same time • these investment flows increase average productivity and hence accelerate growth of the superordinate economy, while • the effect on both labour income and wage share is negative for regions with net investment outflows. S. Sardadvar, Economic Growth in the Regions of Europe, Contributions to Economics, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-7908-2637-1_12, 

117

118

12 Summary

• In the long run, all regions converge to identical steady state levels of factors and output, while • output growth during transition periods is determined by one region’s initial conditions and those of its neighbours, thus • a region’s output may either converge to, or diverge from others, where regions benefit on the one hand • from more highly developed regions in their neighbourhood, and • if their own human capital endowment is relatively high, and • if their neighbours’ human capital endowments are relatively low; but on the other hand • regions suffer • from less developed regions in their neighbourhood, and • if their own human capital endowment is relatively low, and • if their neighbours’ human capital endowments are relatively high. A Taylor approximation of the model’s dynamics can be solved for output growth for each individual region as a function of its own as well as its neighbours’ initial conditions. The growth equation can be expressed either as a function of the two types of capital both within and beyond a region’s borders, or as a function where one factor of production is replaced by a function of output. This allows one to overcome lack of data on regional physical capital stocks, hence the empirical parts’ explanatory variables are output per labour unit and human capital endowment per labour unit. The observation area comprises 255 European regions on the NUTS 2 and equivalent level of the EU and EFTA, over the observation period 1995–2004. The Taylor approximation of the theoretical model provides a testable spatial econometric model specification that may be labelled as a spatial Durbin error model. The specification is tested for (i) growth of gross regional product per capita and (ii) growth of gross value added per gainfully active person, the explanatory variables are according measures of initial output per labour unit and human capital as proxied by educational attainment. Neighbourhood is captured by spatially dependent errors and spatial lags of the explanatory variables, where the spatial weight matrices are based on (i) common borders or (ii) nearest neighbours according to car travel times. The model’s empirical results are in line with the main results from the theoretical model. In particular, they show how regional growth benefits from relatively rich human capital endowments within one region, but at the same time suffers due to high levels of human capital in neighbouring regions. Finally, a closer look shows how convergence as defined by the European Commission has occurred within the observation area as a whole, but within most European countries, disparities between the regions have increased. Neoclassical growth theory is sometimes misunderstood as predicting convergence to identical output levels. This study has shown that an assumption of long run convergence of reasonably similar economic entities is not contradicted by the observation of divergence tendencies during transition periods. The long run may,

12

Summary

119

however, refer to a very long time, which underlines the importance of understanding processes of intermediate regional growth and development. Interdependence and possible backwash effects resulting from different initial conditions have already been identified by Gunnar Myrdal’s (1957) studies on regional inequality, and later by Paul Krugman’s (1991a and b) findings on investment decisions depending on the economic structure of a region. In this study, it has been shown that a neoclassical framework that acknowledges the role of space as well as initial conditions of regional economies leads to comparable results. Future research may build on this work by exploring the role of human capital in more detail, for instance by focussing on migration patterns of human capital. This study’s main contribution to regional growth theory is the description and identification of the influence and interplay of stocks of human capital, investments in physical capital and the relative location in space.

Appendices Appendix 1: Regions of the Observation Area

The study covers the European territory of the European Union and the European Free Trade Association on the NUTS 2 level (EU) and equivalent level (EFTA) as classified by the European Union. Due to lack of data, the classification used in this study deviates from the official classification as of December 2007 in the following cases: Bulgaria, Cyprus, Iceland, Liechtenstein, Malta and Romania have not been included; the NUTS 2 regions Brandenburg-Nordost and Brandenburg-Südwest as well as the NUTS 2 regions Provincia Autonoma Bolzano-Bozen and Provincia Autonoma Trento have been merged to one region, respectively. By focussing on Europe, the study excludes the French regions Guadeloupe, Martinique, Guyana and Réunion, the Portuguese regions Região Autónoma dos Açores and Região Autónoma da Madeira, and the Spanish regions Ciudad Autónoma de Ceuta, Ciudad Autónoma de Melilla and Canarias. The following list contains the official names of all included regions sorted alphabetically by the corresponding nation states: • Austria (9 regions): Burgenland; Niederösterreich; Wien; Kärnten; Steiermark; Oberösterreich; Salzburg; Tirol; Vorarlberg • Belgium (11 regions): Région de Bruxelles-Capitale/Brussels Hoofdstedelijk Gewest; Prov. Antwerpen; Prov. Limburg (BE); Prov. Oost-Vlaanderen; Prov. Vlaams-Brabant; Prov. West-Vlaanderen; Prov. Brabant Wallon; Prov. Hainaut; Prov. Liège; Prov. Luxembourg (BE); Prov. Namur ˇ • Czech Republic (8 regions): Praha; Stˇrední Cechy; Jihozápad; Severozápad; Severovýchod; Jihovýchod; Stˇrední Morava; Moravskoslezsko • Denmark (1 region): Danmark • Estonia (1 region): Eesti • Finland (5 regions): Itä-Suomi; Etelä-Suomi; Länsi-Suomi; Pohjois-Suomi; Åland • France (22 regions): Île-de-France; Champagne-Ardenne; Picardie; HauteNormandie; Centre; Basse-Normandie; Bourgogne; Nord – Pas-de-Calais; Lorraine; Alsace; Franche-Comté; Pays de la Loire; Bretagne; Poitou-Charentes; Aquitaine; Midi-Pyrénées; Limousin; Rhône-Alpes; Auvergne; LanguedocRoussillon; Provence-Alpes-Côte d’Azur; Corse S. Sardadvar, Economic Growth in the Regions of Europe, Contributions to Economics, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-7908-2637-1, 

121

122

Appendices

• Germany (40 regions): Stuttgart; Karlsruhe; Freiburg; Tübingen; Oberbayern; Niederbayern; Oberpfalz; Oberfranken; Mittelfranken; Unterfranken; Schwaben; Berlin; Brandenburg - Nordost & Brandenburg – Südwest; Bremen; Hamburg; Darmstadt; Gießen; Kassel; Mecklenburg-Vorpommern; Braunschweig; Hannover; Lüneburg; Weser-Ems; Düsseldorf; Köln; Münster; Detmold; Arnsberg; Koblenz; Trier; Rheinhessen-Pfalz; Saarland; Chemnitz; Dresden; Leipzig; Dessau; Halle; Magdeburg; Schleswig-Holstein; Thüringen • Greece (13 regions): Anatoliki Makedonia, Thraki; Kentriki Makedonia; Dytiki Makedonia; Thessalia; Ipeiros; Ionia Nisia; Dytiki Ellada; Sterea Ellada; Peloponnisos; Attiki; Voreio Aigaio; Notio Aigaio; Kriti • Hungary (7 regions): Közép-Magyarország; Közép-Dunántúl; Nyugat-Dunántúl; Dél-Dunántúl; Észak-Magyarország; Észak-Alföld; Dél-Alföld • Ireland (2 regions): Border, Midland and Western; Southern and Eastern • Italy (20 regions): Provincia Autonoma Bolzano/Bozen & Provincia Autonoma Trento; Piemonte; Valle d’Aosta/Vallée d’Aoste; Liguria; Lombardia; Veneto; Friuli-Venezia Giulia; Emilia-Romagna; Toscana; Umbria; Marche; Lazio; Abruzzo; Molise; Campania; Puglia; Basilicata; Calabria; Sicilia; Sardegna • Latvia (1 region): Latvija • Lithuania (1 region): Lietuva • Luxembourg (1 region): Luxembourg (Grand-Duché) • Netherlands (12 regions): Groningen; Friesland; Drenthe; Overijssel; Gelderland; Flevoland; Utrecht; Noord-Holland; Zuid-Holland; Zeeland; Noord-Brabant; Limburg (NL) • Norway (7 regions): Oslo og Akershus; Hedmark og Oppland; Sør-Østlandet; Agder og Rogaland; Vestlandet; Trøndelag; Nord-Norge ´ askie; • Poland (16 regions): Łódzkie; Mazowieckie; Małopolskie; Sl ˛ Lubelskie; ´ Podkarpackie; Swie˛tokrzyskie; Podlaskie; Wielkopolskie; Zachodniopomorskie; Lubuskie; Dolno´slaskie; ˛ Opolskie; Kujawsko-Pomorskie; Warmi´nsko-Mazurskie; Pomorskie • Portugal (5 regions): Norte; Algarve; Centro (PT); Lisboa; Alentejo • Slovakia (4 regions): Bratislavský kraj; Západné Slovensko; Stredné Slovensko; Východné Slovensko • Slovenia (1 region): Slovenija • Spain (16 regions): Galicia; Principado de Asturias; Cantabria; País Vasco; Comunidad Foral de Navarra; La Rioja; Aragón; Comunidad de Madrid; Castilla y León; Castilla-La Mancha; Extremadura; Cataluña; Comunidad Valenciana; Illes Balears; Andalucía; Región de Murcia • Sweden (8 regions): Stockholm; Östra Mellansverige; Sydsverige; Norra Mellansverige; Mellersta Norrland; Övre Norrland; Småland med öarna; Västsverige • Switzerland (7 regions): Région lémanique; Espace Mittelland; Nordwestschweiz; Zürich; Ostschweiz; Zentralschweiz; Ticino • United Kingdom (37 regions): Tees Valley and Durham; Northumberland and Tyne and Wear; Cumbria; Cheshire; Greater Manchester; Lancashire; Merseyside; East Riding and North Lincolnshire; North Yorkshire; South Yorkshire;

Appendices

123

West Yorkshire; Derbyshire and Nottinghamshire; Leicestershire, Rutland and Northamptonshire; Lincolnshire; Herefordshire, Worcestershire and Warwickshire; Shropshire and Staffordshire; West Midlands; East Anglia; Bedfordshire and Hertfordshire; Essex; Inner London; Outer London; Berkshire, Buckinghamshire and Oxfordshire; Surrey, East and West Sussex; Hampshire and Isle of Wight; Kent; Gloucestershire, Wiltshire and North Somerset; Dorset and Somerset; Cornwall and Isles of Scilly; Devon; West Wales and the Valleys; East Wales; North Eastern Scotland; Eastern Scotland; South Western Scotland; Highlands and Islands; Northern Ireland.

Appendix 2: Supplementary Econometric Results

At date of print the issue of fine tuning the spatial weight matrix remains controversial. While some claim it to be unnecessary, others point to possible problems with endogeneity. In the latter case one may apply two different weight matrices. In the case of this study, when defining neighbourhood based on car travel times, the errors remain lagged by a spatial weight matrix based on physical contiguity, so we have: 







 ι + β 1 q0 + β 2 Wc q0 + β 3 h0 + β 4 Wc h0 + (I − ρWq )−1 u y= α

where Wc is the weight matrix based on car travel times, as defined by the concept of nearest neighbours, and Wq is the weight matrix based on physical contiguity, with neighbourhood defined by the concept of queen of the first order (because they are considered to be perfectly exogenous, results based on physical contiguity need no refinement). As can be seen from the tables below, the results differ only slightly from those presented in Sect. 11.1. It follows that the study’s empirical results are robust with respect to reasonably similar spatial weight matrices, and the study’s conclusions and interpretations remain unchanged.

125

126

Appendices Alternative estimation results with gross regional product per capita

Constant Initial output W initial output Human capital W human capital Spatial autocorr. Residual variance LR test LIK AIC BP Wald

3 nearest neighbours

4 nearest neighbours

5 nearest neighbours

6 nearest neighbours

7 nearest neighbours

2.576 (0.000) −0.176 (0.000) −0.058 (0.061) 0.130 (0.000) −0.092 (0.016) 0.774 (0.000) 0.009 27.440 (0.000) 210.65 −407.30 6.072 (0.194) 423.09 (0.000)

2.603 (0.000) −0.182 (0.000) −0.051 (0.137) 0.137 (0.000) −0.116 (0.010) 0.775 (0.000) 0.009 25.365 (0.000) 210.64 −407.29 7.562 (0.109) 433.35 (0.000)

2.598 (0.000) −0.189 (0.000) −0.048 (0.191) 0.141 (0.000) −0.103 (0.034) 0.770 (0.000) 0.009 19.969 (0.000) 207.99 −401.97 7.732 (0.102) 415.86 (0.000)

2.643 (0.000) −0.182 (0.000) −0.054 (0.155) 0.137 (0.000) −0.122 (0.018) 0.774 (0.000) 0.009 22.972 (0.000) 209.59 −405.19 8.850 (0.065) 428.87 (0.000)

2.705 (0.000) −0.185 (0.000) −0.061 (0.134) 0.140 (0.000) −0.111 (0.046) 0.772 (0.000) 0.009 19.649 (0.000) 208.15 −402.29 8.773 (0.067) 420.66 (0.000)

Notes: See Table 11.1. Alternative estimation results with gross value added per gainfully active person

Constant Initial output W initial output Human capital W human capital Spatial autocorr. Residual variance LR test LIK AIC BP Wald Notes: See Table 11.1.

3 nearest neighbours

4 nearest neighbours

5 nearest neighbours

6 nearest neighbours

7 nearest neighbours

3.274 (0.000) −0.257 (0.000) −0.036 (0.212) 0.066 (0.000) −0.049 (0.145) 0.771 (0.000) 0.007 80.127 (0.000) 249.63 −485.26 26.785 (0.000) 417.25 (0.000)

3.369 (0.000) −0.266 (0.000) −0.016 (0.616) 0.069 (0.000) −0.079 (0.042) 0.772 (0.000) 0.007 81.615 (0.000) 250.05 −486.10 25.420 (0.000) 423.09 (0.000)

3.327 (0.000) −0.271 (0.000) −0.011 (0.734) 0.070 (0.000) −0.073 (0.070) 0.770 (0.000) 0.007 80.414 (0.000) 249.25 −484.50 25.336 (0.000) 413.97 (0.000)

3.388 (0.000) −0.267 (0.000) −0.019 (0.578) 0.068 (0.000) −0.075 (0.081) 0.771 (0.000) 0.007 80.516 (0.000) 249.50 −484.99 26.632 (0.000) 416.78 (0.000)

3.226 (0.000) −0.277 (0.000) −0.006 (0.881) 0.070 (0.000) −0.060 (0.191) 0.766 (0.000) 0.007 78.326 (0.000) 248.07 −482.14 23.425 (0.000) 400.15 (0.000)

Appendix 3: Intra-state Developments 1995 to 2004 0.3 0.3 0.2 0.2 0.1

0.1

0.0 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004

0.0

Austria

Belgium

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004

0.6 0.3

0.5 0.4

0.2

0.3 0.2

0.1

0.1 0.0

0.0

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004

Finland

Czech Republic 0.3

0.2 0.2

0.1

0.1

0.0

0.0 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004

France

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004

Germany 0.5

0.5

0.4

0.4 0.3

0.3

0.2

0.2

0.1

0.1

0.0

0.0

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004

Greece unweighted standard deviation

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004

Hungary weighted standard deviation

share of capital region

Gini coefficient

127

128

Appendices 0.2 0.3

0.2 0.1

0.1

0.0 1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

Italy

0.0 1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

1997

1998

1999

2000

2001

2002

2003

2004

1997

1998

1999

2000

2001

2002

2003

2004

1997

1998

1999

2000

2001

2002

2003

2004

Netherlands

0.3

0.2

0.2 0.1

0.1

0.0 1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

Norway

0.0

1995

1996

Poland

0.5 0.4 0.4 0.3 0.3 0.2

0.2

0.1

0.1

0.0 1995

0.0 1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

1996

Slovakia

Portugal

0.5 0.3

0.2 0.2

0.1

0.1

0.0 1995

1996

1997

1998

1999

2000

Spain unweighted standard deviation

2001

2002

2003

2004

0.0 1995

1996

Sweden weighted standard deviation

share of capital region

Gini coefficient

Appendices

129 0.3

0.2 0.2

0.1 0.1

0.0 1995

1996

1997

1998

1999

2000

2001

Switzerland unweighted standard deviation

2002

2003

2004

0.0 1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

United Kingdom weighted standard deviation

share of capital region

Gini coefficient

Appendix 4: List of Symbols

Variables in equations refer either to a specific economy if they carry a subscript, or to all economies under consideration if they carry no subscript. The notation follows the common notation in the literature as close as possible, but deviates where necessary to avoid double assignments. The following lists are split into Latin and Greek letters, both sorted alphabetically by the symbols; vectors and matrices are bold and non-italic. A a b C d E e f G g H h h∗ hˆ hˆ ∗ I i J j j K k k∗

Level of technology (stock of knowledge) Output elasticity of physical capital (equilibrium output-share of physical capital) Output elasticity of human capital (equilibrium output-share of human capital) Aggregate stock of physical capital defined by ownership Depreciation rate Expected value Mathematical constant (Euler’s number) A function Total income from capital generated Rate of technological progress Aggregate stock of human capital Human capital per labour unit Steady state human capital stock per labour unit Human capital per unit of effective labour Steady state human capital stock per unit of effective labour Identity matrix Index for an economy under consideration Total income from capital earned Index for an economy under consideration Index for an economy under consideration Aggregate stock of physical capital defined by location Physical capital per labour unit Steady state physical capital stock per labour unit

131

132

Appendices

kˆ kˆ ∗ L N n p Q q q∗ qˆ qˆ ∗ r sK sH T t u V v W wij X x y z

Physical capital per unit of effective labour Steady state physical capital stock per unit of effective labour Labour force (population size) Total number of economies under consideration Labour force growth rate (population growth rate) Number of explanatory variables Aggregate output Output per labour unit (labour productivity) Steady state output per labour unit (steady state labour productivity) Output per unit of effective labour Steady state output per unit of effective labour Interest rate Saving rate (physical capital investment rate) Education expenditure rate (human capital investment rate) End of an observation period (number of periods) Point in time Error term in a regression equation Total wage Average wage Spatial weight matrix (matrix of connectivities) Connectivity between i and j Matrix of explanatory variables An explanatory variable Growth of output per labour unit Percentage deviation from steady state

α  α α˜ β β˜ βˆ  β γ  δij δ∗ ε θ ι  λ μ ν

Intercept in a regression equation Intercept as it results from the model Influence on convergence Coefficient in a regression equation Speed of convergence Estimated regression parameter Regression parameter as it results from the model Degree of technological interdependence Exogenous component of technology Distance from i to j Critical cut-off distance Error Set of neighbours, unspecified Vector of ones Likelihood function Overall degree of integration Magnitude of interaction An integer

Appendices

ξ  π ρ  σ σˆ ς τ υ χ 

Condition for steady state Product of a series Mathematical constant (pi) Parameter of spatial dependence Sum of a series (sigma sign) Standard deviation Estimated standard deviation Balanced growth rate of a physical capital Degree of external effects within one economy Eigenvalue Vector of (spatial) coefficients Variables held constant

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