Growth and international trade : an introduction to the overlapping generations approach [2 ed.] 9783662629437, 3662629437

Table of contents :
Preface to the Second Edition
Preface to the First Edition
Contents
Abbreviations
1: Growth and International Trade: Introduction and Stylized Facts
1.1 Introduction and Motivation
1.2 Definition of Growth Magnitudes
1.2.1 Growth Rates of Products and Quotients
1.3 Kaldor´s ``Stylized Facts´´
1.4 Kuznets´ Facts
1.5 Internationalization Facts
1.5.1 World Trade Is Growing Faster Than World Output
1.5.2 Export and Import Ratios Increase Over Time
1.5.3 Two-Thirds of Foreign Trade Takes Place Between Developed Countries
1.5.4 Neighboring Countries Trade More with Each Other Than Countries That Are Further Apart
1.6 Globalization Facts
1.6.1 Foreign Direct Investment and Financial Investment
1.6.2 Asia Since the 1970s: ``The´´ Dynamic Export Region
1.7 Summary and Conclusions
1.8 Exercises
References
Part I: Growth
2: Modeling the Growth of the World Economy: The Basic Overlapping Generations Model
2.1 Introduction and Motivation
2.2 The Set-Up of the Model Economy
2.3 The Macroeconomic Production Function and Its Per Capita Version
2.4 Structure of the Intertemporal Equilibrium
2.4.1 Intertemporal Utility Maximization of Younger Households
2.4.2 Old Households
2.4.3 A-Temporal Profit Maximization of Producers
2.4.4 Market Equilibrium in All Periods
2.5 The Fundamental Equation of Motion of the Intertemporal Equilibrium
2.6 Maximal Consumption and the ``Golden Rule´´ of Capital Accumulation
2.7 Summary and Conclusion
2.8 Exercises
Appendix
1. Constrained Optimization
2. Walras´ Law
References
3: Steady State, Factor Income, and Technological Progress
3.1 Introduction and Motivation
3.2 The GDP Growth Rate in Intertemporal Equilibrium and in Steady State
Definition 3.1
3.3 Existence and Stability of the Long-Run Growth Equilibrium
3.4 Efficiency of the Steady State
3.5 Comparative Dynamics in the Basic OLG Model
3.5.1 Increase in the Time Discount Factor
3.5.2 Reduction of the Population Growth Rate
3.5.3 Increase in the Rate of Technological Progress
3.6 Real Wage, Real Interest Rate, and Income Shares
3.6.1 Income Distribution along the Equilibrium Growth Path
3.7 Technological Progress in Neoclassical Growth Theory
3.7.1 Hicks-Neutral Technological Progress
3.7.2 Harrod-Neutral Technological Progress
3.7.3 Solow-Neutral Technological Progress
3.7.4 Resumé
3.8 Growth Accounting
3.9 Summary and Conclusion
3.10 Exercises
References
4: Economic Growth and Public Debt in the World Economy
4.1 Introduction and Motivation
4.2 European Debt Statistics
4.3 First-Order Conditions and Market Clearing
4.3.1 Market Equilibrium in All Periods
4.4 Intertemporal Equilibrium Dynamics
4.5 Existence and Stability of Steady States
4.5.1 Existence of a Long-Term Growth Equilibrium
4.5.2 Stability of the Steady States
4.5.3 Analytical Investigation of Dynamic Stability
4.6 Reducing Public Debt Under Dynamic Efficiency
Proposition 4.1
Definition 4.1
Proposition 4.2
Proposition 4.3
4.7 Summary and Conclusions
4.8 Exercises
References
5: ``New´´ Growth Theory and Knowledge Externalities in Capital Accumulation
5.1 Introduction and Motivation
5.1.1 Empirical Shortcomings of the ``Old´´ Growth Theory
5.1.2 Theoretical Shortcomings of the ``Old´´ Growth Theory
5.1.3 Main Approaches of ``New´´ Growth Theory
5.1.4 Aims of Explanation and Preview
5.2 Public Good Characteristics of Knowledge Externalities
5.3 Knowledge Externalities in the Basic OLG Model
5.3.1 The Production Technology with Knowledge Externalities
5.3.2 Inter-Temporal Utility Maximization of Active Households
5.3.3 Profit Maximization of Firms
5.3.4 Market-Clearing Conditions
5.3.5 Structure of the Inter-Temporal Equilibrium
5.4 The Shortcomings of the Old Growth Theory from the Perspective of Romer´s New Growth Model
5.5 Public Debt and Net Deficit in Romer´s New Growth Model
5.5.1 Government Budget Constraint, FOCs, and Market Clearing
5.5.2 Inter-Temporal Equilibrium Dynamics
5.5.3 Existence and Dynamic Stability of Balanced Growth
Proposition 5.1
Proposition 5.2
Proposition 5.3
5.6 Business Cycles and Endogenous Growth
5.7 Summary and Conclusion
5.8 Exercises
Appendix
Stochastic Growth Theory and Stochastic Processes
References
6: Endogenous Technological Progress and Infinite Economic Growth
6.1 Introduction and Motivation
6.2 Monopolistic Competition and Product Innovation in Inter-Temporal Equilibrium
6.2.1 Production Technologies and Innovation
6.2.2 Choice Problems and Market-Clearing Conditions
6.2.3 Structure of the Inter-Temporal Equilibrium
6.3 Unbounded Economic Growth and Increasing Growth Rates
Proposition 6.1
Proposition 6.2
6.4 One-Period Versus Long-Duration Patents
6.5 Summary and Conclusion
6.6 Exercises
References
7: Human Capital, Religion, and Economic Growth
7.1 Introduction and Motivation
7.2 Human Capital Formation in the Basic OLG Model
7.2.1 FOCs for Profit and Inter-Temporal Utility Maximization
7.2.2 Market-Clearing Conditions
7.2.3 The Structure of Inter-Temporal Equilibrium
7.3 Inter-Temporal Equilibrium Dynamics and Steady State
7.3.1 Inter-Temporal Equilibrium Dynamics
7.3.2 Steady State
7.3.3 Predictions of the Basic Human Capital Model
7.4 An OLG Model of Religion and Human Capital Formation
7.4.1 The OLG Set-Up
7.4.2 Religious Participation and Children´s Education
7.4.3 Religion as a Steady-State Phenomenon
7.5 Summary and Conclusion
7.6 Exercises
References
8: Economic Growth with Bubbles
8.1 Introduction and Motivation
8.2 Stylized Bubble Facts
8.3 Bubbles in the Basic OLG Growth Model
8.3.1 Equilibrium with Bubbles and Without Financial Frictions
8.3.2 Equilibrium with Bubbles and Financial Frictions
8.3.3 Where Is the Market for Bubbles?
8.4 Summary
8.5 Exercises
References
9: Involuntary Unemployment in an OLG Growth Model with Public Debt and Human Capital
9.1 Introduction and Motivation
9.2 The Basic Model: The Log-Linear Cobb-Douglas OLG Model with Internal Public Debt and Involuntary Unemployment
9.2.1 The Household Sector
9.2.2 The Production Sector
9.2.3 The Public Sector
9.2.4 Magnani´s (2015) Macro-Founded Investment Function
9.2.5 Market Clearing Conditions
9.2.6 Temporary Unemployment Rate
9.2.7 Intertemporal Equilibrium Dynamics
9.2.8 Existence and Dynamic Stability of Nontrivial Steady States
Proposition 9.1
Remark 9.1
Proposition 9.2
9.2.9 Comparative Steady-State Analysis: How More Optimistic Investor´s Animal Spirits or a Lower Saving Rate Affect the Unemp...
9.3 The Basic Model Extended by Human Capital Accumulation: The Log-Linear Cobb-Douglas OLG Model with Endogenous Growth and I...
9.3.1 The Household Sector
9.3.2 The Production Sector
9.3.3 The Public Sector
9.3.4 Human Capital Accumulation and GDP Growth
9.3.5 Magnani´s Macro-Founded Investment Function
9.3.6 The Intertemporal Equilibrium Dynamics in Terms of Per GDP Ratios
9.3.7 Existence and Dynamic Stability of Nontrivial Steady States
Proposition 9.3
Proposition 9.4
9.3.8 Comparative Steady-State Analysis: How a Larger Government Debt to GDP Ratio Affects the Unemployment Rate
9.4 Summary and Suggestions for Future Research
9.5 Exercises
References
10: Robots, Human Capital Investment, Welfare, and Unemployment in a Digital World Economy
10.1 Introduction and Motivation
10.2 The Basic Model
10.3 Intertemporal Equilibrium Dynamics and Steady State in the Basic Model
Proposition 10.1
Proposition 10.2
Proposition 10.3
Proposition 10.4
10.4 The Basic Model Extended by Involuntary Unemployment
10.5 Intertemporal Equilibrium Dynamics and Steady State in the Extended Model
Proposition 10.5
Proposition 10.6
Proposition 10.7
Proposition 10.8
10.6 Summary
10.7 Exercises
References
Part II: International Trade
11: International Parity Conditions in a Two-Country OLG Model Under Free Trade
11.1 Introduction and Motivation
11.2 The General OLG Model of International Trade in Goods, Financial Assets, and Money
11.3 Young Household´s Choice and International Parity Conditions
11.3.1 Domestic and Foreign Households in the International Equilibrium
11.3.2 Choice-Based Consumer Price Indices (``Ideal´´ Deflators) and Real Consumption Expenditures
11.3.3 Purchasing Power Parity in Its Absolute and Relative Version
11.3.4 The Household´s Choice Problem Using the Deflator and Real Consumption Expenditure
11.4 The Neoclassical Model of International Commodity Trade
11.4.1 Domestic and Foreign Producers in the Intertemporal World Market Equilibrium
11.4.2 Domestic and Foreign Households in the Intertemporal World Market Equilibrium
11.4.3 Terms of Trade in the Neoclassical Basic Model
11.4.4 Market-Clearing Conditions and Current Account Balance
Proposition 11.1
11.5 Summary
11.6 Exercises
Mathematical Appendix
References
12: Factor Proportion, Inter-Sectoral Trade, and Product Life Cycle
12.1 Introduction and Motivation
12.2 Production-Based Equilibrium Conditions in Autarky
Lemma 12.1
Lemma 12.2
12.3 Equalization of Factor Prices in the World Market Equilibrium
Lemma 12.3
12.4 Factor Proportions of Inter-Industrial Trade in the World Market Equilibrium
Proposition 12.1
12.5 The Leontief Paradox and the Neo-factor-proportion Theory
12.6 The ``Product Life Cycle´´ and the Dynamics of Comparative Advantages
12.7 Summary
12.8 Exercises
References
13: Product Differentiation, Decreasing Costs, and Intra-Sectoral Trade
13.1 Introduction and Motivation
13.2 Linder´s Demand-Based Trade Theory
13.3 Monopolistic Competition and Product Differentiation in a Closed Economy
13.3.1 Utility Maximization of Households and the Demand for Differentiated Goods
13.3.2 Profit Maximization of Producers and Short-Term Equilibrium
13.3.3 The Long-Term Market Equilibrium
13.4 Intra-Industry Trade Under Monopolistic Competition
13.4.1 Price Elasticity of Demand Is Independent of the Number of Variants
13.4.2 Price Elasticity of Demand Depending on the Number of Variants
13.5 Summary
13.6 Exercises
Mathematical Appendix
References
14: Globalization, Capital Accumulation, and Terms of Trade
14.1 Introduction and Motivation
14.2 Causes and Consequences of the Globalization of Commodity Markets
14.3 Globalization of Commodity Markets: A Comparative-Static Analysis
14.4 Dynamics of the Region-Specific Capital Intensities and of the Terms of Trade
14.5 Globalization and Inequality Between Nations
14.6 Summary
14.7 Exercises
References
15: Innovation, Growth, and Trade in a Two-Country OLG Model
15.1 Introduction and Motivation
15.2 Intermediate Products in the Two-Country OLG Model
15.3 Agents´ Choice Problems and Market-Clearing Conditions with Intra-sectoral Trade in Intermediates
15.4 Producers´ First-Order Conditions
15.5 The Structure of Intermediate Product Prices and Quantities
15.6 Cost Minimization of Final Good Producers and the Prices of Final Goods in the Industry Equilibrium
15.7 The Growth Rate of Intermediate Product Innovations in the International Equilibrium Versus in Autarky
Proposition 15.1
15.8 Integration, Efficiency, and Economic Growth: Some Comments
15.9 Summary
15.10 Exercises
References
16: Real Exchange Rate and Public Debt in a Two-Advanced-Country OLG Model
16.1 Introduction and Motivation
16.2 Literature Review and Preview
16.3 The Two-Good, Two-Country OLG Model
16.4 Intertemporal Equilibrium Dynamics and Existence of Steady States
Lemma 16.1
Proposition 16.1
16.5 Stability of Steady States and Steady-State Effects of Public Debt Shocks
Lemma 16.2
Proposition 16.2
Proposition 16.3
16.6 Transitional Impacts of Shocks in Home´s Sustainable Public Debt
Proposition 16.4
Corollary 16.1
Corollary 16.2
16.7 Summary and Conclusions
16.8 Exercises
Mathematical Appendix
1. Utility Maximizing Consumption and Saving Functions of Younger Households
2. Proof of Proposition 16.1
3. The Jacobian Matrix and Its Derivation
4. Proof of Lemma 16.2
5. Proof of Proposition 16.2
6. Proof of Proposition 16.4
References
17: Public Debt Reduction in Advanced Countries and its Impacts on Emerging Countries
17.1 Introduction and Motivation
17.2 Literature Review
17.3 Firm FOCs and Intertemporal Equilibrium in a Two-Country OLG Model with Unequal Technologies
17.4 Existence, Dynamic Stability, and Comparative Statics of Steady States
Proposition 17.1
Proposition 17.2
Proposition 17.3
Proposition 17.4
17.5 Steady-State Welfare Effects of a Unilateral Reduction of Public Debt in Home
17.5.1 Steady-State Welfare Effects of Unilateral Debt Reduction in Home
Proposition 17.5
Proposition 17.6
17.5.2 Numerical Illustrations of Welfare Effects for the Leading US-China Case
17.6 Summary and Conclusions
17.7 Exercises
Appendix: Mathematical Appendix
1. Proof of Proposition 17.1
2. The Jacobian Matrix
3. Proof of Proposition 17.2
4. Comparative Steady-State Effects of Debt Changes
5. Proof of Proposition 17.3
6. Derivation of Domestic Welfare Effect of Debt Change
References
18: External Balance, Dynamic Efficiency, and Welfare Effects of National Climate Policies
18.1 Introduction and Motivation
18.2 Literature Review and Preview
18.3 The Two-Good, Two-Country OLG Model with Nationally Tradable Emission Permits
18.3.1 Firms
18.3.2 Households and Governments
18.3.3 Market Clearing and International Trade
18.4 The Steady State and Unilateral Permit Policy
18.4.1 Inter-Temporal Equilibrium Dynamics
18.4.2 Characterization of Steady States
18.4.3 Steady State Effects of Unilateral Permit Policies
Proposition 18.1
18.5 The Steady-State Welfare Effects of Different Permit Policies
18.5.1 Derivation of Welfare Effects
Proposition 18.2
18.5.2 Comparison of Global Welfare Effects of Unilateral Permit Policies in Home and in Foreign
Proposition 18.3
18.5.3 Comparison of Welfare Effects of a Unilateral Domestic and a Multilateral Permit Policy
Proposition 18.4
18.5.4 Comparison of Welfare Effects of a Unilateral Foreign and a Multilateral Permit Policy
Proposition 18.5
18.6 Summary and Conclusions
18.7 Exercises
Appendix: Mathematical Appendix
1. Existence of Steady States
2. Saddle-Path Stability of Steady States
3. Proof of Proposition 18.1
4. Proof of Proposition 18.2
5. Proof of Proposition 18.3
References
19: Nationally and Internationally Optimal Climate Policies
19.1 Introduction and Literature Review
19.2 Internationally Differing Environmental Preferences in the Basic Model
19.3 Nationally Optimal Permit Levels
Proposition 19.1
Proposition 19.2
19.4 Internationally Optimal Permit Policies
Proposition 19.3
Proposition 19.4
Proposition 19.5
19.5 Summary and Conclusion
19.6 Exercises
Appendix
Derivation of Reaction Functions
Cross Welfare Effects
References
20: Modeling the Debt Mechanics of the Euro Zone
20.1 Introduction and Motivation
20.2 Stylized Macroeconomic Facts: Financial Autarky Versus EMU
20.3 The Basic Model
20.3.1 Financial Autarky
Proposition 20.1
20.3.2 International Equilibrium Under Financial Integration
20.3.3 Financially Integrated Versus Financially Autarkic Steady State
Proposition 20.2
Proposition 20.3
Corollary 20.1
20.4 Beyond the Basic Model: From Debt Mechanics Towards Debt Trap for South
20.5 Conclusion
20.6 Exercises
References
21: Financial Integration, House Price Dynamics, and Saving Rate Divergence in an OLG Model with Intra-EMU and Asian-US Trade ...
21.1 Introduction and Motivation
21.2 Stylized Macroeconomic Facts: Financial Autarky Versus Financial Integration
21.3 Model Setting
21.4 Pre-Euro and Asian-US Financial Autarky
21.5 International Equilibrium under Intra-EMU and Global Financial Integration
21.6 National Net Foreign Asset Position, Trade Imbalance, and Aggregate Savings
21.7 Numerical Specification and Model Generated Versus Stylized Macro Facts
21.8 Summary
21.9 Exercises
References
22: Why Religion Might Persist in a Globalized Market Economy: An Economic Modeling Approach
22.1 Introduction and Motivation
22.2 Autarky Equilibrium
22.3 International Equilibrium Under Capital Market Integration
22.4 Summary and Conclusion
22.5 Exercises
References
Index

Citation preview

Springer Texts in Business and Economics

Karl Farmer Matthias Schelnast

Growth and International Trade An Introduction to the Overlapping Generations Approach Second Edition

Springer Texts in Business and Economics

Springer Texts in Business and Economics (STBE) delivers high-quality instructional content for undergraduates and graduates in all areas of Business/Management Science and Economics. The series is comprised of self-contained books with a broad and comprehensive coverage that are suitable for class as well as for individual self-study. All texts are authored by established experts in their fields and offer a solid methodological background, often accompanied by problems and exercises.

More information about this series at http://www.springer.com/series/10099

Karl Farmer • Matthias Schelnast

Growth and International Trade An Introduction to the Overlapping Generations Approach Second Edition

Karl Farmer Department of Economics University of Graz Graz, Austria

Matthias Schelnast Department of Economics University of Graz Graz, Austria

ISSN 2192-4333 ISSN 2192-4341 (electronic) Springer Texts in Business and Economics ISBN 978-3-662-62942-0 ISBN 978-3-662-62943-7 (eBook) https://doi.org/10.1007/978-3-662-62943-7 # Springer-Verlag GmbH Germany, part of Springer Nature 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, 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. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer-Verlag GmbH, DE part of Springer Nature. The registered company address is: Heidelberger Platz 3, 14197 Berlin, Germany

Preface to the Second Edition

The second edition of the textbook at hand contains—compared to the first edition— four more chapters on growth and international trade from the Diamond-type overlapping generations (OLG) perspective. In Part I, “Growth”, involuntary unemployment in intertemporal general equilibrium under flexible prices and wages is the subject of a new chapter (Chap. 9). Robots, investment in human capital, and unemployment in a digital world economy are dealt with in Chap. 10, which now concludes Part I. The empirical data in other chapters of Part I, in particular in Chap. 1, are updated, minor errors are corrected, and the whole text is checked for the sake of internal consistency. Part I of this second edition comprises fifty-fifty introductory and advanced material. This is also true for Part II, “International Trade”, which is also extended by two new Chaps. 21 and 22. In Chap. 21, financial integration and house price dynamics are depicted in a global OLG with intra-EMU and Asian–US trade imbalances. The final chapter (Chap. 22) extends the closed economy of Chap. 7 towards a two-country, open-economy model in order to explain why religion might persist in a globalized world economy. Empirical data in Part II are also updated, minor errors are corrected, and the whole text is examined for internal consistency. As indicated by the book’s subtitle, it was the authors’ intention to introduce the concept of the overlapping generations approach to growth and international trade to a much wider audience. What was said in the preface to the first edition is also true for the new chapters of the second edition. Nonetheless, the approach taken in the new chapters remains relatively introductory in that utility and production functions are specified in such a way that any interested reader should be capable of deriving solutions to the intertemporal general equilibria described. However, our primary intention is not simply to hone skills in general equilibrium solutions but rather to improve the reader’s ability to grasp the analytical significance of the much more advanced dynamic general equilibrium models (on growth and international trade) published in leading journals. The second edition of the textbook at hand represents a thoroughly revised and substantially extended version of Wachstum und Außenhandel, co-authored by Ronald Wendner and first published in German in 1997 and then as a second edition in 1999. The newly added chapters in Part I are based on the first author’s previously published work (in cooperation with Stefan Kuplen), while Chaps. 21 and 22 are v

vi

Preface to the Second Edition

based on previously published work of the first author. While both authors vouch for the accuracy of all the chapters, the major onus of responsibility rests clearly with the first author. Acknowledgment Niko Chtouris, the Associate Editor for books in Economics and Political Science, initiated the revision of Growth and International Trade. We are very grateful to him for this initiative and for his excellent cooperation during the revision process. Graz, Austria September 2020

Karl Farmer Matthias Schelnast

Preface to the First Edition

This textbook contains introductory and rather advanced topics on growth and international trade from the Diamond-type overlapping generations perspective. Part I, “Growth”, comprises mainly, although not exclusively, introductory material. This is also true for Part II, “International Trade”, but to a much lesser extent. In Part I, the unified analytical approach of Diamond’s (1965) overlapping generations model to neoclassical (“old”) and “new” growth theories (R&D and human capital approaches) figures prominently. Chapters 2 and 3 exhibit the basic overlapping generations model of the world economy, its intertemporal equilibrium dynamics, and steady-state growth. Discussion of public debt in Chap. 4 and economic growth under exogenous and endogenous conditions of technological change in Chaps. 5 and 6 and a description of the factors (religion included) determining human capital formation in Chap. 7 are both expected to attract the interest of the reader. Given the present economic climate, the chapter concluding Part I, on “Growth with Bubbles”, will likely be particularly interesting. The first three chapters in Part II are devoted to the presentation of an intertemporal equilibrium version of neoclassical (HeckscherOhlin) trade theory, including a neoclassical model on globalization. Further two chapters focus on international trade under imperfect competition and on product differentiation under exogenous and endogenous technological change. The remaining five chapters of Part II contain recent research results obtained by the first author in cooperation with both Birgit Bednar-Friedl and the second author. The first two of these chapters deal with the existence of limits to national debt and international effects of debt reduction in advanced countries. The international impact of unilateral climate policy on capital accumulation and welfare and the determination of an optimum climate policy from a unilateral and a multilateral perspective are investigated in the next two chapters. The final chapter in Part II looks at the internal debt mechanics leading to the recent crisis in the euro zone. As indicated by the book’s subtitle, it was the authors’ intention to introduce the concept of the overlapping generations approach to growth and international trade to a much wider audience. The approach taken is relatively introductory in that utility and production functions are specified in such a way that any interested reader should be capable of deriving solutions to the intertemporal general equilibria

vii

viii

Preface to the First Edition

described. However, our primary intention is not simply to hone skills in general equilibrium solutions but rather to improve the reader’s ability to grasp the analytical significance of the much more advanced dynamic general equilibrium models (on growth and international trade) published in leading journals. This book represents a thoroughly revised and substantially extended version of Wachstum und Außenhandel, co-authored by Ronald Wendner and first published in German in 1997 and then as a second edition in 1999. As mentioned above, the last four chapters of Part II are based on the first author’s previously published work (in cooperation with Birgit Bednar-Friedl), while Chaps. 8 and 15 are based on joint work of the present authors. While both authors vouch for the accuracy of all the chapters, the major onus of responsibility rests clearly with the first author. Acknowledgments Several people have helped us over the long period of gestation needed to complete this work. First, we are particularly grateful to Laurie Conway and Ingeborg Stadler for their excellent language check. Laurie made such substantial revisions to several chapters that to some extent he could even be seen as co-author. We are also grateful to Anita Schewczik-Pauritsch and Corinna Blasch for careful proofreading. Last but not least, we thank Dr. Martina Bihn and Barbara Fess from Springer for encouraging us to revise Wachstum und Außenhandel and for accepting the manuscript for publication. Authoring a book is a highly demanding exercise for all those concerned. We both owe a considerable debt of gratitude to all those among our family and friends who repeatedly offered us their patience and support over the last few months. Graz, Austria August 2012

Karl Farmer Matthias Schelnast

Contents

1

Growth and International Trade: Introduction and Stylized Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Definition of Growth Magnitudes . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Growth Rates of Products and Quotients . . . . . . . . . . 1.3 Kaldor’s “Stylized Facts” . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Kuznets’ Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Internationalization Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 World Trade Is Growing Faster Than World Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Export and Import Ratios Increase Over Time . . . . . . 1.5.3 Two-Thirds of Foreign Trade Takes Place Between Developed Countries . . . . . . . . . . . . . . . . . . . . . . . . 1.5.4 Neighboring Countries Trade More with Each Other Than Countries That Are Further Apart . . . . . . . 1.6 Globalization Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Foreign Direct Investment and Financial Investment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Asia Since the 1970s: “The” Dynamic Export Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part I 2

1 1 3 6 7 15 15 16 16 18 20 22 22 23 23 26 27

Growth

Modeling the Growth of the World Economy: The Basic Overlapping Generations Model . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Set-Up of the Model Economy . . . . . . . . . . . . . . . . . . . 2.3 The Macroeconomic Production Function and Its Per Capita Version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Structure of the Intertemporal Equilibrium . . . . . . . . . . . . . .

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31 31 32

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34 37 ix

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Contents

2.4.1

Intertemporal Utility Maximization of Younger Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Old Households . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 A-Temporal Profit Maximization of Producers . . . . . 2.4.4 Market Equilibrium in All Periods . . . . . . . . . . . . . . 2.5 The Fundamental Equation of Motion of the Intertemporal Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Maximal Consumption and the “Golden Rule” of Capital Accumulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Constrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 2. Walras’ Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4

. . . .

37 41 41 43

.

44

. . . . . . .

46 48 49 50 50 53 54

Steady State, Factor Income, and Technological Progress . . . . . . . . 3.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The GDP Growth Rate in Intertemporal Equilibrium and in Steady State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Existence and Stability of the Long-Run Growth Equilibrium . . 3.4 Efficiency of the Steady State . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Comparative Dynamics in the Basic OLG Model . . . . . . . . . . 3.5.1 Increase in the Time Discount Factor . . . . . . . . . . . . . 3.5.2 Reduction of the Population Growth Rate . . . . . . . . . 3.5.3 Increase in the Rate of Technological Progress . . . . . . 3.6 Real Wage, Real Interest Rate, and Income Shares . . . . . . . . . 3.6.1 Income Distribution along the Equilibrium Growth Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Technological Progress in Neoclassical Growth Theory . . . . . . 3.7.1 Hicks-Neutral Technological Progress . . . . . . . . . . . . 3.7.2 Harrod-Neutral Technological Progress . . . . . . . . . . . 3.7.3 Solow-Neutral Technological Progress . . . . . . . . . . . . 3.7.4 Resumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Growth Accounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 57

69 72 73 75 77 77 78 79 81 82

Economic Growth and Public Debt in the World Economy . . . . . . 4.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 European Debt Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 First-Order Conditions and Market Clearing . . . . . . . . . . . . . 4.3.1 Market Equilibrium in All Periods . . . . . . . . . . . . . . 4.4 Intertemporal Equilibrium Dynamics . . . . . . . . . . . . . . . . . .

83 83 85 86 89 90

. . . . . .

59 60 64 66 66 68 68 69

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xi

4.5

93 93 95 98 102 108 110 110

Existence and Stability of Steady States . . . . . . . . . . . . . . . . . 4.5.1 Existence of a Long-Term Growth Equilibrium . . . . . 4.5.2 Stability of the Steady States . . . . . . . . . . . . . . . . . . . 4.5.3 Analytical Investigation of Dynamic Stability . . . . . . . 4.6 Reducing Public Debt Under Dynamic Efficiency . . . . . . . . . . 4.7 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

6

“New” Growth Theory and Knowledge Externalities in Capital Accumulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Empirical Shortcomings of the “Old” Growth Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Theoretical Shortcomings of the “Old” Growth Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Main Approaches of “New” Growth Theory . . . . . . . . 5.1.4 Aims of Explanation and Preview . . . . . . . . . . . . . . . 5.2 Public Good Characteristics of Knowledge Externalities . . . . . 5.3 Knowledge Externalities in the Basic OLG Model . . . . . . . . . . 5.3.1 The Production Technology with Knowledge Externalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Inter-Temporal Utility Maximization of Active Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Profit Maximization of Firms . . . . . . . . . . . . . . . . . . 5.3.4 Market-Clearing Conditions . . . . . . . . . . . . . . . . . . . 5.3.5 Structure of the Inter-Temporal Equilibrium . . . . . . . . 5.4 The Shortcomings of the Old Growth Theory from the Perspective of Romer’s New Growth Model . . . . . . . . . . . . . . 5.5 Public Debt and Net Deficit in Romer’s New Growth Model . . 5.5.1 Government Budget Constraint, FOCs, and Market Clearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Inter-Temporal Equilibrium Dynamics . . . . . . . . . . . . 5.5.3 Existence and Dynamic Stability of Balanced Growth . 5.6 Business Cycles and Endogenous Growth . . . . . . . . . . . . . . . . 5.7 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stochastic Growth Theory and Stochastic Processes . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113 113 113 116 116 118 119 120 120 122 123 123 123 126 128 129 130 130 132 134 135 136 136 137

Endogenous Technological Progress and Infinite Economic Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.2 Monopolistic Competition and Product Innovation in Inter-Temporal Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 140

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6.2.1 Production Technologies and Innovation . . . . . . . . . 6.2.2 Choice Problems and Market-Clearing Conditions . . 6.2.3 Structure of the Inter-Temporal Equilibrium . . . . . . . 6.3 Unbounded Economic Growth and Increasing Growth Rates . . 6.4 One-Period Versus Long-Duration Patents . . . . . . . . . . . . . . 6.5 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

8

9

. . . . . . . .

141 141 143 147 150 151 152 152

Human Capital, Religion, and Economic Growth . . . . . . . . . . . . . . 7.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Human Capital Formation in the Basic OLG Model . . . . . . . . . 7.2.1 FOCs for Profit and Inter-Temporal Utility Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Market-Clearing Conditions . . . . . . . . . . . . . . . . . . . 7.2.3 The Structure of Inter-Temporal Equilibrium . . . . . . . 7.3 Inter-Temporal Equilibrium Dynamics and Steady State . . . . . . 7.3.1 Inter-Temporal Equilibrium Dynamics . . . . . . . . . . . . 7.3.2 Steady State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Predictions of the Basic Human Capital Model . . . . . . 7.4 An OLG Model of Religion and Human Capital Formation . . . 7.4.1 The OLG Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Religious Participation and Children’s Education . . . . 7.4.3 Religion as a Steady-State Phenomenon . . . . . . . . . . . 7.5 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153 153 156

Economic Growth with Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Stylized Bubble Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Bubbles in the Basic OLG Growth Model . . . . . . . . . . . . . . . 8.3.1 Equilibrium with Bubbles and Without Financial Frictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Equilibrium with Bubbles and Financial Frictions . . . 8.3.3 Where Is the Market for Bubbles? . . . . . . . . . . . . . . 8.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

157 159 160 160 161 162 163 164 164 165 167 167 168 169

. . . .

171 171 172 177

. . . . . .

178 181 186 189 190 190

Involuntary Unemployment in an OLG Growth Model with Public Debt and Human Capital . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 The Basic Model: The Log-Linear Cobb–Douglas OLG Model with Internal Public Debt and Involuntary Unemployment . . . . 9.2.1 The Household Sector . . . . . . . . . . . . . . . . . . . . . . . .

193 193 195 196

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xiii

9.2.2 9.2.3 9.2.4

The Production Sector . . . . . . . . . . . . . . . . . . . . . . . The Public Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnani’s (2015) Macro-Founded Investment Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.5 Market Clearing Conditions . . . . . . . . . . . . . . . . . . . 9.2.6 Temporary Unemployment Rate . . . . . . . . . . . . . . . . 9.2.7 Intertemporal Equilibrium Dynamics . . . . . . . . . . . . . 9.2.8 Existence and Dynamic Stability of Nontrivial Steady States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.9 Comparative Steady-State Analysis: How More Optimistic Investor’s Animal Spirits or a Lower Saving Rate Affect the Unemployment Rate . . . . . . . . 9.3 The Basic Model Extended by Human Capital Accumulation: The Log-Linear Cobb–Douglas OLG Model with Endogenous Growth and Involuntary Unemployment . . . . . . . . . . . . . . . . . 9.3.1 The Household Sector . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 The Production Sector . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 The Public Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.4 Human Capital Accumulation and GDP Growth . . . . . 9.3.5 Magnani’s Macro-Founded Investment Function . . . . 9.3.6 The Intertemporal Equilibrium Dynamics in Terms of Per GDP Ratios . . . . . . . . . . . . . . . . . . . 9.3.7 Existence and Dynamic Stability of Nontrivial Steady States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.8 Comparative Steady-State Analysis: How a Larger Government Debt to GDP Ratio Affects the Unemployment Rate . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Summary and Suggestions for Future Research . . . . . . . . . . . . 9.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Robots, Human Capital Investment, Welfare, and Unemployment in a Digital World Economy . . . . . . . . . . . . . . . . . 10.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 The Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Intertemporal Equilibrium Dynamics and Steady State in the Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 The Basic Model Extended by Involuntary Unemployment . . 10.5 Intertemporal Equilibrium Dynamics and Steady State in the Extended Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

197 198 198 199 199 201 201

204

205 205 206 206 207 207 208 211

215 220 221 221

. 223 . 223 . 226 . 229 . 240 . . . .

241 251 254 255

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Part II 11

12

International Trade

International Parity Conditions in a Two-Country OLG Model Under Free Trade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The General OLG Model of International Trade in Goods, Financial Assets, and Money . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Young Household’s Choice and International Parity Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Domestic and Foreign Households in the International Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Choice-Based Consumer Price Indices (“Ideal” Deflators) and Real Consumption Expenditures . . . . . 11.3.3 Purchasing Power Parity in Its Absolute and Relative Version . . . . . . . . . . . . . . . . . . . . . . . . 11.3.4 The Household’s Choice Problem Using the Deflator and Real Consumption Expenditure . . . . . . . . . . . . . . 11.4 The Neoclassical Model of International Commodity Trade . . . 11.4.1 Domestic and Foreign Producers in the Intertemporal World Market Equilibrium . . . . . . . . . . . . . . . . . . . . 11.4.2 Domestic and Foreign Households in the Intertemporal World Market Equilibrium . . . . . . . . . . . . . . . . . . . . 11.4.3 Terms of Trade in the Neoclassical Basic Model . . . . . 11.4.4 Market-Clearing Conditions and Current Account Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Factor Proportion, Inter-Sectoral Trade, and Product Life Cycle . 12.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Production-Based Equilibrium Conditions in Autarky . . . . . . 12.3 Equalization of Factor Prices in the World Market Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Factor Proportions of Interindustrial Trade in the World Market Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 The Leontief Paradox and the Neo-factor-proportion Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 The “Product Life Cycle” and the Dynamics of Comparative Advantages . . . . . . . . . . . . . . . . . . . . . . . . . 12.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

259 259 260 261 261 263 265 269 271 272 272 273 274 277 278 279 281

. 283 . 283 . 284 . 288 . 291 . 295 . . . .

297 299 300 300

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13

14

15

xv

Product Differentiation, Decreasing Costs, and Intra-Sectoral Trade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Linder’s Demand-Based Trade Theory . . . . . . . . . . . . . . . . . . 13.3 Monopolistic Competition and Product Differentiation in a Closed Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1 Utility Maximization of Households and the Demand for Differentiated Goods . . . . . . . . . . . . . . . . . . . . . . 13.3.2 Profit Maximization of Producers and Short-Term Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.3 The Long-Term Market Equilibrium . . . . . . . . . . . . . 13.4 Intra-Industry Trade Under Monopolistic Competition . . . . . . . 13.4.1 Price Elasticity of Demand Is Independent of the Number of Variants . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.2 Price Elasticity of Demand Depending on the Number of Variants . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Globalization, Capital Accumulation, and Terms of Trade . . . . . . . 14.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Causes and Consequences of the Globalization of Commodity Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Globalization of Commodity Markets: A Comparative-Static Analysis . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Dynamics of the Region-Specific Capital Intensities and of the Terms of Trade . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Globalization and Inequality Between Nations . . . . . . . . . . . . 14.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Innovation, Growth, and Trade in a Two-Country OLG Model . . 15.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Intermediate Products in the Two-Country OLG Model . . . . . 15.3 Agents’ Choice Problems and Market-Clearing Conditions with Intra-sectoral Trade in Intermediates . . . . . . . . . . . . . . . 15.4 Producers’ First-Order Conditions . . . . . . . . . . . . . . . . . . . . 15.5 The Structure of Intermediate Product Prices and Quantities . . 15.6 Cost Minimization of Final Good Producers and the Prices of Final Goods in the Industry Equilibrium . . . . . . 15.7 The Growth Rate of Intermediate Product Innovations in the International Equilibrium Versus in Autarky . . . . . . . . . . . . .

303 303 306 308 308 312 313 315 316 318 320 322 322 324 325 325 327 328 331 335 341 342 342

. 345 . 345 . 347 . 347 . 351 . 353 . 355 . 358

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15.8

Integration, Efficiency, and Economic Growth: Some Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

17

. . . .

. . . .

. . . .

Real Exchange Rate and Public Debt in a Two-Advanced-Country OLG Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Literature Review and Preview . . . . . . . . . . . . . . . . . . . . . . . . 16.3 The Two-Good, Two-Country OLG Model . . . . . . . . . . . . . . . 16.4 Intertemporal Equilibrium Dynamics and Existence of Steady States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5 Stability of Steady States and Steady-State Effects of Public Debt Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6 Transitional Impacts of Shocks in Home’s Sustainable Public Debt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.7 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Utility Maximizing Consumption and Saving Functions of Younger Households . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Proof of Proposition 16.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The Jacobian Matrix and Its Derivation . . . . . . . . . . . . . . . . 4. Proof of Lemma 16.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Proof of Proposition 16.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Proof of Proposition 16.4 . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Public Debt Reduction in Advanced Countries and its Impacts on Emerging Countries . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3 Firm FOCs and Intertemporal Equilibrium in a Two-Country OLG Model with Unequal Technologies . . . . . . . . . . . . . . . . . 17.4 Existence, Dynamic Stability, and Comparative Statics of Steady States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5 Steady-State Welfare Effects of a Unilateral Reduction of Public Debt in Home . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5.1 Steady-State Welfare Effects of Unilateral Debt Reduction in Home . . . . . . . . . . . . . . . . . . . . . . . . . 17.5.2 Numerical Illustrations of Welfare Effects for the Leading US–China Case . . . . . . . . . . . . . . . . . . . . . . 17.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Mathematical Appendix . . . . . . . . . . . . . . . . . . . . . . . . . .

360 363 364 364 365 365 366 369 373 379 382 387 389 389 389 390 391 392 394 394 395 397 397 398 400 401 407 408 412 416 417 417

Contents

1. Proof of Proposition 17.1 . . . . . . . . . . . . . . . . . . . . . . . . . 2. The Jacobian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Proof of Proposition 17.2 . . . . . . . . . . . . . . . . . . . . . . . . . 4. Comparative Steady-State Effects of Debt Changes . . . . . . 5. Proof of Proposition 17.3 . . . . . . . . . . . . . . . . . . . . . . . . . 6. Derivation of Domestic Welfare Effect of Debt Change . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

19

xvii

. . . . . . .

External Balance, Dynamic Efficiency, and Welfare Effects of National Climate Policies . . . . . . . . . . . . . . . . . . . . . . . . . 18.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Literature Review and Preview . . . . . . . . . . . . . . . . . . . . . . . . 18.3 The Two-Good, Two-Country OLG Model with Nationally Tradable Emission Permits . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.1 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.2 Households and Governments . . . . . . . . . . . . . . . . . . 18.3.3 Market Clearing and International Trade . . . . . . . . . . 18.4 The Steady State and Unilateral Permit Policy . . . . . . . . . . . . . 18.4.1 Inter-Temporal Equilibrium Dynamics . . . . . . . . . . . . 18.4.2 Characterization of Steady States . . . . . . . . . . . . . . . . 18.4.3 Steady State Effects of Unilateral Permit Policies . . . . 18.5 The Steady-State Welfare Effects of Different Permit Policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.5.1 Derivation of Welfare Effects . . . . . . . . . . . . . . . . . . 18.5.2 Comparison of Global Welfare Effects of Unilateral Permit Policies in Home and in Foreign . . . . . . . . . . . 18.5.3 Comparison of Welfare Effects of a Unilateral Domestic and a Multilateral Permit Policy . . . . . . . . . 18.5.4 Comparison of Welfare Effects of a Unilateral Foreign and a Multilateral Permit Policy . . . . . . . . . . 18.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Mathematical Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Existence of Steady States . . . . . . . . . . . . . . . . . . . . . . . . . 2. Saddle-Path Stability of Steady States . . . . . . . . . . . . . . . . . 3. Proof of Proposition 18.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Proof of Proposition 18.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Proof of Proposition 18.3 . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nationally and Internationally Optimal Climate Policies . . . . . . . . . 19.1 Introduction and Literature Review . . . . . . . . . . . . . . . . . . . . . 19.2 Internationally Differing Environmental Preferences in the Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3 Nationally Optimal Permit Levels . . . . . . . . . . . . . . . . . . . . . .

417 419 420 422 422 423 424 425 425 426 428 429 430 432 433 433 434 436 437 437 440 441 442 444 445 446 446 447 450 452 454 455 457 457 460 463

xviii

20

21

22

Contents

19.4 Internationally Optimal Permit Policies . . . . . . . . . . . . . . . . . 19.5 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 19.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Derivation of Reaction Functions . . . . . . . . . . . . . . . . . . . . . Cross Welfare Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

466 473 474 474 474 475 476

Modeling the Debt Mechanics of the Euro Zone . . . . . . . . . . . . . . 20.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 Stylized Macroeconomic Facts: Financial Autarky Versus EMU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3 The Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.1 Financial Autarky . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.2 International Equilibrium Under Financial Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.3 Financially Integrated Versus Financially Autarkic Steady State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4 Beyond the Basic Model: From Debt Mechanics Towards Debt Trap for South . . . . . . . . . . . . . . . . . . . . . . . . 20.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 479 . 479 . 480 . 484 . 486 . 489 . 493 . . . .

Financial Integration, House Price Dynamics, and Saving Rate Divergence in an OLG Model with Intra-EMU and Asian–US Trade Imbalances . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Stylized Macroeconomic Facts: Financial Autarky Versus Financial Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Model Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Pre-Euro and Asian–US Financial Autarky . . . . . . . . . . . . . . . 21.5 International Equilibrium under Intra-EMU and Global Financial Integration . . . . . . . . . . . . . . . . . . . . . . . 21.6 National Net Foreign Asset Position, Trade Imbalance, and Aggregate Savings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.7 Numerical Specification and Model Generated Versus Stylized Macro Facts . . . . . . . . . . . . . . . . . . . . . . . . . 21.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

496 501 502 502

505 505 508 509 518 532 535 537 543 544 545

Why Religion Might Persist in a Globalized Market Economy: An Economic Modeling Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 547 22.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 547 22.2 Autarky Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550

Contents

International Equilibrium Under Capital Market Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 22.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xix

22.3

. . . .

. . . .

561 578 579 580

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581

Abbreviations

CD ECU EMU FOC GATT GDP HOS IIT OLG

Cobb–Douglas European Currency Union Economic and Monetary Union First-Order Conditions General Agreement on Tariffs and Trade Gross Domestic Product Heckscher–Ohlin–Samuelson Intra-industry Trade Overlapping Generations

xxi

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Growth and International Trade: Introduction and Stylized Facts

1.1

Introduction and Motivation

The fact that economic activity in the Western hemisphere has continued to grow from the beginning of the modern era onwards—especially over the past 200 years (since the Industrial Revolution)—is amazing. Even more astonishing is the fact that economic growth does not appear to stop after a certain time interval, but seems to be capable of continuation for centuries. Not even the two world wars and their disastrous destruction of human life, social relationships, houses, machinery, and public infrastructure were able to interrupt this enduring process of growth. What lies behind this astounding phenomenon? Why do self-interest and competition foster persistent economic growth in capitalistic systems? The answer to these questions is the main focus of the first part of the book, i.e., Chaps. 2–10. Numerous explanations for different aspects of economic growth have been provided by economic theory—especially in the period since the end of World War II. All such theories attempt to explain the characteristic features of the economic growth process. The second part of this book is mainly devoted to international trade. International trade theory or more generally international economics goes back to the beginnings of economic research in the mid-eighteenth century. The essay “Of the balance of trade” by the Scottish philosopher David Hume, published in 1758, can be seen as the foundation of international economics as a science, and of an approach based on the manipulation of abstract economic models. The debates on trade policy in the British Empire in the nineteenth century contributed considerably to the status and reputation of economics as a respected field within the social sciences (see Krugman et al. 2018). “Yet the study of international economics has never been as important as it is now” (ibid, 1). This claim by Krugman, Obstfeld, and Melitz can be easily confirmed by a quick glance at the newspapers. For example, the financial crises of 2007/2008, which started in the United States, seriously impacted the economic development of numerous countries in the world. Due to the interdependences of the global # Springer-Verlag GmbH Germany, part of Springer Nature 2021 K. Farmer, M. Schelnast, Growth and International Trade, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-662-62943-7_1

1

2

1

Growth and International Trade: Introduction and Stylized Facts

economy, Greece and Iceland, among others, faced severe financial problems. Another example that illustrates the strong transnational interdependences is the COVID-19 pandemic. At present, long-term impact forecasts of the pandemic on the world economy are highly speculative. However, the first few months after the outbreak of the pandemic were marked by major economic downturns and massively reduced international trade all over the globe. The interdependencies between national economies, so often the subject of discussion among economists, have clearly become part of everyday life. One no longer has to be an expert to be aware of the feedback effects in economic and political processes around the world and of the relationships between the local and the global economy. International economics applies basic economic theory (micro- and macroeconomics) to the economic relations of countries connected through the exchange of goods and services or asset and debt instruments (Ingham 2004, 1). Although there is no need for a separate theory of international interdependencies, but international economics differs clearly from micro- and macroeconomics in a closed economy. The reasons for this, and for the increasing importance of international economics are explained in the next few paragraphs. Modern domestic households and firms buy goods and services from abroad (¼imports) and sell goods and services to foreign countries (¼exports) to an extent never seen before in history. It is now common for mid-sized companies in Europe to sell their products in India and China, and for an average income recipient to holiday in Vietnam and Burma. Although the decision concerning the purchase of foreign or domestic goods is the result of rational choice (utility maximization subject to budget constraints), in such a context the impact of exchange rates is crucial. Even more than the trade in goods and services, trade in foreign financial assets and internationally traded bonds and stocks has also witnessed a dramatic growth (this was especially true for the 1990s). Prior to the 1970s, national investments were financed almost exclusively by domestic savings, and (positive or negative) net investments in foreign countries were negligible. There is clear evidence of a trend toward globalization in real and monetary terms. Neither countries nor individuals or firms are now in a position where they can safely make economic decisions in isolation. Small and medium-sized companies, and multinationals all interact on a global market. Such a market is characterized by global fragmentation of production, outsourcing, global supply management, global economies of scale, internationally integrated production technologies, international direct investment, and by mega-mergers and global acquisitions. These endogenous economic developments have been accompanied by fundamental changes in the national foreign trade policy of emerging and developing countries. While during the first two decades after WWII most developing countries relied on import substitution, during the 1980s several emerging markets in Latin America and elsewhere followed the example of the South East Asian “tiger economies” and opened their fragmented national markets to the world economy—sometimes for “humanitarian” reasons, but more frequently in pursuit of economic advantage. The IMF (International Monetary Fund)—the supranational

1.2 Definition of Growth Magnitudes

3

institution called on to finance foreign exchange deficits—played a central role in this process, for which in some cases, it was heavily criticized. Together with the fall of the Iron Curtain in 1989 and the collapse of the communist regimes of Eastern Europe, trade barriers between Central and Eastern Europe decreased. After half a decade of more or less painful transition from planned to market economies, on May 1, 2004, eight Central and Eastern European countries (CEEC) and in January 2007, Romania and Bulgaria, all joined the EU. Croatia followed in July 2013, and the EU became a union of 28 member countries. Despite the withdrawal of the United Kingdom from the EU in January 2020, there is still some drive toward international integration and transnational cooperation the recent rise toward nationalism and pandemic-related national solo efforts not withstanding. Of particular importance for the world labor supply was the integration of populous, formerly almost autarkic, developing countries such as India and China. This process was particularly evident during the 1990s—and reached a peak in 2001 with the accession of China to the World Trade Organization (WTO). The postwar integration of the world economy was heavily based on the GATT (General Agreement on Tariffs and Trade) and the seven negotiation rounds (the Uruguay Round was the final one) between the signatory countries. Their objective was to reach a multilateral reduction in trade barriers. The WTO, as a successor to the GATT, administers the negotiated contracts, tries to settle disputes between member states, and is committed to further trade negotiations, especially on agricultural products, textiles, and services. The complete liberalization of international capital flows in the late 1980s was decisive for the establishment of global financial and foreign exchange markets in the 1990s. Combined with the dramatic reduction in global communication costs and the revolution in the information technology sector, liberalization has led to an enormous growth in transaction volume on international financial markets. Such transactions now amount to several trillion dollars a day. There were mergers of large banks and national stock exchanges. New mammoth banks and larger stock exchanges emerged, and many of the older patriarchal bodies disappeared overnight.

1.2

Definition of Growth Magnitudes

To aid understanding of global economic growth, we will now define exponential growth and give an introduction to the arithmetic (algebra) of growth variables. The growth rate of a macroeconomic variable x for time period t is denoted by the symbol gxt. It gives the percentage change in the variable x between the beginning of period t and of period t + 1. Conceptually, we can consider a variable in discrete time (intervals of finite length) or in continuous time (intervals of infinitesimal length). In discrete time the growth rate (growth factor) of x is defined as:

4

1

gxt 

Growth and International Trade: Introduction and Stylized Facts

xtþ1  xt ðgrowth rateÞ and Gxt ¼ 1 þ gxt ðgrowth factorÞ: xt

When gxt ¼ gxtþ1 ¼ gx, i.e., the growth rate of the variable x is constant over time (stationary), the evolution of the variable can be calculated. For illustration, let us consider the development of the gross domestic product (GDP) Yt (xt ¼ Yt). First Step: We define the growth rate (by using the growth factor) of GDP. gYt ¼

Y tþ1  Y t Y tþ1 ¼  1 ¼ GYt  1 Yt Yt

Second Step: The GDP can be calculated for any (future) period. The starting point for our calculation is the definition of the growth rate in the first step. Assuming gYt ¼ gYtþ1 ¼ gY , 8t, and solving the equation for Yt + 1, we obtain the equation of motion for Yt:
Y tþ1 ¼ 1 þ gY Y t , t ¼ 0, 1, 2, . . .
For t ¼ 0 : Y 1 ¼ 1 þ gY Y 0 ,



2 for t ¼ 1 : Y 2 ¼ 1 þ gY Y 1 ¼ 1 þ gY 1 þ gY Y 0 ¼ 1 þ gY Y 0 ,




3 for t ¼ 2 : Y 3 ¼ 1 þ gY Y 2 ¼ 1 þ gY 1 þ gY 1 þ gY Y 0 ¼ 1 þ gY Y 0 , for t ¼ T  1 : Y T ¼ 1 þ gY


T

Y 0:

YT represents the value of GDP in period T, Y0 that of the initial period. gY gives the stationary growth rate (e.g., gY ¼ 0.03) and T the number of periods elapsing since the initial period. If the length of each period becomes shorter, so that Δt ! 0, we arrive at continuous time. The growth rate gx(t) (as a function of t) can be written as follows: gx ð t Þ 

d ln xðt Þ d ln xðt Þ dxðt Þ 1 dxðt Þ ¼ ¼ : dt dt xðt Þ dt dxðt Þ

The relation between discrete- and continuous-time growth rates may be expressed as follows: 

gY lim 1 þ τ!1 τ

τt

Y

¼ eg t :

Figure 1.1 shows that a stationary growth rate implies an exponential trend in the level of world GDP. The steepness of the exponential function depends on the magnitude of the growth rate.

1.2 Definition of Growth Magnitudes

5

discrete time

continuous time

Yt

Y(t )

Y(0)

Y0

0

1

t

2

t

Fig. 1.1 Exponential growth of gross world product in discrete and continuous time

Table 1.1 Gross domestic product per capita in various cultures, in US$, real 1990 prices Year Western Europe China

1820 1455 600

1900 2959 545

1950 4517 448

2000 19,298 3421

2010 20,889 8032

Source: Bolt and van Zanden (2014)

Small differences in the growth rate have a huge impact on the level of a variable. This effect is shown in Table 1.1 where GDP per capita of Western Europe and China are compared over nearly two centuries. While the level in China was slightly declining between 1820 and 1950, the annual growth rate in Europe was 1.4%. While the per capita product in Europe rose by a factor of 13, that of China only by a factor of 6 in the period 1820–2000. The “doubling time” concept is useful as a quick means of estimating the impact of different stationary growth rates. It specifies how long (e.g., how many years) it takes for a variable experiencing constant exponential growth to double in size. For instance, we might be interested in knowing how long it will take for world population to double, given a growth rate of 1.6% per annum. The quick answer: about 44 years. How do we arrive at this result? We denote the world’s population in year T (we yet do not know the exact value of T) by L(T ) and the world’s population today by L(0). The question to answer is: When is L(T ) twice as large as L(0) (i.e., L(T ) ¼ 2L(0))? A stationary L population growth rate gL implies: LðT Þ ¼ Lð0Þeg T . If we equate the two L L definitions of L(T ), we obtain: 2Lð0Þ ¼ Lð0Þeg T , eg T ¼ 2 , gL T ¼ ln 2 , T ¼ ð ln 2Þ=gL  70=ð100gL Þ ¼ 70=1:6  44.

6

1

1.2.1

Growth and International Trade: Introduction and Stylized Facts

Growth Rates of Products and Quotients

We take the aggregate labor income as an example for a product, which is defined as the product of the real wage rate wt and the number of employees Nt in period t, i.e., wtNt. As an example for a quotient, we consider per capita income, defined as the quotient of aggregate income Yt and the number of people of a country Lt, i.e., Yt/Lt. Growth rate of a product (discrete time): ¼ GwN 1¼ gwN t t

wtþ1 N tþ1 1¼ wt N t



wtþ1 wt

  N tþ1  1 ¼ Gwt GNt  1: Nt

Growth rate of a product (continuous time): gwN ¼ t

d ln ðwðt ÞN ðt ÞÞ d ln ðwðt Þ þ ln N ðt ÞÞ 1 dwðt Þ 1 dN ðt Þ ¼ ¼ þ dt dt wðt Þ dt N ðt Þ dt

¼ gwt þ gNt : Growth rate of a quotient (discrete time): Y L

Y L

gt ¼ Gt  1 ¼

Y tþ1 Ltþ1 Yt Lt

Y L  1 ¼ tþ1 t  1 ¼ Y t Ltþ1

Y tþ1 Yt Ltþ1 Lt

1¼

GYt  1: GLt

Growth rate of a quotient (continuous time): Y

gtN ¼

d ln ðY ðt Þ=N ðt ÞÞ d ln ðY ðt Þ  ln N ðt ÞÞ 1 dY ðt Þ 1 dN ðt Þ ¼ ¼  dt dt Y ðt Þ dt N ðt Þ dt

¼ gYt  gNt : For illustrative purposes, Table 1.2 provides the average growth rates of GDP, of the population and the GDP per capita in selected countries between 1991 and 2019.

Table 1.2 Average growth rates of GDP (€, current prices), population and GDP per capita (€, current prices) 1991–2019 in selected countries Germany France Austria UK Japan USA Source: AMECO (2020)

GDP 3.0 3.1 3.8 3.3 1.6 4.9

Population 0.1 0.5 0.5 0.5 0.1 0.9

GDP per capita 2.8 2.6 3.3 2.8 1.6 4.0

1.3 Kaldor’s “Stylized Facts”

1.3

7

Kaldor’s “Stylized Facts”

A “stylized” description of the evolution of important macroeconomic variables (¼“stylized facts” à la Kaldor 1961) in advanced industrial economies over several decades reveals that (a) some aggregate variables grow at a positive and on average constant rate, (b) some macro variables grow at the same rate, and (c) some macroeconomic variables do not exhibit a long-term average growth trend at all. Among the variables that exhibit a positive (and constant) growth rate are: the number of (employed) workers, the capital stock, the gross national product, capital equipment per workstation (i.e., capital intensity), and labor productivity. Moreover, GDP and capital stock grow at the same rate. Those variables exhibiting no longterm growth include the real interest rate, the capital–output ratio, and the wage and profit shares. A first impression of the longer-term macro-dynamics can be gained by turning to various empirical facts. First, we turn to those macroeconomic variables exhibiting long-term constant growth. A look back at the last millennium is quite revealing. We start with (shares of) GDP. Table 1.3 shows that Asia dominated world production for a long time in the past and may now be in a position to regain its former position (more details in: Bigsten 2004, 33; Maddison 2007, 340). Table 1.4 gives GDP growth rates of Western industrialized countries for various time intervals between 1820 and 2018. The table shows among other things—that the United States was the growth engine of the world economy for most periods. The level of GDP in selected years (in $, current prices) and the average growth rate (constant prices) for 1980–2000, 2000–2019, and the forecast for 2019–2024 for certain regions are presented in Table 1.5. Another macroeconomic variable that grows at a constant rate over the long run is real income per capita. If 100% of all residents in a country worked (in reality this portion is less than 50%), this variable would also be the same as labor productivity. Table 1.6 presents the development of real income per capita over the course of the twentieth century in selected “rich” countries and illustrates, for purposes of comparison, the evolution of real income in Argentina, Bangladesh, China, and India. The last column gives the average growth rate of real income per capita over the last century. Table 1.7 shows GDP per capita (in 2011 international $) worldwide and in selected regions between 1980 and 2024. Moreover, the average growth rates of GDP per capita between 1980 and 2005, and for 2005–2024 (forecast) are also provided. Again, Table 1.8 gives GDP per capita for selected countries, but now starting with 2012 and ending in 2024 (forecast). The numbers in this table are those found at regular intervals in daily newspapers. Please note: these are not inflation-adjusted income per capita values. Capital intensity is one of the variables that exhibit exponential growth. It gives the capital stock per worker and represents the ratio of capital stock employed to the number of employees in an economy. Table 1.9 gives the capital stock (in billion € at

1500 17.8 6.1 0.5 3.1 61.8 2.9 7.8 100.0

Source: 1000–2003: Maddison (2007, 381); 2018–2024: IMF (2019)

Western Europe Eastern Europe + CIS North America + Australia + New Zealand Japan Asia (without Japan) Latin America Africa World

1000 9.1 4.6 0.6 2.7 67.9 3.8 11.4 100.0

1820 23.0 9.0 1.9 3.0 57.7 2.1 4.5 100.0

1870 33.1 12.0 10.0 2.3 36.0 2.5 4.1 100.0

1950 26.2 13.1 30.6 3.0 15.6 7.8 3.8 100.0

1973 25.6 12.8 25.3 7.8 16.4 8.7 3.4 100.0

2003 19.2 5.7 23.7 6.6 33.9 7.7 3.2 100.0

2024 20.9 16.2 3.5 47.3 6.9 5.2 100.0

2018 23.0 17.7 4.1 42.6 7.6 5.0 100.0

1

Table 1.3 Regional and country shares of world GDP

8 Growth and International Trade: Introduction and Stylized Facts

1.3 Kaldor’s “Stylized Facts”

9

Table 1.4 Average yearly GDP growth (selected countries and time intervals)

Austria Belgium Denmark Germany UK Finland France Italy Netherlands Norway Sweden Switzerland USA

1820– 1870 1.45 2.25 1.91 2.01 2.05 1.58 1.27 1.24 1.70 1.70 1.62 1.85 4.20

1870– 1913 2.41 2.01 2.66 2.83 1.90 2.74 1.63 1.94 2.16 2.12 2.17 2.43 3.94

1913– 1950 0.25 1.03 2.55 0.30 1.19 2.69 1.15 1.49 2.43 2.93 2.74 2.60 2.84

1950– 1973 5.35 4.08 3.81 5.68 2.93 4.94 5.05 5.64 4.74 4.06 3.73 4.51 3.93

1973– 1998 2.36 2.08 2.09 1.76 2.00 2.44 2.10 2.28 2.39 3.48 1.65 1.05 2.99

1998– 2007 2.65 2.51 2.03 1.62 2.99 3.79 2.42 1.50 2.77 2.42 3.45 2.41 3.09

2007– 2018 1.28 1.36 0.94 1.42 1.24 0.70 0.99 0.24 1.24 1.31 1.87 1.79 1.64

Source: 1820–1998: Maddison (2001, 187); 1998–2018: IMF (2020) Table 1.5 Level and growth of GDP of selected countries, 1980–2024 In billion $, current prices

Advanced economies EU Japan USA Emerging + develop. Europe Latin America + Caribbean Sub-Saharan Africa Bangladesh Brazil China India World

Average yearly growth rate (constant prices) 1980– 2000– 2019– 2000 19 24 2.98 1.90 1.60 2.23 1.70 1.59 2.91 0.92 0.56 3.20 2.09 1.82 0.22 3.84 2.39

1980 8454 3805 1105 2857 249

2000 26,788 8919 4888 10,252 914

2019 51,744 18,292 5154 21,439 3826

2024 62,472 22,045 6260 25,793 4996

853

2209

5188

6547

2.55

2.49

2.10

299 23 146 305 189 11,156

393 55 655 1215 477 33,858

1694 317 1847 14,140 2936 86,599

2460 499 2296 20,979 4632 111,569

n.a. 4.36 2.31 9.76 5.54 3.20

4.87 6.30 2.31 8.97 7.07 3.81

3.78 7.41 2.05 5.78 7.13 3.46

Source: IMF (2019) n.a.: not available

constant 2015 prices), the number of employees in millions, and the capital intensity for selected countries. Capital stock and GDP normally tend to grow at the same rate. Table 1.10 shows the (average) growth rates of capital stock in selected countries. Comparing these rates with the respective GDP growth rates reveals some degree of conformity.

10

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Growth and International Trade: Introduction and Stylized Facts

Table 1.6 Income per capita (GDP in €, 2000) in selected countries and years and average growth rate of per capita income Austria Belgium Canada Denmark Finland France Germany Italy Japan Netherlands Sweden Switzerland USA Argentina Bangladesh China India

1900 2462 3188 2488 2578 1426 2457 2550 1526 1008 2925 2188 3275 3496 2355 417 466 512

1913 2961 3606 3800 3343 1804 2978 3117 2191 1185 3459 2646 3645 4529 3245 443 472 575

1929 3160 4319 4328 4337 2322 4025 3462 2643 1731 4861 3306 5410 5895 3732 445 481 622

1950 3167 4667 6231 5933 3634 4504 3316 2992 1641 5124 5759 7745 8170 4261 461 383 529

1987 13,085 13,280 15,678 15,401 13,144 14,144 13,417 12,771 13,887 13,447 14,483 16,912 18,618 6237 515 1484 961

1999 17,145 17,010 18,347 19,017 15,931 17,549 15,737 15,612 17,597 17,966 16,962 18,590 23,669 7443 708 2702 1568

2007 20,289 20,069 21,619 21,414 21,051 19,040 18,066 17,229 19,505 20,854 21,688 21,175 27,052 8803 960 4857 2402

2015 20,824 20,104 22,446 20,552 19,582 18,904 19,420 15,330 19,699 20,715 22,285 21,944 28,065 10,069 1396 8948 3639

gY/L 1.9 1.6 1.9 1.8 2.3 1.8 1.8 2.0 2.6 1.7 2.1 1.7 1.8 1.4 1.1 2.6 1.7

Source: Burda and Wyplosz (2018, 4)

Table 1.7 Real income per capita in selected regions

1980 23,642 21,277 20,769 29,136 10,088

1990 30,501 25,903 30,607 36,750 15,788

2005 40,186 33,267 35,664 49,413 15,714

2019 46,593 38,884 39,763 56,844 23,080

2024 49,569 41,790 41,593 60,255 25,982

Average yearly growth rate 1980– 2005– 2005 2024 2.14 1.11 1.80 1.21 2.19 0.81 2.14 1.05 1.79 2.68

10,826

10,023

12,203

14,280

15,539

0.48

1.28

2790 1160 11,372 719 1297

2646 1312 10,727 1508 1802

2877 2083 12,424 5669 3258

3663 4390 14,372 17,027 7315

3899 5938 15,607 22,213 9762

0.12 2.37 0.35 8.61 3.75

1.61 5.67 1.21 7.45 5.95

In 2011 international PPP $

Advanced economies EU Japan USA Emerging + develop. Europe Latin America + Caribbean Sub-Saharan Africa Bangladesh Brazil China India Source: IMF (2019)

Source: IMF (2019)

Argentina Austria Bangladesh Belgium Brazil Canada China Denmark Finland France Germany Italy Japan Netherlands Norway Sweden Switzerland UK USA

2012 19,764 45,500 2981 41,678 15,583 42,716 11,260 43,966 40,370 39,314 44,028 35,078 37,088 47,145 64,665 44,690 55,782 38,231 51,556

2013 20,366 46,032 3177 42,258 16,196 44,009 12,291 44,977 40,576 40,031 44,870 34,878 38,559 47,771 65,722 45,545 57,223 39,449 53,061

2014 20,000 46,826 3405 43,413 16,438 45,643 13,363 46,347 40,880 40,949 46,520 35,405 39,486 49,202 67,460 47,161 58,988 41,049 55,010

2015 20,538 47,379 3638 44,404 15,881 46,064 14,362 47,651 41,358 41,659 47,411 36,096 40,430 50,473 68,849 49,235 59,638 42,117 56,787

Table 1.8 GDP per capita in selected countries in PPP-$ at current prices 2016 20,105 48,230 3900 45,218 15,387 46,569 15,397 48,889 42,822 42,440 48,577 36,957 41,104 51,873 69,682 50,210 60,633 42,959 57,901

2017 20,800 49,964 4231 46,701 15,716 48,273 16,659 50,570 44,826 44,115 50,522 38,335 42,761 54,062 72,120 51,742 62,231 44,301 60,000

2018 20,551 52,172 4630 48,327 16,146 49,690 18,116 52,279 46,596 45,893 52,386 39,676 44,246 56,489 74,357 53,652 65,010 45,741 62,869

2019 20,055 53,558 5028 49,529 16,462 50,725 19,504 53,882 47,975 47,223 53,567 40,470 45,546 58,341 76,684 54,628 66,196 46,827 65,112

2024 22,854 62,154 7514 56,963 19,751 58,302 28,110 62,792 56,361 55,065 63,281 46,701 52,637 68,542 89,892 63,450 74,356 54,378 76,252

1.3 Kaldor’s “Stylized Facts” 11

12

1

Growth and International Trade: Introduction and Stylized Facts

Table 1.9 Net capital stock at 2015 prices, employees, and capital intensity in selected countries in 2019 Austria Belgium Bulgaria Croatia Cyprus Czechia Denmark Estonia Finland France Germany Greece Hungary Ireland Italy Latvia Lithuania Luxembourg Malta Netherlands Norway Poland Portugal Romania Slovakia Slovenia Spain Sweden EU Euro area Japan Switzerland UK USA

Capital stock (in € billion) 1303.57 1167.57 126.03 121.68 54.31 523.22 718.68 64.42 682.24 7178.48 9088.37 743.53 278.46 659.83 5842.98 23.75 83.82 123.50 21.72 1972.24 1144.70 969.52 544.92 379.14 133.43 79.18 3918.11 1746.27 38,548.99 33,685.97 11,639.01 1824.29 6104.48 40,911.40

Employees (in million) 4.43 4.97 3.53 1.69 0.44 5.40 2.96 0.67 2.65 28.94 45.10 4.30 4.51 2.28 24.97 0.92 1.38 0.27 0.25 9.34 2.84 16.46 4.98 8.83 2.58 1.05 20.29 5.13 208.34 159.82 68.97 4.71 32.79 164.69

Capital intensity (€) 294,153 234,720 35,667 72,075 124,233 96,942 242,717 95,663 257,273 248,030 201,516 172,861 61,714 289,854 234,021 25,854 60,692 453,970 87,278 211,116 403,206 58,898 109,431 42,945 51,642 75,332 193,112 340,297 185,033 210,772 168,757 387,668 186,146 248,416

Source: AMECO (2020), own calculations

Finally, an example of a macroeconomic variable exhibiting no clear growth trend is the capital coefficient. This indicates how much capital is needed to produce one unit of GDP. Table 1.11 presents capital coefficients for various advanced countries in selected years. Table 1.11 illustrates that the capital coefficient (¼capital–output ratio) in similarly developed countries is quite different in size, but relatively stationary for each

1.3 Kaldor’s “Stylized Facts”

13

Table 1.10 Average growth of net capital stock at 2015 prices in selected countries between 2000 and 2019 Austria Austria France Germany UK EU Japan USA

Growth rate of net capital stock 1.7 1.7 0.9 1.3 1.5 0.3 2.2

Growth rate of gross domestic product 1.5 1.3 1.3 1.7 1.5 0.8 2.0

Source: AMECO (2020), own calculations Table 1.11 Capital coefficient in selected countries and years

Capital coefficient Austria France Germanya UK Japan USA

1960 3.0 3.0 3.0a 3.0 3.0 3.0

1980 3.1 2.9 3.1a 2.8 2.7 2.5

2000 3.3 2.8 3.0 2.4 3.1 2.2

2010 3.5 3.0 3.0 2.3 3.0 2.4

2019 3.5 3.1 2.8 2.2 2.8 2.3

Source: AMECO (2020) West Germany

a

country particularly when considered over a longer period (e.g., a century). Given the fact (see Tables 1.4 and 1.10) that GDP and capital stock grow roughly at the same rate, stationarity of the capital coefficient is necessarily implied. The following exercise confirms this algebraically (vt  Kt/Yt): vt ¼ ¼

Kt K v  vt , vtþ1 ¼ tþ1 , gt ¼ tþ1 ) gvt ¼ Yt Y tþ1 vt

K tþ1 Y tþ1 Kt Yt

1¼

K tþ1 Y t 1¼ Y tþ1 K t

K tþ1 Kt Y tþ1 Yt

1

GKt  1: GYt

Now, if GKt ¼ GYt , then it follows immediately that: gvt ¼

GKt  1 ¼ 1  1 ¼ 0: GYt

Another two stationary macroeconomic variables, the labor income share (wage ratio) and the capital income share (profit ratio) should also be mentioned here. The wage ratio is defined as the ratio of wage income wtNt to national income Yt. The real wage rate wt gives the price-adjusted wage of an employee in an economy—i.e., it is the ratio of the nominal wage rate Wt (in € per hour worked) to the consumer price index Pt.

14

1

Growth and International Trade: Introduction and Stylized Facts

Table 1.12 Real interest rate (real short-term interest rates, deflator GDP) and wage ratio (by changes in the ratio of employees and self-employed people adjusted) in the EU 15 and the USA Year 1981–1990 1991–2000 2001–2009 2010–2019

Adjusted wage ratio (in % of GDP) EU 15 USA 59.8 60.6 57.2 60.3 56.0 58.9 56.2 56.5

Real interest rate (%) EU 15 USA 4.2 4.7 4.1 3.1 1.1 0.5 1.1 0.8

Source: AMECO (2020), own calculations

As will be shown in Chap. 3, the long-term real wage increases at the same rate as labor productivity (¼output per employee), i.e., gw ¼ gY/N. If all people willing to work are employed (Nt ¼ Lt), then the growth rate of real wages equals the growth rate of labor productivity (gY/N ¼ gY/L). We already know that the labor productivity (output per employee) grows exponentially at a positive constant rate. Therefore, in the long run also the real wage rate must also grow exponentially. Given the above preliminary considerations, we are now able to show the longrun stationarity of the wage ratio.
wL g Y ¼ gwL  gY ¼ gw þ gL  gY ¼ gw  gY  gL ¼ gw  gY=L If gw ¼ gY/L, then it follows immediately that gwY/L ¼ 0. The total national income Yt is entirely distributed among workers and capital owners (¼exhaustion theorem): Yt ¼ wtLt + qtKt, where qt denotes the real user costs of capital. If there are no differences between the risk-free nominal interest rate of government bonds Rt and the return on real assets, the so-called “no arbitrage” condition holds (see Chap. 2): 1 þ Rt 1 þ Rt ¼ qt þ ð1  δÞ,  1  Rt  ½ðPt =Pt1 Þ  1, ðPt =Pt1 Þ ðPt =Pt1 Þ where δ stands for the depreciation rate of real capital. If we assume a period length of one generation (25–30 years), then δ may be set equal to one. Thus, the real user costs of a unit of capital are equal to the so-called real interest rate factor (¼ 1 + real interest rate) (1 + Rt)/[(Pt/Pt  1)]. Since the real factor payments exhaust the entire product, it follows immediately that 1 ¼ (wtLt)/Yt + (qtKt)/Yt, i.e., the wage and profit share sum to unity. It follows that, if the long-run wage share is stationary, this must also be true for the profit share. The latter is equal to the product of the real interest factor and the capital– output ratio. Given the long-run stationarity of the profit share, it follows that the real interest factor must also be stationary. There is thus a clear difference between the reward of workers and that of entrepreneurs: while wages grow at the rate of labor productivity, the latter remains stationary, implying a constant profit rate. As illustrated in Table 1.12, the above theoretical results can only partly be confirmed empirically for the EU 15 and the USA for the period 1981–2019.

1.5 Internationalization Facts

15

While the wage share in the USA and the real interest rate in the EU 15 remain roughly constant between 1981 and 2000, the wage share in the EU 15 remained constant (albeit on a lower level) while it declined markedly in the USA between 2001 and 2019. The real interest rate before and after 2000 exhibits completely differing levels both in the EU 15 and in the USA. The relative stationarity of the US wage share between 1981 and 2000 may be due to the significantly higher employment rate in the USA compared to Europe (especially to continental Europe) (for more detail see Farmer 2007, 133). The decline of the real interest rate in both areas since 2001 is among other factors due to US central bank policy to combat deflationary tendencies after the dot.com and the US subprime crisis.

1.4

Kuznets’ Facts

We turn now to the so-called “Kuznets facts.” While the Kaldor facts deal with the long-run macroeconomic development of the aggregate economy, Kuznets (1955), among others, pointed to the massive structural changes that accompany growth in aggregate GDP. If we break down aggregate production into the primary sector (agriculture, forestry, and mining), the secondary sector (industrial production), and the tertiary sector (services), we can see that over the past 100 years or more, the proportion of employees in the primary sector (as a portion of total employees) has decreased continuously, the proportion of employees in the industrial sector has remained fairly constant, and the share of people employed in the services sector has increased. The same applies to the sectoral shares in final consumption expenditure. The Kuznets facts are presented schematically in Table 1.13. Figure 1.2 illustrates the Kuznets facts for the US economy. The sectoral development of employment and GDP shares in Western and Central Europe are very similar, although not quite as distinctive as those for the USA.

1.5

Internationalization Facts

So far, we have outlined the development of world production essentially based on the Kaldor and the Kuznets facts for closed economies. The above description is largely consistent with the economic development in the first half of the twentieth century. However, the global economy—in particular higher-income regions—is Table 1.13 Kuznets facts Primary sector Secondary sector Tertiary sector

Portion of total employees Falls Stays constant Raises

Source: Kongsamut et al. (2001, 873)

Portion of total final consumption Falls Stays constant Raises

16

1

Growth and International Trade: Introduction and Stylized Facts Total employees (%)

80 60

Primary sector Industrial sector Service sector

40 20 0 1869

1899

1929

1959

1989

2019

GDP share (%) 80 60

Primary sector Industrial sector Service sector

40 20 0 1869

1899

1929

1959

1989

2019

Fig. 1.2 Sectoral employment shares 1869–2019 and sectoral GDP shares 1869–2017 in the USA (Source: 1869–1970: Kongsamut et al. (2001, 873); 1991–2019: World Bank (2020))

characterized after World War II by ever-increasing globalization and international integration. This is shown in Table 1.14. We now present four so-called “internationalization facts”: world trade is growing faster than world output, the share of foreign trade to GDP steadily increases over time, two-thirds of foreign trade takes place among developed countries, and neighboring countries trade more with each other than countries that are far apart.

1.5.1

World Trade Is Growing Faster Than World Output

Table 1.15 shows that world trade is growing roughly twice as fast as world production when production growth is vigorous.

1.5.2

Export and Import Ratios Increase Over Time

In Table 1.16, the export ratios (¼ratio of a country’s exports to GDP) of various countries and country groups in 1960 are compared with those in 2015. The table clearly indicates that export ratios of all countries and country groups have

4193

35,831 6557

4976

1301

7172

2227

23,178 4864

4306

1376

5200

Source: World Bank (2020) n.a.: not available

Europe + Central Asia East Asia + Pacific North America Latin America + Caribbean Middle East + North Africa Sub-Saharan Africa World

10,330

1695

7584

51,938 9567

9228

GDP per capita ($, constant prices 1990) 1970 1990 2015 11,544 17,582 24,892

13.6

19.0

32.0

7.0 10.6

12.2

19.3

22.4

29.1

10.7 17.3

20.7

29.3

24.2

39.9

14.2 21.0

30.3

Exports of goods and services (% of GDP) 1970 1990 2015 20.3 25.6 41.9

13.7

21.9

23.9

6.5 11.7

12.8

19.5

20.2

32.7

11.9 15.1

19.3

28.6

29.0

40.1

17.0 22.7

27.8

Imports of goods and services (% of GDP) 1970 1990 2015 20.8 26.2 38.3

Table 1.14 GDP per capita, exports, imports, and international migrant stock in selected regions

n.a.

n.a.

n.a.

na. n.a.

n.a.

2.9

2.8

7.2

9.8 1.6

0.6

3.3

1.9

9.4

15.2 1.4

1.1

International migrant stock (% of population) 1970 1990 2015 n.a. 7.0 9.5

1.5 Internationalization Facts 17

18

1

Table 1.15 Growth of world trade and world output between 1870 and 2019 (% average yearly growth rates)

Growth and International Trade: Introduction and Stylized Facts

Period 1870–1913 1913–1937 1913–1929 1929–1937 1950–1990 1950–1973 1974–1990 1991–1999 2000–2009 2010–2019

World output 2.7 1.8 2.3 0.8 3.9 4.7 2.8 3.1 3.9 3.8

World trade 3.5 1.3 2.2 0.4 5.8 7.2 3.9 6.6 5.0 4.5

Source: 1870–1990: Farmer and Wendner (1999, 192), 1991–2019: IMF (2019)

Table 1.16 Ratio of average of exports and imports to GDP in 1960 and 2015

EU USA Japan Belgium Denmark Germany Ireland Netherlands Portugal Spain Sweden Switzerland UK

1960 6.1 4.6 10.5 38.2 33.3 19.0 32.4 47.9 16.7 7.6 23.2 27.3 20.9

2015 12.0 14.0 18.4 83.6 50.1 43.0 111.0 77.2 39.9 31.9 43.1 57.3 28.4

Source: Burda and Wyplosz (2017, 11)

increased, regardless of whether the country had a large or a small internal market. Nevertheless, it holds true that large countries have smaller export ratios than small countries. At the same time, there are countries (like Ireland) whose export ratios amount to more than 100% of GDP.

1.5.3

Two-Thirds of Foreign Trade Takes Place Between Developed Countries

Table 1.17 shows that world exports increased more than 300-fold in the period 1950–2019. On average, the share of trade between industrialized countries amounts to about two-thirds of world trade. In 1960, Europe and North America had a share of almost 70%. After the first oil price shock and the increasing participation of

1.5 Internationalization Facts

19

Table 1.17 Exports of goods by regions in various years

World World North America USA Canada Mexico South and Central America Brazil Argentina Europe Germany France Italy UK Africa Middle East Asia China Japan India Australia and New Zealand Diverse regions EU GATT/WTO members

1950 1960 In US$ billion 62 130 Share in % 100.0 100.0 22.3 20.9 16.1 15.0 4.9 4.5 0.9 0.6 10.8 6.9

1970

1980

1990

2000

2010

2019

317

2036

3490

6454

15,306

18,889

100.0 19.4 13.6 5.3 0.4 5.0

100.0 15.3 11.1 3.3 0.9 4.6

100.0 16.1 11.3 3.7 1.2 3.0

100.0 19.0 12.1 4.3 2.6 3.1

100.0 12.8 8.4 2.5 1.9 3.9

100.0 13.5 8.7 2.4 2.4 3.1

2.2 1.9 37.7 3.2 5.0 1.9 10.2 7.1 2.9 16.3 0.9 1.3 1.8 –

1.0 0.8 46.1 8.8 5.3 2.8 8.1 5.5 3.0 13.4 2.0 3.1 1.0 –

0.9 0.6 50.0 10.8 5.7 4.2 6.1 5.1 3.1 13.3 0.7 6.1 0.6 –

1.0 0.4 44.1 9.5 5.7 3.8 5.4 6.0 10.4 15.9 0.9 6.4 0.4 1.3

0.9 0.4 48.3 12.1 6.2 4.9 5.3 3.0 3.9 22.7 1.8 8.2 0.5 1.4

0.9 0.4 40.8 8.5 5.1 3.7 4.4 2.3 4.1 28.4 3.9 7.4 0.7 1.2

1.3 0.4 36.9 8.2 3.4 2.9 2.7 3.4 5.9 33.2 10.3 5.0 1.5 1.6

1.2 0.3 36.7 7.9 3.0 2.8 2.5 2.4 5.1 35.8 13.2 3.7 1.7 1.6

– –

– –

– –

– 90.1

– 93.1

33.6 98.2

31.2 97.4

30.8 98.2

Source: WTO (2020)

Asian countries in world trade (see Table 1.18), the share of developing countries began to grow in the 1970s and 1980s. During the last decades the fast progress in the industrialization of Asian countries, especially China, shaped international trade. In 2019 Asian countries account for 36% of world trade, while the share of Europe and North America fell to 50%. Looking into the future and trying to predict world trade shares up to 2060 (see Fig. 1.3), it seems that the share of world exports of Europe and North America is likely to fall further, and that of Asia is likely to rise.

20

1

Growth and International Trade: Introduction and Stylized Facts

Table 1.18 Exports of goods from Asian and Western countries in million $ 1870–2003 (constant prices 1990) and 2019 (prices 2019) Japan China India Indonesia South Korea Philippines Taiwan Thailand Total (Asia) France Germany UK USA Total (4)

1870 51 1398 3466 172 6

1913 1684 4197 9480 989 171

1929 4343 6262 8209 2609 1292

1950 3538 6339 5489 2254 112

1973 95,105 11,679 9679 9605 7894

2003 402,861 453,734 86,097 70,320 299,578

2019 705,528 2,499,029 324,163 167,497 542,233

55 7 88 5243

180 70 495 17,266

678 261 640 24,294

697 180 1148 19,757

2608 5761 3081 145,412

27,892 134,884 72,233 1,547,589

70,334 n.a. 246,245 4,555,029

3512 6761 12,237 2495 25,005

11,292 38,200 39,348 19,196 108,036

16,600 35,068 31,990 30,368 114,026

16,848 13,179 39,348 43,114 112,489

104,161 194,171 94,670 174,548 567,550

404,077 785,035 321,021 801,784 2,311,917

569,732 1,489,158 468,817 1,645,625 5,032,144

Source: 1870–2003: Maddison (2007, 170); 2019: WTO (2020), n.a.: not available 0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

Africa Euro area U.K. Canada U.S. China India Indonesia Japan Other Asia Latin America Rest of the world 2012

2030

2060

Fig. 1.3 Regional share of exports in goods 2012–2060 (as percentage of world exports) (Source: Château et al. 2014, 20)

1.5.4

Neighboring Countries Trade More with Each Other Than Countries That Are Further Apart

In the previous section, we have seen that total world trade has increased tremendously and that the respective shares of 2/3 and 1/3 of industrial and developing countries in world exports have reestablished themselves. The next question

1.5 Internationalization Facts

21

Table 1.19 Regional distribution of exports of goods 2000 and 2017 2000 European Exports, thereof: to Africa to Asia to Europe to North America to South and Central America to the rest of world Asian Exports, thereof: to Africa to Asia to Europe to North America to South and Central America to the rest of world N. American Exports, thereof: to Africa to Asia to Europe to North America to South and Central America to the rest of world

US$ billion 2631.7

% 100.0

61.9 200.0 1928.1 275.8 45.0

2.4 7.6 73.3 10.5 1.7

120.9 1835.5 22.1 813.8 289.5 435.3 28.4

4.6 100.0 1.2 44.3 15.8 23.7 1.5

246.4 1225.0

13.4 100.0

12.1 232.6 205.2 682.8 67.9

1.0 19.0 16.7 55.7 5.5

24.5

2.0

2017 European-Exports, thereof: to Africa to Asia to Europe to North America to South and Central America to the rest of world Asian-Exports, thereof: to Africa to Asia to Europe to North America to South and Central America to the rest of world N-American Exports, thereof: to Africa to Asia to Europe to North America to South and Central America to the rest of world

US$ billion 6512.3

% 100.0

182.1 730.8 4482.5 562.8 99.9

2.8 11.2 68.8 8.6 1.5

454.3 6381.4 166.7 3103.5 932.0 1115.1 157.8

7.0 100.0 2.6 48.6 14.6 17.5 2.5

906.2 2376.4

14.2 100.0

25.6 520.4 382.2 1190.5 174.5

1.1 21.9 16.1 50.1 7.3

83.2

3.5

Source: WTO (2020)

is: Have all industrial and developing countries taken part in this overall development equally or are there distinct trading clusters? Which countries are trading more and with whom? An obvious, but not complete answer is: neighboring countries trade more with each other than those far further apart (see Table 1.19). This is not really surprising: if a product of the same quality is offered by a neighboring country and by a more distant country, then it is simply rational to buy the product from the neighbor and not from the distant country since normally transport costs are a positive function of distance. A look at the empirical data confirms this. Each of the three major blocs the Europe, Asia, and North America mostly trades with itself (¼trade between respective trading bloc members). This is mostly true for Europe, less so for Asia and North America. The (almost) universal appeal of North American products has induced intensive trade relations of North America with Asia, Europe, and South and Central America.

22

1.6

1

Growth and International Trade: Introduction and Stylized Facts

Globalization Facts

Closely related to the above internationalization facts are the so-called “globalization facts.” We distinguish between the two since—after an interruption of almost a century—globalization regained momentum in the wake of the collapse of the fixed exchange rate system of Bretton Woods in 1971, the first oil crisis in 1973/74, and the first World Economic Summit of the G6 in November 1975 in Rambouillet, near Paris.

1.6.1

Foreign Direct Investment and Financial Investment

If globalization is primarily understood as the emergence of global markets and firms, then data on the expansion of foreign direct investments and transactions on international financial markets may be taken as indicator for the process. Based on empirical data, an increase in foreign direct investment and a dramatic rise in turnover on financial markets are clearly visible between the end of the twentieth century and the beginning of the twenty-first century (see Fig. 1.4). In 2015, foreign direct investment was about 4000% higher and transactions on financial markets about 6000% higher, compared to the beginning 1980s. This incredible acceleration in international investment was interrupted briefly by the stock market crash of 2000 and the financial crises of 2007/2008, but direct investment and stock market activities gathered pace quickly again after these declines. Thus, phrases such as global, turbo, or casino-capitalism, although somewhat negative, exhibit a fair amount of truth.

7000

GDP Exports of goods and services Foreign direct investment, net outflows Stocks traded

6000 5000 4000 3000 2000 1000 0 1980

1985

1990

1995

2000

2005

2010

2015

Fig. 1.4 Profile of globalization 1984–2015 (US$, current prices; 1984 ¼ 100) (Source: World Bank (2020))

1.7 Summary and Conclusions

1.6.2

23

Asia Since the 1970s: “The” Dynamic Export Region

Turning again to foreign trade data, we find that Asia has managed to achieve an impressive return to world markets since the early 1970s. As Table 1.20 shows, Asia’s share of world exports doubled in the second half of the twentieth century and was still rising in the first two decades of the twenty-first century, while the export shares of Western countries remained virtually unchanged or was even decreasing. Africa, Latin America, and the former COMECON countries lost export shares. Table 1.21 illustrates the (positive) correlation between per capita growth and export growth rates in developed and emerging countries. The root of the increasing importance of (East and South) Asia on world commodity markets is the huge regional annual export growth since the early 1980s (see Tables 1.22 and 1.23). The fast-growing importance of Asia in world trade can also be traced back to the high share of manufactured goods in exports compared to agricultural products and fuels and mining products throughout the first two decades of the current century (see Fig. 1.5).

1.7

Summary and Conclusions

This chapter has outlined some important empirical facts concerning the growth, the internationalization, and globalization of the world economy. The WTO (the successor to the GATT) and the IMF have a leading role in the internationalization process. The rising interdependence of countries and markets is the subject of international economics and trade theory. Section 1.2 was dedicated to defining specific, commonly occurring growth variables. As they are a frequently recurring phenomenon in the course of this book, this section should be a considerable aid to understanding. Table 1.20 Regional shares (in %) of world exports in goods, 1870–2019 Western Europe North America Asia Latin America Form. COMECON Africa World

1870 64.4 7.5 13.9 5.4 4.2 4.6 100.0

1913 60.2 12.9 10.8 5.1 4.1 6.9 100.0

1950 41.1 21.3 14.1 8.5 5.0 10.0 100.0

1973 45.8 15.0 22.0 3.9 7.5 5.8 100.0

1998 42.8 18.4 27.1 4.9 4.3 2.7 100.0

Source: 1870–1998: Maddison (2001, 127); 2010–2019: WTO (2020) Europe b South and Central America n.s.: not available a

2010 36.9a 12.8 33.2 3.9b n.s. 3.4 100.0

2019 36.7a 13.5 35.8 3.1b n.s. 2.4 100.0

24

1

Growth and International Trade: Introduction and Stylized Facts

Table 1.21 Comparison of GDP and export growth of Asian and Western countries, 1950–2019 (in %)

Japan China India Indonesia South Korea Philippines Thailand Taiwan Hong Kong France Germany UK USA

GDP per capita 1950– 1973– 1973 1990 8.1 3.0 2.8 4.8 1.4 2.6 2.6 3.1 5.8 6.8 2.7 0.7 3.7 5.5 6.7 5.3 5.2 5.5 4.0 1.9 5.0 1.7 2.4 1.9 2.5 2.0

1990– 2003 0.9 7.5 3.9 2.6 4.7 1.0 3.4 4.3 2.1 1.3 1.2 2.0 1.7

2003– 2019 0.9 8.4 6.1 4.1 3.1 3.9 3.2 n.a. 3.0 0.8 1.3 0.8 1.2

Export volume 1950– 1973– 1973 1990 15.3 6.7 2.7 10.3 2.5 3.7 6.5 6.0 20.3 13.2 5.9 6.9 4.9 11.5 16.3 12.6 0.6 5.5 8.2 4.2 12.4 4.5 3.9 4.0 6.3 4.9

1990– 2003 2.6 16.5 12.8 8.1 12.5 10.0 5.5 9.2 2.1 5.2 5.1 4.3 5.6

2003– 2019 3.5 10.0 7.4 4.2 6.7 5.9 4.1 n.a. 4.5 2.5 4.2 1.9 4.2

Source: 1950–2003: Maddison (2007, 171); 2003–2019: IMF (2019) n.a.: not available

Table 1.22 Regional (average) yearly export growth of goods in %, 1981–2019 World Africa Asia Australia + New Zealand Europe North America South + Central America + the Caribbean BRIC members Euro Area (19) European Union G-20 WTO Members

1981–1990 3.9 1.6 7.5 n.a. 4.1 4.5 3.7

1991–2000 6.4 3.4 8.2 n.a. 5.5 7.8 5.9

2001–2010 4.6 3.4 9.3 2.9 2.7 2.1 3.4

9.8 n.a. n.a. 3.9 4.1

14.5 n.a. n.a. 6.6 7.0

14.0 2.6 2.9 4.5 4.5

2011–2019 2.7 0.1 3.8 3.3 2.1 3.3 1.0 4.1 2.0 2.1 2.8 2.8

Source: WTO (2020) n.a.: not available

In Sects. 1.3 and 1.4, we presented Kaldor’s and Kuznets’ stylized facts on global economic development. These describe the long-run characteristics of the global economy as a whole. We found a positive and steady growth rate for per capita

1.7 Summary and Conclusions

25

Table 1.23 Average yearly export growth and contribution to world economic growth (in %) of various regions, 2000–2019

World Africa Asia Australia + New Zealand Europe North America South + Central America + the Caribbean BRIC members Euro Area (19) European Union G-20 WTO Members

Export growth (volume) 3.55 1.62 6.44 3.02 2.32 2.52 2.18

Export growth (US$, current prices) 5.81 6.21 7.11 7.61 5.22 3.94 5.85

Contribution to growth 5.81 0.15 2.31 0.11 2.01 0.62 0.18

8.93 2.22 2.40 3.52 3.55

11.32 4.96 5.32 5.72 5.82

1.41 1.36 1.70 4.45 5.71

Fuels and mining products

Manufactures

Source: WTO (2020), own calculations

100% 90%

2000

2017

80% 70% 60% 50% 40% 30% 20% 10% 0% Agricultural products

Fig. 1.5 Asian exports of goods to Europe in % of total exports (Source: WTO (2020))

income amounting to about 2% per year; a similar increase in real wage rates; the stationarity of the real interest rate and of the wage share does not hold any longer, while the capital coefficient (of 2–4) remains roughly stationary at least for advanced countries in continental Europe and in Japan. The basic Kuznets facts indicate that over time the share of the industrial production in GDP remains unchanged (20–25%), the share of the primary sector in GDP converges to zero, and that the service sector continuously increases until it reaches a share of about 80% of GDP.

26

1

Growth and International Trade: Introduction and Stylized Facts

While the Kaldor and Kuznets facts describe the long-run development of the world economy as a closed system, the internationalization and globalization facts address the transition from more or less self-sufficient economies to completely integrated countries in a single world economy. This is a long-lasting and as yet unfinished process. Since in peacetime, world foreign trade grows faster than world production, the national export and import ratios also increase over time. The share of industrial countries in world trade has again settled down to a level of 2/3. But it is also true that trade blocs trade more with themselves than with other trading blocs. Furthermore, North America is a more attractive trading partner for most countries than the EU. Globalization is evident in the worldwide expansion of markets and firms and in the strong rise of foreign direct investment since the 1980s. It is also clearly visible in the spectacular growth of transactions occurring on world financial markets. Regarding the evolution of foreign trade, the doubling of Asia’s share of world exports between 1950 and 1998 and the almost double-digit export growth of East Asia over the past decades is remarkable. Such countries export far more than primary goods and commodities. They mainly export highly developed industrial and technologically advanced products.

1.8

Exercises

1.8.1. Investigate USA’s macro-dynamics from 1970 to 2019 and find out where there is a divergence from Kaldor’s stylized facts. Explain. 1.8.2. Suppose the labor force steady-state growth rate amounts to 1% p.a. while the GDP growth rate is 2%. How large is the growth rate of GDP per capita? The growth rate of the real wage rate is 3% while labor productivity grows by 2% p.a. How large is the growth rate of the labor income share? Is the result consistent with stylized facts? 1.8.3. Search the internet for the USA’s real GDP of 2000 and 2019! How large is the average growth rate between 2000 and 2019? In how many years will 2019 GDP double if the average growth rate for 2000–2019 continues? 1.8.4. Look at the share of developing countries in world exports over the past half century. Why is it not surprising that at the end of 1960s theories claiming economic dependence of developing countries on developed countries were so popular? 1.8.5. Considering the origin of the electronic devices in your pockets, what does this tell you about recent trends in globalization?

References

27

References AMECO. (2020). Annual data set of the macro-economic database AMECO. European Commission. Downloaded on June 6th, 2020, from https://ec.europa.eu/info/business-economy-euro/ indicators-statistics/economic-databases/macro-economic-database-ameco/download-annualdata-set-macro-economic-database-ameco_en Bigsten, A. (2004). Globalisation and the Asia-Pacific revival. World Economics, 5(2), 33–56. Bolt, J., & van Zanden, J. L. (2014). The Maddison project: Collaborative research on historical national accounts. The Economic History Review, 67(3), 627–651. Burda, M., & Wyplosz, C. (2017). Macroeconomics: A European text (7th ed.). Oxford: Oxford University Press. Burda, M., & Wyplosz, C. (2018). Makroökonomie : eine europäische Perspektive (4., überarbeitete Auflage). München: Verlag Franz Vahlen. Château, J., Fontagné, L., Fouré, J., Johansson, Å., & Olaberría, E. (2014). Trade patterns in the 2060 World Economy. OECD Economics Department Working Papers, No. 1142. Paris: OECD. https://doi.org/10.1787/5jxrmdk5f86j-en Farmer, K. (2007). Die wirtschaftliche Zukunft Kontinentaleuropas im weltweiten Systemwettbewerb. In W. Lachmann, R. Haupt & K. Farmer (Eds.). Zur Zukunft Europas. Wirtschaftsethische Probleme der Europäischen Union. Münster: LIT-Verlag Farmer, K., & Wendner, R. (1999). Wachstum und Außenhandel: Eine Einführung in die Gleichgewichtstheorie des Wachstums- und Außenhandelsdynamik (2nd ed.). Heidelberg: Physica. IMF. (2019). World economic outlook database: October 2019 Edition. Downloaded on June 6th, 2020, from https://www.imf.org/external/pubs/ft/weo/2019/02/weodata/index.aspx IMF. (2020). World economic outlook database: April 2020 Edition. Downloaded on June 6th, 2020, from https://www.imf.org/external/pubs/ft/weo/2020/01/weodata/index.aspx Ingham, B. (2004). International economics: A European focus. Harlow: Pearson. Kaldor, N. (1961). Capital accumulation and economic growth. In F. A. Lutz & D. C. Hague (Eds.), The theory of capital. London: Macmillan. Kongsamut, P., Rebelo, S., & Xie, D. (2001). Beyond balanced growth. Review of Economic Studies, 68, 869–882. Krugman, P. R., Obstfeld, M., & Melitz, M. J. (2018). International economics: Theory and policy (11th ed.). Boston: Addison-Wesley. Kuznets, S. (1955). Economic growth and income inequality. American Economic Review, 45(1), 1–28. Maddison, A. (2001). The world economy. A millennial perspective. Paris: OECD. Maddison, A. (2007). Contours of the world economy: I-2030 AD: Essays in macro-economic history. Oxford: Oxford University Press. World Bank. (2020). World development indicators. Downloaded on June 25th, 2020, https:// databank.worldbank.org/source/world-development-indicators WTO. (2020). Statitics database: Time series on trade. Downloaded on June 13th, 2020, from https://timeseries.wto.org

Part I Growth

Part I focuses on economic growth in a fully integrated world economy. It consists of nine chapters. In line with the subtitle of the monograph, the basic Overlapping Generations model in neoclassical growth theory is presented in the second chapter. In Chap. 3, the steady state of the growth dynamics, the evolution of factor incomes and various concepts of neutral technological progress are investigated. Chap. 4 is devoted to the exploration of the relationship between growth and public debt. In Chap. 5, the pioneering approach of “new” growth theory, Romer’s knowledge externalities in private capital accumulation, is presented. Chapter 6 exhibits a neo-Schumpeterian OLG model of self-propelled growth. In Chap. 7 human capital (the second driver of endogenous growth) is introduced in the basic OLG model. In Chap. 8, bubbles and financial frictions are integrated in the basic OLG model in order to explain the economic rationale behind the global financial crisis. Chap. 9 is devoted to the existence of involuntary unemployment in a neoclassical growth model à la Diamond (1965). Robots, human capital investment, welfare, and involuntary unemployment in a digital world economy model conclude Part I.

2

Modeling the Growth of the World Economy: The Basic Overlapping Generations Model

2.1

Introduction and Motivation

The previous chapter provided a detailed introduction to empirical data concerning the long-run growth of the world economy as a whole and to the international economic relations prevailing among nearly 200 nation states. In this chapter, we intend to explore within a simple theoretical model the driving forces behind the apparently unbounded growth of the global market economy, and for the moment simply disregard the international relations among countries. In order to go some way toward addressing the frequently expressed fear that globalization has gone too far we begin by envisioning a world economy where globalization has come to an end. In other words, we assume a fully integrated world economy with a single global commodity market and a uniform global labor and capital market. Although the present world economy is still quite far away from achieving full integration dealing with this commonly cited bogey of globalization critics would nevertheless appear worthwhile. In view of Kaldor’s (1961) stylized facts presented in Chap. 1 it is not surprising at all that economic growth attracted the attention of economic theorists in a postWWII period of the 1950s and 1960s. In contrast to the rather pessimistic growth projections of the leading post-Keynesian economists, Harrod (1939) and Domar (1946), the GDP growth rate, especially in countries destroyed in WWII, dramatically exceeded its long-run average (of about 2% p.a.) and remained at the higher level at least for a decade. As it is well-known, Solow (1956) and Swan (1956) were the first to provide neoclassical growth models of closed economies. These rather optimistic growth models were better akin to the growth reality of the post-war period than the postKeynesian approaches. However, savings behavior in Solow’s and Swan’s macroeconomic growth models lacked intertemporal micro-foundations. In order to address this drawback from the perspective of mainstream growth theory (e.g., Acemoglou 2009) our basic growth model of the world economy is based on Diamond’s (1965) classic overlapping generations’ (OLG) version of neoclassical # Springer-Verlag GmbH Germany, part of Springer Nature 2021 K. Farmer, M. Schelnast, Growth and International Trade, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-662-62943-7_2

31

32

2

Modeling the Growth of the World Economy: The Basic Overlapping. . .

growth theory.1 This modeling framework enables us to study the relationship between aggregate savings, private capital accumulation, and GDP growth within an intertemporal general equilibrium framework. After working through this chapter the reader should be able to address the following questions: • How can we explain private capital accumulation endogenously on the basis of the rational behavior of all agents in a perfectly competitive market economy? • Which factors determine the accumulation of capital (investment) and GPD growth, and how do they evolve over time? • Are there other economic variables determined by the dynamics of capital accumulation? • Is a world economy with a large savings rate better off than one with a smaller savings rate? This chapter is organized as follows. In the next section, the set-up of the model economy is presented. In Sect. 2.3, the macroeconomic production function and its per capita version are described. The structure of the intertemporal equilibrium is analyzed in Sect. 2.4. The fundamental equation of motion of the intertemporal equilibrium is derived in Sect. 2.5. In Sect. 2.6, the “golden rule” of capital accumulation to achieve maximal consumption per capita is dealt with. Section 2.7 summarizes and concludes.

2.2

The Set-Up of the Model Economy

There are two types of households (¼generations) living in the model economy: old households comprise the retired (related symbols are denoted by superscript “2”), and young households represent the “active” labor force and their children (denoted by superscript “1”). Each generation lives for two periods. Consequently, the young generation born at the beginning of period t has to plan for two periods (t, t + 1), while the planning horizon of retired households consists of one (remaining) period only. In each period, two generations overlap—hence the term Overlapping Generations model (or OLG model). The typical length of one period is about 25–30 years. While members of the young households work to gain labor income, members of the old generation simply enjoy their retirement. For the sake of analytical simplicity, we assume a representative (young) household characterized by a log-linear utility function. Moreover, members of the young generation are assumed to be “workaholics,” i.e., they attach no value to leisure. As a

1 Alternative intertemporal general equilibrium foundations are provided by Ramsey’s (1928) infinitely lived-agent approach which is not dealt with at all in this book.

2.2 The Set-Up of the Model Economy

33

consequence, labor time supplied to production firms is completely inelastic to variations in the real wage.2 The utility function of the young generation, born in period t, and at the beginning of period t, is given by: U 1t ¼ ln c1t þ β ln c2tþ1 , 0 < β  1:

ð2:1Þ

In Eq. (2.1), c1t refers to the per capita consumption of the young household when working, c2tþ1 denotes the expected per capita consumption when retired, and β denotes the subjective time discount factor. The time discount factor is a measure of the subjective time preference (consumption today is generally valued more than consumption tomorrow), and specifies the extent to which consumption in the retirement period is valued less than one unit of consumption in the working period. The available technologies can be described by a macroeconomic production function of the form Yt ¼ F(At, Kt), where Yt denotes the gross national product (GDP) in period t, At stands for the number of productivity-weighted (efficiency) employees and Kt denotes the physical capital stock at the beginning of period t. In the absence of technological progress, the actual number of employees is equal to the productivity-weighted sum of employees. Labor-saving technological progress implies that the same number of workers is producing an ever-increasing amount of products. Technological progress thus has the same impact as an increase in workers employed—the number of efficient workers increases (given a constant number of physical employees). A common specification for the production function is that first introduced by Cobb and Douglas (1934): Y t ¼ A1α K αt , 0 < α < 1: t

ð2:2Þ

The technological coefficients α and (1  α) denote the production elasticity of capital and of efficiency employees, respectively. These coefficients indicate the respective percentage change in output when capital or labor is increased by 1%. 0  δ  1 denotes the depreciation rate of capital within one period. The capital stock evolves according to the following accumulation equation: K tþ1 ¼ ð1  δÞK t þ I t :

ð2:3Þ

The labor force Lt (number of young households in generation t) increases by the constant factor GL ¼ 1 + gL > 0. Hence, the parameter gL represents the (positive or negative) growth rate of the labor force. The accumulation equation of the labor force has the following form: Lt + 1 ¼ GLLt. We assume that the efficiency at of employees Nt rises by the constant rate gτ: 2 We make these assumptions to keep the model as simple as possible. They can of course be replaced by more realistic assumptions—e.g., that leisure does have a positive value to households and, thus, labor supply depends on the real wage rate. As, e.g., Lopez-Garcia (2008) shows the endogeneity of the labor supply does not alter the main insights concerning growth and public debt.

34

2

Modeling the Growth of the World Economy: The Basic Overlapping. . .

atþ1 ¼ ð1 þ gτ Þat ¼ Gτ at , a0 ¼ 1,

ð2:4Þ

A t ¼ at N t :

ð2:5Þ

In accordance with the stylized facts of the first chapter, we adopt labor-saving, but not capital-saving technological progress in the basic model. We want to analyze economic developments of the world economy over time. To do this, we have to make an assumption regarding the formation of expectations of market participants with respect to the evolution of market variables.3 As in other standard textbooks, we assume that all economic agents have perfect foresight with respect to prices, wages, and interest rates, i.e., they expect exactly those prices that induce clearing of all markets in all future periods (i.e., deterministic rational expectations). Expectation formation can be modeled in several ways. We may use, for example, static expectations, adaptive expectations, or non-perfect foresight. Such variations, however, are only of relatively minor importance in the growth literature. Under static expectations, households presume that wages, interest rates, and prices in future periods are identical to those existing today. Adaptive expectations mean that expected future prices depend not only on current prices, but also on past price changes. Given non-perfect foresight, expectations regarding prices in some future periods are realized, but after some future period expectations then become “static.” Walras’ law (see the mathematical appendix) implies that the goods market clears—regardless of goods prices—when all other markets are in equilibrium. Hence, goods prices can be set equal to 1 for all periods. Pt ¼ 1, t ¼ 1, 2, . . .

ð2:6Þ

Finally, natural resources are available for free to producers and consumers (“free gifts of nature”). This assumption implies a high elasticity of substitution between capital and natural resources and the possibility of free disposal. If this (up to the early 1970s quite realistic) assumption is dropped, the interactions between the natural environment and production and consumption have to be modeled explicitly. These interactions are the subject of environmental science and resource economics and will not be discussed in this book (see, e.g., Farmer and Bednar-Friedl 2010).

2.3

The Macroeconomic Production Function and Its Per Capita Version

In the basic model of neoclassical growth theory, the technology of the representative firm is depicted by a linear-homogeneous production function with substitutable production factors. It specifies the maximum possible output of the aggregate of all 3 A more thorough discussion of alternative expectation formation hypotheses in OLG models can be found in De la Croix and Michel (2002, Chap. 1).

2.3 The Macroeconomic Production Function and Its Per Capita Version

35

Fig. 2.1 Cobb–Douglas production function

commodities produced in the world economy, Y, for each feasible factor combination. Figure 2.1 illustrates the above-mentioned Cobb–Douglas (CD) production function graphically (for α ¼ 0.3). In general, homogeneous production functions exhibit the following form: Y t ¼ F ðat N t , K t Þ  F ðAt , K t Þ, where F ðμAt , μK t Þ ¼ μY t , μ > 0:

ð2:7Þ

Homogeneity of degree r implies that if all production inputs are multiplied by an arbitrary (positive) factor μ, the function value changes by the amount μr. For linearhomogeneous functions the exponent r is equal to one, i.e., a doubling of all inputs leads to a doubling of production output. Replacing in Eq. (2.7) μ by 1/At attributes a new meaning to the production function: it signifies the production per-efficiency employee (¼per-efficiency capita product ¼ efficiency-weighted average product). It is evident from Eq. (2.8) that the per-efficiency capita product, yt, depends on one variable only, namely the efficiency-weighted capital intensity. The production function Eq. (2.8) is a function of only one variable. The notation used below conforms to the following rule: variables expressing levels are stated using capital letters, per-capita values (and per-efficiency capita values) are depicted using small letters. For example, kt ¼ Kt/At denotes the capital stock per-efficiency employee and is called the (productivity weighted) capital intensity.

36

2

Modeling the Growth of the World Economy: The Basic Overlapping. . .



 Yt Kt Kt  yt ¼ F 1, f ¼ f ðk t Þ At At At

ð2:8Þ

The property of substitutability implies that different input combinations can be used to produce the same output. This is in contrast to post-Keynesian models, where production input proportions are fixed (limitational). Under substitutability, however, the marginal products of capital and of efficiency-weighted labor can be calculated. How is this done? The marginal products equal the first partial derivatives of the production function F(At, Kt) or f(kt) with respect to At and Kt, and with respect to kt, respectively. Thus, the marginal product of capital can be determined as follows: 2

2

∂½At f ðK t =At Þ df ∂F ∂ F d2 f ∂ F ¼ ¼ > 0, ¼ 2 , since < 0: dk t ∂K t ∂K t ∂K t ∂kt dk t ∂K 2t

ð2:9Þ

The derivative of the production function with respect to labor yields: " # ∂F ∂½At f ðK t =At Þ df K t df ¼ ¼ f ðkt Þ þ At , ¼ f ðk t Þ  k t dk t ðAt Þ2 dk t ∂At ∂At

ð2:10aÞ

and 2

2

∂ F ∂ F > 0, since > 0: ∂At ∂kt ∂At ∂K t

ð2:10bÞ

The Eqs. (2.10a) and (2.9) give the marginal products of labor and capital, i.e., the additional output, which is due to the input of an additional unit of capital or efficiency-weighted labor. It is striking that for linear-homogeneous production functions both the average and the marginal products are functions of a single variable, namely the capital–labor ratio. As long as this ratio does not change, the per-efficiency capita product and the marginal products do not change. f ðk1 Þ Y 1 =A1 ∂F df 1 ¼  f 0 ðk1 Þ, tan β ¼ ¼ ¼ where v1 k1 K 1 =A1 v1 ∂K 1 dk 1 K ¼ 1 Y1

tan α ¼

ð2:11Þ

Figure 2.2 shows that if the capital intensity is equal to k1, the per-efficient capita product amounts to f(k1). Moreover, the tangent of the angle α gives the slope of the production function at this point, i.e., the marginal product of capital in k1. The tangent of the angle β denotes the (average) productivity of capital. It gives the amount of output per unit of capital and is the reciprocal of the (average) capital coefficient, which indicates the amount of capital required to produce one unit of output.

2.4 Structure of the Intertemporal Equilibrium Fig. 2.2 Cobb–Douglas per-capita production function

37

y f(k) f(kl) α

β k

kl

2.4

Structure of the Intertemporal Equilibrium

After having presented the basic characteristics of the growth model, we are now able to return to the main question of this chapter: How can we determine the key variables of the model described above, while accounting for all market interactions of economically rational (self-interested) households and firms? The answer to this question is provided in two steps: First, we use mathematical programming (i.e., constrained optimization) to solve the rational choice problems of households and firms (see the Appendix to this chapter for an introduction to classical optimization). Second, to ensure consistency among the individual optimization solutions, the market clearing conditions in each period need to be invoked. To start with, the rational choice problem of younger households is described first.

2.4.1

Intertemporal Utility Maximization of Younger Households

In line with Diamond (1965), we assume that younger households are not concerned about the welfare of their offspring, i.e., in intergenerational terms, they act egoistically. In other words, they do not leave bequests. Thus, consumption and savings choices in their working period and consumption in their retirement period are made with a view toward maximizing their own lifetime utility. Thus, for all t, the decision problem of households entering the economy in period t reads as follows: MaxU 1t ¼ ln c1t þ β ln c2tþ1 ,

ð2:12Þ

c1t þ st ¼ wt ,

ð2:13Þ

subject to:

38

2

Modeling the Growth of the World Economy: The Basic Overlapping. . .

c2tþ1 ¼ ð1 þ itþ1 Þst where 1 þ itþ1  qtþ1 þ 1  δ:

ð2:14Þ

The first constraint Eq. (2.13) ensures that per-capita consumption plus per-capita savings of young households equals their income (the wage rate per employee) and based on the second constraint Eq. (2.14) retirement consumption is restricted by the sum of savings made in the working period and interest earned on savings. Active households save by acquiring capital, and the real interest rate is equivalent to the rental price of capital minus depreciation. Since the no-arbitrage condition it + 1 ¼ qt + 1  δ holds, the real interest rate (¼rate of return on savings) has to be equal to the rental price of capital minus depreciation (¼return on investment in physical capital). Obviously, if the real interest rate were smaller (larger) than the capital rental price minus the depreciation rate, then households would just invest in real capital (savings deposits). The relative prices of assets then would change quickly such that respective rates of return once again equate and the no-arbitrage condition is satisfied. Equations (2.13) and (2.14) can be combined to obtain Eq. (2.15) by calculating st from Eq. (2.14) and substituting st in Eq. (2.13).4 This then leaves Eq. (2.15) as the only constraint in the household’s utility maximization problem. This is known as the intertemporal budget constraint (i.e., all current and present values of future expenses equal all current and present values of future revenues). c1t þ

c2tþ1 ¼ wt 1 þ itþ1

ð2:15Þ

The left-hand side of Eq. (2.15) gives the present value of all spending in the two periods of life; the right-hand side the (present value of) total income. If the objective function Eq. (2.12) is maximized subject to the intertemporal budget constraint Eq. (2.15), we obtain the first-order conditions (¼FOCs) Eqs. (2.16) and (2.17) for household utility maximization. 

dc2tþ1 c2tþ1 ∂ U 1t =∂ c1t  ¼ ¼ 1 þ itþ1 ∂U 1t =∂ c2tþ1 βc1t dc1t

In the  household’s  optimum, the intertemporal substitution dc2tþ1 =dc1t equals the interest factor.

ð2:16Þ marginal

rate

of

4 Here we have to assume that utility maximizing savings per capita are strictly larger than zero. However, this is true since optimal retirement consumption is certainly larger than zero otherwise the marginal utility of retirement consumption would be infinitely large while the price of an additional consumption unit would be finite. This cannot be utility maximizing and thus the optimal retirement consumption must be strictly larger than zero implying, from Eq. (2.14), strictly positive savings.

2.4 Structure of the Intertemporal Equilibrium Fig. 2.3 Graphical illustration of the utility maximizing consumption plan

39

c2

B C

A

c 2t+ l

γ c lt



dc2tþ1  1 ¼ itþ1 dc1t

cl

ð2:17Þ

Equation (2.17) states that at the optimum the marginal rate of time preference (time-preference rate) is equivalent to tomorrow’s real interest rate. What is the rationale behind this result? The left-hand side of Eq. (2.17), the marginal rate of time preference, indicates how much more than one retirement consumption unit the younger household demands for foregoing one working period consumption unit. On the right-hand side, the real interest rate indicates how much more than one unit the household gets in its retirement period, if it forgoes one unit of consumption in its active period (i.e., if the household saves). Utility maximization implies that the lefthand side in Eq. (2.17) equals the right-hand side. However, if the right-hand side were larger than the left-hand side, then the household would receive a higher compensation for foregoing current consumption than it demands. The household would then use these surplus earnings to increase its utility. A situation in which the left-hand side of Eq. (2.17) is smaller than the righthand side can, therefore, never be a utility maximizing situation. The same is true if the left-hand side of Eq. (2.17) is larger than the right-hand side. Maximum utility is thus achieved only if Eq. (2.17) is satisfied. The optimization condition Eq. (2.16) and the intertemporal budget constraint Eq. (2.15) are sufficient to determine the entire optimal consumption plan of a young household. Figure 2.3 illustrates the decision problem and the optimal consumption plan of the young household. Consumption when young is plotted on the horizontal axis (the abscissa), and the retirement consumption of a young household, which enters the economy in period t, is plotted on the vertical axis (the ordinate). The negatively sloped straight line represents the intertemporal budget constraint. This

40

2

Modeling the Growth of the World Economy: The Basic Overlapping. . .

can be obtained algebraically by solving Eq. (2.15) for c2tþ1 : c2tþ1 ¼ ð1 þ itþ1 Þwt  ð1 þ itþ1 Þc1t . If c1t ¼ 0 , we get an intercept of the budget constraint (on the ordinate) of wt(1 + it + 1), and if c2tþ1 ¼ 0, the intercept (on the abscissa) is wt. The negative slope of the budget constraint equals tanγ ¼ 1 + it + 1. The distance between the intersection of the budget constraint with the abscissa and the utility maximizing consumption point gives the optimal savings st of young households. The three hyperbolas in Fig. 2.3 represent intertemporal indifference curves. Along such curves the lifetime utility U 1t of generation t is constant. Analytically, these indifference curves are obtained by solving the intertemporal utility function for fixed levels of utility. The further away an indifference curve is from the origin, the higher is lifetime utility. Accordingly, the consumption plan indicated by point A is associated with a higher utility level than consumption plan B, which is also affordable. The negative slope of the intertemporal indifference curve is a consequence of the intertemporal marginal rate of substitution Eq. (2.16). This rate can be analytically derived by totally differentiating the utility function and setting the total differential equal to zero (because the utility level is constant along each indifferent curve). dU 1t ¼

∂U 1t 1 ∂U 1t 1 β dc þ dc2 ¼ 0 ¼ 1 dc1t þ 2 dc2tþ1 ct ∂c1t t ∂c2tþ1 tþ1 ctþ1 

dc2tþ1 c2 ∂U 1t =∂c1t ¼ ¼ tþ11 1 1 2 ∂U t =∂ctþ1 βct dct

ð2:18Þ ð2:19Þ

Since the intertemporal marginal rate of substitution in Fig. 2.3 corresponds to the negative slope of the intertemporal indifference curve and the interest factor is given by the negative slope of the budget line, the slopes of the intertemporal indifference curve and of the intertemporal budget constraint have to be identical at the optimum—i.e., the intertemporal budget constraint and the indifference curve are at a point of tangency. Point A in Fig. 2.3 represents the tangency point, while point B is a cutting point. Although both consumption plans, A and B, are affordable, the indifference curve associated with the consumption plan A is at a higher utility level. Consumption plan C, which belongs to an even higher indifference curve, is not affordable. Therefore, A gives the optimal consumption plan, i.e., a consumption plan which lies on the indifference curve which is farthest from the origin but still affordable.   By solving Eq. (2.16) for c2tþ1 =ð1 þ itþ1 Þ ¼ βc1t and inserting the result into the intertemporal budget constraint Eq. (2.15) we obtain: c1t þ βc1t ¼ wt . Rearranging yields immediately the optimal (utility maximizing) working period consumption: c1t ¼

wt , t ¼ 1, : . . . 1þβ

ð2:20Þ

2.4 Structure of the Intertemporal Equilibrium

41

Inserting Eq. (2.20) into Eq. (2.13) and solving for st yields utility-maximizing savings per capita: s1t ¼

β w , t ¼ 1, : . . . 1þβ t

ð2:21Þ

Inserting Eq. (2.20) into Eq. (2.16) and solving for c2tþ1 results in the following: c2tþ1 ¼

βð1 þ itþ1 Þ wt , t ¼ 1, : . . . 1þβ

ð2:22Þ

Equation (2.20) reveals that current (optimal) consumption depends only on the real wage rate and not on the real interest rate.5 Equation (2.21) illustrates that the portion of wages not consumed is saved completely. This results from the fact that the public sector is ignored and thus households pay no taxes. Moreover, since current-period consumption is independent of the real interest rate, this is also true for optimal savings. Finally, the amount saved when young (plus interest earned) can be consumed when old Eq. (2.22). As mentioned above savings of retired households (i.e., bequests) are excluded. However, even with the introduction of a bequest motive for old households the following characteristics are still valid.

2.4.2

Old Households

In period 1, the number of retired households equals the number of young households in the previous period, i.e., L0. Their total consumption in period 1 is identical to their total amount of assets (in real terms) in period 1. L0 c21 ¼ q1 K 1 þ ð1  δÞK 1 ¼ ð1 þ i1 ÞK 1

ð2:23Þ

These assets include the rental income on capital acquired in the past, plus the market value of the capital stock (after depreciation). Equation (2.23) assumes that the no-arbitrage condition (2.14) applies.

2.4.3

A-Temporal Profit Maximization of Producers

Besides households, the producers of the aggregate commodity also strive to maximize profits in every period t. By assumption, markets are perfectly competitive. To maximize profits, firms have to decide on the number of employees (labor demand, Nt) and on the use of capital services (demand for capital services, K dt ). The profits, expressed in units of output, are defined as the difference between production output 5 Due to the log-linear intertemporal utility function the substitution effect and the income effect of a change in the real interest rate cancel out.

42

2

Modeling the Growth of the World Economy: The Basic Overlapping. . .

  and real factor costs: π t ¼ F at N t , K dt  wt N t  qt K dt . In the case of a CD produc α tion function, the profit function can be written as π t ¼ ðat N t Þ1α K dt  wt N t  qt K dt . To determine the profit-maximizing input levels, we set the first partial derivatives of the profit function with respect to Nt and K dt equal to zero:   d ∂π t ∂F At , K t ∂At ¼  wt ¼ 0, ∂At ∂N t ∂N t   ∂F At , K dt ∂π t ¼  qt ¼ 0: ∂K dt ∂K dt

ð2:24Þ ð2:25Þ

The first-order condition (2.24) tells us that in each period t firms demand additional workers as long as the physical marginal product is equal to the real (measured in units of output) wage rate. Equation (2.24) is equivalent to:  α ð1  αÞ k dt at ¼ wt :

ð2:24aÞ

In the same manner, one arrives at the decision rule for optimal capital input. Firms have to adjust the capital stock such that the marginal product of capital (yield on capital) in each period t is equivalent to the real capital costs. In the case of a CD production function, Eq. (2.25a) holds.  α1 α kdt ¼ qt

ð2:25aÞ

With exogenously given technological progress, the number of efficient employees At is a direct consequence of the producer demand for labor. A t ¼ at N t

ð2:26Þ

Aggregate production output is determined by the profit-maximizing levels of the capital stock and the number of efficient employees.    d α Y t ¼ F At , K dt ¼ A1α Kt t

ð2:27Þ

Finally, linear-homogeneity implies that the aggregate output is distributed across all production factors. Every factor of production is thus paid according to its marginal productivity. Thus, the sum of factor payments corresponds exactly to the production output and there are no surplus profits. Applying Euler’s theorem to the aggregate  d production function Eq. (2.27) implies: Y t ¼ ð∂Y t =∂N t ÞN t þ d =∂K ∂Y t t K t . Since through Eqs. (2.24) and (2.25) (∂Yt/∂Nt) ¼ wt and  ∂Y t =∂K dt ¼ qt , we obtain: Y t ¼ wt N t þ qt K dt :

ð2:28Þ

2.4 Structure of the Intertemporal Equilibrium

2.4.4

43

Market Equilibrium in All Periods

The second step in delineating the structure of the intertemporal equilibrium is to specify the market clearing conditions. In a perfectly competitive market economy no authority or central administration matches or coordinates individual decisions. The coordination of individual decisions results from changes in market prices such that the supply and demand for each good are equal in all markets (market clearing conditions). In the basic OLG model, there are three markets: the capital market, the labor market, and the commodity market. The clearing of these three markets demands: K dt ¼ K t , 8t,

ð2:29Þ

N t ¼ Lt , 8t,

ð2:30Þ

Y t ¼ Lt c1t þ Lt‐1 c2t þ K tþ1  ð1  δÞK t , 8t:

ð2:31Þ

Due to Walras’ law, the sum of nominal (measured in terms of their prices) excess demands (¼demand minus supply) on all three markets is equal to zero for all feasible prices. Consequently, if two of the three markets are in equilibrium, the third market must also be in balance. Walras’ law is derived for our basic OLG growth model in the mathematical appendix to this chapter. A pivotal equation for the dynamics of the intertemporal equilibrium is implicitly included in the system of equilibrium conditions (2.29– 2.31). This equation becomes immediately apparent when one considers the equilibrium conditions for period t ¼ 1. Equating the left-hand side of Eqs. (2.31) and (2.28) and substituting the budget constraints for both the aggregate consumption of young households (Eq. 2.13 multiplied by L1 on both sides) and for old households Eq. (2.23) into the right-hand side of Eq. (2.31) yields: w1 N 1 þ q1 K 1 ¼ w1 L1  L1 s11 þ ð1 þ i1 ÞK 1 þ K 2  ð1  δÞK 1 :

ð2:32Þ

Since 1 + i1 ¼ q1 + (1  δ) and Eqs. (2.29) and (2.30) also apply for t ¼ 1, this equation reduces to K2 ¼ L1s1. This can be generalized to: K tþ1 ¼ Lt st , t  2:

ð2:33Þ

In an intertemporal market equilibrium, the optimal aggregate savings of all young households in period t correspond exactly to the optimal aggregate capital stock in t + 1. This result becomes immediately apparent when we keep in mind that we have excluded a bequest motive in the basic model. Therefore, in order to consume, old households sell all their assets to the young households of the next generation. The young households save by buying the entire old capital stock plus investing in new capital goods (¼gross investment).

44

2

2.5

Modeling the Growth of the World Economy: The Basic Overlapping. . .

The Fundamental Equation of Motion of the Intertemporal Equilibrium

The next step is to study how the economy evolves over time when in each period households maximize their utility, firms maximize profits, and all markets clear. In order to derive the fundamental equation of motion of the intertemporal equilibrium, we focus on the accumulation of aggregate capital and on the dynamics of the efficiency-weighted capital intensity (capital–labor ratio). To this end, Eq. (2.33) is divided by atLt ¼ atNt ¼ At, to obtain: K tþ1 s ¼ t: At at

ð2:34Þ

If the left-hand side is multiplied by At + 1/At + 1 ¼ 1, we arrive at: K tþ1 Atþ1 A s ¼ ktþ1 tþ1 ¼ t : Atþ1 At At at

ð2:34aÞ

Equation (2.34a) involves the growth factor of efficiency-weighted labor At + 1/At, which is equal to the (exogenous) growth factor of labor efficiency times the population growth factor. The latter is called the natural growth factor, and it is denoted by Gn. When growth rates are not too large it can be approximated by one plus the natural growth rate gn: Atþ1 atþ1 Ltþ1 ¼ ¼ Gτ GL  Gn  1 þ gn : At at Lt

ð2:35Þ

Taking Eq. (2.35) into account, we find that Eq. (2.34) is equivalent to: ktþ1 ¼

st : G n at

ð2:36Þ

By inserting Eq. (2.24a) into Eq. (2.21), we obtain optimal savings per efficiency capita in period t as a function of the capital intensity in the same period: st βð1  αÞk αt : ¼ at 1þβ

ð2:37Þ

Finally, by inserting Eq. (2.37) into Eq. (2.36) we arrive at the following dynamic equation for kt: ktþ1 Gn ¼

βð1  αÞk αt : 1þβ

ð2:38Þ

By introducing the aggregate savings rate σ β(1  α)/(1 + β), we obtain the fundamental equation of motion for our basic OLG growth model:

2.5 The Fundamental Equation of Motion of the Intertemporal Equilibrium Fig. 2.4 The fundamental equation of motion of the basic model

45

kt +l

k4 k3 k2

45° k1

ktþ1 ¼

k2

k3

k

σ α K k , for t  1 and k1 ¼ 1 : Gn t a1 L1

kt

ð2:39Þ

Mathematically, the fundamental equation of motion (2.39) is a nonlinear difference equation in kt (capital per efficiency capita) and determines for each (productivity-weighted) capital intensity kt the equilibrium (productivity-weighted) capital intensity in the next period kt + 1. If the capital intensity of the initial period t ¼ 1 is known, the fundamental equation of motion describes the evolution of kt for all future periods (see Fig. 2.4). Additionally, the fundamental equation of motion allows us to deduce what determines the absolute change of the capital intensity. Equation (2.39) is equivalent to:   ktþ1  kt ¼ ðGn Þ1 σkαt  Gn k t :

ð2:40Þ

Obviously, the capital intensity remains constant if savings per efficiency capita, σk αt , are just sufficient to support the additional capital needed for natural growth, Gnkt. This additional capital requirement arises since the accrued and more efficient workers must be equipped with the same capital per efficiency capita as those already employed. If per efficiency capita savings (¼per efficiency capita investment) exceed (fall short of) this intensity-sustaining capital requirement, the capital intensity of the next period increases (decreases). Thus, there are two types of investments (¼savings): those that are necessary to sustain the current capital intensity and those that increase the current capital intensity. The former are called “capital-widening” investments (savings), the latter “capital-deepening” investments (Müller and Stroebele 1985, 37). A competitive intertemporal equilibrium is completely determined by the abovementioned equilibrium sequence of capital intensities over time. For example, the marginal productivity conditions (2.24a) and (2.25a) immediately determine the period-specific real wage rates and capital rental prices.

46

2

Modeling the Growth of the World Economy: The Basic Overlapping. . .

wt ¼ at ð1  αÞkαt , t ¼ 1, . . .

ð2:41Þ

qt ¼ α kα1 , t ¼ 1, . . . t

ð2:42Þ

The optimal consumption levels for young and old households and the optimal savings can be deduced from the Eqs. (2.20, 2.21, and 2.22). Finally, the aggregate output per efficiency capita is a direct result of the production function Eq. (2.8): yt ¼ f ðkt Þ ¼ kαt :

2.6

ð2:43Þ

Maximal Consumption and the “Golden Rule” of Capital Accumulation

Before closing this chapter, it is interesting to explore whether a world economy with a higher savings rate (¼higher capital intensity) is always better off than one with a lower savings rate (¼lower capital intensity). We begin with the following aggregate accumulation equation: K tþ1  K t ¼ F ðK t , At Þ  C t  δK t ,

ð2:44Þ

where the depreciation rate is not necessarily equal to one. If we divide both sides by At, to arrive at per efficiency capita values, we obtain: k tþ1 Gn  kt ¼ f ðk t Þ  ct  δkt :

ð2:45Þ

Variable ct (with no generation index) denotes total consumption per efficiency employee. Suppose again time-stationary capital intensities, i.e., kt + 1 ¼ kt ¼ k. Equation (2.45) can then be solved for c: c ¼ f ðk Þ  ðgn þ δÞk:

ð2:46Þ

Consumption per efficiency capita is maximized when the savings ratio is such that (dc/dk)(dk/dσ) ¼ 0 holds. This is equivalent to:  
 df ðkÞ dc dk n  ðg þ δ Þ ¼ ¼ 0: dσ dσ ∂k

ð2:47Þ

It is obvious from Eq. (2.47) that c is maximized only if f 0(k) ¼ (gn + δ). In the case of a CD production economy the so-called “golden rule” capital intensity k is equal to:

2.6 Maximal Consumption and the “Golden Rule” of Capital Accumulation Fig. 2.5 Consumption and savings in the basic OLG model

47

f(k)

y c

G nkt

c*

a

σkt

s

k*

k ¼



1=ðα1Þ gn þ δ : α

k

kt

ð2:48Þ

If, in addition, δ ¼ 1 and Eq. (2.39) is taken into account, then the golden rule capital intensity demands σ ¼ α, i.e., a savings rate of about 30% when α ¼ 0.3 is assumed.6 Figure 2.5 illustrates consumption and savings in the basic model under the assumption of δ ¼ 1. In such a case consumption per efficiency capita is equal to c ¼ f(k)  Gnk ¼ kσ  σkα. As shown in the figure, consumption is exactly equal to the difference between output per efficiency capita, f(k), and intensity-sustaining savings Gnk(¼σkα). However, as the consumption level c associated with capital intensity k shows, steadystate (kt + 1 ¼ kt ¼ k) intensity k does not maximize consumption per efficiency capita. In order to obtain maximum consumption c the savings rate must be changed. Hence, we need to search for a savings rate where consumption per efficient capita is maximized. From a purely static perspective, consumption decreases with an increase in the savings rate. But a higher savings rate σ also leads to a higher capital stock in the future and therefore to a greater production capacity and higher potential consumption. In an intertemporal context, we have to weigh short-term consumption losses due to a higher savings rate against the increase in the future capital stock which allows for higher consumption tomorrow. The savings rate which permanently allows for maximum consumption per efficiency capita implies, according to Phelps (1966), the “golden rule” of capital accumulation. This term is borrowed from the “golden rule” of New Testament ethics: “So whatever you wish that men would do to you, do so to them” (Matthew 7, 12). Economically speaking, the “golden rule” consumption level is not only available to currently living generations, but also to all future

6 Empirical values for the other model parameters can be found in Auerbach and Kotlikoff (1998, Chaps. 2 and 3).

48

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Modeling the Growth of the World Economy: The Basic Overlapping. . .

Fig. 2.6 Golden rule savings and consumption

f(k)

y C*

n

G kt

σkt

a

s

k*

kt

generations. Graphically, the “golden rule” capital intensity can be found, by maximizing the distance between f(k) and Gnk (¼ σkα). Figure 2.6 shows the “golden rule” savings rate and “golden rule” capital intensity leading to long-run maximum consumption.

2.7

Summary and Conclusion

In this chapter, the basic OLG growth model of the closed world economy—a log-linear CD version of Diamond’s (1965) neoclassical growth model—was introduced and its intertemporal equilibrium dynamics were derived. In contrast to post-Keynesian growth theory, our basic OLG growth model rests on solid intertemporal general equilibrium foundations comprising constrained optimization of agents and the clearing of all markets in each model period. Regarding production technology, the linear homogeneity of the production function and the substitutability of production factors were emphasized. This is in line with neoclassical growth theory. Factor substitutability enables profit-maximizing firms to adapt their capital intensities (capital–labor ratios) to the prevailing relative wage rate. Another key feature of the basic growth model is the endogeneity of per capita savings. Young households choose savings in order to maximize their lifetime utility. In doing so, they also choose optimal (i.e., utility maximizing) consumption when young, and optimal consumption when old. As in the Solow–Swan neoclassical growth model, the savings rate is constant, and can be traced back to the time discount factor of younger households. The old households consume their entire wealth (bequests are excluded by definition). All market participants (young households, old households, and producers) interact in competitive markets for capital and labor services and for the produced commodity. Supply and demand in each market are balanced by the perfectly

2.8 Exercises

49

flexible real wage and real interest rate. The first-order conditions (FOCs) for intertemporal utility maxima and period-specific profit maxima in conjunction with market clearing conditions yield the fundamental equation of motion for our basic OLG model of capital accumulation together with the equilibrium dynamics of the efficiency weighted capital intensity. The fundamental equation of motion also allows for the determination of the real wage rate and the real interest rate on the intertemporal equilibrium path. Finally, we sought for the savings rate and associated capital intensity that maximizes permanent consumption per-efficiency capita. It turns out that higher savings rates are not in general better than lower savings rates. The golden rule for achieving maximum consumption per efficiency capita demands a capital intensity at which the marginal product of capital corresponds exactly to the rate of natural growth plus depreciation rate. If we assume a depreciation rate of one, the savings rate, leading to the “golden rule” capital intensity, must be equal to the production elasticity of capital.

2.8

Exercises

2.8.1 Explain the set-up of the basic OLG model and provide empirically relevant values for basic model parameters such as β, Gn, and α. Why is α independent of the length of the model period while β and Gn are not? 2.8.2 Use the CD function Eq. (2.2) to show that the marginal product of capital is always smaller than the average product of capital. 2.8.3 Explain in terms of the marginal rate of substitution and the negative slope of the intertemporal budget constraint why point B in Fig. 2.3 is not utility maximizing. 2.8.4 Show that under the CD production function Eq. (2.2) maximum profits are zero. Which property of general neoclassical production functions implies zero profits? 2.8.5 Why must the younger households finance next-period capital stock even when capital does not depreciate completely during one period? 2.8.6 Verify the derivation of the intertemporal equilibrium dynamics Eq. (2.39) and explain why not the whole savings per efficiency capita cannot be used for capital deepening? 2.8.7 Explain the meaning of the golden rule of capital accumulation and provide a sufficient condition with respect to capital production share such that the savings rate is irrelevant for golden rule capital intensity (Hint: See Galor and Ryder 1991).

50

2

Modeling the Growth of the World Economy: The Basic Overlapping. . .

Appendix 1. Constrained Optimization All agents in this chapter aim at optimizing their decisions to reach their goals in the best possible way. However, they are all confronted with various restrictions (constraints)—in some cases they are of a natural or technological nature, in other cases choices are limited due to available income. How can one find the optimum decision in the face of such constraints? The method of mathematical (classical) programming provides a solution. In order to formalize the decision problem we first need to define the following: What are the objectives of the different actors? Which variables are to be included in agent decision making? Which restrictions do they face? The objectives of the agents can be formalized by use of the objective function, Z. This function assigns a real number to every decision (consisting of a list of n decision variables) made by an agent. Z : ℜn ! ℜ1

ð2:49Þ

We introduced two objective functions in the main text of this chapter: one for households whose goal is to act in such a way that their preferences, represented by a utility function, are met best, and one for firms that try to maximize their profit function. Concerning the second and third questions we know that households can determine consumption quantities and the distribution of consumption over time. We also know that producers can determine the demand for labor as well as for capital. These variables are referred to as decision (choice) variables or instrumental variables. The quantities households can consume depend, among other things, on their income. The production cost of a specific quantity of a good depends, among other things, on the technology used in the production process. Such restrictions are represented in the form of constraints. Mathematically speaking, the decision problem is to find values for the instrumental variables which maximize the value of the objective function (profit, utility) or minimize it (cost), subject to all constraints. Formally, the optimization problem can be written as one of the following three programs: Max Z ðxÞs:t: : gðxÞ ¼ b, ðclassical optimizationÞ

ð2:50aÞ

Max Z ðxÞs:t: : gðxÞ  b, x  0, ðnon  linear optimizationÞ

ð2:50bÞ

Max Z ðxÞ ¼ cxs:t: : Ax  b, x  0:ðlinear optimizationÞ

ð2:50cÞ

The objective function Z is a function of n variables, i.e., x is a vector of dimension n (n decision variables). The function g(x) denotes m constraints; b is a column vector of dimension m.

Appendix

51

We now turn to classical optimization and try to find a rule which allows us to unveil the optimal decision-making of agents. An example of the household objective function U(x) is given by Eq. (2.12); the (only) constraint g(x) by Eq. (2.15).   MaxU t c1t , c2tþ1 ¼ ln c1t þ β ln c2tþ1

ð2:51aÞ

subject to (s.t.): c1t þ

c2tþ1 ¼ wt 1 þ itþ1

ð2:51bÞ

The two instrumental variables in this optimization problem are c1t and c2tþ1, and are the (only) variables households can determine. Due to the constraint, future consumption can (under certain conditions) be written as a function of current consumption.   c2tþ1 ¼ ð1 þ itþ1 Þ wt  c1t

ð2:52aÞ

  c2tþ1 ¼ h c1t ,

ð2:52bÞ

∂g=∂c1t dh ¼ : 1 ∂g=∂c2tþ1 dct

ð2:52cÞ

Or, more generally:

e of a single decision The objective function can also be formulated as a function Λ variable:    e ¼ ln c1 þ β ln ð1 þ itþ1 Þ wt  c1 : Λ t t

ð2:53aÞ

Or, more generally:    e¼Λ e c1 , h c1 : Λ t t

ð2:53bÞ

This intermediate step simplifies the search for a value c of the decision variable e (utility) in our decision problem. c1t that maximizes the objective function Λ Obviously, at a maximum, the following condition has to hold: e ð cÞ  Λ eðc þ ΔcÞ: Λ

ð2:54Þ

If we make use of Taylor’s theorem, we can find the maximum of the (modified) objective function Eq. (2.53b). The first-order condition (FOC) of the problem is:

52

2

Modeling the Growth of the World Economy: The Basic Overlapping. . .

e dΛ ∂U ∂U dh ¼0¼ 1þ : 1 ∂ct ∂h dc1t dct

ð2:55Þ

On taking account of Eq. (2.52b), then Eq. (2.55) is equivalent to:     e ∂g=∂c1t ∂U=∂h ∂g dΛ ∂U ∂U ∂U  ¼ 0 ¼ þ þ  : ¼ ∂c1t ∂h ∂c1t ∂g=∂c2tþ1 ∂g=∂c2tþ1 ∂c1t dc1t

ð2:56Þ

We denote the expression in brackets on the right-hand side by λ, so that the maximization problem (2.56) can be written more simply as: e dΛ ∂U ∂g ¼0¼ 1þλ 1: ∂ct ∂ct dc1t

ð2:57Þ

This is the solution to the household’s decision problem. However, a simpler route is provided by a function that leads us directly to condition (2.57). This is:        Λ c1t , c2tþ1 , λ ¼ U c1t , c2tþ1 þ λ w  g c1t , c2tþ1 :

ð2:58Þ

This function is called the Lagrangian function and the variable λ the Lagrangian multiplier. After calculating the first derivative with respect to the two instrumental variables, the first-order (necessary) conditions for the solution of the optimization problem follows. Thus, differentiating Eq. (2.58) with respect to λ results directly in the constraint. The Lagrangian function of young households has the following form: Λ



c1t , c2tþ1 , λ



 ¼

ln c1t

þ

β ln c2tþ1

þλ w

c1t

 c2tþ1  : 1 þ itþ1

ð2:59Þ

The first-order conditions (FOCs) are: ∂Λ 1 ¼ 1  λ ¼ 0, ct ∂c1t

ð2:60aÞ

∂Λ 1 1 ¼β 2  λ ¼ 0, ∂c2tþ1 ctþ1 1 þ itþ1

ð2:60bÞ

c2 ∂Λ ¼ wt  c1t  tþ1 ¼ 0: 1 þ itþ1 ∂λ

ð2:60cÞ

If we solve condition (2.60a) for variable λ and substitute the solution into Eq. (2.60b) then, assuming the constraint Eq. (2.60c) is also taken into account, we can determine the optimal consumption in period t Eq. (2.20), the optimal consumption in period t + 1 Eq. (2.22) and the optimal savings per capita Eq. (2.21).

Appendix

53

To ensure that Eqs. (2.20), (2.21), and (2.22) constitute a maximum (and not a minimum), we have to check the second-order conditions: 2

∂ Λ 1  2 ¼   2 < 0, 1 ∂ ct c1t

ð2:61aÞ

2

∂ Λ 1  2 2 ¼ β  2 2 < 0: ∂ ctþ1 ctþ1

ð2:62bÞ

Both conditions are negative, satisfying the second-order conditions for a strict (local) maximum. One last and very important question remains: What is the meaning of the Lagrange multiplier in this optimization problem? The Lagrange multiplier reflects the sensitivity of the value of the objective function with respect to a marginal change in the constants b (cf. Eq. 2.50a) of the constraints. In the optimization problem of young households the Lagrange multiplier is equal to: λt ¼

∂U t : ∂wt

ð2:63Þ

It indicates the amount by which the optimum value of the utility function increases when disposable income rises by one unit.

2. Walras’ Law Finally, we want to show that our basic growth model satisfies Walras’ law. We, therefore, note the budget constraints of all economic agents for any period t and express all values in monetary units (and not in terms of output units as is done in the main text). In addition, we multiply all per capita values by the number of corresponding number of individuals. Moreover, we indicate what savings of young households are used for, i.e., to buy investment goods and old capital at the reproduction price Pt. Thus, we have: Lt st Pt ¼ Pt I t þ Pt ð1  δÞK t :

ð2:64Þ

This equality implies that the budget constraint of young households can be rewritten as follows: Pt Lt c1t þ Pt I t þ Pt ð1  δÞK t ¼ W t Lt , while the aggregate budget constraint of old households reads as follows:

ð2:65Þ

54

2

Modeling the Growth of the World Economy: The Basic Overlapping. . .

Pt Lt1 c2t ¼ Qt K t þ Pt ð1  δÞK t :

ð2:66Þ

The linear-homogeneity of the production function implies that at a maximum profits are zero: Πt ¼ 0 ¼ Pt Y t  W t N t  Qt K dt :

ð2:67Þ

Adding the left- and the right-hand sides of Eqs. (2.65) and (2.66) yields:  Pt Lt c1t þ Lt1 c2t ¼ W t Lt þ Qt K dt  Pt I t :

ð2:68Þ

Clearing of the labor and capital market (Nt ¼ Lt and K dt ¼ K t ) implies:  Pt Lt c1t þ Lt1 c2t þ Pt I t ¼ W t N t þ Qt K t ¼ Pt Y t :

ð2:69Þ

Since It ¼ Kt + 1  (1  δ)Kt holds, Eq. (2.69) becomes:  Pt Lt c1t þ Lt1 c2t þ K tþ1  ð1  δÞK t  Y t ¼ 0:

ð2:70Þ

Since Pt > 0, the sum of the terms in square brackets in Eq. (2.70) must be zero. Thus, we have shown that the product market clears once the labor and the capital markets clear. Equation (2.31) is thus an identity, not a constraint. Thus, we cannot determine the price level in this economy; it has, therefore, to be set exogenously (e.g.,—and as we have assumed here—it can be set equal to one).

References Acemoglou, D. (2009). Introduction to modern economic growth. Princeton: Princeton University Press. Auerbach, A. J., & Kotlikoff, L. J. (1998). Macroeconomics: An integrated approach (2nd ed.). Cambridge, MA: MIT. Cobb, C. W., & Douglas, P. H. (1934). The theory of wages. New York: Macmillan. De la Croix, D., & Michel, P. (2002). A theory of economic growth: Dynamics and policy in overlapping generations. Cambridge: Cambridge University Press. Diamond, P. (1965). National debt in a neoclassical growth model. American Economic Review, 55, 1126–1150. Domar, E. D. (1946). Capital expansion, rate of growth and employment. Econometrica, 14, 137–147. Farmer, K., & Bednar-Friedl, B. (2010). Intertemporal resource economics: An introduction to the overlapping generations approach. Berlin: Springer. Galor, O., & Ryder, H. (1991). Dynamic efficiency of steady-state equilibria in an overlapping generations model with productive capital. Economics Letters, 35(4), 385–390. Harrod, R. F. (1939). An essay in dynamic theory. Economic Journal, 49, 14–33. Kaldor, N. (1961). Capital accumulation and economic growth. In F. A. Lutz & D. C. Hague (Eds.), The theory of capital. London: Macmillan. Lopez-Garcia, M. A. (2008). On the role of public debt in an OLG model with endogenous labor supply. Journal of Macroeconomics, 30(3), 1323–1328. Müller, K. W., & Stroebele, W. (1985). Wachstumstheorie. München: Oldenburg.

References

55

Phelps, E. S. (1966). Golden rules of economic growth. New York: Norton. Ramsey, F. (1928). A mathematical theory of saving. Economic Journal, 38, 543–559. Solow, R. M. (1956). A contribution to the theory of economic growth. Quarterly Journal of Economics, 70, 65–94. Swan, T. W. (1956). Economic growth and capital accumulation. Economic Record, 32, 334–361.

3

Steady State, Factor Income, and Technological Progress

3.1

Introduction and Motivation

In the previous chapter, the growth rate of the capital stock and of the capital intensity was derived from the FOCs of agents and market clearing conditions in each period (intertemporal equilibrium conditions). As shown, the capital intensity (capital–labor ratio) in period t + 1 depends on the natural growth factor, the savings rate, and the capital intensity of the previous period. What we did not show is how the growth factor (rate) of the capital intensity is related to the GDP growth factor (rate). GDP growth figures prominently in all theories of economic growth. Thus, it is only natural to explore first the relationship between the growth rate of capital intensity and GDP growth rate. Given the dominance of neoclassical general equilibrium theory à la Arrow and Debreu (1954) in the 1950s, equilibrium concepts also permeated growth theory. Indeed, the very notion of a steady state comprises an equilibrium of growth rates in a double sense: (1) the growth rates of output and inputs remain constant over time and (2) the growth rates of output and inputs are equal (“balanced growth”). The second objective of this chapter is to define clearly the steady state in our basic OLG model. While the log-linear intertemporal utility function and the Cobb-Douglas (CD) production function ensure the existence of an intertemporal equilibrium solution (as seen in the previous chapter), the existence of nontrivial steady-state solutions is in general (i.e., in the absence of log-linear utility and a CD production function) by no means ensured. As Galor and Ryder (1989) have forcefully shown, in contrast to the infinitely-lived agent approach in the OLG growth model there is a strong tendency to global contraction.1 While we alert the reader to this unpleasant

1

Galor and Ryder (1989, 368) show in their proposition 1 that for any given set of well-behaved intertemporal utility functions there exists a production function which satisfies the Inada conditions but still result in a steady state with zero production and consumption. # Springer-Verlag GmbH Germany, part of Springer Nature 2021 K. Farmer, M. Schelnast, Growth and International Trade, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-662-62943-7_3

57

58

3

Steady State, Factor Income, and Technological Progress

fact, we also aim to provide sufficient conditions for the existence of a nontrivial steady state, which are trivially satisfied under the log-linear utility function and the CD production function. Assuming the existence of nontrivial steady-state solutions, we then need to explore uniqueness and dynamic stability. While a log-linear utility function and a CD production function ensure a unique and globally stable nontrivial steady state, identifying which conditions are sufficient to guarantee uniqueness and dynamic stability is also of interest. Again, following Galor and Ryder (1989), we provide the general uniqueness and stability conditions in Sect. 3.3. Knowing that a nontrivial steady state is unique and dynamically stable does not ensure that the steady state is Pareto-efficient. Since Diamond (1965) it has been well-known that the steady-state market equilibrium in an OLG model is Paretoefficient only by accident. A necessary condition for intertemporal Pareto efficiency is dynamic efficiency. All these different efficiency concepts and the efficiency of a steady-state market equilibrium will be the subject of Sect. 3.4. Giving a nontrivial steady-state solution it is also interesting to know how the steady-state solution changes when basic parameters such as time preferences, population growth, and labor productivity growth change. This is the subject of Sect. 3.5. While capital intensity represents the pivotal variable in an intertemporal and steady-state equilibrium in the basic OLG growth model, other important economic variables such as the real wage rate, the real interest rate, and the factor income shares are also closely related to capital intensity. The determination of these variables along an intertemporal equilibrium path and in the steady state is described in Sect. 3.6. Growth of per capita income is impossible in the long run without technological progress. There are, however, different notions of so-called neutral technological progress in neoclassical growth theory. Some stress the productivity-enhancing role of capital, others of efficiency labor. The definition of labor and capital-saving technological progress and their relevance for functional income distribution is left to Sect. 3.7. Last but not least the empirical significance of technological progress for GDP growth is investigated in Sect. 3.8. Thus, this chapter seeks to answer the following questions: • How can the GDP growth rate be derived from the growth factor of capital intensity? Which factors determine the GDP growth rate of an economy? Why are there different growth rates across countries? • Does there exist a long-term equilibrium growth path (steady state)? Are steady states unique and is the accumulation dynamics stable? • What is the development path of the real wage, the real interest rate, and income shares of labor and capital in intertemporal equilibrium and in the steady state? • How are the steady-state capital intensity and the steady-state GDP growth rate affected by a change in main OLG model parameters (e.g., in the savings rate or natural growth rate)?

3.2 The GDP Growth Rate in Intertemporal Equilibrium and in Steady State

59

• How does technological progress influence the growth rate of an economy and its long-term growth potential?

3.2

The GDP Growth Rate in Intertemporal Equilibrium and in Steady State

As mentioned above, the most interesting variable in growth theory is not the growth rate of the capital intensity but the output or GDP growth rate. Thus, we first define the GDP growth rate in intertemporal equilibrium and relate it to the growth rate of the capital intensity. Definition 3.1 The output or GDP growth rate along the intertemporal equilibrium path in the basic OLG model is defined and related to the growth factor of capital intensity as follows: gYt
GYt  1with GYt



α 1α α Y tþ1 ðAtþ1 Þ ðK tþ1 Þ n k tþ1 ¼ ¼ G : Yt kt ðAt Þ1α ðK t Þα

ð3:1Þ

Using Definition 3.1 it is easily seen that a stationary state of the capital intensity, i.e., kt + 1 ¼ kt or a fixed point of the fundamental equation of motion (2.39) implies the long-run equilibrium or steady-state GDP growth rate gY ¼ gYtþ1 ¼ gYt : k tþ1 ¼ kt ) gY ¼ Gn  1 ¼ gYt ¼ gYtþ1 :

ð3:2Þ

This is an example of the first of two meanings of equilibrium growth mentioned in Sect. 3.1: i.e., the stationarity of the GDP growth rate over time. The second meaning of equilibrium growth, namely the equality of the growth rates of production factors, is also implied by the fixed point of the above fundamental equation of motion. Thus: k tþ1 ¼ k t ,

K tþ1 K t K A ¼ , tþ1 ¼ tþ1 , GKt
Gn : Atþ1 At Kt At

ð3:3Þ

Both the production factors and output grow at the same rate as the following calculation shows: ðA Þ1α ðK Þα Y tþ1  1 ¼ tþ1 1α tþ1α  1 ¼ Yt ðAt Þ ðK t Þ   α ¼ ðGn Þ1α GK  1

gYt ¼




Atþ1 At

1α


K tþ1 Kt

α

1

60

3

Steady State, Factor Income, and Technological Progress

¼ G n  1 ¼ G K  1 ¼ gn ¼ gK :

ð3:4Þ

Due to this close connection between the time-stationarity of the GDP growth rate and the time-stationarity of capital intensity, it is natural to call kt ¼ kt + 1 ¼ k a steady state. In this steady state, both the GDP growth is time-stationary and the two production factors real capital and efficiency labor grow at the same rate. Therefore, the steady state is called also a “balanced growth” path. Along this path, the GDP growth rate equals the natural growth rate. Does such a state exist? This question is dealt with in the next section.

3.3

Existence and Stability of the Long-Run Growth Equilibrium

In this section, we try to answer the following three questions: Does a steady-state equilibrium exist (existence)? If so, is the equilibrium unique or are there other equilibria (uniqueness)? If an economy is not in its steady-state equilibrium, do market interactions between economic agents guide the economy closer to a steadystate equilibrium (stability) or do these interactions, as in post-Keynesian theory, lead the economy further away from the growth equilibrium (instability)? In the basic OLG model, which is characterized by a log-linear utility function and a CD production function, exactly one non-trivial—i.e., not equal to zero— steady-state growth equilibrium exists. At the fixed point kt ¼ kt + 1 ¼ k the fundamental equation of motion (2.39) simplifies to: Gn k ¼ σk α :

ð3:5Þ

Obviously, this equation has two solutions: the trivial one k ¼ 0 and a unique, positive (non-trivial) and finite solution for α < 1: k¼



σ Gn

1=ð1αÞ

:

ð3:6Þ

Only when capital intensity takes the value in (3.6) have we reached the steady growth equilibrium: The growth rates of the capital stock, of efficiency employment, and of the GDP are time-constant and identical. Figure 3.1 illustrates the log-linear CD case of a unique trivial, and a unique nontrivial steady state, as solutions of Eq. (3.5). The conditions for the existence of a (non-trivial) growth equilibrium where the production function and the intertemporal utility function are not functionally specified (e.g., as a CD production function) are provided by (3.7) and (3.8) (see Galor and Ryder 1989).

3.3 Existence and Stability of the Long-Run Growth Equilibrium

61

n

Fig. 3.1 Steady state and steady state capital intensity

Gk

n

α

Gk

σk

α

σk

k

 lim 

k!0

β kf 00 ðkÞ > Gn 1þβ

k



lim f 0 ðkÞ ¼ 0

k!1

ð3:7Þ ð3:8Þ

Condition (3.7) implies that at the origin of the diagram in Fig. 3.1 the function on the left-hand side of (3.5) has a smaller slope than the function on the right-hand side. Our CD production function satisfies this condition, since:  lim 

k!0

β β kf 00 ðkÞ ¼ lim ð1  αÞαkα1 > Gn , 1þβ k!0 1 þ β lim f 0 ðkÞ ¼ lim αkα1 ¼ 0:

k!1

k!1

ð3:7aÞ ð3:8aÞ

Condition (3.7) can be interpreted economically as follows: A marginal increase in the capital intensity at the beginning of the accumulation process (at low capital intensities) has to raise the per-efficiency capita savings (¼capital supply per efficiency capita), resulting from the additional wage income due to the higher capital intensity, more than the additional capital requirement associated with the growth of the efficiency-weighted population. Condition (3.8) ensures that, if required capital needs can be covered at all, the equilibrium capital per-efficiency employee is finite. Note: (3.7) and (3.8) are sufficient conditions for the existence of a steady-state equilibrium, but do not exclude multiple or unstable growth equilibria. Following Galor and Ryder (1989, 372), the fundamental equation of motion (2.39) has to satisfy two additional requirements in order to guarantee that the growth equilibrium is both unique and (globally) stable. If the right-hand side of (2.39) is written in a more general way as ϕ(kt), these properties are: ϕ0 ðkÞ ⩾ 0 for all k > 0,

ð3:9Þ

62

3

Fig. 3.2 Uniqueness and global stability of growth equilibrium

Steady State, Factor Income, and Technological Progress kt+l

φ(k)

45° k0

k

ϕ00 ðkÞb0 for all k > 0:

kl

kt

ð3:10Þ

These properties are shown graphically in Fig. 3.2. Our basic OLG growth model is consistent with both conditions. If the two conditions (3.9) and (3.10) are met, then the growth equilibrium is unique and globally asymptotically stable. Global asymptotic stability means that irrespective of where the economy starts, i.e., whether k0 < k or k0 > k, the economy always tends towards the steady state k as time approaches infinity (hence the term “asymptotic”). This property is shown for k0 < k in Fig. 3.2: k1 > k0, k2 > k1 . . . and jk2  k1 j < j k1  k0j. Hence, for any period t we have: kt + 2 > kt + 1 and jkt + 2  kt + 1 j < j kt + 1  ktj. Therefore, if the initial capital intensity is too small (compared to the steady-state capital intensity) the intertemporal equilibrium capital intensity increases with decreasing rates until the steady-state value is reached. Per capita production increases with growing capital intensity, but the gains due to a higher capital intensity decrease. Consequently, the capital coefficient vt ¼ kt/f(kt) increases with rising capital intensity: f ðk t Þ  k t f 0 ðk t Þ f ðk t Þ dvt ¼ > 0, since > f 0 ðkt Þ: kt dk t ½ f ðk t Þ2

ð3:11Þ

The steady state is an attraction point. The economy tends to move automatically toward this point since agents in the basic OLG growth model are led by their own self-interest and by market signals in such a way that an accumulation process toward the steady state takes place if k0 < k and a decumulation process sets in if k0 > k. We now assume that k0 < k. How do agents of our decentralized market system know that the initial capital intensity is too low and has to be increased in order to reach a steady state? In a competitive market economy, the agents are guided by changes in market prices, i.e., in our basic OLG model by changes in the relative wage rate wt/qt. The relative wage rate changes if the growth rates in capital stock and efficiency-weighted employment differ. In our basic OLG model, the following relation holds for the growth rate of the capital stock in period 0:

3.3 Existence and Stability of the Long-Run Growth Equilibrium

gK0 ¼ ¼σ

63

K 1 =A1 K1 k1 1¼ 1¼ 1¼ K0 ðA0 =A1 ÞðK 0 =A0 Þ k 0 =Gn


1a  1a 1 1 1>σ  1
gK ¼ gA ¼ gn , since k0 < k: k0 k

In words: If the initial capital intensity is smaller than the steady-state capital intensity, then the current growth rate of the capital stock gK0 is higher than that of the efficiency-weighted population gA ¼ gK. Compared to the balanced growth position, the capital stock is growing too fast: labor relative to capital is becoming increasingly scarce, which leads to an increasing relative wage rate in a perfectly competitive economy with flexible labor and capital markets. As can be seen from the ratio that results when we divide (2.41) by (2.42), profit-maximizing producers respond to a rising w0/q0 by an increase of k1 if α < 1 (decreasing marginal returns to capital). Profit maximizing firms need no conscious knowledge of the fact that in the case of k0 < k the capital intensity ought to be raised to ensure convergence toward the steady state (stability of the market system in general). They are simply led by their own self-interest to do exactly what is needed. Any external, impartial spectator would exhibit a similar perspective. By substituting capital for more expensive labor firms serve their own self-interest and simultaneously serve the common interest by pushing the economy nearer to the steady state. In response to the rising relative wage, the economy is steadily growing toward the steady-state growth equilibrium. Why does capital intensity not continue to rise when the steady state is reached? The answer can be found in the decreasing marginal productivity of capital: With increasing capital intensity the marginal product of capital decreases. Thus, under a constant savings rate σ, the additional savings available for net investment, also decline. The growth rate of capital intensity is positive, but decreasing. In the steady state, the volume of savings is just sufficient to meet the additional capital demand due to population growth and technological progress; thus, the growth rate of capital intensity becomes zero: gkt


 1=ð1αÞ ktþ1  kt σ 1 σ ¼ n 1α  1 and lim gkt ¼ 0, since k ¼ : G kt Gn kt k t !k

At capital intensities higher than the steady state k, lower marginal productivity means that the additional savings are no longer sufficient to cover capital requirements. Thus, the growth rate of the capital intensity declines. This is illustrated in Fig. 3.3. Our analysis so far has brought forth three key results. First, the basic OLG growth model suggests that the world economy tends spontaneously toward a steady state. In this long-term growth equilibrium, the growth rate of world GDP is exactly equal to the growth factor of technological progress times the population growth factor minus one, or approximately speaking, is equal to the sum of the technological progress rate and the population growth rate. The level of the capital intensity and the output per efficiency capita is additionally governed by the time discount factor and

64

3

Fig. 3.3 The growth rate of the (efficiency-weighted) capital intensity

Steady State, Factor Income, and Technological Progress

g

k

k

0

g >0 k

kt

k

g