The mystery of Economic Growth

#709344 9/16/03

ECONOMIC GROWTH Second Edition

Robert J. Barro AND Xavier Sala-i-Martin

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Second Edition Barro AND Sala-i-Martin ECONOMIC GROWTH

This graduate level text on economic growth surveys neoclassical and more recent growth theories, stressing

their empirical implications and the relation of theory to data and evidence. The authors have undertaken a

major revision for the long-awaited second edition of this widely used text, the first modern textbook devoted

to growth theory. The book has been expanded in many areas and incorporates the latest research.

After an introductory discussion of economic growth, the book examines neoclassical growth theories, from

Solow–Swan in the 1950s and Cass–Koopmans in the 1960s to more recent refinements; this is followed by a

discussion of extensions to the model, with expanded treatment in this edition of heterogeneity of households.

The book then turns to endogenous growth theory, discussing, among other topics, models of endogenous tech￾nological progress (with an expanded discussion in this edition of the role of outside competition in the growth

process), technological diffusion, and an endogenous determination of labor supply and population. The authors

then explain the essentials of growth accounting and apply this framework to endogenous growth models. The

final chapters cover empirical analysis of regions and empirical evidence on economic growth for a broad panel

of countries from 1960 to 2000. The updated treatment of cross-country growth regressions for this edition uses

the new Summers–Heston data set on world income distribution compiled through 2000.

Robert J. Barro is Robert C. Waggoner Professor of Economics at Harvard University and senior fellow of

the Hoover Institution at Stanford University. He is the author of Nothing Is Sacred: Economic Ideas for the New

Millennium (MIT Press, 2002), Getting It Right (MIT Press, 1997), and Determinants of Economic Growth (MIT

Press, 1998). Xavier Sala-i-Martin is Professor of Economics at Columbia University and visiting professor at the

University of Pompeu Fabra, Barcelona.

Barro and Sala-i-Martin’s valuable and readable book brings the student quickly to the frontiers of modern

growth theory.” —Robert E. Lucas, Jr., John Dewey Distinguished Service Professor, The University of Chicago,

1995 Nobel Laureate in Economic Sciences

Barro and Sala-i-Martin provide an outstanding and comprehensive treatment of growth theory and empir￾ics—an instant classic! I learn something new every time I pull my copy from the shelf.” —Charles I. Jones,

Department of Economics, University of California, Berkeley

This is an invaluable book for a first graduate course in economic growth. The exposition is clear and easy to

follow, but also rigorous. It is an excellent stepping stone for research in the field.” —K. Daron Acemoglu,

Professor of Economics, MIT

Barro and Sala-i-Martin have done a superb job of synthesizing much of the existing theoretical and empir￾ical research on the mechanisms and determinants of economic growth and convergence. Though it incor￾porates much new material, this updated version is fully accessible to a third year undergraduate student,

while remaining of invaluable use to any research scholar seriously interested in growth and development

economics.” —Phillipe Aghion, Department of Economics, Harvard University

Cover image: Soft Watch by Salvador Dalí, © 2003 Salvador Dalí, Gala-Salvador Dalí Foundation/Artists Rights Society (ARS), New York

The MIT Press Massachusetts Institute of Technology Cambridge, Massachusetts 02142 http://mitpress.mit.edu 0-262-02553-1

ECONOMIC GROWTH Second Edition

Robert J. Barro AND Xavier Sala-i-Martin

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Economic Growth

Second Edition

Robert J. Barro

Xavier Sala-i-Martin

The MIT Press

Cambridge, Massachusetts

London, England

c 2004 Massachusetts Institute of Technology

All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means

(including photocopying, recording, or information storage and retrieval) without permission in writing from the

publisher.

This book was set in Times Roman by ICC Typesetting and was printed and bound in the United States of

America.

Library of Congress Cataloging-in-Publication Data

Barro, Robert J.

Economic growth / Robert J. Barro, Xavier Sala-i-Martin—2nd ed.

p. cm.

Includes bibliographical references and index.

ISBN 0-262-02553-1

1. Economic development—Mathematical models. I. Sala-i-Martin, Xavier. II. Title.

HD75.5.B37 2003

338.9

001

51—dc22

2003059392

To Rachel

—Robert J. Barro

A la mem`oria dels meus estimats Joan Mart´ın Pujol i Ramon Oriol Mart´ın Montemayor

—Xavier Sala-i-Martin

Contents

Preface xv

About the Authors xvii

Introduction 1

I.1 The Importance of Growth 1

I.2 The World Income Distribution 6

I.3 Empirical Regularities about Economic Growth 12

I.4 A Brief History of Modern Growth Theory 16

I.5 Some Highlights of the Second Edition 21

1 Growth Models with Exogenous Saving Rates

(the Solow–Swan Model) 23

1.1 The Basic Structure 23

1.2 The Neoclassical Model of Solow and Swan 26

1.2.1 The Neoclassical Production Function 26

1.2.2 The Fundamental Equation of the Solow–Swan Model 30

1.2.3 Markets 31

1.2.4 The Steady State 33

1.2.5 The Golden Rule of Capital Accumulation

and Dynamic Inefficiency 34

1.2.6 Transitional Dynamics 37

1.2.7 Behavior of Input Prices During the Transition 40

1.2.8 Policy Experiments 41

1.2.9 An Example: Cobb–Douglas Technology 43

1.2.10 Absolute and Conditional Convergence 44

1.2.11 Convergence and the Dispersion of Per Capita Income 50

1.2.12 Technological Progress 51

1.2.13 A Quantitative Measure of the Speed of Convergence 56

1.3 Models of Endogenous Growth 61

1.3.1 Theoretical Dissatisfaction with Neoclassical Theory 61

1.3.2 The AK Model 63

1.3.3 Endogenous Growth with Transitional Dynamics 66

1.3.4 Constant-Elasticity-of-Substitution Production Functions 68

1.4 Other Production Functions ... Other Growth Theories 71

1.4.1 The Leontief Production Function and

the Harrod–Domar Controversy 71

1.4.2 Growth Models with Poverty Traps 74

vi Contents

1.5 Appendix: Proofs of Various Propositions 77

1.5.1 Proof That Each Input Is Essential for Production

with a Neoclassical Production Function 77

1.5.2 Properties of the Convergence Coefficient

in the Solow–Swan Model 78

1.5.3 Proof That Technological Progress Must

Be Labor Augmenting 78

1.5.4 Properties of the CES Production Function 80

1.6 Problems 81

2 Growth Models with Consumer Optimization

(the Ramsey Model) 85

2.1 Households 86

2.1.1 Setup of the Model 86

2.1.2 First-Order Conditions 89

2.2 Firms 94

2.3 Equilibrium 97

2.4 Alternative Environments 98

2.5 The Steady State 99

2.6 Transitional Dynamics 102

2.6.1 The Phase Diagram 102

2.6.2 The Importance of the Transversality Condition 104

2.6.3 The Shape of the Stable Arm 105

2.6.4 Behavior of the Saving Rate 106

2.6.5 The Paths of the Capital Stock and Output 110

2.6.6 Speeds of Convergence 111

2.6.7 Household Heterogeneity 118

2.7 Nonconstant Time-Preference Rates 121

2.7.1 Results under Commitment 123

2.7.2 Results without Commitment under Log Utility 124

2.7.3 Population Growth and Technological Progress 129

2.7.4 Results under Isoelastic Utility 130

2.7.5 The Degree of Commitment 132

2.8 Appendix 2A: Log-Linearization of the Ramsey Model 132

2.9 Appendix 2B: Irreversible Investment 134

Contents vii

2.10 Appendix 2C: Behavior of the Saving Rate 135

2.11 Appendix 2D: Proof That γκˆ Declines Monotonically

If the Economy Starts from kˆ(0) < kˆ∗ 137

2.12 Problems 139

3 Extensions of the Ramsey Growth Model 143

3.1 Government 143

3.1.1 Modifications of the Ramsey Framework 143

3.1.2 Effects of Tax Rates 146

3.1.3 Effects of Government Purchases 147

3.2 Adjustment Costs for Investment 152

3.2.1 The Behavior of Firms 152

3.2.2 Equilibrium with a Given Interest Rate 155

3.2.3 Equilibrium for a Closed Economy with a Fixed

Saving Rate 159

3.3 An Open-Economy Ramsey Model 161

3.3.1 Setup of the Model 161

3.3.2 Behavior of a Small Economy’s Capital Stock and Output 162

3.3.3 Behavior of a Small Economy’s Consumption and Assets 163

3.3.4 The World Equilibrium 164

3.4 The World Economy with a Constraint on International Credit 165

3.4.1 Setup of a Model with Physical and Human Capital 166

3.4.2 The Closed Economy 167

3.4.3 The Open Economy 168

3.5 Variations in Preference Parameters 177

3.6 Economic Growth in a Model with Finite Horizons 179

3.6.1 Choices in a Model with Finite Horizons 179

3.6.2 The Finite-Horizon Model of a Closed Economy 183

3.6.3 The Finite-Horizon Model of an Open Economy 186

3.7 Some Conclusions 189

3.8 Appendix: Overlapping-Generations Models 190

3.8.1 Households 190

3.8.2 Firms 192

3.8.3 Equilibrium 193

3.9 Problems 200

viii Contents

4 One-Sector Models of Endogenous Growth 205

4.1 The AK Model 205

4.1.1 Behavior of Households 205

4.1.2 Behavior of Firms 206

4.1.3 Equilibrium 207

4.1.4 Transitional Dynamics 208

4.1.5 The Phase Diagram 209

4.1.6 Determinants of the Growth Rate 210

4.2 A One-Sector Model with Physical and Human Capital 211

4.3 Models with Learning by Doing and Knowledge Spillovers 212

4.3.1 Technology 212

4.3.2 Equilibrium 216

4.3.3 Pareto Nonoptimality and Policy Implications 216

4.3.4 A Cobb–Douglas Example 217

4.3.5 Scale Effects 218

4.4 Public Services and Endogenous Growth 220

4.4.1 A Public-Goods Model 220

4.4.2 A Congestion Model 223

4.5 Transitional Dynamics, Endogenous Growth 226

4.5.1 A Cobb–Douglas Example 226

4.5.2 A CES Example 230

4.6 Concluding Observations 232

4.7 Appendix: Endogenous Growth in the One-Sector Model 232

4.8 Problems 235

5 Two-Sector Models of Endogenous Growth (with Special

Attention to the Role of Human Capital) 239

5.1 A One-Sector Model with Physical and Human Capital 240

5.1.1 The Basic Setup 240

5.1.2 The Constraint of Nonnegative Gross Investment 242

5.2 Different Technologies for Production and Education 247

5.2.1 The Model with Two Sectors of Production 247

5.2.2 The Uzawa–Lucas Model 251

5.2.3 The Generalized Uzawa–Lucas Model 266

5.2.4 The Model with Reversed Factor Intensities 267

5.3 Conditions for Endogenous Growth 268

Contents ix

5.4 Summary Observations 271

5.5 Appendix 5A: Transitional Dynamics with Inequality Restrictions

on Gross Investment in the One-Sector Model 271

5.6 Appendix 5B: Solution of the Uzawa–Lucas Model 274

5.7 Appendix 5C: The Model with Reversed Factor Intensities 280

5.8 Problems 282

6 Technological Change: Models with an Expanding Variety

of Products 285

6.1 A Baseline Model with a Variety of Products 285

6.1.1 The Producers of Final Output 285

6.1.2 Research Firms 289

6.1.3 Households 295

6.1.4 General Equilibrium 295

6.1.5 Determinants of the Growth Rate 297

6.1.6 Pareto Optimality 297

6.1.7 Scale Effects and the Cost of R&D 300

6.1.8 A Rising Cost of R&D 303

6.2 Erosion of Monopoly Power, Competition 305

6.3 Romer’s Model of Technological Change 310

6.4 Concluding Observations 313

6.5 Problems 313

7 Technological Change: Schumpeterian Models

of Quality Ladders 317

7.1 Sketch of the Model 317

7.2 The Model 319

7.2.1 The Producers of Final Output: Levels of Quality

in the Production Technology 319

7.2.2 The Research Sector 321

7.2.3 Consumers 328

7.2.4 Behavior of the Aggregate Quality Index

and Endogenous Growth 329

7.2.5 Scale Effects Again 331

7.3 Innovation by the Leader 333

7.3.1 Interactions Between the Leader and the Outsiders 333

7.3.2 The Leader as a Monopoly Researcher 336

x Contents

7.4 Pareto Optimality 339

7.5 Summary Observations about Growth 342

7.6 Appendix 343

7.6.1 Intermediates of Various Quality Grades 343

7.6.2 The Duration of a Monopoly Position 345

7.6.3 The Market Value of Firms 346

7.6.4 Research by the Industry Leader 346

7.7 Problems 347

8 The Diffusion of Technology 349

8.1 Behavior of Innovators in the Leading Country 351

8.2 Behavior of Imitators in the Follower Country 352

8.2.1 Producers of Final Output 352

8.2.2 Imitating Firms 353

8.2.3 Consumers 357

8.2.4 Steady-State Growth 357

8.2.5 The Dynamic Path and Convergence 359

8.3 Constant (or Slowly Rising) Costs of Imitation 363

8.3.1 The Steady State 364

8.3.2 Transitional Dynamics 365

8.4 Foreign Investment and Intellectual Property Rights 368

8.5 General Implications for Growth Rates in Follower Countries 370

8.6 Switchovers of Technological Leadership, Leapfrogging 373

8.7 Welfare Considerations 376

8.8 Summary Observations about Diffusion and Growth 379

8.9 Problems 380

9 Labor Supply and Population 383

9.1 Migration in Models of Economic Growth 383

9.1.1 Migration in the Solow–Swan Model 384

9.1.2 Migration in the Ramsey Model 393

9.1.3 The Braun Model of Migration and Growth 398

9.2 Fertility Choice 407

9.2.1 An Overlapping-Generations Setup 408

9.2.2 The Model in Continuous Time 411

9.3 Labor/Leisure Choice 422

Contents xi

9.4 Appendix: The Form of the Utility Function with Consumption

and Work Effort 427

9.5 Problems 428

10 Growth Accounting 433

10.1 Standard Primal Growth Accounting 433

10.1.1 Basic Setup 433

10.1.2 Measuring Inputs 436

10.1.3 Results from Growth Accounting 438

10.1.4 A Note on Regression-Based Estimates of TFP Growth 441

10.2 Dual Approach to Growth Accounting 442

10.3 Problems with Growth Accounting 444

10.3.1 An Increasing-Returns Model with Spillovers 445

10.3.2 Taxes 447

10.3.3 Multiple Types of Factors 449

10.4 TFP Growth and R&D 450

10.4.1 Varieties Models 451

10.4.2 Quality-Ladders Models 454

10.5 Growth Accounting Versus Sources of Growth 457

11 Empirical Analysis of Regional Data Sets 461

11.1 Two Concepts of Convergence 462

11.2 Convergence Across the U.S. States 466

11.2.1 β Convergence 466

11.2.2 Measurement Error 472

11.2.3 σ Convergence 473

11.3 Convergence Across Japanese Prefectures 474

11.3.1 β Convergence 474

11.3.2 σ Convergence Across Prefectures 478

11.4 Convergence Across European Regions 479

11.4.1 β Convergence 479

11.4.2 σ Convergence 482

11.5 Convergence Across Other Regions Around the World 482

11.6 Migration Across the U.S. States 483

11.7 Migration Across Japanese Prefectures 486

11.8 Migration Across European Regions 490

xii Contents

11.9 Migration and Convergence 492

11.10 β Convergence in Panel Data with Fixed Effects 495

11.11 Conclusions 496

11.12 Appendix on Regional Data Sets 497

11.12.1 Data for U.S. States 497

11.12.2 Data for European Regions 500

11.12.3 Data for Japanese Prefectures 506

12 Empirical Analysis of a Cross Section of Countries 511

12.1 Losers and Winners from 1960 to 2000 511

12.2 An Empirical Analysis of Growth Rates 515

12.2.1 Effects from State Variables 517

12.2.2 Control and Environmental Variables 518

12.3 Regression Results for Growth Rates 521

12.3.1 A Basic Regression 521

12.3.2 Tests of Stability of Coefficients 534

12.3.3 Additional Explanatory Variables 535

12.4 Summary and Conclusions about Growth 541

12.5 Robustness 541

12.5.1 Levine and Renelt (1992) 542

12.5.2 Bayesian Averaging of Classical Estimates (BACE) 543

12.5.3 Main Results in Sala-i-Martin, Doppelhofer,

and Miller (2003) 547

12.5.4 Robustness Analysis 556

12.6 Appendix: Long-Term Data on GDP 559

Appendix on Mathematical Methods 567

A.1 Differential Equations 568

A.1.1 Introduction 568

A.1.2 First-Order Ordinary Differential Equations 569

A.1.3 Systems of Linear Ordinary Differential Equations 576

A.2 Static Optimization 597

A.2.1 Unconstrained Maxima 597

A.2.2 Classical Nonlinear Programming: Equality Constraints 598

A.2.3 Inequality Constraints: The Kuhn–Tucker Conditions 600

A.3 Dynamic Optimization in Continuous Time 604

A.3.1 Introduction 604

Contents xiii

A.3.2 The Typical Problem 605

A.3.3 Heuristic Derivation of the First-Order Conditions 606

A.3.4 Transversality Conditions 609

A.3.5 The Behavior of the Hamiltonian over Time 609

A.3.6 Sufficient Conditions 610

A.3.7 Infinite Horizons 610

A.3.8 Example: The Neoclassical Growth Model 612

A.3.9 Transversality Conditions in Infinite-Horizon Problems 613

A.3.10 Summary of the Procedure to Find the First-Order Conditions 615

A.3.11 Present-Value and Current-Value Hamiltonians 616

A.3.12 Multiple Variables 617

A.4 Useful Results in Matrix Algebra: Eigenvalues, Eigenvectors,

and Diagonalization of Matrices 618

A.5 Useful Results in Calculus 620

A.5.1 Implicit-Function Theorem 620

A.5.2 Taylor’s Theorem 621

A.5.3 L’Hˆopital’s Rule 622

A.5.4 Integration by Parts 623

A.5.5 Fundamental Theorem of Calculus 624

A.5.6 Rules of Differentiation of Integrals 624

References 627

Index 641