I Economic Growth 5 Second Edition 1 Robert J. Barro Xavier Sala-i-Martin The MIT Press Cambridge, Massachusetts London, England
Preface About the Authors xv xvii Introduction 1 1.1 The Importance of Growth 1 1.2 The World Income Distribution 6 1.3 Empirical Regularities about Economic Growth 12 1.4 A Brief History of Modern Growth Theory 16 1.5 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 App end i x: Proofs of Various Propositions 1.5.1 Proof Thnf E,«i T_ i T»-,, x T, ^ for Production i a Neoclassical Production Function 1-5.2 Properties of the Convergence Coefficient in the Solow-Swan Model 1.53 Proof That Technological Progress Must Be Labor Augmenting 15.4 Properties of the CES Production Function 1.6 Problems 2 Growth Models with Consumer Optimization (the Ramsey Model) 2.1 Households 2.1.1 Setup of the Model 2.1.2 First-Order Conditions 2.2 Firms 2.3 Equilibrium 2.4 Alternative Environments 2.5 The Steady State 2.6 Transitional Dynamics 2.6.1 The Phase Diagram 2.6.2 The Importance of the Transversality Condition 2.6.3 The Shape of the Stable Arm 2.6.4 Behavior of the Saving Rate 2.6.5 The Paths of the Capital Stock and Output 2.6.6 Speeds of Convergence 2.6.7 Household Heterogeneity 2.7 Nonconstant Time-Preference Rates 2.7.1 Results under Commitment 2.7.2 Results without Commitment under Log Utility 2.7.3 Population Growth and Technological Progress 2.7.4 Results under Isoelastic Utility 2.7.5 The Degree of Commitment 2.8 Appendix 2A: Log-Linearization of the Ramsey Model 2.9 Appendix 2B: Irreversible Investment 77 77 78 78 80 81 85 86 86 89 94 97 98 99 102 102 104 105 106 110 111 118 121 123 124 129 130 132 132 134
2.10 Appendix 2C: Behavior of the Saving Rate 2.11 Appendix 2D: Proof That yg Declines Monotonically If the Economy Starts from (0) < k* 2.12 Problems 3 Extensions of the Ramsey Growth Model 3.1 Government 3.1.1 Modifications of the Ramsey Framework 3.1.2 Effects of Tax Rates 3.1.3 Effects of Government Purchases 3.2 Adjustment Costs for Investment 3.2.1 The Behavior of Firms 3.2.2 Equilibrium with a Given Interest Rate 3.2.3 Equilibrium for a Closed Economy with a Fixed Saving Rate 3.3 An Open-Economy Ramsey Model 3.3.1 Setup of the Model 3.3.2 Behavior of a Small Economy's Capital Stock and Output 3.3.3 Behavior of a Small Economy's Consumption and Assets 3.3.4 The World Equilibrium 3.4 The World Economy with a Constraint on International Credit 3.4.1 Setup of a Model with Physical and Human Capital 3.4.2 The Closed Economy 3.4.3 The Open Economy 3.5 Variations in Preference Parameters 3.6 Economic Growth in a Model with Finite Horizons 3.6.1 Choices in a Model with Finite Horizons 3.6.2 The Finite-Horizon Model of a Closed Economy 3.6.3 The Finite-Horizon Model of an Open Economy 3.7 Some Conclusions 3.8 Appendix: Overlapping-Generations Models 3.8.1 Households 3.8.2 Firms 3.8.3 Equilibrium 3.9 Problems 135 137 139 143 143 143 146 147 152 152 155 159 161 161 162 163 164 165 166 167 168 177 179 179 183 186 189 190 190 1-92 193 200
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
5.4 Summary Observations 5.5 Appendix 5A: Transitional Dynamics with Inequality Restrictions on Gross Investment in the One-Sector Model 5.6 Appendix 5B: Solution of the Uzawa-Lucas Model 5.7 Appendix 5C: The Model with Reversed Factor Intensities 5.8 Problems 6 Technological Change: Models with an Expanding Variety of Products 6.1 A Baseline Model with a Variety of Products 6.1.1 The Producers of Final Output 6.1.2 Research Firms 6.1.3 Households 6.1.4 General Equilibrium 6.1.5 Determinants of the Growth Rate 6.1.6 Pareto Optimality 6.1.7 Scale Effects and the Cost of R&D 6.1.8 A Rising Cost of R&D 6.2 Erosion of Monopoly Power, Competition 6.3 Romer's Model of Technological Change 6.4 Concluding Observations 6.5 Problems 7 Technological Change: Schumpeterian Models of Quality Ladders 7.1 Sketch of the Model 7.2 The Model 7.2.1 The Producers of Final Output: Levels of Quality in the Production Technology 7.2.2 The Research Sector 7.2.3 Consumers 7.2.4 Behavior of the Aggregate Quality Index and Endogenous Growth 7.2.5 Scale Effects Again 7.3 Innovation by the Leader 7.3.1 Interactions Between the Leader and the Outsiders 7.3.2 The Leader as a Monopoly Researcher 271 271 274 280 282 285 285 285 289 295 295 297 297 300 303 305 310 313 313 317 317 319 319 321 328 329 331 333 333 336
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
9.4 Appendix: The Form of the Utility Function with Consumption and Work Effort 9.5 Problems 10 Growth Accounting 10.1 Standard Primal Growth Accounting 10.1.1 Basic Setup 10.1.2 Measuring Inputs 10.1.3 Results from Growth Accounting 10.1.4 A Note on Regression-Based Estimates of TFP Growth 10.2 Dual Approach to Growth Accounting 10.3 Problems with Growth Accounting 10.3.1 An Increasing-Returns Model with Spillovers 10.3.2 Taxes 10.3.3 Multiple Types of Factors 10.4 TFP Growth and R&D 10.4.1 Varieties Models 10.4.2 Quality-Ladders Models 10.5 Growth Accounting Versus Sources of Growth 11 Empirical Analysis of Regional Data Sets 11.1 Two Concepts of Convergence 11.2 Convergence Across the U.S. States 11.2.1 fi Convergence 11.2.2 Measurement Error 11.2.3 a Convergence 11.3 Convergence Across Japanese Prefectures 11.3.1 ft Convergence 11.3.2 a Convergence Across Prefectures 11.4 Convergence Across European Regions 11.4.1 p Convergence 11.4.2 a Convergence 11.5 Convergence Across Other Regions Around the World 11.6 Migration Across the U.S. States 11.7 Migration Across Japanese Prefectures 11.8 Migration Across European Regions 427 428 433 433 433 436 438 441 442 444 445 447 449 450 451 454 457 461 462 466 466 472 473 474 474 478 479 479 482 482 483 486 490
11.9 Migration and Convergence 492 11.10 j8 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.I 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
xm A.3.2 The Typical Problem A.3.3 Heuristic Derivation of the First-Order Conditions A.3.4 Transversality Conditions A.3.5 The Behavior of the Hamiltonian over Time A.3.6 Sufficient Conditions A.3.7 Infinite Horizons A.3.8 Example: The Neoclassical Growth Model A.3.9 Transversality Conditions in Infinite-Horizon Problems A.3.10 Summary of the Procedure to Find the First-Order Conditions A.3.11 Present-Value and Current-Value Hamiltonians A.3.12 Multiple Variables A.4 Useful Results in Matrix Algebra: Eigenvalues, Eigenvectors, and Diagonalization of Matrices A.5 Useful Results in Calculus A.5.1 Implicit-Function Theorem A.5.2 Taylor's Theorem A.5.3 L'Hopital's Rule A.5.4 Integration by Parts A.5.5 Fundamental Theorem of Calculus A.5.6 Rules of Differentiation of Integrals References Index 605 606 609 609 610 610 612 613 615 616 617 618 620 620 621 622 623 624 624 627 641