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
1GrowthModelswithExogenousSavingRates
(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 Inefficiency34
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
2GrowthModelswithConsumerOptimization
(the Ramsey Model) 85
2.1 Households 86
2.1.1 Setup of the Model86
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 Convergence111
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 γˆκ Declines Monotonically
If the Economy Starts from ˆ
k(0)<ˆ
k∗ 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 Investment152
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 Economy167
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 Dynamics208
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 Technology212
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 Model220
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 Example230
4.6 Concluding Observations 232
4.7 Appendix: Endogenous Growth in the One-Sector Model 232
4.8 Problems 235
5Two-SectorModelsofEndogenousGrowth(withSpecial
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
6TechnologicalChange:ModelswithanExpandingVariety
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 Firms289
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
7TechnologicalChange:SchumpeterianModels
of Quality Ladders317
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 Again331
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
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&D450
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 Convergence479
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
AppendixonMathematicalMethods 567
A.1 Differential Equations568
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 SummaryoftheProceduretoFindtheFirst-OrderConditions 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 Theorem621
A.5.3 L’Hoˆpital’s Rule622
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
