本书对OLG模型进行了深入研究。Praise for A Theory of Economic Growth
“The overlapping generations model and the infinitely lived or dynastic model are
the two workhorses of modern macroeconomics. De La Croix and Michel have
written a wonderfully accessible graduate textbook on the overlapping generations
model. They carefully take students through essentially every variant of the model,
prove a large number of known results, and offer a few new ones as well. This book
is an essential addition as a teaching tool and an invaluable reference on every
economist’s shelf.”
– V. V. Chari, University of Minnesota
“In recent decades overlapping generation models have become a central
framework of analysis in the research of economic growth. The authors present a
comprehensive and lucid exposition of the dynamic structure of the basic
overlapping generation features with production. Highly recommended for
researchers and graduate students in the fields of growth theory and dynamic
macroeconomics.”
– Oded Galor, Brown University
“Some of the most hotly debated government policies are those that involve
redistribution across generations, such as social security and public education. De la
Croix and Michel provide a manual of economic tools for evaluating these sorts of
policies based on simple overlapping generations models. Their presentation of
these models strives to combine applicability for policy analysis with a solid
foundation in dynamic general equilibrium theory. Their book should be of use to
economists from the level of advanced undergraduate students to researchers and
teachers.”
– Timothy J. Kehoe, University of Minnesota and Federal Reserve
Bank of Minneapolis
1 Competitive Equilibria 1
1.1 The Model 1
1.1.1 Two-period-lived Individuals 2
1.1.2 Neo-classical Technology 3
1.1.3 Firms 4
1.2 Main Assumptions 4
1.2.1 The Assumptions on the Utility Function 4
1.2.2 The Assumptions on the Production Function 6
1.3 The Behavior of the Agents at Period t 10
1.3.1 The Young Individuals 10
1.3.2 The Inter-temporal Elasticity of Substitution 11
1.3.3 The Properties of the Savings Function 12
1.3.4 The Old Individuals 15
1.3.5 The Firms 15
1.4 The Temporary Equilibrium 16
1.5 The Inter-temporal Equilibriumwith Perfect Foresight 19
1.5.1 Existence of Equilibria 20
1.5.2 Uniqueness of the Inter-temporal Equilibrium 22
1.6 Capital Dynamics at a Rational Inter-temporal Equilibrium 27
1.6.1 Steady States and Stability 27
1.6.2 Dynamics 29
1.6.3 The Behavior Near 0 34
1.6.4 AQuick Look at the Empirics of Growth 37

1.7 Comparison of Myopic and Perfect Foresight 39
1.7.1 The Steady States 41
1.7.2 Local Stability 42
1.7.3 Uniqueness of the Steady State 43
1.8 Applications and Extensions 45
1.8.1 Myopic and Perfect Foresight in an Example 45
1.8.2 ADemographic Shock 50
1.8.3 Non-separable Utility Function 51
1.8.4 Homothetic Preferences 53
1.8.5 Heterogeneous Agents 54
1.8.6 Technical Progress 56
1.8.7 Imperfect Credit Market 57
1.8.8 Three-period-lived Households 64
1.8.9 Borrowing Constraints in the Three-period Model 66
1.9 Conclusion 70
2 Optimality 72
2.1 Optimality of Stationary Paths 73
2.1.1 Feasible Long-run Capital Stock 74
2.1.2 The Optimal Stationary Path: The Golden Age 77
2.1.3 Under- and Over-accumulation of Capital 80
2.2 Optimality of the Dynamics 82
2.2.1 Dynamic Efficiency 82
2.2.2 Pareto Optimality of Dynamics 86
2.3 The Planning Problem 90
2.3.1 The Objective Function 91
2.3.2 Properties of the Value Function 92
2.3.3 Existence and Monotonicity of Optimal Paths 95
2.3.4 Limit of the Optimal Path and Optimal
Steady State 99
2.4 Marginal Analysis of Optimal Solutions 101
2.4.1 The Optimality Conditions 102
2.4.2 The Planner’s Stationary Solution 106
2.4.3 Local Dynamics 106
2.4.4 AGraphical Exposition 108
2.5 Unbounded Optimal Growth 112
2.5.1 Existence of Optimal Paths When σ > 1 112
2.5.2 Existence of Optimal Paths When σ < 1 (and γ ≥ 1) 114
2.5.3 Existence of Optimal Paths When σ = 1 115
2.5.4 General Result 116
2.5.5 The Long-run Growth Rate 116
2.6 Applications and Extensions 117
2.6.1 Properties of the Policy Functions When f (0) > 0 118
2.6.2 Application: The Optimal Speed of Convergence 120
2.6.3 Application: Rise in β 121
2.6.4 AMixed CES–Linear Production Function 122
2.6.5 Optimal Growth in the Ak Model 124
2.7 Conclusion 127
3 Policy 129
3.1 Lump-sumTransfers and the Second Welfare Theorem 129
3.1.1 Equilibriumwith Lump-sumTransfers 129
3.1.2 The Second Welfare Theorem 136
3.1.3 The Direction of Optimal Transfers in the Long Run 138
3.1.4 Reversal of Optimal Transfers Over Time:
An Example 139
3.2 Pensions 140
3.2.1 Fully Funded System 141
3.2.2 Pay-as-you-go System: Existence of Equilibrium 143
3.2.3 Pay-as-you-go Systems with Constant Pensions 144
3.2.4 Capital Accumulation and Pay-as-you-go Pensions 150
3.2.5 Further Comments 152
3.3 Public Spending 155
3.3.1 Public Spending in the Competitive Economy 155
3.3.2 Public Spending: Optimal Financing 158
3.3.3 Second-best Policies 159
3.4 Study of the Second-best Problem 161
3.4.1 Restating the Problem 161
3.4.2 Three Issues 162
3.4.3 AStandard Approach to the Problem 165
3.4.4 An Auxiliary Problem 167
3.5 Applications and Extensions 171
3.5.1 Optimal Growth Rate of Population 172
3.5.2 Application: The Tax on the First Old Generation 173
3.5.3 Application: Financing Future Spending 174
3.5.4 Proportional Government Spending 175
3.6 Conclusion 178
4 Debt 179
4.1 Diamond’s Model with Debt 181
4.1.1 The Model 181
4.1.2 The Temporary Equilibrium 182
4.1.3 The Inter-temporal Equilibrium with Perfect
Foresight 183
4.2 The Inter-temporal Budget Constraint of the Government 184
4.2.1 Debt with the Two Types of Lump-sumTaxes 186
4.2.2 Debt with a Restriction of Only One Type of
Lump-sumTax 190
4.2.3 Ponzi Games 192
4.3 Constant Deficit Policies 193
4.3.1 Balanced Budget Policies: Local Analysis 195
4.3.2 Balanced Budget Policies: Graphical Illustration 198
4.3.3 Non-zero Deficit: Local Analysis 203
4.3.4 Non-zero Deficit: Graphical Illustration 208
4.3.5 Ponzi Debt, Money, and Bubbles 211
4.4 Constant Debt Policies 216
4.4.1 Sustainability in the Short Run 216
4.4.2 Sustainability in the Long Run 219
4.4.3 Characteristics of Inter-temporal Equilibria 223
4.4.4 Policy Implications 226
4.5 Applications and Extensions 230
4.5.1 Constant Debt–Output Ratio 230
4.5.2 Deficits and Cycles 233
4.6 Conclusion 236
5 Further Issues 238
5.1 Dynastic Altruism:ABequest Motive 239
5.1.1 Modeling Voluntary Bequests 239
5.1.2 Marginal Analysis of Bequests 246
5.1.3 Altruismand the Neutrality of Economic Policy 248
5.1.4 When are Bequests Positive? 252
5.2 Human Capital and Education 256
5.2.1 Modeling Education 257
5.2.2 Parental Funding: Private vs Public Education 259
5.2.3 Market Funding 269
5.2.4 The Tradeoff between Studying and Working 274
5.3 Inter-generational Externalities 280
5.3.1 Inter-generational Taste Externalities in the
Competitive Economy 281
5.3.2 The Optimal Allocation 286
5.3.3 Extensions 289
5.3.4 Conclusion 290
5.4 Macro-economics and General Equilibrium 291
5.4.1 Modeling Arrow–Debreu Market Equilibria 292
5.4.2 Arrow–Debreu Market Equilibria from?∞to +∞ 294
5.4.3 Sequence Equilibriumfrom?∞to +∞ 295
5.4.4 Arrow–Debreu Equilibria from0 to +∞ 297
5.4.5 Example 300
5.4.6 Conclusion 304
Technical Appendices 305
A.1 Production Functions 305
A.1.1 Homogeneity