Recursive Methods in Economic Dynamics

PART I The Recursive Approach......Page 16
1.Introduction......Page 19
2.An Overview......Page 24
2.1 A Deterministic Model of Optimal Growth......Page 25
2.2 A Stochastic Model of Optimal Growth......Page 32
2.3 Competitive Equilibrium Growth......Page 38
2.4 Conclusions and Plans......Page 48
2.5 Bibliographic Notes......Page 50
PART II Deterministic Models......Page 53
3 Mathematical Preliminaries......Page 54
3.1 Metric Spaces and Normal Vector Spaces......Page 59
3.2 The Contraction Mapping Theorem......Page 65
3.3 The Theorem of the Maximum......Page 71
3.4 Bibliographic Notes......Page 81
4 Dynamic Programming under Certainty......Page 82
4.1 The Principle of Optimality......Page 83
4.2 Bounder Returns......Page 93
4.3 Constant Returns to Scale......Page 103
4.4 Unbounded Returns......Page 108
4.5 Euler Equations......Page 113
4.6 Bibliographic Notes......Page 116
5.1 The One-Sector Model of Option Growth......Page 119
5.4 Growth with Technical Progress......Page 121
5.6 Learning by Doing......Page 123
5.7 Human Capital Accumulation......Page 125
5.8 Growth with Human Capital......Page 127
5.9 Investment with Convex Costs......Page 128
5.10 Investment with Constant Returns......Page 129
5.11 Recursive Preferences......Page 130
5.12 Theory of the Consumer with Recursive Preferences......Page 132
5.13 A Pareto Problem with Recursive Preference......Page 133
5.14 An (s, S) Inventory Problem......Page 134
5.15 The Inventory Problem in Continuous Time......Page 138
5.16 A seller with Unkonw Demand......Page 139
5.17 A Consumption-Savings Problem......Page 141
5.18 Bibliographic Notes......Page 144
6 Deterministic Dynamics......Page 147
6.1 One-Dimensional Example......Page 149
6.2 Global Stability: Liapounov Functions......Page 155
6.3 Linear Systems and Linear Approximations......Page 159
6.4 Euler Equations......Page 164
6.5 Applications......Page 173
6.6 Bibliographic Notes......Page 177
PART III Stochastic Models......Page 179
7 Measure Theory and Integration......Page 181
7.1 Measurable Spaces......Page 184
7.2 Measures......Page 186
7.3 Measurable Functions......Page 193
7.4 Integration......Page 200
7.5 Product Spaces......Page 211
7.6 The Monotones Class Lemma......Page 215
7.7 Conditional Expectation......Page 217
7.8 Bibliographic Notes......Page 224
8 Markov Processes......Page 225
8.1 Transition Functions......Page 228
8.2 Probability Measures on Spaces of Sequences......Page 236
8.3 Iterated Integrals......Page 241
8.4 Transitions Defined by Stochastic Difference Equations......Page 250
8.5 bibliographic Notes......Page 253
9 Stochastic Dynamic Programming......Page 254
9.1 The Principle of Optimality......Page 257
9.2 Bounded Returns......Page 275
9.3 Constant Returns to Scale......Page 286
9.4 Unbounded Returns......Page 288
9.5 Stochastic Euler Equations......Page 296
9.6 Policy Functions and Transition Functions......Page 299
9.7 Bibliographic Notes......Page 303
10.1 The One-sector Model of Optimal Growth......Page 304
10.3 Optimal Growth with Many Goods......Page 306
10.4 Industry Investment under Uncertainty......Page 308
10.5 Production and Inventory Accumulation......Page 313
10.6 Asset Prices in an Exchange Economy......Page 316
10.7 A Model of Search Unemployment......Page 320
10.8 The Dynamics of the Search Model......Page 323
10.9 Variations on the Search Model......Page 325
10.10 A Model of Job Matching......Page 327
10.11 Job Matching and Unemployment......Page 330
10.12 Bibliographic Notes......Page 331
11 Strong Convergence of Markov Processes......Page 332
11.1 Markov Chains......Page 335
11.2 Convergence Concepts for Measures......Page 350
11.3 Characterizations of Strong Convergence......Page 354
11.4 Sufficient Conditions......Page 360
11.5 Bibliographic Notes......Page 367
12 Weak Convergence of Markov Processes......Page 368
12.1 Characterizations of Weak Convergence......Page 369
12.2 Distribution Functions......Page 380
12.3 Weak Convergence of Distribution Functions......Page 385
12.4 Monotone Markov Processes......Page 391
12.5 Dependence of the Invariant Measure on a Parameter......Page 399
12.6 A Loose End......Page 401
13 Applications of Convergence Results for Markov processes......Page 404
13.1 A Discrete-Space (s, S) Inventor Problem......Page 405
13.2 A Continuous-State (s, S) Process......Page 406
13.3 The One_Sector Model of Optimal Growth......Page 407
13.4 Industry Investment under Uncertainty......Page 411
13.5 Equilibrium in a Pure Cerrency Economy......Page 412
13.6 A Pure Currency Economy with Linear Utility......Page 416
13.7 A Pure Credit Economy with Linear Utility......Page 418
13.8 An Equilibrium Search Economy......Page 419
13.9 Bibligraphic Notes......Page 428
14 Laws of Large Numbers......Page 429
14.1 Definitions and Preliminaries......Page 432
14.2 A Strong Law for Markov Processes......Page 440
14.3 Bibliographic Notes......Page 453
PART IV Competitive Equilibrium......Page 455
15 Pareto Optima and Competitive Equilibria......Page 457
15.1 Dual Spaces......Page 460
15.2 The First and Second Welfare Theorems......Page 467
15.3 Issues in the Choice of a Commodity Space......Page 473
15.4 Inner Product Representations of Prices......Page 478
15.5 Bibligraphic Notes......Page 489
16 Applications of Equilibrium Theory......Page 490
16.1 A One-Sector Model of Growth under Certainty......Page 491
16.2 A Many-Sector Model of Stochastic Growth......Page 497
16.3 An Economy with Sustained Growth......Page 501
16.4 Industry Investment under Uncertainty......Page 503
16.5 Truncation: A Generalization......Page 507
16.6 A Peculiar Example......Page 509
16.7 An Economy with Many Consumers......Page 511
16.8 Bibligraphic Notes......Page 515
17 Fixed-Point Arguments......Page 516
17.1 An Overlapping-Generations Model......Page 518
17.2 An Application of the Contraction Mapping Theorem......Page 524
17.3 The Brouwer Fixed-Point Theorem......Page 532
17.4 The Schauder Fixed-Point Theorem......Page 535
17.5 Fixed Points of Monotone Operators......Page 541
17.6 Partially Observed Shocks......Page 546
17.7 Biliographic Notes......Page 557
18 Equilibria in System with Distortions......Page 558
18.1 An Indirect Approach......Page 559
18.2 A Local Approach Based on First-Order Conditions......Page 563
18.3 A Global Approach Based on First-Order Conditions......Page 570
References Index of Theorems General Index......Page 576
References......Page 578
Index of theorems......Page 590
General Index......Page 594