Dynamic programming in economics /

Preface p. 1
1. Some Growth Models p. 3
1.1 A General Growth Model p. 4
1.2 The Harrod Model p. 6
1.3 The Domar Model p. 7
1.4 The Solow-Swan Model p. 8
1.5 The Frankel Model p. 10
1.6 Some Conclusions p. 11
2. The Ramsey Model p. 13
2.1 The Model p. 13
2.1.1 The Assumptions p. 15
2.1.2 Feasible Paths p. 16
2.2 Existence of Optimal Paths p. 16
2.3 Properties of the Optimal Paths p. 19
2.4 Value Function - Bellman Equation - Optimal Policy p. 21
2.4.1 Some Properties of the Value Function p. 22
2.4.2 Bellman Equation p. 25
2.4.3 Optimal Policy p. 29
2.4.4 Dynamic Properties of the Optimal Path p. 31
2.4.5 Mangasarian Lemma p. 32
2.4.6 On the Continuity of the Value Function and of the Optimal Policy with respect to the Discount Factor and the initial Capital Stock p. 34
3. An Aggregated Optimal Growth Model with a Convex-Concave Production Function p. 37
3.1 The Model p. 38
3.1.1 The Assumptions p. 39
3.1.2 Feasible Paths p. 39
3.2 Existence of Optimal Paths p. 40
3.3 Properties of the Optimal Paths p. 40
3.4 Value Function - Bellman Equation - Optimal Policy p. 41
3.4.1 Some Properties of the Value Function p. 41
3.4.2 Bellman Equation p. 42
3.4.3 Optimal Correspondence p. 43
3.4.4 On the Differentiability of the Value Function p. 45
3.4.5 Dynamic Properties of the Optimal Paths p. 49
3.5 On the Mangasarian Lemma p. 60
4. Multisector Optimal Growth Models With Bounded From Below Returns p. 63
4.1 The General Case p. 66
4.1.1 Existence of an optimal solution p. 67
4.1.2 Value Function and Optimal Correspondence p. 69
4.1.3 Properties of Optimal Paths p. 77
4.1.4 On the Continuity of the Value Function and of the Optimal Correspondence with respect to ([beta], x[subscript 0]) p. 78
4.2 The Case with Concave Return Function and Convex Technology p. 80
4.2.1 The One Dimension Case p. 84
4.3 Examples p. 86
4.3.1 Example 1 (The Ramsey Model) p. 86
4.3.2 Example 2 p. 87
4.3.3 Example 3: A Consumption-Savings Problem p. 89
4.3.4 Example 4: A Two-Sector Model p. 92
4.3.5 Example 5: A Human Capital Model p. 94
4.3.6 Example 5: Learning by Doing Model p. 96
5. Optimal Growth Models With Unbounded From Below Returns p. 101
5.1 The Model p. 102
5.2 Existence of Optimal Solutions p. 103
5.3 Value Function p. 105
5.4 Optimal Policy - Properties of Optimal Paths p. 113
5.5 Examples p. 115
5.5.1 Example 1 p. 115
5.5.2 Example 2: The AK model p. 118
6. Optimal Growth and Competitive Equilibrium p. 121
6.1 The Model p. 121
7. Optimal Growth Models Without Discounting p. 131
7.1 The Model p. 132
7.2 Good programmes p. 132
7.3 Optimal Programmes p. 137
7.4 Value Function - Bellman Equation p. 141
8. Turnpike in Optimal Growth Models with Convex Technology p. 149
8.1 The Visit Lemmas p. 150
8.2 The Neighborhood Turnpike p. 155
8.3 Turnpike Theorems p. 159
8.4 Remarks p. 162
9. Cycles and Chaos in Optimal Growth Models p. 165
9.1 Two-Sector Models p. 165
9.1.1 The model p. 165
9.1.2 Examples p. 167
9.1.3 Optimal interior stationary state p. 168
9.1.4 Difficulties p. 170
9.1.5 Existence of periodic orbits p. 171
9.2 Optimal chaos p. 173
9.2.1 Two definitions of complicated dynamics p. 173
9.2.2 "Rationalizability" of a map p. 175
9.3 Appendix p. 177
A. Appendix p. 181
A.1 Metric Spaces and Normed Spaces p. 181
A.1.1 Distances and Metric Spaces p. 181
A.1.2 Topology on Metric Spaces p. 182
A.1.3 Compact Spaces, Compact Sets p. 185
A.1.4 Normed Spaces p. 185
A.1.5 Product Topology p. 186
A.2 Concave Functions p. 187
A.2.1 Subdifferentiability, Differentiability of Concave Functions p. 188
A.3 Negative Matrix p. 189
A.4 The Implicit Functions Theorem p. 190
A.5 Correspondences, The Maximum Theorem p. 190
A.5.1 Examples p. 191
Bibliography p. 193
Index p. 199