Optimal Control Theory and Static Optimization in EconomicsOptimal control theory is a technique being used increasingly by academic economists to study problems involving optimal decisions in a multi-period framework. This textbook is designed to make the difficult subject of optimal control theory easily accessible to economists while at the same time maintaining rigour. Economic intuitions are emphasized, and examples and problem sets covering a wide range of applications in economics are provided to assist in the learning process. Theorems are clearly stated and their proofs are carefully explained. The development of the text is gradual and fully integrated, beginning with simple formulations and progressing to advanced topics such as control parameters, jumps in state variables, and bounded state space. For greater economy and elegance, optimal control theory is introduced directly, without recourse to the calculus of variations. The connection with the latter and with dynamic programming is explained in a separate chapter. A second purpose of the book is to draw the parallel between optimal control theory and static optimization. Chapter 1 provides an extensive treatment of constrained and unconstrained maximization, with emphasis on economic insight and applications. Starting from basic concepts, it derives and explains important results, including the envelope theorem and the method of comparative statics. This chapter may be used for a course in static optimization. The book is largely self-contained. No previous knowledge of differential equations is required. |
Contents
Static optimization | 1 |
the method of Lagrange | 20 |
13 Comparative statics | 43 |
nonlinear programming | 52 |
15 Economic applications of nonlinear programming | 67 |
16 The special case of linear programming | 70 |
Appendix | 74 |
Exercises | 79 |
64 The maximum principle with inequality constraints | 198 |
the case with inequality and equality constraints | 210 |
66 Concluding notes | 218 |
Endpoint constraints and transversality conditions | 221 |
71 Freeend point problems | 222 |
72 Problems with free endpoint and a scrap value function | 226 |
73 Lower bound constraints on endpoint | 229 |
74 Problems with lower bound constraints on endpoint and a scrap value function | 235 |
Ordinary differential equations | 87 |
22 Definitions and fundamental results | 88 |
23 Firstorder differential equations | 91 |
24 Systems of linear FODE with constant coefficients | 95 |
25 Systems of two nonlinear FODE | 100 |
Appendix | 111 |
Exercises | 113 |
Introduction to dynamic optimization | 117 |
31 Optimal borrowing | 118 |
32 Fiscal policy | 119 |
33 Suboptimal consumption path | 120 |
34 Discounting and depreciation in continuoustime models | 121 |
Exercises | 124 |
The maximum principle | 127 |
42 Derivation of the maximum principle in discrete time | 129 |
43 Numerical solution of an optimal control problem in continuous time | 133 |
44 Phase diagram analysis of optimal control problems | 137 |
45 Economic interpretation of the maximum principle | 151 |
46 Necessity and sufficiency of the maximum principle | 161 |
Exercises | 165 |
The calculus of variations and dynamic programming | 169 |
discretetime finitehorizon problems | 173 |
53 Dynamic programming in continuous time | 182 |
Exercises | 184 |
The general constrained control problem | 187 |
62 Integral constraints | 190 |
63 The maximum principle with equality constraints only | 192 |
75 Freeterminaltime problems without a scrap value function | 240 |
76 Freeterminaltime problems with a scrap value function | 244 |
77 Other transversality conditions | 247 |
78 A general formula for transversality conditions | 248 |
79 Sufficiency theorems | 251 |
710 A summary table of common transversality conditions | 253 |
Exercises | 259 |
Discontinuities in the optimal controls | 263 |
82 The beekeepers problem | 267 |
83 Onesector optimal growth with reserves | 274 |
84 Highest consumption path | 277 |
85 Concluding comments | 281 |
Exercises | 282 |
Infinitehorizon problems | 285 |
92 Necessary conditions | 287 |
93 Sufficient conditions | 288 |
94 Autonomous problems | 289 |
95 Steady states in autonomous infinitehorizon problems | 294 |
96 Further properties of autonomous infinitehorizon problems | 298 |
Exercises | 304 |
Three special topics | 307 |
jumps in the state variables | 310 |
103 Constraints on the state variables | 332 |
Exercises | 342 |
| 345 | |
| 351 | |
Other editions - View all
Optimal Control Theory and Static Optimization in Economics Daniel Léonard,Ngo van Long Limited preview - 1992 |
Optimal Control Theory and Static Optimization in Economics Daniel Léonard,Ngo van Long No preview available - 1992 |
Optimal Control Theory and Static Optimization in Economics Daniel Léonard,Ngo van Long No preview available - 1992 |
Common terms and phrases
Apply the maximum assume assumption autonomous boundary conditions calculus of variations capital stock Chapter concave function Consider control problem control variables convex convex function convex set costate variables current-value curve defined denote derivatives differential equations discount downward jump dt subject dynamic programming economic envelope theorem equilibrium point example exercise exogenously Figure find c(t first-order condition fixed global maximum graph Hamiltonian hence Hessian matrix horizon implies increasing initial input integral Lagrangean linear locus marginal matrix maximize maximum principle multipliers necessary conditions negative nonlinear programming nonnegative notation objective function obtain optimal control theory optimal path optimal policy optimal solution output parameter phase diagram production function profit programming region resource saddle point satisfy scrap value function second-order Section shadow price solve strictly concave Substituting Suppose t₁ terminal Theorem tion transversality condition upward jump vector x₁ yields zero ән