Optimal Control Theory and Static Optimization in Economics

by Daniel Léonard , Ngo van Long

• ISBN13: 9780521337465
• ISBN10: 0521337461
• Format: Paperback
• Copyright: 1992-01-31
• Publisher: Cambridge University Press

Summary

Optimal control theory is a technique being used increasingly by academic economists to study problems involving optimal decisions in a multi-period framework. This book is designed to make the difficult subject of optimal control theory easily accessible to economists while at the same time maintaining rigor. Economic intuition is emphasized, examples and problem sets covering a wide range of applications in economics are provided, theorems are clearly stated and their proofs are carefully explained. The development of the text is gradual and fully integrated, beginning with the simple formulations and progressing to advanced topics. Optimal control theory is introduced directly, without recourse to the calculus of variations, and the connection with the latter and with dynamic programming is explained in a separate chapter. Also, the book draws the parallel between optimal control theory and static optimization. No previous knowledge of differential equations is required.

Table of Contents

Preface	ix
Static optimization	1	(73)
Unconstrained optimization, concave and convex functions	1	(19)
Optimization under equality constraints: the method of Lagrange	20	(23)
Comparative statics	43	(9)
Optimization under inequality constraints: nonlinear programming	52	(15)
Economic applications of nonlinear programming	67	(3)
The special case of linear programming	70	(4)
Appendix	74	(5)
Exercises	79	(266)
Ordinary differential equations	87	(30)
Introduction	87	(1)
Definitions and fundamental results	88	(3)
First-order differential equations	91	(4)
Systems of linear FODE with constant coefficients	95	(5)
Systems of two nonlinear FODE	100	(17)
Appendix	111	(2)
Exercises	113	(4)
Introduction to dynamic optimization	117	(10)
Optimal borrowing	118	(1)
Fiscal policy	119	(1)
Suboptimal consumption path	120	(1)
Discounting and depreciation in continuous-time models	121	(6)
Exercises	124	(3)
The maximum principle	127	(42)
A simple control problem	127	(2)
Derivation of the maximum principle in discrete time	129	(4)
Numerical solution of an optimal control problem in continuous time	133	(4)
Phase diagram analysis of optimal control problems	137	(14)
Economic interpretation of the maximum principle	151	(10)
Necessity and sufficiency of the maximum principle	161	(8)
Exercises	165	(4)
The calculus of variations and dynamic programming	169	(18)
The calculus of variations	169	(4)
Dynamic programming: Discrete-time, finite-horizon problems	173	(9)
Dynamic programming in continuous time	182	(5)
Exercises	184	(3)
The general constrained control problem	187	(34)
The set of admissible controls	187	(3)
Integral constraints	190	(2)
The maximum principle with equality constraints only	192	(6)
The maximum principle with inequality constraints	198	(12)
Necessity and sufficiency theorems: The case with inequality and equality constraints	210	(8)
Concluding notes	218	(3)
Exercises	218	(3)
Endpoint constraints and transversality conditions	221	(42)
Free-endpoint problems	222	(4)
Problems with free endpoint and a scrap value function	226	(3)
Lower bound constraints on endpoint	229	(6)
Problems with lower bound constraints on endpoint and a scrap value function	235	(5)
Free-terminal-time problems without a scrap value function	240	(7)
Other transversality conditions	247	(1)
A general Formula for transversality conditions	248	(3)
Sufficiency theorems	251	(2)
A summary table of common transversality conditions	253	(1)
Control parameters	253	(10)
Exercises	259	(4)
Discontinuities in the optimal controls	263	(22)
A classical bang-bang example	263	(4)
The beekeeper's problem	267	(7)
One-sector optimal growth with reserves	274	(3)
Highest consumption path	277	(4)
Concluding comments	281	(4)
Exercises	282	(3)
Infinite-horizon problems	285	(22)
Optimality criteria	285	(2)
Necessary conditions	287	(1)
Sufficient conditions	288	(1)
Autonomous problems	289	(5)
Steady states in autonomous infinite-horizon problems	294	(4)
Further properties of autonomous infinite-horizon problems	298	(9)
Exercises	304	(3)
Three special topics	307	(38)
Problems with two-state variables	307	(3)
Trade in capital goods: jumps in the state variables	310	(22)
Constraints on the state variables	332	(13)
Exercises	342	(3)
Bibliography	345	(6)
Index	351