Dynamic optimization: the calculus of variations and optimal control in economics and management

Description

The long awaited second edition of Dynamic Optimization is now available. Clear exposition and numerous worked examples made the first edition the premier text on this subject. Now, the new edition is expanded and updated to include essential coverage of current developments on differential games, especially as they apply to important economic questions; new developments in comparative dynamics; and new material on optimal control with integral state equations. The second edition of Dynamic Optimization provides expert coverage on:- methods of calculus of variations - optimal control - continuous dynamic programming - stochastic optimal control -differential games. The authors also include appendices on static optimization and on differential games. Now in its new updated and expanded edition, Dynamic Optimization is, more than ever, the optimum choice for graduate and advanced undergraduate courses in economics, mathematical methods in economics and dynamic optimization, management science, mathematics and engineering. Preface to the Fourth Printing........xi Preface to the Second Edition........xiii Preface to the First Edition........xv PART I. CALCULUS OF VARIATIONS Section 1. Introduction........3 Section 2. Example Solved........12 Section 3. Simplest Problem—Euler Equation........14 Section 4. Examples and Interpretations........21 Section 5. Solving the Euler Equation in Special Cases........30 Section 6. Second Order Conditions........41 Section 7. Isoperimetric Problem........47 Section 8. Free End Value........52 Section 9. Free Horizon—Transversality Conditions........57 Section 10. Equality Constrained Endpoint........65 Section 11. Salvage Value........71 Section 12. Inequality Constraint Endpoints and Sensitivity Analysis........77 Section 13. Corners........86 Section 14. Inequality Constraints in (t , x)........90 Section 15. Infinite Horizon Autonomous Problems........95 Section 16. Most Rapid Approach Paths........97 Section 17. Diagrammatic Analysis........102 Section 18. Several Functions and Double Integrals........112 PART II: OPTIMAL CONTROL Section 1. Introduction........121 Section 2. Simplest Problem—Necessary Conditions........124 Section 3. Sufficiency........133 Section 4. Interpretations........136 Section 5. Several Variables........142 Section 6. Fixed Endpoint Problems........147 Section 7. Various Endpoint Conditions........155 Section 8. Discounting, Current Values, Comparative Dynamics........164 Section 9. Equilibria in Infinite Horizon Autonomous Problems........174 Section 10. Bounded Controls........185 Section 11. Further Control Constraint........195 Section 12. Discontinuous and Bang-Bang Control........202 Section 13. Singular Solutions and Most Rapid Approach Paths........209 Section 14. The Pontryagin Maximum Principle, Existence........218 Section 15. Further Sufficiency Theorems........221 Section 16. Alternative Formulations........227 Section 17. State Variable Inequality Constraints........230 Section 18. Jumps in the State Variable, Switches in State Equations........240 Section 19. Delayed Response........248 Section 20. Optimal Control with Integral State Equations........253 Section 21. Dynamic Programming........259 Section 22. Stochastic Optimal Control........264 Section 23. Differential Games........272 APPENDIX A. CALCULUS AND NONLINEAR PROGRAMMING Section 1. Calculus Techniques........291 Section 2. Mean-Value Theorems........294 Section 3. Concave and Convex Functions........298 Section 4. Maxima and Minima........303 Section 5. Equality Constrained Optimization........307 Section 6. Inequality Constrained Optimization........313 Section 7. Line Integrals and Green's Theorem........320 APPENDIX B. DIFFERENTIAL EQUATIONS Section 1. Introduction........325 Section 2. Linear First Order Differential Equations........328 Section 3. Linear Second Order Differential Equations........332 Section 4. Linear nth Order Differential Equations........339 Section 5. A Pair of Linear Equations........344 Section 6. Existence and Uniqueness of Solutions........350 References........353 Author Index........367 Subject Index........371