Dynamic Optimization, Second Edition: The Calculus of Variations and Optimal Control in Economics and Management

by Morton I. Kamien, Nancy L. Schwartz

Overview

An excellent financial research tool, this classic focuses on the methods of solving continuous time problems. The two-part treatment covers closely related approaches to the calculus of variations and optimal control. In the two decades since its initial publication, the text has defined dynamic optimization for courses in economics and management science.
Simply, clearly, and succinctly written chapters introduce new developments, expound upon underlying theories, and cite examples. Exercises extend the development of theories, provide working examples, and indicate further uses of the methods. Geared toward management science and economics PhD students in dynamic optimization courses as well as economics professionals, this volume requires a familiarity with microeconomics and nonlinear programming. Extensive appendices provide introductions to calculus optimization and differential equations.

Product Details

ISBN-13:	9780486488561
Publisher:	Dover Publications
Publication date:	11/21/2012
Series:	Dover Books on Mathematics
Edition description:	Second Edition
Pages:	400
Sales rank:	1,050,151
Product dimensions:	6.00(w) x 8.90(h) x 0.70(d)

About the Author

Morton I. Kamien (1938–2011) helped to build Northwestern University's Kellogg School of Business, where he served as Associate Dean for Academic Affairs and directed the Heizer Center for Entrepreneurial Studies for nearly 20 years.
Nancy L. Schwartz was the Morrison Professor of Decision Sciences, where in 1970 she was the Kellogg School's first woman faculty member appointed to an endowed chair. She chaired the Department of Managerial Economics and Decision Sciences and served as Director of the school's doctoral program until her death in 1981.

Table of Contents

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