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In this volume Dr. Chiang introduces readers to the most important methods of dynamic optimization used in economics. The classical calculus of variations, optimal control theory, and dynamic programming in its discrete form are explained in the usual Chiang fashion--with patience and thoroughness. The economic examples, selected from both classical and recent literature, serve not only to illustrate applications of the mathematical methods, but also to provide a useful glimpse of the development of thinking in several areas of economics. Outstanding features include: (1) written with clarity and a comparable level of expository patience; (2) reinforces discussions of mathematical techniques with numerical illustrations, economic examples, and exercise problems; (3) presents a simple problem with a well- known solution in several different alternative formulations in the numerical illustrations; and (4) explains economic models in a step-by-step manner (from the initial construction through the intricacies of mathematical analysis to its final solution).
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C1
CHIA
ELEMENTS OF DYNAMIC OPTIMIZATION
1
ELEMENTS OF DYNAMIC OPTIMIZATION
Alpha C. Chiang Professor o f Economics The University of Connecticut
McGraw-Hill, Inc. NewYork St. Louis San Francisco Auckland Bogot6 Caracas Lisbon London Madrid Mexico Milan Montreal NewDelhi Paris SanJuan Singapore Sydney Tokyo Toronto
To my parents Rev. & Mrs. Peh-shi Chiang with gratitude
ELEMENTS OF DYNAMIC OPTIMIZATION Internat~onalEditions 1992 -
Exclus~venghts by McGraw-HIII Book Co-S~ngaporefor manufacture and export. Thls book cannot be re-exported from the country to whlch 11 1s cons~gnedby McGraw-HIII Copyright 1992 by McGraw-H111, Inc. All r~ghtsreserved Except as permitted under the Un~tedStates Copyr~ghtAct of 1976, no part of thls publlcat~onmay be reproduced or distributed In any form or by any means. or stored In a data base or retrieval system, w~thoutthe prlor wrltten permlsslon of the pubhsher.
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L~braryof Congress Catalogrng-~n-PublicatonData Ch~ang,Alpha C., (date) Elements of dynam~coptlmlzatlon I Alpha C. Chlang P cm lncludcs b~blrograph~cal references and Index ISBN 0-07-01091 1-7 I Mathemat~caloptlmlzatlon 2 Economics, Mathemat~cal. I T~tle HB 143 7 C45 1992 9 1-37803 330' 0 1'5 1 ' - dc20 Thls book was set In Century Schoolbook by Sc~enceTypographers, Inc. The ed~torwas Scott D. Stratford; the product~onsupervisor was Den~seL. Puryear The cover was des~gnedby John Hlte. Project supervlslon was done by Science Typographers, Inc. Art was prepared electron~callyby Sc~enceTypographels, Inc. When orderlng this t~tle,use ISBN 0-07-1 12568-X Printed in Singapore.
CONTENTS
Preface
Part 1 Introduction 1 The Nature of Dynamic Optimization 1.1 1.2 1.3 1.4
Salient Features of Dynamic Optimization Problems Variable Endpoints and Transversality Conditions The Objective Functional Alternative Approaches to Dynamic Optimization
Part 2 The Calculus of Variations 2
Alpha C. Chiang received his Ph.D. from Columbia University in 1954 after earning the B.A. in 1946 at St. John's University (Shanghai, China), and the M.A. in 1948 at the University of Colorado. He has been a professor of economics at the University of Connecticut since 1964, but had also taught at Denison University, New Asia College (Hong Kong), and Cornell University. A holder of Ford Foundation and National Science Foundation fellowships, he served as Chairman of the Department of Economics a t Denison from 1961 to 1964. His publications include Fundamental Methods of Mathematical Economics, 3d ed., McGraw-Hill, New York, 1984.
2.1 2.2 2.3 2.4 2.5
The Fundamental Problem of the Calculus of Variations The Euler Equation Some