Lectu re Notes in Operations Research and Mathematical Economics Edited by M. Beckmann, Providence and H. P. KOnzi, ZOrich 11 Mathematical Systems Theory and Economics I Proceeding of an International Summer School held in Varenna, Italy, June 1 -12, 1967 Edited by w. H. Kuhn Princeton University, Princeton, N. J./ USA G. P. Szego University of Milano, Milano/Italy 1969 Springer-Verlag Berlin Heidelberg GmbH Lecture Notes in Operations Research and Mathematical Economics Edited by M. Beckmann, Providence and H. P. Kunzi, Zurich 12 Mathematical Systems Theory and Economics II Proceeding of an International Summer School held in Varenna, Italy, June 1 -12, 1967 . Edited by H. W. Kuhn Princeton University, Princeton, N. J./ USA G. P. Szego University of Milano, Milano/Italy 1969 Springer-Verlag Berlin Heidelberg GmbH ISBN 978-3-540-04635-6 ISBN 978-3-642-46196-5 (eBook) DOI 10.1007/978-3-642-46196-5 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Vedag. © by Springer-Verlag Berlin Heidelberg 1969 Library of Congress Catalog Card Number 70-81409 Tide No. 3761 Preface The International Summer School on Mathematical Systems Theory and Economics was held at the Villa Monastero in Varenna, Italy, from June 1 through June 12, 1967. The objective of this Summer School was to review the state of the art and the prospects for the application of the mathematical theory of systems to the study and the solution of economic problems. Particular emphasis was given to the use of the mathematical theory of control for the solution of problems in economics. It was felt that the publication of a volume collecting most of the lectures given at the school would show the current status of the application of these methods. The papers are organized into four sections arranged into two volumes: basic theories and optimal control of economic systems which appear in the first volume, and special mathematical problems and special applications which are contained in the second volume. Within each section the papers follow in alphabetical order by author. The seven papers on basic theories are a rather complete representative sample of the fundaments of general systems theory, of the theory of dynamical systems and the theory of control. The five papers on the application of optimal control to economic systems present a broad spectrum of applications. The series of lectures given at the school by Prof. A. Strauss on the mathematical theory of control were published as a separate volume in this series, with title "An Introduction to Optimal Control Theory". Five interesting lectures given at the School by Prof. H. Scarf, E. Sheshinsky, R. Radner, J. Florentin and S. Mitter were not made available for publication. All other papers presented appear in this volume. This International Summer School was financially supported by NATO. Fellowships to the participants were in addition given by various organizations, to whom we express our gratitude. Princeton, N.J., USA Milano, Italy R.W. Kuhn December 1968' G.P. Szego Contents Part I Basic Theories N. P. Bhatia: Dynamical Systems.................................................... 1 G. Debreu: Economic Equilibrium •••.•••••••••••••••••••••••••••••••••••••••••••••••• 11 H. Halkin and L. Neustadt: Control as Programming in General Normed Linear Spaces •• 23 R. E. Kalman: Introduction to the Algebraic Theory of Linear Dynamical Systems ••••• 41 H. W. Kuhn: Duality in Mathematical Programming •••••••••••••••••..•••••.•••.••••••• 67 M. D. Mesarovic: Mathematical Theory of General Systems and some Economic Problems. 93 R. T. Rockafellar: Convex Functions and Duality in Optimization Problems and Dynamics •••••.•••••.••••••••••••••••••••••••••••••••••••••••••••••••••• 117 Part II Optimal Control of Economic Systems A. R. Dobell and Y. C. Ho: Optimal Investment Policy •.•••••••••••••••••••••.••••••• 143 M. Kurz: On the Inverse Optimal Problem .•.•••••••••.••••••••••••••••••••••••••••••• 189 L. Markus: Dynamic Keynesian Economic Systems: Control and Identification •••••••••• 203 D. McFadden: On the Controllability of Decentralized Macroeconomic Systems: The Assignment Problem .•••••••••••••••••••••••••••••••••••••••••••••••••••• 221 K. Shell: Application of Pontriagin's Maximum Principle to Economics ••••••••••••••• 241 Part III Special Mathematical Problems F. Albrecht: Control Vector Fields on Manifolds and Attainability •••••••••••••••••• 293 N. P. Bhatia: Semi-Dynamical Systems ••.•••••.•••••••••••••••••••••••••••••••••••••• 303 C. Castaing: Some Theorems in Measure Theory and Generalized Dynamical Systems Defined by Contingent Equations •••.••.•.••••••••••••••••••••••••••••••• 319 A. Halanay: On the Controllability of Linear Difference-Differential Systems ••.•••• 329 W. Hildenbrand: The Core and Competitive Equilibria •••••••••••••••••••••••••••••••• 337 E. B. Lee: Geometric Theory of Linear Controlled Systems •••••.••••••••••••••••••••• 347 J. Nagy: Stability of Sets with Respect to Abstract Processes •••••••••••••••••••••• 355 J. A. Yorke: Invariance of Contingent Equations •••••••••••••••••••••••••••••••••••• 379 J. A. Yorke: Space of Solutions •••••••••••••••••••••••••••••••••••••••••••••••••••• 383 - VI - Part IV Special Applications M. Beckmann: Feedback and the Dynamics of Market Stability •••••••••••••••••••••••• 405 A. J. Blikle: Investigations of Organization of Production Processes with Tree Structure ••.••••••••••••••••••••••••••••••••••••••••••••.••••••••••• 421 cr. Hillinger: Testing Econometric Models by Means of Time Series Analysis ••••••••• 433 R. Kulikowski: Optimum Control and Synthesis of Organizational Structure of Large Scalp. Systems ••••••••••••••••••••••••.•••••••••••••••••••••••••••••• 441 H. E. Ryder: Optimal Accumulation in a Listian Model •••••••••••••••••••••••••••••• 457 A. Straszak: Multi-level Approach to the Large-scale Control Problem •••••••••••••• 481 List of Contributors Albrecht, F., University of Illinois, Department of Mathematics, Urbana, Ill. / USA Beckmann, M. J., Institut fUr Okonometrie und Unternehmensforschung der Universitat, 5300 Bonn / Germany Bhatia, N. P., Case Western Reserve University, Department of Mathematics, Cleveland, Ohio / USA Blikle, A. J., Polish Academy of Sciences, Mathematical Institute, Warszawa / Poland Castaing, C., Universite de Montpellier, Division Mathematiques, Montpellier, Herault / France Debreu, G., University of California, Department of Economics, Berkeley, Calif. / USA Dobell, A. R., Harvard University, Department of Economics, Cambridge, Mass. / USA Halanay, A., Bucharest University, Bucharest / Roumania Halkin, H., University of California, Department of Mathematics, San Diego~ Calif. / USA Hildenbrand, W., University of California, Department of Economics, Berkeley, Calif. / USA and Studiengruppe fur Systemforschung, 6900 Heidelberg / Germany Hillinger, C., Case Western Reserve University, Division of Organizational Sciences, Cleveland, Ohio / USA Ho, T. C., Harvard University, Computation Laboratory, Cambridge, Mass. / USA Kalman, R. E., Stanford University, Department of Mathematics, Stanford, Calif. / USA Kuhn, H. W., Princeton University, Department of Mathematics, Princeton, New Jersey / USA Kulikowski, R., Polish Academy of Sciences, Institute of Automatic Control, Warszawa Poland Kurz, M., Stanford University, Institute for Mathematical Studies in the Social Sciences, Stanford, Calif. / USA Lee, E. B., Center for Control Sciences, Institute of Technology, UniverSity of Minnesota, Minneapolis, Minn. / USA Markus, L., University of Minnesota, Department of Mathematics, Minneapolis, Minn. / USA McFadden, D., University of California, Institute for International Studies, Berkeley, Calif. / USA MesarOvic, M. D., Case Western Reserve University, Systems Research Center, Cleveland, Ohio / USA Nagy, J., Technical University (CVUT), Department of Mathematics FEL, Technicka 2/1902, Prague 6-Deyjvice / Czechoslovakia Neustadt, L. W., University of Southern California, Department of Electrical Engineering, Los Angeles, Calif. / USA Rockafellar, R. T., University of Washington, Department of Mathematics, Washington, Seattle / USA Ryder, H. E., Jr., Brown University, Department of Economics, Providence, Rhode Island / USA - VIII - Shell, K., University of Pennsylvania, Department of Economics, Philadelphia, Pa. / USA Straszak, A., Polish Academy of Sciences, Institute for Automatic Control, Warszawa / Poland Yorke, J. A., University of Maryland, Department of Mathematics, College Park, Maryland / USA Part I Basic ~eories DYNAMICAL SYSTENS * Nam P. B hat i a O. IN'l.'RODUCTION I had planned to give a comprehensive account of the results obtained by me and Dr. Szego during the past few years together with related results of various authors in the field of dynamical systems. With the appear~nce of our lecture notes [lJ this pro- spect is not too attractive now so that I would like to concentrate on a couple of prob- lems and techniques which seem to hold promise. But first the following definition. (0.1) Definition. Let X be a topological space, R the set of real numbers, and w a map from X x R into X satisfying (0.2) w(x,O) = x for every xeX, (identity axiom), (0,3) w(w(x,t),s) = w(x,t+s) for every xeX and every t and s in R, (homomorphism axiom), and (0.4) w is continuous, (continuity axiom). Then the triple (X,R,w) is called a dynamical system. Given HeX and SCR, we shall write lIS for the set {w(x,t) :xel! and teSL If H or S is a singleton, then we use the notation xS or Mt instead of {x}S or M{t}. In this ter minology xt = {x}{t} is the image w(x,t) of the point (x,t) in X x R. The axiom (0.3) then reads xt(s) = x(t+s) • + - + For any xeX, the sets xR, xR , and xR denoted respectively by y(x), y (x), and y-(x) are called the trajectory, positive semitrajectory, and the negative semi-trajec tory through x. Here R+ (R-) is the set of non-negative (non-positive) real numbers. A set MeX is called invariant, positively invariant, or negatively invariant whenever MR = H, 11R+ = H, or HR- = 11, respectively. * Partial support of the author by the National Science Foundation Grant No.NSF-GP-7447 is gratefully acknowledged.