Table Of ContentLectu 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.
