Mathematical Theory of Economic Dynamics and Equilibria
This book is devoted to the mathematical analysis of models of economic dynamics and equilibria. These models form an important part of mathemati cal economics. Models of economic dynamics describe the motion of an economy through time. The basic concept
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v. L. Makarov A. M. Rubinov
Mathetnatical Theory of Economic Dynamics and Equilibria Translated from the Russian by Mohamed EI-Hodiri
Springer-Verlag New York
Heidelberg
Berlin
V. L. Makarov Siberian Branch of the Academy of Sciences
A. M. Rubinov Siberian Branch of the Academy of Sciences
Translation of the Russian edition Matematicheskaia teoria ekonomicheskoi dinamiki i ravnovesia by Mohamed A. EI-Hodiri.
AMS Subject Classification 90A15
Library of Congress Cataloging in Publication Data Makarov, Valerii Leonidovich. Mathematical theory of economic dynamics and equilibria. Translation of Matematicheskaia teoria ekonomicheskoi dinamiki i ravnovesia. Bibliography: p.
Includes index. 1. Economics, Mathematical. 2. Equilibrium (economics) I. Rubinov, Aleksandr Moiseevich, joint author. II. Title. HB135.M3313 330'.01'51 76-15219 All rights reserved. ©1977 by Springer-Verlag New York, Inc. Softcover reprint of the hardcover 1st edition 1977 987654321
ISBN -13: 978-1-4612-9888-5 DOl: 10.1007/978-1-4612-9886-1
e-ISBN-13: 978-1-4612-9886-1
Preface
This book is devoted to the mathematical analysis of models of economic dynamics and equilibria. These models form an important part of mathematical economics. Models of economic dynamics describe the motion of an economy through time. The basic concept in the study of these models is that of a trajectory, i.e., a sequence of elements of the phase space that describe admissible (possible) development of the economy. From all trajectories, we select those that are" desirable," i.e., optimal in terms of a certain criterion. The apparatus of point-set maps is the appropriate tool for the analysis of these models. The topological aspects of these maps (particularly, the Kakutani fixed-point theorem) are used to study equilibrium models as well as n-person games. To study dynamic models we use a special class of maps which, in this book, are called superlinear maps. The theory of superlinear point-set maps is, obviously, of interest in its own right. This theory is described in the first chapter. Chapters 2-4 are devoted to models of economic dynamics and present a detailed study of the properties of optimal trajectories. These properties are described in terms of theorems on characteristics (on the existence of dual prices) and turnpike theorems (theorems on asymptotic trajectories). In Chapter 5, we state and study a model of economic equilibrium. The basic idea is to establish a theorem about the existence of an equilibrium state for the Arrow-Debreu model and a certain generalization of it. Finally, in Chapter 6, the results obtained earlier are applied to a model of economic dynamics with explicit consumption. We study the asymptotes and characteristics of optimal trajectories of this model and study the relationship between these trajectories and the so-called equilibrium trajectories. v
Preface Some parts of the book were used, by us, as materials in special courses at Novosibirsk. The book is intended for readers with some mathematical sophistication and g
