Some Aspects of the Foundations of General Equilibrium Theory: The Posthumous Papers of Peter J. Kalman

173 Pages·1978·6.842 MB·English

by  G. Chichilnisky, P. J. Kalman (auth.), Jerry Green (eds.)

Lectu re Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and H. P. KUnzi
Mathematical Economics
159
Some Aspects of the Foundations
of General Equilibrium Theory:
The Posthumous Papers
of Peter J. Kalman
Edited by Jerry Green
Springer-Verlag
Berlin Heidelberg New York 1978
Editorial Board
H. Albach A V. Balakrishnan M. Beckmann {Managing Editor}
P. Dhrymes J. Green W. Hildenbrand W. Krelle
H. P. KOnzi {Managing Editor} K. Ritter R. Sato H. Schelbert
P. Schonfeld
Managing Editors
Prof. Dr. M. Beckmann Prof. Dr. H. P. KOnzi
Brown University Universitat ZOrich
Providence, RI 02912/USA 8090 ZOrich/Schweiz
Editor
Jerry Green
Harvard University
Department of Economics
1737 Cambridge St.
Cambridge, MA 02138/USA
Library of Congress Cataloging in PubUcation Data
Kalman, Peter Jason.
Same aspects of the foundations of general equilib
rium theory.
(Lecture notes in economics and mathematical systems ;
159 : Mathematical economics)
Bibliography: p.
Includes index.
1. Equilibrium (Economics)--Addresses, ess,,¥s,
lectures. 1. Green, Jerry R. II. Title.
III. Series: Lectures notes in economics and mathe
matical systems ; 159.
HBl45.K34 1978 330'.01'8 78-14520
AMS Subject Classifications (1970): 90 A 15, 90 C50
ISBN-13: 978-3-540-08918-6 e-ISBN-13: 978-3-642-95331-6
DOl: 10.1007/978-3-642-95331-6
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FOREmRD
Kenneth J. Arrow
It is with sadness for a personal and scholarly loss that I write this
introduction to the papers of my late friend, Peter J. Kalman. Peter was
devoted to extending our knowledge of the economic world. He accomplished
much in his short career but had probably never been more productive and
varied in his work than at the moment of his unexpected fatal illness.
The papers that follow illustrate the intensity and variety of his work.
They represent work in progress and in some cases have not yet received the
final polishing they deserve for publication. These papers do not exhaust
his current output, for some papers had already been accepted for publication
in journals and therefore do not fit into a collection of unpublished papers.
To Peter, scholarship was a cooperative endeavor to push back the areas
of ignorance, not a competitive world in which one person's success is
another one's failure. The papers in the present volume reflect this
attitude in that all are collaborative: with his student, Kuan-Pin Lin, his
frequent collaborator over the last few years, Graciela Chichilnisky, and
with his colleague and also frequent collaborator, Richard Dusansky. The
publishers and I are grateful that these coauthors and friends have authorized
reproduction of their joint papers in this volume. Though Peter and I did
not write any articles together, we exchanged many thoughts during the
period that he worked with me on research projects at Harvard University,
from which we both profited.
The range of subjects in the following papers is broad enough, but it
underrepresents the full variety of Peter's interests. The different
approaches to and aspects of the foundations of general equilibrium theory
are found here, including his interest in non-conventional equilibrium
IV
concepts and in the uses of differential topology for the study of
companative statics, but his interesting papers on operations research
and on various applied topics in economics are reflected only in one paper.
The discerning and sympathetic reader will find behind the formulas
and the severe and rigorous scholarship something of the intellectual zest
and warm personality of Peter Kalman.
TABLE OF CONTENTS
AN EXTENSION OF COMPARATIVE STATICS TO A GENERAL CLASS OF
OPTIMAL CHOICE MODELS (with G. Chichilnisky) •••••••••••••••••••
APPLICATIONS OF THOM-S TRANSVERSALITY THEORY AND BROUWER DEGREE
THEORY TO ECONOMICS (with K.-P. Lin) ••••••••••••••••••••••••••• 27
ILLUSION-FREE DEMAND BEHAVIOR IN A MONETARY ECONOMY: THE
GENERAL CONDITIONS (with R. Dusansky) •••••••••••••••••••••••••• 49
COMPARATIVE STATICS OF LESS NEOCLASSICAL AGENTS
(with G. Chichilnisky) •••••••.••••.•.•••••••••••••••••••••••••. 61
ON SOME PROPERTIES OF SHORT-RUN MONETARY EQUILIBRIUM WITH
UNCERTAIN EXPECTATIONS (with K.-P. Lin) •••••••••••••••••••••••• 77
A DIFFERENTIABLE TEMPORARY EQUILIBRIUM THEORY
(wi th K. -P. Lin) .••..•.....••••..•.......•••........•••••.•••.. 115
EQUILIBRIUM THEORY IN VEBLEN-SCITOVSKY ECONOMIES: LOCAL
UNIQUENESS, STABILITY AND EXISTENCE
(with K.-P. Lin and H. Wiesmeth) ••••••••••••••••••••••••••••••• 131
OPTIMAL HOUSING SUPPLY OVER TIME UNDER UNCERTAINTY
(wi th R. Dusansky) ••••••••••••••••••••••••••••••••••••••••••••• 15 1
An Extension of Comparative Statics to
a General Class of Optimal Choice Models
by
*
G. Chichilnisky and P. J. Kalman
Department of Economics
Harvard University
October 1976
Abstract
We study properties of the solutions to a parametrized
constrained optimization problem in Hilbert spaces. A special
operator is studied which is of importance in economic theory;
sufficient conditions are given for its existence, syInmetry,
and negative semidefiniteness. The techniques used are calculus
on Hilbert spaces and functional analysis.
*
This research was supported by NSF Grant GS18174. P. J. Kalman
is visiting Harvard University from SUNY at Stony Brook. The
authors thank K. J. Arrow and 1. Sandberg for helpful suggestions.
2
Introduction
In a wide number of economic problems the equilibrium values
of the variables can be regarded as solutions of a parametrized
constrained maximization problem. This occurs in static as well as
dynamic models; in the latter case the choice variables are often
paths in certain function spaces and thus can be regarded as points
in infinite dimensional spaces.
It is sometimes possible to determine qualitative properties of
the solutions with respect to changes in the parameters of the model.
The study of such properties is often called comparative statics; [15], [2], and
[10]. Certain comparative static properties of the maxima have
proven to be of particular importance for economic theory, since the
works of Slutsky, Hicks, and Samuelson [15]: they have been formu-
lated in terms of synunetry and negative semidefiniteness of a matrix,
called the Slutsky-Hicks-Samuelson matrix. A discussion of this
matrix and its applications is given in Section 1. The study of these
properties in economic theory, however, has so far been restricted
to static models where the choice variable and the parameters are
elements in Euclidean spaces, and where there is only one constraint.
Infinite dimensionality of the choice variables arises naturally from
the underlying dynamics of the models. For example, in optimal
growth models with continuous time and problems of planning with
infinite horizons [4] and also from the existence of infinitely many
characteristics of the commodities indexed, for instance, by states
of nature in models with uncertainty. by location, etc. Many times
these models are formalized as optimization problems with more
than one constraint.
3
It is the purpose of this paper to extend the study of the Slutsky-
Hicks-Satnuelson operator to a general class of paratnetrized.
constrained optimization problems which appear in recent works in
economic theory: the choice variables and paratneters belong to
infinite dimensional spaces. the objective function to be maximized
depends also on parameters. and the optimization is restricted to
regions given by many possibly infinite parametrized constraints.
linear or not. 1 The results provide a foundation for the study of
comparative statics in dynamic models such as optimal growth and
other dynamic models [4].
The derivation of the Slutsky operator is more complicated in
the case of many constraints. and the operator obtained is of a slightly
different nature. One reason is that the "compensation" can be
performed in different manners since there are many constraints. as
becomes clear in the proof of Theorem 1 and the remark following it.
Also. the existence of parameters introduces new effects that do not
exist in the classical models; in general. the classical properties
are not preserved. Further, since the values of the constraints may
be in an infinite dimensional space of sequences (denoted C), the
"generalized Lagrangian multiplier" may also be infinite dimensional.
in effect, an element of the dual space of C, denoted C*. To avoid
the problem of existence of such dual elements which are not represent-
able by sequences (e. g. , purely finite additive measures [8]) and thus
lRelated work in infinite dimensional commodity spaces has been done
for special cases of one linear constraint and no parameters in the
objective function by L. Court [7] and Berger [3]. In finite dimen
sional models, related work for parametrized models with ODe
constraint was done by Kalman [9], and Kalman and Intriligator [10];
Chichilnisky and Kalman studied parametrized multi-constraint
problems in [6].
4
cOIIlplicate the cOIIlputations, we work on a Hilbert space of sequences
c.
Infinite diIIlensional econOIIlic IIlodels where the variables are
eleIIlents of Hilbert spaces have been studied in [4] and [5].
The extens ion froIIl finite to infinite diIIlens ional choice variables
and paraIIleters involves further technical difficulties. In the first
place, existence of optiIIlal solutions is harder to obtain since closed
and bounded sets in infinite diIIlensional spaces are not, in general,
cOIIlpact in certain topologies such as the topology of the norIIl. To
avoid this probleIIl, one usually uses certain weak topologies in which
nOrIIl bounded and closed sets are cOIIlpact. However, in these
topologies, the continuity of the objective functions is IIlore difficult
to obtain, and thus the usual proofs of existence of solutions by
cOIIlpactness-continuity argUIIlents IIlay restrict the class of adInissible
objective functions. However, using the concavity of the objective
function and convexity of the set on which the optiIIlization is perforIIled,
we prove existence of an optiIIlal solution on norIIl bounded closed sets2
or weakly cOIIlpact sets without requiring the objective function to be
weakly continuous, which widens the choice of objective functions.
Thus, the existence of a solution can be obtained in IIlore econoIIlic
IIlodels of this type; a useful tool here is the Banach-Saks theoreIIl [14].
In Section 1 sufficient conditions are given for existence and
uniqueness of a C 1 solution to a general optiIIlization probleIIl and
for existence of a generalized Slutsky-Hicks -Sa=uelson operator
which contains as a special case the operator of classical econoIIlic
2In any reflexive Banach space or Hilbert space, norIIl bounded and
closed sets are weakly cOIIlpact [8].
5
models. In Section 2, properties of this operator are studied: a class
of objective and constrained functions is shown to preserve the
classical properties of symmetry and negative semidefiniteness of
the operator, which are, in general, lost in parametrized models,
as seen in [10].
Section 1
We now discuss the Slutsky-Hicks-Samuelson operator and its
applications. For further references, see, for instance, [15] and [10].
Consider the maximization problem:
(P) max f(x,a)
x
subject to g(x, a) = c
where f is a real valued map defined on a linear space and g is
vector valued, defined on a linear space. Under certain assumptions
the optimal solution vector x denoted h(a, c) is a C 1 function
of the variables a and c, and, as the parameter c varies, the
constraints describe a parametrized family of manifolds on which f
is being maximized. In neoclassical consumer theory, for instance,
f represents a utility function, x consumption of all commodities,
a prices of all commodities and c income. In this theory, h is
called the demand function for commodities of the consumer. In
neoclassical producer theory, f represents the cost function, x
inputs, a input prices, and g a production function constrained
by an output requirement c; in this theory, h is called the demand