Variational Methods in Economics

383 Pages·1971·17.729 MB·English

by  George Hadley, Murray C. Kemp

ADVANCED TEXTBOOKS
IN ECONOMICS
VOLUME 1
Editor:
C. J. BLISS
University of Essex
NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM · LONDON
AMERICAN ELSEVIER PUBLISHING CO., INC-
NEW YORK.
VARIATIONAL METHODS
IN ECONOMICS
G. HADLEY
M. C. KEMP
<q£às?C
NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM · LONDON
AMERICAN ELSEVIER PUBLISHING CO., INC.
NEW YORK.
© North-Holland Publishing Company, 1971
All rights reserved. No part of this publication may be reproduced,
stored in a retrieval system, or transmitted in any form or by any means,
electronic, mechanical, photocopying, recording, or otherwise,
without the prior permission of the copyright owner.
Library of Congress Catalog Card Number: 79-157027
North-Holland ISBN for this series: 0720436001
North-Holland ISBN for this volume: 072043601 X
Elsevier ISBN: 0 44410097 0.
1st edition 1971
2nd printing 1973
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Preface
In this text we have attempted to provide students of economics and
related disciplines with a fairly comprehensive treatment of the classical
calculus of variations and its modern generalizations. It is hardly necessary
to argue the need for such a text. Recent issues of the professional journals,
especially in economics and management, are littered with references to the
Legendre-Clebsch conditions, the DuBois-Reymond equations and the other
paraphernalia of the subject. Nor shall we attempt to justify our selection
of topics and illustrations. However we must describe the class of readers
we have had in mind. We have tried to cater to the rapidly growing number
of students of economics and related disciplines who have a grounding in
the multi-variable calculus and linear algebra, together with some acquaint-
ance with the theory of diiferential equations and with the methods of
ordinary linear and non-linear optimization. No knowledge of functional
analysis, topology, or other advanced mathematical topics is assumed; nor
do we pre-suppose any knowledge of the calculus of variations itself. Perhaps
the most distinctive feature of the text is the inclusion of a large number of
worked examples. Most of the examples are taken from economics and, more
specifically, from the theory of optimal growth. However, while the book
does provide an introduction to the theory of optimal growth, we have not
attempted to make the exposition comprehensive or systematic; we have set
out to write a mathematics text, not an economics text. The reader who,
having mastered the mathematics, wishes to further sample the economic
applications of the variational calculus is referred to Professor Chakravarty's
new book [9] and to the symposia [13] and [34].
The manuscript was begun during the summer of 1969 when both authors
were at the Wenner-Gren Center in Stockholm and Kemp was a guest of
the Institute for International Economic Studies, University of Stockholm.
The manuscript was completed while Kemp was Ford Rotating Professor
of Economics at the University of California, Berkeley. Grateful acknowl-
VI PREFACE
edgement is made to the Wenner-Gren Foundation, to the Institute for
International Economic Studies, to the Ford Foundation and to the Uni-
versity of California, We also acknowledge the helpful remarks of Professor
Ken-ichi Inada (Osaka University) and Professor Yasuo Uekawa (Kobe
University of Commerce) concerning certain problems encountered in
writing Chapter 6. Finally, to Claire Gilchrist go our admiration and
gratitude for the skill and devotion she brought to the task of typing a very
difficult manuscript.
G. H.
M. C. K.
1
Growth models in economics
1.1. Introduction
This book is a text in the calculus of variations. However, it is addressed
to a rather special audience - social scientists, but especially economists and
those interested in management science - and this has determined our choice
of illustrations. Most of these come from a particular variety of growth
theory. Our purpose in this first chapter, therefore, is to provide the necessary
background in that area. We emphasize, however, that nothing in this
chapter is essential to an understanding of our development of the calculus
of variations; the reader who is interested only in the pure mathematics may
begin at once with Chapter 2.
1.2. Growth models in economics
There has in the postwar period been a rekindling of interest in the process
of economic development. Not since the classical economists has the subject
been so far to the fore. In some measure this renewal of interest can be
traced to the problems of reconstruction created by a war of unprecedented
violence and destructiveness. Of much greater importance, however, is the
awakening of a world conscience concerning the plight of the poor countries
of the world. Certain new developments in mathematics also have played
a part. In particular, those developments in the calculus of variations which
are usually referred to as the theory of optimal control have had a consider-
able influence. It is precisely these developments that we seek to expound
in the chapters which follow.
In its first phase theoretical enquiry focused on the task of formulating
highly aggregative descriptive models of the growth process. Roughly
speaking, this phase terminated with the publication in 1956 of the well
1
2 GROWTH MODELS IN ECONOMICS
known papers of Solow [35]* and Swan [38]. The achievement of this phase
was the articulation of a self-contained model of the growth process: given
only their initial values, the model generates the complete subsequent time
path of each variable of interest.
Since 1956 descriptive models of increasing complexity and realism have
been constructed. At the same time, however, there has emerged an interest
in the evaluation of the alternative paths of development available to the
economy. In the simple model of Solow and Swan, for example, there appear
two constant parameters, the savings ratio s and the rate of population
growth y. Each of these parameters can be influenced, directly or indirectly,
as a matter of government policy. To each pair of time dependent functions
[y(t), sit)] there corresponds a different growth path; and the feasible paths
are not all equally attractive. There emerges the problem of finding the
optimal paths οΐ s(t) and y(t). How this problem might be formulated and
solved will occupy us, on and off, throughout this book.
In this very rapid scanning of the horizon many things have been left
unsaid or unclear. We now go over the ground more slowly, beginning with
an exposition of the descriptive model introduced by Solow and Swan. A
single commodity is produced with the services of capital and labor. Choosing
units so that one unit of labor service is provided by one unit of population
in one unit of time, and so that one unit of capital service is provided by
one unit of capital in one unit of time, we may write
7(0 = Ψ(Κ(ή, L(t)) (1-1)
where K(t) is the stock of capital at time t, L(t) is the labor force at time t,
and Y(t) is the output per unit of time at /. In later chapters the production
function Ψ will be restricted in various ways. For the time being, however,
we need assume only that it is homogeneous of degree one in K and L and
that it possesses continuous second derivatives. The population and labor
force are supposed to grow at the steady exponential rate y:
L(i) = L(0)e", y |0 (1-2)
and the accumulation of capital is governed by the constant average savings
ratio s:
K(t) = sY(t), 0 < s < 1 (1-3)
where K(t) = dK(t)/dt.
* Numbers in brackets refer to the bibliography at the end of the text.
GROWTH MODELS IN ECONOMICS 3
In view of the homogeneity of Ψ we may rewrite (1-1) as
y(t) = Y(t)IL(t) = W(K(t)IL(t), 1) ss ψ(Ι<(ή) (1-4)
where y{t) is the average product of labor at time t and k{t) is the capital:
labor ratio. The first derivative r]/'(k) is then the marginal productivity of
capital, and φ — kty' is the marginal productivity of labor. From (1-2)
and (1-3)
k{t) = -^-(K(f)/L(t)) = sy(t) - yk(t). (1-5)
Equations (1-4) and (1-5) are the bare bones of the system; (1-4) is the
static production relationship, and (1-5) tells us how the system behaves
through time. Consolidation of the two equations yields the single differential
equation
k(t) = s\l/(k(t)) - yk(t). (1-6)
If this equation can be solved for k(t), the solution can then be substituted
into (1-4) to yield the time path of y(t) and, using (1-2), of Y(t).
The equations (1-4) and (1-6) cannot be solved explicitly unless \j/{k) is
assigned a particular mathematical form. In the Cobb-Douglas case, for
example, the two equations reduce to
y(t) = Lk(t)Y 0 < α <1 (l-4a)
9
and
k(t) = s[/c(0]a - yk(t) (l-6a)
respectively, and are readily soluble. Figure 1.1 illustrates the behavior of
*- k
4 GROWTH MODELS IN ECONOMICS
the system for y > 0, 1 > s > 0. For values of k less than k* k > 0; and
9
for values of k greater than k* k < 0. Clearly k(t) approaches k* asymp-
9
totically, whatever the initial value of k (provided only that it is positive);
hence the average product of labor approaches (X-*)a, consumption per capita
approaches (1 — s)(k*)a, and in the limit all quantities undeflated by popu-
lation grow at the same rate as the population itself.
However we are not at this stage concerned with specific solutions. The
important thing to notice is that the solutions, if they exist, depend on the
parameters y and s, that these parameters can be influenced by government,
and that there is therefore potentially a problem of choosing the optimal
y and s. More generally, there is potentially a problem of choosing the optimal
time paths y(t) and s(t). But optimal in relation to what criterion? Until the
criterion is specified, the problem is not even defined. Collective decisions
about the temporal distribution of consumption are of course political
decisions, based on compromise. We therefore cannot hope to find a criterion
which will carry conviction to everyone. The most that we can hope is that
any criterion the implications of which we analyze will be found interesting.
It will be assumed, here and later, that it is the stream of consumption C(t)
or the stream of consumption per capita c(t) which 'matters'. What we need
then is a rule for associating with each function C(t) or c(t) a number
^[C(0] 0Γ ^ΚΟ] which provides a numerical evaluation of the entire
consumption stream. The number will be referred to as the utility of the
consumption stream. Even if it is agreed that only consumption matters,
however, there is still no single compelling rule of association, that is,
definition of ^. Among many more-or-less plausible possibilities, economists
have been especially interested in the simple or undiscounted sum or integral,
over some finite or infinite planning period, of some function U of total or
average consumption. The function of consumption is often referred to, not
quite appropriately, as an instantaneous utility function. In a special case,
the function may be a constant positive multiple of total or average con-
sumption so that, in effect, % is simply the integral of consumption. Let us,
for the sake of illustration, choose as our desideratum the simple or un-
weighted integral of the instantaneous utility derived from consumption per
capita. Now consumption per capita at time t is c(t) = (1 — s(t))y(t). Thus
the problem is to find y(t) and s(t) < 1 such that
T
I l/((l - s(t))y(t), t)dt 0 < T < oo
o
is a maximum subject to (1-4) and (1-6) and to the requirement that k(t)
GROWTH MODELS IN ECONOMICS 5
be non-negative, and given k(0) = k. Notice that we have not at this stage
0
imposed the requirement that s(t) > 0.
This problem is typical of those we shall consider in later chapters. To
pose the problem it was necessary to provide: (i) A description of the welfare
criterion. (In the above example the time horizon and the utility function
had to be specified.) (ii) A description of the economy. (In the above
example the description is conveyed by (1-4) and (1-6).) Once the problem
is properly posed there remains the further problem of solving it; and that
is a matter of mathematical technique. The relevant techniques are discussed
in the following four chapters.