Table Of ContentADVANCED TEXTBOOKS
IN ECONOMICS
VOLUME 4
Editors:
C. J. BLISS
M. D. INTRILIGATOR
Advisory Editors:
S. M. GOLDFELD
L. JOHANSEN
D. W. JORGENSON
M. C. KEMP
J.-C. MILLERON
NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM · OXFORD
AMERICAN ELSEVIER PUBLISHING CO., INC.
NEW YORK
CAPITAL THEORY AND THE
DISTRIBUTION OF INCOME
C. J. BLISS
University of Essex
1975
NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM · OXFORD
AMERICAN ELSEVIER PUBLISHING CO., INC.
NEW YORK
© North-Holland Publishing Company, 1975
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: 74-30938
North-Holland ISBN:
for the series: 0 7204 3600 1
for this volume: 0 7204 3604 4 (clothbound)
0 7204 3098 4 (paperback)
American Elsevier ISBN:
for this volume: 0 444 10782 7 (clothbound)
0 444 10865 3 (paperback)
PUBLISHERS:
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM
NORTH-HOLLAND PUBLISHING COMPANY LTD. - OXFORD
SOLE DISTRIBUTORS FOR THE U.S.A. AND CANADA:
AMERICAN ELSEVIER PUBLISHING COMPANY, INC.
52 VANDERBILT AVENUE, NEW YORK, N.Y. 10017
PRINTED IN THE UNITED KINGDOM
To my parents
Preface
When economists reach agreement on the theory of capital they will shortly
reach agreement on everything else. Happily, for those who enjoy a diversity
of views and beliefs, there is very little danger of this outcome. Indeed, there is
at present not even agreement as to what the subject is about. This book aims
at a systematic treatment of those topics which all would recognize to be
central to the theory, and of some others besides. With the needs of students
in mind the treatment of most topics has been made largely self-contained.
Hence, it is hoped that the book will be useful for courses on the theory of
capital and to those who study the subject on their own. In either case there
may well be more in the book than teacher or student can cope with in a
limited time. If so, parts I-IV and VI form a shorter book which is self-
contained. More drastic pruning can be achieved by leaving out ch. 7 and all
or some of part IV.
It is very difficult to know what background and prior knowledge to assume
for a book like this. More students are interested in the theory of capital than
in any other branch of economic theory and many want to get on with studying
it before they have achieved a good grasp of microeconomic theory and general
equilibrium theory. Furthermore, while the subject is naturally and inescap-
ably a mathematical one, at least in part, it attracts students who are not
specialists in mathematical economics. In this situation the author is propelled
in opposing directions by conflicting needs to avoid obvious dangers. To pretend
that a useful knowledge of the theory of capital is available to those without
an adequate background in economic theory, or that what is needed can be
learnt in passing, is to play the role of the charlatan. On the other hand, it is
just as wrong to suppose that only a student highly trained in mathematics
can master the subject. This makes for a presentation which is unnecessarily
difficult and often needlessly obscure.
What has been done here is to eschew any pretensions to a tidy solution to
the problem. In particular, there has been no attempt to make the mathematical
level of different chapters even approximately comparable. Instead, the
mathematical level in each case is that which I judge to be the minimum
viii Preface
necessary for the accurate derivation of results and for the education of an
intuition for what is true and what is not. As a consequence of this approach
the mathematical difficulty varies greatly from one part to another, and from
chapter to chapter. The most difficult mathematical arguments are to be found
in the appendices, which may be missed out, particularly from a first reading,
or in certain chapters, such as ch. 7, which again may be missed out without
undue loss of continuity.
The majority of proofs employ no more than elementary algebra or matrix
algebra, or linear programming theorems that will probably be familiar to
the reader already. Beyond that there is a little calculus (far less than is usual
in books on economic theory), some set theory and occasional use of basic
concepts of mathematical analysis, such as a limit, continuity, differenti-
ability, etc. Hence, the demands made on mathematical knowledge are only
moderate ones. What is required, however, is a certain mathematical maturity
and a familiarity with the mathematical mode of reasoning. This would
normally be acquired from a course on analysis, linear algebra or mathe-
matical economics. Therefore, one or more such courses might be regarded
as a prerequisite.
Where economic theory is concerned the reader is assumed to have studied
microeconomics to the level of a course for final-year undergraduate specialists,
and to have acquired thereby, or otherwise, a sound grounding in the theory
of general equilibrium. Nearly everything else that the reader will require is
provided within the covers of the book.
This volume has been more time in the writing than I care to think about.
For several years I have been working when time allowed on a book which
has eventually turned out to be this one. Many papers that I have presented
to seminars, much informal discussion with my fellow economists and a good
deal of my advanced teaching, has been directly connected with the problem
of sorting out one or other section of the book. As a consequence of this
inordinately long gestation period I have accumulated more debts than I
could begin to enumerate. Chief among them are those which I owe to students
at the Universities of Cambridge and Essex. Whatever shortcomings are to
be found in the following pages were at least as great, and sometimes much
greater, in the earlier drafts which formed the basis of lectures. It has been
my good fortune to have students who took a pride in detecting errors and
obscurities in my arguments. When the reader finds that some of these still
remain he should bear in mind that there would have been many more were
it not for the beneficent effects of this attention to refutation.
My interest in the theory of capital I owe largely to Professor Joan Robinson,
and in arguing with her at Cambridge I received my early education in the
Preface ix
subject. The hours which she devoted to trying to convince me that I was
taking a wrong view of the subject must have been totally unrewarding for
her, but her interest and patience never abated. At the Massachusetts Institute
of Technology, Professor R. M. Solow was likewise unstinting with his time.
He taught me a great deal without seeming to teach, and he tempered his
criticism with kindness and encouragement. In the case of each of these
eminent economists I have departed, sometimes sharply, from views for which
they are noted. But that does not diminish the debt that I owe to them. The
simple fact is that my method of learning the theory of capital has been to
evaluate critically the work which I judged to be most important. Hence,
those I criticize are those from whom I have learnt.
I am grateful for valuable criticism and assistance to, among others, A.
Dixit, F. H. Hahn, P. J. Hammond, M. C. Kemp, J. A. Mirrlees, D. M. G.
Newbery and C. C. von Weizsäcker. Dr. D. Glycopantis rescued me from
serious error in the first draft of the appendix to ch. 10. My co-editor, Michael
Intriligator, gave me valuable advice and was endlessly patient regarding
delays in the completion of the manuscript. Paul Hamelynck of the North-
Holland Publishing Company has proved to be a friend as well as my pub-
lisher, this in spite of being terribly let down with regard to deadlines. The
tedious chore of preparing the bibliography was undertaken cheerfully at
different times by Mrs. Rachel Britton and Mrs. Joanne Davis. The manuscript
was typed by Mrs. Phyllis Pattenden who was my secretary while I was chair-
man of the Department of Economics at Essex. More than anyone else
she made it possible to bring this work to completion; not just by deciphering
my scribbled drafts with wonderful inspiration, but also by lightening my
load and leaving me time to write.
September, 1974 C. /. Bliss
Mathematical notation
Real numbers (scalars) will be denoted by roman or greek letters in normal
type, upper or lower case, according to convenience. Thus a, v, cc and λ are
scalars while U(v) is a real-valued function of the scalar v.
Integers will be denoted by roman letters in normal type, upper or lower
case. The upper case is reserved for the highest value in a sequence of integers.
Hence / is an integer variable which may take any of the values 1,2,...,/.
t is usually an integer-valued index of time, and T the largest number of time
intervals considered.
Real vectors are denoted by roman letters in bold type. The scalars corre-
sponding to individual components of vectors are denoted by subscripted
versions of the same letters in normal type. Hence x is a vector and x is the
t
/th component of that vector.
Matrices are denoted by upper-case roman letters in bold type and the
individual components by doubly subscripted versions of the same letters in
lower case and normal type. Thus A is a matrix and a is the component of
u
that matrix in the /th row and the7 th column.
Sets of vectors are denoted by upper-case script letters, e.g. 2Γ and £f. The
symbol e denotes set inclusion. Hence jce«f means that the vector x belongs
to the set 2Γ. A set will be defined by enclosing between braces the notation
for an element, followed by a vertical line and then the property or properties
that must be satisfied for membership of the set. Thus {x \ x has property P}
is the set of all x such that x has property P.
Implication will be denoted by =>. Thus, A => B is to be read 6A implies B\
The convention for denoting inequalities is the same for vectors and for
matrices. Given two vectors a and b:
a> b means that each component of a is larger than the corresponding
component of b ;
a^b means that each component of a is at least as large as the corre-
sponding component of b;
XIV Mathematical notation
a^b means that each component of a is at least as large as the corre-
sponding component of b and at least one component is strictly
larger (i.e. a and b are not the same vector).
Finally, the symbol · denotes the operation of forming the inner product
of two vectors. Hence a b is the inner product of the vectors a and b.
1
The problem of capital
This method, conclusion first, reasons afterwards, has always been in high favour
with the human race: you write down at the outset the answer to the sum; then you
proceed to fabricate, not for use but for exhibition to the public, the ciphering by
which you can pretend to have arrived at it. The method has one obvious advantage -
that you are thus quite sure of reaching the conclusion you want to reach: if you
began with your reasons there is no telling where they might lead you, and like enough
you will never get to the desired conclusion at all. But it has one drawback; that
unanimity is thus unattainable: every man gives the answer which seems right in his
own eyes.
A. E. HOUSMAN, Introductory Lecture
The subject matter of this book is the theory of capital. We should begin by
explaining what that study concerns itself with; what are the distinctive
problems which the theory attempts to elucidate; to what questions one can
seek an answer in the following pages. Two kinds of definitions readily
suggest themselves : on the one hand there are definitions so all-embracing
that they take in the whole of economics while, on the other hand, there are
definitions which insist on a narrow focus and are merely designed to impose
the conclusions at which someone has already arrived long before framing a
definition of the subject.
However, without making a fetish of precision, one can go a long way with
the following definition. Capital theory is concerned with the implications
for a market economy, for the theory of prices, for the theory of production
and for the theory of distribution, of the existence of produced means of
production. Among the various necessities to produce output by a particular
method will usually be included the use of one or more inputs which them-
selves result from previous production. In other words, the typical cycle of
production is not simply the application of human labour and things provided
by nature to produce output, but it involves as well the use of man-made
inputs which are themselves outputs from other production cycles. Or, to
look at the matter from the point of income distribution, there are people
who derive income from their ownership of produced goods. What determines
how much income they will derive ? How can the ownership of something
