Convex structures and economic theory

CONVEX STRUCTURES AND ECONOMIC THEORY Hukukane Nikaido THE INSTITUTE OF SOCIAL AND ECONOMIC RESEARCH OSAKA UNIVERSITY O...

Author: Hukukane Nikaidō

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CONVEX STRUCTURES AND ECONOMIC THEORY Hukukane Nikaido THE INSTITUTE OF SOCIAL AND ECONOMIC RESEARCH OSAKA UNIVERSITY OSAKA, JAPAN

1968

@

ACADEMIC PRESS

New York and London

COPYRIGHT 0 1968, BY ACADEMIC PRESS,INC. ALL RIGHTS RESERVED. N O PART OF THIS BOOK MAY BE REPRODUCED IN A N Y FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS,

INS. 1 1 I Fifth Avenue, New York, New York 10003

Utritrd Kingdom Editiori published by ACADEMIC PRESS, INC. (LONDON) LTD. Berkeley Square House, London W . l

LIBRARY OF

CONGRESS CATALOG C A R D

NUMBER: 68-23500

PRINTED IN THE UNITED STATES OF AMERICA

To my parents

PREFACE

Modern mathematical methods, particularly those taking advantage of convexity, play a vital part in the study of many basic problems in economic theory and its related subject matter. Recognition of this fact has created an ever-increasing demand among students for a command of the new methods that also keeps pace with the current trend toward the so-called “new mathematics.” This book, with which the author hopes to meet this demand, is designed for anyone who wishes to become familiar with the basic ideas, methods, and results in this modern line of mathematical treatment in economic theory through a detailed exposition of a number of typical representative problems. I t can be understood by anyone having a rudimentary knowledge of calculus and matrix theory, including elementary topological concepts in Euclidean space, even without having much background in economics. It has also been the aim of the author to compile material that may be of interest to more advanced students of mathematical economics and related fields. It is also hoped that the plain mathematical language in this volume will be useful in encouraging experienced research workers outside the realm of economics to acquire intimacy with certain orthodox economic concepts and views. This book is by n o means intended, however, to be either a tract on economic theory itself or a comprehensive handbook of mathematical economics. The material included consists of an account of the theory of convex sets and its application to several basic problems that originate in economic theory and adjacent subject matter. Typical examples are problems pertaining to interesting static and dynamic phenomena in linear and nonlinear economic systems, including the models initiated by Leontief, von Neumann, Walras, and others. More specifically, they are problems centering around the existence, optimality, stability, and uniqueness of solutions to relevant equations or inequalities. A more detailed orientation for the problems treated in individual chapters is given in the Introduction. vii

...

Vlll

PREFACE

Throughout the entire volume, sections are serially numbered by Arabic numerals prefixed by 5. 5x.P means the Pth subsection of Q E ; Theorem u.p stands for Theorem in §x. When Theorem E.P is composed of several propositions numbered by numerals (i), (ii) ,. . ., sole references to proposition (i) in Theorem 3.P appear as Theorem ccP.(i). The same rule applies to a n y cross references. The author owes much t o the works listed in the Bibliography. The present work also includes certain of the author’s work [Nikaido (1954a, 1956, 1957, 1962, 1964a,b,c)] and his joint article with D. Gale [Gale and Nikaido (1965)] in more o r less revised and extended forms, through the cooperation of the joint author as well as the courtesy of Kodai Mathematical Seminar Reports, Metroeconomica, Reciew of Economic Studies, Econometrica, Anncls of Mathematics Studies, and Mathematische Annalen, in which these articles were originally published. The author is particularly grateful for special permission t o include in Chapter VII certain relevant results by Professor J. S. Chipman (1964) and Professor K . Inada (1966) even before their publication. The author is much obliged t o the publishers of his previous two books (in Japanese), “Gendai Keizaigaku no Sugakuteki H o h o ” (Mathematical Methods of Modern Economics), Iwanami Shoten, Tokyo, 1960 and “ Keizai no Tameno Senkeisugaku ” (Linear Mathematics for Economists), Baifukan, Tokyo, 1961, from which part of the ultimate sources of the present work has evolved. Thanks are also due to all the author’s colleagues and friends who have shown an interest in this book. Notably, he wishes t o thank Professors H. Atsumi, I).Gale, K. Inada, D. W. Katzner, H. Kumagai, M. Morishima, T. Negishi, and T. Yasui for their pertinent comments. The author also acknowledges his indebtedness to Professors K . J. Arrow, L. W. McKenzie, P. A . Samuelson, L. S. Shapley, H. Uzawa, and others for the benefits derived from discussions with them on certain of the problems included i n thisvolume. The author is extremely grateful to Professor R. Bellman, who suggested this book and has given encouragement throughout the preparation of the manuscript, and to Professor S. Iyanaga, for his stimulation as the author’s teacher. Conversion of the major part of the original manuscript to a typewritten version was done by Mrs. K. Kasuga, and helpful technical assistance was rendered by Miss M. Nakashima and Mrs. K. Sasaki. For the careful proofreading and valuable suggestions, the author also wishes to thank Miss A. Green and Miss R. Wada. Osaka October. I948

HUKUKANE NIKAIDO

CONTENTS vii xi

PREFACE LISTOF BASICSYMBOLS

1

Introduction 1. Mathematical Analysis in Economic Theory 2. Scope of Mathematical Economic Theory 3. Appeal to Convexity 4. Orientation for Chapters

Chapter I. Mathematical Theorems on Convexity

6 6 15 26 53 74

1. Euclidean Spaces 2. Convex Sets in R" 3. Separation Theorems for Convex Sets 4. Fixed-Point Theorems 5. Homotopy and Extension Theorems

Chapter I I . Simple Multisector Linear Systems 6. 7. 8. 9.

1 2 3 4

87 87

Leontief's Input-Output Systems, Matrix Multipliers Frobenius-Perron's Theorem Imprimitivity and Oscillatory Behavior Other Linear Systems

100 108

Chapter I I I. Balanced Growth in Nonlinear Systems

149

10. The Solow-Samuelson-Type Relative Stability Theorems 11, lncome Propagation under Autonomous Spending

149 162

ix

132

X

CONTENTS

Chapter IV. Efficient Allocation and Growth

180

12. Production Technology Sets 13. Theorems on Self-Sustained Efficient Growth 14. A Consumption Turnpike Theorem

180 194 220

Chupter V. The Working of Walrasian Competitive Economies

234

15. Allocation in Decentralized Systems 16. Existence of a Competitive Equilibrium 17. Normative Implications o f Competition

234 249 219

Chupter VI. Special Features of Competitive Economies: Gross Substitutability

297

19. Stability in Gross Substitutes Systems

291 335

Clmpter VII. The Jacobian Matrix and Global Univalence

355

18. Gross Substitutability and Extreniization

20. Con”ditions on the Jacobian Matrix for Global Univalence 21. Applications to Special Jacobian Matrices

355 385

BIBLIOGRAPHY

393 399 40 I

AUTHORINDEX SUBJECT INDFX

LIST OF BASIC SYMBOLS x is contained in A x is not contained in A

A is included in B (set-theoretic inclusion) A is not included in B A C B, but A # B (proper set-theoretic inclusion)

set of all elements x having a property P empty set set consisting of a single element a set-theoretic union of A , B, C, . . . set-theoretic union of An indexed by h E .\ set-theoretic intersection of A , B, C, . . . set-theoretic intersection of A A indexed by h E A\ set-theoretic difference, namely the set { x I x E A, x $ B } set-theoretic complement of A Cartesian product of sets A , B, C, . . . Cartesian product of sets An indexed by h t A\ closure of A interior of A boundary of A, namely the set n convex hull of A (single-valued) mapping from X into Y set-valued mapping from X into Y composition (product) of two mappings f and g, f followed by 8 inverse mapping of a mapping f Cartesian product of mappings f, g , h, . . . image of a set A under a mapping f inverse image of a set B under a mappingf n-dimensional (real) Euclidean space inner product of x = ( x , ) and y = ( y , ) Euclidean norm row vector column vector standard simplex

2

(n I)-sphere of radius 8 with center at ti-ball of radius 8 with center at a denotes S.(O, l ) , unit (n - I)-sphere denotes CJO, I ) , unit u-ball distance between two points a and h distance between a point a and a set A ~

xi

N

xii

LIST OF BASIC SYMBOLS

€-neighborhood of

(I

sequence of points .Y' sequence of numbers A, limit of a sequence {x') limit of a function (or mapping)f(x) as x tends t o a rightward (leftward) convergence of x to a limit superior of a sequence {A,: of real numbers limit inferior of a sequence {A,.: of real numbers limit superior of a real-valued function / ( x ) as .Y tends t o a limit inferior of a real-valued function /(.Y) as .Y tends t o a least upper bound greatest lower bound transpose of a matrix A determinant of a square matrix A Euclidean norm of an 11 ;. / I matrix A rank of a matrix A semi-order in R", meaning xi 2 J', ( i I , . . . , / I ) for .Y = ( x , ) and L' Lvi) means .Y 2 J', s + y means .yi ( i := I , . . . , / I ) for .Y = ( . Y , ) and J' (.J,,) .Y is at least as preferable as .y ( a preference relation) .Y is preferred t o .J' ( a strong preference relation) an equivalence relation, specifically, the indifference relation associated with 2 o r the honiotopy of mappings a closed interval with end points a , b in the real line, or a segment with end points a, b in R" denotes 10, bl\{b) denotes [N,b ] \ { a ) denotes [a, b]\({oj u { / I ) ) real part of a complex number OL imaginary part of a complex number OL partial increment o f the ith component of a vector p ~

)si

~~

INI’RODlICTION

$1. MATHEMATICAL ANALYSIS I N ECONOMIC THEORY

Economic theory in its early development was seldom based on mathematical formulation. Nevertheless, a number of classical economic doctrines were disguised literary statements of substantially mathematical propositions, although their advocates were sometimes unaware of this fact. As economic theory advanced, these inherent mathematical factors gradually came to the foreground. At a later stage of its history, even an overall mathematization of economic theory was pursued by certain economists. Leon Walras, a French economist, was probably the most eminent representative of the mathematical school in economic theory and he laid the foundation (Walras 1874) of the theory of general equilibrium at the close of the 19th century. The theory is based not on unilateral causality but on the interdependence of relevant magnitudes, such as those represented mathematically by some system of simultaneous equations. I t sets an excellent example for matliematical study of economic plienomena. Much credit may be accorded to Walras for establishing the now dominant fundamental view that so far as their functional working is concerned, economic phenomena are depicted by systems of equations o r inequalities, or et‘eii by mixed systems. In spite of the great achievement of Walras, however, further discoveries were to be made. Subsequent generations became aware that from the modern mathematical point of view, what Walras achieved was only the setting up of several pertinent systems of equations depicting the economic world. He neither gave any satisfactory justification for the existence of solutions to his systems of equations, nor succeeded i n a mathematical consolidation of his other propositions. Responsibility for this fact must be shared by the primitireness of his mathematical consirl~rationand the rather immature development 1

2

INTRODUCTION

of certain fields of 19th century mathematics, especially of topology. The establishment of the existence of solutions as well as the mathematical elucidation of the other important properties of the Walrasian system was actually achieved in the 1950's. This is only one example of the many achievements in economics of recent decades which are based on the advancement of mathematical knowledge. One salient feature of these achievements is the penetrating method of treatment, which is i n striking contrast to the mechanical calculation characterizing the work of mathematical economists in previous times. These achievements seem to herald the birth of a modern mathematical school i t i economic theory .

42. SCOPE OF MATHEMATICAL ECONOMIC THEORY As the stationary state and the dynamic movement in a mechanical or electromagnetic system receive the attention of physicists, so to economists the dynamic, as well as static, phenomena i n a national economy as a system of interdependent magnitudes are major subject matters: The economists are keenly interested in the structural characteristics of an economy, on the one hand, and the evolutionary phenomena in the economy, such as fluctuation and growth, on the other. An economist sets up a mathematical model of an economy, which is either aggregated or disaggregated t o a suitable extent, depending on the major objective of his study. When he tries to obtain a rough sketch of the economy, he works on an extremely aggregated model that is expressed by a few simple relations among some aggreg'ttive magnitudes, such as national income, general price level, etc. But the more deeply he wants to penetrate t o the structural charocteristics of the economy, the more disaggregated his model becomes. I n an extremely detailed analysis, a s exemplified by Walras' approach, the economy is reduced to a gigantic system having numerous relationships among an e,normous number of magnitudes corresponding to its atomistic constituents, namely, individual consumers and producers as well as individual workers and capitalists. An economic theory worked out in terms of nggregativc magnitudes is called a macrotheory, whereas that on a disaggregated system taking into explicit account the behaviors of individual constituents is called a microtheory. These manifold economic systems await the application of modern mathematical methods. There is, however, a methodological dualism i n the mathematical study of economic phenomena. At one extreme, people called ~narlieniaticaleconomists perform mathematical, but not particularly numerical, analysis of deterministic and nonstochnstic systems, whereas strtistically minded economists called c,c,o,iotiic,f~ic,iirli.rare chielly concerned with the application of modern statis-

$3.

APPEAL TO CONVEXITY

3

tical methods to the estimation of parameters relevant to economic relationships at the other extreme. I n the judgment of a majority of mathematical economists, the intrinsic internal mechanisms of economic systems are deterministic in nature, though stochastic factors are not completely lacking. They argue that clear insight into the internal mechanisms can be better obtained by analyzing the effects of several interplaying factors of economic significance by means of mathematical reasoning, without worrying about nonessential stochastic features. Thus, they are primarily concerned with models that are not necessarily posed in a form convenient for statistical testing. This is principally for the purpose of obtaining cognitive insight into the working of an economic system. They leave the statistical testing of their theories to econometricians who, on the basis of empirical data, try to statistically fit numerical equations to economic reality. While econometricians compromise with the limited availability of statistical data as well as with the applicability of statistical estimating methods by employing equations of qualitatively simpler type, equations of fuller economic implication are major subjects in mathematical economic theory. This book is concerned exclusively with the nonstochastic mathematical studies of economic phenomena. Therefore appeal is made to qualitativcly oriented mathematical methods in the following chapters, rather than to statistical and numerical methods. Generally speaking, macroeconomists work on simpler models that can be studied by elementary mathematical reasoning. Intermediately disaggregated models, however, as well as completely disaggregated models, raise various interesting mathematical problems of novel types, which are the main sources of the material presented in this book.

43. APPEAL TO CONVEXITY Mathematical systems appearing in modern economic theory are more or less different in nature from those in the physical sciences. Therefore, it is no wonder that the essential properties of economic systems can hardly be brought to light by mechanical and manneristic application of calculus, which has been developed in a form convenient for applications to the physical sciences. We can penetrate deeply into the intrinsic nature of an economic system by examining carefully and seriously the peculiar mathematical properties that the relevant economic laws impose on the system. This is just the line along which workers in modern mathematical economic theory proceed. These peculiar properties relate more or less to convexity, including linearity, in some mathematical entities. For instance, the laws of constant or

4

INTRODUCTION

diminishing returns in production reduce to the linearity or convexity of some functional relationships or some sets. Other examples of convexity inherent in economic phenomena emerge from the peculiarity of constraints that bind the behavior of economic magnitudes. A typical example is the nonnegativity of prices. Negative prices are unjustifiable in ordinary situations. Constraints of this type are represented generally by inequalities, rather than by equations. I n other words, the movement of economic magnitudes is bound in most cases within a set having a boundary. I n such a situation, it is possible that the relevant solution can be achieved just on the boundary, which makes the conventional methods of solution inapplicable. These situations pertain explicitly to some kinds of convexity and can therefore be studied thoroughly only by means of theorems on convexity. There is, however, still another, more profound, reason for our appeal to convexity. Several important structural properties of the Euclidean space in which we describe economic phenomena are deeply rooted in its topological nature, and some of them are already shared by convex sets. The utilization of these properties is indispensable when we intend to go beyond the conventional, but unjustifiable, method of counting relations and unknowns to prove certain fundamental propositions in economic theory, e.g., the existence of general competitive equilibrium. In classical mathematics, convexity did not receive too much attention either, and results relating to it were few, scattered, and fragmentary. The theory of convexity now covers a newly developed theory of linear inequalities and related systems in its algebraic frontier and various types of powerful existence theorems such as fixed-point theorems i n its topological frontier. The opening chapter of this book discusses useful results in the modern theory of convexity to such an extent as will be necessary for an understanding of the subsequent chapters

$4. ORIENTATION FOR CHAPTERS

This book is intended to be an exposition of the fact that there are many important problems pertaining to convexity in economic theory, and to show how fruitfully a few theorems on convexity can be applied to their solution. However, no claim is asserted as to comprehensiveness in the coverage of material, although a deliberate choice is made of a number of representative problems in which the modern mathematical methods-particularly the theory of convexity-find typical applications. Emphasis is therefore placed on methods of solution, rather than on the profound economic elucidation of the problems in question. Accordingly, the discussion of the economic aspects of the problems will be confined to a necessary minimum and kept

$4.

ORIENTATION FOR CHAPTERS

5

on a primitive level, so that even one who has no preliminary knowledge in economics can readily understand their basic economic motivations and backgrounds. To achieve this goal, the chapters in this book are organized as follows: Chapter I furnishes the mathematical background for all the subsequent chapters. This is done by presenting a detailed exposition of a few concepts and theorems pertaining to convex sets i n the Euclidean space, which are of systematic and frequent use in what follows. In Chapter 11, three basic linear systems, Leontief’s system, linear programming, and von Neumann’s model of an expanding economy are expounded. The mathematical contents of Chapter I I are ( I ) the theory of nonnegative matrices and (2) the examination of properties of solutions to systems of linear equations or inequalities of economic significance. A considerable part of the results established under linearity in Chapter I 1 actually does not rely on linearity and can be extended to nonlinear cases. Chapter 111 presents certain of these extensions, mainly to situations where basic functional relations are similar to those in Chapter 11, but where linearity is discarded. Chapter IV, dealing with efficient allocation and growth, begins with the modern formulation of production in terms of production technology sets, together with an exposition of related concepts. Then, two typical turnpike theorems in planned economic growth are proved, namely, a final-state turnpike theorem and a consumption turnpike theorem. Principal functions of Walrasian competitive economies as extremely disaggregative decentralized systems are elucidated in Chapter V. Their static workability is established by proving the existence of competitive equilibria. Then, the following pages deal with socioeconomic aspects of competitive equilibria, particularly their characterization in terms of the core concept in game theory within the institutional framework of private ownership of production possibilities as well as of goods. [The oxposition qf the foregoing is preceded by some preliminary discussions in the opening part of the cliopter, including a modern treatment of consumer’s choice.] Chapter VI deals with one specific property, gross substitutability. By means of variational methods, both the existence of a competitive equilibrium and the validity of Samuelson’s weak axiom of revealed preference between an equilibrium price vector and a disequilibrium one are established. The remaining part of the Chapter presents an exposition of global stability in systems permitting gross substitutes. With special reference to factor price equalization and the uniqueness of competitive equilibrium, conditions for the global univalence of a mapping on its Jacobian matrix are stated in Chapter VII.

Chapter I

\I.\TIIEMATICAL THEOREMS ON CONVEXITY

Chapter I furnishes the mathematical background for all the subsequent chapters. This is done by presenting a detailed exposition of a few concepts and theorems pertaining to convex sets in Euclidean space, which are of systematic and frequent use in what follows. Concepts and results that are readily available in standard textbooks featuring the rudimentary stages of calculus, matrix theory, and point-set theory are assumed to be known by the reader.

$1. EUCLIDEAN SPACES

31 .IEuclidean Spaces

We are principally concerned with sets of simpler structures in the 11-dimensionalEuclidean space R" formed by all points with n real coordinates x = (.yi). R" is cndowed with two important structures: a linear structure and a topological structure. R 1is furnished with a linear structure by the following conventional vectorial operations: If s = (xi), y = (yi)E R", and CI is a real number, then (i>

x

+ y = ( x i+ yi)

(ii)

ax = (.Xi)

(iii)

(x, y ) = 1x i y i

(addition) (scalar multiplication)

n

(inner product)

i= I

(iv)

Ilxll = (x,x)1/2

(Euclidean norm). 6

$1. EUCLIDEAN SPACES

7

An important relation between the inner product and Euclidean norm is Schwarz's inequality I(X,Y)l

5 llxll Ilrll,

with the equality holding if and only if x and y are linearly dependent. On the other hand, the Euclidean norm satisfies (u)

~~x~~ 2 0, with the equality holding if and only if x = 0,

(PI

IlMXII

(Y)

Ilx + Yll I IIXII

= IaI

IIXII,

+ llyll,

where x, y E R",c i and lsll are, respectively, a real number and its absolute value. The standard topological structure is introduced by the concept of convergence in terms of the Euclidean distance d i s k Y ) = IIX - Yll, based on (M)-(y). Explicitly the convergence of a sequence {x"} in R" to u is denoted by either lim

or

X"=U

x"+u

( v + +a)

V-+m

and is defined by lim dis(x', a ) = 0. V'+m

The same topological structure can be given in alternative ways. For example, either the Euclidean norm or the alternative norms

IIxIIM

= max lxil i

induces the same topological structure in R" through the concepts of convergence in terms of the respective distance functions dis(x, y ) = I1x - yll, dis,(x, y ) = IIx - yllA, dis,(x, y ) = IIx - y/l,. This is clear from the relations (1/&>llxll~ S llxll 5 I l ~ l l A , ilxll,~ 5 I l x l l A 5 H I I x I I ~ . Furthermore, the convergence of a sequence {x") in R" is equivalent to the convergence of the n numerical sequences {x,"} of coordinates. The reader is assumed to be familiar with the most basic facts and concepts that immediately follow from the two aforementioned structures and their interaction. In particular, knowledge of such subjects as linear independence, orthogonality, linear equations, linear subspaces, linear transformations, matrices, determinants, open and closed sets, neighborhoods, closures, relative topology, and continuous mappings is assumed.

8

I . MATHEMATlCAL THEOREMS ON CONVEXlTY

41.2. Compact Sets

One of the topological concepts that is of frequent use in the text is compactness. In a space such as R", in which a topological structure is given in terms of a concept of convergence, compactness is defined as follows: DEFINITION 1.1. A set X is compact if any sequence of its points contains a sub-sequence that conrerges to a point in X . One of the reasons why compactness is an important concept is found in Theorems 1.1-1.3. THEOREM 1.1 (Existence of a Maximum Value). A real-ralued continuousjunction f ( x ) on a cornpact set X is hounded and alct~aystakes on a maximum ( minimum) value. Pvoof. Let sup f ( x ) = a over all x E X . It will be shown that a < + co and a point a E X fulfilling f ( a ) = a exists. The definition of a yields the existence of a sequence {x"} in X such that limv++ f ( x " ) = a. Since X is compact, a convergent sub-sequence { x p }can be extracted from {x"}. Let the sub-sequence converge to a. Because J ( x ) is continuous, f ( x p ) converges to f ( a ) , so that f ( a ) = a. Since a, which is the value of the function f ( x ) at a point, cannot be infinity, a < + 00. This proves f ( a ) = max,..f(x) < + co, Q.E.D. uj

THEOREM1.2. (i) A compact subset A4 of a set X i s closed in X . (ii) A closed subset M of a compact set Xis also compact. Proof. (i) Suppose that M is compact. Let a sequence {x'} in M converge to a point x in X . Since M is compact, we can extract from {x"}a sub-sequence {XI'} converging to a point y of M . But any sub-sequence of a convergent sequence must converge to the very limit of the original sequence. Hence x = y , so that x E M . This proves the closedness of M . (ii) Suppose that a subset M is closed in X . A given sequence {x'j in M , as a sequence in X , contains a sub-sequence {x"} converging to some point x of X , because of the compactness of X . The point x belongs to the closure of M . But M is identical with its closure, since M is assumed to be closed. Therefore x E M and we have obtained { x p j as a sub-sequence converging to x in M, Q.E.D. THEOREM 1.3. The image of a compact set X under a continuous mapping f : X + Y is also compact. Pvoqf: Let { y " } be any sequence in f ( X ) . By definition, there are x" in X such that y" = f ( x " ) ( v = 1,2, . . .). By the compactness of X , a sub-sequence {xp}of {x"} converges to some point x in X . Then, the continuity off implies that the sub-sequence { f ( x " ) )of {y"} converges to f ( x ) inf(X), Q.E.D.

$1.

9

EUCLIDEAN SPACES

Let P be a property such that if a sequence {x"} has P, then every subsequence of {x'}also has P. Suppose that there is a sequence {x"} having P in a compact set X . Then, by compactness, some sub-sequence { x " ~of} {x'} is convergent, so that { x " ~is} a convergent sequence having P. Therefore, we may assume, without loss of generality, the original {x"}in itself to be convergent. This useful procedure will be adopted frequently to avoid complicated notations for sub-sequences. In general, the inverse mapping of a one-to-one continuous mapping need not be continuous. For example, the mapping 0 (cos 0, sin 0) sends the interval [0,271) onto the unit circle continuously and in a one-to-one way. Under the inverse mapping, however, points with negative second coordinates on the unit circle never go into [0, 711. The inverse mapping is therefore discontinuous at the point (1,O). A peculiarity of this kind never occurs if the domain of a mapping is cornpact. --f

o j a one-to-one, onto continuous THEOREM 1.4. The inuerse mapping f mapping f : X 3 Y is continuous if X is compact. Pvoof. Supposethat { y " ]converges t o y in Y . Let x" =f -'(y"),x =f -'@). To show that {x"} converges to x, assume the contrary. Then, IJx"s- xi1 2 E (s = 1,2, . . .) for some sub-sequence {xVs} and some E > 0. Since Xis compact, we may, without loss of generality, assume that {xVs}converges to a limit xo in X . By the continuity off, this implies the convergence of {f(xVT)]to y o =f (xo). Now we observe that { f ( x " * )is} a sub-sequence of { y " ) ,so that the limit yo of the former must coincide with the limit y of the latter. Since f is one-to-one, we have xo = x. On the other hand, we have (/xo- XI( >= E > 0 in the limit, which yields xo # X, arriving at a contradiction. Hence lim f -'(y") = is continuous, Q.E.D. f - i ( y ) , and f ~

In R",

"

compact sets " are synonymous to '' bounded closed sets."

THEOREM 1.5. A subset X of R" is compact ifand only if X is bounded and closed in R". Proof. The " only if " (necessity) part is immediate. In fact, by Theorem 1.2, X i s closed in R".Also, by Theorem 1.1, the norm I/xJ/as a continuous function is bounded in X . The " if " (sufficiency) part proceeds as follows. Since X is bounded, we can take a sufficiently large closed cube (n-dimensional closed interval) A = { ~ I a , j x , S b (~i = 1,2, ..., n ) } containing X. The hyperplane Li depicted by equation x,= $(a, + b,) subdivides A into two closed subcubes. These n hyperplanes L , ( i = I , 2 , . . . ,n) together subdivide A to 2" congruent closed subcubes. Consider a given sequence {x"} in X . Since x"(v = 1,2, . . .) are contained in the union of these

10

I. MATHEMATICAL THEOREMS ON CONVEXITY

subcubes, a sub-sequence {x"} of {x"} lies at least on a subcube, which we denote by A l . I n exactly the same way, we can choose a subcube A, of AI and a sub-sequence {x2"} of {x"} lying on A , . Continuing this procedure, we obtain a descending sequence of subcubes A, 2 A2 3 A3 3 . .. 2 Aa 3..., on each of which such a sequence of points {x"} lies such that {x("+')~)is a sub-sequence of {.xA"). Letting, for each i= 1, 2, . . . ,

A A = { , ~ \ a j , , ~ x i ~(i= b A1,, 2,..., n ) } , we see that for each i,

s . . .s u J . i 5 xia 5 b,, s . . .5 b 2 , 5 b l i ,

(i) a l i 5 a Z i

(ii) O 5 bAi- a,, 5 2-'((hi

-

ui).

The n numerical sequences {aAi} are nondecreasing and bounded from above for all i, so that they are convergent. Similarly {b,;} are convergent for all i. In effect, their limits coincide, respectively for i, by (ii). Therefore the diagonal sequences {x;'}, which are placed between the two convergent sequences with common limits, must be convergent. Hence the diagonal sequence (x"} is convergent. In view of the diagonal sequence being a sub-sequence of the original sequence, this implies that a sub-sequence of {x"} is convergent in R". But, since X is also assumed to be closed, the limit of this convergent sub-sequence must belong to X . This finally proves that the sub-sequence is convergent in X . The following sets, which are familiar and typical examples of compact sets in R", will frequently appear in the sequel: (a) the standard simplext

(b) the n-ball (or n-disk)

(c) the

(17

- 1 )-sphere

(d) a cube

t The definition

of a simplex follows in $4.1.

9 1.

EUCLIDEAN SPACES

It

$1.3. Compact Sets-Continued There is a further elaborated, more modern way of defining compactness. This definition is stated in terms of either families of open sets or families of closed sets. It turns out to be equivalent to the definition in $1.2 for sets appearing in this book, though such is not the case for sets of more general structures. Moreover this mode of formulating compactness is very useful in taking advantage of compactness.

DEFINITION 1.2. A fumi/y {GnI /z E A} of open subsets of X is termed an open corjering of X , if X c G, , i.e., more intuitircly speaking, il. Gh together corjer X .

unEA

DEFINITION 1.3. A famify { F AI /z E A} of subsets of' X is said to h a w ( I Jinite intersection property if any finite number of FA hare a point in common. The modern version of the definition of compactness is stated in the following two definitions, which are mutually equivalent in view of the complementary relation between open sets and closed sets.

DEFINITION 1.4 (Compactness in Terms of Open Coverings). A set X is said to be compact if from any open cotiering of X ~ i ' ecan choose a finite number of its members so that they already cooer X . DEFINITION 1.5 (Compactness in Terms of the Finite Intersection Property). A set X is said to be compact, i f for an),family of its closed subsets kauing a .finite intersection property, there is a point of X belonging to all the members of the family. It is useful in providing for later discussion to note the equivalence of the standard notion (Definition 1.1) of compactness to its modern version (either Definition 1.4 or Definition 1S)for sets of simpler structures. Our R" admits a distance function (often called a metric) that is based on the Euclidean norm. It is therefore a typical example of the so-called metric spaces. A metric spate is endowed with a metric dis(x, y ) 2 0 satisfying the following three conditions: (i) dis(x, y ) = 0 if and only if x = y , (ii) dis(x, y ) = dis(y, x) (iii) dis(x, y ) 2 dis(x, z ) + dis(z, y ) , where x, y , z are points of the space. We can make topological considerations in a metric space by introducing a convergence concept there in terms of the given metric, 21s in R".

12

I . MATHEMATICAL THEOREMS ON CONVEXITY

As a preliminary consideration in establishing the equivalence of both definitions, we note the continuity of the distance of a variable point to a subset in a metric space.

LEMMA 1.1. Let X he a metric space and F its nonempty subset. Then, the distance dis(x, F ) oj'a rwiable point x to F, which is defined by dis(x, F ) = inf dis(x, z) owr all z E F, is continuous with respect to X. Proof. By (iii), for x, y E X , z E F, we have dis(x, z) 5 dis(x, y ) dis(y, z). Hence dis(s, F ) 5 dis(x, y ) dis(y, z) for all z E F. This implies that dis(x, F ) 5 dis(x, y ) dis(y, F ) . Similarly we have dis(y, F ) 5 dis(x, y) + dis(x, F ) . These results combined give Idis(x, F ) - dis(y, F ) ) 5 dis(x, y). This proves the desired continuity, Q.E.D.

+

+

+

It should also be noted that a point x belongs to the closure F of Fif and only if dis(x, F ) = 0. Therefore, in the case of F being closed, x E F i f and only if dis(s, F ) = 0.

DEFINITION 1.6. An &-net of a metric space X is a -finite subset {a' I i = 1,2, . . . , s } of X such that the family of &-neighborhoods{ U(a', E ) I i = 1,2, . . . , s} is a cowring of X . Here, U(a, E ) = {x 1 dis(x, a ) < E } denotes an &-neighborhoodof a in X . LEMMA 1.2. I f a metric space X is compact in the sense of Definition 1.1, it has an &-netf o r any E > 0. Proof. Suppose that X had no &-net.Take any one point a'. Since there is no &-net,the &-neighborhood U ( a ' , E ) cannot cover X , so that some a2 E X does not belong to U(a', 8). Again, for the same reason, there is some a3 belonging to none of U ( a ' , F ) and U(a2,F ) . Continuing this procedure, we obtain a sequence {a"} such that a"" 6 U(a', E ) . By construction, the sequence has the property dis(d', a") 2 t. for p # v. Such a sequence has no convergent sub-sequence, contradicting the compactness of X , Q.E.D.

Ur=,

We are now ready to prove the equivalence of both definitions.

THEOREM 1.6. A metric space X is compact in the sense of Dejinition 1.1 if and only if it is compact in the sense of Dejinition 1.4 (Definition 1.5). Proof(Sufficiency). Let {x"] be any sequence of X . If x" equals a common point for an infinite number of values of v, we can extract a special type of convergent sub-sequence, namely such a sub-sequence that all its terms coincide with each other. Consider the other case. Then, the set M = {x" I v = 1, 2, . . .} is clearly an infinite set. We shall show that there is a limit point of the set. If there were no limit point, each point a of X would have a small neighborhood U(u) such that U(a) contains at most one point of M . Clearly { U ( a )1 a E X } is an open covering of X . Since X is compact in the

$1.

13

EUCLIDEAN SPACES

sense of Definition 1.4, it can be covered by a finite number of these neighborhoods. Therefore, M must be a finite set, which contradicts the assumption on M . Let a be a limit point of M . Let v1 be the first number among all v for which dis(x', a) < 1. Next let v 2 be the first number among all v > v1 for which dis(x', a) < 3. Continuing this way, we obtain a sub-sequence {x';} such that dis(xvi,a) < l/i (i = 1,2, . . .). This sequence converges to a. (Necessity). Now suppose that X is compact in the sense of Definition 1.1. Given an open covering {G, I I E A}, we construct the functions on X , f A W

= dis(x,

FA),

where FA = G,' (=the complement of G,). By Lemma 1.1, these functions are continuous. They also satisfy the relation rl>=fA(X)

2 0 (1E A),

where v] is a common positive bound. Since dis(x,y) is readily seen to be bounded on a compact X , any positive bound to dis(x, y ) can serve as an v ] . Next define a function on X by f ( x ) = supf,(x)

over all

A E A.

In view of the uniform boundedness of f A ,f is well defined and satisfies v] z f ( x ) 2 0. Moreover, f ( x ) > 0 at each x. In fact, since {G,} is an open covering of X , each x is contained in at least one member G,, so that x 6 F, = G,'. Fp is closed so that x 6 F, impliesf,(x) > 0. Whencef(x) > 0. Now let 6 = inff(x) over all x E X . Let {x'} be a sequence for which limv++ ol. f ( x ' ) = 6.Because of the compactness of Xin the sense of Definition 1.l, the sequence has a convergent sub-sequence. Without loss of generality, {x'} may be assumed to converge to some point, say, a. In view of the way of definingj; for any E > 0 there is some p for whichfJa) > f ( a ) - (42). On the other hand, by the continuity of this particularufp,we havef,(x) >f,(u) - ( 4 2 ) for x in a small neighborhood U(a) of a. By combining these two relations, we getf(x) > f ( a ) - E for x E U(u). Noting,f(a) > 0 and considering the special case E =f(u)/2, we see f ( x ) > f ( a ) / 2 for x E U(a). Since {s') converges to a, xV E U(u) for sufficiently large v . Thus, 6 = hmV++ f ( s " ) Z f ( a ) / 2> 0. By Lemma 1.2, there is a (6/2)-net {a' I i = 1,2, . . . , s}. Sincef(a') 2 6 > 6/2, for each u i there is some 1.' for which fi.,(u') > h/2, which implies that U(a', 6/2) c G A i .As these U(u', 6/2) ( i = 1,2, . . . , s) already cover X , it can be covered by GAi(i = 1, 2 , . . . ,s), Q.E.D. ~

01.4. Connected Sets

Connection is also of some importance in what follows, although it will not occur as often as compactness.

14


DEFINITION 1.7. A set X is said to be connected if X can never be clecoinpowd to a union of two intersecting, nonempty open subsets. It should be noted that in the above definition, the connectedness of X as well as the openness of the decomposing subsets is considered with respect to the relative topology of X induced by some embracing space. The following equivalent paraphrase of the definition of connectedness immediately suggests itself.

DEFINITION 1.8 (Connectedness in Terms of Closed Sets). A set X is said to be connected if X cannot be decomposed to a union of two iionintersecting, nonempty closed subsets. EXAMPLE 1. A closed intewal [a,p] in R' is connectcrl. Proof. Suppose that it were not connected; then it would decompose to the union of some two nonempty closed subsets F and G having no intersection. The interval [@. /I] is compact. Since F and G are closed i n [ x , p], they are also compact by Theorem 1.2(ii). The left endpoint r is contained in one of F and G, say, F. Consider the continuous function J'(.Y) = .Y on G. By Theorem 1.1, it takes on a minimum at some y E G. y must be larger than x . Otherwise Fand G would contain r = y in common, contradicting F n G = Therefore s( < y and [ r , y) n G, = 0,so that [ x , 7 ) = [ x , y] n F. Since [IL, y] n F is compact by Theorem 1.2(ii), the same function must take on a maximum on [a, y). This is a contradiction, because the function neLer reaches its least upper bound y in the interval [cc, y) not containing the right endpoint.

a.

THEOREM 1.7 (Invariance of Connectedness). The image of' a connected set X under a continuous mapping f ; X -+ Y is also c~onnected. Proof. Were f ' ( X ) not connected, it would decompose to the union of some nonintersecting, nonempty open subsets F and G of j ( X ) . Then, the continuity offimpliesthat the nonvoid s e t s f - ' ( F ) andf'-'(G) are open in X. Moreover, X = f - ' ( F ) v f - ' ( G ) , and f - ' ( F ) n f - ' ( G ) = 0, contradicting the connectedness of X , Q.E.D.

THEOREM 1.8. Let f ( x ) bP a real-raluecl continuous function on a conrzecled set X . If f ( X ) 3 >.,,u, and i. 4 ) are open in X . Clearly they are disjoint, while P 3 f - ' ( A ) # 0, Q 3f-'(p) # 0. I f 5 $ f ( X ) , these two open sets give rise to a decomposition X = P u Q, contradicting the connectednesj of A', Q.E.D.

$2.

CONVEX SETS IN

R"

15

Application. In Theorem 1.8, a special case where X is a closed interval [or, p] means that any intermediate value between f ( a ) andf(B) is taken by f somewhere in [ M , B]. Another application of Theorem 1.8 is effected by taking a connected set in R' as X and the identity mapping from X into R' asf: The result is that with its any two points, X contains all points between them. --t

EXAMPLE 2. The image of [0, I] under a continuous mapping J': [0, 13 X is called an arc. A n arc is connected. Proof. The desired property follows from Example I and Theorem 1.7.

THEOREM 1.9. Let a family of connected sets {G, I A E A} fu@ll the condition 0, , G, # @. Then, the union G = Q, ,,G, is connected. Proof. Let {P,Q} be any decomposition of G to a union of two nonintersecting open subsets. It will be shown that either P or Q is empty. Let a E nG, . One of P and Q, say P,contains a. For each G, , let P, = P n G,, Q , = Q n G, . P, and Q Aare open in G,, so that { P i , , Q,} is a decomposition of G, t o a union of nonintersecting open subsets. Since P , containing a is nonempty and G, is connected, Q, must be empty, which implies that G, = P, c P. Hence G c P and Q = 0, Q.E.D. EXAMPLE 3. A set X is called arcwise connected, if any two points of it can be joined by an arc. An arcwise connected set is connected. Proof. Take any fixed point a of X . X can be regarded as the union of all arcs in X passing through a. Then, X is connected by Example 2 and Theorem 1.9. 82. CONVEX SETS I N R"

42.1 Convex Sets

The definition of a convex set originates in the intuitive perception that it has no hollows, and it reads: A convex set is such a set that contains any segment joining its twopoints. A square playground (Fig. 1) is convex, whereas a gourd-shaped lake (Fig. 2) is not. A folded balloon is not convex, but

FIG. 1

FIG.2

16


it becomes convex when inflated. The definition of a convex set premises an operation of joining two points by a segment, so that convex sets are conceivable only in a space admitting some sort of linear structure. The simplest kind of these spaces is the Euclidean space R". Formally, a segment with endpoints a, b in R" is denoted by [a, b], and defined as the set of all points of the form cta+Bb,

~ 2 0 DZO, , ct+P=l.

An individual point of [a, b] is termed a concex linear combination of a and 6. Naturally the notation [a, b] is modeled after that for a closed interval in R', a special example of segments. In case a = 6, [a, b] reduces to a single point. Now it is in order to state a formal definition of a convex set. DEFINITION 2.1.

A subset X of R" is termed conzlex i f a , b E X implies

[a, b] c X .

There are many examples of convex sets in R". EXAMPLE 1.

R" is convex

EXAMPLE 2. A linear subspace of R" is conilex. Proof. A linear subspace includes, with any two points, their linear combinations, and a fortiori, their convex linear combinations, since the latter are special cases of the former. EXAMPLE 3. An uflne subspace of R" is contlex. Proof. An affine subspace includes, by definition, with any two points u and h, points of the form xu + fib, a + fl = 1, a fortiori, convex linear combinations of a and b as points of the same form with additional conditions ct 2 0, p 2 0. EXAMPLE 4. The solutions of a system of linear inequalities

+.

a L l - ~+,a,, x, . . + aLnx,,2 6, (A E A) form u conwx set in R". The same assertion holds true, ecen when some or all of the inequality signs 2 are replaced by > andjor = , Pvoof. Verification can easily be done by noting the linearity of the functions on the left-hand side in the defining relations. I t is noted that i n Example 4, the system in question may consist of infinitely many linear inequalities, so that the index set A can possibly be an infinite set. Example 4 is important, in that conversely every convex set can be regarded as the set of solutions to a suitable system of linear inequalities, as will be elucidated in $3.

$2.

CONVEX SETS IN

R"

17

DEFINITION 2.2. A convex linear combination of finitely many points ai E R" (i = 1, 2, . . . , s) is a point of the form

THEOREM 2.1. If X i s a coni)ex set, it contains concex linear combinations of$nitely many, but arbitrary, points of it. Proof. By definition, X contains convex linear combinations of its two arbitrary points. Suppose that X contains convex linear combinations of its s arbitrary points, and let us prove that the corresponding statement is true for its s + 1 arbitrary points x i E X (i = 1, 2, . . . ,s + 1). if some of the s + 1 weights x i are zero, the point in question is in effect a convex linear combination of at most s points of X , so that by the induction hypothesis it belongs to X. Next suppose that none of the weights vanishes. In this case, a simple rearrangement gives

cpi=1. S

i= 1

p i x i E X , so that

By the induction hypothesis, S+l

.

1a,x' E

i= 1

[

p i x i , xS"] c X , Q.E.D. i=

1

52.2. Convex Hulls

An arbitrary set X of R" need not be convex and therefore does not necessarily contain its convex linear combinations. However, one naturally expects the existence of a smallest convex superset of X that may be obtained by filling the hollows of X,or more formally, by adding a minimum number of new points to X .

DEFINITION 2.3. The smallest conL3e.x superset of a set X of R", in the sense of set-theoretic inclusion, is termed the comex hull of X , and denoted by C(X). This definition premises the very existence of such a superset, which can be justified by a simple argument based on the following theorem.

THEOREM 2.2. convex.

The intersection of gicen convex sets X, (A E A) is

18


n,.,,

[.Y,

Proof. If x, y E X,,then x, y E X, for any 2 E A , which implies y ] c X, for any E A in view of the convexity of X,. Hence [x, y ] c X,, proving the theorem, Q.E.D.

nAsh

Now let {X,11. E A} be the family of all convex sets including a given subset X of R". Since R" itself is a convex superset of X,the family is actually nonvoid. By Theorem 2.2, , X , is a convex superset of X . Moreover, it is the smallest among the convex supersets of X because every possible convex superset of X is counted as one X,. Thus Theorem 2.2 justifies Definition 2.3, Q.E.D.

0,

THEOREM 2.3. C ( X ) is equal to the set oJ'all coiwcx linear combinations ofpoints of X . Proof. It is readily seen, by a simple computation, that the set of all convex linear combinations of points of Xis convex. For this reason, we may call the set the COIIL'C.Y set generated (or spanned) by points of X.The convex set generated by points of X is a convex superset of X , whence it includes C(X ) . Conversely, C(X)as a convex set contains convex linear combinations of points of itself, i n particular, convex linear combinations of points of X , so that C ( X ) includes the convex set generated by points of X . Therefore C ( X ) coincides with the convex set generated by points of A', Q.E.D. As a matter of fact, in generating C ( X ) from X , every possible convex linear combination is taken into account, regardless of the number of points selected from X.But in effect the n-dimensionality of R" enables us to confine ourselves to considering convex linear combinations of arbitrary but at most n + 1 points of X,as will be stated in Theorem 2.4. To achieve this sharpening of Theorem 2.3, we shall prove Lemma 2.1 on linear combinations of points of R" with nonnegative coefficients, which for simplicity, will be called nonnegative linear combinations.

LEMMA 2.1. ! f a point s is rcpresentrd as u nonncyutiw linear comhiiration of ajinitr number of points xi( i = 1, 2, . . . , s) of R", a suitable nonncJ~yatii c linear combination of some at most ti points among these x iulreacly rc'prcsent\ c. Proof. If suffices to show that if .Y > n , x can be represented as a n o i i negative linear combination of at most s - 1 points among these xi. In I'aLt. since s > n, these x i are linearly dependent, so that there is a nontrivial linwr relation

c pis'

i- I

Let

= 0.

$2.

CONVEX SETS IN

19

R"

Without loss of generality, some f i i may be assumed to be positive. Now let 8 = min cri/pi over all positive p i . Then, we have

x=

t

(.i

- Ofii)X'.

i= 1

For any positive p i , naturally ui 2 Qg,.But in view of the nonnegativity of 0, for any other pi automatically x i 2 O D , . Moreover, a i 2 flpi reduces to mi = dpi for at least one positive f i i . Hence all the coefficients a i - Opi are nonnegative and some of them vanish, so that x is represented as a nonnegative linear combination of at most s - 1 points among xi. Repeating this procedure, we can remove some of x i one by one until a nonnegative linear combination of some, at most n, remaining points xi turns out to represent x, Q.E.D.

THEOREM 2.4. The convex hull C ( X ) of a set X in R" equals all points represented by "+I

.

the

set of

n+ I

CaixL,

Cai=l,

i= 1

i= I

a i 2 o ( i = 1 , 2, . . . , n + l )

as the n + 1 xi independently range over X and the iveights possible values.

(xi

lake on all

Proof. In the light of Theorem 2.3, we have only to show that if a point x is a convex linear combination of s points xi in R" and s > n + 1, x can be represented as a convex linear combination of at most n + 1 points among these given x i . To see the possibility of removing redundant points, we appeal to Lemma 2.1 on the basis of a simple relation between the two concepts of convex linear combination and nonnegative linear combination of points. Let x denote the point of R"+l whose first n coordinates coincide with those of a given x E R" and whose n Ith coordinate is unity. Then it can be seen immediately that x is a convex linear combination of s points xi in R" if and only if the corresponding x is a nonnegative linear combination of the s corresponding points xi in R"". Therefore, the desired result is obtained by applying Lemma 2.1 to R"+l, Q.E.D.

+

It is obvious that a set X is convex if and only if X = C ( X ) . But it must also be noted that a convex set X can be the convex hull of its proper subset. For example, an n-ball is the convex hull of its (n - ])-boundary sphere. Sometimes a convex set can be generated by some of its finite subsets, as is the case with an n-cube that is the convex hull of the set of its 2" vertices. Such :I convex set as is generated by a finite subset is called a convex polyherlrotr (Definition 3.7).

20


02.3. Topological Structure of Convex Sets Among many properties of a convex set, one that most immediately comes to our attention is its connectedness. Explicitly we have the following. THEOREM 2.5. A concex set is connected. Proof. A convex set is arcwise connected in a very specific way, since its two points can be joined by a segment, which is an arc of a special type. Consequently it is connected by Example 3 in 61.4, Q.E.D. However, convexity is a much more specific structural property than connectedness, and therefore entails the very simple topological properties of convex sets. Nonetheless, their explicit statements are useful in the further utilization of convexity. In particular, we shall note the remarkably simple structures of the closure, interior, and boundary of a convex set. THEOREM 2.6. For a c o n w x set X , we hare the following. (i) Its closure X is also c o n w x . (ii) I f a is an interior point of X , and h belongs to X, ecery point of tlie rrgn7ent [a, h], except possibly b, is an interior point of X . Proof. ( i ) Let S,y E x'.Then there are two sequences {x"] and { y " ) in X with the respective limits x and y . For any cx >= 0, fi 2 0 , a + fi = 1, the continuity of linear operations implies the convergence of ax" + fiy" to u x + /?v. Moreover, by the convexity of X , ax" + fly" belong to X in the course of convergence. This proves that ux + py E (ii) Since a is an interior point of X , there is a spherical neighborhood U(u, E ) c X.Except for h, every point of [a, h] can be put in the form

x.

C,

= (1

-

+

t > ~tb, 0 5 t < 1.

In order to see that c, is an interior point of ,'A it suffices to show that U ( c t ,(1 - t ) c ) c X . In fact, let x E U(c,, (1 - t ) e ) ; then, by definition, we have (/x - ctll < ( 1 - t ) ~ On . the other hand, since EX, we can take a point d i n X s o close to h that tllh - d I / < (1 - t ) e - /Ix- ct/l.Hence 1I-x - [(I - t > a

+ td1II 5 Ilx - c,j/ + t / ( b- dll < (1 - t ) E ,

which, if divided by 1 - t , simplifies to

[I

1

l-t

l

(x - l d ) - Ll < E .

From this, we can conclude that e =[l/(l - t ) ] ( x- t d ) E U(u, E ) c X , which implies in turn that x = (1 - t ) e + t c l E [c, d ] c X . This proves that U ( C , ,( 1 - t ) E ) c X , Q.E.D.

$2.

CONVEX SETS IN

R"

21

From Theorem 2.6 we can derive many important facts.

COROLLARY 1. The interior X o of a concex set X is also C ~ I I L ' L ~ X . Proof. If b in Theorem 2.6(ii) is also an interior point, every point of [a, b],without exception, isan interior point. This establishes that [a, b] c X o for any a , b E X o , Q.E.D. COROLLARY 2. There is at most one boundary point on e i w y ray issuing from an interior point a o j a conivx set X . Proof. Assume that there were two boundary points on such a ray. Of these two boundary points, denote by c the one nearer to u and denote the other by b. Then, c E [a, h]. I n view of the fact that a is an interior point, and b as a boundary point belongs to the closure of X , by Theorem 2.6(ii), c must be a n interior point of X,which contradicts the hypothesis that c is a boundary point of X,Q.E.D.

x x"

x

COROLLARY 3. = for a coni'cs set w/tosc interior X O is nonempty. Proof. From X 2 X o immediately follows 3 p. Conversely, take a fixed interior point a of X . Then, for any h E every point of [a, h], except possibly b, belongs to X o , which implies that we can choose points in X o

x

x,

-

arbitrarily close to 6. Hence

x c X o , Q.E.D.

It should be noted that in the foregoing results we meant by the interior of a convex set that taken with respect to the specific embracing space R". In this setting, the interior of a convex set is empty if it is included in a proper affine subspace of R". For such a convex set, the assertion of Corollary 3 is false. However, if the interior at issue is taken with respect to the smallest affine subspace embracing the convex set in question, this awkwardness disappears. c~ DEFINITION 2.4. A conivs body in R" is clc.finctl as a compact c o n i ~ set whose interior is nonernpty tclitli rc.cpect t o R".

From the topological point of view, convex sets are of a very simple structure, as will be stated in the following theorem.

THEOREM 2.1. A n y ttco coni'ex horlies in R" are Itomronrorpltic ; i.e., there is a one-to-one correspondence. continuous in both directions hctwecn tliern. Proof. It suffices to prove the theorem for the case where one of the convex bodies is the unit n-ball C, = { y I llyll 5 1 ). Let X be a convex body in R", and take an interior point a of X . There is at least one boundary point of X o n any ray L issuing from a because L n Xis nonempty and compact, since L is closed, X is compact, and L n X 3 a. Indeed, the continuous function

22


i(x) = / / x- a / /takes on a maximum on L n X a t 6. By convexity, the segment [a, h] lies completely in X,while x on L satisfying IIx - a / /> l/b - all never belongs to X, however close to b it is. Therefore, b E X n x',and b is a boundary point. Furthermore, by Corollary 2 to Theorem 2.6, there is no boundary point on L other than h. Let X h be the boundary of X. Then, X b # 0, X b $ a. Now define the mapping g : X"+ S,, , where S, = { y / ilyll = I}, by

Here g is clearly continuous. If g(x') = g(,r2)for X I , x2 E X " , then by construction, s' - a is a positive scalar multiple of xz - a, so that xi ,!nd x2 are on a common ray issuing from a. Hence x1 = x2 by Corollary 2 to Theorem 2.6, and y is one-to-one. Moreover, for any y E S,, the ray L = {xy + a I cr >= 0) intersects X h at a single point x, and this x goes to y under g. Therefore g maps X"onto S,, . Since X his closed in Xand therefore compact, g - I : S,, + X b is continuous by Theorem 1.4. Finally we define the mapping J': C, -+ X, in view of the convexity of X,by

+

l ~a) a s y + 0. Hence Sincey-'(jjjly1l) is bounded, (1 - Ilj~ll)n l l ~ l l g - ' ( y / ~ / y+ ,f is continuous at 0. Its continuity elsewhere is obvious by definition. Let s E X. If .r = a, then a = f ( 0 ) . If .t- # N, let h E L n X b , where L = { x ( x - a ) + u I x 2 0 ) ; then .r = f ' ( ( / l x- nll/llrr - hlI)g(b)).Hence f' maps C,, onto X . Moreover, ( 1 - IIj.:l)n ~ i y l ~ y f ' ( ~ / lnever l ~ l l ) equals u for j # 0, while it carries distinct points to distinct images again by Corollary 2 to Theorem 2.6, whencef'is one-to-one. Noting the compactness of C,, and appealing to Theorem 1.4, we see t h a t , f f ' is also continuous. ThusJ'serves as a homeomorphism from C, onto X , Q.E.D.

+

DEFINITION 2.5. The ufji'nc~sirhspcrcc sl~utincclhy n conrrx set X it7 R" is rlc:fiiicti as tlic stnallest ufictie suhspacc qf R including X , and is cienotecl by L(X). The definition of L ( X ) is justified by noting that L ( X ) is obtained as the intersection of all the affine subspaces including X.Now let {.yo, XI, . . .,x k j be a maximal finite subset of X such that X I - so,x 2 - -yo, linearly independent. Here maximality means that XI- s x k - .yo, .rk+'- sobecome linearly dependent with the adjunction of any new point x k + ' of X to ( s o x, i , . . . , s h ;A. maximal set of the above type certainly

92.

CONVEX SETS IN

R"

23

exists because of the n-dimensionality of the embracing space R".Then the reader may easily observe that L ( X ) is nothing but the set of all points of the form x"

+

c k

Ai(X' i= I

- XO),

where are real numbers. By the assumed linear independence, each point x of L ( X ) is uniquely expressed in this form, so that Ai can be regarded as functions A i ( x ) of x.The continuity of these functions is also easily seen. If we let U be the set of all points of the above type satisfying A i > 0 ( i = 1 , 2 , . . . , k ) , l i < 1, U turns out to be an open set of X in the relative topology of L ( X ) ; for U = { X \ . Y E L ( X ) ,3.,(x) > O ( i = 1 , 2 , , . . , k ) , 3.i(x) 0 , p i = 1 by letting po = 1 2,, be put in the form p i = Ai. This result means that X is a convex body of L ( X ) in the relative topology, if X is a compact convex set.

xf=I

cr=o

cf=o

~ set X in R" m<wns DEFINITION 2.6. The dimension d i m ( X ) q f ' conre.\the dimension k qf L ( X ) .

THEOREM 2.8. TN.Oc'ompuct c o n r ~ xsets X in R" and Y it1 R"' urc' homeomorphic i f d i m ( X ) = dim( Y ) = k . Pvoof. It is ;t rudimentary fact that L ( X ) car1 be mapped onto R k i n a one-to-one way by a suitable affine transformation qi. Then, since (i, is continuous in both directions, ( i , ( X ) is ii convex body i n R'. Likewise, Y is homeomorphic to a convex body in R", and therefore X and Y are honieomorphic, as was to be shown. Theorem 2.8 enables us to confine ourselves to the most suitable convex set, depending on the case, when we prove a theorem on a topological nature of convex sets. This will be done in subsequent sections of this book. 02.4. Formation of Certain Convex Sets

Theorem 2.2 gives one of the simplest ways to form a new convex set from given convex sets-namely, the convex set a s obtaincd by taking the intersection of given convex sets. I n this section, other simple methods for forming new convex sets from given ones will be discussed. To this end, let us recall the definition of ;I Cartesian product. The Cartesian product of s Euclidean spaces R'" ( i = I, 2, . . . , .\) is the totality of all s-tuples (XI, x2,. . . ,x i , . . . , where x iE Furthermore, a linear structure is introduced to I R''I by the formulas

n;=

R " l .

, ) ' a \ .

(I)

(XI,

. . . , xi, . . . , s') + ( ) % I , = (x'

. . . , J', . . . , J')

+ y',. . . , s' +yi, . .. , xs+ J');

24


(11) (111)

a(x 1 , . . . , x i , .. ., x") = (ax',

. . . , axi, . . . ,axS); ( ( x i , . . . , x i , . . . ) X S ) , ( y i , ..., y', . . . ) y") S

(xi, y')

=

(inner product);

i=l

and a topological structure is given through (IV)

ll(x1,

.. ., x i , . . ., x y

=

1 llxi1[2

112

(i:l

(norm).

Now let X i be a subset of Rni,respectively (i = 1,2, . . .,s). Then we can form the Cartesian product of X i , ni=I X i , in Rni,so that its linear and topological properties may be studied.

nt=

THEOREM 2.9. X i is a (i) I f X is a convex subset of Rni (i = 1,2, . . . ,s), then convex subset of ni= Rni. X i is (ii) If Xi is a closed (open) subset of Rfll (i = 1 , 2 ,. ..,s), then a closed (open) subset of Rni. X i is also (iii) If X i is a compact subset of Rni(i = 1,2, . . . ,s), then compact. Proof. (i) is obvious by the definition of the linear structure (I), (11). To see (ii) and other similar results (which are not given explicitly here), we observe the simple fact that the mapping 7 c i , which is termed a projection,

nfZl n:= ns=l

ns=

(x',

. . . , xi, .. . , xS)-+ 7ci(x', . . . , x i , . . . , x") = x i :

ir

Rni-i R"'

i= 1

is continuous. Since the inverse image of a closed (open) set is closed (open) under a continuous mapping,

ns=l

Rni as the intersection of a finite number of closed is closed (open) in (open) sets, if X i are closed (open) in Rni,respectively. This completes the proof. (iii) Suppose Xi is compact. We recall the characterization of a compact subset in a Euclidean space as a closed bounded subset (Theorem 1.5). In view of the definition of the norm (lV), X i is bounded if Xi is bounded. Moreover, I X iis closed by (ii) if X i is closed. The proof is thus complete. I t is also possible to prove (iii) more directly, not using the characterization of a compact set in a Euclidean space, but this will be left to the reader.

ns=

n:=

$2.

CONVEX SETS IN

R"

25

COROLLARY. The convex hull C ( X ) of a subset X of R" is compact if X is compact. Proof. We form the Cartesian product Pn+ x XI x X , x . . . x Xnfl, where Pn+l is the standard simplex and Xi= X ( i = 1,2, . . . , n + l), and consider the mapping 4 from the Cartesian product into R",which is given by + ( ( p , x',

2, .. ., X " + ' ) )

=

c pixi,

i=

1

wherep = ( p i ) .Then, by Theorem 2.4, C ( X ) is nothing more than the image of the Cartesian product under 4. The Cartesian product is compact by Theorem 2.9(iii), since Pn+l and Xiare compact. On the other hand, 4 is clearly continuous. Therefore, C ( X ) is compact as the image of a compact set under a continuous mapping by Theorem 1.3, Q.E.D.

DEFINITION 2.7. Given subsets Xiof R" and real numbers Ai (i = 1 , 2 ,

. . . ,s), we can construct a new set

which is called a linear combination of Xiand is denoted by

cS=,l iXi.

The linear combinations of given sets Xi include their vectorial sum X i= X , + X , + . . - + X , as a special case. It should not be confused with their set-theoretic union US= Xi. They also include the vectorial difference X - Y of two subsets X , Y c R" as another special case. It is an entirely different concept from their set-theoretic difference X\ Y .

THEOREM 2.10. (i) If X iare convex subsets in R" ( i = 1 , 2 , . . . ,s), their linear combination AiX iis also convex. (ii) If in the above Xiare compact subsets in R", their linear combination is also compact. R" given ProoJ Consider the mapping 4 : R" x R" x .. . x R" (s R"'s) -i bY

c (i) This 4 is linear. By Theorem 2.9(i), nf=X i is convex. Hence xi= liX the image of a convex set under 4 is also convex. (ii) 4 is also continuous. By Theorem 2.9 (iii), ni=lX i is compact. S

4(2, x2, . ..,X S ) = lix i . i= 1 I

as

Thus

I;=Ai X ias the image of a compact set under 4 is also compact, as was

to be shown.

26


It should be noted, however, that the linear combination of closed sets need not be closed. As an example, we consider the vectorial sum X , + X , of two closed subsets X , = ((x, y ) 1 x > 0, y = 1/x2} and X , = ((x, y ) I x < 0, y = i/xz) in R 2 . The origin (0,O) is the limit of the sequence ((v, l/v2) + ( - v , I/?)} ( v = 1, 2, . . .) in X , + X , , while (0, 0) $ XI X , . Therefore XI + X , is not closed.

+

$3. SEPARATION T HE ORE MS FOR CONVEX SETS §3.1. Supporting Hyperplanes

Let f ( x ) be a real-valued function defined in R". Here f ( x ) is assumed to have continuous partial derivatives dfldx,. We take a point a of R" such that one of the partial derivatives does not vanish at a. Then

is known in calculus as a tangential hyperplane at the point a of the hypersurface f ( x ) = f ( a ) . DEFINITION 3.1. A real-calued function f ( x ) dejned in a convex set X in R" is called conuex on X if

f(Ax + PY) 5 If(4 + P for any x , y X ~and

; I2 0 , p

2 0, I

m

+ p = 1.

To continue the above discussion, we further assume our f(x) to be convex on H",and then consider the set X = { x I f ( x ) 5 f ( a ) } . X is nonempty, since it contains a. By the convexity off(x), f ( I x + Ay) 5 If(x) + iif(y) 5 ( A + p ) f ( a ) = f ( a ) for any convex linear combination I x + ZLy of points x , y in X , so that Xis a convex set. Now let x be an arbitrary point of X , and consider x ( t ) = (1 - t ) a t~ for t with 1 2 t 2 0. Let F ( t ) = f ( x ( t ) ) . By convexity, X 3 x ( t ) , so that F ( t ) = f ( x ( t ) ) 5 , f ( a ) = F(0). It follows that

+

In view of F'(0) = equality, we obtain

xr=,[d~dxi],=,(xi ai), letting

for any arbitrary x E X .

-

t + 0 in the above in-

$3.

SEPARATION THEOREMS FOR CONVEX

sws

27

The above relation means that the set X lies in one side of the tangential hyperplane at a. More formally stated, X is contained in one of the two (closed) half-spaces determined by the hyperplane. In view of this result, it is customary to say that in our present situation the tangential hyperplane at a turns out to be a supporting hyperplane at a of X .

DEFINITION 3.2. Let X be a set in R”. A hyperplane a i x i = is termed a supporting hyperplane of X if ( I ) X lies in one of the two closed halfspaces (either mi x i5 p or a i x i 2 /I) and ( 2 ) the hyperplane has a point in common with X . To be more spt.cific, i f a is one of the common poinfs of the hyperplane and X , we use the expression, a “supporting hyperplane at a. ’’

c;=

I:=,

In classical mathematical economics, problems that reduce to the maximization or minimization of some functions were handled by calculus. Its basic procedure of setting the derivatives equal to zero means essentially to seek for a tangential hyperplane or a supporting hyperplane. The crucial point is not the use of calculus but the existence of these hyperplanes, on which the solution of many problems heavily relies. We can handle by calculus the situation on the left in Fig. 3, in which X has a smooth boundary so that

FIG.3

calculus is applicable. But what can be done when we consider the situation on the right in Fig. 3, in which the boundary is not smooth but has flat parts and kinks? $3 will answer the question by elucidating various features of the separation of a convex set by a hyperplane. 53.2. Separation Theorems A standard general form of the separation theorem that we are proving is the fallowing:

28


THEOREM 3.1. Lef X be a convex set in R".If apoint a is not an interior point of X , then there exists a hyperplane ni(xiai) = 0 passing through a such that X is contained in the half-space ni(xiai) 2 0.

I:=, I;=

Here we follow two different methods of proving Theorem 3.1. The first, which is the classical and best-known one, is immediately suggested by the introductory discussion in $3.1 and consists of a careful application of differential calculus. The other, due to Bourbaki (1955, pp. 69-70), is modern and takes more advantage of the topological structure of a convex set. Theorem 3.1 is, however, not posed in such a form as admits the direct application of the above methods. It is therefore convenient to paraphrase Theorem 3.1 slightly, depending on the method of proof in question. The following two theorems are slightly different versions of Theorem 3.1 that are suitable for the application of the two methods of proof. Theorem 3.2 is suitable for the first method, whereas Theorem 3.3 is convenient for the second method.

THEOREM 3.2. Let X be a closed nonvoid convex set in R". If a point a does not belong to X , then we have the following: (i) There exists a hyperplane x i x i = fi such that niai < /3, while the half-space ni x i 2 b contains X . (ii) For the same ni above the hyperplane, ni(xi - ai) = 0 produces the open half-space ni(xi - ai) > 0 containing X .

1

c

1

1

THEOREM 3.3. Let X be an open nonvoid convex set in R".I f a point a does not belong to X , then there exists a hyperplane ni(xi - a i ) = 0 passing through a such that X is included in the open half-space

I:=,

ni(Xi - Ui)

> 0.

i= 1

We first examine their relations to Theorem 3.1. This will be done by showing the logical implication: Theorem 3.1 Theorem 3.2 -+ Theorem 3.3 -+Theorem3.1. (1) Theorem 3.1 + Theorem 3.2. Let 6 = inf IIx - all over all x E X . The closed set X does not contain a, so that 6 > 0. Form the vectorial sum X U(0, 8 ) = M , where U(0, 6) is the 6-neighborhood of the origin. M is convex from the convexity of X and U ( 0 , 6 ) . M is open, since M = U(x, 6) over all x E X . Then a $ M from the definition of 6, so that a is not an interior point of M . Hence, by Theorem 3.1, there is a hyperplane ni(xi- a i ) = 0 such that the closed half-space ai(x, - ai) >= 0 includes M. Choose a number 0 > 0 for which the vector (-eni) belongs to U(0, 6). Then n i ( x i- Oni - a i ) 2 0 for all x E X . Thus the hyperplane n i x i = p, where b = 1ni(ai Oni), produces a half-space x i x i 2 /? including X . It also --f

+

1

u

+

c

$3.

SEPARATION THEOREMS FOR CONVEX SETS

c

2

29

c

xiai 0. This proves (i} in satisfies Theorem 3.2. Theorem 3.2(ii) immediately follows from what has been proved. In fact, ni(xi- a i ) > nixi - p 2 0 for x E X . (2) Theorem 3.2 Theorem 3.3. The closure X of the convex set X in question is also convex by Theorem 2.6(i). It will be shown that a is not an interior point of X . In fact, otherwise there would exist a neighborhood U(a, E ) included in X . On the other hand, we can choose a point c in U(a, E ) n X from the definition of X . Note that c E X = X o , the interior of X , from the openness of X , and that d = 2a - c E U(a, E ) c X . Consequently, we find

2

C

--f

a = +(c

+ d),

c E XO, d E

x.

The above representation of a as a convex linear combination of c, d with positive coefficients implies, by Theorem 2.6(ii), that a E X o = X , contrary to the assumption a 6 X . From what has just been proved, only two cases are conceivable: Case I : a belongs to, and is a boundary point of, 1;Case 11: a $ X . In Case I, we can choose a sequence { a " } such that a' $ X (v = 1,2, . . .) and lim a' = a. Applying Theorem 3.2(ii) to a", X ( v = 1,2, . . .), we can take a niy(xi - ai") = 0 producing a half-space 1niy(xi - a,") 2 0 hyperplane including X. We may assume that 1ni"12= 1 (v = 1,2, . . .). These normals ( x i v ) belong to the unit sphere, whose compactness enables us to further assume that {(niy)}converges to a normal ( n i )# 0. It defines a hyperplane ni(xi - ai) = 0. For any fixed x EX, we have, by construction, C n,"(xi - a,") 2 0 (v = 1,2, . . .), which becomes C n i ( x i- a i ) 2 0 in the limit for any fixed x E 1 ,a fortiori, for any fixed x E X . Each point of X is, however, an interior point, so that no point of X actually lies in the hyperplane. In effect n i ( x i - a i ) > 0 suffices to include X . Finally, in Case 11, Theorem 3.2(ii) directly applies to a, 1 . (3) Theorem 3.3 + Theorem 3.1. If dim(X) < n , X already lies in a hyperplane 1 n i x i = p, since X generates an affine subspace of dimension lower than n. Then, either n i ( x i - a i ) = 0 or ni(xi - a i ) = 0 is the equation of a desired hyperplane. If dim(X) = n, the interior X o of X is nonvoid. Moreover, X o is convex by Corollary 1 to Theorem 2.6. Since a $ X o , applying Theorem 3.3 to a and X o , an open convex set, we get a hyperplane 1xi(xi - ai) = 0 such that n i ( x i - a i ) > 0 includes - X o . Accordingly, the closed half-space ni(xi - a i ) 2 0 includes X o , a fortiori, X , because X c = X o by Corollary 3 to Theorem 2.6. (I) Proof of Theorem 3.2. From the basic properties of the inner product, it is readily seen that for any x, y E R", we find

1

1

-1

1

x

IIX

+ YIl2 + Ib - YIlZ = 2(11x112 + llvll".

30


Let 6 = infllx - all over all x E X . 6 > 0, since X j a and X i s closed. Taking a sequence (x"} in X fulfilling IirnIlx" - all = 6 and substituting xP - a for x and x" - a for y in the above equation on the norm, we obtain

The convex set X contains +(Y + x"), so that

These results altogether entail 0 j llx@- x"/125 2(llxP - a112

+ llx" -

a112

- 262).

The extreme right-hand side of the inequality tends to zero as p, v tend to infinity, since lirnllx" - all2 = limilx" - all2 = d2. This means that {x"} is a Cauchy sequence, and therefore it converges to a point b . This b must belong to X because X i s closed. Whence 116 - all = 6 by continuity, so that we have I I X - all2 2 !lb - all2 = d2 for x E X . The latter part of the proof begins with putting

c n

f(x)

=

(Xi i= 1

- aJ2.

This function satisfies f ( x ) 2_f(b) for x E X . From here on, we follow the argument in 93.1. In fact, since for any fixed x E X , the convex set X contains x(t) = (1 - t ) b tx (I 2 t 2 0), we havef(x(t)) Z f ( b ) . This entails

+

which implies that the derivative of f ( x ( t ) ) with respect to t is nonnegative at t = 0. I n view of

putting n j= b j - a, ( i = 1,2, .. . , n), we have

c n

Xi(&

- bi) 2 0

i= I

for any fixed x E X. We note that (nil = b - a # 0, because a 6 X , b E X . Therefore ni(xiai)= 0 is a hyperplane passing through a. This hyperplane is a desired one, since for any x E X , we have

2

1n,(s,- a i ) = c n

i= 1

Q.E.D

n

i= 1

I*

.i(Xi

- bi)

+i =

ni2 > 0, 1

$3.


31

(11) Proof of Theorem 3.3. The idea for the following method of proof is due to Bourbaki (1955, pp. 69-70), and is applicable to spaces of a more general type than Euclidean space. For our purpose, however, it suffices to develop it in R". We may assume point a to be the origin because such a situation can easily be achieved by a translation. With this assumption in mind, we consider the set

for the given nonvoid open convex set X . Here Axis open for each A > 0, since the mapping x + Ax is a homeomoprhism, so that it sends the open set X to an open set 1.X. Then M as the union of these open sets is open. M has the following properties: (i) (ii)

x u E M for u E M and a

> 0;

M is convex.

In fact, (i) is obvious by definition. To show (ii), we note that if u, z' E M , they are of the form: u = Ax, u = p y for some x, y E X and positive A, p. Given u 2 0, p >= 0 with c( + P = 1, we have aA + p p > 0, whence

E(Cd

+ Pp)X c M ,

by the convexity of X . We have thereby observed that M is an open convex set satisfying (i), but not containing the origin. To complete the proof, we have only to see the existence of an ( n - 1)dimensional linear subspace not intersecting M , since "such a linear subspace " is synonymous with " a hyperplane passing through the origin, one I whose (open) half-spaces contains M." We consider the collection of all linea: subspaces not intersecting M . The collection is nonempty, because thc 0-dimensional linear subspace consisting of only the origin belongs to it. Since the dimension of a linear subspace in R" is at most n, there must be a linear subspace of the highest dimension in the collection, which we denote by L. R"naturally intersects Ma nd therefore is not a member of the collection, which implies dim(L) 5 n - 1. The purpose of the following discussion is to show that dim@) = n - 1. Suppose now that dim(L) 5 n - 2. Let L' be the orthogonal complement of L, and P be the corresponding orthogonal projection on L'. We consider the image P ( M ) in L'. The linearity of P preserves properties (i), (ii), so that P ( M ) also satisfies (i), (ii). Furthermore P ( M ) is open in L'. To see this in a most elementary way, let c*€P(M),c* = P(c), and c E M , and consider the 1-

32


translation 4(x) = x - (c - c*). It is clear that P(+(x))= P(x) for x E R". If y E 4 ( M ) nL', then y = 4 ( x ) for some x E M , so that y = P(y) = P ( ~ ( x ) ) = P(x)€P(M).Hence 4 ( M ) nL' c P ( M ) . 4 is a homeomorphism and 4 ( M ) is open in R". Therefore + ( M ) n L' is an open set containing c* in L', which proves that P ( M ) is open in L'. It can also be seen immediately that P ( M ) $ 0 , since M n L = $3 and P-l(O) = L. Next it will be shown that L'\{O} is connected. To this end, it suffices to see that it is arcwise connected. In fact, take two distinct points x* and y* o f L'\{O}. If the segment joining x* to y* does not contain the origin, this segment can serve as an arc joining the two points in L'\{O}. On the other hand, if the origin is on the segment, take a nonzero vector z* of L' that is perpendicular to the segment. The existence of such a vector is ensured by the fact that dim(L') 2. Then, the polygonal path consisting of the segment joining x* to z* and that joining z* to y* can serve as an arc joining x* to y* in L'\{O}. Now P ( M ) c L'\{O}. P ( M ) is not empty because M is not empty. Moreover P ( M ) is a proper subset. In fact, if P ( M ) = L'\{O}, we have 0 = *x* + f ( -x*) E P ( M ) for any x* E L1\{O} by the convexity of P ( M ) , which is a contradiction. We have thereby shown that P ( M ) is a nonempty open proper subset of L'\{O}. In such a situation, P ( M ) cannot be a closed subset of L'\{O} at the same time. Otherwise the complement P(M)' is also a nonempty open proper subset of L'\{O}, and L'\{O) is decomposed to the union of P ( M ) and P(M)', which contradicts the connectedness of L'\{O}. Hence there must be a point u* in L1\{O} such that u* # P ( M ) and u* E P ( M ) . With this u*, we construct a linear subspace L* defined by L * = { x + / l u * I x € L ,- o o < / l
= 0 } of R" is called the nonnegative orthant. It is a convex cone. It is evident that a convex cone is a convex set. We can associate with every convex cone, a convex cone called its dual; its formal definition will be given below. DEFINITION 3.4. defined by K*

=

The dual convex cone K * of a convex cone K is { y I (x, y )

>= 0

for any x E K } .

The convexity of K * immediately follows from the bilinearity of the inner product. We observe that K * is always closed, while K need not be so. This is because K * is by definition the intersection of closed half-spaces

34


M , = { y I (x, y ) 2 0 } ,x E K. K* has its dual (K*)*,which will be denoted by K** for simplicity. We shall study below the relation of K** to K.

THEOREM 3.4 (Duality for Closed Convex Cones). For a convex cone K we have (i) K * * = , K ; (ii) K** = K, if and only if K is closed. Proof. (i) The very definition of K * states at the same time that every point x of K satisfies (x, y ) 2 0 for all y E K * , and therefore x E K**. This proves (i). (ii) In view of (i), to prove sufficiency we have only to show that a 6K implies a 4 K**. I n fact, suppose that a 6K. Then, since K is a closed convex set, by virtue of Theorem 3.2(i), there is a separating hyperplane (p, x ) = p such that

(1) (P, a ) < P, (2) (p,x) 2 /j' for all

XE

K.

Note that K 3 Ox for x E K, 8 > 0. Therefore, replacing x by 8x for 8 > 0 in (2), we have (p, Ox) 2 p, which, if divided by 6' > 0, reduces to (p, x) 2 p/O. Letting 0 + +a, we have in the limit ( p , x ) 2 0 for any x E K. This means that p E K * . On the other hand, since K 3 0, (2) for x = 0 implies 0 2 /j', whence, by (l), (p, a ) < p 5 0. Therefore a violates the condition for a point to belong to K**, as ( p , a ) < 0 for thisp E K * , so that a 6 K**. Conversely, necessity is obvious, since K** = (K*)* is always closed, Q.E.D.

A linear subspace L of R" is a special case of convex cones. If (x, y ) 2 0 for any x E L , we also have (-x, y ) 2 0 for any x E L . Hence y EL* if and only if (x, y ) = 0 for all x E L . In the case of a linear subspace L, its dual convex cone is nothing but its orthogonal complement L'. Theorem 3.4 therefore extends the usual duality between a linear subspace and its orthogonal complement to closed convex cones. Let us introduce here some new symbols for simplicity. For any two points x = ( x i )and y = (yi)in R", we define x 2 y , if x i 2 y i (i = 1,2, . . . , n), (P) x 2 y , if x 2 y and x # y , (y) x > y , if x i > y i (i = 1,2, . . . , n).

(ci)

These symbols are frequently used in modern mathematical economics literature. It i s also customary that a point x is called nonnegative, semipositive, or positice, according to whether x 2 0, x 2 0, or x > 0. I n many cases it is very useful to separate a set by a hyperplane (p, x) = P having a scmipositive normal p.

$3.


35

THEOREM 3.5. Let X be a convex set containing no positive point. Then there is a separating hyperplane ( p , x) = 0 hacing a semipositice normalp 2 0 such that the half-space ( p , x) 5 0 contains X . Proof. Consider the vectorial difference set M = R," - X . Here M is clearly convex. Next it will be shown that the origin is not an interior point of M . In fact, suppose the contrary and that the origin, together with an .+neighborhood U(0, E ) , is contained in M . Take a point u in U(0, E ) such that u < 0. By assumption, u E U(0, E ) c M , so that u can be written in the form u = v - x for some U E R , " and X E X . After rearrangement, we have x = u - u. This implies that x > 0 because --u > 0, 2) 2 0. This result contradicts the original assumption on X . Now, since the origin is not an interior point of M , there is, by Theorem 3.1, a separating hyperplane ( p , x) = 0 passing through the origin such that the half-space ( p , x) 2 0 contains M . In view of the definition of M , this results amounts to stating that we have

for any u E R," and x E X . Note that the above situation implies that the hyperplane separates Xfrom R,". From this situation we can derive (i) ( p , x) 5 0 for any x (ii) p 2 0.

E

X,

In fact, R," contains the origin, so that (i) immediately follows upon setting u equal to zero in the above inequality. Furthermore, the above inequality also implies that the linear function ( p , u ) is bounded from below on R,". Let 0 be one of its lower bounds. Then, by definition, ( p , u ) 2 0 for all u E R,". Let e' be the point whose coordinates are eJ' = 0 ( j # i), eg' = 1. Clearly Lei E R," for any I > 0. Hence if p = (7-ci), the lower boundedness of the function on R," implies, for Aei, that Ani 2 0 for all L > 0. This result rules out the possibility 7ci < 0, which proves p 2 0. Finally, since p # 0, we have p > 0, Q.E.D. In Theorem 3.5, we cannot always have a separating hyperplane with a positive normal p > 0. For example, the convex set {(x, y) I x2 + ( y + 1)' 5 1) satisfies the assumption of the theorem. It is tangential to the x-axis at the origin, however, so that the only separating hyperplane (straight line) is the tangent y = 0. But there is an important case i n which the existence of a positive separating normal is ensured. This situation can best be explored by applying Theorem 3.5 to a closed convex cone.

THEOREM 3.6. Let K be a closed conuex cone. I f K contains no semipositive u > 0 , then - K * contains a positive p > 0, and rice versa.

36


Pvoof. Suppose the contrary, so that - K * contains no positive points. Applying Theorem 3.5 to the convex set -K*, we see that there exists a separating hyperplane ( u , y ) = 0 having a semipositive normal u 2 0 such that the half-space ( u , y ) 5 0 contains - K * . This amounts to showing that ( u , y ) 2 0 for all y E K * , whence u E K**. By assumption, K is a closed, convex cone, and therefore Theorem 3.4(ii) entails K** = K . Thus, K 3 u 2 0, a contradiction. Conversely, assume that there is some p > 0 in - K *. Hence -p E K *. Then for any semipositive u 2 0 we have ( u , - p ) = - ( u , p ) < 0 , which implies u q! K**. Since K** 3 K by Theorem 3.4(i), u is not contained in K , a fortiori. Theorem 3.6 states that if a closed convex cone K contains no semipositive point, K is contained in the half-space ( p , x ) 5 0 for a positive p > 0. Again consider the counterexample above. The convex set in question contains no semipositive points, while it has no positive separating normal.

Theorem 3.6 leads to important results on linear inequalities.

THEOREM 3.7 (Stiemke, 1915; Tucker, 1956). Let A be an m x n real matrix, a n d p E R”,x E R“. Then we have (i) Stiemke’s Theorem: The system of linear equations A x = 0 has apositive solution x > 0 if and only if the system of linear inequalities A’p 2 0 has no solution p . Tucker’s Theorem: The system of linear inequalities A’p 2 0 and the (ii) system of linear equations A X = 0 with nonnegativity constraint x 2 0 always fiace a pair of solutions p and x such that A’p + x > 0. Proof. In spite of the seemingly wide generality of Tucker’s theorem (ii), it is equivalent to Stiemke’s theorem (i). More explicitly, the i f ” part of (i) is equivalent to (ii), as will be shown below. ( I ) (ii) the i f ” part of (i). Take a pair ( p , x ) satisfying A’p 2 0, A x = 0 , x >= 0, and A‘p x > 0. Then, since by assumption the possibility A‘p 2 0 is ruled out, A’p turns out to be 0, so that x = A‘p + x > 0. Hence A x = 0 has a positive solution. (11) The i f ” part of (i) + (ii). For any p E R“,let [A’pIj stand for the jth component of A’p. Define the set N ( p ) by N(p) = { j I [A’pl, > O}. Since N ( p ) is a subset of the finite set { 1,2, . . . , n } for any solution p of A‘p 2 0, the number of elements of N ( p ) is maximized at some solution $. If N ( p ) = 0, we have the same situation as in (i), and nothing remains to be proved. On the other hand, if N ( $ ) = ( I , 2, . . . ,n } , A’$ > 0, so that the solutions p = fi and x = 0 form a desired pair. For these reasons, it suffices to consider the intermediate case, namely, the case where N ( p ) contains k integers and n > k > 0. To prove (ii), we have only to find a solution x = (xi) of A x = 0 such that “

“

---f

+

“

$3.


37

x j = 0 f o r j E N(fi), while x j > 0 f o r j 6 N(8). In locating such a solution, we may, without loss of generality, assume that N ( j ) = (1,2, . . . ,k}. This situation can easily be achieved by a suitable renumbering of the columns of A . After this arrangement, we let A = (B, C ) , where B is an m x k submatrix and C is an m x (n - k) submatrix. Let us consider the equation Cy = 0, y 2 0, where y E R‘, I = n - k. C satisfies the Stiemke condition that C’q 2 0 has no solution. In fact, assuming the contrary, take a solution q of C‘q 2 0. Recalling that B’8 > 0, C’$ = 0 , we define a new p E R” as a linear combination of 8 and q, by p = 08 + q, where

Then, by construction, we have B p > 0, whereas Clp = OC’fi + C’q 2 0, so that N(p) contains more integers than N(P). This contradicts the definition of N(P), and C must satisfy the Stiemke condition. By the “ i f ” part of (i), Cy = 0 has a positive solution y = j > 0. Then, 2 = (aj), where )zj = 0 ( j = 1,2, . . . ,k), )zj = j j ( j = k 1, . . . , n), is a desired solution of A x = 0. By virtue of the above discussion, it suffices to prove (i). Before proceeding to the proof, however, it is in order to observe the validity of (111) the “only if” part of (i). In fact, if A x = 0 has a positive solution, x > O , the inner product ( A ‘ p , x ) of A‘p and this x is zero because ( A > , x) = $ A x . Since by definition,, ( A > , x ) = [ A ‘ p l i x j , for any A> 2 0 we have (Alp, X ) > 0 because xi > 0 ( j = 1, 2, . . . ,n). This rules out the possibility of A’p becoming semipositive for some p E R“. We are now ready to proceed to (IV) Proof of the “ i f ” part of (i). Let L = (A’p ( p E R“}. Here L is clearly a linear subspace of R” and therefore a closed convex cone. The Stiemke condition that A’p 2 0 has no solution means nothing except that L contains no semipositive point. Hence, by Theorem 3.6, -L*, its dual multiplied by - 1, contains a positive 9.But, as was noted before, L* is also a linear space, namely, the orthogonal complement of L, which entails L* = -L*. Therefore 2 E L* = L’. It remains to determine L’. It can be seen immediately, however, that x EL’ if and only if A x = 0, Q.E.D. If for any solution x of A x = 0, x 1 0, we also let N ( x ) = { j I xi > O } ; the reader may note that Tucker’s theorem can be paraphrased as follows:

+

(a) N ( p ) n N ( x ) = for any solution of A p 2 0 and any solution x of AX = 0 , x 2 0 ; (b) N ( $ ) u N ( 9 ) = {1,2,. . . , n} for a pair of solutions $ and 9 of the above systems. Another fact to be noted is that although the theorems of Stiemke and Tucker were derived from the separation theorems above, it is possible to

38


prove them in purely algebraic ways without appealing to any topological consideration. The original proofs due to Stiemke (1915) and Tucker (1956) are purely algebraic. Numerous applications of Theorem 3.7 to linear inequalities are possible, but only a few of them will be given here for use in later sections, as well as for illustrative purposes. COROLLARY 1 (Farkas, 1902; Minkowski, 1910). Let A be an m x n real matrix, 6 , p E R", x E R". Then a necessary and suficient condition for the equation A x = b fo hare a nonnegatice sohion x 2 0 is that p'A 2 0 implies p'b 2 0. Proof (Necessity). If A x = b for some x 2 0 and if p'A 2 0, p'b = p'(Ax) = (p'A)x 2 0 , as was to be seen. (Sufficiency). Apply Tucker's theorem to the m x (n + 1) matrix ( A , - 6 ) to obtain a pair of solutions p E R" and y E R"+l of y L 0,

The first inequality can be rearranged to p'A 2 0, p'b S 0. However, we also have p'h 2 0 by hypothesis because p'A 2 0. This gives rise to p'b = 0, which, combined with the last inequality, yields Y , + ~> 0. Finally, the remaining inequalities reduce to A x = b, x 2 0 , if we define x = (xi) by xi = for j = 1,2, ..., n,Q.E.D. Corollary 1 is well known as the Minkowski-Farkas lemma.

DEFINITION 3.5. A square matrix T is called skew-symmetric, if T ' = - T. COROLLARY 2 (Tucker, 1956). Let T be a real skewsymmetric matrix. Then, the system of inequalities Tw 2 0, w 2 0 has such a solution that Tw+w>O. Proof. By Tucker's theorem applied to the matrix ( T ' , I ) , where I is the identity matrix of the same order as T , there are solutions p and x of

$3.


If we let u = (xl, . . . ,XJ,

39

u = ( x , , + ~. ., . ,x~,,)', x = (xl, . . . , xZn)',the above

relations become

T p 2 0 , p>=O,

T ' u + v = ~ ,u ~ O , V ~ O ,

Tp+u>O,

p+u>O.

In view of T' = -T, the third relation entails Tu = u . Therefore, by letting w = p + u, we see

TW= T p + TU= Tp + u 2 0,

+ w = (Tp + + ( p + U) = (Tp + U ) + ( p + > 0.

TW

0)

U)

This completes the proof.

COROLLARY 3 (Tucker, 1956). Let A be an m x n real matrix. There are solutions p of A'p 2 0 , p 2 0 , and x of - A x 2 0 , x 2 0, such that x A'p > 0 , p - A x > 0. Proof. We apply Corollary 2 to the skew-symmetric matrix T of order m n defined by

+

+

where the upper left 0 is the zero matrix of order n and the lower right 0 is the zero matrix of order m. There is w satisfying Tw >= 0, w 2 0 , Tw + w > 0. If we let x = (wl, ... , wJ', p = ( w ~ +..~.,, w,,,+,,)', M' = (wl, . . . , w,,,+,)', the above relations yield the desired result. 93.4. Extreme Points

In this section, attention will be drawn to points of a special nature in a convex set, namely its extreme points. As will be made clear, these points form a minimum subset whose convex hull equals a given convex set.

DEFINITION 3.6. A point x of a convex set X is called an extreme point i f x = Ay + p,2 > 0, p > 0,A + p = 1f o r n o yandz E Xexcept f o r x = y = z. More intuitively, an extreme point is a point that belongs to the (relative) interior of no segment contained in X . The important result due to Krein and Milman (1940) about extreme points of a convex set, though valid generally in an infinite dimensional linear space, will be stated only in R". The finite dimensionality of R" makes the corresponding proof extremely easy.

40


THEOREM 3.8. A nonempty compact convex set X in R" has extreme points and is the convex hull of the set of all extreme points. Proof. The proof will be done by induction on the dimension k of X. If k = 0, then X is a set consisting of a single point, and this single point is clearly an extreme point, which naturally generates X . Assume now that the theorem is true for compact convex sets of dimension lower than k. A compact convex set X of dimension k can be regarded as a convex body in Rk. It has been noted that the set of boundary points of X i s not empty (see the proof of Theorem 2.7). Let a be any boundary point. Then, by Theorem 3.1, there is a supporting hyperplane (p,x - a ) = 0 such that the half-space ( p , x - a ) >= 0 contains X . The intersection X , of X and this hyperplane is a nonvoid compact convex set of dimension lower than k , so that by the induction hypothesis, ,'A has its extreme points, and a can be represented as their convex linear combination. Next we shall see that any extreme point c of X , is also an extreme point of X . In fact, suppose c = I y + p z , I > 0, p > 0, I + p = 1 for some y , z E X . Recalling that ( p , x - a ) = 0 is a supporting hyperplane containing c, we have 0 = ( p , c - a ) = I ( p , y - a ) p(p, z - a ) and (p,y - a ) >= 0, ( p , z - a ) >= 0. In view of the positivity of I , p , however, this implies ( p , y - a ) = ( p , z - a ) = 0, so that y , z E X , .Since c is an extreme point of X , ,we must have c = y = z. We have thereby proved that the set of extreme points of X is not empty, and its convex hull contains the boundary of X . But, because every interior point of X is a convex linear combination of some two boundary points, it is, therefore, a convex linear combination of extreme points. Hence the convex hull of all extreme points contains X . But the latter clearly contains the former, so that they coincide, as was to be shown.

+

It should be noted that the set of extreme points of a convex set need not be a finite set. For example, every boundary point of the n-ball C, = {x 1 ilxll 5 1) is an extreme point. But we have THEOREM 3.9. The convex set X formed by all solutions to a finite system of linear inequalities A x >= b has at most a finite number of extreme points, if any, where A is an m x n real matrix and b E R", x E R". Proof. For any x E X , let [AxIi stand for the ith component of Ax, and N(x-) = ( i I [Ax],- bi > O}. Denoting by X' the set of all extreme points of X , we shall show that X', even though nonvoid, is a finite set. This will be done by seeing that N ( x ) # N ( y ) for any distinct points x,y E X'. Once this fact is proved, it can be seen immediately that X' consists of only a finite number of points. In fact, N ( x ) ranges over distinct sets as x runs over X". But for each x, N ( x ) is a subset of the finite set {1,2, . . . , m},the number of whose subsets is clearly finite. This excludes the possibility for X " to have an infinite number of points.

$3.

41


Now it remains to prove that N ( x ) = N ( y ) for x, y E X' implies x = y . We consider separately the following two cases : (a) N ( x ) = N(y) # @. Define 8 > 0 by

8 = min

AX]^ - bi over all i [Ayli - bi

E N(x)

= N(y).

Then we have [AxIi - bi 2 8( [Ay],- bi) for i E N ( x ) = N(y). But this relation is also valid for i # N ( x ) = N ( y ) because [AxIi = [AyIi = bi for i # N ( x ) = N(y). Hence we have A x - b 2 O(Ay - b) 2 0 , 8 > 0. If 8 2 1 , this relation reduces to A x - b 2 A y - b, which simplifies to A ( x - y ) 2 0. Letting z = y + A(x - y ) for a I > 1, we also have A z = A y + AA(x - y ) 2 b, entailing x = (l/A)z ( 1 - l/A)y, l / A > 0, 1 - 1/1> 0, y , z E X. Since x is an extreme point, we must have x = y = z. On the other hand, if 1 > 8, we have A ( x - 8y) 2 (1 - 8)b, so that A z 2 b, z = ( x - 8y)/(l - 8). This implies x = 8y + ( 1 - 8)z, 8 > 0, 1 - 8 > 0, y , z E X. Again, since x is an extreme point, we must have x = y = z. (b) N ( x ) = N ( y ) = 0.In this case we have A x = A y = b, so that A ( x - y ) = 0. From here, we can proceed similarly as in the discussion for the case 8 2 1 in (a), to conclude that x = y , Q.E.D.

+

I

Theorem 3.9 does not assert the existence of extreme points. A synthesis of Theorems 3.8 and 3.9 will be THEOREM 3.10. If the set X of solutions of A x 2 b in Theorem 3.9 is nonempty and bounded, then X is the convex hull of the nonempty3nite set of extreme points. Proof. Since X i s a nonempty compact convex set under these assump-, tions, by Theorem 3.8, it has extreme points, which together generate X. But by Theorem 3.9, the extreme points are finite in number. This completes the proof. DEFINITION 3.7. A convex set is called a convex polyhedron if it is the convex hull of a finite set. A finite set is compact, so that by the corollary to Theorem 2.9, a convex polyhedron is always compact. In this terminology the X in Theorem 3.10 is a convex polyhedron.

DEFINITION 3.8. A convex cone K is calledpolyhedrd if it is generated by ajinite number of points ai (i = 1, . . . , s), i.e., K is the set of all points

ZAid,

Ai 2 0 ( i = 1 , ..., s ) .

i= 1

Put differently, a polyhedral convex cone K is the set of all points Ap, p ranging over all nonnegative points for a constant n x s matrix A .

42


THEOREM 3.11. Let A be an rn x n real matrix and x E R". The set X 2 0 is a polyhedral convex cone. Pvoof. Denote the rank of A by r(A). We first consider the case r ( A ) = n ; then we extend the result to the general case. (a) r ( A ) = n . If X consists of only the origin, X is clearly a polyhedral convex cone. Let us consider other cases. Choose an rn-dimensional positive vector p > 0. Then the set X, of all solutions to the system of'all solutions fo A x

Ax 2 0,

p'Ax 2 1,

AX 2

-1

is nonempty and bounded. Note first that the last two relations are equivalent to p'Ax = 1 . Also, since p' > 0, for a point x E X, A x = 0 if and only if p'Ax = 0. Moreover, since r(A) = n, Ax = 0 if and only if x = 0. As X contains a nonzero point x, we have p'Ax > 0 for this x, and x/p'Ax E X,. This proves that X , # 0. For x E X p , the ith component fulfills l / p i 2 [AxIi 2 0 (i = 1, . . . , in), where p i > 0 is the ith component ofp. Thus A X , is a bounded set. Since r ( A ) = n, this implies the boundedness of X,. By Theorem 3.10, X,is a convex polyhedron. Let b', . . . ,bs be extreme points of X,. For each nonzero x E X , we have x/p'Ax E X p , so that x/p'Ax is a convex linear combination of b', . . .,bs with suitable coefficients. Thus, X is contained in the polyhedral convex cone generated by b', . . ., b". But clearly the latter is included in X , so that they coincide. (b) r ( A ) < n. Let L = { x 1 A x = O } . L is a linear subspace of dimension t = n - r(A) and has a basis {c', . . . , c'}. Hence L is a polyhedral convex cone generated by 2t points c', . . . , c', -cl, . . . , -c'. Let P be the n x n matrix corresponding to the orthogonal projection to L. Then the rank of the ( m in ) x n matrix

is n. For if A x = 0, Px = 0, then x E Land x = Px = 0, entailing x = 0. Therefore by Case (a), the set of all solutions to A X 2 0, Px 2 0 is a polyhedral convex cone K generated by some b', . . . , 6'. We observe that K c X . Moreover, if x E X , then x - P x E K. For A ( x - Px) = A x - APx = A X 2 0, P ( x - P x ) = P s - Px = O because A P = O , P 2 = P . Hence x - P x is a linear combination of b', . . . , 6' with nonnegative coefficients, while Px is a linear combination of c', . . . , c', -c', . . . , -c' with nonnegative coefficients. From this we can conclude that X is a polyhedral convex cone generated by these h', ..., b', c', .. . , 8,-c', . . . , -8, Q.E.D. Now a complete characterization of a polyhedral convex cone is possible.

$3.


43

THEOREM 3.12. (i) Duality: K** = K holds for a polyhedral convex cone K. (ii) A convex cone K is polyhedral if and only if K is the set of all solutions to a Jinite system of homogeneous linear inequalities. Proof. (i) It is possible to prove directly that K is closed, thereby seeing by Theorem 3.4(ii) that K** = K . But here it is more convenient to appeal to Corollary 1 of Theorem 3.7. Let K = {Bp Ip 2 0, p E R“}), where B is an n x s matrix that spans K. Then, K * = { y 1 B’y 2 0 } because B’y 2 0 if and only if y’Bp 2 0 for allp 2 0. In view of the Minkowski-Farkas lemma, the corollary mentioned above, this amounts to saying that x E K if and only if <x,y> 2 0 for all y E K * , proving (i). (ii) Theorem 3.11 proves the sufficiency part, so that it remains to prove the necessity part. If K = {Bp J p2 0 , p E R”}as in (i), then K = (K*)*. By Theorem 3.1 1, however, K * = { y I B’y 2 0 } in itself is a polyhedral convex cone, so that we may let K * = {Cq 1q 2 0,q E R“},in terms of an n x m matrix C. Clearly K = (K*)* = { x I C ’ x 2 0). Hence K is the set of all solutions to the system A x >= 0, where A = C‘. This completes the proof. In the light of Theorems 3.4 and 3.12, the reader may remark that “ a closed convex cone” is synonymous to “ a system of (possibly infinitely many) homogeneous linear inequalities,” while a polyhedral convex cone is a precise equivalent to a finite set of homogeneous linear inequalities. For an obvious reason, a convex cone has no extreme points except possibly for the origin. The origin may or may not be an extreme point of a convex cone, depending on the structure of the cone. DEFINITION 3.9. A convex cone K is said to be pointed if K does not contain x and -x at the same time for any nonzero X . From the definition it is clear that K is pointed if and only if x + y = 0 for some x, y E K implies x = y = 0. This is equivalent to saying that the origin is an extreme point of K.

THEOREM 3.13. Let K be a convex cone in R“. Then we have (i) If K * has an interior point, K is pointed. (ii) If K is closed and pointed, K * has an interior point. (iii) r f K is closed andpointed, there is somep E R“,such that ( p , X ) > 0for all nonzero x E K. (iv) For any p in (iii) and any number a, the set { x I x E K, ( p , x) 5 M} is bounded. Proof. (i) If K were not pointed, K would contain some nonzero x together with -x. This implies (x, y ) = 0 for all y E K*. K * is therefore contained in a hyperplane, which contradicts the assumption that K* has an interior point.

44


(ii) Assume that the interior of K* is empty, and K * is contained in a hyperplane ( a , y ) = 0. Since ( a , y ) = (--a, y ) = 0 for all y E K*, a and - a must belong to K**. But by Theorem 3.4 for a closed convex K, K** = K. Whence K 3 a, - a and a # 0, implying the nonpointedness of K, which is a contradiction. (iii) In this case, by (ii), the interior of K * is not empty. It therefore suffices to show that every interior point p of K* is a desired element. By definition, there is some E > 0 such that K * 3 p 4- u for all u fulfilling llull < 2e. Hence, if x is a nonzero point of K , letting u = - ~ x / \ l x l l ,we havep + u E K * for this u. Accordingly, 0 5 (p f u, x ) = ( p , x) - E I I x I I , whence (p, x ) 2 ellxll > 0, as was to be shown. (iv) If A = {x I x E K, (p, x ) 5 a } were unbounded, there would exist a co.For large v, llx'll > 0, so that sequence {x'} in A with 1imllx"ll = ix"/IIx"~~ E K and 0 S ( p , x"/Ilx"Il) 5 t(/llx'II. By the familiar reasoning for a compact set, {x"/llx"II} may be assumed to converge to some x . In view o f K being closed, letting v --f + co we end up with a contradiction: ( p , X ) = 0, 0 # x E K, Q.E.D. 03.5. Convex Functions

A category of functions that is closely related to convex sets is that of convex functions. The convexity of a function has already been given in Definition 3.1. It can be defined equivalently as follows:

DEFINITION 3.10. A real-valued function f ( x ) deJined in a convex set X in R" is called convex on X ifthe set X , = { ( x , a) I a 2 f (x),x E X } is a convex set in R" x R. Suppose that f is convex in the sense of Definition 3.1. For any ( x , a), ( y , p) E X , we have a L f ( x ) , /3 2 f ( y ) , and x , y E X , so that, by Definition 3.1, Aa + p p 2 Af(x) pf(y) 1f ( A x p y ) for any A, p 2 0, 1 + p = 1. Whence A(x, a) + p(y, /I) E X , , and f is convex in the sense of Definition 3.10. Conversely, Definition 3.10 implies Definition 3.1. In fact, observing that ( x i , f ( x i ) )E X , for x i E X (i = 1, ...,s), and in view of the convexity of the set X , and Theorem 2.1, we see Ai(xi,f(x'))E X , for Ai 2 0, Ai = 1. Thisimplies C A i f ( x i )2 f ( C A i x i ) by the definition of X , .

+

+

In this section, certain fundamental properties of convex functions will be listed. In the following the domain of definition X of a convex function is a convex set in R", although this basic assumption will not be stated explicitly each time. First of all, we are concerned with the interesting connection between the convexity of a function and its continuity and differentiability.

$3.


45

THEOREM 3.14. L e t f ( x ) be a convex function defined on X , where X is a convex set in R". Then, f ( x ) is continuous at any interior point of X . Proof. The proof will be worked out in such a way that we can take as much advantage as possible of the results already established for convex sets. Since we are concerned with the continuity off at any interior point of X , it will be assumed that X o , the interior of X , is not empty. Under this basic assumption, X , also contains an interior point in R" x R. Otherwise, the convex set X , would be included in a hyperplane p'x

+

.< = y.

Then, the (n + 1)th coefficient .n of the hyperplane must vanish, since X , ,with a point ( x , a), contains all points (x, p) for p 2 a. It follows that X must be included in the hyperplane p'x = y in R", contradicting the assumption X 0 # 0. Now, let us take any fixed interior point a of X . We shall characterize boundary, interior, and exterior points of X , only on the vertical straight line 1 passing through (a, 0). First, we observe that (a,f ( a ) )is a boundarypoint. In fact, (a,f (a) + E ) E X,, (a,f (a) - E ) 4 X , for any E > 0 can approach (a,f (a)) arbitrarily closely as E -+ 0, whence (a,f ( a ) ) belongs to the closure of X, as well as to that of its complement, so that (a,f (a)) is a boundary point of X , . Second, it will be shown that there is at least one interior point of X , on 1. To this end, choose an interior point (e, y) of X, , whose existence has been shown above. Since a E X o , we have b=a

+ d(a - e ) E X

for a small 0 > 0, resulting in the representation of a as a convex linear combination of e and b, i.e., 0 1 11 = a=Ae+pb, I=1 +e' 1+0' If we take some (b, p) in X , , by Theorem 2.6(ii) as applied to (e, y) and (b, p) (4 I v

+

= 4 e , v ) + A b ? B)

is an interior point of X, . This point clearly lies on 1. Let us choose an interior point (a, w ) of X, on 1 once and for all. Since (a, w ) E X, , we have by definition w 2 f (a). But, since (a,f (a))is a boundary point, we must have w >f (a). Then any point (a, a) on I that satisfies a > f ( a ) is also an interior point of X , by Theorem 2.6(ii), since (a, a) is a convex linear combination of (a, w ) and a suitable point in 1 n X , . On the other hand, any point (a, a) on I that satisfies a f ( a ) , (a, a) is a boundary point of X , if a = f ( a ) , an exterior point of X , if a < f ( a ) .

I

The assertion of the theorem is an immediate consequence of this characterization. I n fact, for any E > 0, the point ( a , f ( a ) E ) is an interior point of X , . Whence Xf includes a neighborhood V of ( a , f ( a ) + E ) of radius 6 > 0 in R" x R . Now, consider the neighborhood U(a, 6) of a of the same radius 6 in R". If x E U(a, S), we have (x,f(a) E ) E V c X , . This implies that

+

+

f ( x )s f ( a )

+

E

for x E U(a, 6)

2

X.

Similarly, using the characterization of ( a , f ( a ) - E ) as an exterior point of X , , i.e., an interior point of the complement of X,, we can find a neighborhood U(a, 6') such that f(x) > f ( a ) - E

for x E U(a, S')

3

X.

Therefore, letting 6" = min(b, S'), we have

If(x) -f ( a )1 2 E for x E U(a, 6")

2

X.

This completes the proof. It should be noted that Theorem 3.14 can easily be paraphrased as follows: A convex function defined on a convex set X in R" is continuous in the relative interior of X , which is defined to be the set of all interior points with respect to L ( X ) , the affine subspace spanned by X (Definition 2.5). On the other hand, Theorem 3.14 is no longer true in an infinite dimensional linear space. The following example illustrates such a situation. The set of all polynomials #(f), $ ( t ) , . . . of a real variable t on the unit interval 1 2 t 2 0 is an infinite dimensional linear space with the norm 11411 = supl$(t)l over all t. Defining f(4)= I x , I for # ( t ) = z a i t i , we have a convex function f(4).But d l Y ( f ) = t" - f " + l converges to 0 in the above norm, whilef(4,) = 2 does not converge to 0 = f ( O ) . The source of this trouble is the fact that although X , contains no interior points, it is included in no proper closed linear subspace. Whence neither Theorem 3.14 nor its method of proof as given above is valid in an infinite dimensional space. Finally it should also be noted that Theorem 3.14 does not ensure the continuity of a convex function at a boundary point of X . For example,

1

1

f(f)

= (t2

( t = 0) (1 2 t > 0 )

$3.


47

is a convex function on the unit interval. It is discontinuous at the left endpoint t = 0. Next we consider convex functions with respect to differentiability. As a piecewise linear convex function such as f ( t ) = It I illustrates, a convex function need not be differentiable even at an interior point, but one-sided differentiability at any interior point immediately follows from convexity. THEOREM 3.15. Let f ( t ) be a convex function of a real variable dejined on an interval I. Then we have

(i) I f r , s, t E I and r < s < t, the following inequalities hold:

(ii) I f u is an interiorpoint of I,the right-hand derivative f+'(u) and the lefthand derivative f - ' ( u ) exist. Moreover,f - ' ( u ) 5 f+'(u). (iii) Iff is diflerentiable at v E I, then we have, for u E I ,

f '(v>(u- 4 S f W -f (4. (iv) I f u , v are interior points of I and u < v, then

f-'(4 S f + '(4 s f - '(4S f + '(4. Proof. (i) We have the expression s=-

t-s s-r r+t, t-r t-r

so that by the convexity off,

This, if multiplied by t - Y > 0, becomes ( t - r ) f ( s ) 5 (1 - s)f(r>+ (s - r > f ( t > ,

which can easily be rearranged to the desired inequalities. (ii) Since s is an interior point of I, there is an E > 0 for which (s - E , s + E ) c 1. Applying the first inequality in (i) to r = u, s = u + k,, t = u + k , satisfying 0 < k , < k, < E , we have f ( u + k,) - f ( u > < f @+ k l ) - f ( u ) kz kl

48

1. MATHEMATICAL THEOREMS ON CONVEXITY

Similarly, we obtain

for h,, h, satisfying - E < h, < h, < 0. On the other hand, applying

f ( r ) -f(s> r-s

. (iii) If u > u, putting r

= u,

t = u in (i) and letting s+ u, we have

$3.


49

which, if multiplied by u - v > 0, becomes f’(v)(u - v) S f ( u ) -f ( v ) . Similarly, if v > u, putting r = u, t = v in (i) and letting s -+ v, we have

If multiplied by v - u > 0 and rearranged, this reduces to the same result. (iv) It suffices to see that f+’(u) s f - ’ ( v ) . To this end, we note f ( s ) - f W < f(s> -f(t) s-r s-f ’

a rearranged form of (i). Applying the above inequality to r = u, t = v, and s satisfying u < s < v, and letting k = s - u, h = s - v , we obtain

If we recall the proof for (ii), we see that

Whence f+‘(u)5 f-’(v), Q.E.D.

COROLLARY 1. For a function f ( t ) dejined in an open interval I, we have b II,e f is convex in Z ifand only ifthe derivativef’ is (i) Zf f is d ~ ~ e r e n ~ i ain nondecreasing in I. (ii) r f f is twice dif5erentiable in I, f is convex in I if and only i f f ” is nonnegative in I. Proof. (i) I f f ’ exists, f-’(u) =f+’(u) =f’(u), f-’(v) = f+’(v) =f ’ ( v ) in Theorem 3.15(iii), yielding f ’ ( u ) 5 f ’ ( v ) . This proves necessity. Next suppose that f‘ is nondecreasing in I . To prove the convexity off, it suffices to show that for any a, b E I with a < b, the graph off lies below the segment joining ( a , f ( 4 >to ( b , f ( b ) )in [a, bl, i.e.,

In fact, by the mean value theorem in calculus, there is some 6 between a and b such that

50


Then, sincef’ is assumed to be nondecreasing, we have

Whence the desired majoration is readily seen by noting that f ( a ) = g(a), f ( b ) =g(b), where g ( t ) is the linear function on the right-hand side of the desired inequality. (ii) It is well known in calculus thatf’ is nondecreasing in Z if and only if f” is nonnegative in I, Q.E.D. COROLLARY 2. Let f(x) be dejined and have continuous partial derivatives up to the second order in the interior X of a convex set X in Rn.Then f is convex in X o if and only if the quadratic form

is positive semidefinite in X in the sense that the form takes nonnegative values for any X E X o and any t i (i = 1, . . . , n ) . Proof. Let a E X o . Then we have, for any vector h = (hi),a + th E X o if J t Jis sufficiently small, say, It1 < E. l ffis convex in X o , so is F ( t ) = f ( a + th) in (-8, E ) . Whence by the preceding Corollary l(ii), F”(0) = x f i j ( a ) h i h j2 0. Conversely, let a, b E X o and h = b - a. Since a, b are interior points of X , there is E > 0 such that ci + tlz E X o for - E < t < 1 + E. Then, by the same corollary, F ( t ) = f ( a rh) is convex in ( - 8 , 1 + E ) because F”(t) = ~ f i j ( a + t h ) h i h j ~ O . I f w e n o t e t = ( -l t ) . O + t . l , a + t h = ( l - t ) a + t b , F(0) = f l u ) , F(1) =f(b), the inequality (1 - t)F(O) + tF(1) 2 F((1 - t ) . 0 + t . 1) is easily converted to (1 - t ) , f ( a )+ rJ‘(b)zf((1- t ) a + tb), (a, b E X o , 1 2 t 2 0), Q.E.D. Remark 1. A special version of convexity is strict convexity. A convex functionfon X i s said to be strictly convex, if Af(x) + pf(y) >f ( A x + py) for x, y E X , x # y , A > 0, p > 0, 2 + p = 1. We can adapt the above two corollaries so as to ensure strict convexity. This is effected by replacing the assertion “f is convex ” by f is strictly convex and replacing the conditions, “f ’ is nondecreasing,” “f” is nonnegative,” and “ the quadratic form is positive semidefinite ” by “f’ is strictly increasing,” “f” is positive,” and “ the quadratic form is positive definite,” in the sufficiency parts of the corollaries respectively. Remark 2. As is well known in matrix theory, a quadratic form 1 a i jhi hj with aij = a j i is positive definite if and only if all the principal minors of the

+

“

”

$3.

51


matrix A = ( a i j )are positive. The form is positive semidefinite if and only if the corresponding principal minors are nonnegative. Remark 3. A function f ( x ) defined in a convex set X in R" is said to be concazie if for x, y E ,'A I & 0, p LO,I + p = 1,

f ( 2 x + w )2 Y(x>+ P f ( Y ) . It is said to be strictly concave, if it is concave and for x , y p>O,il+p=I, f(b + PY) I f ( x > + Pf(J9.

E

X, x # y , I > 0,

'

Clearly, the concavity and strict concavity off correspond to the convexity and strict convexity of -f, respectively. Therefore, all the above results can easily be adapted to the corresponding results on concave functions.

83.6. Constrained Extrema Lagrangian multipliers are often used in calculus to convert constrained maxima and minima to unconstrained maxima and minima; but attention is generally confined to local extrema under the differentiability assumption on relevant functions. There has recently been a strong demand among mathematical economists for the use of Lagrangian multipliers in the reduction of global constrained extrema to global unconstrained extrema without differentiability. In this section, a typical result meeting this demand will be stated for the case in which the maximand (minimand) and constraint functions are concave (convex). Consider a maximization problem: maxf(x)

subject to g i ( x ) 2 0 ( i = 1, ...,m),

wheref, g i are real-valued functions on a nonempty subset X of R". Introducing Lagrangian multipliers I i ( i = 1, ... ,m ) , we set up m

f(x) + C Iigi(x), i= 1

the Lagrangian associated with the maximization problem. THEOREM 3.16. If 2 maximizes the Langrangian over X and sutis$es g i ( 2 ) 2 0 , Aigi(2) = 0 ( i = 1, . . . , m ) for a set of nonnegative numbers li2 0 ( i = I, . . ., m ) , then 2 aflords a solution to the original constrained maximization problem. Proof. From the assumptions, we have

52


over all x E X . Whence, if x E X further satisfies gi(x) 2 0 (i = 1, . . . ,m), the extreme right-hand side is not less than f ( x ) , since l i g i ( x )2 0 because of the nonnegativity of l j (i = 1, . . . , m). This proves thatf(9) Z f ( x ) for all x satisfying gi(x) 2 0 (i = 1 , ... ,m), Q.E.D. In Theorem 3.16, specific properties off, g i such as their concavity, even when explicitly assumed, have no bearing upon its validity. But a crucial role is played by the concavity of the relevant functions in the following converse proposition.

THEOREM 3.17 (Uzawa, 1958). Suppose that X i s convex, and that the maximand f and the constraint functions gi (i = 1, . . . , m) are concave in X . Assumefurther the Slater condition that there is a point c in X fu&lling g,(c) > 0 ( i = 1, . . . ,m). Then, i f 2 is a solution to the constrained maximization problem, there are nonnegative l i (i = 1, . . . , m) such that ( a ) l i g i ( 9 )= 0 (i = 1, . . . , m), (p) 2 maximizes the Lagrangian over X for these l i. Proof. We define a mapping # : X + Rm+lby the formulas

40(4 =f ( x )-f@>, 4 i ( ~=)gi(x) (i = 1 , ...,m), where (Pj(x) are the coordinates of #(x). Consider then the set Y = { y Iy 5 4 ( x ) for some x

E

X}

in Rm". First of all, we note that Y contains no positive vector. In fact, if 0 5 y 5 #(x) for some x E ,'A we have f ( x ) 2 f(2), gi(x) 2 0 (i = 1, .. . , m). But, since 2 solves the constrained maximization problem, the first of these inequalities reduces to f ( x ) =f(2), which implies &(x) = 0. Thus, y 2 0 for a y E Y entails yo = 0, so that the inclusion of a positive y is ruled out. Next we verify that Y is con,vex. Using the concavity of # i immediately implied by that of J g i , we see, for a convex linear combination w of two points y l , y 2 E Y with coefficients a1 2 0, a, 2 0. a1 -+ a2 = 1, that since y 1 5 +(xl), y 2 5 4 ( x 2 )for some x', x2 E X ,

x

= a1x1

w

= a,y'

+ a2x2 E X , + a,y2 I $(x') + a, 4(x2)5 #(x). a1

This proves the convexity of Y. By virtue of Theorem 3.5 as applied to Y, there is an (m + 1)-dimensional semipositive vector p = ( x i ) such that p'y 5 0 for all y E Y . This implies in

$4.

53

FIXED-POINT THEOREMS

particular that p'4(x) 5 0 for all x E X , since 4 ( X ) c Y. Recalling the definition of 4, we therefore have m

no(f(x)

-f(a)> + iC Xi gi(x) 5 0 = 1

(X E

W*

It will be noted that no must be positive. Otherwise the above inequality would become c n i g i ( x )5 0 ( X E X ) . Hence, for c fulfilling g i ( c ) > 0 (i = 1, . ..,m), whose existence is ensured by the Slater condition, we have 0 d C nigi(c) 5 0. Whence ni = 0 (i = 1, .. . ,m) because ni 2 0, g i ( c ) > 0 (i = 1, ...,m), yielding a contradiction p = 0. Letting l i = ni/no (i = 1, . . .,m) on the basis of the positivity of no and taking ligi(2)2 0 into account, we can reduce the established inequality to

Thus the proof will be complete, if (a) is shown. (a) is, however, readily seen. In fact, by substituting 2 f o r x in the aboveinequality, we obtain C l i g i ( $ ) 0, which implies (a) because of the nonnegativity of i i ,g i ( 2 ) , Q.E.D. Remark. It is easy to formulate the counterparts of Theorem 3.16 and Theorem 3.17 for a constrained minimization problem, with the concavity of f(x), g i ( x ) replaced by their convexity.

$4. FIXED-POINT THEOREMS

04.1. Simplexes

We begin with the consideration of convex polyhedra of the simplest type.

DEFINITION 4.1. A set of points x i (i = 1, . ..,s) in R" is said to be afinely independent if for s real numbers l i (i = 1, . . . ,s), the relations S

1 l i X i = 0, f,d i = 0

i= I

i= I

imply li=O

( i = l , ..., s).

Letting x denote the point of R"" whose first n coordinates coincide with those of a given x E R" and whose (n + 1)th coordinate is unity, as in the proof of Theorem 2.4, we can immediately observe the following: (a) x i (i = 1, .. . , s) are affinely independent, if and only if the corresponding xi are linearly independent.

54


(b) The maximum number of points forming an affinely independent set in R" is n 1. (c) A point x belongs to the convex hull of { X I , x 2 , . . . , x'} if and only if the corresponding x is a nonnegative linear combination of XI, x2, . . ., xs.

+

From the above remarks, it is obvious that a finite set { X I , x2, . . . ,x ' } spans a k-dimensional convex polyhedron if and only if { X I , x 2 , . . . ,x '} contains a set of k + 1 affinely independent points but no set of m affinely independent points for m > k + I . As a matter of fact, if k is the dimension of the generated convex polyhedron, we have s 2 k + 1. The special case s = k + I therefore entails a convex polyhedron spanned by k + 1 affinely independent points, which is of the simplest kind among k-dimensional convex polyhedra and is called a simplex. Formally we have

DEFINITION 4.2. The convex hull of k + 1 aflnely independent points = 0, 1, . . . , k) that is the set of allpoints expressible in the form

x i (i

k

k

C,Iixi,CAi=l, i=O

,IizO ( i = O , I

,..., k),

i=O

is culled a k-dimensional simplex spanned by these points xi and is denoted by x 0 x 1' .. xk. Each of tliese generating points x i is termed a vertex of the simplex. From the affine independence of its vertices immediately follow these facts : (a) The vertices are extreme points of the simplex, and there are no inore extreme points than the vertices. (p) The expression of a point of the simplex as a convex linear combination of its vertices is unique. The corresponding coefficients ,Ii in the convex linear combination of the point can therefore serve as its coordinates and are termed the harycentric coordinates of the point.

A k-dimensional simplex is a natural analog to familiar configurations in the Euclidean space having dimension of at most 3 ; A 0-dimensional simplex is a point, while a 1-dimensional simplex xoxLis the segment with end points x o , X I . A 2-dimensional simplex xox1x2is the triangle with vertices xo,x l , x2, whereas a 3-dimensional simplex x0x'x2x3 is the tetrahedron with vertices xo, X I , x2, x 3 .

DEFINITION 4.3. A simplex xiOxil . .. xisis called an s-dimensionalface xoxl . . . x k ,ifthe set of vertices of the former { x i O x, i [ ,. . . ,xis} is a subset of that ofthe latter {xo,x l , . . . ,x k } .

of

$4. FIXED-POINT THEOREMS

55

A k-dimensional simplex clearly has + I C, + I s-dimensional faces, where C, + stands for the number of ways to pick up s 1 distinct vertices out of k 1 vertices. It can also be readily seen that the relative interior of a k-dimensional simplex consists of the points all of whose barycentric coordinates are positive, whereas its relative boundary is the union of all its ( k - 1)dimensional faces. If we define the diameter 6 ( X ) of a set X i n R" by 6 ( X ) = supIlx - yII over all x, y E X , we have

+

k+

+

LEMMA 4.1.

6(xox1* * . x k ) = maxoji,j s k I(xi- xjll

for

a

simplex

.

~ 0 x 1 . .x k .

Proof. F o r a n y a = x : = , a i x i , b = ~ j k , O ~ j x j E ~ oxk,we ~ ' . - have -

k

clipjJJxi - xiJJ

_
0, it has an &-netN , = {aE'I i = 1, . . . ,s,}. Next we choose an arbitrary point be' off(a"). This being done, we define s, continuous functions O:(x) on X by O:(x) = max(e - IIx - a"'(l,O) ( i = 1, . . . ,se).

These functions are nonnegative. Moreover, their sum is always positive for the following reason. Since N , is an &-net,for each x we have E > IIx - a"'(1 for

68

1. MATHEMATICAL THEOREMS ON CONVEXITY

some i, so that Uf(x) > 0 for this i. With these functions we can obtain s, weight functions

c s.

WF(X) =

u:(x)/

U,C(x) ( i = 1 ,

. . . , s,).

j= I

Now, using these weight functions wiC(x), a single-valued continuous mapping y ( x ) can be defined by the formula

c SC

f'(x)

=

wf(x)b"'.

i= 1

X 3 P i ( i = I , . . . ,s,) and wiC(x) 2 0, 1W ; ( X ) = 1 imply f"(x) E X because of the convexity of X . We have thereby obtained a single-valued continuous mappingf" : X + X for every E > 0. By the Brouwer fixed-point theorem (Theorem 4.3), there is a fixed point xE,x E =f"(x"). Let us now apply the above results to a sequence ( E ~ }of positive numbers with limit E , = 0. Since X is compact, the corresponding sequence of fixed points {x'~},x E v = f " v ( ~ E i , ) , contains a convergent sub-sequence with a limit 2. Hence, without loss of generality, we may assume that we have chosen a sequence { E , } of positive numbers fulfilling (a)

lim

E,

= 0,

,-+CC

(p)

lim xEY= 2, ,-++a,

(7)

XC" - f E q X E u ) .

We shall show below that 2 is a desired fixed point off. To this end, consider the set 0, =f(2) U , , where U, = { u I llull < S} for a 6 > 0. If 2 E 0, for any 6 > 0, we have dis(i,f(2)) = 0, which entails 2 ~ f ( 2because ) f'(2) is closed in X . The subsequent discussion will clarify that 2 E 0, for any s > 0. First let us note that 0, is an open set containingf'(2). This can be seen by noting that 0, = (x U,), the union taken over all x ~f(.?), and also the openness of U , . The next thing that must be noted is the convexity of 0,. This is true because 0, is the vectorial sum of the two convex setsf(2) and U , (see Theorem 2.10). Now by Lemma 4.4(ii), f is U.S.C.Therefore, since 0, is an open set containingf(2), there is an &-neighborhoodV , = {x I I/x - 211 < E , x E X } of 2 such that f'( V,) c 0,. By ( a ) , (p), we have E , < 4 2 and xEYE V,,, for large v. - xEVII< E , < 4 2 , so that For these large v, M.:'.(X'") > 0 implies

+

u

+

IjO""' -

211 5

IlUC"'

<E/2

-XEq

+E/2 =

+ I ) X E " - 211 E.

$4.

69


In summary, for large v, we have aCviE V, for i with E f ( a E u i ) cf(V,) c 0,. In view of

>

W ~ ~ ( X " * ) 0,

which entails

x Eturns ~ out to be a convex linear combination of only b".' lying in 0, for large v. The convexity of 0, therefore implies xEYE 0, for large v. Letting v tend to infinity in view of (p), we have in the limit 2 E O,, . The replacement of 6 by 26 in the resulting relation is due to the possibility that 9,the limit of { x E W }may , lie on the boundary of U , . In effect, however, 2 E 0,, for any 6 > 0 is equivalent to 2 E 0, for any 6 > 0, whence 2 E f ( 2 ) in the light of the preliminary discussion above. This completes the proof, Q.E.D. It might be of some interest to touch upon the von Neumann version of generalization of the fixed-point theorem. This will be presented here as a corollary to Theorem 4.4. COROLLARY (von Neumann, 1937). Let X a n d Y be nonempty compact convex subsets of R" and R", respectiz>ely. Further, let M and N be closed subsets of the Cartesian product X x Y, such that M , = { y I ( x , y ) E M } and N , = ( x I ( x , y ) ~ N }are nonempty conuex sets for any X E X and y E N , respectively. Then, M n N # @. Pvoof. Consider the set-valued mapping f :X x Y + 2'" defined by f ( x , y ) = N , x M,. If this mapping fulfills the assumptions of the Kakutani fixed-point theorem, it has a fixed point (a,?) E N9 x M 3 . From 2 E N j follows (a, j ) E N . Likewise j E Mt implies (2, j)E M . Therefore (a, j ) E M n N , so that M n N # @. Let us see that all the assumptions of the Kakutani theorem are satisfied. In fact, X x Y is a nonempty compact convex set in R" x R" because it is the Cartesian product of nonempty compact convex sets X and Y in R" and R", respectively. Likewise, from the assumptions on M , and N , , it is clear that f ( x , y ) = N y x M , is convex for each (x, y ) E X x Y. The closedness of the graph G , remains to be shown. The graph is a subset of X x Y x X x Y, defined by G , = ( ( x , y , u, P ) 1 (u, v) E ~ ( xy,) } . Since the relation (u, L-) E f ( x , y ) decomposes into two relations u E N , and E M , , which reduce to (u, y ) E N , (x, D) E M , G , can be characterized as G, = {(x, y , u, r ) I (u, y ) E A', (x, E M } . The mappings 4 : X x Y x X x Y + X x Y given by #(.u,y, u, r ) = ( u , y ) and $ : X x Y x X x Y - t X x Y given by $(x, y , u, = (x, r ) are clearly continuous. I n terms of 4 and $, G , is therefore representable as G , = 4 - ' ( N ) n $ - ' ( M ) . The closedness of N and M implies that of $ - ' ( N ) and $ - ' ( M ) ,whence G , must be closed as the intersection of two closed sets, Q.E.D. zi

21)

13)

70


1t should also be noted that conversely the Kakutani fixed-point theorem in itself can be implied by the above corollary. In fact, the Kakutani theorem is a special case of the corollary where X = Y, M = G,, N = {(x, x) I x E X } . In this case, the points in M n N are nothing but the fixed points off.

34.4. Operations on Mappings

In many instances, it is not only useful but also almost necessary, cspecially in applying the fixed-point theorems, to construct new mappings with given ones through certain operations. The operations on mappings to be discussed here are ( I ) Cartesian product, (2) composition (product), (3) extension, and (4) replacement of image sets by their convex hulls. All the mappings to be considered are assumed to have nonvoid image sets. (1) Cartesian Product. Let f ' : X + Y ; be k set-valued mappings ( i = 1, . . . , k). Then a mappingf': X - t I Y , , which is called the Cartesian product of,f', is defined by

n:=

k

THEOREM 4.5.

(i) (ii)

I f f " are closed, tlicir Cartesian product J' is also close(/. Suppose that the image sets oJ.,f' are compact. Then (f the f are u.s.c., their Cartesian product f is nlso U.S.C.

PVOOJ

'

(i)

Let

(j;1 , y 2 , . . . , yk), lim

(J.", j.", . . . , y k v )E , ~ ( x ' ) , Iim(y'", y2', . . . , y k " )= s" = S . For each component mapping f i , this implies

y J "E,/"(.Y"), lim yJ' = y', lim X" = .v. Whence, by its closedness, we have y' ~f''(.v), which implies (yl, j 2 . . . .,J,') E,/(.Y). This proves the closedness off.

(ii) First we have to show that if A i is ii compact subset of Y i ( i = 1, . . . , k ) , and if 0 is an open set in Y , containing A ;, then there is an open set U iin Y , containing .4; ( i = 1, . . . A-), such that A; c U ; c 0. Lety = ( y ' , y 2 , . . . , y k )represent the generic point of Y i , and let A = A'. Since A c 0, no point of .4 belongs to Or , the complement of 0. Two cases will be considered separately. Case ( x ) : 0 = Y i .I n this case, U i = Y i will do. Case ( p ) : 0 is a proper open subset of Y , . I n this case 0' # 0.

n.

n n n n n

n

n

Since 0' is a closed set, dis(y, 0') > 0 for each y E A . On the other hand, A is compact since A ; is compact for each i. By Lemma 1.1, dis(y, 0') is a continuous function of y , so that it attains a positive minimum 6 > 0 on A . Let U , = ( j iI y i E Y , , dis(y', A ,) < S / k ) . U i is an open subset of Y ; containing A ' . Its openness follows from the continuity of dis(y', A ; ) as a function of j,', Mhereas A i c U isince dis(j%',A , ) = 0 for jiE A ; . Then, in view of the

$4.

71


n

definition of the norm in Y i (see §2.4), we have dis(y, A ) = dis(y', A i ) < 6 for y = ( y ' , y 2 , . . . ,y k ) , yi E U , , which implies y $O', i.e., y E 0. Hence Aic U i c 0. This being done, let 0 be any open set in Y i containingf(x) = n j i ( x ) . By the result above as applied to A , =fi(x), there is an open set U i in Y , containing f'(x). By the upper semicontinuity o f f we can take an open set V , in X containing x such that f '( V,) c U i ( i = 1, . . . , k). Then V = V i is an open set in X containing x such that f i( V ) c U i , which implies f( V ) c f '( V ) c U ic 0, as was to be seen, Q.E.D. (2) Composition of Two Mappings. Let f : X - , 2' and g : Y + 2z be two set-valued mappings. Then their composite mapping g f : X - t 2' is defined by

n n

n

,,

n

n

n

THEOREM 4.6. (i) Suppose that a compact subset M of Y exists such that M 2 f ( X ) = f (x). Then, iff and g are closed, their composite 'napping g j is also closed. (ii) I f f and g are u.s.c., their composite mapping g f is also U.S.C.

u

XE

Proof. (i) Let i"~ g f ' ( x " )(v = 1,2, . . .), lim z" = z, lim x" = x. Then, by definition, there is a sequence { y ' } in Y such that zv ~ g ( y " )y" , Ef ( x " ) (v = 1, 2, . . .). But in reality {y"} is a sequence in M , which is compact. Therefore, without loss of generality, {y"} may be assumed to converge to a limit y in M . Hence from y' E f ( x ' ) (v = 1,2, . . .), limy" = y , lirn x" = x, and the closedness of f follows y E f ( x ) . Similarly, from zv E g(y") (v = 1,2, . . .), lim z' = z, lim y' = y follows z E g(y). These results imply z E g f ( x ) . (ii) It will be shown that the set V = {x 1 g f (x) c U 1 is open in X for any open subset U ofZ. Let W = { y Ig(y) c U } .Then, clearly V = {x I f ( x ) c W } . Since g is u.s.c., the openness of U implies that of W , which in turn entails the openness of V because of the upper semicontinuity off. This completes the proof, Q.E.D.

It should be noted that in special cases eitherf, g , or both are single-valued and continuous. (3) Extension of Mappings. Let two mappings f : X - , 2" and f:r? 2y be given. f is termed an extension off if (a) r? 2 X , (b) f ( x ) =f ( x ) for x E X . Given a mapping f : X + 2', which admits some sort of continuity, and a superset r? of X , can we extend f to r?, without losing its continuity property? Consideration of this problem in general is not easy and calls for much more --f

72


advanced topological considerations. Therefore, as far as the extensibility of a set-valued mapping is concerned, we will limit ourselves to the following theorem dealing with a very simple situation. This theorem should be sufficient, however, for our present purpose. THEOREM 4.7. A closed mapping f : X a closed mapping f : r? + 2' if X is a dense subset o f f , (E) (p) Y is compact.

--f

2' can always be extended to

Proof. Let G, be the graph off. Then, by assumption, G, is closed in X x Y, but need not be closed in r? x Y. Let G, be the closure of G, in 2 x Y . Define for any x E r?,

Since G, is a closed subset of X x Y, we have G, = Gr n ( X x Y ) ,which impliesf(x) = f ( x ) for any s E X.Next let x be a point of r? not belonging to X . Since X is dense in 2,there is a sequence {x")in Xconverging to x.Choosing a pointy' off(.u") for each x", we obtain a sequence { y " }in Y, which, without loss of generality, may be assumed to converge to a limit y because of the compactness of Y . The sequence {(x', y')} in G, therefore converges to (x,y ) , so that (x,y ) E G,, which implies f ( x ) # 0even for x E r? but $ X . We have thereby obtained an extensionfoffto r? such thatf"(x) # $3 for any x E 2.Its closedness is obvious, since G f ,the graph off, is nothing but G,, which is by x Y. This completes the proof. construction closed in

I n the light of Lemma 4.4, Theorem 4.7 also states the extensibility of a U.S.C.mapping to r?, with its upper semicontinuity preserved, whenever its image sets are closed in Y, while X , r?, and Y satisfy the same conditions. (4) Replacement of Image Sets by Convex Hulls. From a given mapping f : X + 2', a new mapping f, : X - t 2", which may be called its coni'ex hull, is obtained by

f,(x) = convex hull of f(x). In the above construction of the convex hull, Y is assumed to be a convex set, so that it makes sense to form the convex hull off(x). THEOREM 4.8. The conoex hulf,f, of a if

U.S.C.,

(a) f (x) is compact f o r cacti x E X , (8) Y is a coflrcx set.

U.S.C.

m a p p i n g f : X-+ 2' is also

$4.

73


Proof. In the light of the Corollary to Theorem 2.9, the compactness of

f ( a ) for each a by ( u ) ensures that of its convex hull, hence the compactness of fc(a).Let 0 be an open set in Y containingf,(a). It will be seen that there is a convex open set U in Y such that 0 3 U 2 fc(a).This is obviously true if 0 = Y , since Y is convex. Next suppose 0 # Y . Then, 0",the complement of 0, is a nonempty closed set in Y, so that dis(y, 0")> 0 for y Ef,(a), Hence the continuous function dis(y, 0")takes on a positive minimum 6 on the compact f,(a). Let U = { y l y E Y , dis (y,f,(a)) < 6/2}. Then, U is open in Y, since dis(y,f',(a)) is a continuous function of y. U clearly containsf,(a). Moreover, U n 0' = implies U c 0. On the other hand, U is convex, since it is representable as the vectorial sum of two convex sets, explicitly U = (f,(a) { u I l\ull < 6/2}) n Y . In summary, U is a convex open set in Y such that 0 2 U 2 h ( a ) . Let us now recall that f,(a) x f ( a ) , whence U ~ f ( a )Sincefis . u.s.c., there is an open set V in X containing a such that f ( x ) c U for any x E V . But, in view of the convexity of U , this implies fc(x) c U c 0 for any x E V , proving the upper semicontinuity off, at an arbitrary point a of X , Q.E.D.

+

Again in the light of Lemma 4.4, Theorem 4.8 also ensures the closedness off, for a closed f'so long as Y is a compact convex set. Finally, useful syntheses of Theorems 4.7 and 4.8 will be given below. COROLLARY 1. Let f : X - + 2' be a closed mapping such that Y is a compact coniiex set, (j?) each image set f ( x ) is convex. Then, if X is a dense subset of 8 , f can be extended to a closed mapping 4 : 8 -+ 2" in sucIi a way that eacli image set of 4 is coni'ex. (a)

Proof. By Theorem 4.7, f can be extended to a closed mapping?: 8 4 2'. But by Lemma 4.4(ii), this f is also U.S.C.and its image sets are compact. Therefore, by Theorem 4.8, 4 = f c , the convex hull off, is a U.S.C.mapping from 8 into Y whose image sets are compact. By Lemma 4.4(i), 4 : 8 2' is closed. This 4 is a desired extension. In fact,fl(x) = f ( x ) for any x E X . But f ( x ) is assumed to be convex, so that its convex hull coincides with itself. Hence, + ( x ) = f,(x) = f ( x ) for any x E X . This completes the proof. --f

Likewise we can derive from Lemma 4.4 and Theorems 4.7 and 4.8 the following corollary, whose proof, however, may be omitted: COROLLARY 2. Let f :X-+ 2' be a U . S . C . mapping sucli that Y is a compact convex set, ( p ) each image set f ( x ) is a closed convex subset of Y . Then, if X is a dense subset of 8, f can be extended to a U . S . C . mapping : r? --f 2' in such a ~ * athat y each image set of 4 is a closed c o n w x subset of Y. (a)

74

1. MATHEMAl ICAL THEOREMS ON CONVEXITY

$5. HOMOTOPY AND EXTENSION THEOREMS $5.1. Homotopy and Extension of Mappings

Here we shall again be concerned with single-valued continuous mappings, with major attention being paid t 3 their homotopyand extensibility. The final objective is to prove a special case of the Kronecker existence theorem.

DEFINITION 5.1. A coniinuous mapping f : X 3 Y is suit1 to be hoinotopic to anotlirr oiic y 1 X + Y if' tlwre is a continuous mapping H : X x [0, 11 --t Y sudi that H ( s , t ), .IE X , I 2 t 2 0 satisfies ( m ) H ( x , 0) =,/(x), s E X . ([I) H ( x , I ) = .Y(x), .Y E X . Tlic .function H is ccrllerl n homotopj bridge hetween f unrl g. Intuitively, the homotopy between two mappings .f and g means that f can be obtained by a continuous deformation of y. I n fact, { H ( s , t ) I t t; [0, I]} is a family of continuous mappings from X into Y , indexed by t . When t varies by a small amount, the corresponding mapping may deform a little, but not too much. Expressed more geometrically,J'aiid g can be joined by an arc in the space of all continuous mappings from X into Y. Homotopy can also be equivalently rephrased in terms of the extensibility of some mapping. With two given ,/; g : X + Y , we construct a continuous mapping h, by li(.\-, 0) =./'(.I-), h ( s , 1) = g ( s ) . The mapping h is defined, with its values in Y , on the subset of X x [0, l]consisting of the points (s,t ) fulfilling either t = 0 or t = 1. The homotopy between faiid y amounts t o stating the extensibility of 11 to the whole X x [0, I], the extension being nothing but a homotopy bridge H between f ' a n d g. Consider the totality of all continuous mappings from X into Y, and denote it by M . A mapping,fin M is clearly homotopic tof'itself. 1f.f; E M , and.f'is homotopic toy, so isg tof: On the other hand, letf; y, / r E M . Iffis homotopic to y and y is homotopic to li, thenf'is homotopic to 11. Of these three statements, the first two are obvious. The last one is true for the following reason. Let P(.I-, t ) and Q(X, 1) be homotopy bridges between .f and y and between y and 11, respectively. Then, J' is homotopic to h by the homotopy bridge R ( s , t ) given by R(x, I ) =

21) I Q ( x , 2t - 1 ) \QX,

if if

$>=t>=O, I 2 t 2 4.

Homotopy is therefore an equivalence relation and gives rise to a classification in M . A homotopy relation depends on the structure of Xand Y . In a trivial case all the mappings in M are mutually homotopic. A typical example is the case

$5.

75

HOMOTOPY A N D EXTENSION THEOREMS

where Y is convex. If Y is convex, any two mappingsf; g E M , are homotopic by the homotopy bridge

H ( x , t ) =( 1 - t ) f ( x )

+ tg(x), 1 >= t

0.

From the homotopy-theoretic point of view, the case where X and Yare spheres is important and interesting. The following theorems are some of the most elementary results (within the scope of this book).

,

xi2 5 1 ] and THEOREM 5.1. Consider in R,the n-hall C,?= {x 1 its boundary ( n - I)-sp/1crc~ S, = {A- I xi2= 1 ) . Then, a continuous mapping 4 : S,, + S, is homotopic to a c~oristantmapping that sencls all points o j S , to a common point of S, iJ'nncl only if 4 can he extended to a continuous mapping @ : C,+S,.

,

Proof. (i) Sufficiency. Let 0 : C,,-, S,, be an extension of 4. Then, 4 is homotopic to the constant mapping sending S, to @(O) by the homotopy bridge H ( x , t ) = @ ( ( I - t ) . ~ ).Y, E S,, , I 2 t 2 0. ( i i ) Necessity. Let H(.Y, t ) be a homotopy bridge between 4 and a constant mapping sending s,,to a single point a of S,,. Define the mapping @ : C,, -,S , by

Its continuity is easy to prove. Moreover, @(.I-) an extension of 4 to C,.

=

4(s) on S,,, whence

@ is

THEOREM 5.2. The B r o i i , ~ ' c r ~ x e ~ l - p otlworPm int (Theorcm 4.3) is equirulent to tlw following statement: The identity mapping s -,s : S, --f S, is not lionlotopic to a constant mapping. ProoJ In the light of Theorem 5.1, it suffices to prove the equivalence of the nonextensibility of the identity mapping to C,, to the Brouwer fixed-point theorem. ( i ) The Brouwer theorem -+ nonextensibility. If the identity mapping had an extension F : C,, -+ S,, the continuous mapping G : C, + C, defined by G(x) = - F ( x ) , x E C, would have no fixed point, contradicting the Brouwer fixed-point theorem. The nonexistence of fixed points of G is seen as follows: Since in effect G sends C, to S,,, a fixed point, if any, must lie in S,. On the other hand, G ( s )= -.Y in S,, i.e., every point of S,, is sent by G to its antipodal point, and therefore cannot be fixed. This proves that no point remains to be fixed under C. (ii) Nonextensibility-, the Brouwer theorem. It suffices to prove the Brouwer theorem for C,,, a special convex body in R". Let f : C,, -+ C,, be a continuous mapping and suppose that f has no fixed point. Consider the

76


mapping F : C,,+ S,, defined by F(x)denoting the point at which the ray issuing fromJ'(x) in the direction of the vector x - f ( x ) meets the boundary sphere S,,. F is continuous and is an extension of the identity mapping from S,, into S, to the whole C,, . This contradicts its nonextensibility. In the terminology of topology, a set X is contractible if the identity mapping from X into X is homotopic to a constant mapping. Intuitively, a contractible set is a set that can be shrunk to a single point by a continuous series of deformations. A spherical rubber membrane, while sticking fast to a solid spherical ball, can deform to some extent, but not to a small flat piece of rubber, without getting torn. Theorem 5.2 is a rigorous statement of this noncontractibility of a sphere. While a sphere, though without holes, is noncontractible because it embraces a solid ball, a set of a much simpler structure as a convex body is contractible. I f CI is an interior point of a convex body X , then X can be shrunk to u by the homotopy bridge H(x, t ) = (I - t ) x + ta, X E X ,

1

ztzo.

We shall next put part of Theorem 5.1 in a slightly more general form, which is useful in the subsequent discussion.

THEOREM 5.3. Let C,, arid S,, he tlie n-hall und its ( n - 1)-boundary splrerc. Supposr that two continuous mappings j ; g : S, -+ X,where X is some set, are lrotnotopic, ana' that J' is c>xtensihleto C, . Tlim g is also extensible to C,, in such u,fa.slrion tliut the two extensions are still homotopic in X. Proof. The idea of the proof is illustrated in Fig. 6. A desired homotopy bridge H(.Y,t ), x E C,,, 1 2 t 2 0, is a continuous mapping, with values in X , defined in the solid cylinder C,, x [0, 11 in R" x R. On the other hand, certain given continuous mappings can be regarded as mappings defined on certain subsets of the solid cylinder. To be explicit,fis defined on the lower boundary sphere S,, x {0), whereasg is defined on the upper boundary sphere S,, x (1). The given extension F of .f' to C,, is defined on the lower n-ball C, x (0). Furthermore, the homotopy bridge h(x, t ) between f and g is defined on the lateral surface of the cylinder S, x [0, I]. As a matter of fact, F and zi coincide on the common portion of their domains of definition s,, x (0) = (C,, x { O l ) n (S,, x [0, l]), the lower sphere. Hence a common extension k of F and h to the union of the lower n-ball and the lateral surface is obtained in a natural way by defining its values as F ( x ) on the lower n-ball and h(x, t ) on the lateral surface. To get a homotopy bridge H between F and an extension G of y to C,,, we need only try to further extend the mapping k defined on (C, x (0))u ( S , x [0, I]) to the whole cylinder C,, x [0, 11. This will be effected by projecting each point (x, t ) of the cylinder from the point (0,2), where 0 is the origin of R", to a point of (C,, x (0)) u (S,, x [0, 11) and then applying k .

95.

HOMOTOPY AND EXTENSION THEOREMS

77

More formally, the projection @ from (0, 2) to (C, x (0)) u (S,, x [O, 11) is given by

H is a homotopy bridge between F(x) and G(x) = H ( x , l), and this G is an extension of g. Also, H is an extension of the homotopy bridge h. This completes the proof. We may observe that the only if part of Theorem 5.1 is a special case of Theorem 5.3, where g = 4 andf’is a constant mapping. “

”

/

R”

FIG.6

Finally it should be noted that although all the propositions are advanced for C, and its boundary sphere S,, they are also true for any convex body C in R” and its boundary C”. For there is a homeomorphic mapping $ between C, and C , which also sends S, onto C bhomeomorphically. Then, the extensibility o f f : C“ -+X to C is equivalent to that of f $ : S, -+ X to C,, whereas the homotopy between 5 g : C“+ X is equivalent to that between f g$ : Sn 4 X . $9

78


55.2. Special Case of the Kronecker Theorem

An important part of pioneering research in topology was initiated by Kronecker (1869), a famous German mathematician in the 19th century. His result is now formulated in a more modern way as the Kronecker existence theorem. To present it in a general form, however, we would need to penetrate more deeply into topology, so we will limit ourselves to touching upon a very special case that will be applied subsequently. Let C, and S, be the n-ball and its boundary ( n - I)-sphere, as before. Let f : C,+ R" be a continuous mapping. With f and a point c of R" such that c $f(S,), we associate a continuous mapping 4,c : S,, + S,, , which is defined by the following rule: 4,c(.r) equals the point at which the ray issuing from the origin in the direction of the vectorf(x) - c meets S , (see Fig. 7).

FIG. I

I n terms of the homotopic property of q5,c, we invcstigate the possibility that the equation f(s)= c

has a solution s i n C,, . Clearly the existence of a solution in C,,to this equation is equivalent to c EJ'(C,,).

THEOREM 5.4 (Kronecker, 1869). //' 4,, : S,, + S,,i s no/ homotopic to a constunt mapping in S, , tlirn c E,f ( C,,). Proof. Ifc $f(C',), wecouldconstructacontinuousmappingcD,, : C, + S, by the same procedure used in obtaining 4sc.It is obvious that is a continuous extension of 4,c to C, . This is impossible, however, by Theorem 5.1 because d f Cis not homotopic to a constant mapping, Q.E.D.

$5.

79


THEOREM 5.5 (Kronecker, 1869). Let ,f; g : C,, + R" he tii'o continuous mappings such that they, as conjined on the boundary sphere S,, are homotopic in R"\{c}. Tlien, if' 4gcis not liomotopic to a constant mapping, it-c hurc c E f (C,,)as well. Proof. Let H ( x , t ) be a homotopy bridge betweenf and y as confined on S, in R"\{c); then the corresponding 4fc and 4,, are homotopic by the homotopy bridge 4,,c, where q5Hc is obtained as follows: Write H , ( s ) = H(x, f ); then H , : 5, R"\(c} is a continuous mapping for each t , so that we can construct q5,,,c. 4Hcis finally defined by letting d H c ( x ,t ) = 4,1,c(x).The continuity of 4Hc: .S, x [0, I ] -+ R"\fc} is readily seen. Therefore 4rc is not homotopic to a constant mapping either; otherwise # g c , which is homotopic to 4rc,would be homotopic to a constant mapping, a contradiction. Thus Theorem 5.4 applies tof, so that c e f ( C , , ) ,Q.E.D. --f

95.3. Differentiable Mappings

We will investigate the homotopic property of dfCfor certain mappings, but only mappings of an elementary type will be considered because of the very limited topological discussion in this book. Let X be a nonempty open subset of R", and let f ( x ) be a real-valued function on X .

DEFINITION 5.2. f ( x ) is called diferentiable at a point a E X if tliere is a linear function of' constant coefficients I uj(xj - a j ) such that

I;=

(p)

lim F ( X , a)/lix - all

as x E X , x # a , lim IIx - all

= 0,

= 0.

x+a

x-ta

Definition 5.2 means that f ( x ) behaves like f ( a ) + aj(xj - a j ) in the neighborhood ofa. The differentiability off(x) immediately entails certain wellbehaved properties. First, limx+aE ( X , a)/llx - all = 0 implies limx+aE ( X , a) = 0, which in turn implies the continuity off(x) at a. Second, for the special mode of convergence lirn xk = ak subject to x j = aj ( j # k ) , ( M ) and (p) imply the partial differentiability of,f, i.e., l i m ( f (a I, ..., a k - l , x k r a k + I ., . . , a , ),

1

Xh-ak

I ? . . . ak- I 2 ak I ? . . . an))/(% - ak) = yk The linear function a j ( x j - a j ) is termed a total diferential, which will be symbolically written as ~ ~ ~ 1Iff(x) . r ~ is. differentiable at every point of X , it is called difierentiahle in X . I n this case, the coefficients aj of the total differential may possibly vary from point to point and are functions of a, a j =fj(a). 7

3

1

1

I

f

80

I . MATHEMATICAL THEOREMS O N CONVEXITY

Sometimes relations (a), Landau's o-symbol o( . ) as

(p) can conveniently be written in terms of n

. f ( x ) = .f(tl)+

C

"j(-xj - aj)

j = I

+ o(IIx - uII).

It should also be noted that Xcan be the closure of an open subset of R" for the most part of the above discussion of differentiability. That is, differentiability can be defined simply as the existence of a total differential. Then differentiability implies continuity everywhere in X , whereas it entails partial differentiability at every interior point of X . Moreover, whenever one-sided partial differentiation either to the left or the right, makes sense at some boundary point of X , it can actually be performed, and the corresponding coefficients of the total differential t u r n out to be the values of one-sided partial derivatives at this point. A closed rectangular region o j R", {.v I p i

5 x i 5 qi

(i = I,

.. . ,n):,

where p i , y i ( p j < y i ) are given constants, is a simple example of X in which partial differentiation can be performed everywhere in the above sense. I n the light of the above argument, one can readily see that most of our discussion of differentiability applies to a function whose domain of definition is any intermediate set X between an open set G of R" and its closure G, G 2 X 3 G. Because of this, we mean by a region any X between G and G for a connected open subset G of R". X automatically turns out to be connected. The proof of the connectedness of such X is easy to work out and is therefore omitted here. We reserve the special names an open region and a clo.sct/ rcyion for the two extreme cases, the former for X = G and the latter for X = G. lff(.y) is defined and has continuous partial derivativesiifldxj(j= 1, . . . ,n ) i n X , it is said to be continuously differentiable in X . Continuous differentiability is often referred to a s a standard differentiability concept for functions of several variables (see 443.1 and 3.5). It implies differentiability in the above sense at an interior point of X . This can easily be seen by means of the meanvalue theorem in calculus. DEFINITION 5.3. A 1nuppi17g ,f : X + R", ivhcwl X is a region qf' R", is c a l l r ~tljffircritit~hlc l i/:f'(.v) = ( j i ( . y ) ) arid tlic coniponentsJ)(x) are r/[fliwntiahle ill

X.

For a differentiable mappingf': X + R", we have the I I corresponding total differentials X./~.;(.Y) d y j . The matrix

$5. HOMOTOPY

AND EXTENSION THEOREMS

81

whose elements are the coefficients of the n differentials, is very important and is termed the Jacobian matrix of the mapping. The determinant of the Jacobian matrix is called the Jacobian or the functional determinant. The differentiability of a mapping can therefore be expressed in terms of the Jacobian matrix as

f ( x >= f ( 4+ J(a)(x - a) + o(llx - all). With the above preliminary consideration in mind, we will study the local properties of a differentiable mapping, in particular, its behavior in the neighborhood of an intericr point. I n what follows, the Euclidean norm 11 . 11 will be used for n x n matrices A , B, . . . . If A = (aij), its norm is defined by

We shall use certain familiar basic properties of this norm, without explicit reference, since they are discussed in any textbook on linear algebra.

LEMMA 5.1. Let an n x II matrix A be nonsingular, i.e., det A # 0. Tlien, there is a positive number q > 0 such that ljAxll 2 qllxll for all x E R". Proof. We have llA-lyll 5 \lA-'ll llyll for all Y E R". llA-'ll > 0 , so that we may let q = l/llA-'ll. Substituting AX for y in qIIA-'ylI 5 Ilyll, we finally have IlAxlI 2 qllxll, Q.E.D. THEOREM 5.6 (Local Univalence). Let f : X X is a region of R". We hare the following:

-+

R" he a mapping, diere

I f f is differentiable at a point a of X and det J(a) # 0, then there is a neighborhood U(a)such that f (x) =f (a),x E U(a) imply x = a. (ii) ,f,f is conrinuously dffferentiable in a neighborhood of an interior point a of X , and det J(a) # 0, tlien there is a neighborhood U(a) wliere f is uniraletit, i.e., f ( x ) = f ( y ) , x , y E U(a) imply x = y . Pvoof. (i) Lemma 5.1 as applied to J(a),a nonsingular matrix, ensures the existence of q > 0 such that \\J(a)x//2 q / / x /for / any x. Take c > 0 subject to '1 > E . By differentiability at a, there is a 6 > 0 such that /Ix - all 6 implies

(i)

Ilf(x) - (f(4+ J(a)(x- a))ll 5 E l l X - all for any x E X . Let x E U(a, 6) = { x 1 /(x- a ( /5 6, x E X ) ; then

Ilf(x) - f ( 4= Ilf(x) - ( f ( 4+ J(a)(x- a ) ) + J(a)(x - all1 2 IIJ(a)(x - a)ll - IIf(x) + J ( a ) ( x- a))ll

>= qllx - all - ~ l l x- all = (vl - E)llX - all.

Whencef(x)

= f ( a ) ,x E

U(a, 6 ) imply x = a.

82


(ii) Choose q > 0 as in (i); then E > 0 such that q > E . Since J ( s ) exists and is continuous in the vicinity of a, there is a neighborhood U(u, 6) = (x I /Ix - a ( /5 S } where J ( x ) exists and I/J(x)- J(a)/I 5 E . Now let x ( t ) = ( I - t ) x ty for any x, y E U(a, 6). Then sinceJ'(x(t)) is differentiable for 0 6 t 5 1 with the derivative J ( x ( t ) ) ( y - x), we have

+

Hence for x, y

E

U(u,S ) ,

lif(.v) - A X )

- ~ ( L l ) ( . v - x)i/ 5 C ' I I J ( ~ (~ J(~>II )) I I -~ xi/ dt 0

5 E IIy - XII, whence, as in (i), it follows that ll,f(y) - f ( x ) [ l 2 ( v - ~ ) / l-y XI[. This proves E U(a, 6), f ( x ) = f ( y ) , Q.E.D.

x = y for x, y

As is well known, Theorem 5.6 generally ensures only local univalence. Global univalence may not prevail, however, even if the Jacobian is nonvanishing everywhere in a region. A familiar example is the mapping (x, y ) 4 (x cos y , x sin y ) . Points with the same y-coordinate mod 27t have a common image, yet the Jacobian is positive everywhere in the half-plane defined by x > 0. To guarantee global univalence, further conditions on the Jacobian, as well as on the structure of the region, are needed. Some of these conditions, which are not available in the ordinary textbook on calculus, will be discussed later (see Chapter V11) with special application to economic problems. Let us t u r n our attention to another feature of the local behavior of a mapping. In Theorem 5.6(ii),f'is univalent in U ( a )= U(a, S ) , so that it has an inverse mappingf-' : f ( ~ ( a + ) ) u(N). j ' is naturally continuous. In view of the compactness of U(u), however,J'-' is also continuous by Theorem 1.4. Therefore f turns out to be homeomorphic in U ( a ) . This by no means immediately implies, however, that the image j'( U ( a ) )includes an open subset of R", while U ( u )certainly does. Although in realityj'(a) is an interior point of f ( U ( u ) ) ,the consolidation of this invariance of interior points needs further topological consideration such as that developed in the foregoing sections. We now proceed to this deeper problem, the invariance of interior points.

LEMMA 5.2 (Rouche's Theorem). tnuppitigs. Let further c E R . Then, if'

ll./'(-v) eiwywhcw

it1

- Y(*t-)II

Let f, g : X

< IMx) - 41

X , j ' i s homotopic to g in R"\{c}.

+ R"

hr tn'o continuous

$5.


83

Proof. f i s homotopic to g in R"\{c) by the homotopy bridge H ( x , t ) = rg(x), x E X , 1 2 t 2 0. In fact, since

(1 - t)f(x)

+

IIW,t ) - CII

=

ll(1

2 2

Ilg(x) - cll - (1

- t ) ( f ( x ) -g(x)) + g ( x ) - cII

- t ) l / f ( x )- g(x)Il

MX ) - cll - llf'(x) - 9(x)ll

> 0, H ( x , t ) # c for x E X , I 2 t 2 0. This completes the proof. In addition to C , , S,, , we shall consider an n-ball and an ( n - 1)-sphere of an arbitrary radius, having an arbitrary point as its center, to which the results in $35.1 and 5.2 are applicable. Henceforth we let C,(a, 6) and S,,(a, 6) stand for the n-ball and the ( n - 1)-sphere of radius 6 with its center at a. We can consider 4Jcrelative to S,,(a, 6), if it is possible to construct ( b J c .

LEMMA 5.3. Let A be a nonsingular n x n matrix andg : R" -+ R" hc the afine mapping defined by g(x) = Ax h. Then the corresponding (bgc (relatire 10 &(a, 6 ) )is not hotnotopic to a constant mappingfor any c E g( Cn(ar6))\g( S,(a, 6)).

+

+

Proof. Let d = A - ' ( c - h). Clearly 4gc(x)= a SA(x - t/)/IIA(x - d)li. Suppose that there would be a homotopy bridge H ( s , t ) between (bgc and a constant mapping from S,,(a, 6) into itself. Then since A is invertible, the mapping x -+a 6(x - c/)/llx - d I( : S,,(a, 6) -+ S,,(u, 6 ) is homotopic to a constant mapping by the homotopy bridge

+

G(x, t ) = a

+

GA-'(H(x, t ) - a ) /lA-'(H(x, t ) - u)ll'

S,(a, d), I >= t 2 0. On the other hand, the former is homotopic to the identity mapping from S,(a, 6) onto itself by the homotopy bridge

x

E

K(x, 1) =u

+ ta)f + 6(x --((((11 --tf)d )d + ' ,,

J/X

tLI)//

x E Sn(u,S), 1 2 t 2 0. Hence, the identity mapping is homotopic to a constant mapping, contradicting Theorem 5.2.

THEOREM 5.7 (Invariance of Interior Points). Let f : X + R" he a continuous mapping, d f f h n t i a h l e at a, d i e r e X is a rcyion of' R". Tlieti,J'(u) is un interior point o j tlie image set ,f( X ) in tlie topolqgy of' R" i f ' u is mi interior point of X and det ,/(a) # 0. As was noted, there is an q > 0 such that I1J(a).uil 2 qlI.ul1 for all E > 0 satisfying 2c < q. We consider, together with ,f; the affine mapping g(x) =J(a) J ( a ) ( x - a). Since .f' is differentiable at an interior point a, there is a 6 > 0 such that (?,,(a, d ) c X and

Proof.

x E R". Take an

+

Ilm - g(-u)ll s

Cl1-Y

-4

84


for x E C,,(a,6).Next, letting 2y = dq in view of the ways of choosing 6, 4 , we find Ilg(x) - f(a)II 2 qllx - all = 6q = 2y over all x E S,,(a, 6). Clearly y > 0. We now assert that {c I llc -j(a)ll < y} c f(C,,(a, 6)), and thereforef(a) is an interior point of f(C,,(a, S)). In fact, if IIc -f(a)il < y , we have Ilg(x) - cII > ll(, - f(a)ll for x E &(a, 6) because lIg(x) - cll = /Ig(x) - f ( a ) + f ( a ) - ell 2 llY(-v) -.f(a)ll - /I f ( a ) - ell > 2y - y = y > IIc -f(a)ll for x E S,,(a, 6). Therefore,

llf(x> - dx)II 5 E IIX - all &

s 4 llJ(a>(x - a>ll -

for x E S,,(a, 6). In view of 2 ~ / 4< 1, this implies llf(x) - 9(x)II < Ildx) - cII

for x E S,,(a, 6), so that by Lemma 5.2 the two continuous mappingsf and g as confined on S,l(a,6) are homotopic in R"\{c). Note that in view of c E g(C,,(a,6))\y(S,,(a,d)), by Lemma 5.3, 4yc(relative to S,,(a, 6)) is not homotopic to a constant mapping. Hence, by Theorem 5.5, we have c ~ f ( C , ( uS)), , completing the proof, Q.E.D. ' COROLLARY. Suppose that X is an open region of R". Then, if the Jacohiari det J ( x ) of a di&>rentiable mapping f : X R" is ei.eryuhere nonr.anis/iiug in X,the image set f ( X ) is also an open region. Proof. First, recall that j is continuous in X because of its differentiability in X.Then, by Theorem 5.7, the nowhere-vanishing Jacobian matrix implies that every point o f f ( X ) is an interior point, whencef(X) is open in R". Finally, the connectedness o f j ( X ) follows from that of X by the continuity of j , Q.E.D. If a mapping f : X -+ R", where Xis a region, is continuously differentiable, with the Jacobian nonvanishing at an interior point a, by Theorems 5.6 and 5.7, c = f ( a ) has a neighborhood U ( c ) such that there is a (unique) mapping --f

$5. HOMOTOPY

85

A N D EXTENSION THEOREMS

h : U ( c ) --+ X fulfillingf(h(y)) = y for y E U(c). Such a mapping h is called a local inverse ofJ at a and is denoted by f -I. We then have

LEMMA 5.4. A local inverse f -' at an interior point a of a mapping f : X -+ R", diflerentiable at the point a, is differentiable at c = f (a), with the corresponding Jacobian matrix J(a)-' if det J(a) # 0. Proof. By the differentiability o f f a t a, we have

f(x) -f(a> - J(a)(x - a) = 4x2 a), limIle(x, a)ll/llx - all = 0 (as I / x - all -+

0).

Whence, multiplying by J ( a ) - ' , we obtain

J ( u ) - ' ( ~ ( x -f(a)) ) - ( X - a ) = J(u)-'E(x,a). If U(c) is a neighborhood of c where f - ' is defined, the above relation reduces to J ( a ) - ' ( y - c) - ( f - ' ( Y ) - f - ' ( c ) ) for y

E

=J(a)-'&(f-'(Y),f-l(C))

U(c). Hence, the lemma will be proved, if we show limIIJ(a)-'&(f-'(y),f-'(c))II/IIy

- cII = 0

(as

IIY - c/I 4 0 ) .

In fact, as was shown before, Ilf(x) -f(a)ll 2 yllx - all for some y > 0, which is rewritten in terms of f-' as Iiy - CII 2 y l ] f - ' ( y ) -f-'(c)Il for y E U(c). Hence limf(y- cI( = 0 implies limllf-'(y) -f-'(c)il = 0. Thus llJ(a)-

' 4 f -' f u > X'(4)ll < llJ(a)- ' It M f - '(y),f-'(c))Ii -

~

IIY - cll y llf-'(Y) - f - ' ( c ) l l as lim((y- c ( ( = 0. This completes the proof.

We may observe that a mapping can have a local inverse at a point even if det J(a) = 0; but such a local inverse may fail to be differentiable, whilefis differentiable. A simple example is the real-valued function f ( x ) = x 3 of a real variable x. It has the global inverseg(y) = y 1 / 3which , is not differentiable at the origin where the derivative offvanishes.

THEOREM 5.8 (Implicit Function Theorem). L e t f ( x , y ) = (f,(x, y ) ) be a mapping dej?ned in a region K of R" x R", with salues in R", where (x, y ) E K .Y E R", y E R". Further let (a. h), where a E R", b E R", be an interior point of K , arulset c =J(a, 6). Then, iJ f is continuously differentiable in a neighborhood oj ( a , b), with det(dj;./ay,) # 0 ai ( a , b), then a neighborhood V(a) of a exists SUCII that (i) there is a unique mapping 4 : V(a)-+ R" satisfying 4(a) = h, (.Y, $(x))E K , f ( ~ q5(x)) , = c identically for x E V(a); (ii) the ahore q5 is continuously differentiable in V(a).

86


Pvoof. (i) Consider the mapping g : K -+ R" x R" defined by g ( x , y ) = (x,f(x, y ) ) for (x, y ) E K where x E R",y , f ( x , y ) E R". g is continuously

differentiable in the given neighborhood of (a, b), with the Jacobian matrix 0

' 1

1

0

~

df"

\ax 1

0

.

1

0

afn ... axrn

afn -

dy,

...

0

...

0

\

d fn ... 6Yn/

Moreover, det J(x, y ) = det(dfi/dy,), which, by assumption, is nonvanishing at (a, b). Therefore, by Theorem 5.7, g(a, b) = (a, c) is an interior point of g ( K ) . This result, in the light of Theorem 5.6(ii), ensures the existence of a neighborhood U(a, c) of (a, c) such that U(a, c) c g ( K ) , and there is a local inverse g - 1 : U(a, c) -+ K fulfilling g-l(a, c) = ( a , b), g - l ( x , w) = (x, $(x, w)), ( x , f ( x , $(x, w))) = g ( g - I ( x , w)) = (x, w) identically for (x, w ) E U(a, c). Then the desired mapping 4 : V(a)-+ R" is obtained by letting V(a) = {x I (x, c ) f?U(a, c)), 4 ( x ) = $(x, c). (ii) Since J(x, y) is continuous and nonsingular at (a, b), we may assume that we have chosen the U(a, c) in (i) in such a way that J(x, y ) is also nonsingular in g - ' ( U ( u , c)). Hence by Lemma 5.4 as applied t o g and its local inverse g - l , 9 - l is differentiable, with its Jacobian matrix J ( x , $(x, w ) ) - ' in U(a, c). Thus the desired implicit function 4(x) = $(x, c) has an n x m Jacobian matrix -(dJ/dyj)-'(dfi/axj) in V(n), Q.E.D.

It is finally noted that the invariance of interior points is ensured not only for differentiable mappings with nonvanishing Jacobians but also for homeomorphic mappings from a region of R" into R" as well. However, its detailed discussion, though within our scope, is omitted here, since the general version of the invariance of interior points is not required in this book.

Chapter II

SIMPLE MULTISECTOR LINEAR SYSTEMS

Three basic linear systems, Leontief’s system, linear programming, and von Neumann’s model of an expanding economy are expounded. The mathematical contents of this chapter are (1) the theory of nonnegative matrices and (2) the examination of properties of solutions to systems of linear equations or inequalities of economic significance.

86. LEONTIEF’S INPUT-OUTPUT SYSTEMS, M A T R I X MU LT IPLI E RS 86.1. Introductory Remarks Since the pioneering presentation of a comprehensive and systematic economic theory by the great 18th-century economist Adam Smith (1 776), the works of eminent economists have gradually elucidated the general laws that govern basic economic phenomena such as resource allocation, production, consumption, income distribution, and capital accumulation. These phenomena are more or less endowed by nature with some mathematical features, and the fundamental views of economic reality by eminent economists are very often strikingly mathematical in character, even if they are not aware of this. Thus, recently an explicit mathematical approach in economic theory has become more broadly accepted. Economic reality is so complicated, however, that no single mathematical model can rightly depict all of its mathematical features. By giving particular attention to some of these features, however, an appropriate mathematical model can be devised and studied by a suitable mathematical method. Generally speaking, dynamic features of economic reality, as well as its more 87

88

11. SIMPLE MULTISECTOR LINEAR SYSTEMS

disaggregative features, are attacked by methods that are more advanced than those for static or less disaggregative ones. This chapter presents the standard results for intermediately disaggregated linear systems, which can be obtained by a rudimentary knowledge of linear algebra, reinforced by topological considerations. The principal methods of proof rely on full use of the linearity that governs vital relationships in the models in question. The results obtained here for intermediately disaggregated models are of value i n three-fold contexts. First, since many existing multisectoral models are intermediately disaggregated as well as linear, they are of independent interest. Second, they can serve as prototypes for corresponding results in nonlinear situations. Third, some, though not all, of them are useful mathematical theorems that can later be fruitfully applied to the solution of problems for nonlinear models in different categories. $6.2. Leontief's Systems

In the 1930s, Leontief (1 949, 1966) initiated a study of the static structure of a national economy i n its intermediately disaggregative features. In his approach, production activities of the economy are disaggregated to n sectors of industries, though not to individual firms in a microscopic sense, and the transaction of goods among the sectors is analyzed. His basic assumptions are as follows: (z) 17 kinds of goods are produced, traded, consumed, and invested in the economy, which are labeled by i = I , . . . , n. (/I) Each sector produces a single kind of good, but no more, so that no joint production prevails. Distinct sectors produce distinct kinds of goods. Thus n sectors and n kinds of goods are in one-to-one correspondence, and the sector producing the ith good is also denoted by i. (y) In each sector, production means the transformation of several (possibly all) kinds of goods in some quantities into a single kind of good in some amount. This pattern of input-output transformation is stable. Explicitly, in a Leontief system, this pattern assumes the following special form: To produce one unit of the j t h good, a,, units of the ith good are needed as inputs for i = I , . . . , n in sector j, and A units of output of the jth good requires ).a,, units of the ith good. These n z magnitudes a,,, called input coeficients, are assumed to be constant. In the economist's terminology, the ratios of inputs are constant, and constant returns to scale prevail.

Let .Y, be the output of the ith good per unit of time, say, per annum. This x, represents gross output. Part of the gross output is consumed as the input needed for production activities. The corresponding net output is

$6.

LEONTIEF’S SYSTEMS

89

obtained by subtracting from it the total amount of the good consumed as input throughout the economy. The latter is

so that the net output of the ith good is n

xi -

1a i j x j . j = 1

This net output is equated t o the final demand c i for the ith good to obtain the system of equations (*)

The system (*) is basic in Leontief’s input-output analysis. Here ci consist of final consumption, export, and investment (i.e., capital formation, meaning the building of new production equipment and plants, as well as the holding of new inventories of finished and unfinished goods). But in the present simplified approach, principles governing them are unspecified, so that the ci are thought of as magnitudes exogenously given from outside the system. Then the corresponding system of solutions xl, x2,. . . , x,,determines the levels of sectoral production activities. The essence of Leontief’s input-output analysis is determination of the gross output levels by the exogenous final demand in the given technological framework, embodied in the input coefficients a i j . Mathematically, it therefore consists of the study of system (*). The system of equations (*) naturally shares in the general properties that any system of linear equations possesses, and there is nothing new to consider i n this respect; but (*) has one special characteristic that captivates mathematically inclined people. This characteristic feature is that for the obvious economic reason, the relevant constants as well as the solutions should satisfy the nonnegativity constraint. That is, the input coefficients a i j , the final demands c i , and the gross outputs x i should be nonnegative. This raises a mathematical question: What is the condition for the existence of a system of nonnegative solutions x, 2 0, x2 2 0, . . . , x, 2 0 for given c i 2 0 under the basic assumption a i j 2 O ? From the economic point of view, the solvability of (*) i n the nonnegative unknowns x i 2 0 means the uwkabilify of a Leontief system. We may observe that (*) is a special case (for p = I ) of the system of equations n

p x i - ~ u . 1J. x J. = c i ( i = I ,..., n), j= I

where p is a parameter.

(1)

90

I I . SIMPLE MULTISECTOR LINEAR SYSTEMS

In relation to (l), let us consider the following system of equations n

1 d i j x j = ci

(i

= 1,

. .., n ) ,

j = 1

where the coefficients d i j satisfy the basic assumption

dij 5 0 (i Zj).

(3)

I f a i j 2 0 (i, j = I , . . . , n), system ( I ) can be converted to (2) by setting dij = p h i , - a i j , where < j i j (Kronecker's deltas) = 0 for i # j , 1 for i = j . Conversely, system (2) can be transformed to ( I ) . In fact, if we take a sufficiently large positive number p such that p > dii (i = I , . . . , n), letting a i j = p a i j - d,,, we have dij = p h i j - a i j with a i j 2 0 for all i, j . Whence (2) reduces to (1). It therefore suffices to look for conditions for the existence of nonnegative solutions to (2). We now list four conditions on system (2):

(I) System (2) has a set of nonnegative solutions xi 2 0 (i = 1, . . . , n ) for some set of positiix ci > 0 (i = 1, . . . , n). (11) System (2) is solruble in the nonnegatiiv unknowns x i 2 0 (i = 1, . . . , n ) for any set of nonnegatitv ci 2 0 (i = 1, . . . , n ) . (111) The square matrix D = (dij) of coeficients has the n posititre upper left-hand corner principal minors, i.e.,

(IV) All the principal minors of D are positire. Condition (111) or (IV) is known as the Hawkins-Simon condition (Hawkins and Simon, 1949; Georgescu-Roegen, 1951, 1966). Clearly (11) implies (I), and (IV) implies (111). On the other hand, ( I ) and (111) seem to be weaker than (11) and (IV), respectively. But, surprisingly, these four conditions are mutually equivalent.

THEOREM 6.1. Conditions (I), ([I), (111) and (IV) are mutually equivalen t.

Proof. We first prove the logical implication: (1) + (Ill) + (11) (I). Then we will establish the equivalence of (IV) to the other three. The proof will be worked out by induction on n, the (i) ( I ) + ( I l I ) . number of equations as well as the number of unknowns. If n = 1, Eq. (2) --f

96.

91


is d,,xl = el. If this equation has a solution x, 2 0 for some c1 > 0, then d,, must be positive. Assume that ( I ) + (111) is true for n - 1. By (I), Eq. (2) has nonnegative solutions x1 2 0, x2 2 0, . . . , x, 2 0 for some set of c1 > 0, c2 > 0, . . . , c, > 0. The first equation in (2) can be rearranged to n

d l 1 x l= c1 -

1d I j x j ,

(4)

j=2

the right-hand side of which is positive because c1 > 0, d l j 5 0, x j 2 0 . . . , n). Thus d, lxl > 0. Then in view of x, 2 0, we have d, > 0. This being done, we apply the Gaussian method of elimination to (2). Explicitly, if we subtract dil/dll times the first equation from the ith one in (2), then (2) reduces to

( j = 2,

O ( i , j = 2 ,..., n ; i # j ) ,

because dij 5 0 (i # j ) , ci> 0. The system of equations

d;,

. * . dTh

. dz2

. * . dzh

>O

(k=2,

..., n).

92


But, from the procedure by which (2) has been transformed into

i12,n2,

d,,tih2

... . ..

(9,it is

tl,,

. . .

. . .

tikh

= dll

whence (2) satisfies (111). (ii) ( l l l ) - + ( l l ) . We also proceed by induction on n. The equation dl Lxl= cI, i.e., (2) for n = I , has the nonnegative solution x, = cl/d,, for a n y c1 2 0, if d , , > 0. Next assume the truth of the implication (111) -+ (11) for n - 1. If (111) is satisfied by (2), we have in particular d,, > 0, so that we can transform (2) into ( 5 ) as was done in (i). It should be noted that this time, however, c i2 0 (i = 1, . . . , n ) are arbitrarily chosen, as opposed to the case in (i). Consider Eq. (7), which fulfils the basic assumption (3). Equation (7) satisfies Condition (I I I) because, by (I 1I) for (2),

Now, relation (6) implies ei* 2 0 (i = 2, . . . , n) for any c i 2 0 (i = I , . . . , n ) and, by the induction hypothesis, (7) has nonnegative solutions x2 2 0 xg 2 0, . . . , x, 2 0 for this set of cI* (i = 2, . . . , n ) . Then we let X, = 1 (c,

dl I

-

$ dljxj). j= 2

using (4). We have thus obtained nonnegative solutions x1 2 0, x 2 2 0, . . . , 2 0 to ( 5 ) . In view of the equivalence of (2) to ( 5 ) , however, these x1 2 0, x 2 2 0, . . . , x, 2 0 are solutions to (2). Hence (2) satisfies (11). (iii) (11) + ( I ) . This is obvious, as was noted previously. (iv). Now that the equivalence of (I), ( I I ) , and (Ill) has been established, it is easy to see the equivalence of (IV) to the other three conditions. First, x,

$6.


93

the implication (IV) -+ (111) is obvious. Conversely, if (11) is satisfied, by an identical and simultaneous renumbering of equations and unknowns, every principal submatrix of the coefficients matrix of (2) becomes one of the n upper left-hand corner principal submatrices of the renumbered system. The determinant of the principal submatrix in question, as well as the truth of (II), is not affected by this kind of renumbering. Hence, by applying the implication (II)+(III) to the renumbered system, we see the positivity of every principal minor of the original coefficients matrix. This proves the implication (11) -+ (IV). Thus the proof is complete, Q.E.D. Next we turn attention to the systems dual to (*), (l), and (2). That is: n

pj

-,Ia i j p i= uj

( j =1 , .

. ., n ) ,

= uj

( j = 1,

. . . , n),

1d i j p i = u j

( j = I,

. ..,n ) .

r=l

n

ppj -

Ca i j p i

I=

1

n

i= I

(24

The systems (**), (Id), and (2d) are dual to (*), ( l ) , and (2), respectively. The dual systems are economically, as well as mathematically, important. Economically they tell about the pricing or value side of the interindustry inputoutput relationship. In (**), pi is the price of the jth good, and z’j is the value added per unit output in thejth sector. Clearly a i j p iis the unit cost, so that the left-hand side of (**) is the net revenue per unit output of the jth sector, which is equated to the right-hand side, the value added per unit output. While the solvability of the original output system (*) in the nonnegative unknowns xi (i = 1, . . . , n) means workability in a Leontief system, that of the dual pricing system (**) in the nonnegative unknown pricespj ( j = I , .. . , n) means profitability. This duality between workability and profitability is something more than conceptual. As will be seen later, one of them ensures the other and uice z’ersa, because of the mathematically intimate linkage of mutually dual systems. Apart from their special relation to the original systems, the dual systems in themselves are of precisely the same structure as the original ones. Explicitly, the basic assumption (3) is met, and the unknowns p i , as well as the constant terms 2 5 , are required to be nonnegative. Hence, Theorem 6.1 also ensures the equivalence of the following four conditions, (Id)-(IVd) on (2d) : (Id) System (2d) has a set of nonnegatioe solutions p j 2 0 ( j = 1,. . . , I?) for some set of positire z;. > 0 ( j = I , . . ., n).

94


(I,, ... d k , .

d,,

..'

>O

( k = 1, ...,H).

llkk

and term r i and sj the ith r o ~ ' s u r nand thejth column sum of A. Then we have

COROLLARY (Brauer-Solow's Condition). Either one of (i) and (ii) is suficient for ~ w k a b i l i t yin ( I ) as well us prqfitability in (1 d) : (i) p > r i (i = I , . . . , n), (ii) p > sj ( j = 1, . . . , n). Proof: (i) implies that xI = x2 = ... = x, = 1 is a set of nonnegative solutions to ( I ) for a particular set of c i , namely, c i= p - ri > 0 (i = 1, . . . , n).

96.

95

LEONTIEF'S SYSTEMS

Thus Condition (I) is met, and the desired result follows by Theorem 6.2. A similar reasoning for (Id) proves the sufficiency of (ii) for workability and profitability, Q.E.D. From the economic standpoint, national income accounting in relation to the output system (*) and the pricing system (**) is of some interest. Since ci represents the net amount produced of the ith good, c c i p i is the national product. On the other hand, z ~ j xisj the national income earned, because v i , the value added per unit output, is distributed among management and labor in thejth sector as profits and wages, and therefore represents the level of the jth sector's income per unit output. These two magnitudes are equal in value, since (*) and (**) imply

1

n

n

1c i p i = C

i= 1

i, j = I

c ujxj. I1

(Sij

-

a i j ) p i x j=

j = I

06.3. Matrix Multipliers We introduce here explicitly the nonnegativity concept for matrices. Paralleling the symbols 2 , 2 , > for semiordering vectors (see Chapter I , §3.3), we define, for any two real matrices A = ( a i j ) B , = ( b i j )of the same size, say, with m rows and n columns: (a) A 2 B , i f a i j 1 b i j ( i = l , . . . , m , j = 1 ,..., n),

(p) A 2 B, if

A 2 B and A # B, A > B, if a i j > bij ( i = 1, . . . , m , j = 1, ..., n). It is also customary for a matrix A to be called nonnegative, semipositire, or positice according to whether A 2 0, A 2 0, or A > 0. We can now give the mutually equivalent statements (I)-(IV) and (Id)(IVd) in the preceding section an alternative expression, which may be referred to as nonnegatire inrertibility : (y)

DEFINITION 6.1. A square matrix T is called nonnegatiiTely inrertible if T is nonsingular and its inrerse T - I is nonnegative. Then the following theorem follows immediately. THEOREM 6.3. The coeflcients matrix D of system (2) with the basic assumption (3) in the preceding section is nonnegatirely invertible if and only if workability (proJitability) as expressed in any one of the equiralent conditions (l)-(lV), (Id)-(IVd) is ensured. Proof: Systcm (2) receives a compact representation Dx = c i n matrix notation, whcre x = ( x i ) ,c = (ci).If D has a nonnegative inverse D - ' 2 0, the equation D x = c has a nonnegative solution x = D - ' c for any c 2 0. Workability condition (11) follows from this. Conversely, if (11) holds, the determinant of D is not zero by ( I l l ) , so that D - ' exists. Then, the nonnegative

96


solution x of Dx = c, whose existence condition (11) asserts, is uniquely represented as x = D - ' c . This implies that D - ' c 2 0 for any c 2 0. If we observe a special case where cj = (r;), c? = 1 for i = j , 0 for i # j , this particular property of D - ' means that thejth column vector of D-'is nonnegative. The above argument is valid for all j = 1, . . . , n. Whence D-' 2 0, Q.E.D. We now put (*) and ( I ) of the preceding section in matrix form: ( I - A)x

= c,

( P I - A ) x = c,

where A = ( a z j )2 0 is the input coefficients matrix and I is the nth order identity matrix. From the above results, the workability of a Leontief system ( I - A)x = c is equivalent to the nonnegative invertibility of I - A . If the system is workable, the gross output vector x is given by ( I - A ) - ' c . ( I - A ) - ' gives us information about the extent to which the exogenous final demand vector c is magnified to the gross output vector x. Moreover, since the system is linear in x and c, the increment Ac of c and the corresponding increment Ax of x are connected through the multiplication by ( I - A ) - ' as well, so that A.Y = ( I - A ) - ' A c . Therefore ( I - A ) - ' also provides us with information about the magnified changes in the gross output levels, resulting from changes in the levels of final demand. For these reasons ( I - A ) - ' is termed a matrix multiplier. It is a matrix version of the numerical multiplier l / ( l - a ) in the simplest Keynesian model of national income determination: Y = a Y + I , where the numerical magnitudes Yand Irepresent the levels ofnational income and investment, respectively, and a stands for the proportion of the part of national income spent for consumption to the total national income, termed the " propensity to consume." Since the propensity to consume a is positive and less than unity, the numerical multiplier I / ( 1 - a) can be represented as a geometrical progression

I

-- - 1 1-a

+ a + a2 + ... + av + ....

This suggests the expression of the matrix multiplier ( I - A ) - ' as a matrix power series : (z-A)-'=I+A+AZ+-..+AV+**..

More generally, the expression of ( P I - A ) - ' in the matrix power series A

suggests itself.

A'

96.

97

LEONTIEF'S SYSTEMS

THEOREM 6.4. Let A be a nonnegatiile square matrix and p a numerical parameter. Then, we have (i), (ii): (i) r f p l - A is nonnegatively invertible, then p > 0, and ( I ) is contiergent, its sum being ( p l - A ) - ' . (ii) I f p > 0 and ( I ) is convergent, then P I - A is nonnegatiaely inwrtible, and the sum of (1) is equal to ( p l - A ) - ' . Proof. (i) If p l - A is nonnegatively invertible, it satisfies the HawkinsSimon condition, so that, in particular, p - a , , > 0. This gives p > a , , 2 0. Now let 1 " T, = A"/p'.

1

p v=o

The equation

T , ( p l - A ) = ( P I - A)T, = I - A " f l / p S + l , is immediately obtained, which implies, by virtue of A 2 0, ( P I - A)T, 5 I .

Premultiplying the above inequality by ( P I - A ) - ' 2 0 yields T, 5 ( P I - A ) - '

(S

= 0, 1,

2, . . .).

This proves that the matrix sequence {T,} is bounded from above. On the other hand, {T,} is, by definition, nondecreasing :

s....

T O I T ,I T 2 1 . . . 5 T , j T s + l

Whence (1) is convergent. Now, since T,, as well as T, converges to a common limit, A S + ' / p S + I= P T , , ~- pT, tends to the zero matrix as s + co. If we let lim T, = T , (2) becomes in the limit

T ( p I - A ) = ( P I - A)T = I ,

(3)

which implies T = ( p l - A ) - ' . (ii) In this case, T = lim T, exists, and likewise ( 2 ) entails (3) in the limit. Whence, p l - A is invertible, and its inverse ( p l - A)-' equals T . Since T, is nonnegative, its limit is also nonnegative. This proves ( P I - A ) - ' 2 0, Q.E.D.

If a Leontief system is workable, the gross output vector x corresponding to a final demand vector c 2 0 is given by the vector series c + Ac + A2c + . . . + A"c + * . . . One can readily understand the intuitive meaning of this result. To obtain the net output c we need Ac as input. Then, to produce Ac within the system we further need A Z c as input, etc. Then, the sum of c and all these required input vectors Ac, A2c, . . . equals the gross output vector x.

98


A Dynamic Multiplier Process. In the framework of a Leontief system, we shall study the dynamic behavior of the sectoral income under a scheme of exogenous expenditure changing over time. Let a Leontief system with an input coefficients matrix A 2 0 be workable. Further let r = ( r j ) > 0 be a vector whose j t h component r j stands for the value added per unit output i n thejth sector. A and I' are assumed to be constant over time. As was mentioned before, the price vector p = (pi) > 0 is given by p' = r ' ( I - A ) - ' . Denote by y j the income of thejth sector. Then, y j = r j - r j , where sj is the gross output level in thejth sector. Thejth sector spends c i j y j out of I;. for the product of the ith sector. These magnitudes cij (i = I , . . . , n), the ,jth sector's propensities to consume, are assumed to satisfy r j j 2 0, cij < 1 and to be constant over time. In formulating a dynamic process of income propagation, our basic assumptions are as follows:

ELl

( a ) Spending of sectoral income is lagged to its receipt by one period.

(b) There is a scheme of exogenous spending (d(t)}over time, where the ith component d i ( t ) 2 0 is the value of spending for the product of the ith sect or. If J(/)is the sectoral income vector at t . the total spending at t + 1 is Cy(t) + cl(t I ) , where C = (cij) is the matrix of propensities to consume. Let LIS define two diagonal matrices

+

+

1 is given by f C y ( t )+ Pd(t + I ) , and Then, the final demand vector at t the corresponding gross output vector a1 t + I equals ( I - A)-'PCy(t) + ( I - A)-'Pd(/+ I). Hence

y(t

+ I ) = H y ( t ) + "t)

( / = 0, I ,

2 , . . .),

(4)

where H = V ( I - A ) - ' P C 2 0, r r ( t ) = K d ( t + I ) , K = V ( I - A ) - ' P . Note that d ( [ )2 O ( t = 0, I , 2 , . . .) implics cr(t) 2 0 over time. If an initial position ~ ( 02) 0 is given, the above recursive relation depicts the propagation of the sectoral incoiney(t). y ( t ) 2 0 ( t = 0. I , . . .) is ensured by H 2 0. Stated i n less mathematical terms, Eq. (4) shows that the income at t + I emerges via the multiplier effects of ( I - A ) - ' from two sources: the lagged spending of the income a t / and the exogenous spending at t + 1 . Equation (4) generates the propagation of income among sectors t i n t k t a n y motlc of cxogenous spending. Cases where the exogenous spendiilp rotif n r n i h 1 1 ) a special mode of change over time, however, are of special intLrc\l.

$6.

99

LEONTIEF'S SYSTEMS

and a case in which all the components a,(t) of a ( [ ) grow exponentially at a common rate is of particular interest to economists. Choose a positive p such that p l - H is nonnegatively invertible, and an arbitrary vector a 2 0. Then it is very easy to trace the dynamic behavior of any solution y ( t ) of Eq. (4) under a ( [ ) = p'a. In fact, let = ( p l - H)-'a. Then, j ( t ) = pry is nonnegative over time and is a solution of Eq. (4). Both j ( r ) and a ( / ) enjoy a special mode of dynamic behavior, namely, balanced growth, meaning the constancy of component proportions over time. j ( t ) is a balanced-growth solution to(4)that is resonant toa(f) = p'a. Once theeconomy starts iit y = ( p l - H ) - ' a , it perpetually enjoys a state of equilibrium over time. j ( t ) is linked to a ( t ) by j ( t ) = ( p l - H ) - ' a ( t ) . Thus some economists term j ( t ) a moving equilibrium solution and ( p l - H ) - ' a matrix supermultiplier. There is, however, more to notice than this equilibrating nature of j ( t ) . The economy, even if it starts at an arbitrary initial position y(O), eventually approaches the moving equilibrium state. Stated formally, we have lim y(t)/pr = y r-+w

for any solution y ( t ) of Eq. (4). This convergence property, termed the r.r/utire stability of j(t). means the convergence of y ( t ) to j ( t ) in the ratio. The proof of relative stability for j ( t ) is immediate. In fact, if a ( t ) = p'a, then Eq. (4), if divided by p', becomes py(t + I)/p'" = H y ( t ) p r + a. On the other hand, y satisfies py = H y a. Whence upon subtraction, we have

+

Y(t

+ 1)

~p'+'

Y

1

= p ( y7-?' . (r)

Repeated application of the above recursive formula allows us to arrive at

+ +

As was made clear in the course of proving Theorem 6.4, Hr,'pr-+ 0 (f + co) whenever p I - His nonnegativelyinvertible. This provesy(r)'p' + y ( t -+ co). In the above discussion, the nonnegative invertibility of pl - H is indis-

pensable, or stated differently, it is almost necessary, as long as a does not vanish but is semipositive. The reader can easily see that if u > 0, a balanced growth solution, j ( t ) = p r y 2 0 of (4), resonant to a ( [ )= pru, exists only if P I - H is nonnegatively invertible. As will be noted explicitly in the section immediately following, the set of real p for which p I - H is nonnegatively invertible is a semiinterval ( i ( H), a), where i ( H )is a nonnegative number uniquely determined by H 2 0. It will also be noted that thisi.(H) is generally positive except for H of a very special structure. T h u s to maintain a moving equilibrium situation under exogenous spending, the common growth rate

+

100


of exogenous spending and endogenous sectoral income generally cannot be arbitrarily low. Exogenous spending, in particular, must be injected into the system at a good pace. Relation to Nutionul Income Approach. Let e be the vector all of whose components are unity. Then Y ( f )= e'y(t) represents the corresponding level of national income when the sectoral income vector at t is y(t). Noting that e ' V = r ' a n d p ' P = e ' , s o t h a t e ' H = v ' ( I - A)-'PC=p'PC=e'Cand,sirnilarly, e'K = e', and premultiplying Eq. (4) by e', we obtain Y(t

+ I ) = x c j y j ( f )+ I ( t + I ) ,

I:=,

where c, = c , ~ I, ( t ) = e'd(t). Therefore, if in particular cj = cx ( j = I , . . . , n ) may be assumed, the propagation of sectoral income is aggregated to a numerical equation Y ( t I ) = c( Y ( t ) I ( t + I ) .

+

+

This establishes the substantial affinity of Leontief's approach to the numerical multiplier approach.

$7. FROBEN IUS-PERRON'S THEOREM 97.1. Frobenius-Perron's Theorem

In the preceding section, we considered a moving equilibrium situation in a Leontief system under exogenous spending. In such a situation, a moving equilibriiini persists as long as exogenous excitation is present. Now a natural question arises: Is a self-sustained moving equilibrium possible without any exogenous spending? This kind of moving equilibrium is represented by a balanced-growth solution of the homogeneous equation y ( t + 1) = Hy(t), y ( t ) 2 0. Clearly the search for such a balanced-growth solution reduces to the study of an eigenvalue problem H y = l.y, I, >= 0, y 2 0 with a nonnegativity constraint for the nonnegative matrix H >= 0. This is one motivation for this section. Let us consider the eigenvalue problem A x = Ix,

A >= 0,

x 2. 0

for an arbitrary nonnegative square matrix A 2 0. As is well known in elementary linear algebra, an eigenvalue o of A, which is generally a complex number, is characterized as a root of the characteristic equation cl,(p) = det(pI - A ) = 0.

If w is a complex number, eigenvectors associated with o,namely, vectors z fulfilling A 2 = to;, 2 # 0 have complex numbers as their components, even

97.

FROBENIUS-PERRON'S

101

THEOREM

though A is a real matrix. A real matrix may possibly not have real eigenvalues, as is exemplified by the matrix

If A 2 0, however, a special nonnegative eigenvalue A and a nonnegative eigenvector associated with A exist, such that A majorizes any other eigenvalues o in modulus, IL 2 lo/. This result, to be proved below, is a remarkable reflection of the nonnegativity of A upon its eigenvalue problem.

LEMMA 7.1. For a nonnegatire square matrix A define the set M ( A ) = { p I P I - A is nonnegaticely inrertible}. Then, M ( A ) is a semiinterral (I., + a), where A is a nonnegatire number. Proof. Take any positive vector x > 0. Then, by choosing a sufficiently large number p , we have a situation where px > A x . Hence (pZ - A ) x > 0, x > 0 for these x, p , so that ( p l - A ) - ' 2 0 exists byTheorem 6.3. This implies M ( A ) # @. Second, ifM(A) 3 p , q 2 p , then M ( A ) 3 q. In fact, since p E M(A) there is an x 2 0 such that ( P I - A ) x > 0. Then q 2 p implies (qZ - A ) x 2 (pZ - A ) x > 0 for the samex 2 0. HenceqZ - A is also nonnegativelyinvertible, and M ( A ) 3 q. Third, if p E M(A), (pZ - A ) x > 0 for some x 2 0 clearly implies p > 0. These three results ensure that A = inf p over all p E M ( A ) is a nonnegative number satisfying M(A)2 (A, + 00). By definition, no real number smaller than /I is contained in M(A).Therefore, the proof will be complete if we see that M ( A ) $ A. To this end, assume that M(A) 3 A. Then there would be some x 2 0 satisfying (AZ - A ) x > 0, and any p that is smaller than, but sufficiently close to, A could satisfy (pZ - A ) x > 0 for the same x 2 0. Whence p E M(A) for some p < A. This contradicts the definition of A, Q.E.D. LEMMA 7.2. For the abore A, there is some x 2 0 fu&illing A x = Ax. Proof. Take any c > 0, and let y ( p ) = ( p Z - A ) - ' c for p E M ( A ) . Let p , q E M ( A ) . p 5 q. Then, by subtracting ( q l - A)y(q) = c from (pZ - A)y ( p ) = c and rearranging, we obtain ( P I - A ) ( y ( p ) - y(q)) = ( q - p)y(q); and premultiplying this relation by (pZ - A)-' we have Y ( P ) - Arl) = (v - P ) ( d - A)-"!)

2 0.

Now take a decreasing sequence { p , } in M ( A ) with lim p, = A. Since { y ( p , ) } is increasing, the component sum a( v) = y , ( p , ) is also increasing. First assume that { ~ ( v ) } is bounded from above. Then each increasing component y,(p,) would be bounded and hence would converge to a nonnegative limit y i . Whence lim y ( p , ) = y = ( y i ) . Then, the equation (p,Z - A)y(p,) = c reduces to (11- A)y = c > 0, y 2 0 in the limit, and M ( A ) 3 i, a contradiction. Therefore in reality we have lim a(v) = 00.

+

102

I[.

SIMPLE MULTISECTOR LINEAR SYSTEMS

Next let s(v)= y(p,)fa(v). Since x(v) E P, = { p I c p i = I , p 2 0}, which is compact, the sequence {x( v) } in itself may be assumed to converge to an x E P,. The equation ( p vI - A)y(p,) = c, if divided by a(v), becomes (p,, I - A).x(v) = c ~ ( v ) , which reduces to (3.1 - A ) x = 0, x 2 0 in the limit, as was to be shown. Alternatively we can prove the lemma by appealing to Chapter I, Theorem 3.7(ii). By the theorem as applied to AI - A , there are p , x E R" such that /)'(>.I - A ) 2 0, ( A I - A ) x = 0, x 2 0, x + (A1- A')p > 0. If this x is semipositive, the lemma is proved. Assume that x = 0; then (AI - A')p > 0. Replacing i, by a p that is larger than, but sufficiently close to A, we can still have ( p l - A')/> > 0. In the light of Theorem 6.2, however, M ( A ' )= M ( A ) = ( 2 , co).Whence ( p I - A ' ) - I 2 0 exists. This implies/> > 0, which, combined with (2I - A'fp > 0, yields a contradiction M ( A )= M(A')3 1,Q.E.D.

+

THEOREM 7.1 (Perron, 1907; Frobenius, 1908, 1909, 1912). Let A be a nonnegatii.e syuarc matrix. Then, (i-iv) hold: (i) A has a rioririegatire eigenralue. A nonnegatire eigenrector is associated with the largest 3. of all the nonnegatice eigenr>alues. (ii) p l - A is nonnegatirely inrertible if and only if p > 3.. (iii) I f Ay 2 / i j , f i ~ ar real number and a semipositire rector y > 0 , then i. 2 ,Li. ( i v ) i. 2 JQI ,for. any eigenralue 0) of A . Pvoof. (i) and (ii) are already established in Lemmas 7.1 and 7.2. The 1, defined i n Lemma 7.1 as the greatest lower bound of p E M ( A ) is a nonnegative eigenvalue of A with which is associated a nonnegative eigenvector x 2 0. Moreover. since p l - A is nonsingular for p E M(A),and therefore no p in M f A ) can be an eigenvalue, A is the largest of all the real eigenvalues. (ii) is a restatement of the structure of M(A). (iii) Supposing that ki > A would entail the existence of ( p l - A ) - ' 2 0; then premultiplying 0 2 (pl - A)y by ( p l - A ) - ' 2 0 leads to 0 2 y , contradicting y > 0. Whence 1. 2 11. (iv) Let z be an eigenvector associated with o,so that Az = o z . We therefore have ti numerical equations n

C a i j z j= o z i j=

( i = I , . . . ,n).

I

In modulus, they reduce to n

1 a i j l i j l 2 101

j= 1

IziJ ( i

=

I , . .., n ) ,

which can again be put in a single matrix relation Alzl 2 IwIIzI, by letting /z/ = ( 1 ~ ~ 1 ) . Since z # 0 so that Iz/2 0, the pair of IwI and izl can be thought

97.

FROBENIUS-PERRON’S THEOREM

103

of as one particular pair of p and y in (iii). Whence I. 2 IwI. The proof is thereby complete. The above A of A is henceforth denoted by A(A) and is called the FrobeniusPerron root or the dominant root of A .

Further useful and remarkable properties of R(A) as a function of A 2 0 will be listed in the following theorem.

THEOREM 7.2. L ( A ) hasproperties (i)-(vi).for A 2 0: (i) A(A) = A(A’). (ii) A(ctA) = ctA(A),for r* 2 0. (iii) L ( A ~=) (A(A))k~forany positire integer k . (iv) L ( A ) 2 A(B), if A 2 B 2 0. (v) A(A) 2 A(C)for any principal minor matrix C of A . (vi) A(A) = 0, if and only if Ak = 0 for some positire integer h Pvoof. (i) By definition, ].(A) = inf p over all p E M ( A ) , A(A’) = inf p over all p E M ( A ’ ) . As was already noted, M ( A ) = M ( A ’ ) , whence ).(A) = ,?(A’). This completes the proof. Alternatively, one may prove (i) by the fact that the characteristic polynomials of A and A’are exactly the same,which is familiar in linear algebra. (ii) The proof is so easy that it may be omitted. (iii) Take a nonnegative eigenvector x 2 0 associated with i, = ).(A). Then, A x = i,x, which becomes Akx = I?x after k times of repeated premultiplication by A . I? is a nonnegative eigenvalue of Ak. It is therefore majorized by I ( A k ) , so that Ak 5 A(Ak).If /Ik < A(Ak), let p = (L(Ak))’Ik.p > i. = A(A) and hence (pZ - A)y > 0 for some y 2 0. This is rewritten as py > Ay, which, if multiplied by p > 0, entails p2;L > Apy 2 A2y. This in turn yields p3y > A3y. In this way we arrive at pky > Aky, y 2 0, after k repetitions, which implies A(Ak) = p k E M ( A k ) , a contradiction. Whence A(Ak) = ( A ( F I ) ) ~ . (iv) Note that M ( A ) c M ( B ) . For if p E M ( A ) , there is x 2 0 satisfying ( P I - A)x > 0. But A 2 B 2 0 implies (pZ - B ) x 2 ( p l - A ) x > 0 for the same x 2 0. Whence p E M ( B ) . Therefore A(A) = inf p over all p E M ( A ) 2 inf p over all p E M ( B ) = ;L(B)because M ( A ) c M ( B ) . (v) Since eigenvalues are invariant under any coordinates transformation, particularly under any identical and simultaneous renumbering of rows and columns, C may be assumed to be an upper left-hand corner principal submatrix of the kth order of A . Let B be the matrix obtained from A by replacing all the elements a i j of A by 0 except for 1 5 i , j 5 k . Then, A 2 B 2 0. Whence A(A) 2 A(B) by (iv). On the other hand, since the characteristic polynomial 4 ( p ) of B equals that of C multiplied by a power of p, E.(B) = I.( C ) . These results, combined, prove A(A) 2 A(C).

104


(vi) Clearly 2(0) = 0. Thus, if A vanishes when raised to the kth power, we have 0 = A(0) = i ( A k )= (),(A))’ by (iii). Whence A(A) = 0. To prove the converse, choose an arbitrary positive vector x and a positive number 0, such that A x 5 Ox. Clearly 0 5 ASf’x5 0A”x (s = 0, 1, . . .), which, if we let N , = {i I the ith component of A“x is positive}, implies No

3

N,

3

N,

3

... 3 N ,

2

N,,,

3

... .

(1)

It will be shown that N , = where ti is the order of A . Denote by [A”xIithe ith component of A”x. If N , # we can define p = min([A”+’x],/[A”x],) over all i E N , . This p satisfies AA“x 2 pA”x, A”x 2 0. Whence A(A) 2 p by Theorem 7.l(iii). As ),(A) = 0, p 2 0, we havep = 0. This implies [A“+’x],= 0 for some i E N , y ,so that if N , # N , , is a proper subset of N,7in ( I ) . As N o contains n elements, N , = @, implying A”x = 0. This, combined with A” 2 0, x > 0, yields A” = 0, Q.E.D.

a,

a, ,

Application 1. We have touched upon a self-scstained balanced-growth situation in the income propagation in a Leontief system. This situation is depicted by a nontrivial balanced-growth solution j ( t ) = A ‘ j 2 0, /I 2 0 of ~ ( +t 1 ) = Hy(r), and one such solution is generated by 2 = A ( H ) and a nonnegative eigenvector J? 2 0 associated with i.(H).From the above results on the dominant root, i ( H ) - I is the greatest rate of self-sustained balanced growth. However, if each sector’s total propensi!? 10 consume ej = c i j is less than unity, as is normally the case, we have ; . ( / I )< 1. Whence any selfsustained balanced-growth solution is damping. A positive rate of balanced growth is therefore possible only when the system is constantly excited by exogenous spending. l ( H ) < 1 is seen as follows: Recalling e‘H = e‘C (see “ Relation to National Income Approach” in $6.3) and premultiplying H j = 1.9 bye‘, we have e‘CJ = le‘j. If we let maxj c i j = y, this reduces to ye’j 2 i.e’J. Since 1 > y, e ‘ j > 0, it follows that A < 1.

1;

xi

Application 2. Let a Leontief system be workable (i.e., profitable). Among possible modes of pricing goods given by the dual equation p’(1- A ) = r’, a special one, such as ensuring a common rate of value added to cost in all the sectors, is of interest. Let 0 be such a common rate of value added to cost, so that r’ = Op’A. Then the dual equation becomes p‘ 2 0.

Thus an eigenvalue problem with a nonnegativity constraint emerges, and a situation where an equal rate of value added to cost prevails is realized by ] , ( A ) = 1 ( 1 + 0 ) and a nonnegative row eigenvector p’ 2 0 associated with ),(A). Note that the workability of the system implies A(A) < 1. Hence a common positive rate O is guaranteed as long as A ( A ) > 0.

57. FROBENIUS-PERRON'S THEOREM

105

Application 3. Let D = ( d i j ) be a real square matrix with nonpositive off-diagonal elements dij 5 0 (i # j ) . We recall that starting at the conditions for the nonnegative invertibility of D, we have explored the nonnegative invertibility of p l - A and the eigenvalue problem A x = 3.x for a nonnegative matrix A . N o w let us consider the eigenvalues of a D. Take a sufficiently large positive p such that pZ - D = A is a nonnegative matrix. It can be seen immediately that y = p - o completely describes the eigenvalues of D when o ranges over those of A . Denote by Re(cc) the real part of a complex number M . Then, p - Re(y) = Re(p - q ) = Re(w) 5 l o 1 5 /I = A(A). That is, Re(q) 2 p - A(A) for all the eigenvalues y of D, with the equality holding when y = p - A(A). If x is a nonnegative eigenvector of A associated with A(A), it is also an eigenvector of D associated with p - A(A) because D = p l - A . Moreover, since D = p l - A is nonnegatively invertible if and only if p > A(A), the nonnegative invertibility of D is equivalent to the condition Re(y) > 0 for all the eigenvalues y of D because of the inequality Re(q) 2 P -4A). 97.2. lndecomposability

The input coefficients matrix A of a Leontief system represents the interrelationships among sectors in connection with the transaction of products. A may or may not have zero elements; in some cases, the layout of these elements allows the system to split into subgroups of sectors where production activities take place more or less independently of each other. A typical concept of splitability is decomposability.

DEFINITION 7.1. A nonnegative n x n matrix A = ( a j j ) is called decomposable if there is a nonempty proper subset J of { I , 2,. . . , n } such that aij = 0

(i 4 J , j

E

J).

(1)

DEFINITION 7.2. A = ( a j j )is called indecomposable, if it is not decomposable and is not the zero matrix of order 1. If A is decomposable with respect to a J consisting of k integers, it is transformed to a matrix

where T is a permutation matrix carrying J to { 1, 2, . . . , k } , and the lower lefthand corner 0 is the ( n - k ) x k zero matrix, the sizes of the submatrices B , , , B , , , BIz being k x k , ( n - k ) x ( n - k ) , and k x ( n - k ) , respectively. Sectors belonging to J use no products of sectors i n J ' as inputs. The group of sectors J is therefore in an autarchy situation as far as the supply of inputs is

106


concerned. But this need not imply that a similar autarchy prevails with regard to the demand for the products of sectors in J . In other words, the decomposability of A does not necessarily mean that a i j = 0 for i E J , ,j 4 J . If aij = 0 for i E J , j 4 J as well as for i 4 J , j E J , the system splits completely into the two groups J and J', each of which is in an autarchy situation, and between which no transaction takes place. B , , = 0 for such an A , if transformed by the same T.

LEMMA 7.3. A

= ( a i j ) is decomposable if and only

if its transpose A'

is decomposable.

Pvoof. If uij = 0 for i 4 J , j E J , where J is a nonempty proper subset of { 1, 2, . . . , n } , let A' = ( b i j ) . Then, b i j = a j i for i 4 J', j E J', and J' is also a nonempty proper subset of {I, 2, . . . , n } . Hence the decomposability of A implies that of A ' . For the same reason, the decomposability of A' implies that of A = A " , Q.E.D. The following lemma gives a useful characterization of decomposability.

LEMMA 1.4. A = ( a i j )2 0 is decomposable if arid only (f there are a notinegatire realnumher anda semipositire r e c t o r s not allqf whose components are positire, ,fulfilling A x 5 /is. Pvoof: (i) Sufficiency. Let J = { j I x j > 0: for an s = ( x i ) satisfying the condition. By assumption, J is a nonempty proper subset of { I , 2, . . ., n ) . A x 5 p x implies n

1

(iijxj

j= I

50

(i

4J)

The summands on the left-hand side of theabove inequalities are nonnegative, S O that ai.;xj = 0 (i 4 J , .j = I , . . . n ) . Then, since x j > 0 for , j E J , we have a i j = 0 for i 4 J , , j E J . (ii) Necessity. A decomposable A is transformed to a matrix of the form (2). Choose a nonnegative eigenvector y' 2 0 of the subinatrix B , , associated with /i = 2 ( B , l ) .Then we have

.

where 0 under y ' is the ( n - k)-dimensional zero vector. This relation is rearranged to

Thus the condition is satisfied by

/i

and x in an equation form, Q.E.D.

lndecomposability entails a sharpening of the general results of nonnegative matrices.

$7.

FROBENIUS-PERRON'S

THEOREM

I07

THEOREM 7.3. Let a nonnegatire square matrix A be indecomposable. Then, (i), (ii) hold. (i) Any nonnegatire eigenrector associated ic,ith A( A ) is positire. Moreover, ).(A) > 0. (ii) The eigenrector of A associated MYth ).(A) is unique up to the multiplication by scalars. Proof. (i) Since A s = i(A)s, x 2 0, Lemma 7.4 rules out the possibility x > 0. Whence s > 0. Next, if A(A) = 0; then A s = 0. Hence x j a i j s , = 0 for all i, which implies a i j = 0 for all i, j because a;, 2 0, x j > 0 for all i, j. Thus A = 0, contradicting the indecomposability of A . This proves A(A) > 0. (ii) Let y be an eigenvector associated with i ( A ) . Without loss ofgenerality y may be assumed to be a real vector. Take a positive eigenvector x associated with ;(A), and let 0 = min(yi!sj) over all i. Then u = y - Ox 2 0 and Au = Ay - OAs = I(A)y - UA(A)s = A(A)u. But the definition of 0 rules out the possibility of u > 0. Whence u = 0 by Lemma 7.4. This proves y = Ox, Q.E.D. THEOREM 7.4. (i) If ( p l - A ) s 2 0 .for an x 2 0 and an indecomposable A 2 0, then p l - A is nonnegatirely inrertible. (ii) I f p l - A is nonnegatirely inrertible for an indecomposable A , its inverse ( p l - A ) - ' is a positire matrix. (iii) I f A 2 B 2 0, and one of A or B is indecomposable, then R(A) > A(B). (iv) Any proper principal minor of A/ - A .for an indecomposable A 2 0, here A = ).(A), is positive. I f A 2 0 is indecomposable, 1 ( A ) is a simple root of the characteristic (v) equation. Proof. (i) Let p' be a positive row eigenvector associated with ;.(A). Then ( p - i.(A))p'.u = p ' ( p l - A ) s > 0, from which follows p > ).(A) because p ' x > 0. Whence p l - A is nonnegatively invertible by Theorem 7.1 .(ii). (ii) Set s = ( p l - A)-'c for any semipositive 2 0; then ( P I - A ) s = c 2 0. That is, Ax 5 px. In the light of Lemma 7.4, s 2 0 must be positive. Therefore ( p l - A)-'c > 0 foranyc L 0. For c j = (ci'), wherecj = 1 for i = j , 0 for i # ,j, this implies that the j t h column vector of ( p l - A ) - ' is positive. Whence ( p l - A ) - ' > 0. (iii) Let C = :(A + B ) , then A 2 C 2 B. It is readily seen that both A and C must be indecomposable. Next let s be a positive eigenvector of A associated with jL(A). Postmultiplying A 2 C by s gives i . ( A ) s = A s 2 Cx, i.e., (i.(A)I - C)x 2 0 for this x > 0. Whence i ( A ) > i ( C ) 2 i ( B ) by (i) and Theorem 7.2(iv). (iv) We proceed in the same way as in the proof for Theorem 7.2(v) up to the inequality A 2 B 2 0 . B is by construction decomposablc, while A is by assumption indecomposable. Hence in reality A 2 8,which implicsl(A) > i.(B) (a

108

11. SIMPLE MULTISECTOR LlNEAR SYSTEMS

by (iii). On the other hand, A(B) = A(C) for the proper principal submatrix C of A from which B has been constructed, as was shown in the proof for Theorem 7.2(v). Whence A(A) > A(C), and therefore A(A)Z - C satisfies the Hawkins-Simon condition. This proves in particular that det(A(A)Z - C) > 0 , (v) The derivative of the characteristic polynomial 4 ( p ) of A is given by @(p) =

2 det(p1 - Ai),

i= 1

where A i is the principal submatrix of A obtained from A by deleting its ith row and column. Then, 4'(A(A)) > 0, since det(A(A)Z - A i ) > 0 by (iv), if A is indecomposable, Q.E.D.

THEOREM 7.5. For A 2 0, let r i and sj be its ith row sum and its jth column sum. Then, we hai'e (i) minlSisn r , 5 A(A) 5 maxlsisn r i . Moreot>er,if A is indecomposable, strict inequality holds unless min r i = max r i . sj . Moreover, if A is indecomposable, (ii) miq, _Ijsn- sj 5 A(A) 5 maxl strict inequality holds unless min sj = max sj . Pvoof. (i) Take a row eigenvector p' = ( p i )2 0 associated with A(A) and normalized in such a way that the components add up to unity. We therefore have

which, upon summation o v e r j = 1, . . . , n, becomes

This shows that A(A) is the weighted average of r i with the weights p i 2 0, p i = 1. Whence the first part of the assertion. If A is indecomposable, these p i are all positive by Theorem 7.3. Hence, if min r i < max r i , none of min r i and max r i equal ],(A), which is a weighted average of r i with positive weights. This proves the latter part of the assertion. (ii) The proof is exactly similar to that for (i), Q.E.D.

2

$8. I M P R l M l T l V l T Y AND OSCILLATORY BEHAVIOR 08.1. Stable Matrices

As was mentioned before, all sectors together compose an indivisible group of industries in a Leontief system having an indecomposable input coefficients

$8.

IMPRIMITIVITYAND OSCILLATORY BEHAVIOR

109

matrix A. They are intimately connected with each other in the transaction of goods. The present section will be devoted to a more detailed analysis of this interindustrial connection in an indecomposable Leontief system. A review of the definition of indecomposability immediately allows us to realize that indecomposability by no means implies the direct connection of all sectors with each other. There is a possibility that some sectors are only indirectly connected to another subgroup of sectors, and this concept of indirect connection gives rise to a characterization of indecomposability (Lemma 8.1).

DEFINITION 8.1. Let A = (aij>2 0. Then, the ith sector is said to be connected to the jth sector if there exists a chain of sectors {ko,k,, k, , . . . , k,} joining k , = i to k , = j such that any two consecutire sectors k, and k,+l are directly connected in the sense that akSk,+,> 0. The intuitive meaning of this connection concept is self-explanatory. If the ith sector is connected to the j t h sector by a chain { k , , k,, k , , . . . , k,}, the j t h sector’s production activity induces the demand for the product of sector i through the successive inducement of demand for inputs j = k , -+ k , - + . . . -+ k , -+ k , = i.

LEMMA 8.1. A = (aij>2 0 is indecomposable if and only ifthe ith sector can be connected to the jth sector for any i, j = 1, . . . , ti. Proof. (i) Necessity. For an arbitraryj,, let Jobe the set of numbers that can be connected to j,. The proof will be complete if it is seen that .Io = { I , 2, . . . , n}. First we observe that Jo# 0. In fact, since A is indecomposable, there is some i # j , for which aijo> 0. Otherwise a,, = 0 for i $ { j , } , j E { j o } , contradicting the indecomposability of A . Now, j in J, , by definition, can be connected t o j , by a chain { j , . . . ,j,}. Then if aij > 0 for some i $ J,, this i could be connected toj, by the extended chain { i , j , . . . ,.jo}, arriving at a contradiction i E Jo. Whence a i j = 0 for i $ J,, j E Jo, so that J , must either be empty or coincide with { 1, 2, . . . , n } because of the indecomposability of A . As the possibility Jo= 0 has been ruled out, we have J , = { 1, 2, . . . , n ) . (ii) Sufficiency. Let J be such a nonempty set of numbers that ai, = 0 for i @ J , j E J . To see the indecomposability of A , we have only to show that such a J m u s t coincide with { 1, 2, . . . , n}. To this end. assume, on the contrary, the existence of some i not belonging to J . Take a j in J ; then i can be connected to j by a chain {i, . . ., k,, k,+,, . . ., j } by virtue of the assumed connection between any two sectors. Since i $ J, j E J, there must be some consecutive numbers k,, k,+, in the chain such that k, $ J, k,+, E J . But this implies akSk,+, > 0, k , $ J, k,+, E J , which is a contradiction. Whence the existence of numbers not belonging to J is ruled out.

110

1 1 . SIMPLE MULTISECTOR LINEAR SYSTEMS

LEMMA 8.2. For A

= (aij)2

0, let A” = (a!;)). Then we have

n

‘,;;+v)

(9

=

C ui/( P )

(v)

> a ! ’ l ) a ( v/) j

a/j =

11

I= I

1, . . . , n ; p , v = 1,2, ... . (ii) A is iridecomyosnble if and only if for any pair ( i , j ) there is v for which a!’:’ > 0. IJ

fori,j,l=

Proof. (i) Since A p f P = ApAVand a$“ 2 0, a!;) 2 0, we have

(ii) In the light of (i), iis connected t o j by a chain { i = k,, k , , . . . ,k , of length v, if and only if a:;) > 0. Hence Lemma 8.1 implies (ii), Q.E.D.

=j

}

In Lemma 8.2(ii), the value of v for which a$) > 0 depends in general on the pair ( i ,j ) , as is the case with the following matrix

However there is a special class of indecomposable nonnegative matrices such that each matrix A = ( a i j )of this class has a positive integer k for which a:;) > 0 for all i,,j,i.e., Ah is a positive matrix. Needless to say, positive matrices belong to this class. Matrices of this class have a special dynamic property, and we begin a detailed exploration of indecomposable matrices by studying this dynamic property. We recall, i n the light of Theorems 6.4 and 7.1, that if p > A(A) for a nonnegative A , we have lim(A/p)”= 0. What if p = A(A)? The behavior of (A,’).(A))’’,as v + co, is in general oscillatory, even if A is indecomposable. I n this respect, it is interesting that the lack of oscillatory behavior characterizes the above special class of indecomposable matrices.

+

THEOREM 8.1. Let A be an indecomposable nonnegatice matrix and i

=

?.(A).

lim,+ + JA,’lL)‘ exists, if and only i f there is a positice integer k such that Ak > 0. (ii) I n the case qf the conrergence of (A/,?)’, et‘ery column of the limit matrix is n positire column eigenrertor of A and erery row is a positke row eigmrector of A , both associated with 1. ProoJ Theorem 7.3(i) ensures /I > 0, so that A//? makes sense. Moreover, by Theorem 7.2(i), the dominant root of A/A is unity. This enables us to assume, without loss of generality, that ,?(A) = 1 . (i)

$8.

IMPRIMITlVlTY AND OSCILLATORY BEHAVIOR

111

First we shall prove (ii). T o this end, let B = lirn A'. Clearly B 2 0. We also have lirn A'" = B, so that A B = A lirn A' = lim A"' = B and, similarly, BA = B. A B = B implies that Ab = b for any column of B. It follows, by Theorem 7.3(i), that b is either a positive column eigenvector of A associated with I = 1 or the column zero vector. Likewise BA = B implies that any row of B is either a positive row eigenvector of A associated with I = 1 or the row zero vector. In reality, B = ( b i j )is a positive matrix. For if some b,, = 0, the above results imply that both the ith row and thejth column of B vanish, and hence all the rows and columns vanish. But the possibility of B = 0 is ruled out, because Bx = lirn A'x = x for a positive column eigenvector x of A associated with I = I . This proves (ii). The " only if" part of (i) immediately follows from (ii). In fact, iflim A' = B exists, B must be positive by (ii). Hence A' must also be positive for t 2 k, whenever k is large enough. For proof of the "if" part of (i), first assume that the assertion is true for any positive matrix. Then, since Ah > 0 and I(A") = A(A)k = 1 by Theorem 7.2(iii), lirn (Ak)' = B > 0 exists. Every positive integer t is uniquely expressible in the form t = p(t)k r ( t ) , where p ( t ) , r ( t ) are nonnegative integers with 0 r ( t ) < k, limp(t) = + co. Therefore, letting ( A k ) p ( ' ) = B + R ( t ) , we have lirn R ( t ) = 0 . Hence A' = Ar(')(Ak)p(') = A'(') ( B + R ( t ) ) = A'(')B A'(')R(t) + B because A'(')B = B and A'(') is bounded. The reason for A'(')B = B is that since A and A k have eigenvectors in common, the columns of B are positive column eigenvectors of A associated with I = 1 ; hence A'(')B = A A ... A . B = B. Thus it remains to prove the assertion for positive matrices. Now let A be a positive matrix with A(A) = 1. To prove the convergence of A', we have only to show that the vector sequence { A ' y } is convergent for any semipositive y 2 0. For, if the convergence of { A ' y } is established, {Arci} is convergent for each ci = (S,,), where S,, are Kronecker's deltas. Whence the convergence of { A ' } follows. Set y ( t ) = A'y for a given y 2 0. y(r) is a solution of the difference equation

+

+

Take a positive column eigenvector x let

= ( x i ) of

A associated with

A = 1, and

112


We let I(t) be an integer for which z ( t ) = O I ( , ) ( t ) . Then, for any i Ji(f

+ 1) = 1

UijJj(t)

j = 1 n

=

1

UijOj(f)Xj

j = 1

5 &(t)atl(r)Xl(r)+ B(t)

1

a i j ~ j

J#1(1) n

( a ( t )- P ( t ) b i I ( t ) =

Xl(t)

+ P ( t )J C =1

~

i

x,,

( 4 0 - B(t))all(l,Xi(') + P ( t ) X l '

Now let

6 = min a,,,

E

= min

1,J

1,

xJ/x,.

i

Then 6 > 0, E > 0, and the above relations, if divided by x,, become y,(r

+ l)/xl 5 ( a ( / )

-

/?(t))~6

+ B(r)

(i = 1, . . . , n);

from which B(l

+ 1) 5 ( a ( / )- B ( t ) ) 66 + B(r),

which can be rearranged to (B(t> - a ( [ > )2

Similarly we have

&a (/?(r)

P(f) - B(r + 1).

- ci(t)) 5 ci(r

+ 1) - N ( t )

Since P ( t ) 2 ci(r), the above two inequalities imply that ( z ( t ) )is nondecreasing and bounded above, and { B ( t ) } is nonincreasing and bounded below. Herice lim a ( [ ) = ci and lim P ( t ) = /? exist. Then, either of the above two inequalities entails CI = p. In view of the definition of a ( j ) , P ( r ) , we have thereby proved lim y ( f )= ax, Q.E.D. Application 1. An economic implication of the above dynamic phenomenon is the relative stability (for its definition see $6.3) of a balanced-growth solution X ( t ) = A'Y, A = A(A), Ax = A x of the self-sustained system x(t

+ 1) = Ax(/),

(1)

where A is a nonnegative matrix such that Ak > 0 for some k . For any solution x ( r ) = A'x(0) of the equation,x(t)//l' converges to Bx(O), as r tends to infinity,

$8.

I M P R I M I T I V I T Y A N D OSCILLATORY BEHAVIOR

1 I3

where B = lim (A/A)'. Stated differently, all the component ratios xi(t)/A'xi converge to a common limit. The latter statement can be justified in the following way: x(t)/A' is a solution of y(t + I ) = (A/A)y(t), with A(Aj3,) = 1, so that xi(t)/A'xicorresponds to di(t) in the proof of Theorem 8.l(i), which converges to a limit common to all i. Application 2. Theorem 8.1 also justifies an iterative solution of the eigenvalue problem Ax = A x for A such that A k > 0 for some k . For any two solutions x ( t ) , y ( t ) of (l), starting at x ( 0 ) 2 0, y(0) 2 0, respectively, let 5 ( t ) = m i n { y l ( t ) / x l ( t ) ,~ z ( t ) l ~ 2 (.t.). ,3 yn(t>/xn(t>)T

~ ( t=) max{yl(t)/xl(t),

YZ(~)/XZ(~),

. . ., Yn(t)/'xn(t>>.

Since Ak > 0, both < ( t ) and q(t) are well defined for t 2 k . A simple evaluation entails n

yi(t

+ 1 ) = j C= 1 aijyj(t>

=

S(t>xi(t+ 1 )

9

+

which implies ( ( I ) 5 5(t 1). Similarly we have q ( t ) 2 q(t {t(t)}and { q ( t ) } converge to a common limit because

+ 1).

Moreover

converge to a common limit t/a, where CT

= lim I'm

xi(t)/A'xi,

z = lim yi(t>/1'xi I

-a

for all i. Now let y ( t ) = x(t + 1) for any solution x(t) of (1) with x(0j 2 0. Clearly y ( t ) is also a solution of (1) with y ( 0 ) 2 0. Since

converge to A for all i, the corresponding { [ ( t ) } and { q ( f ) } tend to 1 monotonically from below and above, respectively. On the other hand, a positive eigenvector is obtained as the limit of

114


for the following reason: O i ( t ) = xi(r)/drxi converge to a common positive limit 0, so that the ith component

C xj(f)

J=1

c

J=

0j(t)xj

1

x,/Cy= ,

converges to xJ for any i. On the basis of Theorem 8.1, we might as well reserve the special name stable matrix for the class of indecomposable matrices considered above. We may call a nonnegative matrix A a stable matrix, if A(A) > 0 and lim ( A ’ i ( A ) ) ‘ exists. Thus for nonnegative indecomposable matrices “ a stable matrix” is synonymous with “ a matrix which, if raised to some positive power, becomes a positive matrix.” “

”

$8.2. lmprimitivity

In a Leontief system having an indecomposable input coefficients matrix A = ( u , ~ )all , sectors are connected with each other, but this interindustry connection js in general indirect, as was explained before. But, if the matrix is unstable, sectors are decomposed to subgroups of sectors, which, if suitably numbered, are connected in a cyclical way. In greater detail, n sectors are decomposed to s subsets (s 2 2) GI, G, , . . . , Gsin such a way that S

G,

where G I t l

=

f Izi ( k

=

i E G,+

for

I

1, . - . )F), 1

G,n G,

fulfilling

=

0

(1 # k ) ,

> 0 ( j G,), ~

( I , ~

(2)

(3)

GI.

Let l, be the number of integers i n G,, and P be a permutation matrix bringing GI to { I , 2, . . . , G, to {/, + 1, I , + 2, . . . , I I + /2}, . . . , G, to It}. Then, A I, + 1, . . . , I I,}, . . . , G, to {CJL I, + I , . . . , transform s to

(c:::

c:=

(4)

F’AP= 0

...

$8.

115

I M P R I M I T I V I T YAND OSCILLATORY BEHAVIOR

where A , , , k is a nonnegative submatrix with Ik +, rows and Ik columns (with the convention of s + 1 regarded as I), and all the other elements are zero. We observe the square zero matrices of order lk ( k = 1, . . . ,s) along the principal diagonal in Eq. (4). For the sake of simplicity, in this section we call { G k }satisfying Eqs. (1)-(3) a cyclical decomposition. DEFINITION 8.2. A nonnegatioe square matrix A primitive

= ( a i j ) is called imhas a cyclical decomposition. Otherwise it is called primitire.

if it

EXAMPLE.

A =

' 0 3 0 0 0 3 4 0 0 0

0 0 3 0 ' 0 0 0 2 0 0 3 0 4 0 0 0 0 0 0 2 , 0 0 0 1 0 0

has a cyclical decomposition G,

/o

= { 1,

1

3, 6}, G, = {2, 4, 5 } , and

0 0(-303(

3l

p-lAp=[w O 0 0O 0O I 30 O1 0

0 0 0 1 0 0

P=

0 1 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0 1

0 0 2 0 0 0

H , = (43, H , position, and

Q-'AQ

=

=

{6}, H ,

=

(2, 5 } , H4

=

0 0 0 O p - r 7 0 0 0 0 0 0 2 0 0 0 0 01210 0 0 0 0 0

0 0

1-1

3 3

3 3

0 0

0 0

{ I , 3) also form a cyclical decom-

P=

0 0 0 1 0 0

0 0 0 0 0

0 1 0 0 0

1

0 1 0 0 0 0 0 0 1 0 0 0 1 0 0

1 0 0 0 0

This example shows that in general there may be many cyclical decompositions. The purpose of this section is first to establish the equivalence of primitivity to stability, and then to clarify various features of imprimitivity, all for indecomposable matrices. To this end we first must consider the following lemmas.

~

116


LEMMA 8.3. Let A 2 0 have a cyclical decomposition (1)-(3), and denote by r(v) the remainder obtained by dividing an integer v by s. Then (i), (ii), (iii) hold for A' = (a@))(v 2 1):

(i) a:;) > 0, ,j E G, imply i E Gr(,+,,),with the convention Go = G, . (ii) IJ'v is not divisible by s, a:;) = 0 for i, j E G, . In particular, a$') = 0 for any i. (iii) If v is dioisible by s, a!;) = 0 for i 4 G, ,j E G, . Proof. (i) The assertion for v = 1 is condition (3) for a cyclical decomposition. Assume the lemma for v and any k (= 1, 2, . . . , s). If a$;+') > 0, by Lemma 8.2, a!;") = a$)alj > 0, so that a$') > 0, alj > 0 for some 1. I f j E G, , then alj > 0 implies, by (3), I E Gk+l, which, combined with a$') > 0, entails i E G r ( k ++,,). l This completes the induction. (ii) r(k + v) # r(k)for v not divisible by s, so that a:;) > 0 f o r j E G, would imply i E Gr(,+,,),r(k + v) # r(k). Hence follows the assertion. (iii) r(k + v ) = r(k) for v divisible by s, so that a$) > 0 for j E G, would imply i E Gr(,+,,),r(k + v) = r(k). Whence (iii), Q.E.D.

c;=l

LEMMA 8.4. Let a l , a 2 , .. ., am be posiiioe integers whose greaiest common divisor is one. Then there is a positive integer h such that any integer v 2 h is representable in the form m

v=

1

cliai,

i= 1

where aiare some nonnegatke integers. PuooJ: We first recall the well-known result in number theory that the greatest common divisor of a, , namely 1, can be put in the form m

where y i are integers, but need not be nonnegative. The proof of this result is easy and can be found in any text book in elementary algebra and number theory. Then, a desired h is obtained by

h

= d2,

d=

1 lyil a,.

i=1

In fact, let v = qd + r, d > r >= 0, for any v 2 h ; then clearly q 2 d. Hence, in view of r = ryi a,, which is implied by ( 5 ) ,

1

v = qd

+r

m

=

1 (9 IyiI + ryi)ai

i =1

5

98.

IMPRIMITIVITY AND OSCILLATORY BEHAVIOR

117

with nonnegative coefficients q ( y i ( + ryi 2 r((yil+ yi) 2 0 because q 2 d > r. This completes the proof.

THEOREM 8.2. Let A be an indecomposable nonnegative square matrix and A' = (a:;)). Denote by n the greatest common divisor of all the integers in the set Mi = {v I a?? > 0). Then two cases (i) and (ii) occur: (i) r f n = 1, A is primitive, and Ak is a positioe matrix for some positice integer k . (ii) If n > 1, there is a cyclical decomposition of { 1, 2, . . . , n} to n subsets, so that A is imprimitive. Proof. Note that MI # @ because a?? > 0 for some v by Lemma 8.2(ii). (i) It suffices to show that the first row of A' is positive for some positive integer t . If this is true, the first column of As is also positive for some positive integer s, since the transpose of A is again indecomposable, with the corresponding M , being the same as that for A . Thus Ak = ASA' > 0 f o r k = s + t . Now, since 7~ = 1, there are a finite number of positive integers v,, . . . , v, in M , whose greatest common divisor is one. By Lemma 8.4 as applied to v,, .. . , v, ,there is an h such that any v 2 h can be put in the form v = ui v i (ui nonnegative integers). In view of Lemma 8.2(i), for any v 2 h, we have

,

a?!

>=

n

(a(i"$l> 0,

ai>O

which says that M , contains all the integers v 2 h. Hence, A can not be imprimitive in the light of Lemma 8.3(ii). The reasoning is carried further by making the first row of some power of A positive. By the indecomposability of A we have a?) > 0 for some positive integers pj ( j = 1, 2, . .., n). With the choice of h, p j , above, we can complete the proof by showing that t = h + max {pl,p 2 , . . . ,p,,} is a desired integer. In fact, by letting t = h + A, + pj for each j , where Aj is a nonnegative integer, we have, by Lemma 8.2(i),

i.e., the first row of A' is positive. (ii) As usual, let r(v) stand for the remainder resulting when v is divided by n. Define 7~ subsets Hk of {I, 2, . . . , n} by

H , = { i I u !;) > 0, r(v) = r ( k ) for some v) with the convention H , = H , , H,, = H I . They form a cyclical decomposition, as will be seen below. First, for any i there is v by Lemma 8.2(ii) such that a!:) > 0. Whence i E H,(,,). That is, any i is included in some of Hk ,so that condition (1) for a cyclical decomposition is fulfilled. Next we consider condition (3). If j E H k ,we have, by definition, a$) > 0, r(v) = r(k) for some v. Hence, if moreover aij > 0, we have a$'+1)2 aija$;)> 0 by Lemma 8.2(i),

118


+

while r(v) = r ( k ) ensures r(v 1) = r(k + 1). That is, a,, > 0, j E Hk imply i E Hk+ . Therefore condition (3) is satisfied. As to condition (2), let us consider the second part, HI n Hk = /zr (I # k). Suppose that i E Hr n Hk (I # k). By definition, there would be p, v such that a$) > 0, a!;) > 0 , r ( p ) = r(l),r(v) = r(k). But, again by Lemma 8.2(ii), we have a\:) > 0 for some w . Whence a:';+-) 2 ay'i)0,' BI ) > 0, a?:") 2 a(c)a:y)> 0, so that p + w , v + w E M,.This implies that p + w and v + o are divisible by n; hence their difference p - v is also divisible by T . This entails r(1) = r ( p ) = r(v) = r ( k ) , which, combined with n 2 Z, X 2 1, implies I = X , a contradiction. This proves the second part of condition (2). Its first part can be derived, however, from conditions (1) and (3) and indecomposability. In fact, by condition (I), some H, is nonempty. Then, HA+ I must be nonempty. Otherwise, by condition (3), a,, = 0 for j E H , and all i = 1, . . . , n. This clearly contradicts the indecomposability of A . Then. similarly, H A + ,# 0implies H k + 2# /zr, etc., until we reach Hr # 0for any I, Q.E.D.

,

For an indecomposable imprimitive matrix, the above n and the corresponding cyclical decomposition given in the proof for Theorem 8.2(ii) completely determine all the possible cyclical decompositions. This result is stated in

THEOREM 8.3. Let A 2 0 be indecomposable and imprimitit'e, with n being the greatest common dii'isor of M , = {vIa~",'> 0 } ,greater than one. Then (i)-(iii) hold. (i) I f a cyclical decomposition consists of s subsets, s dicides n. (ii) For any divisor s of n, there is a cyclical decomposition to s subsets. i t isosubsetsfor n a giren diitwr (iii) There is ctnfy one cyclical ~ e ~ . ~ ~ p o s to s of n. Proof. (i) Since a(,",)> 0 for v E M , , all the integers v in MI are divisible by s by Lemma 8.3(ii), so that their greatest common divisor n is divisible by s. (ii) We recall that there is a cyclical decomposition to n subsets H , , H , , . . . , H , (see the proof for Theorem 8.2(ii)). We construct a cyclical decomposition to s subsets by combining some H k together in the following way: d- 1

G,

=

u

Hm,+k ( k = 1,

. . .,S ) ,

m=O

I t will be noted that these G, form a cyclical decomposition. In fact, of the three conditions for a cyclical decomposition conditions (1) and (2) are obviously satisfied by construction. T o see condition (3), let aij > 0 f o r j E G,.

$8.


119

Then, this j belongs to Hms+,for some m, d - 1 2 m 2 0. Thus aij > 0,j E Hms+k imply i E Hms+k+l c Gk+l,with the convention of G,+, being regarded as G,. (iii) Let {E;, F 2 , . . ., F,} and {G,, G2,. . ., C,} be two cyclical decompositions, both having s subsets. Without loss of generality we may assume Fl n G, # 0. Next let

u 7

J =

(Fk n Gk).

k= 1

Suppose that aij > 0, j E J . This j belongs to Fk n Gk for some k. Hence a i j > 0, j E F,, imply i E F k + , ,with the familiar convention F,+, = F,. Likewise a i j > 0, j E G,, imply i E G k + , ,with G,+l = GI.These results mean that aij > 0,j E J , imply i E J . Whence a i j = Ofor i # J,j E J . Since J 3 F, n GI # the indecomposability of A entails1 = (1, 2, . . . , n } , where n is the order of A . Thus F, n Gk together cover { 1,2, . . . , n}. This is impossible unless F, = G, = F, n G,, completing the proof.

a,

For an indecomposable nonnegative matrix A we term the greatest common divisor n of the set M I = {v I a:",) > 0} theperiodofA. To define the period of A , we could begin with M i = {v 1 a!:' > 01, for we have already characterized TC as the number of subsets in the most refined cyclical decomposition and would arrive at this cyclical decomposition even if we started from M i rather than M , . Whence the greatest common divisor of M iis exactly the same as that of Ml, nLet H,, H 2 , . . . , H , be the cyclical decomposition corresponding to the period 7c. Since a suitabIe renumbering of the rows and columns puts A in the form (4), with s = TC, we may assume, without loss of generality, that A in itself is of the form (4). Then in the light of Lemma 8.3, we have

B,

0

0

B2

- -

.

Bk

'

0

0

where Bk= (a:;'), i, j E Hk , are square nonnegative matrices placed along the principal diagonal, while the other elements of A" are zero. In greater detail, we have

THEOREM 8.4.

Bkin (6) is an indecomposable primitiiie matrix.

120


Proof. Note that B, = (a$)),Bky= (@)), i, j E Hk . By Lemma 8.2(ii), to see the indecomposability of Bk we have only to show that for any i, j E Hk, there is some v for which ajr") > 0. Jn fact, by the indecomposability of A , there is some p for which a$) > 0. But by Lemma 8.3(ii), this p must be a multiple of n, so that p = vn for some v. Whence a$") > 0, and Bk is indecomposable. Next, to see the primitivity of Bk , we consider the set Mi(Bk)= {v I aim) > 0) for some fixed i E Hk. Clearly a positive integer p belongs to M iof the original matrix A if and only if p is divisible by 71 and p / n belongs to Mi(&). Since n is the greatest common divisor of M i , that of M,(B,) is one. Hence Bkis primitive by Theorem 8.2, Q.E.D.

48.3. Alternative Characterization of the Period The period n of an indecomposable nonnegative matrix A is alternatively characterized as the number of the eigenvalues w of A which are equal to I ( A ) in modulus, i.e., 1 0 1 = I ( A ) . In other words, the expressions ok= I ( A ) q k ( k = 0, 1, . . . , n - l), where q is a primitive nth root of unity, completely exhaust these eigenvalues. This section will be devoted to a brief presentation of these results. It should be noted, however, that the complete enumeration of all eigenvalues equal to I in modulus here is of independent interest, since we will not depend on it in the subsequent sections.

LEMMA 8.5. If w is an eigenvalue of an indecomposable primitive nonnegative A such that 1 .01 = d(A), then o = I ( A ) . Proof. Let z be an eigenvector of A associated with w . Clearly we have for any t = 1, 2, . . . , (;)Iz

=

(qz,

1 = I(A).

As was made clear, by primitivity, ( A / I ) fconverges to a limit matrix whose rows are positive multiples Bip' of a positive row eigenvectorp' of A associated with d (see Theorems 8.1 and 8.2). Hence the right-hand side of (1) converges to

so that (w/d)f also converges to a limit yl with lql = 1. From the limit result of (I), it therefore follows that the proportions of the components of z are 8, : B2 : . . . : 0,. Thus oz is a real positive vector for a suitable complex number o.

$8.

IMPRlMlTIVITY A N D OSCILLATORY BEHAVIOR

121

Multiplication by 0 converts wz = Az to w ( m ) = A(rrz). Hence A 2 0, rsz > 0 imply that w must be real and positive. This proves o = A because IwI = A, Q.E.D.

LEMMA 8.6. If an indecomposable nonnegatiue A is imprimitire, with the period rc > 1, every eigenvalue w of A fu@lling Jw I= A(A) satisfies the equation W" = A(A)". Proof. Clearly w" is an eigenvalue of A". We now recall that A" is of form (6) in $8.2. Hence w" must be an eigenvalue of some B, . On the other hand, Theorem 7.2 implies that A(A)" = A(A") and A(A") 2 A(B,). Since JonJ = A(A)" and A(&') >= Jw"Jfor Bk having W" as an eigenvalue, we have A(B,) = A(A)" = 1 0 ~ 1 . Since Bk is indecomposable and primitive by Theorem 8.4, Lemma 8.5 as applied to this particular Bk implies W" = ].(A)", Q.E.D. It now remains to see that every solution of on= A(A)" is an eigenvalue of A .

LEMMA 8.7. Under the same assumption on A as in Lemma 8.6, erery solution w of on= ,?(A)" is an eigencalue of A. Proof. Without loss of generality, A may be assumed to be of the same form as the right-hand side of (4) in $8.2, with s = ri. Let

x=

Xk

\X./

be a positive eigenvector of A associated with 2 = A(A), where {XI, .... x k , .... x " } is the partition of the components of x corresponding to the cyclical decomposition H I , . . . . H,, with xk being associated with H , . Then, the relation Ax = Ax for (2)can be converted to ri relations,

Ax2 = A,,x'

............. AXk = Akk-,Xk-'

.................. AX" = An"- ,x"-'.

( 3)

I22

I I . SIMPLE MULTISECTOR LINEAR SYSTEMS

Let 9 be any rrth root of unity. Multiplying the kth equation in (3) by q l - kand setting o = Rq, we obtain oq-Ixl = A l n q - n x n wq-2x2 = A 2 , q - I x l

................... Wq-kXk

= ~ ~ ~ - ~ . - ( Xk k - l l )

................... w ~ - " x "= A n n - 1q- ( " - I )

I n view of q" relation

=

1, the

TI

X

n-l

resulting relations can again be brought to a single wz = Az,

where

Therefore w = Aq is an eigenvalue of A with an associated eigenvector z. Hence w k = lq' (k = 1, .... TC)are eigenvectors of A for a primitive nth root of unity.q, Q.E.D. From Lemmas 8.5-8.7 immediately follows:

THEOREM 8.5. Let an iiidecomposable nonnegative matrix A hare the period n. Then w satisjjiifg 101 = & A ) is an eigenr.uk of A if and only ifw is a root of the equation on= A(A)n. 88.4. Backward Difference Equations

I n a dynamic process of income propagation as exemplified by (4) in 96.3, there is no lag in the underlying production processes, so that if A is the corresponding input coefficients matrix, the current input A x ( t ) is supplied out of the gross output x ( t ) at the same period, the remaining part of x ( t ) covering the final demand c ( t ) . In this section, we consider a situation in which production takes time, and the fruition of output is retarded to input. N o general situations will be considered, and attention will be confined to a rather specific situation, since we aim only at obtaining results as simple applications of the theorems on indecomposable nonnegative matrices.

48.

lMPRIMlTlVlTY A N D OSCILLATORY BEHAVIOR

123

Let m be a fixed positive integer that denotes the time horizon in the given production technology. This means that the emergence of output is lagged to input by m periods. The given technology is of the Leontief type as before, characterized by nonjoint production and fixed input coefficients, but formulated in a slightly different way, corresponding to the presence of production lags. The formal basic assumptions are as follows : (a) z(t) is an n-dimensional gross output vector at period t . is a square indecomposable nonnegative matrix of order (b) A , = (amij) n, representing an input coefficients matrix whose element amrjdenotes the amount of the ith good needed at period t to obtain unit output of the jth good at period t m. (c) An infinite stream of final demand vectors {d(t)}," is exogenously given, where d ( t ) 2 0 represents an n-dimensional final demand vector at period t = 1 , 2 , . . . . z ( t ) = A,z(t m) d(t), (1) Z ( l ) 2 0 ( t = 0, 1, . . .). The nonnegativity constraint upon z ( f ) in (1) is required for the economic reason that z(t) represents the gross output vector. T o begin with, we should note certain peculiar facts that are unique to backward difference equations with the nonnegativity constraint but are not shared by ordinary forward difference equations such as those considered in $6.3 and elsewhere. First, the stream of exogenous final demand vectors {d(t))oa cannot be completely arbitrary. If the final demand grows very rapidly, (1) cannot have any solutions. Even if the system works for a finite number of consecutive periods, a stream of too rapidly growing final demands eventually consumes the gross output at some period, beyond which the system cannot be workable. To ensure a self-feeding growth (I), the final demand should grow at a moderate pace. This constraint on the stream of exogenous final demand vectors will be given a precise expression in the form of a necessary and sufficient condition for the existence of solutions of (1) at a subsequent stage of discussion. Second, even if the existence of solutions of (1) is ensured, their initial positions z(0) can generally not be arbitrary. For if we start at an arbitrary nonnegative vector z(O), we may eventually encounter a situation at some period t o such that the equation z(t,) = Az + d(to)has no nonnegative solution z to be taken as one value of z(t, m). Such a situation may emerge even at the initial period f, = 0. This peculiar feature can be better observed by considering a special case where A,,, in (1) is invertible. If A,,, is invertible, (1) is rewritten as a forward equation, i.e.,

+

+ +

+

z(t

+ m) = A, z(t)2 0

' z ( t )- A , ' d ( t ) , ( t = 0, 1,

. . .).

124

1 1 . SIMPLE MULTISECTOR LINEAR SYSTEMS

In this equation, the exogenous term - A , ' d ( t ) may have negative components. Moreover, the matrix A , generally has negative elements, simply because A , is nonnegative. It can be seen immediately that the inverse of a nonnegative matrix A is again nonnegative if and only if A is obtained by permuting the columns (rows) of a diagonal matrix. Hence A,'z(t) - A , ' d ( t ) may not be nonnegative even if z(t) is nonnegative. Accordingly, the nonnegativity of z ( t ) over time can be ensured only if we start at a very carefully selected nonnegative vector z(0). This restricts the range of possible initial positions to a proper subset of the set of nonnegative vectors. The situation is somewhat more complicated since the invertibility of A , is not assumed in (1). With these preliminary considerations in mind, we next proceed to the complete determination of all solutions to (1). The first step is to note that Eq. (1) can be converted to an equation of the following type : ~ ( t=) Ax(t

+ 1) + c ( t ) ,

2 0 ( t = 0, 1, . . .).

X(f)

In fact, Eq. ( I ) can be put in form (2) by letting

o..*o A=

z(t

I 0 0 I .. . .

A,

.. . . .. . . . . . . . o..*o I 0o 1 7

(3)

+ 1)

x(t)= z(t

+ m - 1)

In (3), the upper right-hand corner is occupied by A,,,, and m - 1 identity matrices I of order I? are placed just below the sequence of m square zero matrices of order n along the principal diagonal, whereas all the other elements are zero.

LEMMA 8.8. Let A , be a nonnegatice square matrix of order n. Then we hare: (i) A matrix of type ( 3 ) is indecomposable ifand only i f A , is indecomposable. (ii) Let n be the period of A , if A,,, is indecomposable. Then the period of the corresponding indecomposable matrix of type ( 3 ) is mn.

$8.


125

Proof. We take advantage of Lemma 7.4. We first show that the indecomposability of A, implies that of A . To this end, it suffices to see that px 2 Ax, x 2 0 imply x > 0. The relation px 2 A x decomposes to m relations pxl 2 Xm-l if we partition x to m nonnegative vectors A,xm, p x 2 2 XI,. . . , px m > = xl,. . . , xm corresponding to the special structure of (3). Then iteration yields pmxm 2 A,xm, xm 2 0.Thus, Lemma 7.4 as applied to A , , an indecomposable matrix, implies x m > 0, which in turn entails xi < 0 (i = 1, . ..,m - 1). Whence x > 0, and the indecomposability of A follows by Lemma 7.4. Conversely, assume that A is indecomposable. Let x"> 0 be a semi-positive vector that satisfies pxm 2 A,xm for some p 2 0. Clearly, this p can be assumed to be positive. Let o = p"" > 0. Define m - 1 vectors x' = Anixm/o, x2 = x'/o,. . . , x m - l = x,-~/o.Then the combined mn-dimensional vector

X =

j/ x"'

satisfies ox 2 Ax, x > 0,o > 0. By Lemma 7.4, the indecomposability of A implies x > 0. Whence fo!lows X" > 0 in particular, and again by Lemma 7.4, A , is indecomposable. (ii) A matrix A of type (3) becomes

if raised to the mnth power. Since x is the period of A,, we observe mx square nonnegative matrices along the principal diagonal of A"", all of which are indecomposable as well as primitive by Theorem 8.4. This implies that the period of A is mn. Q.E.D. The above preliminary results lead to the study of Eq. (2), whose matrix A is a general indecomposable nonnegative matrix, but need not be of special type (3). We first consider the case with no exogenous demand, i.e., the homogeneous equation u(t) = Au(t

+ l),

u(t) 2 0 ( t = 0, 1,

. . .).

126


Throughout the following A is assumed to be of order n and indecomposable. The first thing to be noted is that Eq. (5) has a nontrivial balanced-growth solution, which is unique up to the multiplication by positive scalars. In fact, a balanced-growth solution u(t) = (l/r])'x2 0 is generated by a solution of the eigenvalue problem qx = A x , x 2 0, q > 0. As was shown in $7.2, the problem is solvable for q = A(A) > 0 and a positive eigenvector x associated with l L ( A ) . Moreover, the asserted uniqueness is ensured in the following way. If q , x are solutions of the eigenvalue problem, we have px 2 A x , x 2 0 for any p > r], however close p may be to r]. Whence p l - A is nonnegatively invertible by Theorem 7.4(i), and hence p > A(A) by Theorem 7.1(ii). This implies q = A(A), which gives rise to the uniqueness of the corresponding x up to the multiplication by positive scalars by Theorem 7.3 (ii). A remarkable result is provided by

THEOREM 8.6. If A is primitiiie, the above balanced-growth solution is the sole possible nontririal solution of (5). Proof: Let u ( t ) be any nontrivial solution, so that u ( t ) 2 0 over time. Take a positive row eigenvector p' of A associated with A = ;(A). Then, premultiplying (5) by p' and iteration give p'u(0) = p'(l.'u(t)). Since p' > 0 and i ' u ( t ) >= 0 over time, the sequence (l.'u(t)} is bounded. Again from (3,it follows that u(0) = ( A 2)'(2'u(t)). Take a convergent sub-sequence {A'w(t,)} with a limit u 2 0. lim(A jL)'= M exists by Theorems 8.1 and 8.2. Hence

u ( 0 ) = (A/A)'Y(A'W(f,)) becomes, in the limit, u(0) = M u , where the columns of M are positive multiples of a positive eigenvector of A associated with A(A). Whence u(0) 2 0 proves to be a positive eigenvector of A associated with ,?(A). For the same reason, u(t) is a positive eigenvector of A associated with A(A), so that u ( t ) = Au(f + 1) holds over time. Therefore, u ( t ) = u(O)/A'.This completes the proof, Q.E.D. THEOREM 8.7. If A is imprimitire ~ Y t hperiod n > I , Eq. ( 5 ) has TI linearly independent oscillatory solutions, and ecery solution of (5) is uniquely expressible as their linear combination ti*ithnonnegative coeficients. Proof. Without loss of generality, A may be assumed to be of the form (4) in $8.2, with s = T I .We recall the partition ( 2 ) of a positive eigenvector x of A associated with 2 = ).(A) to TI positive vectors xk ( k = 1, 2, . . . , n ) and the n relations (3) met by them i n 68.3, both of which were effected in proving Lemma 8.7. I t can also be observed that these xk satisfy

Anxk = B,xk, (6) where B, are the matrices placed along the principal diagonal of A" in (6) of $8.2.

$8.

IMPRIMITIVITYAND OSCILLATORY BEHAVIOR

127

A set of n desired solutions will be constructed by taking advantage of the n vectors xk ( k = 1, . . . , n), As usual let r ( t ) stand for the remainder resulting from the division of a nonnegative integer t by n. Define for k = 1, . . . , n, sk(t) =

I1 if n - r ( t ) = k , LO otherwise.

Then

is a solution of Eq. (5). To verify this we first note that these h,(t) satisfy s,(t

+ 1) =

s,(t

+ 1) = S , ( t ) .

6k+l(f)

( k = 1, 2,

.. .

3

J[

- I),

Then, in view of (3) in $8.3, together with the above property of hk(t),we see that

Moreover, w(t) 2 0 over time, verifying the assertion. Then, clearly, n sequences d i ) ( t ) = ~ ' (+t n - i ) (i = I, . . . , n) are also solutions of ( 5 ) . Clearly d i ) ( t )is oscillatory and traces a cyclical growth with the period n and at the growth rate l/A - 1 because, at each period t , &(t + n - i ) takes on the value 1 for exactly one k, while the other n - 1 h,(r + n - i) vanish. This special k , for which S,(t + n - i) = 1, traces 1, 2, . . . , n cyclically and in the reverse order as t goes on. A similar argument shows that this k, letting 6,(t + n - i) = 1, changes with i, whence follows the linear independence of the n solutions w(')(t) (i = 1, . . . , n). It remains to be seen that the nonnegative linear combinations of d i ) ( f ) completely exhaust all the solutions of ( 5 ) . First, by the linearity of (9,every linear combination of d i ) ( t )with nonnegative coefficients is a solution of (5).

128


Next let u(t) be any solution of (5). Then iteration entails u(vn) = A"u((v

+ 1)n)

( v = 0, 1,

. . .).

(7)

By partitioning u(t) to

in accordance with the partition of x to x k , and in view of the special structure of A" as stated in ( 6 ) of 58.2, we can rewrite (7) as U'(V7t) = k

u

(V.)

BkU k ( ( V

+ 1)n)

(k = 1, . . . , n),

2 0 (v = 0, 1, . . .).

That is, y ( v ) = uk(vn)is a solution of y ( v ) = &.(v + I), y(v) 2 0, whose coefficients matrix Bk is indecomposable as well as primitive by Theorem 8.4. Hence, by Theorem 8.6 as applied to this equation, y ( v ) is a nonnegative multiple of the corresponding balanced-growth solution, which is generated by xk by virtue of (6), with growth rate l/A" - 1. Accordingly, there must be a unique nonnegative number 6 , I 0 such that uk(vn)= ( l / A y 6 k xk. This, combined with

implies R

u(vTI)=

1 eiw(i)(vn)

(v

= 0,

1, . . .).

i= 1

1

We have hereby established that u ( t ) coincides with P ( I ) = Bi d i ) ( f ) for t = vn (v = 0, 1, . . .). Complete coincidence can now be seen immediately. In fact, if we take a v for each f such that vx > t , then u ( t ) = A""-'

~ ( v n=) A""-' ~ ( v n= ) r(t)

follow from u(vn) = r(vn), Q.E.D.

$8.


129

Theorems 8.6 and 8.7 also completely determine the range of possible initial positions of solutions of (5). That is, it is {ex 18 2 0 ) if n = 1, and the set of all vectors

j::l

,

en

ifnB1.

e i 2 0 ( i = l , ..., n ) ,

Xn

We now proceed to consideration of the nonhomogeneous case (2) where a nontrivial stream of exogenous demand c(t) is present.

THEOREM 8.8. The convergence of either one of the tM'o r:ector series for a given srream {c(t)},c(t) 2 0 ( t = 0, 1, . . . )

c a,

IVC(V),

?t

= I(A)

v=o

entails that of the other. Moreover, Eq. (2) has a solution if and only i f ( c ( ) is convergent. If this condition is met, the series

f A'c(v +

(Y)

S)

v=o

is also convergentfor any integers s = 0, 1, . . . ,and the solution of (2) is giuen by m

x(t) = u(t)

+ 2A'c(v + t ) , v=o

where u(t) is any solution of (5). Proof. Since both series have nonnegative terms, their convergence is equivalent to the boundedness from above of their sums up to t = 0, 1, . . . . Choose a positive row eigenvector p' of A associated with il= I ( A ) . Then p'

r

1

v=o

v=o

2 A'c(v) = p' 1 A'c(v)

( t = 0, 1, . ..).

Whence the left-hand side of (8) is bounded from above if and only if the righthand side is bounded from above. In view of the positivity of p ' , this implies that the boundedness (from above) of either one of the two sequences

entails that of the other. Hence (a) is convergent if and only if (j?) is convergent.

I30

I f . SIMPLE MULTISECTOR LINEAR SYSTEMS

Next, suppose the existence of a solution x ( t ) of (2). Then, t f 1 iterations give s(s) = A'+ I X ( S

+ t + 1 ) + 2 A"c(s + v), v=o

which implies

+

because A'+'.u(s + t I ) >= 0. That is, the right-hand side of (9) is bounded above. whence follows the convergence of ( y ) for any s = 0, I , . . . . This implies, i n particular. the convergence of ( x ) for s = 0. Conversely, assume the convergence of ( r ) . Again wc take advantage of p' > 0 satisfying @' = p ' A . Then we have

c r

p'

A'c(v >=a

\if

+ s) = A-'p' 1A"C(V)

([ = 0, I , . . .).

(10)

v=\

The convergence of ( E ) implies that of (p), which entails in t u r n that of the right-hand side of (10). Whence the convergence of its left-hand side follows. The convergence of ( 7 ) can therefore be established as before. Let, on the basis of the convergence of ( y ) , J'(?) =

1 A"C(r + f).

,=o

( 1 1)

I t i s easy to verify that y ( r ) is a special solution of (2).

Finally we establish the formula (6) for the general solution of (2). It is obvious that (6) satisfies (2) for any solution u ( / )of ( 5 ) because of the linearity of (2). On the other hand, by using (9), we see that any solution x(t) of (2) satisfies ~ ( t2)y ( t ) . Hence the nonnegativity of u(t) = x ( t ) - y ( t ) over time is ensured. Then it is obvious from the linearity of (2) that u(t) is a solution of (5), Q.E.D. We observe that the above special solution y ( t ) of (2) is minorant to any solution \ ( t ) of (2). To see the meaning of y ( t ) more concretely, a special case i n which c ( t )grows in a balanced way at a growth rate p - 1 deserves attention. I n this case, c ( t ) = p i c ( t = 0, 1, . . .) for some c 2 0, p > 0, and ( r ) ,(p) converge if and only if p < I 1.The corresponding minorant solution y ( t ) is

(12) I t is also easy to see t h a t (12) is the sole possible balanced-growth solution of (2), since any nontrivial balanced-growth solution x ( t ) of (2) must

98.


131

+

grow at the same rate p - 1 because x ( t ) - A x ( / I ) = p'c. Therefore no balanced growth at a rate higher than or equal to (I/),) - I is possible in system (2) as long as a nontrivial stream of exogenous demand is present. More generally, for any stream of exogenous demand fulfilling (cx), but not necessarily growing in a balanced fashion, the pace at which the minorant solution y ( t ) grows must be slower than that of ( 1 I.)' at which a nontrivial solution u ( t ) of ( 5 ) grows. To be more explicit, we have

THEOREM 8.9. lim

Arj(/) =

0

I-+x

for the tninorant solution y ( t ) of(2).

Proof. Take a positive row eigenvector 11' of A associated with A = i.(A), as before. Then it suffices to show the convergence of I.'p'y(t) to zero as t + +a. In view of(lO), ( I I ) , we have A'p'y(r) =

1 A"+p'c(v + t ) = C

v=o

i'p'c(v).

(13)

"=I

The convergence of the series (B) entails that of ~ extreme right-hand side of (13) tends to zero as t proof.

---f

~ 2'p'c(v). = o Whence the

+ co. This completes the

Theorem 8.9 implies that in a solution u ( t ) of (2) given by (d), the corresponding u(t), if it is nontrivial, is dominant. The minorant solution y(t), a constituent of x(t), eventually becomes negligible relative to ( I , i)', while the remaining constituent u ( t ) = x ( t ) - y ( t ) keeping pace with (1 governs the behavior of x(t). This dominancy of u ( t ) can be expressed as

which is an immediate consequence of Theorem 8.9. We observe that i ' u ( t ) is periodic and takes on at most 7~ values cyclically, a fact that is obvious from the explicit construction of the general solution of ( 5 ) . Equation (14) can be put in a more direct form, particularly if u(0) > 0. In fact, u(0) > 0 implies O i > 0 in the expression u ( t ) = Oi i ~ ( ~ ) ( tso ) , that u ( t ) > 0 over time. Thus,

1

lirn xi(f)/ui(t)= 1

(i

=

'++a

for the components of xu(/),u ( t ) because

where lim i r y i ( t )= 0 and i ' u i ( r ) are periodic.

1,

...,n)

132


The reader now may easily derive the implication of the results on Eqs. (2) and ( 5 ) for Eq. (l), so that its derivation is left to the reader. $8.4 is based on Nikaido (1962).

$9. OTHER LINEAR SYSTEMS $9.1. Linear Programming

Linear programming originated simply as the maximization (minimization) of a linear function subject to linear inequalities in connection with the solution of practical problems in operations research and management science. This has led to the development of useful algorithms to locate numerical solutions, of which Dantzig's (1963) simplex method is the most typical and best known. From the economic point of view, however, some of its qualitative aspects are more interesting than its computational aspects, since a linear program, if coupled with its dual problem, pertains to resource allocation and the valuation of production factors. It gives a most concise depiction of a simplified situation of competitive equilibrium. We are therefore concerned in this book only with these qualitative features of linear programming, leaving the exposition of its numerical solution by several computational algorithms to the relevant literature, which is now comprehensive. The canonical form of a linear maximization program is:

+ urn2x2+ ... + u,,x,

U,,,~X~

XI

20,

5 b,,

X,20, . . . ) X,20,

(3)

where a i j , b i , c j are given constants. Closely related to the above problem is the following linear minimization problem : to minimize b, zll

+ b2 v2 + . . . + b, P,,,

$9.

I33

OTHER LINEAR SYSTEMS

subject to

+

U , ~ V u ~ ,~u,

v120,

+

. a -

+ amjvm2 c j

v 2 2 0)..., v m 2 0 .

(3‘)

For simplicity, we denote the problem of maximizing ( I ) subject to (2), (3) by (M) and that of minimizing (1‘) subject to (2’), (3’) by (m). In view of the array of a,, , b i , cj in both problems, (M) and (m) are said to be mutually dual. We also say that either of them is dual to the other. To pose the problems in a matrix form, let A = ( u , ~ ) : an m x n matrix, b = (b,) : an m-dimensional column vector, c = ( c j ) : an n-dimensional column vector, x = ( x j ) : an n-dimensional column vector, v = (0,) : an m-dimensional column vector.

Then, (M) and (m) can be posed as follows:

(M) (m)

to maximize c‘x subject to A ’ x S b, x 2 0 , to minimize v‘b subject to LI’A2 c’, 1’ 2 0.

The most important relationship between the mutually dual problems (M) and (m) is condensed in

THEOREM 9.1 (Duality in Linear Programming). Let X = { x I A x 5 6, x 2 0}, V = I c’A 2 c‘, v 2 0}, and assume that X # @, V # @. Then, (i) and (ii) hold: (i) u’b 2 c’x for any x E X , u E V, (ii) B’b = c‘2for some 2 E X 8 E V. Proof. (i) Since v >= 0 for u E V , premultiplying A x 5 b for x E X by v’ E V yields u’Ax 2 u‘b. Similarly we have v’Ax 2 c’x. Whence {ti

c’b 2 v‘Ax 2 c‘x

for x E X ,

L’ E

V,

(4)

as was to be shown. (ii) The standard method to prove (ii) is to appeal to the MinkowskiFarkas lemma, namely, Corollary 1 to Theorem 3.7 in Chapter I.

134


Now, in view of (i), it suffices to see that D’b 5 c ‘ i for some i E X and D E V . Thus we have only to show the existence of a solution, consisting of an m-dimensional vector r and an n-dimensional vector x, of the system

Z’

2 0,

x 2 0.

(6)

This system can be converted, by introducing an ( m slack vector 11’ to the system of equations

1’

2 0,

x 2 0,

H‘

2 0,

+ n + I)-dimensional

(8)

where the coefficients matrix of (7) is obtained from that of (5) by adding m + n + 1 new columns that together constitute the identity matrix of order m+n+l. Then, in the light of Corollary 1 to Theorem 3.7 in Chapter I, the system of Eqs. (7), (8) has a solution if

for any solution ( p ’ , q’, 0) of

\

b’

-c’

where the dimensions of the vectors p‘, q’, H are m,n, I , respectively. Let us see that this sufficient condition is fulfilled i n effect whenever X # V # 0. To this end, it is convenient to decompose (10) to the equivalent relations

a,

p’A

>= Oc’,

pzo,

Aq

420,

5 Ob; 820.

(1 1)

(12)

We divide the subsequent discussion into cases (a) and (b), depending on the positivity of 0.

$9.

135


(a) Case for 0 > 0. Dividing ( 1 l), (12) by 0 > 0, we obtain ( p ’ O)A 2 c‘, p,’B2 0, A(ql0) 5 6, q/0 2 0, so thatp,0 E V , q10 E X . Whence b’(p 0) 2 c’(4 0) which, multiplied by 0 > 0, implies (9). (b) Case for 0 = 0. Equations (1 1) and (12) become p ’ A 2 0, p 2 0, A q 5 0 , q >= 0. Choose some arbitrary x from X and 1 from V . Then, p’b 2 p ’ ( A x ) = (p’A)x 2 0, c’q 5 (r’A)q = d ( A q ) 5 0. Hence b’p - c‘4 0 . O 2 0, which implies (9). Therefore (10) implies (9) in both cases (a) and (b), so that the system of Eqs. (7), (8), or equivalently, the system ofinequalities (5), (6), has asolution. This completes the proof.

+

The duality situation established above can be expressed i n a slightly different way. THEOREM 9.2. the linear program ( M ) IIUS N jinrte (i) On the basic ussumption X # 0, maximum i f and only if V # 0. the linear program ( m ) Iias N finite (ii) On the basic assumption V # 0, minimum if and only i f X # 0. Pvoof: (i) If V # 0, then the assumptions of Theorem 9.1 are met, so that the common value of 0’6 and c ’ i in Theorem 9.l(ii) is a finite maximum of (M) as well as a finite minimum of (m). Conversely, let us prove that if V = 0, the function c’s is unbounded from above on the set X , or, more symbolically, max c’x = + co overall x E X . If we use a slack vector u of dimension n, we find that the emptiness of V is equivalent t o the nonexistence of a nonnegative solution (u’, t ‘ ) of (u’,

d)(-;)

= c’,

where I is the identity matrix of order n. By Corollary 1 to Theorem 3.7 in Chapter I , there is some n-dimensional vector 4 such that

( - j 4 2 0,

c’4

< 0.

If the substitution j= -4 is done, these results become A?; 5 0, c ‘ j > 0, j 2 0. Choose now an arbitrary x from X ; then x(p) = x + p j E X for any p 2 0 and c‘x(p) = c’x p c ‘ j + + co (as p -+ +a), as was to be shown. The proof of (ii) is exactly similar, Q.E.D.

+

I t is obvious, in the light of Theorems 9.1 and 9.2, that .? in X and G in V are solutions of ( M ) and (m), respectively, if and only if D’b = c ’ i , this common value being the maximum of c’x in (M) and the minimum of r’b i n (m). In the terminology of linear programming, such 2, D are termed optimal solutions.

136


Moreover, it can also be seen immediately in view of the proof of Theorem 9.1, especially (4), that a pair of optimal 2 and 0 is characterized among points in X x V by the following conditions:

...

i j= O

if

C aijGj> c j , i= I

Relation to National Income Accounting. We now examine an economic implication of the above basic qualitative aspects of linear programming. 1 his will be done by interpreting duality as advanced in Theorem 9.1 in relation to national income accounting. To this end, the relevant constants, namely a i j , b i , c j . are assumed to be nonnegative mostly for economic reasons, although this assumption was unnecessary i n the general discussion above. Consider a national economy in which a finite number of production factors, labeled by i = I , . . . , m, such as several types of capital, labor, land, etc., work together to produce net value added under a given level of technology. Naturally this creation of value added is effected through producing goods as well as by rendering services. For the time being, however, we shall ignore the process by which the creation of value added emerges from the production of goods (services), although this point will be touched upon later. U n d e r a given level of technology, there are several production processes, finite in number, labeled by,j = I , . . . , n. Like the production process in each sector i n a Leontief system, each of them conforms to the law of constant returns to scale and uses the m factors of production in a constant proportion. It should be noted that this factor proportion, though constant for each process, may vary from process to process, so that theremay be substitutability among factors i n the entire economy. It should also be noted that in general a process uses factors of production, not by consuming them as current input, but by binding them for its operation. In other words, a process is operated by utilizing services rendered by factors of production, so that they are not consumed, although they more or less depreciate. Now we interpret a i j , h i , c j , r j , x j in the following way: u i j : The amount of the ith factor of production employed per unit of operation in thejth process, h,

:

the amount of the endowment of the ith factor of production,

('j

:

the amount of value added (e.g., i n terms of dollars) per unit of operation i n the,jth process,

I ' ~

:

the imputed price (or rental rate) of the ith factor of production, i.e., the price of the service rendered by the ith factor of product ion),

99.


I37

xj : the intensity (or activity level) at which thejth process is operated. Then, the ith relation of (2) simply means that the demand for the ith factor of production (the left-hand side) cannot exceed its supply (the righthand side) in equilibrium. To any feasible intensities xj fulfilling (2), (3) corresponds (lj, which stands for the total value added, i.e., the national product. On the other hand, the n relations in (2’) imply that the m factors of production should be so priced that their total remuneration (the left-hand side) exhausts the corresponding value added (the right-hand side) in each process, leaving no surplus. To any feasible imputed prices fulfilling (2’), (3’) corresponds (l’), which represents the national income earned. By Theorem 9.1(i), for any feasible intensities xj and imputed prices L ’ ~ , the national income earned is never less than the national product, although they need not be equal. If the former exceeds the latter, the distribution of national product among the owners of factors ofproduction cannot be effected. This odd situation is due partly to a lower level of national product caused by the inefficient operation of processes in the intensities x j and partly to a higher level of national income claimed for distribution resulting from an inadequate pricing r i . What Theorem 9. I(ii) implies is that the equality of the level of national product and that of national income earned, an equilibrium situation, is brought about by appropriate g j , D i . 7 hus a pair of optimal solutions A of ( M ) and D of (m) can be thought of as an equilibrium in the national economy. Now let us go back to the determination of the value added c j , which has not yet been discussed. The operation of a process means the transformation of current inputs to current outputs in the physical sense through the collaboration of the employed factors of production. If the prices of goods are given, the operation of the process at the unit intensity determines the corresponding current inputs and outputs in value, i.e., the unit cost and revenue. The value added c j is defined as the amount by which revenue exceeds cost per unit operation. Therefore the treatment of cj as given constants here premises a pricing of goods currently produced and consumed, whereas an overall pricing of all goods involved is possible only in a system having a more general setting (see Chapter V). This limits the scope of linear programming as a model for depicting economic equilibrium. We turn our attention to conditions (13), (14), which are also of interest from the economic point of view. Condition (13) implies that thejth process is not operated if it incurs losses, while (14) implies that if the ith factor of production is not fully employed, its price is zero, so that it becomes a free good. I t should be noted, however, that an unoperated process need not suffer from a deficit in equilibrium, although (1 3), (14) completely characterize equilibrium solutions 2 , 6. Likewise, the equilibrium price of a fully employed factor of production may possibly be zero. Therefore, it is worthwhile examining whether there is a special pair of equilibrium .t and 6 fulfilling zli

138

11. SIMPLE h1ULTISECTOR LINEAR SYSTEMS

“,

C a i jOi > c j ,

T j = O if and only if

i= 1

(15)

n

1 a i j i j < bi.

Oi = 0 if and only if

j= 1

THEOREM 9.3. r f X # @, V # 0, there is a solution A of (M) and a solution O of (m) satisfying (15), (16). Pvoof. We recall that the Minkowski-Farkas lemma, a corollary to Theorem 3.7 in Chapter I, suffices to prove Theorem 9.1. But this time we need to rely more heavily on Theorem 3.7 in Chapter I ; we therefore appeal to Corollary 2 of this theorem. Define T, a skew-symmetric matrix of order m + n + 1, by 0 T = (-f,’

--A

0, -%),

where A , b, c are as in (M), (m). By Corollary 2 to Theorem 3.7 in Chapter 1, there is an (m + n + 1)-dimensional vector MI satisfying Tw 2 0, w 2 0, Tw + M? > 0. If we let O = (w,,. . . , A = ( w ~ + . ~. . ,, w m + J ’ ,0 = wm+,e l , the above results decompose to film)’,

8b 2 A 9 ,

D‘A 2 Oc’,

2 2 0, 0 2 0, 0 2 0;

(17)

c‘9 2 O‘b;

(18)

+ a > D’b;

(19)

D+Ub> A 9 ;

(20)

9’+ ;’A > Oc’.

(21)

c‘9

First it will be seen that 0 > 0. Suppose that 0 = 0; then (17), (19) become 0 => ~ 9D‘A , 2 0, 2 2 0, D 2 0, c‘A > 8‘b. On the other hand, since X # @, V# we can take some x E X , 2’ E I/ satisfying c’x = r‘b by Theorem 9. I (ii). Then, 2 x E X , O + 1’ E V , and c’(2 x) > ( 0 r)’b, which contradicts Theorem 9.1(i). Now that 0 > 0 is established, 0 can be assumed to be 1 because of the homogeneity of (17)-(21) in 2 , 0, 0. Then (17), (18), together with Theorem 9. I(;), imply that 2 is a solution of ( M ) and D is a solution of (m). Moreover, (20), (21) ensure the “only if” part of (15), (16), whereas their “ i f ” part was already shown to be true for a n y solutions of (M), (m). This completes the proof, Q.E.D.

a,

+

+

+

All the above discussion along with national income accounting, is, however, something like a castle in the air, unless X , V are nonempty, because

$9.

139


Theorems 9.1 and 9.3, which ensure the existence of equilibrium solutions, premise the assumption X # $3, V # @. We shall therefore complete the discussion by observing that this assumption is fulfilled in a natural way under the nonnegativity of a i j , bi, cj . In fact, if a i j , bi are nonnegative, X contains x = 0, so that X # $3. Moreover, if each process uses at least one factor of production, i.e., if for each j there is some i for which a i j > 0, then positive vectors with sufficiently large components belong to V , ensuring V # @. Note that we can dispense with the nonnegativity of c j . We have almost finished exploring the principal qualitative aspects of linear programming. In what follows, we supplement these by a few remarks. Remark 1. Corresponding to the mutually dual problems (M) and (m), we define +(x,

P ) = C'X

+ L"b - L.'Ax,

which can readily be rearranged to

4(X,

+ c'(b - A x ) = zl'b + (c' - ~ ' A ) x .

P ) = c'x

Consider @(x,L.) as a function of x E R," and u called a saddle point of 4(x, u) on R+" x R," if

2 maximizes @(x,8) over 8

minimizes

4(2, u)

over

E

(22)

R,". A point ( 2 , 0) is

R+",

(23)

R+",

(24)

so that

4(x, 8) 54(2,0) 5 4(2, L')

(x E R+", o E R,").

(25)

Then, following Goldman and Tucker (l956), we may observe that (2,6) is a saddle point of 4(x, L') on R," x R," if and only if 2 , 0 solve (M), (m), respectively. The sketch of the proof is as follows: Necessity. If ( 2 , 0) is a pair of optimal solutions of (M) and (m), we have ~ ' =2 D'b, 2 2 0, 0 2 0, b - A2 2 0, C' - 6'A 5 0, D'(b - A 2 ) = (c' - i?'A)2 = 0. Whence

4(2, 0)

=

0'6

=

0'b + (c' - 0 ' A ) i 2 0'b + (c' - Y A ) x =

4(2, 0)

= c'2 = c'2

(X E

+ D'(b - A,?) 5 c'.? + r'(b =

proving necessity.

4 ( ~6,)

4(.?,L.)

-

(1.

E

R+"),

A,?)

R+"),

140


Sufficiency. If (2,0 ) is a saddle point, (25) entails c'(b - AA) 2 0 (c'

-

for all

L' E

G'A)x 5 0 for all x

E

R+m, R+",

giving rise to A.2 5 b, 2 2 0, D'A 2 c', 0 2 0. Hence A E A', 0 E V . On the other hand, (25) for x = 0, z' = 0 implies D'b = 4(0, 0 ) 5 +(A, 0 ) 5 4(A,O) 5 c'2, which proves the optimality of 9, 8. Therefore 2 , D solve (M), (m), respectively. One may also realize that 4(x, r ) is the Lagrangian of (M) with the multiplier vector c' as well as that of (m) with the multiplier vector x. Expressions (23), (24) signify the reduction of (M), (m) to unconstrained extremum problems through the introduction of Lagrangian multipliers. Remark 2. Let us consider the change of the (common) optimal value of (M), (m) responding to the variation of b or c, while A is kept constant. We write X(b), V ( c ) for X , V , respectively, to make their dependence on b, c explicit. As long as X ( b ) # gi,V ( c ) # we can define the function

a,

f ( 6 , c) = max c'x

over x

E

X(b) = min b'v

over v E V(c).

We observe that if we let B = { b I X(b) # @}, C = {c 1 V(c)# @}, they are closed convex cones having nonvoid interiors in R", R", respectively, since B = { A x + u I x E R+", u E R + m } ,C = { A'v - y I v E R+m,y E R+"},which are polyhedral convex cones including R+m,R+", respectively (see Chapter I, 43.4). Let us note thatf(b, c ) as a function on B x C has the following properties: (i) For each fixed c E C, f(b, c) is concave on B, and for each fixed b E B, f(b, c) is convex on C. (ii) Whencef(b, c) has right-hand and left-hand partial derivatives

in B" x C", where B", C" are the interiors of B, C. (iii) If 2 = (aj),0 = (Oi) are any optimal solutions of (M), (m) corresponding to b E B", c E C", respectively, then

af

~

-

2

dbi-

+ 6.> ;if - (i

' = dbi

af + 2 1.>af -

acj -

J

=

acj

=

1, . , . , m ) ,

( j = 1, . . . ) n ) ,

where the derivatives are evaluated at (b, c).

(27)

99.

141


The proof of (i)-(iii) will be sketched below. To see the b-concavity offin (i), let ,tl, 9 ' be optimal solutions of (M) corresponding to b', b2 E B, respectively, while cis kept fixed in C. Then it can be easily shown that AI2' A2 '1 E X(A,b' + I , b2)for any I , 2 0, A, 2 0, I l + 1, = I . Hencef(A,b' + /I b2, c ) >= Ilc'9' A, c'9' = I,f(b', c) A, f ( b 2 ,c), proving the desired concavity. The c-convexity is similarly seen. Property (ii) is justified by Theorem 3.15 in Chapter I. As to (iii), we derive only (26), since the derivation of (27) is similar. To this end, write Ai b for the vector (0, . . . , 0, A b i , 0, . . . , 0)'. Let D = (ai) be any optimal solution of (m) corresponding to b E B", c E C. Then D need not be an optimal solution of (m) corresponding to b + A i b E B", c E C , although S E V(c). Hence

+

+

,

+

f(b

+ Ai b, c) = min(b + Ai b)'u over t' E V(c) 5 (b + Aib)'D = b'0 + (Aib)'a =f(b, + SiAbi. C)

Therefore for each i,

f(b

+ Ai b,

C)

-f(b,

C)

[ < Di =

>=

Abi

(if Abi > 0), Di (if Abi < 0).

This becomes (26) in the limit as Abi 0. If we refer again to the foregoinginterpretation of linear programming from the viewpoint of national income accounting, (26), (27) can admit economic interpretations. The quantityf(b, c) represents the equilibrium level of national income when the factor endowment vector is b and the value added vector is c. The term farthest to the right in (26) measures the marginal contribution resulting from the infinitesimal increase of the ith factor endowment to raise the level of national income. The term farthest to the left measures the corresponding contribution to the lowering of the level of national income when the ith factor decreases infinitesimally. Equation (26) gives the competitive pricing of the ith factor somewhere in between, and therefore it revives the classical marginal productivity theory in a weakened form. The interpretation of (27) is left to the reader. --f

09.2. Von Neumann's Model of an Expanding Economy A dynamic phenomenon is generally defined as an evolutionary change with respect to time of an economic state, governed by a law relating the state at a period t o that at another or several periods. These intervening periods are discretely distributed or infinitesimally close to each other, depending on

whether the measurement of time is discrete or continuous. The mode of measuring time is very important mathematically, since it has a grave influence on the mathematical nature of the corresponding model representing a dynamic phenomenon, though probably not on its economic characteristic. But in this section, as in the foregoing ones, we shall be concerned only with the discrete time approach. We recall that we have discussed some dynamic phenomena, e.g., income propagation i n a multiplier process ($6.3)and the fluctuation, as well as the growth. of output level under lagged production (48.4). Dimensionally speaking, both income and output are “flow” concepts. From the economic point of view, however. a more important phenomenon is the evolution of a magnitude corresponding to a “stock” concept, a typical example being the accumulation of capital stock. I n the mid 1930s, voii Neuniann (1937) worked out an ingenious analysis of capital accumulation in a unique multisector linear model. It is surprising that his work was done with little influence from the main stream of thought among orthodox economists except for that which he seemed to receive from his close contact with a few economists of the Austrian school. It was, however, ignored because of its mathematical difficulties and because of its earlier publication in a mathematical journal. I t has since become well known to economists, particularly since the publication of its English translation in an economic journal in 1945. Subsequently, it has not only exerted a significant influence on the theory of economic growth and capital accumulation, but also greatly stimulated the modern development of mathematical economics. The major results of the von Ne~ttiiatitimodel to be presented in this section, undoubtedly of independent interest, are also relevant to some of the subsequent chapters. I n the von Neumann model a production proczss means a transformation that carries ccrtain levels of stocks of several goods at the beginning of a period to some alternative. possibly equal, levels at the end of the period, their differences being the corresponding levels of investment of goods. A more explicit statement of the basic assumptions will be given below: (x) There are i n kinds of goods labeled i = I , (/I) There are 17 kinds of basic linear production processes labeled by , 17 having constant coefficients of production. The j t h basic process transforms, per u n i t of operation, the m-tuple ( m , j , a z j , . . . , of aij units of stock of the ith good (i = I , . . . , r n ) to another, though possibly equal, m-tuple ( h ,j . h Z i , . . . , h,,,j) of the same kind. Here more than one b j j may be nonvanishing for each process .j, so that joint production is taken into account, making a sharp contrast to Leontief’s input-output analysis.

These 2 in17 magnitudes o i j , b j j are constants that are determined by the !ken leiel x n d quality of production technology and are assumed to satisfy

$9.

bij>=O

aij>=O,

133


(i=1,

..., m ;

..., n),

j=l,

(1)

m

C aij > O

i =1

( j = 1, . . . , n ) ,

n

j = 1

bij > 0 ( i

=

1, . . . , m ) .

(3)

The implication of condition (1) is self-explanatory. Condition (2) implies that each processj needs at least one kind of input. More concretely, each process either consumes at least one kind of current input or binds at least one kind of capital good to its operation. On the other hand, (3) means that every good can be produced by some process. I n view of its afinity to linear programming, it is useful to start from the formulation ofthe model asa dynamic version oflinear programming,although the original von Neumann work preceded the development of linear programming. We are concerned with the pricing of goods and the determination of intensities at which processes are operated. Let pi = the price of the ith good, ri(t) = p i + the rental rate of the ith good at t , xj(t) = the intensity at which thejth process is operated at t . The working of the system is effected in the following way. In the first place, for the time being, the prices p i will be thought of as given nonnegative constants. Next, suppose that the nonnegative levels of ri(t), xj(t) at tare already determined. Then, the amount of stock of the ith good at the end of period t , which is ready to be available for productive services at period t + 1, is given by n

bi(t) = j= I

bijxj(t) ( i

=

I , .. . , i n ) .

On the other hand, the total value of output in thejth process per unit of operation evaluated at the price system p i is denoted by m

Note that bi(t) depends on t , while cj does not. Now, on the output side, the intensities x j ( t + 1 ) at t be determined subject to

+ 1 are therefore to

n

1 a i j x j ( t + I ) 5 bi(t)

( i = 1, . . . , m ) ,

(4)

..., u ) .

(5)

j = 1

xi(?+

1)ZO

(j= I,

144

11. SIMPLE MULTlSECTOR LINEAR SYSTEMS

On the valuation side, services rendered by goods are to be priced subject to

Ui(t+

1)ZO ( i = l , ..., m),

(7)

so that the value added in each process is exhausted by remunerating their owners at the rental rates vi(t + 1) - p i . In the framework of linear programming, an equilibrium situation results if and only if there are ~ ~x (I), txi(' x 1) that fulfill not only (4)-(7) but also rn

n

i= 1

j= 1

C bi(t)vi(t+ 1) = C c j x j ( t + 1).

Since

C'!!, bi(t)pi =

(8)

c j x j ( t ) , (8) can be rearranged to

stating the equality of the national income earned on the left-hand side to the national product (or the total value added) on the right-hand side. Starting at given initial values si(0)2 0, xj(0) 2 0, sequences {ui(t)}, { x j ( r ) }satisfying (4)-(8) can be obtained because of (I), (2), in the light of the results in 59.1. It should be noted that we have so far regarded the prices p i as given constants, so that alternative levels of p i will result in different behaviors of x i ( [ ) over time. Later we will concentrate on the choice of a particular set of p i and the behaviors of the corresponding vi(f), xj(t), following the idea of von Neumann. The main concern of von Neumann with this model was the possibility of a moving equilibrium situation in (4)-(S), namely, the existence of special sequences { z l i ( t ) } , { x j ( f ) }satisfying (4)-(8) and Lli(r)

=

bpi (i = 1, . . . , m),

(9)

x j ( t ) = d x j ( j = 1, . . ., n) (10) p i , x j . specially chosen nonnegative constants with p i > 0, x j > 0. It should be remarked that (9) implies more than the constancy of zii(t) over time. vi(t) has been defined as p iplus the rental rate of the ith good. This means that the investment of p i dollars for the purchase of one unit of the ith good yields the return of u i ( t ) - p i dollars as the rental for the productive service rendered by this good at period t. Therefore zli(f)/pi- 1 represents the rate of interest for the investment in the ith good, which may generally differ from good to good as well as from period to period. In this respect (9) means a very special situation in which the rate of interest is a constant /j - I , common

for a,

b,

c;=,

z?n=l

99.

145


to all the goods over time. On the other hand, (10)simply means the balanced growth of intensities at the rate ci - 1. Later it will be shown that ci and p are greater than 1 if the underlying technology is sufficiently productive. As can readily be seen by rearranging (4)-(8) for (9), (101,the possibility of such a moving equilibrium reduces to the existence of two vectors p = ( p i ) , x = (xi) and two scalars ci, fi fulfilling

(B - x A ) x 2 0,

(1 1)

p’(B - P A ) 5 0,

(12)

p’(B - ciA)~= 0,

(13)

p’(B - PA)x = 0,

(14)

p20,

(15)

x20,

where A ( = a i j ) ,B = (bi j).It is equally obvious that a quadruplet ( p , x, ci, p) of vectors and scalars satisfying (1 1)-(15) generates a moving equilibrium in (4)-(8). Von Neumann proved the existence of a solution ( p , x, ci, p) to (1 1)-( 15). His proof was done under (l), ( 2 ) ,and a very stringent additional assumption

aij

+ bij > 0

( i = 1, . . . , m ; j

= 1,

. . ., n ) ,

(3‘)

that is hardly acceptable by economists. Furthermore, he appealed to the Brouwer fixed-point theorem (Chapter 1, Theorem 4.3), which has turned out, through the work of later writers, to be a steam hammer used to crack a nut as far as a solution of (1 1)-(15) is concerned. Today it is well known that the separation of a convex set or certain of its equivalent propositions suffices to this end even under (3), a less stringent condition than (3’). The role of (3’) is found in its efficacy in ensuring the uniqueness of c( and rather than the existence of a solution, as will be noted later. Among alternative analogous methods of proof based on the separation of a convex set, the most straightforward seems to be the one appealing to Tucker’s theorem (Chapter 1, Theorem 3.7(ii), or, more directly, its Corollary 3), as stated by Howe (1960).

THEOREM 9.4 (Kemeny, et a/., 1956; Thompson, 1956; Howe, 1960). Assume (l), (2), (3). Then (i), (ii) hold: (i) There is a solution ( p , x, a,fi) o f ( 1 I)-( 15) satisfying

p’Bx > 0, pi = 0 if and only if

n

j= I

j= 1

ci

1a i j x j< 1 b i j x j , m

xj=O LY =

fl for any

m

ifandonlyif P x a i j p i > l b i j p i . i= 1

(ii)

(16)

n

i= 1

( p , x, CI, P) satisfying (13), (14),(16)

146

11. SIMPLE MULTISECTOR L l h E A R StSTEMS

ProoJ: (i) We first consider the set of real numbers r = (01 ( B - w A ) x 2 0 for some x 2 0 i. # 12/ since it contains w = 0. Next it will be shown that r is bounded from above. To this end, let

Then, L > 0 by (2) and M : L >= w for any w E r, showing boundedness from above. Hence 0 = sup w over all w E r is finite. Let {a,}be a sequence converging to 0 in r. For each w,, there is, by definition, an x” 2 0 fulfilling ( B - cc), A)xV 2 0, whose homogeneity enables us to further assume that the components of s)’add up to unity. Thus we may, without loss of generality, assume {x”} itself to converge to some x 2 0. Whence we have ( B - 0 A ) i 2 0, 2 2 0 i n the limit, implying 0 E r, and 0 = max w over all w E I-. Next we shall showthat(3)ensiircsO>O.lnfact,B.u>Oforx = ( I , I , . . . , l)’by(3),whence ( B - wA)Y > 0 for a small positive w . Hence r contains positive numbers and 0 must be positive. By Corollary 3 to Theorem 3.7 in Chapter I as applied to the m x n matrix UA - B, we obtain an m-dimensional vector p and n-dimensional vector x such that (OA’ - B’)p 2 0, p 2 0, (19)

( B - OA)x 2 0, x 2 0,

+ (UA’ - B’)p > 0, p + ( B - OA)x > 0. x

(20) (21) (22)

for SI = p = 0 is a desired solution, as will be seen below. In Then, ( p , s,SI, /i) fact, since 0 E P, there is, by definition, an ,i- L 0 with ( B - UA).? 2 0. I n view of (Ig), this 9 satisfies i ‘ ( O A ’ - B’ )p =p’(UA - B ) i = 0. Premultiplyitig (21) by 2’ therefore yields i ‘ x > 0, which rules out x = 0 and implies x 2 0. We next prove (16), which also rules out p = 0. To this end, suppose that (16) were not valid and hence p‘Bs = 0. Let J = {i I p i > 0 ) . Also let [ A x I i , [ B x ] ,stand for the ith components of A s , B s , respectively. Then (22) implies [B.\-Ji- O[A.u], > 0 for i $ J . On the other hand, since p 2 0, Bx 2 0, the supposition p’5.1- = 0 implies [ E x ] ; = 0 for i E J . I n view of 0 > 0 and (20), this implies i n t u r n [ A s ] , = 0 for i E J . Hence ( B - toA).\- 2 0 is still valid for the same .Y> 0 if 0 is replaced by an 0) that is larger than, but sufficiently close to. 0. This contradicts the property of 0 to be the maximum in r. Thus we have shown that p > 0, x 2 0, and (16). Furthermore, (19), (20) mean ( 1 1 ), (12) for Y = /i= 0. Then ( I 3), (14) are immediate consequences of ( I I), (12) in a situation where Y = /I. It remains to check on (l7), (18). Their “ i f ” part follows from ( II)-( 15). as was the case in the preceding section, while the “only i f ” part is ensui-ed by (21). (22).

$9.


147

(ii) ap'Ax = p'Bx = /?$Ax by (1 3), (14). Since p'Bx > 0, p'Ax does not vanish, so that a = /? = p'Bx:p'Ax, Q.E.D. The remaining part of this section will be devoted to a few supplementary remarks. Remark 1. Although Theorem 9.4(ii) ensures SI = /?,it by no means implies the uniqueness of the common value of a, /?. This is verified by a simple example. For

are solutions of ( 1 1)-( 18). However, this kind of uniqueness obtains under (3'). Explicitly, if ( l ) , (3') are assumed, LY = p, and their common value is uniqueforsolutions(p,x,a,/?)of(11)-(15).Toseethis,let r ={ciJ(B-rA)x~O for some x 2 0}, A = { p 1 p'(B - / ? A ) 5 0 for some p 2 0). For any x E I-, p E A, and p, x associated with them so as to satisfy ( 1 1 ), (1 2), we clearly have pp'Ax 2 p'Bx 2 ap'Ax. By addingp'Ax to these inequalities, we end up with

+ /?)$Ax 2 p'(A + B)x 2 (1 + %)$Ax. (23) A + B > 0 by (3'), which implies p'(A + B ) x > 0 in view of (15). Hence (23) entails p'Ax > 0 and 1 + p 2 1 + a . We have thereby shown that a 5 /? for (1

any a E I-, fl E A. Moreover, if some a E A, p E r fulfill ( I 3), (l4), equality holds in (23), which ensures a = /?.These results mean that if ( p , x, a, p) is a solution of ( 1 I)-( 13,then x = p, and this common value is characterized as the maximum in r and the minimum in r, whence the asserted uniqueness follows. Remark 2. Von Neumann's original method of proof for the existence of solutions to ( I 1)-(15) under ( I ) , ( 2 ) , (3') consisted of consideration of the behavior of the function @ ( p ,X) = ~ ' B ~x ' A . Y

in the framework of game theory. Remark 3. If x = /j > I , the economy in its moving equilibrium (9), (10) grows i n a balanced fashion at the rate x - I , while a positive rate of interest fl - I prevails. Theorem 9.4 does not ensure, however, x > I , p > I , though it does ensure x = /? > 0. The actual excess over 1 of these quantities is guaranteed only when the underlying technology represented by matrices A , B is sufficiently productive. I t may be said to be suficjetit/y prohrctire if there is an intensity vector xo 2 0 such that ( B - A).Y" > 0. This means that it is

148


possible to get positive output of every good at the same time. In this case ( B - oA)x” > 0 for o larger than, but sufficiently close to, 1 for the same xo 2 0. Whence r contains numbers larger than 1, so that c( = p = 0 > 1. Remark 4. I n one sense, the von Neumann model is a very unstable system. Consider a very special case where A is a square nonnegative matrix and B is the identity matrix of the same order. Then, (4)-(8) become, in matrix form,

+ 1) 5 x(t), x(t + 1) 2 0, x(t) 2 0; r‘(t + I ) A Z p ’ , z(t + I ) 2 0, p 2 0; p’x(t + 1) = r’(t + l)x(r). As(t

Now, choose a semipositive 1. 2 0 and let p‘ = r ’ A . Clearly the triplet of p , the sequence { ( ( t ) = z.1, and a solution u(t) of ( 5 ) of $8.4, is a solution of the above system. As was made clear in 58.4, u(t) is in general oscillatory and does not converge to a moving equilibrium. Moreover. if u(t) starts from the outside of the range of admissible initial positions, it cannot remain nonnegative and will protrude beyond the “nonnegativity” barrier in a finite number of periods.

Chapter 111

BALANCED GROWTH IN NONLINEAR SYSTEMS

A considerable part of the results established under linearity in Chapter I1 actually does not rely on linearity and can be extended to nonlinear cases. This chapter presents certain of these extensions, mainly to situations where the basic functional relations are similar to those in Chapter 11, but where linearity is discarded.

$10. THE SOLOW-SAM UE LSON-TYPE

RELATIVE STABILITY THEOREMS

01 0.1. Nonlinear Homogeneous Difference Equations

The theory of nonnegative matrices as developed in Chapter 11 has entailed a complete exploration of a dynamic situation of economic growth that is depicted by a linear forward difference equation with constant coefficients (Eq. (1) in Chapter 11, @.I), namely, x(t

+ 1) = Ax(r),

(1)

where A is a square nonnegative matrix. It should be noted, however, that most of the qualitative properties of the above self-sustained system do not owe their validity to the linearity of the functional relation connecting x(t + 1) to x ( t ) . Concepts about (l), such as indecomposability, primitivity, stability, and balanced growth, as well as results relating to them, can be carried over to a dynamic system x(r 1) = H ( x ( t ) ) , (2)

+

+

where the causal relation x ( t ) + x(t 1) is represented by some vector-valued function H sending a nonnegative vector x to a nonnegative vector H ( x ) of 149

I50

111. BALAKCtD GKOU i H 15 \O\LICLAR

SYSTEMS

the same dimension. Extensions in this line were achieved by Solow and Samuelson (1953), Suits (l954), Muth (1954), and Morishima (1961b, 1964) under considerably weaker assumptions on H. They will be stated in this section, somewhat adapted, simplified, or supplemented. The basic weakened conditions that replace linearity are listed below : ( u ) H ( x ) = ( H i ( . v ) )is defined for all nonnegative x in R", with its values being also nonnegative vectors in R", H ( x ) 2 0. ( f l ) Continuity. H(x) is continuous as a mapping H : R," -+ R," except possibly at x = 0. (y) Homogeneity. H(x) is positively homogeneous of order nz, 1 2 ni 2 0 i n the sense that H(ctx) = ccmH(.u) for

(6,) H , ( j , ) for

(6) (6,)

CY

2 0, x 2 0.

Weak monotonicity. If s = (xi),y = ( y J ,x 2 1' 2 0, then H , ( x ) 2 all i such that x i = y i . Monotonicity. H(x) 2 H ( y ) for Strict monotonicity. H ( x ) > H(y) for

,Y

2 y 2 0.

x 2y

2 0.

In the following lines some or all of conditions (z)-(a5) will be assumed, depending on the proposition to be proved. DEFINITION 10.1. A balunced--growth solution x ( t ) o f (2) is dejtied as sucli a nontririal solutioti that the proportiom x , ( t ) : x , ( t ) : . . . : x,(t) o f the rotiipotietits o f .v( t ) retmiti cotistatit o i w time. It therefore takes the ,form s ( t ) = a(t ) s ,

(3)

ithere s 2 0 is a cotistatit rector and a ( t ) is a positive tiunierical,#iinction of'

t = 0 , I, .... Assuming ( a ) , ( y ) and putting (3) in (2), we get a ( t which, if divided by a(t)m > 0, becomes a(f

+ 1)x = o(r)"H(x),

+ I ) x = H(x).

a( t)"'

(4)

The validity of Eq. (4) over time for a constant x obtains if and only if

this constant 1. being an eigenvalue of H to which x is associated as an eigenvector in the sense that 2x = H(x), x 2 0.

(6)

$10.

THE SOLOW-SAMUELSON

THEOREMS

151

The balanced growth in (2) therefore leads to the nonlinear eigenvalue problem (6). Once i. is given and a(0) is prescribed, the function o ( t ) is uniquely determined by (5). Explicitly, 11' o ~ U = ) 1 , we have o(t , = 1.I + 1112 + . + n1' ( t 2 1). T

III

'

~

1

This formula reduces to a(t)= j.r,a case of customary exponential growth when m = 1. Note that if m is less than I , a(t) converges to A l i ' l a s t - t +a. Therefore balanced growth under homogeneity of order lower than I leads to a stationary state or a stagnation.

THEOREM 10.1. I f (a), (b) are assumed, the eigetiralue problem ( 6 ) is solwble for some 1 2 0. Proof. Let P, = (x I x 2 0, xi = 11, the standard simplex. The mapping 4 given by

c;=

xi

4 L - U ) == I

+

+ Hi(x)

( i = I , . . . , 11)

H,(x) j = I

continuously carries P,, into P, by (a),( p ) and because 4i(x) = I , $,(.Y) 2 0 ( i = I , . . . , t i ) . Hence, by virtue of the Brouwer fixed-point theorem (Chapter I , Theorem 4.3), 4 has a fixed point 2 in P,, so that 2 = d ( 2 ) . Whence

which, if rearranged, become

This implies that 1 = H j ( 2 ) is a nonnegative eigenvalue of H to which is associated 2 2 0 as an eigenvector, Q.E.D. Next, further assuming homogeneity (y) we examine the distribution of eigenvalues.

THEOREM 10.2. Assume (cc), (/I), (y), aiid let A = (1 I H ( x ) = 2.x for some

cS=,

x

E

Pll],

where P , =,(x 1 x >= 0, xi = I is the staiidaarrl sinipkr. Tlien, A contahs a maximum which is dmoted by I.( H ). Furtliermore (i), (ii) liold: (i) I f m = 1, A( H ) is the greatest amoiig all tlie eigeiiralrres of' H . H( ) > 0. crwy positire tiumber is an eiqemulue of' H . (ii) If 1 > rn 2 0 aird i.

152

111. BALANCED GROWTH IN NONLINEAR SYSTEMS

Proof. In view of the proof of Theorem 10.1, A is nonempty. The component sum H,(x) as a continuous function on P, is bounded. Hence, if 1E A and x E P,,are paired so as to satisfy H ( x ) = Ax, we have 1= H i ( x ) , so that A is bounded. Moreover, let { A v } be a sequence in A converging to 1. Choose such a sequence {x'} in P, that xvis paired with 1,to satisfy H(x') = I., x". We may assume, by virtue of the compactness of P, ,that {xv}converges to an x in P, . Then, we have H ( x ) = Ax, x E P, in the limit by (p), implying the closedness of A. Whence A as a closed bounded set of numbers is compact by Theorem 1.5 in Chapter I, so that it contains a maximum 1 ( H ) . It is also clear by ( a ) that A ( H ) 2 0. (i) Suppose nz = 1. Let p be any eigenvalue of H , so that H ( y ) = p y for some y 2 0. Then, H ( x ) = p x for x = y / c y i E P, by homogeneity (7) with M = 1, implying p E A. Whence A ( H ) 2 p by definition. (ii) Suppose 1 > m 2 0. Let x be an eigenvector in P, associated with A = i ( H ) . If 1> 0, we also have H ( y ) = Om-'ly, y = I9x 2 0 for any I9 > 0 by homogeneity ( y ) . Since drn-'A ranges all the positive numbers as 19 ranges likewise, every positive number is a n eigenvalue of H . This completes the proof. I t should be remarked that even in the case of m < 1, no grave difficulty will result from considering only eigenvectors in P,, , since H ( y ) = p y , y 2 0 imply p ( x y,)' - m x = ~ ( x )x ,= y / C y i E P , . The above 1 ( H ) corresponds to 1 ( A ) , the dominant root of a square nonnegative matrix A . To obtain more detailed information about A ( H ) , which justifies our stress on the analogy of A ( H ) to A(A), we need assume some kind of monotonicity. In this respect, the role played by monotonicity (6) is standard. It is interesting that under monotonicity even the existence of an eigenvalue can be established in the course of considering several properties of A ( H ) such as shared by A(A), without appeal to the fixed-point method. To this end define

1

L ( H ) = { p 1 H ( y ) >= py

THEOREM 10.3. i f ( a ) ,

for some

J' E

P,}.

(p), ( y ) , (6) are assuimd, A ( H ) can be dejiriedas

i.(H)= max

ji

over all

11 in

L(H),

(7)

so that it satisjies.

(i) I f H ( y ) 2 pj*,fbr soine J' E P,, tlietl i . ( H ) 2 p. (ii) I f G alsoJulf1ls ( a ) , (p), ( y ) , (a), and H ( x ) 2 G ( x ) uriijormly holds for x E P,,, then l ( H ) 2 A(G). (iii) Deiiote hq' H" the s times iterate of H girlen by H " ( x ) = H ( . * . F f ( H ( x ) ) * * .).

$10. THE SOLOW-SAMUELSON

THEOREMS

153

Then, A ( H ) = 0 if and oniy ifthere is somepositice integerp such that HP(x) = 0 uniformlj)for x 2 0. Furthermore, n can serge as one of such p's, where n is the dimension of x as well as H(x). Proof. Denote by L + ( H ) the subset formed by all nonnegative numbers in L ( H ) . By (a),H ( y ) is nonnegative, so that L + ( H ) 3 0. It is obvious that if L + ( H ) contains a maximum, it is also a maximum in L ( H ) . In effect, the existence of a maximum in L + ( H ) can be proved by slightly adapting the argument used to establish the existence of a maximum in A in Theorem 10.2. Let A ( H ) stand for the maximum in L ( H ) . That 2 ( H ) is actually an eigenvalue of H will be proved after (i), (ii), (iii) are established, for they d o not depend on the property of A ( H ) as an eigenvalue. First of all, (i) is clear from the definition in Eq. (7) of i ( H ) . Next, t o prove (ii), we have only t o show that L ( H ) 2 L(G), which is, however, a n immediate consequence of the uniform validity of H ( x ) 2 G ( x ) on P , . In fact, if L ( H ) 3 L(G), L ( H ) contains in particular 2(G). Hence A ( H ) is, by definition, not less than A(G), proving (ii). We may observe that (iii) is a genera1 version of Theorem 7.2(vi) in Chapter 11. Moreover, the proof of (iii) can also be effected by essentially the same method of proof for a constant matrix. In fact, following the procedure in the proof for Theorem 7.2(vi) in Chapter 11, take a positive vector x in P,,and a positive number 0 such that H ( x ) 5 Ox. It is then clear from (y), (6) that 0 5 H " + ' ( x )5 OmSHS(x) (s = 1, 2, . . .). If we let N , = { i 1 the ith component of H"(x) is positive}, this result implies

N o 3 N,3 N ,

3 .-* 3

N,3 N,,,3 . . . .

For simplicity, we denote by His(x)the ith component of H"(x). When N, 0, equality automatically holds in (8). In fact, (8) decomposes to the n component relations H i ( x ) 2_ Axi ( i = I , . . . , n), which add up t o n

1 H i ( X ) 2 A. i=

(9)

I

Since I. > 0, the sum of H i ( x )is positive by (9). Then, (S), together with ( y ) , (6)

Whence L ( H ) 3 p, so that i. 2 ji by definition in Eq. (7). Hence I, 2 I." which, combined with 2 > 0, entails A ' - m 2 H,(x))'-". Since 1 - m > 0, we finally have

(c

(C H i ( , y ) ) ' - " ' ,

n

1 H i ( x ) 5 A. i= I

Thus, equality holds in (9), so that no component of H ( x ) can be greater than the corresponding one of I.x in (8). This ensures that equality holds in (8). ( 2 ) Case of rn = 1 . I n this case equality need not hold in (8) unless H satisfies a more strict kind of monotonicity, e.g., (6,). Let us therefore suppose for the moment that H satisfies ( h J . Then, if H ( x ) 2 2.y in (8), we would have, by virtue of ( 7 ) with rn = 1 and (6\),

$10.

THE SOLOW-SAMUELSON THEOREMS

155

This implies that L ( H ) 3 A + E for a sufficiently small positive E , contradicting the maximum property of A. Therefore we have seen that equality automatically holds in (8) and hence A is an eigenvalue under (6J. The general case in which (6,) need not be assumed can be reduced to this special case through approximation. Taking a fixed vector c > 0, let H,(x)

= H(x)

+ \E

xi c

X

(E

> 0).

i=l

H , ( x ) clearly satisfies (a), (p), (y) with m = 1, (6,) for each E > 0, since H,(x) = H ( x ) satisfies (u), (p), (y) with m = 1, (6). Moreover, H,(x) 2 H 6 ( x )2 H ( x ) uniformly for x 2 0, if E 2 6 > 0. Whence, by (ii), which is already proved, it follows that A(H,) 2 A(H,) = A ( H ) for E 2 6 > 0. Let ( E , ) be a decreasing sequence of positive numbers converging to 0. Then, { A ( H J } is a nonincreasing sequence of numbers bounded from below by A ( H ) . Hence lim A(HJ exists with a limit p not less than A ( H ) . Now, since He, satisfies (6J, A(HEU)is an eigenvalue of H,, to which is associated an eigenvector x" lying in P , to satisfy H E " ( X V ) = A(HE")XV

(v = 1, 2, . . .).

(10)

In view of xuE P , (v = I , 2, . . .) we may assume, by the compactness of P , , that {x"} in itself converges to some x in P, . Then, the right-hand side of (10) becomes px in the limit, whereas its left-hand side HEu(xv) = H(x') + E , C tends to H ( x ) by (p). Whence we have H ( x ) = p x,x E P,. Here p 2 A ( H ) as was already noted. But, we also have A ( H ) 2 p by definition (7), since L ( H ) 3 p. Hence p = I.(H). This completes the proof. It should be noted that under monotonicity we can dispense with the fixedpoint theorem to solve the eigenvalue problem (6). Moreover, Proposition (iii) of Theorem 10.3, the characterization of the case A ( H ) = 0, seems to be new. None of the previously mentioned authors has remarked on these two points, though they were chiefly concerned with the case where monotonicity prevails. $10.2. Relative Stability

This section is devoted to discussion of relative stability in system (2) in 910.1. In this respect, we anticipate, i n view of the results on matrices in Chapter 11, 97, 8, that some nonlinear versions of concepts such as indecomposability (Chapter 11, 57.2) and primitivity (Chapter 11, 58.2) will be useful. A nonlinear version of indecomposability, which naturally suggests itself, will be the following: For two vectors x,y such that x 2 y 2 0, let N (x,Y ) = { j I xj > Y j } .

N ( x , y ) is a subset of N

=

{ I , 2, . . . , n } .

156


DEFINITION 10.2. H ( x ) = ( H i ( x ) )in ( u ) in $10.1 is said to be indecomposable if for any pair of vectors x , y such that x 2 y 2 0, and N ( x , y ) is a nonempty proper subset of N , we have H i ( x ) # H i ( y ) for some i # N(x, y).

(1)

If H also satisfies weak monotonicity (6,), the above (1) is replaced by H i ( x ) > H,(y) for some i # N ( x , y).

(2)

It is clear that strict monotonicity (6,) is an extreme case of indecomposability. Under the above concept of indecomposability, most of the results on indecomposable matrices stated in Chapter I1 can be extended to nonlinear situations. But here we consider only some of them, as long as they are relevant to the present study of relative stability, without trying to give all the possible extensions. In this respect, the most relevant among them is undoubtedly a counterpart of Theorem 7.3 in Chapter 11, which will be given a5

THEOREM 10.4. Zf (u), ( y ) , (6,) in $10.1 and indecomposability are assumed, we have (i), (ii) for n 2 2 : (i) The eigenvalues and eigenvectors of H are positive. (ii) The eigenvectors of H are unique up to the multiplication by positive numbers. Here uniqueness is asserted over all possible eigenvectors arising in association with any eigenvalues of H . Proof. (i) Let H ( x ) = Ax, x 2 0. First, we have A 2 0 by ( u ) and the semipositivity of x . Next, in view of the definition of N ( x , 0), we have, by (a) and 0 = Axi = H i ( x ) 2 H,(O) 2 0 for i # N ( x , 0).

(a

That is, H i ( x ) = Hi(0) for i 4 N ( x , 0). Whence, by indecomposability, N ( x , 0) must either be empty or must coincide with N = {I, 2, . . . ,n } . Since x 2 0, the first alternative is ruled out, so that N ( x , 0) = N , proving x > 0. Next, to prove A > 0, suppose A = 0. Then H ( x ) = 0 because H ( x ) = Ax. Take any nonempty proper subset J of N , and let y be the semipositive vector obtained from x by replacing xi by 0 f o r j E J while retaining x j f o r j # J . Then, clearly x 2 y 2 0, and in view of H ( x ) = 0, we have, again by (a)and (6,), 0 = H i ( x ) 2 H,(y) 2 0 for

i # N ( x , y ) = J.

That is, H,(x) = Hi(y) for i #N(x, y ) = J, which contradicts indecomposability. Whence A > 0, since 12 0. (ii) Let H ( x ) = qx, x 2 0. Then, by (y), we have H(u) = q(C xi)'-%, u = x / c x i E P , . In this way, any eigenvector is a positive multiple of an eigenvector with the unit component sum. Therefore it suffices to prove the uniqueness for eigenvectors lying in P,,.

157

$10. THE SOLOW-SAMUELSON THEOREMS

Now let H ( x ) = Ax, H ( y ) = py, x, y E P, . We may assume I >= p . Noting that both x and y are positive by (i) which is already proved, we let, as usual, O

Y.

=min2

over all i.

Xi

Then y 2 Ox, but y 3 Ox. First, y 2 Ox decomposes to the n component relations, which sum up to 1 2 O because x and y have the unit component sum. Second, for i $ N ( y , Ox) we have, by (6,) together with A2 p ,

iexi 2 =o

= pyi = ~

~ (2 y~ ~) ( 0 x 1

~ H ~=(w~ x) i >= loxi,

because 1 2 0,1 2 m 2 0 entail Om 2 6. That is, H,(y) = H,(Ox) for i N(y, Ox). On the other hand, y 3 Ox rules out the possibility N(y, Ox) = N = { 1,2, . . . ,n}. Whence the indecomposability of H implies N ( y , Ox) = which proves y = Ox. In this case, 8 must be 1, which follows from summing up y i = Oxi over all i. Hence x = y , Q.E.D.

a,

The following theorem is also a consequence of indecomposability.

THEOREM 10.5. Under (a), ( 6 ) and indecomposability, we have (i), (ii) for n 2 2: (i) x 2 y >= 0 implies H ( x ) 2 H(y). (ii) x > y 2 0 impZies H(x)> H(y). Proof. By (cI), (a), we have H ( x ) 2 H ( y ) 2 0 if x 2 y 2 0. Suppose first that x 2 y 2 0 and x 3 y . Then we have H i ( x ) > Hi(y) for some i 6 N(x, y ) by indecomposability, implying H ( x ) 2 H ( y ) . This proves (i) for the case x 3 y . Thus we have only to prove (ii). Now to this end, for any fixed k chosen x:) by :X = xi for i f k and from N = { 1 , 2, . . . , n), define the vector xk = ( x i = y , . Clearly xk 2 y , and N ( x k ,y ) contains all the integers in N except k if x > y. Hence, H,(xk) > H,(y) by indecomposability. On the other hand, x 2 xk 2 y implies H ( x ) 2 H(xk) 2 H ( y ) ,so that Hk(x)2 Hk(xk)> Hk(y). Recalling the arbitrariness of k , we may conclude that x > y 2 0 implies H k ( x )> Hk(y)for any k = 1, . .., n. This proves (ii). To establish relative stability, we need to strengthen indecomposability further. This will be done by formulating a nonlinear version of primitivity. In view of Chapter 11, Theorem 8.1, rather than the original definition of primitivity stated in Definition 8.2 of Chapter 11, we have

DEFINITION 10.3. H(x) = (Hi(x))in ( a ) in $10.1 is said to be primitive at x = a 2 0 if for any y such that y 2 a there is a positive integer s for which H"(y) > H"(a).

158


I t should be noted that s may generally depend on y as well as on a. Clearly a strictly monotonic system satisfying (6,) is indecomposable and primitive at every point with s = 1.

Let H ( x ) = ( H , ( x ) ) satisfy (a),(p), ( y ) , (6), and indecomposability. To ensure relative stability in system (2) in 910.1, we further assume a certain degree of primitivity, depending on the order rn of homogeneity; as will be seen, we need only a minimum degree of primitivity in the case 1 > m >= 0, whereas a further intensification of primitivity is required in the case m = 1. Therefore these two cases will be considered separately. (a) Case of n? < 1 . Note that indecomposability ensures A(H) > 0 for n 2 2 by Theorzm l0.4(i), so that II = 1 is, by Theorem 10.2(ii), also an eigenvalue of H. Moreover, the eigenvector u associated with I is positive and unique by Theorem 10.4. The last statement, namely the uniqueness of u, is justified as follows. If x = H ( x ) , u = H(u), then Theorem 10.4(ii) entails x = Ou for some B > 0. Whence Bu = x = H ( x ) = H(Bu) = OmH(u), so that O1-mu= H ( u ) = u, yielding = 1. Since 1 > m 2 0, this implies 0 = 1 and x = u. This eigenvector u associated with I generates a stationary solution u ( t ) = u ( t = 0 , 1,2, . . .).

(3)

This is a special case of balanced growth (3) in $10.1 for which a ( t ) = 1 over time. As was noted, a general balanced-growth solution x(t) = a(t)x converges to 2 = II1'(l-m),~, which must equal u, because x ( t + 1) = H(x(t)) becomes 2 = H ( 2 ) in the limit, whence 2 = u by the uniqueness of u. More generally, regarding the convergence of any solution to u, we have

THEOREM 10.6. Assume (a), ( n 2 2), (p), ( y ) (1 > rn 2 0), (a), and indecomposability. Then, if H is prirnitire at 0, every solution x ( t ) of ( 2 ) in $10.1, starting at any initialposition x(0) 2 0, coniierges to the stationary state (3). Proof: Let

P ( t ) = max

xi(t)

- over all i = 1, . . . , n. ui

In order to prove the convergence of x ( t ) to u, it suffices to see that of cc(t), to 1. We have, by definition, x ( t ) 2 a(t)u, and i n view of monotonicity (d),

P(t)

Xi(t

+ 1) = H,(x(t))2 H,(a(t)u) = cc(t)"H,(u) = cc(t)mui,

$10.

159


whence

+ 1) -2 a(t)"

Xi(t

( i = 1, . . . , n ) .

Ui

This implies

a(t + 1) 2

Similarly we have P(t

( t = 0, 1,

. . .).

(4)

( t = 0, 1,

. . .).

(5)

+ 1) s P(t)"

We may observe that P ( t ) > 0 over time for the following reason. First we have x(0) 2 0 by assumption. Moreover, if x(t) 2 0, we have x(t + 1) = H(x(t)) 2 H(0) 2 0 by Theorem lOS(i). Hence x ( t ) 2 0 over time, so that P ( t ) > 0 for any t. On the other hand, a(t) may possibly be zero for a finite number of initial consecutive periods. However, by primitivity at 0, we have x(s) = H"(x(0))> 0 at some s 2 1. Then by Theorem 10.5(ii),we have x ( t ) = H'(x(0)) > 0 for t 2 s. This implies a ( t ) > 0 for t 2 s. Now a repeated application of (4), ( 5 ) leads to a(s)"'s

LY(s

+ t ) 5 P(s + t ) 5 p(s)mt

(t

= 0,1,

. . .).

(6)

Since 1 > m 2 0, a(s) > 0, P(s) > 0, we find ~ ' Iim P(s)"' = I, lim ~ ( s ) = t++m

t++m

which, combined with (6), entails lim a(t) = lim P ( t ) = 1. t++m

f'+m

This completes the proof. Remark. In spite of the valid internal logic of Theorem 10.6, this theorem cannot be applied in effect to the case m = 0 because monotonicity (6) and m = 0 imply the constancy of H ( x ) for all x > 0, which is incompatible with indecomposability. In fact, let x, y be any two positive vectors. Then, Ox 2 y for some 8 > 0, so that H(x) = H(8x) 2 H ( y ) by (6) and nz = 0. Similarly, we have H(y) 2 H(x). Therefore H ( x ) = H(y). (b) Case of m = 1 . Again, 1 ( H ) > 0, as in the preceding case, and system (2) in $10.1 has a balanced-growth solution

u(t) = i ' u

( t = 0, 1 ,

. . .),

(7)

where u is a positive eigenvector associated with A = A(H). u is unique up to the multiplication by positive scalars by Theorem 10.4(ii). To overcome some complication due to homogeneity of the first order, the system H ( x ) is assumed to be primitive not only at 0 but also at u. Incidentally, it can be immediately seen that if primitivity prevails at u, it also prevails at every positive

160


multiple of u. We are concerned below with the convergence of any solution to (7) in system (2) in $10.1 in the ratio, rather than in the absolute level. Explicitly we shall prove THEOREM 10.7. Assume ( a ) ( n 2 2), (p), ( y ) (rn = I), (a), and indecomposability. Then, i f H is primitive at 0 as well as at u, every solution x ( t ) = (x,(t)) of (2) in $10.1, starting at any initialposition x(0) 2 0, converges in the ratio to the balanced-growth solution (7), i.e., . xi(t) lim - = y ( i = 1 , ..., n ) ui(t) the limit y being positioe and common to all i. Pvoof. The method of proof is somewhat intricate, compared with that for Theorem 10.6, although we begin with a similar procedure of letting i++m

a(t) = min xi(t> __ over all i ui(t) P ( t ) = max

x.(t>

ui(t>

over all

i

=

1,

= 1,

. . . , n,

. . . , n,

and getting the immediate results p(0) 2 p( 1)

2 * .* 2 P(t) 2 fi(t + 1 ) 2 * * *

2 a(t + 1 ) 2 a(t) 2 . - * 2 a(1) 2 a(0).

(9)

Relationship (9) follows from the inequalities fi(t)Au(t)= H(fi(t)u(t))2 x(t + 1 ) 2 H(a(t)u(t))2 a(t)Au(t), which are ensured by p(t)u(t)2 x ( t ) 2 a(t)u(t)and (6). By (9), the sequences { ~ ( t ) }{ ,p ( t ) }are bounded. They are also nondecreasing and nonincreasing, respectively. Therefore lim a(t) = a, i++m

lim fi(t) = p

i++m

exist and p 2 a 2 0. If, moreover, a = fi > 0, clearly we have (8) with y being this common positive limit a = fi. To see a = p, suppose the contrary, j? > a. It will be shown that this supposition leads to a contradiction. In fact, if p > a,we have fi >a6 for some 6 > I . Let, for any fixed k , the vector xk = (xi") be defined by xi" = ui (i# k ) , X: = auk. Thus we obtain n special vectors xk ( k = 1, . . . , n). At any t, we have x,(t) 2 a(t)A'ui for all i, while x k ( t )= fi(t)Aiuk for at least one k . In view of the fact that 2 p(t)P 2 a6 2 a(t)6, the second of the above statements reduces to Xk(t) 2 ct(t),?6uk for some k. Therefore, we have at each t x ( t ) 2 a(t)J.'xk for some k , (10) where k may depend on t.

$10.

161


Next, we observe that for each fixed k , u > 0 and 6 > 1 entail xk 2 u. Whence primitivity at u ensures the existence of a positive integer s(k) such that H“(‘)(Xk)> H’(’)(u), which further implies by Theorem 10.5(ii) that H‘(Xk) > H‘(u) for t 2 s(k). Then, if we let s = max s(k) over all

we .have

k

=

1, .. . ,n,

H”(Xk) > H”(u) (k = 1, . ..,n).

Therefore we can take n > 1 in such a way that H“xk) 2 7CHS(U) (k = 1,

. .. ,n).

(1 1)

With these preliminary arrangements, we consider the subsequence {x(vs)};= with the corresponding a-sequence {a(vs)};= . Noting that {x(vs)};=, is a solution of the system y(v + 1) = H”(y(v)),we see, in view of (lo), (1 1) and using some xk,

,

,

x((v + 1)s) = H“(x(vs))

2 If”(.(

VS)2’Xk)

= a(vs)nv”H”(Xk)

2 a(vs)P7rHS(u) = a(vs)nA(”+ %,

and hence xi((v + 1)s) 2 7ra(vs) ( i = 1, . ..,n). Ui((V 1)s) -

+

This yields .(V

+ 1)s) 2 na(vs)

(v

= 0,1,

. . .).

(12)

Finally, if we appeal to primitivity at 0 as in the proof of Theorem 10.6, the original a-sequence is positive from some period I, on, so that r ( t )> 0

( t 2 to).

If we let p be the smallest integer among such of (12) and (13) is

(13) 1’

that vs 2

to,a

a((p + v)s) 2 nVcr(ps) (v = 0,1, . ..).

synthesis (14)

Since 7r > 1, cl(ps) > 0, from (14) follows a contradictory result lim ~ ( v s )=

+ a,

v++w

as was expected. Therefore we must have a = fi, this common limit being positive by (13), Q.E.D.

162

111. BALANCED GROWTH 1N NONLINEAR SYSTEMS

$11. INCOME PROPAGATION UNDER

AUTONOMOUS SPENDING $11.I.Nonlinear Multiplier Processes $ 1 I will be devoted to the study of a dynamic process represented by

Y(t

+ 1) = H ( y ( t ) )+ a ( [ ) ,

(1)

where H ( y ) is a vector-valued function of an argument vector y as considered in $10, and a ( t ) is an exogenously given autonomous term. The formal properties of H are the same as in $10, i.e., H i s assumed to satisfy all or some of (a), (p), (y), ( h J , (h),and (hJ, depending on the case in question. The entire exposition of $1 1 is based on, and improves, Nikaido (1964a). We are mainly concerned with reconsidering the theory of matrix multipliers i n Chapter 11, $6.3 as well as with its extension to nonlinear situations. Nonlinearity intervenes when the matrix H in (4) of Chapter 11, $6.3 is replaced by a vector-to-vector mappingy(t) + H(y(t))to obtain (1). The standard economic interpretation as advanced for (4) in Chapter 11, $6.3 also applies to (1) with some modifications necessitated by the nonlinearity of spending patterns of sectoral income. We recall that the matrix H in (4) of Chapter 11, 66.3 is given as the product V(Z - A ) - ' P C . In the present general setting, the linear spending pattern represented by a matrix C of sectoral propensities to spend is replaced by F ( y ) = (Fi(y)),a vector-valued function, which may be termed a spending function or consumption function. The ith component Fi(y) stands for the spending, one period lagged, for the product of the ith sector corresponding to the sectoral income vector y = (yi)at the preceding period. Thus we obtain a composite mapping H ( y ) = V ( 1 - A)-'PF(y), which is a synthesis of the underlying linear technology and nonlinear consumption functions. In this way (1) admits an interpretation as a nonlinear process of income propagation. The general consumption function F ( y ) more faithfully represents the mode of consumption. For example, the rnth order homogeneity of F ( y ) , which is equivalent to that of H ( y ) , seems to be in better harmony with the familiar hypothesis of diminishing propensities to consume, if 1 > m 2 0. Even if m = 1, the first order homogeneity need not imply additivity in consumption patterns F(x + y ) = F(x) + F(y). A review of Chapter 11, $6.3 and 8.1 lets us realize that the mechanism ensuring the stability of a moving equilibrium as in (4) in Chapter 11, $6.3 differs in nature from that i n a self-sustained system such as (1) in Chapter 11, $8.1. The mode and pace of an exogenous excitation are the major cause of stability in the former, whereas the essential source of stability in the latter is indecomposability reinforced by primitivity. Now that we have seen that the stability of self-sustained growth in nonlinear systems shares essentially the

$11.

INCOME PROPAGATION-AUTONOMOUS

SPENDING

163

same cause with that in linear systems in Chapter 11, it may be in order to see what the situation is with respect to forced growth under an exogenous excitation in (1). This will be done in the present section. One interesting feature of forced growth in system (I), which is, however, not shared by self-sustained growth in system (1) in $10.1, already emerges when balanced growth comes into question. In fact, if the endogenous variable y ( t ) in (1) grows in a balanced way, so does the exogenous variable a(r), and vice versa. But they need not move on a common ray. Thus clear distinction should be made between balanced growth of the$rst kind and that of the second kind. The former is defined to be the case in which y ( t ) and a(t) move on different rays, respectively, while the latter is defined to be the somewhat singular case in which they move on a common ray. This classification, which seems trivial at a glance, will turn out to be relevant to the discussion of the possibility as well as of the stability of balanced growth. It must also be added that although this classification has not been mentioned in Chapter 11, $6.3, it applies equally to linear situations as extremely special cases of (1). Next we shall examine the possibility of balanced growth for the two cases above. To this end, we recall that a sequence of vectors y ( t ) is said to trace out a balanced growth path if the ratios of its components y l ( t ):y 2 ( t ): . . . :y,(c) remain unchanged over time. Whence such a sequence moves on a certain ray emanating from the origin and having nonnegative direction cosines. Let a(t) = r(t)a, a 2 0, (2) z ( t ) > 0, z(0) = 1 (3) be a given exogenous term in (1) which grows in a balanced fashion. We shall check on what balanced growth path y ( t ) in (l), given by y ( t ) = o(t)u, u 2 0, (4) o ( t ) > 0, ~ ( 0= ) 1, (5) is compatible with (2), (3). First it is obvious from the definition stated above that balanced growth is of the first kind if and only if u and a are linearly independent. Otherwise it is of the second kind. Now assuming (a), (y) in 510.1 for the function H ( x ) , we examine separately the two kinds of balanced growth.

( I ) Balanced Growth of the First Kind. If we substitute (2), (4) for a(t), y ( t ) , y ( t + 1) in (1) and take homogeneity (y) into account, (1) becomes o(r

+ 1)u = a(t)"H(u) + z(t)a

which is, by the positivity of a ( t ) in o(t

(9,rearranged

+ 1 ) u - - a .=( t )H ( u )

4t)"

4t)"

( t = 0, 1,

. . .),

(6)

to

( t = 0 , 1 , ...).

(7)

164


Because u and a are linearly independent, (7) holds if and only if the coefficients of u and a in (7) remain constant over time. Hence, in the case of the prevalence of (7), in view of (3), (9,we have a(t

+ 1 ) = po(t)" ( r = 0 , 1,

. . .),

(8)

T ( t ) = O(t)",

where p = a(1)

>0

and u is a solution of the equation pu - H ( u ) = a.

Equations (8), (9) completely determine a(t), ~ ( t )once , p = a( 1) is prescribed under the sole condition that Eq. ( 1 I ) has a solution u. Explicitly, a simple computation yields

Therefore, if for a p > 0 and an a 2 0, Eq. (1 1) has a solution u 2 0, these u and a togcther with ( 12), ( 1 3) generate a balanced growth (4), (5) compatible with (2), (3). Thus the possibility of balanced growth of the first kind reduces to the solvability of Eq. ( I I), which will be studied in $1 1.2. (11) Balanced Growth ofthe Secotd Kind. In this case, because a and u are by assumption linearly dependent, we have a = Ou for a positive scalar 0. Hence Eq. (6) becomes

(14) Equation (14) holds if and only if the coefficient of u remains a constant i over time. This means that u is a solution associated with 2 of the eigenvalue problem i.u = H(EI), u 2 0.

(15)

Correspondingly, the basic recursive relation for the sequence {a(?)} is given by a(t

+ 1) = io(t)" + &(t).

(16)

$11. INCOME PROPAGATION-AUTONOMOUS

SPENDING

165

We recall that a ( t ) and z ( t ) are subject to (12), (13) in the case of balanced growth of the first lund. In this respect, it should be noted that the mode of change over time of the sequence { z ( t ) }is unrestricted in (16). This is a peculiar feature in which balanced growth of the second kind is distinguished from that of the first kind. Once ( r ( t ) }is freely prescribed, any solutions I and u of the eigenvalue problem (1 5), together with the o ( t )determined by ( 1 6), will generate a balanced-growth path of the second kind given by (4), (5) which is compatible with &(t)u. Thus the possibility of balanced growth of the second kind reduces to the solvability of the eigenvalue problem (15), which has already been well explored. Forced balanced growth of the second kind therefore bears some similarity to self-sustained balanced growth. Nevertheless the stability of the former does not reduce to that of the latter because of the presence of z(t)a, as will be seen later. 01 1.2. Positive Resolvents

In matrix theory, the resolvent set of a square matrix A is defined as the set of all numbers p for which p l - A is invertible. Moreover, if A is nonnegative, certain members of the resolvent set for which ( p I - A ) - ' is nonnegative receive special attention, as was expounded in Chapter 11. We recall, as a typical result, that a complete exploration of the structure of the set of p making p l - A nonnegatively invertible has resulted in its characterization as the set of all positive numbers exceeding R(A), the dominant nonnegative eigenvalue of A . In this section, we shall investigate under what conditions an equation of type (1 1) in $1 1.1 is solvable in the nonnegative unknown vector u for a given nonnegative vector a, especially, conditions on p for which the equation has a unique nonnegative solution u for any nonnegative a. If p is one such number, the mapping that sends a 2 0 to the corresponding solution u 2 0 may be regarded as a nonnegative inverse of pu - H ( u ) . For easier reference, we call the positive resol~~ent of H the set of positive numbers p for which px - H ( x ) = a is solvable in the unknown vector x for any given semipositive a. Also, for simplicity, px

-

H(x) = a

(1)

will be referred to as a resolvent equation. As the first step toward investigation of the positive resolvent of H , some sufficient conditions for the solvability of Eq. (1) will be given.

THEOREM 11.1. Let H satisfy (a), (p), ( y ) in 910.1. Then we hare (i) If 1 > rn 2 0, any positice p beloiigs to the positiue resolrent of H . (ii) If m = 1, any p that is larger than A( H ) belongs to the positiile resolrent set of H .

I66

I l l . BALANCED GROWTH IN NONLINEAR SYSTEMS

Pvoof. As usual, let P, be the standard simplex formed by all nonnegative vectors with components summing to unity. The proof will be worked out by applying the Brouwer fixed-point theorem (Chapter I, Theorem 4.3) to certain mappings, suitably constructed for the two cases (i), (ii), respectively, which map P, i n P I , .

(i) 1 > 1y1 2 0. Let p > 0 and a 2 0 be arbitrarily chosen. Then, for any fixed x E P,, the equation

c

+ ic= a, n

n

/KO = 0‘),H i ( X ) i= I

I

is uniquely solvable in the single numerical unknown w > 0, because of the assumption 1 > 111 2 0. This is because the continuous function n

j ( w ) = pw

-

o l ) I

1 H;(x), i= I

(c

which satisfies , f ( w ) 5 0 for 0 5 w 5 wo = Hi(,u)/p)”(’ and is strictly increasing in the interval w 2 coo, with limu+ + “I,f(w) = oo,can be once and only once set to equal to a i at some value of o.Furthermore, since H ( x ) is the above solution o is a continuous function of x defined continuous by (/I), on P I , , which we denote by w ( x ) . This single-valued, positive, continuous, numerical function m(.u) satisfies, by construction, pto(s) = w(x)”l

c H , ( x )+

i= I

-“I)

+

n

i= 1

0;,

which, i n view of ( y ) , becomes po(.u) =

1 Hi(o(x)x) + C ui

i= I

With the aid of Q(.Y),

over all x

E

(2)

P,.

i= I

we construct the continuous mapping

4:P,,

-+

P,,, by

where ~/I,(.Y) stands for the ith component of 4c.y). Its continuity, as well as the nonnegativity of its values, is obvious, while (2) ensures the unit component sum for 4(s).Therefore, by virtue of the Brouwer fixed-point theorem, there is afixedpoint-i- = $(a)that automatically fulfillspw(.?).?=H(o(.?).?) + a. We have thereby obtained one solution x = o(.?).of ? Eq. (1). (ii) nz = 1. Let now p > 2 ( H ) and a 2 0 be arbitrarily chosen. We con-+ PI,by struct another continuous mapping $: P,,

$1 1.


SPENDING

167

where I,

i= 1

O(x) =

The denominator of (4) is positive because it is not less than p. Hence these n functions I+!I~(X), as well as O(x), are clearly continuous. Moreover, $,(x) >= 0 (i = 1, . . . ,n ) . They add up to unity for the following reason:

which is the numerator of the sum of (4) over all i, is easily seen to equal the common denominator of (4), namely,

Thus the mapping $ carries P , in P, continuously, so that there is a fixed point 2 = $(a) by the Brouwer fixed-point theorem. 2 satisfies

Suppose O(2)= 0. Then, in view of definition in Eq. (5), we would have n

while, at the same time, (6) would reduce to

( i $ l ~ i ( a ) ) 2=i ~ ~ ( (2i =) I , . . . , n ) .

(8)

Hi(,?) is an eigenvalue of H, which is majorized Equation (8) means that by A ( H ) because of the very definition of R ( H ) . Then, from (7) follows

c Hi(3 5 4H), I,

pi

i= 1

which contradicts the assumption p > A ( H ) . Whence O(2) > 0. This implies first that p > H i ( 2 ) ,so that the denominator of (6) equals p in effect. Then

1

168

111. BALANCED GROWTH 1N NONLINEAR SYSTEMS

(6), if divided by O(2) on the basis of the assumption of the first order homogeneity, becomes

p ( a i / 0 ( 2 >= ) Hi(9/O(2))+ a, (i = 1, . . . ,n). We have thereby obtained one solution x

=

2/U(2)of Eq. (I), Q.E.D.

$11.3. Further Results under Monotonicity

Theorem 1 I . I completely determines the positive resolvent of H for the case 1 > I?? 2 0. I n the case m = I , however, it does not tell if the positive resolvent contains a positive p that is less than l.(H), nor does it give us any information about the uniqueness of solutions of the resolvent equation. Results along these lines can be obtained by further assuming weak monotonicity ( ~ 3 ~ ) . First of all, we shall prove THEOREM 11.2. Let H satisfj ( a ) , (y) with nz = 1 and (0,) in $10.1. Then, i f the resolrent equation px - H ( x ) = a possesses a solution x 2 0 j o r some a > 0, we hare p > A ( H ) . Proof. Let u 2 0 be an eigenvector associated with I. = j . ( H ) , so that Ru = H(u). Define 0 by U = min s i / u i over all ui > 0. H ( x ) 2 0, which is ensured by ( r ) , entails p.u = H ( x ) + a 2 a > 0. Since we know the nonnegativity of x,this implies x > 0, so that 0 > 0. Next, let k be an integer such that 0 = s k / u k . Then x 2 Ou, xk= Ouk. Therefore, in view of ( a ) , (7) with nz = 1, (cSw), a > 0, we have pouk

= Pxk = Hk(X)

+

ak

2 H k ( 0 ~+ ) ak =O h k

+ ak

> (1 Auk

3

whence ( p - A)Ouk > 0. Since Ouk > 0, this gives p > I., Q.E.D. THEOREM 11.3. Let H satis/j (a), ( 7 ) n3ith I 2 nz >= 0 and(&) in $10.1. Suppose that p , q belong to the positir>errsolcent of H . Let x 2 0, y 2 0 be solutions o f the resolrent equationspx - H ( x ) = a, qy - H ( y ) = b, respectioely. Then, p 2 q , b 2 a 2 0, and b > 0 iniply x 5 y. Proof. Noting the positivity of y = ( y j )resulting from qy = H ( y ) b 2 h 0, we let

+

0

X.

= max 2 Pi

over all

i.

$1 I .


SPENDING

169

Then x 5 8y and xk = 8yk for some k. For this k, we have, using (a), ( y ) , (dW), and taking a 5 b into account, qeyk = )Ix, 5 P x k = Hk(x)

+ a&

Hk(ey)

+ bk

+ 6, .

= OrnHk(y)

If 0 > 1 were true, we would have 0 2 8" because I 2 m 2 0, and the above inequality would become )I8yk 5 0 Hk(y) 6,. This, if rearranged, gives 86, = O(qyk- H,(y)) 5 bk. But 6, 2 06, results in a contradiction 6, > b k , because 8 > 1 , 6, > 0. Therefore we must have 0 5 I , so that x 5 Uy 5 y , as was to be proved.

+

COROLLARY. Under (2) ( y ) (dW), the uniqueness of solutions qf the resoloent Eq. (1) in $1 1.2 obtains if the autonomous term a is positiiv. Proof. Consider a special situation in Theorem 11.3, where p = )I and a = 6. Then, Theorem 11.3 implies x 2 y as well as y 2 x, ensuring x = y , Q.E.D. Remark 1. Let H , ( x , , x 2 ) = H 2 ( x 1 x2) , = xIzx2p,where 2 > 0, > 0, a + /l = 1. This H satisfies (M), ( y ) with m = 1 . Moreover, not only (6,) but also (6) is met. It can be seen immediately that 2 ( H ) = 1. For p = i.(H) = I , however, the resolvent equation p x , - H , ( x , , x2) = 1,

px2 - H,(x,,

x2)

=0

has a solution x1 = 1 , x2 = 0. Therefore the positivity of the autonomous term a in Theorem 11.2 is indispensable. Remark 2. In spite of Remark I , the assertion of Theorem 1 1.2 remains true even for a semipositive autonomous term a if (6,) is reinforced by indecomposability. Justification can be made by a slight modification of the original proof of Theorem 1 I .2 because, by (6,) together with indecomposability (see Definition 10.2), there is some k 6 N ( x , Ou) such that H , ( x ) > H,(Uu). If we evaluate the relevant terms of the inequalities in the original proof for this k , the relation ( p - %)Uu, > 0 can be ensured under the possibility a, = 0. Likewise, similar extensions are possible for Theorem 1 1.3 and the result stated below in Theorem 1 I .4, but their consideration will be left to the reader. The results i n $1 I .2 as well as in $1 1.3 are important to the discussion of stability on the following section, but, apart from it, they have an application of independent interest, which we state in

I70


THEOREM 11.4. Let t i homogeneous ,functions G , ( x l ,x 2 , . . . , x,) of a coninion degree ni n.ith 1 2 171 > 0 he defitiedfor all tiontiegatiiv ralues of the arguments arid be continuous except at the origin. Moreover, it is assumed that they satisfy the , f o l l o ~ ~ i ncotirlitioti: g For each k, xk = y k , arid xi2 y i ( i # k ) imply Gkf*YI, x2 , X , J 5 Gk(Y1, y 2 , . . . , r,J.I f G J U l , 242, . . . , u,) > 0 ( i = I , . . . , n ) f o r some set of ui 2 0, then the systeni of equations 3

G,(x1, x 2 , . . . , x,)

( i = I , ..., t i )

= a,

xr=,

(1)

has notitiegatire solutions for at7y set of a , 2 0 bztith a , > 0. The corresponding solution x = ( x i ) is positirv atid unique if a, > 0 (i = I , . . . , 17). Pvoof. Let F,(xl,x 2 , . . . , x,) = Gi(x:/'",xi/'", . . . , x!,'") (i = I , . . . , t i ) . It can readily be seen that the functions F, satisfy all the conditions originally imposed on G, except for the fact that the degree of homogeneity of F , is exactly one. Clearly, to prove the theorem, it suffices to see that the system of equations F,(xl,x 2 , . . . , x,J = a , (i = 1, . . . , t 7 ) (2)

-

has the desired property. We first take a positive number p fulfilling I

. . . , 0, 1, 0 , . . . , 0 ) ( i = 1,. and define ti functions H i(i = I , . . . , t i ) by p 2 F,(O,

Hi(SI,x*, . . . , x,)

= pxi

-

. . , ti)

F , ( x 1 ,X L , . . . , x,J.

(3)

(4)

I t will be seen that these functions satisfy (a), (p), (y) with n i = I , (6,). First of all, we note that the 11 functions Fivanish at the origin because of their homogeneity of degree one. Then, since (.y1, . . . , x - 1, 0, sk+ 0, . . . , 0) = 0 for any x i 2 0 (i # k) and their kth components are the same, the monotonicity (d,) of - Fk implies F k ( x , .. . . , .yk - 1, 0 , .Y~+

. . . , xI,)5 F,(O,

. . . ,0 , . ..,0 ) = 0

for any si2 0 ( i # k). Whence H k ( s )= p . 0 - F J x ) 2 0 for any x = (xi) 2 0 with ,yk = 0. Likewise monotonicity implies Fk(-y~ 7 .

-

..

7

Xk-I

>

1

9

xk

+ I . . ., xn) 9

k

- F,(O, . . . , 0, 1,0,. I

. . , 0) S p

(5)

f o r a n y s , z O ( i # k ' l by(3). Hence,H,(x)=p.u,-F,(x)=x,(p-F,(x,/x,, ..., X ~ - ~ / . YI ,~ X , ~ + ~ / X . .~. ,, .Y,,/.\*~)) 2 0 for any x = ( x i ) 2 0 with xk > 0 by ( 5 ) and the first degree homogeneity of Fk . Thus H k ( x ) 2 0 ( k = I , . . . , n) for any x 2 0 so that (2) is met.

$1 1.


171

SPENDING

On the other hand, all the remaining conditions (p), ( y ) with m = I , (6,) on H iimmediately follow from the corresponding ones on F , . The results in $ 1 1.2 and $1 1.3 can therefore be applied to this system of H i . In fact, we note that (4) reduces to p ~ -i H i ( x l , x2, . .,., x,) = F i ( x l , ~

2 . .,

. , x,~)

(i = 1 . . . , P I ) .

(6)

. . . , n),

(7)

By assumption, at x i= uim2 0 (i = I , . . . , 11) we have Fi(ulm,uZm,. . . , unm)= Gi(ul, u 2 , , . . , un) > 0

( i = I,

so that puim - H i ( U l r n , U 2 m , . . . , Unm ) > 0 ( i = 1, . . . , n ) by (6), (7). Whence p > 2 ( H ) by Theorem 11.2. Then, again in view of (6), we see that ( 2 ) has nonnegative solutions for any a = ( a i ) 2 0 by Theorem 1 l.l(ii). Finally, the positivity as well as the uniqueness of solutions is ensured for a = (a,) > 0 by the Corollary to Theorem 1 I .3 through (6), Q.E.D. Remark 3. Evidently Theorem 11.4 is a nonlinear counterpart of the equivalence of ( I ) and (11) stated in Theorem 6.1. in Chapter 11. 011.4. Stability in Forced Growth

In this section the relative stability of a balanced-growth solution of Eq. (1) in $1 1.1 will be established. The two kinds of balanced growth will be discussed. separately. ( I ) Stability of Balanced Growth of the First Kind. We recall that a balanced-growth solution of the first kind y ( t ) resonant with a ( { ) is given by (2)-(5) and (8)-( 11) in $1 1.1.

THEOREM 11.5. Suppose thatfor a giren positice autononious term a ( t ) groii.ing in a balanced \c'ajJ,there is a balanced grow*thsolution qf the jirst kind y ( t ) qf y ( t + 1) = H(y(r)) + a ( t ) , y ( t ) 2 0, resonant with a(r). Assunie nonnegatirity ( a ) , homogeneity ( y ) and monotonicity ( 6 ) on H , it~liiclzare stated in $10.1. Then, we hace z.(t) Iim L = I + rn yi(t)

( i = 1, ..., n )

r-

j b r any solution z ( t ) = ( z i ( t ) )starting at any initialposition z(0) 2 0 ofequation z(t

+ 1) = H ( z ( t ) ) + a ( t ) ,z ( t ) 2 0

flaring the same autonomous term a(t).

172


Proof. Since a ( t ) = z(t)a is assumed t o be positive, y ( t ) remains positive. With this in mind, we let zi(t)

~ ( t= ) min - over all Yi(t) zi(t> b ( t ) = max -

Yi(t>

i,

over all

i.

Then, nonnegativity (a), homogeneity (y) of degree m, and monotonicity (6) imply, for each i, zi(t

+ 1) = Hi(z(t))+ ai(t) 2 H i(4t)At)) + ai(f) = a(t)"Hi(y(t)) = W(t)"(Yi(t

+ ai(t)

+ 1) - a,(t)) + a,(t).

This result, when divided by y i ( t + I), will become Zi(t

+ l)/Yi(t + 1 ) 2 N ( t ) " ( l

-

Ui(t)hi(t

+ 1)) + U i ( t ) h J i ( t + 1).

But, ai(t)/-vi(t+ 1) = t(t)ai/a(t+ I)ui = a i / p u iby (2), (4), and (8)-(11) in $1 1.1. Taking this into account, we finally obtain z,(t

+ l)/Yi(t + 1) >= a(t)"( I - a,/pu,) + a,/pu,,

which implies

+ ai/pui].

(3)

+ ai/puil.

(4)

~ ( 0= ) maxCB(O), 11,

(5)

~ ( +t 1) 2 min [a(t)"(l - ai/pui) 1s i s n

A similar argument will also lead to P(t

+ 1) 5 max [P(t)"(l sisn 1

- ai/pui)

Consider the sequences { ( ( t ) } , { ~ ( t ) }starting , at

t(0) = min[40>,11, which are defined by

+ 1) = min [5(t)"(l - a i / p u i )+ ai/pui], sisn (+ i 1) = max [r(t)"(l - a i / p u i )+ a i / p u i ] . sisn

5(t

1

~

1

(6)

(7)

Since u > 0 is a solution of the resolvent equation pu - H(u) = a > 0, we have 0 < a J p u , 5 1 (i = 1, . . . , n), so that M = max(1 - ai/pui) over all i satisfies

9 1 1.


173

SPENDING

Then, in view of (8) and 1 2 m 2 0, we can easily see, by simple arguments based on induction over t , that 5(t)

hold for t

s 40 s P ( t ) s ?(t),

(9)

0 2 &t + 1) - 1 2 M(&t)" - 1) 2 M(&t) - l),

(10)

0 5 q(t + 1) - 1 5 M(?(t)" - 1) 5 M(q(t) - 1 )

(1 1)

= 0,

1, . . . . Hence (8), (lo), (1 1) clearly imply lim t++m

t(t)=

lim q ( t ) = 1. t++m

Whence, by (9), follows lim a(t) = lim P ( t ) = 1. t++m

t++m

This proves the theorem. (11) Stability of Balanced Growth of the Second Kind. Certain subcases of this case will be dealt with separately. In what follows, as in Case I, the autonomous term a(t) = O T ( t ) u is required to be positive over time, so that u, an eigenvector of H , is assumed to be positive. The eigenvalue problem H(u) = Au, u > 0 can be solved under appropriate conditions on H (see $10).Hence, once an arbitrary scalar factor ~ ( t>)0 is given, a balanced growth solution of the second kind y ( t ) = o(t)u resonant with a ( t ) = &(t)u surely exists. We are concerned with the behavior of an arbitrary solution z ( t ) 2 0 of the equation

+

+ eT(t)u

~ ( t 1) = H(z(t))

(12)

under the basic assumptions of nonnegativity (a), homogeneity (y), and monotonicity (6) on H , which are listed in $10.1. Let y ( t ) = o(t)u be the balanced-growth solution of (12). We shall examine below under what conditions any solution z ( t ) = (zi(t)),starting at any initial position z(0) 2 0, of (12) converges to y ( t ) = ( y i ( t ) )in the ratio.

THEOREM 11.6.

If the homogeneity degree m is less than 1, we have

Proof. Define a(t), P ( t ) in the same way as in (l), (2). Then, similarly to Theorem 11.5, we have for each i

174


+

Dividing by y,(t 1) and taking yi(t reduce the above relation to

+ 1) = o(t + l)ui into account, we can

This implies

Similarly we have

} related by (16) in 911.1. We recall that the sequences {a(t)}, { ~ ( t ) are Hence

Let us define a minorant sequence { 5 ( t ) } and a majorant sequence { ~ ( t ) } for t 2 1 by

v ( 1 ) = max "I),

11.

It is easily shown by induction o n t , in the light of (15), that at)

s 4 t )I B(t> I q(t),

5 0 ) 5 a t + 1) 2 1 , q ( t ) 2 q(t

+ 1) 2 1.

(18) (19)

(20) By (19) and (20), lim 5(t) = 5 5 I , and lim ~ ( t=) q 2 1 exist. Next, subtracting (16) from (17) and taking (15) into account, we obtain Y(t

+ 1) - dt)" s 4(t + 1) - 4(t)".

This becomes in the limit 9 - qrn5

5 - 5".

4 11. INCOME PROPAGATION-AUTONOMOUS

SPENDING

175

But, since 5 5 1 m >= 0, from (21) it follows that 0 5 v] - qm 5 5 - 5" 5 0, resulting in 5 = cm, v] = v]". Whence 5 = 1, q = 1 because 1 > m 2 0, if it is seen that 5 > 0, q > 0. In fact, v] is clearly positive, since v] 2 1. On the other hand, the positivity of 5 is seen as follows: z( 1) 2 6r(O)u > O entails ~ ( 1 > ) 0, so that by (19), 5 2 5(1) = min[cr(l), 11 > 0. Thus lim & t ) = lim v](t) = 1, whence lirn a ( t ) = lirn p(t)= 1 by (18). This completes the proof. The case m = 1 needs special treatment. It must be noted that a crucial part is played by the rate at which the autonomous term a ( t ) = 8z(t)u grows. If the rate is sufficiently rapid, relative stability still prevails under the same basic assumptions of nonnegativity ( E ) , homogeneity ( y ) , and monotonicity (6) on H . Otherwise, there is a possibility of instability in general, and additional conditions on H are required to ensure stability. This is because the forced growth is in effect very much like self-sustained growth, when the growth rate of the autonomous term is slow and not stimulant, We regard the infinite series

where I is the eigenvalue of H with which the generator u > 0 of the forced growth in question is associated as an eigenvector. Recalling that by assumption z ( t ) > 0 over time, we see that the terms in (22) are positive, and therefore there are two mutually exclusive situations: (22) either diverges to +co or else converges to a positive limit. We may conventionally let the divergent cases include the case 1 = 0. It naturally suggests itself that we may measure the growth rate of ~ ( t ) relative to I t on the basis of the behavior of (22). Thus we say that { t ( t ) } grows fast if (22) is divergent; it grows slowly if (22) is convergent. Now regarding the scalar factor of the balanced-growth solution y ( t ) = o(t)u resonant with a(t) = &(t)u we have LEMMA 11.1. The sequence {A'/o(t)}has the following properties: (i) Iimt++ a, A'/o(t)= w exists and is nonnegative. (ii.1) w = 0 if ( ~ ( t )growisfast. } (ii.2) w > 0 if ( r ( t ) }grows slowdy. Proof. These propositions immediately follow from (16) in $11.1 with m = 1, namely, o(t + 1) = no(t) + et(t). (23)

+

(i) Equation (23) implies o(t 1) > h ( t )because &(t) > 0. This can be rearranged to It+ 1 I' -5o(t 1 ) - o(t)

+

176


meaning that {A'/o(t)}is nonincreasing. Furthermore, its terms are nonnegative. Whence { I ' / a ( t ) }converges t o a nonnegative limit. (ii) A repeated application of (23) yields

which becomes in the limit 1 -=a(O)+-

w

0

?(k) c7 . i

i k = O

From (24) we can easily deduce (ii.1) and (ii.2), Q.E.D.

THEOREM 11.7. For the case m = I , the following (i), (ii), hold: (i) (f { ? ( t ) > groivsfast, we haae . zi(t) Iim -= 1 (i f+ +m Yi(4

=

1, . . . , 1 2 ) .

(ii) If { T ( t ) } grows slowly and H is primitive at u (see Definition 10.3), we hare

the limit y being positive and common to all i. Proof. (i) We observe that the argument in the proof of Theorem 11.6 is independent of the assumption m # 1 except at the very last step. I'hus (16)(21) are naturally valid for the case m = 1, with which we are now concerned. Therefore the proof will be complete if it is seen that lim t(t)= lim q ( t ) = 1. Since 1 - Os(r)/a(t+ 1) = Lo(t)/a(t+ 1) by (23), we can put (16), with 111 = 1, in the form ((t

+ 1) = o(th (+t )1) ( ( f ) + 1 ~

-

lo( t ) a(t 1)'

~

+

A repeated application of (25) leads to

5(f + 1) Then, since

{T(t)}

=

Ira(1) t(1) a ( t 1) ~

+

A'a( 1) + 1 - ___ a ( t + 1)

grows fast, lim A'/a(t) = 0 by Lemma 1 l.l(ii.l), so that

8 1 1.

INCOME PROPAGATION-AUTOKOMOUS

SPENDING

177

converges to 0 as t tends to infinity. Whence lim c ( t ) = 1 by (26). Similarly we have lim q ( t ) = 1 . (ii) We note again that the argument in the proof of Theorem 11.6 naturally applies to the present case m = 1 . In particular, we can deduce from (13), (14) for m = I the following facts:

+ 1) - a(2) 2 (O?(t)/U(t + 1))(1 - a(t)), a(t + 1) - 1 2 (1 - &(t)la(t + l))(.(t) - I),

a(t

P(r + 1) - P ( t ) 5 ( W ) / a ( t+ 1))(1 - P(t)>,

+ 1) - 1 5 (1 - Bz(t)/a(t + l ) ) ( B ( t ) - I), P(t + 1) - a(f + 1) 5 (1 - eT(t>/.(t+ l))(P(t> - cr(t)). P(t

(27)

(28) (29) (30)

(31)

Using (15), (27), and (28), we see that { o l ( t ) } is nondecreasing as long as stays in the interval ( - 00, 11, and that once a ( t ) steps in the interval [l, + 00) at some period t = t , , it continues to stay there from period t , on. Similarly, we see that { P ( t ) } is nonincreasing as long as P ( t ) stays in the interval [I, +GO),and that once P ( t ) steps in the interval (- co, I] at some period t = t , , it continues to stay there from period t , on. These results imply that at least one of {cr(t)},{ P ( t ) } is convergent. For example, if u ( t ) < 1 for t 2 0, {cr(t)) as a bounded nondecreasing sequence is convergent. If otherwise a(tl) 2 1 at some t,, p ( t ) 2 a ( t ) 2 1 holds for t 2 I,, and hence ( p ( t ) } as a bounded nonincreasing sequence for t >= t , is convergent. On the other hand, (15) and (31) imply that the sequence { P ( t ) - cc(t)}, whose terms are by definition nonnegative, is nonincreasing. Whence ( P ( t ) cr(t)}, is convergent. Thus, in effect, all three sequences { a ( t ) } , { P ( t ) } , { P ( t ) - cr(t)} are convergent, with the respective limits a, P, fi - a. Therefore the proof will be complete if it is shown that c1 = P. To see that c1 = P, suppose the contrary, /3 > a. Then, we have > 6a for some 6 > 1. We construct I? vectors xk = (x:) ( k = 1, . . . , n ) exactly in the same way as in the proof of Theorem 10.7. We also recall that in the proof of Theorem 10.7, the primitivity of H at u entails the existence of a positive integers and 7c > 1 for which

N(t)

H ” 2 ) 2 7cH”(u) (k = 1, . . . , n)

(32)

hold. Now, neglecting the second term in z(t + 1) = H ( z ( t ) ) a(t),we naturally have z(t 1) 2 H(z(t)). After s iterations based on monotonicity (6), this becomes

+

+

+

~ ( t S)

2 Hs(z(t)).

(33)

178


Since the limits

CI,

p of

{ ~ ( t ) }{ P, ( t ) } satisfy P(t)

p > dcr, we have

2 W t ) (t 2 to)

(34)

for sufficiently large t o . At each t we have z ( t ) 2 a(t)a(t)u. Also, Zk(t) = P(t)a(t)ukfor at least onek depending on t , which reduces to zk(t)2 a(t)o(t)bu,if t 2 t o . Whence, at each t 2 to we have z ( t ) 2 x(t)a(t)xk for some k. We now consider the sub-sequence {z(vs)},mO with the corresponding cc-sequence {cc(vs)}~,where vo is a positive integer fulfilling vo s 2 t o . Then, in view of (32) and (33) and using some xk depending on v, we see

+ 1)s) 2 H”(z(vs))

z((v

>= Hs(.(vs)a(vs)xk) = a(vs)o(vs)H”(xk)

2 - CI(vs)a(vs)nHS(u) = .(vs)a(vs)ASnu,

whence zi((v a((v

+ 1)s)

+ 1)S)Ui

2n -

o(vs)X

a((v

+ 1)s)

.(vs)

(2

=

1, . . . , n).

This yields .(V

2n

4- 1)s)

a( vs)2 B((V

+ 1)s)

a(vs) (v 2 vo).

(35)

At this step we finally appeal to the assumption that { z ( t ) } grows slowly. By Lemma 1l.I(ii.2), lirn A‘/o(t)= w is positive. This ensures A > 0 in particular and implies a( vs)A’ i ( V + 1)\ AVF lim -- n. lim n = n lim v-+m a((v 1)s) v++, .((v 1)s) v + + m a(vs)

+

Hence, if we take a number

n

I

+

E

such that n > E > 1, we have

a( V S ) 2 2&>l a((v 1)s) -

+

(v2pIvo)

for sufficiently large p . Using (36), we can reduce (35) to V(.

+ 1)s) 2

E.(VS)

(v

Ip),

which, after iterations, becomes a((v + p)s) 2 EVLY(PS) (v 2 0).

(37)

$11.


SPENDING

179

Since z ( t ) = H(z(t - 1)) + &(t - 1)u 2 &(t - 1)u > 0 for t 2 1, we are sure that a ( t ) > 0 for t 2 1, in particular, a(p) > 0. Whence (37) yields in the limit lim ~ ( v s = ) +a, v-ttoo

because E > 1. This contradicts the convergence of the original @-sequence ( ~ ( t ) > t " Thus , ~ . a = p is proved. ) by (12) and (13) Remark 1. If I u = H(u), u > 0, then the o(t),~ ( tgiven in $1 1 . 1 for p > I satisfy (23). This special situation of balanced growth of the second kind can be treated by the method of proof used for balanced growth of the first kind, but this method cannot apply to genera1 situations where the specification of z ( t ) can be made freely without any restriction of its functional form. Remark 2. In Theorem 11.7(ii), the assumption of primitivity is indispensable. An example can be given even for a linear H. If I > 0, the dominant root of

H= is

(I:

A, with which is associated =

(i)

as a positive eigenvector. If r ( t ) = p', 0 = p, we have the corresponding

If I > p > 0, the solution

z ( t ) = o(t)u

+ (-i)t (II)>O( t = O , l ,

...)

of z(t + 1) = Hz(t) + Qp'u does not converge to the balanced-growth solution y ( t ) = o(t)uin the ratio, since both zl(t)/yl(t)and z2(t)/yZ(t)sufferan undamped oscillation.

Chapter IV

EFFICIENT ALLOCATION AND GROWTH

The modern formulation of production in terms of production technology sets, together with related concepts, is expounded in $12. Two typical turnpike theorems in planned economic growth are proved in the following two sections; a final-state turnpike theorem in 413, and a consumption turnpike theorem in $14.

$12. PRODUCTION TECHNOLOGY SETS $12.1. Technology Sets

One of the most basic features in the working of a national economy is production. In economics, production means the transformation of some quantities of goods as inputs into some amounts of products as outputs under a given technology. Economists are primarily interested not in the physical or chemical features of technology but in the quantitative correspondences of inputs to outputs effected by technology. Outputs are acquired at the cost of consuming inputs. What is economically most vital is, therefore, the amount of products obtained as outputs relative to the amount of goods consumed as inputs. Thus, for example, from the economic point of view, the efficiency of a method of electric power generation is evaluated according to some principle based principally on the information about the quantities of goods consumed as inputs and the electric energy obtained as outputs, regardless of whether electricity is generated by atomic reactors or by hydraulic turbines. We were previously concerned with simple but important examples of production as the transformation of inputs to outputs in Chapter TI, in which we dealt with production under constant input-output coefficients on which

180

912.

PRODUCTION TECHNOLOGY SETS

181

Leontief systems (96), linear programming (99. I), and von Neumann's models (99.2) are based. In modern mathematical economics, each particular realization of production is represented as a pair (x, y ) of an input vector x and an output vector y , which is referred to as a production process or simply as a process. There are as many goods used as inputs as the dimension of x ; each component of x stands for the amount of a certain kind of good used as an input. Likewise, there are as many goods obtained as outputs as the dimension of y ; each component of y stands for the amount of a certain kind of good obtained as an output. Generally there are many processes ( x ,y ) under a given production technology. From the economic point of view, therefore, a production technology is completely determined by enumerating all the processes that are possible under it. Thus a production technology set T is defined as the set of all these possible production processes. Many important properties of a production technology are embodied in the structure of the corresponding technology set, so that the study of a production technology from the economic point of view consists essentially of studying the structural characteristics of the corresponding technology set.

EXAMPLE 1. Let aj be thejth column of the input coeficients matrix A in a Leontief syslem. I f xi 2 0 represents the amount of the jth good produced in the j t h sector, the corresponding technology set is T j = { ( x j a j ,x j ) I x j 2 O } .

Obviously, Ti is thought of as a ray in

R"'l

spanned by

a special polyhedral convex cone. There may or may not be some goods entering a process ( x , y ) as both inputs and outputs, depending on the specific situation. I n this respect, a particularly interesting case is that in which every good has the possibility of serving as an input and of being produced as an output. In this case, x and y are of the same dimension and their corresponding components refer to a common good. Let x = (xi),y = ( y , ) ; then y i - xi represents the amount produced as output minus that consumed as input of the ith good, i.e., the net output of the ith good, a flow magnitude. Sometimes what is relevant is not the pair (x, y ) of input and output vectors x, y but the corresponding net output vector y - x. A technology set is conveniently defined as the set of all possible net output vectors rather than that of all possible pairs of input and

182

IV. EFFICIENT ALLOCATION A N D GROWTH

output vectors. For simplicity, a net output vector is also referred to as a process, or, more precisely, as a j o ~process ) whose positive and negative components represent the actual net outputs and inputs. The definition of a technology set based on the pairing of inputs and outputs is useful i n analyzing production in terms of stock magnitudes, such as several kinds of capital. On the other hand, the definition based on the net output concept is convenient i n analyzing production in terms of flow magnitudes. For easier reference, we term the former the stock version and the latter thepow iiersion of the definition of a technology set.

EXAMPLE 2. With the aid of the gross outpiit i’ector x 2 0 and the input coejjicioits matri.y A , the national econon7y-ii.ide technology set in the flow i.rrsion q j a Leoiiticf system is gicen by

T = { ( I - A ) x I x 2 0}, ii.here I is the identity matrix. T is a polyhedral conoex cone. $12.2. Structural Properties of Technology Sets

Several classical fundamental laws of production can be represented as structural properties of technology sets. They will be enumerated below. A technology set T, whether formulated in the stock version or in the flow version, is obviously a set in a Euclidean space. For this reason, certain of the above structural properties of T can be represented without specifying in which version T is given. So far as possible, advantage will be taken of this fact in what follows. ( a ) The lait* OJ constant returns to scale is said to prevail if T is a cone, i.e.,

T 3 aT for any number

a 2 0.

This means that a proportional increase (decrease) of inputs by some percentage ensures a corresponding proportional increase (decrease) of outputs by the same percentage. Mathematically, multiplication of a process by a number a 2 0 is always possible. But, if the law of constant returns to scale fails to prevail, some positive multiples of a technologically feasible process are not technically feasible, even though they are mathematically conceivable. Under constant returns to scale, a process is technologically feasible if and only if its positive multiples are also technologically feasible. They can be distinguished from each other only by their intensities of operation in terms of an arbitrarily fixed representative process, since they have the same input-output proportions.

$12.


183

(/3) External diseconomies are said to be absent if T is convex (see Chapter 1, Definition 2.1). A convex linear combination of two processes with weights CI >= 0, p >= 0, a + /3 = 1 represents their simultaneous operation in the respective intensities a, p. If it is technologically possible with any two original processes, at least the weighted sum of the original outputs can be produced within the technology set T by using the corresponding weighted sum of the original inputs at most. This phenomenon is usually referred to as the absence of external diseconomies within the system for which T serves as the whole technology set. (y) The impossibility of the land of Cockaigne means that it is impossible to produce something from nothing, or, stated differently, nothing can be acquired without sacrifice. This is formulated i n terms of the technology set T as follows: ( i ) Impossibility of the land of Cockaigne in the stock version : ( x , ~ ) ET and

(ii)

x=O

imply y = O .

Impossibility of the land of Cockaigne in the flaw version : X E T and x

imply O x=O.

~

The meaning of the above formulation of ( i ) in the stock version is selfexplanatory. On the other hand, if a flow process x satisfies x 2 0, some positive amounts of certain products emerge under this process without input. In the flow version, (ii) excludes such a possibility. (6) The irreversibility of aprocess means that if a process is technologically possible, then its reverse process, which produces the original inputs as outputs by consuming the original outputs as inputs, is technologically impossible. This is formulated in both versions of the technology set T as follows:

(i)

Irreversibility in the stock version: ( x ,y ) is irreversible if ( x ,y ) E T but ( y , x) 6 T

(ii) Irreversibility in the j70\i, wrsion: x

is irreversible if x

E

T but

--x

6 T.

One important source of irreversibility is the existence of an indispensable factor of production, e.g., labor. If every nontrivial, technologically feasible process uses a positive amount of labor as an indispensable item of input, no process is reversible, with the sole exception of the zero process, which means inaction. This corresponds to the view in traditional economic theory that labor is an unproducible primary factor of production. However, if, in accordance with another classical view, in which labor is reproducible through

184

1V. EFFICIENT ALLOCATION A N D GROWTH

the consumption of wage goods by workers, we consider a n economy as a closed system with regard to production, some processes may be reversible. Statements (c()-(d) are basic laws of production that implicitly o r explicitly underlie orthodox economic theory. In addition to them, an assumption that is often made for mathematical reasons and is usually acceptable to economists, is

( c ) The closedness ofthe technology set. In most economic situations, it is not unreasonable to assume that a n input-output configuration that can be approximated arbitrarily closely by technologically possible input-output configurations is also technologically possible. From the mathematical point of view, ( E ) is almost indispensable to ensure the existence of desired entities. In what follows, all or some of the above basic structural properties of the technology set will be assumed, depending on the case under consideration. Remark 1. If a technology set T is a subset of R", the origin of R" represents the process of inaction. Naturally, the process of inaction is possible under any technology. Hence it is appropriate to assume that T contains the origin. Remark 2. If T is a technology set in the flow version, the impossibility of the land of Cockaigne is represented by T n R+" = {0}, where R," is the nonnegative orthant and R" Remark 3. For a T i n the flow version,

2

T.

T n ( - T ) = (0)

is a representation of the irreversibility of all processes in T. There is one more special structural property of the technology set T that is assumed in some cases: free disposability. This concept is defined as follows: ([)

(i)

Free disposcrbilitj, iii T : iii

the stock ivrsion: (.Y,

(ii)

ill

>I)

the j701r

E

T,

Li

2 s, y 2

z'

imply

( u , 1 ) ) E T.

/~Prs;ol?:

.x

E

T,

.Y

2

ii

imply

LIE

T.

A disposal process means the process o f disposing o r throwing away some amounts of goods. In the above formulation ( i ) in the stock version, the process ( I / , 1 . ) is operated i n three stages: the disposition of Li - x, the transformation of the input vector .I-t o the output vector y by the process (.Y, y ) , and the disposition o f j . - A corresponding interpretation in the flow version is I?.

$ 1 2 . PRODUCTlON TECHNOLOGY SETS

185

possible. It must be noted that there may be another process ( x * , y * ) i n T satisfying u 2 x*, y * 2 v, by which the process (u, v ) is operated in three similar but alternative stages of action. Free disposability means that disposal activities are costless in the sense that they require no additional consumption of inputs other than the amounts of goods disposed of.

$12.3. Efficient Allocation and Prices Since it is only common sense that homo economicus will aim to acquire as great a return as possible at as little sacrifice as possible, the following definitions suggest themselves:

DEFINITION 12.1. A flow process x is called more eficient than another ,flort'process v i f x 2 v in the sense of the usunl vectorial semiorder (see (ci-(y), Chapter I, 43.3).

DEFINITION 12.2. A j b i *process x E T is culled e ~ c i e i i t(/' there is no more efJicient process in T than S . DEFINITION 12.3. A stock process 11.) if'

(.Y, J,) is culled more eficierit than

another stock process (v,

(-;)

(

-l:).

DEFINITION 12.4. A stock process (x, y ) E T is called efficient i f there is no more efJicient process in T than ( x , y ) . The concept of efficiency does not premise the profit evaluation under a price system. Its definition is based solely on the comparison of physical quantities of inputs and outputs. But profit evaluation is not altogether irrelevant to the efficiency of processes. A bridge is built between the two by the existence of appropriate price systems that make the results of the two methods of evaluation actually equal. The following theorems account for this situation.

THEOREM 12.1. Let T be a technology set in thefloit. version in R'. Then, \tie have: (i) If, under a positive (n-dimensional) price uector p > 0, a process .f maximizes the proJit p'x over all x E T, then 2 is un eficient process. (ii) Suppose that T is convex. I f 2 is an eficient process, there is such a semipositive price vector p 2 0 that the profit p ' x is maximized at x = 2 oiler T. Pvoof. (i) If 4were not efficient, there would exist a more efficient process contradicting the assumed maximum profit at 2.

x in T, so that x 2 ,?. Since p > 0, this yields p ' s >,I>'.?,

186


(ii) Let X = T - {f},the vectorial difference (see Chapter I, Definition 2.7). Clearly the convexity of T implies that of X. On the other hand, X contains no semipositive vectors and, a fortiori, no positive vectors, since i is efficient in T. Therefore, by Chapter 1, Theorem 3.5, there is a semipositive vector p 2 0 such that p‘y 5 0 for all y E X . Putting y = x - 2 for all x E Tin p’y 5 0 yields p’2 2 p‘x. This p therefore serves as a desired price vector, Q.E.D. Remark 1. In Theorem 12.I(ii), the positivity of p is not ensured. The process .4= (.Ci), -4,= I , 2 , = - 1 , R, = 0 is an efficient process in T = {x 1 x = (xi),-yl + s2+ .y3, 5 0, x3 2 0 ) . For, if x = (xi)>= i in T, we have xI 2 I , x2 2 - 1, .Y, >= 0, so that 0 ix1 x2 x32 0, reduces to p 2 x, 2 p 3 2 0. Since s, > 0 is arbitrary, we must have p , = 0. Therefore, no positive price vector exists under which the profit is maximized at i over T. In effect, p’ = ( I , I , 0) is a desired price vector. In spite of Remark I , we can associate positive price vectors to efficient processes in special situations.

+ +

THEOREM 12.2. Let T be u teclinology set iti tlieJlon. iiersion. Then, i1.e liai>e: (i) If T is CI polylieclral c o i i w x colic mid 0 is eficient in T,tlicre is p > 0 siich thut 0 = p’0 2 p’x,for ull x E T. (ii) The S N I I I ~ asserfioii as iii (i) is true prvriclclcd that T is a closed C O I ~ I Y ~ . Y mic. ( i i i ) I f T is o polyliedrul coi1w.v corie mid .4 is efficient iii T, tlierc is p > 0 siich tlint 0 = p’.? 2 p’.~,fbrall x E T. Proof. ( i ) Since T is a polyhedral convex cone, we have T = {Air I LI 2 01, where A is a suitable matrix. The assumption that 0 is efficient i n Tamounts to saying that Au 2 0, u >= 0 imply Au = 0. We appeal to Tucker’s theorem (Chapter I, Theorem 3.7(ii)), which ensures the existence of vectors i?,p, q satisfying A’[.’

+ 4 = 0,

p

2 0, 4 2 0,

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187

The first inequality decomposes to A v 2 0, v 2 0, which, by assumption, yield Av = 0. This, combined with the last inequality, implies p > 0. On the other hand, A'p + q = 0, q 2 0 give p ' A = -q' 5 0. Whence 0 2 p ' A u for any u 2 0, i.e., 0 2 p'x for any x E T under this p > 0. (ii) By assumption, T, which is a closed convex cone, contains no semipositive vector. Hence, by Chapter I, Theorem 3.6, - T * contains a positive p > 0. Then, since - p E T*, we have -p'x 2 0 for x E T. Whence 0 2 p'x for any x E T, and this p > 0 can serve as a desired price vector. (iii) For an efficient 2 in T, let K be the convex cone generated by -R and T. Since Tis polyhedral, K is also polyhedral. It will be shown that 0 is efficient in K. Suppose, on the contrary, that K contains a semipositive vector u 2 0. By definition, u is of the form u = - A i + x for some A 2 0, x E T. This expression of u 2 0 can be rearranged to X +2 -- - 2 + -

l + A

11

l + A

2 2.

The left-hand term of ( I ) belongs to T, since x, iE T, 1 + A > 0, and T is a convex cone. This contradicts the assumption that 2 is efficient in T. Now that K is shown to contain no semipositive vector, there is a positive p , by (i) as applied to K , such that 0 2 p'y fory E K ; in particular, 0 2 p'(x - 9) for x E T. This implies that at this p > 0 we have p ' i 2 p ' x for all

XE

T.

The proof is therefore complete. Remark 2. We may note that in Theorem 12.2, Condition (i) can be reduced to (ii) on the basis of the fact that a polyhedral convex cone is always closed in R". Remark 3. The efficiency of 0 in T as in Theorem 12.2 amounts to saying that the land of Cockaigne is impossible in T. At a glance, the above two theorems seem to pertain only to technology sets in the flow version. In effect, however, they can be applied to technology sets in the stock version, for the following reason. Let a technology set T be given in the stock version, so that T c R" x R", meaning x E R", y E R" for (x,y) E T. We define the mapping 4 : R" x R" 4 R"'" by

Here (b is a one-to-one linear mapping, which is naturally continuous in the both direct ions. 4 ( T ) may conveniently be regarded as an alternative representation of T in the flow formulation. Then the following results are obvious:

188

I V . EFFICIENT ALLOCATION AND GROWTH

(2, j ) is efficient in T if and only if #((a,j ) ) is efficient in # ( T ) . (p) T is convex if and only if # ( T ) is convex. (a)

(y) T is a polyhedral convex cone if and only if 4 ( T ) is a polyhedral convex cone. (6) T is a closed convex cone if and only if # ( T ) is a closed convex cone.

On the basis of (a)-(6), Theorems 12.1 and 12.2 can be converted to the corresponding propositions on technology sets in the stock version. Explicitly we have

THEOREM 12.3. Let T be a technology set in the stock version in R" x R . Then, 1i.e ham (i) IL under two positive price vectors p > 0, 9 > 0, the .former evaluating the input x and the latter evaluating the output y , a process ( 2 , j ) maximizes the profit q'y - p ' x over all (x, y ) E T, ( 2 , j ) is an eficient process. (ii) Suppose that T is convex. If(.?, j ) is an efJicient process, there are nonnegative price vectors p 2 0, q >= 0 such that at least one of them is sernipositive and the profit q'y - p'x is maximized at (x, y ) = ( 2 , j ) over T.

THEOREM 12.4. Let T be a technology set in the stock oersion. Then \iv have (i) If T is a polyhedral convex cone and (0,O) is eficient in T, there are p > 0, q > 0 such that 0 = q'0 - p'0 2 9'y - p'x,for all (x,y ) E T. (ii) The same assertion as in (i) is true provided that T is a closed convex cone.

(iii) I f T is a polyhedral comex cone and (2,j ) is 0, q > 0 such that 0 = q ' j - p'2 q'y - p ' x for all (x, y ) E T. The formulation of production in terms of a technology set as discussed above is rather modern. Alternatively, classical mathematical economists worked on the basis of the concept of production functions. Therefore it may be in order to say a few words to clarify the concept of a classicalproduction firnction and its relation to the modern formulation. Suppose that a technology set T, say, in the stock version, is closed and convex. An efficient process (x, y ) in T is a boundary point of T. Otherwise it would be an interior point, so that (x, y + v) E T for some positive v with sufficiently small norm. Then, (x, y v ) is more efficient than (x, y ) , contradicting the efficiency of (x, y ) . The set of all efficient processes therefore composes a certain portion of the boundary of T,which generally is a hypersurface in R""", where n7 and n are the numbers of input goods and products, respectively . Let F(x,, x2 . . . , x;, y,, Y2 > . . YrJ = 0 (2)

+

2

.?

912.


I89

be an equation representing the hypersurface. Equation (2), if further reinforced by some well-behaved properties such as differentiability up to a desired order, is a typical classical (or, more correctly, neoclassical) production function. Assuming n = 1 and solving (2) for yl, we obtain a most familiar production function of the neoclassical type : y1

=f(x1,

x2 >

. . ., x,).

Classical mathematical economists usually assumed the concavity of the functionf. This assumption is compatible with the convexity of T. In fact, if y , =f(xl, . . . , x,), u, =f(ul, . . . , urn),then ( ( x i ) ,yl), ((ui), vl) E T, so that the convexity of Timplies ((axi+ flui), ay, + f l u , ) E Tfor a 2 0, p 2 0, a + p = 1. Since ((axi + p u i ) , f ( a x , + pu,, . . . , ax, + flu,,)) is by definition an efficient pu,,) 2 ay, + flul. process of T, we havef(ax, + pul, . . . , ax,

+

912.4. A Nonsubstitution Theorem

As was discussed in the foregoing section, there is an intimate connection between the choice of an efficient process and the maximization of profit under a price system. The managements of firms in a capitalist economy and even those of production units in a matured planned economy choose efficient processes by seeking their maximum profits in a prevailing price situation. The efficient processes actually employed therefore depend more or less on the price situation, and switching from one process to another, responding to the change in the price situation, may generally be noticed. For instance, one observes such a switching phenomenon in a linear programming model or a von Neumann model, and hence the variability of input-output coefficients, within a firm or a production sector, resulting from the combined operation of several alternative processes whose input-output coefficients are fixed but may differ from each other. In this respect, the Leontief system is based on the seemingly very stringent premise that essentially one single process is available in each sector. The room for choice open to the management in each sector is confined to determining a level at which the process is operated. This premise of the constancy of input-output coefficients in each sector would be unjustifiable, if it were not for its rationale. The variability of input-output coefficients implies the substitutability among alternative processes, whereas their constancy in the above sense implies the absence of alternative processes, and therefore nonsubstitutability. There are several kinds of rationale justifying the nonsubstitutability premised in the Leontief system; these are referred to as nonsubstitution theorems.” Here we will present a nonsubstitution theorem advanced by Samuelson (1951), Koopmans (1951), Arrow(1951), and Georgescu-Roegen (1951,1966), which is closely related to the results of the preceding sections. “

190

1V. EFFlClENT ALLOCATION AND GROWTH

According to them, even if a set of alternative processes is available in each sector of the Leontief system, one suitable single process can be picked from among processes available in each of n sectors in such a way that in effect the combined operation of these n fixed processes, one for each sector, suffices to ensure all the efficient input-output configurations resulting from all the processes available. Hence the exclusive use of this particular process, meaning the constancy of input-output coefficients, in each sector neither brings about any loss of production efficiency nor impedes a full maximization of profit under the condition that the final net output of every good should be nonnegative. This peculiar situation is due to two important features of the Leontief system : nonjoint production, and the presence of one single indispensable factor of production, namely, labor. To state the problem in a well-posed form, we first recall the basic assumptions (a), (B) in Chapter 11, $6.2. I n addition to them, we assume that labor is the sole indispensable factor of production. Let T j be a production technology set available in the jth sector, which consists of processes ( I b j , xj, y’) in the stock version. Here,

,Ij

=a

nonnegative scalar, representing a nonnegative amount of labor input; .uyi= an n-dimensional (gross) nonnegative input vector; y J = an n-dimensional (gross) nonnegative output vector whose components satisfy (nonjoint production). (a) y j = 0 (i # . j ) Further assumptions on T j are as follows: (b) Ti is a closed convex cone. x j , y j ) E T i , R j = 0 imply y j = 0. That is, labor is indispensable. (c) The technology set of the overall national economy is T = I T i , and

cJ=

is a rearranged representation of the productive portion of T. Since I., y - x stand for the corresponding labor input and net output vector, the evaluation of efficiency in T , can rightly be made on the basis of the semiorder defined by

between certain (I., y

-

x), (it, u

-

u) E T , .

THEOREM 12.5 (A Nonsubstitution Theorem). Jf T isproductiue in the sense thut T , contuins a process having a positiile net output vector, then there are special processes ( I , .Pi, 3’) E Ti ( j = 1, . . . , n) fulfilling the fol/oirYng conditions:

$12.

191


The set T+' of all eficient ( A , y - x) in T+ is completely described by C;= O j (1, j J- i j ) , ivhen n parameters O j ( j = 1, . . . , n ) range all nonnegative numbers. (ii) For any nonnegative jinal demand vector d there is a nonnegative scalar I such that (A, d ) E T + e . Proof. The essential idea of the following proof is due to Arrow (1951). To begin with, it will be shown that for any (o, c) E T+ there is an efficient (X, j - ,t) in T , satisfying 2 5 w , j - ,t 2 c. To this end, we first show that the set (i)

T+(w,C )

=

((2, Y - X)

I I 5 0, Y - x 2 C, (1,X, Y ) E T )

is a nonempty compact set. Its nonemptiness is obvious from the fact that (0, c ) E T+(w,c). Also, clearly, T+ I> T+(w,c). By Chapter I, Theorem 1.5, it suffices to see that T+(w,c) is bounded and closed. If T+(w,c ) were unbounded, there would be a sequence {(A,, y' - x')} in T+(w,c ) such that limlly"l1 = +a.Decomposing each term of the sequence to n

a,

=

n

1i j V ,

yv =

n

1

xV=

yjv,

j= I

j = 1

C xj',

(1)

j = 1

we obtain sequences { ( A j , , xj", y")} in T j . From y' 2 x', A j , 2 0, xi" 2 0, yj" 2 0, and Eqs. (1) follow

O s A j , 5 w ( j = 1 , ..., n),

IIXjVIl, llY"II 5 Ilu"ll =

V Y

(2)

(i= 1 , . ' .

3

n).

(3)

Whence { ( A j J q , , x j y / q y yJ"/qv)} , is a bounded sequence in T j for each j , because Ti is a cone. Hence, without loss of generality, these n bounded sequences may be assumed t o converge to ( p j , ui, d ) , respectively. Note that ( p j , d , d)E T j by the closedness of T j . But, 0 5 i j ,5 o,lim q y = + GO imply p j = 0 ( j = I , . . . , n), so that ui = 0 ( j = I , . . . , n ) by the indispensability of labor in T j . Therefore we must have

1 " . . lim - C ( A j v , x J v ,y") = v++m

qv j

=

~

n

C(pj, j = I

which implies 1 " Iim yj' v+ + m qyj = I

= 0.

uj,

d)

192

I V . EFFICIENT ALLOCATION AND GROWTH

This contradicts

1

‘lv

i;Y’”//

=

1 (v

=

I , 2, .. .),

j = I

proving the boundedness. Next let us prove the closedness of T+(o,c). If {(Av, y” - x”)} is a sequence in T + ( w ,c) converging to (A, rl), the sequences { ( A j v , xjv,yj”)} in Tiobtained by the corresponding decomposition (1) are bounded from the boundedness of { y ” } shown above in conjunction with o 2 Av 2 0, y’ 2 yj” 2 0, y” 2 x” 2 x j ” 2 0. Whence we may assume, without loss of generality, that they are originally convergent, with their respective limits ( A j , xj, yj). Then, we have

(A, cl)

=

lim (A,, y ” - x”) ”++%

n

= liin v-

=

+ %j

1 ( a j v ,y j v - x’”) = I

i:(A’, y’

- 2).

j= 1

We now observe that ( A j , xi, y j ) E T j by the closedness of Ti,Moreover, / I v s ~ , ~ ” - . ~ ” ~ c ( 1 ~ ...) = 1 e, n2 s, u r e C A j s w , C ( y j - x x j ) 2 c . Whence (I., d ) = ( I L jjij , - x j ) E T + ( w ,c), proving the closedness of T+(w,c). Next choose an arbitrary positive price vector p > 0. Then, by its continuity, the function p’(y - x) - 2 is maximized a t some (1,j - X) on the compact set T,(cu, c.). (2, j - X) is efficient in T,. In fact, if

for some (A,y - .u) E T , , we have A 5 2 5 w , y - x 2 j - -X 2 c, so that (IL, - .I-) E T,(w, Then, by a simple reasoning similar to the proof of Theorem l2.l(i), we have J’ - x = j- X,A = 2. This proves that (2, j -X) is efficient in T , . We note an immediate consequence of this result: O(2, j - 2 ) is also efficient i n T , for any 0 2 0. I n fact, suppose that Ox, J’ - .Y 2 O ( j - X) for (i J’, - s)E T+ . If 0 > 0, these relations reduce to A/O 5 2, ( y - x)/O 2 - ?i with ( l / O ) ( i , J’ - x) E T , . Then i,/O = 1, ( y - x)/O = j --X by the efficiency of (17, -F - Y), so that (L,J’ - .u) = a(], j - Z).O n the other hand, if 0 = 0. the corresponding relations reduce t o 0 5 1. 5 0, y - x 2 0. Hence 1, = 0, so that i.j = O( j = I , . . . , H ) in the decomposition (A, X, y ) = (i .,>. . y i i,j),( 21. ’ . Y’, E T i . Whence j-j = 0 ( , j = I , . . . , /7) by the indispensability o f labor in T j . Accordingly, y = y J = 0. Also, from 0 = y 2 .S 2 0, it follows that .Y = 0. Therefore (A,y - x) = (0, 0) = O(2, j - X). In both cases 0 > 0 and 0 = 0, the efficiency of O(2, j - X) is seen. ( 5 ) .

As

2

3

.

j s j )

$12.


193

Now we recall the assumed existence of a process ( p , e ) E T , with e > 0. By the results in the foregoing steps of the proof, T + ( p ,e) contains a process (2, j- X), which, as well as all its nonnegative multiples O(2, j - ,?), is efficient in T , . Since j- X 2 e > 0, we must have 2 > 0 by the indispensability of labor in T j ( j = I , . . . , n). We can therefore normalize these processes to an efficient process with unit labor input, which, without loss of generality, will be denoted by (1, j- X) with the same notations X, j. In the corresponding decomposition

(4) we must have J j > 0 ( j = 1, . . . , n). For, if 2, = 0 for some k , the corresponding jk would also vanish, while the other sectors produce no amount of the kth good, i.e., j k j= 0 ( j # k) by the nonjoint production assumption (a). Whence the kth component of jwould vanish, which contradicts j- ?, > 0, x 2 0. Consequently, we can also normalize ( I j , X’, j j ) to a process with unit labor input in T j , (I,j?j,jj)

( j = 1, ...) n),

(5)

by putting

9= f J / J i ,

j9 = j j / X j ( j = I , . . . , n).

The final step of the proof is to see that the desired set of processes. To this end, consider the

D

=

17 17

(6)

processes in (5) form a 17 matrix

x

d I.J . = j .1 J- , j : . (d..) tJ >

In view of the nonjoint production assumption (a), the matrix D satisfies the basic assumption d i j 5 0 (i # , j ) in (3) in Chapter 11, 46.2. Moreover, the positive vector I = ( A j ) fulfills

DI

=j

-? ,

> 0.

Hence, by Chapter 11, Theorem 6.3, D is invertible and D-’ 2 0. Let f ( s ) be the sum of the components of a vector s. Then the assertion, expressed in terms of D, is that T , “ = { ( , f ( s ) , Ds) I s 2 0 ) . Suppose (L,y - x) E T + “ . Since J - .Y 2 0, it can be put in the form y - x = Ds for some s 2 0. T , 3 (,f’(s), Ds) so that by the efficiency of (IL,y - .Y) we have 1, S,.f’(s). It will be shown that i =,f(s). To this end, suppose, on the contrary, that j. 0, such a situation can be effected by replacing (A,y - .Y) by (A, - s ) / f ( s ) , which belongs to T , . Next, since DI = j- X > 0, we can let

c(c()=(I

-~(cc)DI+ctDs>O

194

IV. EFFICIENT ALLOCATION AND GROWTH

still hold for a negative number a < 0 sufficiently close to 0. Then, ~ ( a=) D(( 1 - sr)/ + as), with (1 - a)[+ c(s = D-'c(c() > 0. Moreover,f'(( 1 - a)l CIS) = (1 - a ) f ( / )+ a f ( s ) = I . Whence (1, ~ ( a )E) T , . We also recall that y - x = Ds,and therefore (A, Ds) E T , . Then, a simple computation will yield

+

(1 - p

+ AP, DI)= (1 - P)( 1, ~ ( a )+) p(A, Ds)E T+, +

where 1 > P = - a / ( l - a) > 0. Since 1 - p +I$ < I , the process(1 - p Ap, D/)is more efficient than ( I , D l ) = ( I , j - X), contradicting the efficiency of the latter. This proves 2 = f ( s ) . Conversely, it will be proved that (,f(s), Ds)E T,' for any s 2 0. By the opening part of the proof, T + ( , f ( s )Ds) , contains an efficient (A, y - x), which is of the form 2. = , f ( t ) , J' - .Y = Dt with a suitable t 2 0 by the result just proved above. Whence Df >= D s , f ( t ) s f ( s ) . Premultiplying the first relation by D-'2 0 entails t 2 s, which, combined with the second relation, yields t = s. This proves (,f(s), Ds) = ( f ( t ) , D t ) E T+e.The proof of (i) is thereby complete. (ii) Condition (ii) is obvious since D-' 2 0. In fact, if d 2 0 is any final demand vector, d can be produced by the process ( f ( s ) ,Ds) in T+",where s = D - ' d . as was to be shown.

$13. THEOREMS ON SELF-SUSTAINED

EFFICIENT GROWTH $13.1. Efficient Growth Paths

I n spite of their seemingly static appearance, the situations considered in $12 need not be static, but may admit dynamic interpretation. A good, if expli-

citly dated, can be further classified into several distinct goods. Thus, from the economic point of view, steel in the present year may be thought of as being distinct from steel a year hence. Therefore, if the components of production processes refer to goods with dating in the above sense, these processes represent dynamic input-output configurations extending over several periods. We shall henceforth pay explicit attention to dynamic situations. To establish definite results, we shall be concerned exclusively with the simplest dynamic situations where input-output configurations extend over just two consecutive periods. Of these situations, $13 deals with one typical situation in which the economy is self-sustained in such a sense that neither inflow of goods to the system nor outflow of goods from the system is present, while every good reproduces itself within the system. $14 discusses another typical situation where there is an indispensable factor of production, namely labor, which is treated as an exogenous inflow into the system.

$13.

SELF-SUSTAINED EFFICIENTGROWTH

I95

Consider a self-sustained economy producing and consuming n (undated) categories of goods. It is assumed that all these goods are reproducible and emerge as both inputs and outputs. Thus, an input vector x and an output vectory, which altogether compose a process (x,y ) in the stock version, are of the same dimension. Their ith components commonly refer to the ith (undated) good. The process (x,y) means that the consumption of x as inputs in the present year yields y as outputs in thefollotising year. The economy is endowed with a technology set T, a set of technically possible processes of the above type. Throughout what follows, the two most basic assumptions on T a r e as follows: (A.l) T is a closed convex cone lying in the Cartesian product R+" x R+", R," being the nonnegative orthant of R". (A.2) (0,y ) E Timplies y = 0 (the impossibility of the land of Cockaigne). Moreover, we often need additional assumptions: (A.3) For any x 0, there is at least one y such that (x, J,) E T. That is, any possible input vector can be transformed to some output vector (transformability). (A.4) (x, y ) E T, u 2 x, y 2 u 2 0 imply (u, u)

E

T (free disposability).

(A.5) If (x, y ) E T, (u, c) E T, and if x and u are linearly independent, then for any positive numbers CI > 0, fl > 0, LY fl = 1, there is an output vector tv such that

+

(ax

+ pu, M') E T,

ay

+ flu < tv

(strict convexity).

Remark 1. In view of the economic implications of (A.l) and (A.2), which have been made clear in 512.2, it is not unreasonable to premise these two basic assumptions. Remark 2. Assumption (A.3), i.e., the transformability of any input configuration to some outputs, is a reflection of the flexibility in some sense of the underlying production technology, and is met in most situations. If free disposability (A.4) is assumed, there arises an extremely special case where (A.3) is prevalent; for, since T contains (0, 0), the process of inaction, T also contains all (x, 0) for x 2 0, so that (A.3) is trivially fulfilled. On the other hand, (A.3) need not imply the presence of costless disposal processes and therefore applies to somewhat more general situations. It is, however, still powerful enough, combined with ( A . l ) and (A.2), to enable us to prove the relevant propositions with which we are concerned in $13. Remark 3. (A.5) is one version of strict convexity and may be thought of as reflecting the presence ofexternal economies. Naturally, this is very stringent, as compared with the mere convexity of T corresponding to the mere absence

I96

1V. EFFICIENT ALLOCATION AND GROWTH

of external diseconomies. We shall assume (A.5) to establish certain decisive results that are useful in providing the so-called “turnpike theorem” in the simplest way. Suppose now that at a given period t there are x , ( t ) ( 2 0 ) units of the ith good in the economy as a whole (i = I , . . . , n). The nonnegative vector x ( t ) = ( s i ( t ) ) represents the state of the economy at period t , as far as the stock of goods is concerned. The economy reaches another state x(t + 1) at the next period t + 1 through a production activity available within the given technology set T if ( x ( t ) ,s ( t I ) ) E T. Then the economy can grow further from the state x ( t 1) to such a state x(t 2) at period t 2 that (x(t + I), x ( t + 2)) E T. In this way, a dynamic movement x ( t ) , x ( t + l), a(t + 2) . . . is generated, whose behavior we shall consider below. This dynaniic movement is self-sustained, since goods are reproduced while no inflow of goods from outside is present. With this i n mind, we have

+

+

+

+

DEFINITION 13.1. A jinite sequence of zvctors {.x(t)}:=, (or it,ritten simply us {x(t)}oN.forb r e r i t y ) is culleda~feasiblepath of (finite) horizon N ifthe pairs of its tbt’o consecictiiv terms belong to T, i.e.,

+

(.X(t), ~ ( t 1))

E

T ( t = 0, 1,

.. ., N

-

1).

DEFINITION 13.2. An infinite sequence of aectors fx(t)},“ is called a feasible path of injitiite horizon if

( x ( t ) ,x(t + I))

E T ( t = 0,

I , . . . , ad inf.).

A feasible path { x ( t ) f o Nrepresents a dynamic movement over N periods. It must be noted that since, for an input vector x 2 0, the set { y I (x, y ) E T } need not be a set consisting of one single element, there may generally be many feasible paths of a common horizon N starting at a common initial state. But the mode and degree of utilization of the growth potentiality embodied in Tare different from path to path, and certain paths on which the potentiality is fully utilized are to be chosen from among feasible paths. The comparison of feasible paths may be made in alternative fashions. A n evaluation of the “efficiency” of feasible paths is done by applying the efficiency comparison stated in $12.3 to the final state vectors x ( N ) . This method of evaluation has been standard, since it was rightly adopted by the initiators of the turnpike proposition (Samuelson, 1960; Dorfman et al., 1958). Let X ( x o , N ) be the set of all feasible paths of horizon N starting at a given initial state .yo 2 0. The nonemptiness of X ( x o , N ) is ensured through a repeated application of (A.3). DEFINITION 13.3. Let { ~ ( t ) ) , {~ ~, ( t ) }E ,X(xo, ~ N ) . Then, ( ~ ( t ) ) ,is, ~ said to be more efficient thun {y(t)},N i f x ( N ) 2 y ( N ) .

$13.

SELF-SUSTAINED EFFICIENT GROWTH

197

~ to be eficient if there is no more DEFINITION 13.4. { ~ ( t ) )is, said ejicient path in X ( x o , N ) than { x ( t ) } o N .

Corresponding to $12.3, a complete association of prices to efficient paths will be effected below. Lemma 13.1 serves for this and other purposes. Given a nonempty set S i n R+",let X(S, N ) be the set of all feasible paths of horizon N starting at some point in S. X(xo, N ) is a special case of X ( S , N ) where S = {x'}. A path { x ( t ) l o Ni n X ( S , N ) can be identified with the point having x ( t ) as the tth coordinate in the ( N + ])-fold Cartesian product R of R". For simplicity, we may therefore regard X ( S , N ) as a subset of R.

LEMMA 13.1. We have (i) I f S is nonempty and bounded, X(S, N ) is a nonempty bounded subset of R. (ii) I f S is nonempty and compact, X ( S , N ) is a nonempty compact subset of R. I f S is convex, X ( S , N ) is aIso convex. (iii) Proof. Nonemptiness in both (i) and (ii) is obvious from (A.3). Also, convexity in (iii) immediately follows from Definition 13.1 and the convexity of T assumed in (A.1). It remains to prove boundedness in (i) and compactness in (ii). Suppose that X ( S , N ) were unbounded in (i), and let k be the smallest integer among such integers I between 0 and N that x(/) is unbounded for { x ( t ) } o Nranging over X ( S , N ) . k > 0, since S is assumed to be bounded. Let {x"(t))oN be a sequence of points in X ( S , N ) with lim Ilx"(k)ll

=

(1)

+a.

V-r+CC

Observing that (x"(k - I), x"(k)) E T by feasibility, and in view of the cone property of T, we have (x"(k- l ) / I l ~ " ( k ) l~~", ( k ) / ~ ~ . ~E" T ( k )( ~v ~=) 1, 2, . . .).

(2)

As x"(k - I ) is bounded by the definition of k , ( I ) implies lim x"(k - l ) / 1\2'(k)\I = 0. On the other hand, x " ( k ) / ~ ~ x " (has k ) ~unit ~ norm so that, without loss of generality, it may be assumed to converge to a limit J' with lly[l = 1. Whence (2) becomes in the limit (0, y) E T by the closedness assumed in (A.1). Hence J* must be 0 by (A.2), which contradicts llyll = I . This proves the boundedness of X ( S , N ) in (i). Let us proceed to compactness in (ii). By Chapter I, Theorem 1.5, it suffices to show the boundedness and closedness of X ( S , N ) ; but boundedness is immediately implied by (ii), since S as a compact set is bounded. To see closedness finally, we consider N + 1 continuous mappings 40({X(m3N) =

40) : R

+

R",

+ t ( { ~ ( t ) } o N= ) ( x ( t - l), s ( t ) ) : Q-+ R" x R"

( t = 1, . . ., N ) .

198

1V. EFFICIENT ALLOCATION A N D GROWTH

Then, clearly

I f it is shown that S is closed in R", and T is closed in R" x R", the N + 1 inverse image sets 4;'(S), 4L1(T) are all closed in R by the continuity of the N + 1 mappings, so that X ( S , N ) as their intersection (3) is also closed in R. But it is obvious that S a s a compact set is closed in R". On the othcf-iiand, T i s closed in R" x R", since it is by assumption (A.l) closed in R," x R+", which . is closed in R" x R". This completes the proof. N o w that Lemma 13.1 is established, it is easy to prove

THEOREM 13.1. Let X ( s o , N ) be as before. Then I I hare ~ If {x(t))," is an eficient path in X ( s o , N ) , tlien there is a semiposrtiiv price vector p 2 0 sirch that

(I)

p ' s ( N ) 2 p'y(N)./br anj. {j*(r)JONin X(.xo, N ) .

(ii) Gicen nriy semipositi1.v price uector p 2 0, there is an eficicnt path N X(xo, N ) . {s(t)>oN that maximizes p ' y ( ~ o)w r a / / { y ( t ) ) O in which Proof: (i) By Lemma 13. I(iii), as applied to X ( S , N ) for S = {so}, is convex, X ( x o , N ) is a convex set in 0. Hence M = { y ( N )- x ( N ) I {y(r))," E X ( s o , N ) } is a convex set in R". By the efficiency of { x ( t ) } o NM, has only the origin as common points with R,". Therefore, by Chapter I, Theorem 3.5, M can be separated by a hyperplane with a semipositive normal p 2 0 from R,", so that p ' z 5 0 for any z E M . This implies p ' x ( N ) 2 p ' y ( N ) , for all { J , ( ? ) } , ~i n X(.Y', N ) , as was to be shown. (ii) By Lemma l3.l(ii), as applied to X ( S , N ) for S = {x0}, which is compact, X(so, N ) is a compact set in R. Accordingly, the continuous function . f ( ( ~ ( r ) ) ~ "=) p ' j ( N ) takes on a maximum 0 over X(xo, N ) . Next consider

E = ( b ( t ) ) o " IP'.Y(N)

=

0, {J'(t)}oN

E

X(xo,N ) ) ,

the set of all paths maximizingp'y(N). E is closed in X(xo, N)and is therefore compact. Choose a positive price vector q > 0. Then the continuous function g({y(r)}oN)= q ' y ( N ) is maximized at a path { x ( t ) } o Nover E. This path is efficient. I n fact, if y ( N ) 2 x ( N ) for some { y ( t ) } O EN X(xo, N ) , then q > 0 implies q ' y ( N ) > q ' x ( N ) . On the other hand, y ( N ) 2 x ( N ) , p 2 0 imply 0 2 p ' j j N ) 2 p'.u(N) = 0, so that p ' y ( N ) = 0. Whence {y(t))," E E, which means q ' x ( N ) 2 q'j(N). This contradicts q ' j ? ( N )> q ' x ( N ) . Thus {.u(t))oNis efficient. Q.E.D. Remark 4. Theorem 13.1 in particular implies the existence of efficient paths in X(.u0, N ) .

$1 3.

199


013.2. Maximal Balanced-Growth Paths We have already considered balanced growth several times in Chapters I 1 and II1. Naturally, the notion of balanced growth is also conceivable in the present setting.

DEFINITION 13.5. A (nontricial) feasible path {x(r )}om of illfinite horizon is called a balanced-growth path if x ( t i- 1) = p x ( t )

( t = 0, 1,

. . ., ad inf.),

where p is a nonnegatice constant. p is termed the gron.tliJuctor, ~ t h i l ep called the rate ojgroit.th.

-

1 is

Clearly, a balanced-growth path can be put in the form x ( t ) = y'x,

and p,

.Y

11)

x = x(0) 2 0 ,

solve the eigenvalue problem (x, p x ) E T.

We are concerned with the existence of balanced-growth paths, particularly that of a maximal balanced-growth path, which is defined as a balancedgrowth path having a maximum growth factor over all balanced-growth paths.

THEOREM 13.2. Assume (A. I), (A.2), and (A.3). Then, there is a tnaxima1 balanced-groirsth path. Pvoof. For convenience, a process (x,y ) in T will be called a nontrivial process if x # 0. Nontrivial processes surely exist by (A.3). Let . over all i such that p(x, y ) = min Y1 xi

xi > 0

for a nontrivial process (x, y ) . p(x, y ) will be called the growth factor of the process (x, y ) . It will be shown that p ( x , y ) takes on a maximum over all nontrivial processes. To begin with, we note that p(ctx, ay) = p(x, y ) for any

c(

> 0.

(3)

This being seen, let /z = sup p(x, y ) over all nontrivial processes.

For convenience, a nontrivial process (x, y ) will be referred to as a normalized process if the components of x sum up to unity. Then, in view of homogeneity (3), we may confine attention to the growth factors of normalized processes to define I.. Thus, there is, by definition, a sequence of normalized processes {(x', y " ) } such that p(xv,y " ) = p , + ias v + + a.

200


Next let S = {x" I 11 = I , 2, . . .}. Then S is bounded, since x" belong to the standard simplex n

i= 1

Whence, Lemma 13. I(i) as applied to X ( S , 1) ensures that {(x', y " ) } is bounded. Clearly, we have by definition y" 2 p v x V ( v

=

I , 2, . . .).

(4)

Summing up the components on both sides of (4), we see

cl=l yi 2 p ,

( v = I , 2, . . .), which implies that { p , } is also bounded. Therefore, we may assume, without loss of generality, that {x"}, { y " } , { p , } are convergent with their respective limits .Y, y , %, where x E P,. Whence (4) becomes i n the limit

(5)

x20,

yl/lx,

whereas, by the closedness of T assumed in (A-I),

(6)

(x, Y ) 6 T.

We have thereby shown the finiteness of 1 as well as the existence of a nontrivial process satisfying (5). But equality need not automatically hold in (5). Thus the next step of the proof is to see that equality holds in (5) for some special nontrivial process. To this end, further let

K

= {(x, y ) I (x, y ) E

T, y 2 A.Y,

x E f,).

By the result just proved, K is nonempty. It will also be shown that K is compact and convex. To this end, define the set Q = ((x, y ) J y>= Ax} in R" x R . Then K = X ( P , , I ) n Q. (7)

By Lemma 13.l(ii) and (iii), X ( f , , , 1) is compact and convex, whereas Q is obviously closed and convex in R" x R". Whence K as their intersection (7) is a compact convex set in R" x R", which is of the same linear and topological structure as R'". The next procedure is to define a mapping from K into itself. For this purpose, put 1

+ ic= y i I

Since for any

(.Y,y) E

K we have

$13.


20 1

Eq. (8) can be put in the form

Now ( x , y ) E T and the cone property of T assumed in (A. 1) imply

1+CYi

1+CYi

i=I

i= 1

On the other hand, there is by (A.3) some u for which (u, u) E T.

(1 1)

Hence, Eqs. (9)-( 1 1) combined with the convex cone property of T assumed in (A.l) yield

On the other hand, in view of y 2 Ax, we have

Therefore, Eqs. (12)-( 14) imply that for any (x, y ) E K there is a u such that

This proves that @(&

Y ) = {(4(x,Y ) , t

4 I(d(x,Y ) , ,\-> E K l

(1 5 )

is a nonempty subset of K for any (x,y ) E K. We have thereby obtained a set-valued mapping @ : K - t 2K defined by (l5), which assigns to (x,y ) E K a nonempty subset @(x, y ) of K. To apply Chapter I, Theorem 4.4 (Kakutani's fixed-point theorem) to this mapping, let us see if its conditions are fulfilled. We know already that K is a nonempty compact convex subset of R" x R" (homeomorphic to R'"), and that @(x, y ) # @ for each (x, y ) E K. In addition to these conditions, it is required that (a) @(x, y) be convex for each (s,1,)E K ;

202


(b) @ is a closed mapping. Condition (a) immediately follows from the convexity of K assumed in (A. I). T o see (b), let ((x”,y’)), { ( z ’ ,it,’)} be a n y two convergent sequences in K with their respective limits (x,y), ( z , IV) satisfying

(z”, 1tXY) E @(.Y”, y ” )

(1, =

I , 2, . ..).

(16)

Equation (16) decomposes into two relations:

Since (i, clearly is a continuous function, Eq. (17) becomes in the limit

z

=

4 k v),

(19)

whereas (18) becomes in the limit, by the closedness of K ,

(4(x,y),

11,)

= (2, I!,) E

K.

This implies ( z , it’) E @(x,y ) , proving the closedness of @. Whence Chapter I, Theorem 4.4 can apply to 0,so that there is a fixed point (a, j )in K fulfilling ( 2 , j ) E 0(2,j),i.e., .??=

2+$

4(2,9)= 1

,

+ iC= IP i

j 2 22. From (20) it follows that

Now, 1ji= p ( 2 , j ) 5 2 by the maximum property C 9, >= 1; hence 9 = ,I.? i.e.,, (2, 22) E T, Q.E.D.

of E,, while (21) entails

A further simplification of the proof of Theorem 13.2 is possible under more stringent assumptions, as will be shown in the following t w o remarks. Remark 1. If free disposability (A.4) is assumed instead of (A.3), Eq. (5) immediately implies (s,2.r) E T, so that the remaining part of the proof is not needed. Remark 2. If strict convexity (A.5) is assumed in addition to (A.3), likewise Eq. (5) immediately implies y = As, provided 1, > 0. In fact, suppose J’ 2 2.u in ( 5 ) and let y = A.Y u, il 2 0. I f u and IL.uwere linearly dependent, so that u = % A x , 2 2 0, we would have J’ = j.(I %)s,which implies CY = 0 by the maximum property of A. This contradicts the supposition u 2 0. Hence u and 1.s are linearly independent. For this I I , there is o by (A.3) such that (u, L’) E T. O n the other hand, ( A x , I.J,) E T. Since I I and 1,x are linearly independent, there

+

+

$13.


203

is w by (A.5) and the cone property of T such that (Ax + u, w) E T, Ay f u < it'. This implies ( y , it') E T and p(y, it,) > A, contradicting the maximum property of L. Therefore y = Ax. DEFINITION 13.6. The ray generated by a uector x 2 0 satisjjing

(x,Ax) E T for the maximum gro\r.th factor 1 is termed a nzaximal balancedgro~~ h or a uon Neumann ray. ray,

i n the above definition, the latter nomenclature is used because von Neumann was the first to study balanced growth deeply (see Chapter I I , $9.2). Remark 3. The maximal balanced-growth ray is generally not unique; but uniqueness obtains under special assumptions. For example, strict convexity (AS) ensures the uniqueness of a maximal balanced-growth ray. In fact, let two semipositive vectors x, y generate maximal balanced-growth rays, so that (x, Ax), ( y , Ayj E T. If x were not proportional to y , there would be, by (A.5) and the cone property of T , an output vector H' such that (x + y , 11,) E T, L(x + y ) < w. This implies p(x + y , \ti) > A, contradicting the maximum property of 1. Hence x must be proportional to y , resulting in the uniqueness of a maximal balanced-growth ray, The maximum growth factor A, which is uniquely determined by the technology set T and may therefore be denoted by I.(T), need not be positive, but can possibly be zero. However, a complete characterization of the case A(T) = 0 is possible, corresponding to Chapter 11, Theorem 7.2(vi) and Chapter I l l , Theorem 10.3(iii). For a given technology set T let

P') = {(x(Oj, x(s)) I {x(t)1oSE X ( R + " ,s)), where X ( R + " ,sj stands for the set of all feasible paths of horizon s, as defined before. Then we have

THEOREM 13.3. I.(T) = 0 if and only if (x, y> E T'") implies y

= 0,

n being the number of goods.

Proof. The proof is completely similar to those for Chapter 11, Theorem 7.2(vi) and Chapter I l l , Theorem 10.3(iii). First, sufficiency can be seen immediately. In fact, since x ( f ) = %'xsatisfies(x(t), x ( t + 1)) E T ( t = 0, I , , . .) for a generator x of a von Neumann ray, (.Y, 1"x) belongs to T ' " ) .I f the condition is satisfied, we have L"x = 0, which implies A = 0 because I # 0. This proves 2 ( T ) = 0. To prove necessity, let M,v = ( i 1 y i = 0 for all (.Y, y) E T " ' } . Then, it suffices to show that M , = { I , 2, . . . , n } . To this end, we shall see that ( a ) M , # 0; ( p ) M , c M s + , ; M , = M,+, implies M , = { I , 2, . . . , n ] . Condition (CY)can be seen as follows. If M , for any i there would exist a

=a,

204


process ( x i ,y ' ) E T such that yi' > 0. Hence T as a convex cone contains the process x i , y'). Since y' > 0 by construction, y' 2 0 xi for a sufficiently small positive(). Moreover, x i # 0 by (A.2), so that xi, y') is a nontrivial process. Then p ( c x i , 2 y ' ) 2 fl > 0 contradicts A(T) = 0. This proves ( x ) . Condition (0)can be seen similarly. Take any i E M,. For any (x, y ) E T ( " ' ) there is by definition a feasible path of horizon s 1 {x(t)}i" fulfilling x(0) = x, x(s + I ) = y . Then, (x(l), y ) E T ' " ) ,so that y i = 0 for this i. Whence M,$c Finally, if M,T= M,,, 5 { I , 2, . . . , n}, similarly as in (a), there would exist (s,y ) E T ( " + ' such ) that y i > 0 for all i M,,, = M , . If we take a feasible path {x(t)}i" satisfying x(0) = X, x(s 1) = y , then ( x ( s ) ,y ) E T. This (x(s), y ) is nontrivial by (A.2), since y # 0. On the other hand, (.~(0),x(s)) E T(.'), so that xi(s)= 0 for all i E M , = M,,,. Whence y i > 0 for any i such that xi(s)> 0. This implies that y 2 8x(s) for a sufficiently small positive 0. Hence p(x(s),y ) 2 0 > A(T) = 0, again contradicting the maximum property of i ( T ) . This proves (p). Clearly ( x ) , (j?) imply M , = ( 1 , 2, ..., n } , Q.E.D.

(1 2

1

1

1

(c

+

+

In the light of Theorem 13.3, we are sure that A(T) > 0 in most cases.

Remark 4. A(T) > 0 for n 2 2 if (A.3) and strict convexity hold. For if we take any two linearly independent semipositive vectorsx, y , to which output vectors u, v are associated by (A.3) to satisfy (x, u), ( y , u) E T, respectively, there is another output vector w,by (A.5) and the cone property of T, such that (x y , w) E T, u u < w . Then, clearly the growth factor of the nontrivial process (x y , is positive, implying A(T) > 0 Although the case A(T) > 0 is completely characterized by Theorem 13.3, no meaningful necessary and sufficient condition to ensure the positivity of the generator of a maxima1 balanced-growth ray is available. But a structural property that suffices to ensure the desired positivity is immediately suggested upon reviewing the indecomposability concept for nonnegative matrices. Consider the technology set T = {(x, y ) I x 1 0, y 1 0 , x 1 A y } , where A is an indecomposable nonnegative square matrix of order n. For each pair (i, j ) , by Chapter 11, Lemma 8.2(ii), there is a positive integer v ( i , , j ) for which the ( i , . j ) element of A"('.j' is positive. Let k = max v(i, j ) for all i , j . If the horizon N of a feasible path { ~ ( t ) } exceeds , ~ k , and t h e j t h component of x ( N ) is positive, then x i ( N - v ( i , j ) ) > 0, since x ( N - v ( i , , j ) ) 2 A'('*j)x ( N ) . This consideration suggests

+

+

+

\$I)

DEFINITION 13.7. A tecl?nology set T is said to he indecomposable if there is a positive integer k such that uriy,feasible path { ~ ( t ) }\tihose , ~ horizon N exceeds k lias the property that if X ( N ) 2 0, ,for any i, the ith component of x ( t ) is positive at some t , N > t 2 0.

$13.

205


THEOREM 13.4. Any generator 2 of a maximal balanced-groii.th ray is a positive vector under indecomposability if1 = I ( T ) > 0. Proof. If 2, = 0 for some i, the feasible path {i'i}oN for N > k would satisfy A ' i i = 0 ( N > t 2 0), t N 22 0, contrary to indecomposability, Q.E.D. Next we consider the important implication of the maximum property of 1(T) in the evaluation of profit. If p is the interest factor (= 1 the rate of interest) and p 2 0 is a price vector, then p'y - pp'x represents the profit (revenue minus cost including interest charges) that results from the operation of a process ( x , y ) in T. In this respect, the case where p = A(T) deserves special attention and is relevant to the discussion of a turnpike theorem in $13.3.

+

THEOREM 13.5. There is a semipositive price cector p associated iiYth

1 = A(T) such that p'y

-

1p'x 5 0 for any (x, y ) E T.

(22)

Proof. Consider the set M = { y - / z x I ( x , y )T~} in R". M is convex because T is convex by (A.1). M contains no positive vector. For otherwise y > Ax for some (x, y ) E T, so that the growth factor of this process would be greater than A, contrary to the maximum property of 1. Then, by Chapter I, Theorem 3.5, M can be separated from R," by a hyperplane 0 i f n 2 2, n being the number of goods. (iii) At most m e component of p can be zero: if the ith ~ o m p o n e i ~oft p vanishes, all the components of ivanish except,for the ith one.

206


Pvoof. (i) If .Y 2 0 generates a maximal balanced-growth ray, it satisfies (x, As) E T, so that p ’ ( 2 x ) - I p ‘ x = 0. Whence, by (*), x must be proportional

to .?. This proves the desired uniqueness. (ii) Take an x 2 0 that is not proportional to 2. This is possible because I I 2 2. Then there is, by (A.3), an output vector y satisfying (x, y ) E T. Moreover, by unique profitability (*), we must havep’y - ig’x < 0 for this process. Whence 0 p’y < ip’x, implying A > 0. (iii) Let the ith component of p be zero. Take a vector x whose components satisfy .yi > 0, xi = 0 ( , j # i). There is, by (A.3), an output vector y such that (x, y ) E T for this x. (x, y) naturally fulfills (22). But, since p ’ x = 0 for this .\-, Eq. (22) reduces to 0 5 p’y = p’y - Ap‘x 5 0. Whence x must be proportional to f,implying that .Pi> 0, .Gj = 0 ( , j # i). Now, if there were another vanishing component ofp, say, the kth one (k # i ) , the components of .? would also satisfy f k> 0, .4,= 0 ( j # k) for the same reason as above, yielding a contradiction, Q.E.D. Unique profitability (*) lets a simple turnpike theorem hold, as will be seen in the following section. One typical structural property ensuring (*) is strict convexity (A.5)

THEOREM 13.7. If strict conr3exity (A.5) is ussumed, unique projitability (*) liolils for ally ron Neirnlurin price vector p . Proof. Choose a fixed generator 2 of a maximal balanced-growth ray once and for all. If .Y is not proportional to 2 for a process (x, y ) E T, strict convexity (A.5) implies the existence of an output vector 11. satisfying

Whence by (22) we have

p‘w

, x + i

-

s

A p - 0, 2

while preniiiltiplying the second relation in (23) by p’ 2 0 gives P‘

4’ ~

+ 2.G < p‘rv. 2

Then, combining (23) with (24), we have p‘y - i p ‘ s < 0, Q.E.D.

513.3. A Turnpike Theorem Starting at any nonzero point ,i- on a maximal balanced-growth ray, the economy can enjoy the balanced growth .r(t) = 2‘2, which steps on the

$13.


207

feasible path {Af2}om. But, since there may be, in general, other feasible paths starting at the same initial position 2, the preference of the balanced-growth path to other paths would be questionable if there were no justification. The path {Af2}oN, consisting of the first N + 1 consecutive terms of {A'2}o.onaturally belongs to X ( i , N ) , the set of all feasible paths of horizon N and starting at 2. If p is a von Neuniann price vector, we have p'y(t + 1) Ap'y(t) 5 0, y(0) = 2 ( t = 0, 1, . . . , N - 1) for any { ~ ( t ) }E ,X~( 2 , N ) . These recursive relations can easily be combined to obtain

The resulting inequality shows that (Af2}," maximizes p ' y ( N ) over all { y ( t ) J OEN X ( 2 , N ) . Moreover, {EL'2}oN is efficient in X ( 2 , N ) , provided p > 0. As the positivity of p is guaranteed in normal situations, this implies a normative significance of a maximal balanced-growth ray, which justifies preference for the balanced-growth over any other feasible modes of growth if the economy starts at i. Nonetheless, efficient paths need not conform to the pattern of balanced growth. Their patterns of dynamic movement or, stated geometrically, the shapes of these paths in space, may be so complicated that it is hard to establish explicitly the equations of the orbit curves on which these paths lie. But certain economists, represented by Samuelson, firmly believe that the normative significance of a maximal balanced-growth ray is not of such limited scope as was stated above. Samuelson (1960) conjectured that efficient paths have an inclination for leaning to a maximal balanced-growth ray at intermediate periods on the way to the respective final goals, and that the longer the time horizons are, the more persistently their proximity to the ray will occur [see also Dorfman et al. (195S)l. Samuelson compares a maximal balanced-growth ray to a turnpike in an automobile highway system. If a person goes from a town to a nearby town, he proceeds directly to the final goal, not using a turnpike. However, when he makes a very long journey to a remote town, he takes a roundabout course. He first reaches a turnpike, then goes on it for most of the time, and finally gets off it to use a local road leading to the goal town. It is conjectured that the strategy of efficient economic growth in the long run is similar to such a plan. Starting at a historically given initial state, the economy first approaches a maximal balanced-growth ray, then grows in a balanced fashion on it or in a nearly balanced fashion closely along it for most of the programming periods, and finally deviates from it toward the goal. Based on this, the proximity of efficient paths stated above has been referred to as the turnpike proposition. If the turnpike proposition holds, a greater normative significance may be accorded to a maximal balancedgrowth ray.

208

LV. EFFICELNT ALLOCATION A N D GROWTH

Recent studies originated by Samuelson have revealed that certain stringent conditions on the structure of the technology set T a r e required to be met in order for the turnpike proposition to hold. Various turnpike theorems have been proposed and proved by several authors (Morishima, 1961a, 1964; Radner, 1961 ; McKenzie, 1963a, b, c ; Inada, 1964; Nikaido, 1964b; Tsukui, 1966). All of them require certain stringent conditions and need more o r less intricate proofs. Here, Radner's result, strengthened by Nikaido (1964b) and lnada (l964), will be stated by following Radner (1961) and Nikaido (1964b), as a turnpike theorem whose assertion and method of proof are both simple and lucid. To begin with, we introduce a n optimality concept that bears a close relevance, in a special case, to the efficiency concept already defined. Let U be a collection of functions U(.Y) defined on R," and having nonnegative values.

DEFINITION 13.9. A feasible path ( ~ ( t )in} X(xo, ~ ~ N ) is said to be u-optilnal i f tliere is a firtictioli ii it? U suclz tliat ( s ( t ) } o Nmaximizes u ( y ( N ) ) orer u / / { ~ ( t ) ) ~in"X(.u0, N ) . Let P,, be the standard simplex formed by all semipositive vectors whose components add up to unity. For each 4 E P, define a function iiy on R+" by the formula u , ( . Y ) = 4's.Then, in the light of Theorem 13. I(i), we see that every efficient path in X ( x o , N ) is u-optimal with respect to the special U = { u y 14 E P,,;. It must be noted that a u-optimal path need not be efficient unless the corresponding 4 is positive; but, if the turnpike proposition is established for u-optimal paths, it automatically holds a .fortiori for efficient paths as zi-optimal paths o f a special type. Based on this idea, we shall prove a turnpike theorem on u-optimal paths for an unspecified U. Once this is achieved, the turnpike theorem on efficient paths will immediately follow upon taking { ziy 14 E P,,] as a special U. We require the basic assumptions (A.1) and (A.2) set forth i n $13.1 a s before. Following Radner, we start from a situation where (a)-(?) hold:

(x) There are an 2 2 0, a price vectorp 2 0, and an interest factor (growth factor at the same time) 1 > 0 such that (.f, 1.f) E T with p'2 > 0; (x.I ) p ' ( y - 2s)5 0 for any (.Y, y ) E T ; (a.2) p'(y - Ls) < 0 for any (.Y, y) E T unless s is proportional to 2. (2.3) A collection U of functions u on R,", having nonnegative values, is (/I) given and satisfies ([1i). Any u E U is positively homogeneous of the first order. (p.2) Functions of U are uniformly majorized by p'x in the sense that there exists a positive number K such that U ( X ) 5 K/>'x uniformly for ii E U, .Y E R+".

$13.


209

(p.3) For some positive number Ci and some nonnegative integer N , , there is a feasible path {\i(f)}:l starting at ~ ( 0 )= i and ending at w(N1) = a such that u(a) 2 li for all u E I/. ( y ) An initial vector xo 2 0 is given, such that for some positive number (r > 0 and some nonnegative integer N o , there is a feasible path { z ( t ) } t o starting at z(0) = xo and ending at z ( N o ) = oi. Remark 1. As was discussed in 913.2, assumption ( a ) is met by a triplet of a generator 2 of 3 maximal balanced-growth ray, a von Neumann price vectorp, and the maximum growth factor I. under (A.I), (A.2) reinforced by (A.3), and (A.5). On the other hand, one can easily verify, without referring back to the structural properties of T, that I. in (m) must be the maximumgrowth factor, and 2 in (a)generates the unique maximal balanced-growth ray. Remark 2. We shall examine assumption ( B ) for the special case U = { u , I q E P,}. (8.1) is obviously satisfied. If p is positive, K = max( lipi) over all i = I , . . . , n satisfies (8.2). I n fact, since the components qi of any q in U never exceed unity, we have q i5 1 5 K p i (i = 1,. . . , n). This implies q'x 5 Kp'x uniformly for q E U , s E R,". The positivity of p is ensured under (R) by Theorem 13.6(iii) if, at most, two components of 2 are positive. Finally let us consider (8.3). If there is a feasible path from .f to a positive u in N , periods, (8.3) is valid for ir = min q'a overq E P,, because this li is positive. We have the simplest case, N, = 0, when iitself is positive. Remark 3. I t must be noted that if transformability (A.3) and strict convexity (A.5) are explicitly assumed, any input vector x having at least two positive components can be transformed to a positive output vector y , i.e., (,y, y ) E T. In fact, such an input vector .Y is representable as a convex linear combination .Y = :(XI+ x'), wheres', x2arelinearly independent semipositive vectors. Then, (A.3) ensures the existence of some output vectors y ' , y2 such that (xi, yi) E T ( i = I , 2). Finally, (A.5) entails the existence of an output vector y satisfying (;(XI + x2), y ) E T, y > t ( y ' y '). Whence (x, y) E T and y > 0. Therefore, if 2 has at least two positive components, there is a feasible path of horizon 1 from 2 to some positive u under (A.3) and (A.5), meaning the validity of (8.3) for N , = 1.

+

Remark 4. Under free disposability, any positive initial vector xo can be transformed to an output vector o.? for some positive number CT, provided A > 0. In fact, we have xo 2 x.? for some positive number x. On the other hand, ( m i , Axf) E T by means of the cone property of T. Whence free disposability (A.4) implies (so,o.f) E T, where (T = i,r > 0. Thus we have a simple situation where (y) is satisfied for N o = 1. I f strict convexity is further assumed,

210

I V . EFFICIENT ALLOCATION A N D GROWTH

any initial vector xn having at least two positive components can be transformed (as was seen in Remark 3) to a positivevector, which can be converted to a.f for some o > 0, as above. Hence (y) is met for No = 2. We now proceed to a discussion of the turnpike property. In discussing the proximity of optimal paths {.u(t)}nN to the (unique) maximal balancedgrowth ray, which may be called the turnpike on the basis of the reference to a highway system above, measure the proximity not in terms of the direct absolute distance from x ( t ) to the turnpike but in terms of the deviation of the direction of the vector ~ ( t from ) that of the turnpike. In other words, we are solely concerned with the extent to which the proportions x,(f) : xz(f) : . . . : x,(t) of the components of x ( t ) approach the corresponding proportions -2, : .C-, : . . . : ,C-,# of the turnpike a i ( + co > c( 2 0). There are several ways to measure the distance of this kind between two nonzero vectors s,y in R'.Here, following Radner, we adopt

as an angular distance between x and y , where I/ . 11 is the Euclidean norm. In defining ( I ) , we can also use alternative norms such as I/ . I/A or 11 . /I defined in Chapter I, $1. I . Throughout what follows, we may assume, without loss of generality, that i is normalized, i.e., Whence the angular distance between x and 2 is given by

First of all, a lemma due to Radner will be stated. The lemma is very powerful and useful in the following argument, although it is easy to prove.

LEMMA 13.2 (Radner, 1961). Assitnie (A.I), (A.2), atid ( M ) . Tlzen, for any E > 0, there is a p sirch tliat (i) 1. > p > 0, ~ .?/I . u2 ~ E~, )(.I-, 1,)E T iniplies p'y - pp'.u 5 0. (ii) ~ ~ ( x ~ Pvoqf. I n view of the cone property o f T, it suffices to prove the assertion ~ 1 1 . Consider the set in R" x R" for processes (s,j>)with ~ 1 . ~ = V = {(s,y ) I (.Y, 1') E T, IIS/I

=

I,

I/.\-

- PI1

2 c).

V may or may not be empty. If V is empty, any p fulfilling (i) will do. I f V is nonempty, consider the expression V = X(P,,, 1) n {(.u, y) I 1 1 s-

2 cj,

$13.

21 1

SELF-SUSTAINED EFFICIENTGROWTH

where P,, is the standard simplex, and X(P,, , 1) is the set of all feasible paths of horizon I as before. Lemma 13.l(ii) for S = P,, N = I implies that X(P,, 1) is compact. On the other hand, {(x, y) 1 Ilx - 211 2 E } is closed in R" x R". Whence V i s compact. It is noted that (A.3), which is not assumed here, was used in the proof of Lemma 13.1 only once when ensuring the nonemptiness of X ( S , N ) . This justifies our appeal here to Lemma 13.1. (x, y ) E Vimplies particularly that s # 0, and .Y is not proportional to .C, so that p'y < Ap'x by (x.3). Whence p ' x > 0 for (x, y) E V , and f ( x , y) = p'y/p'x is a well-defined continuous function on the compact set V. By Chapter I , Theorem 1.1, this function takes on a maximum 5 , which must be smaller than iby (a.3). Then, any p between T and i, is a desired positive number, Q.E.D. We are now ready to state and prove Radner's turnpike theorem.

THEOREM 13.8 (Radner, 1961). Assume (A. I), (A.2), (x)-(y). The/?,f o r every E > 0, there is a positive integer k such that atiy u-optimal path { x ( t ) } o N starting at the given initial position xo satisjes

except possibly,for at most k periods. Proof. Let L denote the set of periods t between 0 and N at which

/I

--

i//

(4)

holds for a feasible path { x ( t ) } o Nstarting at xo.Further, let I be the number of periods in L. L , as well as I, depends on the path in question. The theorem will therefore be proved if I is uniformly bounded for all u-optimal paths starting at xo. Since ( x ( t ) , x( t + 1)) E T ( t = 0, I , . . . , N - 1) by feasibility, we have by (M.2) p ' x ( t 1) 5 Ap'x(t) (5) for any t between 0 and N - I , and, i n particular, by Lemma 13.2,

+ +

iff

E

p'x(r 1) 5 pp'.u(t) L. A repeated application of (5) and (6) therefore yields

(6)

p'x(N) u(x(N)),contradicting the assumption that { x ( t ) ] O Nis maximizing u ( y ( N ) ) over X(xo,N ) , Q.E.D.

+

Remark 7. We assume neither transformability (A.3) nor the monotonicity of u-functions in the Radner weak-turnpike theorem (Theorem 13.8) whose method of proof essentially consists of the comparison of u-optimal paths with the special feasible path { j ( t ) )constructed on the basis of (8.3) and (7). This suggests, in the light of Lemmas 13.3 and 13.4, weakened versions of (p.3) and ( y ) on the explicit assumption of (A.3) and the monotonicity of u-functions. They are obtained by weakening the feasibility of the paths {\v(t)}tl, { z ( t ) } pin (8.3) and (y) with respect to the original technology set T to their feasibility with respect to the disposal hull T They will henceforth be referred to as (p.3') and (y'), respectively. This remark is also relevant to Theorem 13.9, below, which explicitly premises (A.3), (c(.4), (p.4).

THEOREM 13.9 (Consecutive Proximity). Assume (A. l)-(A.3), (e) (p) ((p.I), (p.2), (p.3'), (8.4)), and (7'). Tlzen, for every E > 0, there is apositive integer k such that any u-optimalpath { x ( t ) starting at the given initial position xo consecutively satisjes

((e. I)-(a.4)),

Ilm-,f l < x(r)

E

(N - k 2 t 2 k ) .

Following Nikaido (l964b), the proof of Theorem 13.9 will be worked out in several steps, each of which will be stated as a lemma. First of all, we may assume without loss of generality that 2,p are normalized so that they satisfy (2) and

(18)

p ' i = 1.

As is well known, any vector x in R" can be uniquely decomposed to a sum of its orthogonal projection fl(x)ion the straight line spanned by i and its orthogonal complement e(x). Explicitly, we have x = O(x)i

+ e(x),

O(x)

= .Yx.

(19)

216


LEMMA 13.5. x # 0, l ~ ( x / ~ ~-x2l11~ < ) 6 f b r a 6 > 0 imply Ile(x)ll < 6 llxll.

+

Proof. By decomposing x/llxll to the sum U(x/ilxll)i e(x/llxll), and in view of the orthogonality of e(x/llxll) to O(x/llxll)i - 2, we see

proving the lemma.

LEMMA 13.6. Let x"(xo, N ) be the set of' all paths feasible in starting at xo. Then, It,e have ~ N ) , ii'e h a w (i) For each { ~ ( t ) }E, x"(xo, p'.u(t

+ l)/A"'

rp'x(t)/A'

( t = 0, 1,

. .., N

T and

- I).

(ii) p > 0. (iii) There is a number A > 0 such that Ilx(t)/i.'II 5 A (iv)

( t = 0, I , . . . , N )

holds uniforinly ,for paths iii g(xo,N ) and independently of N . There is a number I- > 0 such that p f x ( ) /~A N

2

r

holds uiiifornily ,for all u-optimal paths in x"(xo, N ) protiidecl N

NO

+ N,.

>=

Pvoof. Some of (i)-(iv) are restatements of facts already noted above. We also note, in the light of Lemma 13.3, that (A,]),(A.2), (A.4), ( r ) , (p),and ( 7 ) are valid for 'Tand the original U . With these in mind. we consider (i)-(iv). ( i ) (x(t), x ( t + 1)) E 7 impliesp'x(t + 1) J.p'x(/) by ( r . 2 ) for t = 0, I , . . . , N - I , which, if divided by A'+', becomes the desired result. (ii) By Theorem l3.6(iii), the vanishing property of any component o f p is incompatible with (cx.4). (iii) By (i), already proved, we have p'xo 2 p's(t)/A' 2 0 uniformly for all paths in x"(xo, N ) . This implies the desired uniform boundedness in the norm, because p > 0 and s(t)/l.' 2 0. (iv) This can be seen by comparing u-optimal paths, as in the proof of Theorem 13.8, with the special path {j(t)}oNwhose feasibility is ensured only with respect to 7, but not with respect to T. In fact, if ( s ( t ) } , N is u-optimal in

$13.


217

z(xo,N ) , there is a function u in U such that u ( p ( N ) )is maximized at {x(t)},N over 8(xo,N ) . Accordingly, we get K p ' x ( N ) 2 u ( x ( N ) )2 u ( j ( N ) ) 2

iN-(No+N

"oii,

as in the proof of Theorem 13.8. Whence p'x(N) AN

oii

2- KANo+NI'

whose right-hand side can therefore serve as r. Naturally, the above comparison with { j ( t ) } o Nis possible only for N 2 N o + N , .

LEMMA 13.7. Let (x(t)};=, E f(xo, N ) . I f x ( t ) # 0 a17d ll(x(t)/llx(t)ll) < S f o r some t , then lIe(x(t)/A')l\< 6AJor the t. Proof. This can be proved by combining Lemma 13.5 with Lemma 13.6(iii). -2 11

LEMMA 13.8. Let {x(t)};=, be u-optimal it7 )?(xo, N ) . ff lle(x(r)/>~)ll < v ] , \le(x(s)/L')ll < v] f o r some v] > 0 and r , s, such that N 2 s > r 2 0, 1t.e have

M>heref i (i = 1 , . . . , n ) are the compoiients of 2. Pvoof. This lemma states the most crucial part in the proof of Theorem 13.9. Suppose, on the contrary, that o 2 max 2v]/-fiover all i. Let

Since, for any i = I , . . . , 17,

we have

218


Clearly 1.1:) E T. Therefore x(r>/n' > G, ensured by (20), implies, by free disposability (A.4) prevailing in 7, that (x(r)/>:, AG)E 7. Whence ( x ( r ) , >:+I$) E Similarly, since G > x(s)/2, we can choose a positive vector 0, proportional to -4,such that Il"G > x(s) 2 0 . Then, since (RS-'L), k'k) E T, free disposability (A.4) for T implies that ( I Y ' G , ~ ( s ) 2 0 ) E 7. Therefore, the path ($1,

+

.Y(O),

. . . , .Y(Y),

i.'+' b?, ;Ir+ *I:,

+

. . ., A' - I8,

+

s(s) 1 3 , . . . , x ( N )

+ AND

belongs to g(so,N ) . Note that there is a function in U for which t/(j(N)) is maximized at the { . Y ( [ ) ; , ' = ~ in question over g(so,N ) . This implies in particular u(.\-(N)) 2 u(.\-(N) + i"i.).On the other hand, we must have u ( s ( N ) + i N D ) > u(.u(N))by (p.4) becaiise s ( N ) + 2°C > .\-(N),arriving at a contradiction.

0 5 P' /or

of?)'

x(t)

x(t

+ 1) < 4r1 max

R'- P' Y p -

t sutisfi'iiig s - 1 2 I 2

i\\p\\,l/min .fi1 i

Y.

ProoJ: In view of Lemma 13.8 and Schwarz's inequality (see Chapter I , $ l . i ) , and using the orthogonal decomposition (19) together with (18), we see. for r , s (s > r ) ,

Therefore, by virtue of Lemma 13.6(i), we have

for t between r and s - 1. as was t o be shown. Proojoj' Theorem 13.9. We shall prove the desired consecutive proximity o f u-optimal paths in X"(.Y', N ) , which implies, in the light of Lemma 13.4, that o f ir-optiinal paths in X ( x 0 , N ) . Given a n y c > 0, we can take a p, by Lemma 13.2 as applied to 7,such that iI i > p > 0; (2) /?'.I, - pp'.v =< o for any (x, E $satisfying l l . ~ / l l . ~ - \ ~ - -4112 c. The application of Lemma 13.2 to 'i= is justified by Lemma 13.3. 19)

$13. Next choose

r]


219

> 0, 6 > 0 subject to

SA < vl, 6 I_ E ,

(23) (24)

where A, r are the bounds considered in Lemma 13.6. Since (A.I), (A.2), (CY)((a.I)-(a.3)), (p) ((p.l)-(p.3)), and (y) are valid in T by Lemma 13.3, Theorem 13.8 (weak proximity) holds for 7, so that for this 6 there is a number k = k ( 6 ) such that

for any u-optimal paths i n Z(xo,N ) except possibly for at most k periods. Suppose N 2 k + 2. Then, (25) holds for at least two periods. Now, let r and s be the first and last periods at which (25) holds. Then

Hence, in view of (22), (23), and Lemmas 13.7 and 13.9, we have, for t satisfying s - 1 >= t 2 r ,

Then, if we had !Ix(t)/\l.x(t)lj - $11 2 E for some t between r and s - 1, the inequality p’x(t + I ) 5 pp’x(r) would obtain, so that by virtue of Lemma 13.6(i) and (iv) we have

This contradicts (27). Whence ~ ~ x ( t ) / ~ ~ .?/I x (< f )F~ ~for t satisfying s - 1 2 I 2 r. Moreover, this proximity also holds for t = s by (24), (26). We have thereby shown

Note that r and s may vary with the u-optimal path in question. But I’ + ( N - s) 5 k by Theorem 13.8. This ensures

uniformly for all 1,-optimal paths starting at the proof.

.I-’

for N 2 2k, which completes

220


$14.A CONSUMPTION TURNPIKE THEOREM $14.1. Optimal Consumption Streams

The system considered in $13 is a closed one, in which there is neither inflow of goods from the outside into the system nor outflow of goods from the system to the outside. Every good reproduces itself within the system. Thus even labor, a nonmaterial good, may be considered as the output of labor feeding processes to which consumption goods enter as inputs. On the other hand, the evaluation of the satisfaction that a worker acquires by consuming goods is almost ignored, although the influence on the rate of population growth of the level of consumption through the change of living standard is taken into account. However, recently there have been studies on optimal growth in open systems in which labor is treated as inflow from the outside and consumption occurs only as outflow from the system, and which revive a pioneering work of Ramsey (1928) on optimal saving. The results of Uzawa (1964), Atsuini (1965% b), Gale (1965, 1967), Koopmans (1965), Samuelson (1969, and Tsukui (1967), among others, are worthy of notice. In particular, the results of Atsumi, Gale, and Tsukui parallel those on closed systems discussed in $13. They culminate in consumption turnpike theorems that assert the proximity of optimal growth paths to a special balanced-growth path along which the maximum utility level of per capita consumption is maintained. In what follows, the essence of the results due to Atsumi, Gale, and Tsukui will be adapted to a typical consumption turnpike theorem under the strict convexity of the underlying technology set. To begin with, we explain and formulate the three major constituents of the system: a technology set, labor force as an exogenous factor, and a utility function. ( I ) Technology Set. There are M items of goods that serve as both inputs for production and materials for consumption. These goods emerge as outputs, one period lagged, from material inputs with the cooperation of the labor force under a given technology. Let 1 = a real number standing for labor input, .Y = an /?-dimensional vector standing for material inputs, 'I, = an mdimensional vector standing for material outputs. Then a triplet (/, x, y ) represents a production process, and all technologically possible processes form a technology set T. Basic assumptions on T are as follows: (B.l) T is a closed convex cone in R,' x R," x R+", including in particular the process ( I , x,y ) , where x = y = 0.

$14.

A CONSUMPTION TURNPIKETHEOREM

22 I

(B.2) (0, x,y ) E T implies y = 0 (the indispensability of labor input). (B.3) (/, x, y ) E T , u 2 x, y 2 v 2 0 imply (I, 11, E T (free disposability). (B.4.i) If (I, x l ,y ' ) E T, (I, x2,y') E T, and i f y ' # y', / > 0, then for any positive numbers P1 > 0, B , > 0, PI P2 = 1, there is an input vector .Y such that (1, x,Ply' + P'Y2> E T, x < A x ' + / j 2 x2 (strict convexity for input). 21)

+

(B.4.o) If ( I , X I , y ' ) E T, (/, x2,y') E T, and if x1 # x2, / > 0, then for any positive numbers a1 > 0, a2 > 0, a1 + a2 = I , there is an output vector y such that (I, alx'

+ a2 x2,y ) E T,

aly'

+ a 2 y 2< y (strict convexity for output).

(11) Exogenous Labor Force. The rate at which the labor force grows is exogenously given and assumed to remain constant over time. If A > 0 is 1 plus the given growth rate, and I, > 0 is the number of workers at the initial period 0, the total labor force at period t is given by l ( t ) = A'/,

.

( I I ) Utility Function. In the modern value theory in economics, it is widely agreed that a consumer's behavior is best predicated on the orthodox assumption that he has a preference for consumption goods. This idea has been stated in certain well-posed mathematical formulations, some of which will be given later in Chapter V, which deals with more disaggregated systems. Here we are contended with a somewhat naive, though widely accepted, hypothesis that this preference is represented by a numerical function u(c), which economists call a utility function (or indicator). The magnitude u(c) orders a consumer's preferences on various n-dimensional vectors c = (c,), whose ith components stand for the corresponding amounts of the ith (consumption) good. c = (c,) is said to bepreferredto d = (d,) if u(c) > u ( d ) and indrfSerent to d if u(c) = u(d). We assume here that (C. 1) Every worker has the same utility function u(c), invariant over time. (C.2) u(c) is a continuous concave function on R,". (C.3) c 2 d implies u(c) 2 u(rI), strict inequality holding if c 2 d. With these data on the system, we consider feasible growth plans maintaining the full employment of the labor force I ( t ) = ;.'I, over time.

DEFINITION 14.1. Girjen an initial stock of goods represented by a semipositive vector y o > 0, a feasible path oJ horizon N is dejned to be a sequence ( x ( r ) ,y ( t ) , c ( t ) ) o N ,iAose tth terrn consists of a triplet of an input i w t o r x ( t ) , an output vector y ( t ) , and a consumption uector c(t ), satisfi~ing

222


(i) y(0) = y o , (ii) y ( t ) - x(t) = c ( t ) 2 0 ( t = 0, 1, . . ., N ) , (iii) ( l ( t ) ,x(t), y ( t + I)) E T ( t = 0, I , . . ., N - I).

DEFINITION 14.2. A feasible path of liorizoii N starting at y o is said to be optinial if it niuxiniizes the sum of utility of per capita consumption stream ouer N + 1 periods N

ocer all feasible p a t h of the same liorizon starting at the same initial position. We can interpret (ii) in the following way. p(t) represents gross national product measured in terms of physical units. This is allocated to consumption c ( t ) and saving x ( t ) , of which the latter forms gross investment. T o save more at the present period ensures more products at the subsequent periods, promising higher levels of future consumption, but at the cost of less present consumption. On the other hand, society can alternatively consume more now at the cost of less future consumption. Thus at each period society faces the choice between these alternatives in order to have a more preferable utility sum of per capita consumption stream as given by ( I ) . Thus originates the prevailing philosophy of optimal saving. Using (B.I), (B.2), similarly as in the proof of Lemma 13.1, one can prove that the set of all feasible per capita consumption streams { c ( t ) / l ( t ) } o Nis regarded as a nonempty compact subset in the ( N + I)-fold Cartesian product of R”. Moreover, since U ( C ) is continuous by (C.2), the utility sum (1) is maximized at some feasible per capita consumption stream { c ( f ) / / ( f ) j Othe N, corresponding feasible path {x(t), y ( t ) , ~ ( t ) }being , ~ optimal. This ensures the existence of optimal paths. $14.2. Optimal Balanced-Growth Paths

We are concerned with the properties of optimal paths and particularly with their proximity to a balanced-growth path of special kind, which is a counterpart of the turnpike in a closed system such as was considered in $13. We first note, in view of the cone property of T ensured by (B. l), that the conditions of feasibility (i), (ii), (iii) i n Definition 14.1 can be put in the form

$14.

223

A CONSUMPTION TURNPIKE THEOREM

Moreover, redefining, if necessary, a new technology set as the set of all ( I , x, y ) satisfying ( N o , x, y ) E T, and a new utility function by u(c/l0),we may henceforth assume, without loss of generality, that 1, = 1. Thus, the feasibility conditions finally take the form Y ( 0 ) = yo>

Y ( t ) / a t - x(t )/,I' (1, s ( t ) / , l f ,ay(t

= c(t

)/af 2 o

+ i)/n'+l) E T

( t = 0, 1 , . . . , N ) , ( t = 0, 1 ,

. . ., N

- I),

(4) (5) (6)

with the corresponding sum of utility of per capita consumption stream being N

C 44t)iA').

t=O

(7)

We shall be concerned below with those balanced-growth paths for which x ( t ) , y ( t ) , c ( t ) all grow exponentially at the growth factor 3.. Therefore such a path is generated by a triplet of vectors x, y , c fulfilling y - x = c 2 0,

(8)

( 1 , x, 2.Y) E T. (9) We call (x, y , c) satisfying (8), (9) aproductiue triplet. If ;C is too large, there may be only a trivial productive triplet x = y = c = 0. Therefore we assume the given technology to be productive enough, relative to 2, to admit a productive triplet whose corresponding c is a positive vector. Then a routine work argument, based on (B.]), (B.2), shows that

r+= { y - x l y

- x 2 0, ( I , x,ay) E T J

is a compact convex subset of R+" including positive vectors. Whence u(y - x) is maximized at some j - 2 = 2 over r+because of the continuity of u.

LEMMA 14.1. W e have (i) A productioe triplet (x,y , c) is either trivial or satisjies x > 0, y > 0. (ii) The productive triplet ( 2 , j, 2) maximiziiig u(y - x) over all productive triplets (x, y , c ) is nontrivial and unique. PuooJ (i) Since]) = x + r, x 2 0, c 2 0, the supposition thaty = 0 entails x = c = 0, implying the triviality of the triplet. On the other hand, suppose y 2 0. We observe that ( I , x, Ay), ( I , 0,O) E T and Ay # 0. Therefore, applying (B.4.i) to ( I , x, Ay), ( I , 0, 0), fil = fi2 = +, we can establish the existence of an input vector z such that x/2> z. (1, 2, Jyy/2)E T, Since z 2 0, we must have x > Zz 2 0. This further implies y = s + c 2 x > 0. (ii) Take a productive triplet (X,jj, T) with T > 0, whose existence has already been assumed. Then, if 2 = 0, we would have u ( j - 2 ) < u ( j - X) by

224


(C.3). This contradicts that (a, j , t ) maximizes u(y - x) over all productive triplets. Hence (i,j; E ) is nontrivial. To see uniqueness, let (x,y , c ) also be a maximizing productive triplet. Suppose j # for the two processes (1, 2, l j ) , (1, x, iLy) in T. Then, by (B.44, there would exist an input vector z such that %

I

,

(1, z , 4 $ / 2

+ y/2))E T, 212 + x/2 > z.

Since $12 + y/2 2 212 + xj2 > z , we obtain a productive triplet (2,912 + y/2, $12 + y/2 - z ) . For this triplet we have, by virtue of (C.2), (C.3),

>u

L (9 -

- a)

+ 2l ( y - x)) -

1 1 2 - .(j - a) + - u ( y - x) -2 2

=

.(j - 2 ) .

This contradicts the property of (2, j , t ) being a maximizing productive triplet. Whence j= y . Similarly, we can see 2 = x,by virtue of (B.4.o). Therefore we also have E = j- 2 = 1' - x = c. Thus, the two triplets (a. .?, t), (x, c) coincide, Q.E.D. J?,

The triplet (2,j , 2 ) generates a ray in the three-fold Cartesian product of R+", on which moves the balanced-growth path ( L ' f , A'j, 3,'?),". Along the path, the economy consumes t and saves (accumulates) ?, for investment of the gross national product f in per capita terms at each period, maintaining the highest possible level of utility of per capita consumption over all balancedgrowth paths at the same pace ).'. Some economists find a normative significance in this fact and often call it the golden rule of accumulation. For this reason, the triplet (.f, f, E ) , the ray generated by it, and the path {A',?, j.'.p, ift)om will henceforth be called thegolden triplet, thegolden ray, and the go1det1 p t h , respectively. The optimality of the golden triplet also reflects on the association of a special price vector with it. Explicitly we have

LEMMA 14.2. Tliere is u price i w t o r p > 0juulJilling tlie follo\i.iiig three conditiorw (a), (p), (y): ( a ) u(c) - zr(t)5 p'(c - ?)for ull c 2 0, (p) p ' ( y - x) 5 p ' ( j - 2) ,for ull processes ( I , x,l y ) in T, IiYth tlie strict iiiequoli fj;

(7) p ' ( y - s)< p ' ( j - a) holdillg, if(., y ) # (f,f).

$14.

A CONSUMPTION 'TURNPIKE THEOREM

225

Proof. Let r = { y - x I ( I , x,21))E T } , which differs in the lack of the nonnegativity condition y - x 2 0 from P, defined above. Clearly r 3 r +. Further let Q = {c I c 2 0, u(c) 2 ~ ( 2 ) ) .Then, M = r - Q is a convex set containing no positive point. Since r, Q are obviously convex, M as their vectorial difference is convex. On the other hand, suppose that a positive z were contained in M . Then, y - x - c = z > 0 for some y - x E r, c E (3, so that y - x > c 2 0 would imply u(y - x) > u(e) 2 u ( t ) by (C.3) and the definition of Q. This contradicts the optimal property of 2 = 3 - f. Thus, there is, by Chapter I, Theorem 3.5, a vector p 2 0 such that p'z 5 0 for all z E M . Whence

p y y - x) 5 p'c

(y -

E

r, c E p).

(10)

From (10) we can derive (p), since 2 = j - .? E Q. (y) is automatically guaranteed for p satisfying (p) by virtue of (B.44, (€3.4.0). The verification of this fact is similar to the proof of Theorem 13.7, so it is omitted here. It remains to ensure (K). To this end, it will be shown that for the above p we can find a positive scalar p > 0 such that

( d ) u(c) - u(2) 5 pp'(c - t ) for all c 2 0. If the existence of such a scalar is seen, we have (x),(p), (y), with p replaced by pp. Therefore we have only to find a p satisfying (a'). Consider now the constrained maximization problem : to maximize u ( c ) on R + N subject to g(c) = p ' t - p'c 2 0. u(c), g(c) are concave on R,". Since r+, which is a subset of r, is assumed to contain a positive y - s, we have p'i. 2 p ' ( y - s) > 0 by (10). This implies g(0) = p't > 0, which means the fulfillment of the Slater condition i n Chapter I. Theorem 3.17. Next we shall show that 2 is a solution of this maximization problem. In fact, if we note E E r, from (lo) we can conclude that g(c) > 0, c E R," imply u(i.) > u(e). On the other hand, for c E R+" satisfying g(c) = 0, we have g(6c) = p'E - 6p'c > 0 whenever 6 fulfills 1 > 6 2 0. Whence u ( t ) > u(Sc) provided 1 > S 2 0. Thus, i n view of the continuity of u, letting 6 -+ 1, we have u(2) 2 ~ ( c in ) the limit. Since 2 has been shown to be a solution of the constrained maximization problem, by applying Chapter I , Theorem 3.17 to it, we see that there is a nonnegative Lagrangian multiplier ,D such that

,Lcg(i.) = 0,

~ (+ t )pg(i.) 2 u(c) + pg(c)

( 1 1) (c E

R+").

(12)

Clearly, (12) can be put i n the form (2'). Moreover, this [ I as well a s p must be positive. For otherwise (x') would be violated by some c satisfying c 2 E in virtue of (C.3), Q.E.D.

226

IV. EFFlClENT ALLOCATION A N D GROWTH

In analogy to Definition 13.8, we may term a price vector satisfying (p) a von Neumann price vector. The existence of a von Neumann price vector having the additional properties (M), (y) is very crucial t o the proof of a consumption turnpike theorem in 514.3. Notably, the following lemma, which was essentially first advanced by Atsumi (1965a) and which is a counterpart of Lemma 13.2, is important. LEMMA 14.3. Let (ji ,, 2 ) be the golden triplet and p > 0 be a von pricc rector sutisjjing ( y ) . Then, for any E > 0, there is a 6 > 0 such thut max(I1-x - i l l , lly - j l l ) 2 F , (1, x, Ay) E T

Neiiiiiutiii

itiiplic~sp ' ( y - .x) 5 p ' ( j - f) - 6. Proof: The proof is also completely analogous t o that of Lemma 13.2. The compactness of the set {(x, y) 1 ( I , x, i y ) E T, p ' y 2 p ' x , max(llx - $11, lly - ?\I) >= c] can be seen immediately, in view of (B.l), (B.2), and the positivity of p. If the set is nonempty, the continuous function f ( x , y ) = p'(y - ,I-) - p ' ( j - i) therefore takes a maximum over the set, which is negative by (7) and hence is of the form -5' for some 6' > 0. Then, 6 = min(6', p'(.i. - 2)) is a desired positive number, as is easily seen. If the set is empty, any 6 > 0 will do, Q.E.D. As a matter of course, a historically given initial stock of goods y o is very much unlikely to equal the golden output vector j. Therefore, in general, an economy is initially situated away from the golden ray. But can an economy, starting at y o , get onto the golden path in a finite number of periods? The fo I lo w i t i g le in ma a t i swers this question .

LEMMA 14.4. Assiiiiit. i. > 0. Theii, gireii aiiy positive initial yo, there is a positire iiitcjger N , siich thut there is u,feasible path { ? ( t ) , j ( t ), c(t)}," starling ut J ( 0 )= otirl coiiicitliiig ii,ith the golden path {Ai-?, A r i , l.ii.}om .for t 2 No . over all components i = 1, . . . , n. From the Prosf: Let (I,, = min assumption on yo, it is obvious that 0 , > 0. Moreover we clearly have I'o 2 0, F. If 0, >= I , we have y o 2 f . I n this special case, the feasible path { . T ( t ) , y ( t ) , c ( t ) ) given by ~

1

'

j l i 0 / j i

qr)

is

;I

desired one with N O = I

=

xc

(t 2

o),

$14.


227

Next suppose 1 > 0, > 0. For any 0 satisfying 1 2 0 2 0, the process (1, UP, A 0 j ) belongs to T because it is a convex linear combination of the two processes (1, 0, O), (1, i, A$) in T with the coefficients (1 - O), 0. Since processes of the type (1, x, Ay), y 2 x form a compact subset in T because of (B.I), (B.2), the set of numbers F, = ( p I(1, OP, 2 p j ) E T } is also compact for each 0 2 0. Moreover, 1 2 0 2 0 implies F, # @ because Fo 3 0. Hence, if 1 2 0 2 0, F, as a nonempty compact set contains a maximum, which will be denoted byf(0). The function f ( 0 ) defined in [0, I] satisfies (a), (b), (c) : (a) 1 2f(0)2 0, (b) f ( 0 ) is strictly concave, (c) f(0) = 0 > 0 implies 0 = 1 . The second half of (a), namely,J’(O) 2 0, is obvious from the definition of f’(0) and the fact that F, 3 0. To show its first half, suppose, on the contrary, ,f(@ > 1 for some 0 between 0 and 1. Then, if we compare the two triplets (2,j , t),(0P,f(0).P, ce) in view of the positivity of j, we have co =f(Q)j$2 > j - P = E, so that u(c,) > u ( t ) by (C.3), contradicting the optimality of (a,?, t).Whence 1 I f ( $ must ) hold. (b) follows from (B.4.o). In fact, let (T, T E LO, 11 and (T # T . Then 0.2 # TP because P > 0. Hence, applying (B.4.o) to the two processes (1, a2, i f ( o ) j ) , ( I , TP,Af(t)j) in T for any a > 0, p > 0, CI + p = 1, we see the existence of an output vector 11%such that

+ p r ) i , An3) E T, > (af(0) + Pf(t))j. Take a p satisfying > p j > (af(a) + f i f ( ~ ) ) j Then . (B.3) entails (1, + fir).?, A p j ) E T. Hence f(.a + PT) 2 p > af(a) + &’(T), proving strict (1, (ao

it’

(CIG

1~’

concavity. (c) is immediately implied by (a) and (b). In fact, from (a) we findf( I ) = 1 , whereas f ( 0 ) 2 0. Therefore, in view of 0 = (1 - 0)0 + 0 . I , we see f ( d ) > (I - O)f(O) + 1) 2 U by (b) whenever 1 > 0 > 0. This proves (c). With the aid of the functionJ’(0). we can define a numerical sequence {Or}, starting at 0, set forth above, by the formulas

ef’(

Or+, =f(0,) (t

= 0, 1 , .

. . , ad inf.).

(1 3)

Condition (a) ensures 1 >= 0,+, =f(0,) 2 O,, so that the sequence is nondecreasing and bounded. Hence the sequence converges to a limit 0 , in [0, I]. The possibility 0, = 0 is ruled out by 0, > 0 and the nondecreasingness of the sequence. We now observe that by virtue of Chapter I, Theorem 3.14, f’(0) as a concave function on [0, I ] is continuous i n (0, I ) . With this in mind, suppose that this limit 0 , is smaller than 1. Then,,/(())is continuous at 0 , , SO

228

1 V . EFFICIENT ALLOCATION AND GROWTH

that (13) becomes 0 , =,f(O,), Therefore we have

I > 0 , > 0 i n the limit. This contradicts (c).

lim 0, = I , t++m

( I , O t a , n o , + , g ) E T ( t = 1 , 2 ) . . .).

(16)

+

We now construct a desired feasible path. Since lim O , j = j = 2 2 > 2, we can surely take No fulfilling O,j > ,i- ( t 2 No - I). Then the path (,T(t), J?(t),T ( t ) ) , m defined by ( t = 0), 2'0 2 ( N o - 2 2 t 2 l),

U,,?

A'.t ( t 2 No), Y o ( t = 01, ( N o - 1 2 t 2 l),

2'0';

( N o - 2 2 t 2 l), ,9 - 2 ) ( t = N o - l),

/lNo-l(ONo-

2'2 ( t >= N o ) is a desired one, Q.E.D.

In the light of Lemma 14.4, we shall henceforth assume 2 > 0.

514.3. A Consumption Turnpike Theorem We are now ready to formulate and prove a consumption turnpike theorem that asserts the proximity otoptimal paths to the golden ray. In order to take full advantage of the preliminary results obtained in the foregoing sections, however, it is appropriate t o reconfirm all the assumptions t o be premissd in the theorem. Throughout what follows we assume (B.l)-(B.3), (B.4.i), (B.4.o), and (C.l)-(C.3) as basic assumptions, and 2 > 0 as an a u xi I ia ry one. It must also be noted that we are concerned only with the proximity of optimal paths to the golden ray i n proportions, rather than in absolute distances, as i n the turnpike theorem in closed systems. Therefore, if { s ( t ) ,~ ( t )~, ( t ) ) ,is, ~an optimal path, we are interested in knowing how closely the triplet ( x ( t ) / A ' , y ( t )/I.', c ( t ) / l G ' )approaches the golden triplet

4 14. A CONSUMPTION TURNPIKE THEOREM

229

(2,j,2) at intermediate periods. The results of Atsumi (1965a) and Gale (1965, 1967) on optimal growth paths of infinite horizon are readily rearranged to the following turnpike proposition. LEMMA 14.5 (Weak Proximity). Suppose that a positive initinl stock of goods y o > 0 is given. Then,for any E > 0, there is a nonnegative integer k such that any optimal path ( x ( t ) , y ( t ) , ~ ( t )satisfies } ~ ~

except possibly for at most k periods. Proof. We take a feasible path ( T ( t ) , j ( t ) , C(t)}omstarting at the same initial position y o and coinciding with the golden path { I f 2 ,I ' j , I'E},") from a period N o on, whose existence is ensured by Lemma 14.4. If we compare the per capita utility sum of any optimal path { x ( t ) ,y ( t ) , c ( t ) } o N'for N 2 N o with that of { F ( t ) , j ( t ) , T ( t ) } o N , we have N

N

C u ( ~ ( t ) / A5~ r=o )C u ( c ( t ) / X )

t=O

because of the optimality of (x-(r), y ( t ) , ~ ( t ) }Hence, ~ ~ . letting No- 1

Y

=

1(Wt)/Lt)

-

42)),

t=O

we obtain

We observe that the feasibility of { x ( t ) , y ( t ) , c(t)}," entails

For any given E > 0 choose 6 > 0, which lets the assertion of Lemma 14.3 hold, and define the set of integers

Then, by Lemmas 14.2 and 14.3, we see

A(+ I

l i

(j-2) 2

0 (t=0,1, ..., N - I ) , -6 ( t E L ) .

(2)

230


Hence, using the relation c ( t ) = y ( t ) - x ( t ) , Lemma 14.2(a), and (2), we have, in view of p ’ x ( N ) / A N2 0, p ’ ( j - 2) > 0,

5 -61

+ p‘y0,

(3) where I is the number of elements in L. Therefore (1) and (3), combined, yield

1s-P‘YO6

-

Y

(4)

.

Since inequality (4) holds uniformly for all optimal paths starting at y o of horizon longer than N o , we are sure that the smallest integer among integers not less than

can serve as a desired k, Q.E.D.

THEOREM 14.1 (Consecutive Proximity). Suppose that a positioe initial stock of goods y o > 0 is giuen. Then, for any E > 0, there is a nonnegative integer k such that

( k - N 2 t 2 k) uniformIy.for aII optimal paths starting at y o . Proof. The proof will be worked out essentially by following Tsukui (1967). Corresponding to a given E > 0, we take a 6 > 0 for which the assertion of Lemma 14.3 is true. Then, since 2 = 9 - 2 > 0, and u(c) is continuous, we can find an > 0 so small that

< E, 4 Ilpll < 1, /Ic - Ell < c1 implies lu(c) - u(2)l < 6/4, (Ix - 211 < E t , Ily - 911 < E , imply j > y > x > 2. El

(5)

(6) (7)

(8)

$14.


By Lemma 14.5, corresponding to this

23 1

> 0, there is an integer k,

is fulfilled by any optimal path starting at y o except possibly for at most k , periods. Let ( x ( t ) , y ( t ) , c ( t ) f o Nbe any optimal path starting at yo whose horizon satisfies N 2 k , + 3. Then, (9) is met by at least three distinct values o f t . Let r and s be the first and last values of t satisfying (9). Whence

By virtue of (S), inequalities (10) and (1 1) entail Y ( r + 1) > R, Ar+ 1

On the basis of (12) and (13), we can construct the following feasible path { Z ( t ) , j ( t ) , C"(t)}oNstarting at y o by

I

x ( r ) ( r 2 t 2 O), % ( t ) = lLfR (s - 1 2 t 2 r x ( t ) ( N 2 t 2 s),

i

+ I),

+

y ( t ) (r 1 2 t 201, j ( t ) = A'j (s L t 2 r + 2), y ( t ) ( N 212 s 11,

+

c ( t ) ( r 2 t 2 0), y ( r + 1) - X+'R ( t = r + I), l i t (s - I 2 t 2 r + 2), nsj - x(s) ( t = s), c ( t ) ( N 2 t 2 s + 1).

Then, since { x ( t ), y ( t ) , c(t)}oNis optimal, we have

which reduces to

232


Let L be the set o f t between r

+ 1 and s which fulfill

Then, denoting by I the number o f t in L, we can evaluate the right-hand side of (14), similarly as i n the proof of Lemma 14.5, to obtain

From ( I4), (1 5 ) we derive

(16) On the other hand, (6), (lo), and ( 1 1) yield

6 0 of a commodity bundle x = (xi)signifies his acquisition of x iunits of the good in question, whereas a negative one indicates his offering of the corresponding units of the good in question. In normal situations, the preference of a consumer has some sort of continuity. The continuity of preference means that if a consumer prefers xo to y o , he also prefers x to y for x, y that are close to xo,y o , respectively. This concept can be directly formulated as follows. DEFINITION 15.3 (Continuity of Preference in the First Form). A preference relation on X is said to be continuous if the set {(x,y ) I x >y } is an open subset of the Cartesian product X x X , where > is the associated strong preference relation.

Clearly, the continuity of preference in the first farm is equivalent to the closedness of the set {(x,y ) I x 2 y } in X x X . We also put forth another, seemingly weaker, concept of the continuity of preference as DEFINITION 15.4 (Continuity of Preference in the Second Form). preference relation 2 on X is said to be continuous if (a) the set { y 1 a > y } is open in X for each a E X , (b) the set {x I x > b} is open in X for each b E X , where > is the associated strong preference relation.

A

It can also be readily seen that the continuity of preference in the second form is equivalent to the closedness of the sets {x I x 2 a}, ( y I b 2 y } in X for each a and each b of X . In spite of their direct appearance, however, Definition 15.3 and Definition 15.4 prove to be equivalent, as will be seen below. LEMMA 15.2. DeJinitions 15.3 and 15.4 are equivalent. Proof. It is obvious that Definition 15.3 implies Definition 15.4. Conversely, suppose that 2 is continuous in the sense of Definition 15.4, and let u s prove that the set G = {(x,y ) I x > y } is open in the Cartesian product X x X . To this end, it will be shown that for each (a, b) E G there are two open subsets U(a) and U(b) of X , containing a and b, respectively, such that x > y for x E U(a),y E U(b). We consider the two cases separately. (i) There is some c in X that satisfies a > c > 6. In this case, U(a) = {x x > c}, U(b) = { y 1 c >y } are desired open subsets by the continuity of 2 in the sense of Definition 15.4. (ii) N o c in X satisfies a > c > b. In this case, let U(a) = {x I x 2 a}, U(b) = { y I b 2 y } . Then X = U(a) u U(b), U(a) nU(b) = 0, U(a)3a, U(b)3 b. U(a)is closed in X by the continuity of in the sense of Definition 15.4,

240

V . WALRASIAN COMPETITIVE ECONOMIES

so that U(b)as the complement of U(a) is open in X . Similarly, U(a) is open in X . Whence U(a), U ( b ) are desired open sets, Q.E.D.

Next we shall establish a simple but useful result on the existence of a most (least) preferable element. DEFINITION 15.5. Given a preference jield ( X , 2) and a subset M of X , an element x of M is said to be a most (least) preferable element of M i f x 2 y ( y 2 x) for all y E M . LEMMA 15.3. Let ( X , k), M be as in Definition 15.5. If k is continuous and M is compact, M contains a most (least)preferable element and the totality of all most (least) preferable elements is a compact set. Proof. We first note that for any finite number of elements x i (i = I , . . . , s) of X , the finite set {xiI i = 1, . . . ,s} contains a most (least) preferable element, which will be denoted by nz. pref(x', x2, . . . ,xs) (1. pref(x', x2, . . . , 2)). The proof of this can be easily worked out by induction on the number of elements. Now let Fy = { x I x k y , x E M } for each y e M . { x I x k y } is closed in X for each fixed y , so that Fy = {x I x 2 y } n M is a closed subset of M for each fixed y. This being seen, we consider the family of closed subsets {FyI y E M } . This family admits the finite intersection property, since

n F,, S

3 m.pref(y',

y 2 , . . .,y s )

i= 1

for any finite number of elements y i (i = 1, . ... ,s) of M . Whence, in view of the compactness of M and by virtue of Chapter I , Theorem 1.6, we have

n F , z ~zr.

Y E M

The left-hand side of the above result obviously equals the totality of all desired elements. Moreover, it is compact because of its representation as the intersection of the closed subsets Fy of the compact set M . The argument is similar for least preferable elements, Q.E.D. In spite of Lemma 15.3, the whole consumption set X need not contain a most preferable element. There may or may not be a most preferable element in X . In connection with this aspect, we have DEFINITION 15.6. For a preference field ( X , k ) ,a most preferable element of X is called a satiation (saturation)point. If X contains no satiation point, we say that insatiability preoails. Another orthodox premise on the consumer's preference (in addition to its continuity) is its convexity, which is most frequently formulated according to

$15.

24 1

ALLOCATION IN DECENTRALIZED SYSTEMS

one of the following conditions (I), (11), or (111) on a preference field (X,2), where the corresponding consumption set X is convex: (I) {x 1 x 2 a } is a convex subset of X for each a E X . (11) x 2 y for x,y E XimpIies cix + p y 2 y (a2 0,p 2 0, a + f l = 1). (111) {x I x > a } is a convex subset of X for each a E X . LEMMA 15.4. Conditions (I), (TI) and (111) are mutually equizialent. Proof. We shall prove the logical implication: (I) + (11) + (111) (I). (i) (I) + (11). If x 2 y for x,y E X , we have x,y E { w I w 2 a } for a = y. Whence ax + b y . { w l w k a } ( a 2 0 , p 2 0,M + p = 1) for a = y by (I), implying ax py 2 y . (ii) (11) + (111). Assuming (II), we have to show that x > a, y > a imply M X + by > a ( a >= 0 , fl 2 0, a + p = 1). To this end, we first note that either x 2 y or y 2 x by (7). Hence, if x 2 y, we have M X + by 2 y by (II), which, combined with y > a, entails ax + fly > a by Lemma 15.1(ii). Similarly, the supposition y 2 x also entails M X + py > a. This proves (111). (iii) (111) +(I). Suppose that x 2 a, y 2 a but not ax + fly 2 a for some x,y, a E X , a >= 0, p 2 0 , a + p = 1. Then a > ax + py, which, combined with x 2 a, yields x > ax + py by Lemma 15.l(ii). Similarly, y 2 a, a > M X + fly imply y > ax + fly. In summary, x,y E { w I w > ax + f l y } , which is a convex set by (111). Whence ax py E {w I w > ax b y } , implying C I X By > ax + by. This contradicts Lemma 15.1(i). Therefore, x 2 a, y 2 a must imply cix py 2 a, proving (I), Q.E.D. --f

+

+

+

+

+

DEFINITION 15.7. A preference field ( X , 2 ) is called convex, if X is a satisfies (I). convex set and the preference relation Remark 3. It should be noted that convexity (11) neither ensures x 2 cix + py nor excludes the possibility ax + py > x for such x, y E X that XXYNext we shall consider some of strengthened versions of convex preference fields. They are formulated in the following alternative ways for a preference field (X,2)whose consumption set X is convex.

(IV) x > y for x,y E X implies ax + /?v > y (ci > 0,p > 0, a + p = 1). (V) x 2 y for distinct x,y E X implies cix + By > y (a> 0, p > 0, a + p = 1 ) (strict convexity).

LEMMA 15.5. We have the following relations (i), (ii), (iii), which are also indicated in the diagram below. (i) (V) implies convexity (11). (ii) (V) impZies (IV). F ( I I) convexity (iii) If 2 is continuous, (IV) impZies (I I).

242

V. WALRASIAN COMPETITIVE ECONOMIES

ProoJ

(i) Suppose x 2 y for x, y

E

X . First, if x

=y ,

we obviously have

+ py = x k y (a 2 0, p 2 0, a + fl = 1). Second, if x # y , we have by (V) EX + py > y and, a fortiori, ax + py 2 y (a > 0, p > 0, o! + j? = l), while ax + fly 2 y trivially holds for a = 1, p = 0, and o! = 0, fl = 1. Therefore (V)

ax

implies (11). (ii) Suppose x > y for x , y E X . Since this supposition rules out y 2 x,we must have x # y . On the other hand, x > y implies x 2 y. Whence by (V) we find ax + py > y (a > 0, /?> 0, a + /? = 1). This proves (ii). (iii) Suppose x 2 y for x , y E X , and denote by [x,y ] the segment with endpoints x, y as in Chapter I, $2.1. Relation (iii) will be proved by showing [x, y ] n (w1 y > w} = 0. Since {w I y > w} is open in X by the continuity of 2,if the above intersection were nonempty it would include a segment and, a fortiori, at least two distinct points a, 6. As x , y $ { w I y > w}, these a, b are different from x , y . We may therefore assume that the notations are so arranged that

a=Ay+pb

(A>O,p>O,A+p=

l),

(1)

b=ax+ra

(~>O,T>O,CT+T=

1).

(2)

Then, y > b and (1) entail a > b by (IV). On the other hand, x 2 y > a implies x > a by Lemma 15.1(ii), which, combined with (2), yields b > a. Thus we have arrived at the mutually contradictory results a > b, b > a. This proves Cx, ul = {w I w 2 ~ 1Q.E.D. ,

DEFINITION 15.8. A preferenceJield ( X , 2)is called strictly convex i f X is coniiex and the preference relation 2 satisfies (V). LEMMA 15.6. Let ( X , 2)be a convex preferencefield and M be a convex subset of X . Then we have (i) Most preferable elements in M , if any, forni a convex subset of M. (ii) Moreocer, $(X,2)is strictly convex, there is at most one most preferable element in M , i f any. In particular, there is at most one satiation point in x. Proof. (i) If a, b are two most preferable elements in M , then a, b 2 x over all x E M , whence the convexity of the preference field implies aa + flb 2 x over all x E M , a 2 0 , a + j? = 1. Hence aa + pb E M is also a most preferable element of M , proving (i). (ii) If the convexity of the preference field is strict, the preference relation a, b a for two distinct most preferable elements a, b of M would entail aa + pb > a for o! > 0, > 0, a + fi = 1, the existence of a more preferable element in M than a, which is a contradiction, Q.E.D. Remark 4. It is obvious that if (X,2 ) is a convex preference field, then (VI)

x

- yfor x, y

E

Ximplies ax + py

2x

-

y(cc 2 0, p 2 0, o!

+ /3 = 1).

$1 5.

ALLOCATION IN DECENTRALIZED SYSTEMS

243

In switching from a menu of consumption x to a weighted combination of menus of consumption ax By, a consumer gives up /?x and substitutes By. Condition (VI) therefore means that if two menus are indifferent, their fractions are mutually substitutable without deteriorating the original level of preference. One may also observe that the combination of the menu CLXwith the menu by is at least as preferable as the original combination with Px because of the absence of mutual deterioration or even because of the presence of a complementing effect. Recognition of the absence of mutual deterioration in combined menus underlies the philosophy of the orthodox economist in assuming the convexity of preference fields in normal situations. This corresponds to the assumption of the convexity of technology sets on the basis of the recognition of the absence of external diseconomies (see Chapter IV, 812.2). Remark 5. For a preference field ( X , k),where X is convex and 2 is continuous, condition (VI) conversely implies convexity (11). In fact, suppose x k y . First, if X - y , we find ax + B y k y by (VI). The case where x t y remains to be considered. Denote by [x, y ] the segment with endpoints x, y and suppose that y t a for some a E [x, y ] . Clearly a # x, y . Then there must be some b in [x, a ] such that b y . Otherwise we would have the decomposition [x, a] = { w I w >y , w E [x, a ] } u { w I y > w, w E [x,a ] } . These two disjoint sets are open subsets of [x, a] by the continuity of 2. Moreover they are nonempty, since they contain x and a, respectively. This contradicts the connectedness of [x, a]. Therefore we can choose one b in [x, a ] that is indifferent to y , On the other hand, a E [ x , y ] , b E [x,a] entail a E [b,y ] . Whence b y , a E [b, y ] imply a 2 b y , contradicting y t a. This completes the derivation of (11) from (VI) under continuity.

+

-

-

-

EXAMPLE 1. The nonnegative orthant R," of R" is the simplest of all consumption sets; it appears most frequently in the economic literature. EXAMPLE 2. Let ( X , 2)be a preference field, and 2 be the semiorder in R" which is defined by (a)in Chapter I, 93.3. Then, 2 is said to be increasing if x 2 y for x,y E X implies x 2 y . It is widely agreed among economists that this compatibility of 2 with 2 is in accordance with consumer's preference in many situations. We obtain a strengthened version of increasingness when x >= y for x, y E X implies x 2 y , with the strong preference relation x ty holding if x > y. A still more strengthened version of increasingness is obtained if we impose the alternative (more stringent) additional condition that x 2 y entails x > y . Now let ( X , 2)be a convex preference field whose preference relation 2 is increasing and ensures the strong preference x > y for x, y E X satisfying x > y . Then, we can associate with each a E X a semipositive vector u such that x 2 a for x E X implies u'x 2 c'a. In fact, consider the subset M = {x 1 x 2 a> of X . A4 is by assumption convex, so that {a> - A4

244


is also a convex set. The set ( a ) - M contains no positive vectors. Otherwise a > x for some x E M , which would entail a > x by the additional condition on increasingness: This contradicts x 2 a implied by x E M . Whence, by Chapter I, Theorem 3.5, there is a separating hyperplane ( u , y ) = 0 having a semipositive normal v 2 0 such that the half-space (v, y ) 5 0 contains { a ) - M . It follows that u’x 2 u’a for x E M . Thus u can serve as a desired vector. It should be noted that the proportions of components of u need not be unique. Competitive Consumer’s Choice. We shall now formulate the behavior of an individual competitive consumer under given budget constraints. Usually the factors that constrain the domain of his choice are budgetary conditions such as prices of goods and his income. Once a price vectorp and a level of his income Z are given, he chooses a most preferable commodity bundle x with respect to his preference relation 2 from among his consumption set X , subject to the budget constraint ( P , x> 5 1.

(3)

Here the most essential feature of his competitive behavior is to be observed in the supposition that prices and income emerge as uncontrollable data to him, whereas he only passively adjusts himself to a given price-income situation in achieving his best satisfaction. It should be noted that the above statement merely formulates the pattern of a competitive consumer’s choice; it by no means ensures even the existence o f commodity bundles in X satisfying the budget constraint (3). Even when the budget constraint is consistent with the consumption set, the existence of a most preferable commodity bundle need not be automatically guaranteed. With the above facts in mind, we shall next give a notion of demand function. To any conceivable pair ( p , Z) of a price vectorp and a level of income Z we associate a subset +(p, I) of X by the following rule:

’”’

=

I

the set of all most preferable elements in X subject to the budget constraint, if such ones ever exist; the empty set, otherwise.

(4)

The above (set-valued) mapping from price-income pairs to commodity bundles will be referred to as a deinandfunction, or, more precisely, as a demand function deriivd.from a preference optimization. Preference Indicators. Let (A’,2) be a preference field. A real-valued function u(x) on X is termed a preference indicator whenever u(x) satisfies: for x,J J E A’, u(x) 2 u ( y ) if and only if

x 2 y.

$15. ALLOCATION IN DECENTRALIZED SYSTEMS

245

A preference indicator therefore affords a numerical representation of the order structure of a preference field. It is obvious that if u(x) is a preference indicator of a preference field (X,k),and iff(t) is a strictly increasing realvalued function on u ( X ) , the range of u(x), then v(x) = f ( u ( x ) ) is again a preference indicator of ( X , 2).Conversely, if u(x) and v ( x ) are preference indicators of ( X , k), there is a strictly increasing real-valued functionf(t) on u ( X ) such that v(x) = f ( u ( x ) ) .In fact, we may define a desired f(t) on u ( X ) in the following way. For each t E u ( X ) , define f ( t ) = v(x) for an x in u - ' ( t ) = {x I u(x) = t}. The value f(t) is uniquely determined since u-'(t) is an indifference class, and u(x) = v(y) for x,y E u-'(t).f(t) is strictly increasing. For if s, t E u ( X ) and s > t, then x > y for x E u-l(s), y E u-'(t), so that f ( s ) = u(x) > u(y) =f(t). Finally, the relation v(x) = f ( u ( x ) ) automatically holds by definition. From the above considerations, we see that there are an infinite number of preference indicators of a preference field, if any at all, and that they are unique only up to transformations by strictly increasing functions. Clearly, linear transformations f ( t ) = at + with positive slopes a > 0 fill out a subset in the totality of all strictly increasing functions. Hence preference indicators split in equivalence classes if we regard two preference indicators as being equivalent when they are connected by a linear transformation of the above type. Therefore we have measurable utility if there is some property of the preference field that is preserved under the linear transformations with positive slopes of preference indicators, yet by means of which we can single out a unique class from among the equivalence classes of preference indicators effected by the equivalence relation based on linear transformability. In the case of measurable utility, a unique preference indicator can be chosen up to linear transformations, so that we may measure preference within the choice of units and origins of measurement. The now classical word utility instead of its modern counterpart preference reflects the strong belief of the advocates of utility theory in its infancy in the cardinal measurability of preference. They conceived of cardinal measurability itself in a more naive way, rather than in the modern sophisticated way as stated above. Advocates of cardinal measurable utility are called cardinalists and are to be distinguished from the so-called ordinalists, who are interested only in the properties of a preference field that are invariant under the transformations of preference indicators by strictly increasing functions. Thus the modern extreme ordinalists ignore even the numerical representation of preference and work directly with the order structure of a preference field. In this book, we try to hold the extreme ordinalist viewpoint so long as it is possible [except for a few sections (e.g., Chapter IV, §14)]. A preference field need not admit its numerical representation by preference indicators. Therefore it is worthwhile to give conditions under which a

246

V. WALRASlAN COMPETlTlVE ECONOMIES

preference field has a preference indicator. A satisfactory solution to this existence problem of preference indicators was given by Debreu (1954, 1959). His general result states: If (X,2 ) is such a preference field that 2 is continuous and X is a connected topological space having a denumerable dense subset, then there is a preference indicator. In the above general theorem, X need not be a subset of R". If X is a subset of R",the existence of a denumerable dense subset is automatically fulfilled, so that merely the connectedness of Xand the continuity of 2 suffice to admit preference indication. 915.3. The Basic Model

In this section, we formulate a basic disaggregated model of a national economy involving many consumers and producers, for which the existence of a competitive equilibrium and some optimal properties will be established on certain additional assumptions in 916 and $17, respectively. The basic model involves I consumers (households) denoted by i = I , . . .,I ; 111 producers (production units) denoted by k = I , . . . , ni, and ti categories of goods denoted by j = I , . . . , ti. , ;), where (C) Consumers. Each consumer i has a preference field ( X i2 X iis his consumption set in R". Moreover, consumer i possesses an initial holding of some amounts of goods denoted by a vector u i= (aj') E R", thejth component u j l standing for the amount of thejth good initially held by him. ( P ) Producers. Each producer k has a production technology set Y, in the flow version (see Chapter IV, 412). I n the above formulation of the basic model, we have not specified further assumptions on the preference fields, technology sets, and initial holdings involved in it. These further assumptions will be specified in the following sections. The basic model is, however, already formulated to such an extent that it enables us to obtain a general view of the principal framework of the economy with which we shall be concerned in the following sections. The principal working of the economy, whether it is achieved in a decentralized mode or not, is to allocate production and the resulting products among consumers. A n allocation of production is effected by choosing a process y k from the technology set Y , for each k = I , . . . , m. Then the sum rn

Y= CYk k= 1

is the corresponding uggregute process. Hence these aggregate processes together f i l l out the aggregate technology set m

y =

1 Yk, k= 1

$15.

ALLOCATION IN DECENTRALIZEDSYSTEMS

247

the vectorial sum of Y , , which serves as a national economy-wide technology set. On the other hand, the sum I

a=

C U i i= I

(3)

over all consumers makes up the aggregate initial holding. Whence the vectorial sum of the aggregate initial holding { a } and the aggregate technology set

{a>+ y

(4)

forms the totality of all aggregate supply vectors available. It should be noted that the initial holding may consist not only of finished consumption goods, which are ready for distribution among consumers, but also of semifinished (i.e., intermediate) goods as well as fixed capital stocks, land, and labor. Let us turn our attention to the demand side of allocation. An allocation of consumption is effected by choosing a menu of consumption (i.e., a commodity bundle) xi from the consumption set X ifor each i = I , . . . ,I. Then the sum 1

x=

1.i i= 1

is the corresponding aggregate demand vector. Here attention should be paid to the fact that some of the components of an aggregate demand vector may be negative. These components correspond to the positive supplies of such goods as labor (a flow magnitude) in the individual consumers’ menus of consumption, which may serve as inputs in production. An allocation of production and consuniptiori together therefore means the choice of an (I + m)-tuple (XI,.

. . , xi , . . . , II,y 1 , . . . , yk, . . . , y“)

(6)

EX, ( i = I ,..., /),

( 7)

of JJk€

Y, ( k = I , ... , m )

( 8)

such that the aggregate demand equals the aggregate supply I xi = a m yk.

1

+1

i= 1

k= I

(9)

Naturally, a single allocation of production alone need not be attainable, since some inputs may be supplied solely by consumers. On the other hand, a single allocation of consumption alone is unattainable, unless there are supplies of goods by production and out of the initial holdings. Therefore, the word “allocation” can be applied aptly only to such a simultaneous allocation of production and consumption together that the equality of supply and demand (9) holds.

248


The principal working of the economy is undoubtedly to allocate production and consumption in the above sense. As a matter of fact, there are many (probably, infinitely many) allocations. Moreover, in allocating production and consumption, the choice of individual menus of consumption xi and production processes J,' need not be performed on the initiative of individual consumers and producers. The planning agency of an extremely centralized economy intends to achieve an allocation by forcing individual consumers and producers to choose menus of consumption and production processes that are not necessarily preferable to them from their own individualistic points of view, but may conflict with their preference judgments and profit evaluations. At the other extreme, a competitive market mechanism tries to achieve an allocation by setting up an equilibrium price system under which the individual allocations of consumption and production are performed solely on the initiative of individuals, and which entails the equality of the corresponding aggregate supply and demand. We shall be concerned with the possibility of a competitive allocation in $16, while in $17 we shall compare one allocation with another, and evaluate competitive allocations in comparison with other allocations either from the welfare economist's viewpoint or from the gametheoretic viewpoint. Remark 1. The individual consumption set X imay be a subset of a proper afine subspace of R". For example, if some categories of goods, say, capital goods, to which a consumer's preference judgment cannot apply, are involved i n the economy, it is reasonable to conceive that his consumption set X ilies in the linear subspace formed by all points x = (x,) whose components sjare zero f o r i corresponding to these categories. Remark 2. Although, in the above basic model, consumers are conceptually distinguished from producers, this formulation by no means excludes the possibility that an individual constituent of the economy is a consumer i, and a prot1ucer.j" at the same time. I t is conceived, however, that he has a dual character, so that he behaves as a consumer to make up a menu of consumption and as a producer to set up a production plan. Remark 3. A special case of the model, namely, the case of no production, deserves attention. I f the aggregate technology set Y consists of only the origin, the sole possible aggregate process is that of inaction, whence production is impossible. In such a special case, (9) reduces to I

2 xi

= a,

i= I

and the corresponding situation is nothing but a pure exchange economy. An allocation in this situation is a distribution of the fixed amounts of goods initially available among consumers.

$16.

EXISTENCE OF A COMPETITIVE EQUILIBRIUM

249

916. EXISTENCE OF A COMPETITIVE EQUILIBRIUM $16.1. Walrasian Competitive Equilibrium

As was pointed out in $15.1, the existence of a competitive equilibrium in the Walrasian system-one of the fundamental propositions in the Walrasian theory of general equilibrium-remained conjectural for more than half a century until its complete mathematical consolidation in the 1950’s. Naturally, the pioneering work of Wald (1935, 1936) which proved the existence of equilibrium for a Casselian system, is remarkable; but, the system with which he was concerned deviates very much from the standard Walrasian system. Apart from the difference in the degrees of disaggregation of both systems, the former lacks the most important characteristic of the latter, namely, a circular relation between supply and demand in national income accounting, which will be touched on below. The mathematical consolidation of the Walrasian conjecture for the existence of a competitive equilibrium was done for the first time in the 1950’s by a-number of authors (Arrow and Debreu, 1954; McKenzie, 1954; Gale, 1955; and Nikaido, 1956). Their penetrating results broke through the major barrier to the solution of the problem and paved the way for subsequent generalizations and elaborations by Debreu (1959, 1962), McKenzie (1959), Nikaido (1959b), Uzawa (1959-1960, 1962a,b), Morishima (l960b, 1964), and Negishi (1961). In what follows, we present a standard proof o f the existence of a competitive equilibrium, not for an extremely general model in which the most recent results are described, but rather for a moderate, though fairly general, model formulated by Arrow and Debreu. This is mainly to bring the major internal mechanism ensuring a competitive equilibrium into relief more vividly than could be done in extremely general models. Explicitly, we shall prove the principal results in Arrow and Debreu (1954), with somemodifications and iniprovements due to Debreu (1959) and Uzawa (1962a,b) taken into account by a method of proof that has proved useful in locating a competitive equilibrium since its first appearance in Gale (1955) and Nikaido (1956). This method consists of the reduction of the search for a competitive equilibriurn in the given disaggregated model to a simple, though powerful, existence theorem in lucid form [Gale (1955), Principal Lemma, p. 159; Nikaido (1956), Section 5. I , p. I41 ; Debreu ( 1 959), Proposition (I), p. 821. The proof of this theorem appeals to Kakutani’s fixed-point theorem (Chapter I, Theorem 4.4) or to its alternative equivalent version, which are extensions of Brouwer’s fixed-point theorem (see Chapter 1, 44.3). With these orienting remarks in mind, we proceed to the explicit formulation of a competitive equilibrium in the Arrow-Debreu Model of the Walrasian economy.

250


The Arvou,-Debveu Model. This model is obtained by imposing the following additional assumptions on conditions (C), (P) in the basic model stated in $15.3: (C.1) The consumption set Xi is a closed convex set in R" for each consumer i. (C.2) Each X i has a lower bound ciwhich satisfies xi

2 ci for all x iE X i

relative to the semiorder based on the component-wise comparison among vectors. (C.3) Each preference field ( X i , 2 ; )satisfies (IV) in $15.2. (C.4) Each preference relation k iis continuous. (C.5) Each consumer i has an initial holding ui,which is a vector of R". (P. I ) Each technology set Y, is a subset of R" containing the origin of R", namely, the process of inaction. (P.2) Y , is convex and closed in R". Y, satisfies (P.3) The aggregate technology set Y =

xr=

Y n R,"

=

(0)

(the impossibility of the land of Cockaigne),

where R+" is the nonnegative orthant of R". (P.4) Y n ( - Y ) = {O} (the irreversibility of aggregate processes). (C-P) There are ltn constants a i j 2 0 ( i = 1, . . . , I ; k = 1, . . . , m ) satisfying 1

1 i=

1

C(ik

=1

(k

=

1, . . . , m ) ,

(1)

with x i h standing [or the relative share of consumer i in the profit x h of producer k , so that the share of consumer i in nk is a;, n k .

Most of the economic implications of (C.1)-(C-P) have been discussed in Chapter IV, $12 and Chapter V, $15. Therefore we have only to make a few supplementary remarks. Remark 1. There may be a minimum subsistence level of menu of consumption for a consumer, below which he starves, while he has no physical potentiality to supply positive infinite amounts of productive services such as labor per ~ n i i i i i i iThis . motivates (C.2),the lower-boundedness of consumption sets. Remark 2. (C.3) and (C.4) imply, by virtue of Lemma 15.5, the convexity o f each preference field ( X i , 2 ;). Remark 3. Equation ( I ) means the exhaustion of the profit nk by its d ist r i b 11ti on a m o n g con s u mers as share h o Ide rs .

$16.

EXISTENCE OF A COMPETITIVEEQUILIBRIUM

25 1

We next state the definition of a competitive equilibrium in the basic model.

DEFINITION 16.1. An ( I

+ m + 1)-tuple

(9, z2, . . . ,P,j l ,j 2 , .. . ,j", j) of menus of consumption 2' E X , ( i = I , . . . , I ) , production processes j k E Y, (k = 1 , . . . , m), and an n-dimensional price rector j 2 0 is called a competitice equilibrium of the basic model if the follow*ing conditions (a)-(y) are fulJilled: (E) The maximum profit of each producer under j3. That is,

7[k(fi)

=

( f i , j k ) = max(j3,y)

over all y E Y,

(k = I,.

. . , m). (2)

(p)

The optimum preference of each cotisumer subject to budget constraints wider j3. That is, for each i = I , . . . , I, 2' is a most preferable menu of consumption among all x in X , fuljilling the budget constraint

(y)

The balance of aggreyutc .wpp/y and demand (i.e., market equilibrium). That is, 1

I

m

i= 1

i= I

k= I

(4)

with equality holding in the jth component relation of (4) i f the corresponding price j j is positire.

DEFINITION 16.2. brium price rector.

Tlie j3 in Dejnition 16.1 is referred to as an equili-

The following remarks might be useful in elucidating the nature of a competitive equilibrium as formulated in Definition 16.1. Remark 4. One important characteristic of a competitive equilibrium is that every member of the economy, whether he is a producer or a consumer, pursues a self-aggrandizing activity on his own initiative as a pricetaker. Thus (a)states that producers maximize their profits solely by controlling input-output configurations at their disposal under the equilibrium price vector j3 conceived as an uncontrollable datum. These profits are distributed as dividends among consumers as shareholders. The right-hand side of (3) represents the income of consumer i under jj, which is composed of ( b , a ' ) , the total value of his initial holding a' and a8knk(j?), his total dividends from producers (firms).

c

252


Condition (b) states that cmsumers achieve their most preferable menus of consumption solely by controlling their possible schemes of demand (and/or supply) subject to their budgets under f i . Remark 5. The maximum profit n,(B) of producer k in ( u ) is nonnegative because the technology set Y , contains the process of inaction 0 by (P.l) so that n,(p) 2 ( p , 0> = 0. However, n n ( j )may or may not be positive, depending on the technology set. One typical example of factors that cause a no-profit situation is the cone property of the technology set. If Y, is a cone, n k ( @must ) be zero. For, if n,(I;) = (I;, jA)> 0, then ( j ? , 09,) = O(B, jk)--f + m, Uyk E Y, as 0 > 0 tends to infinity, contradicting the finite maximum profit n,(B). An alternate, though similar, situation emerges when the aggregate technology set Y = c Y , is a cone. In such a situation, no producer earns a positive profit, i.e., n,(b) = 0 ( k = I , . . . , 171). This is clear from the simple fact that f = j?, where jkE Y , ( k = I , . . . , n i ) , maximizes ( I ; , y ) over Y if and only if jAmaximizes ( j ? , I . ) over Y , for each A-. Remark 6. ( 7 ) means that aggregate demand does not exceed aggregate supply for every good, a less stringent qualification for an equilibrium situation than their equality (9) in $15.3. But the additional qualification in ( y ) requires the equality of aggregate supply and demand for goods whose equilibrium prices are positive, and the positive aggregate excess supply of a good is possible only if its equilibrium price is zero,i.e., onlywhen it becomes a free good at equilibrium. Therefore, if the equilibrium prices of all goods are positive, (y) states the complete equality of aggregate supply and demand for every good. Positive prices of all goods at equilibrium can be ensured, e.g., by specializing consumers' preference fields, as will be touched upon later. Another important special situation where the equality of aggregate supply and demand can be achieved at equilibrium in spite of the presence of free goods is the case in which free disposability prevails. In view of the definition of free disposability in the flow version in Chapter IV, $12.2, assume free disposability for the aggregate technology set Y = Y,, i.e., J ' E Y, p' 2 u imply LI E Y . Then, any competitive equilibrium ( . ? I , P2, j"'fi), can be modified to obtain an alternative competitive equilibrium for which the equality of aggregate supply and demand holds. I n fact, since (4) is met by any competitive equilibrium, free disposability implies that Y s ir = 1.ti - cii. C is, by definition, of the form

1

(c)

c

1

...

C =

1 Gk, k= I

E

Yk ( k == 1 , .. . , i J I ) .

Therefore, if, for each /i, C" also maximizes ( j ? , j,) over Y k , then the ( I + I I I + I)-tuple ( . ? I , .t', . . . , .?', ir'. it2, . . . , El"', I;) is a desired competitive equilibrium. We now recall that goods whose aggregate supplies exceed

$16. EXISTENCE OF

A COMPETITIVE EQUILIBRIUM

253

aggregate demands are free goods at equilibrium. Hence ( f i , ii) = ( B , j ) , where j = jk.It follows that

On the other hand,

( B , j , ) ( k = 1 , . . . )m), (6) since, for each k , jk maximizes ( B , y ) over Y , , which contains G k . Equation (j,2 , ) 5

(5), combined with (6), therefore entails equality in (6), so that iik also maximizes ( j , y ) over Y,, the corresponding maximum profit being nk(@). DEFINITION 16.3. For consumer i, the dth good is ternied a desired good, if j o r an)’ x = (x,) E X , , there is apositire number A > 0 such that X i3 xi - (x,, . . . , Xd- 1, .Yo + 2, xo+,, . . . , s,)’and s* > , x. DEFINITION 16.4. Let D he the set oj’goods that are cotmion desired goods,for all consutiiers i = I , . . . , 1. Then, the 11th good is termed a productire good if,for any y = ( y j ) it7 the aggregate technology set Y , there is another y* E Y satishing yj*

2 yi (.iZ I?),

yd* > y d (for some d E 0).

(7) (8)

Remark 7. In Definition 16.4, y* depends on y . Likewise the desired good for which (8) holds also depends on 1’and need not be common for all y E Y. We are now ready to state principal results in the following three theorems on the existence of a competitive equilibrium for the basic model satisfying ((C.l)-(C.5), (P.l)-(P.4), and (C-P).

THEOREM 16.1 (Arrow and Debreu, 1954). If each consunzer i has a positire initial holding ai in the sense that there is a conimodity bundle b‘ in X i ,fu@lling

a’ > hi,

(9)

and i f he has no satiation point (see Definition I5.6), then there e.uists a conipeti-

tire equilibrium. Although the result of Theorem 16.1 is quite lucid and elegant, the assumption of positive initial holdings is somewhat unrealistic. On the other hand, Theorems 16.2 and 16.3 drop this unrealistic assumption at the cost of imposing further conditions on the preference fields and the aggregate technology set.

254

V . WALRASIAN COMPETlTlVE ECONOMIES

Let D,Q stand for the set of goods unanimously desired by all consumers and that of all productive goods, respectively. THEOREM 16.2. Assume that all goods belong to D. Then, there exists a competitire equilibrium such that the corresponding equilibrium prices are all positire if'the fol1ool;lYng(i), (ii) are satisfied: (i) For each i, consumer i has ail initial holding a' such that there is a comniodity bundle b' iti X i salisfying

ai 2 b'.

(10)

(ii) The aggregate technology set Y contaim a process jsuch that a+j>b,

( 1 1)

iihere 1

1

u

=

Eni,

b

i= I

=

C b' i= 1

THEOREM 16.3 (Arrow and Debreu, 1954). Assume Q # 0(hence also D # @). Tlieii there exists a competitioe equilibrium if the following (i), (ii)

are ,fulJilled: (i) For each i, consumer i has an initial holding ai such that there is a coinmoditqi bundle b' in Xisatisfving i > bi a = , (12) ahi> b,,'

(for some h E Q).

(13)

(ii) It is possible to allocate consumption and production altogether so as to hatie positire excess supplies of all goods, i.e., there are an X E X and a jG Y such that a+j>Z,

(14)

~Aere I

m

I

The proofs of Theorems 16.1-16.3 are not straightforward but rest on $16.2 through $16.4 below. The remaining portion of the present section will therefore be devoted to a few remarks on the above theorems, as well as to a relaxation of the convexity and closedness of individual technology sets, which is due to Uzawa (1962a). Remark 8. Theorems 16.2 and 16.3 assume explicitly the existence of an allocation yielding positive excess supplies of all goods in (ii), while Theorem 16.1 does not. But, in Theorem 16.1 the existence of such an allocation is

$16.

255


automatically implied by the assumption a' > b' ( i = 1, . . . , I ) . For the assumption entails a' > b', which means the validity of (1 1) for j = 0, the process of inaction. Remark 9. Theorems 16.1 and 16.2 include as a special case the pure exchange situation where Y = (0).

1

1

Aggregative Convexity. Assumption (P.2) requires the convexity and closedness of individual technology sets and is often criticized for its divergence from economic reality. Uzawa (1962a) observed that in proving the existence of a competitive equilibrium, this assumption can be dispensed with and replaced by a weaker and more reasonable one: (P.2') The aggregate technology set Y = Yk is convex and closed.

zl=

This useful observation will be considered further in what follows.

LEMMA 16.1. Let Yk(k = 1, . . . ,m) be nonempty subsets of R". Furthermore, let C( Yk),C( Y,) denote the cun~~ex hull of Ykand its closure, respectively. Then, if Y = Yk is convex and closed in R", we also hace

c C(Y,). ni

Y

=

-

k= 1

Proof. If it is seen that m

y

=

1 c(yk),

k= 1

c(

( 1 5) will immediately follow from (16). In fact, since Yk)c ______ C( Y,) c CC( Yk),we have, in view of the closedness of Y ,

1

c(Yk) and ~

which entails (15). Therefore we have only to consider (16). To show (16), however, it suffices to see

since Y is convex. First, it is obvious that the left-hand side of ( 17) is included in its right-hand side. Second, let y be an arbitrary element of the right-hand side of (17), so that

256

V.

WALRASIAN COMPETITIVE ECONOMIES

Then, since

we have

This proves ( 1 7).

THEOREM 16.4 (Uzawa, 1962a). Theorems 16.1-16.3 still hold if(P.2) is replaced by ug.qregatii~coiwexitj (P.2'). Proof. C( Y J is a closed convex subset of R" for k = I , . . . , nz. But (P.2') implies (15) by Lemma 16.1. Hence the modified economy that has the original preference fields (k',,2,)as its preference fields and c(Yk) as its technology sets satisfies (C.l)-(C.5), (P.1)-( P.4), and (C-P), provided the original model satisfies the corresponding assumptions with (P.2) replaced by (P.2'). Moreover, the additional assumption on initial holdings and allocations in Theorems 16.1-16.3 are not affected by the above modification of the original economy. Theorems 16.1-1 6.3 therefore hold for the modified economy so that it has a competitive equilibrium (.?I, . . ., j?', j l , . . . , jm, j). It will be seen that we can obtain a competitive equilibrium for the original economy from the one for the modified economy. In fact, although jkneed not belong to Yk, the aggregate process 9 = C jk can be put in the form rn

-

because j kE c(Yk), Y = 1 Yk = 1 c(Yk). Then, the ( I + nz + 1)-tuple (.?I, . . . , i' i', ,. . . , h", 8) is a competitive equilibrium for the original economy, as will be seen below. First, since j= jk= 1 2k, (4) for the modified economy implies ~

1

1

I

m

with equality holding in thejth component relation of (19) if the corresponding price jj is positive. Whence (7) is satisfied. Next, we observe that m

m

Since jkmaximizes (j,y ) over C( Y J , to whose subset Yk hk (k = I , . . . , m) belongs, (20) implies, as in Remark 6, that ek also maximizes ( @ , y ) over Y, ( k = 1, . . . , n?), with the coinciding maximum value z k ( j ) = (@, gk) = (j,2'). Therefore, (a) is also met. Finally, (p) is not affected by the transition

$16.

257


from the modified economy to the original one. This proves that (%I, .. ., 2', .??I, . . . , 2m, f j ) is a competitive equilibrium of the original economy, Q.E.D.

016.2. Supply and Demand Functions The purpose of this section is to reduce a competitive equilibrium of the Arrow-Debreu model to an equilibrium situation formulated in terms of suitably constructed supply and demand functions. A review of the exposition of competitive consumer's choice in $1 5.2 reminds us of the difficulties involved in getting demand functions that are defined and have nonenzpty image sets for all semipositive price vectors. We have to face similar difficulties in constructing supply functions. The main cause of these difficulties is that consumption and technology sets need not be bounded, and this does not ensure the existence of most preferable menus of consumption and most profitable processes. To overcome such difficulties, we will substitute certain virtual supply and demand functions for the true ones which are difficult to define. The former can be defined by narrowing the ranges of consumers'and producers' choice to suitable bounded subsets. But this approach must be preceded by preliminary considerations on allocations. Let

as before. Then a competitive equilibrium ( 2 ' , . . . , gf,y* l , . . . , j " , j), if any, satisfies (y) and therefore must satisfy the condition m

( u + Y - x ) ~ R + ~ ~ u + k= 1

1

.

Ci'.

(1)

i=l

Let us now define

( i = 1, . . . , I ) ,

( k = 1,

(2)

. . . , rn). (3)

Throughout what follows, the basic assumptions ( C .l)-(C.5), (P.l)-(P.4) and (C-P) are always premised. Moreover, when a proposition is presented, its validity is asserted commonly under the respective conditions of Theorems 16.1-16.3, unless there is a statement to the contrary.

258


LEMMA 16.2. We hare r i, k . (i) X i la, Fh # ~ j o any (ii) I f ( 2 ’ , . . . , ,?I, $ I , . . . , $”, j) is a cornpetitice equilibrium, then ,Qi j hE Y, for i = I , . . . , I, k = I , . . . , m. (iii) x“’, Fk are c0fTre.Y subsets of Xi, Y k , respectirely, for any i, k . F, are bouiideecl,for any i, k . (iv) T i , Proof. (i) In Theorem 16.1, we have b’ E f i , 0 E y, . In Theorem 16.2, we have hi E 8,, jk E F k , where j = j k , j k E Y , . Finally, in Theorem 16.3, S’E jhE y,, where X = X’,.U’ E Xi, j = jk, jkE Y, . This is clear from ( I ) , (2), and (3). (ii) (iii) They are convex, because they are obtained by the combined operations of taking linear combinations of convex sets and letting them intersect with R + “ , a special convex set, as is given in ( 2 ) , (3). Clearly, they are subsets of X i ,Y , , respectively. (iv) I n view of the definitions of Xi,Fk by (2), (3), it suffices to show that if I + m sequences {xi”>in Xi(i = I , . . . , I ) and {y”} in Y, (k = I , . . . , m ) satisfy

+

€xi,

-

zi,

1

2

1

(4)

they must be bounded. To this end, we first prove the boundedness of {yk”)( k = I , . . . ,m). In fact, supposing the contrary, and substituting some sub-sequences for the original sequences if necessary, we may assume, without loss of generality, Iim Y’+

= +GO,

[I,

= maxl/yk”l(

over aII

k = I ,. . . , m.

(5)

m

Then (4) implies

If the lower-boundcdness of Xii n (C.2) is taken into account, (6) reduces to

I!=,

where c = c’. Noting that p,, > 0 for large 1’ by virtue of ( 5 ) and that YLis a convex set including 0 by (P.I), (P.2), and yhv,we have, for large Y,

Dividing (7) by p,, , we obtain

616.


259

because lim pv = + co by (5). On the other hand, the definition of pV in (5) ensures, for large v, kv IlY /p,ll 5 1 ( k = 1, . . . , n7), which implies the inclusion of y""/pVin the unit ball. Hence the compactness of the unit ball enables us to assume, without loss of generality, that the / ? I sequences {y"} ( k = 1, . . . , m) are convergent, with the corresponding limits Iim ykv/pv= y k o . V-*+u3

These limits yko belong to Yk ( k = 1, . . . , m), respectively, because of the closedness of Yk by (P.2). Whence (9) reduces to

The left-hand side of (1 1) belongs t o Y , since yko E Y , ( k = 1, . . . , m). Therefore, (1 1) entails m

C ykoE Y n it+", k= 1

which, by virtue of (P.3), yields m

ykO = 0.

k= 1

On the other hand, we observe that we have, for each t , m- I

Y,

y'O=O+...+O+y'OE

because Y , 3 0 ( k = 1, . . . , m), while (13) implies - y'0 = 0

+ c ykO E

Y.

k+t

Hence, (14), ( I 5 ) together yield J ~ O E

Yn(-Y)

( t = I,

..., m),

which, by virtue of (P.4), gives Therefore we must have y k o= 0 ( k = I , . . . , rn) i n (lo), so that Iiin V + + W

maxl(yk"/p,.j( = 0. k

But (17) contradicts 1

=

max/Iyh"/pvJ/over k

= 1 , . . . , 171,

which is implied by the definition of p , in ( 5 ) .

260


Now that the boundedness of {j~''''} (k = I , . . . , m ) has been shown, that of the / sequences {sI'"}( i = I , . . . , /) can be seen immediately. In fact, using the assumed lower-boundedness of Xiwith lower-bounds ci (i = I , . . . , t ) in (C.2) as well as the boundedness o f { y " ) , we can establish the boundedness of {x"] for each s = I , . . . , I by m

csI - xB"5 - n

+ 1 y k " - 2 xi 5 CI + k= I

c rn

yk"

-

k= 1

i#s

1 ci. i#s

Since the above argument can apply to arbitrary sequences {xi")in X i ,{y,") in Yk satisfying (4), the proof i s thereby complete. This method of proof is due to Arrow and Debreu (1954), Q.E.D. Coiistructioii of' Supply nnd Deninnd Functions. Lemma 16.2 is very important i n that it enables us to lock allocations (i',..., i', j ' , ..., 9") associated with competitive equilibria in the bounded sets g i ,y,, although the existence of competitive equilibria themselves has not yet been established. On the basis of the lemma, we shall next define (virtual) supply and demand functions. We choose a sufficiently large cube

E

= {S

I t j 5 .yj 5 yli

(.j = 1, . . . , / I ) )

such \hat 0, hi E E, T i , ?, c E o ( i = I ,

, I, k

= 1,

. . . , ni),

(13)

where E o denotes the interior o f E. We thereby obtain I + m nonempty compact convex sets Xin E, Y, n E ( i = I , . . . , /, k = I , . . . , MI). With this setup, we define individual supply and demand functions on the set of all semipositive price vectors. This will be done by letting consumers and producers pursue their best satisfaction on the narrowed ranges of choice X i n E, Y , n E.

( I ) Individual Supply Functions. The supply function $,(p) of producer k and his profit function 7ck(p)are defined for all p 2 0 by $'(I)) = {jkI ( p , J-') = max(p, j) over all

nr(p)= max(p,

J.)

over all

y

E

v,

E

Y, n E ] ,

Y, n E ( k = 1 , . . . , n7).

(19) (20)

Since Y, n E i s a notiempty compact set on which the continuous function ( I ) ,1,) takes on ;I maximum, the definitions (19) and (20), make sense. We observe that for cach k , 'I(/ is a set-valued mapping that sends a semipositive p to a notiempty compact convex subset @ ( p ) of Y, n E. On the other hand, 7ck(p) is ;I numerical single-valued function satisfying n k ( p )2 0 for all p 2 0.

(21)

$16.


26 I

Relation (21) is clear from the inclusion of 0, the process of inaction, in Yk n E by (P. 1) and (1 S), for producer k can guarantee at least the zero profit by choosing y = 0 in Yk n E. (11) Individual Demand Functions. The demand function 4 ' ( p ) of consumer i is defined for all p > 0 by

4YP) =

[x l x i ~ X i , x i k xforall

xEXinE

1

rn

subject to

( P , x> 5 ( P ,

0')

+ h1 "iknk(P)) = I (i

=

(22)

1, ..., I ) .

In order that &'(p) # @, the budget constraint

must be consistent with the condition sE

Xin E

I;"=,

(24)

for any p 2 0. T o check this, we first note x i h n h ( p )2 0 for any p 2 0, an immediate consequence of (21) and % & 2 0 in (C-P). Then, since h iE X in E by (18) and u i 2 B' by the respective assumptions on initial holdings i n Theorems 16.1-16.3, .Y = h i fulfills both (23) and (24). Whence all s in X i n E satisfying (23) f i l l out a nonenipty compact convex subset M iof X i for each p 2 0. The compactness and convexity of M ifollow from the linearity of constraint (23) in the compact and convex set X i n E. 4i(p),which is the set of all most preferable elements in M i with respect to k i , is therefore a notiempty compact set by Lemma 15.3 and the continuity of k i in (C.4). Finally, qb'(p) is a convex set by virtueof Lemma I5.6(i) and the convexity of k iobserved in Remark 2 in $ 1 6.1. We have thereby shown that 4' is a set-valued mapping that sends a semipositive 17 to a nonempty compact convex subset of X i n E. (111) The Aggregate Excess Supply Function. The aggregate supply function $(I?), the aggregate demand function &p), and the aggregate excess supply function ~ ( pare ) defined by

262


All of them are defined for any p 2 0 and have nonempty compact convex subsets of R as their image sets. This follows from the same properties of 4', /+!I by virtue of Chapter 1, Theorem 2.10. There is a very important identity, called the Walras law, which connects 4(p)to $ ( p ) . Let .Y E &p) and J' E $ ( p ) , and let I

.Y =

1 .xi,

x i E 4'(p)

(i

=

1, . . ., !),

i= I

be their decompositions. Then, since x i satisfies the budget constraint (23), we have m

On the other hand,

J,' E

@ ( p ) , combined with (19) and (20), gives

Trh(/I)=

(p, yh)

( k = 1,

. . . , n7).

(29)

Hence, summing up (28) and (29) over all i, k and taking ( I ) in $16.1 into account, we obtain

Whence

or, expressed i n terms of the excess supply function ~ ( p ) ,

$16.


263

which we call the Walras law in the general sense. It should be noted that this law states the identical validity of (30) or (31) for any p 2 0, not merely for some special p . When equality holds identically in (30) or (31), i.e., when or

the corresponding identity is referred to as the Walras law in the narrow sense. From the economic point of view, the Walras law in the general sense states that the sole origin of effective demand is the spending of income. The Walras law in the narrow sense expresses the complete circular flow of income, or the equality of supply and demand in value, which need not imply, however, that for individual items of goods. These laws play a central part in letting the Walrasian system admit a competitive equilibrium.

DEFINITION 16.5. A triplet ( 2 , 9 ,fi) of an aggregate demand vector 2, an aggregate supply vector j , and a semipositice price cector j? is said to be an equilibrium solution of the aggregate supply and demand,functions $(p), & p ) if 2

E

4(fi),

9 E W),

j>= 2 .

(34) (35)

DEFINITION 16.6. The price rector fi in Dejnition 16.5 is termed an equilibrium price i w t o r of the aggregate supply and demandfunctions $(p), 4 ( p ) . It is obvious that an equilibrium price vector of $(p), $(p) is characterized in terms of the aggregate excess supply function ~ ( pas) such a price vector j? that

X(B)

n R+"#

0.

(36)

The importance of the aggregate supply and demand functions is found in

THEOREM 16.5. We haiie: (i) Zf an ( I + rn + I)-tupIe (I1, . . . , 2 ' , jl, . .., j m ,fi) of 2' E xi, j k E Y , , fi 2 0 is a competitiiw equilibrium of the basic model, then

2iE 4 ' ( j ) ( i = 1, . . . , 11, and the triplet ( 2 , j?,

$"j)

jk

E

fi),

where

(k = 1 , . .. ,m),

(37) (38)

is an equilibrium of the aggregate supplj~anddeniatid functions I&), 4 ( p ) .

264


+ +

(ii) If a triplet (a, j,j?) is an equilibrium of$(p), q5(p), then any ( I rn 1)tuple ( a ' , . . . , 2', j', . . . , j", fi) obtained by performing the deconiposition (37), (38), (39) is a competitiw equilibrium of the basic model. Pvoof: (i) Condition (35) follows from (y) in Definition 16.1. Moreover, ,ti j k E F, by Lemma 16.2(ii), so that 2' E X in E, jkE Y, n E by (18). Then, jkmaximizes ( j ? , y ) over Y , n E, because it does even over Y, by ( a ) in Definition 16. I . Likewise, 2' is a most preferable element of X i A E subject to the budget constraint (3) in 916.1, because it is even a most preferable element of X i subject to the same budget constraint by (p) in Definition 16.1. Hence .E' E q 5 j ( j ? ) , 9' E $'(j?), which proves condition (34). (ii) We have to show that (a)-(y) in Definition 16.1 are satisfied. First of all, we note that (35) implies

-

t i E x i ( i = 1, . . . , /), jk

E

(40)

F, ( k = 1, . . . ,n l ) ,

(41)

by the definitions of T i F, ,. (ii.a) Proof of (N). Suppose ( f l , y k ) > (8, 9') for some y k E Y,. Then, ( f i , z k ) > ( j ? , j k ) for any zk on the segment with end points y k , j k so long as zk # j '.Clearly z k E Y , by the convexity of Y , . On the other hand, j k must be an interior point of E by (18) and (41). Hence we can take the above z kso close to j ' that zk E Y, n E. This contradicts j k E $,(fi), which proves (a). (ii.P) Proof of (p). Suppose m

xi > 2,

( j ? , xi) 5 (fi, a ' )

+ 2 ai,n,(fi),

I

k= 1

for some xi E X i . Then, ((2.3) ensures

w i> ii I

for any iiyi on the segment with end points x i , ,ti so long as H , ~# 3'. Moreover, ir' belongs to X iby the convexity of X i and fulfills the budget constraint. Since 3' must be an interior point of E by (18) and (40), we can take the above w i so close to 2' that i v i E X i n E. This contradicts ,ti E 4'(p), proving (8). (ii.y) Proof of (y). Since (4) i n 51 6.1 is already ensured by (35), it remains to be shown that ?j

= jj

if

fij > 0.

I t can be readily seen, under ( 3 9 , that (42) is equivalent to

(42)

$16.

EXISTENCEOF A COMPETITIVEEQUILIBRIUM

265

Thus it suffices to prove (43). A review of the derivation of the Walras law in the general sense (30) suggests that t o prove (43) we have only to show that equality holds in the budget constraint in (p) for any consumer i. To this end, we first observe that for each i, there is some r i in Xi satisfying vi

>- .?

(44)

i

because of the insatiability of consumer i. Insatiability is explicitly assumed i n Theorem 16.1, whereas it is implied by the existence of (unanimously) desired goods in Theorems 16.2 and 16.3. Hence, (44) and (C.3) entail xi

>- f i

(45)

i

for any x i on the segment with end points ui, Ai whenever .xi # A'. Now suppose that strict inequality held in the budget constraint of some consumer i. Then we would be able to take the above x i so close to ,ti that xi could satisfy the budget constraint. In such a situation (45) contradicts (p), the already established fact that 2i is a most preferable element of Xi subject to the budget constraint, Q.E.D. 516.3. The Walras Law and Economic Equilibrium

Now that the competitive equilibria of the Arrow-Debreu model have been transformed to equilibrium solutions of the corresponding aggregate supply and demand functions, it is in order t o give a powerful existence theorem formulated in a very simple way in terms of an excess supply function.

THEOREM 16.6 (Gale, 1955; Nikaido, 1956; Debreu, 1959). Put P,, = { p Ip 2 0, C;=,p j = 1 } (the standard simplex of R") and let r be a conipact concex subset of R'.Suppose that there is giren a set-c.aluednxiyying K : P, -+ 2r wihich, j o r easy reference, 1cYll be called an excess supplj).function. It is assunied that the niappiiig satisjies the following conditions (M), ( p ): (a)

(p)

x : P,

2' is a closed mapping that carries each point qf P, to a empty coiirex subset of r. The Walras law in the general sense holds, i.e., ---f

( p , u> 2 0 for Then there is some fi in P,, such that

14

E ,y(p).

11017-

266


Proof. The proof will be worked out by constructing a suitable mapping, and applying Kakutani’s fixed-point theorem to it. To this end, first define a single-valued mapping

o:rxP,,+P,

by the formulas O(u, P )

= (Oi(u, p ) ) ,

11 = (Ui) E

+ max( - ui ,0) 1 + 1 max(-uj, 0) pi

oi(ll,

P> =

n

r,

P

= ( P i ) E Pn 9

( i = l , ..., n ) .

(2)

j = I

Then, by constructing the Cartesian product (see Chapter I, 94.4) of 0 with the mapping x, which can be thought of as a mapping defined on r x P,, , we obtain a mapping f

=x

x

0:r x P,,+2rXPn

P) = X ( P ) x {O(u,PI}.

(3)

Recalling Chapter I , 54.3, we shall show that all the conditions of Kakutani’s fixed-point theorem (Chapter I , Theorem 4.4) are fulfilled by f . This will be done in steps (i)-(iii). (i) Both P and P,, are compact convex sets in R”. Hence r x P,, is a compact convex set in R“ x R” by Chapter I , Theorem 2.9. (ii) ~ ( pis) by assumption a convex subset of P for each p , while {O(u,p ) ) is a special convex subset, consisting of one element, of P,,for each ( u , p ) E P x P,, . Whence for each (u, p ) E x P,, the image,f’(u,p ) defined by (3) is a convex subset of r x P , . (iii) T o prove the closedness of J; it suffices to show, in the light of Chapter I , Theorem 4.5, that the component mappings X, 0 are closed. The closedness of 1,viewed as a mapping on x P,, ,is an immediate consequence of the assumed closedness of x, as a mapping defined on P,,. On the other hand, 0 is obviously a single-valued continuous mapping from r x P, into P,,, so that it is also closed. Thercfore, by virtue of Kakutani’s fixed-point theorem, ,f has a fixed point ( n , so that

a)

(2, B> E f ( C , B>. (4) Relation (4) can be reduced, by using (2), (3), to the relations for the component mappings 2 E XCB),

(5)

;= O(ii, 8).

(6)

$16.


267

The theorem will have been proved if it is seen that ii 2 0. To see ii 2 0, using ( 2 ) , we rewrite (6) to obtain

pi =

+ max( - iii , 0 ) 1 + C max(-iij, 0 ) ji

(i

n

= 1, . . . , n).

(7)

j= 1

From (7), we get n

bizmax(-iij,

0 ) = max(-iii, 0 ) ( i

=

I , ..., n ) .

j = 1

(8)

Multiplying the ith equation in (8) by h i and summing over all i, we have n

n

j= 1

i= 1

(j,ii)

C max( - I j j , 0 ) =

iii max( - iii , 0).

(9)

If we note that t max(t, 0) = (max(t, 0))’ for any real I, we can reduce (9) to n

-

C max( - i i j ,

(j,ii)

j= 1

n

C (max( - iii, 0))’.

0)=

i= 1

We observe (9, ii) 2 0 in (lo) by virtue of the Walras law in the general sense (p), whereas max( -Gj, 0) 2 0 by definition. Hence the right-hand side of (lo) must be nonpositive, so that

C

max(-iii, 0) = 0 ( i = I ,

. . ., n).

( 1 1)

Equation ( 1 I ) implies - h i 5 max( - i i i , 0) = 0 for all i, whence iii 2 0 for all i. This establishes ii 2 0, and the proof is thereby complete. Alternatively, one can also get equilibrium solutions of ~ ( pas) fixed points of the mapping

(u,p ) -+ ~ ( px) V(U) : r x P , -+ 2‘

pn,

(12)

over all q E P,,‘,.

(13)

where q(u) = { r I r

minimizes

(u, q )

The fulfillment of all the conditions of Kakutani’s fixed-point theorem by (12) is easy to see. It is also obvious that a fixed point (a, fi) of (12) satisfies ( 5 ) and

fi E q(Ij).

(14)

If we recall the Walras law in the general sense (/I), then ( 5 ) and (14) imply

05

(Ij,

f i ) 5 ( I j , q ) for all

which entails ir 2 0, Q.E.D.

q E P,, ,

(1 5 )

268

V . WALRASIAN COMPETlTlVE ECONOMIES

Theorem 16.6 is a key theorem to prove the existence of a competitive equilibrium in a Walrasian system such as the Arrow-Debreu model, as will be made clear in $16.4. It presents conditions for the existence of equilibria in a most purified form, namely, the continuity (or, to be more precise, the closedness or upper semicontinuity) of the excess supply function and the Walras law. Remark 1. If ( I j , f i ) = 0 in (/Iat ) equilibrium, we obviously have for j such that p j > 0. In such a case we have, in particular, ii = 0, provided jj > 0. For a single-valued ~ ( p=) ( x j ( p ) ) , Theorem 16.6 therefore sheds light on the system of equations Ijj = 0

X j ( P I , ~2 >

... ,P n )

=0

( j = 1, . . . > n).

(16)

(Economists tried in vain to detect the existence of solutions of (16), for about half a century prior to the 1950’s, by merely counting its equations and unknowns.) Remark 2. Walras conceives the formation of equilibrium prices in the market as a tiitorinen?ent,i.e., a trial and error process. An auctioneer quotes prices, to which every participant in the market adapts himself as a pricetaker and announces his supply and demand. If the balance of the corresponding aggregate supply and demand results, the prices clear the market and are therefore equilibrium prices. If not, the auctioneer quotes alternative prices, and market participants announce their corresponding supplies and demands, etc. This trial and error process continues until finally equilibrium prices are reached. The mappings 0 and q , which appear in the proof of Theorem 16.6, may be interpreted as representing the behavior of the auctioneer who proposes a modification of prices responding to a nonequilibrium market situation (cf. Chapter VI, $19). Use of Kakutani’s fixed-point theorem in the proof of Theorem 16.6 is not a mere technical convenience; it is necessary for an intrinsic reason. We recall that Kakutani’s fixed-point theorem is an extension of that of Brouwer (see Chapter I , $4.3). But it is surprising that Brouwer’s fixed-point theorem is conversely implied by Theorem 16.6. This decisive result, which is due to Uzawa ( I 962b), accounts for the unavoidableness of the fixed-point method in the proof of Theorem 16.6. THEOREM 16.7 (Uzawa, I962b). TlTeorenz 16.6 itnplies Brouwer’sjxedpoint theorem (Clrapter I, Theorem 4.3). Proof; We recall that any two compact convex sets of the same dimension are homeomorphic; it therefore suffices to prove the existence of a fixed point for any given continuous mappingffrom the standard simplex P,,into itself (cf. Chapter I , Theorems 2.7 and 4.3). Let , f ( p ) = ( , f i ( p ) ) for p E Pn,where

416.


f i ( p ) is the ith barycentric coordinate of , f ( p ) . f i ( p ) ( i = 1, tinuous functions satisfying

. . . , I ? ) are con-

f i ( p ) 2 0 ( i = 1, . . . , n ) ,

We always have (IpI(’ fact, let

= (p,p

) > 0 for p

E P,, ,

269

(17)

because p # 0. Noting this

and define n single-valued numerical functions

(i = 1, . . ., 17).

x~(P= ) A(P)Pi - f ; ( p )

(20)

It can be readily seen that these / I functions in (20) are continuous on P,, . If we put ~ ( p =) ( x i ( p ) ) ,we obtain a continuous mapping defined on P,. The compactness of P, therefore ensures the boundedness of x(P,) and hence the. existence o f a closed cube containing x(P,). We shall verify conditions ( a ) , (p) for the single-valued mapping x : P,, + to apply Theorem 16.6 to it. In fact, r, a closed cube, is a special compact convex set. Each image set o f f consists of a single point and is therefore convex. The continuity off implies its closedness. Hence ( u ) is met. Moreover, (19) and (20) yield

(P?X(P)>

=0

GJE P A

(21)

verifying (p) in an equality form, i.e., the Walras law in the narrow sense. Thus, by Theorem 16.6, there is some fi such that

The final step of the proof is to show that 1(,8)= 1 and equality holds in (22). To this end, we note that the validity o f (21) for j? entails equality in the ith relation of (22) for i with Bi > 0. On the other hand, (22) implies,fi(j3)5 0 for i with pi = 0. Therefore, equality holds for all the relations of (22), i.e.,

,?($)Pi =,fi(fi)

( i = 1, . . . , 17).

Then, summing (23) over all i in view o f (1 8) and

A($)

=

1 fii = I , we obtain

I.

Therefore (23) turns out to be

fii =,fi(j?)

( i = 1,

so that ,8 is a fixed-point 0f.f; Q.E.D.

. . ., n),

(23)

270


I n the light oi’ Theorems 16.6 and 16.7, we realize that the Walrasian conjecture of the existence of equilibria is an equivalent alternative form of Brouwer’s fixed-point theorem. The Walrasian general equilibrium theory (Walras, 1874) was published in the 1870’s, while Brouwer’s work on fixed points (Brouwer, 1909, 1910) appeared three decades later. It is therefore no wonder that Walras could not achieve a mathematical consolidation of the conjecture in the days before the advancement of topology; he should certainly not be criticized for his failure to achieve a mathematical solution, but should be admired for his mathematical imagination which let him formulate this well-posed conjecture. Remark 3. I f the given excess supply function ~ ( p is) single-valued, the Cartesian product x x 0 i n the first proof of Theorem 16.6 is also singlevalued. I n such a situation we can appeal directly to Brouwer’s fixed-point theorem to obtain fixed points of x 0. Such is not the case with the x x used i n the the second proof.

x

$16.4. Existence of a Competitive Equilibrium

By the results in $16.2 and $16.3, we have fully paved the way for the proofs of Theorems 16. I , 16.2, and 16.3. In carrying them out, the general idea of the method of proof is to apply Theorem 16.6 to the excess supply function (27) constructed in $16.2. But the reduction to Theorem 16.6 will be somewliat intricate because the excess supply function is not always as well behavcd as that i n Theorem 16.6. This is because of the nonclosedness of the individual demand functions in such situations as those considered in Theorems 16.2 and 16.3, which necessitates a modification of these functions by means of the extension procedure discussed in Chapter I , $4.4. We begin by noting the 0th order homogeneity of the individual supply and demand functions t,bh(p),cfi(p),and hence that of the corresponding aggregate supply, demand, and excess supply functions $ ( p ) , $I@), ~ ( p )all , defined in 916.2. Their 0 t h order homogeneity, which means their having the same image sets for p and its positive multiple 1.1, (A > O), is easy to see. It originates in the linenr homogeneity i n 1) of the maxima { p , J.> of producers, and the budget constraints { p , x i ) 5 ( p , u’> + gil,n k ( p ) o f consumers. This suggests and justifies our conlining ourselves to consideration of these functions within the standard simplex P,, = j p / p 2 0, = I), whose elements can be thought of as normalized price vectors. Next we shall examine the relevant properties of the individual supply and demand functions in turn, in order to obtain an alternative well-behaved excess supply function. ( I ) Individual Supply Functions

916.


27 1

LEMMA 16.3. For @(p) in (19) and 7[k(p)in (20) of 916.2 tve hare (i) 7[k(p)is a continuous function on P,, . (ii) tjk: P,, + 2 E is a closed mapping whose image set $ k ( p )is convex for any PEP,. Proof. We shall prove (i) and (ii) together. In fact, let p , p" E P,, y k , yk"E E, yk"E $"(p') ( v = I , 2, . . .), and p" + p , yk" -+ y k as v + + co.Then we have, by definition, for v = 1,2, . . . , yk'

E

Y, n E,

(1)

(p", y") 2 ( p " , y ) for each y E Yk n E, (2) which become in the limit, by virtue of the closedness of Yk n E in E and the continuity of the inner product, n,(p")

=

y kE

(p,y k )2 (p, y )

Yk

n E,

(3)

for each y E Y, n E.

(4)

Relations (3) and (4) mean that y k maximizes ( p , y ) over y E Y, n E, with ( p , y k ) being the corresponding maximum 7[k(p). This establishes that lim nk(p') = q,(p), y k E $"(p), proving the continuity of q , ( p ) and the closedness of $,. Finally, the convexity of $,(p) is readily implied by the fact that $k(p) is the set of points maximizing the linear function ( p , y ) over the convex set Y, n E, Q.E.D. (11) Individual Demand Functions. Using a' and b' appearing in Theorems 16.1-16.3, we let m

l i ( p ) = ( p , a' - b')

+1

C(ik

nk(p) ( i

=

1,

k= I

.. , j ) , f

(5)

which will be referred to, for convenience, as the income of consumer i at p . We observe that ( p , u i- b') >= 0 for p E P, by a' >= 6' assumed in any of Theorems 16.1-16.3 and that the total dividend of consumer i is also nonnegative for p E P,, by uik >= 0 and (21) in $16.2. Whence I,@) >= 0

( p E P,,) ( i = I , . . . , I).

(6)

Moreover, I i ( p ) is a continuous function on P, since nk(p) is continuous on P,, for each k from Lemma 16.3. By making use of I i ( p ) ,we can put 4 ' ( p ) , the individual demand function of consumer i, in the form 4 ' ( p ) = {xi1 x i is a most preferable element of X in E subject to ( p , x - b') 5 I ; @ ) } .

(7)

Denote by P,O the interior of P, in the relative topology of L(P,,) = { p1 1p j = l}. P," is nothing but the set { p I p E P,, p > O } . Furthermore, let

P,' = { p 1 p E P, ,I,(p) > O} (i = 1 ,

,,

., I ) .

(8)

272


LEMMA 16.4. For 4'(p), P,' (i = 1, . . . , I ) we hace (i) P,' = P , in the case of Theorem 16.1. (ii) P,' is a dense (open) subset of P, for each i in the cases of Theorems 16.2 and 16.3, where the openness is clainierl in the relatice topology of UP,). (iii) 4' : P,' + 2 E is a closed tnapping. (iv) 4 ' ( p ) is a concex set for each p E P, , Proof. (i) Since a' > b' by assumption in Theorem 16.1, we have ( p , a' - b ' ) > 0, and therefore I,(p) > 0 for any semipositivep. This proves (i). (ii) Since a' 2 6' is guaranteed in Theorems 16.2 and 16.3, we assuredly have ( p , a' - b ' ) > 0 for any positivep. Whence Pn0 c P,' ( i = I , . . . , I ) . This proves (ii) because Pnnis dense in P, . Incidentally, the openness of P,' follows from the continuity of I,(p). (iii) Let p, p" E P,', xi, xi' E E, xi"E &'(p") ( v = 1,2, . . .)? and p" -+p, xi"4 xi as v -+ 00. To prove (iii) we shall have t o show that x'E @'(p). Indeed, let us first see that x i belongs to X in E and satisfies the budget constraint under p. The former fact follows from the assumption that x i is a limit of the sequence {xi")in the closed set X i n E. On the other hand, xi"E @'(p") implies ( p " , xi"- 6') 6 Ii(p') ( v = I , 2, . . .), which becomes ( p , x i - b') 5 l i ( p ) in the limit, because of the continuity of the inner product as well as of /,(p). It now remains to be shown that x E X in E, ( p , x - b') 5 I , ( p ) imply xi

x. i

To this end, recalling I,(p) > 0, I,(p") > 0 (v = 1, 2, . . .), we define

A, = Ii(p")/max(fi(pv)),( p " , x - 6 ' ) )

( v = 1 , 2,

. . .).

It can be readily seen, from (10) and the continuity of Z,(p), that 1 >= A, > 0,

Iim 2,

= Ii(p)/max(Ii(p), ( p ,

"- + m because ( p , x - b ' ) 5 l i ( p ) . Then if we let 9"= ( 1

- IJb'

+ I.,x

(v =

x

- b')) = 1,

I , 2, . . .),

we have, by (lo)-( 12), and the convexity of X i n E containing b', x,

x'EXinE ( f , x y - b')

(I!=

5ri(f)

1,2, ...), (v = i , 2 , . . .),

lim x" = x. v++m

$16.

EXISTENCE OF A COMPETITIVEE Q U I L I B R I U M

213

Since xi"E 4'(p") implies, in particular, that xi" is a most preferable element of Xi n E subject to the budget constraint underp", we see from ( 1 3) and (14),

xi"2 x"

( v = 1, 2,

. . .).

(16)

i

Then ( 1 6) becomes in the limit

xikx 1

because of the continuity of 2 i. (iv) The convexity of the image set d i ( p ) was already noted in (11) of $16.2, Q.E.D. Remark 1. In such a situation as was considered in Theorems 16.2 and 16.3, 4 i very often fails to be closed on the (relative) boundary of P,,. This peculiarity occurs even in a pure exchange case. Consider a two-good pure exchange economy in which some consumer i has a convex preference field ( R + 2 ,Xi)where k iis continuous. Assume further that x 2 y for x,y E R + 2 implies x > y i (see Example 2 in $15.2). Consumer i initially holds one unit of .the first good but no quantity of the second. By choosing any sufficiently large E = (x I t j 5 xi 5 vj ( j = I , 2)), where 0 > t i , i f j > I , we see that his corresponding demand function 4 ' ( p ) = { x i I x i is most preferable in R + 2n E subject to budget constraint). x i E 4 ' ( p ) implies that x1 2 0, x 2 i 2 0, pix, + i 1 for x i E $ ' ( p ) as long as p 2 x Z i5 p1 . 1 + p 2 . O = pl. Hence 0 5 x L 5 p 1 > 0. Therefore p" E P, , x i , xi" E E, xi"E $i(p') ( 1 1 = 1,2, . . .), lim p" = (0, I)', lim xi" = x i yield 0 5 x l i 5 1 < q l , x 2 i = 0. But the consumer can buy the commodity bundle (ql, 0)' when p 1 = 0, p 2 = I , which is preferred to x i . Whence x i4 4i((0,I)'), showing the nonclosedness of $ i at (0, I)'. We now proceed to the proofs of Theorems 16.1-16.3. We begin with the simplest of them, 16.1, for which n o modification of the demand functions is required. (a) Proojof Theorem 16.1. In the light of Theorem 16.5, it suffices to see that the corresponding excess supply function x ( p ) has an equilibrium price vector j for which ~ ( jn ) R," # @. T o this end, it will be shown that this ~ ( psatisfies ) all the conditions of Theorem 16.6. In fact, since 4', t+hk are set-valued mappings whose image sets are subsets of the cube E chosen by (18) in $16.2, the excess supply function ~ ( pcan ) be regarded as a mapping whose image sets are subsets of

-1

rn

{a}

+ E + . ' . + E - ( E + ... + E ) .

(17) Denoting by r the set (17), which is a compact convex subset of R' by Chapter I, Theorem 2.10, we shall show that the mapping

X : P,, 42r satisfies conditions (m) and

(fl) of Theorem

16.6.

274


( 2 ) We note that all of q5i, t,hk : P,, + E are closed mappings by Lemma 16.3(ii) and Lemma l6.4(i), (iii). In addition, a constant mapping 4' : P,, + { a ) defined by 4'(p) = a for any p E P, is naturally closed. The mapping ( I 8) can therefore be thought of as the composite of two mappings, i.e., the Cartesian product

40

fi 4'

i= I

of the / + 117 + 1 closed mappings continuous mapping

fj,p k= I

4', 4', i j h followed ,,r

from the Cartesian product

by the single-valued I

1 t ,,I

{[I)

x E x .'.x E

into F. Whence (18) is closed by virtue of Chapter I , Theorems 4.5(i) and 4.6(i). On the other hand, from the convexity of the image sets of do, 4', rc/k ensured in Lemmas 16.3(ii) and 16.4(iv) follows the corresponding property of by Chapter 1 , Theorem 2.10. This verifies (2). (/I) It has been verified i n $16.2 that the Walras law i n the general sense holds for- %( 11). Thus, applying Theorem 16.6 to ~ ( p )we , can guarantee the existence of an equilibrium price vector, Q.E.D. ( b ) Morlijiccl E.~cc.ss Supply Fuiictioii. In view of the possible nonclosedness on the whole P,, in the cases of Theorems 16.2 and 16.3, we are required to modify the individual demand functions. To do this, we shall take advantage of Chapter I , Theorems 4.7 and 4.8 and construct modified individual demand functions 4, : P,, + E ( i = I , . . . , I ) , which will be commonly used in the proofs of Theorems 16.2 and 16.3. Indeed, by Lemma l6.4(ii),(iii), (iv), the mapping (b' : P,'+ Eisclosed and has convex image sets on P,,', which is dense in P,. I n the foregoing considerations, we ignore the values of 9' originally defined for all p in P,, not belonging to P,,'. Then, we extend 4' : P,,'+ E on P, by means of Chapter I , Theorems 4.7 and 4.8 to obtain such a closed extension : P, + E that it has convex image sets. Replacing I$' : P, + E by $ ' ( / I ) : P,, --t E in the construction of the original excess supply function ~ ( p )we , obtain the modified excess supply Tu nc t i o t i

6'

$16.


275

It will be shown that 2 : P, -+ 2r, where r is the same as in (l), satisfies conditions (K), (p) of Theorems 16.6. (E) This is verified similarly as in the case of Theorem 16.1. (p) A review of the derivation of the Walras law in the general sense in 916.2 suggests that in order to verify the corresponding law for the modified excess supply function, we have only to see ( p , x i - 6') 5 z i ( p )

(20)

for any x i E $'(p), p E P,. This verification is done in the following way. We recall that since we have followed the extension procedure stated in the proofs of Chapter I , Theorems 4.7 and 4.8, a point xi in $ ' ( p ) ( p E P,, , 4 PI,') is characterized as a convex linear combination of a finite number of points x i s (s = 1, . . . , t ) for which there are sequences {p""}in Pni, { x i s v }in E fulfilling xis"E (p(p"") (v = 1 , 2 , . . .; s = 1 , . . . ) t ) , lim psv= p ( s = 1, . . . , t ) , v++m

Iim xis" = x L s (s = v-

+m

I,

. . ., t ) .

But we find (p"", xis"- b ' ) 5 Z,(p") from the fulfillment of the budget constraint by the original 4',and obtain, because of the continuity of I i ( p ) on P,, as well as the inner product, ( p , x i s - b') 5 Z,(p) (s = 1,

..., t )

as a limiting result when v -+ + co. Therefore the fulfillment of (20) by a convex linear combination x i of x i s (s = 1, . . . , t ) immediately follows from that by x i s ( S = 1, . . . , t ) . The above discussion justifies the application of Theorem 16.6 to 2. Whence 2 has an equilibrium price vector j E P,,, and

2(B) n R,"

f

a.

(21)

It will be seen, however, that we find ~ E P , ' ( i = 1, ..., I )

(22)

and therefore, in effect,

x(B) n R,"

#

@.

This will be done separately for Theorem 16.2 and Theorem 16.3. (c) Proof'qf' Theorem 16.2. We recall that although (6) holds generally, Z,(p) need not always be positive in PI,. Nevertheless, it should be observed that I

1I i ( P ) > 0

i= I

(P

E

P,).

(24)

276


Inequality (24) is seen from the following simple evaluation. Let j = E Y (k = I , . . . , HI) be the aggregate process whose existence is assumed in ( i i ) of Theorem 16.2. Then it is obvious that j hE Fh and therefore

xr= j h , y"

(k

j ? ~Y, n E

= 1,

..., M?).

(25)

Relation (25) implies Z,(P)

2 ( P , j k > ( k = 1, . . . > In).

(26)

Hence, in view of (26), we have

in

+ 1 ( P , j,) = (p, a - 0+j )> 0 2 (P,

a

- b)

k= I

because of ( 1 1) in 51 6.1. To prove j3 E P,' (i = 1, . . . , I ) , it naturally suffices to show

( j = I , . . . , /?),

bj>O

(27)

which is also an important part of the assertion of Theorem 16.2. To this end, it is first noted that (6) and (24) imply Z i ( j ) > 0 for some i, say s, which ensures l j E P,". (28) Relation (21) ensures the existence of a semipositive i2 in f ( B ) and its decomposition I

in

c=n+

xai20,

E $ h k= I

i= I

,fiE 4 ' ( p ) ( i j k

E

=

I, ...

Ip(j3) ( k = 1, . . . , / ? I ) ,

of which ,c" E $'(j?) reduces to 2' E @ ( B ) because of (28). Suppose then that pi = 0 for some j , say cl. The cAh good is a desired good for consumers, since every good is assumed to be unanimously desired by all consumers. Hence, by Definition 16.3, we find for some 2 > 0, .xi. = .Cs

+ ie" > .c',

sj.

E

X,,

5

where d

ed = (0, .. . ,0, I , 0, .. . ,o)'.

(30)

416.

277

EXISTENCE OF A COMPETITIVE E Q U I L I B R I U M

Let x ( t ) = ( I - t ) P

+ tx'.

(C. 1) and (C.3) ensure x ( t ) E X,,

s(r)> TS ( I > t > 0).

(31)

S

But, in view of (29), we are certain that 2' E 2'.Whence 3' is an interior point of E by (18) in $16.2, and x ( t ) E X,, n E for a sufficiently small positive t . Moreover, x ( t ) differs only in the dtli component from 2', implying (PI x ( t ) - b") = ( b , 2' - by)5 Z5(jj). These results, combined with (31), contradict 2' E 4'(fi). Therefore we must have (27), Q.E.D. (d) Reconsideration of the Original Iiidiridual Demand Finictions. We reconsider the behavior of the original (pi on P,, \ f , t i , which we have ignored so far in constructing its modification $ i . Explicitly, we put forth the following lemma, which will be used to prove Theorem 16.3.

LEMMA 16.5. For each i, closed at a point p in P,,\Pni, tlieti

6'is so related to 4' oii P,,\P,,'

6'(P)

that i f

(pi is ( 32)

($'(PI.

Proof. Let x i E $j(p). Then, xi is, by definition, a convex linear combination of finitely many x i f ( t = I , . . . , r ) for which there are sequences (pfl'j in Pni, (xi'")in E fulfilling Xitv E

#(p'")

(v

lim p" = p

=

1, 2, . . .; t

( 1 = 1,

=

1, . . . , r ) ,

(33)

. . . , r),

(34)

v-+'X

liln

XifV

= Xit

v - + a

( t = 1 , ..., r ) .

(35)

But, since 4' is by assumption closed at this p , relations (33)-(35) imply x i t E 4 i ( p ) ( t = 1, . . . , r ) . The convex set 4 ' ( p ) (see Lemma 16.4(iv)) must therefore contain x i as a convex linear combination of these .yrr, Q.E.D. (e) Proof of Tlieoreni 16.3. The final goal of the proof is to see, as in (c), that fi E P,,' ( i = I , . . . , I ) . To this end, in view of the assumption on the initial holdings, we have only to show that the equilibrium prices of all productive goods are positive. This will be done in steps (i)-(iii). (i). Let (29) be a nonnegative ii in i(jj)and its decomposition, and

be the corresponding aggregate demand and supply. Suppose now that j?, = 0 for some productive good I, say Definition 16.4, there is some J* in Y satisfying .l;* 2 j,

( i # I?),

> jd (for some L I E D).

J,~*

=

11. Then, by (36)

(37)

278


We observe that the method of proving ( a ) for Theorem 16.5(ii) applies equally to the present j k ;hence we realize that 9 maximizes ( B , y ) over all J' E Y, and this implies, i n particular, 9 ) 2 (b, y * ) , from which we derive, in view of y^ 2 0, $/, = 0 and (36),

(a,

Then, (37), (38), and jjd 2 0 entail jjd = 0. We have thereby shown that if the equilibrium price of some productive good were zero, that o f some unanimously desired good would also vanish. (ii). If it is proved that

&(a>

c @($) (for some s), (39) then we have PsE @(b). This can be shown, by exactly the same argument developed in the last half of (c) following formulas (29), to contradict the vanishingness o f the equilibrium price of some unanimously desired good-the conclusion reached in (i). Thus it suffices to prove (39). (iii). In the fight of Lemma 16.5, (39) will have been proved if we see that 4.' is closed at for some s. Using X E X , J E Y satisfying (14) in Theorem 16.3 (whose existence has been assumed) and the corresponding decompositions 1

x=~x',

F ' E X ~ ( i = 1 ,..., 11,

i= 1

c m

y=

ykf

yk,

Yk ( k = 1, ...) m),

k= I

and, noting nA(fi)2

(B,

J ' ) implied by j" E

yk, we obtain

m

I

=

2 ( j , Xi)

i= I

+ L; > S. The above strict inequality implies {B, a') + C cr,,7rk(b)> (a, is) (for some s).

because i, 2 0, a

m

k= I

(40)

$ 17.

NORMATIVE IMPLICATIONS OF COMPETITION

279

If we let

then Js(p) is continuous on P, . Clearly we can put @ ( p ) in the form @(p)

= {x’ I xs

is a most preferable element of

X, nE

(p, xs - 3”) 5 Js(p)).

subject to

(41)

From (40) we also have J,(p) > 0 in a neighborhood U ( @ )of @.Then, using the expression (41) in U ( @ )and applying the method used in proving Lemma 16.4(iii), with i, Zi,b’ replaced by s, J , , X”,we can see that 4” is closed at p. This completes the proof.

617. NORMATIVE IMPLICATIONS OF COMPETITION 817.1. Pareto Optimum and Prices

We shall consider various allocations of production and consumption in the basic model formulated in $15.3, and evaluate them from a normative point of view. In particular, we are interested in showing how competitive allocations rank in the evaluation, for whose feasibility we have specified a set of fairly general conditions in $16. Traditional welfare economists advocate the concept of Pareto optimality as a fundamental criterion to evaluate allocations. Throughout what follows an allocation means that of production and consumption (x’,. . . , XI, y’, . . . ,y”) in the basic model, which is required to satisfy (7), (8), and (9) in $15.3, with (9) replaced by the inequality m

1

k= 1

i= 1

Clearly, this replacement does not enlarge the set of possible allocations in effect as far as the consumption terms (XI, . . . , x’) are concerned, if free disposability prevails in the aggregate technology set Y.

DEFINITION 17.1. An allocation (XI, . . . , XI, y’, . . . ,ym) is said to be better than an allocation ( 2 ’ , . . . , 2’, j ’ , . . . , j ” ) , if xi

>- 2’ (i= I , ..., 1), 1

x i >. ii (for some i). 1

280


DEFINITION 17.2. An allocation is termed a Pareto optinium if there is no better allocation. The philosophy that underlies Pareto optimality is the belief that it is good to make someone better off without making anyone else worse off. In this section, we impose the following additional assumptions on the basic model: (C.i) The consumption set X i is a convex set in R" for each consumer i. (C.ii) Each preference field ( X i , k i )satisfies (IV) in $15.2. (C.iii) Each preference relation k i is continuous. (C.iv) Each consumer i has an initial holding ai, which is a vector of R". Yk is convex. (P.i) The aggregate technology set Y = (C-P) There are h i constant relative shares to profits x i k 2 0 ( i = 1, . . . , I ; k = 1, . . . , m) satisfying 1

C x i k = I ( k = 1 , ..., 11-1). i= I

One may note that (C.i)-(C.iv), (P.i), and (C-P) form a set of weaker conditions on the basic model than (C.l)-(C.5), (P.l)-(P.4), and (C-P) i n $16.1. With the above additional assumptions in mind, we shall state the Pareto optimality of a competitive equilibrium in

THEOREM 17.1. Suppose that I I O coiisutner is satiated. Tlieii, f ( . 2 ' , . . . , i',$ I , . . . , 9"'; p) is a conipetitire equilibrium (see Dejnitioii 16. I ) , the correspoiidiiig aI/ocation ( 2 ' , . . . , ,ti,PI, . . . , 9") is a Pareto optimum. Proof. Assume the contrary. Then we have (2) and (3) for some allocation (x',. . . , X I , yl, . . . , y"). For each consumer i there is, by his insatiability, liti in X i fulfilling wi > i x i . If we let x i ( t ) = ( I - t ) s i + tit,', we have, by (C.ii), (1 > f >O),

??(l)>Xikii i i

which yields, in the light of Lemma 15.1(ii), xi(?)

>9

(1 > t > 0).

(4)

I

Since iiis a most preferable element of X i subject to the budget constraint by condition (b) in Definition 16.1, (4) implies

+c m

( p , x'(t)) > ( p , a')

C(ik 71k($)

(i

k= 1

Letting t -+ 0 in these inequalities, we end up with m

=

1, . . . , 1).

$1 7 .


28 1

of which at least one must be a strict inequality because of (3). Whence sunimation of (5) over all i gives

Noting n,(j) 2 (@, y k ) by condition ( a ) in Definition 16.1, we derive from (6)

On the other hand, the allocation ( X I , . . . , s',y', . . . ,y") must satisfy ( I ) , which, if premultiplied by the transpose of fl 2 0, becomes

This contradicts (7), Q.E.D. A somewhat converse proposition of Theorem 17. I is the following

THEOREM 17.2 (Debreu, 1959). With each Pareto optimur?? allocation ( 2 1 ,. . . , 2', $ I , . . . , 9") in whicl? at least one consumer i is not satiated n,ith g i, we can associate a price i'ector p 2 0 such that (i) j k maximizes ( p , y k ) orer a// y k E Y , (I( = I , . . . , 177). ( p , xi) orer a// x i E {.Y' I xi E X i , .d 2 ;.?I ( i = I , . . . , /). (ii) t i nzi~ii?~izes Without loss of generality, we may assume that consumer I is not Proof: satiated with his allotment 2 ' . Let M~ = {s'l xi E

x i ,x i 2 s'}

(i

= I,

.. . , I ) ,

i

=

{s'I x 1 E X I , X I

>.?I}. 1

M i # @ since M i3 2'. Moreover, M : # 0 because of the insatiability of consumer 1 with 2 ' . Then consider the set

xr=

cf=

I Y,, a = a', as before. G contains no positive vector. In where Y = fact, if some u > 0 belonged to G , u could be put in the form

where x1 E M ? , X ' E M i (i = 2, .. ., I ) , y' E Y, ( k = 1,. . . , 177). Then the (I + m)-tuple (XI, . . . ,XI, y ' , . . . , y") is a better allocation than ( 2 ' , . . . , 2, jl, . . . , j " ) , since X I 2 ' , x i ki2i(i = 2, . . . , /),yielding a contradiction. Moreover, G is a convex set in R', since it is a linear combination of the I + 2

282


convex sets ( a ] , Y , My, M i (i = 2, . . . , I ) . Therefore, C can be separated from R+" by a hyperplane with a semipositive normal p (see Chapter I, Theorem 3 . 9 , so that

(P,u>

s0

( u E GI.

Whence for x1E MT, .yi E Mi ( i = 2, . . . , I ) , y' E Yk ( k = I , ..., in), we find

Moreover, (8) is true even for XI E M , , for the following reason. If we let + tx' for .TI E M,, X I E MT, we also have x ' ( t ) E M : ( I > t > 0) by (C.ii) and Lemma I5.l(ii). Hence (8) holds even if x1 is replaccd by s ' ( t ) ( I > t > 0). Thus the validity of (8) for any bundle in MI results, as we let t + O . Now, from the validity of (8) for any xi E Mi, y k E Yk we can deduce several results. To begin with, a special case of (8), where x i = fi, yk = j k ,is

.xl(t) = ( 1 - I).?'

On the other hand, ( 2 ' , . . . , t', p', . . . , 9") as an allocation must satisfy (I), which, combined with the nonnegativity of p , yields I

+ c 2 ic ( P , $9. = I

m

(p,

[Ii)

i= I

k= I

I

Iience, we find i n effect

Next, eliminating

(17, a ' )

from (8) by means of (9), we have

for .Y' E M s , j ? 'E Y , . Then, we obtain (i) as a special case of (lo), where .us= .? (s = I , . . . , I ) , y t = 9' ( t = I , ... ,in, # k ) , but yk is an arbitrary element of Y k . Likewise, we obtain (ii) as another special case of (lo), where J - I = jr( t = I , . . . , U I ) , .rs= ,Y(s = I , . . . , I, # i ) , but x i is an arbitrary element of Mi, Q.E.D. Remark 1. If, in the conclusion (ii) of Theorem 17.2, for each i, fi happens not to minimize ( p , - x i ) over aII x i E X i , (ii) implies: (ii':') .Ti is ;I most preferable element of Xi subject to (17,

S')

5 ( p , 2i>.

(11)

$17.


283

Choose a bundle 6' in X i for which ( p , 2 ' ) > ( p , bi). Let x ( t ) = (1 - ?)xi+ tbi (1 2 t > 0) for any x i E X,satisfying (1 1). Then x ( t ) E X iby the convexity of X i , and ( p , x ( t ) ) < ( p , ai). Hence Theorem 17.2(ii) implies i i> i x ( t ) . Letting r -+ 0, and in view of the continuity of k i ,we end up with i ikixiin the limit. Remark 2. Suppose that free disposability prevails in the aggregate technology set Y. Then for an obvious reason we have only to consider allocations for which (1) holds in an equality form. If with such an allocation ( i ' ,. . . , if, j i ' , . . . ,$") we can associate a price vector 9 2 0 fulfilling (i) in Theorem 17.2 and (ii*) in Remark 1, then ( i ' .,. ., if,j i ' , _ ., . jj"', fi) can be regarded as a competitive equilibrium for a suitable set of relative shares to profits C(ik and a redistribution of the initial holdings. Indeed, (i', . . . , i' jl, , . . . , j",j) is a competitive equilibrium whenever a reallotment of initial holdings is made to assign with consumer i his new initial holding

and relative shares aik = 1/1 are chosen, as pointed out by Debreu (1959). In view of Remarks 1 and 2, welfare economists attach great importance to a proposition such as Theorem 17.2, which, in their opinion, establishes the attainability of Pareto optimum allocations by a competitive price mechansm. Nevertheless, one may wonder if this attainability is completely competitive simply because this premises a reallotment of initial holdings and a distribution of profits that cannot necessarily be effected through a competitive mechanism. Some Pareto optimum allocations may not be attained by means of a competitive mechanism if individuals cling to their historically given initial holdings and relative shares to profits. It therefore can be said that although the concept of Pareto optimality has a great bearing on the policymaking of a planning agency aiming to make the working of the economy more competitive and decentralized, it is not useful enough to shed light on the true nature of competition in a so-called private ownership economy. $17.2. Game-Theoretic Review of Competitive Equilibrium As is clear in Definitions 17.1 and 17.2, Pareto optimality is too general a concept to characterize the nature of competition in the social framework of private ownership. Since Smith, there has been a firm view among economists that economic society in its bold aspects consists of competition among its individual members in pursuit of their own interests which conflict with each other. Thus economists conceive an equilibrium state of an economic society t o be a balance-of-power situation in the midst of conflict of interests, and a competitive equilibrium to be a special balance-of-power situation that results

284

V. WALRASlAN COMPETITIVE ECONOMIES

when there are a great many competing individuals. It is therefore very important to reconsider, from a more socioeconomic point of view, competitive equilibria that are undoubtedly something more than mere Pareto optimum situations. The first formal work along these lines was done by Edgeworth (1881) in the 1880’s. Although he discussed only a very special case, namely, a two-good pure exchange situation, his result is very illuminating. Quite recently it was re-evaluated and reformulated into more general results from a modern game-theoretic point of view by Scarf (1962), Debreu (1963), and Debreu and Scarf (1963). They borrow the concept of the core from the theory of manyperson games, put it in a form applicable to traditional economic situations involving interpersonally incomparable preferences, and use it as a key for inquiring into the true nature of competitive equilibrium. This methodology is readily acceptable, since game theory is exclusively concerned with situations where individuals with conflicting interests compete. The authors cited regard an economy as a many-person game played by its individual members, in which some members may form a coalition against another group of members to block an allocation. Then, the core of the economy is defined to be the set of such allocations as can never be blocked by any subgroup of members. We may conceive an allocation in the core as a balance-of-power situation where no coalition is effective, and hence there is no inducement to a coalition. A competitive equilibrium belongs to the core, while an allocation in the core need not correspond to a competitive equilibrium. But Debreu and Scarf have proved that the core eventually shrinks to the set of competitive equilibria when the number of members becomes large and tends to infinity in a specific way (based on a many-fold replica of the economy), as might be expected intuitively. In their results, Scarf (1962) and Debreu (1963) are concerned solely with the pure exchange case. Debreu and Scarf (1963) consider not only the pure exchange case but a situation involving production. However, this is still so special a situation that historically specified relative shares to profits have no chance to effect it, because of the zero maximum profits resulting from the assumed cone property of the aggregate technology set. Later, Rader (1964, Section 12, p. 175) remarked the extensibility of the Scarf-Debreu result to general situations involving production. In this section, corresponding results on the core will be proved for a more general situation as represented by the same basic model given in $15.3, with specified relative shares to profits T X , ~ being taken into full account. The method of proof is a modification of but essentially follows theirs, as in Racier’s remark. The same basic model formulated i n $15.3 is used. The additional assumptions, which we specify as ones suitable to this section, are as follows:

$17.

NORMATIVE IMPLICATIONSOF COMPETITION

255

The consumption set X iis a convex set in R“ for each consumer i. Each preference field ( X ik,i )satisfies (IV) in jji5.2. Each preference relation 2 ;is continuous. Each consumer i has an initial holding a i , which is a vector of R’. Each technology set Y, is a convex subset of R” and contains 0. There are lm constant relative shares to profits a i k 2 0 ( i = I , . . . , I ; k = 1, . . . , m ) satisfying 1

c a i k = 1 i= 1

(k=l,

..., t??).

(1)

It should be noted that the present set of additional assumptions (C.1)(C.IV), (P.l), and (C-P) differs from that in $17.1 only in (P.I), which replaces (P.i), We now define the concept of the core, prefaced by a remark on the private ownership of production possibilities and then by the notion of blocking an allocation by a coalition of consumers in the private ownership economy in question. In the private ownership economy, each consumer holds ownership of the initial holding a i , which is completely at his disposal. What about production possibilities ? The relative shares alh govern the distribution of profits in a price situation, but may rightly be conceived as originating in, and reflecting, consumers’ stock holdings, which represent private proprietorships of production possibilities and facilities. Therefore we may understand that consumer i owns the technology set x i L Y, at his disposal in producer (firm or production unit) k . This supposition needs a justification for its consistency by the following LEMMA 17.1. Let m

Zi =

C aikYk

( i = 1,

..., I ) .

k= I

Then we haoe

(9

iiAere

(ii) I j S

5 is an), noneniptj, subset

= L,

we hare

of’ the set L of all cotisim7ers.

ati equalitj?relatioti in I

m

i= I

k= I

(i), i.e.,

286


Proof. (i) An arbitrary y belonging to the set on the left-hand side of (3) can be put in the form

The above result, if rearranged, becomes

Note that for each k we have C(;k

>= 0

( i E As),

1

12

C(jh,

its

while 0, implies

j s i r

E

Yh ( i E S ) for each k . The convexity of Y , by (P.1) therefore

xr=

whence j' E I Yh by ( 5 ) . In the light of (i), it suffices to show that the set on the right-hand (ii) side of (4) is included in the set on the left-hand side. In fact, J' in the aggregate technology set can be put in the form y =

c

yk,

yk E Y,

(k

=

1,

.. . , m ) ,

h= 1

which, by ( I ) , reduces to

Q.E.D. DEFINITION 17.3. BJ,a coalitio~Ii'e tmaii a ~ o t i e i ~ p subset ty S cf the set L of all

COIISIIIII~YS.

Consider two coalitions S , T that are disjoint. The coalitions own at their disposal the technology sets

respectively. The disposability of the technology set by a coalition neither interferes with the other coalition's disposability nor is hindered by it. Any ys in Z , and any y T in Z , , as well as their sum process ys + y T , are feasible and performable independently from each other, because, by virtue of Lemma 17.1(i), we find

$1 7.


c

287

m

zS,zT,zS+zT=zSuTc

yk*

k= 1

Therefore, the consumer i's ownership of aik Y,, which puts C ( j k Yk at his complete disposal, can consistently be conceived. In this context, (1) means that the total technology set Yk of producer k is owned exclusively by consumers in Sk = {iI x i k > 0} and he only manages production activities. We have thereby provided an Idealtypus of a private ownership economy. DEFINITION 17.4. An allocation ( 2 ' , . .., 2', j ' , . . ., 9") is said to be blocked by a coalition S if one can allot a bundle x i in X i to consumer i E S and find a process zs in Z , such that .$+,ti

(iES),

i

x i > ,ti (for some i E S),

(7)

i

Intuitively, the blocking of an allocation by a coalition S means that some consumers, discriminating against the consumers outside the coalition, cartelize an autarchic subeconomy which possesses an initial holding ai and a technology set Z, at their disposal, and in which there is an allocation making no consumer involved worse off while bettering at least one consumer involved than in the original allocation in the entire economy.

xi€,

DEFINITION 17.5. The core of an economy is defined as the set of all allocations that cannot be blocked by any coalition. We note, in view of Lemma 17.l(ii) that a Pareto optimum allocation is nothing more than an allocation that cannot be blocked by the coalition involving all consumers. Therefore, an allocation in the core must be a Pareto optimum, whereas a Pareto optimum allocation need not belong to the core. THEOREM 17.3. Suppose that no consumer is satiated. Then, if (2', . . . , 2', j ' , . . . , j", j3) is a competitioe equilibrium (see Definition 16. I), the corresponding allocation (2', . . . ,2',j l , . . . ,9") is in the core. Pvoof. The proof is similar to that of Theorem 17.1. Assume the contrary. Then, the allocation could be blocked by a coalition S , so that we have (6), (7), and (8) for some x iE X i (i E S ) , zs E Z,. Then by arguing in the same way as was done in obtaining (6) in the proof of Theorem 17.1 while confining the summation to S, we find m

288

V. WALRASJAN COMPETITIVE ECONOMJES

O n the other hand, the zs in Zs can be put in the form m

zs = isS k = l

y i kE Yk (i

aikyik,

E

S, k

=

1 , . . . , ni).

(10)

Noting nk(j3)2 (j3,yik) by condition (LX) in Definition 16.1 and using the expression (lo), we derive from (9)

But ( 1 1) contradicts

which follows from the premultiplication of (8) by the transpose of j3 2 0, Q.E.D. In order to establish a converse connection from the core back to competitive equilibria, we need a concept of a repeated economy, which, for convenience, will be called a replica of a given economy. We denote by 6 an economy fulfilling assumptions (C.1)-(C.IV), (P.l), and (C-P), and consider an r times repetition of G‘ for r , a positive integer or a denumerable infinity. Explicitly, we put forth

DEFINITION 17.6. Ati r - f d d replica G’ o f an econonij G is &$tied to be a compourrci econoniy composed o f r subecoiiomies G ( v ) ( v = I , . . . , r ) icleritical to the original 8. All cotistituents iti G (v), such as coiisuimrs, proclircers, their prefiretice fields, tecliirologj sets, iriitial holcliiigs, and relatitv shares are clistim~guislietlby the tiuniberiiq v froit? the corresponditig iiletitical ones iii other suhecor?ot?iies. Witli this cotiiietitioti, 8‘ as mi econot?iy fulfillitig (C.I)-( C.IV), (P. I), oiid (C-P) is,formulated in the,following way: (C) Consumers. There are lr coiisumers doubly sufixed by (i, v ) , i = I , . . . , I; (C.cc)

Each coiisut?ier ( i , vj’s preference f i e l d is gicer? by k ( i , v\there J,

Xi(vj = X i ,

+ =>

(i,v)

( i = 1 , . . . , I ; v = 1,

. . ., r ) .

(Xi(v), (12)

i

(C.fl) Euclz cotisutiier ( i , v) lias an iiiitial lioldiiig

ai(v)= a’

( i = I , , . ., I ; v = I , .

. ., r ) .

(13)

(P) Protlucecs. T l w e are tiir proclucers doiihlj’ sufi.wd 6). (li, v), k = I , . . ., t ? l ; v = I,...,/”. (P..*) The techiiology set of each producer (I?, v) is g i i w i by Y,(v) = Yk ( k = 1, . . . , 111; v

=

I , . . ., Y).

(14)

$1 7.


289

(C-P), lmr' constant relative shares are gicen by

p , v = 1) . . . )r ) ,

(1 5 )

where C ( i p k v stands for the relative share qf coiwumer (i, p ) in the projit of producer ( k , v). Thus, a replica &' involves r homogeneous consumers of each type i corresponding to consumer i in 8,and r homogeneous producers of each type k corresponding to producer k in G. It is also important to observe that a consumer (i, v)'s ownership of initial holdings and production possibilities is restricted within his subeconomy but does not extend to other subeconomies. We note that condition (Ca) corresponds to (C.1)-(C.1I I ) , and conditions (C.p), (Pa),and (C-P), are counterparts of ( C I V ) , (P.l), and (C-P), respectively. Therefore, an economy 8'and its replica 6' are of the same category as far as the satisfaction of (C.1) through (C.IV), (P.l), and (C-P) are concerned. We denote by Cor. (g)the core of 6. Whence, Cor. (el)is well defined for a finite r . Then we have immediately, as a direct application of Theorem 17.3 to A',

COROLLARY (to Theorem 17.3). Szri7po.w t h t 170 cor~sznner qf' L is satiated, arid that Y is a j n i t e positire integer. Tlien tlic co:iiiwtitiw allocation in 6''effected bj, a conyetitire equilihriinii in 6' is ii! Cor. ( d r ) . Pmuf. Since no consumer of C" turns out to be satiated, this corollary is a restatement of Theorem 17.3 for L', Q.E.D.

I n what follows, Tor the sake of brevity, we occasionally write (xi,y') and (xi,y', p ) for an ( I + nz)-tuple ( X I , . . . , XI, j", . . . ,y"') and an ( I + nz + 1)-tuple (x',. . . , .Y I ,y 1 , . . . , yl", p ) , respectively. Then, an allocation i n d " is given by a sequence {(xi(v), y k (v))>:= I of ( I + nz)-tuples (xi(v), I,'( v)) such that x'(v) E X i ( i = 1 , . . . , I ; yk(v)E

r,

(k

=

11 =

I , .. . ) / , I ; v

1 , ...,r), =

(16)

I , . . . )r ) ,

(1 7)

,

Therefore, in particular, a sequence {(.xi(v), .y'( v))j:= of allocations (xi(\>), yk(v))i n 6(v)is also an allocation in 6'. More especially, if (xi, I,') is an allocation in 6,the sequence {(xi(v), j i k ( v ) ) ) ; = defined by (xi(\!),J.'(Y)) = (xi, y ' ) ( v = I , . . . , r ) , which we call the r_fold repetition of (xi, J") and denote by (xi,yk)r, is an allocation in 8'.

290


LEMMA 17.2.

If

{(xi(v),yk((v))}:, is an allocation in B', the (I

+ m)-

tuple

uhich will he called the average of the original allocation, is an allocation in 8.

Pvoof: We have by assumption ( 1 6), (1 7), and (1 8) for the given allocation in 6'. Then, (16) and (17) entail

because o f the convexity of X i ,Y k . Moreover, (18) can be rewritten

Whence the average (19) is an allocation in 8, Q.E.D. LEMMA 17.3. Let {(.ti(\)), j3'(\!))]:=

be in Cor. (6"). Then,for each i, the

r cotiitiiodity huiiclles allotted to cotisut?iers of t j p e i are mutually itidifferetit, i.e., ,fiw each i HY' hare 2 i ( p ) ti(\!) (,u, v = I , . . . , r ) .

-

Pvoof. For each i, let .2i( be the least preferable among the r commodity bundles i i ( v ) ( v = I , . . . , r ) in the preference k i ,which is common to consumcrs ( i , 11)(\, = I , . . . , r ) . Then, we have, by definition, \qi)

.ti(\!) 32(Vi)

(i

=

I,

. .., I ; v

=

I , . ..,r ) .

(20)

i

Whence

by the con\exity of 2, ensured by (C.11) and (C.111) (see Lemma 15.5). Moreover, if, contrary to the assertion, the strong preference held in (20) for some i , say .P, and some we would have 19,

Consider the coalition S = { ( I , (2, vJ, . . . , (I, in 6'. This coalition involves I types of consumer, exactly one for each type, so that, in the light of Lemtna l7.l(ii), it has ill),

I

ill)}

m

5 17. NORMATIVE IMPLICATIONS OF COMPETITION

29 1

at its disposal as its initial holding and technology set. By Lemma 17.2, the average

is an allocation i n 8,and therefore is attainable by the coalition S. Then (21) and (22) show that S blocks the allocation { ( a i ( v ) , j k ( v ) ) } : = , contradicting its being in Cor. (ar),Q.E.D.

DEFINITION 17.7. A consumer i in & is said to be positiueiy endowed if there are a commodity bundle bi in his consumption set X i and a process w i in his technology set Z i such that ai

+ w i> b'.

(23)

We shall now state the following important result: THEOREM 17.4. Suppose that no consumer in W is satiated and that each consumer in 8 is positively endowed. Let { ( a i ( v ) , j k ( v ) ) } , " = , be at? infinite sequence of allocations ( a i ( v ) , j k ( v ) ) in b ( v ) ( v = I , 2, . . .). Then, if {(P(v), j k ( v ) ) } : = is in the core, Cor. (ar),ofthe r-fold replica 6' o f 8 , f o r each positiile integer r = I , 2, . . . , ad it$, there is a price rector p 2 0 such that all ( i ' ( v ) , Y k ( v ) ,fi) are competitiue equilibria in b. Proof.

In the light of Lemma 17.3, we find a'(p)

-

i'(v)

(i = I , . . . , I ; p , v

=

1,2, . . . ,ad in$).

i

(24)

Let M,(v) = ( x i I xi E xi,xi 2 a i ( v ) ) , i

M ) ( v ) = {xi I x i E x;, xi > a i ( v ) } 1

( i = 1, . . . , 1 ; v = 1 , 2, . . . , ad inf.).

Clearly, insatiability implies Mi *(v) # @. Moreover, (24)ensures that for each i we have M , ( p ) = M i ( v ) , M * ( p ) = M,*(v) ( p , v = 1,2, . . .). Whence we may rightly put for each i M i = Mi( I ) = M i @ ) = . . . = Mi(\,) = . . .,

MT

=

MT( I )

= MT(2) = . . . =

M:(\!) = . . . .

(25)

Recalling (2), we define I nonempty sets

G i= { a ' }

+ Zi

-

MT

( i = 1, . . . , 1).

(26)

292


G iis convex because it is a linear combination of { a ' } , Y, ( k = 1, . . . ,m),Mi" -all

of which are convex. We then consider the convex hull

of the union of G i( i = I , . . . , /). It can be readily seen from the convexity of Gi that is the set of all convex linear combinations of the following form: 1

1

/zizO, u ' E G ;

CAitri, CAi=l, i= I

i= 1

(i=1,

..., I).

Next we shall show that r contains n o positive vector. Otherwise there would be a positive vector of the form (27). This, combined with (26), gives rise to the validity of 1

1 &(a' + zi - xi) > 0 i= 1

(28)

for some ai >= 0, C / l i = 1, zi E Z i , xi E M: (i = 1 , . . . , I ) . It is a well-known fact that Ai (i = 1, . . . , Z) can be arbitrarily closely approximated by I positive rational numbers of the form /

ri/

c rs I

(i

=

I , . . . , 11,

s= I

where ri (i = 1, . . . , I ) are positive integers. This enables us to replace Ai in (28) by r i / c rs to still obtain

for some positive integers r i , some processes z i E Z i , and some commodity bundles x iE Mi*(i = I , . . . , I ) . We decompose the set R of all positive integers from 1 to r , where I

r= Cri, i= 1

to a union of / mutually disjoint nonempty subsets R i( i = I , . . . , /) in such a way that R i contains exactly r i elements. We then consider the coalition S = ({i} x R,) (the union over all i = 1, . . . , I ) , in 8' consisting of ri consumers of type i and therefore r consumers in total. The coalition S has a t its disposal the initial holding

u

5 ria

i= I

$17.


293

and the technology set

ri

c (Zi + . .. + Zi).

=

i= 1

Since

inequality (29) shows the attainability of an allotment within S that assigns xi with every consumer involved in S of each type i. Moreover, xiE M: and (25) imply

x i > i i ( v ) ((i, v)

E

S).

i

Thus the coalition S blocks the allocation {(?'(v), 9'(v))}:= I , contradicting the assumption. The inclusion of a positive vector in r is thereby ruled out. The convex set is therefore, by Chapter I, Theorem 3.5, separated from the nonnegative orthant by a hyperplane with a semipositive normal j? 2 0. It will be shown that j? is a desired equilibrium price vector. We have ( j ?

11)

5 0 ( u E r) which, combined with the inclusion of Gi in

r, implies, in particular,

+ ( b , hc= cxikyk) 5 ( j ? , xi) rn

( j ? . a')

( i = 1,

... 1 ) )

(30)

1

for any siE M T , yh E Y , . But (30) is valid even for xi E M i . This extension of ranges of validity from Mi*' to M i which is justified i n exactly the same way as i n the transition from h4: to M , while keeping (8) remains valid in the proof of Theorem 17.2. We derive several results from (30) that have turned out to hold for any x i E h4i = Mi( r) ( v = I , 2, . . . , ad iuf:),y' E Y , ( k = I, . . . , ni). T o begin with, by substituting a'( v), $'( v ) for xi,y', respectively, in (30), we find

,"

( j ? , a'>

+ (j?, 1 h= I

ril,

jk(v)) 5 (j,a'(\>)>,

which, by virtue of (I),s u m up to I

I

(31)

294


On the other hand, the right-hand side of (32) must not be larger than its left-hand side as is clear from j 2 0 and 1

1

m

c ui + c P ( v ) 2 1ayv). k=l

i=l

i= I

Whence all of (31) and (32) prove to be equality relations. Therefore

Froin (30) and (33), we find m

( B , k1 u i k . y k > 5 (4, = 1

in

1

( y k E yk)

fiikEk(Ll))

k= I

( i = 1, ..., l),

c

(34)

Ill

(6,a ' ) + ( j , h = 1 a ' k j q v ) ) 5 (b,

xi)

(xi E M , ( v ) )

(35) ( i = I,...,[).

Summation of (34) over all i enables us to realize that (j?, maximum of ( b , y) over Yk and therefore to conclude that (2)

j?(v)

j ( v ) ) is a

1 maximizes (a, yk) over Y,, the maximum profit being nh(j?)=

( j , J ' ( V ) )for each k = I , . . . , t n and each v = 1 , 2, . . . , ad in$ Let us now take into account that every consumer in d is assumed to be positively endowed. Then using hi, \vi i n the assumption of positive endowment in view of (34) and (a),just proved above, and (2), we see ( j , b ' ) < (3,

.i)

+ (p, w ' ) m

5 ( j , rr')

+ 1 g i k n k ( j j ) ( i = 1, . . . , I ) .

(36)

h= I

Now for any x iE X isatisfying 111

(jj, x'j 5 .

(p, d) + k1 aiknk(fi), =l

.

(37)

let x ' ( t ) = ( 1 - [ ) x i + th'. Y(f) E X i ( 1 2 t 2 0) by the convexity of X i . Moreover, (36) and (37) ensure

which, combined with (35) and ( a ) , implies .?(v)

> .xi(r) 1

x'(f)

( I 2 t > 0) ( i

$ M j ( v ) .Whence

= 1 , . . . , I),

$17.


which become in the limit

2(v)

+xi

(i

=

I , . .., 1 )

295

(38)

i

as t -+ 0. This proves: (p) a'(\>)is a most preferable element of X i subject to the budget constraint (37) for each i = 1, . . . , I and each v = 1, 2, . . . , adit$: Finally, since (i'(v), jh(v)) is an allocation in 8,for which (32) has proved to be true in an equality form, we also have

with equality holding in thejth component relation of (39) if the corresponding price r j j is positive (v = I , 2, . . . , ad in$). We have thereby shown that all (i'(v), j k ( v ) , r j ) satisfy conditions ( E ) , (p), (7) in Definition 16.1 and hence competitive equilibria in 8.This completes the proof, Q.E.D. From Theorem 17.4 and the Corollary to Theorem 17.3, we can characterize a competitive equilibrium as

THEOREM 17.5. Witli the same assumptions on iiisatiabilitj~and positire eiido\t.ment as in Tlieoreni 17.4, in order that witli an allocation (i', j h )in an economy 6' there be associated a semipositire price rector that lets (i jk, 'r, j) be a cornpetitire eyuilibriun? in 8,it is necessary arid suflcieiit that its r_Jold repetition (i jk)' ', belong to the core oftlie r-fold replica 8''O j G ,for atiy positire r = I , 2, . . . , ad it$ The core of 6"may include noncompetitive allocations; but, as r increases, every individual becomes more and more fractional in er, and more noncompetitive allocations are excluded from the core of 6' until only competitive ones remain. These results, which have been proved above, justify what economists have in mind as a rationale underlying the hypothesis of pure competition. Finally, it is argued that we could conceive of the equivalence of competitive equilibria and allocations in general in the countably infinite-fold replica P , not confining ourselves to allocations of a special type as considered in Theorem 17.4. This might cause some difficulty, however, because we are confronted with infinite series problems in connection with definitions of aggregate supply and demand. In order to consider a general situation in which every individual is infinitesimally fractional, it is better to conceive the set of individuals as a nieasure space, and an allocation as a mapping satisfying some measure-theoretic conditions, rather than to adhere to 6 %. Decisive results on the core of an economy in this measure-theoretic line have been obtained by Aumann (1964, 1966), and Vind (1964).

296


Remark 1. Two allocations j k ) ,(u', r') may be referred to as equivalent allocations if .Y' - , u i for each consumer i. Clearly, any one of two equivalent allocations is in the core if and only if the other is in the core. Remark 2. I n Lemma 17.3, we can also readily see that (XI,

,I; p

=

I , ... , r ) .

To this end, we observe that Lemma 17.3 and the convexity of X i ensure 1

r

On the other hand, by Lemma 17.2, the r-fold repetition

of the original allocation is an allocation in 8'. Therefore, if strong preference prevailed in (40) for some (i, p ) , the original allocation could be blocked by the set of all consumers in G', contradicting the belongingness of the original allocation to Cor. (nr). I f we then also take Remark 1 into account, we realize that with any allocation i n the core of 6' is associated an equivalent allocation of a special type, namely the r-fold repetition of its average, which must also be in the core. Remark 3. Alternative approaches to game-theoretic elucidation of the Walrasian competitive equilibrium have been under way by Shapley and Shuhik. One typical result is that of Shapley (1964) and Shapley and Shubik (1966) on the convergence of the Shapley value solution (Shapley, 1953) of a decentralized economy to i: competitive equilibrium as the economy becomes more and more replicated.

Chapter VI

SPECIAL FEATURES OF COMPETITIVE ECONOMIES : GROSS SUBSTITUTABILITY

This chapter deals with one specific property, gross substitutability, possessed by certain special types of competitive economies. Special results, static as well as dynamic, are proved under gross substitutability. By means of extremization methods, $1 8 establishes the existence of a competitive equilibrium, and the validity of Samuelson's weak axiom of revealed preference between an equilibrium price vector and a disequilibrium price vector. $19 presents an exposition of global stability in gross substitutes systems, based on the results of Arrow, Block, Hurwicz, and Uzawa.

$18. GROSS SUBSTITUTABILITY A N D EXTREMIZATION

$18.1. Gross Substitutability

In this section, we shall pay attention to a special preference field, ( X , 2) where X = R+",and 2 is assumed to satisfy: (s()

k

(y)

k

is increasing, that is, .Y 2 j' for x, y

E

R," implies s 2 y.

(p) 2 is strictly convex in the sense of (V) in Chapter V, $15.2, that is, x 2 y for distinct x, y E R," implies E X + py > y (u > 0, p > 0, E + p = I). is continuous.

A preference field of this type is standard in traditional economic theory. We write p , I for an dimensional price vector and a level of the income

of a consumer. If his preference field is of the above type, his individual demand function 4 ( p , I ) , defined by (4) in Chapter V, 31 5.2, has the following properties in the set of all p > 0, I > 0: 297

298

VI. GROSS SUBSTITUTABILITY

(i) Each image set q!I(p,I ) is nonempty and consists of a single commodity bundle. (ii) + ( p , l ) is continuous with respect to the argument ( p , I ) , where p > 0, I > 0.

A justification for the statements (i), (ii) is as follows. A consumer's range of choice M ( p , I ) = {x I x 2 0, ( p , x) 5 Z } is a nonempty compact, convex subset of the consumption set R," for each ( p , Z), so that by Chapter V, Lemma 15.3, M ( p , Z) contains a most preferable element. Moreover, by Chapter V, Lemma 15.6(ii), there is at most one more preferable element in M ( p , I ) . Whence (i) follows. On the other hand, (ii) can be readily seen by the same argument for the proof of Chapter V, Lemma 16.4(iii), although in this case the range of choice is confined to no cube, and I is an independent variable. We have thereby obtained a single-valued continuous mapping q!I(p,I ) from the set of ( p , Z) with p > 0, Z > 0 into R,", with whose behavior responding to the change of the argument value we shall be especially concerned below. T o this end, we define a useful auxiliary mapping a(q, a), defined for any n-dimensional positive price vector q and any commodity bundle a E R+", which is also of independent economic significance. This is defined by the rule

o(q, a ) = the unique element of R," minimizing (q, x) on R," subject to x k a .

(1)

It should be noted that M(q, a ) = {x I x 2 0, x 2 a, ( q , x) 5 (4, a ) } is a nonempty compact set, and therefore the continuous function (4, x) is minimized at some of its points, say x = 2. (q, 2 ) is obviously the minimum of ( q , x) over all x 2 0, x 2 a. Moreover, 2 is uniquely determined. Let 2, j E R,", 2 # and suppose that both of them minimize ( q , s) on R," subject to x 2 a. Without loss of generality, we may assume 2 2 j . Then 2 2 j , 2 # j entail t(2 9) > a by strict convexity (/,I). f ) is not zero, because 2 2 0, f 2 0, X # f. Hence, if we choose .Y i n R," subject to +(a + j ) 2 x and sufficiently closely to + f), we still have .Y > a by the continuity of 2. For such an x, we find ( q , x) < ( q , 2 ) = ( q , j ) because of the positivity of q, which contradicts the minimizing property of 2, f. These results allow the definition of a(q, a ) as a single-valued mapping by ( I ) to make sense.

+

+

LEMMA 18.1. a(q, a ) ( q > 0, a 2 0 ) Ims tlie,folloiviirgproperties. (i) d q , a ) a. (ii) a(q, a ) is coiitiiiuous ii 0 unless a = 0. 1 ) ) = 4 h 0 ,J ( P , 1 ) ) = I f o r P > 0, 1 > 0. (v) (vi) +(q, J ( q , a ) ) = a(q, a ) prouided J ( q , a ) > 0. (vii) ( A q , A a ) 5 0, where q > 0, q Aq > 0, ACT= a(q + Aq, a ) - a(q, a). Proof. (i) Naturally, we have by definition o(q, a) a. Now suppose o(g, a ) > a. This supposition rules out the possibility that 4 9 , a ) = 0, because a(q, a ) = 0 implies a 2 o(q, a ) , which entails a 2 a(q a ) by ( a ) . Thus, we must have a(q, a ) 2 0. Hence, if we choose x in R," subject to o(q, a) 2 x and sufficiently closely t o o(q, a), we still find x > a for such an x by the continuity of k . But the positivity of q yields ( q , 4 9 , a ) ) > ( q , x), contradicting ( I ) . (ii) Consider arbitrary sequences {q"},{ a " } with lim q' = q, lim a" = a. we shall have t o show

do,

do, +

z

lim a(q", a") = a(q, a ) .

(4)

v++m

To this end, we first note that {o(q", a")} is bounded. We have by definition (q", c(q",a')) 5 (q", a). It follows that 0 5 a,(@, a) 5 (1/4,y)(4",a>

+

(l/qJ(q, a>

(\J

-+

+ m),

ensuring the boundedness of {a(q",a")}. In order t o prove (4), we therefore have only to show that every convergent sub-sequence of {o(q",a ' ) } converges to a(q, a). Hence it suffices to consider the case where the original sequence {o(q',a')} itself is convergent. Next we note that x 2 y for x, J' E R," implies x > y by virtue of (a) and (p), and hence insatiability prevails. Relying on insatiability we choose such a 11' once for all that H' > a. For any x satisfying x 2 a , we let x ( t ) = ( 1 - t ) x t w . Then,

+

x(t)>a

(1 > t > 0 )

(5)

by (V) in Chapter V, $15.2 implied by (/I). If we consider ( 5 ) for each value o f t and take lim a' = a into account, from the continuity of 2 we see x(t)

> a'

(6)

for this value of t and sufficiently large 1'. Whence. by the definition of o(q",a'), we find for t and sufficiently large v (4">4 t ) ) 2 (q", d q " , a")).

(7)

300 Letting


ti-

+ co in (7), we obtain the limit

= o,(q, Ql as was to be seen. (iv) The supposition a(9, a ) = 0 would imply a 2 0 = a(q, a ) and hence a > o(q, a ) by (a), (/I). This contradicts a(q, a ) a. Thus

J ( q , a ) = ( q , 4%a ) ) > 0 because q > 0, ~ ( qa ,) 2 0, whenever a 2 0. (v) First we note that insatiability ensures

$18. GROSS SUBSTITUTABILITYAND EXTREMIZATION

30 1

Otherwise an x satisfying x > 4 ( p , I ) and sufficiently close to 4 ( p , I ) would also fulfill the budget constraint. On the other hand, we must have x > (p(p, I ) for such an x by (cL), (p). This contradicts the definition of I$@,f). We must now show that x 2 $(p, I ) implies ( p , x) 2 ( p , 4 ( p , I ) ) . Taking a NJ satisfying I$’> 4 ( p , I ) and letting x ( t ) = (1 - t ) x + t w for each x 2 4 ( p , I ) , we obtain, as in the proof of (ii), X(t)

>4(P, 1)

(1 > t > 0).

(13)

4 ( p , I ) and (12), (P,x(t)> > I = ( P , 4 ( P J ) ) (1 > t > 01,

Relation ( I 3) yields, by the definition of

which reduces, in the limit, t o

.>

2
$ ( / I , I > >

as t 0. Finally, Jb,&P, 1))= ( P , o h 4 ( p , 1 ) ) ) = =I.

( 4 , -y> 5 J ( q ,4,

we let .r(t) = t s . Then, sinceJ(y, a ) > 0, we find (y, ~ ( t ) t which rules out x ( t ) 2 a. Hence

>= 0),

o(q, a) 2 a > x ( t ) (1 > t 2 01,

which becomes by continuity 4 % a) 2 a 2 x

as t + I . (vii) Similarly to the argument for (iii), we find by definition (9,4Y (9 +

&?

+ Ac73 0 ) ) 2 < 4 , 4 Y >a)), 4 % a ) ) 2 ( 4 + Aq,CT(q + Aq, 0 ) ) .

Whence ( 4 , A o ) 2 0 2 ( q + Aq, ACT).

Hence, by subtracting the leftniost term from the rightmost one, we arrive at ( A q , ACT)5 0, Q.E.D. From Lemma 18.I , the economic significance of o ( q , a ) is now clear. For an arbitrarily fixed a, J(q, a ) is the income level that guarantees under the price vector q the same preference level as that of the commodity bundle a, the corresponding (unique) optimum commodity bundle being n(q, a). Using o(q, a ) and J ( y , a), we can decompose the change of demand A4 = 4(P

+ AP>I ) - 4 ( P >I ) ,

302


responding t o a change Ap in price to two terms in the following way. Performing the decomposition

A4

= 4(P

+ AP, I ) - 4(P, I )

+ A/]> I ) - 5(P + AP, 4(P>1 ) ) + 4 P + AP>4(P, 1))- 4 ( P >I ) ?

= 4(P

and noting

4 ( P + AP, I ) = 4(P + 4 1 , J(P> 4@,I ) ) ) , + AP, 4(/& 1 ) ) = 4@ + A/& J ( P + AP>4(P> I))), +(/& 1 ) = 4 / 1 7 4 h 111,

4 P

ensured by Lemma 18.1, we finally find

A4 = 4 ( P + AP? 4 4 =

-(P7

1)

+ A/& J ( Y >4(P>1 ) ) )- 4 ( P + AP>J(P + AP, 4(P, 1 ) ) ) + 0+ AP? 4 h 1 ) )- 4 P >4(P, 1)).

(14)

4 P + AP, 4(LJ+ AP, J ( P , 4(/& 1 ) ) )- 4(P + AP? J(P + AP, 4 h 1 ) ) ) and 4(p,I )) - o(p, 4(p, I ) ) are called the inconie eflect term and the suhstitutioii

ejjGcr term, respectively. Equation (14) is the Slutskj. equation in the finite increment version. A change Ap in price affects the behavior of a consumer in two ways. It entails a change of price ratios that induces substitution among goods i n the optimum men~iof consumption, and it causes a change i n his real income, or his purchasing power, although the nominal level of income remains unchanged. Therefore, the substitution effect represents the shift of the optimum menu of consumption within the indifference class to which the original + ( / I , I ) belongs. J ( p Ap, $(p, I)) is the corresponding (virtual) level of nominal income that keeps the consumer within the same indifference class as before the price change, even in the new price situation p + Ap. Thus, the income effect represents the shift of the optimum menu of consumption, in the price situation 1) Ap, responding to the change of income level, from the virtual J ( p Ap, (p(p, I ) ) t o the actual J ( y , 4(p, I ) ) .

+

+

+

THEOREM 18.1 (Slutsky, 19 15). Assume fhar tlie deemand function $ ( p , I ) ( p > 0, I > 0 ) //as partial c/erii.atii.es (!4;/?pj, (i,j = I , . . . , t i ) atid continuous partial dcrii.atii.e.s &bi/?I ( i = I , . . . , n). Then, we kaiv the Slutsky equutio~iin tlie d(f~erentia1i-evsioii

4 18.

GROSS SUBSTITUTABILITY AND EXTREMIZATION

303

where

A j p = ( 0 , . . . , O , A p j , 0 , . . . ,O)'

By the mean-value theorem in calculus, and Lemma lS.l(iii), the income effect term can be rewritten as

where

#i,

is by assumption continuous, whereas aj is also continuous by Lemma

18. I(ii). Hence, in view of (IS), we have

because of Lemma lS.l(v). On the other hand, we have by definition

304

V1. GROSS SUBSTlTUTABlLlTY

Therefore the limit o i j of substitution effect term

as Apj 3 0 also exists and equals

This proves Eq. ( 1 5). In the differential version of the Slutsky equation (15), as in its finite increment version, we also call - $ j d q 5 J d l and crij the income effect term and the substitution effect term. We observe, in the light of Lemma lS.l(iii), that if oi(q,a ) have partial derivatives (i,j = I,

. . . ,n )

that are continuous with respect to q for each fixed a, we find

( i , j = 1 ,..., n ) .

In such a case, we have the symmetry of the substitution terms in Theorem 18.1 0 i.J.

= 0.J 1

=

'5

" ' 3

">

from (17). On the other hand, we can say nothing definite about the signs of uij except for crii 5 0 ( i = I , . . . , I?), which is immediately implied by Lemma lS.l(vii) for q = y , Aq = A i p and (17). Based on the symmetry of the substitution terms, Hicks (1939) put forward definition of substitutability and complementarity as follows. The ith and j t h goods are substitutes or are complementary according to o i j > 0 or oij < 0. As a matter of fact, the intervention of the income effect makes the general determination of the sign of ?4J?pjdifficult, even if the sign of oij is known. In this connection, the concept of gross substitutes, which was advanced by Hicks (1939) and Mosak (1944), is of interest. The ith good is termed a gross substitute for the jth good if ?4J?pj > 0. Here the word gross means the actual total effect, including the income effect, as opposed to the (net) substitution effect along an indifference surface (class).

6 18. GROSS SUBSTITUTABILITY AND EXTREMIZATION

305

It should be noted that substitutability and complementarity, as well as gross substitutability, are concepts whose definitions depend on the individual consumer and the price-income situation ( y , I ) . However, we can also define counterparts of these three concepts for the aggregate demand function by means of the signs of the sum of the individual substitution effects and the sum of the individual total effects, respectively. I n particular, gross substitutability is conceivable for the aggregate excess demand function, even if supplies of goods by production intervene, because this concept does not premise a decomposition to the income effect and the substitution effect of the change of the aggregate excess demand but is based on the total change itself. Moreover, this concept may be intensified to its overall prevalence throughout a range of price-income situations. Usually the individual incomes enter an aggregate excess demand function as functions of prices rather than as independent variables. More recently, a weakened version of gross substitutability, called weak gross substitutability, requiring only d4JdyJ 2 0 (i # j ) has received much attention by several authors, particularly Hahn (1958), Negishi (1958, 1962), Arrow er al. (1959), Nikaido (1959c, 1964c), Arrow and Hurwicz (1960, 1962), McKenzie (1960a, b), Morishima (1960a, 1964), Uzawa (1961) and Kuga, 1964, 1965). Their results seem to suggest that it might be better to modify the terminology in such a way that we refer to weak gross substitutability simply as gross substitutability and the original version of gross substitutability as strong gross substitutability. With the foregoing as background, we now have DEFINITION 18.1 (Gross Substitutability in the Finite Increment Version). Giiwi ti excess deematid functions E l ( p ) ( i = I , . . . , ti)

(19)

ti-ciiiiieiisional price rector p = ( p , ) , dejned oii a conitiioti doimiti P, that gross substitutabi/itji prerails iii P if for any y , q E P satisfying p 2 q we Iiaw of the

1i.e SUJ~

E L p ) 2 E,(q) (i # N P> 4 ) = { iI PI > q , } ) .

(20)

DEFINITION 18.2 (Gross Substitutability in the Differential Version). (19) hare partial dericatires dE,/c?p, (i,j = 1, . . . , t i ) in P, tie SUJ~ that gross substitutability precails iti P if tile haue

If E,(p) in

everywhere in P.

306


DEFINITION 18.3 (Generalized Gross Substitutability). If there exist partial tieo.iiwtir.es i E l / 2 p , ( i ,j = I , . . . , n ) in P as in Dejnition 18.2, tt.e say that yeneralizetl gross suhstitutabilitj>prevails i i i P i j itse hai3e (22) ererj'ii l7rre in

P. Remark 1. Gross substitutability in the finite increment version is nothing but an alternative nomenclature of weak monotonicity (6,) in Chapter 111, $10.1. But the former seenis to fit the present economic background better.

Remark 2. I t is easy to verify that in the case of the existence of the partial derivatives ?El,'?/), , Definitions 18.1 and 18.2 are equivalent. Verification will be left to the reader. Remark 3. Generalized gross substitutability includes as special cases gross siibstitutability i n the differential version as well as sh e~~,-sj,i?uiietricifj., which means

Nikaido (I964c) seems to be the only source dealing with generalized gross substitutability. It is difficult, however, to conceive a workable finite increment version of this concept. Remark 4. I n view of Remarks I and 2, we may reasonably carry over the concept of indecomposability (Chapter I I I , Definition 10.2) to gross substitutes systems. Explicitly, a gross substitutes system (a system satisfying Definition 18. I ) is said to be ii~tI~~c~oiiipo.vcihle if for any 17, 4 E P satisfying p 2 (1 but / I 3 q, at least one inequality of (20) is a strict inequality for some i. When strict inequality holds for all i i n (20), the system is called a strong gross substitutes system. For differentiable cases, a gross substitutes system (a system satisfying Definition 18.2) and a generalized gross substitutes system (a system satisfying Definition 18.3) are said to be indecomposable if the matrices (?E,/?pj)and (?Ei/pj + ? E j / ? p t are ) each everywhere indecomposable i n P. The corresponding strong versions of gross substitutability are defined as the positivity of all off-diagonal entries of the above matrices.

EXAMPLE 1. A simple nontrivial gross substitutes system satisfying(]), >= O), ( I l l ) i n $18.2, and continuous differentiability can be given on the set of a11 positive price vectors p = (11;) by

( 1 1 ) ( 1 > nz

I

1

PiJ=l

where 1 > 117 2 0 and

n

"

Ei(p) = -

N,,

t il .J . p Jl . + " l-

1

LljiPi))l

( j = I , .. . , n),

j = 1

( i , j = I , . . . , n) are nonnegative constants.

(24)

$1 8. GROSS

307

SUBSTITUTABILITY A N D EXTREM!ZATION

If there are / consumers s = I , . . . , indicator

I, each of whom has

a preference

on the set of all nonnegative commodity bundles and a nonnegative initial holding vector a" = (a,"), the (aggregate) excess demand functions in the corresponding pure exchange situation are of the above type, with m = 0 and

The reader may readily verify these facts.

EXAMPLE 2. If n? = I , we can have nontrivial generalized gross substitutes systems satisfying (I), (11) (n7 = I ) , (111) in $18.2, and continuous differentiability on P = R+"\{O). The simplest example is a skew-symmetric system with

c aijpj n

E;(p) =

( i = 1, . . . , 17);

j= 1

where ( a j j )is a nonzero skew-symmetric matrix. An equilibrium price vector (see Definition 18.4, below) of this system is nothing but a synonym for an optimal mixed strategy of the two-person game having ( a i j )as its payoff matrix. Every game having a general payoff matrix reduces to a game with a skew-symmetric payoff matrix [cf. Gale et al. (1 950)]. Gross substitutes systems have various interesting properties of economic and mathematical significance, which are not shared by general competitive systems such as those considered in Chapter V. These properties have been brought to light by the results of At-row, Hurwicz. and the other authors mentioned above. It is difficult to present all of them in full detail here, however, because of limited space. We shall therefore confine ourselves to describing two lines of results. One is the establishment of both the existence of a competitive equilibrium and the prevalence of Samuelson's weak axiom of revealed preference between an equilibrium price vector and a disequilibrium price vector, by means of further intensified utilization of extremization methods such as those given in Nikaido (1964~).The other is the presentation of a few representative results in 519 from among the numerous results on global stability in gross substitutes systems by several authors mentioned i n the paragraph preceding Definition 18.I . 918.2. Extremization and the Existence of Equilibrium

In this section, as well as in all the following ones of the present chapter, we are concerned with a system where there are given 17 single-valued numerical functions E j ( p ) (i = I , . . . , 17) of an n-dimensional argument vector

308

V1. GROSS SUBSTITUTABILITY

p = (I, 0: P is open in R," in the relative topology of K,". The excess demand functions are homogeneous of a common order

1 2 171 2 0. That is, Ei(ip) = i,"'E,(p) for p (111)

E

P , 1, > 0 (i

=

I,

. . . , 17).

The Walras law in the narrow sense holds. That is. n

2 piE,(p)= 0

everywhere in

P.

i= I

Remark 1. The simplest example of P in assumption (1) is the set of all semipositive vectors. The second simplest example may be the set of all positive vectors. In general, howeker, P can be of more complicated structure than is usually premised. P need not be convex and may even have holes in its intersection with the normalizing hyperplane p , = 1 . As will be shown later, if the excess demand functions are continuous as well a s homogeneous of order 0, there is no gross substitutes system in the whole set of all semipositive price vectors other than the trivial one consisting of the identically vanishing excess demand functions. On the other hand, under gross substitutability or generalized gross substitutability, we can obtain interesting results even for P of more general structure. These facts motivate a drabtic relaxation of the conventional restriction on the structure of P in ( I ) . Remark 2. I n traditional economic situations, of course we have i n assumption (11) (see Chapter V, $16.4).

iu =

0

Remark 3. Proposition ( 1 1 1 ) is one of the most important features of a Walrasian competitive economy, as was explained i n Chapter V. As a counterpart of Definition 16.6 in Chapter V i n the present setting of the problem, we have

$1 8. GROSS SUBSTITUTABILITY

A N D EXTREMIZATION

309

DEFINITION 18.4. A price rector 8 = ( p i ) is an equilibrium price uertor of a system of’ excess demand functions E i ( p ) on P if j? E P a i d

E , ( j ) 5 0 (i = 1, . . . , M), E , ( j ) = 0 provided p i > 0. It should be noted that (2) is implied by ( I ) on the basic assumption (111). Recall Theorem 16.6 in Chapter V, which proves, by means of Kakutani’s fixed-point theorem (a generalized version of that of Brouwer), the sufficiency of certain conditions of broad applicability for the existence of an equilibrium, and which is equivalent to Brouwer’s fixed-point theorem by virtue of Chapter V, Theorem 16.7. There seems t o be little need to reconsider the existence of equilibrium here. But the situation is a bit different. Note that though the mere imposition of the Walras law and the closedness (continuity in a general sense) o n excess demand functions in Chapter V, Theorem 16.6 is indeed hardly restrictive, the theorem does restrict the domain of thc excess demand functions t o an extremely special set, namely, the whole standard simplex, or-what amounts to the same thing under the homogeneity assumption-the set of all semipositive price vectors As was already noted in Remark 1 and will be explicitly shown later, there is no other gross substitutes system of continuous excess demand functions homogeneous of order 0 apart from the trivial system. Nontrivial gross substitutes systems can be conceived only on its proper subsets. This necessitates a reconsideration of the existence of equilibrium i n situations in which fixed-point methods of proof such as that for Chapter V, Theorem 16.6 w e not applicable because of the general structure of the domain of the excess demand functions. I n this connection, it is of interest that gross substitutability enables us to prove the existence of equilibrium in a domain of general structure as formulated in the basic assumption ( I ) . Results along this line, which are due to Nikaido ( 1 9 6 4 ~ )and Kuga (1965) will be given in more general forms in this section. Nikaido ( 1 964c) dealt with generalized gross substitutes systems. Later, Kuga (1965) gave an alternative proof for gross substitutes systems under 0th order homogeneity, relaxing the differentiability assumption in Nikaido ( 1 9 6 4 ~ )to continuity. Though Kuga’s result is interesting, it premises an extremely stringent boundary condition that is vital to his method of proof but that is invalidated even in some familiar traditional pure exchange situations (see Remark 6, below). Therefore we shall treat all interesting cases, including a generalization of his result, by the extremization methods of Nikaido ( 1 9 6 4 ~ ) .For most cases, the principal extremand will be the sum of squared positive excess demands, namely,

c O,(P)*, I,

@(P) =

(3)

, = I

O , ( p ) = max(E,(p), 0) ( I = I , . . . , 17).

(4)

310


Clearly we have, by definition, @(p)>= 0. Therefore, if I", is an equilibriuni price vector, @(@) = 0, and @ ( p )takes on a minimum 0 at p. It is also obvious that satisfies ( I ) if and only if @(p) = 0. Our method of proof is to see, by taking advantage of gross substitutability, that the extremum of @(p) (Theorems 18.2, 18.4, and 18.5) or an alternative extremand (Theorem 18.3), if any, is zero. As a matter of course, in proving the existence of an equilibrium in P, we need a sort of well-behavedness of the excess demand functions in P , such as their continuity or differentiability. In addition to this requirement, we need a further requirement for their behavior. We realize that P is a nonvoid open subset of R+"\(O), and, in view of the connectedness of R+"\(O), that P is not closed in R+"\{O) unless P = R+"\{O). Whence P has in general boundary points in R+"\{O}, which are not included in P, and on whose vicinities the behavior of the excess demand functions are further required to be nonviolent. These sorts of well-behavedness, which may be referred t o as boundary conditions, will be given below. In what follows, we mean by a boundary point of P that in the relative topology of R+"\{O],so that it cannot be the origin, but is a semipositive vector. Boundary Condition A . For atiy boundary point p of P and any sequence { p ' } in P that coniwges to p , there is an element i j in P , dependent possibly ON p atid { p " ) , ~ c that h p, = p, and

C:=

lim sup @(p") >= @(p). Y

+

(5)

+a

Boundary Condition B. For any boundary point p of P and afiy sequence { p ' ) in P which coiiiwges to p , 1t'e hare lim sup E,(p") =

+ co

(,for at least one

i).

v - + r

(6)

Boundary Condition C. In the case of a special P none of whose boundary points hare all con7potieiits positire, ,for any boundary point p , let K = {iI p i = 0). Then, ,for any sequence { p " ) in P contierging to p , we haw lim sup E i ( p v ) = +co

(,for at least one

i E K).

(7)

Y - + L

Boundary Condition D. I n the same setting of Boundary Condition C , ( 7 ) is replaced bj*

lirn sup E,(p") > 0 (,for at least one i E K ) .

(8)

Y'+T.

Remark 4. I n the above alternative boundary conditions, C is obviously a special case of B. One may readily observe that (6) amounts to the requirement that lim sup @(p")= +GO. Whence B is a special case of A. O n the

$18. GROSS SUBSTITUTABILITY AND EXTREMIZATION

31 1

other hand, C is also a special case of D. it is of vital importance to note that the i in (6), (7), and (8) may depend not only on the boundary point but also on the sequence { p " } (cf. Remark 6 , below). Remark 5. If the excess demand functions are bounded from below, as is often the case, Eq. (6) occurs only when the corresponding ith component p i of the boundary point in question is zero, and hence B is equivalent to C. In fact, if Ei (p) 2 y ( p E P, i = 1, . . . , n ) by lower boundedness, the Walras law implies pkvEk(pv)

5

-

c

1piv. lim sup Ek(p")= + co. We

i#k

pivEi(p") 5 -7

ifk

Suppose that limp" = p = ( p i ) , may assume lim E,(p") = +m, since this can be achieved by choosing some sub-sequence. > 0 for large 11,the above inequality reduces to Now, since

which becomes in the h i i t 0 5 pk 0. Hence p k = 0 if lim sup &(p") = +a, limp" = p . Tn proving the main results in $18.2, we need only Boundary Condition A for Theorems 18.2 and 18.4, whereas Theorem 18.3 premises Boundary Condition D. Before stating alternative existence theorems for price domains of general structure under gross substitutability, we note several simple facts, given in the following lemmas.

LEMMA 18.2. Let f ( x ) be a convex,fimction diflerentiable on a convex = f i ( x ) dxi . subset X of R", with the corresponding total diflerential being Then we haue

cy

Proof. We first observe that if x ( t ) is a differentiable function o f t on an interval of real numbers, with all its values in ,'A and if , f ( x ) has a total f i ( x )dxi in A', then 4 ( t ) =f(x(t)) is also differentiable in the differential interval, with the derivative

Applying this result to the given convex functionf(x) and x ( t ) = ( I - t)x + fy (1 >= t 2 0), we have

312


Clearly d ( t ) is convex on the unit interval. Thus by Theorem 3.15(iii) in Chapter 1 as applied to u = 0, v = 1, we have (9), Q.E.D.

LEMMA 18.3. The function

has the following properties (i), (ii): (i) f(x) is a convex function and has continuous partial derivatives

af

axi = 2 max(x,, 0) -

(i = 1 , .

. . , n).

(10)

(ii)

Proof. (i) It suffices to show that the function a ( t ) = (max(t, 0))2 has a continuous derivative 2 max(t, 0) in (- co, + co). First, o ( t ) as a function having a nondecreasing derivative is convex by Corollary l(i) to Theorem 3.15 in Chapter I. This implies that each (max(x,, 0))’ is convex on R”, whence the convexity off’(x) follows. Second, we find af/dx, = o’(xi) = 2 max(x,. 0). Now, since a(t) = 0 for t < 0, we have ~ ’ ( t=) 0 = 2 max(t, 0) in ( - co,0). On the other hand, a(t) = t 2 for t > 0, so that a’(t) = 2t = 2 max(t, 0) in (0, + a).At t = 0, o(t)has a left-hand derivative and a right-hand derivative, both of which are zero and therefore equal 2 max(0, 0). Hence o’(t)= 2 max(t, O), which is continuous. (ii) Since continuous differentiability implies differentiability, and continuous partial derivatives can serve as coefficients of a total differential, (ii) is implied by (i) and Lemma 18.2, Q.E.D.

We now proceed to the main results. The following theorem weakens the assumptions of Kuga (1965) in several respects, except for a slight strengthening by assumption (6) below.

THEOREM 18.2. There is an equilibrium price rector fi in P on the basic assumptions (I), (11) (in = 0), and (111) if ( a) Boundary condition A is assumed. (p) The excess demand,functions E,(p) (i = 1 . . . , t i ) are continuous in P. ( y ) Gross substitutability in the Jinite increment version (Definition 18.1) holds in P . ( 6 ) For arij. i, j = 1, . . . , n we haw

$18.


313

-

at a n y p in P, where

i

A j p = (0,. . . , 0, A p j , 0 , . . .,O)'. ProoJ

For the function @(p)in (3), let

6

= inf

@(p) over all p

E

P

satisfying

C p i= 1,

(1 1)

i= 1

and choose a sequence {p"} in P which fulfills

c piy n

=

1 (v

= 1,

2,

. . .),

(12)

i=l

lim @(p')

= 6.

"++a0

Clearly 6 2 0 by the definition of @. From ( 1 2) and the nonnegativity of p", (p'f is a sequence of the standard simplex P,, . In view of the compactness of Pn, we may therefore assume, without loss of generality, that { p " ] converges to a limit p in P,. This p may or may not belong to P . If p q! P, then p is a boundary point of P. Whence there is p , by boundary condition A, such that p E P n P,, and ( 5 ) holds. But, in the present case the left-hand side of (5) equals 6 by (13). Thus we end up with 6 2 @(p), p E P n P,, , which, combined with the definition of 6, implies @(p) = 6. On the other hand, if the limit p belongs to P , we must also have @ ( p ) = 6 from (13) and the continuity of @. We can therefore conclude that in both cases @ ( p ) actually takes on a minimum over P n P , . It should be noted that the minimum of @ ( p ) over P n P, is even the minimum over P because of the 0th order homogeneity (11) ( n 7 = 0). We have thereby shown that @(p) is minimized at some j? over all p E P. It will be seen that fi is an equilibrium price vector. To this end, it suffices to show the fulfillment of ( I ) by j?, since (2) follows from (1) by the Walras law (111). Suppose that ( I ) were violated by j?. Then, if we let K ( j ) = {i I E,(j?)> 0}, we have K ( j ) # @. Since P is open in R,", we find j? + A,p E P for m a l l Apj > 0. If we make Ap] > 0 sufficiently small, we still have by continuity

E,(j? + A j p ) > 0 (i E K ( $ ) , j = 1, . . . , n ) . Let us now consider the Walras law (Ill), which ensures

2 fii E,($ + A j p ) + A p j E j ( f i + A, P)

i= I

= 0.

(14)

314


Thus, upon subtraction, we find t l j j ( A p j ) l j j = E j ( B + A j ~ ) + C(-dij(Apj))$i i#j

( j = 1, . ., .

where dij(Apj1 =

Ei(lj) - Ei(B + Aj P> ( i , j = 1, AP j

..., n ) .

We may observe, from gross substitutability,

which, combined with assumption (d), implies the local boundedness of the left-hand side of ( 1 8), i.e., the existence of some positive constant M satisfying Iu‘ij<Apj>I

5 M (i # j )

(19)

for sufficiently small Apj > 0. Resting on (19), for each j we may choose a sequence ( A p y } of positive numbers satisfying lim A p j v = 0

(20)

V-t+CC

and ensuring the existence of n - 1 limits (211

lim d i j ( A p Y )= d i j ( i # j ) . V-+m

If we let J = { j l p j > O}, then (16), (20), and (21) together also ensure the existence of limits lim djj(Apj”)= d j j ( j E J) v+

+

(22)

rT

by virtue of the continuity of Ej at lj. Hence, substituting Ajp”, Ap: for A j p , Apj, respectively, in ( I 6), and letting 11+ + co, we find, after rearrangement,

c L l i j pi

=

Ej@) ( j E J).

(23)

ieJ

Clearly, the coefficients d i j i n (23) fulfill dij 5 0 ( i # j , i , j E J )

(24)

from (18) and (21). Next, recalling that @(jl) is the minimum of @(p) over all p the light of Lemma l8.3(ii), we have

,,

0 5 @ ( j + A j p ) - @(fi) S 2 C Oi(j

E

P , and in

+ A j p ) ( E i ( l j + A j p ) - Ei(lj))

, = I

( j = 1, . . . , n ) .

(25)

$18. GROSS SUBSTITUTABILITY AND EXTREMIZATION

315

Note that (16) and E j ( j + A j p ) > 0 ensure rij > 0 because of the nonnegativity of Ci+j( -dij(Apj))fii( j = 1, . . . , n) implied by pi >= 0 and (18). Whence we realize J

= K ( j ) # @.

(26)

With these remarks in mind and using ( 1 7) and (25), as well as the positivity of Ap:, we find n

The above relations become in the limit

from (21), (22), continuity, and the fact that Oi(fi)

A.

= max

Qi($>

- pi

over all

= 0 (i 4

J ) by (26). If we let

iEJ,

A. must be positive by (26), and Oi(j3) 5 A.ji (i E J ) , with equality holding for some i, say k . Then, for this k , we have

=

/lo,($) = A.2$,

>0

from (23), (24), and (27), which is a contradiction. This proves that K ( $ ) must be empty and $ is an equilibrium price vector, Q.E.D. As a matter o f course, assumption (6) in Theorem 18.2 by no means requires the existence o f finite-valued derivatives, but merely rules out the existence of vertical tangents to the graph of Ei(p) as a function of t h e j t h component pi o f p . It says nothing about the existence of nonvertical tangents. This assumption can be dispensed with, however, for price domains of a special and familiar structure under an alternative boundary condition, as is the case in the following theorem, which gives a direct generalization of Kuga's result (1965).

THEOREM 18.3. There is an equilibrium price vector 3 in P if we preniise the saiiie assicniptions as in Theorem 18.2, except for (6) i i o ~ 'dropped and ( a ) replaced by (a') :

(d) Boundarj>condition D holds. Proof. Let

Q ={PIP

~f',

P > 01.

316


Since P satisfies the basic assumption (I), so does Q, in which the original set of excess demand functions Ei(p) (i = 1, . . . , n ) fulfills the basic assumptions (11) ( m = 0) and (111). Taking n2 constants aij having the properties

( i , j = i ,..., n ) ,

aij>O n

1a , . = i= I

IJ

1 ( j = 1, ..., n ) ,

arbitrarily once for all, we let 11

1

F,(p) =

aij P j

j = I

, n). - 1 ( p ~ Q , i = l ...,

~

Pi

(28)

As a matter of course, (28) is a system of excess demand functions of the types (24) in $18.1 considered in Example I , which fulfills the basic assumptions (I), (11) ( m = O), ( I l l ) , continuity, and strong gross substitutability in Q. One can readily verify that this system has a very gentle boundary behavior. ( I ) Any boundary point p ( = p i ) of Q has at least one zero component because of the assumed same property of P , and (2) we have lirn Fi(q) =

+ co

(for any i E K = {i ( p i = 0 ) ) .

(29)

4-P

With these facts in mind, for any

E,“(q) = Ei(q)

E

> 0, we define the functions

+ EFi(q)

(i = 1, . . . ,n),

(30)

which obviously form a strong gross substitutes system of excess demand functions on Q satisfying the basic assumptions (I), (11) (m = 0), ( I l l ) , and continuity. Moreover, this system fulfills boundary condition C. In fact, let K = {i ( p i = O} for a boundary point p of Q. We have K # 0, as noted above. Let { 4 ” ) be any sequence converging to p . p may or may not belong to the original P . We consider two cases separately. Case (i) p E P . In this case Ei(q) are well-defined and continuous at p , so that lim Ei(q”)= E i ( p ) (i = 1, . . . , n ) . This, combined with (29), implies lim E,”(q’) = + co for any i E K. Case (ii) p $ P . In this case, we find Jim sup Ei(q”)> 0 for some i E K, say k , from boundary condition D for the original E i . Then, in view of (29) again, we have lim sup Eke(q’)= + co.We have thereby shown that boundary condition C holds on Q for system (30). The remaining part of the proof consists of the minimization of a suitable function over Q. In doing this, we have several useful minimands at hand, e.g., the sum of the squared positive excess demands (3), the sum of the positive excess demands, and the maximum of the excess demands $(q) = max E:(q)

over all i = 1 , . . . , n.

$18. GROSS SUBSTITUTABILITY A N D EXTREMIZATION

317

Although we shall work out the remaining part of the proof in terms of the latter, it should be noted that essentially the same argument can be made in terms of any of the three alternative minimands. We first observe that at every point 4 E Q we have

from the Walras law (III), which rules out the possibility of having negative excess demands for all goods. Since boundary condition C, which is a very stringent version of boundary condition A , holds in Q, we can show, in the same way as in the minimization of @ in the proof of Theorem 18.2, that @(q)is minimized at some 4 = (4,) having unit component sum over Q. We now assert that

q"4)

= 0.

T o this end, suppose that ~'((4) > 0. Then, we must have E,"(4) < 0 for some i, say k , again from the Walras law (111) and 4 > 0. If for such a fixed k , we make Aq, > 0 sufficiently small, we find

4 -'Ah 4 E Q, E,"(4 - Ah 4 ) < from the positivity of $, the openness of Q in R,", and the continuity of E,"(q). Moreover, the strong gross substitutability of E,'((q) (i = 1, . . . , n) implies

E,'(Lj - A k q ) < E,'(Q) (for all

i # k).

Whence a contradiction @(@) > q'(4 - A k 4 )follows. This proves

~ ' ( 4 )= 0.

Next, choose a sequence ( E , } of positive numbers converging to zero and a corresponding sequence (4') in Q n P,, satisfying q"u(Lj') = 0

(v

=

1, 2, . . .)

This is possible by the result just proved for any E > 0. Moreover, as usual, we may assume by virtue of the compactness of P,,that (4") converges to 3 in P,. It will be shown that fi E P and is a desired equilibrium price vector of the original E,(p). In fact, noting that (28) satisfies F,(p) 2 - 1 ( p E Q, i = 1 , , . . , n), we obtain for any i, v , 0

= q"(4')

L E:"(4")

+ CvF,(qV)2 E,(Q')

= E,(Q")

which entails E,

2 E,(4") ( i = I , . . . , n ; v

=

I , 2 , . . .).

- E,,

318


Since lim E , = 0, the above result rules out the possibility of any i having lim sup Ei(q^")> 0, so that fi cannot be a boundary point of P but must belong to P by virtue of boundary condition D. Whence the above relations become in the limit

0 = lim E,, ,,- + x

>= E,(q^)

(i

=

1, _ .., n )

because of the continuity of E,(p) in P, showing that fi is an equilibrium Q.E.D. price vector. Incidentally, fi need not belong to Q, while P 3 /I, Next we shall establish a similar result for generalized gross substitutes systems, for which n i need not be zero. T o this end, we recall a well-known result due to Euler.

LEMMA 18.4 (Euler's Theorem on Homogeneous Functions). Let,f(p) be a cliffircntiahlc, firtictioii, liotiiogetic~ousof order t n , oti P satisjjitig (I). Then, i f C ,fi(p) chi is a total ciijierentiaI qf',f, we liai-e

Proof. For each p E P , we find tp E P (0 < t < + m). Then, by the differentiability of,/; the function 4 ( t )= f ' ( t p ) is differentiable, and its derivative is given by

On the other hand, since

4(t)= t'';f(p) by

homogeneity, we find

f$'(t) = i71tn1- If ( p).

Hence the above alternative representations of d ' ( j ) yield the desired result, particularly for t = 1. Note that./(/)) has right-hand partial derivatives everywhere in P and partial derivatives at any positive p of P, all of which coincide with,fi(p), Q.E.D.

THEOREM 18.4. Tlicrc i s (I), ( I I ), at1d ( I I I ) if

aii eqirilibriittii

price rector

fi

iii

P

011

tlie basic

n.ssLll1lptioll.s

(2)

(/I)

Boirtitlur~,coiiditioti A is ass~tt~ictl. Tlic e.\-c'c.rsik~~iiiutic/,Jiciic.ti~ll.(. E i ( p )( i = I , . . . , t i ) arc rJ(fuwitiable n,itli total cii/firoitials gii.cti I ~ J -

dE, =

Eij(p) tlp,.

iii

P,

0 18.


319

Proof: We note that the differentiability of Ei implies its continuity and right-hand partial differentiability in P, with E i j being its right-hand partial derivative with respect to p i . Then, it can be seen, in a similar way as in the first part of the proof for Theorem 18.2, that @ ( p ) is minimized at j? over P n Pn. It is not readily shown, however, that @ ( j is ) also the minimum of @ ( p )over P because this time m need not be zero. Thus we will proceed somewhat differently. In what follows, it should be understood that all Api > 0 are small enough to ensure j + A j p E P. Since j E P,,the component sum of fi + A j p equals 1 + A p j , so that

320

VI. GROSS SUBSTlTUTABlLlTY

from the definitions of the respective derivatives, and lim oi(P(APj)) = Oi(lj), Ap,-

+O

litn @(p(Apj)) = @ ( l j ) Ap,-+O

from the continuity of E,(p). Therefore, (33) becomes in the limit

C Oi(lj)Eij(P)2 in@(/?) ( j

I , . . . , n).

=

r = I

Multiplying the ,jth relation of (34) by U j ( l j )

>= 0

(34)

and summing over all

j = I , . . . , n, we find It

n

i.j= 1

j = 1

We have thereby obtained (35) as a necessary condition for @(B) t o be a minimum of @ ( p ) over P n P,. At this stage of the proof, we need digress to the following lemma.

LEMMA 18.5. Under the same assuniptions as in Theorem 18.4, except that ( a ) is noit' dropped, i1.e have (i) ff 1 > n z 2 0, /et K(p) = {i 1 E,(p) > O} ,for a p E P . Then, i f K ( p ) # @, the quadratic,forni

C

Eij(P)Si<j i.jaK(p)

is negatire definite.

(ii) I/ m

=

1, the quadratic form

is tiegatii.e semidefinite. Pvooj: By the Walras law ( I I I ) , we have p i E,(p) = 0, which, upon right-hand partial differentiation with respect to p j , yields

C;=

,I

C p j E , j ( p ) + E j ( p )= 0 (,j= 1, i= I

. . . , n).

On the other hand, Euler's theorem (Lemma 18.4) gives n

1 EJi(P)Pi = rllEj(p) i= I

( j = 1, . . . , n ) .

(37)


32 1

from generalized gross substitutability. Then, adding (36) to (37) for each j we end up with

Henceforth we will consider (i) and (ii) separately. (i) Rearranging (39) in a similar way as in the proof of Theorem 18.2, and using (38) as well as the nonnegativity of p i , we find

Since 1 > M and Ej(p) > 0 for j E K ( p ) , the above relations reduce to

Pi

2 0 (i E K(P)).

(41)

By Chapter 11, Theorem 6.2, we deduce from (38), (40), and (41) that the matrix ('ij(P>)

('7

j

E

K(P))

must have all its principal minors positive. Moreover, the matrix is symmetric by construction. These results imply that the quadratic form

is positive definite. Hence

is negative definite. (ii) Since rn = I , (39) becomes n

dij(p)pi= 0 ( j = I,

..., n ) .

i= I

Using Kronecker's deltas rewrite (42) as n

(E&j

i= I

and an arbitrary positive number

+ d i j ( p ) ) p i=

with the new coefficients still satisfying

"j

(j

=

1,

. .., n),

8,

we can

(43)

322

V I . GROSS SUBSTITUTABILITY

Hence, if p j > 0 ( j = I , . . . , i?) in (43), the matrix (cSij + dij(p))has all principal minors positive by Chapter I I , Theorem 6.2, so that the matrix (dij(p)) has all principal minors nonnegative at any positive p E P because of the arbitrariness of c > 0. Since P is an open nonvoid subset of R+",P obviously contains positive vectors that f i l l out a dense subset of P . Thus, the continuity of Eij(p) at any point of P having at least one zero component ensures that the matrix (dij(p))has all principal minors nonnegative everywhere in P . Since the matrix i s symmetric, this implies the negative semidefiniteness of

2

i. j = I E i j ( p ) t i t j =

-;

i,z,dij(~)titj.

The proof of the lemma is thereby complete. We now return to the proof of Theorem 18.4. To see that j satisfying (35) must also fulfill ( I ) , let K ( b ) = { i 1 E i ( j ) > 0), as in Lemma 18.5. It will be shown that the supposition K ( j ) # 0 will lead to a contradiction. (a) The case 1 > nz 2 0. Since Ui(fi) = 0 for i $ K ( j ) , the left-hand side of (35) equals i, j

1 E

Eij(B)oi(fl>oj(P),

K(P)

a.

which must be negative by Lemma 18.5(i) if K ( j ) # On the other hand, the right-hand side is nonnegative. Whence K ( B ) = 0. (b) The case m = 1. By Lemma 18.5(ii), the left-hand side of (35) is nonpositive. On the other hand, if K ( j ) # we have the right-hand side positive in (35) becauseti? = I , @ ( $ ) > 0,cOi(b) > 0,yieldingacontradiction. Therefore K(fi) = 0. Thus, in both cases (a) and (b), j satisfies Eq. ( I ) . Equation (2) follows from ( I ) and the Walras law ( I l l ) , as already noted. Incidentally, the extremization method of proof stated here can apply mutatis titutaiidis to a general minimax theorem on zero-sum two-person games played on compact convex strategy spaces which need not be finite-dimensional [cf. Nikaido ( I 959a)l. The treatment of further generalized types of games seems to be possible, however, only by means o f fixed-point theorems and related concepts [cf Nikaido (1954b, 1959b). Sion (1958)], Q.E.D. Under differentiability, we can readily obtain an interesting result o n the maximization of cP(p).

a,

THEOREM 18.5. Uiider the saiiie assimiptiotis as iti Theoreni 18.4, ercept that ( x ) is dropped. n'e hare (i), (ii) ,for the case 1 > m 2 0: ( i ) I f ' a riia.\-iiiiiiiii of' @(pi i s attaiiietl at sortie point of P n P,, , tfie excess tkwiaiirl jirrtctioiis r.aiiish itleiiticall~iii P. (ii) There is iio s j ~ s t e r i i of e.ycess ~~et?iatid,fiitictiotis 011 P = R+"\{O) othcr thnir the tririal S ) ~ . S ~ C I I IOf icletitically i~anisliitigoties.

$1 8. GROSS SUBSTITUTABILITYA X D

EXTREMIZATION

323

Proof. (i) It will be shown that if 8 maximizes @ ( p )over all p in P n P,, we must find @(fi) = 0. Once this is seen, @ ( p )= 0 everywhere, which proves that Ei (p) 5 0 ( i = 1, . . . , / I ) . This ensures E,(p) = 0 at any positive p in P by (111). Since any p in P can be approximated arbitrarily closely by positive vectors in P , we finally have E i( p) = 0 (i = 1, . . . , n ) everywhere in P. To this end, we note that as a counterpart of (31) we find, for such a j,

0 L t ( @ ( p ( A p j ) ) - WP)) n

L

1 Oi(fi)(Ei(p(Apj))- E;(B))

(j=

1 9

.. . n ) .

i= 1

Then, using an argument similar to the one by which (34) is derived from (31), we obtain

c Oj(j?)Eij(fi)5 m @ ( j ) n

( j = I , . . .,n ) .

i= I

(44)

Since we have n

1~

~ ~ = ( 1j 7 >7 ~ j~ (~j ()i = I , . . . , n )

j = I

from the nith order homogeneity of E , and Euler’s theorem, and O;(j)E,(fi)= 0 , ( $ ) 2

( i = I , . . . ,n )

from the definition of O,(p), multiplying the,jth inequality in (44) by f i j 2 0 and summing over all j will give rise to the same result n7@(fi) on both the left-hand and the right-hand side. Whence equality holds in (44) if j j > 0. Now, let K ( B ) = {i 1 E,(fi) > 0) as before and suppose K(fi) # 0. We recall (40) and (41), together with (38), which ensure p i > 0 for i E K(p). If we apply this result to K(fi), we find that Oj(fi) > 0 ensures jj > 0 and hence equality in thejth relation of (44). From these results it follows that

Then the left-hand side of (45) is negative by Lemma l8.5(i), whereas its right-hand side is nonnegative. This is a contradiction. Therefore K(fi) = and @(fi) = 0. (ii) In this special case, we have P n P,, = P,, which is compact. Since @ ( p ) is continuous, it must take on a maximum on P,. Therefore the assertion follows from (i), completing the proof. Remark 6. Economic intuition seems to claim the plausibility of the hypothesis that the convergence of the price of a good to zero will result in

a

324


either a finite o r an infinite excess demand for the good. Explicitly, it presumes that if p converges to a boundary point p, we may. have a well-defined limit, finite or infinite, lirn E,(p). (46) P+B

B u t intuition is deceptive in this case. There is often a possibility that for two sequences { p ” ) , 14’) converging to b, we have a finite lirn E j ( p v )but an infinite We may even have a n undamped oscillation of E,(r”) lirn Ei(4“)as I’ + +a. for a third sequence { r ’ } converging to j?. Consider (24) in Example 1 in $18.1 for I I = 3, a I 3= 0, and a i j > 0 for all other pairs ( i , j ) . When a sequence { p ” } of positive price vectors converges t o the boundary point fi = (0, 0, l)’, we find, for example, lirn El(/?”)= a, I if limp,”/p,” = 0, whereas lirn E,(p”) = + co if Iimp2’/p,’ = +a.The first situation occurs for the sequence { p ” = ( I / \ * , ijr2, I)’), and the second for { p ”= I/$,, I)‘). Diversity in the mode of convergence to fi makes this point a maligrant singular point of E l @ ) . This suggests that a boundary condition requiring (7) for all i E K , as adopted by K U ~ (1963, I is too stringent. Remark 7. I n Theorem 18.2, boundary condition A is indispensable. The following example, which satisfies all the assumptions except that boundary condition A is violated a t B = ( I , O)’, has n o equilibrium price bector i n the set of all positive vectors:

E z ( p ) = - 1. The boundary condition is violated because @ ( p )> 0 for p > 0 but Q ( p ) -+ 0 ( p + 9). Nonetheless, if this system is considered in the set of all p = (pi), pl > 0, p2 2 0, it satisfies all the assumptions of Theorem 18.2 in the extended price domain and has 3 as an equilibrium price vector. El(/]) = PZ/l’,>

$18.3. Extremization and a Revealed Preference Relation

I n this section, we are concerned exclusively with gross substitutes systems homogeneous of order 111 = 0, a typical situation i n economic theory. Arrow ct ul. (ISSS), and Arrow and Hurwicz (1960) have proved that if E,(p) (i = I , . . . , 1 1 ) are the excess demand functions of a gross substitutes system, we find, for any equilibrium price vector fi = (8,) and any disequilibrium price vector p,

This relation is very important because it enables LIS to establish the convexity of the set of all equilibrium price vectors (Corollary to Theorem 18.6)a fact observed for the first time by McKenzie (1960~1)-as well as global stability (519.2).

$1 8.


325

Samuelson’s weak axiom of revealed preference (Samuelson, 1948a, Chapter 5 ) means, if stated in terms of E i ( p ) , that for any two semipositive vectors p = ( p , ) ,q = (qi),the relations n

and Ei(p ) # Ei(9) (for some i)

(3)

entail

If the Walras law is explicitly premised, (2) is obviously true for q = j?, an equilibrium price vector, because in (2) we then have zero on the left-hand side and a nonpositive number on the right-hand side. Furthermore, if p is a disequilibrium price vector, we also have (3) for 9 = jj, since Ei(p) > 0 for some i but Ej(q) 5 0 for all i. Then (4) for 9 = 6,whose right-hand side vanishes by the Walras law, is nothing but ( I ) . For this reason, ( I ) will henceforth be referred to as a revealed preference relation. The purpose of the present subsection is to prove this revealed preference relation ( I ) between any equilibrium price vector and any disequilibrium price vector in a novel way by a method that differs completely from that of Arrow and Hurwicz. The principal idea of our method of proof is as follows. We regard the left-hand side of (1) as a function n

of two argument vectors j?, p . To establish the revealed preference relation ( I ) between any equilibrium price vector j? and any disequilibrium price vector p , it suffices to prove the following: (i) When j?, p range over all equilibrium price vectors and over all price vectors, respectively, the function ~ ( j ? , p ) takes on a minimum; (ii) The corresponding minimum must be zero and is attained only if p is an equilibrium price vector. Throughout this subsection, the basic assumptions (I), (11) (117 = 0), and (111) stated in $18.2 are premised, with (I) specialized to (1’) P = { p J p=(pi) 2 0, p i > 0 (i E M ) ] ,where M is a given fixed subset of { I , 2 , . .., 1 7 ) . M may be empty, nonempty, proper, or improper.

The additional assumptions are as follows: (IV) The excess demand functions E i ( p ) (i = 1, . . . , 1 7 ) are continuous in P .

326

VI. CROSS SUBSTITUTABILITY

(V) Gross substitutability in the finite increment version (Definition 18.1) prevails in P. (V1) The excess demand functions are bounded from below, i.e., there is a lower bound y such that E,(p) 2 y ( p E P, i

=

I , . . . , 11).

(VII) There is at least one equilibrium price vector in P.

We shall begin with preliminary results, which may also be of independent interest. Lemma 18.6, whose proof does not rest on (VI) and (VII), will be frequently used, not only in this section but also in $19.

LEMMA 18.6. The ,/ollowiiig itenis hokd: (i) For each Ic # M ,we hare €,(p) 5 0 eivrywliere iii P. 1ii

particrrlar, if' M

= @, ire

hare

reP Ei(p) = 0 r w r ~ ~ i c h e it7 (ii)

For ecicIi.fised k # M , let P ( k ) = { p IpL = 0, p E,(p)

(iii)

(i = 1 , . . . , 11).

Fwflieriiiore,

= constaiit

if' E6(y)= 0 at

soiiic

q

E

P I . Tlieii,

in P ( k ) . = (qi)E

P

iii tlie

,foregoing (ii),

theii

E,(p)

=0

ercrjmhcrc~iii

P(k).

ProoJ (i) Let p = ( p i )E P. Two cases (i.x), (i./l) will be considered sc pa r a t e I y . (i.2) p i = 0 ( i # k ) . (This case can occur only when M = 0.) Then, / i n > 0 and p6 EL(/>) = p i € i ( p ) = 0 by the Walras law (111). Whence E6(p) = 0. (i./l) pj > 0 (for some ,j # 1,). Denote by p ( t ) = ( p i ( t ) )the vector whose components are given by p,(t) = p i ( i # k ) , pi(!) = t . Clearly p ( t ) E P for any t >= 0. For any t > 0, we lind p , ( t )= p i (0 ) = p i ( i # k ) , p r ( t ) > pA(0),so that by gross sii bs t it 11t a hi I it y

E , ( p ( t ) )2 Ei(P(0)) (i # k ) .

(6)

Multiplying the ith ineqiiality in (6) by p i ( t )= p i ( 0 ) and summing over all i # k , we see

from the Walras law. Also from the Walras law,

$18. GROSS

SUBSTITUTABILITY A N D EXTKEMIZATION

327

Whence pk(t)Ek(p(t)) 5 0, that is, tE,(p(t)) 5 0 for any t > 0. This implies E,(p(t)) i0 for t > 0 and hence also for t 2 0 by continuity. Thus the desired result follows upon equating t to p k . Finally in the extreme case M = @, the above result implies E,(p) 5 0 everywhere in P (i = I , .. ., / I ) . Then, Ei(p) = 0 (i = I , . . . , n) for any positive p in P from the Walras law. Since positive vectors f i l l out a dense subset in P, we must have Ei(p) = 0 everywhere in P (i = I , . . . , n ) by continuity. (ii) Let p = ( p i ) , q = (4,) be in P(k), and p i > 0, qi > 0 ( i # k ) . Let . 9i cx=min-. i#A

Pi

Then, cx > 0, q 2 my, qk = upk = 0. Whence, by gross substitutability and homogeneity, we find Ekh) 2 Ek(V) = Ek(P),

and hence also Ek(p)2 Ek(q) from the symmetry between p and 4. Thus, E,(p) = Ek(q) for p , 4 in P(k) as long as their components are all positive except for the vanishing kth one. Since such p , q form a dense subset in P(k), we also have Ek(p)= Ek(q)for any p , q i n P ( k ) by continuity. (iii) Since Ek(p)equals a constant in P(k) by (ii), it suffices to see that Ek(p)= 0 at some p in P(k). If qk = 0, this q belongs to P ( k ) and can serve as a desired vector. The case qk > 0 will be divided in two subcases (iii.x) and (iii.p). (iiicx) If q has all components zero except for the kth one, M must be empty. Therefore, all E,(p) identically vanish by (i), so that we have, in particular, E k ( p ) = 0 in P(k). (iii.P) Suppose now that t h e j t h component of q is positive for some,j # k . Let the vector q ( t ) = ( q i ( f ) be ) defined by 4 i ( t )= qi (i # k ) , q k ( t )= t. Then, q ( t ) E P ( f 2 0), q(q,) = q. Clearly, if 4,, 2 t 2 0, we have q 2 q ( t ) , qi = qi(r) ( i # k), so that gross substitutability ensures Ei(q) 2 E M [ ) ) ( i # k , q k 2 t 2 0). Multiplying the ith inequality above by qi ( = q i ( t ) ) and summing over all i # k, we find

But the left-hand side of (7) must vanish because q, E,(q) = 0 by assumption, and the sum of all q , € , ( q ) ,including q, €,(q), is zero from the Walras law. Whence the right-hand side of (7) is nonpositive. Then, again by the Walras law, we must have q,(t)E,(q(f)) 2 0, that is, t€,(q(t)) 1 0 (4, 2 t 2 0). Whence it follows that E,(q(t)) 2 0 (4, 2 f 2 0, f > 0), and hence E,(q(t)) 2 0 (qr 2 t 2 0) by continuity. We have thereby found, in particular, E,(q(O)) 2 0. Note that q(0) E P(k) and therefore E,(q(O)) 5 0 by (i). These results combined imply E,(q(O)) = 0 at q(0) E P ( k ) , Q.E.D.

328


Remark 1. An indecomposable, homogeneous, gross substitutes system can exist only when P in ( I ) contains no vector having some components zero; hence A4 = { I , 2, . . . , / I ) in the case of (1’). Suppose that there were a vector p = ( p i ) , with pn = 0, in P. Then, ocp 2 p , and N(ocp,p) = { i I @pi> p i ) would be a nonempty proper subset of { I , 2, , . . , t i } provided M > I . Hence, by gross substitutability and indecomposability, we must have E , ( M ~2) E i ( p ) ( i 4 N ( x p , p ) ) , with a strict inequality for at least one i. On the other hand, E,(xp) = E i ( p ) ( i = I , . . . , 11) from homogeneity, arriving at a contradiction. Throughout the following, we shall frequently use the notation f ( p ) to denote the set defined for a price vector p by

I ( p ) = { i I p i > 01.

(8)

LEMMA 18.7. Let = ( B i ) he ail arbitrarj~$xedequilibrium price vector. Suppose tliat $(b,p ) iii ( S ) , n ~ l ~ e rieii.ed ti as a ,fuiictioti of a single argument rector p, is riiiiiitiiizctl o i w P at .such a price rector p tliat l ( P ) ={(it).

(9)

T / m , if’ I I V let w(p, p ) = min

Pf

i E f(B),

over all

(10)

Pi

sea, P ) = { i I P i ’W ( B >P ) B i ) ,

(11)

we liarv, for aiiy price rector q jiiljilliiig p 2 q 2 ~ ( p)P, j ,

k ( q ) 5 0 ( i !$ I ( l i ) ) ,

(12)

E;(q) = 0 (i E I(B)\S(B, PI).

(1 3)

Pr*oof. We first note that p i p

= q i = w(b, p ) b i

for i

4 S(B,p ) . Whence

2 q 2 w ( j ,p ) b , gross substitutability, and homogeneity entail E;(P)= E,(wB)

where w

= w(B, 17)

s Ei(9) 5 Ei(P)

( i 4 S(P>P))>

(14)

> 0 by (9). Relation (14) implies, in particular, 0 5 E i h ) 5 E;(P) ( i fzI(B)\S(B, P ) ) ,

(15)

since Ei(fi) = 0 for i E f ( b ) . Note that (12) holds trivially by Lemma 18.6(i), since f(j) 3 M . Equation ( 1 3) is also trivially true by ( I 5 ) if p is an equilibrium price vector. Therefore, it remains to consider ( 13) for a disequilibrium price vector p. I n what follows we use, as before, the notations K(cr) = { i I E , ( L I > ) 01, i

Aju

= (0,. . . ,0, A l l i ,

0 , . .. , 0 )


32 9

for a price vector u. Then, since p is a disequilibrium price vector, we must have K ( P ) f 0, and hence also, by homogeneity,

0-

S(B, P ) #

Moreover, for sufficiently small Apj > 0 ( j = I , . . . , t i ) , we find K(p

+ Ajp)

3

K ( p ) ( j = 1, . . . , / I )

(16)

from the continuity of E i . Then, setting

+

djj Ei(p) - E;(p Ajjp) ( i , j = 1 , . . . , n) and using the Walras law, we obtain as counterpart relations of (16) in $18.2 djjpj=Ej(p+Ajp)Apj- xdijpi (j= l,...,~). i+ j

Since$ is minimized over P at p , we must have $ ( @ , p + Ajp) that is, n

$(B,p),

n

1 Di Ei(P + Aj p ) 2k1= Bi E j h )

i= I

1

( j = I , . . ., n ) ,

which can be put in the form d J. J. *P J< = - Ci +dj i j p i

( j = J , ..., n).

Multiplying the ,jth relation in (IS) by w the j t h relation in ( 1 7), we find

d j j ( p j- o b j ) 2 E j ( p

= w@,p)

and subtracting it from

+ A j p ) A p j C d i j ( p i - obi) -

(18)

( j = I , . . . , n).

(19)

i#j

Since p i - wBi 2 0 ( i = 1, . . . ,n ) by definition, and dij 5 0 (i # , j ) from gross substitutability, (16) and (19) imply

which, in conjunction with (1 5), yields (1 3), Q.E.D. We are now ready to prove, with the vital aid of Lemmas 18.6 and 18.7,

THEOREM 18.6. (Arrow et al., 1959; Arrow and Hurwicz, 1960). W e haoe, for any equilibrium price rector j? and any diseyuilibriuni price rector p , the rerealed preference relation ( 1 ), namely,

330


Proof. The principal idea of our novel method of proof is the minimization of the function cc/(j,p) over all equilibrium price vectors /j and all price vectors p , as already noted. The proof will be worked out in steps 1-4. Step I . This step establishes that $ is actually minimized at some pair ( 8 , ~of) an equilibrium price vector j3 and a price vector p , and the corresponding minimum value is zero. We note that the minimization problem is well posed because of assumption (VII). To accomplish step I , we need only show that for each fixed equilibrium price vector fi, we always have cc/(jl, PI 2 0 (Y

E

(21)

PI,

which is part of the assertion of Theorem 18.6. For, if (21) holds for each equilibrium price vector 8, we find, in conjunction with the Walras law,

$ ( AP ) 2 0 = $ ( A li) for any pair (b,p ) of an arbitrary equilibrium price vector 8 and an arbitrary price vectory, so that cc/ is minimized at special pairs (b, j?), with the minimum value being zero. To prove (21) for any fixed equilibrium price vector 8 = (BJ, whose component sum may be assumed to be unity because of the first-order homogeneity of the relation (21) in /j, let

P ={pIp

E

P , pi > 0 (i E I@))),

where f(b) is defined by (8) f o r p = j?. Clearly, Z@) M , so that Q is a subset of P and satisfies in itself assumption (f'). Define auxiliary excess demand functions Fi(p) on Q by =J

n

This being done and taking an arbitrary Gic(p,)= E,(p)

+ cF,(p)

E

> 0, let

( p E Q , i = 1, . . . , 1 7 ) .

The system of excess demand functions C,'(p) (i = 1 . . . , 17) is readily verified to fulfill assumptions ( 1 ' ) and (11)-(VII) in Q , with f(b) as an M, y - E as a lower bound. and fi as a n equilibrium price vector. Next let n

6

=infx 1=

1

a, G , ' ( p )

over all

p

E

Q n P,,

$18.

GROSS SUBSTITUTABILITY A K D EXTREMlZATlON

33 1

where P, is the standard simplex as before. Since 6 2 'J - E , (5 is finite. Moreover, 6 5 0 = fiiCt(j3) from the Walras law. Let {p'] be a sequence in Q n Pn such that

c

'-+I,

1=1

We may assume, as usual, that { p ' ) converges to a vector p = ( p i ) in P,, . Suppose that this p were a boundary point of Q. Then clearly p, = O for some i E I(/?),say k. Note that from (22) we find litn F,(p') = +m V++CL

because k E I @ ) . Whence, using (VI) and F,(p') 2 = I , 2, . . .), we see

-

1 (i = I,

. . . ,n ;

n

This contradicts 6 5 0. Hencep cannot be a boundary point of Q but belongs to Q. It follows that

from the continuity of G,', and p is a minimizer. Then, by virtue of Lemma 18.7 as applied to GIE,b, q

G,"(p)= 0 (i E

r(B)\s(P,PI>.

= p-

we find (23)

We now assert that I($) n Sib, P> =

Izr.

(24)

Suppose (24) t o be false. Then, since by definition (10) we cannot have Z(B) c S(b,p), the sets /(B)\S(fi,p ) and I @n ) S(B, p ) would be nonempty proper subsets of /(,6). These results, combined with p 2 o($, p ) p , give rise to

'G,F(P)

G,W

=0

(i E

/iB)\S(B,/?)I

by gross substitutability, homogeneity, and definition (22) of F l . which contradicts (23). Whence (24) must hold, so that (23) turns out to be G,'(p) = 0

/(F,,.

(I E

Hence we find

s=

n

piG,'(p) = i= I

f i i G , " ( p )= 0. i

E

ICp)

(25)

332

VI. GROSS SUBSTITUTABILII~Y

If we take the 0 t h order homogeneity of G;' into account, the above result implies n

n

;= 1

;= I

2 B; E ; ( l / )+ 2 B; F ; ( 4 ) 2 0

(26)

( Y E Q)

for any c > 0 and therefore for E = 0. Finally, since Q is dense in P, we have (26) for c = 0 and 4 E P by continuity. SIP/)2. Let E be the set of all equilibrium price vectors. If we write A for the set of all pairs ( f i , p ) minimizing $ over E x P, we have seen that A # 0, and that the corresponding minimum value is zero, in step I . Therefore if it is shown that A contains no pair p ) for which y is a disequilibrium price vector, the theorem will have been proved. p ) i n E x P an e-d pair when is S / q 3. For convenience, we call an equilibrium price vector and p is a disequilibrium price vector. Denote by r the subset of A. formed by all e-d pairs i n A. The final goal of the proof is to see that assuming r # 0 will lead to ;I contradiction. To this end, we pay attention to special e-d pairs. called c ~ a i ~ o ~ ~ones. i c u l A pair p ) is said to be canonical if their components a;. p i satisfy I ( p ) = I(/?). We write C for the set of all canonical e-cl pairs i n r. The purpose of step 3 is to see that I- # 0 implies C # @. To this end. let (a, p ) E r be an e-d pair. We have

(a,

(a.

(a.

K ( p ) = ( i I E ; ( p )> 0; # @, because 1'

= ( p i ) i q ;I

disequilibriuni price vector. Note

I

=

/(B)

nI ( p ) 3 M.

E;(q)5 0 ( 4 E P, i

(27)

4 I),

(28)

from Lemma 18.6(i), so t h a t

I With ihe\e remarks in mind, let

= K ( p ) # @.

fi*

=

{/?TI,

/I* =

{ p : ] be defined by

The above definitions make sense because of (27) and. (29). It will be shown that (,V, p * ) E C and hence C # Clearly, it sufices to show that (fi:*. p * ) E r, since it is canonical by construction. I n fact, we find >= and l;; = i,: ( i E I ) so that Ei(l;") 5 E ; ( f i )5 0 ( i E I ) from gross substitutability. On the othcr hand. E;(fi")5 0 ( i 4 I ) by (28). Therefore a* is an equilibrium price vector. Regarding / I * . we first observe E;(fi)= 0 ( i E from the fact that fi is an eqiiilibrium price \'ector. Hence. by Lemma l8.6(iii),

a.

a*

/(a))

E ; ( / l )= E , ( / P )= 0 ( i E I ( / ? ) \ / ) .

(30)

5 18. GROSS SUBSTITUTABILITY Moreover, we find p 2 p* and p i

= p'

A N D EXTKtMIZATION

333

(i E I ) , whence

Ei(P) 2 Ei(P*) (i 1).

(31)

In view of (30) and (31) we see that n

1 B,Ei(P) = U P ) 2 1 l i m p * ) = $(B*, P*) 2 0,

0 = $(A P> =

Bi

i= I

is1

(32)

i t 1

whence equality turns out to hold in all the relations in (32). This also entails the validity of equality in (31) because p , = j?: > 0 (i E I ) . That is. E i ( / l )=

$(p. p ) =

.€;(p$:) (i E I ) ,

(33)

$(P*. p * ) = 0.

(34)

Equation (33) ensures that / I : % is a disequilibrium price kector. since Ei(p*)= E i ( p ) > 0 for i E K ( p ) c I . Then. (34), in cotijunction with the fact that B* E E, shows (p*, p * ) E P. We have thereby proved that (j?*,p*) is a canonical e-d pair in and hence C # Step 4. This step deduces a final contradiction from the conclusion C # fa in step 3. For any pair (fi, p ) in C, we can define the set S(b. p ) in Lemma 18.7 by (9), (10) and ( I I ) . Since S($,p ) c { I . 2, . . . , 1 7 ) for any (P, p ) E C, the number of elements i n S(p,p ) attains a minimum over C at some pair, say (b,p ) . We observe that Lemma 18.7 applies to this pair p). The canonicality of ( i , p ) , homogeneity, and the fact that ( b , p ) is an e-d pair ensure

a.

(a,

I

=

I ( b ) = /(PI = s(p,p ) #

a.

(35)

Now, letting

2

=

min

Pi

over all

i E SO, p)

Pi

(36)

and noting 1. > w = w(p, / I ) > 0,

we define a new price vector y^ A

qi =

=

(37)

(4;) by

\).fi,

( i E S(B, p ) ) , lop, ( i $ S(F, P ) ) .

Then / ( / I ) = f ( p ) = I ( y ^ ) , so that (4, p ) is canonical. Relations (36) and (37) imply p 2 4 2 o p , whence from ( 12) and ( 13) for q = 4 in Lemma 18.7, we find E,(y^)5 0 ( i $l

PI).

(38)

334

V I . GROSS SUHSTlTUTAHlLlTY

On the other hand, appealing to (37), we have 4iljb =

Bi

(i

E

s(fi,PI),

Lc)

gill. = fii 5 pi ( i !$ S(fi, p ) ) . l. Whence

Ei(4) = E i ( 4 / j . )5 E i ( j ) 5 0 ( i E S(fi,p ) )

(39)

from homogeneity, gross substitutability, and the fact that fi is a n equilibrium price vector. Therefore 4 is also an equilibrium price vector by (38) and (39). Finally, again from ( I 31, for 4 = / J in Letntna 18.7, we find Ei(/7)= 0 ( i E f\S(fi, I))).

(40)

Then, letting S = S(p. / I ) , for simplicity, and taking the definition of (35) into account, we have $( g, p ) =

14; Ei(p ) = i:c f i i Ei( p ) = I.$(

its

4

and

fi, p ) = 0.

iSS

Therefore, (4, / I ) E C. But S(4,p ) is ;I proper subset of S(fi,p ) , which contradicts the minimality of S(p, / I ) over C with respect to the number of its elements. This completes the proof, Q.E.D. Si/,i/i/ificu/io/iw d c r I/rc/c,c.oi,r/io.sLihi/ir.i.. I n special cases, certain steps in the proof of Theorem 18.6 can be dispensed with, so that the essence of the method of proof t-e\,eals itself more clearly. For example, if the price domain P consists exclusively of positive price vectors. we can dispense with Lemma 18.6, step 3, and all about canonicality i n step 4, since any e-d pair is canonical i n these cases. Moreover, it should be noted that further simplification of the proof is effected f o r indecomposable gross substitute systems. First, as was already noted in Remark I , such systems can be conceived on the basic assumptions (1’) and ( 1 1 ) ( n i = 0) only f o r A4 = [ I , 2, . . . , n ) , whence the general simplification of the proof noted above applies. Second, step 4 can be consiclet-ably simplified by dispensing with thc minimization of S(p, p ) with respect t o the tiumber of its elements. In fact, any e-d pair (fi, /I) satisfies

Ei(/i)= 0 ( i $ scp, /7)) f r o m ( l 3 ) in Lcmma 18.7 f o r q = / I . l ( f i )= i l , 2 , . . . , 1 1 ) . However this result clearly contradicts inciecomposability, since S(fi,/ I ) is ;I nonempty proper subset of { I , 2, . . . . / I ) from the hypothesis at the outset of the proof. Therefore, the argument i n terms of 4 i n step 4 is not needed under indecomposability. I t i general cases, the proof is made somewhat intricate by corn hi ti at or ia I jobs wit 11 c;i ti o ti ica I i t y and in i t i i ma I i t y .

$19.

STABILITY IN GROSS SUBSTITUTES SYSTEMS

335

The following corollary, which was first discovered by McKenzie (19604, is an important, though immediate, consequence of Theorem 18.6.

COROLLARY. On the sanie assumptions as in Tlieoreni 18.6, let E be the set of all equilibriuin price rectors. Then, E u ( 0 ) is a coiii~e.~ cone. Proof. We first note that the price domain P in (1’) is a convex set. Consider any convex linear combination ir = (ir,) = zfi + p$ (a 2 0, p >= 0 , LY + p = 1) for any fi = ( P I ) , (i = (4,) E E. If ir E, we would have

+

n

n

1

i= 1

i)i

E,(ir) > 0,

c

i= I

lji E,(ir) > 0 ,

by Theorem 18.6. Whence

i= I

i= I

i= I

which contradicts the Walras law. This proves that ir E E. Finally, the cone property of E u (0) is obvious from homogeneity, Q. E. D. Remark 2. Under indecomposability, the E in the above corollary is a ray generated by a positive price vector, that is, we have the uniqueness of equilibrium price vectors up to the multiplication by positive scalars. This can easily be seen as follows. Let fi, 4 be two arbitrary equilibrium price vectors. As noted in Remark I , fj, 4 are positive vectors. Whence E,(fi) = E,(Lj) = 0 ( i = I , . . . , n ) . Consider S(fi,Q)= { i 14, > toi),), where (u = min g,/i), for all i = I , . . . , 11, as in step 4 in the proof of Theorem 18.6. Suppose S(fi,Lj) # @. Then, S(fi,4) is a nonvoid proper subset of { I , 2, . . . , n } , so that homogeneity and indecomposability imply E,(lj) # E , ( o f i ) = E,(fj) = 0

(for some i

+ S(fi,$)),

contradicting E , ( $ )= 0 (for all i = I , . . . , 1 1 ) . Whence S(i),$) = @ and = wi), which proves uniqueness. In an indecomposable case, the revealed preference relation ( I ) therefore holds between an equilibrium price vector i ) and any p not proportional to fi.

4

619. STABILITY IN GROSS SUBSTITUTES SYSTEMS

$19.1. Dynamic Processes of Price Formation

I ti the foregoing sections, we have explored such fundamental static features of competitive equilibria i n a Walrasian decentralized economy as their existence (Chapter V, $ 1 6 and Chapter VI, $ 1 8) and their socioeconomic

336


implications (Chapter V, $1 7). Nevertheless these results say little about how an equilibrium price system is established in a competitive market. Walras, who was content with counting equations and unknowns to justify the existence of a competitive equilibrium, conceived of a competitive market as being a giant computer locating an equilibrium by virtue of the so-called law of supply and demand. He attached great importance to this practical solution of the corresponding huge system of equations of general equilibrium by the market mechanism. One typical process of competitive price formation that he and his followers had in mind is as follows, To a current quoted price system correspond aggregate supplies of, and aggregate demands for, goods which originate in the integration of the behaviors of individuals as price-takers. The classical law of supply and demand presumes that prices respond at the next instant to the current levels of supplies and demands in such a way that there will be a rise i n the price of every good with positive excess demand and a fall i n the price of every good with negative excess demand. Then the modified price system is met by new levels of excess demands for goods, which give rise to another price change, etc. Thus a trial and error process of price adjustment, called a tiitonnernent, is generated. It should be noted that no transaction takes place during the process until a general equilibrium situation is reached. Mathematically, a tiitonnen?ent process is formulated in terms of differential equations, difference equations, and other types of functional equations. This clear formulation of tiitonnenient processes from a dynamic viewpoint is due to Samuelson (1948a). For example, a familiar typical formulation of a tiitonnetnent process by a system of differential equations is d Pi

- = /ZiEi(p) (i = 1,. nt

..

where p = (11,) is the price vector; E,(p) is the excess demand function for the ith good ; 1, is a positive constant, called the coefficient of price adjustment of the ith good. Note that E,(p) > 0 entails a rise in the price p I , and E,(p) < 0 implies a fall in the price p , in ( I ) . The system of differential equations ( I ) may therefore be conceived as a representation of the classical law of supply and demand, and its solution p ( t ) = ( p r ( t ) )is expected to simulate the process of price change by tiitonnement. More recently, the forniulation of a ~iitonnernentprocess by ( I ) was generalized by Arrow et al. (1959) and others to the system

(!Pi -= H,(p) ( i ri t

=

1 , ...

$19.

STABILITY IN GROSS SUBSTITUTESSYSTEMS

337

where H,(p) (i = 1 , . . . , n) are functions defined on the price domain P and are of the same sign as the excess demand functions E , ( p ) , i.e., sgn H i ( p ) = sgn Ei(p) ( p E P , i = 1,

. . . , n),

(3) in the sense that for each i, H i ( p ) is positive, negative, or zero according as E i ( p ) is positive, negative, or zero. When there is a numhraire good, whose price is always kept at unity because of its special role as a unit of accounting and transaction, the requirements of the positivity of A i in ( I ) and the sign-preserving property of H , ( p ) are imposed on all goods other than the numhraire good, for which we set 2, = 0 and H , ( p ) = 0. In this section, we shall be concerned with properties of the solutions p ( t ) of ( I ) or ( 2 ) , particularly with their asymptotic properties as t tends to infinity. In view of the Walrasian concept of tbtonnement, economists are very interested in the convergence of a solution p ( t ) to an equilibrium price vector 1 satisfying ( I ) , (2) in 518.2 as t tends to infinity. Mathematically, this convergence property means the following: ( a ) Existence of Local Solutions. Given any initial price vector p o , there is a local solution p ( t ) of (1) (or (2)) defined on an interval [0, a), a > 0 and satisfying the initial condition p ( 0 ) = p o . (p) Possibility of Continuation of Local Solutions. Any local solution p ( t ) of ( I ) (or ( 2 ) ) on [0, a) can be continued to a global solution defined on [O, +a>. any global (y) Convergence to an Equilibrium Price Vector. As t --+ +a, solution p ( t ) of ( I ) (or (2)) on [0, +a)converges to an equilibrium price vector 1 characterized by (l), (2) in $18.2.

DEFINITION 19.1. Global stability is said to prwail in systen7 ( I ) (system ( 2 ) ) if(@), (fl) and (7) hold. Undoubtedly, (a) is a prerequisite to the possibility of tiitonnernent. If a local solution starting at some initial position cannot be continued beyond a moment t = z, the tbtonnenient breaks down at T and is not workable. (p) therefore guarantees the workability of tbtonnen7ent. Finally, (y) ensures that a tlitonnement eventually leads to an equilibrium. As a matter of fact, global stability is so special a dynamic property that contrary to the Walrasian view, one can hardly expect it to be shared by all competitive economies. As Scarf's interesting examples of instability (Scarf, 1960) suggest, instability seems to be a universal phenomenon in competitive economies, rather than an exceptional one, whereas global stability is expected to prevail only in very well-behaved systems. It is important as well as interesting, however, to single out special types of competitive economies that enjoy global stability. I n other words, it is worthwhile finding conditions of economic significance that ensure global stability. In this respect, recent

338


studies have concentrated on gross substitutes systems and revealed the prevalence of global stability in them. On the other hand, no drastically different kinds of competitive economies have been known to parallel gross substitutes systems with regard t o this global stability aspect. The purpose of $19.2 is to discuss global stability under gross substitutability. This will be done, however, by presenting only certain results by Arrow et a/. (1959), by Arrow and Hurwicz (1960), and by Uzawa (1961) as representative examples from among the bulky literature on the subject. This opening section lists a few important basic results i n the theory of ordinary differential equations and formulates a useful lemma on majorization.

THEOREM 19.1 (Special Form of the Cauchy-Peano Local Existence Theorem). Let tixi -

(it

=

G,(x)

(i

=

I , . . ., n i )

(4a)

he a .Y.I..Y t el 7 i 9f rh/ferei i t i d equations, 1I here (i) s = (xi)stands,for the in-ditiietrsiorial argrrnient cector. , n i ) are continuous functions defined on a coninion (ii) G,(x) ( i = I , open subset A of R". Then, ,for an?' .yo E A and an)' time poitit t o , there is a set of continuously dijerentiahle ,functions of' t '

.Yi(t)

( i = I , ...,nl),

(4b)

which are &fined in an interra1 It - t o / < E ( I : > 0 ) nYt/i x ( t ) = (.yi(t))E A, and satisfj, (4a) iti the i i i t c r r d as \ i d 1 as the initial condition ~ ( t , = ) xo.

For the proof of Theorem 19.1 the reader may refer t o a standard textbook on ordinary differential equations (e.g., Coddington and Levinson (1953, Chapter I ) . Henceforth the system of functions (4b) formulated in Theorem 19.1 will be referred to as a local solution of (421). It should be notcd that Theorem 19. I says nothing about the uniqueness of local solutions o r their continuity with respect to the initial value .yo. All the proofs that will be stated i n $19.2 below need no reference to the continuity of solutions with respect to the initial value, as opposed to the proof in Arrow and Hurwicz (19621, so that Theorem J9.1 will suffice. Incidentally, Theorem 19.1, by not having G i depend on t , is distinguished from the general form of the Cauchy-Peano existence theorem in the cases where Gi are functions not only of s but also of t . of Continuation). Suppose that G,(x) are cotititiiious in a subset A o f R, , and that ,for each poitit o f A, s l ~ s t o i i(4a) has a local ~Y#lliti#l7starting at the point. I f anj' local solution p ( t )

LEMMA 19.1 (Possibility

(i = 1. . . .

.

111)

$19.


339

of (4a) in [ t o ,a) is contained in a compact subset r of A, which may depend not only on its initial position but also on the solution itself, any local solution of(4a) can be continued to [ t o , 03).

+

Proof. Take any local solution (4b) in an interval [ t o ,a) a > t o . To prove the possibility of its continuation, consider the set of numbers z such that T > to and the given local solution can be continued to the interval [ t o ,T). This set is nonempty, since it contains a. Now let w be the least upper bound of the set. It will be shown that Q = +a. Suppose w < fco. The solution can be continued to the interval [ t o ,w ) . Let x ( t ) = (x,(t)) be a continued solution to [ t o ,0). Then, x ( t ) E r (w > t 2 to)for a compact subset r of A by assumption. Thus there is a bound M from the compactness of r and the continuity of G,(x) ( u = I , . . . , m ) such that JG,(x)I 5 A4 (x E r, i = I , . . ., m). Now, by integrating (4a), we find

Whence for i = I , . . . , n7, S2

Ixi(s1)

j

- x i ( s z ) l ~ IGi(x(t))l dt 5 M IS^

- s21

51

(w > s 2 ,

s1

2 to),

so that xi(sI)- xi(s2)-+ 0 ( i = 1, . . . , m ) as sl, s2 w . In view of the Cauchy criterion of convergence, this implies the existence of --f

lim xi(t) = xiw ( i

= 1,

. . . , m).

r-w-0

As a matter of course, x"' = (x,"') E r by the compactness of r. Since x" E A, there is by assumption a local solution y ( t ) = (yi(t))of (4a) on an interval [w, 41, 4 > w satisfying the initial condition y(w) = x"). Define z ( t ) = (zi(t)) on [to, 4 ) by

i

z(t) = x(t) Jft)

t 2 to), t 2 0).

(0>

(V

It can be readily verified that this ~ ( t is) a continuation of the given solution beyond Q, which contradicts the definition of w . Whence w must be +03 Q.E.D. Next we proceed to a majorization lemma that will play a vital role in the proofs of stability.

340


LEMMA 19.2. Let f ( t ) be a real-calued ,function on a semiir.tterra1 [ t o , a), a > t o , irhere a niay or niay not be co. Then (i) I f a real nimiber (5 satisfies

+

we hare

(ii) l f a real number S satisJies lim sup f ( t '

+

h-0

I?

-

f

(

f

)

s- 6

(.for all t E [ t o , a)),

vve Iiaoe .f(t)

5 &t

- I,)+ f ( t , )

(a > f 2 to).

Proof. Clearly either of (i) o r (ii) follows from the other. It therefore suffices to prove only (i). Moreover, if it is seen that , f ( t ) 2 (6 - ~ ) (-t 1,) +,f(to) (a > t 2 r,) for any E > 0, the arbitrariness of E ensures the result of (i). Thus letting L

=

{ t I f ( t > I (6 - E ) ( t - to) + f ( t o ) , a > t

2

to>,

we shall show L = [ t o , a). To this end, let w be the least upper bound of all numbers T such that 7 2 to and [ t o ,T ] c L. w is well-defined because [ t o ,T] c L for T = t o . Then, clearly CJ 2 w 2 t o , and proposition (i) is proved if w = a. With this in mind, suppose a > o.Then, let us first show that f ( w ) 2 (6

-

1(

-

to)

+ .f(to).

Relation (5) is true in an equality form if w > o > t o . Since

= to.

Consider the

5

by assumption, we must have / ( o - 11) - f ( w )
0. Moreover,

>= ( ( j - c)((o - / I

- t o )+ f ( t O ) (I? > 0,

-

from the definition of w . Relations (6) and (7) combined entail

T(w) - (0 - & ) / I 2 (6 - &)(a- I?- t o )+ f ( r o )

2

to)

$19.

STABILITY I N GROSS SUBSTITUTES SYSTEMS

34 1

for sufficiently small h > 0. This becomes (5) in the limit, as 42 + +O. On the other hand, there must be a sequence { h v ] of positive numbers converging to 0 and satisfying

+

.f(o h,)

< (4 - E 1 ( 0 + h,

-

to) +. / U O ) ,

(8)

again from the definition of 0).Then, subtracting ( 5 ) from (8) and dividing the resulting relation by 11, > 0, we find

f ( w + k )- f ( w ) < h"

- E.

Whence

which contradicts the assumption on 6. This proves o = n, Q.E.D.

A special case of Lemma 19.2 deserves attention. This will be stated as follows.

LEMMA 19.3. Suppose that 6 (i) (ii)

f(t)

=0

in Lemma 19.2. Theri we hare

in Lemma 19.2(i) is nondecreasitig if7 [ t o , n). 19.2(ii) is nonincreasing in [ t o , 0).

f ( t ) in Lemma

Proof. (i) One can readily see that Lemma 19.2(i) ensures f(t)

2 6(t - s) +.f(s)

(a > t 2 s 2 to).

If we take the assumption S = 0 into account, the above inequality implies t 2 s 2 to), as was to be shown. The proof of (ii) is completely similar, Q.E.D.

f ( t ) z f ( s ) (n >

Remark. In view of the fact that in Chapters 11, 111, and IV we worked on the dynamic aspects of intermediately disaggregated systems by using formulations in terms of difference equations, the reader may wonder why we employ differential equation formulations here. Most of the results of dynamic nature in Chapters I I through IV can also be established i n terms of differential equations, so that the choice of types of equations there is mainly a matter of convenience. Here the situation is different. Almost all of the results achieved so far on stability i n Walrasian competitive economies are in terms of differential equations [except for that of Uzawa (1959-1960)], and it seems to be impossible to establish their difference equation versions. This is the main reason for employing differential equations in the present chapter.

342


$19.2. Stability under Gross Substitutability

I t i this section, global stability will be established under gross substitutability for system ( I ) or its more general version ( 2 ) in $19.1, namely, “Pi

-=AiEi(p) rlt

( i = 1,

or

P

2=

rlt

Hi(p) ( i = 1, . . . , I ? ) .

The basic assumptions premised in this section are (l’), (11) (rn = 0), ( I l l ) , (IV), and ( V ) given in $18.3. For easier reference they are restated below: (1’) P = { p 111 = ( j j l ) 2 0, p l > 0 ( I E M ) ] ,where M is a given subset of { I . 2. . . . , I t ) . (11) The excess demand functions E,(p) (i = I , . . . , 11) are homogeneous of order zero ( n i = 0). ( 1 1 1 ) The Walras law i n the narrow sense holds everywhere i n P. ( I V ) E , ( p ) ( i = I . . . .. 1 1 ) are continuous in P. ( V ) Gross substitutability in the finite increment version (Definition 18.1) prevails in P.

We also assume (Vll’), a stringent version of (VII) in 418.3: (VII‘)

There is a positive equilibrium price vector p

=(PI).

A < noted several times before, I, must satisfy

E,(p)=O ( i = l , . . . , / I )

(3)

from Definition 18.4 and the Walras law. Ac to system (2). we premise (a)?(b): (a) H , ( p ) (i = I , . . ., 1 7 ) are continuous in P . (b) For each i = I , . . . , 11. H , ( p ) and E , ( p ) are of the same sign,

sgn H , ( p ) = sgn E , ( p ) ( p E P, i

=

I , . . . , n),

(4)

with the understanding that in the case of a tdtoIitieInnit where a good, say the Ith one, is a I I I I I I I ~ ; Y C I ~ Ithe . ~ ’ , Ith relation i n (4) is removed and replaced by

H,(p)

=0

everywhere in P.

(5)

If system (2) is specialized to system ( I ) , assumption (a) is implied by assumption ( I V ) , whereas assumption (b) means

2, > 0 (i = I , . . ., I ? )

(6)

$19.


343

except for a numgraire good, if any, say the lth good for which /I1 = 0.

(7)

We begin with the following two lemmas. LEMMA 19.4. Any equilibrium price rector j?

=

(Fi) satiJfies

Ei(fl) = 0 (i = 1, ..., 11). Proof. l j I > 0 implies E,(P) = 0 by (2) in $18.2. Moreover, since E,(p) = 0 ( i = 1 , . . . , t i ) for the equilibrium price vector p in assumption (VII’), Lemma 18.6(iii) encures E,@) = 0 for i such that j3, = 0. This completes the proof. LEMMA 19.5. L e t ji = (p,) be the equilibriuni price w c t o r in assumption (VII‘) and I be an arbitrary nonroid subset of { I , 2, . . . ,n } such that I 3 M. Then the price rector p ( I ) = ( P I ( / ) ) in P, kchich is dejined bJ-

is again an eqirilibriuni price rector. PvooJ First we have E , ( p ( I ) )5 0 ( i ef I ) by Lemma 18.6(i). Second, p ( l ) 5 p , p i ( / ) = Pi (i E I ) imply E i ( p ( / ) )5 Ei(p) = 0 ( i E I ) from gross substitutability. Thus, p ( / ) satisfies ( I ) in $18.2. Hence (2) in $18.2 is also ensured by the Walras law. Therefore p ( I ) is a n equilibrium price vector, Q.E.D. THEOREM 19.2. On the basic assuniptions (l’), (11)-(V), (Vll’), (a) and (b), i1.e haw (i) rf’ asswiiptioii (VI) in $18.3, namely, the I o n w boundedness of the excess deniand functions, is also premised, global stability precuils in s-btstetn ( I ) (Arroic. and Hurnicz, 1960). (ii) rf’ gross substitutabilitji (V) is strengtllenetl to strong gross substitutability in the finite increnient rwsion (see Reniark 4 in $18. I), then global stahilitj. prerbils in system ( 2 ) (Arrow et al., 1959). (iii) Specialize assuriiptions (IV), (V), (a), and (b) to (IV’), (V’), (a’), and (b’), respectiivly: (IV’) E i ( p ) ( i = I , . . . , n ) are drflerentiable in P. (V‘) Gross substitutabilitj. in the cf(fcvential rersion prerails in P (see Defiriitioii 18.2). (a’) H i ( p ) are of’the ,form Hi(p) =.A.(Ei(/))) ( i = I , . . , I ] ) , ichere , f ; ( t ) (i = I ,

able

t.

. . . , t i ) are cli~i~rctitiahle,~irIlc.riolls o / a real rari-

344


(b')

f,'(t)

ij

> 0 for all t (i = 1, . . . , n), and each f , ( t ) canislies i f and onl,v f i ( t ) = 0 identically for a nuniPraire good I, if

t = 0, except j o r

any. Then, global stability preoails in system ( 2 ) (Uzawa, 1961).

Pvoof. Recalling Definition 19.1 of global stability, we shall have to prove (a), (p), and ( y ) in $19.1. In doing this, (a ) and (8) can be proved for system (2) on the basic assumptions, without appealing to the specific additional assumptions in (i)-(iii). Moreover, the respective proofs of ( y ) for (i)-(iii) commonly rest on a fundamental relation, derived from gross substitutability, appearing in the course of the proof of (p), while they proceed differently only in the remaining parts. The proof is also intended to be worked out in as unified a way as possible for tfitonnenient processes both with and without numPraire. Readers who are exclusively interested in processes without nuniPraire may skip the passages pertaining to a iiuniPraire good. (a ) Existence of Local Solutions. Let p o = (p,") be any price vector in P. It will be shown that there is a local solution of (2) starting at p a . Let Z ( po) = { i I p," > 0}, as before (ste (8) in $18.3). Since there is a positive equilibrium price vector p by (VII'), which satisfies (3), we find

from Lemma 18.6(iii) and assumption (b). Let

Consider the system of differential equations

If Z ( p o ) consists of ni integers, H i ( p ) (i E Z(po)) are continuous on A(po), which can obviously be regarded as an open subset of R". Since p o E A(po), there is, by virtue of Theorem 19.1, a local solution p ( t ) = ( p i ( t ) ) of (10) defined in an interval [0, a), o > 0. This solution satisfies At)E by definition, so that Pk(t) = 0 (k

# I ( p o ) ) in [O, 0), and hence automatically

9d3t = H , ( p ) = 0

( k # Z(p0))

from (8). Whence p ( t ) is a desired local solution of (2).

$19.


345

Incidentally, it should also be noted that any solution p ( t ) of (2) in an interval [0, a), a > 0 is of the above type, whether 0 < +co or 0 = + a.In fact, upon integration, we find Ph(') = Pko

+fHL(ds))

ds = ? X H k ( d s ) ) d s

( k $ I(pO>, a > t 2 0). (1 I )

The integrands f f k ( p ( t ) ) must be nonpositive by Lemma 18.6(i) and assumption (b), whereas p k ( t ) must be nonnegative by definition. The integrals in the rightmost of (1 1) therefore vanish and hence Pk(f) = 0

(k

# r ( p o ) , a > t 2 0).

(12)

(p) Possibility of Continuation. Let p ( t ) be any solution of (2) in [0, a), a > 0, local (a < +a),or global (a = 03). For any equilibrium price vector

+

B such that I(B)

=

4P0)?

(13)

define x(t,

j?)

=

min -T over all

Pi

i E f(pO).

Let CI) be the least upper bound of all 7 such that 0 > 7 > 0 and I ( p ( t ) ) 3 Z(po) (7 > t 2 0). w is well defined and positive, since i ( p ( t ) )=I /(PO) for a small 7 > 0 because ofp(0) = po and the continuity o f p ( r ) .Clearly, a >= o > 0 and P(r) 2 ~ ( t>) 0 in [0, w ) . Let J&,

B) = iiI P i t ) = d t , m,>.

(16)

Then, p , ( t ) 2 ~ ( f B)B, , (i E I ( p o ) ) , while p k ( t )= cc(t, fi)j$ = 0 ( k $ / ( P O ) ) by (I 2) and (13). Hence p ( t ) 2 cc(t, B)B in [0, a). Moreover, a(t, B ) > 0; hence cc(r, p)fi E P in [0, 0). Consequently, E,(p(t))2 E,(cc(r,j?)fi) = E , @ ) = 0 (i E J,(t, j), t E [0, w ) ) from homogeneity, gross substitutability, and Lemma 19.4. Therefore, in view of assumption (b), we find

H,(p(O) 1 0 (i E J,@,

B),

t E [O, a)).

(17)

It should be noted that (17) holds even if a nume'raive good happens to be listed in J J t , B) because H , ( p ( t ) )= 0 by definition when the ith good is a numkraire good.

346

V I . GROSS SUBSTITUTABILITY

We now assert

In fact, at each t E [0, w), there is a sequence {h,) of nonzero numbers converging to zero such that lirn inf

4 tf

h+O

h, B ) - 4 h

L B) =

lim v- + m

+ hv B ) - 44 B)

4 t

5

hV

.

(19)

By definition of a({, fi), we can choose integers k(v) satisfying 4t

+ h v ,ri) = ~ ! + ) ( t+ hvYBh(Y)( v

= 1,

2, . . .),

k ( v ) ~ , I ( p ’ ) n J , ( t + h , , f l ) ( v = l , 2 ,...).

(20) (21)

For all values of v = 1, 2, . . . , k(v) lies in { I , 2, . . . , n}, so that k(v) takes on a common value, say k , for an infinite number of We may therefore assume, without loss of generality, that for a fixed k we have $1.

cc(t+h,,B)=p,(t+h,)/B,, k

Moreover, as v

4

E

( v = 1,2,...),

+

I ( p o ) n ~ , ( t h v , 9) (v

=

1,2,

...).

(22) (23)

+ co, (22) becomes in the limit B ) =P k ( t ) / b k

(24)

from the continuity of p k ( f ) and u(t, P), and hence k

E

I(pO) n JA,B).

(25)

Taking (22), (24), and (25) into account, we find lim V ’ i o o

cr(t

+ h , , j3) - a(t, j3) = -I hV

Bk

lim Pk(t v*+m

+ hv) - P d t ) hV

from (2). The rightmost of (26) is nonnegative by (17) and (25). This, combined with (19), ensures

I n exactly the same way, we find

$19.


347

Whence a(t, #) is nondecreasing and P(t, fi) is nonincreasing in [0, o) by Lemma 19.3. This implies, in particular, p i ( t ) >= a(0, fi)fii > 0 (i E I@'), o > t 2 0). If o > o,we would have pi(w) 2 a(0, fi)fii > 0 ( i E Z(p')) and hence also pi(o + h) > 0 ( i E Z ( p o ) ) for any small h > 0 satisfying o + h E [0, o), both from continuity at o,thus contradicting the definition of w. Therefore o = 0 . In summary, we have shown that 0 < 40,B) 5 a(s,

3) 5

fi) 5 Pi(t)/'fii

I P(r, B) 5 B(s, B) 5 B(0, B) (i E I(po), o > r

1 s 2 0). (29)

This result, in conjunction with (12), is of vital importance in proving (p) as well as (y). If we denote by j j = (pi) the special p( I) in Lemma 19.5 for I = I ( p o ) , then (12) and (29) for # = p imply in particular that any solution p ( t ) of (2), whether it is local or global, remains in the cube r(po), TCP')

== {P

I pi

=0

(i $ Z(Po)>, a(0, B ) i i i 5 P i

5 P(0, P ) B i (i E I(po)>>>.

(30)

T(po) is a compact subset of P. Moreover, for each point of P, system (2) has a local solution, starting at the point, by (a). Hence any local solution of (2) can be continued to [0, + a)by Lemma 19.1 as applied to system (2). ( y ) Convergence to an Equilibrium Price Vector. The basic idea common to all the methods of proving (i), (ii), and (iii) is as follows. We first note that the convergence of a global solution p ( t ) of (2) to some vector p ( a ) occurs if and only if p , ( t ) + p i ( a ) ( t -+a)for i € I ( p ' ) and p k ( a ) = 0 for k 4 I ( p o ) , since the convergence of p k ( t ) ( k I ( p o ) ) to zero is always ensured by (12). On the other hand, for any equilibrium price vector fi satisfying I @ ) = I ( p o ) and any global solutionp(t) of (2) starting at p o , we have by (29), f$

( I ) a(t, fi) is nondecreasing and bounded from above in [0, + 0 0 ) ; (2) P (t, fi) is nonincreasing and bounded from below in [0, + a). Hence lim a(t, fi) = a(#), (31) t++m

lim P(4 t++m

#I = P(#>

(32)

exist and P(P) >= a(P) > 0. As a matter of fact, .(B) need not equal /I@). But, if there is an equilibrium price vector fi satisfying = I(PO)>

a@) = P(#),

(33)

(34)

348


we find lim pi(t) = u(fi)ji = p ( j ) j i ( i E I(p'))

(35)

f'+cc

from (29), and

+

pi(t) = fii = o ( i $ ~ ( p ' ) , co > t 2 0)

(36)

from (1 2). That is,

as was to be proved. Therefore, it suffices to have (33) and (34) for some equilibrium price vector j. To d o this, however, we have only to show that there is a sub-sequence {tvf,"=l of time points t L , t , , ..., t , , . . ., tending to +a,for which the sequence { p ( t ' ) } converges to an equilibrium price vector p ( w ) . First, we find Z(p(00)) =I@') because p ( t ) E T(p'), and hence p(co) = lim p ( t , ) E T(p'). Second, for j = p(co) we have lim a ( t , , p ( c c ) ) =

lim P ( t , , p ( c o ) ) = l , v-+m

V-'+a,

so that lim

cc(t,

p ( c o ) ) = lim P ( t , p ( c o ) ) = 1

t++m

f'+m

from their monotonic convergence ensured by ( I ) and (2) for j = p ( c o ) . The proof will henceforth proceed differently for (i), (ii), and (iii). (y.i) Case of (i). We recall Theorem 18.6, which, under assumptions (1') and (II)-(VI I), ensures, in particular, n

for a positive equilibrium price vector P = (pi) whose existence is ensured by (VI1') and any price vector p E P, with strict inequality holding if p is a disequilibrium price vector. With these remarks in mind, we first consider the non-nurnr'raire case. Let " lim inf C piE i ( p ( f ) = ) 6. f++o,

Then 8 2 0 by (38). If 6 which

= 0,

i=l

there is a sub-sequence

Iim x p i E i ( p ( t v ) ) = 6 Y+

+m

= 0.

{I,}

tending to

+ co for (39)

$19. 5-1- W I L I T \

Since p ( t , ) E r ( p o ) ( v

=

1, 2,

GKC)SS SUBSTITUTESSYSTEMS

I\

349

. . .), which is compact, we may assume that

lim P(f,> = p ( . o ) ,

p(m>E Up").

, - + 3

Then we find n

1

i= 1

Pi

Ei(p(co)) = 0

from (39) and continuity. Whence p ( c o ) must be an equilibrium price vector because otherwise we would have n

1

Pi

i=1

E i ( p ( a > >> 0.

In the light of the conclusion of the general remark for (y), this proves (i). Thus, it remains to be shown that the supposition 6 > 0 would lead to a contradiction. Suppose 6 > 0. Then

for some t , > 0. Hence we find for f 2

I,,

from ( I ) , the Walras law, and (40). Then, by Lemma l9.2(ii), we must have for t 2 t o ,

whose right-hand side tends t o - co as t n

This contradicts p ( t ) E r(po) ( t 2 0).

-+

+ co. Whence

350


I n the case where the Ith good is a numr‘raire, by using a normalized positive equilibrium price vector p with its fth component PI = 1 and noting p l ( t ) = PL = I over time, we find

n

=

-2

C Pi E i ( P ( 0 )

i= 1

from ( I ) and the Walras law. Thus, the counterpart of (41) is

so that the proof proceeds in the same way. (y.ii) Case of (ii). Since a strong gross substitutes system is a further stringent version of an indecomposable gross substitutes system, we must have M = { I , 2, . . . , 1 1 ) by Remark I in $18.3, and the uniqueness of equilibrium price vectors up to multiplication by positive scalars by Remark 2 in $18.3. The case without /i/mi&aire will be discussed first. Taking a positive equilibrium price vector p stated in ( V l l ’ ) , let

where J,(t, p ) is defined by ( I 6) for substitutability and (b), and hence lim inf r-+

fi = P.

((2)

Then we find

= 20

t(t)>= 0 from gross

2 0.

7

Then, supposing ii = 0, and taking, as in (y.i), a sub-sequence {t,,) tending to +cc and fulfilling liin ( ( t , ) = \-t

26

= 0,

liin p(t,) = p(c0)

,-+

(44)

I

r

in

qpO),

(45)

$19.

STABlLlTY IN GROSS SUBSTITUTES SYSTEMS

35 1

we shall show that p(m) is a positive multiple of P . In fact, as in the argument leading to (22), (23) from (20), (21), we may further assume that for a common k,

k E J,(t,, I j )

(V =

1, 2,

. . .).

Relation (47) implies P k ( t v ) = a(t,, p)&, p ( t v ) 2 a(t,, pk(co) = n P k , p(co) >= np by (45) and continuity, 72 = or@)

=

(47)

P)p (v

=

1, 2,

. . .), so that

lim a ( t v , Is). V++cO

Hence k E J = {i I p i ( a ) = npi},

(48)

P(W>2

(49)

nP.

Moreover, from (44), (45), and (46), and continuity, Hk(p(oo))

=

lim H k ( p ( t v ) )

=pk

v++m

lim ((t,)

= 0,

V’+m

which entails, by (b), & ( P ( ~ ) >= 0.

(50)

Now, if we had p ( c o ) 2 np in (49), we would find E,(p(m)) > 0 from strong gross substitutability, homogeneity, and (48), thus contradicting (50). Hence p ( m ) = np, the desired situation, as expected in the concluding part of the general remark for (y). Finally, to deduce a contradiction from the supposition 6 > 0, we use (19), (25), and (26) for f i = p and rest on Lemma 19.2(i), to obtain for some I, > 0, a((, F )

2 6(t - to) + a(to,

( t 2 to),

as in (7.i). This results in a contradiction a(/, p ) -+ + 00 ( t -+ + co). Next it will be observed that the above argument for the nonnormalized case applies directly to the case where the Ith good is a numhraire, provided the argument is worked out in terms of a normalized positive equilibrium price vector with its fth component p1 = I . Then, p l ( / ) = p1 = 1 over time, so that

@, PI 5 1 5 bv? Is). With the monotonicity of a(t, p ) , P(t, p ) in mind, consider the case where ~ ( tp ,) < 1 over time. Since the nunihraire good is never listed in J,(t, p ) , the

352


argument in terms of eft) in (43) for the non-numkraire case is directly applicable. The case where /I(?,p) > 1 over time can be handled by the corresponding argument in terms of /I(t, p) and q ( t ) = max H k ( p ( t ) ) over all ~

k

E J,(t,

p)

Pk

I Pi(t) = P ( t , F > P i > . Finally, if u(t,, p ) = /I(t,, p) = 1 at some moments ={i

4 4 PI

= B(t, P ) =

t, , r, , we find

1 (t 2 max(t,, t,))

by monotonicity, whence the convergence of p ( t ) to p takes place in a finite period. (y.iii) Case of (iii). Recall that any solution p ( t ) of ( 2 ) satisfies Z ( p ( t ) ) = Z(po). With this in mind, for any global solution p ( t ) of (2) starting at po, let

Z(p(t)) = Z(po) justifies the definition of p ( t ) by (51). At each t , we find E i ( p ( t ) )2 0 for at least one i in I ( p o ) , from (8), (12), and the Walras law, so that p ( t ) 2 0 ( t 2 0) (52)

from assumptions (a’) and (b’). Equation (51) implies H,(p(t)) 5 p ( t ) p i ( t ) ( i E I ( p o ) ) .On the other hand, Hk(p(t))= p ( f ) p k ( r )= 0 ( k $ Z(po)) by (8), and Z(p(t)) = Z(po). Hence Hi(P(t)) I P(tlPi(t) (i

=

1 , . . . , 4.

(53)

Then, in a way similar to the derivation of (26), if we let J ( t ) = ( i I H i ( A t ) ) = p(tlpi(t>>,

we find

Taking assumption (a’) into account, and performing differentiation, we have

”

where p i ( t ) = d p i / d t , Eij(p) = d E i / d p j ( i , j = 1, . . . , n).

$19.

353


Note fk’(E,(p(f)))2 0 by assumption (b’). Moreover,

n

2 A t ) C E,j(p(t))pj(t) j = 1

from pj(t) = Hj(p(t)), gross substitutability Ekj(p(t))2 0 (k Z j ) , (53), and H,(p(t)) = p(t)p,(t). The rightmost of (56) vanishes by homogeneity and Lemma 18.4. Since pk(t) > 0, the first term on the right-hand side of (55) is nonpositive by the above remarks. On the other hand, its second term reduces to -p(t)’ because d,(t) = H,(p(t)) and k E J ( t ) . These results, combined with (541, imply

By (57) and Lemma 19.3, the function p ( t ) is nonincreasing. This, together with the lower boundedness by (52), ensures the monotonic convergence of p ( t ) to a nonnegative limit lim p ( t )

=

t++m

inf p ( t ) = 6 2 0. t20

If 6 > 0, and hence p ( t ) 2 S > 0, we would find, by (57) and Lemma 19.2(ii),

+ p(0)

p ( t ) 5 -6’t

which results in a contradiction: p ( t )

--f

( t 2 O),

- 00 ( t --t

+ co). Hence

lim p ( t ) = 0 (monotonically). r++m

In view of the boundedness p ( t ) 5 p(0, ,7)@ by (29) for n

lim sup t++m

6 = ,7,

we find

n

C ffi(p(t)) = E 5 p(O, j )iC= l pi lim i=l

p(t) = 0

t++m

from (53) and (58). Suppose imply

(59)

< 0. Then, (59), combined with (2), would

E

C hi(2) 5 4 2 < 0

i= 1

(t

2 to)

for some t o , which yields, by Lemma 19.2@), a contradiction n

lim t++m

C pi(t) = - 00. i=l

Hence E = 0, so that there is a sub-sequence (t,,} tending to lim v++m

1 H.i ( p ( t v ) )= E = 0 ,

i=l

+ co and satisfying

354


where {p(t,)} may be assumed t o converge to p ( m ) in r(po) from the fact that p(t,) E l-(po) (v = I , 2, . . .), a compact set. It will be shown that p ( m ) is an equilibrium price vector fulfilling I ( p ( c 0 ) ) = Z(po), and we therefore have a situation such as was desired in the conclusion of the general remark for (7). In fact, f ( p ( o 0 ) ) = f(po) is obvious from p(m) E Tfp'). On the other hand, substituting t , for t in (53) and letting v + + m , we find

H i ( p ( c o ) ) 5 p i ( . o ) . lini p ( t , ) v- +

=0

(i

=

1,

..., n )

(61)

rl)

from continuity and (58), whereas n

from continuity and (60). Thus (61) and (62) entail

H , ( p ( m ) ) = 0 (i = I , . . . , 17).

(63)

. . . , n)

(64)

Equation (63) implies E ; ( p ( m ) )= 0 (i = I ,

by assumptions (a') and (b'), except possibly for a numr'raire, if any, say the Ith good, whose excess demand E,(p(co))must also vanish following (64) for i # I and the Walras law, as was to be shown. This completes the proof. The general idea of the method of proof stated above is due t o Arrow et al. ( I 959) and Uzawa. Of its particular parts, (y.i) is due t o Arrow et al. (1959) and to Arrow and Hurwicz (1960); (y.ii) is due to Arrow et al. (1959);

and (y.iii) is due to Uzawa (1961). An alternative result by Arrow and Hurwicz (1962), which is not presented here, can claim extreme generality among numerous results on global stability i n gross substitutes systems. Unlike the results given here, it takes explicit advantage of the continuity of solutions of differential equations with respect to the initial position.

Chapter VII

THE JACOBIAN MATRIX AND GLOBAL UNIVALENCE

I n special reference to factor price equalization and the uniqueness of competitive equilibrium, conditions Lor the global univalence of a mapping on its Jacobian matrix are stated. 920 expounds univalence theorems. 921 explains their applicability to special Jacobian matrices.

520. CONDITIONS O N T H E JACOBIAN M A T R I X FOR GLOBAL UNIVALENCE 020.1. Underlying Motivations

Consider a system of n equations in n unknowns , f i ( x l ,x2,. . . , x,) = p i (i = 1,

. . . , n),

(1)

where ( a ) J;(x,, x2, . . . , x,) (i = 1, . . . , n ) are real-valued differentiable functions on a region X of R” (as to differentiability, cf. Definition 5.2 in Chapter I) with total differentials

dfi =

C fij(x,, x 2 , . . ., x,)

dxj

j= 1

( i = I , . . ., n ) ,

(2)

(p) p i (i = I , . . . , n) are given constants. The Jacobian matrix (3) fnl

fn2

355

356

VII. THE JACOBIAN MATRIX A N D GLOBAL UNIVALENCE

provides us with some information about the properties of solutions of (I). I t is a well-known classical result that the nonvanishingness of the determinant of J at a solution of ( I ) rules out the existence of a second solution of ( I ) in the vicinity of the former (see Chapter I , Theorem 5.6(i)). More specifically, the mapping ( x i )+ (fi(xl, x ~ . ., . , x,)) : X + R" defined by the functions on the left-hand side of ( I ) is univalent in the vicinity o f a point if the elements f i j of J are continuous and the determinant of J does not vanish at the point (see Chapter I, Theorem 5.6(ii)). These classical results are all local and ensure only the local uniqueness of solutions or the local univalence of the mapping. There is often a demand i n economic theory and related fields, however, for conditions on J that ensure this global uniqueness of solutions of ( I ) or the global univalence of the mapping in a given whole region; and little related material seems to be available in the orthodox mathematical literature. The present chapter is designed to fill this gap to some extent by giving certain results ensuring global univalence and their applications on the basis of recent work by Gale and Nikaido (1965) and lnada (1966). Of various motivations for the study of global univalence in terms of the Jacobian matrix, a typical and most stimulating one is that from the proposition of factor price equalization in the theory of international trade. A controversial theme concerns whether or not rewards to production factors equalize by trade, without their international movements, among countries having the same technological possibilities but different factor endowments. The equalization thesis has both advocates and antagonists. Samuelson (1948b, 1949, 1953-1954), an advocate of the thesis, bases his argument on a serious consideration of the important, though special, case where the number of goods equals the number of production factors. This case is formulated mathematically as follows. Suppose that there are / I goods and 17 factors of production, and that all the countries involved possess the same technological possibilities in which constant returns to scale prevail, but from which both joint production and external diseconomies (within each sector producing a single good in each country) are assumed to be absent. Recalling the formulation of production possibilities in terms of technology sets (cf. Chapter IV, especially i n reference to (A.I)-(A.5) in sI3.l), for each i = I , Ti: the technology set of the sector producing the ith good, where ( A . l ) Tiisaclosedconvexconelyinginthe Cartesian product R," x R + ' . I f (xi, 1.J E T i ,then xi = (xji)and R i represent the input vector and the corresponding output of the ith good, respectively, with x i i standing for the required amount of thejth factor of production. (A.2) (0, 1.;) E Tiimplies 1.; = 0 (the impossibility of the land of Cockaigne). (.4.3') There is a process in Tithat produces a positive output.

$20.

CONDITIONS FOR UNIVALENCE

357

For any given factor price vector w 2 0, the value of the (unit) cost function ci(w) of the ith good is defined to be the minimum of the inner product ( w , x i ) over all processes

(.xi,1) E Ti.

The reader can readily verify that the cost function ci(w) is well defined for w 2 0 under the above assumptions. Now let us further assume that (A.5) If ( x i , Li) E Ti,(ui, pi) E T i , and if x i and u i are linearly independent, then for any positive numbers a > 0, p > 0, a + /3 = 1, there is a level of output of the ith good q i such that

aLi + ppi < q i (strict convexity).

(axi + /hi, qi) E Ti,

Then, in essentially the same way as in the treatment of J(q, a) = ( q , o(q, a ) ) in Chapter VI, $18.1, the reader can also verify the following facts: (i) For any positive w > 0, the unit input vector minimizing the unit cost of each good i is unique and can therefore be regarded as a single-valued function ai(w) = (aji(w))of w > 0. Hence 0

Ci(W)

=

.

1Ujt(W)Wj

(w > 0).

(4)

j = 1

(ii) aji(iv) ( i , j = 1, . . . , n) are continuous and nonnegative for w > 0. (iii) ci(w*)(i = 1, . , n) have continuous partial derivatives for M' > 0 and % ,

meaning the equality of the partial derivatives of the (unit) cost functions to the (unit) factor intensities. As a matter of course, the cost functions are common to all countries from the hypothesis of their common production possibilities. Therefore, if goods are traded in the competitive world market under a price system p = (pi),the world price p iof the ith good must equal its unit cost ci(ii,) in equilibrium, thus resulting in a system of 17 equations in the single unknown factor price vector it-, = p ; ( i = 1, . . . , n).

(6) It should be noted that (6) must hold in equilibrium, however unevenly endowed countries may be with factors of production. Samuelson argues that if (6) admits only a single solution i t ' , then the equalization of factor prices results without international factor movements. System (6) is undoubtedly a typical example of (1). The elements of the Jacobian matrix of system (6) are factor intensities. Whence conditions on the Jacobian matrix of ( I ) for the global uniqueness of solutions of (1) can be translated into conditions on factor intensities for that of (6) over the whole region of positive factor price vectors it', ensuring the factor price equalization in Samuelson's sense. Ci(ll.)

358

VII. THE JACOBIAN MATRIX AND GLOBAL UNIVALENCE

Another motivation for the study of global uniqueness in system (1) in terms of its Jacobian matrix comes from the system of equations in general equilibrium. Let E,(p) = 0 (i = 1, . . . , n)

(7)

be a system of n equations, where p = (pi)is the positive unknown n-dimensional price vector, p i standing for the price of the jth good, and E , ( p ) (i = I , . . . , n) are excess demand functions for the ith good. Suppose that the Walras law in the narrow sense holds, and E,(p) are homogeneous of order zero. Then, if the nth good is taken as a numeraire, the determination of equilibrium price ratios p 1 : p 2 : . . . : p , through the solution of (7) reduces to the study of the system of n - I equations in the positive unknowns p l , p z ,. . . , Pn-1:

E i ( p , ,p 2 , . . . ,p n - l , 1) = 0 ( i = I , . . . , n - 1).

(8)

If the excess demand functions are differentiable in the region of positive p i - l), conditions on its Jacobian matrix ( E i i ) of (8) for the global uniqueness of solutions, if any, will be of great importance in traditional economic theory. System (8) is obviously of type (1). Remark 1. The cost functions ci(w) have the following interesting properties : (i) Homogeneity of the First Order. That is (i = 1, . . . , n

ci(cut') = ctc,(~v) (a 2 0,

11'

20, i

. . . , n).

= 1,

This is obvious by definition. (ii) Nonnegativity of Function Values. That is Ci(lt.)

20

(11%

2 0, i = I , . . . ,n),

and they are strictly positive if 11' > 0. Their general nonnegativity is obvious by definition. Their strict positivity for I(' > 0 is implied by (A.2) and iv > 0. it** 2 0, I 2 r 2 0, we have (iii) Concavity. That is, for 111,

ci(( 1

- t ) n , + m*)2 ( I

- r)ci(rr.)

+ rc,(iv*)

(i = I , . . . , n).

Let (x, I ) , (x+,I), and ( y , I ) be unit cost minimizing processes in Ti associated with II'* and ( I - t ) ~+, t i t , * . Then, one finds, for each i, 111,

Ci((

1

proving concavity.

- t)lV

+ m*) = (( 1 - t)ll, + t w * , y ) = (1 - t)(Il., y ) + f ( W * , 2 ( 1 - t)(W, x) + /(It'*, = (1 - t)Ci(ll.) + fC,(N'*),

y) x*)

520.


359

(iv) Monotonicity. That is, w* 2 u’ 2 0 implies Ci(W*) 2 Ci(u’) (i = 1, . . . , n), which is obvious by definition and from the fact that x i 2 0 for ( x i , I ) E Ti by (A.1). For positive w > 0, this monotonicity is represented as the nonnegativity of the partial derivatives c i j ( w ) . Remark 2. The classical result for n = 2 by Samuelson on the factor price equalization, in his sense as global uniqueness in system ( 6 ) , is decisive and suggestive, although it is simple and easy to prove. Assume that the Jacobian determinant is nonvanishing in the region of all positive two-dimensional factor price vectors u’. This assumption ensures by virtue of the continuity of cij(w) that the determinant is either everywhere identically positive or everywhere identically negative. Whence the factor proportions of both sectors never reverse. In the light of the homogeneity and positivity of c,(\ti1, w2) (i = I , 2) noted in Remark 1, the uniqueness of positive solutions wl, 1 1 ’ ~ of (6) (n = 2) reduces to the uniqueness of positive solutions of the equation

in the single unknown w (= w2/wl), the factor price ratio. Using (4) and (5), we can compute the derivative of the function on the right-hand side of (9), obtaining

which is of the same sign for all o > 0. Whence the function c2(1, o)/c,( I , w ) is either strictly increasing or strictly decreasing, so that it can be equated to p 2 / p 1only once at most. Remark 3. The classical result for M = 2 in Remark 2 takes considerable advantage of certain of the properties of cost functions. The same conclusion of global uniqueness can also be reached by merely relying on the nonnegativity of the elements of the nonsingular Jacobian matrix, without appealing to properties (i) and (ii) in Remark I (see $20.5). The first higher dimensional determinantal condition on the Jacobian matrix for global uniqueness was proposed by Samuelson (1953-1954). This condition is as follows: If its equations and unknowns are suitably renumbered, the Jacobian matrix of system ( I ) has all its upper-left-hand-corner principal minors nonvanishing everywhere in the region X , that is,

lfkl

.’‘

fkk

I

360


The condition has not yet been proved to ensure global uniqueness for system (6), which has cost functions on its left-hand side and is therefore endowed with certain special properties not shared by ( I ) in general. But it has been shown (Gale and Nikaido (1965) that condition (10) alone is not sufficient for global uniqueness in system (1) in general, even in a rectangular region. This is verified by the following example. The system of equations

+ 3 = 0,

f l ( x , , x2)= exp(2x,) - x 2 2

f2(x,, x2) = 4 exp(2x,)x2 - x23 = 0 has two solutions x, = 0, x2 = k2, although fl ,(XI> x2)

=2

exP(2x*)> 0,

the whole plane R2. Remark 4. Let J ( w ) = (cJN~))be the Jacobian matrix of system (6). If J ( w ) is a constant matrix independent of MI,(6) is in effect a system of linear equations, so that the nonsingularity of J ( w ) entails the uniqueness of solutions of (6) in this extremely special case. McKenzie (1955) observed that if one takes the concavity of cost functions ((iii) in Remark I ) into account, the uniqueness of solutions by the constancy of J ( w ) can be extended to a somewhat general nonlinear situation whereJ(1v) need not be constant, but is nearly constant in the following sense. For a square matrix A of order n, let us denote by K ( A ) and K(A)* the convex cone generated by the n column vectors of A and its interior in R". Then, J ( w ) may be said to be nearly constant if for any H!' > 0, i t 2 > 0, we have in

a.

~ ( ~ ( r t , ' ) ' ) ' n K ( J ( M , ~ ) '# )"

(1 1)

It can be readily seen that if J ( w ) is nearly constant, the uniqueness of solutions obtains in system (6). Let IV', w2 be two solutions of (6). Using the vectors c(w) = (cl(it-)), p = ( p , ) , we can put (6) in the form ~ ( 1 1 3 ) = p , so that c(d)

= c(w2) = p .

(12)

By Lemma 18.2 in Chapter VI, we find J(\tJ)(td

- I?)

5 C(lV1)

J(IP)(lV' - w') 2

C(W')

- c(,v') = 0,

- C(M'2)

=0

from (12) and the concavity of cl(it>). Note that (1 1) implies in particular the nonsingularity of J(it1). Hence, if either J(wl)(wl- w 2 ) = 0 or J(w2)

$20.


36 1

w2)= 0, we have w1 = w2 from nonsingularity. It therefore remains to consider the case where

(wj’ -

J(W’)(W’

- d)5 0,

(13)

J(w2)(w’

-

2 0.

(14)

By (1 I), there are two positive vectors

WJ) X I

> 0, x2 > 0, for which

J(W”)’X’ = J(WJ)’X2 = u.

(15)

Forming inner products between x i and (13) and between x2 and (14), we obtain (u, w1 - Wl’) < 0 < (u, w1 - UJ), yielding a contradiction. Despite motivations from many disciplines, however, there are only a few works in the orthodox mathematical literature that deal with conditions on the Jacobian matrix for global univalence; and even these are quite recent. The joint research of Gale and Nikaido (1965), which considers the case of the Jacobian matrix having positive principal minors as well as other cases, seems to be the first general work along this line. Later, an interesting related result on the case of the Jacobian matrix having negative principal minors was obtained by Inada (1966), who applied a similar method of proof. The main purpose of the present chapter is to present these results (420) with applications (421). 020.2. P-Matrices and Related Concepts

For a real square matrix A of order n, we present a few definitions.

DEFINITION 20.1 (Gale and Nikaido, 1965). A is termed a P-matrix if all the principal minors of A are positive. DEFINITION 20.2 (Inada, 1966). A is termed an N-matrix if all the principal minors of A are negative. Moreover, N-matrices are classiJied according to the fof/owYng cutegories: (i) A n N-matrix A is said to be of the Jirst category if A has at least one positive element. (ii) A n N-matrix A is said to be of the second category if all the elements of A are nonpositive. DEFINITION 20.3. A is termed an N-P-matrix if it has all the principal minors of odd orders negative and those of even orders positive. DEFINITION 20.4. A is termed a P-N-matrix if it has aN the princ@al minors of odd orders positive and those of even orders negatice.

362

VIJ. THE JACOBIAN MATRIX AND GLOBAL UNIVALENCE

We begin by stating several lemmas and theorems that are of fundamental importance in what follows. As will be made clear, the role played by P-matrices is so decisive that the handling of N-matrices not only depends on the results on P-matrices, but is also very similar in treatment. Unless otherwise stated, in what follows, we mean by a matrix a real square matrix of order n. LEMMA 20.1. The follon.ing (i), (ii) hold: (i) A is an N- P-matrix if and only if - A is a P-matrix. (ii) A is a P-N-matrix ij' and only if - A is an N-matrix. Proof. Propositions (i) and (ii) are clear from the basic properties of the determinant, Q.E.D. LEMMA 20.2. A simultaneous and identical reordering of rows and columns converts P-matrices, N-matrices, N-P-matrices, and P-N-matrices again to P-matrices, N-matrices N-P-matrices, and P-N-matrices, respectively. Proof. By such a reordering, the values of the principal minors of a matrix do not change, whence follows the assertion, Q.E.D. LEMMA 20.3. Let L be a (proper, nonproper, or empty) subset of N = (1, 2, . . . , n}, and D be the matrix obtained by replacing the j t h column ej of the identity matrix by -e' f o r j E L. Then, D - ' A D remains to be a P-matrix, an N-matrix, an N-P-matrix, or a P-N-matrix, respectively, accordittg as A is a P-matrix, an N-matrix, an N-P-matrix, or a P-N-matrix. Proof. Every principal minor of D - ' A D is derived from the correspond ing original principal minor of A by multiplying the latter by an even power of - 1 ; therefore, they must be equal. This proves the lemma. Also note that D-' = D, an obvious but important fact, Q.E.D. Remark. It should be noted that if A is an N-matrix i n Lemma 20.3, A and D - I A D need not be of the same category. LEMMA 20.4. Every principal submatrix of a matrix A is a P-matrix, an N-matrix, an N-P-matrix, or a P-N-matrix, according as A is a P-matrix, an N-matrix, an N- P-matrix, or a P- N-matrix. Proof. This is clear by definition, Q.E.D. THEOREM 20.1. For a given matrix A , the system of linear inequalities

A x S 0,

x>,o has only the triuial solution x = 0 if either (i) or (ii) holds: (i) A is a P-matrix (Gale and Nikaido, 1965); (ii) A is an N-matrix o j thejirst category (Inada, 1966).

$20.


363

Proof. (i) We proceed by induction on n. The result for n = 1 is obvious. Assuming the result for n - 1, consider the case for n. Since A as a P-matrix has all its diagonal elements positive, we can increase some components of x, if necessary, until we obtain AY 5 0, (3)

with equality to zero for at least one component of A y , say the kth one, and

(4)

y?x_20.

Clearly, (i) will have been proved if it can be seen that y = 0. Now, in the light of Lemma 20.2, we may assume, without loss of generality, that k = 1. Explicitly, we find ally1 + a , 2 ~ 2 + . . . + a , n ~ n = O

+

~ 2 1 ~ ~1 2

2

+~ * 2. * + a2nyn 5 0 (5)

only1

+ a n 2 ~ 2+

+ Q n n Y n 5 0.

We note a,, > 0 and perform the following elimination: the ith relationai,/al times the first relation in ( 5 ) (i = 2, . , . , n) as in the proof of Chapter 11, Theorem 6.1. Thus we obtain

,

ally,

+ ... + Q l " Y " = 0 4 2 y 2 + ... + az*yy,5 0

-I-a , , y ,

a:, y2

+ . . . + a:n y , 5 0.

Let A* = (a:) ( i , j = 2, .. ., n) be the coefficients matrix of the system, consisting of all the relations in (6) except the.first one. If C* = (a,*,)( i , j E K ) , where K is a nonempty subset of ( 2 , . . ., n), is an arbitrary principal submatrix of A*, then C = (f . j ) (i, j E { 1) w K ) is a principal submatrix of A and

det C =

from the method of elimination. Moreover, a, > 0, det C > 0 because A is a P-matrix. Whence (7) implies det C* > 0, showing that A* is a P-matrix of order n - 1. Then the fulfillment of the last n - 1 relations in (6) by y i 2 0 (i = 2, . . . , n) entails y i = O ( i = 2 , ..., n) (8)

364


from the induction hypothesis. Finally, we find y , = 0 from the first relation in (6) and (8) in conjunction with a, > 0. Hence x = y = 0, Q.E.D. (ii) Since the diagonal elements of an N-matrix are negative, the order of an N-matrix of the first category must be at least two. With this in mind, consider (I), (2) for an N-matrix A of the first category. Since A has positive elements by assumption, we can have the same situation (3), (4) as in (i). Moreover, the argument for (i) leading from (5) to (7) applies directly to the present case, since a,, # 0 ( a l l < 0 ) and the order n of A is at least two, as already noted. The only difference is that we have a , , < 0, det C < 0 in (7) from the property of A being an N-matrix. Nonetheless (7) obviously ensures det C* > 0, whence A* is a P-matrix of order n - 1. Hence (i), as applied to the last n - 1 relations in (6) and y i 2 0 (i = 2, . . . ,n), entails y i = 0 (i = 2, . . . , n), which also implies y , = 0 by the first relation in (6) and a, # 0. Hence x = y = 0, Q.E.D.

,

,

COROLLARY 1. For a given matrix A , the system of linear inequalities A x > 0,

(9)

x>O

(10)

has a solution x i f A is either a P-matrix or an N-matrix of the first category. Proof. The transpose A’ of A is clearly either a P-matrix or an N-matrix of the first category, since A and A’ have the same principal minors in common. Therefore, Theorem 20.1 as applied to A’ implies that A’p 5 0, p 2 0 entail p = 0. Expressed in terms of the n x 2n matrix (Z, - A ) , where I is the identity matrix of order n, this means that the system of linear inequalities p’(Z, - A ) 2 0

has only the trivial solution p’ = 0, - p ’ A = 0. This is nothing but the Stiemke condition (Chapter I , Theorem 3.7(i)) for the existence of positive n-dimensional solution vectors u, x of the system of linear equations A)(:) = 0.

From (1 l), combined with the positivity of u, x, we have Ax=u>O,

x>O;

hence x is a desired solution of (9) and ( l o ) , Q.E.D. COROLLARY 2. 0 1 1 the same assumptions as in Corollary I , it’e catifitid a positice number 6 > 0 such that the maximum of components of A x is as large as 6 f o r all x 2 0 with unit norm (Ix/I= 1.

$20.


365

Pvoof. Write [ A x l i for the ith component of Ax as before, and let 6 ( x ) = max[Ax], over all i. 6 ( x ) is continuous in R" and therefore takes a minimum 6 on the compact set of all x 2 0 with unit norm j/x//= 1. This minimum 6 must be positive by Theorem 20.1. Clearly, 6 is a desired positive

number, Q.E.D. Geometric Characterization. From the geometric point of view, Theorem 20.1 means that a P-matrix A induces a linear mapping that does not send a point from the positive orthant R," into the negative orthant - R," except for the origin. But this property alone by no means characterizes P-matrices, since it is also shared by N-matrices of the first category, as shown i n Theorem 20.1 (ii). The following theorem provides us with a complete characterization in terms of the concept of sign-reversibility.

DEFINITION 20.5. For a matrix A and a vector x,denote by [ A x l i the ith component of A x , as before. Then A is said to reverse the sign of x if x i [ A x l i 5 0 ( i = 1, . . . , n).

(12)

THEOREM 20.2 (Gale and Nikaido, 1965). A is a P-matrix ifand only i f A does not reverse the sign of any vector except that of the zero .vector. Proof. (i) Necessity. Suppose that (12) holds for a P-matrix A and a vector x . Let L = { j I x j < O), where x = (xi), and D be the matrix in Lemma 20.3 associated with the present L. Then, D - I A D , which is again aP-matrix by Lemma 20.3, clearly reverses the sign of Dx 0. It therefore suffices to deduce the conclusion x = 0 from (12) for x 2 0, so that henceforth we may assume x 2 0. Next, let K = ( k 1 xk > o}. Assume K # 0. Then, let A be the principal submatrix of A obtained by deleting its ith rows and columns for i $ K , and R be the vector obtained from x by deleting its ith components for i 6K. Then A^ is again a P-matrix by Lemma 20.4 and reverses the sign of 2 . Moreover, we observe that 22 can have no positive component, since 2,[A^,t], 5 0, 2k > 0 ( k E K ) . That is, A^.? 5 0, 2 > 0, which contradicts Theorem 20. I(i). (ii) Sufficiency. Let A^ = ( a i j ) (i,,jE K ) be an arbitrary principal submatrix of A , where K is a nonempty subset of N = { 1, 2, . . . , n ). We shall have to show that det A^ > 0. In fact, if det A^ 5 0, A^ would have a real nonpositive eigenvalue 15 0 and an associated real eigenvector R # 0. This is obvious from the fact that det A^ equals the product of all the eigenvalues of A, with their multiplicities taken into account, and complex eigenvalues, if any, have their conjugate counterparts. We may conveniently number the components of 2 in a natural order by using integers in K. Then define an n-dimensional vector x = (xi) by x i = 2i (i E K ) , = 0 (i $ K ) . A reverses the sign of x # 0, since x i [ A x l i = 2i[A^2]i = Hi25 0 (i E K ) , xi[Axi] = 0 (i 4 K ) . This contradicts the assumption. Therefore det A^ > 0, and hence A is a P-matrix. This completes the proof.

366


920.3. Univalence Theorems Let f : X - t R"

(1)

be a differentiable mapping, where (E)

X is a region of R";

(b) f ( x )= (f i ( x ) )( x E X , i = 1, . . . , n),f i ( x )being differentiable functions on X , with total differentials n

dfi

=

C fij(x)dxj

(i = 1,

. . ., n ) .

i =1

Coupling the mapping (1) with a constant vector p

f (4 = P?

= (pi) E

R", we obtain (3)

an equation in the unknown vector x in X , which is a vector form of (1) in $20.1. The uniqueness of solutions of (3) for the unspecified (i.e., indefinite)p implies the (global) univalence of the mapping (l), meaning that (1) is one-toone in X , and vice versa. We are concerned with conditions on the Jacobian matrix J ( x ) = CfijCX)) (4) of (1) that ensure the univalence of (1). Special attention will be paid to univalence theorems that are derived by extending Theorem 20.1 to nonlinear situations.

THEOREM 20.3. Assume that X is a rectangular region of R". Then, for any fixed a E X , the inequalities

f(4 5 f (a),

(5)

X Z U

(6)

have only the trivial solution x = a in X if either (i) or (ii) holds: (i) The Jacobian matrix J ( x ) of the mapping (1) is everyctihere a P-matrix in X(Gale and Nikaido, 1965); (ii) The Jacohian matrix is continuous and everyicyhere an N-matrix of the first category (Inada, 1966). Proof. (i) The proof will be worked out by induction on M, the common dimension of both the argument and the image vectors x, f ( x ) of ( I ) . The result for n = 1 is obvious. Assuming the result for M - 1, let us consider the case for 17. Denote by M the set of all solutions x of (5) and (6) in X . Since the result holds trivially in the case where s 2 a for no x E X , we have only to consider

$20.

367


the nontrivial case where the opposite situation prevails. With this premise in mind, it will be shown first that a is an isolated point of M . In fact, differentiability (see Definitions 5.2 and 5.3 in Chapter I) implies

Since J(a) is a P-matrix, there is a positive number S > 0, by Corollary 2 to Theorem 20.1, such that for any x in X fulfilling x 2 a, some component of J ( a )

x-a ~

IIX

- all

2s>o. -

From (7) and (8), it follows that in the vicinity of a, some component of f ( x ) - f ( a ) must be positive for x 2 a in X . This proves that a is an isolated point of M . What we shall have to prove is that A4 contains only the single point a. To this end, suppose that A4 contained a second point b # a, which will be fixed once for all in the following argument. Clearly b 2 u, and X includes the bounded closed cube {x I a 5 x 5 6). Consider the subset of X :

Q = {x I a S x S 6,

A x ) Sf(a)>.

We recall the remark immediately following Definition 5.2 in Chapter I that differentiability implies continuity; hence mapping ( I ) is continuous. Whence Q is a closed subset of the compact cube, and is compact by Theorem 1.2(ii) in Chapter I. Moreover, Q\{a) is a closed set in Q , since a is an isolated point of M . Hence Q\{a} is also compact, again by Chapter I, Theorem 1.2(ii), while it is nonempty because of the inclusion of b. The sum of the components of x is therefore minimized at some X over Q\{a}. Clearly, this X is minimal in Q\{a} in the sense that no other element x i n Q\{a} fulfills x X. Now, in view of the obvious fact X 2 a implied by X E Q\{a}, only two possible situations are conceivable: (i.1) X > a ; (i.2) X 3 a. It will be shown separately in what follows that either situation entails a contradiction, thereby proving A4 = { a } . (i.1) X > a. Corollary 1 to Theorem 20.1, as applied to J ( X ) , a P-matrix, ensures the existence of a vector u satisfying u

< 0,

J(X)u < 0.

Then, noting .? > a, u < 0, we can choose a positive number we can let the vector-valued function

x(t) = ,u

+ tu

of a real variable t satisfy

x > x(t) > a

(e > t > 0).

(9) E

so small that

368


Since the rectangular region X contains X as well as a, we find X 3 x(t)

(E

> t > 0)

from (10). Moreover, by differentiability, we have

. f ( x ( t ) >= f G >+ j J ( +

+ 4Ilull)

(E

> t > o>,

so that

can be made as small as we like by letting t > 0 approach zero. This, in conjunction with the second inequality in (9), implies that for a sufficiently small positive I, f ( 4 t >I < f ( 32 f ( 4 . (12) It follows that x ( t ) E Q\{u} for this value o f t . In view of (lo), however, this contradicts the minimality of X. (i.2) X 3 a. In this case a t least one component of X = (Xi)equals the corresponding component of a = (ai). We may assume, renumbering the equations and variables if necessary, that X1 = a,, without invalidating the basic premise of a P-Jacobian matrix. With this understanding, define a new differentiable mapping?: 2 + R"-', where

(4) 8 = {(xz,xj , . . . , x,,)' I ( u l , x 2 , xj , . . . , x,,)' ion in R " - ' ; (/I> f(.x2,

-Yj,

. . . , .x,) = ( f i ( X 2 ,

Sj,

. . . , x,) =.f;.(u,, s 2 , x3,

x3,

E

x } ,a rectangular reg-

. . . , x,))

and fl(.Y2,

. . . , x,J (i = 2,

. . . ,n).

The Jacobian matrix of this new mapping is a principal submatrix of order I o f J(.x), evaluated for x 1 = ctI, which is a P-matrix by Lemma 20.4. Moreover,

17 -

- Y ~ ,. . . ,

x,> z f i ( u 2 , u 3 , . . . , a,) -Yi

(i = 2, . . . , n),

a , ( i = 2, . . .,1 7 ) .

Therefore we must have Xi = a , ( i = 2, . . . , n ) by the induction hypothesis. This contradicts S # a. (ii) The proof of (ii) almost parallels that of (i) except for the need to supplement the latter because of the classifications of N-matrices into the two categories. The supplementary remark is due to the fact that although a principal submatrix of an N-matrix is also an N-matrix, they need not be of the same category.

$20.

369


Now, in view of the indiscriminate validity of Theorem 20.1 and its two corollaries for N-matrices of the first category as well as for P-matrices, we can observe the direct applicability of the argument in the proof of (i) to the present case (ii), except for two points: (a) The induction must start at the case for n = 2, since N-matrices of the first category are at least of order 2; (b) The Jacobian matrix of the new mappingf : r? + R"-' should be shown not only to be an N-matrix but also to be of the first category. The proof of (ii) will therefore be complete if these two points are thoroughly discussed. (a) The Treatment of the Case for n = 2. We have the inequalities fi(x1, x 2 )

S f i ( a l , a21 ( i = 1, 21,

(13)

xi 2 a, ( i = I , 2)

(14)

in a rectangular region in R2 with the N-Jacobian matrix of the first category. We find

from theN-matrix property ofthe Jacobian matrix. Whencef,, f 2 , >fl f Z 2> 0, so thatf,, , f 2 1 are of the same sign. Note that the Jacobian matrix has positive elements by its first categoricity, while its diagonal elements are negative. Hence f 1 2 > 0, f 2 1 > 0. (15) Therefore, interchanging the numbering of equations, we can reduce (ii) for n = 2 to a case with a P-Jacobian matrix, while essentially preserving (13) and (14). (b) It should first be noted that under continuous differentiability, as is the case with (ii), the elements of an N-Jacobian matrix J ( x ) = ( f i j ( x ) ) do not change signs but are of definite signs throughout the whole region. In fact, the diagonal elements must be negative by the very definition of an N-matrix. Next, for i, j ( i # j ) we find

from the N-matrix property ofJ(x). Whencefij,fji >fii,fjj > 0, which implies that.fij ,.fji are not zero and that they are of the same sign. Since they are continuous by assumption,fij,fji ( i # , j ) are of the same and definite sign in the whole region (a connected set) for each pair (i, j ) , i # j , by Chapter I, Theorem 1.8. Let us now see that the new mapping f : 8 R"-' i n (i.2) has an NJacobian matrix of the first category in the present case (ii). As a matter of course, the Jacobian matrix j(x,, x3, . . . , x n ) o f f is the principal submatrix --f

'

370


J,(al, x,, . . . , x,,)obtained by deleting the first row and column of the original Jacobian matrix J(x) and evaluated for x1 = a,. It therefore suffices to see that J,(x,, x2, . . . , xn)is an N-matrix of the first category in X . That J 1 ( x l ,x2,.. . , x,,)is an N-matrix is obvious by Lemma 20.4. Next, suppose .fi,(Xl,

Xz . . . , X") > 0 (.i= 2, . . ., n). 3

(16)

Then,

f,,(x) > 0 (for any x E X , j = 2, . . . , n),

(17)

by the preliminary remark in (b). But, since XI = a , , X i 2 ai (i = 2, . . . , n), with strict inequality for at least one i 2 2, (17) must entail fi(X) >fi(a), contradicting (5). Hence (16) is ruled out, so that the left-hand side of (16) must be nonpositive for at least o n e j 2 2, say k , obtaining fik(X1, 2 2

9

.. . > Xn) I 0.

Then, if J,(X,, X, , . . . , 2,) were of the second category and hence had only the nonpositive elements, for the N-matrix of the first category J(X), the vector u = (vi) defined by i l k = 1, v i = 0 (i # k ) would be a nontrivial solution of the inequalities J(X)v 5 0, v >= 0. This contradicts Theorem 20.1(ii). Thus J,(X,, X,, . . ., X,) has some positive elements, which must remain positive in the whole region, again by the preliminary remark on their definite signs. Hence, J,(x,, x, , . . . , x,) is everywhere of the first category. The supplementary remarks are thereby complete. The reduction of the proof of (ii) to that of (i), as well as the indispensable supplementary remarks, are essentially due to Inada (1966). It is now possible to derive useful univalence propositions from Theorem 20.3. THEOREM 20.4. The mapping ( I ) , it-here X is a rectangular region, is univalent if either (i) or (ii) holds: (i) The Jacohiari matrix J ( x ) o f tlie mapping is euery,ihere a P-matrix in X (Gale and Nikaido, 1965); (ii) The Jacobiaii matrix is coiitiiiuous and is everyic,here an N-niatrix in X (Inada, 1966). Pvoof. We shall have to prove that f(a)

= f ( b ) , a, b E X

(18)

imply a = 6. To this end, let L = { , j I a j > b j } and D be the matrix corresponding to this L in Lemma 20.3. We define a new mapping g : Y + R" by the following formulas: (a") Y = D - ' X is again a rectangular region in R"; (Po) g ( y ) = D - ' f ( D y ) , with its Jacobian matrix being

J&)

=

D-'J(Dy)D.

$20. For a*

=

D-'a, b*

=


37 I

D - ' b in Y, clearly we find s(a*) = g(b*),

(20)

a* 5 b*.

(21)

Equations (20) and (21) imply a* = b*, and hence a = b, as will be shown separately for (i) and (ii), below. (i) J,(y) is everywhere a P-matrix in Y by (19) and Lemma 20.3. Hence, Theorem 20.3(i) as applied to (20) and (21)-a special situation of the inequalities g(a*) 2 g(b*), b* 2 a*-ensures a* = b*. (ii) J,(y) is everywhere an N-matrix in Y by (19) and Lemma 20.3. Jg(y)is continuous in Y from expression (19) and the assumed continuity of J(x). Therefore, J,(y) is of one and the same category throughout Y for the reason stated in the preliminary remark in (b) in the proof of Theorem 20.3(ii). The discussion henceforth will deal separately with (ii. I ) and (ii.2), depending on the category of Jg(y). (ii.1) The first category case. a* = b* is implied, as in (i), by Theorem 20.3(ii) as applied to g(a*) 2 g(b*), b* 2 a*. (ii.2) The second category case. All the elements of Jg(y)are everywhere negative by the preliminary remark in (b) in the proof of Theorem 20.3(ii). Thus, the supposition a* # b* in (20) and (21) would entail a contradiction g(a*) > g(b*). This proves a* = b*, Q.E.D.

COROLLARY. The mapping (I), where X is a rectangular region, is univalent i f either (i) or (ii) holds: (i) The Jacobian matrix J ( x ) of the mapping is everywhere an N-P matrix in X ; The Jacobian matrix is continuous and is everywhere a P-N matrix in (ii) X. Proof. This is immediately implied by Lemma 20.1 and Theorem 20.4. Univalence by a P-Jacobian matrix in Theorem 20.3 (i) can be put in a more general form, which is a nonlinear counterpart of the necessity part of Theorem 20.2.

THEOREM 20.5.

For the mapping (I), the inequalities

(xi- a,)(fi(x) -fi(a)) 5 0 (i = I , . . . ,n)

(22)

have only the trivial solution x = a if the region of dejnition X is rectangular and the Jacobian matrix J ( x ) is every\z>herea P-matrix in X . Proof. For a solution x of (22), letting L = { j I x j < a j } , we can define the corresponding coupled mapping g : Y + R" exactly as in the proof of Theorem 20.4. The Jacobian matrix J,(y) is again everywhere a P-matrix in Y.

312


After the transition from the original f to the new g, (22) obviously takes the form (Xi*- a,*)(g,(x*)- g,(a*)) 5

0 ( i = 1, . . . , n),

(23)

with x* 2 a*,

(24)

where x* = D - ' x = (xi*), a* = D - ' a = (ai*). Next, let K = { j 1 xj* > a j * ) .The proof will be complete, if it is shown that K = @. With this in mind, suppose K # @. We may assume, by virtue of Lemma 20.2, that K = { I , 2, . . . , k } while (23), (24), and the P-matrix property of J&y) are still valid. Consider the mapping h : Z + Rk defined by (E) and (8):

(G) z = { z I z = ( z i ) , (zl, . . . ,z k , a,*+l,. . . , a,*)' tangular region in R k ; (8) h(z) = (hi(z>>, hi(z)=gi(ZI,

.. . ) z k , a,*,,, .. . ) a,*)

E

Y } is again a rec-

...

( i = 1,

9

k),

(25)

with the Jacobian matrix Jh(z)being the principal submatrix of Jg(zl, . - . >z k , a:+,,

. - . )a n * )

(26)

formed by its first k rows and k columns. Then, we find hi(xl*, . . . , xk*) 5 hi(al*, . . . , ak*) (i = 1, xi* > a,*

. . . ,k),

(i = 1, . . . , k )

(27) (28)

from (23), (24), and the definition of K . On the other hand, Jh(z)is everywhere a P-matrix in Z , by Lemma 20.4. These results contradict Theorem 20.3(i). hence x* = a*, which implies x = a, Q.E.D. This proves K = 0; COROLLARY. (Xi

For the mapping ( I ) , the inequalities - Ui)(fi(X)-.f,(a))

1 0 (i = I , . . . , n)

have only the trivial solution x = a if the region of definition X i s rectangular and the Jacohiun matrix J ( x ) is everywhere an N-P-matrix in X .

Proof. This is immediately implied by Lemma 20.1 and Theorem 20.5.

The remaining part of $20.3 will be devoted to special situations where the Jacobian matrix is a particular type of P-matrix.

$20.

373


THEOREM 20.6 (Gale and Nikaido, 1965). Assume that the Jacobian matrix J ( x ) = (f i j ( x ) )of the mapping (1) has nonpositive of-diagonal elements fij(x)

2 0 ( i Zj)

(29)

in the region X . Then (i) and (ii) hold: (i) I f X is rectangular and Jex) is everywhere a P-matrix in X , the mapping (1) is unicalent, and its inverse mapping f is monotonic increasing. That is,

f (a) 5 f (b)

for a

= (a,),

b

= (bi)E

X

(30)

implies a 5 b. (ii) Suppose that the mapping (1) is univalent in an arbitrary open region X , not necessarily rectangular. I f its inverse mappjng f is diferentiable and monotonic increasing, then the Jacobian matrix of the mapping (1) is everywhere a P-matrix in X . Proof. (i) Univalence is already proved in Theorem 20.4(i). Monotonicity remains to be proved. Noting obvious monotonicity for n = 1, we proceed by induction on n. Assume monotonicity for n - 1, and consider it for n. It should be noted first that (30) rules out a > b, by Theorem 20.3(i), and hence we must have ak 5 b, for some k. By Lemma 20.2 and the invariance of condition (29) under a simultaneous and identical renumbering of equations and unknowns, we may assume that k = 1. As was noted before (Chapter I, §5.3), differentiability implies partial differentiability, whenever partial differentiation can be performed. In reality, partial differentiation can be performed everywhere in a rectangular region provided one-sided partial differentiation is applied to boundary points, if any. Under this convention, we find fi(b,,a2, . . . , a,) 5 f i ( a l ,a,, . . . ,a,) s f i ( b , , b,, .. ., b,)

( i = 2, . . . , n),

(31)

the first inequality holding from-f,, 5 0 (i > 1) and a, S b,, and the second being implied by (30). We next define a rnapping.7: ??-+ R"-' by the rules:

- X- = {(x,,

(ct)

- -

x 3 , . . . , x,)' I ( b l ,x2, x3, . . . , x,)'

E

X } , a rectangular re-

gion in R"-' ;

(P)

f(X2

2

x3 2

' ',>

x,) =

(.hx3 , 3

fi(b,, x2 , x 3 , . . . , x,) ( i = 2, . . . , n).

In terms o f x (31) is written as

-

.f(az, . . . ,a,)

3

' . . , x,)>

-

and

5f@, , . . . ,b,,h

icy,

>

-y3,

. . . , x,) =

(32)

374

VII.

THE JACOBIAN MATRIX AND GLOBAL UNIVALENCE

The Jacobian matrix off^is again everywhere a P-matrix having nonpositive off-diagonal elements in r?, by Lemma 20.4 and (29). Then (32) implies ai5 bi (i = 2, . . . ,n) from the induction hypothesis. This, in conjunction with a1 0 that for a real variable t we find x*(t) =a*

+ tu* E f ( X )

(It1

< E).

(33)

Let us consider the vector-valued function x ( t ) = f - ' ( x * ( t ) ) in the interval E ) . Then, sincef-' is assumed to be differentiable as well as monotonic increasing, x ( t ) is differentiable and

(-E,

d dr

-x(t) = u(t) 2 0

(It1

< E).

Whence, differentiating x*(t) = f ( x ( t ) ) at t = 0, we find

J(a)u(O) = u* > 0, u(0) 2 0. (34) Since J ( a ) has nonpositive off-diagonal elements, (34) implies, by Theorem 6.1 in Chapter 11, that J(a) satisfies the Hawkins-Simon condition and hence is a P-matrix. This completes the proof. Another noteworthy special situation is the case with a quasidefinite Jacobian matrix. This enables us to establish global univalence even in convex regions, and in a simpler alternative way. DEFINITION 20.6. A matrix A is said to be positive (negative) quasidejinite if the quadratic f o r m ( A u , u ) is positive (negative) definite. Clearly, any principal submatrix C of a positive (negative) quasidefinite matrix A is again positive (negative) quasidefinite. A matrix A is positive quasidefinite if and only if its symmetric part +(A + A') has all principal minors positive. Let 2 be any eigenvalue of A , and p,,,, p M be the minimum and the maximum of the eigenvalues of * ( A + A ' ) . Then it is well known that Prn 5 R e ( 2 ) ~S M . If A is positive quasidefinite, and hence 11, > 0, the real parts of all the eigenvalues 2 of A are positive from the above inequality, which implies that det A > 0. Since this argument applies to any principal submatrix C of A , a positive quasidefinite matrix is a P-matrix. Hence global univalence is ensured in rectangular regions by Theorem 20.4(i) if the Jacobian matrix is everywhere positive quasidefinite. Nonetheless quasidefinite cases deserve a reconsideration.

$20.


375

THEOREM 20.7 (Gale and Nikaido, 1965). The mapping ( I ) is univalent is convex and if its Jacobian matrix J(x) = (fi,(x))is everywhere positive (negative) quasidejinite in X .

if X

Proof. For obvious reasons, it suffices to prove a positive quasidefinite case. For a, b E X with a # b, let x ( t ) = a + th, where h = (hi)= ( b - a) # 0. By convexity we have x ( t ) E X ( l >= t 2 0). Consider the function

Then, upon differentiation, we find

which is identically positive by positive quasidefiniteness. Since (D(0)= 0, we must have @(1) # 0 and hence ,fi(b)-fi(a) # 0 for some i. This proves univalence, Q.E.D. The following application of Theorem 20.7 to analytic functions is of some interest, although it has no special bearing on economic theory.

COROLLARY (Noshiro, 1934). Let f ( z ) be an analytic (complex) function of a complex variable z in a convex region X of the complex plane. Then, f ( z ) is univalent in X ij'the derivativef ' ( z )lies in a half-plane whose interior does not contain the origin. Proof. It will be noted first that it suffices to consider the case in which the real part of the derivative is positive in X . In fact, putf(z) = u(x, y ) + iv(x, y ) , where z = x iy, u(x, y ) = Re(f(z)), u(x, y ) = Im(f(z)). Hence f'(z) = &/ax + i(av/ax),and there are, by assumption, some real numbers a,p, not all of them zero, such that

+

all

av ax

a-+D->O

ax

(inx).

Then, h(z) = (IX- ip)f ( z ) is analytic and its derivative has a positive real part in X . Clearly, f ( z ) is univalent if and only if h(z) is univalent. Now, let us assume Re(f'(z))

aLl

= -> 0

ax

(in X ) ,

(35)

and consider the Jacobian matrix J(x, y ) of the mapping

( x , y)'

-+

Y ) , v(x, Y))' :

x

-+

R2.

376

V11. T H E JACOBIAN MATRIX AND GLOBAL UNIVALENCE

J(x, y ) is positive quasidefinite in X because

from the Cauchy-Riemann equations

au ay

au -_ -a u _ ax

-=--

ay’

aU ax

and from (35). Whence univalence follows from Theorem 20.7, Q.E.D. Remark. The convexity assumption in Theorem 20.7 is indispensable. The following example shows that Theorem 20.7 is no longer true in a nonconvex region. In the light of the basic relation (36), the existence of a nonunivalent analytic function f ( z ) having the real part of its derivative positive in a nonconvex region will be shown. An example is given by the Joukowski mapping 1 f ( z )= z - ( 2 # O), z

+

when it is considered in suitable regions. Let R be the common exterior portion of two circles of radius having their centers at and -+, respectively, on the real axis. The real part of the derivative f ’ ( z ) = I - 1/z2 can be written as

+

+

which is positive in R. Butf(i) = f ( - i ) = 0 and i, - i E R. Moreover, one can easily construct a simply connected subregion X of Q containing i, - i .

$20.4. Weakening of Conditions The conditions on the Jacobian matrix for univalence in Theorems 20.4(i) and 20.7 can be relaxed to some extent. The purpose of $20.4 is to prove univalence under slightly weaker conditions with the aid of Kronecker’s existence theorem (Chapter I, Theorem 5.5). The results will be considered only for open regions, however, in order to avoid complications that might arise on the boundary.

DEFINITION 20.7. A matrix A is called a weak P-matrix if det A > 0 and all its proper principal submatrices have nonnegative determinants. A M9eak N-P-matrix is dejined likewise.

$20.


377

DEFINITION 20.8. A matrix A is called weakly positive quasidefinite i f det A > 0 and the quadratic form ( A x , x ) is positive semidefinite. A weakly negative quasidefinite matrix is defined likewise. THEOREM 20.8 (Gale and Nikaido, 1965). The mapping ( 1 ) in 620.3 is univalent i f either (i) or (ii) holds: (i) X i s an open rectangular region of R" u*herethe Jucobian matrix J ( x ) is a weak P-matrix everywhere; (ii) X is an open convex region of R" where the Jucobian matrix J ( x ) is weakly positive quasidefinite everywhere. Proof. The method of proof is exactly the same for (i) and (ii). We first prove that for any fixed a in X there is no x in X such that

f ( x )-f(a)

=

-A(x - a),

x # a (for some A > 0).

(1)

Suppose that (1) holds for some b in X . Then, by rearrangement, we find

Aa + f ( a ) = Ab + f ( b ) . Using this A, define a new mapping f : X

+ R"

(2)

by the formula

f ( x ) = Ax + f ( x ) .

(3)

The mapping f is clearly differentiable, with its Jacobian matrix being J,(x)

= AZ

+ J(x),

(4)

where Z is the identity matrix. Regarding the properties of J,(x), we have (lo), (2') below: (lo) J,(x) is everywhere a P-matrix in X in case (i); (2') J,(x) is everywhere positive quasidefinite in X in case (ii).

To prove (1') for a weak P-Jacobian matrix, let K , be an arbitrary principal submatrix of J,(x) of order k . Then, K, = AZ, + K for the identity matrix Zk of order k and a principal submatrix K of J(x). Whence we find det K,

= det(A1,

+ K ) = Ak + &(A),

(5)

where &(A) is a polynomial of degree k - 1 at most, having coefficients equal to certain sums of the principal minors of K . Since these principal minors are assumed to be nonnegative for a weak P-Jacobian matrix, we have det K, > 0 from (9, proving (1'). On the other hand (2') is obvious from A > 0 and (4), since (J,(x)u, u ) = A(u, u ) (J(x)u,u ) > 0 for u # 0. From (1') and (2'), it follows by virtue of either Theorem 20.4(i) or 20.7 that the mappingfis univalent. But (2) impliesf(u) = f ( b ) , and hence a = b, contrary to the assumed fulfillment of (1) by b.

+

378


Henceforth we will rely on the concepts and result in Chapter I, 55. For each a E X , let

X,= {x IfW # f (a)>.

(6)

Clearly Xu # 0 from the nonvanishingness of det J(x). Moreover, the mapping (1) in 520.3, if confined on X,,is homotopic to the mapping 9, : X ,

+

R"\{f(a)),

(7)

+x -a

(8)

g"(x) = f ( a )

by the homotopy bridge

H(x,t ) = (1 - t),f(x) + @(x)

(x E X , ,

1 2 t 2 0).

(9)

Note thatf(XJ $ f (a)from the definition (6). We also have g"(X,) $ f ( a ) , since x E Xu must imply x # a. To prove the asserted homotopy, it therefore suffices to see that (9) never coincides with ,f(a) for any x E X , and any t, 1 > t > 0. Suppose on the contrary that

H(x,t ) = f ( a ) (for some x E ,'A , t , 1 > t > 0).

(10)

Rearranging (lo), we would then find

x # a,

t > 0, 1-t

which contradicts the result in the foregoing step of the proof. This proves the homotopy. We are now ready t o prove univalence. To this end, if we suppose that

f(4=f@)

(1 1)

for some two distinct a, b in ,'A we shall have to deduce a contradiction. If we write 4 $ to mean that a mapping (b is homotopic to another $, from the above results on homotopy, we have the two homotopy relations

-

f - 9" : X,

-+

R"\{f(a)l,

f- gb : x b * R"\{ f ( b ) ) . But, since X ,

= Xb

by ( I l), we can reduce the above relations to

f

N

g" : M

-+

R"\(c),

f - g b : M - + R"\{c), where A4 = X , = Xb, c = f ( a ) = f ( b ) . Then, (12) and (13) imply, by the transitivity of homotopy, ga -gb : M+R"\{c}.

(14)

$20.


379

In view of the openness of X , det J(a) # 0, and a # b, we can find, by Chapter I, Theorem 5.6(i), an n-ball C,,(a, 6) lying in X such that b 4 C,(a, 61, f ( x ) #f(4 ( x E C,(a, 61, x

(15)

+ 4.

(16)

Note that (16) ensures, for the boundary sphere &',,(a, S) of C,,(a, S), S,(a, 6) c M .

(17)

From (14) and (17), it follows that

-

gQ gb : S,(U,S ) + R"\{c).

(18)

Consider dymc : S,,(a, S) &(a, 6), the mapping associated with g" (see Chapter I, $5.2 and the remark preceding Lemma 5.3 in Chapter I). Clearly, bgacis the identity mapping and hence, by Theorems 4.3 and 5.2, in Chapter I, is not homotopic to a constant mapping. Therefore, applying Kronecker's existence theorem (Chapter I, Theorem 5.5) to g', go asf , g , we can conclude that --f

c E g"C,(a, 6)).

(19)

But (19) implies b E C,,(a, 6) because c = gb(x) only for x = b. This contradicts (15). We have thereby proved that f(a) = f ( b ) for a, b E X entails a = b, Q.E.D.

COROLLARY. The mapping (1) in $20.3 is univalent i f either (i) or (ii) holds: (i) X i s an open rectangular region of R" where the Jacobian matrix J(x) is a weak P-N-matrix everywhere; (ii) X is an open convex region of R" tchere the Jacobian matrix J(x) is weakly negative quasidejinite everywhere. Proof. This follows by applying Theorem 20.8 to - f : X - + R", Q.E.D. 620.5. Miscellaneous Results for n = 2

The univalence proposition for a special two-dimensional case in Remark 2 in $20.1 rests not only on the information about the Jacobian matrix but also on certain properties of the cost functions, namely, their homogeneity and the positivity of their values. While this proposition is undoubtedly of great relevance in so far as factor price equalization is concerned, the special lower dimensionality of two-dimensional cases enables us to establish global univalence without relying heavily on the specific properties of cost functions.

380

VlI.

THE JACOBIAN MATRIX AND GLOABL UNIVALENCE

The results for two-dimensional cases in this section, although apparently fragmentary, may be useful. Throughout this section, f : X+R2

(1)

is a differentiable mapping, where

X is a rectangular region in R 2 ; xZ) = (,fi(xi, ~ 2 1 fi(x1, 7 x2))’, fi(x1, able functions on X , with total differentials (LX)

(B)

f(x17

d f i =fi1(x,,

x2)

dx,

~

+ f i 2 ( x , , x 2 ) dx2

2

(i) = 1, 2) are differenti-

(i = 172).

We shall consider two kinds of conditions on the Jacobian matrix for global univalence: (I) the one-signedness of principal minors and (2) the onesignedness of a row. ( I ) One-Signed Principal Minors. In the two-dimensional case, global univalence by a P-Jacobian matrix (Theorem 20.4) essentially implies global univalence by a Jacobian matrix having one-signed principal minors, as will be stated in the following theorem.

THEOREM 20.9 (Gale and Nikaido, 1965). Suppose that the Jacobian matrix

of the mapping ( I ) is continuous in X . Then, the mapping (1) is univalent in X if either (i) or (ii) holds: (i) N o principal minors of J ( x , , x2) vanish in X ; (ii) det J ( x , , xz)# 0 and no diagonal elements of J(x,, xZ) change signs in

an open X . Proof. (i) By virtue of the continuity of J(x,,x2) on a connected set X , Theorem 1.8 (Chapter 1) ensures that each principal minor is everywhere positive or everywhere negative in X . Multiplying some or all of the functions fi(xl, .rz)(i = 1, 2) by - 1 if necessary, we may assume that all the diagonal elements f j j ( x l ,x2) ( i = 1 , 2) are everywhere positive in X . Note that det J(xl, x 2 ) is also everywhere positive or everywhere negative for the same reason. Then if detJ(x,, x2) is everywhere positive in X , the present (i) is a special case of Theorem 20.4(i). Next consider the case where det J(x,, x 2 ) is everywhere negative. In this case, we find h 2

from

f i l f Z 2> 0, ,fil f 2 2

SZl > 0

-fi2f21

(in X)

= detJ(x,, x2) < 0,

(3) as in the preliminary

$20.


38 1

remark in the discussion of (b) in the proof of Theorem 20.3(ii). Inequality (3) implies, again from continuity and connectedness, that either

or In the case of (4), a new mapping f : X + R2,where f(xl, x2) = (f2(x1, x2), fi(x,, x2))', has a P-Jacobian matrix in X , and hence is univalent by Theorem 20.4(i). In the case of (5), another new mappingf : X + R2,wheref(x,, x2) = (-f2(x1, x2)> -fl(xl, x,))', has a P-Jacobian matrix in X , and hence is univalent by Theorem 20.4. (i) Clearly, in both cases the univalence o f f or implies that of the original mapping. (ii) If we proceed in the same way as in (i), we have three situations to consider: (m) the Jacobian matrix of the original mapping is a weak P-matrix; (fl)fhas a P-Jacobian matrix; (y)rhas a P-Jacobian matrix. Since Xis further assumed to be open this time, univalence obtains for any of the above three mappings by Theorem 20.8(i). This always ensures univalence for the original mapping, Q.E.D. ( 2 ) One-Signed Elements in a Row. We shall pay attention to the implication of the presence of a one-signed row in the Jacobian matrix, or, put in a seemingly more general form though amounting to the same thing, that of the existence of two real numbers n1,n 2 , not both zero, such that qfij+ n2 f i j ( j = 1,2) are one-signed.

THEOREM 20.10 (Gale and Nikaido, 1965). Suppose that det J(xl, x2) is one-signed and nonvanishing in X . Then the mapping (1) is univalent i f (i) There are two real numbers n,, 7 c 2 , not both of them zero, such that nlflj(x1,

~

2

+) 712 fij(x13 x2)

( j = 122)

(6)

are one-signed in X and at least one of the functions of ( 6 ) never vanishes in X . (ii) The nonvanishingness of one of the functions of ( 6 ) in (i) can be dispensed with, provided X is further assumed to be open. Proof. (i) Suppose n1 # 0. Then, the original mapping is obviously univalent if and only if the mappingf: X - t R2 defined by

f<x,,x 2 ) = (%fl(Xl, x 2 ) + 7(2 f i ( X 1 , X 2 ) > f 2 ( X l > X Z ) ) , is also univalent. The Jacobian matrix of,f, which is obtained by premultiplying that of the original mapping by the constant matrix

382


has one-signed elements in its first row. Thus, we can assume that the original Jacobian has one-signed elements in its first row. Moreover, by tra'nsforming some or all of the variables x l , x 2 to - x l , - x 2 , respectively, if neeessary, we may also assume that fij(xl, x 2 ) 2 0 ( j = I , 2) in X . Note that one of flj ( j = I , 2) is by assumption everywhere positive. Thus, renumbering the variables if necessary, we may finally assume that f l I ( X 1 , x2)

> 0,

f i 2 ( X I , x2)

2 0 (in

w.

(7)

The case n2 # 0 can be also reduced to a situation fulfilling (7). It therefore suffices to prove univalence for a situation where (7) is satisfied. Let a = (a,), b = (bi)E X be two points that are mapped under (1) to the same point, say p = ( p , ) , so that fj(a1, a2)

=fi(bl, b2) = p i

(i

=

1, 2).

(8)

We shall deduce a contradiction from the supposition a # b. We first observe that a2 # b, . Otherwise we would also have a, = b, byf,, > 0 in ( 7 ) . Hence u2 # 6 , , and we may assume that a2 < b2 .

(9)

Next, it will be shown that there is a function 4(x2) (b, 2 x2 2 a21

(10)

fulfilling f 1 ( 4 ( x 2 ) , x 2 ) = p1

@2

L x2 2 a2).

(1 1)

Indeed, for each x, in the interval [ a z , b2] we find fl(%

x2)

f l ( b 1 , x2)

2fl(%a21 = P1, S f , @ , , b2) = P1

fromf,, 2 0 in ( 7 ) and 6 , 2 x 2 2 a 2 . Then, from continuity and connectedness, f l ( x , , x 2 ) can be equated to p , at some x1 in the interval [min(a,, bl), max(a,, b,)] (cf. Example 1 in Chapter I, 9 1.4 and Theorem 1 .8). Such a value of x1 is unique for each x 2 E [ a 2 ,b,] byf,, > 0 in (7), and can therefore be regarded as a single-valued function (10) satisfying ( I 1). As will be noted at the end of the proof, the function 4 ( x z )is differentiable in [a,, b 2 ] , with its derivative +'(x2) satisfying fll(4(X,>,

xz)4'(x,) + h 2 ( 4 ( x 2 ) , x2) = 0.

(12)

Then the function b+z)

=f2(4(x2),x2)

is also differentiable in [a,, b 2 ] , and its derivative $'(x2) is given by V(x2)

= det 4

4 W ?x2>/fi1(4(x2),x2)

(13)

@0.

383


from (12) and .fil # 0 ensured by (7). Equation (13) implies that $'(x,) is nonvanishing and one-signed fromf,, > 0 in (7) and the nonvanishingness and one-signedness of det J(x,, x,). Whence $(x2) is monotonic. But this monotonicity contradicts $(a2)=f2(a,,a 2 )= p 2 =f,(b,, b,) = $(b2),a, f b, because a, = 4(a2),b, = 4(b2). The differentiability of 4(x2) and (12) remains to be shown. From (1 1) and the differentiability off,(x,, x,), we find

E(h) = fl l ( d ( X 2 1 , x2)A4

((A4)2

+fl2(4(XA

+ h2)1i2

x2)h

~

(h

+

O),

(14)

where

We first show

h+O

Suppose, contrary to (15), that there were a sequence {h,} such that

v- + ( r l 1 A4V

lim

where

A4v = 4(xz + h,)

- =

hv

+ 00,

- 4(x2) ( v = 1,2,

lim v-+

+

lim h,

= 0,

v++m

. . .). Equation (16) implies

hV

((A4,,),

+ hy2)l/, = O'

which, combined withf,, > 0 in (7) and (14), entails

But (17) and (18) contradict

This proves (15). Relation (15) ensures the existence of a constant M such that (for sufficiently small h).

384

V11. THE JACOBIAN MATRIX A N D GLOBAL UNIVALENCE

Then, using (14), (19), andf,, > 0 in (7), we have

whence

l i m Ad, -- - .fl2(d,(X,), 11-0 h fl I ( 4 ( X Z ) >

x2) x2)‘

This proves the differentiability of (P(x,) and (12). (ii) This case can also reduce t o the case where

.f;j ( d ~ l x2) , 2 0 (.i= 1 , 2) in X , as in (I). Suppose that there were two distinct points a = (a,), b = (b,) in X satisfying (8). Since X I S assumed to be open, we can find real numbers a t , p, ( i = I , 2) such that

a, b E R = {(xl,x,)’ 1 c[, < x,< B, (i = 1, 2)) c X . Then, choosing a sufficiently small positive number a2

- EM1

B2

E,

(21)

we can let

- EB1,

and the parallelogram subregion of R, R* = { ( X I , x,)’

1011

< x] < /II, c(,

< fi2 - E D 1 } ,

- &a, < x2 - E X ,

contain the two points a and h. Let us now define a new mapping g: Y+R2,

where

(4 (P)

y

= {(YI

1

YJ

I

< Y1 

+ Y 2 h 92(1’13 Y,) =fz(Y,,

EYl

+ Y2).

R* for (yl, y 2 ) ’E Y. The Jacobian matrix

3

i Whence

/ I i(1’1,

~ J ’ I J 2)

/,I(.Yl?

CJ’l

(22)

+ cfi ~ ( J ’ IEYI + Y 2 ) 1

+ Y 2 ) + c/22(L’1.

EY1

+ 1’2)

fi

+ Yz) V l + 1’2)

~ ( Y I EYi ,

f22(1’1?

$21.

APPLICATIONS TO SPECIAL JACOBIAN MATRICES

385

from (23), and is therefore nonvanishing and one-signed in Y . On the other hand, the nonsingularity ofJ(x,, x2)rules out flj(xl,x2) = 0 ( j = 1, 2) in (20), so that fil(Yl7 EYl

+ Y2) + Efi2(Yl. Vl + Y2) > 0,

f12(Y1r

W l

+ Y2) 2 0

in (23). Thus Sll(Y1, Y 2 )

> 0,

912(Yl, Y2)

2 0 (in Y ) ?

(24)

which is a counterpart of(7). But, Ycontains two distinct points ( a l , a2 - &a,)’, ( b l , b, - ~ b , ) ’that are mapped under g to the same point p . This contradicts (i), Q.E.D. Remark. Theorems 20.9 and 20.10 include as special cases situations where the corresponding Jacobian matrices have all their elements nonnegative. These cases can be handled by Theorem 20.9 since the diagonal elements of a nonnegative Jacobian matrix are one-signed. They can also be handled by Theorem 20.10 since all the rows of a nonnegative Jacobian matrix are one-signed. Factor price equalization for n = 2 is therefore already achieved by the irreversibility and nonnegativity of factor intensities before other specific properties of cost functions take effect (cf. Remark 2 in $20.1).

021. APPLICATIONS TO SPECIAL JACOBIAN MATRICES 021 .I.Special P-Matrices

In applying the univalence theorems proved in $20, in most of which P-matrices play a central part, it may be useful to enumerate certain specific types of P-matrices. This will be done in the present section. All matrices that appear in the following are real square matrices unless otherwise stated. ( I ) Matrices Having a Positive Dominant Diagonal. Let us put forward

DEFINITION 21.1. A complex matrix A = ( a l j )qf order n is said to have a dominant diagonal, if‘ there is a set of n positive numbers n i ( i= 1, . , . , n ) such that (i = I, . . . , n). luiil ni > ]aij(TC, (1)

C

jSi

Remark 1. It is clear from Definition 21.1 that a complex matrix A = ( a l j ) has a dominant diagonal if and only if the matrix D = (d,.,) defined by dii = laii(( i = I , . . . , n), dij = - laijl (i #j) satisfies the Hawkins-Simon condition or an equivalent (cf. Chapter 11, Theorem 6. I). This implies in particular that a matrix has a dominant diagonal if and only if its transpose has a dominant diagonal.

386

V11. THE JACOBIAN MATRIX A N D GLOBAL UNIVALENCE

Remark 2. It is.also clear from Definition 21.1 that every principal submatrix of a complex matrix having a dominant diagonal has a dominant diagonal.

THEOREM 21.1 (Ostrowski, 3937-3938, 1956; McKenzie, 1960b). A matrix A = ( a i j )liming a positive dominant diagonal is a P-matrix.

Puoof; First, it will be shown that a (complex) matrix B = (bij)having a dominant diagonal is nonsingular. Let n

2b i j Z j = 0

(i

=

I , .. . , n )

j = 1

for (complex) numbers zi( i = 1, . . . , 17). To prove the nonsingularity of B, we have only to see that all the z i must be zero. There are a set of positive numbers x i ( i = 1, . . . , n ) satisfying

from the dominancy of the diagonal. Let

lzil 0 = inax - over all

i

=

1,

. . . , n.

(4)

xi

Thus 0 >= 0, and it clearly suffices to show 0 = 0. Suppose 0 > 0 and this maximum is attained for i = k . Then lzll = Ox,,

lzil

5 Oxi

( i # I 0, and (5), implies

5 Ox IbkjI " j j#k

arriving at a contradiction. Whence 0 = 0, so that z i = 0 (i = 1, . . . , n). Now let us consider a (real) matrix A = ( a i j )having a positive dominant diagonal. Since every principal submatrix of A also has a positive dominant diagonal (cf. Remark 2), to verify the P-matrix property of A, we need only show that det A > 0. To this end, we observe that in the present case (1) becomes a i i n i > Iai,ilxj ( i = 1 , .. ., n )

C

.j # i

$2 1.

387


by virtue of the positivity of the diagonal elements, and hence (p

+ aii)ni> C

laijl xi

(i

=

I, . . . , n ; p 2 0 ) .

(6)

j+i

Inequality (6) implies that p Z + A , where Z is the identity matrix of order has a dominant diagonal for any p 2 0. Therefore det(p1

+ A) # 0

(p

2 0)

ti,

(7)

from the nonsingularity established above. Recall that det(pZ + A ) is a polynomial of degree n, with the coefficient of p” being 1 ;hence det(pZ A ) + 00 ( p -+ + a).Then if det A 5 0, we could let det(pZ + A), a continuous function, vanish for some p 2 0 by Theorem 1.8 in Chapter I, and Example 1 in Chapter I, 91.4. This contradicts (7). Whence det A > 0, Q.E.D. Remark 3. Matrices having nonpositive off-diagonal elements and satisfying the Hawkins-Simon condition form a special subclass in the set of all matrices having dominant diagonals (cf. Chapter 11, $6, Theorem 20.6, and Remark 1). ( 2 ) Nonsingular Skew-Symmetric Matrices. It is well known that the determinant of a skew-symmetric matrix of order n (see Chapter I, Definition 3.5) vanishes for an odd n and is the square of a polynomial of the matrix elements for an even n. Hence a skew-symmetric matrix can be nonsingular only when it is of an even order.

+

--f

THEOREM 21.2. A nonsingirlar skewsymmetric matrix A is a weakly positive quasidefinite matrix (see Definition 20.8) as itsell as a iiseak P-matrix (see Definition 20.7). Proof. Note that principal submatrices of a skew-symmetric matrix are also skew-symmetric. From the foregoing, it is therefore obvious that a principal minor of order k of A vanishes for an odd k and is nonnegative for an even k , and positive for k = n, the highest even order. This proves that A is a weak P-matrix. On the other hand, the quadratic form ( A x , x) identically vanishes by virtue of skew-symmetricity and hence is positive semidefinite. Since det A > 0 as noted above, A is weakly positive quasidefinite, Q.E.D. (3) Hicksian Matrices. Hicks (1939) put forward a condition on the Jacobian matrix of a system of excess demand functions which was asserted to be necessary for a kind of stability of equilibrium. A Hicksian matrix is a matrix whose principal minors of odd orders are negative and whose principal minors of even orders are positive. A Hicksian matrix is therefore nothing but an N-P-matrix (see Definition 20.3). It should be noted that the results in $20 have brought to light the implication of the Hicksian condition for the uniqueness of equilibrium in a competitive price formation, rather than the stability implication originally presumed by Hicks himself. This will be explained below.

388

Vll. THE JACOHIAN MATRIX AND GLOBAL UNIVALENCE

Let us recall (8) in $20.1. This is a system of 11 - 1 equations of general equilibrium in the 11 - 1 unknown levels of equilibrium prices of goods apart from therith one, which is anumeruirr. If the Jacobian matrixJ(p,,p,, . . . ,p,-,) of order n - 1 is Hicksian i n a rectangular region of prices, the Corollary to Theorem 20.4 ensures the uniqueness of solutions of (8) in $20.1. What is more interesting is the fact that the Hicksian condition entails the uniqueness of equilibrium price vectors determined, as in Chapter VI, Definition 18.4, by a system of inequalities E,(P,, P 2 >

. . . , P,) 5 0

( i = 1, . . . > n),

(8)

. . ., p,) = 0 (if P , > O), (9) ( i = I , . . . , n) are excess demand functions. We shall

€,(PI,

p2

3

where E,(pl,p 2 , . . . , p,) now consider a situation in which the excess demand functions are homogeneous of order zero in the region of all semipositive price vectors and are differentiable for p , 2 0 ( i = I , . . . , 17 - I), p , > 0. Assume further the Walras law in the narrow sense n

in the price region. For the case where the nth good serves as a numgraire, the system of inequalities (8) and (9) is equivalent to the normalized system (1 1)

E i ( ~ I , ~ 2 , . .1.) 5,0~ n( i = 1l,, . . . , n - l ) , p i E i ( p l , p 2 ,. . . ,p n p l ,1) = 0 (i = 1, . . . , 17

-

I),

p i 2 0 ( i = 1, ..., n - 1).

(12) (13)

It will be shown that if J ( p , , p r , . . . ,pnp1)= ( E i j ( p l p, 2 , . . . , P , , - ~ ) ) ( i , , j = 1. . . . , 17 - I ) is Hicksian, the uniqueness of solutions obtains for the system of inequalities(ll)-(l3). L e t q i ( i = 1, . . . , n - l)alsosatisfy(ll),(12)and(13), so that E i ( 4 , . 4 2 , . . ., Yn-], 1) 5 0 (i = 1, . . . , f l - 11, (14) qiE,(ql,q, ,..., q n - l , I ) = O

( i = I , ..., n - l ) ,

y i 2 _ o ( i = I , ..., n - I).

(1 5 )

(16)

It is then immediately possible to reduce (1 I)-( 16) to (pi -qi)(Ei(Pl>

~2

7

. . . > P n - 1 , 1) - Ei(q1>42 . . . > 7

(;=I

1)) 2 0 , . . . , 17-1).

qn-1,

(17)

In the light of the Corollary to Theorem 20.5, the Hicksian Jacobian matrix of system (17) i n the region of all (ri - 1)-dimensional nonnegative vectors therefore implies p i = q i ( i = 1, . . . , n - l), as was to be shown.

$21.


389

021-2. The Stolper-Samuelson Condition

In a special reference to the protection of domestic factor owners by tariff, Stolper and Samuelson (1941) studied the effect on factor rewards of a change in the world good prices. They consider a simple two-good two-factor situation. As in the consideration of factor price equalization, they rely on (6) in $20.1 for n = 2 as the fundamental relationship between the world good prices p i and domestic factor prices w j . Explicitly

where ci(wl, w2) is the cost function of the ith good (i = I , 2) defined for positive factor prices wl, w 2 . As was already noted in 520.1, the elements cij(wl, w2) of the Jacobian matrix of (I),

are factor intensities, which are assumed to be positive for simplicity. If the factor proportions satisfy CIl(W,,

w2)

> C2I(Wl,

C 1 2 ( W I , w2)

w2)

C 2 2 ( W I , w2)'

(2)

we say that the first good is intensive in the use of the first factor, and the second good is intensive in the use of the second factor. On the other hand, if

we say that the first good is intensive in the use of the second factor, and the second good is intensive in the use of the first factor. When the factor proportions never reverse in the region of all positive factor prices I I ' ~ , H ' ~one , can, in a unique way, associate with each good the factor in the use of which it is intensive. For easier reference, we mean by the associated factor of a good, the factor that is associated with the good in the above good-factor association. Then the noted Stolper-Samuelson proposition asserts that a rise in the price of a good entails a rise in the price of its associated factor and a fall in the price of the other factor. The proposition is verified as follows. Since case (3) can be put in form ( 2 ) by renumbering either goods or factors, it suffices to discuss case (2). From (2), we find detJ(ttr,,

1it2)

> 0,

(4)

390


which, in conjunction with the positivity of c i j , implies that J(w,, w2) is a P-matrix in the region W of all positive two-dimensional factor price vectors w = (wi). Denote by P the image of W under the mapping w + c ( w ) = (ci(w)) induced by the cost functions. Then, by the Corollary to Theorem 5.7 in Chapter I, P is an open region of RZ,from (4), whereas, by Theorem 20.4(i), the mapping is invertible from the P-matrix property of J ( w l , wz). Let the inverse mapping be p+w(p):P+

w.

(5)

By Lemma 5.4 in Chapter I, the mapping ( 5 ) is also differentiable because of (4), its Jacobian matrix being the inverse matrix J(kvI, ivJ1 of J(w,, N , ~ ) .The Stolper-Samuelson proposition will be established if it can be verified that .f(wl, w2)- has positive diagonal elements and negative off-diagonal elements. In fact,

which has the desired property from (4) and the positivity of c i j ( w , , w z ) . In the general n-good n-factor case, there seem to be no well-accepted factor intensities conditions that make possible an n-dimensional version of the above good-factor association. I n what follows, we will therefore only touch upon the implication of an n-good n-factor version of the StolperSamuelson proposition for factor price equalization. Let us reconsider the original Stolper-Samuelson proposition. We recall that the factor intensities condition (2) first ensured the P-matrix property of the Jacobian matrix and hence the global invertibility of the mapping i t ' + c(iis), before the signs of the Jacobian matrix of the inverse mapping were detected. It seems to be better, however, to reverse our procedure in the general n-good n-factor case. Consider the system of equations (6) in 520.1, which governs the fundamental relation between good prices p i and factor prices in the general case. We shall consider the mapping It' + c(w)

: W + P,

(6)

where W is the region of all positive factor price vectors; W under (6); ( y ) c(it.) = (ci(ii,)), with cj(ii.) ( i = I , . . . , n ) being the cost functions. (x)

( p ) P is the image o f

Instead of advancing a faithful n-dimensional version of the StolperSamuelson proposition, we put forth a condition on the Jacobian matrix J(It7)

= (Cjj(Ilj)

(7)

$21.


39 1

of the mapping (6), which is suggested by the proposition, and we examine the implication of the condition for factor price equalization. The condition is formulated as follows :

DEFINITION 21.2. A matrix A is said to satisfy the Stolper-Samuelson condition if (i) A is nonnegative; (ii) A is nonsingular; (iii) its inverse A - ' = ( d i j ) has all its ofl-diagonal elements nonpositive, that is, dij 5 0 (i # j ) .

THEOREM 21.3 (Chipman, 1964). condition, A must be a P-matrix. Proof. Let D as A , we have

= A-'

= (dij).If

(8)

If A satisjies the Stolper-Samuelson

I is the identity matrix of the same order

D A = I.

(9)

Further let c be an arbitrary positive vector. Then, premultiplying c by (9), we find D ( A c ) = c > 0,

(10)

while Ac

>= 0,

(1 1)

from (i) in Definition 21.2. By virtue of Theorem 6.1, in Chapter 11, it follows from (8), (lo), and (11) that D satisfies the Hawkins-Simon condition. We shall now prove the assertion by induction on the order n of A . Equation (9) for n = l is d,,a,, = l . This implies a,, > 0 because d,, > 0 by the Hawkins-Simon condition. Next, assuming the theorem for 17 - I , consider the case for n 2 2. First, we find det D . det A = 1 from (9). This implies det A > 0,

(12)

since det D > 0 by the Hawkins-Simon condition. To prove the P-matrix property of A , it therefore suffices to show that every principal submatrix of order 17 - I of A is a P-matrix. Let A* be any of such principal submatrices. Performing an identical and simultaneous renumbering of the rows and columns of A , D, f, if necessary, we may assume that A* is located in the lower right-hand corner of A . I n view of d,, > 0 implied by the HawkinsSimon condition. define the matrix

392


1

0

0

1

T=

:I

0 0

0

an 1

1

BIBLIOGRAPHY

I. Books for General Mathematical Reference ANALYSIS Courant, R. (1934). “Differential and Integral Calculus,” Vols. I, I1 [transl. from the original German ed.]. Wiley (Interscience), New York. ELEMENTARY TOPOLOGY Alexandroff, P., und Hopf, H. (1935). “Topologie,” Vol. I . Springer, Berlin. Lefschetz, S. (1949). “Introduction to Topology.” Princeton Univ. Press, Princeton, New Jersey. Pontryagin, L. S. (1952). “Foundations of Combinator-ial Topology” [translated from the first (1947) Russian ed.]. Graylock Press, Rochester, New York. Berge, C. (1959). “ EspacesTopologiques, Fonctions Multivoqucs.” Dunod, Paris [English transl. “Topological Spaces, including a Treatment of Multi-Valued Functions, Vector Spaces and Convexity.” Oliver & Boyd, Edinburgh and London, 19631. MATRIX THEORY Bellman, R . (1960). Introduction to “ Matrix Analysis.” McCraw-Hill, New York. Gantmacher, F . R. (1960). “The Theory of Matrices,” Vols. 1, 11. Chelsea, New York.

DIFFERENTIAL EQUATIONS Coddington,E. A.,and Levinson, N.( 1955). “ Theoryof Ordinary DifferentialEquations.” McGraw-Hill, New York. MATHEMATICAL ECONOMICS Gale, D. (1960). “The Theory of Linear Economic Models.” McCraw-Hill, New York. Karlin, S. (1959). “Mathematical Methods and Theory in Games, Programming and Econonomics.” Addison-Wesley, Reading, Massachusetts. 11. References

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Ostrowski, A. (1937-1938). Uber die Determinanten niit iiberwiegender Hauptdiagonale. Comment. Muth. Helu. 10, Ostrowski, A. (1956). Determinanten niit iiberwiegender Hauptdiagonale und die absolute Konvergenz von linearen Iterationsprozessen. Comment. Mrrth. Helu. 30. Perron, 0 . (1907). Zur Theorie der Matrizen. Muth. Aim. 64. Rader, J. T. (1964). Edgeworth exchange and general economic equilibrium. Yule &on. Es.srrys 4, No. I . Radner, R. (1961). Paths of economic growth that are optimal with regard only to final states: A turnpike theorem. Rev. Econ. Studies 28, No. 2. Ranisey, F. P. (1928). A mathematical theory of saving. €con. J . 38, No. 152. Saniuelson, P. A. ( I 948a). “Foundations of Econoniic Analysis.” Harvard Univ. Press, Cambridge, Massachusetts. Saniuelson, P. A. (1948b). International trade and the equalisation of factor prices. €con. J. 58, No. 230. Samuelson, P. A. (1949). International factor-price equalisation once again. Econ. J . 59, No. 234. Saniuelson, P. A. (1951). Abstract of a theorem concerning substitutability in open Leoiitief Models. In “Activity Analysis of Production and Allocation ” (T. C. Koopmans, ed.). Wiley, New York. Saniuelson, P. A. (1953-1954). Prices of factors and goods in general equilibriuni. Rev. Ecorr. Strrrlies 21, No. 1. Sanitielson, P. A. (1960). Efficient paths of capital accumulation in ternis of the calculus of variations. I n ‘* Mathematical Methods i n the Social Sciences” ( K . J. Arrow, S. Karlin, and P. Suppcs. eds.). Stanford Univ. Press, Stanford, California. Saniuelson, P. A. (1965). A Catenary turnpike theorem involving consumption and the golden rule. Amer. Eeon. Reu. 55, No. 3. [Saniuelson (1948b, 1949, 1951, 1953-1954, 1960), Solow and Samuelson (1953), and Stolper and Saniuelson ( 1941) are included in “The Collected Scientific Papers of Paul A. Samuelson,” Vols. I , 2. M.I.T. Press, Canibi-idge, Massachusetts, 19661. Scarf, H. (1960). Some examples of global instability of the competitive equilibrium. I/rtertrrrt. Eco/r. Re[.. 1 , No. 3. Scarf, H. (1962). An analysis of markets with a large number of participants. In “ Recent Advances in Game Theoi-y.” Princeton Univ. Conference Rept. Shapley, L. S. (1953). A value for n-pcrson games. I / / ”Contributions to the Theory of Games ” ( H . W. Kuhn and A. W. Tucker, eds.). Vol. 11. Pi-inccton Univ. Press, Princeton, New Jersey. Shapley, L. S. ( I 964). Value of large games-VII : A general exchange cconomy with money. Memo. RM-4248-PR. RAND Corp. Santa Monica, California (cited with the kind permission of the author). Shapley, L. S . , and Shubik, M . (1966). Purc conipctition, coalitional power and fair division. Memo. RM-4917-PR. RAND Corp. Santa Monica, California (cited with the kind permission of the authors). Sion, M . (1958). On general minimax theorems. Poci$c J . Mrrth. 8, No. I . Slutsky, E. ( I 9 15). Sulla tcoria del bilancio del consumatore. Giom. E m / . 51. Smith, A. (1776). “An Inquiry into the Nature and Causc of the Wealth of Nations,” 2 Vols. Strahan and Cadell, London. Solow, I.5 , N o . 2. von Neuniann, J. ( 1928). Zur Theorie der Gesellschaftsspiele. Moth. Ann. 100. von Neumann, J. (1937). Uber ein okononiisches Gleichungssysteni und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes. Ergehni.Pse eines Mathematischen Kolloc/rrcirrni.r, No. 8, 1935-1936. Franz-Deuticke, Leipzig and Wien, 1937 [English transl. A model of general economic eqiiilibrium. ReC. Econ. Stirdies 13 (1945-1946)]. \on Nc Liniann, J., and Morgcnstern, 0. ( I 947). “Thcory of Games and Econoniickhavior.” Pi-inceton Univ. Press, Princcton, New Jersey. Wald, A . ( 1935). U ber die cindeiitige positive Losbarkeit der neuen Produktionsgleichungen. Ergeln~.Mrrt/7. Colloq. No. 6. (1933-1934). Wald, A. ( 1936). Uber die Produktiongsleichiingen der okonomischen Wertlehre. Ergebr7. Motli. Colloq. No. 7, (1934-1935). Wall-as, L. ( I 874) ‘* EICmcnts d’Economic Politique Pure.”Corbaz, Lausanne [Englishtransl. ‘* Elements of Pure Economies” (W. Jafle-, transl.). Allen & Unwin, London, 1954; Irwin, H omewood, Illinois, 19541.

Author Index Numbers in italic refer to pages where thc complete reference is listed.

F

A Alexandroff, P., 393 Arrow, K. J., 189, 191, 236, 249, 253, 254, 260, 305, 324, 329, 336, 338, 343, 354, 393 Atsumi, H., 220, 226, 229, 394 Aumann, R. J . , 295,394

B Begle, E. G . , 64, 394 Bellman, R., 393 Berge, C . , 393 Block, H. D., 305, 324, 329, 336, 338, 343, 354, 393 Bourbaki, N., 28, 31, 394 Brouwer, L. E. J., 63, 270, 394

C Chipman, J. S . , vii, 391, 394 Coddington, E. A., 338, 393, 394 Courant, R., 393

Farkas, J., 38, 394 Frobenius, G., 102, 394

G Gale, D., vii, 220, 229, 236, 249, 265, 307, 356, 360, 361, 362, 365, 366, 370, 373, 375, 377, 380, 381, 393, 394, 395 Gantmacher, F. R., 393 Georgescu-Roegen, N., 90, 189,395 Goldman, A. J., 139, 395 H

Hahn, F., 305, 395 Hawkins, D., 90, 395 Hicks, J . R., 304, 387, 395’ Hopf, H.. 393 Howe, C. W.. 145, 395 Hurwicz, I.., 305, 324, 329, 336, 338, 343, 354, 393 1

D Dantzig, G. B., 132, 394 Debreu, G., 236, 237, 246, 249, 253, 254, 260, 265, 281, 283, 284, 393, 394 Dorfman, R., 196, 207, 394

E Edgeworth, F. Y . , 237, 284, 394 Eilenberg, S., 64, 394

Inada, K., vii, 208, 356, 361, 362, 366, 370, 395

K Kakutani, S., 64, 67, 395 Karlin, S., 393 Kemeny, J . G., 145, 395 Knaster, B., 63, 395 Koopnians. T. C., 189, 220, 395 Krein, M., 39, 395 399

400

AUTHOR INDEX

Kronecker, L., 78, 79, 395 Kuga, K., 305, 309. 312, 315, 324, 395 Kuhn, H. W., 307, 394 Kuratowski, C . , 63, 395

L

R Rader, J. T., 284, 397 Radner, R., 208, 210, 21 1,397 Ramsey, F. P., 220, 397

5

Lefschetz, S., 393 Leontief, W. W., 88, 395 Levinson, N., 338, 393, 394

M McKenzie, L. W., 208, 236, 249, 305, 324, 335, 360, 386, 395, 396 Mazurkiewicz, S., 63, 395 Milman, D., 39, 395 Minkowski, H., 38, 396 Montgomery, D., 64, 394 Morgenstern. O., 145, 395, 398 Morishima, M., 150, 208, 249, 305, 396 Mosnk, I. L., 304, 396 Muth, J . F., 150, 396

Samuelson, P. A., 150, 189, 196, 207, 220, 325, 336, 356, 359, 389, 394, 397, 398 Scarf, H., 237, 284, 337, 394, 397 Shapley, L. S., 296, 397 Shubik, M., 296, 397 Simon, H. A., 90,395 Sion, M., 322, 397 Slutsky, E., 302,397 Smith, A,, 87, 397 Solow, R. M., 150. 196, 207, 394, 397 Sperner, E., 61, 397 Stiemke, E., 36, 38, 398 Stolper, W. F., 389, 398 Suits, D. B., 150, 398

T N Negishi. T.. 249, 305, 396 Nikaido. H., vii, 67, 132, 162, 236, 249, 265, 305, 306, 322, 356, 360, 361, 362, 370, 373, 375, 377, 380, 396 Noshiro, K., 375, 396

0 Ostrowski, A,, 386, 397 P Perron, O., 102, 397 Pontryagin, L.. S., 393

208, 307, 365, 381,

215, 309, 366, 395,

Thompson, G . L., 145,395, 398 Tsukui, J., 208, 220, 230, 398 Tucker, A . W., 36, 38, 39, 139, 307, . 394, 395, 398

U Uzawa, H., 52, 220, 249, 254, 255, 256, 168, 305, 338, 341, 344, 354, 398 V Vind, K., 295, 398 von Neumann, J., 64, 67, 69, 142, 398

W Wald, A., 236, 249, 398 Walras, L., 1, 235, 270, 398

Subject Index A Additivity, 214 Affine subspace, 16 spanned by convex set, 22 Affinely independent points, 53 Allocation, of production and consumption, 247 Arrow-Debreu model, 250 Average, of allocation, 290

B Balanced growth, 99, 199 of first kind, 163 of second kind, 164 Balanced-growth solution. 99, 100, 112, 126, 130, 150, 159, 171, 179 Barycenter, 55 Barycentric subdivision, 55 Brauer-Solow’s condition, 94 Budget constraint. 344

C Cardinalist. 245 Cartesian product of mappings, 70 of sets, 24 Cauchy-Peano local existence theorem, 338 Cauchy-Riemann equation, 376 Characteristic equation, 100 Closed interval, /I-dimensional. 9 Coalition, 286 blocked by, 287 Commodity bundle, 238 indifferent to, 221, 238 preferred to, 231, 238 Competitive equilibrium. 251

Composition, of set-valued mappings, 71 Concave function, 51 strictly, 51 Constant returns to scale, 88, 182 Consumer, positively endowed, 291 Consumption function, 162 set, 238 Continuation, possibility of, 338 Convex body, 2 1 Convex cone, 33 dual, 33 pointed, 43 polyhedral, 41 Convex function, 26. 44 strictly, 50 Convex hull of mapping, 72 of set, 17 Convex polyhedron, 41 Convex set, 16 dimension of, 23 Convexity, aggregative, 255 of function, 26, 44 of preference field, 241 of set, 16 Core, of economy, 287 Cube. 9, 10 Cyclical decomposition, 115

401

D Demand function, 244 individual, 261 Diameter, of \et, 55 Differenti tbility of function, 79 of mapping, 80 Dimension, of convex set, 23 Disposal hull, 213

402

SUBJECT INDEX

Dominant diagonal, 385

Dua I it y for closed convex cones, 34 in linear programming, 133 for polyhedral convex cones, 43

E c . 4 Pair, 332 canonical, 332 E-Ncighborhood, 12 €-Net, 12 Eipcnvalue problem, 100 nonlinear, 151 Equilibrium price vector, 251, 263, 309 Equiljbrium solution, 263 Euclidean distance, 7 Euclidean norm of matrix, 81 of vector, 6 Euler's theorem, on homogeneous functions. 315 Excew \upply function apprcpate, 261 modilied, 274 Extcrnal diseconomies, 183 Extreme point. 39

F Feasible path of finite horizon N , 196, 221 of infinite horizon, 196 Finite intersection property, I 1 Fixcd point of set-valued mapping, 67 of singlevalued mapping. 60 Fixed-point theorcm Rrouwcr's, 63 Kakutani's, 67 Free disposability, 184 Frobenilis-Perron's theorem, 102

G Golden rule, of accumulation, 224 Good desired. 153 productive, 253

Gross substitutability, 305 in differential version, 305 generalized, 306 strong, 305 Gross substitute, 304 Growth fast, 175 slow, 175 Growth factor. 199

H Hawkins-Simon condition, 90 Homogeneity. 150, 270, 358 Homotopy, 74, 378 bridge, 74, 378 Hyperplane separating, 32 supporting, 27

I Implicit function theorem, 85 Income effect term, 302, 304 Indecomposability, 156, 306 of rnattix. 105 for technology set, 204 Indifference class, 238 relation, 238 Individual supply function, 260 Inner product, 6 Input coefficients, 88 Insatiability, 240 Invariance of interior points, 83 11-revcrsihility of agpi-egate processes, 250 o f all processes, 184 of process, 183

J Jacobian, 81 Jacobian matrix, 81, 355 nearly constant, 360

K Kronccker's existence theorem, 78, 79

SUBJECT INDEX

L Lagrangian multiplier, 51, 140, 225 Land of Cockaigne, impossibility of, 183, 184, 250, 356 Leontief's system, 88 profitability of, 93 workability of, 89 Linear combination convex, 16 nonnegative, 18 of sets, 25 Local inverse, 85 Local univalence theorem, 81

M Majorization lemma, 339 Mapping differentiable, 80 point-to-set, 64 set-valued, 64 Marginal productivity theory, 141 Matrix decomposable, 105 Hicksian, 387 imprimitive, 115 indecomposable, 105 Jacobian, 81, 355 multiplier, 96 N-, 361 N-P-, 361 nonnegative, 9.5 P-,361 P-N-, 361 period of, 119 positive, 95 positive quasidefinite, 374 primitive, 1 15 semipositive, 95 skew-symmetric, 38 stable, 114 super-multiplier, 99 weak N-P-, 376 weak P-, 376 weakly positive quasidefinite, 377 Metric, 11 , Metric space, 11 Minkowski-Farkas lemma, 38

403

Monotonicity, 150 strict, 150 weak, 150 Moving equilibrium,99, 144 Multiplier, matrix, 96 numerical, 96 Multiplier process, dynamic, 98

N ri-Ball, 10 rz-Disk, 10 (rr-1)-Sphere, 10 Nonnegative invertibility, 95 Nonnegative orthant, 33 Nonsubstitution theorem, 190 Nrrniiraire, 337, 358, 388

0 Open covering, 11 Ordinalist, 245 Orthogonal complement, 3 1 , 37 of vector, 2 15

P Pareto optimum, 280 Path balanced-growth, 199, 223 maximal, 199 efficient, 197 feasible, 196, 221 golden, 224 optimal, 222 rr-optimal, 208 Period, of matrix, 119 Point nonnegative, 34 po$itive, 34 semipositive, 34 Preferable element least, 240 most, 240 Preference cardinal measurability of, 245 continuity of, 239 field, 238 indicator, 244

404

SUBJECT INDEX

Preference ( c o n ! . ) relation, 238 weak axiom of revealed, 325 Preference relation, 238 increasing, 243, 297 strong, 238 Primitivity, 157 of matrix, 115 Process, 181 aggregate, 246 disposal, 184 efficient, 185 flow, 182 of inaction, 184 irreversibility of, 183 production, 181 Production function classical, I88 neoclassical. 189 Productive technology, 147 Projection, 24 orthogonal, 31, 42, 215 Proximity consecutive, 212, 215, 230 weak, 229

Q Quadratic form, positive semidefinite, 50

R R;iy golden, 224 maximal balanced-growth. 203 von Neuninnn, 203 Region closed. 80 open. 80 rectangular, 80 Relation eq ti i va 1elice, 2 38 indifference, 238 preference, 238 Repetition, r-fold, 289 Replica, r-fold, 288 Resolvent equation. 165 positive. 165 set. 165 Reverie, \i:n of vector, 365

Root dominant, 103 Frobenius-Perron, 103 RouchC’s theorem, 82

S Satiation point, 240 Scalar multiplication, 6 Schwarz’s inequality, 7 Segment, 16 Separation theorems, 27 Set arcwise connected, 15 compact, 8, I 1 connected, 14 contractible, 76 convex, 16 resolvent, 165 Set-valued mapping closed, 65 graph of, 65 upper semicontinuous, 65 Simplex, 54 derived, 57 face of, 54 regular derived, 61 regular derived face, 61 standard, 10 Skew-symmetricity, 306 Slater condition, 5 2 Slutsky equation, 302 Sol LIt i on niinorant, 130, 131 optimal, 135 stationary, 158 Spending function, 162 Spel-ner’s lemma, 61 Stability global, 337 relative. 99, 112, 155, 171 Stiemke’s theorem, 36 Stolper-Samuelson condition, 391 Strict convexity of function, 50 of preference field, 242, 297 of technology set, 195, 221, 357 Sribstitution effect term, 302, 304 Super-multiplier, matrix, 99 Symnielry, of substitution terms, 304

405

SUBJECT INDEX

T TPtonnement, 268, 336 Technology set, 181 aggregate, 246 flow version of, 182 production, 181 stock version of, 182 Total differential, 79 Transformability, 195 Triplet golden, 224 productive, 223 Tucker’s theorem, 36 Turnpike theorem consumption, 228 Radner’s, 21 1 strong, 212 weak, 212

U Unique profitability, 205 Utility function, 221 indicator, 221

V von Neumann model, 142 von Neumann price vector, 205 von Neumann ray, 203

W Walras law in general sense, 263 in narrow sense, 263

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