Equilibrium, Stability and Growth: A Multi-Sectoral Analysis

EQUILIBRIUM STABILITY, AND GROWTH

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EQUILIBRIUM STABILITY, AND GROWTH A Multi-sectoral Analysis BY

MICHIO MORISHIMA

OXFORD AT THE C L A R E N D O N PRESS

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OXFORD UNIVERSITY PRESS

Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Bangkok Buenos Aires Cape Town Chennai Dar es Salaam Delhi Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Sao Paulo Shanghai Singapore Taipei Tokyo Toronto with an associated company in Berlin Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Oxford University Press 1964 The moral rights of the author have been asserted Database right Oxford University Press (maker) Reprinted 2002 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose this same condition 011 any acquirer ISBN 0-19-828145-5

TO MY PARENTS

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PREFACE

THIS book brings together my recent papers published in several journals. They have been revised in various places and contain some new material. The main model carried through the whole book is Professor Leontief's input-output system; and it is dynamized from various points of view. Chapter I discusses formal similarities between the exchange equilibrium under weak gross substitutability and the static input-output system. By algebraic operations only (no infinite processes being used) it is seen that the Hicksian laws of comparative statics and the LeChatelier-Samuelson principle hold in both systems. Chapter II is concerned with the stability of the mixed WalrasLeontief system, to which I apply the conventional analysis of stability of equilibrium. The use of this method may be j ustified by the Walrasian device of tdtonnement, or by making the assumptions that consumption goods and raAV materials are perishable and not used more than once, and that capital goods are not subject to purchase and sale, only their services being traded on the market. There does not occur any redistribution of durable goods among individuals throughout the whole process of trading and pricing. On the other hand, Note to Chapter II shows that a pure exchange equilibrium is stable under some fairly reasonable assumptions, even if transactions of durable goods and, therefore, redistributions of the stocks of goods take place in the midst of pricing. In Chapter II, I also explore the laws of comparative statics which can be established for the Walras-Leontief system. Chapters III and IV are companion chapters dealing with a mixture of the dynamic Leontief system and the Walrasian model of capital formation. They, however, assume different types of production functions: neo-classical production functions or discrete 'activities'. They discuss various effects of technological improvements and correlations among prices, wages,

viii

PREFACE

and the interest rate. It is seen that there exists an efficient balanced growth path. Chapter IV, in particular, examines the stability of the balanced growth path as well as that of the long-run equilibrium prices. I am also concerned with the possibility of the golden age in that model, a multi-sectoral version of Mrs. Robinson's problem. (Note to Chapter III establishes the existence of a solution to the Walrasian equations of capital formation and credit.) Chapters V and VI are devoted to an analysis of the von Neumann model of economic expansion, which may be considered as a variant of the dynamic Leontief system. As Professor Champernowne pointed out, the original von Neumann model is a slave-system, where the object of production is mere enlargement at the subsistence level of wages. In Chapter V, I generalize it so as to allow consumers' choice. I present two versions of the generalizations, called the Marx-von Neumann and Walrasvon Neumann models respectively, each of which is shown to have an efficient balanced growth solution. In Chapter VI the Turnpike Theorem (asserting that there is a long-run tendency for the optimum growth path of an economy to approximate to the von Neumann path of steady proportional growth) is proved for a von Neumann-Leontief model and for a von NeumannGale model. Finally, an Appendix generalizes the classical theorems on non-negative matrices to systems of non-linear and homogeneous functions. Some of these results are repeatedly used in the text, and all of them are of great importance to workers in the field of multi-sectoral analysis. Other mathematical theorems and concepts which are not familiar to economists are explained in footnotes of the chapters where they are used for the first time. I wish to express my indebtedness to Professors J. R. Hicks, F. H. Hahn, and L. R. Klein, who read the manuscript in a nearly finished form, and made a number of most valuable suggestions. Professor Nikaido, my colleague in Osaka, often helped me out of mathematical difficulties. I owe the idea of Note to Chapter III to a conversation with Professor Kinzo

PREFACE

ix

Saito. The acknowledgements which I have to make are not only to them but also to Professors K. J. Arrow, N. GeorgescuRoegen, L. W. McKenzie, T. C. Koopmans, P. A, Samuelson, F. Seton, R. M. Solow, G. L. Thompson, H. Uzawa, and T. Yasui, by personal discussion with whom I have much benefited. Thanks are also due to the editors and publishers of Econometrica, Economica, the International Economic Review, the Journal of the London Mathematical Society, the Review of Economic Studies, Stanford Mathematical Studies in the Social Sciences, and Zeitschrift fur Nationalokonomie for permission to include in this book the articles originally published in them. My final debt is to my wife who typed the whole manuscript and prepared the diagrams. M.M. Osaka, Japan July 1962

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CONTENTS

I. C O M P A R A T I V E STATIC A N A L Y S I S OF THE SIMPLEST I N P U T - O U T P U T S Y S T E M 1. Introduction 2. Laws of Comparative Statics under Weak Gross Substitutability: The Hicksian Laws and the LeChatelierSamuelson Principle 3. Properties of the Input-output System IT. S T A B I L I T Y A N A L Y S I S OF THE W A L R A S L E O N T I E F SYSTEM 1. Introduction 2. Semi-global Stability of Equilibrium 3. Global Stability of Equilibrium 4. An Alternative System 5. Comparative Statics 6. A Mathematical Note N O T E TO C H A P T E R II.

III. A D Y N A M I C L E O N T I E F SYSTEM W I T H N E O CLASSICAL PRODUCTION FUNCTIONS 1. Introduction 2. The Determination of Production Coefficients 3. The Balanced Growth of Outputs 4. The Demand-Supply Balance of Labour CAPITAL FORMATION

3 14

23 23 25 31 33 36 38

S T A B I L I T Y OF A N O N -

TATONNEMKST P R I C I N G

N O T E TO C H A P T E R III.

1 1

43

54 54 56 69 76

W A L R A S ' s T H E O R Y OF 83

IV. AN A L T E R N A T I V E D Y N A M I C S Y S T E M WITH 93 A S P E C T R U M OF T E C H N I Q U E S 1. Introduction 93 2. The Time Path of Prices 94 3. The Time Path of Outputs 105 4. The Rate of Interest 112 5. Technological Invention 116 6. Mechanization of Technique 122

xii

qONTENTS

V. W O R K A B I L I T Y OF G E N E R A L I Z E D VON N E U M A N N M O D E L S OF B A L A N C E D GROWTH 131 1. The Original von Neumann Model 131 2. A Model of an Expanding Capitalist Economy : Marx—von Neumann 136 3. Existence and Efficiency of the Balanced Growth Path 140 4. Alternative Model: Walras-von Neumann 148 VI. B A L A N C E D G R O W T H AND EFFICIENT P R O G R A M M E OF V E R Y - L O N G - R U N GROWTH 1. Introduction 2. Proof of a Turnpike Theorem : The 'No-joint-production Case' 3. Cyclic Exceptions 4. A Generalization of the Radner-Nikaid6 Theorem : Cyclical Growth around the Turnpike APPENDIX. G E N E R A L I Z A T I O N S OF THE PERRON-FROBENIUS THEOREMS FOR NON NEGATIVE SQUARE MATRICES 1. Introduction 2. Non-linear Eigen-vector Problems 3. Balanced Growth 4. Stability 6. Non-linear Resolvent Problem

154 154 156 171 174

195 195 196 202 204 211

REFERENCES

216

INDEX

225

I COMPARATIVE

STATIC

ANALYSIS

OF

THE SIMPLEST INPUT-OUTPUT SYSTEM

1. Introduction I N T E R I N D U S T R Y analysis of the Leontief type is concerned with systems in which the products of economic factors (materials, machines, labour, etc.) are themselves used as factors to produce further goods. Various Leontief-type models have been presented, the simplest of which will be discussed in this chapter. The economy is divided into n industries, each producing one commodity and consuming the products of the other industries as well as its own product. There is no good entering production repeatedly in more than one period (capital good), and the model contains only goods which cease to exist once they are used up in production (current goods). Furthermore, it is assumed that there is no production-lag. Let xi be the total output of industry i, and x^ the amount of output of industry i absorbed by industry j. The net output of each industry, i.e. the excess of' xi over

, is available for

outside use and will meet the final (or exogenous) demand. Then the overall input-output balance of the whole economy can be expressed in terms, of n equations:

where ci represents the final demand for output i. If we assume that only one method of production is available to each industry, we have x{i — a^Xj (i,j = l,...,n), the aif being constant and standing for the input coefficient of commodity i used in industry j. The above system of equations now becomes:

2

C O M P A R A T I V E STATIC A N A L Y S I S OF

which are the basic equations of the simplest input-output model. The price-valuation subsystem of the Leontief system consists of n identities each saying that the value of output equals that of inputs plus profits, or that the price of every good produced equals its unit cost of production plus the profits earned per unit of output. Let pi be the price of product of industry i, w the wage rate, and qi the profits per unit of output in industry i. Let an+li be the labour-input coefficient of industry i. Since the unit cost of good i is denoted by we obtain the following system of equations:

In particular, if the long-run competitive equilibrium prevails, profits vanish in all industries, and the above equations are reduced to:

Note that there is a duality relation between outputs and prices in the Leontief model in the sense that the coefficients (a^) of the price-determining subsystem are obtained by transposing the coefficients (a^) of the output-determining subsystem. Algebraic properties are common to both the subsystems. The Leontief system described above descends from the Walrasian system of general equilibrium which consists of the four sets of equations: (i) market supply functions for factors, (ii) market demand functions for goods, (iii) the demand-supply equations for goods and factors, and (iv) the price-cost equations. In the Walrasian system the supply functions for factors and the demand functions for goods are given in terms of prices by the marginal utility theory. In the Leontief model, however, there is no counterpart to them, and the final demands (clt..., cn) and the values added (am+1-1 w+qv..., a m + i, n w+q n ) are taken as variable constants or parameters. Outputs are determined independently of the prices given by the price-cost equations. On

THE SIMPLEST I N P U T - O U T P U T SYSTEM

3

this separability and the assumption of fixed technical coefficients is based the simplicity of the Leontief model. As a^ I> 0, the model has a close formal relation to a competitive equilibrium model in which all goods are gross substitutes ; so the results obtained for the latter can be applied to the former mutatis mutandis. It is instructive to devote one section to the competitive equilibrium model before the input-output analysis is taken up in detail. 2. Laws of comparative statics under weak gross substitutability: the Hicksian laws and the LeChatelierSamuelson principle We are indebted to Professor Hicks for showing the kinds of propositions that can be derived from a system in which all goods are gross substitutes for each other. Recently, the stability properties of that system were discussed by Arrow and Hurwicz [2, 3], Arrow, Block, and Hurwicz [1], Hahn [1], McKenzie [2], Negishi [1, 2], Uzawa [3], and others. Hicks himself [1, pp. 7275] was concerned with the comparative static analysis of the system; he showed that when the demand for a good, say good j, increased, (i) the price of good j would rise, (ii) all other prices would rise too, and (iii) the prices of all the other goods would rise proportionately less than the price of good j. Mosak [1, pp. 42-51], Okishio [1], and others proved these laws mathematically, but their proofs were not elementary in the sense that differential calculus and determinants could be dispensed with. It should be remarked additionally that their proofs are valid if and only if the change in the demand for good j is infinitesimal. In this section we give alternative proofs of the Hicksian laws; no differential calculus, no theorems of determinants are used; and the theorems below hold true even if the demand for good j is increased drastically. The second great law of comparative statics to be discussed below is that which may be called the LeChatelier-Samuelson principle. Mathematically speaking, it is a theorem originally concerned with the definite matrices associated with maximizing problems and then extended (by Samuelson) so as to hold for

4

C O M P A R A T I V E STATIC ANALYSIS OF

non-negative matrices which need not be definite or symmetric. It has found many applications in economic theory (see Samuelson [1, pp. 36-39] and [6]). In the application to a system of multiple exchange in which no complementarity is present, the principle states: the change in the price of a good (say i) with respect to a shift of demand from the numeraire to any good is most positive when all prices are permitted to vary; only less so when the price of one other good is held constant, but all other prices are adjusted; and so forth, until all prices, excepting that of good i, are held constant. In this section a simple, but nonlocal, proof of this principle will be given which does not require any theorem on determinants, though it is usually proved by use of the Jacobi determinant theorem. Let us consider an economy in which n-{-\ goods, labelled 0, 1,..., n, are exchanged for each other. The non-normalized price of good i is denoted by Pi and is always assumed to be nonnegative. Let Xt(P) be the excess demand for good i at price vector P = (P0,P1,...,Pn). Throughout this section the following are assumed: (B) Xt(P)'s are single-valued and bounded from below; (C) X^Pys are continuous in an extended sense such that, for any sequence {Pr} converging to P, if Xt(P) is finite, if Xt(P) is not finite;*1' (H) Xf(PYs are positively homogeneous of degree zero in P, i.e. XAXP) = XAP) for a-ny A > 0; (W) Xi(P)'a satisfy the Walras law, i.e. Arrow and Hurwicz [2], Arrow, Block, and Hurwicz [1], Hahn [1], Hicks [1], Mosak [1], Negishi [1], and others were concerned with the case where all goods are strong gross substitutes at all prices, that is, (i,j = 0,1,...,n; i ^ j) (1

> Negative infinite values of Xj(P)'a are excluded by (B).

THE SIMPLEST I N P U T - O U T P U T SYSTEM

5

for all P, or

for all P' and P" such In this section, however, attention is given to the more general case, discussed by Arrow and Hurwicz [3], McKenzie [2], and Uzawa [3], where all goods are weak gross substitutes at all prices, that is, where it is assumed only that (i,j = 0, l,...,n; i ^ j) for all P, or that (i = 0, l,...,n) for all P' and P" such

(S)

that The assumption (S) asserts that, with price increases in any goods, the excess demands for those goods whose prices remain unchanged move upwards, if at all. The excess demand function vector, X = (X0, Z'1,...,^'n), is said to possess an equilibrium price vector P if the conditions are satisfied. If -X^(P) < 0 for i such that I\ > 0, we would have PX(P) < 0, a contradiction to (W)._ Thus X^P) = 0 if Pt > 0. Conversely, if X^P) < 0, then Pt = 0. A good i is said to be free if X.t(P) < 0. We use the following lemmas due to Arrow and Hurwicz [3, 4], LEMMA 1. Continuity, (C), in the extended sense, single-valuedness and the boundedness from below, (B), of the excess demand function [together with homogeneity (H) and the Walras Law (W)] imply the existence of a set of equilibrium price vectors; i.e. the set^ is not empty. (1 ! We adopt the following conventions for vector inequalities. Let a — {a(} and 6 = {bt}, a 3: b means a^ £ b^ for all i; a > b means a § b and a ^ b; a > b means ai > 6j for all i, < 2 > E = {P | X(P) g 0, P > 0} denotes the set of all P's such that X(P) g 0 and P > 0.

6

C O M P A R A T I V E STATIC A N A L Y S I S OF

LEMMA 2. // (B), (C), (H), and (W) hold, weak gross substitutability (S) implies that the weak axiom of revealed preference holds for any pairs of price vectors, one of which is an equilibrium price vector, i.e. where P e E and P $ E Let us now assume that the Oth component, PQ, of any P e E is strictly positive. Suppose that there is an increased desire for a particular good j, which the persons trading are prepared to satisfy by diminishing their demand for the good 0, the demands and supplies for all other goods being unaffected. The problem at hand is to show what change in the equilibrium price Pj will result from such an increase in desire. We can establish THEOREM 1 (the first Hicksian law). Given a shift in tastes (excess demand] from good 0 to good j, the price of j in terms of 0 must increase unless j is a free good.W Proof. Let a change in tastes give rise to changes. in the Oth and the jth excess demand functions, such that W This inequality establishes the stability of the non-normalized process as well as that of the normalized process where Ki and ki are positive constants, and pi — Pj/P0. See Arrow and Hurwicz [3], Arrow, Block, and Hurwicz [1], Karlin [2, pp. 305-20], and Uzawa [3]; also Note to Chapter II of the present volume. £ is the usual symbol for settheoretic membership. If both P and P' belong to E, Q = aP+ (1 - a)P' with 1 > a > 0 also belongs to E; because otherwise we have yielding

, a contradiction to (W). Thus£ is convex when all

goods are weak gross substitutes. This fact was observed by McKenzie [2] and Arrow and Hurwicz [3]. See p. 62, n. 1. < 2 > In Theorems 1-4, assertions (B), (C), (H), (W), and (S) are supposed to hold before and after a shift in tastes; so that it does not affect the stability of the system. When tastes change, prices will depart from the old equilibrium position, converging to the new equilibrium.

THE SIMPLEST I N P U T - O U T P U T SYSTEM

7

X0(P) > X'0(P) and Xj(P) < X'^P) at an old equilibrium price vector P, other functions X^P) (i = 1,..., j— I , ji'+l,..., n) remaining unchanged. Thus, for all t's other than 0 and j, the new and the old excess demand functions are identical, i.e. Xi(P) =X{(P)foTi = l,...,j-l,j+l,..., n. Let E' be th e s e t of new equilibrium price vectors; i.e. Since good j is a non-free good, Xj(P) is equal to zero, so that X'j(P) > 0; therefore P does not belong to E'. Hence, by Lemma 2. (1) where P' e E'. Taking into account that (2)

we find: (3)

We also_obtain PfX^P) = 0 for i = 1,..., j - l , j+l,..., n, because .E is zero if in (2) strict inequality applies; this, together with

(the Walras law), yields (4)

Subtracting (4) multiplied by P'0 from (3) multiplied by P0, we From

it follows

that

In deriving Theorem 2 (the second and third Hicksian laws in weak form) below we use the condition ofindecomposability.M (I) Let E be any set of indices {i0,i1,...,im}. The relations P'i = PI for ie R and Pk < P"k for k $ R imply that there exists at least one i e R such that Xf(P') =£ X^P"). ^ The corresponding definition of indecomposability in terms of partial derivatives is as follows: If there is no permutation matrix II such that

with square submatrices, X] and -X^n> on ^ne diagonal, then the system is called indecomposable. See Solow [1] and Debreu and Herstein [1].

8

C O M P A R A T I V E STATIC A N A L Y S I S OF

In words, no set of goods can be found whose price increases would not affect the excess demand for at least one of the remaining goods in the system. We first obtain LEMMA 3. // (B), (C), (H), (W), (S), and (I) hold, every equilibrium price vector is strictly positive, and the normalized equilibrium price vector (or the equilibrium price ray) is unique, Proof. Let P' be a price vector such that P't — 0 for i in R = {i0,i1,...,im}. Suppose all X^P'Ys are finite at P'. It follows from weak gross substitutability (S) and indecomposability (I) that there exists at least one i in R such that where A > 1. This, of course, contradicts the positive homogeneity (H). Hence at least oneXt(P') must be plus infinity, which implies that P' cannot be an equilibrium price vector. Let P and P' be two equilibrium price vectors. The above argument shows that they are strictly positive. Suppose •nP ^ P' for all TT > 0. Let A = min (JP-/IJ). Then we find t(P') must be plus infinity,

that \PI = P'i for i in some R — {i0, *i,.--, v} and XPk < P'k for k$R. Therefore, it follows from (S), (I), and (H) that (5)

for at least one i e R. Since P is an equilibrium price vector and is positive, excluding the possibility of free goods, the righthand side of (5) is equal to zero. Thus Xt(P') > 0. This contradicts the fact that P' is an equilibrium price vector. Hence •nP = P' for some -rr > 0. Now we shall prove THEOREM 2 (the second and third Hicksian laws in weak form). Given a shift in tastes (excess demand) from good 0 to goodj in an indecomposable system (I), the price, of any good in terms of 0 will move upwards if at all, and there are no goods whose prices (in terms of good 0), increase proportionately more than that of good j. Proof. Let the excess demand functions after a change in tastes be denoted by X't(P) as in Theorem 1. Let the old and

THE S I M P L E S T I N P U T - O U T P U T S Y S T E M

9

the new normalized equilibrium price vectors be p — (pv..., pn) and p' = (p'i,-.-,p'n), respectively, where From Lemma 3, p and p" are unique and strictly positive, so that ratios ^i = pjp'i are well defined. Write fj, = max fj.t. Suppose it is greater than 1; without j=l

n

loss of generality we may assume that p, = ^k (k— 1, 2,..., r 1, pp'r+1 > pr+l,..., p,pn > pn into account, it follows from (S) and (I) that there exists at least one good i e {!,...,r} such that (7)

Together with (6), this leads to for at least one

(8)

On the other hand, it follows frocti and

that

(9)

(10)

It is evident that (8) contradicts (9) and (10). Hence for all JNext write mm Reorder the goods so that the first r values of fj{ are the smallest. Suppose the set {!,...,»•}. does not contain j. By (H) we obtain (6). Since fj, < 1, PPr+i < Pr+n — ' PPn < Pn> weak gross substitutability (S) and indecomposability (I) imply that there exists at least one good ie{l,...,r} such that X'i(fi,p1,...,pr,np'r+1,...,tip'n) < X'j(l,p

10

C O M P A R A T I V E STATIC ANALYSIS OF

Therefore we obtain Xi(l,p') < jq(l,p)

for at least one i ^ OJ,

(11)

which contradicts (10). Hence j belongs to {l,...,r}, i.e. /^ pn, it follows (S) and (I') that there exist at least two goods i e {0, l,...,r} satisfying (7). Therefore, -X^(l,p') > X'i(l,p) for at least one ie{l,.,.,r}, which contradicts (9) and (10). Therefore, /j. ^ 1. Since p, m) will increase the price of any good i (i > m) less if the supplies of goods 1,..., m are adjusted so as to keep their prices constant than it will if the supply of one of those m goods, say good m, is not adjusted, its price being permitted to vary.(l) Proof. Let a shift in tastes from good 0 to good j give rise to shifts of excess demand functions, Xi(l,p) -> X'i(l,p) (i — 1, ...,n), when the supplies of goods 1...., m are adjusted, and shifts, X{(l,p) -> Xl(l,p) (i = l,...,n), when goods 1,..., m are the only goods whose supplies are adjusted. As the excess demand functions for goods m+1,..., j—l, j +1,..., n remain unchanged in both cases, we have

(12) (13)

We also have for all

and

(14) at the equilibrium price set p before the shift in tastes. The functions X'i(l,p) and X ' - ( l , p ) as well as Xt(l,p) fulfil (B), (C) (H), (S+), and (W). Let us denote the equilibrium prices after the shift in tastes by p" = (Pi,...,pm,p'm+1,...,p"n), when the supply adjustment are made in the markets l,...,m, andp" = (plt..., P,n~i,Pm>-->Pn)> when they are made in the markets 1,..., m—1 only. Under strong gross substitutability (S+) we have pi > 0 for i = I,..., n, p'i > 0 for i = m-j-1,..., n, and pi > 0 for i = m,..., n (by f 1 ' From this theorem we may find that the long-run equilibrium prices are less flexible than those in the short-run.

12

C O M P A R A T I V E STATIC ANALYSIS OF

Lemma 3). Thus goods 1...., n are not free goods at all. so that for all i'a (15)

(16) (17) An argument parallel to the proof of Theorem 3 yields Pm < Pm- That is, suppose p = p". Then it follows from (17) that X"j(l,p) = 0. This and (15) are incompatible with (14 Hence p ^ p". Next, define max p^p] = Pk/Pk == V-- If i=m

n

(j. ^> 1, we have (p,,^") > ( l , p ) . Homogeneity and strong gross substitutability imply that k(l,p") = Xk(p, p,p") > Xk(l,p). Taking (12) and (14) into account, we have Xk(l,p") > Xk(l,p). By (15) and (17), both sides of this inequality are zero. This is a contradiction. Hence /j. < 1, so that pm < p"n. Let us write min (Pi/p't) = p'klP'k = ^- ^Y definition, By homogeneity (H), for

(18) By homogeneity (H), for

(19) Now suppose that Q xm, which is a consequence of Theorem 6. Inequalities (33) and (34), together with (32), imply

This contradicts (31). Hence An exactly similar argument shows that the weak LeChatelierSamuelson principle [which states that x\ ^L x'i (i = m-^-l,...,n)] holds for a system with non-negative coefficients.

THE SIMPLEST I N P U T - O U T P U T SYSTEM

21

Since the coefficients of the price-valuation subsystem are obtained by transposing those of the output-determining subsystem, comparative static laws similar to Theorems 5-8 are derived from the price-cost equations. We can show that when the value added in industry j is increased cet. par., the prices of the products of all other industries will rise in no greater proportion than the price of the output of industry j. We can also show that the rise will be thinly spread over the whole system because the elasticities of prices, p1}..., pn, with respect to the value added in industry j are not greater than unity.(1) Furthermore, the LeChatelier-Samuelson principle holds for the price-valuation subsystem. Finally, we are concerned with a simple dynamic system discussed by Goodwin [1], Chipman [1], Solow [1], and others, i.e. a system consisting of n difference equations:

or, in matrix form,

(35) where x^t) stands for output of industry i in period t, and x(t) and c are the column vectors {^(i),..., #„(£)} and {cj,...,^}. By iteration, the general solution of (35) is seen to be (36)

Let us now examine the convergence of (36). We shall prove the following theorem. THEOREM 9. // a non-negative, indecomposable matrix A satisfies the Hawkins-Simon condition, then (37)

Proof. Let d be an n-dimensional column vector which has only positive components. The Hawkins-Simon condition implies that there exists a positive vector y such that y = Ay-\-d. Therefore « > Ay. Hence there exists an f, with 0 < f < 1, such that (*> These laws of change of the price system were empirically found by Leontief [1, pp. 192-201] without any algebraic proofs.

22

C O M P A R A T I V E STATIC A N A L Y S I S

From this it follows that f'y ^ A'y ^> 0. Since 0 < / < 1, we have lim A'y = 0. Together with y > 0 and A1 ^> 0, this leads /—>00

to (37). Since

we obtain

It at once follows from (37) that

(38) By taking (37) and (38) into account, we find from (36) that x(t converges to the stationary solution

II STABILITY ANALYSIS OF THE

W A L R A S - L E O N T I E F SYSTEM

1. Introduction IN the previous chapter we were concerned with the simplest Leontief model in which the input-output subsystem and the price-valuation subsystem were separated from each other, and all final demands were treated as parameters. But if final demands are not parameters but variables depending on prices, the model will lose its simplicity. In fact a change in the price of a primary factor gives rise to various repercussions on prices which in turn induce changes in final demands and hence changes in outputs. In this chapter we deal with the Walras-Leontief general-equilibrium system where prices and outputs are interdependently determined by the whole system consisting of the household demand functions for goods, the supply functions of productive factors, the price-cost equations, the input-output equations, and the demand-supply equations for productive factors. We shall examine the comparative static properties of the system as well as its stability properties .(1) Much work has been done recently on the problem of stability of equilibrium. The work may be classified into two categories, according to the author's approach. The first approach consists in constructing a dynamic model that satisfies a number of given assumptions as to the microeconomic behaviour (e.g. utility maximization for consumers and profit maximization for producers) and in examining its stability properties, while in the second analysis it is the assumed properties of the aggregate < J > Its static aspects have been elucidated by Wald [1], Kuhn [1], Dorfrnan, Samuelson, and Solow [1, chap. 13], Fukuoka and Koyama [1], and others, but with regard to its dynamic aspects, much remains to be done. The original idea of this chapter is due to Yasui's article [1], in which he is concerned with the local stability of the Walras-Leontief system.

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STABILITY A N A L Y S I S OF

excess demand functions that determine whether stability of equilibrium will prevail or not. The former was adopted by Hicks [1], Arrow and Hurwicz [2], and Morishima [2], and the latter by Samuelson [1], Lange [1], Metzler [1], Yasui [1], and others. In this chapter we shall follow the second approach. With regard to stability theory, at least two important problems remain to be solved. First, the global stability of economic equilibrium (i.e. Samuelson's 'Stability of the first kind in the large' [1, p. 261]) has not been completely elucidated as yet, because most treatments, with several exceptions such as Arrow and Hurwicz [2], Arrow, Block, and Hurwicz [1], and others, have been concerned with local stability (i.e. stability of the first kind in the small) only. Secondly, in dynamics as well as in statics, numbers that are to serve as prices and quantities must be non-negative; but there have been very few stability theories that take explicit account of non-negativity restrictions on the solutions, although it has recently become customary to prove the existence of an equilibrium under these restrictions. In this chapter, pursuing the ideas of Arrow and Hurwicz [1], Kose [1], Uzawa [1], and others who have developed a method for switching a system from one regime to another, we shall deal with the global stability of an economic equilibrium from the non-negativity viewpoint. Samuelson's 'correspondence principle between comparative statics and dynamics' [1, pp. 5, 258] shows how the problem of deriving operationally meaningful theorems in comparative statics is closely tied up with the problem of stability of equilibrium. But since his principle enunciates the relationship between stability of equilibrium in the small and changes in the position of equilibrium due to a small change in a datum, it may be called the correspondence principle in the small. In the last section of this chapter we shall study the correspondence principle in the large, whereby various comparative statical laws of non-infinitesimal character are seen to be intimately related to the global stability properties of the system.

THE W A L R A S - L E O N T I E F SYSTEM

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2. Semi-global stability of equilibrium Consider an economy with n goods (consumption goods and raw materials) and m+1 factors of production (labour, landservices, and capital-services). We assume that all consumption goods as well as raw materials are perishable, being unable to be used more than once. We also assume that consumption goods are used up as soon as they are delivered to consumers. Furthermore, it is assumed that not only persons but also land and capital goods are not subject to purchase and sale, although their services are offered and demanded on the market. We contemplate a products market and a services market to determine the prices of products and services, but not a land market and a capital goods market, the latter being taken into consideration in Chapter III where the problem of capital formation is discussed. Under these assumptions, no consumer has any stock of consumption goods which may be offered for further exchange; the distribution of land, personal faculties, and capital goods among individuals is independent of the dynamic process of trading and pricing, so that we may assume that it is constant, as we do throughout the following discussion of stability of equilibrium/1' Let xi be the total output of good i, c1 the final demand for the good, and rl the total supply of factor i. The price of good i is denoted by p{ and that of factor i by vi. Column vectors {xl,...,xn}, {c1,...,^}, {r1,..., rm+i}, {pl,...,pn}, and {v\...,vm^} a represented by x, c, r, p, and v, respectively. Let A be a matrix of material-input coefficients whose element ay denotes the quantity of good i used up per unit of good j; and let B be a matrix of factor-input coefficients whose element fry stands for (1 > If, however, some consumption goods are durable, or land and capital goods are exchangeable, we are confronted with redistribution troubles. Walras avoided them by using the device of 'tickets' and by assuming the tdtonnement process where no exchange transaction will be undertaken until an equilibrium is established in every market. Most contemporary economists implicitly or explicitly follow his line of approach. But Negishi [2], Hahn [2], and Uzawa [4] are concerned with non-tdtonnement processes, showing that competitive equilibrium is stable in a class of cases even if redistributions of goods among individuals occur in the midst of trading and pricing. See the note to this chapter, below.

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STABILITY A N A L Y S I S OF

the quantity of factor i consumed per unit of good j. An accent applied to a vector denotes the transposition of that vector. The following assumptions*1' are used in this chapter. ASSUMPTION 1. A is non-negative and indecomposable. ASSUMPTION 2. A fulfils the Hawkins-Simon conditions.*2' ASSUMPTION 3. B is non-negative, and for each i there is at least one j such that bti > 0. That is, each factor is used for production of some goods. ASSUMPTION 4. Each