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. /^ <^ p{ (i = l,...,n). We thus find that there are no goods whose prices increase proportionately more than that of good j. Let us now introduce the condition of strong indecomposability: (I') The relations and P'k < P"k for Ic $ R imply that there exist at least two i e R such that ZAP') ^ XAP"). This strong condition implies that no set of goods can be found whose price increases would not affect the excess demands for at least two of the remaining goods in the system. It is obvious that (I') is more stringent than (I). We can now strengthen Theorem 2 to THEOREM 3 (the second and third Hicksian laws in strong form). Given a shift in tastes (excess demand) from good 0 to goodj in a strongly indecomposable system (I'), the price of any good in terms of 0 must increase proportionately less than that of good j. Proof. Write max/.^ = n, and assume that p, = 1. Let r be the number of ju/s that equal the maximum. If r equals n, we obtainp = p". Accordingly, 0 = X'.,(l,p") = X'.,(l,p) > X , j ( l , p ) . This, however, contradicts Lemma 3, which asserts that there are no free goods in an indecomposable system. Hence r < n. Let us now reorder the goods so that goods 1, 2,...,r have the greatest p.t. Since juj?J.+1 > pr+l,..., fj.p'n > 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, <1 1 by Theorem 2, we obtain / / , < ! . Hence p'{ > pt for all i = I,..., n. Next reorder the goods so that goods l,2,...,r have the smallest p.^. Write min/x £ = /x. By Theorem 2, j belongs to the set {!,.... r}. Suppose now it contains i other than j. By (H)
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
11
the equation (6) holds true. As /j, < 1, i^p'r+l < pr+1,..., ^p'n < pn, it follows from (S) and (I') that there exist at least two goods ie {!,...,r} such that X'i(fj.,p1,...,pr, /j.pr+1,...,^p'n) < X'f(l,p). Therefore we obtain (11) which contradicts (10). Hence the set {!,...,r} does not contain any * other than^'. Finally, we establish the following theorem: THEOREM 4 (the LeChatelier-Samuelson principle in strong form). Suppose the strong gross substitutability (S+) prevails. A shift of demand from good 0 to good j (j > 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
pm ^ 6pm. Under (S+) these inequalities, together with (18), yield Equations (12), (13), (17), and (19) reduce this inequality to 0 > 0, a contradiction. Hence 6 > 1, i.e. By a similar argument we find that weak gross substitutability implies p'l^. fy (i = m-\-l,...,n)\ the weak LeChatelierSamuelson principle is established. On the assumption that only three goods are traded, Theorems 1-4 are well illustrated in Fig. 1. Measure along two axes the prices of goods 1 and 2 (in terms of good 0) respectively. Corresponding to any arbitrary price of good 2, we have the price of good 1 which brings the market for good 1 into equilibrium. A series of price sets thus obtained will form a curve, A A orpl = f(p2), on the price plane. In an exactly similar
THE SIMPLEST I N P U T - O U T P U T SYSTEM
13
we can draw the curve, BB or pz = g(pl],representing the sets of prices which will bring the market for good 2 into equilibrium.
FIG. 1
As was shown by Hicks [1, pp. 67-71 and 317], the assumption of gross substitutability, together with that of homogeneity, implies that the curve AA slopes upwards with an elasticity less than unity, i.e.
Similarly, Thus AA and BBhave such forms as we depict in Fig. 1; they intersect in a point Q — (Pi,p^),at which both the market for good 1 and that for good 2 are in equilibrium. Let us now suppose that there is an increased desire for good 1, which is satisfied by diminishing the demand for good 0. Then the curve AA will shift rightwards, intersecting with BB
H
C O M P A R A T I V E STATIC A N A L Y S I S OF
at a new equilibrium point Q' = (p^, p'2). If, however, the supply of good 2 were adjusted so as to keep its own price constant the curve BB would shift downwards. This, together with the shift of A A, would give rise to an increase pl -> pl, in the price of good 1. From Fig. 1 we at once see that (1)pt < pl (the first Hicksian law), (2) p2 < p'2 (the second Hicksiari law), (3) p'1/p1 > p'z/Pz (the third Hicksian law), and (4) pl > pl (the Le Chatelier-Samuelson principle).
3. Properties of the input-output system Let us now turn to the Leontief model which is similar to the Hicksian model discussed above in fulfilling the assumption of gross substitutability (a^ ^ 0), but is different from it in assuming fixed coefficients instead of homogeneity (H) and the Walras law. It is obvious that we cannot follow the exact lines of Section 2. But the following independent argument shows that the counterparts of Theorems 1-4 are obtained for the outputdetermining subsystem as well as for the price-valuation subsystem. We begin by proving a necessary and sufficient condition due to Hawkins and Simon [1] that the equations describing the balance of input and output for all goods be satisfied only for outputs all positive. As before, we define afj- as the quantity of good i used up per unit of good j, Xj as the total production of good j, and Cj as the quantity of good j available for final demand. The input-output system is written as:
(20) where ai} I> 0 and ct ^> 0. We make the following assumption: (I) The n X n matrix A = (atj) is indecomposable in the sense that A cannot be transformed, by permutations of the same rows and columns, into the form
with square submatrices, A1 and A3, on the main diagonal.
THE SIMPLEST I N P U T - O U T P U T SYSTEM
15
Then we obtain THEOREM 5 (the Hawkins-Simon theorem). A necessary and sufficient condition that the x satisfying (20) be all positive for any c such that c I> 0 and c =/= 0 is that all principal minors of the determinant \I—A be positive;^ i.e.
where A$ stands for the principal minor of \I—A\, which consists of the first i rows and columns of I—A. Proof. Our proof is obtained by induction, and the theorem is assumed to hold for a system containing only n—1 goods. Necessity. Consider the following system:
(21)
from which we obtain®
The first n— 1 equations of (21) can be written as:
where aln,..., am_1>re are non-negative and the assumption (I) requires that at least one of them be positive, i.e.
As it is assumed that the Hawkins-Simon theorem holds for n—1, all xv..., #„_!, xn are of the same sign if and only if AJ > 0,..., A n _! > 0. In order that xn should be positive, it is necessary to have A n _ 1 /A M > 0. Thus we find that (HS) is necessary for xi > 0 (i = I,..., n). (1 > A sufficient condition that a non-negative, indecomposable matrix satisfy the Hawkins-Simon condition is that all column sums of the matrix are not greater than one, and at least one of them is less than one. See Solow [I]-2 < > By Cramer's rule.
16
C O M P A R A T I V E STATIC ANALYSIS OF
Sufficiency. Consider the following two systems:
(22) and
(23) From these we find
so that are tions to (20). Suppose (HS) holds. As the theorem is assumed to be for Ti—1, we find from (22) that (x*,...^*^) ^> 0 for any negative Cj,..., c n _ 1 , the strict inequality holding when at one of the c's is positive. Since ani ^> 0, one obtains
therefore
soluvalid nonleast
at least
one of c,,..., cn_i is positive, so that x* > 0; hence because at least one of anl,..., a ;i>ri _ 1 must be positive by virtue of the assumption (I). Thus, even if cn = 0, we also have
Solving (23), we get
As A n _j > 0, and A n > 0, a;** is positive. Let the first n—l equations of (23) be rewritten as
THE SIMPLEST I N P U T - O U T P U T SYSTEM
17
Since aln,..., an_l>n are non-negative and indecomposability (I) requires that at least one of them be positive, we find that if (HS) holds af,..., a** x are positive. Thus (HS) leads to x* I> 0 and a;** > 0. Hence x > 0. So far we have seen that, if the theorem holds for a system containing only n—l goods, it holds also for one with n goods. It can at once be verified that the theorem holds for n = 1. Next we show that a system satisfying the Hawkins-Simon condition (HS) obeys the following: THEOREM 6. Given an increase in the final demand for good j, with the final demand for all other goods held constant, in an indecomposable system, the output of all goods must increase, and that of j will do so by the largest percentage. Proof. Let an increase in the final demand for good j be i ~* c'i> sucn that the new system has a solution, giving rise changes, xi-^-x'i (i = l,...,n), in outputs. Let Cj — VjC'j and xi = Xix'i. Clearly, v} < 1, and since the Hawkins-Simon condition is satisfied, Xt > 0. c
Suppose maxA^ = Ax == A2 = ... = Ar I> 1. It follows from and the definitions of x\ an
Now, suppose r ^>_j. Since Xk/X3- <^ 1 and i^/A3- < 1, the righthand side of the above equation is less than
This contradicts so that As
we have (24)
18
C O M P A K A T I V E STATIC A N A L Y S I S OF
On the other hand, from (20) we obtain
( t = l , . . . , r ) . (25) In order for (24) to hold true, i (i = l,...,r) must equal one [orX ci (i = l,...,r) must vanish], and aik (i = l,...,r; k = r-\-l,...,n) must vanish. Clearly the latter contradicts indecomposability (I). Hence max A; < 1; this implies that the output of all goods must increase when, the final demand for good j is increased. Let us now reorder the goods so that the first r goods give the smallest A i; and suppose that r < j; that is, mm
Then an argument similar to that above leads to equations like (25), in which A,- < 1 and A^/Aj > 1 (i = l,...,r; k — r-\-l,...,n). Therefore, the only case compatible with (24) is the one which satisfies ci = 0 and aik = 0 (i = l,...,r; k = r-\-l,...,n). This contradicts indecomposability (I) again. Hence minA^ = A,,-; i.e. the output of good j increases by the largest percentage/1' As the element in the ith row and jth column of the Leontief inverse [I—A]-1 is (x'f — x^Kc^ — Cj),^ we find from Theorem that an indecomposable input-output system satisfying the Hawkins-Simon condition (HS) has an inverse whose elements are all positive, i.e. We also have COROLLARY (Metzler's theorem).(3> If A is a non-negative, indecomposable matrix none of whose row [or column] sums is greater than one, and at least one of whose row [or column] sum is less than one, then the diagonal elements of [I—A]-1 are not less than the off-diagonal elements of the corresponding columns [or rows]. ll > If some c's are zero, it is shown that Aj g A 4 (i = I,..., n), but if all c's are positive, we find that Aj < A,- for all i ^ j. < 2 ' Here the final demands for all other goods are held constant. < 3 > See Metzler [2, p. 12] and [3, p. 437].
THE SIMPLEST I N P U T - O U T P U T S Y S T E M
19
Proof. Suppose that none of the row sums of A is. greater than one, and at least one of them is less than one. Write
(26) We obtain c{ I> 0 (i = 1,..., n), the strict inequality holding for at least one i. For ci defined by (26), (20) has a solution xi = (i = l,...,n). Taking xi = A^ into account, we get
(27) (28) As Xj f^ A4 (by Theorem 6), (27) cannot be greater than (28). The case where at least one column adds up to less than one and no column adds up to more than one may similarly be treated. By multiplying (x'i—Xj)j(c'] — c;.) by Cjfxi we obtain the elasticity of output xi with respect to final demand Cj:
It is of interest to derive the following: THEOREM 7. Given a solitary increase of the final demand for good j in an indecomposable system, the output of all goods will increase with elasticities not exceeding unity. Proof. From the equations xi = \x\ and c,. = >>•<;'•, we have
(29) We also have from the jth equation of (20)
(30) Suppose Vj > AJ. Since A{ i> A;. for all i (Theorem 6), and We obtain
20
C O M P A R A T I V E STATIC A N A L Y S I S OF
which contradicts (30). Hence v3- ^ Ay. Since 1 ~^> Af ^> A;., it follows directly from (29) that Exf/Ecj <^ 1. Next we prove THEOREM 8 (the LeChatelier-Samuelson principle in strong form). Let a^'s be all positive. The increase in any output xt resulting from an increase in the final demand for good j is less if m other outputs are kept constant than if one of those m outputs is permitted to vary. Proof. Let xi-^x'i (i = m-f 1 ,...,n) denote changes in out puts resulting from an increase, c^ -» c], in the final demand for good j when x^..., xm are kept constant; let xi ~> x^(i = m,...,n) denote changes in outputs when only xv..., xm_1 are kept constant. We have
(31)
(32) Write
By definition, (33)
we also have the last strict inequality immediately following from the fact, x"m > 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.
24
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
25
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.
26
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 0, v j> 0, and (p, v) =£ 0. For any non-negative p and v such that v =£ 0, all cl are nonnegative, and at least one of them is strictly positive/3' ASSUMPTION 5. Each r* is a single-valued and continuous function of p and v, where p jj> 0, v ^ 0, and (p, v) ^ 0. Moreover, if v* = 0, r* is zero. ASSUMPTION 6. Each c1 and rl is homogeneous of degree zer in variables « and v. and the following identitv (the Walras Law) is satisfied: (1)
Now a general equilibrium system of the Walras-Leontief type is formulated by the following three sets of equations, first: (2)
expressing equality between the demand and supply of each good; secondly:
(3)
expressing equality between the quantities of productive factors employed and the quantities offered; thirdly: (4)
expressing equality between the prices of goods and the cost of producing them. By using the same method as that adopted in the Note to Chapter III below, we can prove, under Assumpl1' Our assumptions are somewhat different from those in Wald [1]. < 2 ' For various implications of the Hawkins-Simon conditions, see Chapter I, section 3. < 3 ' For definitions of ' > ' and ' jg ', see p. 5, n. 1.
THE W A L R A S - L E O N T I E F SYSTEM
27
tions 1-6, that equations (2)-(4) have at least one solution such that x > 0, p > 0, and v > 0 with vf+1 = I/1' Let us now fix vmJrl at 1, and write v = {v1,...,$/"*} and f = {r1,...,rm}. Make the following additional assumptions: ASSUMPTION 7. Each component of c(p, v, 1) and r(p, v, 1) satisfies the Lipschitz condition® at each point (p, v) such that (p,v) >0. ASSUMPTION 8. The (ra+ra)x(w + ra) functional matrix
is negative quasi-definite for any set (p, v), i.e. X'J(p, v) A is negative for all A = {A 1; ...,A n+m } ^ 0 and (p, v) > 0. Since each element of J(p,v) splits into two terms, due respectively to the income effect and the substitution effect and, as was shown by Hicks [1, pp. 310-11], the matrix of the substitution terms is negative definite. Assumption 8 implies that the substitution effects are dominant. There is no reason why the demand and supply functions should satisfy this assumption. But once it is assumed, we can prove that the equilibrium is unique. In what follows, the unique solution of (2)-(4) is denoted by (XO,PO,VO), where v™+l = 1. LEMMA 1. If Assumption 8 holds, then
^ See Yasui [1] and Fukuoka and Koyama [1]. Another proof was given in Kuhn [1J and Dorfman, Samuelson, and Solow [1]. See Hicks [2] also. < 2 > A function f ( x l , . . . , xn) is said to satisfy the Lipschitz condition (with constant f}) at a point (x^, x2,..., xn) if l/(*i. xn »»)-/(*i. *i xn)\ < P(\zl-x1\ + \xi-x1\ + ... + \a:n-xn\) t xn] in some neighbourhood of (£,,..., xn). If all partial derivatives of a function are continuous at each point of a closed region, it satisfies a Lipschitz condition.
28
STABILITY A N A L Y S I S OF
Proof. Let Consider
Then we find $(0) = 0. Differentiating <£(T) with respect to T, we obtain where By virtue of Assumption 8, Hence Now we shall turn to dynamics. We make the additional assumption: ASSUMPTION 9. (a) If excess demand for good i is positive (or negative), the price of that good is increased (or decreased). (6) When excess demand for factor i other than the (m+l)th factor is positive (or negative), the price of that factor is increased (or decreased); the (wi+l)th factor is called the numeraire and its price is set at 1. (c) The output of good i is increased (or decreased) when the price of good i exceeds (or falls short of) its cost of production. Let H, K, and M be diagonal matrices whose diagonal elements ht, ki; and m{ are all positive. Then the dynamic system based on Assumption 9 may be formulated as follows: (6) (6) (7)
where B = (bti) (i = l,...,m; j = !,...,«), and bm+1 is the (m+l)th row of B, i.e. (6m+u,...,6m-n,n).
THE W A L R A S - L E O N T I E F SYSTEM
29
Let us now give definitions of stability. The equilibrium point (pQ,vQ,xa)—an interior point of the non-negative orthant of the Euclidean space of 2n-|~m dimensions—is said to be stable with respect to a neighbourhood N(pQ, VQ, xa) of the point (p0, v0, x0) if every path starting from any point of N(p0,v0,x0) converge to the equilibrium point. If N(p0, VQ, a;0) is infinitesimally small so that the functions c(p, v, 1) and f(p, v, 1) can safely be approximated by linear expressions, then the equilibrium point is locally stable, and if N(p0, v0,x0) coincides with the non-negative orthant of the (2w-j-m)-dimensional Euclidean space, it is globally stable. Furthermore, the equilibrium point is said to be semi-globally stable if N(p0, v0, x0) is so large as to contain at least one boundary point of the non-negative orthant, although it may not be large enough to coincide with the whole nonnegative orthant. Obviously, global stability implies semiglobal stability which, in turn, implies local stability. In this section we establish a semi-global stability theorem. We have (8) (9) (10) at the equilibrium point. Equations (5)-(7) may be written (")
(12) (13) As Assumption 7 is satisfied, equations (5)-(7) [or (11)-(13)] have, for any initial position, a unique solution continuous in time and in the initial position.(1> Let us now define Vt and V2 as follows:
(14) (15) <'I See, for example, Coddington and Levinson [1, pp. 1-20].
30
S T A B I L I T Y A N A L Y S I S OF
the dot denotes differentiation with respect to time. Write
(16) Differentiating T^ and V% with respect to time and taking (11)(13) into account, we obtain
By virtue of Lemma 1 dVJdt is negative unless (p, v) — (p0, v0); and we find from Assumption 8 that dV^dt <1 0, the strict inequality holding if (p,v) = (p0,$0) and x ^ x0. Thus dVjdt is always negative unless (p, v,x) — (PO,VO,XO). Hence lim V = 0. <->•«)
When V = 0, (p,v,x) = (PQ,VO,XO) and (p,v,x) = 0. Thus and x approach j00, v0, and x0, respectively, as t tends to infinity.(1> Next we show that if the initial position satisfies a condition specified below, p, v, and x are always non-negative on the way to equilibrium point (PO,VO,XO). Let U be
Define N(p0, #n> xo) as a se* of all points (p, v, x) suc
If the initial position belongs to N(p0,va,x0), the initial valu of 2Fi is less than or equal to t/2. Once 2^(0) < U2, 2FX(0 is U) Write y = (p, v, x} and ya = (pa, va, x0). As p and v depend continuously on y, dVfdt is a continuous function of y. Considering the fact that dV/dt is strictly negative unless y = ya, it takes on a negative maxiniurn on any bounded closed set of i/'a not containing j/0. As one of these sets we may take the set of y's fulfilling the inequalities ff1 a V § 0. Since V converges «->co
to V* in a decreasing fashion, we have 2F* ^ V(t) > V* for sufficiently large t ; so that it is clear from the above that AVjdt :£ — a < 0 for sufficiently large t, say t Si T. Therefore it follows that V(t) 5S — o(t — T) + V(T) foi't g T, which yields lim V(t) = -co, contradicting V* > 0. Hence V* — 0, that is, y t~>00
approaches j/0 as t tends to infinity.
THE W A L R A S - L E O N T I E F SYSTEM
31
2
always no greater than U because dV1/dt ^ 0. On the other hand, it is obvious that if 27^) <: U2, then ^(^(O—Po) 2 ^ U2 and m^(i!)-4)2 va, x0). Accordingly, the equilibrium is semi-globally stable. This result may be dignified with the name of a theorem. THEOREM 1. If the Walras-Leontief system fulfils Assumptions 1-9, the equilibrium is semi-globally stable. 3. Global stability of equilibrium If the initial position, however, does not satisfy the condition some components of (p,v,x) may be negative on the way to the equilibrium point. In order for any path starting from an arbitrary initial point to be non-negative through time, we must strengthen some of Assumptions 1-9. In the following we shall deal with a system in which Assumptions 7 and 9 are replaced by: ASSUMPTION 7'. At each point (p, v) such that (p, v) > 0, each component of c(p,v, 1) and f(p,v, 1) satisfies the Lipschitz condition and has all partial derivatives of any order. ASSUMPTION 9'. (a) If excess demand for good i (or factor i) is negative and its price is zero, then the price does not change. (6) If output of good i is zero and its price is less than its cost, the output remains zero, (c) The assertions 9 (a), 9 (b), and 9 (c) hold true in all other cases. Now let the column vector Ax-\-c(p,v, l)—x be denoted by D = {d1,..., dn}, the column vector Bx—f(p, v, 1) by E = {e1,..., em},
32
S T A B I L I T Y A N A L Y S I S OF
and the row vector p'— p'A — v'B — bm+l by F = (f1,...,/"). Then the system based on Assumption 9' may be written:
(17) (18) (19) where Gw, G(Z\ and G(3) are diagonal matrices whose diagonal elements are g(l), g(f}, and gf} such that(1
and
Let us now assume that Assumption 7' is satisfied and that the system of differential equations (17)-(19) has no singular point in the sense that is defined in Section 6 below. Since
the lemma that will be proved in Section 6 guarantees that the system has a solution, for any initial position, non-negative and continuous with respect to time. Let us define V as in (16). The derivatives of V1 and V2 exist almost everywhere. Considering (8)-(10) arid (17)-(19) we find that, at every point where I1' When Assumptions 3 and 5 are made, the rule of switching g ( ?> from 1 to 0 is reduced to a mere formality. Because they imply that e* 2: 0 if v* = 0.
THE W A L R A S - L E O N T I E F SYSTEM
33
the derivative dV^dt exists,
On the other hand, if g(l) = 0, p,t = 0, so that pOm = p. Similarly, vG(2) = v, and G(S>x = x. We, therefore, obtain from (15) and (17)-(19)
at every point where dV2/dt exists. Assumption 9' and Lemma 1 imply that 'dV^/dt is non-positive for all (p, v, x) > 0, and strictly negative unless (p,v) = (p0,v0). Assumption 8 implies that dVzjdt
4. An alternative system So far we have been concerned with the orthodox adjustment process: (1) Excess demand for a good (or a factor) makes its price rise and excess supply makes it fall. (2) The existence of positive (or negative) profits will lead to an increase (or a <>' This theorem is similar to Theorem 4 of Arrow and Hurwicz [2].
34
S T A B I L I T Y A N A L Y S I S OF
decrease) in output of each industry. An alternative system is obtained by replacing Assumptions 8 and 9 by: ASSUMPTION 8*. The demand functions c* (i = 1,..., n) and the supply functions r* (i = l,..,,m-f-l) satisfy the weak axiom of revealed preference, i.e. if (Pi,i\) ^ A(f> 2 ,v 2 ) for all A, then PiC(pz,v2) <; v'^p^Vz) implies p'^ip^v^ > v'^p^vj. ASSUMPTION 9*. (a) The outputs of goods are instantaneously adjusted so as to maintain equilibrium in the markets for them. (b) The assertion 9(6) holds true, (c) The prices of goods are instantaneously adjusted so as to maintain equality between the prices of goods and the costs of production. By virtue of the Walras law the weak axiom of revealed preference may be paraphrased as follows: If (p^v^) ^h(p2,v2) for all A, then pi Ac—t^Ar f^ 0 implies p'2 Ac— VjAr < 0> where Ac = c(p^vz) — ^(Pvvi) and Ar = r(p2,v2)—r(pvv1). We find from Lemma 1 that the axiom is weaker than Assumption 8, although there is no reason why the collective demand and supply functions should satisfy the former as well as the latter.^ A system based on Assumption 9* may be formulated as follows: (2) (20) (4)
Clearly, there is an asymmetric treatment between goods and factors; the price of each good is assumed to be adjusted with infinitely high speed, while the speeds of adjustment in the factor markets are assumed finite. Since we have, from (2) and (4), (20) (20)
respectively, we can eliminate x and p from equation (20); so that we get (20)
where c(v) — c(p,v, 1), and f(v) = r(p,v, 1), with p defined by < J > See Dorfraan, Samuelson, and Solow [1, pp. 367-8].
THE W A L R A S - L E O N T I E F SYSTEM
35
(22). By virtue of Assumption 7, (23) has a unique solution, for any initial position, continuous with respect to time and with respect to variations in the starting point. Since Assumptions 1 and 2 are made, it can be proved that [/—A]- 1 > 0. Therefore, taking Assumptions 3, 4, and 5 into account, it follows from (23) that dvi\dt > 0 if v* = 0. Thus, even if vi vanishes at a certain point of time, it becomes positive again. This shows that v1,..., vm are non-negative through time. The non-negativity of x and p follows from (21) and (22), respec tively. Next we shall prove that the equilibrium point is globally stable. Since (24) (25) (26)
at the equilibrium point (PO,VO,XO), it is readily seen that B[I—A]-lc(vQ}-f(v0} = 0. It.follows, therefore, from (23) that
(27) where Ac = c(v)—c(v0) and Af = f(v)—f(v0).
Let us now define V as 2V = (v—v0)'K(v—v0). Differentiating V with respect to time and substituting from (27), we have
(28) which, as will be shown below, is negative at any v ^ VQ if the weak axiom of revealed preference is satisfied. Thus V(t) is decreasing, unless v ^ v0, so that it approaches V* when t tends to infinity. It is proved that V* = 0; hence v -> VQ as t -> oo. (Seep. 30, n. 1.) Let us prove that dV/dt< 0 at anyv -=f=- v0. It is obvious from (1) and (22) that This, together with (24) and (25), leads to
(29)
36
S T A B I L I T Y A N A L Y S I S OF
In the particular case where equations (22) and (26) hold, the weak axiom of revealed preference implies that
(30) for
(31) Since (30) contradicts (29), the left-hand side of (31) must be positive, i.e. (31) for v T£ v0. Subtracting this from (29), we find that the right hand side of (28) is negative. Hence dV/dt < 0 at v = The above analysis may be summarized in the following theorem: THEOREM 3.(1) // Assumptions 1-7, 8*, and 9* are satisfied, prices and outputs are always non-negative, and the unique equilibrium point has global stability.
5. Comparative statics In what follows we shall consider how equilibrium prices tend to change in response to changes in demand functions c(p, v, 1), supply functions r(p, v, 1), material-input coefficients A, and factor-input coefficients B. Let some changes,
be imposed, such that the new system has an equilibrium, giving rise to change in outputs and prices, respectively. Then we have
(33) (34) (35) I1' This theorem corresponds to Theorem 2 of Arrow and Hurwicz [2].
THE W A L R A S - L E O N T I E F SYSTEM
37
Since (PO,VO,XO) i® a solution of equations (2), (3), and (4), we find, by subtracting (2), (3), and (4) from (33), (34), and (35), respectively, that
(36) (37) (38) where Ac = c(pl,vl, l) — c(p0,v0,1), Af = r(p^i\, I}~r(pn,v0,1), &p = P!~PO, Ax = xl—x0, and Av = vl — v0. From these equations we have the following results. First suppose that all technological coefficients remain constant, i.e. that SA = SB = 8bm+1 = 0. It then follows from (3 and (37) that
By virtue of
Let us now assume that the functions c and r satisfy the weak axiom of revealed preference. Let v = vi in (29); then subtract (32) from (29). We find that the left-hand side of the above equation is negative, and therefore — A^'Sc+AjJ'Sf < 0. We may now consider, as an interesting special case, the shift of a single demand function 8c{ > 0, where all other functions (other than the supply of numeraire) remain unchanged, i.e. 8c* = 0 (j = l,...,i—l,i+!,...,») and Sr = 0. Then we find A^ > 0. Another interesting case is Sr* > 0, SrJ' = 0 (j = 1,..., i—I,i-\-l,...,m), and Sc = 0, i.e. when the only shift is in the supply function rl. Here we obtain Aw1 < 0. Hence the following theorem is proved. THEOREM 4. In a system ivhere Assumption 8* holds true, an upward shift in the demand function c* raises the equilibrium price of the i-th good, and an upward shift in the supply function rl diminishes the, equilibrium 'price of the i-th factor.
38
STABILITY ANALYSIS OF
Secondly, suppose that Sc = Sf = 0. Then it follows from (36)-(38) that
If we make Assumption 8, which is more powerful than 8*, it follows from Lemma 1 that the left-hand side of this equation must be negative, so that we can conclude as follows. (1) If Oy is decreased, all other coefficients remaining unchanged, we have the inequality obtained bv Beckman [11 and Bailey [1], i.e.
(2) Similarly, if b.-,- (i < m) is the sole coefficient that diminishes, then
(3) If only one technological coefficient bm+1j is decreased and all other coefficients remain constant, we find Aa;3' > 0. Thus we obtain the following theorem: THEOREM 5. Suppose that a system satisfying Assumption 8 makes a technological improvement so as to save the i-th good (or the i-th factor) required to produce a unit of the j-th good. Then the induced relative change in the equilibrium price p* (or v*) of the input directly involved falls short of the relative change in the output x> directly produced. If the quantity of the (m-\-l)-th factor (numeraire) used up per unit of good j is reduced, the output of good j is necessarily increased.
6. A mathematical note It remains to establish the lemma used in proving Theorem 2. In order to solve the differential equations (17)-(19) [or (40) below], it seems necessary to use some special assumptions, in addition to the usual Lipschitz condition; accordingly, we use the notion of 'singular point', which is somewhat complicated. Let (/^(z!,...,;^),..., ^(z^.-.jZjJ have all partial derivatives of
THE W A L R A S - L E O N T I E F SYSTEM
39
any order. Consider a sequence {z, a(1), 2(2),...} defined as follows:
(39)
where when max(2t-, ^(z)) > 0, when zi = 0 and ^(z) < 0, (or 0) when zi = 9^(2) = 0 and the first non-vanishing term $^(2) of the sequence {^'(z), ^(z),...} is positive (or negative). A point z = (2 1 ,...,2 n ) is called a singular point if for at least one i. An equilibrium point z where all j(z) vanish may be a singular point, because, since all <j>(/](z) (i = l,...,n; j = 1,2,...) vanish at the point 2, it is singular if it contains at least one z{ that equals zero. We can prove the following lemma: LEMMA 2/1' Let there exist a unique (equilibrium) point 2 > 0 such that Az) — 0 (i = I,..., n), and let
Moreover, let <^(z) have all partial derivatives of any order. Finally, let there exist no singular point other than the equilibrium point. Under these conditions the system of differential equations (40)
with
(40)
I1) Another theorem similar to this lemma was proyed by Uzawa [1, pp. 129-30].
40
S TA B ILITY ANAL YSIS OF
has a solution zlt, z°) > 0 for any initial point If is strictly negative for all z ^> 0 such that z ^ z, then the equilibrium point z is stable. Proof. Define the following sets of indices:
Since 2° is a non-singular point, the set U° is divided into the following two subsets:
Let us now consider an auxiliary system:
(42) As this system satisfies the Lipschitz condition, there exists, for the initial position 2°, a unique solution continuous in 2° and int. It is seen that, for a sufficiently small positive number t, we have It is also seen from (39) and (42) that
it follows that for i in M°,
for i in T°.
THE W A L R A S - L E O N T I E F SYSTEM
41
Hence when t (> 0) is sufficient^ small, 2f(£,z°) > 0 for i in R° and fci(t, 2°)] < 0 for i in T°. Let t1 be the smallest upper bound of t such that, for all 0 < r < t,
It is clear that, in the interval [0, t1], the solution z(t, z°) of the auxiliary system satisfies the original equations (40), and conversely. Hence in [O,^1] (40) has a solution, continuous in time. 'As will be shown below, no component of z(t, z°) goes to in finity as t tends to t1, so that we may take z(t l , 2°) as a new starting point. Since 2(^,2°) is not singular, we obtain, by the same method as above, a new interval [tl,tljrtz] where the original system has a solution. Thus the interval in which there exists a solution of (40) is extended from [0,tl] to [0, <1 + <2]. Repeating this procedure, we get a sequence Let
Taking (40) into account, we find that, for all t such that (43)
We know from the assumption that the first term on the righthand side is non-positive and from (41) we know that the second term is non-positive. Hence we find dV^dt <^ 0, which implies that no component of z(t, 2°) tends to infinity as t approaches t1. Let (40) have a solution z(t,z°) in [0,t], and let t* (< oo) be the smallest upper bound of 1. Since dVJdt <^ 0 on each interval [tP, fi+1], no component of z(t, 2°) goes to infinity, so that each t[z(t, 2°)] is at no time greater in absolute value than a properly chosen constant L. Then
Therefore,
S T A B I L I T Y A N A L Y S I S OF
hence Now suppose 2* is a non-singular point, and let Q*, R*, S*, T* be denned in terms of z* in the same way that Q°, JR°, S°, T°, respectively, are denned in terms of z°. By taking 2* as a new starting point, we obtain, by the same method as above, an additional interval [^,^#] on which the solution to (40) can be continued. Clearly, this contradicts the supposition that t* is the smallest upper bound of I. Next suppose 2* is a singular point. By the assumption that there exists no singular point other than the unique equilibrium point 2, 2* is nothing else than z. Therefore, (40) has a solution such that 2 = z(t, 2°) for 0 ^ t < t^ and 2 = z for 11> t*. This implies that t% is not the smallest upper bound of t, a contradiction. Hence t% must be infinity. Thus we have a solution to (40) that is valid from t = 0 to
is strictly negative, dl^jdt is negative on each interval [^,^+1]. This implies the existence of lim Vl = Ff > 0, which, in turn, <-»00
implies that d\\jdt approaches zero as t tends to infinity. Suppose F* > 0. Since }\ converges to V* in a decreasing fashion, we have 2F? ^ V^t) >_ F? for sufficiently large t. As is strictly negative unless 2 = z, it takes on a negative maximum on any bounded closed set of z not containing z. Hence there exists a a > 0, such that fulfilling
For allz's For
allz's
count, we find that
for sufficiently large ^. This is a contradiction. Hence F* = 0, i.e. (44)
THE W A L R A S - L E O N T I E F SYSTEM
43
An alternative stability argument can be made in terms of
Namely, ifdVjdt is negative on each interval [ 0 for any t ^> 0.
Note to Chapter II STABILITY OF A NQN-TATONNEMENT PRICIN 1. Various writers have recently studied the dynamic stability of systems where only instantaneously perishable goods are traded or that of tdtonnement processes during which no transactions actually take place until equilibrium is reached/1' In this note we show that under some not too restrictive assumptions non-tdtonnement processes of pricing are stable. 2. Let us consider an economy in which m individuals exchange n goods under conditions of perfect equilibrium. We denote the quantity of the ith good at the disposal of the ath individual by xai. Write xa = {xal,...,x^; x = {fj,...,£„,} shows the distribution of stocks among m individuals, and x* = 2 ^a the total stocks of the n goods available in the a
economy. We make the following assumption. ASSUMPTION 1. (i) The utility function ua(xa), of each individual is strictly quasi-concave(2) and continuously differentiable over the non-negative orthant of the 7i-dimensional Euclidean space R+. (ii) du^x^jdx^ > 0 at any point xa of J?+. (1) See Arrow, Block, and Hurwicz [1], Arrow and Hunvioz [2], [3], [5], Hahn [1], Karlin [2], Lange [1], MoKenzie [2], Negishi [1], Newman [1], Nikaido and Uzawa [1], Samuelson [1], Uzawa [2], [3], and Yasui [1]. The stability of non-tdtonnement processes has, however, been discussed by a few writers such as Hahn [2], Negishi [2], and Uzawa [4]. < 2 > A function u(x) is said strictly quasi-concave if u[Xx + (1 — \)x°] > u(x) for all A such that 0 < A < 1 and for all non-negative vectors x and x° such that u(x) = u(x").
44
STABILITY ANALYSIS OF
Let p = {pi,...,pn} be a price vector. Let xa(p, «„) be a point where ua(x) is a maximum, subject to p'x <jl ^'£a and x ^> O/1* Since wa is strictly quasi-concave, xa(p,xa) is uniquely determined once p and xa are given. Assumption 1 (ii) implies p'xa(p, xa) = p'xa for all a. Write z = x*(p, x) —x* where x*(p,x) = 2 xa(P>%a)> itg ith component z^ stands for the exa
cess demand for good i. It is seen that z/s are homogeneous functions of degree zero in p and fulfil p'z = 0. We make ASSUMPTION 2. The weak axiom of revealed preference holds for collective demand functions: If p°'x*(p1,x) ^ p°'x* for any two sets of prices p° and p1 such that p1 ^ hp° for all h > 0, then the inequality p1'x*(p°,x) > jj1'^* is fulfilled. Define E(x) = {p > 0 | M a (x) is a maximum at xa subject to E(x) may be an empty set. A point (p, x) such that p e E(x) is called an equilibrium point; because at (p,x) every individual enjoys a maximum utility subject to his budget constraint, so that z = 0. 3. Let us now consider the non-normalized process of pricing, which may be defined, by a suitable choice of units of measurement for each of the n commodities, by the differential equations
1)
Additional assumptions to be made are: ASSUMPTION 3. (i) No individual can hold negative amounts of goods, i.e. x(t) = {x^(t),...,x sumption and production of goods do not take place during the process of pricing and trading, so that x*(t) = x*(0) for all t^> 0. ASSUMPTION 4. dxjdt ^ 0 at p and x if and only if there is an i = {%!,...,%m} ^ 0 such that wa(£a) ^ wa(^a) for all a, with strict inequality for at least one a, p'^a ^ p'xa for all a, and '*' A prime denotes transposition. For definitions of ' > ' and ' S ', see p. 5, n. 1.
THE W A L R A S - L E O N T I E F SYSTEM
where
for some a, no ASSUMPTION 5. If
When i
45
then
being negative. is sufficiently small, then
ASSUMPTION 6. For any positive initial position [p(0), x(0)], there exists a unique path, [p(t,p(0), x(0)), x(t,p(Q), x(Q))], which is continuous in t and in [p(0),x(0)]. Furthermore, the family of functions {pi(t,p(0),x(Q)) (i = l,...,n) \t ^> 0} are equicontinuous in [;p(0),x(0)]; that is, for arbitrary e > 0, there exists a 8 (dependent on e) such that \pi(t,p0,x0)—pi(t,p(0),x(0))\ < e for all t ;> 0 and i, when \p°—p(Q)\ < 3 and \x°—x(Q)\ < 8. Assumption 3 requires no comment. Assumption 4 implies that trade takes place if and only if some individuals gain by exchange, while no individual loses. Assumption 5 makes prices remain strictly positive. Assumption 6 is a mathematical assumption which does not seem too restrictive. 4. We can now establish the following: THEOREM 1. If Assumptions 1-6 are fulfilled, and if p(0) > 0, and xa(0) > 0 for all a, then the system will approach an equilibrium arbitrarily closely as t -> oo. Proof. From Assumption 1 (ii) we have already obtained the Walras law: (2)
This, together with (1) and Assumption 5, implies that
The boundedness of x(t,p(Q),x(Q)) follows from Assumption 3. Hence p(t,p(0),x(Q)) and x(t,p(0),x(Q)) have limit points ^a and x1 as t tends to infinity.(1) for Let f 1 ' By the Bolzano-Weierstrass theorem which states that a bounded set containing infinitely many points has a limit point, i.e. a point which is the limit of a sequence of points of the set.
46
S T A B I L I T Y A N A L Y S I S OF
some sequence {£,,} such that Consider a path [p(r,p(tv),x(tv)), x(r,p(tv},x(tv))~\ starting from [jp(£,,), £(£„)]• The uniqueness of the path assures us that
that is, the new path coincides with that part of the original path which is beyond [p(tv),£(<„)]. As p(r) and x(r) are continuous in the initial position, we have
Assumption 4 implies that every u^xj^t,^(0),xa (0))] never decreases, and Assumption 3 implies that it is bounded. Therefore, it converges as t tends to infinity ;(1) we have for all a
Thus, (3) Assumption 4 implies that 2, ^a increases 11 and only if trade a
takes place; hence, it follows from (3) that there is no trade during the process starting from (p1, xl), i.e. x(r,pl, xl) = xl i> 0 for all T :> 0. Let pl be a set of prices such that z(pi, x1) ^ 0 and
Such a set of prices exists, because z/s are continuous functions of p fulfilling (2). (2) Assumption 5 implies pl > 0. Therefore, I1' A monotonic sequence, if it is bounded, necessarily converges, because it has one and only one limit point. < 2 > See, for example, Debreu [2, p. 82]. Consider the following continuous mapping of
) into itself (Continued on next page)
THE W A L R A S - L E O N T I E F SYSTEM
47
z(pl, xl) — 0; because otherwise we would obtain pvz(p1, x1) < 0, contradicting (2). The uniqueness of pl follows from Assumption 2; because if z(p°, x1)
but p^^p1, then p1'z(p°,x1) ^0; so that p°'z(pl,xl) > 0. Clearly, this contradicts z(p1, x1) = 0. Let us define V as
Differentiating it with respect to r, and substituting from (2) and dpf/dr = zt> we obtain dV/dr = —pl'z(p,x1), which, by Assumption 2, is negative unless p = hp1 for some h > 0; because, as p'z(p1,xl) = 0 follows from z(p1,xl) = 0, the weak axiom of revealed preference implies pl'z(p,x1) > 0. Since p'z = 0, we find that
for all r I> 0. Therefore, p(r,pl, xl) j^ hp1 for any h > 0 unless ^(r,^1,^1) = pl. Hence,
unless p(r,pl, x1) = pl. This implies that V converges to 0 as r -> oo, so that lim^T.p1,^1) = p:.(1) T—>QO
Since by Assumption 6 the family of functions
Brouw
there exists a point y in <S such
Brouwer (see p. 62, n. 2) there exists a point y in <S such that If &(?/) < 1, then max{0, zt(y, x1)} should be positive for all i with y^ > 0. Hence y'z(y, x1) > 0, contradicting (2). Therefore, k(y) = 1, i.e.
Hence,
<') See p. 30, n. 1.
48
STABILITY A N A L Y S I S OF
is equicontinuous, there exists a S (dependent on arbitrary e) such that \p(tv)—pl\ < S and \x(tv) — xl\ < 8 im On the other hand, there exist a sequence {£„} (tv -> oo as v -> oo) and a number As (dependent on 8) such that for all tv larger than As. Hence, for all (r, tv) such that T > A € , and tv > As,
this implies (4)
Let us next show that p1 e E(xl). Suppose the contrary, that (5) (5')
and for some
We find that
for at least one a; otherwise, we would have the following inequalities : u^x^p1, a;*)] i> u^x^) for all a, with strict inequality for at least one a, pirYx^p1,^) <^ ^(r)'^'for all a, and
Hence, by Assumption 4, trade would take place at P(T); this contradicts the fact that there is no trade during the process starting from ( p l , x l ) . Suppose
where /3 <1 y. (5') and z(p1,x1) = 0 yield /3 < y. We may, without loss of generality, assume that * a < (P 1 ,*i)>0
(« = &_! + !,...,&),
THE W A L R A S - L E O N T I E F SYSTEM
where
49
Write
By definition, we have
the second equality following from
there exist non-negative vectors 2/p+1,..., yy such that
Let us now write
It can be seen that (6) (7)
Moreover, it follows from and
that lim eni = 0. Therefore, for sufficiently large T, (8)
50
STABILITY A N A L Y S I S OF
We also find from (5) and the continuity of uJx) that (9)
when T is sufficiently large. Assumption 1 (n) and the nonnegativity of «„ assure (10) By virtue of Assumption 4, inequalities (6)-(10) assure that trade takes place at sufficiently large T, a contradiction. Hence Finally, let us show that xl is unique. Suppose x(t,p(0), x(G)) had another limiting point x2 such that We would have
(11) From Assumption 3 we have
(12) where
Define
then it follows from (11) and Assumption 1 (i) that with strict inequality for at least one a. It is clear from (12) that On the other hand,
with equality for
implies that
Hence, we have
(13) equality holding for i with #L > 0. Inequalities (13) imply that, at is a maximum subiect to This contradicts the result obtained above. Thus, x1 is unique. Hence lim
which
together with (4), shows that the system approaches an equilibrium (p1, x1) arbitrarily closely as t -> oo.
THE W A L R A S - L E O N T I E F SYSTEM
51
5. An alternative system which may deserve attention is obtained by replacing Assumptions 1 and 2 by the following assumptions. ASSUMPTION 1*. (a) The assertions (i) and (ii) of Assumption 1 hold only in the interior of the non-negative orthant of the n-dimensional Euclidean space. (6) All indifference surfaces are asymptotic to the axes, so that x > 0 if «„(#) > w a (x a ) and xa > 0 . ASSUMPTION 2*. Let Define
(i) If i e F at (p,x) > 0, then z{(p,x) > 0. (ii) If ieG at (p,x) > 0, then z^p.x) < 0. Assumption 1* (6) is a rather strong restriction, which says that every good is indispensable for every individual. As will be seen below, however, this assures the positivity of x throughout the whole process of trading. Since xai may be arbitrarily small, it is not very far-reaching. Assumption 2* also is strong, but it may be rationalized as follows.<2> If i e F, the marginal rates of substitution of i for any other goods do not fall short of their relative prices for all individuals, while the marginal rate of substitution of i for some good j exceeds their relative prices for some individuals; it seems reasonable to assume that, in such a situation, no one wants to exchange good i for any other good, but some are willing to exchange j for i, so that zi > 0. A similar rationalization may be given to 2* (ii). (1
> The symbol n denotes set-theoretic intersection. Logically speaking, i e F (or i e O) does not imply z4 > 0 (or < 0). It is possible to give an example which shows that the utility maximizing vector x a(P> ®a) mav have a component such that x^(p, xa) < g(or > ) xai for i e F (or ieG). (2)
52
STABILITY A N A L Y S I S OF
In deriving Theorem 2 below we use: LEMMA.(1> A necessary condition for no trade to be possible at (p,x) > 0, given Assumptions 1* and 4, is that both F and O are not empty. Proof. Suppose F is empty. Denote the empty set by 0. Without loss of generality, we may assume that for some a (1 a. This assumption is justified because each Fa ^ 0, while F = 0. Take any i e Jf^ n Fz n ... n Fa and any j e Fp for an arbitrarily chosen ft > a. Since j ^ 2^ n .F2 n ... n jf^, there exists some a with 1 <[ a <1 cr such that j ^ .Fa. Hence, in view of the fact that i e Fa, we have On the other hand, as i $ Fo and j E Fa, we have These two relations imply that the marginal rate of substitution of i for j exceeds their relative prices for individual a. and falls short of their relative prices for j3. Since x > 0, and the utility functions are assumed continuous, some exchange is possible between a and /?. An exactly symmetrical argument is applicable to O. Hence for no trade to be possible, it must be true that F and G are not empty. We can now establish: THEOREM 2. If Assumptions 1*, 2*, and 3-6 are fulfilled, then [p(t),x(t)] starting from [p(Q), x(0)] > 0 converges loan equilibrium as t tends to infinity. Proof. An argument parallel to the proof of Theorem 1 shows that p(t,p(Q),x(Q)) and x(t,p(0),x(0)) have limit points p x1 as t -> on. Bv Assumption 1* (b). x1 > 0. because for all a. We can, again by the same argument as in the proof of <*' This lemma can be derived from the necessity part of Uzawa's Lemma 2 as a corollary. See Uzawa [41.
THE W A L R A S - L E O N T I E F SYSTEM
53
Theorem 1, show that no trade takes place throughout the new process starting from (pl, x1), i.e. ce^jp1, a;1) = x1 > 0 for all r ^> 0. Moreover, by Assumption 5, p(r,pl, x1) > 0 for all r ^ 0. Therefore, by the Lemma, the sets F and G are not empty at any point of time, r. Let us now define
where / and g are the numbers of members of F and G respectively. Since it satisfies a Lipschitz condition, it is absolutely continuous.(1) It can be shown that the derivative of V* exists almost everywhere. It can also be shown that, at almost every point where the derivative exists,
It follows from Assumption 2* and (1) that dV*/dr < 0. We find that V* converges on 0 as T -> oo. Let -p1 be a vector of prices such that p1 e E*(x1) and
it is clear that pl is unique. As V* is zero if and only if p e E*(xl), and as
by (1) and (2), we get lim p(r,pl, x1) = p1. By use of AssumpT—»-oo
tion 6 we finally obtain lim p(t,p(Q),x(Q)) = p1. £—>co
The uniqueness of x1, which may be established by an argument exactly parallel to the last part of the proof of Theorem 1, yields lim x[t,p(0),x(0)] = x1. Thus [p(t,p(Q),x(Q)), x(t,p(0), t—>03
o;(0))] converges to an equilibrium (pl,xl) as t tends to infinity. <*' See p. 27, n. 2. Let xi— (x1 »B) and yi == (ylt..., yn) be points in a region. Let dj be the distance between x1 and y\ A function (fx) is absolutely continuous in the region if for any positive number e another positive number i) can be determined so that if the sum of the distances rf^..., dr is less than 77, then 828145
E
Ill A DYNAMIC LEONTIEF
SYSTEM
WITH NEO-CLASSICAL PRODUCTION
FUNCTIONS
1. Introduction THE so-called Leontief model, originally static in character, is usually dynamized by introducing consumption- and productionlags, the length of life of capital goods, accelerators, growth of final demand, structural changes due to technological innovation, and so on. In this chapter we are not concerned with consumption- and production-lags. Instead, we shall concentrate on a multi-sectoral model with accelerators; it contains at least one good which enters production more than once (capital good); the final demand for goods is assumed to depend on the disposable income of consumers. We shall also discuss how prices and outputs are influenced by technological change. A typical dynamic Leontief system is presented as follows. Let there exist n industries whose outputs in period t are denoted by X^t), X2(t),..., Xn(t) respectively. Let ai} be the current input of good i used per unit of output j (the current input coefficient), and b^ the quantity of good i invested in industry j/' in order to. increase the output of that industry by one unit (the investment- or capital-coefficient). Then the basic balance equations can be written :(1)
where ci stands for the final demand for good i. These n difference equations of the first order relate the outputs in period t to those in period <+l; given c/s, they will determine a time path of outputs. Next let an+1>i be the labour-input coefficient of industry i, (D See Leontief [3].
NEO-CLASSICAL PRODUCTION FUNCTIONS
55
Pi the price of good i, pn+1 the money wage rate and r the rate of interest. In the long-run competitive equilibrium of the dynamic Leontief model, price must equal unit cost, which includes an interest charge on the value of the unit capital requirements, so that we have:(1)
These equations contain n-\-l unknowns, i.e. n wage prices These equations contain
These equations contain
so that they have one degree of freedom. One more equation fixing, say, the real wage rate is required to determine all the unknowns uniquely. The output-determining system is a genuine dynamic system in the sense that once the output levels and the final demand for goods are given in period t, it determines the output levels in period t-\-l. But the price-determining system is not genuinely dynamic; it is true that it can yield a time path of prices if the time path of the real wage rate is given, but it yields constant prices and interest rate so far as the real wage rate remains unchanged. For this reason Solow [3] rejected the traditional price-determining system and proposed a dynamic model of price change. In this chapter, however, we confine ourselves to the study of a state of steady, balanced-growth of outputs, and as he [3, p. 31] recognizes, prices and interest rate remain unchanged in a constant-return-to-scale economy undergoing balanced growth. Thus so far as this chapter is concerned, there is no need to replace the traditional set of price-determining equations by that of Solow. In the usual Leontief system it is assumed that coefficients of production, a^ and b^, are constant, so that changes in prices can influence outputs of goods only through the change in the final demand for goods. In this chapter, however, we start from neo-classical production functions and determine a^ and 6y as functions of prices and wage rate. Whenever positive or negative (1)
See Georgescu-Roegen.[l] arid Zarnowitz [1],
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A D Y N A M I C L E O N T I E F SYSTEM WITH
excess demand for labour occurs, the wage rate will be adjusted so as to maintain equilibrium in the market for labour; this, as a rule, causes prices to change. Coefficients a y and btj will be influenced, not only directly by the change in the wage rate, but also indirectly through changes in prices. Since the rate of balanced growth is determined by the values of ay's and by'a, we find that the ultimate determinants of the rate are the supply of labour and technological knowledge. The price-valuation side of our system will be discussed in Section 2, and the output-balance side in Section 3. The effects of population growth and technological change on the balanced growth rate will be dealt with in Section 4.
2. The determination of production coefficients The Walrasian theory of general equilibrium (Walras [1, pp. 267-83]) assumes that capital goods and non-capital goods are distinct, while Leontief [3, pp. 53-58] assumes that each good may serve the capital requirements as well as the current production requirements of all the various industries of the economy. In the present chapter, however, we limit ourselves to the Walrasian case and denote industries producing noncapital goods by numerals 1,2,...,TO and industries producing capital goods by numerals m-\-l,...,n. Next we assume constant returns to scale or homogeneous production functions of the first order. Let xi (i = 1, 2,..., n) be the volume of output of good i by industry i and let xit (i = l,2,...,n; j = 1, 2,...,?i-f-l) be the volume of input of factor J into industry i. The (w-f-l)th factor stands for labour which is thought of as the sole 'primary factor' or 'nonproduced good'. Under the assumption of constant returns to scale the traditional production functions for industries i (i = l,2,...,n) can be written in the form: (1)
where represent the production coefficients of industry i.
NEO-CLASSICAL PRODUCTION FUNCTIONS
57
Let pi (i = l,2,..,,m) be the price of non-capital good i, pk (k = m+l,...,n) the price of capital-service k and pn+1 the wage rate. The coefficients ait or bki will be determined by the condition that the average cost
be a minimum subject to (1). As is well known, each ait or bki is non-negative and a homogeneous function of zero degree in the prices p^p^-^n^ and at least one of aii>---,ami>bm+i,i,---, bni,an+li is strictly positive/1* That is:
(2)
Denote the price of new capital good k by Pk and the rate of interest by r. Let ftk Pk and vkPk(k = m-f-1,..., n) be the portions representing, respectively, the depreciation charges and the insurance premiums to be deducted from the gross income, pk, realized from the capital good k (k = m-\-\,...,n). The rate of net income, i.e. the ratio of the net price of capital service, pk~^kPk—vkPk, to the price of capital goods, will tend to coincide with the monetary rate of interest. Accordingly, we have equations Pk = Pkl(rJrlJLk'^vk) (k = m-\-l,...,n).w Denoting, for simplicity, ^k-\-vk by qk, we express the above equations as loilows: (3)
Under a regime of free competition in production, entrepreneurs will make neither profit nor loss. That is, the existence of positive profits will lead to entry of new firms into industry and these firms bid up the prices of factors and force down the prices of products until profits vanish. We have, therefore, the ('I If they are all null, then xi = 0. < 2 > See Walras [1, pp. 267-72 and 289-90.]
58
A DYNAMIC L E O N T I E F SYSTEM WITH
following equilibrium conditions:
We can rewrite equations (4) in the following alternative and equivalent forms. First, substituting for pk (k = m+l,...,n) from (3) and dividing by pn+v (4) becomes:
where a^ (j = m-\-l,...,n; i = l,,..,ri) stand for q^b^. Second, eliminating Pk (k = TO+!,...,«-) from (4), we have
where
Note that fk = qkgk for k = m-\-\,...,n. A price-interest set satisfying (5) or (5') is called a long-run equilibrium set. Since equations (2) and (3) allow us to put aH and bki in the following forms
NEO-CLASSICAL PRODUCTION FUNCTIONS
59
equations (5) can be solved for
once r is known. In this section, however, we shall solve equations (5') for p1,...,pn, r on the basis of the following three assumptions: ASSUMPTION 1. Labour is an indispensable factor of production of any good, i.e. all an+li (i = 1, 2,..., n) are positive. ASSUMPTION 2. At least one capital good is used in the production of every non-capital good, i.e. for each i (i = l,2,...,m), at least one of bm+lti,bm.^ti,...,bni is positive. ASSUMPTION 3. For some pn_tl > 0, there exists a positive vector p such that (6)
Professor Samuelson [3, pp. 15-16] has proved that
(1>
W By consulting the Wong-Viner envelope-type theorem, which is based on Hicks's first law of substitution andEuler's theorem on homogeneous functions. In the present case, the former implies that, for all i = 1,..., n,
See Hicks [1, p. 321]. The latter asserts that
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A D Y N A M I C L E O N T I E F SYSTEM WITH
It follows from (2) that ft (i = I, 2,..., m), and fk and gk (k — m-\-1,..., n) are non-decreasing functions of p1,...,pn+1; Assumption 1 means that they are strictly increasing functions of pn+i, while Assumption 2 means that /f (i = l,...,m) are strictly increasing functions of at least one ph (h = m-\-l,...,ri). Assumption 3 implies that, for some pn+l, there exists at least one set of prices at which all industries gain positive profits when the rate of interest is zero. Since the //s and g{'s are homogeneous (of the first degree) in plt p2,..., pn+i, we may normalize the prices by requiring that the sum of them be 1; in what follows, all prices are taken to be such normalized prices. In order to show that there exist solutions to (5'), we first investigate the following equations:
where i = l,...,m, and/fc = m+l,...,n. This system of equations, though it is a purely technical system which is introduced as an auxiliary system for the purpose of establishing the existence of long-run equilibrium prices, may be interpreted as follows. Consider an economy where 'entrepreneurs' borrow capital goods from their owners ('capitalists') in order to combine capital services with labour and various materials. We may classify 'competition' in two categories: weak and strong. When weak competition prevails, entrepreneurs who specialize in the production of a particular good switch from one line with low profit margin to another line with high profit margin. Different profit margins are equalized by means of competition into a uniform profit margin. Equations (7) are established when weak competition prevails; we may interpret TT as a uniform profit margin. On the other hand, if strong competition prevails, the existence of positive profits gives rise to entry of new firms which bid up the prices of factors and force down the prices of products until profits vanish; a reverse process takes place when
NEO-CLASSICAL PRODUCTION FUNCTIONS
Gl
negative profits exist. Equations (5) or (5') are, therefore, fulfilled when the economy is strongly competitive. Let us assume that a capitalist maximizes his income which is got from his capital goods; he then holds only those capital goods which have the greatest rate of net income. Capitalists' selection among capital goods rules out discrepancies among rates of net income; the common rate of net income thus established equals the rate of interest, as is described by (3). Substituting from it, (7) may be rewritten as (7')
where A state described by (7') will be referred to as a quasi-equilibrium state.(1) We can now prove LEMMA 1. The equations (I1) determine a unique set o$ unknowns (T7,p1}...,pn) corresponding to a preassigned set of parameters (r°>Pn+i) such Mat r° ^> 0 and 1 > p°l+1 > 0. All pi,p2,-'-,Pn are strictly positive.<2) Proof. We demonstrate the lemma by the same method that Solow and Samuelson [1, pp. 414-16] have adopted to find a balanced growth solution to difference equations. The vectors^ with I1' We do not necessarily maintain that quasi-equilibrium precedes long-run equilibrium in time, although the notion of quasi-equilibrium is used below as a logical means of ascertaining the existence of long-run equilibrium. < 2 ' Our analysis which follows is different from those of Arrow and Debreu [1] and Debreu [2] in two points. First, Arrow and Debreu are concerned only with the existence of an equilibrium for a competitive economy, but we investigate the uniqueness of the solution given y n+I . Secondly, according to Arrow and Debreu, prices and quantities are interdependently determined by the whole system of simultaneous equations representing the demand for goods by consumers, the supply of goods by producers and the equilibrium condition that supply equals demand in every market, which is an idea originated with Walras. In Leontief's theory, however, the mechanism of price formation is purposely separated from that of output determination and vice versa. In our system, both mechanisms are unilaterally coupled; the influence of the latter on the former is completely nonexistent, whereas changes in prices have effects on outputs through changes in production coefficients. Samuelson [3] has also been concerned with such unilateral coupling of prices and outputs.
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A D Y N A M I C L E O N T I E F SYSTEM WITH
define a closed bounded convex set in Euclidean n-space.(1) Write
Since aj{ ^> 0, a)1+1>i > 0, and b^ ^> 0, we fmdft > 0 and gi > 0. It follows from ft > 0, gi > 0, 1 > p°n+l > 0, and r° I> 0 that the points y = (y^,...,«/„) determined by
have the properties We have a transformation of the closed convex set into itself; moreover, it is continuous. Therefore, by the Brouwer fixedpoint theorem,(2:) there is at least one^° = (pi,Pz,...,pn) carried into itself by this transformation of coordinates. For this vector p° we can write: where 1+77° = (1— K-u)/(/°+^V)> the/?'s and g$'a representing the values of the ft's and gt's at the point (Pi,p°,.~,p%+1). Thus p° is a solution to our problem and gives us the quasiequilibrium prices; TT° is the uniform profit margin. Asf- > 0, each component of p° is strictly positive; but TT° may be positive, zero, or negative, although I+TT° is necessarily positive. Next we shall prove the uniqueness of the quasi-equilibrium prices and the uniform profit margin. Suppose the contrary, namely, that there exist two distinct sets of the quasi-equilibrium prices y = ( p i , p z , — ,Pn) and^ 1 = (p{,pl,---,Pn)- Let n° and TTI be the uniform profit margins corresponding to p° an tively. We mav assume nl > TT° without loss of generality. Since p° ^ pl and I1' Let x and x' be any two elements of a set S. S is said to be convex if ux-}-(l — a)x' belongs to S for any oc such that 0 ^ ct ^ 1. < 2 > The Brouwer fixed-point theorem states that if a set S is non-empty, closed, bounded, and convex, a continuous transformation / of S into itself has a fixed-point, i.e. a point x e S such that x = f(x). See, for example, Lefschetz [1, pp. 117-19], and Karlin [2, pp. 273-7].
NEO-CLASSICAL PRODUCTION FUNCTIONS
63
there exists at least one p\lp\ such that 1 > p\]p\
Since fh and gh are non-decreasing functions of the variables Pi,-.-,pn and strictly increasing functions of pn+i, the inequalities P°M^Pi/Ph (* = 1,2,...,71), pl^lpl < pl^lpl and 77° ^TT1 imply that the right-hand side of (8') must be greater than that of (8). This contradicts equations (8) and (8'). Hence p° = p1. The uniqueness of the quasi-equilibrium prices implies the unique uniform profit margin. Lemma 1 does not assert that the profit margin TT° determined by equations (7') is necessarily non-negative. But the case in which TT° < 0 is meaningless, because if TT° < 0, all production will be zero. Lemma 2 below holds for p°t+l not exceeding the least upper bound, pn+1> of pn+l in Assumption 3. LEMMA 2. // the preassigned rate of interest r° does not exceed a certain critical upper limit r*, the profit margin TT° corresponding to r° is necessarily non-negative. Proof. Let p = (Pi,...,pn,pn+i) be an arbitrary set of pri n
such that ^ ^> 0 and 2 Pi = 1—pn+v By hypothesis,/(y) and tttt g(p) are positive for all p. Therefore min g(p) > 0. Hence, if j> r° is sufficiently large, (1— Pn+i)l[f(p)Jrr°(j(p)'\ ig IGSS than 1 at any pointy; hence ( l ~ P n + i ) l ( f 0 j i - r ° g 0 ) i s , o f c o u r s e , l e s s t h a n 1 and accordingly TT° takes a negative value. Next denote by (Pi,...,pn) the quasi-equilibrium set of prices corresponding to the nil rate of interest: r° = 0. Then we have p°t ^ (l + AM,...,^)
(i = l,2,...,n).
Let A be an arbitrary number which is positive; if we define
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A D Y N A M I C L E O N T I E F SYSTEM WITH
Aft as vt and *p°l+l as vn_rl,-we get from (6), for p°n+1 <
(9) because each /f is homogeneous of the first order. Now define A; as pi = A;'^ (i -— l,...,n), and let A = max A;. Then A ^> 1; t and *p°l+l as vn_rl,-we get from (6), for p°n+1 <[ pn^,
so
(10)
where at least one equality holds. Taking into account that eachjf,; is a non-decreasing function, we find (11)
It follows from (9) and (11) that (12)
Since (10) contains at least one equality, we find from (12) that l-f-7T° is greater than 1, i.e. TT° > 0. Thus, 77° < 0 if r° is large enough, while TT° > 0 if r° is nil. Since, as Lemma 3 below will show, TT° is a continuous and decreasing function of r°,there exists a unique rate of interest r* corresponding to zero profits; and if r° < r*, TT° > 0. LEMMA 3. Let (-n°,p^,...,p^) and (Trl,pl,...,Pn) be solutions to (!') corresponding to preassigned sets of parameters (r°,pn+i) and (rl>Pn+i> respectively. Let r1 > r° I> 0 and 1 > p%+l > 0. Then TT1 is less than rr°, and as r1 approaches r°, -n1 approaches TT°. Proof. Since /f and gi are homogeneous functions of degree one, we have from (7') the following two sets of equations:
(13)
(14)
for non-capital goods and capital services respectively. Let us assume that all prices are unchanged in spite of an increase in
NEO-CLASSICAL PRODUCTION FUNCTIONS
65
the rate of interest, i.e. p\ = p\ (i = l,...,n). Then it follows from (13) that n° = TTI, whereas we find from (14) that TT° > TTI. This is a contradiction, so that prices must be changed. Since
constant, at least one price must be
reduced. If»,, (h < m) is the most reduced price (in percentage), we have
so that Hence it follows from (13) that 77° > TTI. Similarly, if p./ (j > m) is the most reduced price, we find from (14) that TT° > irl. Thus 77 is a decreasing function of r. Let n be the value of -n at r = 0. Clearly, TT <^ -n. On the other hand, I+TT ^> 0, so that TT I> — 1. Let r^ be any sequence of r that converges to r°, and let Tri'',p^,...,p1^ be the corresponding profit margins and prices. Since (•nil,pt{,...,p^Ys are contained in a bounded portion of an (n+l)-dimensional Euclidean space, we can pick a convergent subsequence (-7rv,p\,...,n} with limit (tf, y>i,...,pm).(1) The fact that Upi,...,pn,Pn+i) and gi(p1,...,pn,p°n+l) are co tinuous functions implies that
and
Since
must hold for every v, we find
so that (TT, p^,...,pn) = (TT°,pl,...,p^); otherwise, the uniqueness of the solution to (7') [ensured by Lemma 1] would be violated. If the sequence (nv-,pi{,...,p£) itself does not converge, then we <» Seep. 45, n. 1.
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A DYNAMIC L E O N T I E F SYSTEM WITH
can pick two different convergent subsequences with different limits which would also violate the uniqueness of the solution to (7'). Hence lim n^ = -n°, i.e. tr depends continuously on r. /4-+CO
From the above we find that for each p^+1 not exceeding pn+1 there exists a unique set (p*,.--,p%,r*) satisfying (7') with zero profit margin. Clearly, this is one and only one solution to (5'). Hence we have proved THEOREM 1. A long-run equilibrium set of prices and rate of interest, p*,...,p*,r*, is uniquely determined for any preassigned value Pn+i such that pn^1 ^> p°l+1 > 0. We shall next be concerned with relations between the normalized price of labour, pn+1, and the rate of real wages, w. Let a1,aa,...,am be quantities of goods l,2,...,m which enable a worker to maintain a given standard of living. Some of these quantities may equal zero. We define the real-wage rate, w, as
(15) Generally speaking, pn+1 bears no unique relationship to w; but we can prove: THEOREM 2. // the long-run equilibrium prices prevail, there is a one to one correspondence between the normalized wage rate, Pn+i> and ^e real-wdge, w. Proof. A unique set of long-run equilibrium prices (p*,.-.,p%) is obtained forp^+1 [Theorem 1]. Hence w is uniquely determined Suppose that p<>+1 and p^+1 (p*l+1 > Pn+i) correspond to the same real-wage rate w*. Let (p*,...,p*, r*) and (p**,---,Pn*, r**) be two sets of long-run equilibrium prices and of the long-run equilibrium rate of interest corresponding to Pn+i and Pn+i, respectively. Writing pf* = X{p*, we find that at least one \ is less than one, because Let minAj = \h. Then (16)
NEO-CLASSICAL PRODUCTION FUNCTIONS
67
Now consider (17.1) (17.2) It follows from (16) thatft and gh of (17.2) are greater than th of (17.1). In order for these equations to hold simultaneously, h must be greater than m [because if h <1 m, Sh = 0; therefore (17.1) is riot compatible with (17.2)], andr** must be less than r*. Next suppose that maxAi = \h is not less than p\+i/pn+iThen (18) so that/ft and gh of (17.2) are not greater than those of (17.1). First, let h > m; then Sh = 1. Since r** < r*, the right-hand side of (17.2) is less than that of (17.1). This is a contradiction. HenceA < ptl+1/p°n+1. Secondly, let h <1 m. Since Xk < Aft for all Ic > TO, inequalities (18) are turned into
Since fh is increased when all pk's (k = m-\-l,...,n) increase [Assumption 2], we find that fh of (17.1) is greater than that of (17,2). As 8ft = 0, equations (17.1) and (17.2) cannot hold simultaneously. Hence Xh < pn+i/Pn+i-
simultaneously. Hence X
which contradicts our assumption w* — w**. We conclude our long investigation of the existence of the long-run equilibrium prices by comparing Theorem 1 with Samuelson's substitution theorem/1' The latter which has been < l > See Samuelson [2], Arrow [1], Goorgescu-Roegen [2], Karlin [2, pp. 2614], and Koopmans [1].
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A DYNAMIC L E O N T I E P SYSTEM WITH
proved for a static Leontief system may be stated as follows: In a competitive economy a set of equilibrium prices (including the real-wage rate) is uniquely determined by a system of simultaneous equations expressing the fact that the prices of goods are equal to their unit costs; if we assume constant returns to scale or homogeneous production function of degree one, the unit cost for each good is independent of the scale of production;
Fm. 2< 1 >
consequently no variation in the consumption of final output will give rise to price change, so that only the circled points in Fig. 2 will ever be observed; hence all the production coefficients will remain constant under pure competition. It is evident, however, from Theorem 1 that our system (5') for the determination of prices possesses one degree of freedom. That is to say, in order to determine all the unknownsPi,---,p n+1 , r uniquely, we must make an additional constraint fixing, say, the rate of real wages w or the normalized wage rate pn+1. If a change in the final demand for goods disturbs the constraint, it (1 > Curves and lines stand for equal output curves and price lines respectively. X represents the level of output.
NEO-CLASSICAL PRODUCTION FUNCTIONS
69
will give rise to a change in the real-wage rate, which induces in turn changes in prices. The optimal current input coefficients dji and the optimal capital coefficients bki will be affected by the changes in prices brought about in this way. Accordingly, in order that the coefficients should remain constant regardless of the values of final demands for goods, it is necessary that the constraint fixing the real-wage rate is not disturbed by any variation in final demands. If we adopt the Ricardian hypothesis that the rate of real wages is determined quite independently of the marginal productivity by the supply of labour, or the Marxian hypothesis that the 'reserve army' of labour prevents the rate of real wages from rising above the minimum that must be paid to enable the labourers to perform the work,(1) the rate of real wages w may be regarded as an exogenous factor whose value is determined independently of the level of activity. But, on the basis of the (neo-classical) assumption that the demand and supply of labour jointly determine the rate of real wages, a change in the final demand for goods will affect the real-wage rate by influencing the demand for labour. This proves that in a dynamic Leontief system the substitution theorem holds true within narrow limits. Even under the Ricardian or Marxian hypothesis which makes the optimal coefficients of current and investment inputs inflexible with respect to changes in the final demands for goods, they are not constant in an absolute sense, but may be affected by causes other than changes in the final demands. For instance, if the subsistence level to which the rate of real wages is tied shifts upwards, w increases from w° to w1; this autonomous growth of w will again result in changes in the optimal production coefficients.
3. The balanced growth of outputs Let us now turn from prices to the determination of outputs. Let Xt(t) (i = l,2,...,m) represent the rate of output of noncapital good i in period t; Xk(t) (k = m-\-\,...,ri) the rate of output of new capital good k in period t; and ci the final demand for non-capital good i. If the long-run equilibrium prices prevail, C> See Kaldor [1, pp. 84-85 and 87]. 828145
B1
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A DYNAMIC L E O N T I E F SYSTEM WITH
the demand-supply balance of non-capital goods can be described in terms ofTOlinear equations:
(19) where the a*t's are the optimal current input coefficients determined by (5') (i.e. the a^ evaluated at the long-run equilibrium prices pf,-,P%,Pn+i]Let Yh(t) be the existing quantity of capital good k in period t. Throughout the following we assume capital to be freely transferable from one industry to another. We also assume that all the existing capital goods are fully employed, i.e.
(20) where the 6^'s are the optimal capital coefficients determined by (5').(1) New capital goods are devoted to the replacement of the obsolete and destroyed capital and to the net creation of capital. Since the obsolete and destroyed portions of capital goods are and the net creation 01 capital is we find Taking (20) into account and remembering the definition we obtain:
(21) Let us now assume that the final demands for goods consist of workers' consumption only. Then the total value of the final demands for goods cannot exceed the total income of workers, so that we obtain (22) where E(t) denotes the total employment of labour, i.e.
(23) * l > Note that fcj^'s as well as ajj's are independent of the X£a because of the assumption of constant returns to scale.
N E O - C L A S S I C A L P R O D U C T I O N FUNCTIONS
71
According to the traditional theory of consumer behaviour, workers' demand for consumption goods depends on prices and their incomes in such a way that variables Pi,...,pn, pn+i are not necessarily separable from E(t). In the following, however, we assume as a first approximation that each c,. is of the form :(1) (24) We obtain from (23) and (24)
(25) where d* stands for dj at the long-run equilibrium point (P*,->P*>Pn+i)- Then it follows from (19), (21), and (25) that
(26)
where Sj = 0 if j m. These equations can, more simply, be written in matrix form: (27) where
"> Cf. Chapter V, section 2, below.
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A D Y N A M I C L E O N T I E F SYSTEM WITH
A* is the matrix of current-input coefficients, B* that of capital coefficients, and L* the vector of labour-input coefficients, all of them being evaluated at the long-run equilibrium prices. The equation (27) is a variant of Leontief 's dynamic system, and determines the outputs of non-capital goods and capital goods. Now write where P£'s are the long-run equilibrium prices of new capital goods, i.e. P% — p*l(r*-\-qk). We at once find from (22) that if
the average propensity to consume of workers (evaluated at the long-run equilibrium prices) is less than unity, and if it is unity. We can prove LEMMA 4. Suppose that A*-\-D*L* is indecomposable.(1) // the average propensity to consume of workers is not greater than unity, i.e. Q*D* <^ Pn+i, fhzn A*-\-D*L* satisfies the HawkinsSimon condition which may be formulated in the following equivalent forms:
where A f is the determinant obtained by striking off the last n—i rows and columns of \I—A*—D*L* ; ^ is a characteristic root of A*-\-D*L* which is real, positive and not less in absolute value than any other root. Proof. Since (A*-\-D*L*) is non-negative and indecomposable, it follows from the Frobenius theorem'21 that the eigenvector X* associated with the root f x is positive. Since £,X* = (A*+D*L*)X*, we easily find: "' For the definition of indecomposability, see, for example, Chapter I, section 3. W See Debreu and Herstein [1], Frobenius [3], Karlin [2, pp. 246-56], Schwartz [1], and Appendix below.
NEO-CLASSICAL PRODUCTION
FUNCTIONS
73
Premultiplying this equation by Q* we obtain
(28) On the other hand, it is shown that (5) is equivalent to the following matrix equation: (29) hence It follows from (28V and the above equation that In this equation: B* 2> 0, L* > 0, r* > Q,Q* > 0,andZ* > 0. Moreover, by virtue of Assumption 2 at least one element of B* is strictly positive. Accordingly, the right-hand side is positive if P°n+i ^ Q*D*; hence 1 > &. Equation (27) can be solved by the usual method. Since any element of the first m rows of B* is zero, its rank is at most n—m. For simplicity's sake we assume that it is n — m. Partition A*, B*, D*L* and X, into:
Then (27) can be written in the form:
Eliminating X,, and X - . , , , from these equations, and writing
we get
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A D Y N A M I C L E O N T I E F SYSTEM WITH
Its solution takes the form (30)
where f]^..., Vn-m are latent roots of the equation
(31) and the vectors X depend upon the matrices, and £7n-m upon the initial conditions. X,, is obtained as 1
We can prove: LEMMA 5. // the average propensity to consume of workers is not greater than unity and A*-\-D*L* is indecomposable, equation (27) has one and only one balanced-growth solution. Proof. If ^ = 0, it follows from (31) that This implies \I—A*~-D*L*\ = O,'1' so that ^ = 1, contradicting Lemma 4. Hence we find -r){ ^ 0 (i = l,...,n—m). Therefore (31) can be written as: whereof = (I—M^M^. By assumption ^ < l,i.e..4*+Z>*£* fulfils the Hawkins-Simon condition; it is, therefore, evident that .4 ?,+//?, also satisfies that condition; hence We also find, by applying the identity in footnote (1) to the principal minors of 7—A*—D*L*, that the principal minors of I—M± are positive if those of /—A*—D*L* are all positive; thus M1 satisfies the Hawkins-Simon condition if A*-\-D*L* "' For any matrix we have an identity,
where R, S, T, and U are subraatrioes. Put R = ^ft + HJi, S = A& + H&, T = At, and V = A&. We then get M See Chapter I, section 3. Note that the indecomposability of -4*-f D*L* does not necessarily mean the indecomposability of /IJ^ + HJi.
NEO-CLASSICAL PRODUCTION FUNCTIONS
75
does; hence (/—M^)"1 > O.(1) We have, furthermore, B^ I> 0, A*2-\-H*2 ^ 0, and B*z ^> 0. Therefore, from the assumption that the rank of J3|3 is n — m, it follows that M > 0. It is obvious that 1/^j,..., 1/!?„_,„ are all possible characteristic roots of M and that Xi is the eigen-vector associated with 1/17^; we have (\lrii}Xt = MX' (i = 1,..., n—m}. It is also obvious from the Frobenius theorem that there exist a positive number £ and a positive vector X° such that £X° = MX0. The number £ is unique, and the vector .X"0 is unique up to a scalar factor. Therefore, among (1/ij,-, X1) (i = I,..., n—m), there is one and only one pair, say (l/iy^^f 1 ) such that I/T ?I = £ and X1 = 8X°, where 6 is a positive number. The particular solution
gives a unique balanced-growth path; •TJI is the rate of balanced growth. We can now establish: THEOREM 3. // the average propensity to consume of workers is less than 1, the rate of balanced growth is greater than the rate of interest, while if it is 1, the two rates are equal. Proof. Since 77-, is a root of (31) and X1 is an eigen-vector associated with it, we have By writing
this becomes Premultiplying this equation by Q* and subtracting (29) post multiplied by X, we have Since Q*B*X I> 0, and L*X > 0, we find that if pl^ > Q*D*, the rate 77j is greater than r*, and if pfl+l = Q*D*, •r]1 = r*. M It can be shown that this holds with strict inequality as the indecomposability of Ml follows from that of A*-\-D*L*.
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A D Y N A M I C L E O N T I E F SYSTEM WITH
Thus the rate of interest gives us a lower bound of the rate of balanced growth. Now let us assume g2 = g3 = ... = gn-m = 0; then all the terms of (30) other than the first vanish and the outputs of goods grow in balance at the rate T^. If it is further assumed that Q*D* = jp» n (i.e. workers only consume and capitalists only save), the rate rll equals the rate of interest r*, so that the balanced-growth solution is reduced to Mrs. Robinson's solution or to von Neumann's solution:'1' (l+r*)^X 1 . In Chapter V we deal with a generalized von Neumann model in which capitalists consume a constant proportion of their income. In that case total interest payments exceed saving, and it is shown that the rate of balanced growth is less than the rate of interest. In the first case treated in Theorem 3, where Q*D* < Pn+i, workers save, so that saving exceeds total interest payments; and the rate of balanced growth is greater than the rate of interest. 4. The demand-supply balance of labour In the previous section we have implicity assumed that the supply of labour adapts itself to the demand for labour. In this section we remove this assumption and deal with a process by which the demand-supply balance of labour is established. The following assumption due to Mrs. Robinson [1] is made: ASSUMPTION 4. Workers only consume and capitalists only save. Accordingly there is no discrepancy between the balanced growth rate rj x and the rate of interest r*. We also follow Mrs. Robinson and von Neumann in concentrating our attention on the balanced growth solution of outputs, and in neglecting all the other particular solutions. Suppose the supply of labour grows at the rate p.(2) If p < rjj, i.e. if outputs are growing faster than available labour, sooner or later entrepreneurs will find difficulty in employing more workers W See Mrs. Robinson [1, p. 76] and von Neumann [1]. ^ We assume that p is constant. The following argument holds mutatis mutandis when p is an increasing function of the income per capita.
NEO-CLASSICAL PRODUCTION FUNCTIONS
77
to man new equipment, and offer higher money wages. If the prices of non-capital goods and capital services increase in exactly the same proportion as the money-wage rate, the optimal current- and capital-input coefficients remain unchanged. But if a rise in the money-wage rate gives rise to an increase in the real-wage rate, iv, we have the following: THEOREM 4. // the rate of real wages is increased,w then we find that (1) the wage price (i.e. price in terms of labour) of each commodity is reduced, (2) the most reduced wage price is of a capital service, and (3) the rate of interest declines.(2) Proof. Let the wage price of good i be denoted by zi = p.tlpn+1. If the rate of real wages goes up, it follows from (15) that at least one wage price goes down. We denote sets of solutions before and after the increase in the real-wage rate by (z\,...,z°n,r°) and (z\,...,z^,rl), respectively. Let the most reduced wage price be zh; then
(32) On the other hand, by taking into account the definition of zi and the homogeneity property of/ £ and g{, we get from (5'): (33.1) (33.2) It follows from (32) that/ ft and gh of (33.1) are less than those of (33.2) respectively. In order for equations (33.1) and (33.2) to hold simultaneously, h must be greater than m (because Sh — 0 for h <^ m), and rl must be less than r°. Let us assume that there is a capital service whose wage price <" An increase in the real wage rate may be due to a change inCT.Jor a change in pi- Throughout this chapter, however, alt..., am are assumed constant. < 2 ' In Theorem 4 we are concerned with the prices of non-capital goods and of capital services, but not with the prices of new capital goods. Taking equations (3) into account, however, we can deduce some comparative-static laws of the prices of new capital goods.
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A D Y N A M I C L E O N T I E F S Y S T E M WITH
is not reduced. Then there exists a capital service h such that (34)
As rl < r°, (34) makes the right-hand side of (33.1) greater than that of (33.2). This is a contradiction. Hence the wage price of each capital service is reduced. It follows from Assumption 2 that the wage price of each non-capital good is reduced. Therefore, remembering that the balanced growth rate T]I is assumed to be equal to the rate of interest, we find that TJJ, will be reduced. If 7^ is still greater than p, the scarcity of labour will sooner or later emerge again, and the whole process is repeated until p and •ql are brought into harmony. At last we shall reach Mrs. Robinson's golden-age equilibrium [1, pp. 63176] where outputs of all goods and services grow together at the rate of increase of the labour force. In the contrary case where the labour force is increasing at a faster rate than outputs are growing, the ratio of the number of unemployed to the number of employed workers continues to rise. The workers' power to resist a cut in wage rates will be weakened, so that the money-wage rate will be lowered. If a fall in the money-wage rate leads to a fall in the real-wage rate, it results in an increase in the rate of interest. The rate of balanced growth of outputs is increased in step with the rate of interest. Hence the ratio of unemployment to employment will fall. It will continue to fall until a golden age is realized. This result, though it is based on Assumption 4 and neglects all the particular solutions other than the balanced growth path, may be dignified as a theorem. THEOREM 5. If a balanced growth of output, whose rate ^ is greater (or less) than that of population growth p, gives rise to an increase (or a decrease) in the money-wage rate such that it in turn results in a rise (or fall) in the real-wage rate, then the discrepancy between the rates r]1 and p will eventually disappear; that is, the golden equilibrium is stable. We shall next be concerned with the effects of an invention of labour-saving techniques on the rate of balanced growth of
NEO-CLASSICAL PRODUCTION FUNCTIONS
79
outputs. In the subsequent discussion we use ASSUMPTION 5. The system is indecomposable in the sense that the relations z'a = z"s for s e 8 = {i1,i2,...,il} and z'u < z"u for u $ S imply the existence of at least one s e S such that Note that/j. — gj.<7j. for k — m-\-l,..., n. The indecomposability implies that no set of goods can be found whose wage price increases would not affect the unit cost of at least one of the remaining goods in the system. We begin with the following THEOREM 6. Suppose our system contains at least two capital good industries and is indecomposable.^ If an industry h (which may be a non-capital good industry or a capital good industry) makes an improvement in the technical methods of production so as to reduce the labour-output ratio, and if the real-wage rate does not change, then: (1) at least one wage price is changed; (2) the wage price of the h-th good (or the h-th capital service) is not increased, and if there exist wage prices which are reduced, the most reduced wage price is of the h-th good (or of the h-th capital service); (3) the wage price of at least one capital service is not reduced, and if there exist wage prices which are increased, the most increased wage price is of capital service; (4) the rate of interest increases. Proof. Let a (> 0) be a parameter representing a reduction of labour per unit of the hth output due to the development in technology. We denote the long-run equilibrium wage prices and the long-run equilibrium rate of interest before and after the technical development by (zj,...,z°, r°) and (z\,...,z*, rl) respectively. In case h <1 m, we have
(35) If 2? = z] (j = l,2,...,n) equations (35) do not hold simul<*' In case there exists only one capital good industry, we obtain THEOREM 6*. // the real-wage rate is constant, a technical improvement which saves labour required to produce a unit of the capital good, gives rise to an increase in the rate of interest, wage prices of all goods remaining unchanged. When a non-capital good industry makes an improvement reducing the labour-output ratio, all the assertions (l)-(4) of Theorem 6 hold true.
Theorem 6* also assumes the indeoomposability of the system.
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A D Y N A M I C L E O N T I E F SYSTEM WITH
taneously. Clearly this is a contradiction. On the other hand, in case h > ra, we have
(36)
and (37)
for i such that i > m and i ^ h. If z° = z} (_?' = l,...,n), it follows from (36) that r° < rl. This contradicts, however, equations (37). Hence at least one wage price must be changed. Next suppose that all wage prices are changed proportionately, i.e. let z\ = Xz°t (i — l,...,n), where A ^ 1. Then we find that
This contradicts the hypothesis that the real-wage rate remains unchanged. Therefore max A^ > min A^, where z\ = Xtz^. 4 = As (= A* ^ 1) for s e 8
Furthermore, it is at once shown that if maxA £ < 1, w° < w1. Hence max A^ ^ 1. Similarly, min At- ^ 1. Let maxA4 = As (= A* ^ 1) for s e 8 = {«1,«2,...,it,}. z\ = \*z°a for s e S and z\ < A*z° for u <£ S, we find that fa(z\,...,zl, 1)
becausefa = qsgs by definition. On the other hand, since A* > 1, we have inequalities (40) (41)
N E O - C L A S S I C A L P R O D U C T I O N FUNCTIONS
81
As ss and and ggss are arehomogeneous homogeneous of degree one, we we can can writef write
(42) (43) 1 Now suppose r <^ r°- From (38)-(43) we obtain one of the following inequalities: (44)
(45)
Any of (44)-(46) leads to z* < A*z°, contradicting z* = A*z°. Hence rl > r°. Next suppose S = {i^i^,...,^} does not contain any i such that i > m. Then ineciualitv (44} or ineaualitv must hold for at least'one s e S. Ihis is a contradiction, so that S must contain at least one i > m. For i e S, we have A^ ^> 1. If A* > 1, (40) hold with strict inequality; so that z] < A*z° for s e S such that s <^ m. This is a contradiction. Therefore, if A* > 1,8 contains no non-capital good. Finally, let minA { = As = A** for s e 8 — {i^i^,...^^. By an exactly symmetrical argument we find that
for at least one s e S. Therefore, if s ^ h. we have
because rl > r°. This contradicts zj = A**z^- Hence he S, i.e. A** = AA <; 1. In particular, if A** < 1, we get S = {h}; otherwise we would have a contradiction to z] = A** zjj. Now we find from Theorem 4 that the long-run equilibrium rate of interest is a decreasing function of the real-wage rate. This
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A D Y N A M I C L E O N T I E F SYSTEM WITH
relationship is shown graphically as the solid curve rr' of Fig. 3. By Theorem 6 (or 6*) the curve rr' shifts upward when an industry experiences a technical improvement which saves labour. Suppose a new method of production which saves labour is invented in an industry. Before the invention the economy is
w°
Fro. 3
assumed to be in a golden age, so that the rate of balanced growth of outputs is equal to the rate of population growth. They are also equal to the rate of interest by the assumption that the average propensity to consume of workers is unity. If the real-wage rate remained unchanged in spite of the invention, the rate of interest would rise, and hence the rate of balanced growth would rise too. It is impossible for the rate of growth of outputs to exceed the rate of population growth for a long time, because the scarcity of labour will sooner or later emerge. Hence the realwage rate should be increased from w° to wl as shown in Fig. 3. In the Marxian case, however, where there is a large reserve army of labour, the rate of real wages is prevented from rising above the subsistence level. If a higher rate of growth of outputs due to a technical improvement is still less than the rate of
NEO-CLASSICAL PRODUCTION FUNCTIONS
83
population growth, the reserve army will continue to e'xpand. Thus a technical invention results in a faster rate of growth of outputs without affecting the real-wage rate (see Fig. 4). Some backward countries are experiencing this pattern of growth; in order to attain a golden state, they must first raise the rate of
FIG. 4 balanced growth to the level of the population growth rate by importing advanced methods of production from developed countries or cut the rate of population growth. Only after that, will they succeed in raising the real-wage rate above the subsistence level. Finally, it is remarked that even if a technical improvement which saves materials or capital goods (instead of labour) is experienced, the same results as above are obtained.
Note to Chapter III WALRAS'S T H E O R Y OF CAPITAL FORMATION 1. Leon Walras [1] developed four models of economic equilibrium: a model of exchange, a model of production, a
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A D Y N A M I C L E O N T I E F SYSTEM WITH
model of capital formation and credit, and a model of circulation and money. Our model in Chapter III is closely related to his third model, but there are some differences between them. In this note we describe Walras's analysis, though not in his own terms but in a modified form, to facilitate a clearer comprehension of the text. 2. Consider an economy with I consumption goods, m primary factors of production and n capital goods. Let xt be the demand for the ith consumption good, ft the supply of the ith primary factor, and gi the amount of the ith capital-service supplied. Let ht be the initial endowment of the ith primary factor and k^ that of the ith capital good. Prices are denoted by pi (for one unit of the ith consumption good), by wi (for one unit of the ith factor), and by vi (for one unit of the ith capital-service). We assume that all consumption goods are not durable, being used up as soon as they are delivered to consumers. Column vectors x = {#!,...,z,}, / = {/!,...,/,„} and g = {ft,...,^} are related to row vectors, p = [Pi,...,pi], w = [wv...,wm], v = [vv...,vn~\, H = [&!,...,&„,], and k — [&!,...,&„], and to the rate of interest rH = [&!,...,&„,], and k — [&!,...,&„], and to the rate of interest r by the following demand and supply functions: (1) (2) (3)
where z stands for the reciprocal of r, i.e. z = \\r. Next let e be the excess of the value of the services offered over the value of the consumers' goods demanded, and suppose s defined as ejz is a well-defined function of p. w, v, z, h, and K, i.e. (4)
By definition the functions (1 )-(4) satisfy an identity, (5)
an identity (called the Walras law) in p, w, v, z, h, and k. Let us now assume fixed coefficients of production. Let A = (dij) and B = (b^) be matrices of factor-input coefficients whose element a^ [or btj] denotes the quantity of the ith factor
NEO-CLASSICAL PRODUCTION FUNCTIONS
85
consumed per unit ofthejth consumption good [or thejth capital good]; and let (7 = (cy) and D = (dy) be matrices of capitalinput coefficients whose element Cy [or dy] stands for the amount of the ith capital-service used up per unit of the jth consumption good [or the jth capital good]. If we denote by qi the price of the ith capital good and by jttf qi and vi qi the portions representing, respectively, the depreciation charges and the insurance premiums to be deducted from the gross income, v{, of the ith capital good, we have equations so that
(6)
Finally, let y = {?/,•} be a column vector whose element yt represents the amount of the ith new capital good produced. Just as Zeuthen [1], Schlesinger [1], Wald [1], and others have replaced the Walrasian equations of production by inequalities, we may modify the Walrasian equations of capital formation and credit(1) as follows:® (7) (8) (9) (10)
(11) These inequalities have the following interpretations. First, (7) and (8) express that the quantities of productive services employed cannot exceed the quantities effectively offered. Secondly, (9) and (10) imply that in equilibrium the selling prices of the consumers' goods or of new capital goods cannot exceed their costs of production (or else prices would fall). Thirdly, (11) means that saving cannot exceed investment. Walras showed that one of his equations follows from the rest. <» See Walras [1, pp. 279-81]. < 2 ) For definitions of ' > ' and ' ^ ', see p. 5, n. 1. 828145
G
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A DYNAMIC L E O N T I E F SYSTEM WITH
Similarly, we can show that one inequality, say (11), must be satisfied with equality if all the inequalities hold. In fact, by premultiplying (7) and (8) by w and v respectively and by taking account of (5), (9), and (10), we get qy <^ zs. Thus, if there is equilibrium in the markets for goods and services, there can be no excess of investment over saving. (11) now implies qy = zs. The modified Walrasian system (1)-(11) and the model in the text are different from each other in the following points: (i) It is assumed in the text that the coefficients of production are determined by the condition that the cost of production of the products be a minimum. In the modified Walrasian system, however, they are assumed to be determined a priori, though Walras himself recognized in the fourth edition of his Elements the variability of the coefficients and developed the marginal productivity theory. (ii) In the model in the text non-capital goods may be distributed in part as intermediate products to all the various industries as well as in part as final products to consumers. But in this note non-capital goods (or consumption goods) are assumed not to be used as inputs in further production. (iii) The Walrasian model (l)-(l 1), like the model in Chapter II, assumes that the demand for consumption goods and the supply of primary factors and capital services depend on prices. On the other hand, the model in Chapter III assumes that capital services are fully supplied irrespectively of prices and the rate of interest, i.e. gi = kt (i = I,..., n) in the present notation. 3. Walras did not give any conclusive arguments to show that the equations describing his models have solutions. The existence of solutions was first investigated by Wald [1] and then by Arrow and Debreu [1], Gale [2], McKenzie [1], Debreu [1,2], Nikaido [1], Kuhn [1], Dorfman, Samuelson, and Solow [1] and others. They, however, focused attention on the model of exchange or on the model of production. To the author's knowledge, there does not seem to be any established proof of the existence of a solution to the Walrasian system of capital formation and credit. It is the main purpose of this note to prove the following theorem.
N E O - C L A S S I C A L P R O D U C T I O N FUNCTIONS
87
THEOREM. Given h and k, there exists an economically meaningful solution to the system (l)-(ll). To prove this theorem, we shall use the following assumptions. ASSUMPTION 1. Given h and k, each of the demand functions and the supply functions (i.e. each-of x{, fit and gt) is a singlevalued and continuous function of p, w, v, and z, where z 1> 0, and (p, w, v) > O.(1> For any non-negative p, w, v, and z such that (w, v) ^ 0, every component of x , f , and g is non-negative and at least one component of x is strictly positive. ASSUMPTION 2. If wt = 0, then ft = 0, and if v{ = 0, the g{ = 0, i.e. if the price of the tth primary factor (or capital service) is zero, there is no supply of the factor (or the capital service). ASSUMPTION 3.(2) Given h and k, the saving s is a singlevalued and continuous function of p, w, v, and z, where z I> 0, and (p, w, v) > 0. If z is not less than a positive number z, i.e. the rate of interest r is not greater than a positive rate f, then s(p, w, v, z, h, k) is non-positive for all values of the variables, p, w, and v. If z = 0, i.e. the -rate of interest is infinitely large, then s is positive for all p, w, and v such that (w, v) =£ 0. ASSUMPTION 4. x, f, and g are positively homogeneous of degree zero in p, w, and v, while s is positively homogeneous of degree one in the same variables, i.e. for any A > 0,
ASSUMPTION 5. All of the coefficients of production, aijt biit Cy, and dy, are non-negative. The mth primary factor (1 > If no consumer can be satiated by exchange, x^ is positively infinite at p^ — 0. We may, however, extend the usual definition of continuity so as to be consistent with the insatiability of consumers; that is, we may say that x$ is continuous at PI = 0, if lim Xi — -\-co for Xi such that it is positive, but
})(-»0
not finite at p^ = 0. Even if a^'s are continuous in this extended sense, it can still be shown that Theorem 1 holds true. ' 2) This assumption was explicitly made by Walras [1, D. 275].
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A D Y N A M I C L E O N T I E F SYSTEM WITH
(labour) is an indispensable factor of production of any good, i.e. amj >Q(j= l,2,...,l) and bmi > 0 (j = 1,2,...,»). 4. We are now in a position to prove the theorem. We can always choose # so as to make (12) so that p/s are not independent variables and can be eliminated. Also, by (6) we can eliminate g^'s from the list of independent variables. Remembering the fact that h and k are taken as given we find that our independent variables are w, v, y, and z (or r). Define ttj as Write:
(13) (14) (15) (16) where u is the row vector with components ut (i = l,...,n) and denends on v and 2. "Further, we write
where max(a, b) means the larger of the numbers in parentheses. Consider the points ( W , V, Y, Z] determined by
NEO-CLASSICAL PRODUCTION FUNCTIONS
89
where (w, v, z) belongs to the set
and y ^> y ^> 0; z is defined by Assumption 3; yt 's are sufficiently large so that for all^p satisfying (12) and for all (w, v, z) in the set R. It is clear that (W, V, Z} e R and y I> 7 I> 0. Hence by the fixed-point theorem of Brouwer,(1> the above mapping (or transformation) has a fixed point, ( w , v , y , z ) , for which we can write
(17) (18) where
being the values
at the fixed point (w, v, y, z). Let us now prove that E* — F* = G* = a* = 0. Suppose E* is positive; then AVO find that wi = E*^*. Therefore, Similarly,
As at least one of wi and vi is positive, we obtain this contradicts
(19)
which follows from (12)-(16) and (5). Hence S* = 0, so that
8* = f* = o. I1' See p. 62, n. 2, above.
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A D Y N A M I C L E O N T I E F SYSTEM WITH
Next suppose Q* is positive. Since S* = 0, we find from (17) that 0* yi = y{ Gf, so that yi = y{. It follows from the definition of y that E^ > 0. Of course this contradicts 2* = 0, so that (?? = 0. If fl* > 0, it follows from 2* = 0 and (18) that z = z. Therefore, from Assumption 3 and inequality uy ^ 0 we find that ft — s — uy <1 0. This is a contradiction. Hence 3* — 0. Thus there exists a fixed point (w, v, y, z) such that E <1 0, f1 <; 0, Q <; 0, and # <: 0. We can prove that
(20)
Because if, for example, w^ were positive when E{ < 0, we would have wE+vfi+Gy+zd < 0, contradicting (19). Next let us show that Ei = 0. If we suppose the contrary, i.e. that Ei < 0, then we get w{ = 0; therefore by virtue of Assumption 2 we obtain ji = 0. This leads to Ei I> 0, because it follows from x > 0,$^> 0 and Assumption 5 that^ai-f--^ ^ 0. Thus f l i < 0 i s r e d u c e d t o a n a b s u r d i t y , s o t h a t $ i < 0 i s i m possible. By a similar argument it can be shown that f'i — 0. As for the mth primary factor which is indispensable to produce any good, we obtain
(21) because ami > 0, bmi > 0, x ^> 0, y I> 0 and at least one com ponent of x is assumed to be strictly positive. Now suppose wm = 0; then/ m == 0. Together with (21), this leads to Eni > 0, contradicting the above. Hence wm > 0. Let us now suppose z = 0. Since B ^> 0, D ^> 0, w ^ 0, ^ ^ 0 , bmi > 0 and wm > 0, we get wB+vD > 0, so that wB+vD > zu = 0. Therefore, from (20) it follows that y = 0. This leads to 3 = s—uy > 0, because, by Assumption 3, s > 0 when z = 0. Of course fi > 0 contradicts # ^ 0. Hence z
NEO-CLASSICAL PRODUCTION FUNCTIONS
91
must be positive (i.e. the equilibrium rate of interest must be finite and positive). Taking (20) into account, we find that H = 0. Since A ^> 0, C ^> 0, w ^> 0, v I> 0, amj > 0, and wm > 0, it is shown that wA-\-vC > 0. Therefore it follows from (12) that p > 0. We can show that if vi = 0, then yi = 0. Because if •Si = 0, then Uf = 0, so that Gt < 0, yielding yi = 0. Since q = zu, we have proved that for p > 0, w > 0, v ^> 0, y ^ 0, and z > 0 the following conditions are satisfied:
Note that wm > 0, and also that if some of the conditions wB-}-vD ^> q are satisfied with strict inequalities, the corresponding dual variables yt are zero. By virtue of Assumption 4, (Xp, Xq, \w, Xv, y, z) is a solution so long as A is positive. 5. Throughout the fiUments Walras asserted that any good could serve as numeraire. But, as was shown above, the prices of some primary factors and capital goods may be zero, while all consumption goods and the with primary factor (labour) are always sold at positive prices. Thus there may be some goods which cannot serve as numeraire. Our argument supports the classical or neo-classical idea of fixing the prices of goods and services in terms of unskilled labour or a representative consumption good. The following special case deserves attention. Let us make two additional assumptions: ASSUMPTION 6. Every component of x is strictly positive for any non-negative p, w, v, and z;, such that (w, v) ^ 0. ASSUMPTION 7. Some amount of every primary factor (resp. every capital service) is required to produce at least one
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A D Y N A M I C L E O N T I E F SYSTEM
consumption good; there can be no i for which a,^ = 0 (resp. Cy = 0) for all j. We then find Ax+By > 0 and Cx+Dy > 0, so that/ and $ must be positive. Hence we obtain w > 0 and v > 0. Thus any good can serve as numeraire in the case where Assumptions 6 and 7 hold.*1). 6. So far we have been concerned with what happens in a particular period; the prices of goods and services, the rate of interest, and the quantities of capital goods produced are determined correspondingly to the initial endowments of primary factors and capital goods. When the economy enters upon the second period, the situation will be different from that in the first. The quantities of primary factors available at the beginning of the second period may exceed (or fall short of) those of the first period owing to the natural growth (or decay) of the economy; and the capital equipment existing at the beginning of the second period will be affected by the activities in the first period. It is likely that the distribution of primary factors and capital equipment among individuals will change from period to period. The new Walrasian equilibrium which will be established in the second period will be different from that which is established in the first period. Going on in like mariner, we have a process of change through periods. I1' If we make, instead of Assumption 7, a more stringent assumption that [A, C] > 0,
(*)
Assumption 6 can be dispensed with. I think, however, that the set of Assumptions 6 and 7 is more plausible than the sole assumption (*).
IV AN A L T E R N A T I V E D Y N A M I C SYSTEM
W I T H A S P E C T R U M OF T E C H N I Q U E S
1. Introduction IT is of some interest to re-examine, in terms of discrete 'activities' or processes, the problems discussed in the preceding chapter. This chapter deals with a system where a number of discrete activities, instead of the neo-classical production function, are available to each industry and each good may serve, as in Leontief's own system, capital requirements as well as current production requirements of all the various industries. The present model is more dynamic in character than that in Chapter III. Section 2 of this chapter discusses the global stability of the long-run equilibrium prices. Section 3 presents the condition for stability of the balanced growth of outputs. Our analysis of structural changes due to technological invention (Section 5) is dynamic, and departs from the comparative static approach to structural change as used by Leontief [2], Simon [1], N0rregaard Rasmussen [1], and others. Mrs. Robinson [1] has constructed her theory of accumulation of capital in terms of discrete activities; those economists who are interested in Game Theory, Linear Programming, and Input-Output Analysis may be charmed by her borderline model between von Neumann's theory of growth [1] and Marx's theory of reproduction. But they may have an urge to reconstruct her theory because her two-sectoral analysis pays only the minimum regard to the inter-sectoral phenomena with which they have been concerned. Our model in this chapter may be considered as a multi-sectoral generalization of her Simple Model.'1' Section 4 shows that a 'golden' balanced growth, i.e. a balanced growth maintaining full employment of labour, is impossible in an economy with a very high rate of growth of the (1)
See Mrs. Robinson [1, chapters 7, 8, and 9].
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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
labour force. This conclusion may be worth noticing, though it is intuitively apparent, for some writers have been concerned with a 'golden age' on a tacit assumption that it is compatible with any rate of growth of the labour force. Section 6 shows that the essential part of Mrs. Robinson's argument holds good in our model. It contains a restatement (in terms of multisectoral analysis) of the production function proposed by her. 2. The time path of prices We assume that an economy is divided into n industries producing one commodity each, 'and that industry j can choose between m3- different manufacturing processes ('activities') to produce good j. Let bisj be the quantity of good i used per unit of output j by the 5,,-th activity of industry j, and let lSj be the labour-input coefficient of that activity. As usual we shall denote row vectors by square brackets, [ ], and column vectors by braces, { }. The activity vectors of industry j are now represented by: where the bisj are non-negative and the lsj are strictly positive (on the assumption that labour is indispensable for each activity), i.e. (1)
Let us assume that the production period for each output is one unit of time. Now if p{ is the price of commodity i in terms of labour (wages = 1), r the rate of interest, and risj the length of life of factor i (when used by the s?th activity), then the unit cost of good j produced by the s,.th activity is written: (2)
If factor i is a current good (fuel, material, etc.), risj = 1; if a capital good, riSj > 1. We shall split the first term of (2) into two terms: £ (bhsjlrhsj}ph and £ (bksjTkSj)pk, the A-summation h
k
taking place over all current goods and the ^-summation over all capital goods. The former represents the value of current inputs
A S P E C T R U M OF T E C H N I Q U E S
05
and the latter the depreciation of capital goods. Similarly the second term of (2) can be split into: r( ^ bhsjph\ and r(2 bksipk h
k
the latter being interest on fixed capital, and the former, together with rlsj, the interest charge on working capital; L, of course, represents wage costs. Let the total sets of activities in the economy, numbering be arranged in the ordei
Define A, B, and L as:
respectively, where aisj — bis)lrisj.m The matrix [A,B,L] may be referred to as the 'catalogue of activities'. Now let P — [Pi,p2,...,pn] be the vector of prices; then the costs (2) of various activities per unit output can be written: this being the 'catalogue of unit costs'. Now assume that each industry selects a single activity from among those available to it.(2) Then there are m1 X m2 X ... X mn activity sets which could be adopted by the economy. These I1' In the literature on input-output analysis the current-input coefficients ais, are usually treated as independent of the capital-input coefficients &l8j. But bisj is equal to ais, times ris.. This fact was emphasized by Lange. See Lange [2, p. 314]. ( 2 ) In reality; of course, each industry may choose a mixture of activities, i.e. any point of the convex hull of activities. We assume for simplicity here that it chooses a ' pure' activity, but it can be shown that our results are not tlff«ot.«rj Vnr +.ViiQ rlortnr't.nvo frfiTn crpr\f>rnllt.\7
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AN A L T E R N A T I V E D Y N A M I C SYSTEM WITH
may be arranged in a certain order and denoted by a, /?,..., jit. Let a be an activity set (s1,s^,...,sn)in which industry j selects its Sjth activity (j = 1,2,...,«,). The 'selector of activity set' is denned (in transposed form) as:(1)
where S's are the Kronecker deltas, i.e. 8SlSl = 1 and all other S's = 0. When activity set a is selected, the cost structure of the economy can be written: or
where A(a) = A\a), B(a) = 5A(ff), and L(a) = L&M. If prices are equal to costs, we must have Solving this equation, we obtain a set of prices, P(a), which we shall call the price set 'proper' to the activity set a. Conversely, the activity set a and each of its activities will be described as 'proper' to the price set P(a). Several activity sets maybe 'proper' to a given price set. By analogy with Mrs. Robinson's terminology, a price set may be called the alpha-beta price set if both activity sets a and /8 are proper to it.(2) Now the possible activity sets, a, /?,..., p, may be divided into those with all positive prices 'proper' to them (the first type), and into those whose prices are not all positive (the second type). Throughout this chapter we shall use the following ASSUMPTION 1. When an appropriately small (positive) value r is preassigned to the rate of interest, the range of choice open to our industries includes at least one activity set of the first type.(3) I1' A prime applied to A denotes its transpose. < 2 > See Mrs. Robinson [1, pp. 108-9 and 413-14]. < 3 > Assumption 1 is equivalent to the postulate that there exists at least one activity set satisfying the Hawkins-Simon conditions.
A S P E C T R U M OF T E C H N I Q U E S
07
With this assumption the following theorem may be derived from the substitution theorem for Leontief models.(1) THEOREM 1. Among activity sets of the first type there always exists one set e with a price set P(e) proper to it such that P(e)5i P(#), where 8 is any other activity set of the first type.(y> Proof. Let v and be two sets of activities (uls u2,,,., un) and (vltvz,...,vn) which belong to the first class; and let P(v) and P^j be the price-sets proper to them; then: (3) (4)
Suppose that P(l/) is not equal to P(<^ and that P(v) is neither greater nor less than P^j in the sense of vector comparison; then we may order the p's in such a way that and
Denning
where
we have P* < Pw and P* < P^,. Since AM, A(^, BM, and B($ are non-negative matrices, it follows from (3) and (4) that: (5) (6)
Let Ij be the unit matrix of order j and In_j that of order n—j. Post-multiply (5) by
and (6) By
and (6) By
result is: (7)
where a stands for the set of activities: (ulL,u2,...,ui,v^l,...,vn}. It follows from (1) that Lia) is a strictly positive vector; hence we obtain Let A be the maximal characteristic root oi A^^-rB^), and X I1' See Arrow [1], Georgescu-Roegen [2], Koopmans [1], and Samuelson [2]. (2 > See p. 5, n. 1, for definitions of ' > ' and ' > '.
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AN A L T E R N A T I V E D Y N A M I C SYSTEM WITH
be the eigen-vector associated with A, i.e. XX = (A^-\-rB(a))X This, together with the above inequalitity, leads to P*X > AP*X, because X > 0 by the Frobenius theorem/1' Hence 1 > A. Accordingly, again by the Frobenius theorem, we have: [I-AM-rBM]-i
> 0,
(8)
where 1 is the unit matrix of order n. Hence from (7) it follows that P* > (l+r)LM[I—AM-rBM]-1; the right-hand side o this inequality is a solution, P(a), of the system: (8) Therefore we find PM < PM and P((J) < P^j. Since LM > 0, it follows from (8) that P(CT) is non-negative. If P(a) contains some zero elements, equation (9) reduces to a contradiction. Hence P(ff) > 0. Thus there exists an activity set a whose equilibrium price-set, P(o), is positive and less than P(v) and P ( J). If P(CT) 5^ P(#) for any activity set 9 of the first type, then a = e. If there exists an activity set d for which P^ <J[ P(^ does not hold, the above procedure is repeated. We at last find a such that P(CT) <^ P(0) for all 8 of the first type. The P((J) will be referred to as the long-run equilibrium price set.< 2 > Let us now turn from statics to dynamics. Let pt(t) be the price of good i in period t. Suppose the rate of interest and the rate of wages are held constant. Without loss of generality, the latter may be fixed at 1. We consider an entrepreneur who has a given sum of money M available for expenditure. He can lend M to someone else for one period at the prevailing rate of interest r; he also has opportunities for spending it upon m;different activities {blsj,...,bnsi,lsj} (s^ = lj,...,mj) If he chooses the first option, he may expect to enter the next period with sum equal to (l-{-r)M. If he chooses the second option and spends the money exclusively on the Sjth activity, he will at the beginning of the next period get zs. units of output I1' See Debreu and Herstein [1], Schwartz [1], a.nd Appendix below. ' a > In the following we assume that only one activity set is 'proper' to the long-run equilibrium price set. The results are unaffected even if several activity seta are proper to it.
A S P E C T R U M OF T E C H N I Q U E S
of commodity j
99
he will still
own an outfit of capital equipment, after deducting the portions worn in the process of production. By selling products on the market he will gain a sum of money equal to pj(t-\-l)zaj, and his outfit of capital equipment will be n
worth ^ (^isl~ais)zsjPi(tJr^-}- From among m;-possible activitie •i = l
he will choose the w^th activity such that
As Solow [3, pp. 32-34] has claimed, in equilibrium neither option can have an advantage over the other; therefore, we have
Considering
and elimi-
nating M and;
This may be rewritten in the form of
the first term and the sum of the last three terms of the right-hand side standing for capital losses due to price changes and the unit cost of production respectively, We can verify that the longrun equilibrium price-set P(e) is a stationary solution to (10).
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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
It is to be noted that in (10) we implicitly regard one unit of an old capital good i, a certain portion, say l/q, of which has been worn away, as being equivalent to the quantity (1 — l/q) of a new capital good i. An entrepreneur carrying forward, from period t to period t-\-l, such a capital good for use in further production may be considered as if he can sell it at the price (1 — l/q)pi(t+l) Sit the beginning of period (+1. In case of his selling it at the end of period t, he would receive the amount (1 — l/q)pi(t), so that the capital loss due to holding good i is reckoned as (l — l/q)[pi(t)—pi(t-\-l)]. Equations (10) state that prices are determined so as to cove capital losses as well as the unit cost of production. A change in the price of good j affects the prices of various goods through changes in their costs of production and by giving rise to capital losses or capital gains. The changes in prices thus induced will bring about a further change in the price of good j. Moreover, they may cause structural changes to happen by making an activity set cheaper than that which was selected as the cheapest in the previous period. We can show that the long-run equilibrium price set P(e) is not always stable; a numerical example can be obtained which generates a path of prices not converging on P(e). We can, however, also show that the long-run equilibrium may be unstable only when depreciation and appreciation of the stock of capital goods have strong influence on price determination. In fact, we can prove THEOREM 2. Suppose the rate of interest is forever constant. If the capital losses (or capital gains) due to price changes are always negligible, the long-run equilibrium price set P(f) is globally stable, i.e. any path starting from an arbitrary initial position of prices eventually reaches P^. Proof. Under the assumption of negligible capital losses, equations (10) are reduced to
A SPECTRUM OF T E C H N I Q U E S
which may be written in matrix form:
101
(1)
(10') Let 01 be a set of activities such that
of course, o> depends on P(t); we therefore denote u> during period i by u>/. From (10') we obtain:
(11) Since Pi,) is the solution of
(12) it follows from (11) that this yields we have
Now by writing
(13) Since it follows from (1) that L(t) is a strictly positive vector, we obtain from (12): P(£) > P{e)(A(()-}-rB^)). Hence the maximal characteristic root of the non-negative matrix A^-rrS^) is less than unity; and any other characteristic root must be a fortiori less than unity in absolute value. Accordingly we have lim(A(e)~\-rB(f})t = 0, from which it follows that R(t) tends to tf—>00
zero when t tends to infinity. Next let us consider the following auxiliary system: (14)
which, as will be shown below, gives a never-decreasing sequence f 1 ' Equation (10') implies that the given prices of period ( determine the cost-minimizing activities which will be adopted in that period, and the products will be sold in the next period (since the production of each output is assumed to take one unit of time) at the minimal cost arising in this way. 823145
H
102
AN ALTERNATIVE DYNAMIC SYSTEM WITH
such that Q(t) <[ P(t) for all t. By the same argument as above, we have Putting Q(0) = 0, we find that the second term of the righthand side must always be nonpositive; accordingly Q(t-\-\), starting from the initial position 0. satisfies the inequality:
(16) Since
it follows from (14) and this inequality that Also, since A^-^-rB^ ^> 0, it follows that Q(t-\-l) ^> Q(t) when Q(t) ;> Q(t— 1). Putting Q(0) = 0, the inequality Q(l) I> Q(0) is satisfied, because Q(l) = (1+J")AA0) > ®- Hence Q(t), starting from the initial position 0, is non-decreasing. Taking (15) into account we find that Q(t) converges*1) and lim Q(t) <^ P^. i—>CO
It is clear from (14) that
holds for some a. Thus, lim Q(t) is a price set proper to a. By t-t-ao
definition of P(e), we have lim Q(t) ^ P(f). Hence, lim Q(t) — P(€). (-*oo
(->co
Suppose, now, that Q(t) ^ P(t). Since A(Mi)-\-rB(Wi) ^> (16)
On the other hand, (14) implies
This, in combination with (11) and (16), leads to holds
Of course W See p. 48, n. 1.
A S P E C T R U M OF T E C H N I Q U E S
103
for any value of t. Remembering (13), we have Q(t+l)^P(t+l)^PM+E(t). As lim Q(t) = P(e) and lim R(t) == 0, we find lira P(t) = P((}. t-+QO
t—»•«>
t—>00
Theorems 1 and 2 may be illustrated by Figs. 5 and 6 respectively. Let there be two industries. Let activities l a , 2 X , 3j be
Fio. 5
available to the first industry and activities 12, 22, 32 available to the second. For simplicity, we assume that these activities are 'net' activities, i.e.aisi = bist = 0 (st ~ lit 2it 3^; i = 1, 2), so that the unit cost of the first product depends on the price of the second only, and conversely. In Figs. 5 and 6 the coordinates represent the prices (or unit costs) of product 1 and 2 respectively. The solid lines (llt 2lt and 3J stand for the unit cost of the first product when produced by activities llt 2V and 3X. In the same
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AN A L T E R N A T I V E DYNAMIC SYSTEM WITH
way the dotted lines (1 2 , 22, and 32) trace the unit cost of the second product. Possible sets of non-negative equilibrium prices are represented by P(a), P^,..., P^), the points of intersection of all pairs of cost lines in the non-negative quadrant; and P(e) gives minimal equilibrium prices or the long-run equilibrium prices.
At P(e), activities 1L and 3t are more expensive than 21; the activity 'proper' to P(e) in the first industry. Similarly 12 and 32 are more expensive than 22, the activity proper to P(ej in the second industry. Fig. 6 illustrates the convergence of prices. When the initial point of prices is P(0), activity 3X of the first industry and l a of the second industry are selected as the cheapest in period 0, activities 21 and 12 in period 1, and 21 and 22 in period 2 and all subsequent periods. When the initial point of prices is P(0)*,
A S P E C T R U M OF T E C H N I Q U E S
105
the pairs of activities selected will be l: and 32 in period 0; ll and 22 in period 1; 2l and 22 thereafter. The path of prices starting from any initial point is seen to converge on P(e). Thus the longrun equilibrium price set is stable so long as price changes arising on the way to it give rise to only negligible capital losses or gains; a sole source of instability is strong effects of capital losses on price formation, whose absence is assumed in the following.
3. The time path of outputs The output-determination side of our system is described as follows. Let xaj(t) be the level of the s^th activity of industry j during period t. As the period of production of each good is one mj
unit of time, x^t) = 2 xs,(t) is the output of good j available at 8)=l
the beginning of period t-\-l. Now define an m-dimensional column vector Xm and an w-dimensional column vector X as:
respectively. Further, let kj(t) be the total stock of good j available at the beginning of period t, and define: Since no more capital goods can be used during any period than are available at the beginning of the period, we have In the following, however, we assume full capacity use of goods, so that we obtain No industry uses any activity other than the cheapest one, and the set of the cheapest activities during period t is hence xsj(t) = 0 (s3- ^ wj) and xwj(t) = Xj(t); or, more concisely, Xm(t) = AwZ(i). Therefore we have (17) Let us assume that consumers can buy goods only at the beginning of each period. Goods are delivered during the period; but
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AN A L T E R N A T I V E DYNAMIC SYSTEM WITH
no new contracts can be made until the beginning of the next period. Let C(t) = [c^t),^),...,^)} be a vector whose element, ct(t), represents consumers' demand for good j at the beginning of period t. The stock of good j made available as productive capital at the beginning of period t-\-1 is the stock of goodj at the end of period tplus the output of good j (as available at the beginning of period <+1) minus the consumption of good j (as effected at the beginning of period t-\-l); and the stock of good j at the end of period t is the stock of good j at the beginning of period t minus the productive consumption of good j during the period. Since the productive consumption of goods is A(tul]X(t) we have
(18) If C(t) is a given function of t, the three sets of equations (11), (17), and (18) determine the behaviour through time of P(t), X(t), and K(t) from arbitrary initial positions P(0) and K(Q). As was discussed in Theorem 2, prices change on the way to the long-run equilibrium point P(f), and the cheapest activity adopted by each industry may also change from one period to another. At the later stage of the process of convergence, however, changes in prices become sufficiently small, so that they give rise to no further changes in the selected activities, and therefore the economy settles down to activity set, e, which is 'proper' to the long-run equilibrium price set P(e). In what follows we shall be concerned with this later stage only. Putting u>t = e in (17) and writing, for simplicity's sake, JD = £f(t), we obtain
(19)
Taking (19) into account, we have from (18)
(20) where A = A(f). System (20) is a dynamic input-output system; A is the matrix of current-input coefficients; .Bthe matrix of capital coefficients; and C(t) the vector of consumers' demands. Consumers' demands are sometimes regarded as some known function of time, but in the following we assume, as in Chapter III, that it is
A S P E C T R U M OF T E C H N I Q U E S
107
proportional to the total employment in period t denoted by L(e]X(t], i.e.
(21)
Here we assume instantaneous payment of wages and no consumption-lag. Let us write for simplicity P = P(f). Then PD is the average propensity to consume of workers evaluated at the long-run equilibrium prices, which will be referred to as the long-run average propensity to consume of workers. Taking (21) into account, we can rewrite (20) in the form:
(22^ L denoting i(e). The matrix (A-{-DL) may be called the matrix of 'augmented' current-input coefficients, since it comprises both the current-input matrix A and the 'labour feeding' input matrix DL.m Similarly, (B-\-DL] may be called the matrix of 'augmented' capital coefficients. In the following we assume the indecomposability of A-\-DL. Equation .(22) is examined as (27) in the previous chapter. If (-B+DZ/) is of rank n, the general solution of (22) can be written: (23) where T^,..., ^ are latent roots of the following equation:® the .X^'s are n-dimensional column vectors, [X\,..., Xln}, such that
24 and gl,...,gn are determined by the initial conditions. We can show that if the propensity to consume, PD, is not greater than unity, and r > 0, the matrix of augmented current-input coefficients satisfies the Hawkins-Simon conditions; they imply (I—A—DL)-1 > 0 as A-\-DL is indecomposable. We can also show that the general solution (23) contains a steady balanced growth solution, say (l-j-ijj/gfj.X' 1 , uniquely determined by the initial conditions, wherer]1 and X1 are strictly positive. If the long-run propensity to consume is less than unity, the rate of W SeeSeton[l, pp. 150-1]. < 2 > For simplicity we assume that ijlt..., rjn are distinct. This assumption is not essential for the following argument.
108
AN A L T E R N A T I V E D Y N A M I C SYSTEM W I T H
balanced growth, T^, is greater than the rate of interest, r, while if it is unity, the two rates are equal.(1) Let us now be concerned with the stability problem of the balanced growth path which has recently been discussed by many writers such as Solow and Samuelson [1], Solow [3], Jorgenso [1], Tsukui [1], and others. The balanced growth solution is said to be stable when any path starting from an arbitrary initial position of outputs will expand asymptotically at the balanced growth rate rjv The stability is said to be regular if the level of the balanced growth path eventually generated from an initial position is higher than that from a lower initial position, i.e. if the inequality X ( 0 ) < X*(0) implies gl < g*. There is no a priori reason for expecting the balanced growth in our system to be stable, but if it is stable at all, it must be regularly stable. This can be shown as follows. From (24) we have X1 = ^(I—A—DL^B+DfyX1. It is shown that As X1 > 0, it follows from the Frobenius theorem that I/T/J (the characteristic root of (/—A—DL)~l(B-\-DL) associated with X1) is greater in modulus than any other roots.(2> Since (I—A—DL^B+DL) and (B+DL^I—A—DL)-1 have the same characteristic roots, 1/ijj is also greatest in modulus among the characteristic roots of (B-}-DL)(I—A—DL)~l. Therefore, an w-dimensional row vector U1 such that
(25) is positive. After a little manipulation (25) turns into Similarly, from (24) Therefore,
(26) 1
f ' See Chapter III, Lemmas 4 and 5, and Theorem 3. I 2 'Thus Tt < 17^| for all i. The balanced growth is stable if 1 + ijl > 11 + ijj|j (i = 2,..., n) and unstable if l + ?h 5» U + 'Jil for at least one i. Since neither of these conditions follows from iji < |i)j (i = I,..., n), it cannot be decided on a priori grounds whether the balanced growth path is stable or not.
A S P E C T R U M OF T E C H N I Q U E S
109
Bv taking account of the fact that the conditions are necessary and sufficient for the balanced growth solution to be stable, we find from (26) that the stability implies Hence, from the relation (23) at t = 0, i.e. from we get Similarly,
As has been assumed, B-\-DL is a non-negative matrix of rank n, and as has been shown, Ui is positive, so that Ul(B-}-DL} > 0. Hence we find that if X(0) < X*(0), then g1 < g%. This proves the regularity. Let us now rewrite (22) as: (27) By Theorem 13 of Appendix the balanced growth is regularly stable if and only if there exists a (finite) positive integer h such that (28) From the fact that if the balanced growth in Leontief 'a dynamic system (22) is stable it is regularly stable, we find that the balanced growth is stable if and only if (28) holds for some h. It follows from (27) that where h > 0. This system may be referred to as the normalized dynamic input-output system of order h, and elements of [(B+DL)-l(I—A-\-B)]h as normalized input-coefficients (including labour-feeding coefficients) of order h.m The balanced growth of outputs is stable if and only if the system (22) can b reduced to a 'normalized' system of finite order with all its (normalized) input-coefficients positive. (D X(t+l) = (B+DL)-1(I—A+B)X(t) is said to be the normalized form of the dynamic input-output equation.
110
AN A L T E R N A T I V E D Y N A M I C SYSTEM WITH
An example of the instability is obtained if A = B, i.e. all T^ == 1. In fact, if A = B, (22) is reduced to Let •rj1,...,r]n be latent roots of \ I — ( l + T])(B+DL)\ = 0. It is shown that, among ijj,..., t\n, there exists a root, say rjv s that 0 < 1 + 1?! ^ ll + ^i f°r au i> and a non-negative eig vector is associated with this root. Thus the balanced growth (l + ^1)tglX1is unstable. If the balanced growth is unstable, there are economically meaningful initial points, the paths starting from which will sooner or later cease to be non-negative. Thereafter, the system (22) has no economically meaningfulsolution.(1) Let us now return to our previous notation. We write {A, B, L} for the catalogue of activities, arid [A(e), B(f), L^)} for the long-run equilibrium activity set. Let E be the matrix which has 1 in place of 8 of the matrix A'(CT), i.e. an n X m matrix such that its tth row has, first,(ml-\-...-}-mi__l) zeros, and then m{ units followed by (mi+1 + ...+wi n ) zeros. The balanced growth {(1-frj1)tg1 X1} starting from g^-X1 enables consumers to consume various goods by the amounts (29) at time t. (This at once follows from (20).) Let (Xm(t)} be a path starting from Xm(Q) such that (30)
This inequality requires that, at the initial point of time, the path does not use so much capital (evaluated at the long-run equilibrium prices) as the balanced growth does. When the economy expands along {Xm(t)}, the quantities of goods equal to (30)
are made available for consumption at time t; because EXm(t—l)—AXm(t—l) is the vector of net outputs at t and B[Xm(t) — Xm(t—1)] the vector of investments at t. (1 ' The difficulties associated with the possibility that the usual dynamic input-output system may generate negative output levels have been discussed by Leoritief [3], McManus [1], Dorfman, Samuelsoii, and Solow [1], Jorgoneon [2], and others.
A S P E C T R U M OF TECHNIQUES
111
Let us now define the efficiency of the balanced growth as follows: The balanced growth path is said to be efficient of order T if there is no feasible path fulfilling (30) and
(32) (33) (34)
with at least one strict inequality. Inequality (32) means that the present value of the stream of consumption {F/}, evaluated at the long-run equilibrium prices and discounted by the prevailing rate of interest, is at least as much as the corresponding value of the stream {F*}. (33) and (34) mean that along the path {Xm(t)} at least as much capital is used in period T (for the outputs in period T-\-l)and at most as much labour is used in periods 0,l,...,T— 1 as are used along the balanced growth path in those periods. We can prove THEOREM 3: The balanced groivth, X(t) = (1 -j- rj1)'gl X1, has the efficiency of any order. Proof. It follows from (31) that
This Yields
112
AN A L T E R N A T I V E D Y N A M I C SYSTEM WITH
Substituting from (29), (32), (33), and (34), we have
(35)
The first term on the right-hand side of this inequality may be written:
(36)
As P(f) is the solution to (12), the first term on the right-hand side of (36) vanishes; the second term may be written:
Hence (35) yields (37)
There is no possibility that (30) holds with strict inequality. Moreover, we can show that if (32), (33), and (34) hold with at least one strict inequality, then (37) holds with strict inequality It is clear that this contradicts (30). Thus the efficiency of the balanced growth path is established. 4. The rate of interest Enough has already been said about the dynamic properties of our system to show that if the rate of interest remains unchanged and the effects of capital losses on prices are negligible, prices eventually approach the long-run equilibrium prices, and that if the system can be normalized into a system with all its input-coefficients positive, outputs will finally grow proportionally at a steady rate. In this section we drop the assumption of constancy of the rate of interest, but throughout the rest of
A SPECTRUM OF T E C H N I Q U E S
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this chapter we keep the assumption of negligible capital losses; we shall be concerned with effects of a,change in the rate of interest on long-run equilibrium prices, on the long-run equilibrium structure, and on the balanced-growth rate. We begin with proving the following: THEOREM 4. // the rate of interest is reduced, the long-run equilibrium prices (in terms of labour] are reduced, and therefore real wages are increased. Proof. Let the rate r be reduced by Ar; since (1) is assumed, some elements o f r B are decreased, others remaining unchanged, and all the elements of rL are reduced. Hence the equation (12) becomes By a method similar to that used in the later part of the proof of Theorem 1, we can prove that the equation has, after the reduction of the rate of interest, an equilibrium price-set P*E), such that P*€) < P(f); of course, the new long-run equilibrium price-set is not greater than the price-set P*e). As prices in terms of labour decrease, real wages increase. Now let the new long-run equilibrium price-set after the reduction of the rate of interest be P(S), which satisfies the equation The activity-set S may or may not be identical with e; if 3 = e, then P(8) = P*e); but if 8 ^ e, i.e. if a reduction in the rate of interest changes the long-run equilibrium activity-set, we shall speak of a 'long-run structural change' being present. We can prove the following: THEOKEM 5. A reduction in the rate of interest gives rise to a long-run structural change if and only if at the price set P*f) the old long-run equilibrium activity set e is not the cheapest possible, i.e. if and only if there exists an activity-set a such that
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AN A L T E R N A T I V E D Y N A M I C SYSTEM WITH
Conversely, a rise in the rate of interest increases the long-run equilibrium price of each good, and hence the real-wage rate falls. But if the interest rate rises sufficiently, the greatest characteristic roots of all matrices A(g)-\-(r-\-/^.r)B(^, a = a,/?,...,^, become greater than unity; so that no activity-set has the 'proper' positive price-set. This contradicts Assumption 1; the rate of interest is thus allowed to change only within a certain range. If D is constant in spite of changes in prices caused by a fall in the rate of interest, we say that the demand schedules of workers are rigid; in such a case a fall in the rate of interest diminishes the long-run marginal propensity to consume of workers, because P(E) > P(S>. If workers change D in such a way as to maintain the same long-run marginal propensity to consume as before, we shall speak of the demand schedules of workers being perfectly flexible. We can establish: THEOREM 6. Suppose the long-run marginal propensity to consume of workers is unity be/ore a fall in the rate of interest. If the demand schedules of workers are rigid or imperfectly flexible, the rate of balanced-growth becomes greater than the interest rate after the fall. But if the demand schedules are perfectly flexible, the two rates are equal to each other after the fall as well as before. Let the range within which the rate of interest is consistent with Assumption 1 be denoted by a semi-open interval [0, r). We assume that the balanced growth of outputs is stable for all r such that 0 0. It follows from Theorem 4 that the I1' It is true that there may be an economy where any rate of interest within the range [0, f] gives the real-wage rate below the subsistence level. In the following, however, the assumption r° > 0 is made.
A SPECTRUM OF TECHNIQUES
115
real-wage rate associated with the rate of interest r such tha r > r° is below the subsistence level. If the rate of growth o the labour force, p, is not greater than r°, the horizontal lin through point 6 with coordinates (0,p) intersects the 45° line i a point (p,p); the rate of interest corresponding to the intersec tion will eventually realize the state of golden balanced growth
FIG. 7
where annual outputs of all goods and the stocks of capita goods grow together at the rate of increase of the labour forc and the rate of real wages is not below the subsistence level. O the other hand, if the rate of growth of the labour force exceed the rate of interest r°, there is no intersection of the horizonta line and the 45° line within the range [0, r°],so that the rate o growth of the labour force is always greater than the rate o balanced growth unless the real-wage rate drops below th subsistence level. It will be expected that the ratio of un employment to employment continues to rise. Thus a countr whose population is growing at a very high rate is not capabl of golden balanced growth. Her only hope for help is technologi cal invention.
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AN A L T E R N A T I V E D Y N A M I C SYSTEM WITH
5. Technological invention Assuming again that the rate of interest is constant, we shall analyse effects of a technological invention. Technological inventions are divided into two categories: inventions of new ways of making goods, and inventions of new goods. Among the effects of technological inventions we can distinguish long-run and short-run effects. Let us start with the long-run effects of inventions of the first category. Suppose a new activity for making an existing commodity, say the product of the first industry, is invented in period t°; and let the catalogue of activities after the invention be {A*, B*, L*}. In period t° the dynamic equations will be switched from (11) and (17) to:
where ^ is selected from among (m1-\-l)Xin!lx...Xmnpossible sets of activities. It is to be noted that effects of capital losses (or capital gains) due to price changes are assumed negligible. Two cases are possible: (1) the invention decreases the longrun equilibrium prices and (2) it has no effect on them. We shall call an invention 'effective' if the newly invented activity valued at the old long-run equilibrium prices (before the invention) is cheaper than any of the old activities. We can prove: THEOREM 7. If an invention is effective, the new long-run equilibrium price-set (after the invention) will be less than the old; and hence the real-wage rate will be increased; the newly invented activity, say the (?%-)-!)-£& activity of industry 1, is necessarily the activity proper to the new long-run equilibrium price-set.^ Proof. Let e = (elt ez,,..,en) and v = (m 1 +l,e 2 ,...,e n ), where et is the activity of industry i proper to the old long-run equilibrium price-set P(f}. Since the invention is effective, we have <" From Theorem 7 we find that the rate of balanced growth of a system where the long-run propensity to consume of workers was unity before an effective invention is increased by the invention if the demand schedules of •workers rigid or imperfectly flexible, but it remains unchanged if the schedules are perfectly flexible.
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Hence we find that the new long-run equilibrium price-set, P^, is less than the old; i.e. P< P(0. Let £ = (-"> P^; hence the (mj+l)-th activity is proper to P{^. Long-run effects on prices will operate when an invention is effective; if the long-run effects exist, there must also be shortrun effects appearing on the way to long-run equilibrium. But if it is not effective, the new long-run equilibrium prices are the same as the old, and hence the invention has no long-run, effect; but if the system was not in long-run equilibrium before the invention, there may be short-run effects even in such a case, producing transitory structural changes which, however, will disappear in the long-run. If an effective invention of an industry gives rise to changes in the long-run equilibrium activities of other industries, we shall speak of a 'trigger effect' being present/1' Consider an activityset v = (mljrl,e2,...,en)in which industry 1 adopts the newly invented activity and all the other industries adopt the old longrun equilibrium activities ez, ea,..., en; and let P(v) be the price-set proper to v. We can prove: THEOREM 8. (1) The invention in industry 1 will have no trigger effect on the other industries if at the set of prices, PM> the new activity of industry 1 and the old long-run equilibrium activities of the other industries are the cheapest possible, i.e. if (38)
where a, fi,...,x exhaust possible activity sets after the invention. (2) //, as a result of the new invention in industry 1, a number of the remaining industries find that activities other than those previously cheapest have come to be the cheapest (at the price-set <" Trigger effects of a technological change were first analysed by Simon [1, p. 268].
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AN A L T E R N A T I V E D Y N A M I C SYSTEM WITH
P(v)}, then the invention must have a trigger effect on at least some of the industries so affected.
Proof. Let A be an activity-set whose prices are non-negative. Taking (1) into account, we have P^ > P^A^-^-rB^), so that [/—A(x>—r-B^)]"1 > 0. The effectiveness of the invention and (38) imply, for all A other than v, the semi-strict inequality
Therefore PM < (l+r)L(A)[/—Af^—rB^]-1 = PM. Hence v is the new long-run equilibrium activity set after the invention. Suppose a number of industries 2, 3,...,j find, but all the other industries j-\-l,...,n do not find, activities which are cheaper than their old long-run equilibrium activities at the price-set Pw. Let (f> be an activity-set (mi-{-l,ez,...,ei,si+1,...,sn); then Suppose the price-set, P^, proper to ^ is positive; then Hence we find Next let s2 be an activity of industry 2 which is cheaper than its old long-run equilibrium activity e2 at the price-set P(vy Let TT = (m 1 +l,5 2 ,e 3) ...,e n );thenP w >P w [J (w) +rS( w) ]+(l+r)L (w) ; hence we have P(v) > Pfe). Therefore P^ > P(7r). Thus the priceset P(j) proper to the activity set cj>, in which the activities of industries 2, 3,..., j are e2,e3,...,e,!, cannot be the minimal priceset; consequently the invention has a trigger effect on at least one of those industries. Fig. 8 illustrates that if the long-run effects exist there must also be short-run effects appearing on the way to long-run equilibrium. Suppose the initial position of prices is P(0). The dotted staircase describes the path of prices with no invention. Let activity 4X of industry 1 be invented in period 0. Before the invention the first industry had adopted activity 2t and the second 32; but in period 0 when activity 4t is invented the first
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119
industry adopts it while the second sticks to 32. Thereupon activities 4j and 22 are selected in periods 1 and 2, activities 4j and 12 in period 3 and all subsequent periods. It is clear that the invention has a trigger effect on the second industry, but its
FIG. 8
full effect is not exhausted in one period. The first industry adopts the newly invented activity without delay, but the second industry adopts activities 32 and 22 before it finally settles on 12 in period 3. It may well be that empirical tables of input coefficients (say the 1939 table for the United States) do not represent a long-run equilibrium corresponding to activities 4X and 12 (Fig. 8), but merely transitional situations corresponding to periods such as those in which activities 4X and 22 are adopted. Even if there is no new invention in the period represented, the table might still
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AN A L T E B N A T I V E D Y N A M I C SYSTEM W I T H
be in process of change as a result of an invention in the past. Thus some of the changes undergone by the 1939 input coefficients in later years may be interpreted as after-effects of old inventions in, say, 1919, though others may be due to 1939 inventions.
FIG. 9
Even an invention which is not effective may have short-run effects producing transitory structural changes if the system was not in long-run equilibrium before the invention. This is illustrated in Fig. 9, where the dotted staircase stands for the time path of prices in the case of no invention, and the solid staircase for the path after an invention has been made. Let us now turn to the invention of new goods. Suppose a number of activities producing a new consumer good (say good n-\-l) are invented in period 0. After that time, a new industry (industry n-\-l) comes into existence, so that the
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121
Leontief system is augmented to size (n-\-1). It is shown, however, that if a new industry is a pure consumer good industry whose joutput is not used in production, its appearance has no effect on the time path of the prices of existing goods and hence on their long-run equilibrium prices. The invention has no effect on the input coefficients of the existing industries; the structure of the existing industries remains unchanged in spite of the appearance of a new industry. Next we assume that a newly invented good (say good n-\-1) is not a consumer good but an intermediate good; then activities of two different kinds will be invented: first, activities to produce the new good; and, second, activities to use the new good. Let be the activities of the first kind; the 6 n+ i Sn4 .,'s vanish for some s n+i> because it must be possible to produce the new good without any use of itself. We assume, for simplicity, that bn+ls = 0 for all s n + J , i.e. the new industry does not use the new good. Define:
where P(e) = [p^\---,Pn^] is the old long-run equilibrium price set. Due to the appearance of the new industry the system is augmented to size n-\-1. An invention which uses the new good in production of an existing good is called 'effective' if the newly invented activity evaluated at [jP( e ),Pn+i] ig cheaper than any of the corresponding old activities at P(e). It can be shown that if and only if an invention of a new good is accompanied by an effective invention to use this good in production of an existing good, it has trigger effects on some of the existing industries, and the new minimal equilibrium prices of the existing goods are less than their old minimal prices. Suppose there is no trigger effect. Then all the existing industries adopt old activities which do not use the new good. By assumption, the new industry does not use the new good. Thus none of the industries in the new situation makes use of good
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AN A L T E R N A T I V E DYNAMIC SYSTEM WITH
n-\-1; all the elements of the row w-f 1 of the new Leontief table vanish, so that it can be reduced to the old n-industry system. But if the appearance of a new industry does have trigger effects, some industries will use the new good, and the new table cannot be reduced to an n-industry table, except in a few special cases, such as the following. Let an existing good n as well as the new good n-\-\ be not a consumer good but an intermediate good. (1) If the new activities adopted by industries' n and n-\-l have an identical cost structure, these two industries can be consolidated into one industry. (2) If good n+1 is a 'complete substitute' for good n, in the sense that the newly invented activities in the other industries use it as a substitute for that good (without the latter entering into its production), it can easily be shown that in spite of the appearance of the new good, the tables of currentinput and capital-input coefficients will again reduce to size n once the trigger effects on industries, 1, 2,...,n— 1 have worked themselves out.(1> Thus, if a new industry has the same cost structure as an old one, the latter can be merged with the former; and if a new industry is a complete substitute for an old one, the former drives out the latter. In these cases, the tables of current-input and capital-input coefficients are not enlarged in spite of the appearance of a new industry. In almost all other cases, however, they must be enlarged when a new industry comes into existence. 6. Mechanization of technique In this section we translate Mrs. Robinson's analysis of mechanization of technique into the language of this chapter.<2) If the economy always adopted an activity-set A with nonnegative proper prices, the time path of outputs would be given by: where the rj^'a are latent roots of <*> Because all the elements of the row n of the new Leontief table vanish in this case. < a > See Robinson [1, pp. 411-20].
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123
XVs are n-dimensional vectors such that and g^'s are determined by the initial conditions. Assume that PdbD 0 arid X^ > 0. For initial conditions such that g^ = 0 (i = 2,...,n), outputs would grow at the common rate -n^, i-e. (39) When the balanced growth outputs (39) are generated, the net output per man evaluated at the long-run equilibrium price-set P(e) and the real-capital ratio at P(e) would be respectively: (40)
and
(41)
If we measure the net output per man (40) along the vertical axis and the real-capital ratio (41) along the horizontal axis, we obtain a point A in Fig. 10. Similarly, we have points a, /?, y,..., /a corresponding to activity-sets a, j3, -y,,.., /u. respectively.(1) The curve a/Jy.../* is Mrs. Robinson's productivity curve showing the relation between net output and the realcapital ratio when capital is reckoned at the given interest rate corresponding to the long-run equilibrium price-set P(). (42) Since arm
we have Let OB (in Fig. 11) be 1 -fr, and let the curve a/?... p, in the same figure be the productivity curve corresponding to the rate of interest r. It is evident that the slope of EX is the solution r^ of the following equation:
(1 > We assume that the price-sets proper to the activity sets a, /3,..., fj, are non-negative.
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AN A L T E R N A T I V E DYNAMIC SYSTEM WITH
FIG. 10
FIG. 11
We find that rA fg r. As
the slope of Re equals r. Thus R\ cannot be steeper than He.
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125
The same result is obtained for a,£,..., ju,; so that the line Re is tangent to the productivity curve at the point e. If the rate of interest falls, the long-run equilibrium prices P(f) are reduced, so that both net output per man and the realcapital ratio are reduced. Accordingly points a, /3, y,...,/x for lower rates of interest lie to the south-west of those for higher rates. In Fig. 12 there are three productivity curves, a1j81y1...ju1,
FIG. 12
<x 2 /? 2 y 2 .../x 2 , and a 3 /3 3 y 3 .../^ 3 corresponding to three rates of interest, rlt r2, and ra ,where r1 > rz > r3. If the interest rate is reduced, the point R shifts downward and the point e to the south-west; the slope Re declines. Now suppose X$ > 0; then at a rate of interest at which the slopes of -Re and R8 are the same, we have P(E) = P(g),(1) and the activity-set § is the long-run activity-set as well as the set e. Fig. 12 illustrates that the set e is still chosen at the rate of interest r2, and a mixture of e and S at (1) Because if the slope of-Re equals that of RS, (42) holds with strict equality for A = 8 with X^ > 0.
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AN A L T E R N A T I V E DYNAMIC SYSTEM
the rate r3. Imitating Mrs. Robinson's terminology, the rate of interest r3 may be called the epsilon-delta interest rate.(1) Thus we shall obtain a real-capital ratio curve, i.e. the thick line in Fig. 13, representing all the positions of long-run equilibrium which are possible in the given technical conditions with a range of the interest rates from the epsilon-delta interest rate to the beta-alpha rate. Usually a lower rate of interest will
Fio. 13
correspond to a more mechanized activity-set, as is shown in Fig. 13; but as in those situations illustrated in Fig. 14a and 146 a lower interest rate may perversely correspond to a less mechanized activity-set. Now suppose a new activity is invented in an existing industry; then we have a number of new activity-sets. If the invention is not effective, net output per man and the real-capital ratio of each new activity set are evaluated at the same long-run equilibrium price-set P(e) as before. In Fig. 15 the productivity I1' This is the case in which the two activity-seta e and S are proper to the sameP u) (= P(6)).
FIG. 14a
Fia. 146
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AX A L T E R N A T I V E D Y N A M I C SYSTEM WITH
curves before arid after the invention are represented by the solid and the dotted curves respectively. It is seen that the set e remains to be the long-run equilibrium activity-set after the invention for so long as the interest rate is constant. When the invention is effective, it gives rise to a decrease in the long-run equilibrium prices; hence, in case of an effective
FIG. 15
invention, all the activity-sets including old sets must be evaluated at the new long-run equilibrium prices. Since the new price-set is less than the old, the points representing the old activity-sets, 8, e, A,..., will shift to the south-west. In Fig. 16, v and £ are new activity-sets evaluated at the new long-run equilibrium prices; S,e,A,..., the old activity-sets evaluated at the old long-run equilibrium prices; and 8', e', A',..., the old activity-sets at the new prices. The solid curve stands for the productivity curve before the invention and the dotted one for the curve after it. If the new long-run equilibrium activity-set £ lies on e, the invention is called neutral, and iff lies to the right (or left) of e, the invention is of a capital-using type (or a capital-saving type).
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If the invention is neutral, net output per man and the realcapital ratio remain unchanged; but since the new long-run equilibrium price-set is less than the old, the invention increases physical net output per man and physical capital. With a capital-using bias in technical progress net output per man and the real-capital ratio corresponding to a constant rate of interest
Fio. 16
are increased, and hence physical net output per man and physical capital increase also. With a capital-saving bias in technical progress, however, net output per man and the realcapital ratio corresponding to a given interest rate are reduced, but the latter decreases more than in proportion to the former. Finally, to facilitate readers' comprehension, let us concentrate our attention on the following special case. Let the activity-set € = (e 1 ,e 2 ,...,e n ) be the long-run activity-set before inventions; and let each industry i make an improvement on the activity et which reduces the coefficient of labour input le. in the same proportion. Then e continues to be the long-run equilibrium activity-set after the inventions also; the long-run equilibrium prices fall in proportion to the labour-input coefficients. On
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the other hand, however, these technical improvements will affect the eigen-vector X\, the proportions in which outputs grow steadily. If, after the inventions, those industries whose real-capital ratios are greater (or less) than the average dominate others, then the net output per man (40) arid the real-capital ratio (41) become greater (or smaller). Thus the improvements are of the capital-using type (or the capital-saving type) in the sense denned above.
V WORKABILITY OF GENERALIZED VON NEUMANN MODELS OF BALANCED GROWTH
1. The original von Neumann model THE model discussed in the previous chapter has been described, on the assumption that there is no joint production, in terms of the current-input coefficients and the capital-input coefficients. In this chapter, however, we follow von Neumann [1] in assuming that each good may be produced jointly with certain others, viz. its joint products. This enables us to treat capital goods as follows :H) We describe capital goods at different stages of wear and tear as different goods, and define a process of production as an operation which converts one bundle of goods—including capital equipment left over from the preceding period—into another bundle of goods—including the capital goods (at appropriate stages of wear and tear) which are carried forward for use in further production. Thus the fixed capital equipment employed is included both in the bundle of inputs and again in the bundle of outputs. On the von Neumann convention, therefore, the 'process' of production which uses at least one capital good should inevitably be multi-product. The von Neumann way of approach enables us to proceed in terms of two matrices (i.e. the output and the input matrices), instead of three matrices (i.e. the output, the current-input, and the capital-input matrices) used in the previous chapter, though we are confronted with a new trouble—the 'joint-production' trouble. In von Neumann's model of economic expansion it is assumed that 'consumption of goods takes place only through the processes of production which include necessities of life consumed by workers';(2) wages are held at the subsistence level; all (1 (2
> See von Neumann [1, p. 2] and Hicks [3, p. 78]. > See von Neumann [1, p. 2].
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income in excess of necessities of life, i.e. all capitalists' income is reinvested. Each input coefficient ci;- is defined as the sum of the following two quantities: the quantity, a^, of good j technologically required per unit of process i, and the minimum of good j needed to persuade people employed per unit of process i to work. It is the same as the 'augmented' input coefficient in our previous terminology. The output coefficient b^ simply represents the quantity of good j produced per unit of process i. Let us assume that all 6y's and Cy's (i = l,...,m;j = l,.,.,n) are constant and that each process is of unit time duration. The former implies that constant returns to scale prevail, and the latter that processes of longer duration are broken down into a number of processes of unit duration introducing, if necessary, intermediate products as additional goods. Let p(t) be the interest factor in period t (= 1-fthe rate of interest in period t). Let q(t) and P(t) be an m-dimensional row vector \qi(t},...,qm(t)] and an ?i-dimensional column vector [Pl(t),..., Pn(t)} standing for the intensity and the price vectors (in period t) respectively. With given prices and rate of interest, some processes are likely to be more profitable than others. In equilibrium, however, there can be no process which yields a return greater than that yielded by the prevailing interest rate; for under perfect competition positive profits would attract competitors to use the same process, so that prices of factors would rise. Thus we obtain (1)
where B = (b^), and 0 = (c y ). Next, if a process yields negative profits after payment of interest, it will not be used, and its intensity is zero; in equation form this reads (2) Since each process is of unit time duration, the components of the vector q(t—1)J5 give the amounts produced at time t, while those of the vector q(t)G give the amounts of input used up in production (including workers' consumption) at time t. It is impossible to consume more of a good in the production processes
VON N E U M A N N M O D E L S OF B A L A N C E D GROWTH 133 Than is available, so that we have (3)
Finally, in equilibrium those goods that are overproduced will be free goods, and zero prices are charged for them. This implies (4)
Von Neumann defines equilibrium to be a state of balanced growth, where prices and the rate of interest are constant over time, and the intensities of production grow or decay at a constant geometric rate; it is Champernowne's 'quasi-stationary' state(1) where the economy changes only in scale, but not in composition. Thus balanced growth implies (5)
a being 1+the rate of balanced growth. Inserting (5) in (l)-(4) and dividing them by
P^t) and/or
(t—1), we obtain (»)
(7) (8) (9) where x and y are the normalized intensity vector and the normalized price vector respectively, i.e.
obviously (5) implies that x and y are constant over time. It is crucial and fundamental in von Neumann's theory of balanced growth to examine that the system (6)-(9) has meaningful solutions such that x > 0, y > 0,<2> a I> 0 and /S ^ 0. Approaching this question, he makes the following assumption:
(1) (2
See Champernowne [1]. > See p. 5, n. I, for definition of ' > ' and ' 3:
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GENERALIZED
Obviously, the former is reasonable, because the output coefficients by and the augmented input coefficients cti are always non-negative. The latter, however, is a strong and unrealistic assumption implying that every good is involved (either as input or as output) in every process, although it ensures the uniqueness of a and /?. It is observed by Kemeny, Morgenstern, and Thompson [1] that there are examples which fit into the von Neumann model but do not correspond to economic reality; in a special limiting case where C > 0 and B = 0, nothing is produced. They therefore imposed an additional condition
(10) which implies that the total value of all goods produced must be positive. They also replaced the assumption (f) by: (**) every column of .6 has at least one positive entry, (***) every row of C has at least one positive entry. Intuitively, (**) means that every good can be produced by some processes, and (***) that every process uses some inputs. Since (*) holds, we find that (**) and (***) are equivalent to the following game-theoretical conditions: respectively, where v(B) and v( — C) are the values of each of the matrix games B and — C in which the maximizing player controls the rows and the minimizing player the columns/1' Kemeny, Morgenstern, and Thompson [1] have proved that there exist I 1 ) Let H be an mxn payoff matrix. Let
be the space of strategies of the maximizing player, and let
be that of the minimizing player. The value of the matrix game H is defined as v ( H ) = max min xHy — min max xliy, xeX veY v£Y xeX
i.e. the expected payoff for the game when the players choose their respective optimal strategies. See, for example, McKinsey [1, p. 24] and Gale [4, pp. 193-6].
VON NEUMANN MODELS OF BALANCED GROWTH 135
meaningful solutions to equations (6)-(10) if Assumptions (*), (**), and (***) are made. (See also Karlin [2, pp. 335-40] and Gale [4, pp. 311-19].) Furthermore it can be shown that the rate of balanced growth, a— 1, equals the rate of interest, ft— 1, in the Kemeny-Morgenstern-Thompson model as well as in the original von Neumann model. Von Neumann's dynamic model summarized above is based on the following assumptions: (a) there are constant returns to scale; (6) the primary factor of production (labour) can be expanded indefinitely; (c) wages are held at the subsistence level (d) all capitalists' income is automatically reinvested. As was pointed out by Champernowne [1], the model ignores the role played by consumers' choice in the determination of the intensities of production of the individual commodities and their relative prices; both of them are determined solely by the technological conditions of production. In what follows we shall relax the last three assumptions and deal with two generalized models, both of which assume that (6') the working population grows at a finite rate, (c') workers' demand for consumption goods depends not only on wage income, but also on prices, and (d1) capitalists consume a constant proportion of their income, and their demand for consumption goods is such as to allow substitution in response to price changes. Thus the models developed below explicitly allow for consumers' choice though, as will be shown below, it is subject to a stringent assumption of income-elasticity. As for the unit cost of production there are at least two definitions which are historically important; the first definition (used by Marx) assumes that wages are paid in advance (i.e. at the beginning of each period), and the second (by Walras) assumes that they are paid after work (i.e. at the end of eac period). In Section 2 we are concerned with a model in which the Marxian assumption of equal rates of profit (based on his definition of unit cost) is satisfied, while in Section 4 we deal with a model where the Walrasian definition prevails. The former may be regarded as a generalization of Marx's reproduction scheme (see Marx [1], vol. ii, chaps, xx and xxi) in terms of th
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von Neumann apparatus,'1* and the latter as a von Neumann version of Walras's theory of capital formation and credit.(2) Section 3 offers a proof for the existence of economic solutions to the model presented in Section 2; it also discusses whether any of these solutions gives an efficient path of economic growth. 2. A model of an expanding capitalist economy: Marxvon Neumann Let us consider an economy in which there are a finite number m of technically possible processes of production operating at discrete time intervals and producing n different goods with a lag of one time period of production. Let A be a matrix of material-input coefficients whose element a^ (in the ith row and the jth column) denotes the quantity of good j used up per unit level of process i; and let L be a column vector of labour-input coefficients, the ith component (lt) of which represents labour employed per unit level of process i. Finally, the element, bijt in the ith row and the jth column of the output-coefficient matrix B represents the quantity of good j produced per unit level of process i. We make the following assumptions concerning the technological coefficients A, B, and L:
where v(B), as before, stands for the value of the matrix game B in which the maximizing player controls the rows and the minimizing player controls the columns. It should be noted that although the assumption v(—A) < 0 implying that every process uses some inputs is economically plausible, we need not assume it in the following discussion. The meaning of the first assumption is quite obvious. The second assumption implies that every process must consume a positive amount of labour. This rules out the possibility of <*> Its special case preserving the assumption (c) fits in with Marx's idea that the capitalist economy is a slave-system. (•> Walras [1, pp. 267-312].
VON N E U M A N N M O D E L S OF B A L A N C E D G R O W T H
137
automatic processes. The third assumption is the same as (**) in Section 1 and implies that every good can be produced by some processes. Next, let w(t) be the money wage rate in period t. As before, )8(<) stands for the interest factor in period t (= 1+the rate of interest in period t), and an m-dimensional row vector and an n-dimensional column vector represent the intensity and price vectors (in period t), respectively. Of course, /?(£), w(t), q(t), and P(t) are non-negative. At the position of equilibrium no process must yield positive supernormal profits, i.e.
(11) AP(t) and w(t)L are the Marxian 'constant capital' and 'variable capital', respectively. Since wages are paid in advance at the beginning of each period, the unit cost of production should include not only the interest charge on constant capital but also the interest charge on variable capital. It is worth mentioning that (11) is an inequation interpretation of Marx's assumption of general equality in the rates of profit/1' Following von Neumann and others, we assume that if process i is unprofitable, it will not be used and its intensity qAt) equals zero; therefore we find (12) Capitalists' income in period t-\-\ is now equal to
which may be positive or negative. In what follows it is assumed that when capitalists' income is non-positive, their consumption is zero, and when it is positive, it is proportional to their income. We shall denote by c (or s) capitalists' average propensity to <"•' Cf. Marx [1], vol. 3, pp. 182-203, Morishima and Seton [1], and Samuelson [4, p. 887]. ji(t)~\ may be interpreted as the rate of profit in Marx's sense. See below, Section 3.
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consume (or to save) in case of positive incomes. Obviously, 0 < c < 1 and c+s — 1. Let d(t) be an n-dimensional row vector, the jtii element of which represents capitalists' consumption of good j in period t; then we have the capitalists' consumption function, cE(t) = d(t)P(t), provided that capitalists' income in period t, E(t), is assumed to be positive. According to the pure theory of consumer's behaviour, d(t) is a homogeneous function of degree zero in the variables P(t) and E(t); so that d(t) can be written in the form:
where y(t) stands tor the normalized price vector in period t. There is no a priori knowledge of consumer's behaviour that n
where y(t) stands tor the normalized price vector in period t. h=1
may generally depend on prices, y(t), and may be different from 1. Throughout this chapter, however, we assume that the Engel-elasticity of capitalists' consumption of each good is unity for all values of their income. Then d(t) is of the following simple iorm:(1)
where f(y) is an n-dimensional row vector such that/(i/)?/ = c. Next let an n-dimensional row vector e(t) represent workers' consumption of goods in period t. If we assume that workers do not save, we obtain where W(t) designates workers' income, q(t)w(t)L, in period t. It is evident that e(t) is a homogeneous function of degree zero in I" This assumption is assured if tho utility functions are identical for all capitalists, and are strictly quasi-concave and quasi-homogeneous in their arguments. For the definition of strict quasi-concavity, see p. 43, n. 2, above. The quasi-homogeneity means that, for all non-negative vectors x and x', and for all positive numbers A, U(x) 5: V(x') if and only if U(Xx) & V(\x'). See Radner [1],
VON N E U M A N N M O D E L S OF B A L A N C E D G R O W T H 139
P(t) and W(t); and we assume that the Engel-elasticity of workers' consumption is unity, so that
•where g(y)y = 1. As for f(y) and g(y) we assume that (iv) f(y) and 9(y) are single-valued, non-negative, and continuous for all y in the set Goods are used up either in the processes of production or through workers' or capitalists' consumption. The demand for goods for production in period t is q(t)A, and for workers' consumption is e(t). Remembering that capitalists' consumption, d(t), does not take place when their income is non-positive, we find that the total consumption of goods amounts to
where 'max
means the larger of the number
in parentheses. Since no more goods can be consumed during any time period than were produced during the preceding one, we obtain:
(13) We assume that if there is excess production of good j, its price P.j(t) becomes zero, and therefore, taking into account that g(y)y = I andf(y)y = c, we find from (13) that (14) The final condition requires that the total value of all goods produced must be positive, i.e. (15) Let us now assume that the money wage rate w(t) is adjusted so as to maintain a given level, O, of the real wage rate, and let n
units be chosen in such a way that w(t) 2 -^(0 ig equal to O. g(y) = I andf(y) = c, we find from (13) that
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We concentrate our attention on the state of balanced growtl where equations (5) are satisfied. Considering n.nrl
we find that our expressions (!!)-(15) now become
(16) (17) U8) (19)
(20) where x and y stand for the normalized intensity vector and the normalized price vector respectively.
3. Existence and efficiency of the balanced growth path In order to prove the existence of solutions to (16)-(20), given Assumptions (i)-(iv), we shall consider the following auxiliary system: (21)
(22) (23) (24) (25) where x e X (the set of all m-dimensional non-negative vectors with unit sums), y e Y (the set of all ^-dimensional non-negative vectors with unit sums), a and y are non-negative scalars, and
If this system has a solution (x,y,oL,y), we see from (22), (24 and (25) that axM(y, a,)y — yxM(y, <x)y > 0. Since xM(y, a.}y ^> 0, we obtain a — y > 0. Thus we need look only for solutions in which a equals y. Under the assumption that a = y, equations (22) and (24) are direct consequences of (21) and (23) and of th
VON N E U M A N N MODELS OF B A L A N C E D GROW-TH 141 fact that x e X and y e Y; hence we shall show that inequalities,
have a solution. Divide B—aM(y,a.) by 1 + a and set A = a/(l-fa); the above expressions then become (26) (27) (28) where (29) From Assumptions (i)-(iv) we find at once:
the inequality N(y,A) _^> 0 holds, (iii') N(y,X) is continuous for all (y, A) in S. (iv') L > 0. Let us now choose a point («/*, A*) in the set S, and consider the following expressions: (30) (31) In view ofg(y)y = l , f ( y ) y = c, A ^> 0 and O > 0, we find from (29) that (iv') implies N(y,X}y > 0 for all (y,\) in S. Together with (ii'), this implies that every row of N(y,X) is non-negative and has at least one positive entry. Hence, (32) From (i') and (32) we find that there is at least one A, 0 < A < 1, such that (33) (') Because it is clear from the definition of the value of a matrix game that it is a continuous function of the elements of the payoff matrix.
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If, for such a A, there were no y fulfilling (30), the maximizing player could always find a strategy x such that This contradicts (33). An exactly symmetric argument is applicable to (31). Hence there is a pair (x, y) which satisfies (30 and (31). Let T(y*,X*) be the following set: W,A*) = {(y, A) I (30) and (33) hold}. We can establish LEMMA l.(1) The mapping (y*, A*) -> T(y*, A*) is an upper semicontinuous mapping of Fx[0,1] (the cartesian product of Y and [0,1]) into itself, where Y is the set of all n-dimensional non-negative vectors with unit sums.(V> Furthermore T(y*,X*) is contradible.w Proof. Let (yk,Xk) and (yk,\k) be any sequences of points in Yx [0,1] that converge to the points (y*, A*) and (y,A), respectively. Suppose (yk,'\k) e T(yk,Xk). By (iii') we have Inequalities for every k imply therefore, (y, A") e T(y*, A*). Thus the mapping (y*, A*) -> T(y*, A*) is upper semicontinuous. Let both (y',X) and (y",X) belong to T(y*,X*). We have
so that W I am indebted to H. Nikaid6 for the following proof of this lemma. "> Let {x1} and {j/*} be any sequences with limits x and y respectively. A multi-valued mapping (or correspondence) x -> F(x) is said to be upper semicontinuous at a; if t/* 6 F(x%) for all i implies y e F(x). A mapping is upper semicontinuous on a region R if it is so at each point of R. < a > A set F is said to be contractible or deformable into a point y1 e F, if there exists a continuous function G(y, ^) such that, for any y e F and for any ^ 6 [0, 1], 0(y, p.) e F, G(y, 0) = y, and G(y, 1) = y1.
VON NEUMANN MODELS OF BALANCED GROWTH 143
for all p, in the interval [0,1]. This, together with (33), implies that Next let A° be the largest of those A's which fulfil (33). Let x° be a non-negative vector such that (34)
and let (y,\) e T(y*, A*). Write It follows from (34) and A ^ A° that (34)
(36)
for all /ji in [0,1]. (35) and (36) imply v(H) 2: 0 and v(H) S 0 respectively. Hence v(H) = 0. This, together with (36), implies [y, (l-^A+juA 0 ] e T(y*,A*) for all ju in [0, !].»> Now let A1 be the smallest among the A's satisfying (33). Let (fAA 1 ) e T(y*,X*), and let (y,X) be any pair in T(y*,X*). Consider the following operator: where 0 ^ p <1 1. It is clear that G[(y, A), 0] = (y, A) and G[(y,X), 1] = (yl,X°) e T(y*,X*). Furthermore, from the above argument, we see that [y, (1— ^)A+M°] e T(y*} A*) and [y1, (1— ju,)A+^A°]e T(y*,X*) for all /* in [0,1]; therefore, [(l-^y+M1,(1— ja)A+ jaA°] e T(y*, A*) for all /LI in [0,1]. Hence r(j!/*,A*) is deformable into the point (^A 0 ) e T(y*,X*), We can now prove the following theorem. THEOREM 1. A solution exists for inequalities (26), (2 and (28). Proof, By the Eilenberg-Montgomery fixed-point theorem'2* (1 > Of. Kemeny, Morgenstern, and Thompson [1], Lemma 2. < 2 ' See Eilenberg and Montgomery [1], In a special ease, the Eilenberg— Montgomery fixed-point theorem may be stated as follows: Let S be a nonempty, closed, bounded, convex subset of a Euclidean space. Let x -> F(x)
(Continued on next page)
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the mapping (£/*,A*)-> T(y*,X*), defined in Lemma 1, has a fixed point, say (y**,A**) e T(y**, A**). Let U(y**, A**) be the set {x | x e X, x[(l-\**)B-X**N(y**, A**)] I> 0}. As
Therefore, we have It follows from (i') that A** > 0. It also follows from (32) that A** < 1, and from (iv') that x**N(y**, X**)y** > 0. Hence, we finally obtain x**By** > 0. Let us define a** = y** = A**/(l— A**). We can show that x**,y**,a**, and y** satisfy inequalities (21)-(25). Write (37)
We can also show that x**, y**, a**, and ^** satisfy our original inequalities (16)-(20). Itfollowsfrom (37) that if j9**-l <1 0,thena**-l = ]8**—1, and that if/3**— 1 >0,thena**—1 = s(0**—1). Consequently, we may make the following statement: // the rate of interest is negative (or zero), the output of each commodity decreases by a constant percentage per unit of time equal to the rate of interest (or remains unchanged); if the rate of interest is positive, the output of each commodity increases by a constant percentage equal to the product of the rate of interest and capitalists' average propensity to save. The solid line in Fig. 17 represents the relation between the interest factor and the expansion factor. For comparison we also show the original von Neumann relation in the form of a dotted line. Let us write xAy as the Marxian 'constant capital' C, x£lL as be a multi-valued upper semicontiiiuous mapping from S to S such that for all x 6 S the set F(sc) is contractible. Then F has a fixed point, i.e. a point x such that x e Fix).
VON N E U M A N N MODELS OF B A L A N C E D G R O W T H 145 'variable capital' V, and xBy—xAy—xQL as 'surplus value' S. Then it follows from (17) that the Marxian 'rate of profit'. Since capitalists spend cS for consumption and sS for capital accumulation, s represents the ratio of capital accumulation to
Fio. 17
surplus value. Thus the conclusion above can be stated in terms of Marx's terminology as follows: In the case of expanded reproduction, the rate of growth of the economy equals the rate of profit multiplied by the ratio of capital accumulation to surplus valued So far we have obtained x**, y**, a**, and ft** corresponding to a given real-wage rate Q. It remains to determine the rate O that makes the demand and supply of labour equal. This problem can be solved by an argument which is identical to that discussed in Section 4 below. Let us now turn to our second problem: we shall show that the balanced growth, whose existence has been established, is an efficient path. Let (x,y,ot.,fi) be a solution to (16)-(20), and let W A similar result was obtained by Morishima [1, pp. 179-84].
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q(G) be a vector proportionate to x. A balanced growth, at the rate a, starting from q(Q) produces output of the amount a'"1g'(0)£ at time t, parts of which are used in further production, the rest being devoted to consumption in period t. As the value of input in period t is dq^Ay, arid, by virtue of (17), the value of output in period t equals a'-^OX/J'Ay-\-fiQ.L), the value of consumption in the same period may be written: It is evident that the employment of labour in period t amounts to:
Let {q(0),q(l),...,q(t),...} be any path starting from the same initial point q(0) as that of the balanced growth path. Write
(38) ana tne components 01 tne vector Jt give tne quantities 01 goods made available for consumption at time t when the economy expands along the path {q(t}}. The balanced growth path is said to be efficient of order T if there is no possible path {q(t)} fulfilling
(39) (40) (41) with at least one strict inequality. Accordingly, if the balanced growth is efficient of order T, it is shown that no possible path can provide us with a stream of consumptions, whose present value, evaluated at the equilibrium prices y and discounted by the equilibrium interest factor /J, is greater than the present value of the corresponding stream along the balanced growth path, unless it produces less output in period T-\-1 or uses more labour in some periods than the balanced growth does in those periods. We have THEOREM 2. The balanced growth path determined by inequalities (16)—(20) is efficient of any order among all technologically feasible paths.
VON N E U M A N N MODELS OF BALANCED GROWTH 147 Proof. It follows immediately from (38) that
T-Z Subtracting T fi-lq(t-\-l)£lL from both sides, we have (=0
Considering (16) and the fact that q(t)'s are non-negative vectors, we get
(42) Let us suppose the contrary, that the path {q(t)} fulfils conditions (39), (40), and (41). Then it is clear from (42) that the following inequality a fortiori holds:
As
and
we obtain
Together with (17), this inequality leads to (43)
Furthermore, the inefficiency of the balanced growth path implies that at least one of (40), (43), and q(T)L<^(xTq(0)L holds with strict inequality. Hence, we get As the right-hand side of this inequality vanishes because of (17),
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W O B K A B I L I T Y OF G E N E R A L I Z E D
the left-hand side is strictly positive. This fact contradicts q(T) ;> 0 and (16). 4. Alternative model: Walras-von Neumann We shall now turn from Marx's reproduction scheme to Walras's theory of capital formation and credit. Let ay be the quantity of good j used per unit level of process i, and leti;r. be the length of life of good j (when used by process i); then the Walrasian unit cost is written :(1)
where r(t) is the rate of interest in period t, i.e. fj(t)—l. It should be noted that the unit cost defined by Walras does not include the interest charge on the working capital w(t)L, because wages in period t are assumed to be paid at the end of period t after workers have done their work in that period. Since we follow von Neumann in treating capital goods at different stages of wear and tear as different goods, all T^'S are equal to unity. Thus the Walrasian unit cost becomes:
so that we obtain (44) (45)
At the beginning 01 each period t, wages paid at the end oi the previous period, q(t—l)w(t—l)L, and profits, are available for consumption. Define W(t) as q(t—\)w(t—1)L. Assume that workers spend their whole income W(t) for consumption in period t, while capitalists spend the amount max(0, E(t))c for their consumption. Moreover, capitalists' demand for goods, d(t), and workers' demand for goods, e(t), are assumed to be of the f1' T is the reciprocal of the rate of insurance and amortization, i.e. the reciprocal of the sum of Walras's /j. arid v. Using the equations (8) oil p. 281 in Walras []], his equations (6) on p. 280 can be rewritten in our form.
VON N E U M A N N M O D E L S OF B A L A N C E D G R O W T H 149 following respective forms:
where
We then have: (46)
(47) (48) Let us now fix our thoughts on the state ot balanced growth
where equations (5) hold. Wwrite. Write then we find:
From (45) and the definition of E(t), we at once find that so that we obtain
828H5
L
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W O R K A B I L I T Y OF G E N E R A L I Z E D
Therefore (44)-(48) can be written in the form:
(49) (50) (51) (52) (53) where x stands for the normalized intensity vector. Let us suppose for a while that a> is a given constant. Consider the following auxiliary system:
(54) (55) (56) (57) (58) where
Since A ^> 0, B ;> 0, L > 0, and v(B) > 0 hold, we can show, by the same argument as in Section 3, that the system (54)-(58) has at least one solution (x**,y**,x**,y**)such thatx** e X, y** 6 7 and ex** = y** > 0. Let
(59) Taking account of the fact that c-f-s = 1, we easily find that x**, y**, a.**, and/3** satisfy the original inequalities (49)-(53). Clearly, from (59) it follows that if fi**-l<^(), then a **-l =0**—1, and that if / 8**-l > 0, then Let us now write xBy — xAy—xQ.L as 'profit' P and xAy as capital' K. Since fi = /3oj, we have from (50) that/?— 1 = PjK.
VON N E U M A N N M O D E L S OF B A L A N C E D G R O W T H 151 Hence we may conclude that: Barring negative expansion, the economy grows at a rate equal to the product of the, capitalists' average propensity to save and the rate of profit on capital. This conclusion is identical to that reached by Kaldor in his recent model of economic growth. m We can also show, by the same argument as in Section 3, that the balanced-growth solution thus obtained is efficient. In the above we started from an arbitrarily given constant w and then determined the /3** corresponding to the co.' If the rate of growth of population were a passive factor, the real-wage rate would continue to be /3**o>; but in reality the growth of population is an independent datum. It remains, therefore, to solve one more problem: how is the real-wage rate Q determined ? Let us now make a simple assumption that the real-wage rate is determined by the demand and supply of labour. Suppose the supply of labour in period t is p'N, where the rate of growth of population (or labour force) p is assumed constant. Since the pulation (or labour force) p is assum
the
demand-supply balance of labour is described by
which is equivalent to the following two equations:
(60) (61) It is worth mentioning that a — 1 is a variant of Harrod's 'warranted rate of growth' and p — 1 a variant of his 'natural rate' (under the hypothesis of no technological progress)/ 2 ' Our equation (61) implies that the 'natural' and the 'warranted' rates are equal. Let Sa(w) be a set of a's which fulfil (49)-(53) for a given w. We can find that it depends on w in an upper semicontinuous way. It can also be shown that even if a> tends to zero, any <x(w) ") See Kaldor [•_>, pp. 613-141. < 2 > See Harrod [1, pp. 03-100].
FIG. 18a
FIG. 186
VON NEUMANN MODELS OF BALANCED GROWTH 153
belonging to S a (w) remains positive, and if oj tends to infinity, it tends to zero. Let us now denote the maximum element of $a(w) by a(oi), and suppose 0 < p <^ a(a>) for some w. If Sa(w) is convex for every a>, there is at least one a> such that p e S^uj), as is illustrated by Fig. 18ft.(1) But, in case of $a(oj) being not convex for some o>, there may be no o5 such that p e Sa(&), even if p does not exceed a(w) for som.e tu (see Fig. 186). Thus the convexity of Sa(w) for all a> is sufficient for establishing (61), once the condition 0 < p <^ 5(o») is fulfilled for some a>. Let us now assume that, for w, a(<3) equals p. Write & = a(a>) and /3 = a4- m ax(0, (a — l)/s)c. Then O, being equal to /3oj, is the real-wage rate at which the 'natural' arid the 'warranted' rates of growth are equal. The solution (x(w),y(a>), &,$} corresponds to the state which Mrs. Robinson describes as a golden age, and the absolute level of activity in the initial period can be derived with the aid of equation (60) as Nj[x(w)L]. If, however, p is very high so that p > 5(w) for all to, then, for any balanced-growth solution to (49)-(53), we have
Thus, in such an economy with a very high rate of growth of population, unemployment of labour is inevitable as long as all the sectors grow in balance: it would be expected that the rate of real wages will be pushed down to the subsistence level. < l > For the definition of convexity, see p. 62, n. 1.
VI BALANCED
GROWTH AND
EFFICIENT
PROGRAMME OF VERY-LONG-RUN GROWTH
1. Introduction O N E of the most fascinating propositions recently advanced in the field of growth economics is that due to Dorfman, Samuelson, arid Solow 11],(1) asserting that there is a 'long-run' tendency for the efficient path of economic growth to approximate to the von Neumann path of steady balanced growth. In Figs. I9a and 196 let ON be the von Neumann ray, and suppose the system starts from an initial position P which may be off the ray. The terminal ray OR at which the system aims is specified, and the programming period T at the end of which, it reaches the ray OR is also specified. When T increases without bound, the efficient path leading from P to a point on OR will converge to the von Neumann path, ON. Thus when T is sufficiently large, the efficient path will remain most of the programming period within a prescribed neighbouring cone of ON, though it will have to leave it, sooner or later, in order to reach the objective. Since the von Neumann path may be characterized as a path maximizing the speed of growth of the economy, so it may be nicknamed "Turnpike'; and, therefore, the DorfmanSamuelson-Solow proposition may be referred to as the 'Turnpike Theorem'. A historical description of the studies of this subject may be given as follows. Dorfman, Samuelson, and Solow first presented three models of capital accumulation formulated in terms of linear programming, difference equations, and calculus of variations respectively. Except for the linear programming model, the authors linearized the systems by expanding them "' See Doi'finan, Samuelson, and Solow [1], chapter J 2 .
FIG. li)a
FIG. U)fc
156
B A L A N C E D G R O W T H AND E F F I C I E N T
in Taylor's series around the von Neumann ray and by keeping only the linear terms. Unfortunately they made some careless, but fatal, mathematical mistakes in expanding the systems in Taylor's series. If the authors had linearized them in a correct way, they would have obtained the Turnpike Theorem only in some special cases, i.e. the systems could not have got rid. of certain 'cyclic' exceptions such as those discussed later. We can also give a counter example showing that their linear programming model is not free from that exception.(1) For their difference equation model, however, a correct but local proof of the Turnpike Theorem has recently been given by McKenzie [5] on an assumption that rules out the cyclic phenomena. After the publication of the book by Dorfman, Samuelson, and Solow, several Turnpike Theorems have been established by various writers, e.g. for the von Neumann-Leontief model by Hicks [3], Morishima [i], and McKenzie [3, 6], for the von Neumann-Gale model(2) with strongly super-additive production processes by Radner [ 1 ] and Nikaido [4], and for a generalize Leontief model with capital goods by McKenzie [4]. A model with a neoclassical transformation function has been examined by Lancaster [1] by use of classical calculus of variation methods. Section 2 of this chapter reproduces my paper [i], which was originally written as a companion to a paper by Hicks [3]. Section 3 elucidates cyclic exceptions to the Turnpike Theorem established in Section 2. Section 4 extends the results by liadner [1] and Nikaido [4j to a more general model where the condition of strong super-additivity is replaced by a more plausible one. We shall show that once such a generalization is made, the model will suffer from cyclic exceptions. 2. Proof of a Turnpike Theorem: the 'No-joint-production case' In this section, we are concerned with an economy consisting of n industries whose outputs are themselves used as factors I" Samuelson has privately told me that Harold Kuhn and David Gale have also discovered the cyclic phenomena. <"> See Gale [3J.
PROGRAMME
OF V E R Y - L O N G - R U N
GROWTH
157
to produce further goods. We assume (like von Neumann) that the role of the worker-consumer can be reduced to that of a farm animal and all income from property is saved/1' and (like Leontief) that there is no joint supply of outputs. The von Neumann model based on the assumption of no joint production is not a probable system. Because, in the von Neumann treatment of capital goods, the process of production which uses at least one capital good should inevitably be multi-product (see Chapter V). We shall also make other restrictive assumptions among which the assumption of primitivity will be farthest from reality. Under this set of assumptions, the Turnpike Theorem is shown to hold with prescribed (terminal) quantity ratios without any 'cyclic' exception. It will, however, be seen in Section 3 that, once we admit joint productions or the assumption of primitivity is removed, we must be confronted with the 'cyclic' trouble. In our model, which we may call the von Neumann-Leontief model, industry i can choose between mi different manufacturing processes or activities in producing good i. An activity, say the Sjth activity of industry i, is defined by an w-dimensional column vector, as. = {aj,, a|(,...,a"(}, stating the inputs of n commodities per unit output.(2> The total set of activities or the matrix of input coefficients is denoted by an nxm non-negative matrix:
n
where m = ^ mi- As it is assumed that there is no joint supply, ? =i the matrix of output coefficients (an nxm matrix) may be written as:
(1)
See Chapter V, section 1. > The input, aj ( , consists of the technological input and the 'wage input'. Thus it corresponds to the c^ in the previous chapter. (z
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the tth row of'which has, first, (m1-\-ma-\-...-\-mi_l) zeros, an then mi units followed by (»r m --)-...-f-m rt ) zeros. Next let yst be the output of good i produced by the stth activity of industry i, and pt the price of good i. Let (an m-dimensional column vector), and P = [pi,p-i,---,Pn\ (an %-dimensional row vector). Assume that the production period for each output and the length of life of each factor are one unit of time. The von Neumann model of economic equilibrium is then stated in terms of the following five inequalities: (1) (2) (3) (4) (5)
where / and g are (1 4-the rate of interest) and (1+the rate of growth) respectively. Inequality (1) makes activities profitless, (2) assures that unprofitable activities are not used, (3) implies that it is impossible to consume more of a good than is being produced, (4) makes overproduced goods free, and (5) requires that pricing and production schemes are such that something of value is produced/ 1 ' Let us now find solutions to (l)-(5). If each industry selects a single activity from among those available to it, there are m1xm2x...xmn possible sets of activities which could be adopted by the economy. They are arranged in a certain order and denoted by «, j3,..., fj.. Let a be the activity set (slts2,...,sn) in which industry i selects its s^th activity (i = l,...,n), and define: Since Aa is non-negative, it has a characteristic root ha which I 1 ' The condition (5) not contained in the original von Neumann model was imposed by Kemeriy, Morgenstern, arid Thompson [1].
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is non-negative and not Jess in absolute value than any other root.(1) Let Af — [a(,1,aC2,...,a(,J be an activity set such t hf 5^ ha (a = a,/?,...,/i); and let Ae's column and row eigenvectors (associated with the characteristic root he) be denoted by xc and P( respectively, i.e. hexc = Aexf and hePe — P(Af. Let xei be the ith component of xe, and let Ye be an m-dimensional column vector such that its s^th component is x f i w h e n si = e.t, and zero when st ^ et. We make the following two assumptions: ASSUMPTION 1. Ae is indecomposable. ASSUMPTION 2. Af is unique, i.e. there is no Ag other tha Ae such that kg = h£. As will be shown by Lemma 1 below, Ac is that activity set which corresponds to the vori Neumann configuration and generates maximum growth. Assumption 1 implies that when industry i selects its e^th activity (i = l,...,n), the system does not break up into separate subsystems, so that commodities belonging to a subsystem may be produced only from themselves. Thus, once the activity set e is adopted, no subset of industries can be found, which may infinitely grow at a common rate, independently of the rest of the economic system. On the other hand, Assumption 2 postulates that there can exist only one set of input coefficients—one activity in each industry— which is consistent with maximum growth. It is evident that these two assumptions are very restrictive. But we introduce them, the effect of which is to guarantee a unique von Neumann ray. We can now establish LEMMA 1. Let rl and r2 be any positive numbers. The system (l)-(5) has solutions such that Y = r l Y f , P — r 2 P f f ~ ff — h~l; conversely, ifY, P,f, g are solutions to (l)-(5), then Y = r±Yf, P = r2Pe, and f = g = h~l; i.e. the equilibrium output and price rays and the equilibrium interest and growth, rates are uniquely determined. (1
> See Debreu and Herstein [1], Schwarz [1], and Appendix.
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Proof. Let Ag be an activity set which does not contain any of the activities in Ae. We first show that (6)
Suppose the contrary, that some components of h ( P f , s a y t h e first j components, are greater than or equal to the corresponding components of Pe A%. Replacing the first j columns of Af by those of Ae, we get Aa such that (7)
Let kg be the maximal characteristic root of A$, and xg its column eigen-vector associated with kg. Since Ag is non-negative, xg is non-zero and non-negative; Assumption 1 implies that P( is strictly positive.11' Therefore, it follows from (7) that Hence he ^> kg. This contradicts the definition of Ae and Assumption 2. The relation (6), together withhfPf = PfAf, leads to where rz > 0. Since Y( is an m -dimensional column vector with the e^th components x (i = l,...,n) and zeros everywhere else, we obtain and
where r1 > 0. As rzPf > 0 and rlxf > 0, it is easily verif that (r 2 PJ£(r,7 (; ) > 0. Thus are solutions of (l)-(o). Let Y, P , f , g be any solutions to (l)-(5). From (2), (4), and (5) we immediately see that/ = g. Hence from (3) Since (6) holds, we get hePe B < PA, so that hePf BY < PfAY. Therefore (8) (1)
Debreu and Hersteiri [1, pp. 698-600] arid Appendix below.
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As I\ is positive, while Y is non-negative and non-zero, Pf BY must be positive. Hence f-1 ^> he. On the other hand, we have hf £Y( = AYe. Since £YC > 0, we get: 0 < htPBYe = PAYf. From (1) fo\\ovrsf-1PBYf <; PAYf. Therefore (9)
irom which it follows that he ^ / ~ . Hence hf = f . Once he = /~ 1 , (9) must hold with equality. Together with the definition of Ye and (1), this leads to keP = PAf. Since Ae is indecomposable, P must be proportional to Pe, i.e. P = r 1 P f . Similarly, (8) holds with equality. As (6) holds, we find that the s^th component of Y is zero for all st- ^ c £ . Hence heX ^ A€X, where X is an n-dimensional vector so chosen that its t'th component equals the e,.th component of F. As Pe > 0, hf X = J£Z; otherwise we would have a contradiction to (4). The indecomposability of Af implies that X is proportional to xf. Hence Y = rzYe. By virtue of Lemma 1 it may be justified to call xe and Pf the von Neumann output and price configurations respectively; he is called the von Neumann discount factor, and h~l the von Neumann growth factor. When an activity set Aa (instead of Ae) is adopted, the growth factor will be h~l. As he < ha, we find that the von Neumann activity set generates maximum growth. Let Y(t) be an intensity vector at time t—\ (an m-dimensional column vector), and X(0) an initial stock vector (an Ti-dimensional column vector). A sequence {Y(l),...,Y(T)} of intensity vectors is called feasible, given X(Q), if
Let x* be a given stock structure (an n-dimensional unit-sum column vector) at a prescribed point of time T. Let q be a number defined by q ~ min(.ri(T)/a;f), where x^T] is the ith i
component of BY(T), and x* is that of x*. As x* is non-
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negative and non-zero, q is non-negative; we have
A feasible path is said to be efficient if there is no feasible path { Y ' ( l ) , . . . , Y ' ( T } } such that q' > q. Thus an efficient path of outputs starting from X(Q) and reaching x* is found by solvi the following linear-programming problem (referred to as LPl): Find a non-negative set of values ( Y ( l ) , . . . , Y ( T ) , q ) such that h^q is as great as possible, subject to the requirements,
The dual linear-programming problem (referred to as LP2) is stated as follows: Find a non-negative set of values (-P(O), P(l),..., P ( T ) } which makes P(0)X(Q) as small as possible while satisfying the inequalities
By the duality theorem of linear programming, we find that if LP2 is solved, the minimum value P(0)X(Q) that is obtained is equal to the maximum value h^q associated with LP1.(1) We also see that the minimal solution to LP2 gives a time path of imputed prices corresponding to the efficient growth of outputs (the maximal solution to LP1). The following assumptions will be used in the proof of the following lemmas: ASSUMPTION 3. Ae is primitive. ASSUMPTION 4. For each good i there is at least one industry j which uses good i indispensably. <" See, for example, Charnes, Cooper, and Henderson [1, pp. 72-74], Dorfmari, Samuelson, and Solow [1, pp. 174-8], and Gale [4, pp. 78-82].
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Assumption 3 needs a little discussion. The nxn matrix Ae is said to be primitive if it cannot be transformed, by permutations of the same rows and columns, into the form
with square submatrices on the main diagonal.(1) Thus, if Ae were imprimitive, there would be simultaneous permutation of rows and columns by which A\ could be transformed into
Assumption 3, therefore, may be translated as follows :(2) 'If this assumption did not hold, the commodities in the system could be divided into (say) k mutually disjoint groups, such that the commodities of the first group produced only those of the second, those of the second only those of the third (and so on), until the commodities of the kth group produce only those of the first group; so that after k periods the inputs of commodities of any group would be converted, independently of other groups, into outputs of commodities of the same group. Accordingly, though the system is indecomposable, so that in any single period it does not fall apart into sub-systems, we could make it decomposable, if we were prepared to construct new period that consisted of k of our previous periods, and look only at the initial inputs and final outputs of these compound periods. Thus we may say that while Assumption 1 rules out simple independence, Assumption 3 rules out compound independence also.' Let (P(0),P(l),...,P(T)) be the minimal solution to the dual linear programming LP2 stated above. Let a;.( be an activity of industry i such that P(t)ajt = min[P(«)alj, P(t)a2t,..., P(t)ami]; O See Debreu and Herstein [1, pp. 599-600]. (=) Hicks [3, p. 88].
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of course, ji depends on l'(t); we therefore denote [a}i, ah,..., ajn] by At. We have LEMMA 2. Let the, initial stock vector X(0) and the terminal output configuration x* of the original problem LPl be strictly positive. Then the solution, P(0), P(\),..., P(T), of the dual problem LP2 satisfies Proof. We first show that if there are no zero outputs in the final period, there can be no zero outputs in any period. Let (Y(l), F(2),..., F(T),g) be the maximal solution to the problem LPl. As X(0) > 0, q > 0. This together with x* > 0 results in BY(T) > 0; so that for each j, at least one ysj is strictly positive. On the other hand, Assumption. 4 implies that for each i there is at least one j such that a* > 0 for all s;- = l j , . . . , HJ. Hence, AY(T) > 0, which in turn, implies BY(T-l) > 0. By the same argument it is shown that BY(t) > 0 for all * = • ! , 2, ..., T. It results from the duality theorem of linear programming that price equals cost unless the corresponding output is zero. Therefore, the fact established above implies that equality of costs and prices holds throughout. That is, for each i (i = 1, ..., n) and each t (t = 0,1,..., T—1), there exists at least one au such that P(t)ai{ = pt(t-\-l), where Pi(t4-l) is the ith component of P(t+l). From the constraints in LP2 we at once find that Hence, for each i, that P(t)(tjt which equals p[(t-{-l) is the smallest of P(t)alt, P(t)a2.,..., P(t)am.. LEMMA 3. Let alt be an activity of industry i such that and denote [a^,«,„,...,a ( J at time t by At. Consider a dynamic equation: 10
Any path (10) starting from an arbitrary non-negative position
P R O G R A M M E OF V E R Y - L O N G - R U N G R O W T H P(0) ^ 0 eventually approaches the von Neumann price ray,
165 i.e.
(11)
where Pi(t) and pei are the i-th components of P(t) and Pf respectively.1^ Proof. We prove the theorem by a method similar to that which Solow and Samuelson used to show the relative stability (in the large) of the balanced growth of outputs.®. Let Pe be an n x n diagonal matrix whose diagonal elements are pfl,..., pfn, ana write
(12) because A~'P;~ I> 0. On the other hand, recalling (6), we can vprifv tVia.t, (13) where an %-dimensional vector. Next, define the two sequences and 1
we have trom (12) and ( i d ) Therefore, C(t) ^ C(t+l). Similarly, from (13) and (10) which implies c(t) <± c(t-\-l). Thus C(t) is never increasing, c(t) never decreasing, and C(t) >_ c(t), so that the two limits exist:
W Theorem 2 of Chapter IV is concerned with a similar dynamic process gullet ttctJU uy
At nrst sight, (1U) may lomnally be considered as a special case or the above equation where all La'a vanish. There exists, however, a noticeable difference between them: Assumption 1 of Chapter IV reminds us that the HawkinsSimon conditions hold for at .least one ofA^ + rB^ A^ + rB^, while Lemma 3 above is completely independent of them. (2) Solow and Samuelson [1, pp. 419-21]. 828145
M
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Now suppose that C > c. Given any positive number 8, we consider the following n row-vectors: cl = [c, (7+8,..., <7 + S], ca = [C+S,c,C+8,...,C+8],..., cn = [C+8,...,C+S,c]. Corresponding to S, we can find a t so large that Z ( t ) O.(1) Taking account of the fact that Pf is a diagonal matrix whose diagonal elements are all positive and he > 0, we find that Therefore, if § is taken very small, each component of the extreme right-hand side of (14) is less than C', because C > c. Hence we finally get contradicting the fact that C(t-{-v) I> C. Hence C = c. This means lim zt(t) = c. t-^x,
The positiveness of c is shown as follows. It is evident that c is non-negative. We get from (10) An argument similar to the proof of Lemma 1 yields Hence, As Pfhexe > 0 and Z(0) _^> (=£)0, Z(t) cannot converge to 0. Thus limzt(t) = c > 0, i.e. (11) is obtained. <->CO
Lemma 3 may equivalently be stated as follows: Let N be a prescribed neighbouring cone of the von Neumann price ray, and let P(t, P(0)) be the path (10) starting from a point P(0); for P(0) > 0, there exists a finite integer t(P(Q)) (dependent on N) such that, for all integers t being not less than t(P(Q)), P(t, P(0)) remains within N. Let <(P(0)) be the smallest of such (1
> See Gale [4], chapter 8.
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integers. It gives the time required for P(t, P(0)) to come into N; the smaller the integer t(P(0)) is, the greater the speed of convergence of P(t, -P(O)) to the von Neumann price ray becomes. Let us now sharpen Lemma 3 to the following lemma establishing the uniformity of convergence. LEMMA 4. For any specified N and any point P* =£ 0, there exists a neighbourhood of P* such that, for all t being not less than tj. (an integer dependent on N and P*), the path P(t, P(0)) starting from any P(0) =t 0 in the neighbourhood of P* remains within N; that is, i(P(Q)) <1 t^for all P(0) in that neighbourhood of P*. Proof. Let A{ be any activity set containing none of the activities in A f . S i n c e h f P e — P e A e , a n d P e A f < . P f A g ( s e e (6)), we have PfAse < PeAs£-1A^ for all s (s = 1,2,..., ad inf.); therefore, there exists a neighbouring cone H of the von Neumann price ray Pe such that for any point P in H,
(15) where r is an integer to be defined below. Let P(t, P*) be the path (10) starting from P*. By Lemma 3 we can find a minimum integer t1 such that when t ^L tlt P(t, P*) always belongs to the interior of H. Let A *, A*,..., A * _l be activity sets (square matrices) so chosen from among possible activity sets Aa (a = a,^8,...,//) that:
(16) where
Next let
then a finite sequence (A
where P(0, P(0)) = P(0). Note that A°, A\,..., A f i _ 1 a r e a l s o chosen from among An (a = a, 5 M). From (16) we obtai (18)
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and from (17)
(19) Let us now classify all possible activity sets Aa (a = a,/?,..., m) into two groups, G* and Gt: any activity set Aa such that P(t, P*)Aa = P(t, P*}A* is a member of G*, and all other activity sets belong to Gt. Suppose A\ belongs to Gt. As P(t, P*)Af ^ P(t, P*}t and P(t, P*)Af £ P(t, P*)A<}, wecanAn show that if P(t, P(0)) is sufficiently near to P(t, P*), then where Q ^ R means that at least one component of vector E exceeds the corresponding component of vector Q. This contradicts (17). Hence, if
(20) In the following, P(0) is taken so near to P* that (20) holds for Since Irom 20) that
it follows
Since P(t1: P*) belongs to the interior of H, it is seen from (18) and (19) that if the norm of the vector 8 is sufficiently small, P(tv P(0)) also belongs to //. Once P(tlt P(0)) is a member of H, we have, from (15),
From this we find that Ae is the cheapest activity set at P(^P(0)), P(^+1,P(0)),..., P(VH'~-1,P(0)), such that P(^ + s,P(0)) = P^+s-l.P^A,
(s = l,...,r).
It is clear that the finite sequence {P(t, P(0))} (t = t1,...,t1-\-r) obtained in this way satisfies equation (10). Let us now consider a linear difference equation:
(21) A little calculation will confirm that for any neighbouring cone,
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K, of the von Neumann ray there exists an integer tf(K) > 0, such that for t ^> te(K), any path (21) starting from an arbitrary initial position stays within K.m Let K be contained in N, and take it so small that PA( < P-4^ holds for any P in K and for any A% containing none of the activities in Af. Put r = tc(K). We then find so that, from (10), P^ + r + l , P(0)) = P(tv + r, P ( 0 ) ) A f . Hence P(£]-f-r+l, P(0)) belongs to K. In repeating this procedure it is found that Pfa + r', P(0)) belongs to K for all r' ;> r. Thus, if P(0) belongs to a sufficiently small neighbourhood of P*, P(«, P(0)) stays in K (and hence in lY) when t^t* = t^+r. Hence <(P(0)) ^ «*. LEMMA 5. Le£ (0). T/ien every component of Py(0) does no£ gro to infinity even if T tends to infinity. Proof. Let Pe be an eigen-vector of A f such that he Pe = PeAe and Pe .T* = 1. Let
(22) From (6) we find that (22) is a feasible solution to LP2. Clearly PT(0)X(0) 0) to the terminal ray x* (> 0) will remain most of the period T within '" Let te(P) be the smallest value of te(P) such that, for t S t((P), the path (21) starting from the initial point P remains within K. Let P' = AP, where A is any positive scalar; then we find that tc(P) = lt(P'). Write P1 = [1, 0,..., 0], P2 = [0, 1, 0,..., 0],..., P« = [0,..., 0, 1], all of which are n-dimensional row vectors. Let tt(K) be the maximum among ^(P1),
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an assigned (arbitrarily small) neighbourhood of the von Neumann output ray. Proof. It is seen from Lemma 2 that for t <^ T the solution P(t) to LP2 coincides with the solution of (10) from the initial price set PT(0). We can show that PT(0) ^ 0 for all T. By virtue of Lemma 5 we can take, within the non-negative orthant, a large cube F which contains all P2.(0) (T = 1, 2,...). Let us assign to each point P* of F a neighbourhood U(P*) for which the assertion of Lemma 4 is valid. Then F is completely covered by the totality of these U(P*)'s. As F is closed and bounded, it can be covered by a finite number of U(P*)'s, say [/(P,*) (i= 1,2,...,£).<« Denote <„. associated with P* in Lemma 4 by t^P*), and let T* = max(^(P*),^(PJ),...,< :(! (P^)). It is clear that for t ^> T* the path (10) starting from any initial point P(0) of F remains within the prescribed neighbouring cone K of the von Neumann ray. Hence for a sufficiently large T, P(t, P T (0)) remains within K for t in [T^., T]. (Thus the uniform convergence established by Lemma 4, together with the boundedness by Lemma 5, enables us to determine a finite time, that is independent of the length of the programming period, such that after that time the sequence of prices will be within the prescribed neighbouring cone of the von Neumann price-ray.) As PAe < PA^ in K, it follows from the duality theorem of linear programming that, in the solution to LPl, yfi(t) corresponding to any activity other than those in Ac is zero for all t in [T^ + l, T]; that is, the same activities as on the von Neumann ray will always be chosen after a finite time T%. Hence
where X(t) = {yei(t),ye!(t),...,yen(t)}—an n-dimensional vector. Because of the fact that the von Neumann price vector Pe is positive, it is always possible to take K so small that any point P of K is strictly positive. Then P(t,PT(0)) > 0 for all t in [T*, T]. Therefore, it follows again from the duality theorem (*) By the n-dimensional version of the Heine-Borel theorem. See Lefschetz [1, pp. 34-36], also.
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of linear programming that
(23)
Let us now write (24)
Then (23) can be written in the form: (25)
where W(Q) = x*. As Af is indecomposable and primitive, the same argument as that of Lemma 3 can be applied: The solution to (25), if the range of s is extended without limit, converges to the von Neumann output ray. i.e. there exists a finite intege such that, for all s being not less than that integer, W(s) remains within an assigned neighbourhood of the von Neumann ray. Hence when T is sufficiently large, the solution to (25) remains within the neighbourhood for all s in [v'71, T—7^—1]. This, together with (24), implies that the efficient path X(t) will remain, for all t in [J^-fl, T— \T], in the predetermined neighbourhood of the von Neumann ray. Furthermore, note that lim [(T—^'T) — ( T ^ + l ) ] / T = 1. This completes the proof. T-s-co '
3. Cyclic exceptions In proving the Turnpike Theorem above, we have assumed that the von Neumann activity set Ae is primitive (or acyclic). It is clear from general economic considerations that this assumption is not a highly probable condition. For example, the two-sector system where the sole input of each sector is the output of the other sector is imprimitive. As will be shown below, however, the assumption of primitivity is necessary for the theorem. Let us suppose, for simplicity's sake, that each industry has only one activity available to it, i.e. A = Af and B ~ I. Suppose also that A is imprimitive (or cyclic). Our linear programming LP1 is now stated as:
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Maximize Ji,Tn subiect to (26)
where h is the von Neumann discount factor, and X(Q) and x* are given positive vectors standing for the initial capital stocks and the terminal output configurations respectively. Let Wq be a maximum at q = q,r. Consider a difference equation: (27) Since A is non-negative, it iollows irom (26) and (iJ7) that the path W(s) starting from W(0) = x* satisfies qTW(s) <1 X(T—s) (s = 0,1,..., T). From the relation qTW(T)
Thus the sequence is a maximum solution to our linear programming problem. (29) Next write Then (27) can be written (30)
It is clear that the dominant characteristic root of the nonnegative matrix h~lA is 1. As h~lA is indecomposable, but imprimitive it has at least two characteristic roots of absolute value 1, one of them being 1 and the other not l.(1) All other characteristic roots do not exceed 1 in absolute value. Hence V(s) determined by (30) will oscillate around the von Neumann configurationxf = {xel,...,xen}, and the oscillations are not damped. <»> See Dobreu and Herstein [1, pp. 599-600].
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It is at once seen that (28) and (29) yield This implies X^/x^) = I'C-T-tyv^T-t), where x^t) is the ?'th component of X ( t ) , and v;(T — t) that of V(T — t). As V(s) traces out undamped oscillations around the von Neumann configuration,X ( t ) / x i ( t ) m d o e s n o t a p p r o a c h xjxfi, however large we make T. Thus, the Turnpike Theorem does not hold when A is imprimitive. Finally, the following example will serve to illustrate that a system where outputs are jointly produced cannot get rid of the cyclic exception, even if the matrix of input coefficients A satisfies Assumptions 1-4. Let(2)
Then both the von Neumann growth factor g and the von Neumann output ratio xcljxe2 are \'l-5, and the von Neumann price ratio PJPeZ is l/v'1-5. Let X(0) = {1,1} and x* = (0-5,0-5}. In case T is even, the efficient path X ( t ) will be:
If T is odd, it will be
It is evident that A a (<)/A 2 (/) does not converge to the von Neumann output ratio v'1-5, however large T is taken; we have undamped oscillations around the von Neumann configuration. (3) From this example and the above discussion of <» Note that X(t) depends on T and t. ('-> It is noticed that
an imprimitive matrix.
(3)
This results from the imprimitivity of the matrix B~1A. Note,' however, t h a t the average rate of growth over T periods is 1-S 1 ' 2 — 1 if T is even, and ] .-,(3'--i);27'__ j if rj' is 0 Jrt p n. approaches the von Neumann rate l-S 1 ' 8 —1 as T becomes infinite. Thus even in cyclic eases the Turnpike Theorem holds in (Continued on next page)
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imprimitivity we may conclude that the Turnpike Theorem can only hold for extremely special von Neumann models. 4. A generalization of the Radner-Nikaido theorem: cyclical growth around the turnpike Let us now turn to the Radner-Nikaido theorem, which states that any optimal path must remain near the von Neumann ray except possibly for a finite number of periods in the beginning and in the end, the number being independent of the length of the path. (1) This theorem differs from the previous Turnpike Theorem in the following two points. First, in Section 2 we have been concerned with a system where the desired proportions in which outputs of various commodities are to emerge at the end of T periods are specified, while Hadner and Nikaido dealt with a system where, instead of the proportions of the commodities at the terminal, a well-defined function u(y] of the final states y was given, which described the society's preferences among the final states of all feasible paths. In this respect, however, our system may be reduced to a special case of the Radner-Nikaido system, because to find a solution to the linear programming LPl in Section 2 amounts to maximizing the preference function of the form, u(y(T)) = max{g ! qx* <^ y(T)}, subject to the requirements of that problem, where x*,..., x* are the desired proportions of the n goods, and y(T) is the output vector at the end of T periods, i.e. y(T) = BY(T) by the previous symbols. Second, the Radner-Nikaido analysis (based on the assumption that the von Neumann balanced growth path is the unique profit-maximizing direction of growth under the von Neumann equilibrium prices) applies to the case in which production processes are strongly super-additive. (2) As Radner explicitly the sense that the average rate of growth along the efficient path leading to a prescribed terminal output configuration will approximate to the von Neumann rote of balanced growth as the programming period T is very long. A conversation with Professor Georgescu-Roegen brought this to my knowledge. (11 Radner [1] did not say what parts of the path the exceptional periods belonged to. The specification, was made by Nikaido [4]. < 2 ' The definition of strong super-additivity will be given below.
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recognizes, however, it does not apply to the production set which is a polyhedral cone (the case treated by von Neumann and including our model in Section 2 as a special case). In fact, let A and B be matrices that fulfil Assumptions 1—4. Let e = (gj, ez,..., en) be the von Neumann activity set. Denote the von Neumann output and price configurations by xe and Pf respectively, and the von Neumann interest factor by /. Radner's assumption implies, in this case, that PfBY—fPeAY be strictly negative for all Y I> 0 that are not proportional to Yf, where1 Yf is a vector such that its ^th component is x(i if st = ei and zero if si =£ e,t. It follows, however, from the definition of Pf and/ that Pe BY-fPe AY = 0 for all Y = {?/!,,..., 2/m,} such that ys. = 0 for st =£ e{. Thus the case discussed in Section 2 cannot be covered by the Radner-Nikaido theorem. In this section we mainly follow the argument by Radner and Nikaido but depart from it in the following respects. First, we assume that u(v) > u(y) for any vectors v and y such that v > y ^> 0 (Assumption 13 below). It is true that this assumption is stronger than Nikaido's (1) and rules out the case of fixed desired proportions, u(y) = max{g qx* <^ y}, which is covered by the latter. But from the economic point of view it may be accepted as a very reasonable assumption to be made. Second, the strong super-additivity of processes means that from any two different processes which produce the commodities in the amounts y = {y\,...,y,} and y* •= {y*,...,y*} respectively, we can form a third process with the output vector y** such that y** > y-\~y*- This is so implausible a property that actual production sets may not fulfil it; in fact, it is not realistic to assume that any two processes producing entirely different commodities, say a process producing rabbits and a process producing airplanes, can be amalgamated to form a third process which is more effective than the simultaneous operations of the two. In the following, therefore, we shall replace the assumption of strong super-additivity by a weaker and more plausible assumption, i.e. by Assumption 10 below. Finally, we assume that commodities are partitioned into m groups, so that (1
> He merely assumes that v > y implies u(v) > u(y).
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commodities of the first group produce only those of the second, which in turn produce only those of the third and so on, until commodities of the mth group produce only those of the first (Assumption 9 below). When the commodities cannot be partitioned into groups, m is considered to be unity. Under these assumptions, we shall observe cyclical growth (with period m) around the von .Neumann ray, and it will also be seen that the Radner-Nikaido Turnpike Theorem is obtained in the special case of m = 1. It should be mentioned that our analysis below much depends on the results obtained by Gale [1, 3], Karlin [2], Furuya and I'nada [1], and Fisher [1] as well as those by Radner [1] and Nikaido [4]. The model to be considered involves n goods, numbered from 1 to n. At the beginning of each period t, some amounts x^t),..., xn(t) of the various goods are used as inputs, and at the end of the period amounts ?/,(£+ !),•••> J/»('+1) are produced as outputs. Denote column vectors {x 1 (<),...,x rt (<)} and {j/j(i-|-!))•••>?/»('+1)} by x(t) and y(t-\~l) respectively. The set of technologically possible pairs (x(t),y(t-\-l)) is assumed to be the same for every period, and is denoted by //. When no confusion is expected from the context, we simply write (x, y] e H without letters t and t-\-l denoting dates. We make the following assumptions. ASSUMPTION 5. If (x, y) e //, then x = 0 implies y = 0. ASSUMPTION 6. H is a closed cone in the non-negative orthant of 2/i-dimensional Euclidean space RZn, that is, (i) H is closed in RZn, (ii) if (x,y) e H, then x ^> 0 and y ~,> 0,(1) (iii) if (x, y) e //, then (A.r, \y) e H for any A ^> 0. ASSUMPTION 7. For any x ^> 0, there exists a y I> 0 such that (x,y) e //. ASSUMPTION 8. If (x, y) e H, then x* ^> x and y* <^ y imply (x*,y*)eH.™ Assumption 5 precludes the possibility of the Land of (1)
For definitions of ' § ' and ' > ', see p. 5, n. 1. < 2 ' Assumptions 5 and 8 imply Assumption 7. Nikaido [4] has shown that the Radner-Nikaido theorem holds \vithout Assumption 8.
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Cockaigne, so that a positive output requires some positive input. Condition (i) of Assumption 6 states that every limit point of// is a point of H; (ii) is obvious; (iii) asserts that ther are constant returns to scale. Assumption 7 makes possible a production with any initial endowment of available stocks. Assumption 8 implies that disposal activity is costless. Let us now divide the n commodities into the following m mutually disjoint groups: l,...,^; w t -f 1,..., nz\ ...;nm_l-\-l,...,n. Write x(i) = {xn^1+l,...,xn,} and y(i} = {y m _ 1 + 1 > ...,y n t }, where i = ] , . . . , m, nu = 0, and nm = n. The vectors x and y may then be partitioned into {x(l),x(2),...,x(rn}} and {y^y^-^M} respectively. The following assumption will play an important role in a later discussion of cyclic exception to the Turnpike Theorem. ASSUMPTION 9. The commodities of each group i produce only those of the succeeding group i-f-1 (i = ],...,m — 1), and the commodities of group m produce only those of group 1; that is, if (a'd),...,.r (ffl) ,y (1) ,...,w (m) ) e H, then
(31)
The assumption of strong super-additivity made by Furuya and Inada [1] states that, for any (x, y) eH and (x*,y*) eH such that x ^ 0, and x* ^= Xx for any non-negative A, there is a process (x-\-x*,y**} e H such that y** > y+y*', that is, any two processes can jointly be operated to form a third process in which the output for any commodity is not less than the sum of the outputs for that commodity in the two given processes, with at least one good whose output in the third process is greater than the sum of the corresponding outputs in the two. In the following, however, we assume that the strong superadditivity prevails between two different processes with common products; but for those two which have no common product we merely assume that they can simultaneously be operated without either one of them affecting the outcome of the other.
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More exactly the assumption may be stated as follows: ASSUMPTION 10. Let and
be any two processes with at least one subgroup j such that, for any non-negative A j; (£*•_!),?/*•,) ^ A^,^..-^//,;,). If both yi and j/fare positive for { = ilt..., irbelonging to the sainegroup j, there exists a process (x-\-x*,y**) e H such that y** ^> y-{-y* and y** > ?/,- + ?/* for at least one i in the set {ilt...,ir}. But if it is empty, i.e. if processes ( x , y ) and (x*,y*) have no common products in group j, then (x-\-x*,y-4-y*) e //. This assumption, together with Assumption 8, implies that (x-\-x*,y-\-y*) e // for any ( x , y ) e H and (x*,y*) e H; that is, the set H of all technologically possible processes is convex. Let us consider a time sequence of successive transformations ( x ( t ) , y ( t - \ - \ ) ) e H (t = 0, l , , . . . , s — 1 ) such that all outputs produced at the end of each period t—1 are used as inputs in the next period t, i.e. y(t) = x(t) (t = l , . . . , s — 1 ) . After s periods the initial inputs x(0) will be converted into outputs y(s). Such a compound transformation of x(0) into y(s) may be called a transformation of order s, and the sequence ( x ( t ) , y ( t - \ - l ) } (t = 0, l , . . . , s — 1 ) a feasible path of the same order. The set of technologically possible transformations of order s is denoted by Ha. Throughout the rest of this section we shall, for simplicity, denote vectors {0,..., 0,.ra), 0,..., 0} and {0,...,0,y(i),0,...,Q} by the corresponding capital letters X(i} and Y(i). By definition,
It is seen from (31) that a time sequence, (^ (1) (0), F (2 )(l)) (XM(l),Y(3)(2)),..., (XM(m-l), Y(l)(m}}, such that is a feasible path of order m. We have (X{l)(0),Y(1](m)) similarly,
e Hm;
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That is, for all i = 1,..., m, (0,...,0,x(i}(0),0,...,0, 0 , . . . , Q , y ( i } ( m ) , Q , . . . , Q ) e H m . Thus, if Assumption 9 is fulfilled, the compound production set of order m, Hm, is decomposable; that is, the commodities can be divided into m groups, such that the initial inputs of commodities of any group are converted, independently of other groups, into the final outputs of commodities of the same group. According to the traditional definition, the technological possibility set // is called primitive if and only if any non-zero input vector can yield a strictly positive output vector after at most some fixed number of periods, i.e. if and only if, for any x(0) > 0 and for any integer t being not less than some fixed integer Jc, there exists a transformation of order t, (x(Q),y(t)) e H', such that y(t) > 0. The primitivity of // has explicitly or implicitly been assumed by various writers. But it is incompatible with Assumption 9 (unless m = 1), because, as is easily verified, the latter implies that (Xli}(0),Yti)(s)) e H*, where s is any multiple of' m. Thus the traditional primitivity assumption amounts to postulating that commodities cannot be divided into cyclic groups, i.e. m = l. (1> Throughout the rest of this section, we accept Assumption 9 for m ^> 1, and make the following assumption which is weaker than the traditional assumption of primitivity. ASSUMPTION 11. For each i = \....,m and for any X(i)(0) > 0, there is a transformation (X(i)(Q], Y(i)(s)) e IIs such that t/(,-)(s) > if s (a multiple of m) is not less than some finite number &. (2) As is seen immediately, this amounts to postulating that for each i = 1,..., m, the set //$, of all (.c(l.)(0),«/(l.)(w,)) such that (0,..., 0, .rw(0), 0,..., 0; 0,..., 0, y(i)(m), 0,..., 0) e Hm is primitive. The expansion factor a of the process (x.y] e H is defined by the largest value of A such that y ^> \x; a is defined for any process (x, y) ^ 0 and is a non-negative number. It is evident I 1 ' Footnote 2, p. 182, below shows that the assumption of strong superadditivity also implies m = 1. |2 > It is noted that k can be selected independently of X^(0) and i.
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from the definition that y^ = aXj for at least one non-zero component Xj. Let H be the set of all (x, y) e II such that
It is clear that the origin of the 2n-spa.ce does not belong to H and that by Assumption 5 no point (0, ?/) is an element of //. Therefore, the factor <x is well defined on H. Moreover, it is seen that a is continuous at every point of H. As H is closed H is compact, so that a attains a maximum a on H. As H is a cone, we find that a ^> a at all points of the whole production set. The maximum expansion factor a will be referred to as the von Neumann expansion factor, and any process (x, y) e H satisfying y ^> Sex as a von Neumann process. Under Assumption 9, x and y_are grouped into {f(1),...,.r(,nj} and {?/(!),.••,?/(,„)}• By writing X(i) = {0,...,0; ;r(i),0,..., 0} arid Y(!) = (0,..., 0,y(a, 0,..., we obtain
We now define: DEFINITION 1. Let J(x, y} = [j\xi — 0 or yt = 0}. T set H is called indecomposable at (x,y) if J(x,y) is empty or if J ( x , y ) contains at least one j such that x^ > 0 or y^ > 0. Our final assumption on H is: ASSUMPTION 12. H is indecomposable at every point (x, y}. In other words, there is no division of the n commodities into sets J ( x , y ) and J ( x , y ) so that for any j e J ( x , y ) both x^ and y.j vanish, but for any i e J(x,y) both xi and yi are positive.(1) It is noted that this assumption is weaker than the von Neumann assumption that every good is either consumed or produced by every process, i.e. if xi = 0, yi > 0, and vice versa. We can now prove the following lemmas. LEMMA 6. Let (x, y} be a von Neumann process; then y = otx and x > 0. There is no other von Neumann process (x*, y*) such that x* ^ Xx for any non-negative A. (1
> J(x, y) is the complement of J(x, y).
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Proof. Partition x and y into {x(l},...,x<m)} and {&D,. ••,$„,)} respectively, buppose (32) By Assumption 9, x(l) produces y(2), x(2) produces y^, and so on. By feeding back y^ as inputs we can produce the commodities of the third group in the amounts a$(3), because it follows from j7(2) ^> aX(Z), (-X"(2), F(3)) e J/, Assumption 6 (iii) of homogeneity, and Assumption 8 of free disposability that (F(2),aF(3)) e H. Similarly, aj/(3) produces a2^), and so forth, until the commodities of the first group are produced in the amounts otm~ly(DWrite F(1)(m) = S"l^Y(l). We have, from (32), F(1)(m) > 5tmX(1). Thus there is a process of order m, (X{1}, y(1)(m)) e Hm, such that F(1)(m) > M(1). Next, define X*l} as .X*) = Y{l](m)~-~5imX(l}; we have X*D > 0. By feeding back S"1^) as inputs Ave can, after m periods, produce the commodities of the first group by the amounts 5m!/(1)(m). Similarly, after s periods (s being a multiple of m) we have the outputs oiey(l)(m); that is, (ctm.Ya), asF(1)(w)) e Hs. On the other hand, Assumption 11 implies that for a sufficiently large multiple s of TO, there is a transformation (Jf* ) ,F( 1 )(s)) e #s such that 2/(i)(s) > 0. As the convexity of the set(1) Hs follows from the convexity of H, we get (F(1)(m), a.sYti)(m)+Y(l}(s)} e Hs. Together with (X(l), F(1)(m)) e //'", this leads to (J(1),F(1)(s+m)) e Fs+m, where Y(1)(s+m) = <JsF(1)(m)+r(1)(s). As F(1)(m) ^ amJ(l) and yd)(s) > 0, we find ^(s-f-m) > a*'+mf-(1). We may, therefore, choose 9 > 0 so that (33)
Let (X(l}, Y(z)), (X(2},Y(3)),..., (X(m},Y(1)(m)), (X(l)(m),Y^(m}},..., (X.(m)(s),Y^(s-\-m)) be a feasible path of order s-\-m starting from -X(1) and terminating in Y(l}(s-\-m), where
(34)
d' That is, if (x, y) E Hs and (a;*, y*) 6 H3, then (x + x*, y + y*) 6 H'. 828145
N
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r being any multiple of m which does not exceed s. Define
(35) As H is a convex cone, we find that (x,y) e //. From (33), (34) and (35) it follows that y I> (a-f-0)a;. Thus there exists a process (x,y) which has an expansion factor not smaller than a+0. This contradicts the fact that 5 is the greatest possible factor. Hence Y^ = 5-X(1). Similarly, Y(fl = aiX^ for other i. Therefore, y = Six. By definition, a ^> 0. By virtue of Assumption 5 it is verified that 5 = 0 implies Y^(s) = 0 for all multiples s of m, provided that (.X"(j), Y^s)) 6 Hs. This contradicts Assumption 11, so that 5 > 0. Hence yi = 0 if and only if xi = 0. Therefore, if J ( x , y ) = {j x-j = 0 or y^ = 0} were not empty, both Xj arid y^ would vanish for all j in J ( x , y ) . This is a contradiction to Assumption 12. Hence J(x,y) is empty; that is, x > 0. Finally, let (x,y) and (x*,y*) be two von Neumann processes such that (x*,y*) ^ X(x,y) for any A ^> 0. Then there is an i such that (a-f,-), 2/* +1) ) ^ A^^,^^!,) for any A; ^ 0. Because if (£*>,2/* +1) ) = A^^.^+D) for all i, but A; =£ A i+1 for some i, then (a;*, y*) cannot be the von Neumann process. As y > 0 and y* > 0, Assumption 10 assures that there is a third von Neumann process (x-\-x*,y**) e H such that y** >y+y* = a.(x-}-x*). Clearly, this contradicts the above argument; the von Neumann ray is uniquely determined. LEMMA 7. Let (x,y) = (x(l),...,x(ni), y(l),...,y(m}) be the (unique,} von Neumann process. (I) There exists a price vector p > 0 such that p'y = 5.p'x, and p'y < J ' In this section, primed letters denote row vectors. ' 2 > TO+1 is interpreted as 1. This convention will be retained throughout the rest of this section. We can see that if the strong super-additivity of processes prevails, then the commodities cannot be partitioned into groups, i.e. m — 1. Suppose the (Continued on next page)
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Proof. We define the set C = {c c = ax—y, (x,y) e H}. As H is a closed convex cone in 2w-space, it is verified that C is a closed convex cone in w-space. Let C+ be the positive polar of C, i.e. the set of all vectors p such that p'c ^> 0 for all c E C. We shall first establish the following proposition due to Gale [1]: (A) If C+ contains no p > 0, then C contains a vector c < 0. We shall use the following properties of convex cones:
where C and D are convex cones in m-space, C-\-D is the set of all vectors expressible as the sum of vectors from C and D, and N is the negative orthant, i.e. the set of all vectors v <^ 0. The property (i) is verified as follows: By definition, (C-\-D)+ is the set of all vectors w sucli that w'(c-}-d) ^> 0 for all c e C and d e D. As 0 e D, any w e (C'+D)+ satisfies w'c ^> 0 for all c e C; that is, iv e C+. Similarly, any w e (C-\-D)+ belongs to D+. Hence w e C+ n D+. Next suppose v e C+ n Z>+. Then *>'c ^> 0 for all c e C and i/cZ 2?_ 0 for all d e D. Therefore v'(c+d) ;> 0 for all c e C and d e D, i.e. v e (C+D)+. In order to get (ii), we first show C++ D C. Let c e C and w) e <7+. By definition, i^'c ^> 0 for any c and w, and 61++ is the set of all vectors v such that v'w I> 0 for all w; E C*+. As w'c = c'w ^> 0, c 6 (7++. Next we show that the converse is true, i.e. C++ c C. Let 2 be a vector which is not contained in C. As C is a closed convex cone, there is a hyperplane h'x = £ contrary, m > 1. As (A" (1 |, ?(%)) eH and (^( a ), ?< 3 )) e H, the strong euporadditivity implies the existence of a process (^"^j-f-A'( 2 ), y] e / / such that y > ?(2)+-^(3>- By Lemma 7 (I), wo have p'?/ £ 5;1'(^?(,) + X'( 2 )), and by Lemma 7 (II), ju'P( 2 ) = a-jj'X^ and j5'F(D) = dj)'X( 2) . Therefore, This, together with •(/ & J'(2) + ^"(3) 5 0 &nd j" > 0, implies y — y(2i+P(3), a contradiction.
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separating z and <7,(1) so that If h'c < 0 for some c e C, we would have h'\c < £ for Ac e C1 with sufficiently large A. This is a contradiction, so that we have h'c ^> 0 for all c e G. Therefore h e C+. Since it is clear from 0 e C that £ <^ 0, we have h'z < 0. As h e C+, we find z £ C++. Thus if 2 £ C1, then 2 £ C++. The contra/positive of this result implies C++ c C. Hence C++ = C".<2> The property (iii) is clear, because N (the set of all nonpositive linear combinations of the unit vectors) can alternatively be described as the set of all vectors whose inner product with all v e N is non-negative; the latter is N+ by definition. Let us now proceed to prove (A). By hypothesis C+ contains no vector p > 0. Therefore, C+-\-N also contains no p > 0. As C++N = (C+ + N) + + by (ii), this means that (C++N)++ is not the whole space. This, in turn, implies that(C+-^-N)+ contain a vector c 7= 0, because othenvise (C'++A r ) 4 + would be the whole space. We obtain from (i), (ii), and (iii) Hence, the fact that (C+J\~N} + a c 7^ 0 implies that C contains a vector c < 0. The contrapositive of (A) may be stated as follows: If C contains no c < 0, then C+ contains p > 0. From Lemma 6 we at once find that C contains no c < 0. Hence, there is a price vector p > 0 such that pc ^> 0 for any c e C. It is clear that pc = 0 when c = ax~y = 0. The proposition (I) is thus established. By Assumption 9 all (X^, Y(i+l)) (i = \,...,m) belong to H. We, therefore, have (36)
Taking
into account, we
By the separation theorem : Let C be a convex set. If a point z does not belong to (7, then there exists a hyperplane which contains z in one halfspace produced by the hyperplane and all of C in the other halfspace. Sec, for example, von Neumann and Morgenstern [1, p. 134], and Koopmans [2, p. 29]. < a > See Nikaido [2].
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find that inequalities (36) must hold with strict equality. Hence the assertion (II) of the lemma is easily verified. Suppose contrary to (III), i.e. that p'y = ap'x for some process (x,y) such that, say, (:%>, y(2)) ^ M#d)> #(2)) f°r anY nolv negative \t. Obviously, (%>, y^)) ^ 0; :r(1) = 0 and y^) =£ 0 contradict Assumption 5; a;(1) ^ 0 and 7/(3) == 0 can be shown to yield p'y < ap'x. Therefore, x(l) ^ 0 and 7/(a) 7^ 0. Hence, by Assumption 9 there is a process (x-\-x, y*) e H with y* > y-}-y, strict inequality holding for at least one output in the second group. As p > 0, it follows from p'y = ixp'x and p'y = Hp'x that p'y* > <5p'(z4-£), a contradiction to the assertion (I). The vector p will be referred to as the von Neumann prices. LEMMA 8. Let ( x ( t ) , y ( t + l ) ) (t = 0,l,...,T—l) be a feasible path of order T, so that y(t) = x(t) (t = 1,..., T— 1). Then (37)
Write sucn mat
ft depending on the initial point x(Q) but being independent of the path as well as the time span T. Proof. As and
Hence, bv Lemma 7. By dividing these inequalities by S'+1 and S.T respectively, we get (37). From (37) we have p'x(t)/of <1 p'x(O). As p > 0 by Lemma 7 and x(t) > 0, we find
Similarly
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LEMMA 9. Define the distance between two vectors x and v as
For any e there is a positive 8 (depending on e) such that for any process (x,y) = (x(1),...,x(m),y(1),...,yim)) for which
Proof. Suppose on the contrary that there are an e > 0 and sequences (Xf ;) , rf i+1) ) e // (i = l,...,m) such that
and where
(38)
Without loss of generality, we may normalize xk so as to fulfil \\xk\\ = 1 (k = 1,2,..., ad inf.). It is shown that the sequences (^fi),yfi+i)) are bounded;*1' therefore, ( x k , y k ) has a limit point (x*,y*). From (38) we have
(39) As H is closed, (x*, y*) belongs to H. It follows from (39) and Lemma 7 that each (#(*), 2/fi+i)) is proportional to (x(i), y~(i+1)) (i = l,...,m). Hence
Clearly, this is a contra-
diction. Let us now introduce a preference function u(y) in terms of which an optimality concept is defined. We make: (1
> Suppose on the contrary that there is a group j such that [|F(k + 1)|| -> oo butrS+D/liyg + DlKi = 0, l , . . . , m - l ; f c = 1, 2,...) are bounded. The normalized setjijt;iluca
are bounded in II and have a limit point (0, F$ ,_])) in //, because Ha^U = 1. (Se p. 45, n. 1.) Therefore, (0,..., O.j/ft y*m)}eH, where yft^ g 0 (i ^ j) and 2/(* + i ) > 0. This contradicts Assumption 5. See Radner [1].
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ASSUMPTION 13. (i) The preference function u(y) is nonnegative, continuous, and quasi-homogeneous on the commodity space; (ii) if (x,y) is the von Neumann process, then u(y) > 0; (iii) there is a number y such that (40)
and (iv) if We define: DEFINITION 2. A feasible path (x(t),y(t+l)) (t = 0,1,..., T—l) which, given z(0) and T, maximizes u(y(T}} in the set of all feasible paths is called optimal of order T. We now have: LEMMA 10. Suppose an initial stock vector
is given such that every x(i)(Q] > 0 (i = I,..., TO). Then there are positive numbers a and ^ such that, if T > T, then for all optimal paths of order T where a depends on the initial point x(0) but is independent oj T. Proof. As .r(,-)(0) > 0 (i = 1,...,TO) it follows from Assumption 11 that there is a feasible path (x(t),y(t-\-l)) (t — 0, I,...,T—1) of some finite order r such that y(r) > 0. This path may be connected with the von Neumann balanced growth path of order T — T starting from qy, where q .= max!*? j y(r) ^> qy}, and yielding, at the termination, output by the amount 5iT-Tqy. Thus there is a feasible path of order T giving u(y(T)} = u(a.T~Tqy). Next, let ( x ( t ) , y ( t + l ) ) (t = 0,1,..., T—l) be an optimal path of order T, which starts from cr(0) = x(Q). By definition, (y(^)) ;=5 u(®T~T
u(y(T)/a.T) ;> u(qyl&7). This together with (40), leads to It is evident that a, y, and y are independent of the initial position a;(0) and the programming period T; -T may also be (^ The properties other than (iv) arc assumed by Radnor [1]. An assumption similar to (but weaker than) (iv) IR made by Nikaido [4|. For definition of 'quasi-homogeneity', see p. 138, n. 1.
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taken independently of them, while q is so defined as to depend on x(0) but to be independent of T. By writing a = u(qyl&')ly, we obtain the desired result. LEMMA 11. Let (x, y) e H be the von Neumann process. Let r and s be multiples of m such that 0 ^ r < s < T. If any feasible path (x(t), y(t-{-!)) (t = 0,1,..., T— 1) satisfies, for some i, r, and s, (41) with some A > 0, then it cannot be optimal. Proof. Since ( x ( t ) , y ( t + l ) ) (t — 0,1,..., T—l) is feasible and r is a multiple of m, it follows from Assumption 9 that the following path of order r starting from X^G) is feasible:
(42) w As (X({), Y({+1)) e //, we have (\arX^),Aa.rY(.i+1)) e H by Assumption 6. The second inequality of (41), together with Assumption 8 of free disposability, implies (43)
This may be followed by the path: (44) where On the other hand, it follows from the first inequality of (41) that
for sufficiently small
and
we find from Assumptions 8 and 10
(45) It is clear that we have the following two paths starting from Ya-u)(s+l) = X(i+1}(s+l) and ^5s7(m) = ja5s+1l:(<+1) respectively :
P R O G R A M M E OF V E R Y - L O N G - R U N GROWTH
189
and
where h is the remainder when T is divided by m. Hence by Assumption 10 the following path is feasible:
(46)
By connecting paths (42)-(46), we obtain a feasible path of order T starting from -X^O) and terminating with As ( x ( t ) , y ( t + l ) ) (t = 0,1,..., T—l) is feasible, it follows from Assumption 9 that the path,
Is feasible for eachYa-u)(s+l) = X(i+1}(s+l) and ^5s7(m) = ja5s+1l:(<+1) respecPath,
is also assured. By Assumption 10 a feasible path starting from
and terminating with
may be composed of paths (42)-(46) and path (47). As the original path cannot be optimal. We can finally establish the following theorem.
190
B A L A N C E D G R O W T H AND E F F I C I E N T
THEOREM 2. Suppose the initial stock vector is given such that each a;(i)(0) > 0 (i = 1,..., m). Let (x(t), y(t-\-1)) (t = 0,1,..., T — 1) be an optimal path of order T with the beginning point x(Q). Let ( x , y ) be the von Neumann process, and let d(x(i)(t), X(i)) be the distance between vectors x^t) and x^), where x(t) = {x(l)(t),...,x(m)(t)}andx = {x(l},...,x(lu}}. Then, for any e > 0 there is a positive number t1 such that, for any sufficiently large T and any optimal path,
for all t fulfilling Proof. We first show that, for any e > 0, there is a number p such that for any T and any optimal path, the number of periods in which (48) cannot exceed p. For any period t in which (48) holds, it follows from Lemma 9 that there is a S > 0 such that Dividing by a(+1, this may be put in the form:
We obtain from (37) p'y(T)/aT <: p'x(t)/a< (t = l,...,T-l). We, therefore, have by virtue of Lemma 10
Write p = p'x(0)l(Sal5c). Taking y(t) = x(t] (t == 1,..., T—l) into account, we find that the number of periods in which (48) holds cannot exceed p, because otherwise p'y(T)j5.T < 0, contradicting Lemma 10.
Next let
and take
0 so small as to fulfil (49)
P R O G R A M M E OF V E R Y - L O N G - R U N G R O W T H
191
where 8 is given by Lemma 8, and
(50) (50') Let us consider a sequence of periods, 0, m, 2m,..., vm, where vm is the largest multiple of m not exceeding T. Among them, let r and s be the first and the last periods in which
(51) holds. If T is very large, then the existence of r and s such that s > r is assured.*1' Obviously, from the definition of d(x(i}(t},x(i)} we have for t = r and s
(52) where /^ is an (nt—wi_1)-dimensional column vector with components 17, i.e. /(f) = (17,..., r)). Since p(i) > 0, we get from (52)
Hence we have
(53) Let us now write
(54) As (52) holds, we have
(55) W The number of periods in which finite number p^.
(aW.*!!)) = 1 cannot exceed a
192
B A L A N C E D G R O W T H AND E F F I C I E N T
(56)
Define: Then (54) may be put in the form
(57) (58) Suppose
for some i. We then have, from (57) and
(58) respectively,
By virtue of Lemma 8, both \\x({)(r)\\jar and ||%)(s)||/as are not greater than /3. As x > 0, o> is less than any ayp^jl, where j belongs to the group i. It follows, therefore, from (55) that
Hence so that by Lemma 11 ( x ( t ) , y ( t + l ) ) (t = 0 , 1 , . . . , T — 1 ) cannot be optimal. This is a contradiction. Therefore,
(59) We now find from (50'), (53), (56), and Lemma 8
(60) Since, by Lemma 8,
P R O G R A M M E OF V E R Y - L O N G - R U N GROWTH
193
for r
(61) As x(t+l) = y(t+l), we have from (60) and (61)
Suppose
for some t such that r ^ t < s.
It then follows from Lemma 9 that $p'x(t)/&l+l < 2j8[(u+w7r)/w]7j. As p'x^/af ^> a from Lemmas 8 and 10, we finally get
This is a contradiction to (49). Hence
or all
t between r and s. Since for any T and any optimal path, the number of periods in which (51) does not hold does not exceed a finite number p , we have because in the sequence (0, m,..., vm] periods r and s are the first and the last in which (51) holds. Therefore, r <^ p^m and T— (p -f-l)m < s. Writing t1 = (p +l)m, we have
for any optimal path starting from x(0). This completes the proof. Let r be any multiple of m such that t1 <1 r <^ T — t^ where tl is determined by Theorem 2. If e is taken very small, it is found from (49) and (52) that x^r) may be approximated as follows: Next let s be any other multiple of m such that tl <^ s <1 T — t^. It then follows from (49), (56), and (59) that when e is sufficiently small, ||:%)(»")||/a'' and ||:%)(s)j /5s approximate to each other. We may, tnerelore, write (62) where tf>t — |J£ (i )(?-)||/(a r p( i )i|), a constant.
194
B A L A N C E D G R O W T H AND E F F I C I E N C Y
As (\X(i), XY,i+1)) e // for all A > 0, we have Thus (62) implies
Taking account of we rind
or
By proceeding on this line of argument, we obtain the table below on the assumption that m equals three. The immediate implication of the table is that x(l) traces a path of cyclical growth around the von Neumann ray unless ^1 = 2 = <j)3 or m = 1. The period of cycles is m = 3, and the amplitude, when
the growth is eliminated, is max 1^ —(^ 1 +
is a time lag between phases of x^}(t) and X(ilrl)(t) (i = 1, 2, 3). It is, however, seen that, in particular cases where Theorem 2 amounts to establishing the Turnpike Theorem: given an arbitrary small neighbourhood of the von Neumann ray, any optimal growth path starting from any initial point stays in the neighbourhood, except possibly for a finite number of periods in the beginning and in the end, when the programming period is taken long. The equalities j = 2 = ••• — <j>m are, however, carried out only if the economy is initially furnished with capital stocks of various commodities in some specific proportions. The alternative condition m = 1 implies that the commodities in the system cannot be divided into cyclic groups. I11 In fact, Radrier arid Nikaid6 implicitly assume m = 1. Recollect, however, Georgescu-Roegen's comment stated in p. 173, n. 3 above.
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 IN various places of this book we use the classical PerronFrobenius theorems for non-negative square matrices and their generalizations to systems of non-linear and homogeneous functions. The original theorems, which were discussed by a number of mathematicians and mathematical economists such as Perron [1], Frobenius [1, 2, 3], Wielandt [1], Solow [1], Debreu and Herstein [1], Schwartz [1], and others, may be stated as follows: THEOREM I. Let A be an n x n matrix, all of lohose elements are non-negative. Then: 1. A has a non-negative characteristic root A with ivhich a nonnegative eigen-vector V can be associated; 2. A does not decrease when an element of A increases; 3. A is not less in modulus than any other characteristic root of A; 4. (pi—A)-1 > 0 if and only if p > A. THEOREM II. Let A be an nxn matrix which is non-negative and indecomposable. Then: 1. A has a positive characteristic root A with which a positive eigen-vector V can be associated ;(1) 2. A increases when any element of A increases; 3. A is a simple root; 4. if A is primitive, A is greater in modulus than any other characteristic root of A; 5. (pi—A)~* > 0 if and only if p > A. Recently Solow and Samuelson [1] have been concerned with <'> It is also seen that eigen-vectors associated with other characteristic roots are not non-negative.
196
APPENDIX
a non-linear eigen-vector problem, Wi — //^T^,...,^) (i = l,...,n), where the functions Ht are all homogeneous of the first degree. They have, furthermore, supposed the H^s to be continuous and strictly increasing. In this appendix, however, we merely assume them to be continuous and monotonic non-decreasing functions of each argument, and generalize Theorems I and II to non-linear eigen-vector problems/1' In fact, Theorems 1, 4, 9, and 14 below will cover the four assertions of Theorem I respectively, and Theorems 3, 5, 6, 10, and 15 the five assertions of Theorem II respectively.
2. Non-linear eigen-vector problems Let X be an ?i-dimensional vector (X-i,X2,...,Xn), and let H also be an w-dimensional vector (H1} H2,..., Hn), such that each Hf is a continuous function of X. We use the following notation for vector comparisons. Let a = {at} and b = {6J; a ^> b means i ^ ^t f°r a^ *> a ^ ^ means a ^> b and a 7^ b', a > b means
ai ~> bi for all i. We make the following assumptions. ASSUMPTION 1. Each Hi is positively homogeneous of degree one in X, i.e. H^X) — pH^X) for any p. > 0 and any X ^> 0. ASSUMPTION 2. If X ]> 0, then H(X) ^> 0. We can prove THEOREM 1. The non-linear eigen-vector problem (1)
can be solved with non-negative A, Kj,..., Vn where we can choose
Proof. Vectors
define
f 1 ' It has been pointed out by Solow and Samuelson [1, p. 413] that if the system is indecomposable, many of their results can be generalized to include the case where the H^s are non-decreasing. But, in order to establish Theorem 10 below, the indecomposability assumption is insufficient, and we must assume the system to be primitive. A reference should also be made to Karlin's [1] which provides generalizations of the Perron-Frobenius theorems.
THE P E R R O N - F R O B E N I U S T H E O R E M S
197
a closed simplex S in Euclidean w-space. Let
Clearly, Y -> (f>(Y) is a continuous point-to-point mapping of S into itself. Hence, there exists a V e S such that V = (F).(1) We obtain XV = H(V), where Throughout this appendix the vectors designated V, V1, V2, etc., belong to 8, while vectors X, X1, etc., do not necessarily belong to S. Next let us make ASSUMPTION 3. (the Solow-Samuelson assumption in weak form). H(X) ^ H(X') for all X and X' such that X ^ X'. We can then prove THEOREM 2. // Assumptions 1-3 hold, there are at most a finite number of non-negative Xr's such that ATr = H(V)for nonnegative vectors Vr with Proof. Let there exist two non-negative A's, say A1 and A8, with non-negative eigen-vectors
respectively. Assume that
We have
(2)
('' Cf. p. 62, n. 2, above.
(3)
198
APPENDIX
By virtue of Assumptions 1 and 3, (2) and (3) lead to
respectively. Hence A1 = A 2 . Thus, if two non-negative eigenvectors have the same zero components, they give the same A. There is at most a finite number of ways in which some components of F are zeros. From this fact we find that the number of A's is finite. Throughout this appendix the greatest A is referred to as A. The property asserted by the following Assumption 4 is analogous to indecomposability in linear systems: An nxn matrix A (n ^> 2) is said to be indecomposable if there is no permutation matrix n such that
with square submatrices on the diagonal. See Debreu and Herstein [1], Frobenius [3], Schwartz [1], and Solow [1]. ASSUMPTION 4 (indecomposability). For any set of indices R = {f1( i2,..., im}, the relations Xi = X't for i e R and Xh < for h $ R imply that there exists at least one i e R such that Ht(X) *Ht(X'). We have THEOREM 3. // Assumption 4 holds in addition to Assumptions 1-3, A and V are strictly positive. Moreover, A and V are unique. Proof. The theorem is proved by reductio ad absurdum. Let us assume that Vt = 0 for i e R = {ii,iz,---,im} and Vh > 0 for h $ R. It follows from Assumptions 3 and 4 that if ^ > 1, then #j(/j,F) > PIt(V) for at least one i e R. Since H^V) = pH^V), we obtain H^V) > 0, which contradicts Wi = Ht(V) = 0. Hence V > 0. Next suppose
and
we find
that Ht(V) = 0 (i = l,...,n). Let X1 = l\ and O^Xi
THE PERRON-FROBENIUS THEOREMS
199
(i = 2,...,n). Then it follows from Assumptions 3 and 4 that H^X) < H^V) = 0. Clearly this contradicts Assumption 2. Hence A > 0. The uniqueness of V is proved as follows. Suppose there exist two different eigen-vectors V1 and F2 such that (4)
respectively. Without loss of generality we may assume that A2 ;> A1. Let p = max (Ff/FJ), and let p = Ff/Fl > Fj[/FJ <=i,...,)t for i e R and h $ R. We immediately find that /u,F£ = F2 for i e -R, and p.V\ > V\ for h £ R. It follows, therefore, from Assumptions 3 and 4 that H^/j-V1) > H{(VZ) for at least one i e -if?. Taking Assumption 1 arid equations (1) into account, we obtain for at least one i e R. As jnF| = Ff, we find that A1 > A 2 . This is a contradiction. Hence F1 = F2. is a contradiction. Hence F Let us now assume that junctions /^(A.a) (i = l,...,n) iulnl Assumptions 1-3 with respect to X. Make also ASSUMPTION 5. If a1 < a2, then //(F 1 , a1)
Then we can assert THEOREM 4. // a increases, the greatest non-negative A" does not decrease. /TOO/. Let K(X) be a strictly increasing function such that K(X) :> 0 for X ;> 0, and /f(^Z) = /^(Z) for any positive p. Let e > 0. Consider an eigen-vector problem: Denote a solution to this problem by (Ae, Fe). Since is indecomposable in the sense of Assumption 4, A6 and FE are positive and unique.
200
APPENDIX
Next let A* be the greatest A such that AF = H(V, at), and Vi the eigen-vector associated with A% where i = 1,2. Let VHVl be min(Ff/F|). It is clear that F*/F£ ^ Ff/F| for all i
F* =£ 0, and Ff/F| < F|/F| for all FJ = 0. Therefore, we obtain
(5)
Since Ve is a non-negative vector with unit sum, the set {Fc,€ = e1,e2,...} is bounded for a sequence {ej such that lim e^ = 0. Therefore, it has at least one limit point F*.(1) We i—> co
get A*F* = H(V*,e?}. Clearly, A* < A2. As A1 <; A* from (5), we obtain A1 <^ A 2 . Theorem 4 can be strengthened to THEOREM 5. Let F1 be, the non-negative eigen-vector of H(X, a1) associated with the greatest non-negative eigen-value A1. Let H(Vl, a1) < H(Vl, a 2 ) for a1 and a2. // H(X, of] fulfils Assumption 4, the greatest non-negative eigen-value A2 of H(X, of) is greater than A1. Proof. The indecomposable system, AF = H(V, a 2 ), can be solved with a positive scalar A2 and a positive vector F2, both of which are unique. Since A1F1 = H(Vl, a1) < H(Vl,ef) and #(F2,a2) = A2F2, we find that if F1 = F2, then A1 < A2. If F1 ^ F2, the following argument holds true.
By virtue of the assumption that H(X, a 2 ) is indecomposable, <» See p. 45, n. 1,
THE PERRON-FROBENIUS THEOREMS
201
there is at least one i (
where T * and H^ stand for (n—l)-dimensional vectors (Vv..., Vn_-^ and (H1,...,Hn_1) respectively. We have THEOREM 6. The greatest non-negative A of the problem (1) is not less than that of (6). // H(V) is indecomposable, the former is greater than the latter.1-1'1 Proof. Equations (6) and (1) can be written in the form: (6')
(!')
respectively, where Since Theorem 4 can be applied, so that A1 <^ A 2 . Next let H(V) be indecomposable; then (F|,y 2 ) must be positive. Suppose // m (F*,0) in (6') equals zero. This leads to I" From this theorem follows the assertion 3 of Theorem II quite easily. Let O(v) = \vl—A\ and A* be a principal (n— l ) x ( n — 1) submatrix of A. Theorem 6 ^mplies that if A is indecomposable, vlt — A+ cannot vanish for v S A, where /„, is a unit matrix of order n—1. Hence |A/» — At\ > 0. As '(A) is the sum of the principal (n~ l ) x ( n — 1) minors of
202
APPENDIX
a contradiction, because it is shown that (!') could be solved with A1, Fi, 0. Therefore, HJVLQ) > 0, so that
Hence Theorem 5 is applicable, i.e. A1 < A2. 3. Balanced growth So far we have been concerned with the non-negativity or positivity of a solution (A, F) to (1). It is well known that X(t) = xV\l is a particular solution of the difference equation (7)
where x is a positive scalar determined by the initial conditions. Under what conditions will this solution give a balanced growth (or decay) of the X ? The fact of significance from this viewpoint is not the relationship of the A to zero but that to one. We may offer THEOREM 1. Let H(X) be indecomposable; A is greater (or less) than one if and only if there is at least one X° > 0 such that Proof. Sufficiency.
Let a1 be a diagonal matrix such that (8)
Then X° < H(X°) may be written in the form (9)
where / is the unit matrix. Let a.} be the ith diagonal element of a1. It is non-negative and does not exceed one if H^X0) > 0, but indefinite if H^X0) = 0. We may, however, set a| = 1 for those i's such that H^X0) = 0. We have Let F1 be the eigen-vector associated with the greatest nonnegative A1 of the system A^1 = a^H(Vl). Write / = of. We then have <J> If H(X) is linear, this necessary and sufficient condition for A < 1 is equivalent to the Hawkins-Simon conditions.
THE PERRON-FROBENIUS THEOREMS 1
203
2
Hence, by Theorem 5, A is less than the A determined by A 2 F 2 = « 2 //(F 2 ). Since this equation is nothing else but AF = H(V), we obtain A > A1. On the other hand, we find from (8) that (A1, F1) = (l,X°) is a non-negative solution to the problem AT1 = a^F1); so that A11> 1. Hence A > 1. Next consider the case where X° > H(X°). Let a2 be a column vector such that
After normal-
izing X°, this equation may be written as
with A2 = 1 and F2 = X°/ % X*}. As H(X) is indecomposable, (A2, F2) is the unique positive solution to the above equation. From X° > H(X°) it follows that a 2 > 0. Let (A, F) be the positive solution of AF = H(V). We have
(10) for a1 = 0. Hence, by Theorem 5, A2 is greater than the A, i.e. 1 >A. Necessity. From AF = H(V) and F > 0 we see that F < H ( V ) or F > H(V) according as A > 1 or A < 1. By a similar argument we can prove THEOREM 8. // there is at least one X° > 0 such that X° < H(X°) [or X° > H(X°)], the greatest non-negative eigenvector A of H(X) is greater (or less) than one. Proof. Let a1 be a diagonal matrix such that X° = <x1H(X°). Write A1 = 1, and a2 = /. Considering X° < H(X°), we get Hence, by Theorem 5*, the greatest non-negative eigen-value A3 of oc2H(X) is greater than A1. As a2H(X) = H(X), A2 equals A. Next suppose X° > H(X°). Let a2 be a column vector such that It follows that a2 is strictly positive; therefre,
204
APPENDIX
is indecomposable even if H(X) is decomposable. Then the non-negative eigen-value of
with non-negative
eigen-vector is unique, and we find from the above equation that it is unity. On the other hand, from the definition of A we see that
^holds for a1 = 0. Clearly, (10) holds
at F = F. Hence, by Theorem 5, A~ < 1. Note that this theorem remains true even if H(X) is decomposable. 4. Stability Let X[t, X(0)] or simply X(t) be a solution to the difference equation (7), starting from the initial position X(Q). Solution xVX, with A ^> 0 andV > 0, is the balanced growth solution of (7), where a; is a positive constant determined by the initial conditions. The xV\' is said to be weakly stablem if there exist finite numbers, c and c, such that for all 11> 1, and strongly stable(2) if Furthermore, the balanced growth is said to be regularly stable if
for the initial positions Z(0) and Z'(0) such that Z(0) > Z'(0). THEOREM 9. Let (A, F) be a non-negative solution of XV — H(V), and let V = {F^.^F^O....^}, where V{ > 0 (i — l,...,m). Then xVX' is locally weakly stable in the sense that if X^Q) = 0 (i = m-\-\,...,n),tliencV or some finite numbers c and c, and for all t ^> 1. Proof, (i) If A = 0, AF = H^.^V^O,...^) = 0. By homogeneity /Z(/iF1;..., pVm, 0,..., 0) = 0 for any ^ > 0. Since it follows from Assumptions 2 and 3 that I1' Let H(X) be linear, i.e. H(X) = AX. Let i) be any characteristic roo of Al other than A, the greatest non-negative root. The weak stability of xV\z implies that |ij| S A, the assertion 3 of Theorem I. < > In case of H(X) being linear, the strong stability implies that |i;| < A, the assertion 4 of Theorem II.
THE P E R R O N - F R O B E N I U S THEOREMS
205
for sufficiently large /u, we obtain The same argument shows that H[X(l)] = X(2) = 0, and so on. (ii) Next suppose A > 0. Since Vh = 0 (h = m+l,...,n), the same argument as in (i) will lead to Write may write
Considering in the form
(H) we
By use of the homogeneity property this becomes
(12) Let max
We get
This, together with (12), leads to x^t+l) <: M(t). Similarly Hence, from which we obtain
(13) where c = m(0) and c = M(0). It is clear from (11) and Vh = 0 (h = m-\-l,...,n] that (13) holds for i = m-\-l,..., n also. Let us consider two paths X[t, X(Q)] and X[t, X*(Q)] generated by the system (7) from the respective initial positions X(0) and X*(Q). If there exists a finite number 5 for which for all X(0) such that X(0) > X*(0), then H(X) is said to be primitive at J£*(0). In particular, H is said to be globally primitive if it is primitive at any point. Our definition of primitivity is analogous to the following definition in linear systems: an
206
APPENDIX
n X n matrix A is said to be primitive if there is no permutation matrix II such that
with square submatrices on the diagonal. In fact, if A is primitive, there is an s such that As > 0: therefore for all X(0) such that X(0) > X*(Q). We get THEOREM 10. // H(X) is indecomposable and primitive at 0 and V, then xVX is strongly stable with respect to any X(0) ~> 0. Proof. By the indecomposability, A > 0 and V > 0. Define Substituting in (7) and using the homogeneity of the //^ we obtain As two sequences,
are bounded and monotonic, they have limits: lim M(t) = M t—*•«>
and lim m(t) — m. Clearly M ^> m. As x(0) > 0 and H(X) is t—>OC
primitive at 0, we get m > 0. Now suppose that M > m. Given any small positive number e, we can find that m—e^Lx^t] (i = l,...,n) for sufficiently large t. Some x^t) are greater than or equal to M, because maxa;i(<) = M(t) ^> M. Without loss of generality we may assume that Xj(tv) ^ M (j = l,...,r) for some sequence {<J such that tv < ^,,+j^ and lim tv = oo. Then we shall have V—>co
for sufficiently large tv. Define two sequences
THE P E R R O N - F R O B E N I U S T H E O R E M S
207
starting from and
respectively. As
we find that
15 for all s. Since M > m, we have 7(0)(0) > Vm, so that it follows from the primitivity of H that
for at least one finite s. 15 If we let e tend to zero (and hence tv tend to infinity), it follows from (14) and (15) that«
As lim min xt(t) = m, we get finally m > m, a contradiction. <-><»
i
Hence M = m. i.e.
THEOREM 11. IfH(X) is indecomposable and globally primitive, the balanced nrnwth if! reaularlij .itable. Proof. Define and
respectively. Suppose X(Q) > X'(0). From the assumption of global primitivity it follows-that
for at least one finite s and all i. This implies that there <" As x^(t) (i — 1,..., n) are bounded, they have limit points. (See p. 45, n. 1.) lim Xj(<) denotes the limit inferior o f ' x ^ t ) , i.e. the least of the limit 2->00
points of x^(t).
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APPENDIX
exists a p > 1, such that a^s+l) ^> px'^a+l) (i = l,...,n). Then by Assumption 3
i.e. Xf(a-\-2) ^> px'{(a-\-2) (i = l,...,n). Repeating the identical argument, we prove that Hence lim
we have
Let Ot(i = 1,..., n) be arbitrary functions which are positively homogeneous of degree one in variables X — (Xlt Xz,...,Xn). Denne
(16) where 0 — (G^..., On). If Assumptions 1-4 are made for H(X), the non-linear eigen-vector problem (17) can be solved witn positive A, V^..., Vn. We can prove THEOREM 12. Assume that H defined as (16) satisfies Assumptions 1-4. Assume that H = (Hlt..., Hn) is primitive at 0 and V. Then to the system (*•*)
there exists a balanced-growth solution which is unique and strongly stable. Proof. From (16) and (18) we have
(19) By Theorem 3 the A and the V determined by (17) are unique. By Theorem 10 the unique balanced-growth solution xV\r to (19) is strongly stable. That is,
(20) for any solution
THE PERRON-FROBENIUS THEOREMS
209
By
so that by the homogeneity property of G On the other hand, by definition (16) we have
Hence
(21)
As (17) uniquely determines the proportions K : fj,:... :K, we find from (21) that (22) Clearly, xV/jf is a balanced-growth solution to (18). From definition (16), equations (17) and (22), and the homogeneity of G, it follows that Therefore, /* = A1'*. Thus (18) has a balanced-growth solution xV(Wky. Next let (v, U), with be a .non-negative solution to the non-linear eigen-vector problem vU = G(U). Then we have
As (17) is uniquely solved with (A, V), it is shown that v = A1/fc and U = V. Hence (18) has no other balanced-growth solution than xV(Wk)1. Let X(t) be generated by the system (18) from the initial position X(0) > 0. Then it is shown that X(rk+s) (r = 0,1, 2,...), where s is any integer such that 0
Write
210
APPENDIX
and substitute into (18) to obtain
(24) Clearly (23) implies lim eis(r) = 0 (a = 0, l,...,k—l), so that by r-*oo
the homogeneity property (24) becomes
Therefore, Substituting into (23) we get finally,
i.e. This means that the balanced growth solution to (18) is strongly stable. Note that the strong stability established by Theorem 12 is sharpened to the regular stability if H is globally primitive. When G(X) is linear, i.e. 0(X) = AX, we have the following theorem due to Tsukui [1]. THEOREM 13. Let A be cm nxn matrix being not necessarily non-negative. Let X(t+l) — AX(t) have a balanced-growth solution. It is regularly stable if and only if there exists a finite integer k such that Proof. Sufficiency
is obvious from Theorem 12.
Necessity. Let X\ > X I . Define where
be solutions to
starting from X' and X" respectively. As the balanced growth is regularly stable, there exists a finite positive integer ui such that X'(t) > X " ( t ) for all 11> ut. Hence A'X' > A'X" for all
THE PERRON-FROBENIUS THEOREMS
211
i ^> u{. This implies that all components of the ith column of A1 (t ^ ut) are positive. Let ki be the smallest value of such a ut. If we define k as max(A;1, kz,..., kn), then Ak > 0.
5. Non-linear resolvent problem Finally, let us be concerned with the non-linear resolvent problem, (25) which has recently been dealt with by Nikaido [3J. We begin with discussing the general case where H(X) is not necessarily indecomposable. We can prove THEOREM 14. Suppose H(X) fulfils Assumptions 1-3. (i) For any c > 0, the resolvent equation (25) has a non-negative solution U > 0 if p is greater than the greatest non-negative eigen-value A of H(X). (ii) // c > 0, the relation p > A is necessary for the existence of a non-negative solution to (25). (iii) Let Ul and U2 be solutions to (25) when c = c1 and c = cz respectively. Then p > A and c1 > c2 > 0 imply U1 > C/2. Proof, (i) Let us, lor a while, assume that c is strictly positive. Write where a is a scalar being positive. As c > 0, H(X, a) is indecomposable. Therefore, by Theorem 3, the eisen-vector problem can be solved with [A(a),F(a)] > 0 such that
(26) Moreover, A(a) and F(a) are unique. Thus we have
(27) It follows from (26) and (27) that
(28) As F(a) is always contained in a closed simplex
212
APPENDIX
and H(X) depends continuously on
is found to
be bounded. Therefore, we see that A(a) is also bounded so long as a is bounded. Let a^ be any sequence that converges to a, and let [A^), F(af)] be the corresponding solutions to (27). As A(cx'1)'s are bounded and F(a^)'s belong to 8, we can pick convergent subsequences X(x") and V(a.v) whose limits are A* and F* respec tively. The continuity of H(X) implies that Hence,
From this we see that [A*, F*] = [A(a), F(a)], or else the uniqueness of the solution to (27) would be violated. If the sequenc [A^), V(uP)] did not converge, we could pick two subsequences with different limits which, of course, would also violate uniqueness. Hence lim A(a't) = A(a) arid lim F(a^) = F(a), i.e. the /i-»CO
^->-CO
solution to (27) depends continuously on a. It follows from (27) that A(a) approaches an eigen-value of H(X) when a approaches 0, and from (28) that A(a) tends t infinity when a increases indefinitely. Furthermore, by Theorem 5, it is an increasing function of a. We find, therefore, that for any p > A there exists one and only one a such that p = A(a). Since H(X) is homogeneous of degree one in X, we obtain from (27)
(29) Thus the resolvent equation (25) has a unique solution F(a)/a as long as c > 0. Next suppose c > 0, and let c,v- be any sequence such that c^ > 0 and lim cv- = c. Clearly, corresponding to each c^, there U-^-QO
is a unique solution F(a^)/a'i to pU = H(!})-{-&. As each F(<x^) is a non-negative vector with unit sum, we have
THE PERRON-FROBENIUS THEOREMS
213
1
from which the boundedness of at follows immediately. We can pick convergent subsequences a" and V(<xv) with limits a* and V*. Taking the continuity of H(X) into account, we have
i.e.
(30)
If a* = 0, p itself would be an eigen-value of H(X), so that it would not exceed A. This is a contradiction. Hence, a* > 0. In view of the fact that H(X] is homogeneous of degree one, (30) can be written in the form (29). The resolvent equation is thus solvable for c > 0. (ii) Let V be an eigen-vector of H(X) associated with A, i.e. AF = H(V). If c> 0, it follows from (25) that U is strictly positive. Define 6 = max (Vf/Uf). We have 6U I> V, so that i=1....n
H(0U) ;> H(V). Therefore""
(31)
Hence,
Since GUi = Vi for at least one i and ci > 0 lor all %, we see from (31) that p > A. (iii) We start from the case in which c1 and c2 are strictly positive. It follows from (25) that U1 and t/2 are strictly positive. Define 6 = max (t/f/f/J), so that 8U\ I> V\ (i = 1, i = l,...,n
...,n), the equality holding for at least one i. Let us suppose that 9 > 1 when c1 > c2. Then dc1 > c2. As H(9Ul) ;> H(UZ), it follows from equations and
that 6U1 > U2, a contradiction. Hence, ( 9 ^ 1 , i.e. Ul ^ U2. For the case where some components of c1 and c2 are zero, we consider sequences c^ and c^, which satisfy c^ > c^ > 0 for all ju, and converge to c1 and c2 respectively. Corresponding to them, there are sequences C7J, and Uz fulfilling p U = H(U)-\-c^, and pU = H(U)-\-c* respectively. By (i) some subsequences
214
APPENDIX l
Ul and U\ converge to U and £/2 when v increases indefinitely. As for all v, we get We can show that c1 ^ c2 implies ence f/1 > U2 if c1 > c2 > 0. The indecomposability of H(X) enables us to sharpen Theorem 14 to the following: THEOREM 15. Suppose H(X) fulfils Assumption 4 as well as Assumptions 1-3. (i) For any c > 0, the resolvent equation (25) has a positive solution U > 0 if and only if p > X, where A is the positive eigen-value of H(X), with which a positive eigen-vector V is associated, (ii) Let Ul and U2 be solutions to (25) corresponding to c1 and c2 respectively. If c1 > c2 > 0, then Ul > U2. Proof, (i) Sufficiency.
Define:
Since H(X) itself is indecomposable, H(X, a) is so for any c > 0. Therefore, for any c > 0, the eigen-vector problem can be solved with [A(a), F(a)] > 0. Note that A(a) and F(a) are unique. By an argument similar to the proof of Theorem 14(i), it is shown that, for any c > 0, there is an a° such that A(a°) = p so long as p > A. We have (29) for such an of. Thus the resolvent equation with c > 0 can be solved with U = F(<x°)/a° > 0 if p > A. Necessity. As H(X) is indecomposable, the eigen-vector F is strictly positive (Theorem 3). DefineCT= min (UJVi); we have i=1....,n aV <; U. When aV = U, we obtain so that p > A. If aV < U, the indecomposability of H(X) implies that there is at least one i such that crFf = Ut and HAoV)
THE P E R R O N - F R O B E N I U S THEOREMS
215
(ii) From we get
(32)
We also have f/1 > U2 by Theorem 14 (iii). If for some i, there is at least one i such that U\ = C7f and H^U1) > H{(U2), because H(X) is indecomposable. For such an i, we have This contradicts (32). Hence
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INDEX
Activities, 93 (see also Catalogue), proper to a price set, 96. selector of, 96. Arrow, K. J., 3, 4, 5, 6n., 24, 33 n., 36 n., 43n., 61 n., 67 n., 86, 97 n., 216, 217, 218.
Eilenberg, S., 143, 217, 220. Equilibrium: existence of, 5, 26, 61, 87. long-run, 58, 66. quasi-, 61, 61 n. uniqueness of, 62. uniqueness of, 8.
Bailoy, M. J., 38, 216. Balanced growth, 74-78, 107, 140 (see also Quasi-stationary state). efficiency of, 111, 146. golden, 115. stability of, see Stability. Beckman, M. J., 38, 216. Block, H. D., 3, 4, 6 n., 24, 43 n., 216, 218. Bolzano-Weierstrass theorem, 45 n.
Fisher, F. M., 176, 217. Fixed-point theorem: Brouwer, 62 n., 88. Eilenberg-Montgomery, 143 n. Frobenius, G., 72 n., 195, 198, 217. Frobenius theorem, 72, 75, 98, 195, 196 n. Fukuoka, M., 23 n., 27 n., 217, 219. Furuya, H., 176, 177, 217, 218.
Catalogue: of activities, 95. of unit costs, 95. Champernowne, D. G., 133, 135, 216. Charnes, A., 162 n., 216, 217, 218. Chipman, J. S., 21, 216. Coddington, E. A., 29 n., 217, 219. Competition: weak and strong, 60. Continuity: absolute, 53 n. equi-, 45. in an extended sense, 4. upper semi-, 142 n. Cooper, W. W., 162 n., 216, 217, 218. Correspondence principle in the large, 24. Cramer's rule, 15. Debreu, G., 7 n., 46 n., 61 n., 72 n., 86, 98 n., 159 n., 160 n., 163 n., 172 n., 195, 198, 216, 217, 218. Demand schedules (see also Propensity to consume of workers), rigidity of, 114. flexibility (perfect or imperfect) of, 114, 117. Dorfman, R., 23 n., 27 n., 34 n., 86, 110, 154, 156, 162 n., 217, 221, 222.
Gale, D., 86, 134 n., 135, 156, 162 n., 166 n., 176, 183, 217. Georgescu-Roegen, N., 55 ri., 67 n., 97n., 174 n., 194, 218. Golden age, 78, 82 ,93. Goodwin, R. M., 21, 218. Gross substitutability of goods: strong, 4. weak, 5. Hahn, F. H., 3, 4, 25 n., 43 n., 218. Harrod, R. F., 151, 218. Harrod's warranted rate of growth, 151-3. natural rate of growth, 151-3. Hawkins, D., 14, 74, 202 n., 218, 222. Hawkins-Simon conditions, 15, 26, 72, 74, 96 n., 107, 165 n., 202 n. Heino-Borel theorem, 170 n. Henderson, A., 162 n., 216, 217, 218. Herstein, I. N., 7 n., 72 n., 98 n., 159n., 160n., 163 n., 172 n., 195, 198, 217, 218. Hicks, J. R., 3, 4, 13, 24, 27, 59 n., 131 n., 156, 163 n., 218. Hicksian laws, 3. first law, 6. second and third laws: in weak form, 8. in strong form, 8.
226
INDEX
Homogeneity: positive: of degree one, 87, 196. of degree zero, 4, 87. quasi-, 138 n, 187. Hurwicz, L., 3, 4, 5, 6 n., 24, 33 n., 36 n., 43 n., 216, 218. Inada, K., 176, 177, 217, 218. Indecomposability, property of a set of functions, 7, 14, 26, 72, 79, 159, 198. strong, 8. of a technological possibility set, 180. Input coefficient: augmented, 107, 132. capital-, 54. current-, 54. normalized, 108. optimal, 69. Input-output system, 1. dynamic, 54. normalized, 108. Jorgenson, D. W., 108, 110, 218. Kaldor, N., 69 n., 151, 218. Karlin, S., 6n.,43n., 62n., 67n., 72n., 135, 176, 196 n., 219. Kemeny, J. G., 134, 135, 143 n., 158 n., 219, 220, 222. Koopmans, T. C., 67 n., 97 n., 184 n., 219. Kose, T., 24, 219. Koyama, A., 23 n., 27 n., 217, 219. Kuhn, H. W., 23 n., 27 n., 86, 156 n., 219. Lancaster, K., 156, 219. Land of Cockaigne: impossibility of, 176. Lange, O., 24, 43 n., 95 n., 219. LeChatelier-Samuelson principle, 3-4, 11-14, 20. Lefachetz, S., 62 n., 170 n., 219. Leontief, W. W., 1, 18, 21 n., 54 n., 56, 61 n., 93, 110, 157, 219. .Levinson, N., 29 n., 217, 219. Limit inferior, 207 n. Limit point, 45 n. Linear programming: duality theorem of, 162, 164, 170. Lipschitz condition, 27 n., 31.
McKenzie, L. W., 3, 5, 43 n., 86, 156, 220. McKinsey, J. C. C., 134 n., 220. McMonus, M., 110, 220. Marx, K., 69, 82, 93, 135, 136, 137 n., 148, 220. Marxian: assumption of equal rates of profit, 61-62, 135, 137. constant capital, 137, 144. reproduction scheme, 135. reserve army of labour, 69, 82-83. surplus value, 145. variable capital, 137, 145. Metzler, L. A., 18 n., 24, 220. Metzler's theorem, 18. Montgomery, D., 143, 217, 220. Morgenstern, O., 134, 135, 143 n., 158 n., 184 n., 219, 220, 221, 222. Morishima, M., 137 n., 145 n., 156, 220, 222. Mosak, J. L., 3, 4, 220. Negative quasi-definite, property of a matrix, 27. Negishi, T., 3, 4, 25 n., 43 n., 220. von Neumann, J., 76, 93, 131, 133, 135-7, 148, 154, 157, 184 n., 220, 221. von Neumann ray, 154 (see also Turnpike theorem), convergence to, 154. cyclic growth around, 194. Newman, P., 43 n., 221. Nikaid6, H., 43 n., 86, 142 n., 156, 174-6, 184 n., 187 n., 194, 211, 221, 223. N0rregaard Rasmussen, P., 93, 221. Okishio, N., 3, 221. Optimal path: of order T, 187. Outputs and prices: duality relation between, 2. separability between, 2. unilateral coupling of, 61 n. Perron, O., 195, 196 n., 221. Prices (see also Outputs and prices), alpha-beta, 96. long-run equilibrium, 58, 66. proper to an activity set, 96. Primitivity, property of a set of functions, 162-3, 171, 205-6. of a technological possibility set, 179.
INDEX Productivity curve, 123. shift of, 124. Propensity to consume of workers, 72-75, 107 (see also Demand schedules). Quasi-concavity, property of a function, 43, 138n. Quasi-stationary state, 133 (see also Balanced growth). Kadner, R., 138 n., 156, 174-6,187 n., 194, 221. Real-capital ratio curve, 126. Resolvent problem, 211-15. Revealed preference: weak axiom of, 6, 34, 44. Ricardo, D., 69. Robinson, J., 76, 78, 93, 94, 96, 122-6, 153, 221.
227
of the normalized and non-normalized processes, 6 n. of the Walras-Leontief equilibrium of production: global, 24, 31-36. local, 24, 29. semi-global, 24-31. Strong super-additivity, 174-7. Structural change (see also Technical invention). long-run, 113. Substitution theorem, 67, 97.
Technical invention (see also Structural change). of the capital-saving type, 128-30. of the capital-using (or laboursaving) type, 79, '128-30. effectiveness of, 116. long-run effects of, 116-18. neutral, 128-30. of new goods, 120-1. of a new method of production, 116. Samuelson, P. A., 3, 4, 23 n., 24, short-run effects of, 117-18. 27 n., 34 n., 43 n., 59, 61, 67, 86, trigger effects of, 117-22. 97, 108, 110, 137 n., 154, 156, 162 n., 165, 195, 196 n., 197, 217, Technological possibility set, 176 (see also Indecomposability and 221, 222. Primitivity). Schlesinger, K., 85, 222. Schwartz, J. T., 72 n., 98 n., 159 n., of order s, 178. Thompson, G. L., 134, 135, 143 n., 195, 198, 222. 158 n., 219, 220, 222. Separation theorem, 184n. Set, 5 n. Tsukui, J., 108, 210, 223. contractibility or deformability Turnpike theorem, 154 (see also von into a point, 142 n., 143. Neumann ray), cyclic exceptions to, 171. convexity of, 6 n., 62 n., 153. positive polar of, 183. Uzawa, H., 3, 5, 6 n., 24, 39 n., 43 n., set-theoretic intersection, 51 n. membership, 6 n. 52 n., 221, 223. Seton, F., 107 n., 137 n., 220, 222. Simon, H. A., 14, 74, 93, 117, 202 n., Value of a matrix game, 134, 136, 218, 222. 141 n. Solow, R. M., 7 n., 15, 21, 23 n., 27 n., 34 n., 55, 61, 86, 99, 108, 110, Wald, A., 23 n., 26 n., 85, 86, 223. 154, 156, 162 n., 165, 195, 196 n., Walras, L., 25 n., 56,57 n., 61 n, 83, 85, 197, 198, 217, 222. 86, 87 n., 91, 135, 136n., 148, 223. Stability Walras law, 4, 14, 26, 84. of the balanced growth path, 108. Wielandt, H., 195, 223. Wong-Viner envelope-type theorem, regular, 108-9, 204, 207-11. strong, 204, 206-7. 59 n. weak, 204-5. of the long-run equilibrium price Yasui, T., 23 n., 24, 27 n., 43 n., 223. set,
100.
of non-tdtonnement pricing, 25 n., 43-53.
Zarnowitz, V., 55 n., 223. Zeuthen, F., 85, 223.
