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1 The Review of Ecoomic Studies Ltd. Walras' Tâtoemet i the Theory of Exchage Author(s): H. Uzawa Source: The Review of Ecoomic Studies, Vol. 27, No. 3 (Ju., 1960), pp Published by: The Review of Ecoomic Studies Ltd. Stable URL: Accessed: 29/12/ :07 Your use of the JSTOR archive idicates your acceptace of JSTOR's Terms ad Coditios of Use, available at. JSTOR's Terms ad Coditios of Use provides, i part, that uless you have obtaied prior permissio, you may ot dowload a etire issue of a joural or multiple copies of articles, ad you may use cotet i the JSTOR archive oly for your persoal, o-commercial use. Please cotact the publisher regardig ay further use of this work. Publisher cotact iformatio may be obtaied at. Each copy of ay part of a JSTOR trasmissio must cotai the same copyright otice that appears o the scree or prited page of such trasmissio. JSTOR is a ot-for-profit service that helps scholars, researchers, ad studets discover, use, ad build upo a wide rage of cotet i a trusted digital archive. We use iformatio techology ad tools to icrease productivity ad facilitate ew forms of scholarship. For more iformatio about JSTOR, please cotact The Review of Ecoomic Studies Ltd. is collaboratig with JSTOR to digitize, preserve ad exted access to The Review of Ecoomic Studies.

2 Wairas' Tatoemet i Theory of Exchagel the I Walras' theory of geeral equilibrium, a importat role is played by the cocept of tatoemet. I spite of may cotributios to the theory of tatoemet,2 there are still iterestig problems which have ot bee satisfactorily solved. I the preset paper, we ited to fill some of the gaps i that theory; especially with regards to the stability problem of tatoemet processes. I [14], Walras first cosiders a ecoomic system i which oly exchage of commodities betwee the idividuals takes place, ad the proceeds to hadle more complicated systems i which productio of commodities or capital goods becomes possible. I ay ecoomic system, however, he shows by coutig the umbers of the ecoomic variables (ukows) ad the relatios (equatios) which prescribe those variables, that it is theoretically or mathematically possible to determiequilibrium values of the ecoomic variables. He the shows that the problem of determiatio of equilibrium values of the ecoomic variables is empirically, or i the market, solved by the tdtoemet process which represets the mechaism of the competitive market. I a exchage ecoomy, the competitive market process cosists of a price adjustmet by which the price of a commodity will rise or fall accordig to whether there is a positive excess demad or a positive excess supply of the commodity. Walras himself, however, does ot make clear what is meat by his tatoemet process. I particular, he has two distict tatoemet processes i mid: the oe with simultaeous adjustmet, ad the other with successive adjustmet, both with respect to prices of commodities. For example, the passages o pp , [14], i which he attempts to prove the stability of the process, show that his process is oe of successive price adjustmets, while the summary o p. 172, [14], seems to suggest that his tatoemet process is a simultaeous price adjustmet. Let us cosider a competitive exchage ecoomy with + I commodities ad R participats. Commodities will be deoted by i = 0, 1,...,, while idividual participats will be deoted by r = 1,..., R. At the begiig of the market day, each idividual has certai amouts of commodities, ad durig the market day the exchage of commodities betwee the idividuals will take place. Let the amout of commodity i iitially held by idividual r be y'i, i = 0, 1,..., ; r = 1,..., R. Ugig vector otatio, we may say that the vector of the iitial holdigs of idividual r is yr = (YOS ir Y.... j, Yp') r = 1,.I. R. It will be assumed that each idividual has a defiite demad schedule whe a market price vector ad his icome are give. Let the demad fuctio of idividual r be xr(p, Mr) = [X?o(Ps M)..., x(p, Mr)], * This work was supported by the Office of Naval Research uder Task The author is idebted to Professors K. J. Arrow, R. Dorfma, L. Hurwicz, ad G. Titer for their valuable commets ad suggestios. 2 The reader is i particular referred to a excellet ote by Professor Patiki for the most complete treatmet of the theory; [8], pp

3 WALRAS' TATONNEMENT IN THE THEORY OF EXCHANGE 183 where p = (Po, pl..., p) is a market (accoutig) price vector ad Mr a icome of idividual r. By a price vector p is meat a o-zero vector with o-egative compoets. Each deiad fuctio Xr(p, Air) is assumed to be cotiuous at ay price vector p ad icome Mr, ad to satisfy the budget equatio E p X'r(p, Air) i=o = Air. If a price vector p = (Po, p,. p) has bee aouced to prevail i the whole market, the the icome Mr (p) of iidividual r is Mr (p) = YE piy,, r = 1,. i=o,r. Hece the demad fuctio of idividual r becomes xj[p, Mr(p)], r 1,..., R. The aggregate fuctio is ow defied by R xi (p) = E x [p, Mr(p)J, i-0 1, I X., i ad the excess demad fuctio z(p) (z0(p),..., z(p)) by where zi (p) = xi (p) -yi R yi - Sy~, i- 0,1,... The excess demad fuctio z(p) is cotiiuous at ay price vector p, ad homogeeous of order zero. It may also be oted that the Walras lawv holds (1) X pi zi(p) = 0, for ay price vector p = (po, Pi...,p) i=o A price vector p (Po, Pfi, p5) is defied to be a equilibrium price vector if zi(p) < 0, i = 0, 1,...,, which i view of the Walras Law (1), may be writte, z (fp) 0, if p3 > 0, zt( 6-< 0, ifp = 0. The mai problem i the theory of exchage of commodities is ow to ivestigate whether or ot there exists a equilibrium, ad if such exists, how oe ca determie a equilibrium. I what follows, we will rigorously show that the problem is solved by the Walras' tatoemet process.' 1 The existece of a equilibrium for the Walrasia modcl of geeral equilibrium was rigorously proved by, e.g., Wald [13] ad Arrow ad Debreu [2]. The competitive ecoomy of exchage with which we are cocered i the preset paper is a special case of the Arrow-Debreu model hece, their existece proof ca be applied. We are, however, iterested i applyig the Walrasia tztoemet process i order to give a proof of the existece of a equilibrium, which will be doe i the Appedix.

4 184 REVIEW OF ECONOMIC STUDIES Let us iterpret the competitive exchage ecoomy as a game which R idividuals ad a fictitious player, say a Secretary of Market, play accordig to the followig rules: (i) Secretary of Market aouces a price vector. (ii) Each idividual submits to Secretary of Market a " ticket" o which the quatities of demad ad supply made by the idividual accordig to the aouced price vector are described. (iii) Secretary of Market calculates the quatities of aggregate excess demads from the tickets submitted to him by the idividuals. (iv) Secretary of Market aouces a ew price vector such that prices of commodities which have a positive excess demad will rise, ad prices of commodities which have a egative excess demad (a positive excess supply) will fall. Moves (ii) ad (iii) are repeated at the ew price system. The game is cotiued util Secretary of Market aouces a equilibrium price vector. Let p(t) = (po(t),..., p4l(t) ) be the price vector aouced by Secretary of Market at the t-th stage, t = 0, 1, 2,...., ad if. The the rule (iv) may be mathematically formulated as follows :1 (2) pj(t + 1) = max {0, pi(t) + fi(t)}, t = 0, 1, 2,. i = 0, 1,... where fi(t) has the same sig as z[p(t)]. The tdtoemet process (2) leaves the price vector p(t) ivariat if ad oly if it is a equilibrium price vector.2 The process (2), as will be show i the Appedix, eables us to prove the existece of a equilibrium price vector pi = (Po0, j31... f i). Now we take as a ume'raire a commodity, say commodity 0, which has a positive price at a equilibrium, ad suppose that prices are expressed i terms of the ume'raire good, i.e., we cosider oly those price vectors p = (po, Pi,..., p) for which Po 1. I what follows, by a price vector p is meat a -vector (P,..., p) with o-egative compoets. The simultaeous tatoemet process (2) accordigly may be reformulated by (3) pi(t + 1) = max { 0, pi(t) + fi(t) }, t =,1, 2,...; i = 1,...,. We ow ivestigate the stability problem3, i.e., whether or ot the price vector p(t) determied by the tatoemet process (3) coverges to a equilibrium price vector.., 1 It may be oted that the system of differetial equatios correspodig to (2) is give by (2)' Jh = max {O, pi + fi(p)} -p, i = O, 1,...,. The process of price adjustmet represeted by (2)', is a special case of the oes discussed i Arrow, ad Hurwicz [1], p. 94. The solutio to the process (2)', however, remais positive wheever the iitial positio is positive. 2 See Walras [14], p For recet cotributios to the stability problem, see, e.g., Arrow ad Hurwicz [3] ad Arrow, Block, ad Hurwicz [1].

5 WALRAS' TATONNEMENT IN THE THEORY OF EXCHANGE 185 It is oted that the Lyapuov stability theorem' may be modified so as to be appropriately applied to our tdtoemet process ; amely we have the followig: STABILITY THEOREM.2 Let p(t; p ) be the solutio to the process (3) with iitial price vector po. If (a) the solutio p(t; po) is bouded for ay iitial price vector p0; ad (b) there exists a cotiuous defied for all price vectors p such that p(t) = ( [p(t; p0)] is strictly decreasig uless p(t; po) is a equilibrium; the the process is quasi-stable. Here the process (3) is called quasi-stable if, for ay iitial price vector p0, every limitig-poit of the solutio p(t; po) is a equilibrium. I the case i which the set of all equilibria is fiite, the quasi-stability of the process (3) implies the global stability. The fuctio ()(p) satisfyig the coditio (b) will be referred to as a Lyapuov fuctio with respect to the process (3). Let us cosider the case i which the excess demad fuctio z(p) = [z0(p), z1(p),.., z,(p)] satisfies the weak axiom of revealed preferece3 at equilibrium: (4) zo(p) + ; pizi (p) > 0, 1=1 for ay equilibrium p ad ay oequilibrium p. The the simultaeous tdtoemet process (3) is globally stable, provided the equilibrium vector p is uiquely determied; fi(t) = Pi [zi(p(t) ) ], P > 0 sufficietly small, i = 1,..., ; ad zi(p) are cotiuously twice differetiable with a osigular matrix [ a2z' - at the equilibrium p. io api pj apk, k I fact, as will be rigorously show i the Appedix, the weak axiom of revealed preferece implies that the fuctio (5) O(p) = i (p--pf)2 i=1 is a Lyapouov fuctio; ad the Stability Theorem above may be applied. The weak axiom of revealed preferece at equilibrium is satisfied if, e.g., all the commodities are strogly gross substitutes4 or there is a commuity utility fuctio. 1 We are cocered with the so called Lyapuov secod method. See Lyapuov [6] or Malki [7]. The first applicatio of the Lyapuov secod method to the ecoomic aalysis was doe by Clower ad Bushaw [4]. 2 The preset Stability Theorem is a differece equatio aalogue of the oe which was proved i Uzawa [12] for systems of differetial equatios. For the proof, see the Appedix. 3 The weak axiom of revealed preferece which was origially itroduced by Samuelso [9] ad Wald [13] is usually formulated as follows: (4)' For ay two price vectors pi ad p2, pi z(pl) 2 pl z(p2), z(pl) : z(p2) imply p2 z(pl) > p2 z(p2). It is easily see that (4)' implies (4). 4 See Arrow, Block, ad Hurwicz [1], Lemma 5, p. 90.

6 186 REVIEW OF ECONOMIC STUDIES We shall ext be cocered with the successive tdtoemet process which was itroduced by Walras i [14], pp The process uder cosideratio cosists of a price adjustmet which successively clears the markets of commodities. Let p(0) = [pi(0),..., p(o)] be a price vector iitially aouced. Secretary of Market first pays attetio to the market of commodity 1, ad determies the price pl(l) of commodity 1 at the ext stage so that there will be o excess demad at price vector [p(l)), (0), * * *(0)]. I other words, he solves the equatio Z[IP1, P2(0),..., p(0)] = 0, with respect topi. He the goes o to cosider the market of commodity 2 ad determies the price p2(l) of commodity 2 by solvig the equatio z2[p(1), P2, p3(0),..., p (0)] -= 0, with respect to P2. The prices P3 (1),..., p(l) are determied successively. I geeral, the price pj (t + 1) of commodity j at stage t + 1 is determied so as to satisfy the equatio zj [pi(t + 1),..., Pji-i (t + 1), pj(t + 1), pj+,(t),..., p(t))] = O, t = 0, 1,2,.. ;j = 1,...,. I takig ito cosideratio the requiremet that prices should be o-egative, we may modify the above process as follows: pj(t + 1) is determied so as to satisfy the relatio that' r= 0, if pj(t + 1) > 0, (6) z[p(t 1), 1), pj(t..., pl(t + ),..., p(t)] [ 0, if pj(t + 1) = 0, t = 0, 1,2,.. ;j = 1,...,. I order for the successive tdtoemet process to be possible, it is required that, for ay o-egative p(t + 1),..., pj_(t + 1), p+(t),..., p(t), the relatio (6) should have a uique o-egative solutio pjtt + 1). It is evidet that a price vector p(t) = [pl(t),..., p(t)] is left uchaged by the successive tdtoemet process if ad oly if p(t) is a equilibrium price vector. We shall ow cosider the stability of the process defied by (6). Theorem remais valid for the successive tdtoemet process (6). The Stability Let us first cosider the case i which all commodities are strogly gross substitutes; i.e., for ay commodityj, icludig the umeraire, the excess demad fuctio zj (pi..., p) is a strictly icreasig cotiuous fuctio of prices of commodities other tha j. I this case, the equilibrium price vector P = (.,... ) is uiquely determied ad is positive. We have the followig theorem: If all commodities are strogly gross substitutes, the successive tdtoemet process (6) is globally stable. 1 The system of differece equatios (6) defies a iterative method for solvig systems of equatios which is kow as the Gauss-Seidel method i umerical aalysis. The case hadled by Seidel [11] is the oe i which z(p) is liear, [azi/apj]i, j is symmetrical, ad azi/?pp are all positive. Aother formulatio of the successive tdtoemet process was discussed by Professor M. Morishima i a upublished paper.

7 WALRAS' TATONNEMENT IN THE THEORY OF EXCHANGE 187 For the case : = 2, the global stability of the process (6) may be easily see from the Figure. The geeral case will be proved i the Appedix. p2 zt(p).o z2(p)o 'Pi PI We ext cosider the stability of the process (15) for the case i which the followig two coditios are satisfied : (7) For ay fixed p..., j-, p+1,.., P, the equatio Z(pi,..., p-1, pj, P+i,.., S,) == 0 has at most oe o-egative solutio with respect to pj, ad Zj(pi.,* I P-i +- P... P) < 0, for sufficietly large p;. (8) There is a differetiable fuctio (I(p) defied for all positive price vectors p such that, for some positive fuctios X(p),..., (p), - = - Xj(p) Zj(p), j = 1,...,. The coditio (8) is satisfied if, e.g., there is a commuity utility fuctio or the excess demad fuctio has a symmetric matrix of partial derivatives. I this case, the successive tdtoemet process (6) is quasi-stable. The fuctio N(p) satisfyig the coditio (8) becomes, as will be show i the Appedix, a Lyapuov fuctio for the process (6). We thirdly cosider the two-commodity case i which the coditio (7) is satisfied. The successive tdtoemet process (6) the is trivially stable. I fact, the relatio (6) for t = 0 is reduced to z[pi(l)] = 0, which, by the Walras law, implies that pl(l) is a equilibrium price of commodity 1.

8 188 REVIEW OF ECONOMIC STUDIES It may be fially oted that the above three cases, together with the domiat diagoal case, exhaust the cases essetially for which the stability of the Samuelso processa differetial equatio formulatio of the successive tdtoemet process-is kow." Staford. See Samuelso (101, part II; Arrow ad Hurwicz [31; Arrow, Block, ad Hurwicz [1]. H. UZAWA. 1. Proof of the Existece Theorem APPENDIX We first ormalize price vectors such that the sum of prices of all commodities is oe. Let P be the set of all price vectors thus ormalized: P ={p = (Po,P1,..,pt): pt 2i= 0,i=0 1,...,, ad E pt = 1} I-0 The simultaeous tttoemet process, iduced o the set P of ormalized price vectors, defies the followig mappig p T(p) = (TO(p))..., T(p) (9) T,(p) = m()max {O,p' + f zi(p)}, i=0,1,..., l, p gp, where A is a positive umber ad X (p) = max {0, pt + zi(p)} lio It is evidet that ) (p) > 0, ad T(p) is a cotiuous mappig from P ito itself. Hece, by applyig the Brouwer fixed-poitheorem,' we kow that there exists a ormalized price vector = p (p... fi) such that p-=t(a). which, i view of (9), may be writte (10) X i = max {0jp +, zto ) i = 0, 1,...,. Multiplyig (10) bypi ad summig over i = 0, 1,..,, we get I ) 1f0 1-0 f-0 which, by the Walras law, implies Hece, p is a equilibrium price vector. Q.E.D. 1 Cf., e.g., Lefschetz [5], p. 117.

9 WALRAS' TNTONNEMENT IN THE THEORY OF EXCHANGE Proof of the Stability Theorem Let p(t) be the solutio to the process (3) with a iitial price vector p(o) which is arbitrarily give, ad let p* be ay limit-poit of p(t), as t teds to ifiity; i.e. p* = lim p(tv), V --00 for some subsequece { tv }. Cosider the solutio p*(t) to the process (3) with iitial price vector p* ad defie the fuctio qp*(t) by p*(t) = 4D[p*(t)]. Sice 9p(t) = OD[p(t)] is oicreasig ad {p(t)}is bouded, lim cp(t) exists, ad is t oo equal to, say, p* =* = lim 9(t). t --* O the other had, sice the solutio p[t; p(o)] to the system (3) is cotiuous ad uique with respect to iitial positio p(o), we have p*(t) = p(t;p*) = lim p[t ;p(t; ) V = lim p[t + tv; p(o)]. V --00 Hece, we have 9 *(t) = D(p*(t)) = lim D(p(t+tv)) = lim cp(t +tv) v---). 00 V--* 00 which, by the coditio (b), shows that p* = p*(0) is a equilibrium. = *, for all t, Q.E.D. 3. Stability of the Simultaeous Tdtoemet Process: The Weak Axiom Case We may without loss of geerality assume that fi(t) fzi (p(t)), i = 1,...,, ad - f is a sufficietly small positive umber. For the solutio p(t) = p[t; p(o)] to the system (3) with a arbitrarily give iitial price vector p(o), we cosider the fuctio qp(t) defied by 9(t) = (D[p(t)], t = 0, 1, 2. We first show that (11) 9(t + 1) g 9(t) - f3 {2[zo(t)+ ± pfizi(t)] - z2(t) where z(t) = zip(t)]. From the relatio (3), we have

10 190 REVIEW OF ECONOMIC STUDIES (12) p](t + 1) < [pi(t) + 3 Zi(t)]2 = p2(t) + 2 p{(t) zt(t) + P2 z(t), i = 1,...,. Summig (12) over i = 1,...,, we get (13) pp2(t + 1) ; p2(t) + 2p pi(t) z,(t) + 12 z?(t). i=1 i-l i=l 1= Substitutig the Wahllas law (1) ito (13), we have (14) Z p2(t + 1) < 2 p2(t) - 2p zo(t) + 2 z2(t). 1=1 /=1 i=1 O the other had, multiplyig (3) by pi ad slummig over i = 1,..,, we get (15) S pipi(t + 1) 2 ~ fipi(t) + P3 ~ pt zi(t). /=1 /-=1 /=1 Subtractig two times (15) from (14) ad addig Z p2, we ca derive the iequality (11). i=1 Let S be the set of the commodities such that, at equilibrium jp, they have egative excess demad : S = {i: zi(p) < 0}. Equilibrium prices of the commodities i S are zero: (16) pi = 0, for i S. We shall ow show that, if p is a sufficietly small positive umber, there exists to such that pi(t) = 0, for all t 2 to, i e S. Let e be a positive umber such that (17) Zi(p) - c < 0, for all i e S ad ay price vector p such that ()(p) _ e, where c is a positive umber. Sice the excess demad fuctio z(p) is cotiuous, it is possible to choose a positive umber e which satisfies (17). We may assume without loss of geerality that e< c?(0) = 0[p(0)]. Let p be ay positive umber smaller tha the followig two umbers: e/2 if (18) if e / L (/22 (p) ()] 2[zo(p) + : pi z(p)] The weak axiom (4) ad the cotiuity of the excess demad fuctio z(p) ow imply that the two umbers defied by (18) are positive, so that the choice of p is possible. The, by the iequality (11), we have (19) qp(t + 1) _?(t) + 2 e,' if?p(t)! 2'

11 WALRAS' TATONNEMENT IN THE THEORY OF EXCHANGE 191 ad (20) p(t + 1) < (t), ifj 2< 9(t)?(0), t = 0, 1, 2... The iequalities (19) ad (20) imply that there exists t, such that (21) 9(t)! e, for ay t 2 tl. Otherwise, p(t) is strictly decreasig ad the solutio p(t) is bouded, while o limit-poit of p(t) is the equilibrium, thus cotradictig the Stability Theorem. The relatio (3) together with (17) ad (21) implies that there exists to for which (16) is satisfied. The relatio (16) ad a argumet similar to the oe by which we derived (11) lead to the followig : (22) p(t + 1) 9p(t) 13 {2[zo(t) We shall ow show that + Z pizi(t)] -P z2(t) ii=1 i- S J. if [zo(p) + Z PiZi(p)] (23) 0 < AD(p) _ C Z z2(p) ifs is positive. Expadig the umerator ad the deomiator of (23) i the Taylor series at the equilibrium p, ad otig the Walras law (1), we get zo(p) + pfizi(p) = S ajk(pj - fj) (pk - k) + 0[0(p)], 1=1 j,k==l where S z(p) = Z bjk (pj - P) (pk - pk) + 0[D(p)],, ios j,k=l a2 Zo - a2 ajk (= p i Azp ) apb apk apz ( 1 apk ibjk - (P' ad 0[(D(p)] correspod to terms such that a j, k - 1,..., lim O[OD(p)] = O l0im) 0 0(p) The weak axiom together with the o-sigularity of (ajk)j, k =,..., implies, however, that the matrix (ajk)j, k i,..., is positive defiite. Hece

12 192 REVIEW OF ECONOMIC STUDIES Zo(p) + f P zi(p) lri= > 0. (D P) 0 YS z (p) iqos which implies that the umber (23) is positive. Now let 3 be ay positive umber smaller tha the umbeis defied by (18) ad (23). The, by the relatio (22), we have that y(t + 1) < y(t), wheever p(t) is ot a equilibrium. Applyig the Stability Theorem, we have the stability of the process (3). 4. Stability of the Successive Tdtoemet Process: The Gross Substitute Case Cosider the fuctios A(p) ad X(p) defied by Q.E.D. (24) A(p) maxi 1, ' -- (25) A (P) mi {19 L PI) Pi''P **s We shall show that, for ay solutio p(t) to the process (6), (26) A[p(t + 1)] ; A[p(t)], with strict iequality uless A[p(t)] = 1, ad (27)?[p(t + 1)] 2 4[(t)], with strict iequality uless 4[p(t)]J 1. Sice the relatio (27) is proved similarly to (26), we shall give a proof for (26). I order to prove (26), it suffices to prove the followig: (28) A[p1(t + 1),..., pl(t + 1), p;+1(t). * 1 p(t)] with strict iequality uless A[pl(t + 1),..., p;_.(t + 1), p;(t),.., pm(t)] 1, t-=0,1,2;j= 1,...,. Gross substitutability, homogeeity of order zero, ad the Walras law (1) imply that' (29) z;[p1(t + 1),..., pj-.(t + 1), p1, p;+l(t), *. *, pi(t)j] 0, wheever Pi > A[pl(t + 1),..., pj.1(t + 1),pj(t),... p(t)]. 1 See Arrow, Block, ad Hurwicz [1], Lemma 3, p. 89.

13 WALRAS' TATONNEMENT IN THE THEORY OF EXCHANGE 193 ad, if A[pl(t + 1),..., p1_(t + 1), pj(t),..., p(t)] > 1, (29) holds with strict iequality. But, by gross substitutability ad the Walras Law (l), zj[p1(t + 1),..., Pj- (t + l), pj pj+1(t),.., p(t)] is strictly decreasig with respect to pj. Hece, (29) implies that (30) p t A[pl(t + 1),..., p (t + 1), (t),... p(t)], with strict iequality if A[pl(t + 1),..., p-(t + 1), pj(t),..., p(t)] > 1. The iequality (28) is easily implied by (30). The relatios (26) ad (27) i particular show that the solutio p(t) to the process (6) is i a bouded set {P (Pi,... ) : (0) g A(0), j 1..., of positive price vectors. Sice A(p) = X(p) = 1 if ad oly if p is a equilibrium, the relatios (26) ad (27) show that the fuctio D(p) defied by 0(p) = A(p)- X(p), is a Lyapaov fuctio for the process (6). Q.E.D. 5. Stability of the Successive Tdtoemet Process; The Quasi-Itegrable Case Let D(p) be the fuctio satisfyig (8) ady j(pj) be the fuctio defied by (31) Vi(PJ) = I[p1(t + 1),..., pgj-(t + 1), pj, pj+l(t),... p,(t)]. The defiig relatio (6), together with (8), shows that pj(t + 1) is a solutio to the equatio (32) = 0, if p > 0, (32) s(pi)\ 2 0, if pj = 0. The relatios (7) ad (32) show that pj(t + 1) uiquely miimizes unj(pj) subject to pj _ 0. Hece, i particular, we have (33) vj[pj(t + 1)] < (t)], with strict iequality if pj(t + 1) = pj(t). By (31), we may rewrite (33) as follows: (34) $[pl(t + 1),..., pj(t + 1), pj+1(t),..., (t)] [pl(t- + 1),..., p-(t + 1), pj(t),... p(t)], with strict iequality if pj(t + 1) = pj(t). By summig (34) over j D[p(t + 1)] 1 D[p(t)], = 1,..,, we get with strict iequality if p(t + 1) = p(t).

14 194 REVIEW OF ECONOMIC STUDIES Hece, the fuctio (D(p) is a Lyapouov fuctio; ad applyig the Stability Theorem, we kow that the process (6) is quasi-stable i the preset case. REFERENCES [1] Arrow, K. J., H. D. Block, ad L. Hurwicz, " O the stability of the competitive equilibrium : II," Ecoometrica 27 (1959), [2] Arrow, K. J., ad G. Debreu, " Existece of a Equilibrium for a competitive ecoomy," Ecoometrica 22 (1954), [3] Arrow, K. J., ad L. Hurwicz, " O the stability of the competitive equilibrium: I," Ecoometrica 26 (1958), [4] Clower, R. W. ad D. W. Bushaw, " Price determiatio i a stock-flow ecoomy," Ecoometrica 22 (1954), [5] Lefschetz, S., Itroductio to Topology. Priceto : Priceto Uiversity Press, [6] Lyapuov, A. "Probleme geeral de la stabilite du mouvemet," Aales de la Faculte des Scieces de l'uiversitg de Toulouse, (2), 9 (1907), [7] Malki, I. " O the stability of motio i the sese of Lyapuov," Recueil Math6- matique [Math. Sborik], N. S., 3 (45) (1938), Traslatio: America Mathematical Society Traslatio No. 41 (1951). [8] Patiki, D., Moey, Iterest, ad Prices. Evasto: Row, Peterso ad Co., [9] Samuelso, P. A., " A ote o the pure theory of cosumers' behaviour," Ecoomica, Vol. 18 (1938), 61-71, ad [10] Samuelso, P. A., Foudatios of Ecoomic Aalysis. Cambridge: Harvard Uiversity Press, [11] Seidel, P. L., " tber ei Verfahre die Gleichuge, auf welche die Methode der kleiste Quadrate fiihrt sowie lieare Gleichuge uberhaupt durch sukzessive Aahrug aufzulose," Abhadluge der bayersche Akademie der Wisseschafte, Math-pysik. Klasse, 11 (1874), [12] Uzawa, H., "O the stability of dyamic processes," Ecoometrica, to be published. [13] Wald, A., "Uber die Produktiosgleichuge der okoomische Wertlehre," Ergebisse eies mathematische Kolloquiums, No. 7 (1934-5), 1-6. [14] Walras, L., Elkmets d'ecoomie Politique Pure, Paris et Lausae, 1926: traslated by W. Jaff6, Elemets of Pure Ecoomics, Homewood: Richard D. Irwi, Ic., 1954.