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NOVEMBRE 1990 . MARKETS WITH COUNTABLY MANY PERIODS • Mon~que FLORENZANO CNRS-CEPREMAP N° 9019 1 MARCHES SUR UNE INFINITE DENOMBRABLE DE PERIODES Résumé • Un des premiers résultats d'existence de l'equilibre pour une économie avec une infinité de biens est celui où Peleg et Yaari (1970) considèrent une économie d'échange définie sur l'espace des suites réelles ; les consommateurs y sont supposés avoir des préférences totalement préordonnées, convexes, continues et strictement monotones, tandis que l'offre totale de ressources est supposée stictement positive. Une abondante littérature a depuis été consacrée aux économies définies sur un espace de biens de di~ension infinie. La théorie des espaces de Riesz et une hypothèse dite de "propreté uniforme" des préférences et/ou de la production se sont avérées jouer un rôle crucial dans les théorèmes d'existence. L'objectif du papier est de trouver l'hypothèse cachée de propreté uniforme à l'oeuvre dans Peleg et Yaari et, grâce aux espaces de Riesz, de généraliser les résultats aux économies intertemporelles admettant l'incertitude à chaque période. Espaces de clés Mots - symétrique Economie Riesz localement - d'échange convexes Equilibre - Systéme de Riesz dual intertemporel - Préférences uniformément propres . • MARKETS WITH COUNTABLY MANY PERIODS Abstract The existence of equilibria is proved in a stochastic finite number of agents and countably many periods. words Key Riesz dual : Locally convex-solid systems Exchange Uniformly proper preferences. JEL : 021 market with a topological vector lattices economy Intertemporal - Symmetric equilibrium - 2 MARKETS WITH COUNTABLY MANY PERIODS Monique FLORENZANO CNRS-CEPREMAP, 75013 Paris, France The existence stochastic market with a of equilibria is proved in a finite number of agents and countably many periods. 1. Introductio n This note is intended to reconsider, at the light of some recents works, an equilibrium existence result obtained by Peleg and Yaari (1970) for an exchange economy with IR'° (the space of all real sequences) as commodity space, IR'° as price space and no explicit commodity- price duality. We first restate the topological vector lattice properties of the commodity space to be considered in this model. A limited attempt in this direction is made by Besada et al. (1988) who introduce Kothe perfect spaces as commodity spaces which arise in a natural way in connection with go further and do a similar constructio n in the case where, at each period, the consumption choice of each agent may be stochastic. The resulting commodity- price duality is described by what is called a symmetric Riesz dual system by Aliprantis et initial the endowments the of agents. We al. (1989), We then look establish the relevance pointed satisfied in a it is at the out by hidden uniform properness assumptions which would of this vector lattice theoretic constructio n. As Aliprantis et al. (1989), uniform properness is subspace of the commodity space,conve niently topologize d; in this subspace, an equilibrium existence theorem can be deduced from MasColell's theorem (1986). The main result of this note is to show that under to assumptions similar equilibrium is associated with prices in its dual. the ones used an equilibrium by Peleg in the and Yaari, to this initial commodity space 3 2. The economic model and Burkinshaw Aliprantis period. time the denotes used terminology and notations The in et al. (1989). The (1978), Aliprantis duality commodity-price The note are borrowed from this at index t period t is represented by a symmetric Riesz dual system <Et, E;>.If xtE Et and PtE E;, we writè Pt·xt for <xt, Pt>. We consider m infinitely lived agents i = 1, ... ,m, each of which has at m oo L oo wi E nE; initial endowment Wi= (w;) E nE;. Let w = t=l i=l t=l be the total endowment. its disposal an 00 00 P = {p E nEt L lpt l .wt t=l t=l < +oo } and 00 00 !\(P) = {x E n Et will The I L lpt 1. lxt 1 < +oo for all p E P} t=l t=l be respectively the price space and the commodity space of our model. commodity-price duality is represented by the bilinear form: <x,p> = 00 L Pt .xt. If x E /\(P) and p E P, p.x is the value of x reckoned at t=l the prices p = (pt). The construction of /\(P) makes it natural to consider the topology T p.x = defined on /\{P) by the Riesz seminorms 00 L lpt 1. lxt 1 , p E P • t=l Similarly, r'is the topology defined on P by the Riesz seminorms Pp (x) = 00 L lpt 1. lxt 1 , x E /\{P) . t=l As it is proved in the propositions 1 and 2 of the appendix, <A(P),P> is a d ual pair; T ( resp. T t) is a Hausdorff locally convex-solid topology on Px(p) = !\(P) (resp. P), consistent with the duality. Moreover, let a(/\(P),P) and a{P,/\{P)) be the weak topologies associated with the duality; every orderinterval of /\(P) (resp.P) is a(/\(P),P){resp. a(P,/\{P)))-compact. In other words, <A(P),P> is a symmetric Riesz dual system. As Peleg and Yaari (1970), Besada et al. (1988), we consider an exchange economy, to be called i:E. A commodity bundle is an element of /\(P)+ .Each agent i has indifference reflexive, an initial relation ~i transitive and bundle on wi > 0 /\(P)+ complete. In in The /\{P)+ relation and a preference or ~i is assumed to be addition, the initial bundles and preference relations are required to satisfy the following assomptions : 4 X For i = 1, ... ,m, if x,y monotonicity 1) Strict E A(P)+ and x >y, then >iy. { y i = l, ... ,m, For 2) Convexity x each for E A(P) + the set A{P) + 1 Y ~ix } is convex. E the sets For i = 1, ... ,m, for each X E A(P) + • r- continuity { y E A(P}+ I y ~ix } and { z E A(P}+ 1 X ~iz } are both r-closed in A(P}+. The r-continuity assumption is a myopy assumption on preferences. A fourth assumption corresponds to the positivity of totai supply with an additional requirement which is automatically satisfied if each Etis an 3) Euclidean space: 4) For every period t, wt is an order-unit of Et and Et= (Et)~ (the order- continuous dual of Et). yi{p} = { x E A(P)+ 1 p.x ~ p.wi} c5i{p} = { x E A(P)+ 1 p.x < p.wi }. Let (resp. a quasi-equilibrium) for economy c8 is an A competitive equilibrium -m - -1 - - = (x 1 •••• ,xm)) (m+l)-tuple (x , ... ,x ,p} such that p E P, p m m . a) L xi= L wi i=l b) budget set of i and set be the (attainability of X ~ 0 and i=l and y >i xi implies each i = 1, ... ,m xi E yi(p) for y$ yi(p} {resp. y$oi{p}). 3. Existence theorem A{P} is Dedekind-complete. Let Aw be the principal ideal of A{P} generated by w, Aw = { x E A(P) 1 there exists À> 0 with lxl ~ ÀW }. Recall that under the Riesz norm llxll 00 = inf{ . Aw is an by . > 0 1 lx 1 À ~ ÀW } ' AM-space with unit w. The norm dual of (Aw, 11.11cc) will be denoted AC:,. According to Mas-Colell (1986), an economy is said to satisfy the closedness condition whenever for every sequence of attainable allocations (x 1 ·v, ... ,xm,v) which satisfies xi,v+l ~i xi,v for all v and all i=l, ... ,m, there exists another attainable allocation (x 1 , ••• ,xm) satisfying xi~ xi,v i=l, ... m. If viand viare respectively a vector and a 0neighborhood of the commodity space L, ~i is said to be (vi, Vi)-uniformly proper if for all x EL+, x + ÀVi + z EL+, À> 0 and z E ÀVi imply for all v and all x + ÀVi + z >i x. w is said desirable for every i = 1, ... ,m, if x EL+ 5 x + implies >i x for all œ > O. In a topological vector lattice, if an a!.() condition, the closedness satisfies endowment total desirable a and preferences proper uniformly continuous, monotone, convex, with economy exchange has a quasi-equilibrium with prices in the topological then it dual of the commodity space. In order to apply Mas- Colell's existence result, we will first consider ©w, the same economy than ©, but restricted to the commodity space Aw. It is worth noticing that © and ©w have the same attainable allocations. p) quasi-equilibriwn. (x, Proof. It the asswrrptions Under 1. Proposition E (A:)m x A~+ with assumption from follows 2) p (w) = 1. w is that 1) ©w has a 3), and 1), desirable for every also well known that, under assumption 1), every point of i=l, ... ,m .It is 1 int (A:) ( the Il. 11-interior of A:), in particular - w, is a properness vector m every ~i in ©w. As it is pointed out by Aliprantis et al. (1989), the for the order-convergence 11.11 00- convergence implies the order-convergence topology, continuity of satisfies the it a Lebesgue is So the 11.11 00- assumption 3), Finally, ©w follows the from easily 3) and the (a(A(P}, P))m-compactness of the set of all 2) and assumptions from as condition closedness T r-convergence. implies the ©w follows in ~i each ; since attainable allocations. The conclusion follows from Mas-Colell's theorem. D Coroiiary. Let p be the normal component of p (x, p) is a quasi-equili- briwn. of ©w and p(w) > O. pE (A~)+ c (A;)+ ,so by the Riesz decomposition theorem, there is a the disjoint unique decomposition of p, p E (Aw);: and q E ((Aw);:)d = complement of (Aw);:, p ~ 0, q ~ 0, such that p = p + q. We first remark that (Aw);: separates the points of Aw; indeed, as it is shown immediately after the Proposition 2 of the appendix, P c (A{P}};; c (Aw);: and P separates the points of Aw • Then let Nq= { x E Aw I lq 1 (x) = 0 } be the Proof. null q in ideal of Aw.It follows from Theorem 2.2 in Aliprantis et al. (1989) that Nq is order dense. Then that suppose for some xi >i i, xi 0 in . -i (xia) c Nq, xia ~xi.For large enough œ, xia >i x (p + i such ©w· There exists a net - - . . and p(xia) = p(xia) ~ q)(wi) ~ p(wi}, By passing to limit, p(xi) ~ p(wi). Since there exists that (p + q)(wi) > 0, it is easily deduced from the strict 6 monotonicity of preferences ~ that p O. From the strict monotonicity of . m . we get that for each i, p(:i=1') 2:: p(wi). Finally, since = Ix'I, preferences, i=l Lm wi, = -i p(x ) = p(wi) holds for all i. i=l Recall now that Theorem 6.4 (see Aw is a Banach lattice and thus that A~ is a band of A: in Aliprantis w is p E A~; since in the and Burkinshaw li.li- interior of (1978)). It follows that A_;,, p(w) > 0 and p can be re-normalized so that p(w) = 1. D Proposition (1989), 1 is nothing else reproved there construction (1972). To used in go in order than Theorem to keep 5.6 in Aliprantis et al. this note self-contained. The the proof of the corollary is classical since Bewley further, we need to exploit Assumption 4), through the proposition 3 of the appendix. Proposition 2. Under the asswrrptions 1), 2), 3) and 4), (x, p) is an equitibrium of <5 and pis a strictiy positive eiement of P. Proof. From Proposition 3 in the appendix, we know that Aw is order dense (x, p) is a quasi-equilibrium of <5w as in the corollary of Proposition 1, p E P. Suppose now that for some i, xi >i xi in <5. There exists a net (Xia) C Aw, Xia: xi. For large enough œ, Xia >i xi and p(xia) = p .Xia 2:: p(wi) = p.wi. By passing to limit, p.xi 2:: p.wi, which shows that (x, p} is a quasi equilibrium of <5. Since p. w > 0, there exists i such that p. wi > O. For all z > 0 in and thus r- dense in À(P) and that P = (Aw);. Then if -i -i À(P), by the strict monotonicity of preferences, X + z >i X and by the T-i -i = = We get continuity of preferences, p= X (x + z) > p . wi = p :i p .z > 0, which shows that pis strictly positive. Hence p= . wi > 0 for all . . . i and (x, p) is an equilibrium. D Proposition of 2 is the existence (1988). a straight generalization (as to the commodity space) statements in Assumptions 1-4 Peleg and Yaari (1970) or Besada et al. on the economic model are exactly the assumptions made in these two papers. By adding an explicit desirability assumption for the total assumption endowment w, it would on preferences. be easy to The quasi-equilibrium weaken the monotonicity price would not need be 7 assumption would be necessary to an irreducibility positive and strictly guarantee the existence of an equilibrium. assumption 4) allow A(P) space commodity together the with to consider at each period a commodity-price duality the idea of a stochastic choice at each period. A <L00, L1 > which captures the case where at each period, the commodity space application is simpler the of construction The is an Euclidean space, of which the dimension may depend on time. Appendix Let <A(P),P>, and T T' be defined as in the beginning of Section 2. resp. T') is a Hausdorff locally Proposition 1; <A(P),P> is a dual pair.T convex-solid topology consistent with this duality. 00 00 Proof. Let cp (resp. 4>' )denote the set of all elements of nEt (resp. nE;) t=l t=l and Pare with only finitely many non-zero coordinates. Obviously A{P) 00 00 nEt spaces, more precisely ideals of Riesz and nEt containing 4> et 4>' • t=l t=l 00 properties separation The <x, p> = p.x = bilinear form of the L Pt .xt t=l follow from the last remark. 00 Each i.e. a Px(P) ~ L lptl· lxtl seminorm Pp (x) (resp.px{p)) = seminorm Riesz Px(q) ) ;thus r lxl ( resp. ~ t=l ~ lyl r')is Pp(x) ~ is obviously monotone, Pp(Y) (resp. lpl ~ lql ~ a locally convex-solid topology on A(P){resp. P). clear that P c (A{P),r)', the topological dual of A{P).To should be It the converse prove with coincides seminorms : xt Pt E E; ~ the inclusion, let absolute weak that on us remark each factor Et, r lal(Et,Et) generated by the topology lptl· lxtl· Hence, if f E (A(P),r)' ,for each t there exists such that f(O, ... ,O,xt,O, ... ) = Pt·xt. Since for all x in A{P), 00 (x 1 ,x2 that ,xt,O, ... ) r-converges to x, f(x) = LPt·xt· It remains to prove t=l p = (Pt) E P. lfl {0, ... , 0, xt, 0, ... ) = lpt l .xt, so that If 1{w) = , ••• 00 L lpt i .wt < +oo. From a similar proof, it follows that A(P) = (P ,r')'. t=l Finally, if a(A(P) ,P) and a{P,A{p)) are the weak topologies associated with 8 duality <A(P),P>, remark that a(A(P),P) c the proves that T and a(P,A(P)) cr', which and r' are Hausdorff. T D Proposition 2. Each order-interval of A(P) (resp. P) is a(A(P) ,P) (resp. a(P,A(P))-compact. Proof. é If u > 0 in A(P), let [-u,+u] be an order-interval of A(P). For any > 0, Pu(P) < and x E [-u,+u] imply E 00 L Pt .xt This shows compactness now 00 L L lpt .xt 1 ::;; lpt 1. lxt 1 ::;; Pu(P) < E. t=1 t=1 r'- equicontinuity of [-u,+u]. The a(A(P),P)-relative t=l the 1 ::;; of [-u,+u] that each 00 follows,from the Alaoglu-Bourbaki theorem. Recall <Et,Et> is a symmetric Riesz dual system. Each [-ut,+ut] is a(Et,E;)-compact. It follows easily that I-u,+u] is a(A(P},P)-closed. The a(P,A(p)}-compactness of order-intervals of Pis proved in a similar way. D ',, Applying Theorem 22.1 of Aliprantis and Burkinshaw(1978}, we see that A(P) (resp.P) is Dedekind-cornplete and, (resp. ,') is a Lebesgue topology. If (A(P)};; (resp. P;;) linear order-bounded denotes forms the on collection A(P} (resp. of all order-continuous, P), P c (A(P));; (resp. A(P) c P;;). Assume now, as in assumption 4), that for every t, wtis an order-unit of Et and that E; = (Et)~.Recall that Aw is the principal ideal generated by w in A(P). Proposition 3. Under assumptions 4), Awis order dense in A(P) and (Aw)~ CP C (A(P))~ C (Aw)~. Proof. From the first part of the assumption, it follows that each factor Et can be considered as a Riesz subspace of Aw. For all x in A(P}, (x 1 ,x2 , ••• ,xt,O, ... ) î x. This proves the first part of the statement. To second prove the part of first inclusion, the assumption, let f be any element of (Aw)~. From the one deduces that there exists PtE (Et)~ 00 = E~ such that j(x) = LPt·xt and that p = (pt} E P. t=l The second inclusion was proved above; the third one is obvious. D ~ 9 References economies Brown and O. C.D.,D.J. Aliprantis, with countable a number Analysis and Applications 142, and C.D. Aliprantis, Burkinshaw,1978, O. New York and London). Besada, Esteves and M. countably many commodities, Bewley, T.F., many commodities, Këthe, 1969, G., Mas-Colell, lattices, Peleg, B. A., Journal Herves, M. Economies Letters Solid Riesz 26,203-207. 514-540. (Springer-Verlag, Berlin). The price equilibrium problem in topological Yaari, Spaces equilibria in economies with infinitely vector 1039-1053. 1970, Markets with coutably many commodities, International Economie Review 11,369-377. ,. of Mathematical Equilibria in economies with 1988, Topological Vector Spaces 1986, Locally of Economie Theory 4, Econometrica 54, and C. Existence of 1972, Journal 250-299. (Academic Press, M., agents, of Equilibria in exchange 1989, Burkinshaw,