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On the existence of equilibria in economies with an infinite dimensional commodity space
Journal
of Mathematical
Economics
12 (1983) 207-219.
North-Holland
ON THE EXISTENCE OF EQUILIBRIA IN ECONOMIES WITH AN INFINITE DIMENSIONAL COMMODITY SPACE* Monique FLORENZANO CNRS, CEPREMAP, Received
September
75013 Paris, France
1982, final version
received
September
1983
We prove an infinite dimensional extension of the Gale-Nikaido-Debreu lemma which includes all necessary limiting processes and allows a proof of the existence of equilibria under standard assumptions in an economy with infinitely many commodities which exactly parallels the proof of Debreu (1959) for the finite dimensional case.
1. Introduction
The assumption that the commodities are not in finite number agrees with many classical situations for economic theory: intertemporal equilibrium with an infinite horizon, uncertainty with an infinite number of states, differentiation of commodities. Since the paper of Bewley (1972) which proved the existence of a Walrasian equilibrium for an economy with a finite number of agents and L,(M, A, p) for commodity space [let us recall that M is a state space, J%’a o-field of subsets of M, p a a-finite measure on J%!, and L,(M,A, p) the Banach space of all measurable functions such that Ilf((, = sup
-While the proof of Bewley is based on a limiting process which deduces the existence of an equilibrium in the infinite dimensional case from the existence of an equilibrium in the finite dimensional case, Bojan (1974), Toussaint (1981), Magi11 (1981), Aliprantis and Brown (1982) look for a direct proof which is the analogue of one of the proofs in the finite dimensional case. *The main result of this paper was achieved when I was visiting the group for the Applications of Mathematics and Statistics to Economics at the University of California, Berkeley, Summer 1982. The support of National Science Foundation grant SES 79-12380 Debreu to the University of California is gratefully acknowledged. 1 would like to thank Gerard Debreu for his hospitality and his advice. I also wish to thank Bernard Cornet who followed the successive versions of this paper and made many suggestions, and Andreu Mas-Cole11 and Karl Vind for helpful discussions. 03044068/83/$3.00
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M. Florenzano, Equilibria with infinite dimensional commodity space
-At the same time, Aliprantis and Brown (1982) try to set the most general framework in which their results can be achieved. Before them, Elbarkoky (1977) extended the equilibrium existence theorem of Bewley to an outline which is essentially the same as this which will be used here. This paper corresponds to both objectives. It states an extension of the Gale-Nikaido-Debreu lemma, in order to give a direct proof of the existence of an equilibrium which exactly parallels that of Debreu (1959) for the finite dimensional case, in a general framework which is to be defined in section 2. The idea of the proof is to use the duality between L,(M, .,&‘,p) and L,(M, &,,u) and to truncate consumption sets to weak-star compact sets, in order to define excess demand correspondences when the consumer’s preferences satisfy a weak-star continuity assumption; Debreu’s argument is applied to the truncated economy and then a limit argument allows the truncation to go to infinity. The object of the generalization to Banach spaces is to point out the assumptions on the commodity space which prevent from applying our approach (and also those of Bewley, Brown and Alipantis,. . .) to some Banach spaces other than L,(M, ~#,,u) which have been found useful in practice; namely ca(K), the space of countably additive signed measures on a compact metric space [see Mas-Cole11 (1975)] and L&M, &,p), 1 g’p< co [see Ostroy (1982)]. 2. The model In this paper, the space of commodities is a Banach space E, the conjugate space of another Banach space G, i.e., E = G’ where G’ is the vector space of all continuous linear functionals on G and the norm on E is the conjugate norm to the norm on G. G is ordered by a closed convex positive cone G+ satisfying: G = G+ - G ‘. E itself is ordered by the positive cone:
where (p,x) is the value at p of the continuous linear functional XE G’. We shall denote by 2 the orders on G and E. The first assumptions guarantee that in an exchange economy, under the usual conditions of lower boundedness of consumption sets, the attainable sets are compact for the weak topology a(E, G) for E associated with the duality (E, G). This will be deduced from the following elementary proposition: Proposition 1. IfG is a real topological vector space ordered by a positive cone G+ satisfying: G=Gf -G’, and if E is the conjugate space of G, ordered
M. Florenzano, Equilibria with infinite dimensional commodity space
209
by the cone of all positive continuous linear functionals on G+, let a and b be two elements of E and let A= {x E EJa 5 x 5 b}. Then A is a(E, G)-bounded.
Proof: Let PEG, ol~R, cc>0 and V={x~E/((p,x)l~a). absorbed by K If PEG+ and if P=sup{I(p,a)(,(
If pE -G+,
if
fi>O
and
AcV
if
We claim that A is then
/3=0.
one obtains the same result.
where p+ and p- belong to G+, then I’1 T/+ n I’- where and I’- =(x E E/l(p-, x)1 5 42). Let p and v be real numbers such that AcpV+ and AcvV-. Then Aclv where I= sup c/4 v>. Cl If
p=p+-p-
V+=~x~:E/l(p+,x)l~o1/2}
Prices belong to E’, the conjugate space of E. E’ is ordered by the positive cone: E’+ = {n~E’/(x,rr) ~OVXE Ef), where as above we denote by (x, rc) the value at x of the continuous linear functional rr E E. In order to define a price simplex compact for the weak topology a(E’, E) for E’ associated with the duality (E, E’), we need that E+ has a non-empty interior. For example, if, as in Aliprantis and Brown (1982), E is an A.M. space with unit (i.e., a Banach lattice, the lattice order and the norm of which are related by the two relations: ((x v y((=max {\IxI\,/y\I} an d t h ere exists e E E+ such that the order interval C-e, +e] coincides with the closed unit ball (XE E/llxIlz 11) this assumption is satisfied. The following proposition is classical: Proposition 2. Let E be a real Hausdorff locally convex topological vector space, E’ the conjugate space of E and Y be a convex cone of E such that Y” = {II E E’/(y, IC) 50, Vy E Y) # (0). If u belongs to the interior of I: the set A=(nEY0/(u,71)=
-l}
is equicontinuous and a@‘, E)-compact and satisfies Y” = Ulzo AA. Proof.
Let V be a closed circled neighborhood v7t E YO,
vxey
of 0 such that
(u+x,?t)~O.
From
210
M. Florenzano, Equilibria with infinite dimensional commodity space
We consider an exchange economy a=((X’, R’, oi)i = 1,, ,m, Y) consumers and, for each i= 1,. . . , m, a consumption set Xi cE, a preorder& R’ on Xi and an initial endowment oi E E; in addition, activities are described by a convex cone T: contained in (-E+). preordering R’, we associate the strict preference relation defined by xipixtioxiRixfi
and not
x”R’x’.
Let xi EX’; we introduce the following notations: pi(xi) = {,fi E Xi/xfipixi}, Ri(xi) = {,Ji EXi/xfiRixi}, (pi) - ‘(xi) = {x’i E Xijxipixfi), (Ri)-l(xi)=
cxri EXi/xiRixri}.
For each II E E’, we define
y’(n)= {xiEXi/(Xi, n)
5
i.e., the budget correspondence
(co’,7c)}, for consumer i.
6’(n) = {xi EXi/(Xi, 7c) < (CO’,n)),
~‘(7c)={x’EXi/X’Eyi(n) i.e., the demand correspondence ~‘(n)={X’EXi/XiEyi(?l)
and
y’(z) n P’(x’) = a},
for consumer i. and
i.e., the quasi-demand correspondence
~i(~)nPi(x’)=P)), for consumer i.
i.e., the excess quasi-demand correspondence. An allocation x = (xi) E ny= 1Xi is attainable if
with m complete disposal To the
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211
An equilibrium of E is a point (X,y,5) ~(nT=r Xi) x Y x (E’\(O)) satisfying the conditions: Vi=1 ,..., m,Zic[‘(ji),
71~Y”={z~E’/(y,~)~O,Vy~ p=
(1)
Y)
and
(jj,%)=O,
(2)
F o’+y. i=l
A quasi-equilibrium of E is a point (2, jj,iz) E(~Y= I X’) x Y x (E’\(O)) satisfying the above conditions (2) and (3) and the condition (1’): Vi=1 ,..., m,%‘E#(jl).
(1’)
In fact, we will prove the existence of a quasi-equilibrium from an extension of the Gale-Nikaido-Debreu lemma (or market equilibrium theorem) that we state in the next section.
3. The extension of the Gale-Nikaido-Dehreu lemma In the literature, one can find two extensions of the finite dimensional Gale-Nikaido-Debreu lemma to correspondences defined on a compact subset of the space E’ of all continuous linear functionals on a real Hausdorff locally convex topological vector space E, with convex values in a compact subset of E. The first is from Bojan (1974), the second from Toussaint (1981); the common defect of the two statements is that they cannot be applied without a limit operation to prove the existence of equilibrium under standard assumptions in a transitive economy with a finite number of agents and a commodity space as in section 2. Here we prove an infinite dimensional analogue of the Gale-NikaidoDebreu lemma which is the most convenient extension for an application to a proof of existence of equilibrium which exactly parallels in the infinite dimensional case the proof of Debreu (1959) for the finite dimensional case. First, let us introduce some definitions: I. Let $ be a correspondence from a topological space X to a topological space Y and let x0 be an element of X; we define the subset lim supx +x0 6(x) as the set of all elements y of Y such that for all neighborhoods W of y and V of x0, there exists x E V such that 4(x) n W # $I In other words, lirn~up,,,~ 4(x) is the set of all y in X such that (x,,y) belongs to the closure of the graph of b, in the product topology for X x Y Definition
212
M. Florenzano, Equilibria with infinite dimensional commodity space
Definition 2.
Given a convex subset X of a vector space E, let d be the collection of all finite subsets CIof X. If CI= {x’, x2,. . . , xna} EL&‘, let SnQdenote the unit simplex of the Euclidean space R”* and pa,:S”QX the map defined by if
p=
2 pie’,
pi20
Vi=l,...,n,,
i$JI Pi = l,
i=l
then
The finite topology [see Aubin (1979, p. 21 l)] defined on X is the strongest topology for which the maps fib are continuous for all c1E d. A correspondence 4 from X to a topological space Y is upper semicontinuous when X is equipped with the finite topology if and only if the correspondences 4 0j3,: Pa-+ I: defined by
4 oP,(P)= WdP)) are upper semi-continuous for all tl Ed. I.e., when X is equipped with the finite topology for verifying upper semicontinuity of the correspondence 4, it is necessary and sufficient to verify upper semi-continuity of all the restrictions ~/CO c( of the correspondence C$ to the convex hulls co c1of the finite subsets of X. The finite topology on X is at least as strong as any topology induced on X by a vector space topology on E. Thus we can state: Lemma 1. Let E be a real Hausdorfl locally convex topological vector space, E’ the vector space of all continuous linear functionals on E, Y a convex cone of E such that the polar cone Y” = {z E E’/(y, 71) 5 0 Vy E Y} is # {O], u a point in the interior of Y, A = (rc E Y’/(u, rt) = - l}, and i a correspondence from A to E. Let 5 be a Hausdorff locally convex topology for E at least as weak as a(E, E’), and D={~cE A/n is continuous zf E is equipped with F-). We make the following additional assumptions: If E is equipped with 9 and if D is equipped with the finite topology, the restriction c/D of [ to D is #upper semi-continuous with non-empty closed convex values in a compact set Z of E. (2) Vrt~ A,Vz’z~(n), (z,~)~Oo. (3) Y is F-closed.
(0
M. Florenzano, Equilibria with infinite dimensional commodity space
213
Then there exists it E A such that Y n lim SUP,,~ c(z) #$3, where the lim sup is taken for the topology on A induced by a(E’,E) and the topology 9 on E.
The idea of the proof we give is borrowed partly from Bojan (1974), partly from Aliprantis and Brown (1982). Proof
Let d be the collection of all finite subsets of D and, for each ME&, cX={7c’,712,...,x”“}, A, be the convex hull of a; fi,, as in Definition 2, the map ‘fp(~~ “n:; onto A,: ~~&)=CYL 1 pi 7-c’;and ya the map from E to R”q y,(z) (z, nna)). The correspondence i 0 p, is upper semi-continuous from )snu id’; equipped with y. The map yI1is continuous from E equipped with y to RnU.Thus the correspondence (y, 0 i/D 0 8,) from S”a to R”= is upper semi-continuous with non-empty convex values in a compact subset of Rna. Furthermore, for all p E S”” and for all y E y,(c(&(p)), if y = y,(z), one has: P’Y=
2 Pi
i=l
From the finite dimensional Gale-Nikaido-Debreu lemma [see for example Debreu (1959, p. 82)], there exists p, E S”” and Z,= [(/?,(pJ) such that: n,. Thus ii,= b&J and Z, satisfy: 5,~ [(it,) n Y,, where (?,,n’)~O Vi=l,..., Y,={y~E/(y,n)50, Vn~d,j. Now the collection d is directed
(Proposition 2) and 2 is r-compact. we can assume that
We have to prove that 5~ Y If Z# I: since Y is y-closed,
by inclusion. A is a(E’,E)-compact By passing to two subnets, if necessary,
there exists 7~E D such that (2, 7~)> 0.
Let M.E & be such that k E A,. Since (5, X) > 0, Z$ Y,. Furthermore, Y, is yclosed. Thus, there exists p such that y 2 P=c-5,$ Y,; and y 2 sup (fl, cc)=>Z,tf Y,, which is a contradiction proving that 5~ Y 0 In particular, if E is as in section 2 a Banach space, the conjugate space of some other Banach space G, we can take for 5 the weak topology a(E,G). If we take y = a(E, E’), we deduce the following result, proved by Bojan when E is a normed space and extended here for any real Hausdorff locally convex topological vector space: Corollary 1. Let E be a real Hausdorff locally convex topological vector space, E’ the vector space of all continuous linear functionals on E, Y a closed
M. Florenzano, Equilibria with infinite dimensional commodity space
214
convex cone of E such that the polar cone Y” is # {0}, u a point in the interior of E: A = (7~E E’/(u, 7~)= - l} and i a correspondence from A into E, satisfying the following assumptions:
with non-empty closed convex values in a compact subset Z of E, when E’ is equipped with the topology o(E’, E) and E with the topology o(E, E’). (2) Vx E A, Vz E i(z), (z, 7~)S 0. (I) [ is upper semi-continuous
Then there exists ti E A such that Y n c(5) # $I Proof: Since the correspondence c is upper semi-continuous with closed values in a compact subset of E, one easily deduces that c(E) =lim SU~,,~[(Z). Then, the proof follows easily from the fact that the finite topology on A is stronger than the topology induced on A by a(E’, E). 0 Corollary 2 [Toussaint (198Z)l. The statement of Corollary 1 is true with the topology o(E, E’) replaced by the initial topology of E. Proof:
Corollary 2 is easily deduced from Corollary 1.
Note that in the statement of Corollary 2, Toussaint adds an assumption of joint continuity of the restriction to 2 x A of the canonical bilinear functional: (z, rr)+(z, rr), not needed for the result and which limits the topology to be considered for E to the Mackey topology z(E,E’) of the pairing (E, E’). 4. The existence theorem
Let us come back to the exchange economy S=((Xi, R’, w’)~ =1,, ,nr, I’) defined in section 2. The consumption set, the preference preordering, the initial endowment of each consumer i and the cone Y are subjected to the following assumptions: (1) Vi=l,..., m, Xi is convex, bounded from below, a(E, G)-closed. (2) Vi=l,..., m, Vx’ eXi, R’(x’) is convex and o(E, G) closed; furthermore, y’ E P’(x’) and 0 5 1< 1 imply: Ax’ + (1 - ;l)y’ E P’(x’). (3) Vi= 1,. . . , m, wi~Xi-Y (4) If x = (xi) E ny= r xi is an attainable allocation, then for all i= 1,. . . , m, (5)
P’(x’) # 8. Y is a o(E, G) closed convex cone, contained empty interior for the norm topology of E.
in (-E+)
and has a non-
From assumption (5), it follows that Y” is not reduced to (0). If u belongs
M. Florenzano,
Equilibria
with infinite dimensional
commodity
space
to the interior of Y as above we define A to be the g(E’,E)-compact set: d={z:E’/(u,~~)= -l}. Let 8’ denote the attainable set of consumer i: xi=
x’~X’/x’+
c xj=xoi+y,xjEXjVj#i j+i
215
convex
and
j
Each 2’ is convex, bounded from below and from above [assumption (l)] and since G is a Banach space, it follows from Proposition 1 that each attainable set xi is relatively o(E, G)-compact. In fact, each attainable set Xi is o(E, G)-compact and non-empty [assumption (3)]; it follows from the completeness of preordering R’, the assumption (2) and the finite intersection property for compact sets that there exists ci E w’ n (nxiezi R’(x’)) and therefore, there exists d’ E P’(c’), or d’ E P’(x’) Vx’ E r?‘. We choose a a(E,G)-compact convex subset K of E containing all the attainable sets and, for each i = 1,. . . , m, ai E Xi, d’ E Xi satisfying ai=(-Jj+b’, d’ &(xi)
For each economy
b’ E Y
[assumption (3)],
Vx’ ~8’.
i= 1, . . . ,m, let X’=X’
n K and consider
the new exchange
where each consumer retains his original endowment and preferences. We shall begin by proving the following existence theorem for the truncated economy 0: Proposition 3. Zf d = ((Xi, R’, oi)i = 1,, , m, Y) satisfies the assumptions (I), (2), (3), (4), (5), b has a quasi-equilibrium (jZ,j,%)~fi8%.YxA. i=l
Proof
Let us consider, as in section 2, the correspondence yi, 8, Pi, [‘, 4’ and q now defined for the economy E. We have to verify the assumptions of Lemma 1 in the case where F is the topology a(E, G) and for the restriction to A of the correspondence q:
216
M. Florenzano, Equilibria with infinite dimensional commodity space
From assumption (5), Y is o(E, G)-closed and obviously correspondence satisfies the Walras identity:
v
So that only the condition (1) of Lemma 1 has to be proved true. Let D denote the subset D = A n G. On D, the correspondences C$ are nonempty and convex valued. Indeed, for all ZE D, y’(n) is non-empty [assumption (3) and definition of K] and a(& G)-compact [assumption (1) and definition of 8’1. For all xi ~$(rc), r’(n) A @(xi) is a(E, G)-closed in $(rc) [assumption (2)J and it follows from the completeness of preordering R’ and the finite intersection property for compact sets that t’(n) is non-empty; a fortiori c#J’(z)#@ That for all rc in D, #(Tc) is convex follows from completeness and transitivity of the preference preordering. Thus the correspondence r is non-empty and convex valued with values in the a(& G)compact subset of E: 2 =c”= 1 8’ - {cr= 1 hi}. Let d be the collection of all finite subsets of D and, for each a~&‘, CI= (rc’, 7r2,. . . ) TP}, A, = co {d, r2, . . . , TP} be the convex hull of CCFor a fixed, since A, is finite dimensional, the canonical bilinear form (x, rc)-+(x, rr) is jointly continuous on 2’ x A, equipped with the topology induced by the product of topologies a(& G) and a(E’,E); thus the restriction y/A, of the budget correspondence of any consumer i on A, has a closed graph in A, x 8’. It easily follows from continuity properties of Pi in assumption 2 that the same is true for the restriction $/A, of # to A,. Finally, for all c1E .@‘,the restrictions q/A, are correspondences with a closed graph in A, x Z and it follows from Definition 2 that the restriction of [ to D is upper semicontinuous and closed valued from D with finite topology to E with a(E, G). One can apply Lemma 1 and there exist EEA, a net (E,) and a net (ZJ such that:
and, for all ~1,&EY](ZJ. Let Z,=~~zlZ~-_ci, with x~~#(E,),Vi=l,..., generality, we can suppose that o(E, G)
Z;~xiE~i,Vi=l
m.
Without
a loss in
,..., m,
and thus, since Z= cy= 1 X’ -I”== 1 oi E I: the allocation X= (Xi) is attainable. Then let x’~E Pi(zZi). From assumption (2) there exists c1E d such that U’-2 a==~” E P’(x$) and, since XL,E #(5@,), (x’~, it,,) 2 (oi, &). Passing to the
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with infinite dimensional
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217
limit, one sees that (x’i, ii) 2 (CJ, 7-Q. The previous relation being true for all i = 1,. . . , m and for all xii E I”(%‘), one deduces from assumption (4) and assumption (2): Vi=1 ,...,
(X’,71)~(~J,jl)
m.
On the other hand, C~=‘=((Xi,71)-_~=“=(~i,~)=(Z,~) deduces:
and since ZET: one
Vi=1 ,..., m,
(X’,71)=(~1+,5) (Z, it) =o.
- - Thus, (x,z, rc) satisfies all the conditions equilibrium of k? (see section 2). 0
of the definition
of a quasi-
One obtains the existence theorem for I by increasing the convex a(& G)compact set K and by passing to limit. Proposition
4.
If 8 = ((Xi, R’, Oi)i = 1,. , ,,,, Y) satisjies
the assumptions
(I),
(2), (3), (4), (5), d has a quasi-equilibrium (X,j,jl)E
fi
xi x Y x A.
i=l
Proof: The collection Z of the convex, o(E, G)-compact subsets K of E containing all attainable consumption sets and for each i= 1,. . . , m, ui E Xi, d’ EX’ defined as above, is directed. For each K E x, the economy Y), where 2: =X’ n K, has a quasi-equilibrium ~K=((r2t,Ri,oi)i=,,...,,, x Y x A. Since each xi is o(E, G)-compact and since A is (Xk,jk, 7%)E j--P: 1 xi _a(E’, Q-compact, by passing to subsets if necessary, we can assume: a(E,G) x;
-2‘
w EXi
_ dE’,E) 7LkAEE
Vi=l,...,m,
A
and since for all K of Z?, CT= 12: =c”= 1 cd + yk and Y is o(E, G) closed, Yk.
UC&G) jEY;
218
M. Florenzano, Equilibria with infinite dimensional commodity space
By passing to limits in relations I”= i 2: =c”=r oi +yk, we see that Z=(Z’) is an attainable allocation: EYE 1 Xi = C cd + y. If i belongs to {l,... , m} and x’~EP~(?), there exists K’E%? such that xii EK for all K 2 K’. Arguing as in the previous proof, one deduces from assumption (2):
and from the assumptions (2) and (4): (Xi,E)=(cOi,E)
Vi=1 ,..., m,
- - Thus (x, y, n) satisfies all the conditions equilibrium of 6. 0
of the definition
of a quasi-
5. Remarks (I) The assumptions (l)--(5) of Proposition 4 are standard, once the a(& G) topology on E has been accepted. When the commodity space E is &(M,J&‘,~), the continuity property of assumption (2) can be interpreted as an impatience assumption on preferences [see Brown and Lewis (1981)]. Note also that if E is L,(M,&,p), assumption (2) implies that, as in the finite dimensional case, each of preorderings R’ on Xi is representable by a g(E, G)-continuous utility function, since E is o(E, G)-separable. (2) The equilibrium price, the existence of which is proved in Propositions 3 and 4, is an element of E’, the conjugate space of the commodity space E. It does not have a very natural economic interpretation. If one wants a price equilibrium in G, one needs a decomposition theorem like the Yosida-Hewitt theorem. (3) The main limitation of the result proved in this paper arises from the assumption that the disposal cone has an interior point. This prevents any extension of the result to the existence of equilibria without the free-disposal assumption. Furthermore, this stringently limits the Banach spaces to be used as commodity spaces to spaces with a positive cone with a non-empty interior. Since this paper was written, Mas-Cole11 (1983) has obtained by an other proof the existence of equilibria in Banach lattices without using this assumption but by adding additional assumptions on consumers’ preferences.
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space
219
References Aliprantis, CD. and D.J. Brown, 1982, Equilibria in markets with a Riesz space of,commodities, Social Science working paper no. 427 (California Institute of Technology, Pasadena, CA). Aubin, J.P., 1979, Mathematical methods of game and economic theory (North-Holland, Amsterdam, New York, Oxford). Bewley, T.F., 1972, Existence of equilibria in economies with infinitely many commodities, Journal of Economic Theory 4, 514-540. Bojan, P., 1974, A generalization of theorems on the existence of competitive economic equilibrium in the case of infinitely many commodities, Mathematics Balkanica 4, 491494. Bourbaki, N., 1966, Espaces vectoriels topologiques, 2nd ed. (Hermann, Paris). Brown, D.J. and L.M. Lewis, 1981, Myopic economic agents, Econometrica 49,359-368. Debreu, G., 1959, Theory of value (Wiley, New York). Elbarkoky, R.A., 1977, Existence of equilibrium in economies with Banach commodity space, Akad. Nauk. Azerdaijan SSR Dokl. 33, 8-12 [in Russian]. Kelley, J.L., J. Namioka and coauthors, 1963, Linear topological spaces (D. Van Nostrand Co., Princeton, NJ). Magill, M.J.P., 1981, An equilibrium existence theorem, Journal of Mathematical Analysis and Applications 84, 162-169. Mas-Colell, A., 1975, A model of equilibrium with differentiated commodities, Journal of Mathematical Economics 2, 236-296. Mas-Colell, A., 1983, The price equilibrium existence problem in Banach lattices, Mimeo. Ostroy, J.M., 1982, On the existence of Walrasian equilibrium in large-square economies, Mimeo. Toussaint, S., 1981, On the existence of equilibria in economies with infinitely many commodities, Discussion paper no. 174/81 (Mannheim University, Mannheim).
of Mathematical
Economics
12 (1983) 207-219.
North-Holland
ON THE EXISTENCE OF EQUILIBRIA IN ECONOMIES WITH AN INFINITE DIMENSIONAL COMMODITY SPACE* Monique FLORENZANO CNRS, CEPREMAP, Received
September
75013 Paris, France
1982, final version
received
September
1983
We prove an infinite dimensional extension of the Gale-Nikaido-Debreu lemma which includes all necessary limiting processes and allows a proof of the existence of equilibria under standard assumptions in an economy with infinitely many commodities which exactly parallels the proof of Debreu (1959) for the finite dimensional case.
1. Introduction
The assumption that the commodities are not in finite number agrees with many classical situations for economic theory: intertemporal equilibrium with an infinite horizon, uncertainty with an infinite number of states, differentiation of commodities. Since the paper of Bewley (1972) which proved the existence of a Walrasian equilibrium for an economy with a finite number of agents and L,(M, A, p) for commodity space [let us recall that M is a state space, J%’a o-field of subsets of M, p a a-finite measure on J%!, and L,(M,A, p) the Banach space of all measurable functions such that Ilf((, = sup
-While the proof of Bewley is based on a limiting process which deduces the existence of an equilibrium in the infinite dimensional case from the existence of an equilibrium in the finite dimensional case, Bojan (1974), Toussaint (1981), Magi11 (1981), Aliprantis and Brown (1982) look for a direct proof which is the analogue of one of the proofs in the finite dimensional case. *The main result of this paper was achieved when I was visiting the group for the Applications of Mathematics and Statistics to Economics at the University of California, Berkeley, Summer 1982. The support of National Science Foundation grant SES 79-12380 Debreu to the University of California is gratefully acknowledged. 1 would like to thank Gerard Debreu for his hospitality and his advice. I also wish to thank Bernard Cornet who followed the successive versions of this paper and made many suggestions, and Andreu Mas-Cole11 and Karl Vind for helpful discussions. 03044068/83/$3.00
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M. Florenzano, Equilibria with infinite dimensional commodity space
-At the same time, Aliprantis and Brown (1982) try to set the most general framework in which their results can be achieved. Before them, Elbarkoky (1977) extended the equilibrium existence theorem of Bewley to an outline which is essentially the same as this which will be used here. This paper corresponds to both objectives. It states an extension of the Gale-Nikaido-Debreu lemma, in order to give a direct proof of the existence of an equilibrium which exactly parallels that of Debreu (1959) for the finite dimensional case, in a general framework which is to be defined in section 2. The idea of the proof is to use the duality between L,(M, .,&‘,p) and L,(M, &,,u) and to truncate consumption sets to weak-star compact sets, in order to define excess demand correspondences when the consumer’s preferences satisfy a weak-star continuity assumption; Debreu’s argument is applied to the truncated economy and then a limit argument allows the truncation to go to infinity. The object of the generalization to Banach spaces is to point out the assumptions on the commodity space which prevent from applying our approach (and also those of Bewley, Brown and Alipantis,. . .) to some Banach spaces other than L,(M, ~#,,u) which have been found useful in practice; namely ca(K), the space of countably additive signed measures on a compact metric space [see Mas-Cole11 (1975)] and L&M, &,p), 1 g’p< co [see Ostroy (1982)]. 2. The model In this paper, the space of commodities is a Banach space E, the conjugate space of another Banach space G, i.e., E = G’ where G’ is the vector space of all continuous linear functionals on G and the norm on E is the conjugate norm to the norm on G. G is ordered by a closed convex positive cone G+ satisfying: G = G+ - G ‘. E itself is ordered by the positive cone:
where (p,x) is the value at p of the continuous linear functional XE G’. We shall denote by 2 the orders on G and E. The first assumptions guarantee that in an exchange economy, under the usual conditions of lower boundedness of consumption sets, the attainable sets are compact for the weak topology a(E, G) for E associated with the duality (E, G). This will be deduced from the following elementary proposition: Proposition 1. IfG is a real topological vector space ordered by a positive cone G+ satisfying: G=Gf -G’, and if E is the conjugate space of G, ordered
M. Florenzano, Equilibria with infinite dimensional commodity space
209
by the cone of all positive continuous linear functionals on G+, let a and b be two elements of E and let A= {x E EJa 5 x 5 b}. Then A is a(E, G)-bounded.
Proof: Let PEG, ol~R, cc>0 and V={x~E/((p,x)l~a). absorbed by K If PEG+ and if P=sup{I(p,a)(,(
If pE -G+,
if
fi>O
and
AcV
if
We claim that A is then
/3=0.
one obtains the same result.
where p+ and p- belong to G+, then I’1 T/+ n I’- where and I’- =(x E E/l(p-, x)1 5 42). Let p and v be real numbers such that AcpV+ and AcvV-. Then Aclv where I= sup c/4 v>. Cl If
p=p+-p-
V+=~x~:E/l(p+,x)l~o1/2}
Prices belong to E’, the conjugate space of E. E’ is ordered by the positive cone: E’+ = {n~E’/(x,rr) ~OVXE Ef), where as above we denote by (x, rc) the value at x of the continuous linear functional rr E E. In order to define a price simplex compact for the weak topology a(E’, E) for E’ associated with the duality (E, E’), we need that E+ has a non-empty interior. For example, if, as in Aliprantis and Brown (1982), E is an A.M. space with unit (i.e., a Banach lattice, the lattice order and the norm of which are related by the two relations: ((x v y((=max {\IxI\,/y\I} an d t h ere exists e E E+ such that the order interval C-e, +e] coincides with the closed unit ball (XE E/llxIlz 11) this assumption is satisfied. The following proposition is classical: Proposition 2. Let E be a real Hausdorff locally convex topological vector space, E’ the conjugate space of E and Y be a convex cone of E such that Y” = {II E E’/(y, IC) 50, Vy E Y) # (0). If u belongs to the interior of I: the set A=(nEY0/(u,71)=
-l}
is equicontinuous and a@‘, E)-compact and satisfies Y” = Ulzo AA. Proof.
Let V be a closed circled neighborhood v7t E YO,
vxey
of 0 such that
(u+x,?t)~O.
From
210
M. Florenzano, Equilibria with infinite dimensional commodity space
We consider an exchange economy a=((X’, R’, oi)i = 1,, ,m, Y) consumers and, for each i= 1,. . . , m, a consumption set Xi cE, a preorder& R’ on Xi and an initial endowment oi E E; in addition, activities are described by a convex cone T: contained in (-E+). preordering R’, we associate the strict preference relation defined by xipixtioxiRixfi
and not
x”R’x’.
Let xi EX’; we introduce the following notations: pi(xi) = {,fi E Xi/xfipixi}, Ri(xi) = {,Ji EXi/xfiRixi}, (pi) - ‘(xi) = {x’i E Xijxipixfi), (Ri)-l(xi)=
cxri EXi/xiRixri}.
For each II E E’, we define
y’(n)= {xiEXi/(Xi, n)
5
i.e., the budget correspondence
(co’,7c)}, for consumer i.
6’(n) = {xi EXi/(Xi, 7c) < (CO’,n)),
~‘(7c)={x’EXi/X’Eyi(n) i.e., the demand correspondence ~‘(n)={X’EXi/XiEyi(?l)
and
y’(z) n P’(x’) = a},
for consumer i. and
i.e., the quasi-demand correspondence
~i(~)nPi(x’)=P)), for consumer i.
i.e., the excess quasi-demand correspondence. An allocation x = (xi) E ny= 1Xi is attainable if
with m complete disposal To the
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211
An equilibrium of E is a point (X,y,5) ~(nT=r Xi) x Y x (E’\(O)) satisfying the conditions: Vi=1 ,..., m,Zic[‘(ji),
71~Y”={z~E’/(y,~)~O,Vy~ p=
(1)
Y)
and
(jj,%)=O,
(2)
F o’+y. i=l
A quasi-equilibrium of E is a point (2, jj,iz) E(~Y= I X’) x Y x (E’\(O)) satisfying the above conditions (2) and (3) and the condition (1’): Vi=1 ,..., m,%‘E#(jl).
(1’)
In fact, we will prove the existence of a quasi-equilibrium from an extension of the Gale-Nikaido-Debreu lemma (or market equilibrium theorem) that we state in the next section.
3. The extension of the Gale-Nikaido-Dehreu lemma In the literature, one can find two extensions of the finite dimensional Gale-Nikaido-Debreu lemma to correspondences defined on a compact subset of the space E’ of all continuous linear functionals on a real Hausdorff locally convex topological vector space E, with convex values in a compact subset of E. The first is from Bojan (1974), the second from Toussaint (1981); the common defect of the two statements is that they cannot be applied without a limit operation to prove the existence of equilibrium under standard assumptions in a transitive economy with a finite number of agents and a commodity space as in section 2. Here we prove an infinite dimensional analogue of the Gale-NikaidoDebreu lemma which is the most convenient extension for an application to a proof of existence of equilibrium which exactly parallels in the infinite dimensional case the proof of Debreu (1959) for the finite dimensional case. First, let us introduce some definitions: I. Let $ be a correspondence from a topological space X to a topological space Y and let x0 be an element of X; we define the subset lim supx +x0 6(x) as the set of all elements y of Y such that for all neighborhoods W of y and V of x0, there exists x E V such that 4(x) n W # $I In other words, lirn~up,,,~ 4(x) is the set of all y in X such that (x,,y) belongs to the closure of the graph of b, in the product topology for X x Y Definition
212
M. Florenzano, Equilibria with infinite dimensional commodity space
Definition 2.
Given a convex subset X of a vector space E, let d be the collection of all finite subsets CIof X. If CI= {x’, x2,. . . , xna} EL&‘, let SnQdenote the unit simplex of the Euclidean space R”* and pa,:S”QX the map defined by if
p=
2 pie’,
pi20
Vi=l,...,n,,
i$JI Pi = l,
i=l
then
The finite topology [see Aubin (1979, p. 21 l)] defined on X is the strongest topology for which the maps fib are continuous for all c1E d. A correspondence 4 from X to a topological space Y is upper semicontinuous when X is equipped with the finite topology if and only if the correspondences 4 0j3,: Pa-+ I: defined by
4 oP,(P)= WdP)) are upper semi-continuous for all tl Ed. I.e., when X is equipped with the finite topology for verifying upper semicontinuity of the correspondence 4, it is necessary and sufficient to verify upper semi-continuity of all the restrictions ~/CO c( of the correspondence C$ to the convex hulls co c1of the finite subsets of X. The finite topology on X is at least as strong as any topology induced on X by a vector space topology on E. Thus we can state: Lemma 1. Let E be a real Hausdorfl locally convex topological vector space, E’ the vector space of all continuous linear functionals on E, Y a convex cone of E such that the polar cone Y” = {z E E’/(y, 71) 5 0 Vy E Y} is # {O], u a point in the interior of Y, A = (rc E Y’/(u, rt) = - l}, and i a correspondence from A to E. Let 5 be a Hausdorff locally convex topology for E at least as weak as a(E, E’), and D={~cE A/n is continuous zf E is equipped with F-). We make the following additional assumptions: If E is equipped with 9 and if D is equipped with the finite topology, the restriction c/D of [ to D is #upper semi-continuous with non-empty closed convex values in a compact set Z of E. (2) Vrt~ A,Vz’z~(n), (z,~)~Oo. (3) Y is F-closed.
(0
M. Florenzano, Equilibria with infinite dimensional commodity space
213
Then there exists it E A such that Y n lim SUP,,~ c(z) #$3, where the lim sup is taken for the topology on A induced by a(E’,E) and the topology 9 on E.
The idea of the proof we give is borrowed partly from Bojan (1974), partly from Aliprantis and Brown (1982). Proof
Let d be the collection of all finite subsets of D and, for each ME&, cX={7c’,712,...,x”“}, A, be the convex hull of a; fi,, as in Definition 2, the map ‘fp(~~ “n:; onto A,: ~~&)=CYL 1 pi 7-c’;and ya the map from E to R”q y,(z) (z, nna)). The correspondence i 0 p, is upper semi-continuous from )snu id’; equipped with y. The map yI1is continuous from E equipped with y to RnU.Thus the correspondence (y, 0 i/D 0 8,) from S”a to R”= is upper semi-continuous with non-empty convex values in a compact subset of Rna. Furthermore, for all p E S”” and for all y E y,(c(&(p)), if y = y,(z), one has: P’Y=
2 Pi
i=l
From the finite dimensional Gale-Nikaido-Debreu lemma [see for example Debreu (1959, p. 82)], there exists p, E S”” and Z,= [(/?,(pJ) such that: n,. Thus ii,= b&J and Z, satisfy: 5,~ [(it,) n Y,, where (?,,n’)~O Vi=l,..., Y,={y~E/(y,n)50, Vn~d,j. Now the collection d is directed
(Proposition 2) and 2 is r-compact. we can assume that
We have to prove that 5~ Y If Z# I: since Y is y-closed,
by inclusion. A is a(E’,E)-compact By passing to two subnets, if necessary,
there exists 7~E D such that (2, 7~)> 0.
Let M.E & be such that k E A,. Since (5, X) > 0, Z$ Y,. Furthermore, Y, is yclosed. Thus, there exists p such that y 2 P=c-5,$ Y,; and y 2 sup (fl, cc)=>Z,tf Y,, which is a contradiction proving that 5~ Y 0 In particular, if E is as in section 2 a Banach space, the conjugate space of some other Banach space G, we can take for 5 the weak topology a(E,G). If we take y = a(E, E’), we deduce the following result, proved by Bojan when E is a normed space and extended here for any real Hausdorff locally convex topological vector space: Corollary 1. Let E be a real Hausdorff locally convex topological vector space, E’ the vector space of all continuous linear functionals on E, Y a closed
M. Florenzano, Equilibria with infinite dimensional commodity space
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convex cone of E such that the polar cone Y” is # {0}, u a point in the interior of E: A = (7~E E’/(u, 7~)= - l} and i a correspondence from A into E, satisfying the following assumptions:
with non-empty closed convex values in a compact subset Z of E, when E’ is equipped with the topology o(E’, E) and E with the topology o(E, E’). (2) Vx E A, Vz E i(z), (z, 7~)S 0. (I) [ is upper semi-continuous
Then there exists ti E A such that Y n c(5) # $I Proof: Since the correspondence c is upper semi-continuous with closed values in a compact subset of E, one easily deduces that c(E) =lim SU~,,~[(Z). Then, the proof follows easily from the fact that the finite topology on A is stronger than the topology induced on A by a(E’, E). 0 Corollary 2 [Toussaint (198Z)l. The statement of Corollary 1 is true with the topology o(E, E’) replaced by the initial topology of E. Proof:
Corollary 2 is easily deduced from Corollary 1.
Note that in the statement of Corollary 2, Toussaint adds an assumption of joint continuity of the restriction to 2 x A of the canonical bilinear functional: (z, rr)+(z, rr), not needed for the result and which limits the topology to be considered for E to the Mackey topology z(E,E’) of the pairing (E, E’). 4. The existence theorem
Let us come back to the exchange economy S=((Xi, R’, w’)~ =1,, ,nr, I’) defined in section 2. The consumption set, the preference preordering, the initial endowment of each consumer i and the cone Y are subjected to the following assumptions: (1) Vi=l,..., m, Xi is convex, bounded from below, a(E, G)-closed. (2) Vi=l,..., m, Vx’ eXi, R’(x’) is convex and o(E, G) closed; furthermore, y’ E P’(x’) and 0 5 1< 1 imply: Ax’ + (1 - ;l)y’ E P’(x’). (3) Vi= 1,. . . , m, wi~Xi-Y (4) If x = (xi) E ny= r xi is an attainable allocation, then for all i= 1,. . . , m, (5)
P’(x’) # 8. Y is a o(E, G) closed convex cone, contained empty interior for the norm topology of E.
in (-E+)
and has a non-
From assumption (5), it follows that Y” is not reduced to (0). If u belongs
M. Florenzano,
Equilibria
with infinite dimensional
commodity
space
to the interior of Y as above we define A to be the g(E’,E)-compact set: d={z:E’/(u,~~)= -l}. Let 8’ denote the attainable set of consumer i: xi=
x’~X’/x’+
c xj=xoi+y,xjEXjVj#i j+i
215
convex
and
j
Each 2’ is convex, bounded from below and from above [assumption (l)] and since G is a Banach space, it follows from Proposition 1 that each attainable set xi is relatively o(E, G)-compact. In fact, each attainable set Xi is o(E, G)-compact and non-empty [assumption (3)]; it follows from the completeness of preordering R’, the assumption (2) and the finite intersection property for compact sets that there exists ci E w’ n (nxiezi R’(x’)) and therefore, there exists d’ E P’(c’), or d’ E P’(x’) Vx’ E r?‘. We choose a a(E,G)-compact convex subset K of E containing all the attainable sets and, for each i = 1,. . . , m, ai E Xi, d’ E Xi satisfying ai=(-Jj+b’, d’ &(xi)
For each economy
b’ E Y
[assumption (3)],
Vx’ ~8’.
i= 1, . . . ,m, let X’=X’
n K and consider
the new exchange
where each consumer retains his original endowment and preferences. We shall begin by proving the following existence theorem for the truncated economy 0: Proposition 3. Zf d = ((Xi, R’, oi)i = 1,, , m, Y) satisfies the assumptions (I), (2), (3), (4), (5), b has a quasi-equilibrium (jZ,j,%)~fi8%.YxA. i=l
Proof
Let us consider, as in section 2, the correspondence yi, 8, Pi, [‘, 4’ and q now defined for the economy E. We have to verify the assumptions of Lemma 1 in the case where F is the topology a(E, G) and for the restriction to A of the correspondence q:
216
M. Florenzano, Equilibria with infinite dimensional commodity space
From assumption (5), Y is o(E, G)-closed and obviously correspondence satisfies the Walras identity:
v
So that only the condition (1) of Lemma 1 has to be proved true. Let D denote the subset D = A n G. On D, the correspondences C$ are nonempty and convex valued. Indeed, for all ZE D, y’(n) is non-empty [assumption (3) and definition of K] and a(& G)-compact [assumption (1) and definition of 8’1. For all xi ~$(rc), r’(n) A @(xi) is a(E, G)-closed in $(rc) [assumption (2)J and it follows from the completeness of preordering R’ and the finite intersection property for compact sets that t’(n) is non-empty; a fortiori c#J’(z)#@ That for all rc in D, #(Tc) is convex follows from completeness and transitivity of the preference preordering. Thus the correspondence r is non-empty and convex valued with values in the a(& G)compact subset of E: 2 =c”= 1 8’ - {cr= 1 hi}. Let d be the collection of all finite subsets of D and, for each a~&‘, CI= (rc’, 7r2,. . . ) TP}, A, = co {d, r2, . . . , TP} be the convex hull of CCFor a fixed, since A, is finite dimensional, the canonical bilinear form (x, rc)-+(x, rr) is jointly continuous on 2’ x A, equipped with the topology induced by the product of topologies a(& G) and a(E’,E); thus the restriction y/A, of the budget correspondence of any consumer i on A, has a closed graph in A, x 8’. It easily follows from continuity properties of Pi in assumption 2 that the same is true for the restriction $/A, of # to A,. Finally, for all c1E .@‘,the restrictions q/A, are correspondences with a closed graph in A, x Z and it follows from Definition 2 that the restriction of [ to D is upper semicontinuous and closed valued from D with finite topology to E with a(E, G). One can apply Lemma 1 and there exist EEA, a net (E,) and a net (ZJ such that:
and, for all ~1,&EY](ZJ. Let Z,=~~zlZ~-_ci, with x~~#(E,),Vi=l,..., generality, we can suppose that o(E, G)
Z;~xiE~i,Vi=l
m.
Without
a loss in
,..., m,
and thus, since Z= cy= 1 X’ -I”== 1 oi E I: the allocation X= (Xi) is attainable. Then let x’~E Pi(zZi). From assumption (2) there exists c1E d such that U’-2 a==~” E P’(x$) and, since XL,E #(5@,), (x’~, it,,) 2 (oi, &). Passing to the
M. Florenzano,
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with infinite dimensional
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217
limit, one sees that (x’i, ii) 2 (CJ, 7-Q. The previous relation being true for all i = 1,. . . , m and for all xii E I”(%‘), one deduces from assumption (4) and assumption (2): Vi=1 ,...,
(X’,71)~(~J,jl)
m.
On the other hand, C~=‘=((Xi,71)-_~=“=(~i,~)=(Z,~) deduces:
and since ZET: one
Vi=1 ,..., m,
(X’,71)=(~1+,5) (Z, it) =o.
- - Thus, (x,z, rc) satisfies all the conditions equilibrium of k? (see section 2). 0
of the definition
of a quasi-
One obtains the existence theorem for I by increasing the convex a(& G)compact set K and by passing to limit. Proposition
4.
If 8 = ((Xi, R’, Oi)i = 1,. , ,,,, Y) satisjies
the assumptions
(I),
(2), (3), (4), (5), d has a quasi-equilibrium (X,j,jl)E
fi
xi x Y x A.
i=l
Proof: The collection Z of the convex, o(E, G)-compact subsets K of E containing all attainable consumption sets and for each i= 1,. . . , m, ui E Xi, d’ EX’ defined as above, is directed. For each K E x, the economy Y), where 2: =X’ n K, has a quasi-equilibrium ~K=((r2t,Ri,oi)i=,,...,,, x Y x A. Since each xi is o(E, G)-compact and since A is (Xk,jk, 7%)E j--P: 1 xi _a(E’, Q-compact, by passing to subsets if necessary, we can assume: a(E,G) x;
-2‘
w EXi
_ dE’,E) 7LkAEE
Vi=l,...,m,
A
and since for all K of Z?, CT= 12: =c”= 1 cd + yk and Y is o(E, G) closed, Yk.
UC&G) jEY;
218
M. Florenzano, Equilibria with infinite dimensional commodity space
By passing to limits in relations I”= i 2: =c”=r oi +yk, we see that Z=(Z’) is an attainable allocation: EYE 1 Xi = C cd + y. If i belongs to {l,... , m} and x’~EP~(?), there exists K’E%? such that xii EK for all K 2 K’. Arguing as in the previous proof, one deduces from assumption (2):
and from the assumptions (2) and (4): (Xi,E)=(cOi,E)
Vi=1 ,..., m,
- - Thus (x, y, n) satisfies all the conditions equilibrium of 6. 0
of the definition
of a quasi-
5. Remarks (I) The assumptions (l)--(5) of Proposition 4 are standard, once the a(& G) topology on E has been accepted. When the commodity space E is &(M,J&‘,~), the continuity property of assumption (2) can be interpreted as an impatience assumption on preferences [see Brown and Lewis (1981)]. Note also that if E is L,(M,&,p), assumption (2) implies that, as in the finite dimensional case, each of preorderings R’ on Xi is representable by a g(E, G)-continuous utility function, since E is o(E, G)-separable. (2) The equilibrium price, the existence of which is proved in Propositions 3 and 4, is an element of E’, the conjugate space of the commodity space E. It does not have a very natural economic interpretation. If one wants a price equilibrium in G, one needs a decomposition theorem like the Yosida-Hewitt theorem. (3) The main limitation of the result proved in this paper arises from the assumption that the disposal cone has an interior point. This prevents any extension of the result to the existence of equilibria without the free-disposal assumption. Furthermore, this stringently limits the Banach spaces to be used as commodity spaces to spaces with a positive cone with a non-empty interior. Since this paper was written, Mas-Cole11 (1983) has obtained by an other proof the existence of equilibria in Banach lattices without using this assumption but by adding additional assumptions on consumers’ preferences.
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References Aliprantis, CD. and D.J. Brown, 1982, Equilibria in markets with a Riesz space of,commodities, Social Science working paper no. 427 (California Institute of Technology, Pasadena, CA). Aubin, J.P., 1979, Mathematical methods of game and economic theory (North-Holland, Amsterdam, New York, Oxford). Bewley, T.F., 1972, Existence of equilibria in economies with infinitely many commodities, Journal of Economic Theory 4, 514-540. Bojan, P., 1974, A generalization of theorems on the existence of competitive economic equilibrium in the case of infinitely many commodities, Mathematics Balkanica 4, 491494. Bourbaki, N., 1966, Espaces vectoriels topologiques, 2nd ed. (Hermann, Paris). Brown, D.J. and L.M. Lewis, 1981, Myopic economic agents, Econometrica 49,359-368. Debreu, G., 1959, Theory of value (Wiley, New York). Elbarkoky, R.A., 1977, Existence of equilibrium in economies with Banach commodity space, Akad. Nauk. Azerdaijan SSR Dokl. 33, 8-12 [in Russian]. Kelley, J.L., J. Namioka and coauthors, 1963, Linear topological spaces (D. Van Nostrand Co., Princeton, NJ). Magill, M.J.P., 1981, An equilibrium existence theorem, Journal of Mathematical Analysis and Applications 84, 162-169. Mas-Colell, A., 1975, A model of equilibrium with differentiated commodities, Journal of Mathematical Economics 2, 236-296. Mas-Colell, A., 1983, The price equilibrium existence problem in Banach lattices, Mimeo. Ostroy, J.M., 1982, On the existence of Walrasian equilibrium in large-square economies, Mimeo. Toussaint, S., 1981, On the existence of equilibria in economies with infinitely many commodities, Discussion paper no. 174/81 (Mannheim University, Mannheim).