GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS

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1 GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS C. D. ALIPRANTIS 1, M. FLORENZANO 2, V. F. MARTINS DA ROCHA 3 AND R. TOURKY 4 1 Department of Economcs, Krannert School of Management, Purdue Unversty, W. Lafayette IN , USA; alprants@mgmt.purdue.edu 2 CNRS CERMSEM, Unversté Pars 1, boulevard de l Hôptal, Pars Cedex 13, FRANCE; monque.florenzano@unv-pars1.fr 3 CERMSEM, Unversté Pars 1, boulevard de l Hôptal, Pars Cedex 13, FRANCE; martns@unv-pars1.fr 4 Department of Economcs and Commerce, Unversty of Melbourne, Parkvlle 3052, Melbourne, AUSTRALIA; rtourky@unmelb.edu.au Abstract. An F -cone of a real vector space s a ponted and generatng convex cone, the unon of a countable famly of fnte dmensonal polyhedral convex cones such that each one s an extremal subset of the followng one. In ths paper, we study securty markets wth countably many securtes and arbtrary fnte portfolo holdngs. If, under the portfolo domnance order, the postve cone of the portfolo space s an F -cone, then Edgeworth allocatons and nontrval quas-equlbra do exst. Ths result extends the case where, as n Alprants Brown Polyraks Werner (1998), the postve cone s a Yudn cone. Keywords: Securty markets; Edgeworth equlbrum; nontrval quas-equlbrum; nductve lmt topology; F -cone; Resz Kantorovch functonal. 1. Introducton Let us consder a fnance model wth m nvestors tradng securtes and havng dentcal expectatons on the securty payoffs. Let E be a portfolo (vector) space. Gven an ordered vector payoff space X, typcally some L p (Ω, Σ, P ) for 1 p and an underlyng probablty space (Ω, Σ, P ), let a lnear operator R : E X defne for each portfolo x the random payoff R(x) an nvestor expects to receve when holdng the portfolo. If we suppose that R s one-to-one, t s natural to order the portfolo space E by the portfolo domnance orderng: z R z whenever R(z) R(z ). Let us now assume that nvestors are constraned to have non-negatve end-of-perod wealth or, equvalently, that ther portfolo set s equal to the cone K = {x E : x R 0} of postve payoff portfolos. If each nvestor has a strct preference over K descrbed by the correspondence P : K K and an ntal endowment of securtes ω K, then a fnancal economy s gven as: E = ( (E, K), (K, P, ω ) m ). An equlbrum defnton for ths economy requres the defnton of a securty prce space E, n dualty wth E. Relatve to ths dualty, an equlbrum s classcally a par (p, x) of a nonzero prce p E and a market clearng allocaton x = (x ) m Km of nvestors optmal portfolos gven Date: September 2,

2 GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS 2 prces of the securtes. Our paper s a contrbuton to the equlbrum exstence problem n ths economy. The exstence of equlbrum n such a model was frst studed by Brown Werner [6], then by Alprants Brown Polyraks Werner [2], who ntroduced n the same paper the noton of portfolo domnance orderng. As n Brown Werner [6], Alprants Brown Polyraks Werner assume that there are countably many securtes, but n [2], each nvestor s restrcted to portfolos wth nonzero holdngs of only fntely many securtes. The portfolo space of the model s thus the vector space Φ of all eventually zero sequences. Assumng that the cone K of postve payoff portfolos s a lattce cone, Alprants Brown Polyraks Werner prove that the portfolo space Φ, equpped wth the nductve lmt topology 1 relatve to the famly of all fnte dmensonal subspaces of Φ, s a topologcal vector lattce. As the portfolo sets of nvestors concde wth the postve cone K of the portfolo space, n the lne of Mas-Colell [10], the exstence of a quasequlbrum wth a prce n R N, the topologcal dual of Φ, s manly based on assumptons of unform properness on K of the preferences of the nvestors. Our purpose n ths paper s to show that, n the same settng and under comparable properness assumptons, the key assumpton that the cone K of postve payoffs portfolos s a lattce cone can be serously weakened. As n [2], we assume that the portfolo space s Φ, ordered by the portfolo domnance orderng and equpped wth the nductve lmt topology. As n [2], we assume that the agents have the postve cone as portfolo sets and make properness assumptons on ther preferences, but we replace the lattce orderng hypothess by the strctly weaker assumpton that the postve cone s an F - cone. By ths assumpton, on whch we wll comment later, we mean that, under the portfolo domnance orderng, the postve cone of the portfolo space s a convex, ponted and generatng cone K = n=1 Kn, the unon of a countable famly of fnte dmensonal polyhedral convex cones K n such that each K n s an extremal subset (or a face) of K n+1. Under ths assumpton, equlbrum exsts. Moreover, the countably many extremal vectors of the postve cone have the same economc nterpretaton as the vectors of the Yudn bass assumed n [2]. The correspondng portfolos can be thought of as mutual funds that nvestors trade at equlbrum. The man tool of ths extenson s a result of Alprants Florenzano Tourky [3, Theorem 5.1] gvng condtons for decentralzng Edgeworth allocatons of a proper exchange economy of whch the commodty space s not a vector lattce. The exstence of Edgeworth allocatons s obtaned under classcal contnuty assumptons on the preferences of the agents. Ther decentralzaton as nontrval quas-equlbra requres properness assumptons that we shall precse later. The double assumpton on the topology of the portfolo space and ts order structure guarantees that we can apply the result n [3], after verfyng the suffcent condton for ths decentralzaton as stated n [3]. Ths condton, expressed n terms of properness of the Resz Kantorovch functonal assocated to a lst of contnuous lnear functonals on Φ, should be understood as a condton (here verfed) of compatblty between the topology and the order structure of the portfolo space. The paper s organzed as follows: In Secton 2, we recall the defnton and propertes of the nductve lmt topology ξ on Φ generated by the famly of all ts fnte dmensonal vector subspaces. We also defne F -cones and establsh some propertes of ξ when Φ s ordered by an F -cone. Edgeworth and nontrval quas-equlbra of our economy are studed n Secton 3. The properness of the Resz Kantorovch functonal s proved n the last secton. Wth the appendx, ths secton s the only techncal part of the paper. 1 The defnton and propertes of ths topology wll be stated later n Secton 2.

3 GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS 3 2. Inductve lmt topology and F -cones 2.1. The nductve lmt topology. Let E be a real vector space and let F be the famly (drected by ncluson) of all fnte dmensonal vector subspaces F of E. The nductve lmt topology ξ on E generated by F s the fnest locally convex topology on E for whch, for each F F, the natural embeddng F : F E s τ F -ξ contnuous, where τ F s the unque Hausdorff lnear topology on F. The fact that, as a fnte dmensonal space, each F F admts a unque Hausdorff lnear topology τ F has several remarkable consequences easy to derve: 2 If (F γ ) γ Γ s a famly of fnte dmensonal vector subspaces such that each F F s contaned n some F γ, then ξ s also the fnest locally convex topology on E for whch, for each γ Γ, the natural embeddng Fγ : F γ E s τ Fγ -ξ contnuous. The topology ξ s also the fnest locally convex topology on E. Consequently, ξ s Hausdorff and the topologcal and algebrac duals, E and E concde. If (e ) I s a Hamel bass of E, E s the drect sum of the one dmensonal vector spaces Re and ξ s the fnest locally convex topology on E for whch each natural embeddng Re E s τ Re -ξ contnuous. For each ε = (ε ) I ]0, + [ I, let V ε := {x E : I x < ε }. The famly of all subsets V ε s a base of neghborhoods of 0 for the topology ξ. The followng result s also classcal (see [5, Chap. II, 4, exerc. 8]). We gve ts proof for the sake of completeness of the paper. Proposton 2.1. Assume that E s a real vector space wth a countable Hamel bass. A subset A of E s ξ-closed f and only f for each fnte dmensonal vector subspace F of E, the set A F s closed for the unque Hausdorff lnear topology on F. In partcular, f F s the famly of all fnte dmensonal subspaces of E, then ξ = {V E : F F V F s open n F }. Proof. Let A be a ξ-closed subset of E. For each fnte dmensonal subspace F of E, 1 F (A) = A F s closed n F. Conversely, let A be a subset of E such that for each fnte dmensonal subspace F of E, A F s closed. Let (e n ) n N be a countable bass of E and let x E \ A. We propose to prove that there exsts ε = (ε ) N ]0, + [ N such that x + V ε E \ A. For each n N, we note E n the vector space spanned by the vectors {e 1,..., e n }, and for each fnte famly (B 1,..., B n ) of subsets of R, we let { } n B := x E n : x = x e wth x B = 1,..., n. Wthout any loss of generalty we can suppose that x E 1, then for each n 1 x E n \ (A E n ). As A E n s closed n E n, t follows that there exsts (ε 1,..., ε n ) ]0, + [ n such that (2.1) x + ] ε, +ε [ E n \ (A E n ). 2 For detals about nductve lmt topologes, see the books [5], [9], [12].

4 GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS 4 Let α > 1. Clam 2.1. There exsts ε n+1 > 0 such that x + n+1 ] ε [ α, +ε E n+1 \ (A E n+1 ). α Proof. Let α > 1; suppose that Clam 2.1 s not true. Then we can construct a sequence (v k ) k N of E n+1 such that for each k N, ] v k ε [ ] α, +ε 1 α k, + 1 [ and x + v k A E n+1. k The sequence (v k ) k N les n a compact subset of E n+1. Passng to a subsequence f necessary, we can suppose that (v k ) k N converges to some v E n+1. Note that [ v ε ] α, +ε {0}. α It follows that Ths contradcts (2.1). x + v A ( x + [ ε ] ) ( α, +ε A x + α ) ] ε, +ε [. In order to apply the prevous clam, we consder a sequence (α n ) n N ]1, + [ N such that 3 lm N n=1 N α n = α n <. Applyng nductvely Clam 2.1, we can construct a sequence (ε n ) n N ]0, + [ N such that for each n N, ] (2.2) x + ε n α, + ε [ n α E n \ (A E n ). Now consder β R such that β > n=1 α n and let V be the followng ξ-neghborhood of 0, { } V := x E : x = N n=1 x e, x < ε β N We assert that x + V E \ A. Indeed, suppose that (x + V ) A. Then there exsts n N such that (x + V ) A E n. But ] (x + V ) E n x + ε [ ] β, +ε x + ε β n α, + ε [ n α. Ths contradcts (2.2). As a consequence, 3 For nstance, let αn = exp(1/(n 2 ))..

5 GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS 5 Corollary 2.2. Let E be a real vector space wth a countable Hamel bass. A subset A of E s ξ-closed f and only f there exsts a famly (F γ ) γ Γ of fnte dmensonal subspaces of E such that: (1) each fnte dmensonal subspace F of E s contaned n some F γ, (2) for every γ Γ, A F γ s closed n F γ. Proof. Assume frst that A s ξ-closed. It follows from Proposton 2.1 that the famly F of all fnte dmensonal vector subspace of E satsfes the condtons 1 and 2. Assume conversely that a famly (F γ ) γ Γ satsfes the condtons 1 and 2 of the present corollary. In order to apply Proposton 2.1, we propose to prove that for each fnte dmensonal vector subspace F of E, A F s closed n F. Let F F; then there exsts γ Γ such that F F γ. Note that A F = F (A F γ ). Snce A F γ s closed n F γ, the subset A F s closed n F F -cones. Defnton 2.3. Let E be a real vector space. A (convex) cone K of E s called an F -cone f K s ponted (.e. K ( K) = {0}), generatng (.e. K K = E) and f K = n=1 Kn s the unon of a countable famly of fnte dmensonal polyhedral convex cones K n such that each K n s an extremal subset 4 of K n+1. Remark 2.4. Note that f K s an F -cone of E, then E has a countable Hamel bass. Moreover, observe that an F -cone K s generated by a countable famly of vectors whch s the unon of the extremal drectons of each K n. Countably generated Yudn cones 5 are a partcular case of F -cones, but an F -cone s not necessarly a Yudn cone. For example, let E be a vector space wth a countable Hamel bass (e n ) n N. Let K be the cone generated by the famly {e 3 + e 1, e 3 + e 2, e 3 e 1, e 3 e 2 } {e n } n 4. Then K s not a Yudn cone, but t s an F-cone. Proposton 2.2. Let E be a real vector space and let K be an F -cone of E. Then K s ξ-closed. Proof. Snce K s generatng, for each fnte dmensonal subspace F of E, F (K n K n ) for some n. Moreover, t s easy to see that each K n s an extremal subset of K n for every n > n. Now we propose to prove that for each n N, K (K n K n ) = K n. Indeed, let x K (K n K n ), then there exsts y, z K n and n > n such that x K n and x = y z. It follows that y = x + z. But K n s an extremal subset of K n ; then x K n and K (K n K n ) K n. The converse s obvous. Now recallng that a fnte dmensonal polyhedral convex cone s closed, Corollary 2.2 ends the proof. We now assume that E s ordered by the F -cone K. Proposton 2.3. The order ntervals of E are ξ-compact. 4 A convex subset F of a convex set C s an extremal subset (or a face) of C f for each lne segment [y, z] of C satsfyng ]y, z[ F then y and z belong to F. If K s a convex cone, the convex subset F s an extremal subset of K f and only f x = y + z, x F and y, z K mply that y, z F. It follows that F s tself a convex cone. 5 A convex cone C generated by a famly (e ) I of elements of a vector space s called a Yudn cone f each x of C has a unque representaton of the form x = I x e, where x 0 and x = 0 for all but fntely many I. C s countably generated f I s countable.

6 GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS 6 Proof. Let a K, a 0. We only have to prove that the order nterval [0, a] = K ({a} K) s ξ-compact. Let n be such that a K n. For every x [0, a], let n be such that x and (a x) le n K n. If n n then x K n. If n > n, from a = x + (a x) and snce K s an F -cone, we deduce that x K n. Thus [0, a] K n, hence s ncluded n the fnte dmensonal space (K n K n ) (ordered by the polyhedral convex cone K (K n K n ) = K n ). It follows that [0, a] s a compact subset of K (K n K n ) = K n and, n vew of the defnton of ξ, a ξ-compact subset of E. Remark 2.5. Let E be a vector space wth a countable Hamel bass (e n ) n N. Let K be the cone generated by the famly {e 1 e n, e n } n 2. Snce the order nterval [0, e 1 ] contans the famly {e n } n 1, t s not ξ-compact. In vew of Proposton 2.3, the cone K s not an F -cone. 3. Equlbrum n the securty market Let us now come back to the fnancal economy E = ( (Φ, ξ, K), (K, P, ω ) m ) of whch the portfolo space s Φ, equpped wth the nductve lmt topology ξ, whle the postve cone K of the portfolo domnance orderng s an F -cone. A Hamel bass of Φ conssts of the vectors (0,..., 0, 1, 0,...) correspondng to each one of the countably many securtes defnng the model. The algebrac and topologcal dual of Φ s R N of whch each element can be thought of as a lst of prces for each securty. We frst observe that t follows from Propostons 2.2 and 2.3 that (Φ, ξ, K) s an ordered lnear vector space equpped wth a Hausdorff locally convex topology satsfyng the propertes: A1: The postve cone K s generatng and ξ-closed; and A2: The order ntervals of Φ are ξ-bounded requred for an economc model n Alprants Florenzano Tourky [3]. Let ω = m ω be the total ntal endowment of securtes,.e. the market portfolo. A portfolo allocaton s any m-tuple x = (x ) m Km such that m x = ω. Defnton 3.1. A par (x, p) consstng of an allocaton x and a non-zero lnear functonal p s sad to be (1) a quas-equlbrum f for every, p(x ) = p(ω ), and y P (x ) mples p(y ) p(x ), (2) an equlbrum, f t s a quas-equlbrum and f y P (x ) actually mples p(y ) > p(x ). Defnton 3.2. A quas-equlbrum (x, p) s sad to be nontrval f for some we have nf{p(z ): z K} < p(ω ). In ths paper, we wll be nterested only n nontrval quas-equlbra. 6 We now ntroduce some specal propertes of feasble allocatons. Defnton 3.3. A portfolo allocaton x s sad to be: (1) ndvdually ratonal f for each, ω / P (x ), 6 If (x, p) s some trval quas-equlbrum, then for every allocaton y, the par (y, p) s also a quas-equlbrum. If the quas-equlbrum (x, p) s nontrval, then t s well-known that (under some addtonal contnuty condton on preferences or concavty for utlty functons, and some rreducblty assumpton on the economy) (x, p) s actually an equlbrum.

7 GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS 7 (2) a weakly Pareto optmal allocaton f there s no portfolo allocaton y satsfyng for each, y P (x ), (3) a core allocaton, f t cannot be blocked by any coalton n the sense that there s no coalton S {1, 2,..., m} and some y K S such that: (a) S y = S ω and (b) y P (x ) for all S, (4) an Edgeworth equlbrum, f for every nteger r 1 the r-fold replca of x belongs to the core of the r-fold replca of the economy E. 7 As well-known, an equlbrum allocaton of E s an Edgeworth equlbrum, hence a core, weakly Pareto optmal and ndvdually ratonal allocaton. Recall that n the fnancal economy E, each nvestor has the cone K of postve payoffs as her portfolo set and an ntal endowment of securtes ω K. We post on E the followng assumptons: C1: For each, for every x K, (a) P (x ) s convex and x / P (x ); (b) (P ) 1 (x ) := {y K : x P (y )} s ξ-open n K. C2: The total ntal endowment ω of securtes s such that ω > R 0 (.e. ω K and ω 0). In vew of Proposton 2.3 and the prevous assumptons, t follows from Florenzano [8, Proposton 3] that Edgeworth equlbrum exsts for E. We now ntroduce the addtonal assumpton on E whch wll allow to decentralze wth prces n R N, the topologcal dual of Φ, any Edgeworth equlbrum. C3: For each and every weakly Pareto optmal allocaton x = (x ) m, we have x cl P (x ) and: (a) P (x ) s ξ-open n K or P (x ) = {y K : u (y ) > u (x )} for some concave utlty functon u : K R; (b) There s a convex set P (x ) Φ such that the vector x + ω s a ξ-nteror pont of P (x ) and P (x ) K = P (x ). Assumpton C3 (b) states that for each, the preference correspondence P s ω-proper at every component of a weakly Pareto optmal portfolo allocaton. Ths properness assumpton was frst ntroduced by Tourky [13] and proved by hm to be strctly weaker that the ω-mas-colell unform properness assumed by Alprants Brown Polyraks Werner [2] for preferences defned on K by utlty functons. The local non-sataton assumed n C3 s mpled by ther assumpton that the market portfolo ω s desrable. Let x = (x ) m be an Edgeworth equlbrum of E. In vew of Assumptons C1(a), C2, C3, t follows from Alprants Florenzano Tourky [3, theorem 5.1] that there exsts p R N such that (x, p) s a nontrval quas-equlbrum, provded the followng asserton s satsfed: 7 The deas n ths defnton go back to Debreu Scarf [7]. An mportant reference s also [4]. Edgeworth equlbra were frst ntroduced and studed n [1].

8 GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS 8 B: If f = (f 1, f 2,..., f m ) s a lst of ξ-contnuous lnear functonals such that f (ω) > 0 for each, and R f (ω) = m f (x ) for some x = (x ) m Km such that m x = ω, 8 then the Resz Kantorovch functonal 9 R f s ω-proper at ω. Checkng B when K s an F -cone of Φ and ξ s the nductve lmt topology generated by the fnte dmensonal vector subspaces of Φ wll be done n Secton 4. Ths wll prove the followng theorem whch extends n several respects Theorem 6.1 n Alprants Brown Polyraks Werner [2]. Theorem 3.4. Assume that Φ s equpped wth the nductve lmt topology ξ and that the cone of postve payoff portfolos K s an F -cone. Under the assumptons C1, C2, C3 on E = ( (Φ, ξ, K), (K, P, ω ) m ), there exsts a nontrval portfolo quas-equlbrum. Remark 3.5. Usng Remark 2.5, t s easy to construct economes E = ( ) (Φ, ξ, K), (K, P, ω ) m satsfyng C1, C2, C3, whose postve cone s countably generated but s not an F-cone, and for whch there exsts nether weakly Pareto optmal allocaton nor nontrval quas-equlbrum. 4. Properness of the Resz Kantorovch functonal Let E be a real vector space and let K be an F -cone of E. Recall that for a fnte lst f = (f 1,..., f m ) of contnuous lnear functonals on (E, ξ), the Resz-Kantorovch functonal R f s defned on K by m m R f (x) = sup{ f (x ): x K for each and x = x}. If we let, for each ω K, P (ω) = {ω K : R f (ω ) > R f (ω)}, then the Resz-Kantorovch functonal R f s sad ω-proper at ω f there exsts a convex subset P (ω) of E such that 2ω s a ξ-nteror pont of P (ω) and P (ω) K = P (ω). We propose to prove n ths secton that for ω > 0 (.e. ω K and ω 0) and any fnte lst f = (f 1,..., f m ) of contnuous lnear functonals on (E, ξ) such that f (ω) > 0 for each, the Resz-Kantorovch functonal R f s ω-proper at ω The fnte dmensonal case. Suppose n ths subsecton that E s fnte dmensonal. Then the nductve lmt topology ξ concde wth the unque Hausdorff lnear topology on E. Moreover the F -cone K s a ponted and generatng polyhedral convex cone. Proposton 4.1. For ω > 0 and any fnte lst f = (f 1,..., f m ) of lnear functonals on ω such that f (ω) > 0 for each, the Resz-Kantorovch functonal R f s ω-proper at ω. 8 It follows from the defnton of Rf and the ξ-compactness of the order ntervals of Φ that the condton R f (ω) = m f (x ) for some x = (x ) m Km such that m x = ω s satsfed for any lst f = (f 1, f 2,..., f m) of ξ-contnuous lnear functonals. 9 The precse defnton of R f s gven n the followng secton.

9 GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS 9 Proof. The functonal R f s contnuous on K (for ths concluson we need Theorems 10.2 and 20.5 n [11]). The set R(ω) := {ω K : R f (ω ) R f (ω)} s a polyhedral convex set (for ths concluson see Theorem 19.3 and Corollary n [11]). Express R(ω) as the set of solutons to a certan system R(ω) = {x E : I a (x) α }, where the fnte famly (a, α ) I (E ) I R I s mnmal. In vew of the postve homogenety of R f, one can assume α 0 for each I. Set I := { I : α > 0}. Snce R f (ω) > 0, t follows that 0 R(ω) and hence the set I s nonempty. Let us denote by I(ω) the level set and defne Clam 4.1. I(ω) = J(ω). I(ω) := {ω K : R f (ω ) = R f (ω)} J(ω) := {ω R(ω): I a (x) = α }. Proof. Assume frst z I(ω) and z / J(ω). So for each I, we have a (z) > α. Ths mples that there s some 0 < λ < 1 such that a (λz) > α for all I. Now f / I, then α = 0 and so a (λz) 0 and therefore λz R(ω). But z I(ω) mples R f (ω) = R f (z) > λr f (z) = R f (λz) R f (ω), whch s a contradcton. Conversely, assume z J(ω) and z / I(ω). It follows that R f (z) > R f (ω). So f λ = R f (ω)/r f (z), then 0 < λ < 1 and R f (λz) = R f (ω). Hence λz I(ω) R(ω) and hence a (λz) α for all I. Now notce that snce z J(ω), there exsts some such that a (z) = α > 0. In partcular for the vector λz R(ω) we have a (λz) = λa (z) < α, contrary to λz R(ω). Clearly, Clam 4.1 mples that We now defne P (ω) = R(ω) {z E : I a (z) > α }. P (ω) = {z E : I a (z) > α }. It follows from ths defnton that P (ω) s a nonempty convex open set and that P (ω) P (ω) K. Clam ω P (ω). Proof. Snce ω R(ω), for each I we must have a (ω) α > 0 and consequently a (2ω) = 2a (ω) > α. Clam 4.3. For each I, there exsts some u R(ω) such that a (u) = 0. Consequently, for any such I, we have for each z K, a (z) 0. Proof. Recall frst that f for some I on has a (z) > α for each z R(ω), then the nequalty a (z) α can be removed from the expresson of R(ω). Snce the famly (a, α ) I s supposed to be mnmal, ths proves the frst asserton. To prove by contraposton the second asserton, assume that I, z K and a (z) < 0. Let u R(ω) be such that a (u) = 0. Snce R f (u) R f (ω) > 0, t follows from the contnuty of R f at u, that there exsts z K such that a (z ) < 0 and R f (z ) > 0. Usng the postve homogenety of R f, we can fnd some λ 1 such that R f (λz ) R f (ω). Hence a (λz ) 0, whch mples that a (z ) 0, contrary to a (z ) < 0.

10 GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS 10 Clam 4.4. P (ω) K P (ω). Proof. Let z P (ω) K. In vew of the defnton of P (ω), we have only to prove that z R(ω),.e. that z satsfes all the nequaltes defnng R(ω). For I, ths follows from z P (ω); for I, ths follows from z K and the prevous clam. We have thus proved that P (ω) s an open convex set wth 2ω P (ω) and such that P (ω) K = P (ω) The general case. In ths subsecton, the space E s not supposed to be fnte dmensonal. Accordng to Defnton 2.3, the cone K s ponted, generatng and K = n=1 Kn s the unon of a countable famly of fnte dmensonal polyhedral convex cones K n such that each K n s an extremal subset (or a face) of K n+1. Recall that, as notced n Remark 2.4, E has a countable Hamel bass. Proposton 4.2. For ω > 0 and any fnte lst f = (f 1,..., f m ) of contnuous lnear functonals on (E, ξ) such that f (ω) > 0 for each, the Resz-Kantorovch functonal R f s ω-proper at ω. Proof. Let ω > 0 and let f = (f 1,..., f m ) be a fnte lst of contnuous lnear functonals on (E, ξ) such that f (ω) > 0 for each. Wthout any loss of generalty, we can suppose that ω K 1. Let n 1, we defne f n be the fnte lst (f1 n,..., fm), n where f n (E n ) s the restrcton of f to the subspace E n = K n K n. Snce each K n s a face of K, we frst have that R f n concde wth the restrcton to K n of R f, that s x K n R f n(x) = R f (x). In partcular, f we let P n (ω) := {ω K n : R f n(ω ) > R f n(ω)}, then P n (ω) = P (ω) K n. Now f we let R n (ω) := {ω K n : R f n(ω ) R f n(ω)}, then as n the proof of Proposton 4.1, there exsts a fnte lst (a n ) I n of lnear functonals and postve scalars (α n ) I n such that R n (ω) = {x E n : a n (x) α n I n }. Moreover f the famly (a, α ) I s chosen mnmal, then P n (ω) = P n (ω) K n, where P n (ω) := {x E n : a n (x) > α n I n}, and where I n := { I n : α n > 0}. Note that P n (ω) s convex, open n E n and 2ω P n (ω). Moreover, followng Lemma A.1 n Appendx, the constructon of P n (ω) s ndependent of the choce of the mnmal famly (a, α ) In. 10 Clam 4.5. The sequence ( P n (ω)) n 1 s ncreasng, that s n 1 P n (ω) P n+1 (ω). Proof. Let n 1, note that followng the defnton of K, R n+1 (ω) E n = R n (ω). In partcular R n (ω) = {x E n : [a n+1 ] E n(x) α n+1 I n+1 }, 10 Indeed, snce K n s generatng n E n, t follows that R n (ω) has an nteror pont n E n. Hence, from Lemma A.1 n Appendx, the faces of R n (ω) wth dmenson dm K n 1 are exactly the convex sets H R n (ω), I where H := {x E n : a (x) = α }.

11 GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS 11 where [a n+1 ] E n (E n ) s the restrcton of a n+1 to E n. Consder a subset J n+1 of I n+1 such that the famly ([a n+1 j ] E n), α n+1 j ) j Jn+1 s mnmal n the defnton of R n (ω). Followng the constructon of P n (ω), we have P n (ω) = {x E n : [a n+1 j ] E n(x) > α n+1 j j J n+1} where J n+1 = {j J n+1 : α n+1 j > 0}. In partcular J n+1 I n+1, and P n (ω) P n+1 (ω). Now let P (ω) = n 1 P n (ω). P (ω) s obvously convex. Snce 2ω P 1 (ω) t follows that 2ω P (ω). For each n 1, P (ω) E n = P k n k (ω) E n, wth P k (ω) E n open n E n. Applyng Corollary 2.2, to the famly (E n ) n N of fnte dmensonal vector subspaces of E, t follows that P (ω) s ξ-open. Now we assert that P (ω) K = P (ω). Indeed, P (ω) K = P n 1 (ω) K n, and for each n 1, P (ω) K n = ( P k (ω) K k ) K n = P k (ω) K n = P n (ω). k n k n It follows that P (ω) K = n 1 P n (ω) = P (ω). Appendx A. Note on the faces of a polyhedral convex set Let E be a non-trval fnte dmensonal vector space and A a polyhedral convex set. By defnton, there exsts a fnte famly of lnear functonals (a ) I and of scalars (α ) I such that A = {x E : a (x) α I}. The famly (a, α ) I s sad mnmal f for each I, the set A := {x E : a j (x) α j j } s strctly bgger than A. Recall that a convex subset F s a face (or an extremal subset) of A f for each lne segment [x, y] of A satsfyng ]x, y[ F, then x and y belong to F. Lemma A.1. Let F be the set of faces of A wth dmenson dm A 1. If the famly (a, α ) s mnmal and f A has an nteror pont then where H = {x E : a (x) = α }. F = {H A: I}, Proof. We frst prove that for each I, F := H A s a face of A wth dmenson dm A 1. Let I and let [x, y] A be a lne segment such that ]x, y[ F, that s, there exsts λ ]0, 1[ such that a (λx + (1 λ)y) = α. Snce a (x) α and a (y) α then a (x) = α and a (y) = α. We have thus proved that F s a face of A. Now we propose to prove that dm F = dm A 1. Let A := {x E : a j (x) α j j }; then nt A = {x E : a j (x) > α j j }. Note that (nt A ) H F. Snce the famly (a k, α k ) k I

12 GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS 12 s mnmal then 11 (nt A ) H. In partcular, dm [ nt(a ) H ] = dm H = dm A 1. But ( (nt A ) H ) F H, hence dm F = dm A 1. Now we prove that F {F : I}. Let F be a face of A wth dmenson dm A 1. Then there exsts a lnear functonal b and a scalar β such that In partcular F = {x E : b(x) = β} A and A {x E : b(x) β}. {x E : a (x) α I} {x E : b(x) β}. It follows from Rockafellar [11, Theorem 22.3] that there exsts a famly (λ ) I of non negatve scalars λ 0, such that b = λ a and β λ α. I I But f x F, for each I, a (x) α and λ α λ a (x) = b(x) = β λ α. I I I It follows that β = I α. Hence F = {x E : I a (x) α and λ a (x) = λ α }. I I That s, F = F where I := { I : λ > 0}. I Let I ; then F F. Snce F and F have the same dmenson, then the relatve nteror of F s contaned n the relatve nteror of F, whch mples (see [11, Corollary ]) F = F. References [1] C. D. Alprants, D. J. Brown, and O. Burknshaw, Edgeworth equlbra, Econometrca 55 (1987), [2] C. D. Alprants, D. J. Brown, I. A. Polyraks and J. Werner, Portfolo domnance and optmalty n nfnte securty markets, Journal of Mathematcal Economcs 30 (1998), [3] C. D. Alprants, M. Florenzano and R. Tourky, General equlbrum analyss n ordered topologcal vector spaces, Workng-paper, Unversty Pars 1 (2002). [4] J. P. Aubn, Mathematcal Methods of Game and Economc Theory, Amsterdam and New York: North Holland, [5] N. Bourbak, Espaces vectorels topologques, 2nd ed., Pars: Herman, [6] D.J. Brown and J. Werner, Arbtrage and exstence of equlbrum n nfnte asset markets, Revew of Economc Studes 62, (1995), [7] G. Debreu and H. Scarf, A lmt theorem on the core of an economy, Internatonal Economc Revew 4 (1963), [8] M. Florenzano, Edgeworth equlbra, fuzzy core, and equlbra of a producton economy wthout ordered preferences, Journal of Mathematcal Analyss and Applcatons 153, (1990), [9] J. L. Kelley and I. Namoka, Lnear Topologcal Spaces, Prnceton, New Jersey: D. Van Nostrand Company, Inc, [10] A. Mas-Colell, The prce equlbrum exstence problem n topologcal vector lattces, Econometrca 54 (1986), If (nt A ) H =, then nt A {x E : a (x) > α }. It follows that nt A A. But A s strclty bgger than A.

13 GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS 13 [11] R. T. Rockafellar, Convex Analyss, 2nd ed., Prnceton: Prnceton Unversty Press [12] H. H. Schaefer, Topologcal Vector Spaces, New York, Heldelberg, Berln: Sprnger Verlag [13] R. Tourky, A new approach to the lmt theorem on the core of an economy n vector lattces, Journal of Economc Theory 78 (1998),