No-arbitrage condition and existence of equilibrium in asset markets with a continuum of traders

No-arbitrage condition and existence of equilibriumin asset markets with a continuum of traders

Cuong Le Van∗ and Francois Magnien†

In the present paper, we prove that a no-arbitrage condition (a la Werner) is necessaryand sufficient for the existence of an equilibrium with a continuum of traders and afinite number of assets. As in Aumann (1966), Hildenbrand (1974) and Schmeidler(1969), preferences are not assumed to be convex. We do not use Fatou’s Lemma anddo not assume that the consumption sets are compact.

Key words equilibrium in asset market, no-arbitrage condition, no-arbitrage price,continuum of trader, closed convergence topology, Fatou’s Lemma

JEL classification C62

Accepted 27 July 2004

1 Introduction

The present paper deals with the existence of an equilibrium in asset markets with a contin-uum of traders. The fundamental characteristics of asset markets is that the consumptionsets are not bounded from below.

When the number of assets are finite, different conditions of non-arbitrage are in-troduced in order to obtain existence of equilibria results (see e.g. Werner 1987; Page1987; Nielsen 1989; Dana et al. 1999). When the number of assets is infinite, generally,no-arbitrage conditions are not sufficient to guarantee existence of equilibria. It is usualto assume that the individually rational utility set is compact (see e.g. Brown and Werner1995; Dana et al. 1997).

In the presence of a continuum of agents, existence theorems seem to have beenproved only for bounded from below consumption sets (Aumann 1966; Schmeidler 1969;Hildenbrand 1970a,b). Later, Khan and Yannelis (1991), Rustichini and Yannelis (1991)and Noguchi (1997) gave an extension to infinite dimensional consumption goods underthe assumption that the consumption sets are bounded or weakly compact.

In the present paper, we assume that the number of assets is finite in order to show theusefulness of no-arbitrage conditions for the existence of equilibria, even with a continuumof agents. We do not require Fatou’s Lemma and do not assume that the consumptionsets are compact. More precisely, we prove that there exist no-arbitrage prices (a la Werner

∗ Center for Mathematical Research, Statistics and Mathematical Economics, National Center for ScientificResearch, University of Paris 1, Paris, France. Email: [email protected]

† National Institute for Statistics and Economic StudiesThe authors would like to thank two anonymous referees for valuable comments and suggestions.

International Journal of Economic Theory 1 (2005) 43–55 C© IAET 43

No-arbitrage condition and existence of equilibrium in asset markets Cuong Le Van and Francois Magnien

1987), iff there exists an equilibrium. When the consumption sets are the positive orthantsand preferences are increasing, the set of no-arbitrage prices is equal to the interior ofthe positive orthant and, hence, non-empty. We obtain a proof of existence of equilibriawithout using Fatou’s Lemma.

As in Aumann (1966), Hildenbrand (1974) and Schmeidler (1969), we do not assumeconcavity of preferences. We assume that the characteristics of the economy; that is, initialendowments, consumption sets and preferences, are measurable with respect to the agents.

The present paper is organized as follows: in Section 2, we present the model and theno-arbitrage condition. In Section 3, we show that, without any loss, we can assume that theinitial endowments, the consumption sets and the preferences are continuous with respectto the agents. Section 4 summarizes the properties of the individual demands. The proofof existence of equilibria is given in Sections 5 and 6.

2 No-arbitrage condition

We consider an economy with l goods: the commodity space is Rl . The set of consumersis described as a complete measure space (A, A, µ) where µ is an atomless σ – additivepositive measure such that µ(A) = 1. Each agent a ∈ A has a consumption set Xa ⊆ Rl

which is non-empty, closed and convex (for a.e. a ∈ A), an initial endowment ea ∈ Xa anda complete, reflexive and transitive preference relation �a on Xa. Let P denote the set ofpreferences relations. We endow it with the closed convergence topology (see Hildenbrand1974, p.14). We suppose that:

(H 1) The function a → ea is µ − integrable.(H 2) The correspondence a → Xa is measurable.(H 3) The correspondence a → (Xa, �a ) ∈ P is measurable1.(H 4) For a.e. a ∈ A, the preference relation �a is continuous.

For a.e. a ∈ A, Let P a = {x ∈ Xa : ea �a x}. Let Pa be the closed convex hull of P a andWa the recession cone of Pa:

Wa = {w ∈ Rl : ea + λw ∈ Pa for all λ ≥ 0

}.

We can now introduce the set of no-arbitrage prices for every agent a ∈ A:

Sa = {p ∈ Rl : p.w > 0 for all w ∈ Wa\{0}}.

Obviously, Sa (respectively Wa) is an open (respectively closed) convex cone. We definethe set of no-arbitrage prices (for the whole economy) as S = ∩a∈A Sa. When S = ∅,we say that the no-arbitrage condition is satisfied. For example, when short-sales are notpermitted, in other words, when Xa ⊆ Rl

+ for every a ∈ A, then the no-arbitrage conditionis satisfied, because Wa ⊆ Rl

+ for every a ∈ A and, hence, (+1, +1, . . . , +1) ∈ S.We introduce the strong survival assumption:

inf p.Xa < p.ea for all p ∈ clS\{0} and all a ∈ A.

1 The set P, with the closed convergence topology, is metrizable and compact. The Borel σ -algebra on P isgenerated by the open subsets of P.

44 International Journal of Economic Theory 1 (2005) 43–55 C© IAET

Cuong Le Van and Francois Magnien No-arbitrage condition and existence of equilibrium in asset markets

An equilibrium of economy (ea, Xa, �a )a∈A is a pair (x∗, p∗) where x∗ : A → Rl isµ–integrable, p∗ ∈ Rl\{0} and:

(i)∫

A x∗a µ(da) = ∫

A eaµ(da);(ii) x∗

a is a maximal element for �a in {x ∈ Xa:p∗.x ≤ p∗.ea} for a.e. a ∈ A.

Remark 1 One has inf p.Xa < p.ea for all p ∈ Rl\{0} if ea ∈ intXa.

Remark 2 Alternatively to the no-arbitrage condition, one can consider another no-arbitragecondition (introduced by Page (1987) and used by Nielsen (1989)): there is no unboundedarbitrage if for any finite subset E of A,

0 =∑a∈E

wa , wa ∈ Wa =⇒ wa = 0 for all a ∈ E .

Proposition 1 If Wa contains no line for all a ∈ A, then S = ∅ iff, there exists no unboundedarbitrage.

PROOF: See Appendix. �

3 From measurability to continuity

The measurability of the characteristics (initial endowments, consumption sets and pref-erences) with respect to the agents (assumptions (H 1), (H 2) and (H 3)) might appear as aweak assumption. In fact, one can obtain continuity by an appropriate “change of variable”.First, we need two lemmas.

Lemma 2 Let B denote the unit-ball of Rl . There exists a measurable function ε : A → Bsuch that µ({a ∈ A : εa = x}) = 0 for all x ∈ B.

PROOF: See Appendix. �Lemma 3 Consider economy (e ′

a , X ′a , �′

a )a∈A where e ′a = εa , X ′

a = Xa − ea + εa andwhere �′

a is defined by x ′ �′a y ′ iff x ′ + ea − εa �a y ′ + ea − εa . If economy (e ′

a , X ′a , �′

a )a∈A

admits an equilibrium then the initial economy (ea, Xa, �a )a∈A also admits an equilibrium.

PROOF: Let (x∗, p∗) be an equilibrium of (e ′a , X ′

a , �′a )a∈A. Then (x∗ + e − ε,

p∗) is an equilibrium of (ea, Xa, �a )a∈A. Indeed,∫

A x∗a µ(da) = ∫

A εaµ(da). Hence,∫A(x∗

a + ea − εa )µ(da) = ∫A eaµ(da). Moreover, if x∗

a + ea − εa ≺a x ∈ Xa, thenx∗

a ≺′a x − ea + εa and, hence, p∗.e ′

a < p∗.(x − ea + εa ); that is, p∗.x > p∗.ea. �Obviously, economy (e ′

a , X ′a , �′

a )a∈A, satisfies assumptions (H 1), (H 2), (H 3) and(H 4). Observe that Wa = W ′

a and, hence, Sa = S ′a for every a ∈ A. Moreover, (e ′

a , X ′a ,

�′a )a∈A satisfies the strong survival assumption if (ea, Xa, �a )a∈A satisfies it.

Now consider the following function:

τ : A → B × P

a → (e ′a , (X ′

a ,�′a ))

where P is the set of preference relations. Denote by � the closure of τ (A) in B × P and bym the image measure of µ under τ . Note that m(�\τ (A)) = 0. Observe that for any a , X ′

aintersects the unit-ball of Rl . Therefore, endowed with the closed convergence topology,



P is compact and metrizable (see Hildenbrand 1974, p. 97 ). Hence, � is compact andmetrizable. Moreover, m is atomless. Indeed, if T is an atom, then the measurable function(e , (X , �)) → e is constant on T (see Hildenbrand 1974, p. 44). Hence, there exists x ∈ Rl

such that εa = x for all a ∈ A satisfying τ (a) ∈ T . From Lemma 2, µ(τ−1(T)) = 0; that is,m(T) = 0, a contradiction.

For all t = (e , (X , �)) ∈ �, we define e ′′t = e , X ′′

t = X and �′′t = �.

Lemma 4 If the “image economy” (e ′′t , X ′′

t , �′′t )t∈� admits an equilibrium, then (e ′

a , X ′a ,

�′a )a∈A also admits an equilibrium.

PROOF: Let (x∗, p∗) be an equilibrium of (e ′′t , X ′′

t , � ′′t )t∈�. We obtain an equilibrium

(y∗, p∗) of (e ′a , X ′

a , �′a )a∈A by defining y∗

a = x∗τ (a) for a.e. a ∈ A. Indeed:

∫A

y∗a µ(da) =

∫A

x∗τ (a)µ(da)

=∫

�

x∗t m(dt) (from the change of variable formula)

=∫

�

e ′′t m(dt) (because (x∗, p∗) is an equilibrium of (e ′′

t , X ′′t ,�′′

t )t∈�)

=∫

Ae ′′τ (a)µ(da) (from the change of variable formula)

=∫

Ae ′

aµ(da)(e ′′τ (a) = e ′

a because τ (a) = (e ′a , (X ′

a ,�′a ))

).

Moreover, if y∗a ≺a y ∈ Xa, then x∗

t ≺t y. Hence, p∗.e ′′t < p∗.y and p∗.y > p∗.e ′

a wheret = τ (a). �

Because � is compact, it is convenient to consider an economy where the set of agentsis �. So, the remainder of the present paper is devoted to the proof of existence of anequilibrium in economy (e ′′

t , X ′′t , � ′′

t )t∈�. For simplicity, we denote by (et , Xt , �t )t∈� thiseconomy. The following proposition summarizes its properties.

Let Pt denote the closed convex hull of {x ∈ X : et �t x}, and Wt the recession coneof Pt .

Proposition 5 The economy (et , Xt , �t )t∈� satisfies the following properties:

(P1) The function t → et is continuous in �.(P2) The consumption set Xt is non-empty, closed and convex for all t ∈ �.(P3) The correspondence t → Xt is closed.(P4) The correspondence t → Xt is lower semi-continuous.(P5) For all t ∈ �, the preference �t is continuous.(P6 ) The function t → (Xt , �t ) ∈ P is continuous.2

(P7 ) The correspondence t → Wt is closed if the correspondence t → Pt is closed.

PROOF: (P1) and (P2) are obvious; (P3) and (P4) follow from Corollary 1, p. 98 in Hilden-brand (1974); (P5) follows from the definition of P; (P6) is obvious (the projection map-ping is continuous); (P7): Let λ > 0, tn → t in � and wn → w in Rl with wn ∈ Wtn

2 For the closed convergence topology.



for all n. Because et + λw = limn→∞ etn + λwn and etn + λwn ∈ Ptn , it follows thate + λw ∈ Pt . �

Remark 3 If (ea, Xa, �a )a∈A satisfies the strong survival assumption, then (et , Xt , �t )t∈�

also satisfies the strong survival assumption.We add the following assumptions:

(H 5) For all t ∈�, all x ∈ Xt such that et � x and all w ∈ Wt\{0}, there exists λ > 0such that x + λw ∈ Xt and x ≺t x + λw.

(H 6) For all t ∈� and all x ∈ Xt such that et � x , there exists w ∈ Wt\{0} such thatx + λw ∈ Xt and x ≺t x + λw for all λ ∈ [0, 1].

(H 7) The correspondence t → Pt is closed.(H 8) For any measurable subset T of � such that m(T) = 1, one has ST = S, where

ST = ∩t∈T St .

We give two examples where assumptions (H 5), (H 6) and (H 7) are satisfied. Moreover,assumption (H 8) is satisfied for the second example.

Example 1 Suppose that the preferences �t are convex for every t . Assumption (H 5)means that there exists uniformity of the preferences and no half-line (see e.g. Werner1987). Assumption (H 6) is satisfied if we assume that the preferences are strictly convex,or if there exists an extremely desirable riskless direction; that is, a vector w ∈ Wt\{0} suchthat x ≺t x + λw for all λ ∈ [0, 1], and all x ∈ Xt . Assumption (H 7) is satisfied, because itis obvious that the correspondence t → P t is closed and because, by the convexity of thepreferences, P t = Pt .

Example 2 Suppose that Xt = Rl+ and that the preferences �t are increasing. In this case,

Wt = Rl+. It is obvious that (H 5) and (H 6) are satisfied. It is also obvious that (H 8) is

satisfied because St = intRl+, for every t ∈�. Let us check (H 7).

Consider a sequence (xtn ) in Rl+ verifying

xtn = λn ytn + (1 − λn)ztn ,

where ytn ∈ P tn , ztn ∈ P tn and λn ∈ [0, 1]. Assume xtn → x . Let λ = lim λn. One has:

xtn ≥ λn ytn ≥ 0,

xtn ≥ (1 − λn)ztn ≥ 0.

Hence,

(i) If λ = 0, 1, then ytn → y ∈ P t, ztn → z ∈ P t , and x = λy + (1 − λ)z ∈ Pt : the claimis true.

(ii) Assume λ= 0. In this case, ztn → z ∈ P t and λn ytn → ξ ∈ Rl+. Because P t + Rl

+ ⊆ P t ,we have x = ξ + (1 − λ)z ∈ P t ⊆ Pt : the claim is true.

We can now state our main result.

Theorem 6 Assume (H 1), . . . , (H 8) and the strong survival assumption. There exists anequilibrium iff , S = ∅.



4 Properties of the individual demand

Because x ≺t et if x ∈ Pt , we define the individual demand by:

D(t, p) = {x ∈ B(t, p) : there exists no z ∈ B(t, p) with x ≺t z}where

B(t, p) = {x ∈ Pt : p.x ≤ p.et}for all (t, p) ∈ �× Rl .

Lemma 7

(i) The correspondence B is non-empty, convex valued and closed in �× Rl .(ii) The correspondence B is bounded (hence, upper semi-continuous) and is compact valued

in a neighborhood of every (t, p) ∈ �× Rl such that p ∈ St .

PROOF:

(i) Obviously, the set B(t , p) is non-empty, closed and convex for every (t, p) ∈ �× Rl .The closedness of the correspondence B follows from properties (P 1), (P 3) andassumption (H 7).

(ii) The range of the correspondence B is locally bounded. Otherwise, there exist asequence (tn, pn) → (t, p) in ∈ �× Rl and an unbounded sequence (xn)n suchthat xn ∈ B(tn, pn) for all n. One might suppose that xn/‖xn‖ converges to somew ∈ Sl−1 (the unit-sphere of Rl). Moreover, p.w ≤ 0 because, from (P 1), (pnetn )is bounded. Because λxn/‖xn‖ + (1 − λ/‖xn‖)etn ∈ Ptn for all λ > 0 and n largeenough, it follows from (P 6) that w ∈ Wt . That is a contradiction with p ∈ St . �

Proposition 8 Define �= {(t, p) ∈ �× Rl : p ∈ St} and �′ = {(t, p) ∈ � : infp.Xt < p.et}. Then:

(i) D(t, p) is non-empty iff (t, p) ∈�.(ii) � and �′ are open subsets of � × Rl .

(iii) D(t , p) is compact for all (t, p) ∈�.(iv) The correspondence D is locally bounded in � and closed in �′; hence, it is upper

semi-continuous in �′.(v) We have p.D(t, p) = {p.et} for all (t, p) ∈�.

PROOF:

(i) Let p ∈ St . From Lemma 7(ii), B(t, p) is compact. Hence, from (P 5), D(t, p)is non-empty. Conversely, suppose that x ∈ D(t, p) exists as well as W \{0}such that p.w ≤ 0. From (H 5), there exists λ > 0 such that x ≺t x + λw whilep.(x + λw) ≤ p.x ≤ p.et : a contradiction with the optimality of x.

(ii) � is open from Lemma (ii) and (i). Then �′ is also open. Indeed, we have just toprove that V = {(t, p) ∈ � × Rl : inf p.Xt < p.et} is open. Suppose (tn, pn) →(t, p) ∈ V and (tn, pn) ∈ V ; that is, inf pn.Xtn ≥ pn.etn for all n. Consider x ∈ Xt .From (P 4) there exists xn ∈ Xtn such that xn → x . Because pn.xn ≥ pn.etn for all n,one has p.x ≥ p.et , which contradicts (t, p) ∈ V .

(iii) Follows from D(t, p) ⊆ B(t, p) and Lemma (ii) and (P 5).



(iv) From D(t, p) ⊆ B(t, p) and Lemma (ii), the correspondence D is bounded ina neighborhood of every (t, p) ∈�. The set D is closed in �′. Indeed, suppose(tn, pn) → (t, p) ∈�′ and consider xn ∈ D(tn, pn) such that xn → x . Pick anyy in Xt which satisfies p.y ≤ p.et . We just have to show that y �t x . Let z ∈ Xt

verify p.z < p.et and consider (from (P4))yn, zn ∈ Xtn such that yn → y andzn → z. For λ ∈ [0, 1], we have p.(λy + (1 − λ)z) < p.et . For n large enough,pn(λyn + (1 − λ)zn) ≤ pn.etn and, hence, λyn + (1 − λ)zn �tn xn. From Theorem1(c) in Hildenbrand (1974), by passing to the limit, one gets λy + (1 − λ)z) �t x .Let λ → 1. We obtain y �t x .

(v) Suppose p.x < p.et for some x ∈ D(t, p). From (H 6), there exists w ∈ Wt such thatx + λw ∈ Xt and x ≺t x + λw for all λ ∈ [0, 1]. That contradicts the definition ofD(t , p) because p.(x + λw) < p.et for λ sufficiently small. �

Proposition 9 Let (tn, pn)n be a sequence in � and (xn)n a sequence with xn ∈ D(tn, pn)for all n.

1. If tn → t ∈�, then the following assumptions are equivalent:(i) The sequence (xn)n is bounded.

(ii) For all p ∈ St , there exists α ∈ R such that p.xn ≤α for all n.(iii) For some p ∈ St , there exists α ∈ R such that p.xn ≤ α for all n.

2. Suppose moreover that pn → p ∈ Rl with inf p.Xt < p.et , where t = lim tn.(i) If p ∈ ∂St , then the sequence (xn)n is unbounded.

(ii) If p ∈ St , then there exists a subsequence (xnk )k of (xn)n, which converges to somex ∈ D(t, p).

PROOF:

1. (i) ⇒ (i i) follows from Schwarz’s inequality: p.xn ≤ ‖p‖ ‖xn‖.(ii) ⇒ (i i i) is obvious.

(iii) ⇒ (i): suppose that (xn)n is unbounded. Because p.xn ≤α for all n, there ex-ists a subsequence (x ν)ν such that x ν/‖x ν‖ → w = 0, and p.w ≤ 0. Becauseλxν/‖xν‖ + (1 − λ/‖xν‖)etν ∈ Ptν for all λ > 0 and ν large enough, it followsfrom (P 7) that w ∈ Wt , a contradiction with p ∈ St .

2. (i) If (xn)n is bounded, then one can suppose that xn → x ∈ Xt and p.x ≤ p.et . Usingthe same technique as in the proof of Proposition 8(iv), one gets x ∈ D(t, p) incontradiction with Proposition 8(i).

(ii) Follows from Proposition 8(iv). �

5 The global excess demand

We will examine some properties of the set S of no-arbitrage prices on which the globalexcess demand will be defined.

Proposition 10

(i) S is an open convex cone of Rl .(ii) S is included in an open half-space.

PROOF:

(i) Let p0 ∈ S. Because � is open, for any t ∈�, there exist open subsets Ut and Vt in �

and Rl , respectively, such that (t, p0) ∈ Ut × Vt ⊆ �. From the compactness of �,



there exists a finite subcovering Ut1, Ut2, . . . , Utn of �. Let V = ∩i=1,...,nVti . V is anopen neighborhood of p0 in Rl . Now, it suffices to prove that V ⊆ S. Let p ∈ V . Forevery t ∈�, there exists i ∈ {1, 2, . . . , n} such that t ∈ Uti . Because p ∈ Vti , one has(t, p) ∈ Uti × Vti ⊆ �. Hence, p ∈ St for all t ∈�; that is, p ∈ S.

(ii) First, observe that 0 ∈ ∂S because 0 ∈ S and 0 ∈ clS. Consider now p∗ = 0 such thatp∗.p ≥ 0 for all p ∈ S. If p ∈ S then, from (i), p − ε p ∈ S for ε small enough. Hence,p∗.(p − ε p) ≥ 0 which implies p∗.p ≥ ε‖p‖2 > 0. �

Lemma 11 Let p ∈ S. Then there exists a neighborhood V (p) of p and a real number M(p)such that ‖x‖≤ M(p) for all x ∈ D(t, p′), all t ∈� and all p′ ∈ V(p).

PROOF: Let t ∈�. From Proposition 8(iv), D is bounded in a neighborhood of(t , p). Hence, there exists a real number M(t , p) and open subsets Ut and Vt in � andRl , respectively, such that ‖x‖≤ M(t, p) for all x ∈ D(t ′, p′) and all (t ′, p′) ∈ Ut × Vt .Because � is compact, one can select a finite subcovering Ut1, Ut2, . . . , Utn of �. LetV(p) = ∩i=1,...,nVti .V(p) is an open neighborhood of p0 in Rl . To end the proof, defineM(p) = max {M(ti, p) : i = 1, . . . , n}. �

Proposition 12

(i) The correspondence:

p ∈ S → Z(p) =∫

�

[D(t, p) − et ]m(dt)

is non-empty and convex-valued. Moreover, p.Z(p) = {0} for all p ∈ S.(ii) Assume inf p.Xt < p.et for all p ∈ S and all t ∈�. Then Z is upper semi-continuous.

PROOF:

(i) Z(p) = ∅ follows from theorem 2, p. 62 in Hildenbrand (1974) and from Lemma 11.The convexity of Z(p) follows from theorem 3, p. 62 in Hildenbrand (1974). Finally,from Proposition 8(v), we have p.Z(p) = {0}.

(ii) If inf p.Xt < p.et for all p ∈ S and all t ∈�, then �′ = � × S. Hence, from Lemma11, Proposition 8(iv) and proposition 8, p. 73 of Hildenbrand (1974), Z is uppersemi-continuous in S. �

The correspondence Z is called the global excess demand. There exists an equilibrium if0 ∈ Z(p∗) for some p∗ ∈ S.

The next section is devoted to the proof of Theorem 6.

6 Proof of theorem 6

We first prove that S = ∅ implies existence of an equilibrium.

Lemma 13 For all p ∈ Rl the set:

A(p) = {t ∈ � : p ∈ St}is an open subset of �.



PROOF: The result follows immediately from Proposition 5(ii). �Lemma 14 There exists a sequence (Cn)n of closed convex cones with vertex 0 in Rl suchthat Cn ⊆ C n+1 for all n and S ∪ {0} = ∪nCn.

PROOF: From Proposition 10 (ii), there is p0 = 0 such that, for each p ∈ S, there existsa unique λ > 0 satisfying p0.(λp) = 1. Denote p′ = λp for such a p and observe thatp, q ∈ S satisfy p = αq for some α > 0 iff p′ = q ′. Define

Cn = {p ∈ S : B(p′, 1/n) ⊆ S} ∪ {0}where B(p′, 1/n) denotes the open ball with center p′ and radius 1/n. For all p ∈ S, p ∈ Cn

iff dist (p′, Rl\S) ≥ 1/n. Clearly, Cn is a closed cone, Cn ⊆ C n+1 and S ∪ {0}= ∪nCn. Letus prove that Cn is convex. Let r = α p + (1 − α)q , α ∈ [0, 1], p′ = λp, q ′ = µq , r ′ = σ r .We have:

1

σr ′ = α

λp′ + (1 − α)

µq ′.

Because p0.p′ = p0.q ′ = p0.r ′ = 1, we have:

1

σ= α

λ+ (1 − α)

µ.

This shows that r ′ ∈ [p′, q ′]. Because B(p′, 1/n) ⊆ S, B(q ′, 1/n) ⊆ S and S is convex,one has B(r ′, 1/n) ⊆ S; that is, r ∈ Cn. �Lemma 15 Let p, p∗ ∈ S such that inf p∗.Xt < p∗.et for all t ∈�. Define

�(t, p, p∗) = min{p.z : z ∈ D(t, p∗)}for all t ∈�. Then �(t, p, p∗) = p.z for some z ∈ D(t, p∗) and the function t ∈� → �(t,p, p∗) is lower semi-continuous.

PROOF: Because p∗ ∈ S, D(t, p∗) is compact for all t ∈� (Proposition 8 (ii)). Hence, thereexists zt ∈ D(t, p∗) such that �(t, p, p∗) = p.zt . Let tn → t in �. From Proposition 9,2-(ii),there exists a subsequence (tnk )k such that ztnk

→ x ∈ D(t, p∗). One has:

p.x ≥ min{p.z : z ∈ D(t, p∗)} = �(t, p, p∗).

In other words:

limk→∞

�(tnk , p, p∗) ≥ �(t, p, p∗)

and this shows that � is lower semi-continuous. �Lemma 16 [Gale–Nikaido–Debreu’s lemma]

Let C be a closed convex cone with vertex 0 in Rl and ξ an upper semi-continuous, convex,compact valued correspondence from C ∩ Sl−1 (the unit-sphere of Rl ) into Rl . If

(i) C is not a linear subspace of Rl ,(ii) p.z ≤ 0 for all p ∈ C ∩ Sl−1 and all z ∈ ξ(p),

then there exists p∗ ∈ C ∩ Sl−1 and z∗ ∈ ξ(p∗) such that p.z∗ ≤ 0 for all p ∈ C .



PROOF: See Florenzano and Le Van (1986). �PROOF OF THEOREM 6: The Gale–Nikaido–Debreu’s lemma cannot be directly appliedwith the cone S, because it is not closed (Proposition 9). Hence, we first consider the closedconvex cones, Cn, introduced in Lemma 14. From Proposition 10 (ii), Proposition 12 (i)and the Gale–Nikaido–Debreu’s lemma, there exists p∗

n ∈ Cn ∩ Sl−1 and z∗n ∈ Z(p∗

n) suchthat:

p.z∗n ≤ 0, for all p ∈ Cn. (1)

Without loss of generality, one might suppose that (p∗n)n converges to p∗ ∈ Sl−1. We now

prove that p∗ ∈ S.Let p ∈ S. One has:

p.z∗n ≥

∫�

min{

p.[z − et ] : z ∈ D(t, p∗

n

)}m(dt) (2)

≥∫

�

[�

(t, p, p∗

n

) − p.et

]m(dt). (3)

Denote by K the complement of A(p∗) in �. From Lemma 13, K is compact. Hence,from Lemma 15, there exists tn ∈ K , zn ∈ D(tn, p∗

n) such that:

min{�

(t, p, p∗

n

): t ∈ K

} = �(tn, p, p∗

n

) = p.zn.

Without loss of generality, one may suppose that tn → t ∈ K . Hence (from the definitionof K ), p∗ ∈ ∂St\{0}. From Proposition 9, one might suppose that p.zn → +∞. Hence,because: ∫

K

[�

(t, p, p∗

n

) − p.et

]m(dt) ≥ p.znm(K ) −

∫K

p.et m(dt),

one has: ∫K

[�

(t, p, p∗

n

) − p.et

]m(dt) → +∞ if m(K ) > 0 (4)

= 0 if m(K ) = 0. (5)

By the same way, there exists t ′n ∈ clA(p∗) and z′

n ∈ D(t ′n, p∗

n) such that:∫

A(p∗)

[�

(t, p, p∗

n

) − p.et

]m(dt) ≥ p.z′

nm(A(p∗)) −∫

A(p∗)p.et m(dt). (6)

One might suppose that t ′n → t ′ ∈�.

First case: t ′ ∈ K .Then p∗ ∈ ∂St ′ \{0}; hence, p.z′

n → +∞. Therefore:

∫A(p∗)

[�

(t, p, p∗

n

) − p.et

]m(dt) → +∞ if m(A(p∗)) > 0 (7)

= 0 if m(A(p∗)) = 0. (8)



Second case: t ′ ∈ K .Then t ′ ∈ A(p∗); that is, p∗ ∈ St ′ . Hence, from Proposition 9, one might suppose that

z′n → z ∈ D(t ′, p∗). Hence, from (6), there exists M ∈ R such that

∫A(p∗)

[�

(t, p, p∗

n

) − p.et

]m(dt) ≥ Mm(A(p∗)) for all n. (9)

Relations (1),(2), (3), (4), (5), (7), (8) and (9) imply that m(K ) = 0; that is,m(A(p∗)) = 1, and t ′ ∈ K . From (H8), SA(p∗) = S. From the definition of A(p∗), wehave p∗ ∈ SA(p∗) and, hence, p∗ ∈ S. Therefore, Z(p∗) is defined and z∗

ν → z∗ ∈ Z(p∗) for asubsequence (z∗

ν)ν of (z∗n)n. Obviously, p∗.z∗ = 0. From (1), p.z∗ ≤ 0 for all p ∈ S. Because

S is open, we have [p∗ + εz∗].z∗ ≤ 0 for ε > 0 small enough. Hence, 0 = p∗.z∗ ≤−ε‖z∗‖2.This implies z∗ = 0.

We now prove the converse part: existence of an equilibrium implies existence of no-arbitrage prices. Indeed, consider an equilibrium (x∗, p∗) where x∗ is a mapping from �

into Rl . One has:

for a.e. t ∈ �, xt �t x∗t =⇒ p∗.xt > p∗.et = p∗.x∗

t .

From (H 5), for every t , every w ∈ Wt\{0}, there exists λ > 0 such that x∗t + λw �t x∗

t .Hence, λp∗.w > 0. In other words, p∗.w > 0 for every a.e. t , and for every w ∈ Wt\{0}.From assumption (H 8), p∗ ∈ S. �

Remark 4 From Example 2, we observe that, without using Fatou’s Lemma, we have givena proof of existence of equilibria for an economy with a continuum of agents, increasingpreferences and consumption sets equal to the positive orthant.

7 Appendix

PROOF OF PROPOSITION 1: Obviously, if S = 0, there is no unbounded arbitrage. To prove the converse, con-

sider the cone L = conv{Wt : t ∈�}. It is closed. Indeed, from the Caratheodory theorem, every z ∈ L can

be written z = ∑i=1,...,l+1 wi , with wi ∈ Wi , ∀i . Let (zn)n be a sequence in L which converges to z ∈ Rl .

Then zn = ∑i=1,...,l+1 wn,i , with wn,i ∈ Wtn,i , ∀ n, ∀i . We can suppose that limn→∞t n,i = ti ∈�, for all i.

If the l + 1 sequences (wn,i )n are bounded, then one can suppose that limn→∞ wn,i =wi ∈ Wti . Hence,

z = ∑i=1...,l+1 wi ∈ L . Otherwise, there is i∗ such that limn→∞ ‖wn,i∗ ‖ = +∞ and ‖wn,i ‖ ≤ ‖wn,i∗ ‖, ∀n, ∀i .

Then limn→∞ wn,i /‖wn,i∗ ‖ =wi ∈ Wti . Indeed, for all λ > 0, eti + λwi = limn→∞ etn,i + λwn,i /‖wn,i∗ ‖ ∈ Pti ,

from (H 7). Moreover,∑

i=1,...,l+1 wi = limn→∞ zn/‖wn,i∗ ‖ = 0. Therefore, from the no unbounded arbitrage

condition, we have wi = 0, ∀i , which is a contradiction. Because Wt contains no line, no unbounded arbitrage

implies that L also contains no line. Hence, if L 0 denotes the polar cone of L, then int(−L 0) is non-empty (see

Rockafellar 1970 (Corollary 14.6.1)). Now, if p ∈ int(−L 0), we have p.w > 0, ∀ w ∈ L\{0}; that is, p ∈ S.

PROOF OF LEMMA 2: Because µ is atomless, from Neveu (1965, I.4.3), there exist A0, A2 in A such that

A = A0 ∪ A2, A0 ∩ A2 =∅ and µ(A0) =µ(A) = 1/2.

Define σ 11 = (0), σ 2

1 = (2), and∑

1 = {σ 11 , σ 2

1 }.By the same way, there exist A00, A02, A20, A22 such that A0 = A00 ∪ A02, A2 = A20 ∪ A22, A00 ∩ A02 = ∅,

A20 ∩ A22 = ∅, µ(A00) = µ(A02) = µ(A20) = µ(A22) = 122 .

Define σ 12 = (0, 0) = (σ 1

1, 0), σ 22 = (0, 2) = (σ 1

1, 2), σ 32 = (2, 0) = (σ 2

1, 0), σ 42 = (2, 2) = (σ 2

1, 2) and∑2 = {σ 1

2 , σ 22 , σ 3

2 , σ 42 }.



By induction, on constructs a sequence (∑

m)m where:

�∑

m = {σ 1m, σ 2

m, σ 3m, . . . , σ 2m

m }� ∀i = 1, . . . , 2m, card(σ i

m) = m

and if∑

m+1 = {σ 1m+1, σ

2m+1, σ

3m+1, . . . , σ

2(m+1)m+1 }, then

� σ 1m+1 = (σ 1

m, 0)� σ 2

m+1 = (σ 1m, 2)

� σ 3m+1 = (σ 2

m, 0)� σ 4

m+1 = (σ 2m, 2)

� σ 2m+1m+1 = (σ 2m

m , 0)� σ 2m+2

m+1 = (σ 2mm , 2)

We also have:

(i) ∀m, A = i = 12m∪ A

σ im

.

(ii) ∀m,∀i = j , Aσ i

m∩ A

σj

m= ∅.

(iii) ∀m,∀i , µ(Aσ i

m) = 1

2m .

(iv) ∀m, ∀n ≥ m, ∀i = 1, 2, 22, . . . , 2m, ∀ j = 1, 2, . . . , 2n , and has two alternate cases:� either A

σj

n⊆ A

σ im

: in this case, we say that σjn extends σ i

m� or A

σj

n∩ A

σ im

= ∅.

For every m, write σjm = (σ j,1

m , . . . , σ j,mm ) where ∀k = 1, . . . , m, σ j,k

m = 0 or 2.

Define xσ

jm

= ∑mi=1

σj ,i

m3i ≤ ∑ 2

3i = 1.

For a ∈ A, there exists a unique σjm such that a ∈ A

σj

m. Therefore, we can define a measurable function

εm : A → [0, 1] by εm(a) = xσ

jm

. One has: either εm+1(a) = εm(a), or εm+1(a) = εm(a) + 23m+1 . The sequence

of measurable functions (εm) is increasing and bounded from above. Hence, it converges to a measurable function

ε : A → [0, 1].

Let x ∈ [0, 1] and C = {a ∈ A : ε (a) = x}. We want to prove that µ(C ) = 0. Assume the contrary: µ(C )

> 0. Let ν be such that 12ν < µ(C ). From (i), (ii) and (iii), there exist σ

jν , σ

j ′ν in

∑ν such that C ∩ A

σjν

= ∅,

C ∩ Aσ

j ′ν

= ∅.

Let a ∈ Aσ

jν, a ′ ∈ A

σj ′ν

. We claim that:

|εν(a) − εν(a ′)| = |xσ

jν

− xσ

j ′ν

| ≥ 1

3k

where k is the first index such that σj ,k

ν = σj ′,k

ν .

Assume first σj,kν = 2 and σ

j ′,kν = 0. The difference x

σjν

− xσ

j ′ν

is minimal if ∀k′ > k, σj ,k′

ν = 0, σj ′,k′

ν = 2.We have:

xσ

jν

− xσ

j ′ν

≥ 2

3k−

ν∑i=k+1

2

3i≥ 2

3k−

+∞∑i=k+1

2

3i≥ 1

3k.

Symmetrically, if σj ′,k

ν = 2, σj ,k

ν = 0, we have xσ

j ′ν

− xσ

jν

≥ 13k .

We have proved the claim.

Obviously, for n > ν, if σ lν (respectively σ l ′

ν ) extends σjν (respectively σ

j ′ν ) then:∣∣∣xσ l

ν− x

σ l ′ν

∣∣∣ ≥ 1

3k. (10)

Let α > 0 verify 2α < 13k . For n large enough, one has |εn(a) − ε(a)|<α and |εn(a ′) − ε(a ′)|<α. Because

ε(a) = ε(a ′) = x , we have:

|εn(a) − εn(a ′)| < 2α <1

3k. (11)



But εn(a) = xσ l

n, εn(a ′) = x

σ l ′n

, where σ ln, σ l ′

n are such that a ∈ Aσ l

n, a ′ ∈ A

σ l ′n

. In other words, a ∈ Aσ l

n∩

Aσ

jν, a ′ ∈ A

σ l ′n

∩ Aσ

j ′ν

. From (iv), σ ln extends σ

jν and σ l ′

n extends σj ′

ν . We have a contradiction between (10) and

(11). Therefore, µ(C ) = 0.

Because [0, 1] may be indentified to a subset of the unit-ball B of Rl , we have found a measurable mapping

ε : A → B verifying ∀x ∈ B, µ({a ∈ A : ε(a) = x}) = 0.

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No-arbitrage condition and existence of equilibrium in asset markets with a continuum of traders

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