Asset market equilibrium with short-selling and ... fileW. Daher et al./Asset market equilibrium...

Asset market equilibrium withshort-selling and differential

information∗Wassim Daher

Cermsem, Universite Paris-1e-mail: [email protected]

V. Filipe Martins-da-Rocha

Ceremade, Universite Paris–Dauphinee-mail: [email protected]

and

Yiannis Vailakis

Cermsem, Universite Paris-1e-mail: [email protected]

Abstract: We introduce differential information in the asset market mo-del studied by Cheng (1991), Dana and Le Van (1996) and Le Van andTruong Xuan (2001). We prove an equilibrium existence result assuming thatthe economy’s information structure satisfies the conditional independenceproperty. If private information is not publicly verifiable, agents may not haveincentives to reveal truthfully their information, and therefore contracts maynot be executed. We show that conditional independence is crucial to ad-dress the issue of execution of contacts, since it ensures that, at equilibrium,contracts are incentive compatible.

JEL Classification Numbers: C61, C62, D51Keywords: Asset Market, Differential Information, Conditional Indepen-dence, Competitive Equilibrium, Incentive Compatibility

∗We are grateful to Francoise Forges and Nicholas Yannelis for valuable comments and sugges-tions. Thanks are also due to Rose-Anne Dana, Cuong Le Van, M. Ali Khan, Paulo K. Monteiroand Frank Riedel. An earlier version of the paper has been presented in the first General Equilib-rium workshop in Rio, as well as in the MED seminar in Paris-1. We thank the participants fortheir valuable comments. Yiannis Vailakis acknowledges the financial support of a Marie Curiefellowship, (FP6 Intra-European Marie Curie fellowships 2004-2006).

1

W. Daher et al./Asset market equilibrium with short-selling and differential information 2

1. Introduction

There are many examples in the literature of models with unrestricted consump-tions sets. These include among others, temporary equilibrium models (Green(1973) and Grandmont (1977)) and equilibrium models of asset markets (the clas-sical CAPM of Sharpe (1964) and Lintner (1965)). It is well known that in suchmodels equilibrium prices may fail to exist, since unbounded and mutually com-patible arbitrage opportunities can arise.

The first equilibrium existence result for asset market models was proven by Hart(1974). In a specific context of an economy with finitely many securities, Hart hasexhibited a list of conditions sufficient to limit the arbitrage opportunities presentin the economy. Much later, Hammond (1983), Page (1987), Werner (1987), Nielsen(1989), Chichilniski (1995) and Page, Wooders and Monteiro (2000) reconsideredthe problem by providing variations of Hart’s list of conditions. In all cases, the roleplayed by those no-arbitrage conditions was to bound the economy endogenously.For a review and comparison of the different concepts of no-arbitrage conditions,refer to Dana, Le Van and Magnien (1999) and Allouch, Le Van and Page (2002).

Developments in continuous trading models and portfolio analysis (Black andScholes (1973), Kreps (1981) and Duffie and Huang (1985)) have simultaneouslymotivated the study of existence problem in infinite dimensional economies withunbounded sets. In the infinite dimensional case, the notions of no-arbitrage are notsufficient to ensure the existence of equilibrium prices. An alternative assumptionin this context that ensures existence, is to assume that the individually rationalutility set is compact (in the finite dimensional case, this assumption is equivalentto no-arbitrage conditions, see Dana et al. (1999)). This is the case in Brown andWerner (1995), Dana, Le Van and Magnien (1997) and Dana and Le Van (2000).The problem then turns out to find conditions under which the compactness ofthe utility set holds. When markets for contingent claims are complete, agents’consumption sets coincide with the space Lp(Ω,F ,P), and preferences take theusual von Neuman–Morgenstern form, Cheng (1991) and Dana and Le Van (1996)have proved (using two different approaches) that this assumption is satisfied. AsLe Van and Truong Xuan (2001) have shown, the result is still valid in economieswith separable but not necessarily von Neuman–Morgenstern utility functions.

Economies with uncertainty are encompassed by the general equilibrium modelof Arrow-Debreu as long as there is a complete market of elementary securitiesor contingent claims. The formal extension of the Arrow-Debreu model to includeuncertainty has been criticized for requiring a formidable number of necessarymarkets. Introducing differences among the informational structures of the several


agents is a way to diminish the force of the above criticism, since it has the effectof drastically reducing the number of required contracts present in the economy.

The first study that introduced differential information in a general equilibriumset up was Radner (1968). Radner studies an environment where the structure ofinformation is fixed in advance and all contracts are negotiated at the beginning ofthe history of the economy. For such an economy Radner defined a notion of equi-librium (Radner equilibrium), an analogous concept to the Walrasian equilibriumin Arrow-Debreu model with symmetric information. As in Debreu (1959, Chap-ter 7), agents arrange contracts that may be contingent on the realized state ofnature at the second period, but after the realization of the state of nature they donot necessarily know which state of nature has actually occurred. Therefore, theyare restricted to sign contracts that are compatible with their private information.It is important to notice that in sharp contrast with the rational expectations equi-librium model (see Radner (1979)), prices do not reveal any private informationex ante. They rather reflect agents’ informational asymmetries as they have beenobtained by maximizing utility taking into account the private information of eachagent.

Recently there has been a resurgent interest in models with differential informa-tion. At issue are questions concerning the existence and characterization of Radnerequilibrium by means of cooperative solutions (see Allen and Yannelis (2001), Einy,Moreno and Shitovitz (2001), Herves-Beloso, Moreno-Garcia and Yannelis (2005a)and Herves-Beloso, Moreno-Garcia and Yannelis (2005b)).

The paper follows this growing literature. Its aim is the exploration of Radnerequilibrium concept in infinite dimensional economies with a complete asset marketstructure and possible short-selling. We introduce differential information in theasset market model studied by Cheng (1991), Dana and Le Van (1996) and Le Vanand Truong Xuan (2001). We prove an equilibrium existence result assuming thatthe economy’s information structure satisfies the conditional independence prop-erty, i.e. individuals’ information are assumed to be independent conditionally tothe common information.

Our existence proof follows in two steps. In a first place, we show that theindividually rational utility set associated with our differential information econ-omy, coincides with the individually rational utility set of a symmetric informationeconomy. Using this fact and the results established in Le Van and Truong Xuan(2001), we subsequently show that the individually rational utility set of our differ-ential economy is compact. In that respect, we show that the compactness resultestablished for symmetric information economies (Cheng (1991), Dana and Le Van(1996) and Le Van and Truong Xuan (2001)) is still valid in a differential informa-


tion setting.It is known, due to Allouch and Florenzano (2004), that the compactness of

the individually rational utility set implies the existence of an Edgeworth equi-librium. The second step then amounts to show that there exist prices supportingEdgeworth allocations as competitive equilibria. In that respect, we provide a char-acterization of Radner equilibrium by means of cooperative solutions.

In addition to the equilibrium existence result, another issue we deal with, con-cerns the execution of contracts. If private information is not publicly verifiable,agents have incentives to misreport their types and therefore contracts may not beexecuted in the second period. We show that conditional independence is crucialto address this issue. Indeed, it ensures that equilibrium allocations are individu-ally incentive compatible. This in turn implies that, at equilibrium, contracts arealways executed.

The structure of the paper is as follows. In Section 2 we describe the environ-ment and give various definitions. In Section 3 we prove the compactness of theindividually rational utility set. Section 4 deals with the decentralization of Edge-worth allocations. Section 5 is devoted to economies with finitely many states ofnature. The reason for paying special attention to the finite case, stems from thefact that the conditional independence property is no more needed to prove exis-tence of equilibrium prices. In this case, under our assumptions, it can be showndirectly that a no-arbitrage price always exists. Existence of a supporting pricesystem then follows from standard arguments in the literature (see Allouch et al.(2002) and the references therein). However, it should be stressed that droppingout conditional independence leaves open the question of execution of contracts.In that respect, we believe that conditional independence is a relevant restrictioneven in the finite dimensional case.

2. The Model

We consider a pure exchange economy with a finite set I of agents. The economyextends over two periods with uncertainty on the realized state of nature in thesecond period.

2.1. Uncertainty and information structure

Each agent i ∈ I, has an individual perception of the uncertainty at the secondperiod, represented by a probability space (T i, T i,Pi). Each element ti in T i maybe interpreted as an individual type or signal, observed at the second period. We do


not assume that all type profiles in∏

i∈I Ti are possible at the second period. This

allows to capture aspects of uncertainty which may be commonly known. The setof possible types is captured by a couple (Ω,F ,P), (pri, i ∈ I) where (Ω,F ,P) isa probability space representing the underlying uncertainty, related to individualperception through the measurable mappings pri : (Ω,F) −→ (T i, T i), satisfying

∀i ∈ I, ∀A ∈ T i, Pi(A) = P[(pri)−1(A)

].

A profile of types (ti, i ∈ I) ∈ ∏i∈I T

i is said to be possible if there exists ω ∈ Ω suchthat ti = pri(ω) for each i ∈ I. It should be clear that this framework capturesthe one in which agent i’s private information is formulated by a sub σ-algebraof F . Indeed, let us denote by F i the sub σ-algebra of F generated by pri. Forevery x ∈ L1(T i, T i,Pi), denote by ∆i(x) the mapping in L1(Ω,F i,P) defined by∆i(x) = x pri. The mapping ∆i is a bijection from Lp(T i, T i,Pi) to Lp(Ω,F i,P)satisfying

∀z ∈ Lp(T i, T i,Pi), EPi

[z] = EP[∆i(z)].1

By abusing notations, we assimilate a function x ∈ Lp(T i, T i,Pi) with ∆i(x),writing indifferently x(ti) or x(ω) with ω ∈ (pri)−1(ti).

Remark 2.1. In what follows, we provide a general framework where the underly-ing uncertainty is derived from the individual perception of uncertainty. Considerprobability spaces (S,S,Q) and (N i,N i, ηi) for each i ∈ I. The probability space(S,S,Q) represents uncertainty that is commonly observed, while each N i repre-sents agent i’s private information. We do not specify if agent i knows the privateinformation Nk of another agent k 6= i. Agent i’s perception of uncertainty isdefined by

T i = S ×N i, T i = S ⊗N i, and Pi = Q⊗ ηi.

The underlying uncertainty (Ω,F ,P) is then represented by

Ω = S ×∏i∈I

N i, F = S ⊗ ⊗i∈IN i, P = Q⊗⊗i∈Iηi,

and pri(s, (nk, k ∈ I)) = (s, ni). Observe that if agent i is not aware of the privateinformation

∏k 6=iN

k of other agents, then he is not aware of Ω.

Remark 2.2. In what follows, we provide a general framework where the individ-ual perception of uncertainty is derived from the underlying uncertainty. Consider

1If (A,Σ, µ) is a probability space and z ∈ L1(A,Σ, µ) then Eµ[z] denotes the expectation ofz under µ.


the probability space (Ω,F ,P) representing states of nature that are commonlyknown to be possible at the second period. Each agent i has an incomplete infor-mation about the true state of nature, in the sense that he will only observe theoutput of a random variable τ i : Ω → R. Denote by T i the range τ i(Ω) of τ i. Thenthe individual perception of uncertainty is represented by (T i,B,Pi) where B is theBorelian σ-algebra and Pi is defined by

∀A ∈ B, Pi(A) = Pτ i ∈ A.

Note that here, the mappings τ i and pri coincide.

2.2. Contracts and incentive compatibility

In the first period, agents sign contracts contingent to their perception of uncer-tainty. In the second period, individual uncertainty is resolved and contracts areexecuted. Since agents trade with an anonymous market, they only sign contractsthat are contingent to events they can observe at the second period. To be spe-cific, for given p ∈ [1,+∞], a contract for agent i is represented by a function inLp(T i, T i,Pi) or Lp(Ω,F i,P).

Each agent i ∈ I is endowed with a type dependent initial endowment rep-resented by a contingent claim ei ∈ Lp(T i, T i,Pi) and a type dependent utilityfunction U i : T i × R → R. Alternatively, we may consider ei to be a function inLp(Ω,F i,P) and U i to be a function from Ω × R → R such that for each z ∈ R,the function ω 7→ U i(ω, z) is F i-measurable.

Following our equilibrium concept defined below, each agent i signs a contractxi ∈ Lp(T i, T i,Pi) such that the allocation (xi, i ∈ I) is exact feasible, i.e.∑

i∈I

xi(ω) =∑i∈I

ei(ω), for P-a.e. ω ∈ Ω.2

If private information does not become publicly verifiable, then there is an issueconcerning the execution of contracts. In order to clarify this issue, we need tointroduce the σ-algebra F c defined by

F c =⋂i∈I

F i

which represents the common knowledge information to all agents. An event Abelongs to F c if and only if it belongs to each F i, i.e. the σ-algebra F c is thecollection of all events that are observable by all agents.

2Formally, we should write∑

i∈I xi(pri(ω)) =∑

i∈I ei(pri(ω)).


2.2.1. Finitely many types

We first consider the particular case where each agent has finitely many types, i.e.the set T i is finite for each i ∈ I and T i is the collection of all subsets of T i. Theinduced sub σ-algebra F i is defined by the partition (pri)−1(ti) : ti ∈ T i. Theatom (pri)−1(ti) is denoted by Ωi(ti) and Ωc(ti) is defined as the unique atom inF c containing Ωi(ti).

Assume that the realized state of nature at the second period is σ ∈ Ω. Eachagent i has an incomplete information in the sense that he only observes his typeti = pri(σ). Individual information may not be publicly verifiable. Therefore, anagent i ∈ I may have an incentive to misreport his type by announcing a type ζ i

instead of ti if the following condition is satisfied

U i(ti, ei(ti) + xi(ζ i)− ei(ζ i)) > U i(ti, xi(ti)).

Since we allow for aspects of uncertainty to be known to all agents, agent i canonly misreport if ti and ζ i are compatible with respect to the common knowledgeinformation, i.e.

Ωc(ti) = Ωc(ζ i).

Assume that each agent i ∈ I reports type ζ i. If there exists ω ∈ Ω (ω may bedifferent from the realized state of nature σ) such that pri(ω) = ζ i for each i ∈ I,then the profile of types (ζ i, i ∈ I) is such that contracts can be executed, i.e. eachagent i will receive xi(ζ i) and markets clear. Indeed,∑

i∈I

xi(ζ i) =∑i∈I

xi(ω) =∑i∈I

ei(ω) =∑i∈I

ei(ζ i).

However, if there does not exist ω ∈ Ω such that pri(ω) = ζ i for each i ∈ I, thencontracts may not be executable.

A sufficient condition to ensure that an allocation (xi, i ∈ I) is executable, isthat each contract xi is incentive compatible as defined below.

Definition 2.1. Given an agent i ∈ I, a contract xi ∈ Lp(T i, T i,Pi) is said to be(individually) incentive compatible if the following is not true: there exist ti andζ i in T i with Piti > 0, Piζ i > 0 such that

(a) agent i has an incentive to misreport ti by announcing ζ i, i.e.

U i(ti, ei(ti) + xi(ζ i)− ei(ζ i)) > U i(ti, xi(ti)) (1)


(b) the misreporting is compatible with the common knowledge information, i.e.

Ωc(ti) = Ωc(ζ i). (2)

Remark 2.3. Similar concepts of incentive compatibility have been proposed inthe literature (see for instance the excellent surveys by Hahn and Yannelis (1997)and Forges, Minelli and Vohra (2002)).

Remark 2.4. To clarify condition (2), return to the framework studied in Re-mark 2.1. In that case, given an agent i ∈ I, a contract xi ∈ Lp(T i, T i,Pi) isincentive compatible if and only if there do not exist s ∈ S, ni and µi in N i withηi(ni) > 0, ηi(µi) > 0, Q(s) > 0 such that

U i((s, ni), ei(s, ni) + xi(s, µi)− ei(s, µi)) > U i((s, ni), xi(s, µi)). (3)

Observe that in this framework, if an allocation (xi, i ∈ I) is exact feasible thenfor each i ∈ I, there exists a function f i ∈ Lp(S,S,Q) such the net trade functionzi = xi−ei satisfies zi(s, ni) = f i(s) for Q⊗ηi-a.e. (s, ni) ∈ S×N i. It then followsthat each contract xi is incentive compatible. Indeed, if it is not the case, thereexist s ∈ S, ni and µi in N i with ηi(ni) > 0, ηi(µi) > 0, Q(s) > 0 such that

ei(s, ni) + xi(s, µi)− ei(s, µi) > xi(s, µi).

A contradiction to the fact that zi(s, ni) = f i(s) = zi(s, µi).

2.2.2. The general case

We do not anymore assume that each agent has finitely many types. In order toextend the concept of incentive compatibility to the general case, we first introducesome notations. For each i ∈ I, we denote by T i

c the σ-algebra of events in T i thatare common knowledge to all agents, defined by

T ic = H ∈ T i : (pri)−1(H) ∈ F c.

Observe that if xi is a function in Lp(T i, T i,Pi) then

xi ∈ Lp(T i, T ic ,Pi) ⇐⇒ xi pri ∈ Lp(Ω,F c,P).

We may assume that each i ∈ I only knows his individual perception of uncertainty(T i, T i,Pi) and what is commonly observed T i

c . In that respect, we do not restrictthe model to the case where each agent i ∈ I knows the underlying uncertainty(Ω,F ,P).

We propose now a generalization of Definition 2.1 to the general case.


Definition 2.2. Given an agent i ∈ I, a contract xi ∈ Lp(T i, T i,Pi) is said to be(individually) incentive compatible if the following is not true: there exist Ei ∈ T i

and Gi ∈ T i with Pi(Ei) > 0, Pi(Gi) > 0 such that

(a) for every (ti, ζ i) ∈ Ei × Gi, agent i has an incentive to misreport ti byannouncing ζ i, i.e.

ui(ti, ei(ti) + xi(ζ i)− ei(ζ i)) > ui(ti, xi(ti)) (4)

(b) the misreporting is compatible with the common knowledge information, i.e.

1

Pi(Ei)EPi

[1Ei|T i

c

]=

1

Pi(Gi)EPi

[1Gi|T i

c

]. (5)

Remark 2.5. Observe that when agents have finitely many types, then Defini-tion 2.2 reduces to Definition 2.1. Indeed, if T i is finite and T i is the σ-algebra ofall subsets of T i, then for every ti ∈ T i with Pi(ti) > 0, one has

EPi[1ti|T i

c

]= EP

[1Ωi(ti)|F c

]=

P(Ωi(ti))

P(Ωc(ti))1Ωc(ti).

2.2.3. Conditional independence: a sufficient condition

Example in Remark 2.4 highlights that some kind of restriction in the informa-tion structure is sufficient to ensure that an exact feasible allocation is incentivecompatible.

Definition 2.3. The information structure F• = (F i, i ∈ I) of an economyE = (U i,F i, ei, i ∈ I) is said to be conditionally independent (given the com-mon information), if for every i 6= j in I, the σ-algebra F i and F j are independentgiven F c.3

Example 2.1. The information structure in Remark 2.1 is conditionally indepen-dent.

Remark 2.6. The information structure is conditionally independent if and onlyif for every pair i 6= j in I we have

∀x ∈ L1(Ω,F i,P), E[x|F j] = E[x|F c].

3That is, for every pair of events Ai ∈ F i and Aj ∈ Fj , we have that P(Ai ∩ Aj |Fc) =P(Ai|Fc)P(Aj |Fc) almost surely.


Conditional independence appears in the literature of rational expectations modelswith asymmetric information (see Polemarchakis and Siconolfi (1993)). It has alsobeen used to address existence issues in continuous-time semimartingale models offinancial markets studied in stochastic finance (see Pikovsky and Karatzas (1996)and Hillairet (2005)). It also appears in game-theory literature dealing with incom-plete information (see Radner and Rosenthal (1982), Milgrom and Weber (1985)and Khan, Rath and Sun (2006)).

The main result of this section is the following proposition.

Proposition 2.1. If the information structure is conditionally independent thenevery exact feasible allocation (xi, i ∈ I) with xi ∈ Lp(Ω,F i,P) is incentive com-patible.

Proof. Let (xi, i ∈ I) be an allocation with xi ∈ Lp(Ω,F i,P) such that∑i∈I

xi(ω) =∑i∈I

ei(ω), for P-a.e. ω ∈ Ω.

We claim that under conditional independence of the information structure, foreach i ∈ I, the net trade zi := xi − ei is F c-measurable.4 Indeed, for each i ∈ I,one has zi = −∑

j 6=i zj and in particular

zi = E[zi|F i] = −∑j 6=i

E[zj|F i] = −∑j 6=i

E[zj|F c].

Fix i ∈ I and assume that the contract xi is not incentive compatible. Then thereexist Ei ∈ T i andGi ∈ T i with Pi(Ei) > 0, Pi(Gi) > 0 and satisfying conditions (4)and (5). Then

Pi(Gi)EPi

[zi1Ei ]− Pi(Ei)EPi

[zi1Gi ] =∫

Ei×Gi[zi(t)− zi(ζ)]Pi ⊗ Pi(dt, dζ) < 0.

Since zi is F c-measurable, it follows from (5) that

Pi(Gi)EPi[zi1Ei

]= Pi(Gi)EPi

[ziEPi

[1Ei|T i

c

]]= Pi(Gi)EPi

[zi Pi(Ei)

Pi(Gi)EPi

[1Gi|T i

c

]]= Pi(Ei)EPi

[zi1Gi

].

which yields a contradiction.

4When there are only two agents, the conditional independence of the information structureis no more needed.


2.3. Equilibrium concept

To relate our equilibrium concept to the traditional one proposed in the literaturewith symmetric information (see Cheng (1991)), we choose to reprensent agent i’sindividual uncertainty by (Ω,F i,P).

The preference relation of agent i is represented by the utility function ui fromLp(Ω,F i,P) to R defined by

∀x ∈ Lp(Ω,F i,P), ui(x) =∫ΩU i(ω, x(ω))P(dω).

If x ∈ Lp(Ω,F i,P) is a contingent claim then P i(x) is the set of strictly preferredcontingent claims for agent i, i.e.

P i(x) = y ∈ Lp(Ω,F i,P) : ui(y) > ui(x).

An economy E is defined by a family (U i,F i, ei, i ∈ I). A vector x• = (xi, i ∈ I)with xi ∈ Lp(Ω,F i,P) is called an allocation. The space A of individually rationalattainable allocation is defined by

A =

x• = (xi, i ∈ I) ∈

∏i∈I

Lp(Ω,F i,P) :∑i∈I

xi = e and ui(xi) > ui(ei)

where e =∑

i∈I ei is the aggregate initial endowment. If ψ ∈ Lq(Ω,F ,P)5 then we

can define the value functional 〈ψ, .〉 on Lp(Ω,F ,P) by

∀x ∈ Lp(Ω,F ,P), 〈ψ, x〉 =∫Ωψ(ω)x(ω)P(dω).

The vector ψ is called a price (system) and 〈ψ, x〉 represents the value (in the senseof Debreu (1959)) at the first period of the contingent claim x under the price ψ.We also denote the value 〈ψ, x〉 by E[ψx].

Definition 2.4. A competitive equilibrium is a pair (x•, ψ) where x• = (xi, i ∈ I)is an attainable allocation and ψ is a price such that each agent maximizes hisutility in his budget set, i.e.

∀i ∈ I, xi ∈ argmaxui(y) : y ∈ Lp(Ω,F i,P) and 〈ψ, y〉 6 〈ψ, ei〉

.

An allocation x• is said to be a competitive allocation if there exists a price ψ suchthat (x•, ψ) is a competitive equilibrium.

5The real number q ∈ [1,+∞] is the conjugate of p defined by 1/p+1/q = 1 with the conventionthat 1/ +∞ = 0 and 1/0 = +∞.


To prove the existence of a competitive equilibrium, we first show that, un-der suitable assumptions, the individually rational utility set (as defined below) iscompact.

Definition 2.5. We denote by U the individually rational utility set defined by

U = v• = (vi, i ∈ I) ∈ RI : ∃x• ∈ A, ui(ei) 6 vi 6 ui(xi).

For any 1 6 p 6 +∞ and for any sub σ-algebra G ⊂ F , the space Lp(Ω,G,P) isdenoted by Lp(G).

Definition 2.6. A function U : Ω× R → R is said

(i) integrable if for every t ∈ R, the function ω 7→ U(ω, t) is integrable,(ii) p-integrable bounded if there exists α ∈ L1

+(F) and β > 0 such that |U(ω, t)| 6α(ω) + β|t|p for every (ω, t) ∈ Ω× R.

Remark 2.7. Obviously, every p-integrable bounded function is integrable. Fol-lowing Krasnoselskii (1964), if p ∈ [1,+∞) and (Ω,F ,P) is atomless then a func-tion U is p-integrable bounded, if and only if, for every x ∈ Lp(F), the functionω 7→ U(ω, x(ω)) is integrable. If p = ∞ then a utility function is integrable if andonly if ω 7→ U(ω, x(ω)) is integrable for every x ∈ Lp(F).

Remark 2.8. Observe that if E = (U i,F i, ei, i ∈ I) is an economy with an atomlessmeasure space (Ω,F ,P) then a necessary condition for the separable utility functionui to be well-defined is that for each i ∈ I, the function U i is p-integrable bounded.

Definition 2.7. An economy E = (U i,F i, ei, i ∈ I) is said standard if, for everyi ∈ I, the following properties are satisfied:

(S.1) the function t 7→ U i(ω, t) is differentiable, concave and strictly increasingfor P-a.e ω ∈ Ω;

(S.2) the function U i is p-integrable bounded;(S.3) the function t 7→ U i

?(ω, t) is continuous for P-a.e ω ∈ Ω, where

U i?(ω, t) =

∂U i

∂t(ω, t);

(S.4) there exist two functions αi and βi from R to R such that for every t ∈ R,

αi(t) < t < βi(t),

ess supω∈Ω

U i?(ω, β

i(t)) < ess infω∈Ω

U i?(ω, t)


andess sup

ω∈ΩU i

?(ω, t) < ess infω∈Ω

U i?(ω, α

i(t)).

Remark 2.9. If the information structure is symmetric, then an economy satisfiesAssumptions S.1–S.4 if and only if it satisfies assumptions H.0–H.4 in Le Van andTruong Xuan (2001). In particular, all economies considered in Cheng (1991), Danaand Le Van (1996) and Le Van and Truong Xuan (2001), satisfy Assumptions S.1–S.4.

As in Le Van and Truong Xuan (2001, Section 2.1), we provide hereafter exam-ples of utility functions satisfying Assumptions S.1–S.4.

Example 2.2. If for every i ∈ I, there exists F i : R → R concave, continuouslydifferentiable and strictly increasing and such that the function U i(ω, t) = F i(t) isp-integrable bounded6, then the economy is standard.

Example 2.3. If for every i ∈ I, there exists F i : R → R concave, continuouslydifferentiable, strictly increasing satisfying

limt→−∞

F?(t) = +∞ and limt→+∞

F?(t) = 0;

and hi ∈ L∞++(F i) such that the function U i(ω, t) = F i(t)hi(ω) is p-integrablebounded, then the economy is standard. In this case, agents’ preference relationsare represented by von Neumann–Morgenstern utility functions with heterogenousexpectations. Indeed, if we denote by Pi the probability measure on (Ω,F) definedby dPi = hidP, then the preference relation defined by ui : x 7→ E[U i(x)] coincidewith the von Neumann–Morgenstern preference relation defined by the felicity (orindex) function F i and the private belief Pi, i.e.

∀x ∈ Lp(F i), ui(x) = EPi

[F i(x)] =∫ΩF i(xi(ω))Pi(dω).

Contrary to the literature in continuous time finance (see Pikovsky and Karatzas(1996) and Hillairet (2005)), we do not assume that aggregate initial endowmentis measurable with respect to the common information F c. As the following tworemarks show, in our framework such an assumption makes the existence issue atrivial one.

Remark 2.10. Consider a standard economy E = (U i,F i, ei, i ∈ I) such thatevery initial endowment ei is measurable with respect to the common information,

6Such a utility function has the well known von Neumann–Morgenstern form.


i.e. ei belongs to Lp(F c) and such that each utility function ui is von Neumann–Morgenstern. In this case the existence of a competitive equilibrium follows in astraightforward way. Indeed, let Ec denote the (symmetric) economy (U i,F c, ei, i ∈I). This economy is standard and following Cheng (1991) we can show that thereexists an equilibrium (x•, ψ) with

ψ ∈ Lq+(F c) and xi ∈ Lp(F c), ∀i ∈ I.

We claim that (x•, ψ) is a competitive equilibrium for the original economy E . Forthis, we only need to prove that xi is optimal in the budget set of the economy E .Assume by way of contradiction that there exists yi ∈ Lp(F i) such that

E[ψyi] 6 E[ψei] and ui(yi) > ui(xi).

Since ψ and ei are measurable with respect to F c, the consumption plan E[yi|F c]satisfies the budget constraints associated with the symmetric economy Ec. Butfrom Jensen’s inequality, we also have that

ui(E[yi|F c]) > ui(yi) > ui(xi)

which yields a contradiction to the optimality of xi in the symmetric economy.

Remark 2.11. Now consider a standard economy E = (U i,F i, ei, i ∈ I) such thatthe aggregate endowment e =

∑i∈I e

i is measurable with respect to the commoninformation, i.e. e belongs to Lp(F c), each utility function ui is von Neumann–Morgenstern and the information structure is conditionally independent. In thiscase, the existence of a competitive equilibrium follows also in a straightforwardway. Indeed, the F c-measurability of e together with conditional independenceimply that individual endowments ei are also F c-measurable.7 Existence followsfrom the previous remark.

3. Compactness of the individually rational utility set

Theorem 3.1 stated below, proves the compactness of the individually rationalutility set associated with our differential information economy. In that respect itgeneralizes the results established for symmetric information economies found inCheng (1991), Dana and Le Van (1996) and Le Van and Truong Xuan (2001).

7We have e =∑

i∈I ei. Fix j ∈ I, taking conditional expectations with respect to F j , we get

e = E[e|Fj ] = ej +∑i 6=j

E[ei|Fj ] = ej +∑i 6=j

E[ei|Fc].


Theorem 3.1. For every standard economy with a conditionally independent in-formation structure, the individually rational utility set is compact.

Proof. We let θc be the linear mapping from Lp(F) to Lp(F c) defined by

∀z ∈ Lp(F), θc(z) = E[z|F c]

and we let ξi be the linear mapping from Lp(F) to Lp(F i) defined by

∀z ∈ Lp(F), ξi(z) = E[z − θc(z)|F i].

Observe that

∀i ∈ I, θc ξi = 0 and ∀zi ∈ Lp(F i), ξi(zi) = zi − θc(zi).

Since the information structure is conditionally independent, we have

∀i 6= j ∈ I, ∀zj ∈ Lp(F j), ξi(zj) = E[zj|F i]− E[zj|F c] = 0. (6)

We let Z be the set of allocations defined by

Z =

z• ∈

∏i∈I

Lp(F c) :∑i∈I

zi = θc(e)

,

where e =∑

i∈I ei is the aggregate initial endowment. From (6) we have that

ξi(e) = ξi(ei).

Claim 3.1. An allocation x• is attainable (feasible) if and only if there existsz• ∈ Z such that

∀i ∈ I, xi = zi + ξi(e).

Proof. If x• be an attainable allocation, then

∀i ∈ I, ξi(e) =∑j∈I

ξi(xj) = ξi(xi) = xi − θc(xi).

If for every i ∈ I, we let zi = θc(xi) then z• belongs to Z and satisfies xi = zi+ξi(e).Reciprocally, if x• is an allocation such that xi = zi + ξi(e) for some z• ∈ Z, then∑

i∈I

xi = θc(e) +∑i∈I

ξi(e) = θc(e) +∑i∈I

ei − θc(ei) = e

which implies that x• is attainable.


Now we consider the function V i : Ω× R → R defined by

∀(ω, t) ∈ Ω× R, V (ω, t) = E[U i(ξi(e) + t)|F c](ω).8

By the Disintegration Theorem we have

∀z ∈ Lp(F c), E[U i(ξi(e) + z)] =∫ΩU i(ω, ξi(e)(ω) + z(ω))P(dω)

=∫ΩV i(ω, z(ω))P(dω)

= E[V i(z)]. (7)

Combining Claim 3.1 and equation (7), the individually rational utility set U co-incide with the set V defined by

V =w• ∈ RI : ∃z• ∈ Z, vi(θc(ei)) 6 wi 6 vi(zi)

where vi is the utility function defined on Lp(F c) by vi(z) = E[V i(z)].

Observe that the set V corresponds to the individually rational utility set of asymmetric information economy E? defined by

E? := (V i,F c, θc(ei), i ∈ I).

In addition, as the following result shows, this symmetric economy is standard.

Claim 3.2. For every i ∈ I, the utility function V i satisfies Assumptions S.1–S.4.

Proof. Fix i ∈ I and denote by ζ i the function ξi(e). Since the function U i is p-integrable bounded, it follows that for each t ∈ R, the function ω 7→ U i(ω, ζ i(ω)+t)is integrable and the function V i(ω, t) is well-defined. It is straightforward to checkthat the function t 7→ V i(ω, t) is concave and strictly increasing for P-a.e. ω. Wepropose now to prove that t 7→ V i(ω, t) is differentiable for P-a.e. ω. Let (tn) be asequence converging to t ∈ R with tn 6= t and let

∀ω ∈ Ω, fn(ω) :=U i(ω, ζ i(ω) + tn)− U i(ω, ζ i(ω) + t)

tn − t.

The sequence (fn) converges almost surely to the function f defined by f(ω) =U i

?(ω, ζi(ω)+ t). Moreover, since the function t 7→ U i(ω, t) is concave, differentiable

and strictly increasing, for every n large enough9 we have

|fn(ω)| 6 U i?(ω, ζ

i(ω) + t− 1), for P-a.e. ω ∈ Ω.

8In the sense that if t ∈ R then we let f it (ω) = U i(ω, ξi(e)(ω) + t). The real number V i(ω, t)

is then defined by V i(ω, t) = E[f it |Fc](ω).

9More precisely, for every n such that |tn − t| 6 1.


Observe that since the function t 7→ U i(ω, t) is concave, differentiable and strictlyincreasing, we have for P-a.e. ω,

0 < U i?(ω, ζ

i(ω) + t− 1) 6 U i(ω, ζ i(ω) + t− 1)− U i(ω, ζ i(ω) + t− 2),

but since the function U i is p-integrable bounded, we get

0 < E[U i?(ζ

i + t− 1)] 6 |E[U i(ζ i + t− 1)]|+ |E[U i(ζ i + t− 2)]| < +∞.

Therefore the sequence (fn) is integrably bounded. Applying the Lebesgue Domi-nated Convergence Theorem for conditional expectations, we get that the functiont 7→ V i(ω, t) is differentiable and for every t ∈ R,

V i? (ω, t) :=

∂V i

∂t(ω, t) = E[U i

?(ξi(e) + t)|F c](ω), for P-a.e. ω ∈ Ω.

We have thus proved that the function V i satisfies Assumption S.1. Assumption S.2follows from the p-integrability of U i and the disintegration theorem. We proposenow to prove that the function t 7→ V i

? (ω, t) is continuous for P-a.e. ω. Let (tn)be a sequence converging to t ∈ R and let gn be the function defined by gn(ω) :=U i

?(ω, ζi(ω) + tn). The function (gn) converges almost surely to the function g

defined by g(ω) := U i?(ω, ζ

i(ω) + t). Moreover, for n large enough we have

0 < gn(ω) 6 U i?(ω, ζ

i(ω) + t− 1), for P-a.e. ω ∈ Ω.

Applying the Lebesgue Dominated Convergence Theorem for conditional expecta-tions we get that the function t 7→ V i

? (ω, t) is continuous for P-a.e. ω.We propose now to prove that

∀t ∈ R, ess supω∈Ω

V i? (ω, βi(t)) < ess inf

ω∈ΩV i

? (ω, t).

Fix t ∈ R. Since the function U i satisfies Assumption S.4, we trivially have that

U i?(ω, β

i(t)) 6 ess supω∈Ω

U i?(ω, β

i(t)), for P-a.e. ω ∈ Ω.

Therefore for P-a.e. ω,

V i? (ω, βi(t)) = E[U i

?(ξi(e) + βi(t))|F c](ω) 6 ess sup

ω∈ΩU i

?(ω, βi(t))

which implies that

ess supω∈Ω

V i? (ω, βi(t)) 6 ess sup

ω∈ΩU i

?(ω, βi(t)). (8)


Symmetrically we can prove

ess infω∈Ω

U i?(ω, t) 6 ess inf

ω∈ΩV i

? (ω, t). (9)

Since U i satisfies Assumption S.4, we have that

ess supω∈Ω

U i?(ω, β

i(t)) 6 ess infω∈Ω

U i?(ω, t). (10)

The desired result follows from (8), (9) and (10).Following similar arguments, it is straightforward to prove that

∀t ∈ R, ess supω∈Ω

V i? (ω, t) < ess inf

ω∈ΩV i

? (ω, αi(t)),

and then we get that V i satisfies Assumptions S.1–S.4.

Claim 3.2 implies (see Remark 2.4) that the symmetric information economy E?

satisfies Assumptions H.0–H.4 in Le Van and Truong Xuan (2001). Based on theirarguments we can directly conclude that the set V (and therefore the set U) iscompact.10

4. Decentralization of Edgeworth allocations

We now recall the definition of different optimality concepts for feasible allocations.

Definition 4.1. A feasible allocation x• is said to be:

1. weakly Pareto optimal if there is no feasible allocation y• satisfying yi ∈ P i(xi)for each i ∈ I,

2. a core allocation, if it cannot be blocked by any coalition in the sense thatthere is no coalition S ⊆ I and some (yi, i ∈ S) ∈ ∏

i∈S Pi(xi) such that∑

i∈S yi =

∑i∈S e

i,3. an Edgeworth allocation if there is no 0 6= λ• ∈ Q∩ [0, 1]I and some allocationy• such that yi ∈ P i(xi) for each i ∈ I with λi > 0 and

∑i∈I λ

iyi =∑

i∈I λiei,

4. an Aubin allocation if there is no 0 6= λ• ∈ [0, 1]I and some allocation y• suchthat yi ∈ P i(xi) for each i ∈ I with λi > 0 and

∑i∈I λ

iyi =∑

i∈I λiei.

10Actually in Le Van and Truong Xuan (2001) the probability space is the continuum [0, 1]endowed with the Lebesgue measure, but the arguments can straightforwardly be adapted toabstract probability spaces. We refer to Appendix B for a formal discussion.


Remark 4.1. The reader should observe that these concepts are “price free” inthe sense that they are intrinsic properties of the commodity space. It is proved inFlorenzano (2003, Proposition 4.2.6) that the set of Aubin allocations and the setof Edgeworth allocations coincide for every standard economy.

Corollary 4.1. If E is a standard economy with a conditionally independent in-formation structure then there exists an Edgeworth allocation.

Proof. Since the individually rational utility set is compact, we can apply Theo-rem 3.1 in Allouch and Florenzano (2004).

It is straightforward to check that every competitive allocation is an Edgeworthallocation. We prove that the converse is true.

Theorem 4.1. If E is a standard economy with a conditionally independent in-formation structure then for every Edgeworth allocation x• there exists a price ψsuch that (x•, ψ) is a competitive equilibrium. Moreover, for every i ∈ I there existsλi > 0 such that

E[ψ|F i] = λiU i?(x

i).

In particular, we have

∀(i, j) ∈ I × I, E[λiU i?(x

i)|F c] = E[λjU j? (xj)|F c].

Proof. Let x• be an Edgeworth allocation. From Remark 4.1 it is actually an Aubinallocation, in particular

0 6∈ G(x•) := co⋃i∈I

(P i(xi)− ei

). (11)

The set Lp(F i) is denoted by Ei, the set Lp(F c) is denoted by Ec and we letE =

∑i∈I E

i. Observe that E may be a strict subspace of Lp(F). We endow Ewith the topology τ for which a base of 0-neighborhoods is∑

i∈I

αiB ∩ Ei : αi > 0, ∀i ∈ I

where B is the closed unit ball in Lp(F) for the p-norm topology.11 From theStructure Theorem in Aliprantis and Border (1999, page 168) this topology is welldefined, Hausdorff and locally convex. Observe moreover that the restriction ofthe τ -topology to each subspace Ei coincide with the restriction of the p-norm

11We refer to Appendix A for precise definitions.


topology. From Proposition A.1, each utility function ui is p-norm continuous. Itfollows that G(x•) has a τ -interior point. Applying a convex separation theorem,there exists a τ -continuous linear functional π ∈ (E, τ)′ such that π 6= 0 and foreach g ∈ G(x•), we have π(g) > 0. In particular it follows that

∀i ∈ I, ∀yi ∈ P i(xi), π(yi) > π(xi). (12)

Since preference relations are strictly increasing, we have that xi belongs to theclosure of P i(xi). Therefore π(xi) > π(ei). But since x• is attainable, we musthave π(xi) = π(ei) for each i ∈ I. The linear functional π is not zero, thereforethere exists j ∈ I and zj ∈ Ej such that π(zj) < π(ej). We now claim that ifyj ∈ P j(xj) then π(yj) > π(ej). Indeed, assume by way of contradiction thatπ(yj) = π(ej). Since the set P j(xj) is p-norm open in Ej, there exists α ∈ (0, 1)such that αyj +(1−α)zj ∈ P j(xj). From (12) we get απ(yj)+(1−α)π(zj) > π(ej):contradiction. We have thus proved that

∀yj ∈ P j(xj), π(yj) > π(xj).

By strict-monotonicity of preference relations, we obtain that π(1) > 0 where 1is the vector in Ec defined by 1(ω) = 1 for any ω ∈ Ω. Choosing zi = ei − 1 forevery i 6= j, we follow the previous argument to get that

∀i ∈ I, ∀yi ∈ P i(xi), π(yi) > π(ei),

or equivalently

∀i ∈ I, xi ∈ argmaxui(zi) : zi ∈ Ei and π(zi) 6 π(ei)

. (13)

In order to prove that x• is an equilibrium allocation, it is sufficient to prove thatthere exists a price ψ ∈ Lq(F) which represents π in the sense that π = 〈ψ, .〉.

Claim 4.1. For every i ∈ I, there exists ψi ∈ Lq(F i) such that

∀zi ∈ Ei, π(zi) = 〈ψi, zi〉.

Moreover, there exists λi > 0 such that ψi = λiU i?(x

i).

Proof. Fix i ∈ I and let zi ∈ Ei such that 〈U i?(x

i), zi〉 > 0. From Claim A.3, thereexists t > 0 such that xi + tzi ∈ P i(xi). Applying (13) we get that π(zi) > 0. Wehave thus proved that

∀zi ∈ Ei, 〈U i?(x

i), zi〉 > 0 =⇒ π(zi) > 0.


It then follows12 that there exists λi > 0 such that the restriction π|Ei of π to Ei

coincides with the linear functional 〈λiU i?(x

i), .〉.

Since Ec ⊂ Ei ∩ Ej the family ψ• = (ψi, i ∈ I) is compatible in the sense that

∀(i, j) ∈ I × I, E[ψi|F c] = E[ψj|F c].

We denote by ψc the vector in Lq(F c) defined by ψc = E[ψi|F c] and we let ψ bethe vector in Lq(F) defined by

ψ = ψc +∑i∈I

(ψi − ψc). (14)

Claim 4.2. The vector ψ represents the linear functional π.

Proof. Since the information structure is conditionally independent, we can easilycheck that

∀i ∈ I, E[ψ|F i] = ψi.

If z is a vector in E, then there exists z• = (zi, i ∈ I) with zi ∈ Ei such thatz =

∑i∈I z

i. Therefore

〈ψ, z〉 =∑i∈I

〈ψ, zi〉 =∑i∈I

E[ψzi] =∑i∈I

E[ψizi] =∑i∈I

π(zi) = π(z),

which implies that π = 〈ψ, .〉.

It follows from (13) and Claim 4.2 that (x•, ψ) is a competitive equilibrium.

Remark 4.2. Observe that in the proof of Theorem 4.1, the conditional inde-pendence assumption on the information structure is only used in Claim 4.2 toprove that the linear functional π is representable by a function ψ in Lq(F). Inparticular if the set Ω is finite, then Theorem 4.1 is valid without assuming thatthe information structure is conditionally independent.

Due to Corollary 4.1 and Theorem 4.1 we are able to generalize Theorem 1 inCheng (1991), Theorem 1 in Dana and Le Van (1996) and Theorem 1 in Le Vanand Truong Xuan (2001) to economies with differential information.

12Let 〈X, X?〉 be a dual pair. Assume that there exists x? ∈ X? and 0 6= π a linear functionalfrom X to R such that for every x ∈ X, if 〈x, x?〉 > 0 then π(x) > 0. Let ∆ = (a, b) ∈R+ × R+ : a + b = 1 be the simplex in R2, we then have

∆ ∩ (〈x, x?〉,−π(x)) : x ∈ X = ∅.

Applying a convex separation argument, we get the existence of λ > 0 such that 〈λx?, .〉 = π.


Corollary 4.2. Every standard economy with a conditionally independent infor-mation structure has a competitive allocation.

Remark 4.3. Observe that when p = +∞, contrary to Cheng (1991) and Le Vanand Truong Xuan (2001), we do not need to establish Mackey-continuity of theutility functions to guarantee that equilibrium prices belong to L1(F). As in Danaand Le Van (1996), our equilibrium prices are related to individual marginal utili-ties. The continuity requirement (i.e. prices belong to L1(F)) follows then directlyfrom the standard assumptions on the utility functions.

5. Finitely many states of nature and no-arbitrage prices

In this section we assume that the set Ω is finite. Without any loss of generality,we may assume that Pω > 0 for every ω ∈ Ω. For every 1 6 p 6 +∞, the spaceLp(F i) coincides with L0(F i) the space of F i measurable functions from Ω to R.

In Theorem 4.1 we proved an equilibrium existence result for economies in whichthe information structure satisfies the conditional independence property. We showbelow that, when Ω is finite, the existence of equilibrium prices can be establishedwithout assuming conditional independence of the information structure. Indeed,in this case one can show that for standard economies a no-arbitrage price alwaysexists. It is well-known in the literature (see Allouch et al. (2002) and the referencestherein) that in the finite dimensional case, the existence of a no-arbitrage price issufficient for the existence of an equilibrium price system. However, it should bestressed that dropping out conditional independence leaves open the question ofexecution of contracts. In that respect, we believe that conditional independenceis a relevant restriction even in the finite dimensional case. We impose hereafterthe following assumption which is a weaker version of Assumption S.4.

(wS.4) There exist three numbers αi < τ i < βi such that,

supω∈Ω

U i?(ω, β

i) < infω∈Ω

U i?(ω, τ

i)

andsupω∈Ω

U i?(ω, τ

i) < infω∈Ω

U i?(ω, α

i).

For each i ∈ I, we let Ri be the subset of L0(F i) defined by

Ri = yi ∈ L0(F i) : ∀t > 0, ui(ei + tyi) > ui(ei).


Since the function ui is concave, it is standard (see Rockafellar (1970, Theorem 8.7))that the set Ri satisfies the following property:

∀xi ∈ L0(F i), Ri = yi ∈ L0(F i) : ∀t > 0, ui(xi + tyi) > ui(xi). (15)

In our framework, we can prove that 1 is a no-arbitrage price, as defined inWerner (1987).

Proposition 5.1. For each i ∈ I, if yi is a non-zero vector in Ri then 〈1, yi〉 > 0.

Proof. Let xi ∈ Lp(F i) be defined as follows

xi = βi1yi>0 + αi1yi<0 + τ i1yi=0.

Since yi ∈ Ri, one has ui(xi + yi) > ui(xi). Following standard arguments, one has

〈U i?(x

i), yi〉 > ui(xi + yi)− ui(xi) > 0.

Let γi ∈ R be such that

infω∈Ω

U i?(ω, τ

i) 6 γi 6 supω∈Ω

U i?(ω, τ

i).

Observe that γi > 0 and for almost every ω ∈ Ω, one has

γiyi(ω) > U i?(ω, x

i(ω))yi(ω).

From Assumption wS.4,

γi > U i?(ω, x

i(ω)), ∀ω ∈ yi > 0

andγi < U i

?(ω, xi(ω)), ∀ω ∈ yi < 0.

Since yi 6= 0, it follows that

γi〈1, yi〉 > 〈U i?(x

i), yi〉 > 0.


Appendix A: Properties of separable utility functions

The space Lp(F) is endowed with the p-norm topology defined by the sup-norm‖.‖∞ if p = ∞, i.e.

∀x ∈ L∞(F), ‖x‖∞ = ess supω∈Ω

|x(ω)|

and the p-norm ‖.‖p if p ∈ [1,+∞), i.e.

∀x ∈ Lp(F), ‖x‖p =[∫

Ω|x(ω)|pP(dω)

]1/p

.

Proposition A.1. Let p ∈ [1,+∞] and U : Ω× R → R be a p-integrable boundedfunction such that the function t 7→ U(ω, t) is continuous and increasing for P-a.e.ω ∈ Ω. Then the separable function u : Lp(F) → R defined by

∀x ∈ Lp(F), u(x) =∫ΩU(ω, x(ω))P(dω)

is p-norm continuous.

Proof. The arguments of the proof are similar to those in Aliprantis (1997, The-orem 5.3). Let (xn) be a sequence in Lp(F) p-norm converging to x ∈ Lp(F). Toshow that the sequence (u(xn)) converges to u(x), it suffices to establish that everysubsequence (yn) of (xn) has in turn a subsequence (zn) such that (u(zn)) convergesto u(z). To this end, let (yn) be a subsequence of (xn). Applying Theoreme IV.19in Brezis (1993) there exists a subsequence (zn) of (yn) such that

(a) the sequence (zn(ω)) converges to x(ω) for P-a.e. ω ∈ Ω,(b) there exits h ∈ Lp

+(F) such that |zn(ω)| 6 h(ω) for P-a.e. ω ∈ Ω.

Since the function t 7→ U(ω, t) is continuous for P-a.e. ω ∈ Ω, then the sequence(U(ω, zn(ω)) converges to U(ω, x(ω)) for P-a.e. ω ∈ Ω. Moreover, since the functiont 7→ U(ω, t) is increasing for P-a.e. ω ∈ Ω, we have that

|U(ω, zn(ω)| 6 maxU(ω, h(ω)), U(ω,−h(ω)), for P-a.e. ω ∈ Ω

so, by the Lebesgue dominated convergence theorem, we get limn→∞ u(zn) = u(z).


Proposition A.2. Let p ∈ [1,+∞] and U : Ω× R → R be a p-integrable boundedfunction such that the function t 7→ U(ω, t) is continuous, increasing and differen-tiable for P-a.e. ω ∈ Ω. For every x ∈ Lp(F) the function ω 7→ U?(ω, x(ω)) belongsto Lq(F) and

∀y ∈ Lp(F), 〈U?(x), y〉 = limt↓0

1

tu(x+ ty)− u(x).

Proof. Since the function t 7→ U(ω, t) is continuous, increasing and differentiable,then

∀τ > 0, 0 6 U?(ω, t)τ 6 U(ω, t)− U(ω, t− τ) (16)

and∀τ 6 0, U(ω, t+ τ)− U(ω, t) 6 U?(ω, t)τ 6 0. (17)

Now let x and y be two vectors in Lp(F), applying (16) and (17) we have almostsurely that,

|U(x)y| 6 |U(x)|+ |U(x+ y)|+ |U(x− y)|. (18)

Therefore we have for every y ∈ Lp(F),

E|U?(x)y| 6 E|U(x)|+ E|U(x+ y)|+ E|U(x− y)| < +∞. (19)

Claim A.1. If p = ∞ then the function U?(x) belongs to L1(F).

Proof of Claim A.1. From (19) we have that E|U?(x)y| <∞ for every y ∈ L∞(F).If we let y1 and y2 be the two functions in L∞(F) defined by

y1 = 1U?(x)>0 and y2 = 1U?(x)<0

thenE|U?(x)| = E|U?(x)y1|+ E|U?(x)y2| < +∞

which implies that the function U?(x) belongs to L1(F).

Claim A.2. If 1 6 p <∞ then the function U?(x) belongs to Lq(F).

Proof of Claim A.2. Since the function U is p-integrable bounded, there exists α ∈L1

+(F) and β > 0 such that for P-a.e. ω ∈ Ω,

∀t ∈ R, |U(ω, t)| 6 α(ω) + β|t|p.

It then follows from (19) that for every y ∈ Lp(F),

E|U?(x)y| 6 3 ‖α‖1 + β(‖x‖p)p + (‖x+ y‖p)

p + (‖x− y‖p)p

which implies that the linear operator y 7→ E[U?(x)y] is p-norm continuous. There-fore the function U?(x) belongs to Lq(F).


Claim A.3. For every x and y in Lp(F), we have

〈U?(x), y〉 = limt↓0

1

tu(x+ ty)− u(x).

Proof of Claim A.3. Let (tn) be a sequence in [0,+∞) converging to 0 and let fn

be the function defined by

∀ω ∈ Ω, fn(ω) =1

tnU(ω, x(ω) + tny(ω))− U(ω, x(ω)).

The sequence (fn) converges almost surely to the function f = U?(x)y. Moreover,we have

U?(ω, x(ω) + tny(ω))y(ω) 6 fn(ω) 6 U?(ω, x(ω))y(ω) for P-a.e. ω ∈ Ω,

in particular for n large enough,

|fn| 6 U?(−|x| − |y|)|y|.

The desired result follows by the Lebesgue dominated convergence theorem.

Proposition A.2 follows from Claims A.1 and A.3.

Appendix B: Economies with symmetric information: from atomlessto general probability state space.

We consider a symmetric economy E = (U i,F , ei, i ∈ I), i.e. F i = F for everyagent i ∈ I. It is proved in Le Van and Truong Xuan (2001) that if E is standardand if the state space (Ω,F ,P) is the continuum [0, 1] endowed with the Lebesguemeasure then the individually rational utility set is compact. The arguments inLe Van and Truong Xuan (2001) are still valid for any atomless probability space.We propose to extend this result to general probability spaces.

Proposition B.1. If E = (U i,F , ei, i ∈ I) is a standard symmetric economy thenthe individually rational utility set is compact.

Proof. Let (Ω,F ,P) be an abstract probability space. Observe that Ω = Ωna ∪ Ωa

where Ωa ∈ F is the (purely) atomic part of Ω and Ωna ∈ F is the atomless partof Ω.


• First step: assume that (Ω,F ,P) is purely atomic. Since P(Ω) = 1, thereexists at most countably many atoms in Ω. Without any loss of generality,we may assume that Ω has at most countably many elements. In particular

∀A ∈ F , P(A) =∑ω∈A

Pω.

We denote by L the Lebesgue σ-algebra on [0, 1].

Claim B.1. The probability space (Ω× [0, 1],F ⊗ L,P⊗ λ) is atomless.

Proof. Let B ∈ F ⊗L such that [P⊗ λ](B) > 0 and fix α ∈ [0, 1]. Since Ω isat most countable, for every ω ∈ Ω, there exists Cω ∈ L such that

B =⋃

ω∈Ω

ω × Cω.

Observe that[P⊗ λ](B) =

∑ω∈Ω

Pωλ(Cω).

Since ([0, 1],L, λ) is atomless, for every ω ∈ Ω, there exists Dω ∈ L such thatDω ⊂ Cω and λ(Dω) = αλ(Cω). We now let A be the set in F ⊗L defined by

A =⋃

ω∈Ω

ω ×Dω.

The set A is such that

A ⊂ B and [P⊗ λ](A) = α[P⊗ λ](B).

Since the information is symmetric, the space Lp(F) is denoted by Lp(P) andthe space Lp(Ω× [0, 1],F ⊗ L,P⊗ λ) is denoted by Lp(P⊗ λ). If x ∈ Lp(P)then we let Fx be the function in Lp(P⊗ λ) defined by

∀(ω, s) ∈ Ω× [0, 1], Fx(ω, s) = x(ω).

If y ∈ Lp(P⊗ λ) then we let Gy be the function in Lp(P) defined by

∀ω ∈ Ω, Gy(ω) =∫[0,1]

y(ω, s)λ(ds).


We propose to prove that the individually rational utility set U is compact.Let (w•

n) be a sequence in U , i.e. for each n ∈ N, there exists an attainableallocation x•n satisfying

∀i ∈ I, ui(ei) 6 win 6 ui(xi

n).

We let V i be the function from (Ω× [0, 1])× R to R defined by

∀((ω, s), t) ∈ Ω× [0, 1], V i((ω, s), t) = U i(ω, t),

and we let vi be the utility function defined on Lp(P⊗ λ) by

vi(y) = EP⊗λ[V i(y)] =∫Ω

P(dω)∫[0,1]

U i(ω, y(ω, s))λ(ds).

Observe that for every z ∈ Lp(P), we have ui(z) = vi(Fz). It then followsthat

∀i ∈ I, vi(Fei) 6 win 6 vi(Fxi

n).

We let E ′ be the symmetric economy defined by (V i,F ⊗ L,Fei, i ∈ I).Observe that for every n ∈ N , the family (Fxi

n, i ∈ I) is attainable for E ′. Inparticular, the sequence (w•

n) belongs to the individually rational utility setU ′ of the economy E ′. The economy E ′ is standard and the probability space(Ω × [0, 1],F ⊗ L,P ⊗ λ) is atomless. Applying Le Van and Truong Xuan(2001), there exists w• in U ′ such that

∀i ∈ I, limn→∞

win = wi.

We claim that w• actually belongs to U . Since w• belongs to U ′, there existsan allocation y• attainable for the economy E ′ such that

∀i ∈ I, vi(Fei) 6 wi 6 vi(yi).

Since U i is concave, we have

∀i ∈ I, vi(yi) 6 ui(Gyi).

Since the allocation (Gyi, i ∈ I) is attainable for the economy E , we haveproved that w• belongs to U .


• Second step: the general case. We let Ξ be the subset of Ω× [0, 1] definedby

Ξ = (Ωa × [0, 1])⋃

(Ωna × 0)and we let G be the trace of the σ-algebra F ⊗ L on Ξ. On the measurablespace (Ξ,G) we defined the probability Q by

∀A ∈ G, Q(A) = [P⊗ λ](Aa) + P(Ana)

where A = Aa∪Ana is the (unique) decomposition defined by Aa ⊂ Ωa× [0, 1]and Ana ⊂ Ωna × 0 and where P(Ana) = P(B) with B ∈ F satisfyingAna = B × 0. Observe that the probability space (Ξ,G,Q) is atomless.Following almost verbatim the arguments of the first step, we can prove thatthe individually rational utility set U is compact.

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