Contract enforcement and incentive compatibility in large economies with differential information: the role of exact feasibility

Contract enforcement and incentive

compatibility in large economies with

differential information: the role of exact

feasibility

Laura Angeloni, Victor Filipe Martins-da-Rocha

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Laura Angeloni · V. Filipe Martins-da-Rocha

Contract enforcement and incentive

compatibility in large economies with

differential information: the role of exact

feasibility.

June 15, 2007

Abstract We consider exchange economies with a continuum of agents and dif-ferential information about finitely many states of nature. It was proved in Einy, Moreno and Shitovitz (2001) that if we allow for free disposal in the market clear-ing (feasibility) constraints then an irreducible economy has a competitive (or Walrasian expectations) equilibrium, and moreover, the set of competitive equi-librium allocations coincides with the private core. However when feasibility is defined with free disposal, competitive equilibrium allocations may not be incen-tive compatible and contracts may not be enforceable (see e.g. Glycopantis, Muir and Yannelis (2002)). This is the main motivation for considering equilibrium so-lutions with exact feasibility. We first prove that the results in Einy et al. (2001) are still valid without free-disposal. Then we define an incentive compatibility prop-erty motivated by the issue of contracts’ execution and we prove that every Pareto optimal exact feasible allocation is incentive compatible, implying that contracts of a competitive or core allocations are enforceable.

This work was partially done while V.F. Martins-da-Rocha was visiting the Dipartimento di Matematica e Informatica of the Universit degli Studi di Perugia. We thank the audience of the First General Equilibrium Workshop at Rio. Section 6 dealing with contract enforcement and coalitional incentive compatibility has benefited from discussions with J. Correia-da-Silva, W. Daher, F. Forges, C. Herv`es-Beloso, E. Moreno-Garc´ıa, K. Podczeck, Y. Vailakis and N. Yan-nelis.

Laura Angeloni

Dipartimento di Matematica e Informatica, Universit degli Studi di Perugia, Via Vanvitelli 1, 06123 Perugia, ITALY

Tel.: +49-228-735747 Fax.: +49-228-731785

E-mail: [email protected] V. Filipe Martins-da-Rocha

Graduate School of Economics (EPGE), Getulio Vargas Foundation (FGV), Praia de Botafogo 190, 22.250-900 Rio de Janeiro, BRAZIL

Keywords Large exchange economies·Differential information·Incentive

compatibility·Contracts enforceability

JEL Classification Numbers D50·D82

1 Introduction

Radner (1968) introduced differential information in the general equilibrium model of Arrow and Debreu (1954). He considers an economy which extends over two time periods with uncertainty at the second period represented by a finite set of states of nature. Each agent is characterized by a random utility function, a random initial endowment, a prior belief and private information. The private information is represented by a partition of the set of states. At the second period, a state of nature is realized but each agent has incomplete information in the sense that he only knows to which atom of his partition the true state belongs but he cannot dis-criminate states inside this atom. At the first period, a complete set of contingent contracts is available for trade and before they obtain any information about the realized state of nature, agents arrange contracts which are assumed to be consis-tent with respect to their private information. At the second period, uncertainty is resolved, information is observed, contracts are executed and consumption takes place.

In this framework, Radner (1968) introduced a competitive equilibrium con-cept (Walrasian expectations equilibrium) which is an analogue concon-cept to the Walrasian equilibrium in Arrow–Debreu model with complete (symmetric) infor-mation. He proved that, under standard assumptions (similar to those used in the existence results by Arrow and Debreu (1954)) on agents’ characteristics, a Wal-rasian expectations equilibrium always exists. Recently, this existence result was generalized in several directions: infinitely many commodities (see Podczeck and Yannelis (2007)), infinitely many states (see Herv`es-Beloso, Martins-da Rocha and Monteiro (2007)) and unbounded consumption sets (see Daher, Martins-da Rocha and Vailakis (2007)).

Yannelis (1991) introduced a cooperative equilibrium concept, called the pri-vate core, which is an analogue concept to the core for an economy with complete (and symmetric) information, and proved that under appropriate assumptions the private core is always non-empty. In the definition of the private core, when a coalition blocks an allocation, each member in the coalition uses only his own private information. This cooperative concept of equilibrium has some interesting properties: Koutsougeras and Yannelis (1993) proved that allowing individuals to make redistributions of their initial endowments, based only on their own private information, results in equilibrium allocations that are always Bayesian incentive compatible and also take into account the informational advantage of an individ-ual. The private core is the appropriate notion of core when the traders do not want to exchange information or when they do not have access to any communi-cation system. When traders are allowed to fully share their information, the weak fine core introduced by Koutsougeras and Yannelis (1993) is the appropriate core concept.

The existence of such allocations was studied by Aumann (1966) and Hilden-brand (1970). An extension of these results to economies with differential

infor-mation was proposed by Einy et al. (2001). They show that, if an economy is

ir-reducible, then a competitive (or Walrasian expectations) equilibrium exists and, moreover, the set of competitive equilibrium allocations coincides with the private core. However, to obtain these results they allow for free disposal on the feasibil-ity (market clearing) constraints. This was motivated by an example provided by Einy and Shitovitz (2001) of an economy with differential information which has a competitive equilibrium with free disposal, but if the feasibility constraints are im-posed with an equality, then the economy does not have a competitive equilibrium where the prices of all contingent contracts for future delivery are non-negative.

We claim that this is not economically inconsistent. If there is a state sthat no

agent can identify, then the contract delivering one unit of a goodℓ contingent

to the realization of the stateswill not be purchased by any trader. The fact the

pricep(s, ℓ)may be negative is irrelevant, what matters are the prices of contracts

that can be traded. Another reason for considering feasibility constraints with free disposal is the version of Fatou’s Lemma used in Hildenbrand (1970) to prove ex-istence of competitive equilibrium. The argument is based on a version of Fatou’s Lemma proved by Schmeidler (1970) where free disposal plays a crucial role.

The aim of this paper is to investigate whether the results in Einy et al. (2001) are still valid if the feasibility constraints are imposed with equality. Using a more general version of Fatou’s Lemma (proved by Balder and Hess (1995)) and a gen-eralization of Hildenbrand’s result by Cornet, Topuzu and Yildiz (2003), we prove

that if an economy isirreducible, then a competitive (or Walrasian expectations)

equilibrium exists and moreover, the set of competitive equilibrium allocations coincides with the private core. We also deal with another issue: contracts enforce-ment at the second period. There is a detailed discussion in Daher et al. (2007) (see also Podczeck and Yannelis (2007, Section 4)) about the relationship between the execution of contracts and incentive compatibility properties. On the definition of the feasibility constraints with free disposal, the total amount to be disposed of might not be measurable with respect to the information partition of a single

agent.1This is the main reason why competitive allocations with free disposal may

not be incentive compatible (see Glycopantis et al. (2002) for an example). We de-fine a notion of coalitional incentive compatibility that guarantees the execution

of contracts2and we prove that every Pareto optimal allocation, satisfying

feasi-bility constraints with equality is incentive compatible, implying that contracts of a competitive or core allocations are enforceable.

The structure of the paper is as follows. Section 2 presents the theoretical framework and outlines the basic model. In Section 3, we introduce assumptions under which existence of competitive allocations and core equivalence will be proved. The proofs follow in Section 4 and Section 5 respectively. Section 6 is devoted to the issue of contract enforceability and its relationship with coalitional incentive compatibility.

1 _{More precisely, the total amount to be disposed of may not coincide with the sum of private}

measurable contingent contracts.

2 _{Our coalitional incentive compatibility condition is related to the conditions introduced in}

2 The model

We consider a pure exchange economy with a continuum of agents represented by

a finite positive measure space(T,Σ,µ), whereTis a set which represents agents,

Σ is aσ-algebra onT which represents coalitions, andµis a non-atomic positive

and finite measure onΣ satisfyingµ(T) =1. IfE∈Σ is a coalition then µ(E)

represents the fraction of agents which belong toE.

The economy extends over two time periodsτ∈ {0,1}. There is uncertainty

over the possible state of nature that may realize atτ=1 represented by a finite set

Ω. Consumption of a finite setLof goods takes place atτ=1 but agents arrange

contingent contracts atτ=0 where there is a complete set of contingent contracts

for future delivery of each good.

Atτ=1 an agentthas an incomplete information about which state of nature

actually occurred. This information is described by a partitionΠtofΩ: ifωis the

true state of nature, agentt cannot discriminate the states in the (unique) element

ofΠt containing ω. The σ-algebra generated by Πt is denoted by Ft and we

denote byXt the set ofFt-measurable functionsx:Ω→RL+. For everyω ∈Ω

we letEt(ω)be the unique atom ofF_{t(or unique element ofΠt) containingω.}

Following the model introduced by Radner (1968) (see also Radner (1982)) the information of an agent places a restriction on his feasible trade in the sense

that each agentt is constrained to choose a contingent contractx:Ω→RL₊

mea-surable with his private informationF_{t. In other words he chooses plans in the}

consumption setXt. Agenttknows atτ=0 that atτ=1 and stateωhe will have

an initial endowmentet(ω)∈RL+. We assume that he can observe his initial

en-dowment, i.e. the functionet isFt-measurable. The ex-ante preference relation

about contingent plans atτ=0 is represented by a correspondencePt:Xt→2Xt.

Ifx∈Xt is a contingent plan thenPt(x)represents the set of plansy∈Xtthat are

strictly preferred tox. An economyE _{is then defined by a family}

E _{= (}F_{t,Pt,et)t∈T.}

Remark2.1. An economy is said to have preference relations represented by ex-pected utilities if for each agentt∈T, there exist

1. a strictly positive3 _{probability measureq}

t on Ω which represents his prior

beliefs, and

2. a state dependent utility functionut:Ω×RL+→Rsatisfying the following

properties

(a) the mapping(t,x)7→ut(ω,x)isΣ×B(RL+)-measurable;

(b) the mappingω 7→ut(ω,x)isF_{t-measurable for everyx∈}RL_+;

(c) the mappingx7→ut(ω,x)is continuous and strictly increasing4onRL_+;

such that

∀x∈Xt, Pt(x) ={y∈Xt : ht(y)>ht(x)} where

ht(x):=

∑

qt(ω)ut(ω,x(ω)).

3 _{In the sense that for eachω∈Ω, we haveq} t(ω)>0. 4 _{A mapping f:}RL

Since the spaceΩis finite, there exists a finite collection(Fi_)i

∈Iofσ-algebra

onΩsuch that

{F_{t : t∈T}={}Fi _{: i∈I}.}

We assume that the setTi⊂T defined by

Ti:={t∈T : F_t=Fi_}

belongs toT _{and that the family(T}i_)i

∈Iforms a partition ofT satisfyingµ(Ti)>

0 for eachi. Therefore there is a finite setIofinformation typesand every agent

t∈Ti_{is of information typeiin the sense thatF}

t=Fi.

Throughout the paper we use the following notations. For eachi∈I, the space

ofFi_{-measurable functionsx:Ω→R}L

+ is denoted byE+i and the linear space

E+i −E+i is denoted byEi. Observe that for eacht∈Ti, the consumption setXt

coincides with the setE+i . The space ∑i∈IEi is denoted byE and is called the

commodity space. The (positive) cone∑_i∈IE+i is denoted by E+. SinceE may

be identified with a vector subspace ofRΩ×L_{, it may be endowed with the cone}

E∩RΩ₊×L_{. Observe thatE+is a subset ofE∩}RΩ₊×L _{but in general,}5_{it is a strict}

subset. A vectorx∈E+ is said strictly positive, denoted byx≫0, if for every

ω∈Ω, the vectorx(ω)belongs toRL_++.

Definition 2.1. An integrable function fromT toRΩ₊×L _{is called an assignment}

and the space of assignments is denoted byS_{. The space of integrable selections}

of the correspondenceX, i.e. the space of integrable functionsx:t7→xtfromT to

Esuch thatxt∈Xt forµ-a.e.t∈T, is denoted bySX. A vectorxinSX is called

aprivate assignmentand it is said

1. feasibleif

Z

Txd µ=

Z

Ted µ

2. free-disposal feasibleif

∃z∈E∩RΩ₊×L_,

Z

T

xdµ+z= Z

T edµ.

3 Assumptions

Throughout the rest of the paper we only consider standard economies in the sense given by the following definition.

Definition 3.1. An economyE _{is saidstandardif}

1. the initial endowment assignmente:t7→et belongs toSX and the aggregate

initial endowmente(T):=R

Tedµis strictly positive, i.e.

∀ω∈Ω, e(T)(ω) = Z

Te

t(ω)µ(dt)∈RL_++;

5 _{If for eachω∈Ω, there exists an information typei∈Isuch that{ω}belongs toF}i_{, then}

2. preference relations are measurable,6irreflexive,7transitive,8continuous9and

strictly monotone.10

Remark3.1. If preference relations are represented by expected utilities then they are automatically measurable, irreflexive, transitive, continuous and strictly mono-tone.

The following irreducibility condition was introduced in McKenzie (1959) for economies with finitely many agents. It was extended to large economies by

Hildenbrand (1974).11

Definition 3.2. An economy is saidirreducibleif for every feasible private

assign-mentx∈S_{X and for every two disjoints coalitionsA,B∈Σ such thatµ(A)>0,}

µ(B)>0, andA∪B=T, there exist two private assignmentsy,z∈S_{Xsuch that}

Z

Axd µ+

Z

Bed µ=

Z

Ayd µ+

Z

Bzd

µ and yt∈Pt(xt) forµ-a.e.t∈A.

We letFc_:=V

i∈IFibe the meet12σ-algebra representing thecommon

knowl-edgeinformation of the grand coalitionI, and we denote byEc(E+c) the space of

Fc_{-measurable functions fromΩtoR}L_(resp.RL

+). Observe thatEcis a subspace

of eachEi. We propose hereafter a condition on the initial endowment assignment

which implies the irreducibility condition.

Proposition 3.1. LetE _{be a standard economy such that there exists a private}

assignment a∈SXsatisfying forµ-a.e. t∈T ,

06=at∈E+c and et(ω)≥at(ω), ∀ω∈Ω.

Then the economy is irreducible.

Proof. Letx∈S_{Xbe a feasible private assignment and let two disjoints coalitions}

A,B∈Σ such thatµ(A)>0,µ(B)>0, andA∪B=T. We leta(B):=R Badµ,

observe thata(B)∈E+c. We define the functiony:T →Eby

∀t∈T, yt=

xt+_µ(A)1 a(B)ift∈A et ift∈B. 6 _{In the sense that}

{(t,x,y)∈T×E×E : x,y∈Xt and y∈Pt(x)} ∈Σ×B(E)×B(E).

7 _{In the sense that for everyx∈X}

t,x6∈Pt(x). 8 _{In the sense that for everyx,y∈X}

t, ify∈Pt(x)thenPt(y)⊂Pt(x). 9 _{In the sense that for everyx∈X}

t, the setPt(x)and the setPt−1(x):={y∈Xt:x∈Pt(y)}are

open inXt.

10 _{In the sense that for everyx,y∈X}

t, ify6=0 thenx+y∈Pt(x).

11 _{Einy et al. (2001) proposed an extension of Hildenbrand’s irreducibility condition to}

economies with differential information which is slightly different from the one we introduced below.

12 _{IfJis a subset ofIthen}V

We define the functionz:T →Eby

∀t∈T, zt=

et ift∈A et−at ift∈B.

Observe that the functionsyandzbelong toS_{X and satisfy}

Z

Axd µ+

Z

Bed µ=

Z

Ayd µ+

Z

Bzd

µ and yt∈Pt(xt) forµ-a.e.t∈A.

⊓ ⊔

Remark3.2. Assume that forµ-a.e.t∈T, we haveet(ω)∈RL++for everyω∈Ω.

Then13 _{there exists a measurable functionε(ω):T →(0,1] such thatet(ω)≥}

εt(ω)1L.14Letat be the vector inEdefined by

∀ω∈Ω, at(ω) =

min ω∈Ωεt(ω)

1L,

then we haveat∈E+c,at6=0 andet≥at, implying that the economy is irreducible.

4 Competitive allocations

At the first periodτ=0 there is a complete set of contingent contracts for future

delivery of each good. Therefore a price system is a functionp:Ω →RL _where

p(ω, ℓ)represents the price atτ=0 of the contract delivering one unit of goodℓ

if the state of nature atτ=1 isω. The budget setBt(p)for agenttis then defined by

Bt(p):={xt∈Xt : E[p·xt]6E[p·et]} where

∀x∈E, E_[p·x]:=

∑

ω∈Ω

p(ω)·x(ω).

We denote byE_[p|F_{t]the function fromΩto}RL_{+defined by}

∀ω∈Ω, E_[p|F_{t](ω) =} 1

#Et(ω)_σ∈

∑

t(ω)

p(σ).

This function is called the conditional price with respect toF_{t or agentt’s}

con-ditional price. Since agentt’s choices are constrained by his information, what

matters is the functionE_[p|F_{t]in the sense that}

Bt(p) =Bt(E[p|Ft]).

In this section we extend to our model the definition of competitive equilibrium (or Walrasian expectations equilibrium) introduced by Radner (1968), and discuss conditions under which its existence can be guaranteed.

13 _{We can apply the von Neumann–Aumann measurable selection Theorem (see Aumann}

Au-mann (1965) or Castaing–Valadier Castaing and Valadier (1977, Theorem III.22)).

14 _{IfAis a subset ofLthen1}

Adenotes the vector inRLdefined by1A(ℓ) =1 ifℓ∈Aand 0

Definition 4.1. A pair(x,p)of a private assignmentx∈S_{Xand pricep:Ω→}RL

is a competitive equilibrium if

(a) forµ-a.e.t∈T, the planxt belongs to the budget setBt(p)and is optimal in

the sense thatBt(p)∩Pt(xt) =/0; and

(b) x is feasible, i.e., for each possible realization ω ∈Ω, markets clear in the

sense that

∀ω ∈Ω, Z

T

xt(ω)µ(dt) = Z

T

et(ω)µ(dt).

A competitive allocation is a feasible private assignmentx∈S_{X for which there}

exists a pricepsuch that(x,p)is a competitive equilibrium.

Remark4.1. If(x,p)is competitive equilibrium of a standard economy then the following properties are satisfied:

1. forµ-a.e.t∈T the budget set restriction is binding, i.e.E_{[p·xt] =}E_[p·et];

2. conditional prices are strictly positive, i.e.,

∀i∈I, ∀ω∈Ω, E_[p|Fi_](ω)∈RL_{++. (1)}

IfS

i∈IFi=2Ωthen (1) implies that for everyω∈Ω, the spot pricep(ω)∈RL++. This property is not true in general: Einy and Shitovitz (2001, Example 2.1) provide an example of a standard economy with differential information (and

finitely many agents) for which there is no competitive equilibrium (x,p)

satis-fyingp(ω)∈RL_{+for everyω∈Ω.}

In Einy et al. (2001) it is the definition of competitive equilibrium with free-disposal (used by Radner (1982)) that was extended to large economies.

Definition 4.2. A pair(x,p)of a private assignmentx∈S_{Xand pricep:Ω→}RL₊

is a competitive equilibrium with free-disposal if it satisfies the previous property (a) together with the following

(b’) xis free disposal feasible, i.e., for each possible realizationω ∈Ω, markets

clear with free-disposal in the sense that

∀ω∈Ω, Z

T

et(ω)µ(dt)−

Z

T

xt(ω)µ(dt)∈RL_+. Remark4.2. Observe that property (b’) can be rewritten as

∃z∈E∩RΩ₊×L, Z

Txt

µ(dt) +z= Z

Tet µ(dt).

In particular the planzis not required to be compatible with the information

avail-able in the market, i.e., it is not imposed thatzbelongs toE+. Observe moreover

that in the definition of a competitive equilibrium, every spot price p(ω) is

re-quired to be nonnegative.

It was proved in Einy et al. (2001, Theorem A) that every irreducible economy has a competitive equilibrium with free-disposal. We may think that in the context of a pure exchange economy with differential information it is more reasonable to assume inequality in the feasibility constraints since it allows to prove the

each agentt∈T can only makeF_{t-measurable plans, it seems natural to only}

re-quire that the conditional priceE_[p|F_{t]with respect to the available information}

F_{t is nonnegative, i.e., for eacht∈T, the vector}E_[p|F_{t](ω)belongs to}RL_{+. We}

assert that if we replace the requirement thatevery equilibrium prices p(ω)are

nonnegativeby the requirement thatall conditional prices are nonnegative, then it is possible to prove that every irreducible economy has a competitive equilibrium (with an equality in the feasibility constraints).

Theorem 4.1. Every irreducible economy has a competitive equilibrium.

The existence result in Einy et al. (2001) follows as a corollary of the ex-istence result in Hildenbrand (1974) which is based on a multidimensional Fa-tou’s Lemma provided by Schmeidler (1970). In Schmeidler’s version of FaFa-tou’s

Lemma the positive cone is the lattice coneRn

+of a finite dimensional Euclidean

vector spaceRn_{. In order to deal with our positive coneE}

+ we propose a proof

which relies on a generalization of Hildenbrand’s existence result provided by Cornet et al. (2003). The later existence result is based on a generalization of Schmeidler’s version of Fatou’s Lemma due to Balder and Hess (1995).

Proof of Theorem 4.1. Let E _{= (}F_{t,Pt,et)t∈T be an irreducible standard}

econ-omy. We recall thatE+denotes the cone∑i∈IE+i andE denotes the linear space

generated byE+.

Claim 4.1. The cone E+is convex, pointed and closed in E.

Proof of Claim 4.1. It is straightforward to check thatE+is convex and pointed.

Let (an)be a sequence in E+ converging toa inE. For eachn∈N, there

ex-ists(ain)i∈I∈∏i∈IE+i such thatan=∑i∈Iain. Hence for eachi∈I, we have 06 ain(ω, ℓ)6an(ω, ℓ)for every(ω, ℓ)∈Ω×L. Since the sequence(an(ω, ℓ)) con-verges toa(ω, ℓ)then the sequenceain(ω, ℓ)is bounded. Passing to a subsequence if necessary, we can assume that there exists(ai)i∈I∈∏i∈IE+i such that the

se-quence(ain)converges toai. Thereforea=∑i∈Iaibelongs toE+. ⊓⊔

Following the notations in Cornet et al. (2003), we consider the following coalitional production economy

E_{C={E,(T,Σ,µ),(Xt,≻t,et,Y(t))t∈T}}

whereC=E+,≻t is the binary relation defined byPt andY(t) =−Cfor every

t∈T. Applying Corollary 3.1 in Cornet et al. (2003)15there exists a triple(x,z,π)

wherex∈S_{X is a private assignment,z∈ −Candπ:E→}R_{is a non-zero linear}

functional such that

(a) forµ-a.e.t∈T,π(xt)6π(et)andy∈Pt(xt)impliesπ(y)≥π(xt); (b) z∈argmax{π(z′): z′∈ −C}; and

(c) R

Txdµ= R

Tedµ+z.

15 _{In Cornet et al. (2003) the spaceEis an Euclidean space}RH_{for some finite setH. But their}

It follows from (c) thatzbelongs to−C=−E+. In particular there exists(zi)i∈I such thatzi∈ −E+i andz=∑i∈Izi. Moreover it follows from (b) thatπ|E+≥0 in the sense that for everyy∈E+, we haveπ(y)≥0. In particular we haveπ(zi)60

for eachi∈I andπ(z)60. But from (b), we must haveπ(z)≥0, implying that

actuallyπ(z) =0 andπ(zi) =0 for eachi. It follows then from conditions (a) and (c) that forµ-a.e.t∈T, we haveπ(xt) =π(et).

Sincee(T)belongs toRΩ₊₊×L_{then it belongs to intEE+, the relative interior of}

E+inE. SinceE=E+−E+, we must haveπ|E+6=0, implying that

π(e(T))>0.

LetA={t∈T:π(et)>0}, this set belongs toΣ andµ(A)>0. Claim 4.2. For every t∈A, y∈Pt(xt)impliesπ(y)>π(et).

Proof. Lett∈Aandy∈Pt(xt). From property (a) we already know thatπ(y)≥

π(et). Assume by way of contradiction thatπ(y) =π(et). SincePt(xt)is open in Xtthere existsα∈(0,1)such thatαy∈Pt(xt). Then applying property (a) we get

απ(y)≥π(et). This yields a contradiction sinceπ(et)>0. ⊓⊔

Claim 4.3. The set A is of full measure, i.e.µ(A) =1.

Proof of Claim 4.3. Assume by way of contradiction thatB:=T\Ais such that µ(B)>0. Let ˜x:T →E+be defined by

∀t∈Ti, x˜t=xt− 1 µ(Ti₎z

i_.

The function ˜xis a private assignment, i.e. ˜x∈S_{X, it satisfiesπ(x˜t) =π(xt) =}

π(et),Pt(x˜t)⊂Pt(xt)and it is feasible. Then applying the irreducibility condition

to ˜x, there exist two private assignmentsy,w∈S_{X such that}

Z

A ˜

xdµ+ Z

B edµ=

Z

A ydµ+

Z

B

wdµ and yt∈Pt(x˜t) forµ-a.e.t∈A.

Sinceπ(et) =0 forµ-a.e.t∈B, we get Z

A

π(x˜t)µ(dt) = Z

A

π(yt)µ(dt) + Z

B

π(wt)µ(dt).

Sinceyt∈Pt(x˜t)we get from Claim 4.2 thatπ(yt)>π(et) =π(x˜t)forµ-a.e.t∈A,

and sinceπ|E+≥0 we haveπ(wt)≥0. This implies that

Z

A

π(x˜t)µ(dt) = Z

A

π(yt)µ(dt) + Z

B

π(wt)µ(dt)> Z

A

π(x˜t)µ(dt)

Since preference relations are strictly monotone, we get from Claims 4.2 and 4.3 that

∀i∈I, ∀y∈E₊i , y6=0=⇒π(y)>0.

Sinceπ(zi) =0 we get that for eachi∈I,zi=0 andz=0. We have thus proved

that the feasibility constraints is satisfied with an (exact) equality, i.e.,R

Txdµ= R

Tedµ.

The linear functionalπ can be extended toRΩ×L_{. In particular, there exists a}

functionp:Ω→RL_{such that}

∀y∈E, π(y) =E_[p·y].

It is now straightforward to prove that(x,p)is a competitive equilibrium ofE_{. ⊓⊔}

5 Core allocations

Yannelis (1991) adapted the standard concept of the core to economies with dif-ferential information by defining the private core.

Definition 5.1. A feasible private assignmentx∈SX is aprivate core allocation

for the economyE _{if there do not exist a coalitionS∈Σ withµ(S)>0 and a}

private assignmentysuch thatyis feasible for the coalitionS, i.e.

Z

S

ytµ(dt) = Z

S etµ(dt)

and forµ-a.e.t∈S, the planytis strictly preferred toxt, i.e.yt∈Pt(xt).

The equivalence theorem in Aumann (1964) still prevails in the framework of differential information. The proof follows the standard arguments in Hildenbrand (1974) and is omitted.

Theorem 5.1. The sets of competitive and private core allocations coincide.

We know adapt the definition of the weak fine core introduced by Koutsougeras and Yannelis (1993) to the framework of exact feasibility constraints. We first

observe that without any loss of generality, we can assume that the coarsestσ

-algebra containing eachFi_{coincides with 2}Ω_{. Now ifJis a subset ofI, we denote}

byF_{(J)the coarsestσ-algebra containing each}Fj_{, j∈J. In particular we have}

F_{({i}) =}Fi_{andF(I) =2}Ω_{. Similarly, ifS∈Σis a coalition then we denote by} F_{(S)theσ-algebra}F_(I(S))where

I(S):={i∈I : µ(Ti∩S)>0}.

The setI(S)represents the informational types that are present in the coalitionS

and theσ-algebraF_{(S)represents the information available to each agent ofSif}

they share their information.

Definition 5.2. A feasible assignment x is aweak fine core allocation for the

economyE _{if there do not exist a coalitionS∈Σ and an assignmentysuch that}

2. the assignmentyis feasible for the coalitionS, i.e.

Z

S

ytµ(dt) = Z

S

etµ(dt),

3. forµ-a.e.t∈S, the planytis strictly preferred toxt, i.e.yt∈Pt(xt).

Observe that no measurability constraints are imposed on a weak fine core allocation. As in Einy et al. (2001), if we can extend the preference relations then

the weak fine core of an economyE _{coincides (and is thus non-empty) with the}

private core of the symmetrized economyE∗_.

Definition 5.3. The preference relations of an economyE _{are saidextendableif} forµ-a.e.t∈T, there exists a multifunctionPt∗fromRΩ+×LintoRΩ+×Lsuch that

1. the preference relations defined byP∗ are measurable, irreflexive, transitive,

continuous and strictly monotone;

2. the correspondenceP_t∗ extends Pt in the sense that Pt(x)⊂Pt∗(x)for every x∈Xt.

Remark5.1. If the preference relations are represented by expected utility func-tions then they are automatically extendable.

Definition 5.4. IfE _{is an economy with extendable preference relations, then we}

letE∗_{be the economy defined by}

E∗_{= (}F∗

t ,Pt∗,et)t∈T

whereF∗

t =2Ωfor everyt∈T. The economyE∗is called the symmetrization of

E_.

Observe that the economyE∗_{is symmetric in the sense that every agent has}

the same information. The proof of the following theorem is based on a result by Vind (1968) and follows almost verbatim the proof of Proposition 5.1 in Einy et al. (2001).

Theorem 5.2. IfE_{is an irreducible economy with extendable preferences then the}

weak fine core ofE _{coincides with the private core of the symmetrized economy} E∗_.

Remark5.2. It follows as a corollary of Theorems 4.1, 5.1 and 5.2 that the weak fine core of an irreducible economy with extendable preferences is non-empty.

6 Contract enforcement and incentive compatibility

A competitive allocation as well as a private core allocation are ex-ante solutions

corresponding to actions taken atτ=0. In order to address the issue of execution

(or enforcement) of contracts atτ=1, we assume that there is an intermediary

(a ”government institution” or a ”market institution”) that is responsible for the

execution of contracts. In this section we assume that the family(Fi_)i∈Iis

Atτ=1 a state of nature is realized. If the intermediary is able to identify the true state then he can enforce the receipts and deliveries of commodities specified

by the contracts made at the previous date, i.e., each agenttreceives the net trade

zt(ω):=xt(ω)−et(ω). This is possible since it follows from the (exact) feasibility

constraints that _Z

t∈Tzt

(ω)µ(dt) =0.

This implies that they are no issues concerning execution of contracts.

More interesting is the situation where the intermediary has an incomplete

information concerning the true state of nature. In that case, each agentt has to

report his information and claims for the corresponding net trade. However, agents

may have incentives to misreport their information. Ifω is the realized state of

nature, agenttshould report his informationEt(ω), but if for some other stateσt

we have

xt(σt)−et(σt)+et(ω)∈RL+ and ut(ω,xt(σt)−et(σt)+et(ω))>ut(ω,xt(ω))

then agenttgains by reporting the (uncorrect) informationEt(σt). If

Z

t∈T

zt(σt)µ(dt)6=0

then the intermediary cannot execute contracts. In general it is not possible to avoid such a situation. Now assume that there is a legal procedure that agents can use to prove that they are not misreporting their information. Assume that this procedure is costly but that the cost of such an action is repaid by agents whose misreporting can be revealed. This has two consequences: first an agent uses this legal procedure only if he is sure that he can detect a misreporting and second an agent decides to misreport only if he is sure that he cannot be detected by other agents. This leads us to the following definitions.

Definition 6.1. Letω∈Ω, ifi∈Iis an information type, then we letEi(ω)be

the atomic event inFi_{which containsω. IfJ⊂Ithen we denote byF}J_{the meet}

σ-algebra∧j∈JFjwhich is interpreted as thecommon knowledgeinformation of

the coalitionJof types. Ifω ∈Ω we letEJ(ω)be the atomic event inFJ_which

containsω.

Definition 6.2. A private assignmentx∈S_{X is said coalitional incentive}

com-patible if there is no coalition S∈Σ with µ(S)∈(0,1)that has an incentive to

misreport a state of nature. A coalitionS∈Σhas an incentive to misreport a state

of nature if there exist statesω6=σ such that

1. forµ-a.e.t6∈S, for every a∈EI(S)(ω),16 the statesσ andaare not

distin-guishable, i.e.,σ∈Et(a); in other words

σ∈ \

a∈EI(S)_(ω) \

{Ej(a): j∈I(T\S)}; (2)

16 _{We recall that for each coalitionE∈Σ, the setI(E)is the set of information types present}

in the coalition, i.e.

2. forµ-a.e.t∈S, we have

∀a∈EI(S)(ω),

    

   

et(a) +xt(σ)−et(σ)∈RL+ and

ut(a,et(a) +xt(σ)−et(σ))>ut(a,xt(a)). (3)

The interpretation is as follows. Assume that the realized state of nature isω. A

coalitionShas an incentive to misreport by announcing the stateσ17_{if it increases}

each agent’s utility (condition (3)) and if the misreport cannot be detected by the

agents inT\S(condition (2)). We assume that agents do not want to share or to

reveal information, therefore when agents in a coalitionSagree in an action, they

have to do so on common knowledge events. More precisely, ifω is the realized

state of nature then each agentt∈Sobserves the eventEt(ω). In order to misreport

they have first to agree on what is the set of possible states of nature. Since they

do not share information, this set isEI(S)(ω). Now if they decide to announce the

stateσ, they must check if any agentτ6∈Scan detect thatσ is not the realized

state of nature. But to do so, agents in the coalitionSneed to agree on what is

agentτ’s information. The common knowledge of the coalitionSis that any state

a∈EI(S)(ω) is a possible candidate for the realized of nature. Therefore, they

must check that for everya∈EI(S)(ω), every agentτ6∈Scannot discern statesa

andσ, i.e.

σ∈ \

a∈EI(S)_(ω) \

{Ej(a) : j∈I(T\S)}.

Similarly, when agents in a coalitionSagree to misreport by announcing the state

σ, it should be the case that almost every agentt∈Sgains by doing so for every

possible state inEt(ω)according to his private information, i.e.,

    

   

et(ω) +xt(σ)−et(σ)∈RL+ and

ut(ω,et(ω) +xt(σ)−et(σ))>ut(ω,xt(ω))

but also for the other states in the common knowledge eventEI(S)(ω), i.e.,

∀a∈EI(S)(ω),

    

   

et(a) +xt(σ)−et(σ)∈RL+ and

ut(a,et(a) +xt(σ)−et(σ))>ut(a,xt(a)).

In other words every agent of the coalition must discern that every agent of the coalition gains by misreporting for every possible state every agent in the coalition can discern. Otherwise some agents may reveal information to other agents of the coalition about what are the possible states of nature.

Remark6.1. A necessary condition for a coalitionS∈Σ to have an incentive to misreport a state is that

I(S)∩I(T\S) =/0. (4)

Condition (4) comes from the fact that an agentt with information typeicannot

misreport a state to another agentt′whose information type is the same.

Remark6.2. A cautious reader will notice that our concept of coalitional incentive compatibility is slightly different than the weak coalitional Bayesian incentive compatibility introduced in Koutsougeras and Yannelis (1993) (see also Herv`es-Beloso, Moreno-Garc´ıa and Yannelis (2005)) where it is imposed that in order to

misreport stateω by announcingσ, agents in a coalitionSshould have the same

information about the realized state of nature, i.e., for a.e.t∈S

Et(ω) =EI(S)(ω).

We consider that coalition could misreport even if they do not have the same infor-mation. What matters is that they agree to misreport on the common knowledge information. In Krasa and Yannelis (1994) it is condition (3) that is used (see Sec-tion 3.4 before Lemma 2). But condiSec-tion (2) is replaced by

σ∈\

{Ej(ω) : j∈I(T\S)}.

Definition 6.3. A feasible assignmentx∈S_{Xis saidPareto optimalif there does}

not exist a feasible assignmenty∈SX such that

µ{t∈T:ht(yt)≥ht(xt)}=1 and µ{t∈T: ht(yt)>ht(xt)}>0. Remark6.3. It is straightforward to check that competitive and private core allo-cations are Pareto optimal.

The main motivation in considering (exact) equality in the feasibility con-straints is the theorem below where we show that if free disposal is not allowed in the feasibility constraints then every Pareto optimal assignment is coalitional incentive compatibility. A straightforward consequence is that contracts of every competitive or private core allocations are enforceable.

Theorem 6.1. Every (exact) feasible Pareto optimal assignment (and thus every competitive or private core allocation) is coalitional incentive compatible.

Proof of Theorem 6.1. Letx∈S_{X be a feasible assignment that is Pareto optimal}

and assume by way of contradiction that there exist a coalitionS∈Σwithµ(S)∈

(0,1)and two statesξ6=σ such that

1. forµ-a.e.t6∈S, for everya∈EI(S)(ξ), the statesaandσ are not distinguish-able, i.e.σ∈Et(a);

2. forµ-a.e.t∈S, for everya∈EI(S)(ξ), we have

wherezt(ω):=xt(ω)−et(ω)for everyω∈Ω.

We consider now the functiony:T →RΩ×L_{defined by}

∀t∈S, ∀ω∈Ω, yt(ω) =

et(ω) +zt(σ)ifω∈EI(S)(ξ) xt(ω) ifω6∈EI(S)(ξ) and

∀t6∈S, ∀ω∈Ω, yt(ω) =xt(ω).

The functionytisFt-measurable forµ-a.e.t∈T. Moreover for eachω∈Ω, the

vectoryt(ω)belongs toR+L. Thereforeyis a private assignment, i.e.y∈SX. We

haveut(ω,et(ω) +z(σ))>ut(ω,xt(ξ))forµ-a.e.t∈Sand everyω∈EI(S)(ξ), which implies thatht(yt)>ht(xt). We have thus proved that

µ{t∈T:ht(yt)≥ht(xt)}=1 and µ{t∈T:ht(yt)>ht(xt)}=µ(S)>0.

In order to get a contradiction, it is now sufficient to prove thaty is a feasible

assignment. We first have that

∀ω∈Ω, Z

T\S

y(ω)dµ= Z

T\S

x(ω)dµ,

and

∀ω 6∈EI(S)(ξ), Z

Sy

(ω)dµ= Z

Sx (ω)dµ

implying that for everyω6∈EI(S)(ξ)we have

Z

T

y(ω)dµ= Z

S

x(ω)dµ= Z

T

e(ω)dµ.

Now fixa∈EI(S)(ξ)then

Z

Sy

(a)dµ= Z

Se

(a)dµ+ Z

Sz (σ)dµ.

Recall thatxis a feasible allocation, in particular

Z

S

z(a)dµ+ Z

T\S

z(a)dµ=0= Z

S

z(σ)dµ+ Z

T\S

z(σ)dµ.

But for µ-a.e.t6∈Swe have thatσ ∈Et(a)and thenzt(a) =zt(σ). As a conse-quence we get that

Z

S

z(a)dµ= Z

S z(σ)dµ

which implies that for eacha∈EI(S)(ξ),

Z

S

y(a)dµ= Z

S

{e(a) +z(σ)}dµ= Z

S

{e(a) +z(a)}dµ= Z

S

x(a)dµ.18

⊓ ⊔

Remark6.4. Observe that the exact feasibility constraints plays a crucial role in the proof of Theorem 6.1.

18 _{In this argument it is crucial that for every information type j∈I(T\S)and for every}

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Krasa and Yannelis (1994) it is only assumed that for every information typej∈I(T\S)agents with information type jcannot discernσandξ. In other words in Krasa and Yannelis (1994) condition (3) is replaced by

σ∈\

{Ej_{(ξ): j∈I(T\S)}. (5)}