Munich Personal RePEc Archive
Edgeworth equilibria: separable and
non-separable commodity spaces
Bhowmik, Anuj
Economic Research Unit, Indian Statistical Institute, 203
Barackpore Trunk Road, Kolkata 700108, India, School of
Computing and Mathematical Sciences, Auckland University of
Technology, Private Bag 92006, Auckland 1142, New Zealand
7 May 2013
Online at
https://mpra.ub.uni-muenchen.de/46796/
NON-SEPARABLE COMMODITY SPACES
ANUJ BHOWMIK
Abstract. Consider a pure exchange differential information economy with an atomless measure space of agents and a Banach lattice as the commodity space. If the commodity space is separable, then it is shown that the private core coincides with the set of Walrasian expectations allocations. In the case of non-separable commodity space, a similar result is also established if the space of agents is decomposed into countably many different types.
1. Introduction
One of the classical result in economic theory is Aumann’s equivalence theorem in a deterministic economy with a contimuun of agents and finitely many commodi-ties, see [2]. Many extensions of this result have been obtained in the literature. Firstly, an extension of this result to an economy with an atomless measure space of agents and finitely many commodities can be found in [14]. In the context of infinite dimensional commodity space, the relation between the core and the set of Walrasian allocations are more interesting, since preferences and endowments are more diverse, and thus blocking become more difficult, refer to [11]. Rustichini and Yannelis [21] extended this result to an economy whose commodity space is a separable Banach lattice. In [2], Aumann also pointed out that many real markets are indeed far from being perfect; such a market is probably best represented by a mixed model, in which some agents are points in a continuum and others are indi-vidually significant. One of the key results on the equivalence between the core and the set of Walrasian allocations in a mixed economy was established by Shitovitz in [22]. To be precise, he showed that if there exist at least two large agents and all of them have the same initial endowment and preference, then the core coincides with the set of Walrasian allocations. Similar results in mixed economies also came out in [6, 9, 10]. In all of these results, the feasibility was defined without free disposal and in terms of Bochner integrable functions. In contrast to so far mentioned positive results, Podczeck [17] and Tourky and Yannelis [23] constructed counterexamples of atomless economies to show that the classical core-Walras equivalence theorem in [2] may fail under desirable assumptions when the commodity space is a non-separable ordered Banach space and the feasibility is defined by Bochner integrable functions. However, when feasibility is defined in terms of Pettis integral, Podczeck [69] obtained a positive result for a ceratin class of commodity spaces without re-quiring that those commodity spaces are separable.
JEL classification: D41; D51; D82.
Keywords.Differential information economy; Extremely desirable bundle; Private core.
In [20], Radner extended the notion of Walrasian allocation to the case of dif-ferential information and it is known as Walrasian expectations allocation. Due to different information and communication opportunities among agents, several alternative core concepts have been introduced, refer to [25, 27]. In [27], Yannelis introduced the notion of private core, which was based on the fact that agents have no access to the communication system, that is, each member of the coalition uses only his own private information whenever a coalition blocks an allocation. It is also essential to mention that under standard assumptions, the private core is non-empty, Bayesian incentive compatible and rewards the information superiority of agents (see [15, 27]). Dealing with differential information, Einy et al. [7] first extended Aumann’s equivalence theorem to the case of the private core and the set of Walrasian allocations, where the free disposal feasibility assumption was used. Later, this result was further generalized to a differential information economy with an atomless measure space of agents and an ordered separable Banach space having an interior point in its positive cone as the commodity space in [8]. In addition to these equivalence results with free disposal, Angeloni and Martins-da-Rocha [1] obtained an equivalence result between the private core and the set of Walrasian allocations in an atomless economy with finitely many commodities and without free disposal feasibility assumption.
The paper is organized as follows. In Section 2, a general description on the model of this paper is given. In Section 3, the main result of this paper is presented, where an extension of the main result in [21] to a differential information economy with a separable commodity space and the free disposal feasibility assumption is given. Section 4 deals with the equivalence theorem in an economy with a non-separable commodity space and an atomless measure space of agents with only countably many different types.
2. Differential information economies
An atomless model of pure exchange economyE with differential information is
presented. The exogenous uncertainty is described by a measurable space (Ω,F),
where Ω is a set of states of nature containingmelements and theσ-algebraF of
Ω denotes the set of possible events. The economy extends over two time periods
τ= 0,1. Consumption takes place atτ= 1. Atτ= 0, there is uncertainty over the states and agents make contracts that are contingent on the realized state atτ= 1. Let the space of agents be a measure space (T,Σ, µ) with a complete, finite and positive measureµ, where T is the set of agents, Σ is a σ-algebra of measurable subsets of T whose economic weights on the market is given by µ. Throughout the paper, the commodity spaceY ofE is a Banach lattice having a quasi-interior
point. The order on Y is denoted by a partial order ≤ and the positive cone
Y+ ={x∈ Y : x≥0} of Y denotes theconsumption set of each agent t in each stateω.The symbol x ≫ 0 (resp. x > 0) means that xis a quasi-interior (resp. a non-zero) point of Y+. Let Y++ = {x ∈ Y+ : x ≫ 0}. Each agent has some private information, which is described by a partition Πtof Ω. The interpretation
is that ifωis the true state of nature then agenttcannot discriminate the states in the unique element Πt(ω) of Πt containingω. LetFtbe theσ-algebra generated
by Πt. Each agent t is also associated with a state-dependent utility function
called thecharacteristicsof the agentt∈T. Thus,E can be defined by E ={(Ω,F); (T,Σ, µ); Y+; (Ft, Ut, a(t,·),Qt)t∈T}.
A functionx: Ω→Y+ is interpreted as a random consumption bundle inE. The ex ante expected utility of an agent t for a given random consumption bundle x
is defined byEQt(U
t(·, x(·))) =Pω∈ΩUt(ω, x(ω))Qt(ω). An assignment in E is a
functionf :T×Ω→Y+ such thatf(·, ω) is Bochner integrable for allω∈Ω. It is assumed thatais an assignment. An assignment f in E is called an allocation if
f(t,·) ∈Lt µ-a.e., whereLt =
x∈(Y+)Ω:xisF
t-measurable . An assignment
isS-feasible ifRSf(·, ω)dµ≤RSa(·, ω)dµfor all ω ∈Ω. This feasibility condition is also known asS-feasibility with free disposal. However, if the last inequality is replaced with an equality, then it is named asS-feasibility without free disposal or
S-exact feasibility. For simplicity, if f is T-(exactly) feasible then it is termed as (exactly) feasible. Throughout the paper, the following assumption on the initial endowments is posed.
(A1)a(t,·)∈Ltanda(t, ω)≫0 for all (t, ω)∈T×Ω.
This is a standard assumption and has been used in many references, see [3, 4, 8, 12]. LetPdenote the family of all partitions of Ω. For any Q∈P, letTQ={t∈T :
Πt = Q}. It is assumed that TQ ∈ Σ for all Q ∈ P. For any coalition S, put
PS = {Q ∈ P : S ∩TQ 6= ∅} and P(S) ={Q ∈ PS : µ(S∩TQ) > 0}. Since
Lt=Lt′ ift, t′ ∈TQ, the notationLQis employed to denote the common value of
Ltfor allt∈TQ. For any allocation f, define a function Pf :T →Y+Ωby
Pf(t) =
x∈Lt:EQt(x)>EQt(f(t,·)) .
For anyn≥1, the (n−1)-simplex ofRn is defined as
∆n =
(
x= (x1, ..., xn)∈Rn+:
n
X
i=1
xi= 1
)
.
Consider a function ϕ: (T,Σ, µ) →∆m defined by ϕ(t) = Q
t for allt ∈ T. For
each ω ∈ Ω, define a function ψω : T ×Y+ → R by ψω(t, x) = Ut(ω, x). The
following assumptions on the priors and the state-dependent utilities of agents will be needed, the last two of which can be found in [3, 4, 8].
(A2) For each (t, ω) ∈ T ×Ω, Ut(ω,·) : Y+ → R is monotone in the sense that
Ut(ω, x+y)> Ut(ω, x) ifx, y∈Y+ withy >0.
(A3) The functionϕis measurable, where ∆mis endowed with the Borel structure.
(A4) For eachω∈Ω, the functionψωis Carath´eodory, that is,ψω(·, x) is
measur-able for allx∈Y+, andψω(t,·) is norm-continuous for allt∈T.
The graph ofPfis defined by GrPf =
(t, x)∈T ×YΩ
+ :x∈Pf(t) , which is the
in-tersection ofS{TQ×LQ:Q∈PT}and(t, x)∈T×Y+Ω:EQt(x)>EQt(f(t,·)) .
Under (A3) and (A4), the function (t, x)7→EQt(x) is Carath´eodory. Thus, the set
(t, x)∈T×YΩ
+ :EQt(x)>EQt(f(t,·)) is Σ⊗B(YΩ)-measurable, whereB(YΩ) is the Boreσ-algebra ofYΩ. This further implies Gr
Pf ∈Σ⊗B(YΩ).
As usual, a coalition of E is a set S ∈ Σ with µ(S)>0. If S and S′ are two
coalitions of E with S′ ⊆ S, then S′ is called a sub-coalition of S. Further, a
coalitionS privately blocks an allocation f inE if there is anS-feasible allocation
ofE, denoted byP C(E), is the set of feasible allocations which are not privately
blocked by any coalition. Aprice system is a non-zero functionπ: Ω→Y∗
+, where
Y∗
+ is the positive cone of the norm-dualY∗ ofY. Thebudget set of agenttwith respect to a price systemπis defined by
Bt(π) ={x∈Lt:E[hx, πi]≤E[ha(t,·), πi]}.
where for allx∈YΩ +,
E[hx, πi] =X
ω∈Ω
hx(ω), π(ω)i.
A Walrasian expectations equilibrium of E is a pair (f, π), where f is a feasible
allocation andπ is a price system such thatf(t,·)∈argmaxEQt(x) :x∈B t(π)
µ-a.e., and
E
Z
T
f dµ, π
=E
Z
T
adµ, π
.
In this case, f is called a Walrasian expectations allocation and the set of such allocations is denoted by W(E). It is well known that W(E) ⊆ P C(E). The
opposite inclusion requires some assumptions on the characteristics of agents and it will be derived in the next two sections.
3. The Edgeworth Equilibria with Separable Commodity Spaces
In this section, it is assumed that Y is separable. When intY+ 6=∅, the equiv-alence betweenW(E) and P C(E) was established in [8] under (A1)-(A4). The purpose of this section is to explore this result in an economy withY+has an empty interior. However, such a result is not true under (A1)-(A4), in general, as follows from the following example in [21].
Example 3.1. Consider the deterministic economy
E =(T,Σ, µ);ℓ+
2; (Ut, a(t))t∈T
where (i) T = [0,1] and Σ is the σ-algebra of Lebesgue measurable subsets of T
with the Lebesgue measureµ; (ii)ℓ+2 is the consumption set of each agent, where
ℓ+2 is the positive cone of the space ℓ2 (the space of real sequences {an : n ≥ 1}
equipped with the normk{an : n≥1}k2= (Pn≥1|an|2)
1
2 <∞); and (iii) for all
t∈T, Ut(x) =Pn≥1n−2(1−exp(−n2xn)) and a(t) =
1
n2 :n≥1 . Let We(E)
andP Ce(E) be the set of Walrasian expectations allocations and the private core
when theS-exact feasibility condition is used. Assume (A1)-(A4). So, We(E) =∅
andP Ce(E) = {a}, refer to [21]. It is now claimed that P C(E) ={a}. To see
this, let f ∈ P C(E). By (A2), one can show that RTf dµ = RTadµ and thus,
f ∈P Ce(E). This implies thatf =aand the claim is verified. If a∈W(E), the
only candidate as a supporting price for the allocationais the multiple of (1,1, ...) which are not inℓ2
+. So,W(E) =∅.
To establish the equivalence theorem, similar assumptions and techniques used in [21] are employed in the rest of this section.
Definition 3.2. [21, 26] Letω ∈Ω,v > 0 andU be an open convex solid neigh-borhood of 0 inY. Suppose thatC is the open cone spanned byv+U. The bundle
vis called anextremely desirable bundle with respect toU at stateωifx∈Y+and
(A5) For eachω∈Ω, there is a v(ω)>0 such thatv(ω) is an extremely desirable bundle with respect to some open convex solid neighborhoodU(ω) of 0 inY. (A6) Suppose that δ1, ..., δm are positive numbers with Pim=1δi = 1. If xi ∈ Y+ andxi∈/δiU(ω) for all 1≤i≤m, then Pmi=1xi∈/ U(ω).
LetU = m1 Tω∈ΩU(ω)m, andCandC(ω) be the open convex cones spanned by
P
ω∈Ωv(ω)1Ω+U andv(ω) +U(ω) respectively for allω∈Ω.
Theorem 3.3. Assume (A2)-(A6) and that f ∈P C(E). Let g:S×Ω→Y+ be
defined byg(t, ω) =yi(ω)if(t, ω)∈Si×Ω, where for each1≤i≤mthere is some
Q∈P(S)such that Si ⊆S∩TQ andµ(Si) =η. Assume further thatg(t,·)∈Lt
andEQt(g(t,·))>EQt(f(t,·))µ-a.e. onS. Then R
S(g−b)dµ /∈ −C, where
b= m X i=1 1 η Z Si adµ
χSi.
Proof. Assume the contrary. Then
m
X
i=1
(yi−ai)η∈ −α
X
ω∈Ω
v(ω)1Ω+U
!
,
whereai=η1
R
Siadµandα >0. So there is an elementw∈ α
ηU such that m
X
i=1
yi+u+w= m
X
i=1
ai ≥0,
whereu= α η
P
ω∈Ωv(ω)1Ω. Since
Pm
i=1yi+u≥0 andU is solid, one hasw−∈ αηU
andPmi=1yi+u≥w− . For anym-tupleσ= (σ1,· · ·, σm) of positive real numbers
withPmi=1σi = 1,
w−≤
m
X
i=1
(yi+σiu).
By the Riesz decomposition property, one obtains a finite set{wσ
1,· · ·, wmσ} such
that w− = Pm
i=1wσi and 0 ≤ wiσ ≤ yi +σiu for all 1 ≤ i ≤ m. Let IQ = {i :
Si ⊆ S ∩TQ} for all Q ∈ P(S). Pick an i ∈ IQ and note that yi+σiu is Q
-measurable. Definedσ
i : Ω→Y+ bydσi(ω) = sup{wσi(ω′) :ω′ ∈Q(ω)}. Obviously,
dσ
i isQ-measurable anddσi ≤yi+σiu. Let
zσ
i =yi+σiu−dσi andcσ=
σiα
η
X
ω6=ω′∈Ω
v(ω′)1Ωfor allω∈Ω.
Fix anω∈Ω. Put
δiσ= dist (ziσ(ω),(C(ω) +cσ(ω) +yi(ω))∩Y+),
and consider a continuous functionf : ∆m→∆m defined by
f(σ) = σ1+δ
σ
1 1 +Pmj=1δσ
j
,· · ·, σm+δ
σ m
1 +Pmj=1δσ j
!
.
By Brouwer’s fixed point theorem, one obtains aσ∗= (σ1∗,· · · , σm∗)∈∆msatisfying
δσ∗ i =σi∗
Pm j=1δσ
∗
j for all 1≤i≤m. It is claimed that
Pm j=1δσ
∗
j = 0. Otherwise,
δσ∗
i = 0 if and only ifσi∗= 0. Define the setJ ={i:δσ ∗
zσ∗
i (ω) =yi(ω)−dσ ∗
i (ω). Ifdσ ∗
i (ω)>0, then one hasUt(ω, yi(ω))> Ut(ω, zσ ∗ i (ω))
fort∈Si and hence
zσ∗
i (ω)∈/cl
C(ω) +cσ∗
(ω) +yi(ω)
∩Y+
.
By definition, δσ∗
i > 0, which is a contradiction with the fact that i ∈ J. Thus
dσ∗
i (ω) = 0 for alli∈J. Pick ani /∈J, then
zσ∗ i (ω)∈/
C(ω) +cσ∗
(ω) +yi(ω)
∩Y+. Consequently,
yi(ω) +
σ∗
iα
η v(ω)−d
σ∗
i (ω)∈/C(ω) +yi(ω),
which further implies thatdσ∗ i (ω)∈/
σ∗ iα
η U(ω) for alli /∈J and so
Pm i=1dσ
∗ i (ω)∈/ α
ηU(ω). Note thatdσ ∗ i (ω)≤
P
ω∈Ωwσ
∗
i (ω) and so m
X
i=1
dσ∗ i (ω)≤
X
ω∈Ω
w−(ω).
Since Pω∈Ωw−(ω) ∈ α
ηU(ω) and α
ηU(ω) is solid,
Pm i=1dσ
∗
i (ω) ∈ αηU(ω), which
is a contradiction. Thus the claim is verified, which means that δσ∗
i = 0 for all
1≤i≤m. Hence,
zσ∗
i (ω)∈cl ((C(ω) +c(ω) +yi(ω))∩Y+)
for all 1 ≤i ≤ m. By (A5), one has Ut(ω, zσ ∗
i (ω)) ≥Ut(ω, c(ω) +yi(ω)), which
together with (A2) givesEQt(zσ∗
i )>EQt(yi) for all t∈Si and 1≤i≤m. Define
h:T×Ω→Y+ by
h(·, ω) =
m
X
i=1
zσ∗ i (ω)1Si.
Clearly,h(t,·)∈LtandEQt(h(t,·))>EQt(f(t,·))µ-a.e. onS and
Z
S
hdµ≤η
m
X
i=1
yi+u−w−
! ≤η
m
X
i=1
yi+u+w
!
=η
m
X
i=1
ai=
Z
S
adµ.
This contradicts withf ∈P C(E) and the proof has been completed.
Theorem 3.4. Assume (A1)-(A6). If the correspondenceF :T ⇒YΩ
+ defined by
F(t) ={x−a(t,·) :x∈Pf(t)} ∪ {0} for allt∈T, thenclRTF dµ∩ −C=∅.
Proof. It is sufficient to prove thatRTF dµ∩ −C=∅. Assume the contrary and let
R
Thdµ∈
R
TF dµ∩ −C. Put
S={t∈T :h(t,·)6= 0 andh(t,·) =g(t,·)−a(t,·) for someg(t,·)∈Pf(t)}.
ThenS∈Σ,µ(S)>0 andRS(g−a)dµ∈ −C.Without loss of generality, one can assume thatP(S) = PS. Pick an Q ={A1, ..., Ak} ∈P(S) and let ωj ∈Aj for
all 1≤j ≤k. Sinceg(·, ωj)∈L1(µS∩TQ, Y+), there is a monotonically increasing
sequencenh(nQ,ωj):n≥1
o
of simple functions converging pointwise tog(·, ωj) such
that
lim
n→∞
Z
S∩TQ
g(·, ωj)−h(nQ,ωj)(·)
Define gn : S×Ω → Y+ by gn(t, ω) = h
(Q,ωj)
n (t) if (t, ω) ∈ (S∩TQ)×Aj. So,
{gn:n≥1} is a monotonically increasing sequence of simple functions converging
pointwise togand limn→∞RSkg(·, ω)−gn(·, ω)kdµ= 0 for allω∈Ω. Let
e
Sn={t∈S:gn(t,·)∈Pf(t)} andSn=
[
{Sen∩TQ:Q∈P(Sn)}.
By the continuity and the monotonicity ofEQt, one obtainsS
n⊆Sn+1for alln≥1 andµ(S\Sn≥1Sn) = 0. Pick an n≥ 1. Letgn|Sn =Pmi=1yiχRi, where for all
1 ≤i ≤m there are some Q ∈ P(Sn) and ηn > 0 such that Ri ⊆Sn ∩TQ and
µ(Ri) =ηn. Assume that
bn= m
X
i=1
1
ηn
Z
Ri
adµ
χRi.
By Theorem 3.3,RSn(gn−bn)dµ /∈ −C.Note thatk
R
Sn(gn−bn)dµ−
R
S(g−a)dµk →
0 asn→ ∞. SinceC is open,RS(g−a)dµ /∈ −C, which is a contradiction. Next, the main result of this section is presented. In its proof, the following result in [13] is employed: Ifϕ:T×Y →Ris Carath´eodory, then
Z
T
inf
x∈F(·)ϕ(·, x)dµ= inf
Z
T
ϕ(·, f(·))dµ:f is an integrable section ofF
.
Theorem 3.5. Assume (A1)-(A6). Then W(E) =P C(E).
Proof. By Theorem 3.4 and the separation theorem, there is a non-zero element
π∈(Y∗
+)Ω such thatE[hy, πi]≥0 for ally∈
R
TF dµ. Since GrF ∈Σ⊗B(Y
Ω),
Z
T
inf{E[hz, πi] :z∈F(·)}dµ= inf
E[hy, πi] :y∈ Z
T
F dµ
≥0.
Since inf{E[hz, πi] :z∈F(t)} ≤0 for allt∈T, one has inf{E[hz, πi] :z∈F(t)}= 0 µ-a.e. Thus, E[hx, πi] ≥ E[ha(t,·), πi] for all x ∈ Pf(t) and µ-a.e. By (A2),
one obtainsE[hf(t,·), πi]≥E[ha(t,·), πi]µ-a.e. This together with feasibility of f
further yieldE[hf(t,·), πi] =E[ha(t,·), πi]µ-a.e. Thus,
E
Z
T
f dµ, π
=E
Z
T
adµ, π
.
To complete the proof, one needs to verify thatf(t,·)∈argmaxEQt(x) :x∈B t(π)
µ-a.e. By (A1), Pω∈Ωha(t, ω), π(ω)i>0 for allt ∈T. Select some t ∈T satisfy-ingE[hx, πi]≥E[ha(t,·), πi] for all x∈Pf(t). IfE[hx, πi] =E[ha(t,·), πi] for some
x∈Pf(t), thenλx∈Pf(t) andE[hλx, πi]<E[ha(t,·), πi] for some 0< λ <1, which
is a contradiction. So, (f, π) is a Walrasian expectations equilibrium ofE. 4. The Edgeworth Equilibria with non-separable commodity spaces
Walrasian allocations under some properties of the commodity space. In contrast, the equivalence theorem in this section does not require such properties. To see the equivalence theorem, let {(Fi, Ui, a(t,·),Qi) : i ≥ 1} be the set of different characteristics available inE andTibe the set of agents inT having the same
char-acteristics as (Fi, Ui, a(t,·),Qi). Suppose thatTi∈Σ for alli≥1. Note that the
measurability conditions in (A3) and (A4) are trivially satisfied. For any allocation
f in E, let ˆf = Ξ(f) be an allocation defined by
ˆ
f(t, ω) =
(
f(t, ω), if (t, ω)∈Ti×Ω, µ(Ti) = 0;
1
µ(Ti)
R
Tif(·, ω)dµ, if (t, ω)∈Ti×Ω, µ(Ti)>0.
Fix an i with µ(Ti) >0, and define ˆfTi(ω) = ˆf(·, ω) for all (t, ω)∈Ti×Ω. The
following lemma is similar to Theorem 3.5 in [6], and is essential for the equivalence theorem.
Lemma 4.1. Suppose (A1)-(A2)and thatUt is continuous and concave for each
t∈T. Iff ∈P C(E), thenEQt(f(t,·)) =EQt( ˆf(t,·))µ-a.e.
Proof. By ignoring a µ-null subset ofT, one can choose a separable closed linear subspaceZofYΩsuch thatf(T,·)⊆Z. Assume that there exist ani0, a coalition
D⊆Ti0 such thatE
Qt( ˆf Ti0)>E
Qt(f(t,·)) for allt∈D. For anyr∈Q∩(0,1), let
Dr=
n
t∈D:EQtrfˆ Ti0
>EQt(f(t,·))o.
Note thatDr is the projection of
D×nx∈Y+Ω:E
Qtrfˆ Ti0
>EQt(x)o∩ {(t, f(t,·)) :t∈D}
onD. By the projection theorem, one hasDr∈Σ, andD=S{Dr:r∈Q∩(0,1)}.
Thus, one can find ar1∈Q∩(0,1) satisfyingµ(Dr1)>0. Letr2 =
µ(Dr1)
µ(Ti0)
. Then 0< r2≤1. For eachω∈Ω, put
υ(ω) =r1r2
Z
T
f(·, ω)dµ− Z
T
a(·, ω)dµ
−r2(1−r1)
Z
Ti0
ai0(ω)dµ.
Clearly,υ(ω)∈ −Y++ for allω∈Ω. So, there is anε >0 such that
υ(ω) +B(0,2ε)⊆ −Y++
for allω∈Ω. Applying Lemma 3.3 in [3], one has a coalitionE⊆T\Ti0 ofE such
thatµ(E)< µ(T\Ti0) andkd(ω)k< εfor allω∈Ω, where
d(ω) =
Z
E
(f(·, ω)−a(·, ω))dµ−r1r2
Z
T\Ti0
(f(·, ω)−a(·, ω))dµ.
LetS=Dr1∪E. Pick anu∈B(0, ε)∩Y++ and defineg:T×Ω→Y+ by
g(t, ω) =
(
f(t, ω) +µ(uE), if (t, ω)∈E×Ω;
r1fˆTi0, otherwise.
Then,g(t,·)∈Pf(t)µ-a.e. onS, and
Z
S
g(·, ω)dµ=
Z
E
f(·, ω)dµ+r1r2
Z
Ti0
for allω∈Ω. It can be easily verified that for allω∈Ω,
−v(ω) +
Z
S
(g(·, ω)−a(·, ω))dµ=d(ω) +u∈B(0,2ε).
Hence,
Z
S
a(·, ω)dµ− Z
S
g(·, ω)dµ≫0
for allω∈Ω, which contradicts with the fact thatf ∈P C(E). Thus,EQt(f(t,·))≥ EQt( ˆf
Ti) µ-a.e. onTi for all i ≥1. Suppose that there is a coalitionR ⊆Ti′ for
somei′≥1 such thatEQt(f(t,·))>EQt( ˆf
Ti′) for allt∈R. By Jensen’s inequality,
one has
EQt
1
µ(R)
Z
R
f(·,·)dµ
>EQt( ˆf Ti′)
and
EQt 1
µ(Ti′\R)
Z
Ti′\R
f(·,·)dµ
!
≥EQt( ˆf Ti′).
Letδ=µµ((TR)
i′). Since
ˆ
fTi′ =
δ µ(R)
Z
R
f(·,·)dµ+ 1−δ
µ(Ti′\R)
Z
Ti′\R
f(·,·)dµ,
EQt( ˆf
Ti′) >E Qt( ˆf
Ti′), which is a contradiction. Thus, E
Qt(f(t,·)) = EQt( ˆf(t,·))
µ-a.e.
Theorem 4.2. Suppose (A1)-(A2)and thatUtis continuous and concave for each
t∈T. ThenW(E) =P C(E).
Proof. To complete the proof, one only needs to verify thatP C(E)⊆W(E). Pick
an elementf ∈P C(E) and note that
H = cl[ Z
S
Pfdµ−
Z
S
adµ:S∈Σ, µ(S)>0
is a non-empty convex subset ofYΩ, refer to [5, Theorem 1]. SinceH∩ −YΩ ++=∅, by the separation theorem, there is a non-zero elementπ∈(Y∗
+)Ωsuch that for any coalitionS,
E[hy, πi]≥E
Z
S
adµ, π
for ally∈RSPfdµ. By Lemma 4.1, EQt(f(·,·)) :Ti→Ris constantµ-a.e. for all
i≥1. Pick an i≥1 and a yi ∈Pf(t) µ-a.e. onTi. If yi ∈Bt(π) for t∈Ti, then
one can construct somezi ∈Bt(π) such thatzi∈Pf(t)µ-a.e. onTi and
E
Z
Ti
zidµ, π
<E
Z
Ti
aidµ, π
,
which is a contradiction. Thus, one hasE[hyi, πi] >E[hai, πi], which further
im-pliesE[hf(t,·), πi]≥E[hai, πi]µ-a.e. Using the feasibility of f, one can show that E[hf(t,·), πi] =E[ha(t,·), πi] µ-a.e. Thus, (f, π) is a Walrasian expectations
References
[1] L. Angeloni and V. Filipe Martins-da-Rocha, Large economies with differential information and without disposal, Econ. Theory38(2009), 263–286.
[2] R.J. Aumann, Markets with a continuum of traders, Econometrica32(1964), 39–50.
[3] A. Bhowmik, J. Cao, Blocking efficiency in mixed economies with asymmetric information, J. Math. Econ.48(2012), 396–403.
[4] A. Bhowmik, J. Cao, Robust efficiency in an economy with asymmetric information, J. Math. Econ.49(2013), 49–57.
[5] A. Bhowmik, J. Cao, On the core and Walrasian expectations equilibrium in infinite dimen-sional commodity spaces, Econ. Theory, DOI 10.1007/s00199-012-0703-5.
[6] A. De Simone, M.G. Graziano, Cone conditions in oligopolistic market models, Math. Social Sci.45(2003), 53–73.
[7] E. Einy, D. Moreno, B. Shitovitz, Competitive and core allocations in large economies with differentiated information, Econ. Theory18(2001), 321–332.
[8] ¨O. Evren, F. H¨usseinov, Theorems on the core of an economy with infinitely many commodi-ties and consumers, J. Math. Econ.44(2008), 1180–1196.
[9] J.J. Gabszewich, J.F. Mertens, An equivalence theorem for the core of an economy whose atoms are not “too” big, Econometrica39(1971), 713–721.
[10] J. Greenberg, B. Shitovitz, A simple proof of the equivalence theorem for olipogolistic mixed markets, J. Math. Econ.15(1986), 79–83.
[11] C. Herv´es-Beloso, E. Moreno-Garc´ıa, C. N´u˜nez-Sanz, M.R. P´ascoa, Blocking efficiency of small coalitions in myopic economies, J. Econ. Theory93(2000), 72–86.
[12] C. Herv´es-Beloso, E. Moreno-Garc´ıa, N.C. Yannelis, Characterization and incentive compati-bility of Walrasian expectations equilibrium in infinite dimensional commodity spaces, Econ. Theory26(2005), 361–381.
[13] F. Hiai, H. Umegaki, Integrals, conditional expectations, and martingales of multivalued functions, J. Multivariate Anal.7(1977), 149–182.
[14] W. Hildenbrand, Core and equilibria in large economies, Priceton University Press, 1974. [15] L. Koutsougers, N.C. Yannelis, Incentive compatibility and information superiority of the
core of an economy with differental information, Econ. Theory3(1993), 195–216.
[16] M. Pesce, On mixed markets with asymmetric information, Econ. Theory45(2010), 23–53.
[17] K. Podczeck, Core and Walrasian equilibria when agents’ characteristics are extremely dis-persed, Econ. Theory22(2003), 699–725.
[18] K. Podczeck, On Core-Walras equivalence in Banach spaces when feasibility is defined by the Pettis integral, J. Math. Econ.40(2004), 429–463.
[19] K. Podczeck, On core-Walras equivalence in Banach lattices, J. Math. Econ.41(2005), 764–
792.
[20] R. Radner, Competitive equilibrium under uncertainty, Econometrica36(1968), 31–58.
[21] A. Rustichini, N.C. Yannelis, Edgeworth’s conjecture in economies with a continuum of agents and commodities, J. Math. Econ.20(1991), 307–326.
[22] B. Shitovitz, Oligopoly in markets with a continuum of traders, Econometrica 41 (1973),
467–501.
[23] R. Tourky, N. C. Yannelis, Markets with many more agents than commodities: Aumann’s “hidden” assumption, J. Econ. Theory101(2001), 189–221.
[24] J.J. Jr. Uhl, The range of a vector valued measure, Proc. Amer. Math. Soc. 23 (1969),
158–163.
[25] R. Wilson, Information, efficiency, and the core of an economy, Econometrica46 (1978),
807–816.
[26] N.C. Yannelis, W. R. Zame, Equilibria in Banach lattices without ordered preferences, J. Math. Econ.15(1986), 85-110.
[27] N.C. Yannelis, The core of an economy with differential information, Econ. Theory1(1991),
183–197.
School of Computing and Mathematical Sciences, Auckland University of Technol-ogy, Private Bag 92006, Auckland 1142, New Zealand
