Asset market equilibrium with short-selling and

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Asset market equilibrium with short-selling and
Asset market equilibrium with
short-selling and differential
information∗
Wassim Daher
Cermsem, Université Paris-1
e-mail: [email protected]
V. Filipe Martins-da-Rocha
Ceremade, Université Paris–Dauphine
e-mail: [email protected]
and
Yiannis Vailakis
Cermsem, Université Paris-1
e-mail: [email protected]
Abstract: We introduce differential information in the asset market model studied by Cheng (1991), Dana and Le Van (1996) and Le Van and
Truong Xuan (2001). We prove an equilibrium existence result assuming that
the economy’s information structure satisfies the conditional independence
property. If private information is not publicly verifiable, agents may not have
incentives to reveal truthfully their information, and therefore contracts may
not be executed. We show that conditional independence is crucial to address the issue of execution of contacts, since it ensures that, at equilibrium,
contracts are incentive compatible.
JEL Classification Numbers: C61, C62, D51
Keywords: Asset Market, Differential Information, Conditional Independence, Competitive Equilibrium, Incentive Compatibility
∗
We are grateful to Françoise Forges, Nicholas Yannelis, an anonymous referee and especially
an Associate Editor for valuable comments and suggestions. Thanks are also due to Rose-Anne
Dana, Cuong Le Van, M. Ali Khan, Paulo K. Monteiro and Frank Riedel. An earlier version of
the paper has been presented in the first General Equilibrium workshop in Rio, as well as in
the MED seminar in Paris-1. We thank the participants for their valuable comments. Yiannis
Vailakis acknowledges the financial support of a Marie Curie fellowship, (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
It is well known that in asset market models equilibrium prices may fail to exist,
since unbounded and mutually compatible arbitrage opportunities can arise. The
first equilibrium existence result was proven by Hart (1974). In a specific context
of an economy with finitely many securities, Hart has exhibited a list of conditions sufficient to limit the arbitrage opportunities present in the economy. Much
later, Hammond (1983), Page (1987), Werner (1987), Chichilnisky (1995) and Page,
Wooders and Monteiro (2000) reconsidered the problem by providing variations of
Hart’s list of conditions. In all cases, the role played 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 and
Scholes (1973), Kreps (1981) and Duffie and Huang (1985)) have simultaneously
motivated the study of existence problem in infinite dimensional economies with
unbounded sets (due to the possibility of short-selling). In the infinite dimensional
case, the notions of no-arbitrage are not sufficient to ensure the existence of equilibrium prices. An alternative assumption in this context that ensures existence,
is to assume that the individually rational utility set is compact (in the finite dimensional case, this assumption is equivalent to no-arbitrage conditions, see Dana
et al. (1999)). This is the case in Dana, Le Van and Magnien (1997) and Dana
and Le Van (2000). The problem then turns out to find conditions under which
the compactness of the utility set holds. When markets for contingent claims are
complete, agents’ consumption sets coincide with the space Lp (Ω, F , P), and preferences take the usual von Neuman–Morgenstern form, Cheng (1991) and Dana and
Le Van (1996) have proved (using two different approaches) that this assumption
is satisfied. As Le Van and Truong Xuan (2001) have shown, the result is still valid
in economies with separable but not necessarily von Neuman–Morgenstern utility
functions.
Economies with uncertainty are encompassed by the general equilibrium model
of Arrow-Debreu as long as there is a complete market of elementary securities
or contingent claims. The formal extension of the Arrow-Debreu model to include
uncertainty has been criticized for requiring a formidable number of necessary
markets. 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 effect
of drastically reducing the number of required contracts present in the economy.
The first study that introduced differential information in a general equilibrium
W. Daher et al./Asset market equilibrium with short-selling and differential information
3
set up was Radner (1968). Radner studies an environment where the structure
of information is fixed in advance and all contracts are negotiated at the beginning of the history of the economy. For such an economy Radner defined a notion
of equilibrium (Walrasian expectations equilibrium), an analogous concept to the
Walrasian equilibrium in Arrow-Debreu model with symmetric information. As in
Debreu (1959, Chapter 7), agents arrange contracts that may be contingent on the
realized state of nature at the second period, but after the realization of the state
of nature they do not necessarily know which state of nature has actually occurred.
Therefore, they are restricted to sign contracts that are compatible with their private information. It is important to notice that in sharp contrast with the rational
expectations equilibrium model (see Radner (1972)), markets are open only at the
first period and prices are observed and not expected.1 There prices do not reveal
any private information ex ante. They rather reflect agents’ informational asymmetries as they have been obtained by maximizing utility taking into account the
private information of each agent.
Recently there has been a resurgent interest in models with differential information (see Einy, Moreno and Shitovitz (2001), Herves-Beloso, Moreno-Garcia and
Yannelis (2005a) and Herves-Beloso, Moreno-Garcia and Yannelis (2005b)). At
issue are questions concerning the existence and characterization of Walrasian expectations equilibrium in terms of the private core introduced by Yannelis (1991).
The paper follows this growing literature. Its aim is the exploration of Walrasian
expectations equilibrium concept in infinite dimensional economies with a complete asset market structure and possible short-selling. We introduce differential
information in the asset market model studied by Cheng (1991), Dana and Le Van
(1996) and Le Van and Truong Xuan (2001). We prove an equilibrium existence
result assuming that the economy’s information structure satisfies the conditional
independence property, i.e. individuals’ information are assumed to be independent
conditionally to the common information.
Our existence proof follows in two steps. In a first place, we show that the
individually rational utility set associated with our differential information economy, coincides with the individually rational utility set of a symmetric information
economy. 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 differential economy is compact. In that respect, we show that the compactness result
established for symmetric information economies (Cheng (1991), Dana and Le Van
1
We believe that the term Walrasian expectations equilibrium may be misleading since, contrary to the rational expectations equilibrium concept (Radner (1972)), agents do not form expectations about prices.
W. Daher et al./Asset market equilibrium with short-selling and differential information
4
(1996) and Le Van and Truong Xuan (2001)) is still valid in a differential information 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 equilibrium. The second step then amounts to show that there exist prices supporting Edgeworth allocations as competitive equilibria. In that respect, we provide
a characterization of Walrasian expectations equilibrium by means of cooperative
solutions.
In addition to the equilibrium existence result, another issue we deal with, concerns the execution of contracts. If private information is not publicly verifiable,
agents have incentives to misreport their types and therefore contracts may not be
executed in the second period. We show that conditional independence is crucial
to address this issue. Indeed, it ensures that equilibrium allocations are individually incentive compatible. This in turn implies that, at equilibrium, contracts are
always executed.
The structure of the paper is as follows. In Section 2 we describe the environment and give various definitions. In Section 3 we prove the compactness of the
individually rational utility set. Section 4 deals with the decentralization of Edgeworth allocations. Section 5 is devoted to economies with finitely many states of
nature. The reason for paying special attention to the finite case, stems from the
fact that the conditional independence property is no more needed to prove existence of equilibrium prices. In this case, under our assumptions, it can be shown
directly that a no-arbitrage price always exists. Existence of a supporting price
system then follows from standard arguments in the literature (see Allouch et al.
(2002) and the references therein). However, it should be stressed that dropping
out conditional independence leaves open the question of execution of contracts.
In that respect, we believe that conditional independence is a relevant restriction
even in the finite dimensional case.
2. The Model
We consider a pure exchange economy with a finite set I of agents. The economy
extends over two periods with uncertainty on the realized state of nature in the
second period.
W. Daher et al./Asset market equilibrium with short-selling and differential information
5
2.1. Uncertainty and information structure
Each agent i ∈ I, has an individual perception of the uncertainty at the second
period, represented by a probability space (T i , T i , Pi). Each element ti in T i may
be interpreted as an individual type or signal, observed at the second period. We do
Q
not assume that all type profiles in i∈I T i are possible at the second period. This
allows to capture aspects of uncertainty which may be commonly known. The set
of possible types is captured by a couple {(Ω, F , P), (pri , i ∈ I)} where (Ω, F , P) is
a probability space representing the underlying uncertainty, related to individual
perception through the measurable mappings pri : (Ω, F ) −→ (T i , T i ), satisfying
∀i ∈ I,
∀A ∈ T i ,
h
i
Pi (A) = P (pri )−1 (A) .
Q
A profile of types (ti , i ∈ I) ∈ i∈I T i is said to be possible if there exists ω ∈ Ω such
that ti = pri (ω) for each i ∈ I. It should be clear that this framework captures
the one in which agent i’s private information is formulated by a sub σ-algebra
of F . Indeed, let us denote by F i the sub σ-algebra of F generated by pri . For
every 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
i
∀z ∈ Lp (T i , T i , Pi), EP [z] = EP [∆i (z)].2
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 underlying uncertainty is derived from the individual perception of uncertainty. Consider
probability spaces (S, S, Q) and (N i , N i, η i ) for each i ∈ I. The probability space
(S, S, Q) represents uncertainty that is commonly observed, while each N i represents agent i’s private information. We do not specify if agent i knows the private
information N k of another agent k 6= i. Agent i’s perception of uncertainty is
defined by
T i = S × N i , T i = S ⊗ N i , and Pi = Q ⊗ η i .
The underlying uncertainty (Ω, F , P) is then represented by
Ω=S×
Y
N i,
F = S ⊗ ⊗i∈I N i ,
P = Q ⊗ ⊗i∈I η i ,
i∈I
2
If (A, Σ, µ) is a probability space and z ∈ L1 (A, Σ, µ) then Eµ [z] denotes the expectation of
z under µ.
W. Daher et al./Asset market equilibrium with short-selling and differential information
6
and pri (s, (nk , k ∈ I)) = (s, ni ). Observe that if agent i is not aware of the private
Q
information k6=i N k of other agents, then he is not aware of Ω.
Remark 2.2. In what follows, we provide a general framework where the individual perception of uncertainty is derived from the underlying uncertainty. Consider
the probability space (Ω, F , P) representing states of nature that are commonly
known to be possible at the second period. Each agent i has an incomplete information about the true state of nature, in the sense that he will only observe the
output of a random variable τ i : Ω → R. Denote by T i the range τ i (Ω) of τ i . Then
the individual perception of uncertainty is represented by (T i , B, Pi ) where B is the
Borelian σ-algebra and Pi is defined by
∀A ∈ B,
Pi (A) = P{τ 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 uncertainty. In the second period, individual uncertainty is resolved and contracts are
executed. Since agents trade with an anonymous market, they only sign contracts
that are contingent to events they can observe at the second period. To be specific, for given p ∈ [1, +∞], a contract for agent i is represented by a function in
Lp (T i , T i , Pi) or Lp (Ω, F i, P). This implies that short-selling is allowed in the sense
that agents can sell an arbitrary large number of contracts.
Each agent i ∈ I is endowed with a type dependent initial endowment represented by a contingent claim ei ∈ Lp (T i , T i , Pi ) and a type dependent utility
function U i : T i × R → R. Alternatively, we may consider ei to be a function in
Lp (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 contract
i
x ∈ Lp (T i , T i , Pi ) such that the allocation (xi , i ∈ I) is exact feasible, i.e.
X
xi (ω) =
i∈I
X
ei (ω),
for P-a.e. ω ∈ Ω.3
i∈I
If private information does not become publicly verifiable, then there is an issue
concerning the execution of contracts. In order to clarify this issue, we need to
3
Formally, we should write
P
i∈I
xi (pri (ω)) =
P
i∈I
ei (pri (ω)).
W. Daher et al./Asset market equilibrium with short-selling and differential information
7
introduce the σ-algebra F c defined by
Fc =
\
Fi
i∈I
which represents the common knowledge information to all agents. An event A
belongs to F c if and only if it belongs to each F i , i.e. the σ-algebra F c is the
collection of all events that are observable by all agents.
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 . The
induced sub σ-algebra F i is defined by the partition {(pri )−1 (ti ) : ti ∈ T i}. The
atom (pri )−1 (ti ) is denoted by Ωi (ti ) and Ωc (ti ) is defined as the unique atom in
F c containing Ωi (ti ).
Assume that the realized state of nature at the second period is σ ∈ Ω. Each
agent i has an incomplete information in the sense that he only observes his type
ti = pri (σ). Individual information may not be publicly verifiable. Therefore, an
agent 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 can
only misreport if ti and ζ i are compatible with respect to the common knowledge
information, i.e.
Ωc (ti ) = Ωc (ζ i ).
Assume that each agent i ∈ I reports type ζ i . If there exists ω ∈ Ω (ω may be
different 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. each
agent i will receive xi (ζ i) and markets clear. Indeed,
X
i∈I
xi (ζ i) =
X
i∈I
xi (ω) =
X
i∈I
ei (ω) =
X
ei (ζ i).
i∈I
However, if there does not exist ω ∈ Ω such that pri (ω) = ζ i for each i ∈ I, then
contracts may not be executable.
A sufficient condition to ensure that an allocation (xi , i ∈ I) is executable, is
that each contract xi is incentive compatible as defined below.
W. Daher et al./Asset market equilibrium with short-selling and differential information
8
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 and
ζ i in T i with Pi {ti } > 0, Pi {ζ 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. Alternative concepts of incentive compatibility have been proposed
in the literature (see for instance Koutsougeras and Yannelis (1993), Krasa and
Yannelis (1994) and Hahn and Yannelis (1997)).
Remark 2.4. To clarify condition (2), return to the framework studied in Remark 2.1. In that case, given an agent i ∈ I, a contract xi ∈ Lp (T i , T i , Pi ) is
incentive compatible if and only if there do not exist s ∈ S, ni and µi in N i with
η i (ni ) > 0, η i (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 then
for each i ∈ I, there exists a function f i ∈ Lp (S, S, Q) such the net trade function
z i = xi − ei satisfies z i (s, ni ) = f i (s) for Q ⊗ η i -a.e. (s, ni ) ∈ S × N i . It then follows
that each contract xi is incentive compatible. Indeed, if it is not the case, there
exist s ∈ S, ni and µi in N i with η i (ni ) > 0, η i (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 z i (s, ni ) = f i (s) = z i (s, µi ).
2.2.2. The general case
We do not anymore assume that each agent has finitely many types. In order to
extend the concept of incentive compatibility to the general case, we first introduce
some notations. For each i ∈ I, we denote by Tci the σ-algebra of events in T i that
are common knowledge to all agents, defined by
Tci = {H ∈ T i : (pri )−1 (H) ∈ F c }.
W. Daher et al./Asset market equilibrium with short-selling and differential information
9
Observe that if xi is a function in Lp (T i , T i , Pi ) then
xi ∈ Lp (T i , Tci , 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 Tci . In that respect, we do not restrict
the 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 E i ∈ T i
and Gi ∈ T i with Pi (E i ) > 0, Pi (Gi ) > 0 such that
(a) for every (ti , ζ i) ∈ E i × Gi , agent i has an incentive to misreport ti by
announcing 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.
h
h
i
i
1
1
i
i
Pi
Pi
i
i
1
1
|T
=
|T
E
E
E
G
c
c .
Pi (E i )
Pi (Gi )
(5)
Remark 2.5. Observe that when agents have finitely many types, then Definition 2.2 reduces to Definition 2.1. Indeed, if T i is finite and T i is the σ-algebra of
all subsets of T i , then for every ti ∈ T i with Pi ({ti }) > 0, one has
i
h
i
h
i
EP 1{ti } |Tci = EP 1Ωi (ti ) |F c =
P(Ωi (ti ))
1Ωc (ti ) .
P(Ωc (ti ))
2.2.3. Conditional independence: a sufficient condition
Example in Remark 2.4 highlights that some kind of restriction in the information structure is sufficient to ensure that an exact feasible allocation is incentive
compatible.
Definition 2.3. The information structure F • = (F i, i ∈ I) of an economy
E = (U i , F i, ei , i ∈ I) is said to be conditionally independent (given the common information), if for every i 6= j in I, the σ-algebra F i and F j are independent
given F c .4
That is, for every pair of events Ai ∈ F i and Aj ∈ F j , we have that P(Ai ∩ Aj |F c ) =
P(Ai |F c )P(Aj |F c ) almost surely.
4
W. Daher et al./Asset market equilibrium with short-selling and differential information 10
Example 2.1. The information structure in Remark 2.1 is conditionally independent.
Remark 2.6. The information structure is conditionally independent if and only
if for every pair i 6= j in I we have
∀x ∈ L1 (Ω, F i , P),
E[x|F j ] = E[x|F c ].
Conditional independence also appears in McLean and Postlewaite (2002) and Sun
and Yannelis (2006).
The main result of this section is the following proposition.
Proposition 2.1. If the information structure is conditionally independent then
every exact feasible allocation (xi , i ∈ I) with xi ∈ Lp (Ω, F i , P) is incentive compatible.
Proof. Let (xi , i ∈ I) be an allocation with xi ∈ Lp (Ω, F i , P) such that
X
xi (ω) =
i∈I
X
ei (ω),
for P-a.e. ω ∈ Ω.
i∈I
We claim that under conditional independence of the information structure, for
each i ∈ I, the net trade z i := xi − ei is F c -measurable.5 Indeed, for each i ∈ I,
P
one has z i = − j6=i z j and in particular
z i = E[z i |F i] = −
X
E[z j |F i] = −
j6=i
X
E[z j |F c ].
j6=i
Fix i ∈ I and assume that the contract xi is not incentive compatible. Then there
exist E i ∈ T i and Gi ∈ T i with Pi (E i ) > 0, Pi (Gi ) > 0 and satisfying conditions (4)
and (5). Then
i
i
Pi
i
i
i
Pi
i
P (G )E [z 1E i ] − P (E )E [z 1Gi ] =
Z
[z i (t) − z i (ζ)]Pi ⊗ Pi (dt, dζ) < 0.
E i ×Gi
Since z i is F c -measurable, it follows from (5) that
i
h
Pi (Gi )EP z i 1E i
i
i
h
i
h
= Pi (Gi )EP z i EP 1E i |Tci
Pi
= Pi (Gi )E
i
"
h
ii
#
i
Pi (E i ) i h
z i i i EP 1Gi |Tci
P (G )
i
= Pi (E i )EP z i 1Gi .
which yields a contradiction.
5
When there are only two agents, the conditional independence of the information structure
is no more needed.
W. Daher et al./Asset market equilibrium with short-selling and differential information 11
2.3. Equilibrium concept
To relate our equilibrium concept to the traditional one proposed in the literature
with symmetric information (see Cheng (1991)), we choose to reprensent agent i’s
individual uncertainty by (Ω, F i , P).
The preference relation of agent i is represented by the utility function ui from
Lp (Ω, F i , P) to R defined by
p
i
∀x ∈ L (Ω, F , P),
i
u (x) =
Z
Ω
U 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 preferred
contingent 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 rational
attainable allocation is defined by
(
•
i
A = x = (x , i ∈ I) ∈
Y
p
i
L (Ω, F , P) :
i∈I
P
X
i
i
i
i
i
x = e and u (x ) > u (e )
i∈I
)
where e = i∈I ei is the aggregate initial endowment. If ψ ∈ Lq (Ω, F , P)6 then we
can define the value functional hψ, .i on Lp (Ω, F , P) by
p
∀x ∈ L (Ω, F , P),
hψ, xi =
Z
Ω
ψ(ω)x(ω)P(dω).
The vector ψ is called a price (system) and hψ, xi represents the value (in the sense
of Debreu (1959)) at the first period of the contingent claim x under the price ψ.
We also denote the value hψ, xi 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 his
utility in his budget set, i.e.
∀i ∈ I,
n
o
xi ∈ argmax ui (y) : y ∈ Lp (Ω, F i, P) and hψ, yi 6 hψ, ei i .
An allocation x• is said to be a competitive allocation if there exists a price ψ such
that (x• , ψ) is a competitive equilibrium.
6
The real number q ∈ [1, +∞] is the conjugate of p defined by 1/p+1/q = 1 with the convention
that 1/ + ∞ = 0 and 1/0 = +∞.
W. Daher et al./Asset market equilibrium with short-selling and differential information 12
To prove the existence of a competitive equilibrium, we first show that, under suitable assumptions, the individually rational utility set (as defined below) is
compact.
Definition 2.5. We denote by U the individually rational utility set defined by
U = {v • = (v i, i ∈ I) ∈ RI : ∃x• ∈ A,
ui (ei ) 6 v i 6 ui (xi )}.
For any 1 6 p 6 +∞ and for any sub σ-algebra G ⊂ F , the space Lp (Ω, G, P) is
denoted 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)| 6
α(ω) + β|t|p for every (ω, t) ∈ Ω × R.
Remark 2.7. Obviously, every p-integrable bounded function is integrable. Following Krasnoselskii (1964), if p ∈ [1, +∞) and (Ω, F , P) is atomless then a function U is p-integrable bounded, if and only if, for every x ∈ Lp (F ), the function
ω 7→ U(ω, x(ω)) is integrable. If p = ∞ then a utility function is integrable if and
only 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 atomless
measure space (Ω, F , P) then a necessary condition for the separable utility function
ui 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 every
i ∈ I, the following properties are satisfied:
(S.1) the function t 7→ U i (ω, t) is differentiable, concave and strictly increasing
for 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
(ω, t);
∂t
(S.4) there exist two functions αi and β i from R to R such that for every t ∈ R,
αi (t) < t < β i (t),
ess sup U⋆i (ω, β i(t)) < ess inf U⋆i (ω, t)
ω∈Ω
ω∈Ω
W. Daher et al./Asset market equilibrium with short-selling and differential information 13
and
ess sup U⋆i (ω, t) < ess inf U⋆i (ω, αi(t)).
ω∈Ω
ω∈Ω
Remark 2.9. If the information structure is symmetric, then an economy satisfies
Assumptions S.1–S.4 if and only if it satisfies assumptions H.0–H.4 in Le Van and
Truong Xuan (2001). In particular, all economies considered in Cheng (1991), Dana
and 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 examples of utility functions satisfying Assumptions S.1–S.4.
Example 2.2. If for every i ∈ I, there exists F i : R → R concave, continuously
differentiable and strictly increasing and such that the function U i (ω, t) = F i (t) is
p-integrable bounded7 , then the economy is standard.
Example 2.3. If for every i ∈ I, there exists F i : R → R concave, continuously
differentiable, strictly increasing satisfying
lim F⋆ (t) = +∞ and
t→−∞
lim F⋆ (t) = 0;
t→+∞
i
i
i
i
and hi ∈ L∞
++ (F ) such that the function U (ω, t) = F (t)h (ω) is p-integrable
bounded, then the economy is standard. In this case, agents’ preference relations
are represented by von Neumann–Morgenstern utility functions with heterogenous
expectations. Indeed, if we denote by Pi the probability measure on (Ω, F ) defined
by dPi = hi dP, then the preference relation defined by ui : x 7→ E[U i (x)] coincide
with the von Neumann–Morgenstern preference relation defined by the felicity (or
index) function F i and the private belief Pi , i.e.
p
i
∀x ∈ L (F ),
i
Pi
i
u (x) = E [F (x)] =
Z
Ω
F i (xi (ω))Pi(dω).
We do not assume that aggregate initial endowment is measurable with respect to
the common information F c . As the following two remarks show, in our framework
such an assumption makes the existence issue a trivial one.
Remark 2.10. Consider a standard economy E = (U i , F i, ei , i ∈ I) such that
every initial endowment ei is measurable with respect to the common information,
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 a
7
Such a utility function has the well known von Neumann–Morgenstern form.
W. Daher et al./Asset market equilibrium with short-selling and differential information 14
straightforward way. Indeed, let E c denote the (symmetric) economy (U i , F c , ei , i ∈
I). This economy is standard and following Cheng (1991) we can show that there
exists 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. For
this, 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 y i ∈ Lp (F i) such that
E[ψy i ] 6 E[ψei ] and ui (y i) > ui (xi ).
Since ψ and ei are measurable with respect to F c , the consumption plan E[y i |F c ]
satisfies the budget constraints associated with the symmetric economy E c . But
from Jensen’s inequality, we also have that
ui (E[y i|F c ]) > ui (y i) > 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 that
P
the aggregate endowment e = i∈I ei is measurable with respect to the common
information, i.e. e belongs to Lp (F c ), each utility function ui is von Neumann–
Morgenstern and the information structure is conditionally independent. In this
case, the existence of a competitive equilibrium follows also in a straightforward
way. Indeed, the F c -measurability of e together with conditional independence
imply that individual endowments ei are also F c -measurable.8 Existence follows
from the previous remark.
3. Compactness of the individually rational utility set
Theorem 3.1 stated below, proves the compactness of the individually rational
utility set associated with our differential information economy. In that respect it
generalizes the results established for symmetric information economies found in
Cheng (1991), Dana and Le Van (1996) and Le Van and Truong Xuan (2001).
8
We have e =
P
i∈I
ei . Fix j ∈ I, taking conditional expectations with respect to F j , we get
e = E[e|F j ] = ej +
X
i6=j
E[ei |F j ] = ej +
X
i6=j
E[ei |F c ].
W. Daher et al./Asset market equilibrium with short-selling and differential information 15
Theorem 3.1. For every standard economy with a conditionally independent information 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 ξ i be the linear mapping from Lp (F ) to Lp (F i ) defined by
∀z ∈ Lp (F ),
ξ i(z) = E[z − θc (z)|F i ].
Observe that
θc ◦ ξ i = 0 and ∀z i ∈ Lp (F i ),
∀i ∈ I,
ξ i (z i ) = z i − θc (z i ).
Since the information structure is conditionally independent, we have
∀i 6= j ∈ I,
∀z j ∈ Lp (F j ),
ξ i (z j ) = E[z j |F i ] − E[z j |F c ] = 0.
(6)
We let Z be the set of allocations defined by
(
•
Z= z ∈
P
where e =
ξ i (e) = ξ i (ei ).
i∈I
Y
p
c
L (F ) :
i∈I
X
i
c
)
z = θ (e) ,
i∈I
ei is the aggregate initial endowment. From (6) we have that
Claim 3.1. An allocation x• is attainable (feasible) if and only if there exists
z • ∈ Z such that
∀i ∈ I, xi = z i + ξ i(e).
Proof. If x• be an attainable allocation, then
∀i ∈ I,
ξ i(e) =
X
ξ i (xj ) = ξ i (xi ) = xi − θc (xi ).
j∈I
If for every i ∈ I, we let z i = θc (xi ) then z • belongs to Z and satisfies xi = z i +ξ i(e).
Reciprocally, if x• is an allocation such that xi = z i + ξ i(e) for some z • ∈ Z, then
X
i∈I
xi = θc (e) +
X
ξ i (e) = θc (e) +
i∈I
which implies that x• is attainable.
X
i∈I
{ei − θc (ei )} = e
W. Daher et al./Asset market equilibrium with short-selling and differential information 16
Now we consider the function V i : Ω × R → R defined by
∀(ω, t) ∈ Ω × R,
V (ω, t) = E[U i (ξ i (e) + t)|F c ](ω).9
By the Disintegration Theorem we have
p
c
∀z ∈ L (F ),
i
i
E[U (ξ (e) + z)] =
=
Z
Ω
Z
Ω
U i (ω, ξ i(e)(ω) + z(ω))P(dω)
V i (ω, z(ω))P(dω)
= E[V i (z)].
(7)
Combining Claim 3.1 and equation (7), the individually rational utility set U coincide with the set V defined by
n
V = w • ∈ RI : ∃z • ∈ Z,
v i (θc (ei )) 6 w i 6 v i (z i )
o
where v i is the utility function defined on Lp (F c ) by v i (z) = E[V i (z)].
Observe that the set V corresponds to the individually rational utility set of a
symmetric 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 ξ i (e). Since the function U i is pintegrable 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 check
that the function t 7→ V i (ω, t) is concave and strictly increasing for P-a.e. ω. We
propose now to prove that t 7→ V i (ω, t) is differentiable for P-a.e. ω. Let (tn ) be a
sequence 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, ζ i(ω) + t). Moreover, since the function t 7→ U i (ω, t) is concave, differentiable
and strictly increasing, for every n large enough10 we have
|fn (ω)| 6 U⋆i (ω, ζ i(ω) + t − 1),
for P-a.e. ω ∈ Ω.
In the sense that if t ∈ R then we let fti (ω) = U i (ω, ξ i (e)(ω) + t). The real number V i (ω, t)
is then defined by V i (ω, t) = E[fti |F c ](ω).
10
More precisely, for every n such that |tn − t| 6 1.
9
W. Daher et al./Asset market equilibrium with short-selling and differential information 17
Observe that since the function t 7→ U i (ω, t) is concave, differentiable and strictly
increasing, 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 Dominated Convergence Theorem for conditional expectations, we get that the function
t 7→ V i (ω, t) is differentiable and for every t ∈ R,
V⋆i (ω, t) :=
∂V i
(ω, t) = E[U⋆i (ξ i (e) + t)|F c ](ω),
∂t
for P-a.e. ω ∈ Ω.
We have thus proved that the function V i satisfies Assumption S.1. Assumption S.2
follows from the p-integrability of U i and the disintegration theorem. We propose
now 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, ζ i(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 expectations 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 sup V⋆i (ω, β i(t)) < ess inf V⋆i (ω, t).
ω∈Ω
ω∈Ω
Fix t ∈ R. Since the function U i satisfies Assumption S.4, we trivially have that
U⋆i (ω, β i(t)) 6 ess sup U⋆i (ω, β i(t)),
for P-a.e. ω ∈ Ω.
ω∈Ω
Therefore for P-a.e. ω,
V⋆i (ω, β i(t)) = E[U⋆i (ξ i(e) + β i(t))|F c ](ω) 6 ess sup U⋆i (ω, β i(t))
ω∈Ω
which implies that
ess sup V⋆i (ω, β i(t)) 6 ess sup U⋆i (ω, β i(t)).
ω∈Ω
ω∈Ω
(8)
W. Daher et al./Asset market equilibrium with short-selling and differential information 18
Symmetrically we can prove
ess inf U⋆i (ω, t) 6 ess inf V⋆i (ω, t).
ω∈Ω
ω∈Ω
(9)
Since U i satisfies Assumption S.4, we have that
ess sup U⋆i (ω, β i(t)) 6 ess inf U⋆i (ω, t).
ω∈Ω
ω∈Ω
(10)
The desired result follows from (8), (9) and (10).
Following similar arguments, it is straightforward to prove that
∀t ∈ R,
ess sup V⋆i (ω, t) < ess inf V⋆i (ω, αi(t)),
ω∈Ω
ω∈Ω
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 their
arguments we can directly conclude that the set V (and therefore the set U) is
compact.11
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 y i ∈ P i (xi )
for each i ∈ I,
2. a core allocation, if it cannot be blocked by any coalition in the sense that
Q
there is no coalition S ⊆ I and some (y i, i ∈ S) ∈ i∈S P i (xi ) such that
P
P
i
i
i∈S e ,
i∈S y =
3. an Edgeworth allocation if there is no 0 6= λ• ∈ Q∩[0, 1]I and some allocation
P
P
y • such that y i ∈ P i(xi ) for each i ∈ I with λi > 0 and i∈I λi y i = i∈I λi ei ,
4. an Aubin allocation if there is no 0 6= λ• ∈ [0, 1]I and some allocation y • such
P
P
that y i ∈ P i(xi ) for each i ∈ I with λi > 0 and i∈I λi y i = i∈I λi ei .
11
Actually 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 to
abstract probability spaces. We refer to Appendix B for a formal discussion.
W. Daher et al./Asset market equilibrium with short-selling and differential information 19
Remark 4.1. The reader should observe that these concepts are “price free” in
the sense that they are intrinsic properties of the commodity space. It is proved in
Florenzano (2003, Proposition 4.2.6) that the set of Aubin allocations and the set
of Edgeworth allocations coincide for every standard economy.
Remark 4.2. In our model, agent i’s consumption set coincides with Lp (F i ) the
space of contracts measurable with respect to agent i’s information. Therefore the
core concept introduced above is related to the private core concept introduced by
Yannelis (1991) (where agent i’s consumption set is Lp+ (F i )).
Corollary 4.1. If E is a standard economy with a conditionally independent information structure then there exists an Edgeworth allocation.
Proof. Since the individually rational utility set is compact, we can apply Theorem 3.1 in Allouch and Florenzano (2004).
It is straightforward to check that every competitive allocation is an Edgeworth
allocation. We prove that the converse is true.
Theorem 4.1. If E is a standard economy with a conditionally independent information structure then for every Edgeworth allocation x• there exists a price ψ
such that (x• , ψ) is a competitive equilibrium. Moreover, for every i ∈ I there exists
λi > 0 such that
E[ψ|F i ] = λi U⋆i (xi ).
In particular, we have
∀(i, j) ∈ I × I,
E[λi U⋆i (xi )|F c ] = E[λj U⋆j (xj )|F c ].
Proof. Let x• be an Edgeworth allocation. From Remark 4.1 it is actually an Aubin
allocation, in particular
0 6∈ G(x• ) := co
[
P i (xi ) − {ei } .
i∈I
(11)
The set Lp (F i ) is denoted by E i , the set Lp (F c ) is denoted by E c and we let
P
E = i∈I E i . Observe that E may be a strict subspace of Lp (F ). We endow E
with the topology τ for which a base of 0-neighborhoods is
(
X
i∈I
i
i
i
α B ∩ E : α > 0,
∀i ∈ I
)
W. Daher et al./Asset market equilibrium with short-selling and differential information 20
where B is the closed unit ball in Lp (F ) for the p-norm topology.12 From the
Structure Theorem in Aliprantis and Border (1999, page 168) this topology is well
defined, Hausdorff and locally convex. Observe moreover that the restriction of
the τ -topology to each subspace E i coincide with the restriction of the p-norm
topology. From Proposition A.1, each utility function ui is p-norm continuous. It
follows 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 for
each g ∈ G(x• ), we have π(g) > 0. In particular it follows that
∀i ∈ I,
∀y i ∈ P i (xi ),
π(y i) > π(xi ).
(12)
Since preference relations are strictly increasing, we have that xi belongs to the
closure of P i (xi ). Therefore π(xi ) > π(ei ). But since x• is attainable, we must
have π(xi ) = π(ei ) for each i ∈ I. The linear functional π is not zero, therefore
there exists j ∈ I and z j ∈ E j such that π(z j ) < π(ej ). We now claim that if
y j ∈ P j (xj ) then π(y j ) > π(ej ). Indeed, assume by way of contradiction that
π(y j ) = π(ej ). Since the set P j (xj ) is p-norm open in E j , there exists α ∈ (0, 1)
such that αy j +(1−α)z j ∈ P j (xj ). From (12) we get απ(y j )+(1−α)π(z j ) > π(ej ):
contradiction. We have thus proved that
∀y j ∈ P j (xj ),
π(y j ) > π(xj ).
By strict-monotonicity of preference relations, we obtain that π(1) > 0 where 1
is the vector in E c defined by 1(ω) = 1 for any ω ∈ Ω. Choosing z i = ei − 1 for
every i 6= j, we follow the previous argument to get that
∀i ∈ I,
∀y i ∈ P i (xi ),
π(y i) > π(ei ),
or equivalently
∀i ∈ I,
n
xi ∈ argmax ui(z i ) : z i ∈ E i
o
and π(z i ) 6 π(ei ) .
(13)
In order to prove that x• is an equilibrium allocation, it is sufficient to prove that
there exists a price ψ ∈ Lq (F ) which represents π in the sense that π = hψ, .i.
Claim 4.1. For every i ∈ I, there exists ψ i ∈ Lq (F i) such that
∀z i ∈ E i ,
π(z i ) = hψ i , z i i.
Moreover, there exists λi > 0 such that ψ i = λi U⋆i (xi ).
12
We refer to Appendix A for precise definitions.
W. Daher et al./Asset market equilibrium with short-selling and differential information 21
Proof. Fix i ∈ I and let z i ∈ E i such that hU⋆i (xi ), z i i > 0. From Claim A.3, there
exists t > 0 such that xi + tz i ∈ P i (xi ). Applying (13) we get that π(z i ) > 0. We
have thus proved that
∀z i ∈ E i ,
hU⋆i (xi ), z i i > 0 =⇒ π(z i ) > 0.
It then follows13 that there exists λi > 0 such that the restriction π|E i of π to E i
coincides with the linear functional hλi U⋆i (xi ), .i.
Since E c ⊂ E i ∩ E j the family ψ • = (ψ i , i ∈ I) is compatible in the sense that
∀(i, j) ∈ I × I,
E[ψ i |F c ] = E[ψ j |F c ].
We denote by ψ c the vector in Lq (F c ) defined by ψ c = E[ψ i |F c ] and we let ψ be
the vector in Lq (F ) defined by
ψ = ψc +
X
(ψ i − ψ c ).
(14)
i∈I
Claim 4.2. The vector ψ represents the linear functional π.
Proof. Since the information structure is conditionally independent, we can easily
check that
∀i ∈ I, E[ψ|F i ] = ψ i .
If z is a vector in E, then there exists z • = (z i , i ∈ I) with z i ∈ E i such that
P
z = i∈I z i . Therefore
hψ, zi =
X
hψ, z i i =
i∈I
X
i∈I
E[ψz i ] =
X
i∈I
E[ψ i z i ] =
X
π(z i ) = π(z),
i∈I
which implies that π = hψ, .i.
It follows from (13) and Claim 4.2 that (x• , ψ) is a competitive equilibrium.
13
Let hX, X ⋆ i be a dual pair. Assume that there exists x⋆ ∈ X ⋆ and 0 6= π a linear functional
from X to R such that for every x ∈ X, if hx, x⋆ i > 0 then π(x) > 0. Let ∆ = {(a, b) ∈
R+ × R+ : a + b = 1} be the simplex in R2 , we then have
∆ ∩ {(hx, x⋆ i, −π(x)) : x ∈ X} = ∅.
Applying a convex separation argument, we get the existence of λ > 0 such that hλx⋆ , .i = π.
W. Daher et al./Asset market equilibrium with short-selling and differential information 22
Remark 4.3. Observe that in the proof of Theorem 4.1, the conditional independence assumption on the information structure is only used in Claim 4.2 to
prove that the linear functional π is representable by a function ψ in Lq (F ). In
particular if the set Ω is finite, then Theorem 4.1 is valid without assuming that
the information structure is conditionally independent.
Due to Corollary 4.1 and Theorem 4.1 we are able to generalize Theorem 1 in
Cheng (1991), Theorem 1 in Dana and Le Van (1996) and Theorem 1 in Le Van
and Truong Xuan (2001) to economies with differential information.
Corollary 4.2. Every standard economy with a conditionally independent information structure has a competitive allocation.
Remark 4.4. Observe that when p = +∞, contrary to Cheng (1991) and Le Van
and Truong Xuan (2001), we do not need to establish Mackey-continuity of the
utility functions to guarantee that equilibrium prices belong to L1 (F ). As in Dana
and Le Van (1996), our equilibrium prices are related to individual marginal utilities. The continuity requirement (i.e. prices belong to L1 (F )) follows then directly
from 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 P{ω} > 0 for every ω ∈ Ω. For every 1 6 p 6 +∞, the space
Lp (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 which
the information structure satisfies the conditional independence property. We show
below that, when Ω is finite, the existence of equilibrium prices can be established
without assuming conditional independence of the information structure. Indeed,
in this case one can show that for standard economies a no-arbitrage price always
exists. It is well-known in the literature (see Allouch et al. (2002) and the references
therein) that in the finite dimensional case, the existence of a no-arbitrage price is
sufficient for the existence of an equilibrium price system. However, it should be
stressed that dropping out conditional independence leaves open the question of
execution of contracts. In that respect, we believe that conditional independence
is a relevant restriction even in the finite dimensional case. We impose hereafter
the following assumption which is a weaker version of Assumption S.4.
(wS.4) There exist three numbers αi < τ i < β i such that,
sup U⋆i (ω, β i) < inf U⋆i (ω, τ i)
ω∈Ω
ω∈Ω
W. Daher et al./Asset market equilibrium with short-selling and differential information 23
and
sup U⋆i (ω, τ i) < inf U⋆i (ω, αi).
ω∈Ω
ω∈Ω
For each i ∈ I, we let Ri be the subset of L0 (F i) defined by
Ri = {y i ∈ L0 (F i ) : ∀t > 0,
ui (ei + ty i ) > 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 = {y i ∈ L0 (F i ) : ∀t > 0,
ui(xi + ty i) > ui (xi )}.
(15)
In our framework, we can prove that 1 is a no-arbitrage price, as defined in
Werner (1987).
Proposition 5.1. For each i ∈ I, if y i is a non-zero vector in Ri then h1, y ii > 0.
Proof. Let xi ∈ Lp (F i) be defined as follows
xi = β i 1{yi >0} + αi 1{yi <0} + τ i 1{yi =0} .
Since y i ∈ Ri , one has ui(xi + y i) > ui (xi ). Following standard arguments, one has
hU⋆i (xi ), y ii > ui (xi + y i ) − ui (xi ) > 0.
Let γ i ∈ R be such that
inf U⋆i (ω, τ i ) 6 γ i 6 sup U⋆i (ω, τ i ).
ω∈Ω
ω∈Ω
Observe that γ i > 0 and for almost every ω ∈ Ω, one has
γ i y i(ω) > U⋆i (ω, xi(ω))y i(ω).
From Assumption wS.4,
γ i > U⋆i (ω, xi(ω)),
∀ω ∈ {y i > 0}
γ i < U⋆i (ω, xi (ω)),
∀ω ∈ {y i < 0}.
and
Since y i 6= 0, it follows that
γ i h1, y ii > hU⋆i (xi ), y ii > 0.
W. Daher et al./Asset market equilibrium with short-selling and differential information 24
Appendix A: Properties of separable utility functions
The space Lp (F ) is endowed with the p-norm topology defined by the sup-norm
k.k∞ if p = ∞, i.e.
∀x ∈ L∞ (F ),
kxk∞ = ess sup |x(ω)|
ω∈Ω
and the p-norm k.kp if p ∈ [1, +∞), i.e.
p
∀x ∈ L (F ),
kxkp =
Z
Ω
p
|x(ω)| P(dω)
1/p
.
Proposition A.1. Let p ∈ [1, +∞] and U : Ω × R → R be a p-integrable bounded
function 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) =
Z
Ω
U(ω, x(ω))P(dω)
is p-norm continuous.
Proof. The arguments of the proof are similar to those in Aliprantis (1997, Theorem 5.3). Let (xn ) be a sequence in Lp (F ) p-norm converging to x ∈ Lp (F ). To
show that the sequence (u(xn )) converges to u(x), it suffices to establish that every
subsequence (yn ) of (xn ) has in turn a subsequence (zn ) such that (u(zn )) converges
to u(z). To this end, let (yn ) be a subsequence of (xn ). Applying Théorème IV.19
in 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 function
t 7→ U(ω, t) is increasing for P-a.e. ω ∈ Ω, we have that
|U(ω, zn (ω)| 6 max{U(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).
W. Daher et al./Asset market equilibrium with short-selling and differential information 25
Proposition A.2. Let p ∈ [1, +∞] and U : Ω × R → R be a p-integrable bounded
function such that the function t 7→ U(ω, t) is continuous, increasing and differentiable for P-a.e. ω ∈ Ω. For every x ∈ Lp (F ) the function ω 7→ U⋆ (ω, x(ω)) belongs
to Lq (F ) and
∀y ∈ Lp (F ),
1
hU⋆ (x), yi = lim {u(x + ty) − u(x)}.
t↓0 t
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)
p
Now let x and y be two vectors in L (F ), applying (16) and (17) we have almost
surely 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 = 1{U⋆ (x)>0}
and y2 = 1{U⋆ (x)<0}
then
E|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 kαk1 + β{(kxkp )p + (kx + ykp )p + (kx − ykp )p }
which implies that the linear operator y 7→ E[U⋆ (x)y] is p-norm continuous. Therefore the function U⋆ (x) belongs to Lq (F ).
W. Daher et al./Asset market equilibrium with short-selling and differential information 26
Claim A.3. For every x and y in Lp (F ), we have
1
hU⋆ (x), yi = lim {u(x + ty) − u(x)}.
t↓0 t
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
{U(ω, x(ω) + tn y(ω)) − U(ω, x(ω))}.
tn
The sequence (fn ) converges almost surely to the function f = U⋆ (x)y. Moreover,
we have
U⋆ (ω, x(ω) + tn y(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 atomless
to general probability state space.
We consider a symmetric economy E = (U i , F , ei , i ∈ I), i.e. F i = F for every
agent i ∈ I. It is proved in Le Van and Truong Xuan (2001) that if E is standard
and if the state space (Ω, F , P) is the continuum [0, 1] endowed with the Lebesgue
measure then the individually rational utility set is compact. The arguments in
Le 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 then
the 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 part
of Ω.
W. Daher et al./Asset market equilibrium with short-selling and differential information 27
• First step: assume that (Ω, F , P) is purely atomic. Since P(Ω) = 1, there
exists 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) =
X
P{ω}.
ω∈A
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 is
at most countable, for every ω ∈ Ω, there exists Cω ∈ L such that
B=
[
{ω} × Cω .
ω∈Ω
Observe that
[P ⊗ λ](B) =
X
P{ω}λ(Cω ).
ω∈Ω
Since ([0, 1], L, λ) is atomless, for every ω ∈ Ω, there exists Dω ∈ L such that
Dω ⊂ Cω and λ(Dω ) = αλ(Cω ). We now let A be the set in F ⊗ L defined by
A=
[
{ω} × Dω .
ω∈Ω
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) and
the 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) =
Z
[0,1]
y(ω, s)λ(ds).
W. Daher et al./Asset market equilibrium with short-selling and differential information 28
We propose to prove that the individually rational utility set U is compact.
Let (wn• ) be a sequence in U, i.e. for each n ∈ N, there exists an attainable
allocation x•n satisfying
∀i ∈ I,
ui(ei ) 6 wni 6 ui (xin ).
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 v i be the utility function defined on Lp (P ⊗ λ) by
v i(y) = EP⊗λ [V i (y)] =
Z
P(dω)
Ω
Z
[0,1]
U i (ω, y(ω, s))λ(ds).
Observe that for every z ∈ Lp (P), we have ui(z) = v i (Fz). It then follows
that
∀i ∈ I, v i (Fei ) 6 wni 6 v i (Fxin ).
We let E ′ be the symmetric economy defined by (V i , F ⊗ L, Fei , i ∈ I).
Observe that for every n ∈ N, the family (Fxin , i ∈ I) is attainable for E ′ . In
particular, the sequence (wn• ) belongs to the individually rational utility set
U ′ 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,
lim wni = w i .
n→∞
We claim that w • actually belongs to U. Since w • belongs to U ′ , there exists
an allocation y • attainable for the economy E ′ such that
∀i ∈ I,
v i (Fei ) 6 w i 6 v i (y i).
Since U i is concave, we have
∀i ∈ I,
v i (y i) 6 ui (Gy i).
Since the allocation (Gy i, i ∈ I) is attainable for the economy E, we have
proved that w • belongs to U.
W. Daher et al./Asset market equilibrium with short-selling and differential information 29
• Second step: the general case. We let Ξ be the subset of Ω × [0, 1] defined
by
[
Ξ = (Ωa × [0, 1]) (Ωna × {0})
and we let G be the trace of the σ-algebra F ⊗ L on Ξ. On the measurable
space (Ξ, G) we defined the probability Q by
∀A ∈ G,
e na )
Q(A) = [P ⊗ λ](Aa ) + P(A
where A = Aa ∪Ana is the (unique) decomposition defined by Aa ⊂ Ωa ×[0, 1]
e na ) = P(B) with B ∈ F satisfying
and Ana ⊂ Ωna × {0} and where P(A
na
A = B × {0}. Observe that the probability space (Ξ, G, Q) is atomless.
Following almost verbatim the arguments of the first step, we can prove that
the individually rational utility set U is compact.
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