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Article Excerpt
Abstract Let ([E.sub.k])[.sub.k[member of]N] be a sequence of differential information economies, converging to a limit differential information economy [E.sub.[infinity]] (written as [E.sub.k] [right arrow] [E.sub.[infinity]]). Denote by [C.sub.[epsilon]]([E.sub.k]) the set of all [epsilon]-private core allocations, [epsilon] [greater than or equal to] (for [epsilon] = we get the private core of Yannelis (1991), denoted by C([E.sub.k])). Under appropriate conditions, we prove the following stability results:
(1) (upper semicontinuity): if [E.sub.k] [right arrow] [E.sub.[infinity]], [f.sub.k] [member of] C([E.sub.k]), and if [f.sub.k] [right arrow] [f.sub.[infinity]] [L.sup.1]-weakly, then [f.sub.[infinity]] [member of] C([E.sub.[infinity]]).
(2) (lower semicontinuity): if [E.sub.k] [right arrow] [E.sub.[infinity]], [f.sub.[infinity]] [member of] C([E.sub.[infinity]]), [epsilon] > 0, then there exist [f.sub.k] [member of] [C.sub.[epsilon]]([E.sub.k]), with [f.sub.k] [right arrow] [f.sub.[infinity]] [L.sup.1]-weakly.
Keywords Private core * Upper semicontinuity * Lower semicontinuity * Weak [L.sup.1]-convergence * Martingales
JEL Classification Numbers D82 * D50 * D83 * C62 * C71 * D46 * D61
1 Introduction
Let ([OMEGA], F) be an exogenously given measurable space. Here [OMEGA] is the set of all states of nature. Let S be a separable and reflexive Banach space. The space S is endowed with the Borel [sigma]-algebra B(S). Let I [colon, equals] {1,..., m} be a set of agents (or players).
We study the following notion of a differential information economy E [colon, equals] {([[SIGMA].sup.i], [u.sup.i], [e.sup.i], [F.sup.i], [[mu].sub.i]) : i [member of] I}. For every i [member of] I in the above expression [F.sup.i] [subset] F denotes agent i's private information [sigma]-algebra (1) and [[mu].sup.i] is agent i's prior probability measure on [F.sup.i], reflecting agent i's personal beliefs about the probability distribution of the state of nature. Evidently there exists a finite measure [mu] on ([OMEGA], F) such that for every i [member of] I the probability measure [[mu].sup.i] has a probability density [[phi].sup.i] with respect to [mu]. Thus, we have [[mu].sup.i](A) = [[integral].sub.A] [[phi].sup.i] d[mu] for all A [member of] [F.sup.i] and [[phi].sup.i] : [OMEGA] [right arrow] [R.sub.+] is [F.sup.i]-measurable and [mu]-integrable. Also, in the above expression for E, [[SIGMA].sup.i] is a multifunction [[SIGMA].sup.i] : [OMEGA] [right arrow] [2.sup.S] with a [F.sup.i] x B(S)-measurable graph, which denotes the random consumption set of agent i. Contingent upon the realized state of nature [omega] in [OMEGA], agent i must select his/her consumption bundle in the subset [[SIGMA].sup.i]([omega]) of S. This leads to the following notion: an allocation for agent i is a function [f.sup.i] : [OMEGA] [right arrow] S that is [F.sup.i]-measurable, [mu]-integrable, and such that [f.sup.i]([omega]) [member of] [[SIGMA].sup.i]([omega]) for [mu]-a.e. [omega] in [OMEGA] (and hence also for [[mu].sup.i] -a.e. [omega] in [OMEGA]). Thus, with probability 1 the consumption choice of each agent is contingent upon the state of nature that is realized. The set of all allocations for agent i is denoted by [L.sub.[[SIGMA].sup.i].sup.1] [colon, equals] [L.sub.[[SIGMA].sup.i].sup.1]([OMEGA], [F.sup.i], [mu]). An allocation for the economy E is defined to be a vector function ([f.sup.i])[.sub.i[member of]I] such that [f.sup.i] [member of] [L.sub.[[SIGMA].sup.i].sup.1] for every i [member of] I; thus [[PI].sub.i[member of]I][L.sub.[[SIGMA].sup.i].sup.1] is the set of all allocations. For every i [member of] I let [e.sup.i] [member of] [L.sub.[[SIGMA].sup.i].sup.1] be agent i's initial endowment; we observe that also initial endowment is contingent upon the state of nature in a way that is compatible with the information [sigma]-algebra [F.sup.i]. An allocation ([f.sup.i])[.sub.i[member of]I] [member of] [[PI].sub.i[member of]I][L.sub.[[SIGMA].sup.i].sup.1] for the pure exchange economy E is said to be attainable or feasible if [[summation].sub.i[member of]I] [f.sup.i] = [[summation].sub.i[member of]I] [e.sup.i] [mu]-a.e. The graph of the multifunction [[SIGMA].sup.i] is denoted by gph [[SIGMA].sup.i]. For every i [member of] I let [u.sup.i] : gph [[SIGMA].sup.i] [right arrow] R be a [F.sup.i] [cross product] B(S)-measurable function. This is agent i's utility function and it is state-contingent: if state [omega] is realized and agent i chooses bundle s [member of] [[SIGMA].sup.i]([omega]), then his/her resulting payoff is [u.sup.i]([omega], s). The (ex ante) expected utility of agent i's feasible consumption allocation [f.sup.i] [member of] [L.sub.[[SIGMA].sup.i].sup.i] is given by
[U.sup.i]([f.sup.i]) [colon, equals] [[integtal].sub.[OMEGA]] [u.sup.i]([omega], [f.sup.i]([omega]))[[mu].sup.i](d[omega]) = [[integral].sub.[OMEGA]] [u.sup.i]([omega], [f.sup.i]([omega]))[[phi].sup.i]([omega])[mu](d[omega]),
provided that the integral exists. Here [[phi].sup.i] is the probability density defined above.
Following Yannelis (1991), where this notion was defined for [epsilon] = 0, the private [epsilon]-core [C.sub.[epislon]](E) of the exchange economy E is defined as follows for any [epsilon] [greater than or equal to] (see also Koutsoungeras and Yannelis 1999):
Definition 1.1 The private [epsilon]-core [C.sub.[epsilon]](E) of E is the set of all attainable allocations ([f.sup.i])[.sub.i[member of]I] in [[PI].sub.i[member of]I][L.sub.[[SIGMA].sup.i].sup.1] for which the following is not true: There exist a nonempty J [subset] I and ([g.sup.j])[.sub.j[member of]J] [member of] [[PI].sub.j[member of]J][L.sub.[[SIGMA].sup.j].sup.1] such that
(1) [[summation].sub.j[member of]J] [g.sup.j] = [[summation].sub.j[member of]J] [e.sup.j] [mu]-a.e.
(2) [U.sup.j] ([g.sup.j]) > [U.sup.j] ([f.sup.j]) + [epsilon] for all j [member of] J.
For [epsilon] = 0, the private 0-core will simply be called the private core; it is denoted by C(E) and most of this paper will concentrate on this notion. Clearly, a private core...
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