Coalitional extreme desirability in finitely additive exchange economies

Abstract

We define a new notion of extreme desirability for economies in coalitional form. Through this, we obtain a finitely additive core-Walras equivalence theorem for an exchange economy with a measure space of agents and an infinite dimensional commodity space, whose positive cone has possibly empty interior.

Coalitional extreme desirability in finitely additive economies with asymmetric information

“…Our next technical results and main theorems require some continuity-like assumptions. Here, we employ assumptions similar to those in [13]. Given these assumptions, our proof for the next lemma exactly follows analogous arguments of the final step of Lemma 3.3 in [13], taking into account Lemma 3.1.…”

“…Consequently, this result and the corresponding individualistic result are the first infinite dimensional extensions to an asymmetric information framework with exact feasibility. We also point out that our results are not mere adaptations of the original definitions and results in [13], as the introduction of asymmetry and informational constraints makes it necessary to adopt new assumptions and techniques.…”

“…Assumption [P.2] claims that as almost all member of F prefer what they get from α than what they get from β (refer to [15] for a deterministic economy), they do the same under G. Assumption [P.3] is similar to Assumption (VI) in [15]. The monotonicity assumption is discussed in [P.4], which is analogous to the assumption (WM) in [13]. It is worth pointing out that our monotonicity assumption is weaker than the one in [5] for a deterministic economy.…”

“…Anyway, at this point, we must note that, also for the asymmetric information framework, Vind's ( [28]) motivations to the use of coalitional models instead of individualistic ones, and Armstrong and Richter's ( [5]) ones to the extension to a finitely additive context, are very applicable (for a complete overview, refer to [7,13]). Given these frameworks, Forges et al ( [19]) criticised on the intrinsical coalitional nature of various core notions under asymmetry of information system, which then lead Basile et al [7] to develop a notion of the core for coalitional economies, based on Yannelis's ( [30]) private information sharing rule.…”

“…Indeed, Basile et al [7] work with a finitely additive Boolean algebra of agents, but with an Euclidean space of commodities. The aim of this work is to try to fill this gap, by introducing in the asymmetric context with a Banach lattice as the commodity space, the notion of coalitional extremely desirable commodity, which is the extension of that in [13] given for complete information. We obtain a coalitional asymmetric core-Walras equivalence result in a framework whose commodity space is X + , the positive cone of a Banach lattice X having the Radon-Nikodym property (see [16]) and feasibility is defined as free disposal; note that this allows for a great variety of infinite dimensional commodity spaces interesting in economics and finance, for example, all the L p spaces for p > 1.…”