Relaxed large economies with infinite-dimensional commodity spaces: The existence of Walrasian equilibria

Abstract

Whereas "convexification by aggregation" is a well-understood procedure in mathematical economics, "convexification by randomization" has largely been limited to theories of statistical decision-making, optimal control and non-cooperative games. In this paper, in the context of classical Walrasian general equilibrium theory, we offer a comprehensive treatment of relaxed economies and their relaxed Walrasian equilibria: our results pertain to a setting with a finite or a continuum of agents, and a continuum of commodities modeled either as an ordered separable Banach space or as an L ∞ -space. As a substantive consequence, we demonstrate that the convexity hypothesis can be removed from the original large economy under the saturation hypothesis, and that existing results in the antecedent literature can be effortlessly recovered.

Relaxation and Purification for Nonconvex Variational Problems in Dual Banach Spaces: The Minimization Principle in Saturated Measure Spaces

“…The extension formula in (5.2) conforms to the relaxation technique investigated in Section 3. As observed in [28], relaxed preferences are also consistent with the axioms for the "expected utility hypothesis" and the continuous function ϕ(t, ·) corresponds to the "von Neumann-Morgenstern utility function" for Denote by E R = {(T, Σ, µ), Π(X), R , δ ω(·) } the relaxed economy induced by the original economy E = {(T, Σ, µ), X, , ω}, where the initial endowment ω(t) ∈ X of each agent is identified with a Dirac measure δ ω(t) ∈ ∆(X), and hence, δ ω(·) ∈ R(T, X). Let ı X be the identity map on X.…”

“…The existence of (relaxed) Walrasian equilibria with free disposal is investigated in [28] for the commodity space with the dual space of L ∞ = (L 1 ) * . The crucial argument for the proof is the nonemptiness of the norm interior of the positive cone of L ∞ , which fails to hold for general dual spaces.…”

The necessity of nowhere equivalence

“…It shows that if pure strategy equilibria exist in all large games with a fixed uncountable compact metric action space, then for any n ≥ 1, the existence result can be extended to large games with any uncountable compact absolute retract of R n as the action space. 19 Though the argument is very much similar to the one in [33,Section 6], we provide a detailed proof of Lemma 6 for the sake of completeness. Lemma 6.…”

“…As noted in[15], atomless Loeb probability spaces are saturated. For some other applications of Loeb and saturated probability spaces, see, for example,[5],[7],[8],[10],[16],[19],[24],[34],[35], and[36].3 When the target space is a Banach space, one can also consider Bochner and Gelfand integration of correspondences. The same kind of regularity properties were shown to hold under Loeb/saturated probability spaces in[30],[38] and[39], while the necessity of saturation for these properties was indicated in[30] and[39].…”

A bang-bang principle for the conditional expectation vector measure

“…However, an infinite dimensional version of such result established by Knowles [29] can be used to establish a purification principle as in Askoura [2]. Furthermore, many works succeeded to recover this property and to establish adequate bang-bang and purification results by using Maharam types and saturated measure spaces; see Greinecker and Podczeck [10], Khan and Sagara [22,23,24,25], Sagara [32] for infinite dimensional Lyapunov convexity theorems, purification processes and applications to the integration of set-valued mappings, equilibrium theory and control systems. Refer to Keisler and Sun [19] and the literature therein for further prevalent studies on this direction.…”