Fuzzy Core Equivalence in Large Economies: A Role for the Infinite-Dimensional Lyapunov Theorem

TL;DR

This paper establishes the equivalence of the fuzzy core and the core in large economies using the infinite-dimensional Lyapunov theorem, showing that minimal assumptions suffice when the measure space is saturated.

Contribution

It demonstrates the equivalence of the fuzzy core and the core under minimal assumptions, leveraging the infinite-dimensional Lyapunov theorem in economic models.

Findings

Fuzzy core and core are equivalent under minimal assumptions.

Coincidence of core, fuzzy core, and restricted core in large economies.

Minimal structural assumptions are sufficient for core equivalence.

Abstract

We present the equivalence between the fuzzy core and the core under minimal assumptions. Due to the exact version of the Lyapunov convexity theorem in Banach spaces, we clarify that the additional structure of commodity spaces and preferences is unnecessary whenever the measure space of agents is "saturated". As a spin-off of the above equivalence, we obtain the coincidence of the core, the fuzzy core, and the Schmeidler's restricted core under minimal assumptions. The coincidence of the fuzzy core and the restricted core has not been articulated anywhere.