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

TL;DR

This paper develops relaxation and purification methods for nonconvex variational problems in dual Banach spaces, applying them to large economies with infinite-dimensional commodities to establish Pareto optimality without convexity.

Contribution

It introduces a new relaxation technique for large economies in infinite-dimensional spaces and characterizes the saturation property of measure spaces.

Findings

Established the saturation property characterization.

Proved existence of Pareto optimal allocations without convexity.

Applied relaxation techniques to economic models.

Abstract

We formulate bang-bang, purification, and minimization principles in dual Banach spaces with Gelfand integrals and provide a complete characterization of the saturation property of finite measure spaces. We also present a new application of the relaxation technique to large economies with infinite-dimensional commodity spaces, where the space of agents is modeled as a finite measure space. We propose a "relaxation" of large economies, which is regarded as a reasonable convexification of original economies. Under the saturation hypothesis, the relaxation and purification techniques enable us to prove the existence of Pareto optimal allocations without convexity assumptions.