The Role of the Assumptions for the Existence of a General Equilibrium

TL;DR

This paper explains how various assumptions about individual behavior and exchange structure underpin the proof of the existence of a general equilibrium in economic theory, using fixed-point theorems.

Contribution

It clarifies the specific assumptions necessary for proving equilibrium existence in general equilibrium theory in a simple, accessible framework.

Findings

Assumptions on individual behavior are crucial for fixed-point theorem application.

The structure of exchange systems influences equilibrium existence.

Simplified models can effectively illustrate fundamental assumptions.

Abstract

General Equilibrium Theory is the benchmark of economics, especially its results concerning the efficient allocation of resources, known as the First and Second Welfare Theorems. Yet, General Equilibrium Theory is beyond the scope of most economists. This paper is pitched as the first entry point into the theory. General Equilibrium Theory proves that at least one state of equilibrium always exists. In its most general approach, it uses fixed-point theorems to this end. This paper discusses the assumptions on individuals' behaviour and the structure of the system of exchange that guarantee that the conditions of the fixed-point theorems are satisfied. The purpose is to lay bare the role each plays in proving the existence of equilibrium and provide a clear picture of the relationship between the assumptions and the result. The discussion is presented in the simplest possible setting that captures the fundamental features of commodity exchange.