Equivalence between the Existence of an Approximate Equilibrium in a Competitive Economy and Sperner's Lemma: A Constructive Analysis

Yasuhito Tanaka

doi:10.5402/2011/384625

Abstract

Brouwer's fixed point theorem cannot be constructively proved, so the existence of an equilibrium in a competitive economy also cannot be constructively proved. On the other hand, Sperner's lemma which is used to prove Brouwer's theorem is constructively proved. Some authors have presented a constructive (or an approximate) version of Brouwer's fixed point theorem using Sperner's lemma. In this paper, I prove the existence of an approximate equilibrium in a competitive economy directly by Sperner's lemma. Also I show that the existence of an approximate equilibrium leads to Sperner's lemma. I follow the Bishop style constructive mathematics according to Bishop and Bridges (1985), Bridges and Richman (1987), and Bridges and Vîţă (2006).

Highlights

It is often demonstrated that Brouwer’s fixed point theorem cannot be constructively proved
I follow the Bishop style constructive mathematics according to Bishop and Bridges 3, Bridges and Richman 4, and Bridges and Vıta 5
In a competitive exchange economy, if the excess demand functions for the goods are uniformly continuous about their prices, there exists an approximate equilibrium and one can constructively find the prices at the approximate equilibrium

Summary

Introduction

It is often demonstrated that Brouwer’s fixed point theorem cannot be constructively proved. Sperner’s lemma which is used to prove Brouwer’s theorem is constructively proved. Some authors have presented a constructive or an approximate version of Brouwer’s fixed point theorem using Sperner’s lemma. Brouwer’s fixed point theorem can be constructively proved in its constructive version. This paper presents such a proof and shows that the existence of an approximate equilibrium leads to Sperner’s lemma. The proof of Sperner’s lemma will be presented. This proof is a standard constructive proof. I follow the Bishop style constructive mathematics according to Bishop and Bridges 3 , Bridges and Richman 4 , and Bridges and Vıta 5

Sperner’s Lemma

Approximate Equilibrium in a Competitive Exchange Economy

Since n j

From the Existence of an Approximate Equilibrium to Sperner’s Lemma

Concluding Remark