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

Abstract EN

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).