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

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 whichis used to prove Brouwer's theorem is constructively proved. Some authorshave presented a constructive (or an approximate) version of Brouwer's fixedpoint theorem using Sperner's lemma. In this paper, I prove the existence ofan approximate equilibrium in a competitive economy directly by Sperner'slemma. Also I show that the existence of an approximate equilibrium leads toSperner's lemma. I follow the Bishop style constructive mathematics accordingto Bishop and Bridges (1985), Bridges and Richman (1987), and Bridges and Viţă (2006).