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Direct Proofs of the Existence of Equilibrium, the Gale-Nikaido-Debreu Lemma and the Fixed Point Theorems using Sperner’s Lemma

Le, Thanh and Le Van, Cuong and Pham, Ngoc-Sang and Sağlam, Çağrı (2020): Direct Proofs of the Existence of Equilibrium, the Gale-Nikaido-Debreu Lemma and the Fixed Point Theorems using Sperner’s Lemma.

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Abstract

In this paper we use only Sperner’s lemma to prove the existence of general equilibrium for a competitive economy with production or with uncertainty and financial assets. We show that the direct use of Sperner’s lemma together with Carathéodory’s convexity theorem and basic properties of topology such as partition of unit, finite covering of a compact set allow us to bypass the Kakutani fixed point theorem even in establishing the Gale-Nikaido-Debreu Lemma. We also provide a new proof of the Kakutani fixed point theorem based on Sperner’s lemma.

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