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

Thanh Le, Cuong Le Van (), Ngoc-Sang Pham and Çağrı Sağlam
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Cuong Le Van: PSE - Paris School of Economics - UP1 - Université Paris 1 Panthéon-Sorbonne - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris Sciences et Lettres - EHESS - École des hautes études en sciences sociales - ENPC - École nationale des ponts et chaussées - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement, IPAG Business School, TIMAS - Institute of Mathematics and Applied Science, CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique

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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.

Keywords: Gale- Nikaido-Debreu Lemma; Fixed Point Theorem; Subdivision; Simplex; Sperner lemma (search for similar items in EconPapers)
Date: 2021-12-04
Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-02993655v2
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Related works:
Working Paper: Direct Proofs of the Existence of Equilibrium, the Gale-Nikaido-Debreu Lemma and the Fixed Point Theorems using Sperner's Lemma (2021)
Working Paper: Direct Proofs of the Existence of Equilibrium, the Gale-Nikaido-Debreu Lemma and the Fixed Point Theorems using Sperner’s Lemma (2020)
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