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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4 décembre 2021
- ISIDORE Id: 10670/1.6ef4b0...
- halshs: halshs-02993655
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https://hal.science/hal-03960256v1
info:eu-repo/semantics/OpenAccess
Mots-clés
Gale- Nikaido-Debreu Lemma Fixed Point Theorem Subdivision Simplex Sperner lemma C - Mathematical and Quantitative Methods/C.C6 - Mathematical Methods • Programming Models • Mathematical and Simulation Modeling/C.C6.C60 - General C - Mathematical and Quantitative Methods/C.C6 - Mathematical Methods • Programming Models • Mathematical and Simulation Modeling/C.C6.C62 - Existence and Stability Conditions of Equilibrium D - Microeconomics/D.D5 - General Equilibrium and DisequilibriumSujets proches
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Thanh Le et al., « Direct Proofs of the Existence of Equilibrium, the Gale-Nikaido-Debreu Lemma and the Fixed Point Theorems using Sperner's Lemma », HAL SHS (Sciences de l’Homme et de la Société), ID : 10670/1.6ef4b0...
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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.
