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Understanding Fixed Point Theorems

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  • Arnold, Lutz G.

Abstract

Fixed point theorems are the standard tool used to prove the existence of equilibria in mathematical economics. This paper shows how to prove a slight generalization of Brouwer's and Kakutani's fixed point theorems using the familiar techniques of drawing and shifting curves in the plane and is, therefore, intelligible without advanced knowledge of topology. This makes proofs of fixed point theorems accessible to a broader audience.Kurvenverschieben in der Ebene' beweisen lassen. Das wichtigste Instrument für Existenzsätze in der mathematischen Volkswirtschaftslehre sind Fixpunkttheoreme. Dieses Papier zeigt, wie sich Brouwers Fixpunktsatz und einige geringfügige Verallgemeinerungen durch 'Kurvenverschieben in der Ebene' beweisen lassen. Weil keine tiefen topologischen Vorkenntnisse notwendig sind, ist der grafische Beweis für einen breiten Adressatenkreis zugänglich.

Suggested Citation

  • Arnold, Lutz G., 2003. "Understanding Fixed Point Theorems," University of Regensburg Working Papers in Business, Economics and Management Information Systems 379, University of Regensburg, Department of Economics.
  • Handle: RePEc:bay:rdwiwi:139
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    File URL: https://epub.uni-regensburg.de/4494/1/Understanding_fixed_point_theorems.pdf
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    Keywords

    Brouwerscher Fixpunktsatz ; Allgemeines Gleichgewicht ; Nash-Gleichgewicht; ; Fixed Point Theorems ; General Equilibrium ; Nash Equilibrium;

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium

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