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Excess demand approach with non-convexity and discontinuity: a generalization of the Gale–Nikaido–Kuhn–Debreu lemma

Author

Listed:
  • M. Ali Khan

    (Johns Hopkins University)

  • Richard P. McLean

    (Rutgers University)

  • Metin Uyanik

    (University of Queensland)

Abstract

This paper provides a three-fold generalization of the Gale–Nikaido–Kuhn–Debreu lemma, a fundamental result for classical Walrasian general equilibrium theory. It weakens the (i) convexity assumption on the given (excess demand) correspondence, and (ii) the continuity assumption by synthesizing both the majorization and inclusion approaches, and uses these results to derive a (iii) theorem on the existence of a Walrasian equilibrium in an exchange economy with externalities, and non-ordered, non-convex, price-dependent preferences under a weaker continuity assumption that is assumed in the literature. It also highlights a trade-off between the weakened continuity and convexity assumptions.

Suggested Citation

  • M. Ali Khan & Richard P. McLean & Metin Uyanik, 2025. "Excess demand approach with non-convexity and discontinuity: a generalization of the Gale–Nikaido–Kuhn–Debreu lemma," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 79(4), pages 1167-1190, June.
    • Handle: RePEc:spr:joecth:v:79:y:2025:i:4:d:10.1007_s00199-025-01641-9
    DOI: 10.1007/s00199-025-01641-9
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    Keywords

    GNKD lemma; Non-convex; Majorization; Inclusion;
    All these keywords.

    JEL classification:

    • C00 - Mathematical and Quantitative Methods - - General - - - General
    • D00 - Microeconomics - - General - - - General
    • D01 - Microeconomics - - General - - - Microeconomic Behavior: Underlying Principles
    • D90 - Microeconomics - - Micro-Based Behavioral Economics - - - General
    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • O21 - Economic Development, Innovation, Technological Change, and Growth - - Development Planning and Policy - - - Planning Models; Planning Policy

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