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Tropical analysis: with an application to indivisible goods

Author

Listed:
  • Bedard, Nicholas Charles

    (Department of Economics, Wilfrid Laurier University)

  • Goeree, Jacob K

    (Business School, University of New South Wales)

Abstract

We establish the Subgradient Theorem for monotone correspondences -- a monotone correspondence is equal to the subdifferential of a potential if and only if it is conservative, i.e. its integral along a closed path vanishes irrespective of the selection from the correspondence along the path. We prove two attendant results: the Potential Theorem, whereby a conservative monotone correspondence can be integrated up to a potential, and the Duality Theorem, whereby the potential has a dual whose subdifferential is a conservative monotone correspondence that is the inverse of the original correspondence. We use these results to reinterpret and extend BaldwinKlemperer2019's BaldwinKlemperer2019 characterization of demand in economies with indivisible goods.

Suggested Citation

  • Bedard, Nicholas Charles & Goeree, Jacob K, 0. "Tropical analysis: with an application to indivisible goods," Theoretical Economics, Econometric Society.
    • Handle: RePEc:the:publsh:5831
    as

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    References listed on IDEAS

    as
    1. Manelli, Alejandro M. & Vincent, Daniel R., 2007. "Multidimensional mechanism design: Revenue maximization and the multiple-good monopoly," Journal of Economic Theory, Elsevier, vol. 137(1), pages 153-185, November.
    2. Jacob K. Goeree & Alexey Kushnir, 2023. "A Geometric Approach to Mechanism Design," Journal of Political Economy Microeconomics, University of Chicago Press, vol. 1(2), pages 321-347.
    3. Krishna, Vijay & Maenner, Eliot, 2001. "Convex Potentials with an Application to Mechanism Design," Econometrica, Econometric Society, vol. 69(4), pages 1113-1119, July.
    4. Harold Hotelling, 1932. "Edgeworth's Taxation Paradox and the Nature of Demand and Supply Functions," Journal of Political Economy, University of Chicago Press, vol. 40(5), pages 577-577.
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    More about this item

    Keywords

    Conservative correspondences; subgradient theorem; potential theorem; Fenchel duality; Fenchel's duality theorem; tropical geometry; convex analysis; normally labeled polyhedral subdivisions; subdifferentials; indivisible goods;
    All these keywords.

    JEL classification:

    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools

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