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From Ordients to Optimization: Substitution Effects without Differentiability

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  • Ludovic Renou
  • Karl H. Schlag

Abstract

This paper introduces the concept of ordient for binary relations (preferences), a relative of the concept of gradients for functions (utilities). The lexicographic order, albeit not representable, has an ordient. Not only binary relations representable by differentiable functions have an ordient, but also preferences representable by non-differentiable functions might. We characterize the constrained maxima of binary relations through ordients and provide an implicit function theorem and an envelope theorem. Ordients have a natural economic interpretation as marginal rates of substitution. We apply our results to the classic problem of maximizing preferences over budget sets.

Suggested Citation

  • Ludovic Renou & Karl H. Schlag, 2009. "From Ordients to Optimization: Substitution Effects without Differentiability," Discussion Papers in Economics 09/6, Division of Economics, School of Business, University of Leicester.
  • Handle: RePEc:lec:leecon:09/6
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    File URL: https://www.le.ac.uk/economics/research/RePEc/lec/leecon/dp09-6.pdf
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    References listed on IDEAS

    as
    1. Mas-Colell,Andreu, 1990. "The Theory of General Economic Equilibrium," Cambridge Books, Cambridge University Press, number 9780521388702, December.
    2. Paul Milgrom & Ilya Segal, 2002. "Envelope Theorems for Arbitrary Choice Sets," Econometrica, Econometric Society, vol. 70(2), pages 583-601, March.
    3. Ernst Fehr & Klaus M. Schmidt, 1999. "A Theory of Fairness, Competition, and Cooperation," The Quarterly Journal of Economics, Oxford University Press, vol. 114(3), pages 817-868.
    4. Apesteguia, Jose & Ballester, Miguel A., 2013. "Choice by sequential procedures," Games and Economic Behavior, Elsevier, vol. 77(1), pages 90-99.
    5. Paola Manzini & Marco Mariotti, 2007. "Sequentially Rationalizable Choice," American Economic Review, American Economic Association, vol. 97(5), pages 1824-1839, December.
    6. Debreu, Gerard, 1972. "Smooth Preferences," Econometrica, Econometric Society, vol. 40(4), pages 603-615, July.
    7. Koichi Tadenuma, 2005. "Egalitarian-equivalence and the Pareto principle for social preferences," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 24(3), pages 455-473, June.
    8. Houy, Nicolas & Tadenuma, Koichi, 2009. "Lexicographic compositions of multiple criteria for decision making," Journal of Economic Theory, Elsevier, vol. 144(4), pages 1770-1782, July.
    9. Tadenuma, Koichi, 2002. "Efficiency First or Equity First? Two Principles and Rationality of Social Choice," Journal of Economic Theory, Elsevier, vol. 104(2), pages 462-472, June.
    10. Driffill, John & Rotondi, Zeno, 2004. "Monetary Policy and Lexicographic Preference Ordering," CEPR Discussion Papers 4247, C.E.P.R. Discussion Papers.
    11. Luca Rigotti & Chris Shannon & Tomasz Strzalecki, 2008. "Subjective Beliefs and ex ante Trade," Econometrica, Econometric Society, vol. 76(5), pages 1167-1190, September.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Binary relation; ordient; maxima; envelope theorem; implicit function theorem;
    All these keywords.

    JEL classification:

    • C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling
    • D01 - Microeconomics - - General - - - Microeconomic Behavior: Underlying Principles

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