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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
    as

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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
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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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