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Recoverability revisited

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  • Hosoya, Yuhki

Abstract

This study considers the uniqueness problem of the preference relation corresponding to a demand function, which is called the “recoverability problem”. We show that if a demand function has sufficiently wide range and is income-Lipschitzian, then there exists a unique corresponding upper semi-continuous preference relation. Moreover, we explicitly construct a utility function that represents this preference relation. Compared with related research, a feature of our result is that it ensures not only the uniqueness, but also the existence of the corresponding upper semi-continuous preference relation. Further, we introduce two axioms related to demand functions, and show that these axioms are equivalent to the continuity of our preference relation in the interior of the consumption set. In addition to these results, we present three examples that explain why our requirements (including the upper semi-continuity of preference relations and the wide range requirement and income-Lipschitzian property of demand functions) are necessary, and a further two examples in which there is no continuous preference relation corresponding to the given demand function.

Suggested Citation

  • Hosoya, Yuhki, 2020. "Recoverability revisited," Journal of Mathematical Economics, Elsevier, vol. 90(C), pages 31-41.
    • Handle: RePEc:eee:mateco:v:90:y:2020:i:c:p:31-41
    DOI: 10.1016/j.jmateco.2020.05.009
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    References listed on IDEAS

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    1. Mas-Colell, Andreu, 1977. "The Recoverability of Consumers' Preferences from Market Demand Behavior," Econometrica, Econometric Society, vol. 45(6), pages 1409-1430, September.
    2. Chipman, John S. & Moore, James C., 1977. "Continuity and uniqueness in revealed preference," Journal of Mathematical Economics, Elsevier, vol. 4(2), pages 139-162, August.
    3. Andreu Mas-Colell, 1978. "On Revealed Preference Analysis," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 45(1), pages 121-131.
    4. Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, number 9780195102680.
    5. Hosoya, Yuhki, 2017. "The relationship between revealed preference and the Slutsky matrix," Journal of Mathematical Economics, Elsevier, vol. 70(C), pages 127-146.
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    Cited by:

    1. Yuhki Hosoya, 2021. "Consumer Optimization and a First-Order PDE with a Non-Smooth System," SN Operations Research Forum, Springer, vol. 2(4), pages 1-36, December.
    2. Yuhki Hosoya, 2022. "Non-Smooth Integrability Theory," Papers 2203.04770, arXiv.org, revised Mar 2024.

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