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The relationship between revealed preference and the Slutsky matrix

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  • Hosoya, Yuhki

Abstract

This paper presents a method of calculating the utility function from a smooth demand function whose Slutsky matrix is negative semi-definite and symmetric. The calculated utility function is the unique upper semi-continuous function corresponding with the demand function. Moreover, we present an axiom for demand functions. We show that under the strong axiom, this new axiom is equivalent to the existence of the corresponding continuous preference relation. If the demand function obeys this axiom, the calculated utility function is also continuous. Further, we show that the mapping from the demand function into a continuous preference relation is continuous, which ensures the applicability of our results for econometrics. Moreover, if this demand function satisfies the rank condition, then our utility function is smooth. Finally, we show that under an additional axiom, the above results hold even if the demand function has corner solutions.

Suggested Citation

  • Hosoya, Yuhki, 2017. "The relationship between revealed preference and the Slutsky matrix," Journal of Mathematical Economics, Elsevier, vol. 70(C), pages 127-146.
    • Handle: RePEc:eee:mateco:v:70:y:2017:i:c:p:127-146
    DOI: 10.1016/j.jmateco.2017.03.001
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    References listed on IDEAS

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    1. P. K. Newman & R. C. Read, 1958. "Demand Theory without a Utility Index—A Comment," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 25(3), pages 197-199.
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    Cited by:

    1. Simon Alder & Timo Boppart & Andreas Müller, 2022. "A Theory of Structural Change That Can Fit the Data," American Economic Journal: Macroeconomics, American Economic Association, vol. 14(2), pages 160-206, April.
    2. Yuhki Hosoya, 2022. "Non-Smooth Integrability Theory," Papers 2203.04770, arXiv.org, revised Mar 2024.
    3. Hosoya, Yuhki, 2020. "Recoverability revisited," Journal of Mathematical Economics, Elsevier, vol. 90(C), pages 31-41.
    4. 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.
    5. Yuhki Hosoya, 2024. "Non-smooth integrability theory," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 78(2), pages 475-520, September.
    6. Victor H. Aguiar & Roberto Serrano, 2018. "Classifying bounded rationality in limited data sets: a Slutsky matrix approach," SERIEs: Journal of the Spanish Economic Association, Springer;Spanish Economic Association, vol. 9(4), pages 389-421, November.

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