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Non-Smooth Integrability Theory

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  • Yuhki Hosoya

Abstract

We study a method for calculating the utility function from a candidate of a demand function that is not differentiable, but is locally Lipschitz. Using this method, we obtain two new necessary and sufficient conditions for a candidate of a demand function to be a demand function. The first concerns the Slutsky matrix, and the second is the existence of a concave solution to a partial differential equation. Moreover, we show that the upper semi-continuous weak order that corresponds to the demand function is unique, and that this weak order is represented by our calculated utility function. We provide applications of these results to econometric theory. First, we show that, under several requirements, if a sequence of demand functions converges to some function with respect to the metric of compact convergence, then the limit is also a demand function. Second, the space of demand functions that have uniform Lipschitz constants on any compact set is compact under the above metric. Third, the mapping from a demand function to the calculated utility function becomes continuous. We also show a similar result on the topology of pointwise convergence.

Suggested Citation

  • Yuhki Hosoya, 2022. "Non-Smooth Integrability Theory," Papers 2203.04770, arXiv.org, revised Mar 2024.
    • Handle: RePEc:arx:papers:2203.04770
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    References listed on IDEAS

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    1. Hosoya, Yuhki, 2020. "Recoverability revisited," Journal of Mathematical Economics, Elsevier, vol. 90(C), pages 31-41.
    2. Debreu, Gerard, 1974. "Excess demand functions," Journal of Mathematical Economics, Elsevier, vol. 1(1), pages 15-21, March.
    3. Mas-Colell, Andreu, 1977. "The Recoverability of Consumers' Preferences from Market Demand Behavior," Econometrica, Econometric Society, vol. 45(6), pages 1409-1430, September.
    4. Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, number 9780195102680.
    5. Hosoya, Yuhki, 2013. "Measuring utility from demand," Journal of Mathematical Economics, Elsevier, vol. 49(1), pages 82-96.
    6. 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.
    7. 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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