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.
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At the 2022 Fall Meeting of the Japan Economic Association, the author received helpful comments and suggestions from Hisatoshi Tanaka. The author is grateful to him. The author is also grateful to an associate editor for his/her kind comments and suggestions. This work was supported by JSPS KAKENHI Grant Number JP21K01403.
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Hosoya, Y. Non-smooth integrability theory. Econ Theory 78, 475–520 (2024). https://doi.org/10.1007/s00199-024-01564-x
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DOI: https://doi.org/10.1007/s00199-024-01564-x
Keywords
- Integrability theory
- Locally Lipschitz demand function
- Rademacher’s theorem
- Completeness of the space of demand functions
- Consistency of the estimation