Measuring utility from demand

Hosoya, Yuhki

doi:10.1016/j.jmateco.2012.10.001

Cited by 5 publications

(6 citation statements)

References 15 publications

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“…Actually, the requirement that R(f ) is open in Ω does not used in the proof, and thus we can show by the same proof as this theorem that under assumptions of our Theorem If f is a continuously differentiable demand function such that the rank of S f (p, m) is always n − 1, then we can show that the inverse demand correspondence G f (x) is a single-valued continuously differentiable function, and thus the C axiom is automatically satisfied. For a proof, see Proposition 1 of Hosoya (2013). In this regard, if (f k ) is a sequence on F L such that every f k is continuously differentiable and the rank of…”

Section: Second Result: Continuity Of Calculationmentioning

confidence: 99%

Non-Smooth Integrability Theory

Hosoya¹

2022

Preprint

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We study a calculation method for utility function from a candidate of a demand function that is not differentiable but locally Lipschitz. Using this method, we obtain two new necessary and sufficient condition for a candidate of a demand function to be a demand function. First is the symmetry and negative semi-definiteness of the Slutsky matrix, and second is the global existence of a unique concave solution for some partial differential equation. Moreover, we present under several assumptions, the upper semi-continuous weak order that corresponds to the demand function is unique, and this weak order is represented by our calculated utility function. We also 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 a metric corresponds to the topology of compact convergence, then the limit function is also a demand function. Second, the space of demand functions that have uniform Lipschitz constant on any compact set is complete under the above metric. Third, under some additional requirement, the mapping from a demand function into a utility function we calculate becomes continuous. This implies that a consistent estimation method for the demand function immediately defines a consistent estimation method for the utility function by using our calculation method.

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Section: Second Result: Continuity Of Calculationmentioning

confidence: 99%

Non-Smooth Integrability Theory

Hosoya¹

2022

Preprint

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“…Next, we can consider using the extension of Frobenius' theorem treated in this paper for the integrability problem in economics. Frobenius' theorem was classically used by Antonelli [18] to solve this problem, and recently, Hosoya [7] submitted a complete solution to the continuous differentiable problem. However, this is an area where no one has yet attempted to solve the problem without differentiability.…”

Section: Discussionmentioning

confidence: 99%

“…Now, suppose that V ⊂ ℝ n is star-convex centered at x * . 7 If u ∶ V → ℝ is a solution to (6), then for every x ∈ V , the following function c(t;x, u * ) = u((1 − t)x * + tx) satisfies the following ordinary differential equation (ODE):…”

Section: Recall the Pde (2)mentioning

confidence: 99%

“…17 There is another treatment of integrability theory that uses the Frobenius theorem. See, for example, Hosoya [7] for detailed arguments.…”

Section: Examplementioning

confidence: 99%

“…to verify this claim rigorously, we need extended Young's theorem. See Lemma 3.2 7. A set V in some linear space is star-convex centered at x if and only if for every y ∈ V , the segment[x, y] = {(1 − t)x + ty|0 ≤ t ≤ 1} is included in V.…”

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Consumer Optimization and a First-Order PDE with a Non-Smooth System

Hosoya

2021

Oper. Res. Forum

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We study a first-order nonlinear partial differential equation and present a necessary and sufficient condition for the global existence of its solution in a non-smooth environment. Using this result, we prove a local existence theorem for a solution to this differential equation. Moreover, we present two applications of this result. The first concerns an inverse problem called the integrability problem in microeconomic theory and the second concerns an extension of Frobenius’ theorem.

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A Theory for Estimating Consumer’s Preference from Demand

Hosoya

2015

Advances in Mathematical Economics

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Measuring utility from demand

Cited by 5 publications

References 15 publications

Non-Smooth Integrability Theory

Non-Smooth Integrability Theory

Consumer Optimization and a First-Order PDE with a Non-Smooth System

A Theory for Estimating Consumer’s Preference from Demand

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