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Yuhki Hosoya

Publications and source records attributed to Yuhki Hosoya.

15 recordsLinked to original sources
arXiv2026

An Equivalence Result on the Order of Differentiability in Frobenius' Theorem

This paper examines the simplest case of total differential equations that appears in the theory of foliation structures, without imposing the smoothness assumptions. This leads to a peculiar asymmetry in the differentiability of solutions. To resolve this asymmetry, this paper focuses on the differentiability of the integral manifold. When the system is locally Lipschitz, a solution is ensured to be only locally Lipschitz, but the integral manifolds must be $C^1$. When the system is $C^k$, we can only ensure the existence of a $C^k$ solution, but the integral manifolds must be $C^{k+1}$. In addition, we see a counterexample in which the system is $C^1$, but there is no $C^2$ solution. Moreover, we characterize a minimizer of an optimization problem whose objective function is a quasi-convex solution to a total differential equation. In this connection, we examine two necessary and sufficient conditions for the system in which any solution is quasi-convex.

math.AP↗
arXiv2025

The Relationship between Consumer Theories with and without Utility Maximization

To study the assumption that the utility maximization hypothesis implicitly adds to consumer theory, we consider a mathematical representation of pre-marginal revolution consumer theory based on subjective exchange ratios. We introduce two axioms on subjective exchange ratio, and show that both axioms hold if and only if consumer behavior is consistent with the utility maximization hypothesis. Moreover, we express the process for a consumer to find the transaction stopping point in terms of differential equations, and prove that the conditions for its stability are equal to the two axioms introduced in the above argument. Therefore, the consumer can find his/her transaction stopping point if and only if his/her behavior is consistent with the utility maximization hypothesis. In addition to these results, we discuss equivalence conditions for axioms to evaluate their mathematical strength, and methods for expressing the theory of subjective exchange ratios in terms of binary relations.

econ.TH↗
arXiv2025

On Gale's Contribution in Revealed Preference Theory

We investigate Gale's important paper published in 1960. This paper contains an example of a candidate of the demand function that satisfies the weak axiom of revealed preference and that is doubtful that it is a demand function of some weak order. We examine this paper and first scrutinize what Gale proved. Then we identify a gap in Gale's proof and show that he failed to show that this candidate of the demand function is not a demand function. Next, we present three complete proofs of Gale's claim. First, we construct a proof that was constructible in 1960 by a fact that Gale himself demonstrated. Second, we construct a modern and simple proof using Shephard's lemma. Third, we construct a proof that follows the direction that Gale originally conceived. Our conclusion is as follows: although, in 1960, Gale was not able to prove that the candidate of the demand function that he constructed is not a demand function, he substantially proved it, and therefore it is fair to say that the credit for finding a candidate of the demand function that satisfies the weak axiom but is not a demand function is attributed to Gale.

econ.TH↗
arXiv2025

The Hamilton-Jacobi-Bellman Equation in Economic Dynamics with a Non-Smooth Fiscal Policy

We consider a class of economic growth models that includes the classical Ramsey--Cass--Koopmans capital accumulation model and verify that, under several assumptions, the value function of the model is the unique viscosity solution to the Hamilton--Jacobi--Bellman equation. Moreover, we discuss a solution method for these models using differential inclusion, where the subdifferential of the value function plays an important role. Next, we present an assumption under which the value function is a classical solution to the Hamilton--Jacobi--Bellman equation, and show that many economic models satisfy this assumption. In particular, our result still holds in an economic growth model in which the government takes a non-smooth Keynesian policy rule.

econ.TH↗
arXiv2025

A Note on the Continuity of Expected Utility Functions

In this paper, we study the continuity of expected utility functions, and derive a necessary and sufficient condition for a weak order on the space of simple probabilities to have a continuous expected utility function. We also verify that almost the same condition is necessary and sufficient for a weak order on the space of probabilities with compact-support to have a continuous expected utility function.

econ.TH↗
arXiv2024

On the Uniqueness and Stability of the Equilibrium Price in Quasi-Linear Economies

In this paper, we show that if every consumer in an economy has a quasi-linear utility function, then the normalized equilibrium price is unique, and is locally stable with respect to the tâtonnement process. Our study can be seen as that extends the results in partial equilibrium theory to economies with more than two dimensional consumption space. Moreover, we discuss the surplus analysis in such economies.

econ.TH↗
arXiv2024

Non-Smooth Integrability Theory

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.

econ.TH↗
arXiv2024

On the Fragility of the Basis on the Hamilton-Jacobi-Bellman Equation in Economic Dynamics

In this paper, we provide an example of the optimal growth model in which there exist infinitely many solutions to the Hamilton-Jacobi-Bellman equation but the value function does not satisfy this equation. We consider the cause of this phenomenon, and find that the lack of a solution to the original problem is crucial. We show that under several conditions, there exists a solution to the original problem if and only if the value function solves the Hamilton-Jacobi-Bellman equation. Moreover, in this case, the value function is the unique nondecreasing concave solution to the Hamilton-Jacobi-Bellman equation. We also show that without our conditions, this uniqueness result does not hold.

econ.TH↗
arXiv2023

Utilitarian Theorems and Equivalence of Utility Theories

In this paper, we consider an environment in which the utilitarian theorem for the NM utility function derived by Harsanyi and the utilitarian theorem for Alt's utility function derived by Harvey hold simultaneously, and prove that the NM utility function coincides with Alt's utility function under this setup. This result is so paradoxical that we must presume that at least one of the utilitarian theorems contains a strong assumption. We examine the assumptions one by one and conclude that one of Harsanyi's axioms is strong.

econ.TH↗
arXiv2023

On the Basis of the Hamilton-Jacobi-Bellman Equation in Economic Dynamics

We consider the classical Ramsey-Cass-Koopmans capital accumulation model and present three examples in which the Hamilton-Jacobi-Bellman (HJB) equation is neither necessary nor sufficient for a function to be the value function. Next, we present assumptions under which the HJB equation becomes a necessary and sufficient condition for a function to be the value function, and using this result, we propose a new method for solving the original problem using the solution to the HJB equation. Our assumptions are so mild that many macroeconomic growth models satisfy them. Therefore, our results ensure that the solution to the HJB equation is rigorously the value function in many macroeconomic models, and present a new solving method for these models.

econ.TH↗
arXiv2023

A Rigorous Proof of the Index Theorem for Economists

This paper provides a rigorous and gap-free proof of the index theorem used in the theory of regular economy. In the index theorem that is the subject of this paper, the assumptions for the excess demand function are only several usual assumptions and continuous differentiability around any equilibrium price, and thus it has a form that is applicable to many economies. However, the textbooks on this theme contain only abbreviated proofs and there is no known monograph that contains a rigorous proof of this theorem. Hence, the purpose of this paper is to make this theorem available to more economists by constructing a readable proof.

econ.TH↗
arXiv2022

Differential Characterization of Quasi-Concave Functions without Twice Differentiability

This paper presents a necessary and sufficient condition for a real-valued function defined on an open and convex subset of a Banach space to be quasi-concave, and a sufficient condition for such a function to be strictly quasi-concave. These conditions are applicable to continuously differentiable functions that satisfy a mild additional assumption, and do not require the functions to be twice differentiable. Because this additional assumption is trivially satisfied for twice continuously differentiable functions, our results are pure extensions to classical results.

math.OC↗
arXiv2022

On the Approximate Purification of Mixed Strategies in Games with Infinite Action Sets

We consider a game in which the action set of each player is uncountable, and show that, from weak assumptions on the common prior, any mixed strategy has an approximately equivalent pure strategy. The assumption of this result can be further weakened if we consider the purification of a Nash equilibrium. Combined with the existence theorem for a Nash equilibrium, we derive an existence theorem for a pure strategy approximated Nash equilibrium under sufficiently weak assumptions. All of the pure strategies we derive in this paper can take a finite number of possible actions.

econ.TH↗
arXiv2021

An Axiom for Concavifiable Preferences in View of Alt's Theory

We present a necessary and sufficient condition for Alt's system to be represented by a continuous utility function. Moreover, we present a necessary and sufficient condition for this utility function to be concave. The latter condition can be seen as an extension of Gossen's first law, and thus has an economic interpretation. Together with the above results, we provide a necessary and sufficient condition for Alt's utility to be continuously differentiable.

econ.TH↗