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On the fragility of the basis on the Hamilton–Jacobi–Bellman equation in economic dynamics

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  • Hosoya, Yuhki

Abstract

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.

Suggested Citation

  • Hosoya, Yuhki, 2024. "On the fragility of the basis on the Hamilton–Jacobi–Bellman equation in economic dynamics," Journal of Mathematical Economics, Elsevier, vol. 111(C).
    • Handle: RePEc:eee:mateco:v:111:y:2024:i:c:s0304406824000028
    DOI: 10.1016/j.jmateco.2024.102940
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    References listed on IDEAS

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    1. Yuhki Hosoya, 2022. "On the Fragility of the Basis on the Hamilton-Jacobi-Bellman Equation in Economic Dynamics," Papers 2203.10595, arXiv.org, revised Jan 2024.
    2. Benveniste, L M & Scheinkman, J A, 1979. "On the Differentiability of the Value Function in Dynamic Models of Economics," Econometrica, Econometric Society, vol. 47(3), pages 727-732, May.
    3. David Cass, 1965. "Optimum Growth in an Aggregative Model of Capital Accumulation," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 32(3), pages 233-240.
    4. Yuhki Hosoya, 2021. "On the Basis of the Hamilton-Jacobi-Bellman Equation in Economic Dynamics," Papers 2102.07431, arXiv.org, revised Feb 2023.
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    Cited by:

    1. Yuhki Hosoya, 2024. "The Hamilton-Jacobi-Bellman Equation in Economic Dynamics with a Non-Smooth Fiscal Policy," Papers 2405.16643, arXiv.org, revised May 2025.

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