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

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  • Yuhki Hosoya

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

  • 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.
    • Handle: RePEc:arx:papers:2203.10595
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    References listed on IDEAS

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    1. 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.
    2. Olivier Jean Blanchard & Stanley Fischer, 1989. "Lectures on Macroeconomics," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262022834, December.
    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.
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    Cited by:

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

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