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

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  • Yuhki Hosoya

Abstract

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.

Suggested Citation

  • Yuhki Hosoya, 2021. "On the Basis of the Hamilton-Jacobi-Bellman Equation in Economic Dynamics," Papers 2102.07431, arXiv.org, revised Feb 2023.
    • Handle: RePEc:arx:papers:2102.07431
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    References listed on IDEAS

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    1. Olivier Jean Blanchard & Stanley Fischer, 1989. "Lectures on Macroeconomics," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262022834, December.
    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.
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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 May 2025.

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