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

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  • Yuhki Hosoya

Abstract

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.

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  • 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.
    • Handle: RePEc:arx:papers:2405.16643
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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. Yuhki Hosoya, 2021. "On the Basis of the Hamilton-Jacobi-Bellman Equation in Economic Dynamics," Papers 2102.07431, arXiv.org, revised Feb 2023.
    3. Olivier Jean Blanchard & Stanley Fischer, 1989. "Lectures on Macroeconomics," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262022834, April.
    4. 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.
    5. 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).
    6. 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.
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