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

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