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

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.