  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.
