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

TL;DR

This paper examines the limitations of the Hamilton-Jacobi-Bellman equation in economic growth models, showing that solutions may not always correspond to the original problem and highlighting conditions for solution uniqueness.

Contribution

It provides an example where multiple solutions exist but do not satisfy the original problem, and establishes conditions for the equivalence and uniqueness of solutions.

Findings

Existence of multiple solutions to the HJB equation without matching the original problem.

Conditions under which the value function solves the HJB equation and is unique.

Without these conditions, the solution's uniqueness does not hold.

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.