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

TL;DR

This paper examines the conditions under which the Hamilton-Jacobi-Bellman equation accurately characterizes the value function in economic growth models, proposing a new solution method applicable to many macroeconomic models.

Contribution

It identifies mild assumptions that make the HJB equation both necessary and sufficient for the value function, and introduces a new method for solving macroeconomic models using HJB solutions.

Findings

HJB equation is not always necessary or sufficient for the value function.

Under mild assumptions, HJB becomes necessary and sufficient.

A new method for solving macroeconomic models using HJB is proposed.

Abstract

We consider the classical Ramsey-Cass-Koopmans capital accumulation model and present three examples in which the Hamilton-Jacobi-Bellman (HJB) equation is neither necessary nor sufficient for a function to be the value function. Next, we present assumptions under which the HJB equation becomes a necessary and sufficient condition for a function to be the value function, and using this result, we propose a new method for solving the original problem using the solution to the HJB equation. Our assumptions are so mild that many macroeconomic growth models satisfy them. Therefore, our results ensure that the solution to the HJB equation is rigorously the value function in many macroeconomic models, and present a new solving method for these models.