The Hamilton-Jacobi-Bellman Equation in Economic Dynamics with a Non-Smooth Fiscal Policy

TL;DR

This paper analyzes economic growth models with non-smooth fiscal policies, demonstrating that the value function uniquely solves the Hamilton-Jacobi-Bellman equation as a viscosity solution, and explores solution methods including differential inclusion.

Contribution

It establishes conditions under which the value function is a classical solution even with non-smooth policies, extending the applicability of HJB equations in economic models.

Findings

Value function is the unique viscosity solution to HJB equations in these models.

Differential inclusion methods are effective for non-smooth policy models.

Classical solutions exist under specific assumptions, including non-smooth Keynesian policies.

Abstract

We consider a class of economic growth models that includes the classical Ramsey--Cass--Koopmans capital accumulation model and verify that, under several assumptions, the value function of the model is the unique viscosity solution to the Hamilton--Jacobi--Bellman equation. Moreover, we discuss a solution method for these models using differential inclusion, where the subdifferential of the value function plays an important role. Next, we present an assumption under which the value function is a classical solution to the Hamilton--Jacobi--Bellman equation, and show that many economic models satisfy this assumption. In particular, our result still holds in an economic growth model in which the government takes a non-smooth Keynesian policy rule.