A utility maximization problem with state constraint and non-concave technology

TL;DR

This paper studies an economic growth control problem with a non-concave, unbounded state dynamics and static constraints, proving the existence of optimal controls and analyzing the value function as a viscosity solution.

Contribution

It introduces weaker assumptions on the dynamics and constraints, providing new existence results and a qualitative analysis of the value function using viscosity solutions.

Findings

Existence of optimal control for any initial state.

Value function is a continuous viscosity solution.

Model accommodates non-concave, unbounded dynamics.

Abstract

We consider an optimal control problem arising in the context of economic theory of growth, on the lines of the works by Skiba (1978) and Askenazy - Le Van (1999). The economic framework of the model is intertemporal infinite horizon utility maximization. The dynamics involves a state variable representing total endowment of the social planner or average capital of the representative dynasty. From the mathematical viewpoint, the main features of the model are the following: (i) the dynamics is an increasing, unbounded and not globally concave function of the state; (ii) the state variable is subject to a static constraint; (iii) the admissible controls are merely locally integrable in the right half-line. Such assumptions seem to be weaker than those appearing in most of the existing literature. We give a direct proof of the existence of an optimal control for any initial value of the state variable and we carry on a qualitative study of the value function; moreover, using dynamic programming methods, we show that the value function is a continuous viscosity solution of the associated Hamilton-Jacobi-Bellman equation.