Optimal Consumption in the Stochastic Ramsey Problem without Boundedness Constraints

TL;DR

This paper develops a novel framework for the stochastic Ramsey problem allowing unbounded consumption, characterizes the value function via nonlinear elliptic equations, and demonstrates that removing boundedness constraints improves expected utility.

Contribution

It introduces a new approach to the stochastic Ramsey problem without bounded consumption constraints, using probabilistic and viscosity solutions techniques for characterization.

Findings

Removing boundedness constraints increases expected utility at all wealth levels.

The value function is uniquely characterized as a classical solution to a nonlinear elliptic equation.

The controlled state process is shown to be non-explosive and strictly positive.

Abstract

This paper investigates optimal consumption in the stochastic Ramsey problem with the Cobb-Douglas production function. Contrary to prior studies, we allow for general consumption processes, without any a priori boundedness constraint. A non-standard stochastic differential equation, with neither Lipschitz continuity nor linear growth, specifies the dynamics of the controlled state process. A mixture of probabilistic arguments are used to construct the state process, and establish its non-explosiveness and strict positivity. This leads to the optimality of a feedback consumption process, defined in terms of the value function and the state process. Based on additional viscosity solutions techniques, we characterize the value function as the unique classical solution to a nonlinear elliptic equation, among an appropriate class of functions. This characterization involves a condition on the limiting behavior of the value function at the origin, which is the key to dealing with unbounded consumptions. Finally, relaxing the boundedness constraint is shown to increase, strictly, the expected utility at all wealth levels.