Unrestricted Consumption under a Deterministic Wealth and an Ornstein-Uhlenbeck Process as a Discount Rate

TL;DR

This paper derives the optimal unrestricted consumption strategy for an individual with wealth and income modeled linearly, where the discount rate follows an Ornstein-Uhlenbeck process, enabling explicit solutions unlike restricted cases.

Contribution

It provides the first explicit solution for the optimal consumption and value function under a stochastic discount rate modeled as an Ornstein-Uhlenbeck process.

Findings

Explicit optimal consumption strategy derived.

Value function explicitly characterized.

Applicable to models with stochastic discount rates.

Abstract

We consider an individual or household endowed with an initial capital and an income, modeled as a linear function of time. Assuming that the discount rate evolves as an Ornstein-Uhlenbeck process, we target to find an unrestricted consumption strategy such that the value of the expected discounted consumption is maximized. Differently than in the case with restricted consumption rates, we can determine the optimal strategy and the value function.