An Optimal Consumption-Investment Model with Constraint on Consumption

TL;DR

This paper develops a continuous-time consumption-investment model with a wealth-dependent consumption constraint, providing explicit strategies and revealing how consumption varies with financial status, differing from classical models.

Contribution

It introduces a novel model with a wealth-dependent consumption constraint and derives explicit optimal strategies, notably showing investment strategies align with Merton's model regardless of wealth.

Findings

Optimal investment strategy matches Merton's model regardless of wealth.

Optimal consumption depends on financial situation, increasing with wealth.

Value function is smooth with a unique exception point.

Abstract

A continuous-time consumption-investment model with constraint is considered for a small investor whose decisions are the consumption rate and the allocation of wealth to a risk-free and a risky asset with logarithmic Brownian motion fluctuations. The consumption rate is subject to an upper bound constraint which linearly depends on the investor's wealth and bankruptcy is prohibited. The investor's objective is to maximize total expected discounted utility of consumption over an infinite trading horizon. It is shown that the value function is (second order) smooth everywhere but a unique possibility of (known) exception point and the optimal consumption-investment strategy is provided in a closed feedback form of wealth, which in contrast to the existing work does not involve the value function. According to this model, an investor should take the same optimal investment strategy as in Merton's model regardless his financial situation. By contrast, the optimal consumption strategy does depend on the investor's financial situation: he should use a similar consumption strategy as in Merton's model when he is in a bad situation, and consume as much as possible when he is in a good situation.