An optimal consumption-investment model with constraint on consumption

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 Brow- nian motion uctuations. 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 the total expected dis- counted utility of consumption over an infinite trading horizon. It is shown that the value function is (second order) smooth everywhere but a unique (known) possibly exception point and the optimal consumption-investment strategy is provided in a closed feedback form of wealth. According to this model, an investor should take the similar investment strategy as in Merton's model re- gardless his financial situation. By contrast, the optimal consumption strategy does depend on the investor's financial situation: he should use a similar con- sumption 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.