Consumption-investment decisions with endogenous reference point and drawdown constraint

We study a consumption-investment decision problem related to the past spending maximum. In the problem, we consider two crucial consumption levels: the lowest constrained level and a reference level, and both levels are fractions of the past spending maximum. The decision-maker has different risk aversions on different sides of the reference level. We solve this stochastic control problem and derive semi-explicit forms of the value function, optimal consumption plan, and optimal investment strategy. We find five important wealth thresholds which are nonlinear functions of the past spending maximum. Based on numerical results and theoretical analysis, we also find that the model has significant economic implications. There are at least three important predictions: the marginal propensity to consume out of wealth is generally decreasing but can be increasing for intermediate wealth levels, and it varies inversely with risk aversion at the reference level; the implied relative risk aversion is roughly a smile in wealth; the welfare is much more vulnerable to wealth shocks when the reference level is not reached.