Consumption-investment decisions with endogenous reference point and drawdown constraint
We propose a consumption-investment decision model where past consumption peak $h$ plays a crucial role. There are two important consumption levels: the lowest constrained level and a reference level, at which the risk aversion in terms of consumption rate is changed. We solve this stochastic control problem and derive the value function, optimal consumption plan, and optimal investment strategy in semi-explicit forms. We find five important thresholds of wealth, all as functions of $h$, and most of them are nonlinear functions. As can be seen from numerical results and theoretical analysis, this intuitive and simple model has significant economic implications, and 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 jumps inversely proportional to the risk aversion at the reference point; the implied relative risk aversion is roughly a smile in wealth; the welfare of the poor is more vulnerable to wealth shocks than the wealthy. Moreover, locally changing the risk aversion influences the optimal strategies globally, revealing some risk allocation behaviors.