Consumption-leisure-investment strategies with time-inconsistent preference in a life-cycle model

This paper investigates a consumption-leisure-investment problem, where the object of an economic agent is to maximize the expected value of discounted lifetime utility in a life-cycle model. The agent is allowed to have considerable labor flexibility and the date of retirement is fixed. To incorporate some well-documented behavioral features of human beings, we consider the situation where the discounting is non-exponential. This situation is far from trivial and renders the optimization problem of the agent to be a nonstandard one, namely, a time-inconsistent stochastic control problem. The extended HJB equation for the time-inconsistent control problem is given. A verification theorem is proved for a general discount function and a general utility function. Explicit-form solutions are presented for the logarithmic utility with exponential discounting, pseudo-exponential discounting and hyperbolic discounting.