Life-cycle planning with CEV model and time-inconsistent preferences

In this paper, we investigate an optimization problem for a wage earner seeking to maximize expected utilities until retirement by choosing optimal consumption, investment, and life insurance purchase strategies. The constant elasticity of variance (CEV) model is adopted to describe the price process of the risky asset. Additionally, we assume that the wage earner has time-inconsistent preferences. This makes the wage earner discount her payoff by a non-constant discount rate. Applying the dynamic programming principle, we have derived the Hamilton-Jacobi-Bellman (HJB) equation corresponding to the optimization problem. Furthermore, we present semi-analytical expressions for optimal strategies and value functions in three cases: the benchmark model with time-consistent preferences, the naive and sophisticated wage earners with time-inconsistent preferences. Finally, illustrations of the optimal solutions and some economic insights are provided in the numerical examples.