Finite-Horizon Optimal Consumption and Investment with Time-Varying Job-Switching Costs

Abstract

In this paper, we study the finite-horizon problem of an economic agent's optimal consumption, investment, and job-switching decisions. The key new feature of our model is that the job-switching cost is time-varying. This extension leads to a novel mathematical characterization: the agent's dual problem reduces to a parabolic double obstacle problem with time-dependent upper and lower obstacles. By employing rigorous PDE theory, we establish not only the existence and uniqueness of the solution to this double obstacle problem, but also the smoothness of the two free boundaries that emerge from it. Building on these results, we characterize the agent's optimal consumption, portfolio, and job-switching strategies.