Labor Supply Flexibility and Portfolio Selection with Early Retirement Option

In this paper, we study the optimal consumption, investment, and life insurance problem of an economic agent who can choose a flexible labor supply and has an option to retire early with the existence of a mandatory retirement date. We model the agent’s preference as the Cobb–Douglas utility, which is a function of consumption and leisure, and consider the agent’s unit wage rate as a stochastic process. The optimization problem has a feature of combining both stochastic control and optimal stopping. To attack this problem, we adopt a dual-martingale approach and derive a dual problem, which is a finite-horizon optimal stopping problem choosing the early retirement date. Based on the partial differential equation techniques, we fully analyze the variational inequality arising from the dual problem. We show that the optimal early retirement time is characterized as the free boundary of the agent’s wealth-to-wage ratio. Finally, we establish a duality theorem and obtain an integral equation representation of optimal strategies.