سرمایه گذاری بهینه، مصرف، اوقات فراغت، و مشکل بازنشستگی داوطلبانه با مطلوبیت کاب داگلاس: روش برنامه نویسی پویا

We consider an optimal consumption, leisure, investment, and voluntary retirement problem for an agent with a Cobb–Douglas utility function. Using dynamic programming, we derive closed form solutions for the value function and optimal strategies for consumption, leisure, investment, and retirement.

We consider an optimal consumption, leisure, and investment problem with voluntary retirement for an agent whose period utility function is a Cobb–Douglas utility function of consumption and leisure. In this model the agent can flexibly choose her leisure amount before retirement above a certain minimum labor requirement, and will receive labor income proportional to the amount of labor supplied. Upon retirement, the agent will enjoy full leisure, at the cost of forgoing all labor income. Using the dynamic programming method pioneered by Merton [1] and [2] and Karatzas et al. [3] we find closed form solutions to the value function and find the optimal consumption, leisure, and portfolio policies. Barucci and Marazzina [4] consider this consumption, leisure, investment, and retirement problem in the case of stochastic labor income. Choi et al. [5] solve a similar problem for an agent who has constant elasticity of substitution (CES) period utility. Farhi and Panageas [6] also consider such a problem, where the choice of leisure is confined to only two values: l1l1 while working and View the MathML sourcel̄ after retirement. In all of these papers, the authors use the martingale method to solve their optimization problems (see also [7] and [8]). Shin [9] extends the results of Farhi and Panageas [6] by solving their problem using the dynamic programming method, and shows the equivalence of the solutions obtained through the martingale method and the dynamic programming method. Likewise, we provide a methodological contribution by solving our optimization problem using the dynamic programming method. The work is organized as follows. Section 2 provides information on the financial market. Section 3 lays out and solves our optimization problem, with detailed proofs provided.