Optimal portfolio, consumption-leisure and retirement choice problem

Abstract

We consider an optimal consumption and portfolio selection problem of an infinitely-lived agent under various conditions. First we solve the problem of the agent whose consumption rate process is subjected to downside constraints with no retirement time $\\tau$. Second we solve the problem of the agent whose consumption rate process is subject to subsistence constraints with retirement time $\\tau$ which is considered as the first hitting time when her wealth exceeds a certain wealth boundary which will be determined by the free boundary value problem and the duality approach. Third we study optimal portfolio, consumption-leisure and retirement choice of the agent whose instantaneous preference is characterized by a constant elasticity of substitution(CES) function of consumption and leisure. For each case we obtain the optimal policies in explicit forms using a martingale method and a variational inequality arising from the dual functions of the optimal stopping problem. We also derive the optimal wealth processes before and after retirement in closed forms. We provide the critical wealth level for retirement in closed forms. We present some numerical results of optimal consumption and portfolio.