Optimal investment, heterogeneous consumption and the best time for retirement

In this talk we discuss an optimal investment and consumption problem with heterogeneous consumption of basic and luxury goods, together with the choice of time for retirement. The utility for luxury goods is not necessarily a concave function. The optimal heterogeneous consumption strategies are shown to consume only basic goods when the wealth is small, to consume basic goods and make savings when the wealth is intermediate, and to consume small portion in basic goods and large portion in luxury goods when the wealth is large. The optimal retirement policy is shown to be universal, in the sense that all individuals should retire at the same level of marginal utility. This level is determined only by income, labor cost as well as market parameters, but independent of individual's utility, which could have important policy implications for government in making pension and retirement age decisions. It is also shown that individuals prefer to retire as time goes by if the marginal labor cost increases faster than that of income. The main tools used in analyzing the problem are from PDE and stochastic control theory including viscosity solution, variational inequality and dual transformation. (Joint work with Zuo Quan Xu of Hong Kong Polytechnic University.)