  This paper studies 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 for a class
of non-homothetic utility maximizer 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 almost all in luxury goods when the wealth is
large. The optimal retirement policy is shown to be both universal, in the
sense that all individuals should retire at the same level of marginal utility
that is determined only by income, labor cost, discount factor as well as
market parameters, and not universal, in the sense that all individuals can
achieve the same marginal utility with different utility and wealth. 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 variational
inequality and dual transformation. We finally conduct the simulation analysis
for the featured model parameters to investigate practical and economic
implications by providing their figures.
