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.