Optimal consumption, labor supply and portfolio rules in a continuous-time life cycle model

The continuous-time intertemporal consumption-portfolio maximization problem was pioneered by Merton (1969, 1971) by implementing the method of dynamic programming. In the 1980s, Pliska (1986), Karatzas /Lehoczky/Shreve (1986) and Cox/Huang (1989) developed an alternative approach to the similar problem using the martingale technique. The main advantage of the latter over the former is that the artingale approach only involves linear partial differential equations, unlike the nonlinear partial differential equation of the dynamic programming. in this paper, we consider the problem maximizing a specified lifetime utility of the consumption and labor supply of an infinitely-lived individual who works when young and consumes when old. And his labor income is invested into a risk-free bond and a risky asset. By means of the martingale approach, the formula of the optimal amount he works when young in order to have his best life when old is obtained. And a closed form of his optimal investment strategy is found.