Life-cycle saving. (Modigliani and Brumberg, 1954.) Consider an individual who lives from 0 to T, and whose lifetime utility is given by U =  T t =0 u(C(t))dt, where u

(•) > 0, u(•) < 0. The individual’s income is Y0 + gt for 0 ≤ t < R, and 0 for R ≤ t ≤ T. The retirement age, R, satisfies 0 < R < T. The interest rate is zero, the individual has no initial wealth, and there is no uncertainty.

(a) What is the individual’s lifetime budget constraint?

(b) What is the individual’s utility-maximizing path of consumption, C(t)?

(c) What is the path of the individual’s wealth as a function of t?