23.10 Optimal control theory can be used to generalize the model of intertemporal consumption choice contained in Example 23.1. Consider the following simple life cycle model: An individual receives wages (w) each period and a return on his or her invested capital. Let k =

capital, r = market interest rate at which the individual can borrow or lend. During each period, the individual chooses consumption

(c) to maximize f U(c)e-p'dt, where p is the individual's rate of time preference. Given these assumptions, the intertemporal budget constraint for this problem is k = w + rk — c with constraints on initial and final k of the form k(0) = k(T) = 0.

a. What are the necessary conditions for a maximum for this problem?