Exercise 13.5 Consider the following variation of the optimal consumption problem described in Example 13.3. At each stage t the amount of nonconsumed wealth Wt Ct can be split between treasury bills and an index fund. Funds placed in treasury bills earn an annual riskfree return 3*" > 0 whereas the funds placed in the index fund earn a risky return T: with expected value Eta) : p > 'r 2 and variance var(rt) : or > 0. Assume the returns are i.i.d. across different periods. Use dynamic programming to formulate the following optimal investment and consumption problem: Determine the consumption 0: E [0, Wt] and fraction of wealth act 6 1R invested in the index fund at stage t : 0, 1, . . . , T that maximize the total expected utility of consumption T max E E U (Ct) Co,...,CT 1'0 ..... IT i=0 over the next T years. Proceed as follows. (a) Write the law of motion; that is, the equation that describes the state WHl in terms of the state W: and decisions 0:, act at stage t. (b) Write the Bellman equation for the value-togo function thl/Vt)- (C) Consider the special case of logarithmic utility of consumption U (0:) : log(Ct). Use the Bellman equation and induction to determine the optimal consumption C; and investment fraction :13: as well as the valuetogo func tion JtU/Vt) at stage t for t : 0,1,...,T. Indicate how your model changes if the fraction of wealth act invested in the index fund at each stage t is subject to the constraint 3:: E [0, 1]. (That is, no leverage is allowed in the investment portfolio.)