For Questions 4-6, consider the Ramsey-Cass-Koopmans model. Note: As- sume the depreciation rate, 6 =0, unless otherwise stated. 4. (16 points) Suppose that the instantaneous utility function is u(C) = In(C). Let W denote the household's initial wealth plus the present value of its lifetime labor income. Find the utility-maximizing path of C, given r, W, and the utility function. (Note: 1. Under log-utility, 0 =1. p-n - (1 -0)g > 0 becomes p - n > 0. 2. Assume that the interest rate is constant atr. Therefore, R(t) = for(T)dT = rt.) 5. (24 points) Describe and support your argument with figures how each of the fol- lowing affects the c = 0 and k = 0 curves in the main phase diagram figure with saddle path, and thus how they affect the values of c and & on the new balanced growth path: (a) (8 points) A permanent increase in the level of productivity that shifts the production function upward. (b) (8 points) A decrease in households' risk aversion. (c) (8 points) A change in the rate of depreciation rate from zero to some positive level. (Hint: Consider the impact of a change in o on both loci.) 6. (15 points) Consider an economy on its balanced growth path, and suppose there is a permanent rise in the growth rate of productivity, g. (a) (5 points) How, if at all, does this affect the k = 0 curve? (b) (5 points) How, if at all, does this affect the c = 0 curve? (c) (5 points) What happens to c at the time of the change