Question

Problem 1. (Consumption smoothing) A consumer who lives for four periods have the following path of income y 60 0 60 0 Assume the consumer has log utility, a ct) 0 so that the real rate of return is 1 Inq, and is infinitely patient, β-1. Also aKsune the interest rate is (a) What is the optimal consumption profile of the consumer? (b) What is the value of assets, a, of the consumer at the beginning of period 47 (c) If the consumer is not able to borrow at any point in time, what is the optimal consumption profile? (d) What is the value of assets of the consumer at the beginning of period 4 in this case if the consumer cannot borrow? Problem 2. (Impatient consumers) Consider an conomy where agents live for two periods. There are two types of agents A and B. Both types of agents are impatient, so value of consumption in period 2 is less than value of consumption in period 1. Their preferences are described by the utility function: where 0 < β < 1 (β being less than 1 reflects impatience). Type A's endowment pattern is (圻4,9 l)-(Ea 0) where EA > 0 Type B's endowment pattern is (9.9')- (0, Eg) where Eg >0. Let 1 +r be the real rate of return (a) What is the marginal utility of consumption in period 1 and period 2? (b) Write down the intertemporal budget constraints for type A and type B agents (c) Write down the Euler equations for type A and type B agents (d) Find the equilibrium real rate of return (Hint: the answer shaould be expressed in terms of EA-Ea and β)

Answer 1

a) The budget constraint of the consumer c_1 + c_2 + c_3 + c_4 = y_1 + y_2 + y_3 + y_4 = 120 At equilibrium: Mu (v_t)/Mu(c_t + 1) = 1 + r a, 1/c_t/1/c_t + 1 = 1 Therefore c_t + 1 = c_t Therefore c_1 = c_2 = c_3 = c_4 = c bar Therefore 4 c bar = 120: c bar = 30 b) Asset at the beginning of t = 4: a_4 = (y_1 - c_1) + (y_2 - c_2) + (y_3 - c_3) = 60 - 30 + 0 - 30 + 60 - 30 = 30 c) In this case, the consumer in a under or saver. As he is not borrowing he will consume c bar = 30 at each period. d) The asset position remain same at a_4 = 30 a) The utility function is given as: u = ln (a) + beta ln (c_2) Marginal utilities: Mu (c_1) = 1/c_1 Mu (c_2) = beta/c_2 b) The intertemporal budget constraint of type A