Question

Problem 2. (Impatient consumers) Consider an economy 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< 81 ( being less than 1 reflects impatience). Type A's endowment pattern is (n,ni)-(EAN) where Ea > 0. Type B's endowment pattern is u'of)- (0, Ep) where Eg >O. Let1+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 should be expressed in terms of EA, EB and β).

Answer 1

The utility function is given as u = m(c_1) + beta ln (c_20 Therefore mu (c_1) = partial differential u/partial differential c_1 = 1/c_1 Therefore mu(c_2) = partial differential /partial differential c_2 = beta/c_2 at equilibrium, mu(c_1)/mu(c_2) = 1 + r_1 or, 1/c_1/beta /c_2 = 1 + r_1 or, c_2/beta c_1 = 1 + r_1 Therefore c_2 = (1 + r_1) beta c_1..(1) the budget constraint of a c_1 + c_2/1 + r_1 = x..(a) the budget constraint of B c_1 + c_2/1 + r_1 = z/1 + r_1..(b) at equilibrium: c_2 = (1 + r_1) beta c_1 (a): c_1 + (1 + r_1) beta c_1/1 + r_1 = x or, (1 + beta)c_1 = x c_2 = x beta (1 + r_1)/1 + beta }for a s_a = y_1^a - c_1^a = x beta /1 + beta