Exercise 4 Barney lives in two periods (period 1 and period 2). In period 1 his endowment is Y = 1 while in period 2 is Yg = 0. Barney's utility function is: u(G.C) = InC+ Inc 2 where and C, are the consumptions in period 1 and period 2 respectively. Barney can report some consumption in the future through a production function with decreasing marginal return (K) = K where K is the investment. a) Write down the intertemporal budget constraint faced by Barney. b) Prove that the optimal investment is K = 1/3. c) Find out the optimal consumptions. Now suppose that in addition to the production function, Barney can report the present consumption in the future through the access to the market interest rater. d) Write down the new intertemporal budget constraint faced by Barney. e) What is the optimal investment? f) Suppose that r = 0, what are the optimal consumptions, saving and investment? g) Present the Fisher separation theorem. Exercise 4 Barney lives in two periods (period 1 and period 2). In period 1 his endowment is Y = 1 while in period 2 is Yg = 0. Barney's utility function is: u(G.C) = InC+ Inc 2 where and C, are the consumptions in period 1 and period 2 respectively. Barney can report some consumption in the future through a production function with decreasing marginal return (K) = K where K is the investment. a) Write down the intertemporal budget constraint faced by Barney. b) Prove that the optimal investment is K = 1/3. c) Find out the optimal consumptions. Now suppose that in addition to the production function, Barney can report the present consumption in the future through the access to the market interest rater. d) Write down the new intertemporal budget constraint faced by Barney. e) What is the optimal investment? f) Suppose that r = 0, what are the optimal consumptions, saving and investment? g) Present the Fisher separation theorem