Exercise 4: Consider a two-period economy: A firm produces output (Y) with capital (K") according to the following production function Y, = 22:K? in which =; is an index of total factor productivity in period t. A portion & of the capital stock used in the production func- tion needs to be repaired each period and new capital installed in any period t becomes productive a period later only. Profits in period 0 and in period 1 are respectively given by no = Yo+ (1 - 8)Ko - Ki and by my = Yi + (1 - 8) 21. The goal of this firm is to maximize the present value of profits no + (1/(1 + r))#1 by choosing K1. given Ko, r, 20. and =1. Suppose that Ko = 16, that 20 = 21 = 1, that 8 = 0.10, and that r = 0.10. a) Explain what the marginal product of capital measures. Explain what the user cost of capital is. Why is it optimal for the firm to choose hi so that the marginal product of capital in period 1 is equal to the user cost of capital? b) Find the optimal value of K1. How much does this firm need to invest in period 0 to achieve this value of A1? Calculate mo. What do you conclude? c) Suppose that =, is expected to fall. How will investment be affected in period 0? Explain. d) Suppose that zo falls. Explain why a fall in zo does not directly affect investment. Explain why investment may nevertheless fall in equilibrium (hint: what happens to r when zo falls?). e) Suppose that investment is added to aggregate demand in period 0 (i.e. AD. = Co(r) + lo + Go). Use a graph of the goods market to describe what would happen to investment if the government increases Go. Will Yo increase? What about Y