1 Firm Investment with Uncertainty (10 points) Parts of this problem are based on question 4 from chapter 11 in the book. Suppose we have a rm which seeks to maximize the present value of its prot across two periods: I 71' +1+r where n is the real prot in period 1, ir'is the real prot in period 2, and r is the real interest rate. 0 In the rst period, the rm is given exogenously determined capital K, and chooses how much labor N to hire at wage to, and how much to invest I. Output is given by zF(K, N) : zKC'N1_\" and Investment tales; the form of output which is not sold. Thus revenues are zh'c'N1 '\" I and so rst period prots are: 7r: zKaN1_\" IwN 0 Between periods, some portion of prexisting capital 6 depreciates, and investment is added to the capital stock, so the formula for second period capital is Kerr-(175\"; 0 In the second period, the rm chooses how much labor N ' to hire at the second period's real wage w'. And because time ends after the second period, the rm will sell off any remaining capital stock K' - (l 5) and so second period prots are given by: 1T]: 21(Kr)a(Nr)1a+ K: . (l_)_qur Problems: (3) Set up the rm's present-value prot maximization problem. (b) Set up the Lagrangian and derive equations describing the rm's optimal labor-hiring and investment decisions. { These equations should show the optimal relationship between the fim's costs and the margins! product of labor or capital.) Now suppose that the firm is uncertain about future productivity. The firm believes that there is only a p 6 [0,1] chance that they will be able to produce anything tomorrow. That is, there is a p chance that z' : 21;! > 0 and a (1 p) chance that z': 0. (c) What is the expected value of n', prots tommorrow? (Hint: If 'JT'H is the profits the firm will earn. if things are good, and ir'L their profit if things are bad, thenlE (ir') = pir'H+(1 *pyri. The firm can choose how much labor to hire in the second period after observing whether it's a good or o. bad day.) . . . E 1" Suppose the firm seeks to maxrrnrze the present value of its expected profits: n + 41+?) (d) Set up the new rms problem. (e) Determine how this change affects the optimal investment rule for the rm. (1') How does investment demand change when 1) changes? Interpret