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 firm whicli seeks to maximize the present value of its profit across two periods: where is the real profit in period 1. 1'is the real profit in period 2, and r is the real interest rate. In the first period, the firm is given exogenously determined capital K, and chooses how much labor N to hire at wage w, and how much to invest I. Output is given by F(KN): -* and Investment takes the form of output which is not sold. Thas revenues are :KN--- and so first period profits are *=KON-I-N 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 KEK (1-0)+1 In the second period, the firm chooses how much labor ' to hire at the second period's real wage w". And because time ends after the second period, the firm will sell of any remaining capital stock K. (1-5) and so second period profits are given by: ??=?'(K")" ()-* + K*-(1-3) -w'N letus: (a) Set up the firm's present-value profit maximization problem. (b) Sut up the Lagrangian and derive equations describing the firm's optimal labor-hiring and investment decisions. These exptations should show the aptimal relationship between the firm's cols and the marginal product of labor or capital.) Now suppose that the firm is uncertain about future productivity. The firm believes that there is only a p [0.1 chance that they will be able to produce anything tomorrow. That is, there is a p chance that ?? = 2x >0 and a (1-P) chance that ? = 0. (e) What is the expected value of profits tomorrow? (Hint: if it is the profits the firm will earn if things are good, and their profit if things are bed, then 2 (3) = pry+(1-P=. The firm can choose how much labor to hire in the second period after abocring whether it's a good or a bad day.) 1+r Suppose the firm seeks to maximize the present value of its expected profits: *+ () Set up the new firm's problem. (e) Determine how this change affects the optimal investment rule for the firm. (1) How does investment demand change when p changesInterpret. 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 firm whicli seeks to maximize the present value of its profit across two periods: where is the real profit in period 1. 1'is the real profit in period 2, and r is the real interest rate. In the first period, the firm is given exogenously determined capital K, and chooses how much labor N to hire at wage w, and how much to invest I. Output is given by F(KN): -* and Investment takes the form of output which is not sold. Thas revenues are :KN--- and so first period profits are *=KON-I-N 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 KEK (1-0)+1 In the second period, the firm chooses how much labor ' to hire at the second period's real wage w". And because time ends after the second period, the firm will sell of any remaining capital stock K. (1-5) and so second period profits are given by: ??=?'(K")" ()-* + K*-(1-3) -w'N letus: (a) Set up the firm's present-value profit maximization problem. (b) Sut up the Lagrangian and derive equations describing the firm's optimal labor-hiring and investment decisions. These exptations should show the aptimal relationship between the firm's cols and the marginal product of labor or capital.) Now suppose that the firm is uncertain about future productivity. The firm believes that there is only a p [0.1 chance that they will be able to produce anything tomorrow. That is, there is a p chance that ?? = 2x >0 and a (1-P) chance that ? = 0. (e) What is the expected value of profits tomorrow? (Hint: if it is the profits the firm will earn if things are good, and their profit if things are bed, then 2 (3) = pry+(1-P=. The firm can choose how much labor to hire in the second period after abocring whether it's a good or a bad day.) 1+r Suppose the firm seeks to maximize the present value of its expected profits: *+ () Set up the new firm's problem. (e) Determine how this change affects the optimal investment rule for the firm. (1) How does investment demand change when p changesInterpret