Question

6. Suppose the hourly wage is $40, the price of each unit of capital is $50, and the price of output is $100 per unit. Assume that the firm cannot affect any of these prices. The production function of the firm is so that the marginal product of labor is MPE 2E a. If the current capital stock is fixed at 1,600 units, how many hours of labor should the firm hire in the short run (i.e., what should E be)? How much profit will the firm earn (hint: remember that profit is just pricexoutput - wagexE -price of capital«K)? (3 points total 2 for the value of E and 1 for the profit).

Answer 1

f(E, K) = squareroot E times K (a) In the short run, K is fixed at K^- = 1600 M P_E = squareroot K/2 squareroot E Prof pi = f(E, K) - W E - r K^- partial differential pi/partial differential E = 0 for profit maximization (also partial differential^2 pi/partial differential E^2 < 0) partial differential pi/partial differential E = P M P_E = W rightarrow 100 times squareroot K^-/2 squareroot squareroot E = 40 squareroot E = 50/40 times squareroot 1600 = 50/40 times 40 = 50 rightarrow = 2500 Profit earned, pi = 100 squareroot E times K - 40 E - 50 K^- = 100 squareroot 2500 times 1600 - 40 times 2500 - 50 times 1600 = 200000 - 100000 - 80000 = 20000 (b) in the long run, K is not fixed pi = Pi squareroot E times K - W E - r K Firm�s objective, max_E, K pi = squareroot E times K - W E - r K rightarrow max_E, K 100 squareroot E times K - 50 E - 50 K