The firm's production function is f(K.L)=K^1/2+L^1/2. In the short run, it has one unit of capital.
(a) Does this firm experience diminishing marginal product of labor? Justify your answer
(b) Find the firm's short run unconditional demand for labor.
(c) Find the firm's short run profit function. Verify that it has the four properties of profit functions we discussed in class.
(d) Use the firm's profit function to find the firm's supply function.
(e) Suppose that p 4, w=1, and r=1. Use your work from above to find the amount of K and L will the firm use in the short run. What is the MRTS at this point?
(f) From the last question, you can see that the MRTS is not equal to the input price ratio. This is because the firm's amount of capital is fixed in the short run. Instead, the first order condition for profit maximization in part (a) says that the derivative of total revenue with respect to L is equal to the derivative of total cost with respect to L. Give intuition for why satisfying this condition will help the firm maximize profit.
(g) Suppose the output price changes from 4 to 2 (the input prices are both still w r 1). What is the change in producer surplus?