A firm produces output according to the following production function:

q = L^1/2K^1/3

where L is the amount of labor and K is the amount of capital. The cost of labor is $9 per hour and the rental cost of capital is $4 per hour.

a. What is the optimal cost-minimizing capital to labor ratio (K/L) for the firm?

b. Suppose the firm wishes to produce 100 units of output. How much capital and labor does the firm employ to minimize the cost? What is the total cost?

c. The firm decides to double the quantity of output in a short notice, and the amount of capital is fixed but labor hours are not. How much will it cost to produce 200 units of output in the short run, assuming the capital is fixed at the level of question (b)? How much would it cost if the firm could also vary capital in the long run? Compare the short run and long run costs.