A price taking firm chooses its inputs to maximize short-run profits. Its Cobb-Douglass production function has the following form: q(L, K) = L ^(1/2)K^(1/3) . The output price is 1,000 per unit and the cost of each unit of input is 10. In the short-run, capital is fixed at 27 units.
(a) Set up the profit function in terms of labor only.
(b) Find the optimal choice of labor, L* .
(c) Given your answer to part (b), do you think that there is excess capital compared to the optimal level of quantity?