Let Q = L1/3K2/3, where Q is output, L is labor and K is capital.
(1) Let w = 1 and r = 2, where w is the wage rate and r is the rental rate of capital. Solve for the optimal input combination to produce 10 units of output.
(2) Now suppose that the firm's capital is fixed in the short run at 5 units with w = 1 and r = 2. Find the input of labor to produce 10 units of output.
(3) For arbitrary values of w, r and Q, solve for the (long run) input demand curves. (4) Now suppose that the firm's capital is fixed in the short run at K ¯ . Find the firm's short run demand for labor.