Consider the Cobb-Douglas production function Q = 6 L^ K^ and cost function C = 3L + 12K. a. Optimize labor usage in the short

Consider the Cobb-Douglas production function Q = 6 L^½ K^½ and cost function C = 3L + 12K.

a. Optimize labor usage in the short run if the firm has 9 units of capital and the product price is $3.

b. Show how you can calculate the short run average total cost for this level of labor usage?

c. Determine “MP per dollar” for each input and explain what the comparative numbers tell in terms of the amount of labor and capital being used in the short run.

d. Optimize for both inputs L & K for a long-run output constraint of Q = 192 units and prove that the average total cost is lower after long-run optimization.