A firm owns two production plants that make widgets. The plants produce identical products and each plant (i) has a production function given by
\(Q_{i} = \sqrt{K_{i}L_{i}}\)
,for i = 1,2.Theplants differ, however, in The amount of capital equipment in place in the short run. In particular, plant 1has K1 = 25,where as plant 2 has K2 = 100. Input prices for K and L are w = r = 1.

a) Suppose the production manager is told to minimize the short-run total cost of producing Q units of output. While total output Q is exogenous, the manager can choose how much to produce at plant 1 (Q1) and at plant 2 (Q2), as long as Q1 + Q2 = Q. What percentage of its output should be produced at each plant?

b)When output is optimally allocated between the two plants, calculate the firms short-run total, average, and marginal cost curves. What is the marginal cost of the 100th widget? Of the 125th widget? The 200th widget?

c)How should the entrepreneur allocate widget production between the two plants in the long run? Find the firms long-run total, average, and marginal cost curves.