3. (Long-run costs, 36 pts) Carmela's pasta factory employs workers and pasta machines according to the following production function f\": K) = [0'5K0'5 The hourly cost of capital is $10 and the hourly cost of workers is $40. a. Write out the Lagrangian for the costminimization problem. (4 pts) b. Using the rst order conditions of the Lagrangian, derive the optimal capital to labor ratio. Describe the long-run output expansion path. (8 pts) c. Suppose Carmela wishes to produce 1000 units of pasta. How much labor and capital should she employ? How much will it cost to produce? (8 pts) d. An order arrives doubling the amount of pasta Carmela needs to produce. Assuming she is unable to purchase more capital, how much will it cost to meet the new production level? (8 pts) e. In the longrun, Carmela will be able to employ more capital as well as labor. If Carmela continues to produce 2000 units of output, how much will it cost in the long run? How is it comparing to your answer in part d? (8 pts)