3. Suppose that a firm has a production function given by Q = 1 K/2 1 1/2, where capital, K, is 10 measured in machine-hours, labor, L, is measured in person-hours, and y denotes the yearly output. The hourly real wage rate w = 10 and the hourly real rental rate of capital is r = 20. (a) Show that this production function displays increasing returns to scale. (b) Compute the marginal products of labor and capital. (c) Suppose that at the end of 2021 the firm has signed a contract to rent K = 100 machine hours over the course of the year 2022. Determine the amount of labour employed to produce any given output, Q. (d) Noting that the firm's total costs are the sum of its labour and capital costs, derive the firm's short run cost function (i.e. C(Q) = C(Q) + F) in the year 2021. (d) What is the firm's short run marginal cost function? What is the firm's short run average total cost function? At what output do the two functions intersect? What is the relationship between average and marginal cost? (e) Suppose the firm is producing 10 units of output in the short-run. In the long-run, the firm can vary all of its inputs. If it continues to produce 10 units of output, will the firm want to change its employment of L and K in the long-run? If so, would it use more or less labour and more or less capital? Use a diagram to help explain your answer.