Chapter 7B Word Problem Empire Industries produces red lightsabers with the production function Q = 4KL. The wage rate is $8 and capital rental rate is $10. a. In the short-run, capital is fixed at 10. How much labor does Empire Industries need to minimize its cost of producing 200 lightsabers in the short-run? How much will it cost? b. A peace treaty decreases the demand for red lightsabers. Empire Industries only wants to produce 120 now. They still employ 10 units of capital. In the short run, how much labor will they hire? How much will it cost? c. Find Empire Industries’ short-run total cost function (TC = ___ where the only variable is Q). Assume capital still is fixed at 10. d. Empire Industries is looking forward and projects they will produce 200 red lightsabers on average. How much capital and labor will they employ? How much will it cost? e. Find Empire Industries’ long-run demand for capital (K = ___ where the only variable is Q). f. Make a graph of K vs L. Include the bundles of K and L you have found above in (a), (b), and (d), their associated isoquants, and their associated isocost curves. Your graph should be about 1?2 a page. Think carefully about what lines should be tangent to what curves! g. Make a graph of the short-run total cost function you found in (c) by graphing TC (y-axis) versus Q (x-axis). Graph the short-run fixed cost and short-run variable cost functions as well. On a separate graph, plot the marginal cost function (y-axis) versus Q (x-axis). The x-axis on both graphs should go from 0 to 200. These graphs can each be about 1?4 of a page. h. Rebellion Inc. is a competitor of Empire Industries but produces blue lightsabers. Rebellion Inc has a long-run total cost function of TC = 115Q – 0.5Q2 + 0.001Q3 . What is Rebellion Inc’s marginal cost function? What quantity of output minimizes Rebellion Inc’s long-run average total cost? i. Make a graph of Rebellion’s long-run TC function (y-axis) versus Q (x-axis); include FC and VC as well. Make a separate graph of ATC, AVC, AFC, and MC. The x-axis on both graphs should go from 0 to 500. You can plot the points by finding the values of the functions for Q = 1, 50, 100, 150,..., 500 and connect the dots. These graphs should each be 1?2 a page. j. Which of the following are true? Explain your answer. i. In the short-run, it costs more for Rebellion Inc to produce 200 lightsabers than for Empire Industries. ii. The long-run average variable cost function for Rebellion Inc is equal to its average total cost. iii. Empire Industries’ production function exhibits increasing returns to scale. iv. Empire Industries’ long-run total cost function exhibits economies of scale for all levels of output. k. Disney purchases Empire Industries and Rebellion Inc. Disney is deciding between producing both red and blue lightsabers in Empire’s factory only, red and blue in Rebellion’s factory only, or to produce red in Empire’s and blue in Rebellion’s factory. What should Disney do? Does Disney exhibit economies or diseconomies of scope? • Empire’s factory can produce 100 blue and 100 red for $7,000 • Empire Industries’ long-run total cost is 8.98Q0.5 • Rebellion’s factory can produce 100 blue and 100 red for $7,600 • (Rebellion’s long-run total cost curve is given above.)