Suppose you manage a plant that produces engines by teams of workers using assembly machines. The technology is summarized by the production function Q = 4KL where Q is the number of engines produced per week, K is the number of assembling machines, and L the number of labor teams. Each assembly machine rents for r = $12,000 per week and each labor team costs w = $3,000 per week. Total engine costs are given by the cost of labor teams and assembling machines plus $2,000 per engine for raw (component) materials. Your plant currently has a fixed installation of 10 assembly machines as a part of its design. a. What is the total cost function [TC = f(Q)] for your plant- namely, how much will it cost to produce Q engines given the production function and input costs above? What are the average [ATC = f(Q)] and marginal costs [MC = f(Q)] of producing Q engines? How do average costs vary with output? TC = 3,000L + 12,000K + 2000Q = 3,000L + 12,000K + 2,000(4KL) Ac = TC/Q = 3,000L/Q + 12,000K/Q +2,000 Ideally Ac functions are only a function of Q. As Q rises economies of scale set in reduce AC till a point, after which they rise again, MC = dTC/dQ again MC will not be in Q terms alone. b. How many labor teams are required for producing a batch of 80 engines given the plant’s current makeup? What is the average total cost per engine? From the production function, we can see that the ratio of MP is K/L. W =3000 and r = 12000 So ¼ must equal K/L—this is the optimal ratio. Put L = 4K and Q= 80 in production function to get K = 2.236 and L=8.944 AC will be 4997.5 (put K = 2.2, L =9 and Q =80 in the AC function to get this value) c. You are now asked to make recommendations for the design of a new production facility. What would you suggest? What capital/labor ratio should the new plant accommodate? If lower average cost per engine was your prime consideration, what would be the optimal capital/labor levels to produce 80 engines? What then would be the average cost per engine? And lastly, should you suggest that the new plant have production capacity than the plant you currently manage? Explain why you chose as you did and the circumstances by which you answered as you did.