I think I have figured out the answers to a. and b., but I am having a bit of tr
ID: 382545 • Letter: I
Question
I think I have figured out the answers to a. and b., but I am having a bit of trouble answering c.
3. 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.
Explanation / Answer
1. You are asked to find a short-run cost function i.e. where the quantity of one input is fixed; here the question tells you that K = 10. This means that the short-run production function can bewritten as q = 40L. This implies that for any level of output q, the number of labor teams hired will be L = q/40 . The total cost function is thus given by the sum of the costs of capital, labor and raw materials:
T C(q) = rK + wL + 2000q
T C(q) = 12, 000*10 + 3, 000 * q/40+ 2000q
T C(q) = 120, 000 + 2, 075q
The average cost function is given by : AC(q) = T C(q)/q = 120,000/q + 2, 075
and the marginal cost function is just MC(q)=2, 075. Indeed, when q increases by one unit, the total cost increases by 2,075 irrespective of the level of production; in this sense marginal costs are constant. On the other hand, average costs will decrease as quantity increases; this is due to the fixed cost of capital getting divided among more and more units.
2. To produce q = 80 engines we need 2 labor teams as L = q/40 = 80/40 = 2. Average costs are given by AC(80) = 120,000/80 + 2, 075 = 3575.
3. We no longer assume that K is fixed at 10. We need to find the combination of K and L which minimizes costs at any output level q. Remember that the cost-minimization rule is given by
MPL/w = MPK/r . To find the marginal product of capital, observe that increasing K by one unit increases
q by 4L so PMK = 4L. Similarly, increasing L by one unit increases q by 4K so MPL = 4K. Using these formulas in the cost-minimization rule, we obtain :
4K/w = 4L/r
K/L = w/r
,
K/L = 3, 000/12, 000 = 1/4
.
Thus the new plant should accommodate a capital/labor ratio of 1 to 4. The firm’s capital-labor ratio is currently 10/2=5. To reduce average cost, the firm should either use more labor and less capital to produce the same output or it should hire more labor (keeping the same level of capital) and increase output.