1 Econ 100A Spring 2001 HW 3: Production, Costs, Competition 1. A firm produces a recorded CD, Q = 1, by using one blank disk, D = 1, and the services of a recording machine, M = 1, for one hour. Write the formula for the production function. Draw an isoquant for this production process. Explain why it has that shape. As one unit of blank disk and one unit of recording service produces one unit of recorded CD, these two inputs are perfect complements for the production. Put differently, two inputs are used in a fixed proportion, one by one. Figure 1 below shows how the isoquants look like: each isoquant of L-shape lies along 45 degree line with its corner on the line. For instance, 2 blank CDs and 1 record service or 1 blank CD with 2 recording service produce only 1 recorded CD. Alternatively, production technology takes the form of Leontief function such as },min{ MDqCD= . (The expansion path passes the corner of each isoquant because extra unit of input is useless for production.) Figure 1) M 45 (M = D) 2 q = 2 1 q = 1 1 2 D 2. Michelle runs a business that produces ceramic cups using labor, clay (a free good, available in unlimited quantities), and a kiln. With this kiln, she can produce 25 cups a day with 1 workers and 35 with 2 workers. Does her production process illustrate diminishing returns to scale or diminishing marginal returns to scale? What is the likely explanation for why output doesn't increase proportionately with the number of workers? Let f be the production function such as ),,( KCLfqCUP= where L, C and K denote labor, clay and kiln respectively. The function f is virtually black-box except two pieces of information: ),,1(25 KCf= and ),,2(35 KCf= . Note that returns to scale literally measures the change in production when all inputs are scaled up/down. Accordingly, the information above is not sufficient enough to figure out returns to scale. However, diminishing marginal product of labor may be an attribute of the production technology. The production process illustrates diminishing marginal returns to scale. Output doesn’t fall as she increases number of workers but marginal returns decreases.2 3. Suppose the production function is q = L2 + LK. (A) What is the average product of labor, holding capital fixed at K*? (B) What is the marginal product of labor? [Hint: calculate how much q changes as L increases by one unit or use calculus.] (C) Does this production function have increasing, constant, or decreasing returns to scale? (A) Holding capital fixed at K*, average product of labor = **2KLLLKLAPL+=+≡ (B) Marginal product of labor = *2*)(2KLLLKLMPL+=∂+∂≡ Alternatively, 1*2*][*])1()1[(22++=+−+++ KLLKLKLL by approximation. As LLAPMP ≥ for all),0[∞∈L, LAP is monotonically increasing. Figure 2 (C) Returns to scale: )()())(()(2222LKLLKLKLL +>+=+ααααα if 1>α. Put differently, doubling both inputs together produces more outputs than twice as many as the original quantities, which entails increasing returns to scale. 4. A badminton set includes two rackets and one net. What is the long-run expansion path for a firm that assembles such sets (at no additional cost), if the firm buys rackets and nets at market prices? How does the expansion path depend on the relative prices of rackets and nets? Once again rackets and net are perfect complements for badminton. As two rackets and one net constitutes a badminton, a Leontief function such as }1,min{21NRqB= represents the production technology. See figure below. As each isoquant of L-shape lies along the line RN21= (each corner on the line), the minimum amount of rackets and nets for a given number of badminton is the corner point itself regardless of market prices for inputs. The long-run expansioin path for a firm doesn’t change because if the price changes, the budget constraint either shifts up or down and consumers purchase less or more with the same combination. Figure 2) N N = ½ R q = 2 2 1 q = 1 2 4 R3 5. If we plot the profit of a firm against the number of vacation days taken by its owner, we find that profit first rises with vacation days (a few days of vacation improves the owner's effectiveness as a manager the rest of the year), but eventually falls as the owner takes more vacation days. If the owner has usual shaped indifference curves between profit and vacation days, will the manager take the number of vacation days that maximize profit? If so, why? If not, what will the owner do, and why? We can consider a unimodal profit curve with respect to the number of vacation days by the owner (figure) With the assumption that indifference curves are downward-sloping and convex to the origin, the profit curve touches an indifference curve at some point (V = V*) to the right of its summit (V=Vmax). Figure 3) profit maximum profit q = 2 Vmax V* #V 6. During recessions, American firms lay off a larger proportion of their workers than Japanese firms. (It has been claimed that Japanese firms continue to produce at high levels and store the output or sell it at relatively low prices
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