Chapter 7 The Cost of Production

CHAPTER 7 THE COST OF PRODUCTION TEACHING NOTES In this chapter, it is easy for the students to concentrate too much on definitions and geometry and lose focus on the economics. Therefore, keep in mind the key concepts: opportunity cost, short-run average and marginal cost, cost minimization, and long-run average cost. These concepts can be illuminated with the supplementary material provided at the end of the chapter, which includes sections on economies of scope, learning curves, and estimating and predicting costs. The Appendix presents the calculus of constrained optimization, as applied to cost minimization. All exercises involve some algebra or geometry: Exercises (12) and (13) are time consuming, but rewarding. Opportunity cost is the conceptual base of this chapter. While most students think of costs in accounting terms, they must develop an understanding of the distinction between accounting, economic, and opportunity costs. One source of confusion is the opportunity cost of capital, i.e., why the rental rate on capital must be considered explicitly by economists. It is important, for example, to distinguish between the purchase price of capital equipment and the opportunity cost of using the equipment. The opportunity cost of a person’s time also leads to some confusion for students. Following the discussion of opportunity cost, the chapter diverges in two directions: one path introduces types of cost and cost curves, and the other focuses on cost minimization. Both directions converge with the discussion of long-run average cost. The geometry of total, fixed, variable, average, and marginal costs can prove to be tedious. An emphasis on the following issues helps students master this topic: 1) the relationship between the production function, diminishing returns in the short run, input prices, and the shapes of the various cost curves; 2) the distinction between total, average, and marginal; and 3) the reasonableness of the assumption of constant input prices (note that this assumption will be relaxed in Chapter 10’s discussion of monopsony). The determination of the cost-minimizing quantity is crucial to understanding Chapters 8 and 10. The concept of duality (minimizing cost subject to a given level of production) is equivalent to maximizing output subject to a given level of total cost) clarifies this concept for students. A clear understanding of short-run cost and cost minimization is necessary for the derivation of long-run average cost. With long-run costs, stress that firms are operating on shortrun cost curves at each level of the fixed factor and that long-run costs do not exist separately from short-run costs. Exercise (6) illustrates the relationship between long-run cost and cost minimization, with an emphasis on the importance of the expansion path. Stress the connection between the shape of a long-run cost curve and returns to scale. While Section 7.7 is starred, it does not require calculus. Example 7.5 “Cost Functions for Electric Power,” gives students another view of long-run average cost and allows for discussion of minimum efficient scale, an important determinant of industry structure. REVIEW QUESTIONS 1. A firms pays its accountant an annual retainer of $10,000. implicit cost? Is this an explicit or Explicit costs are actual outlays. They include all costs that involve a monetary transaction. An implicit cost is an economic cost that does not necessarily involve a monetary transaction, but still involves the use of resources. When a firm pays an annual retainer of $10,000, there is a monetary transaction. The accountant trades his or her time in return for money. Therefore, an annual retainer is an explicit cost. 2. The owner of a small retail store does her own accounting work. How would you measure the opportunity cost of her work? 66 Chapter 7: The Cost of Production Opportunity costs are measured by comparing the use of a resource with its alternative uses. The opportunity cost of doing accounting work is the time not spent in other ways, i.e., time such as running a small business or participating in leisure activity. The economic cost of doing accounting work is measured by computing the monetary amount that the time would be worth in its next best use. 3. Suppose a chair manufacturer finds that the marginal rate of technical substitution of capital for labor in his production process is substantially greater than the ratio of the rental rate on machinery to the wage rate for assembly-line labor. How should he alter his use of capital and labor to minimize the cost of production? To minimize cost, the manufacturer should use a combination of capital and labor so the rate at which he can trade capital for labor in his production process is the same as the rate at which he can trade capital for labor in external markets. The manufacturer would be better off if he increased his use of capital and decreased his use of labor, decreasing the marginal rate of technical substitution, MRTS. He should continue this substitution until his MRTS equals the ratio of the rental rate to the wage rate. 4. Why are isocost lines straight lines? The isocost line represents all possible combinations of labor and capital that may be purchased for a given total cost. The slope of the isocost line is the ratio of the input prices of labor and capital. If input prices are fixed, then the ratio of these prices is clearly fixed and the isocost line is straight. Only when the ratio or factor prices change as the quantities of inputs change is the isocost line not straight. 5. If the marginal cost of production is increasing, does this tell you whether the average variable cost is increasing or decreasing? Explain. Marginal cost can be increasing while average variable cost is either increasing or decreasing. If marginal cost is less (greater) than average variable cost, then each additional unit is adding less (more) to total cost than previous units added to the total cost, which implies that the AVC declines (increases). Therefore, we need to know whether marginal cost is greater than average cost to determine whether the AVC is increasing or decreasing. 6. If the marginal cost of production is greater than the average variable cost, does this tell you whether the average variable cost is increasing or decreasing? Explain. If the average variable cost is increasing (decreasing), then the last unit produced is adding more (less) to total variable cost than the previous units did, on average. Therefore, marginal cost is above (below) average variable cost. If marginal cost is above average variable cost, average variable cost is also increasing. 7. If the firm’s average cost curves are U-shaped, why does its average variable cost curve achieve its minimum at a lower level of output than the average total cost curve? Total cost is equal to fixed plus variable cost. Average total cost is equal to average fixed plus average variable cost. When graphed, the difference between the Ushaped total cost and average variable cost curves is the average fixed cost curve. If fixed cost is greater than zero, the minimum of average variable cost must be less than the minimum average total cost. 8. If a firm enjoys increasing returns to scale up to a certain output level, and then constant returns to scale, what can you say about the shape of the firm’s long-run average cost curve? When the firm experiences increasing returns to scale, its long-run average cost curve is downward sloping. When the firm experiences constant returns to scale, its long-run average cost curve is horizontal. If the firm experiences increasing returns to scale, then constant returns to scale, its long-run average cost curve falls, then becomes horizontal. 67 Chapter 7: The Cost of Production 9. How does a change in the price of one input change the firm’s long-run expansion path? The expansion path describes the combination of inputs for which the firm chooses to minimize cost for every output level. This combination depends on the ratio of input prices: if the price of one input changes, the price ratio also changes. For example, if the price of an input increases, less of the input may be purchased for the same total cost. The intercept of the isocost line on that input’s axis moves closer to the origin. Also, the slope of the isocost line, the price ratio, changes. As the price ratio changes, the firm substitutes away from the now more expensive input toward the cheaper input. Thus, the expansion path bends toward the axis of the now cheaper input. See Exercise (7.6). 10. Distinguish between economies of scale and economies of scope. Why can one be present without the other? Economies of scale refer to the production of one good and occur when proportionate increases in all inputs lead to a more-than-proportionate increase in output. Economies of scope refer to the production of more than one good and occur when joint output is less costly than the sum of the costs of producing each good or service separately. There is no direct relationship between increasing returns to scale and economies of scope, so production can exhibit one without the other. See Exercise (13) for a case with constant product-specific returns to scale and multiproduct economies of scope. EXERCISES 1. Assume a computer firm’s marginal costs of production are constant at $1,000 per computer. However, the fixed costs of production are equal to $10,000. a. Calculate the firm’s average variable cost and average total cost curves. The variable cost of producing an additional unit, marginal cost, is constant at $1,000 Q $1,000, so the average variable cost is constant at $1,000,  $1,000 . Q $10,000 . Average total cost is the sum of average variable cost Average fixed cost is Q $10,000 and average fixed cost: ATC  $1,000  . Q bg b. If the firm wanted to minimize the average total cost of production, would it choose to be very large or very small? Explain. The firm should choose a very large output because average total cost decreases with increase in Q. As Q becomes infinitely large, ATC will equal $1,000. 2. If a firm hires a currently unemployed worker, the opportunity cost of utilizing the worker’s service is zero. Is this true? Discuss. From the worker’s perspective, the opportunity cost of his or her time is the time not spent in other ways, including time spent in personal or leisure activities. Certainly, the opportunity cost of hiring an unemployed mother of pre-school children is not zero! While it might be difficult to assign a monetary value to the time of an unemployed worker, we can not conclude that it is zero. From the perspective of the firm, the opportunity cost of hiring the worker is not zero, and the firm could purchase a piece of machinery rather than hiring the worker. 68 Chapter 7: The Cost of Production 3.a. Suppose that a firm must pay an annual franchise fee, which is a fixed sum, independent of whether it produces any output. How does this tax affect the firm’s fixed, marginal, and average costs? Total cost, TC, is equal to fixed cost, FC, plus variable cost, VC. Fixed costs do not vary with the quantity of output. Because the franchise fee, FF, is a fixed sum, the FC  VC firm’s fixed costs increase by this fee. Thus, average cost, equal to , and Q FC FF average fixed cost, equal to , increase by the average franchise fee . Note Q Q that the franchise fee does not affect average variable cost. Also, because marginal cost is the change in total cost with the production of an additional unit and because the fee is constant, marginal cost is unchanged. 3.b. Now suppose the firm is charged a tax that is proportional to the number of items it produces. Again, how does this tax affect the firm’s fixed, marginal, and average costs? Let t equal the per unit tax. When a tax is imposed on each unit produced, variable costs increase by tQ. Average variable costs increase by t, and because fixed costs are constant, average (total) costs also increase by t. Further, because total cost increases by t with each additional unit, marginal costs increase by t. 4. A recent issue of Business Week reported the following: During the recent auto sales slump, GM, Ford, and Chrysler decided it was cheaper to sell cars to rental companies at a loss than to lay off workers. That’s because closing and reopening plants is expensive, partly because the auto makers’ current union contracts obligate them to pay many workers even if they’re not working. When the article discusses selling cars “at a loss,” is it referring to accounting profit or economic profit? How will the two differ in this case? Explain briefly. When the article refers to the car companies selling at a loss, it is referring to accounting profit. The article is stating that the price obtained for the sale of the cars to the rental companies was less than their accounting cost. Economic profit would be measured by the difference of the price with the opportunity cost of the cars. This opportunity cost represents the market value of all the inputs used by the companies to produce the cars. The article mentions that the car companies must pay workers even if they are not working (and thus producing cars). This implies that the wages paid to these workers are sunk and are thus not part of the opportunity cost of production. On the other hand, the wages would still be included in the accounting costs. These accounting costs would then be higher than the opportunity costs and would make the accounting profit lower than the economic profit. 5. A chair manufacturer hires its assembly-line labor for $22 an hour and calculates that the rental cost of its machinery is $110 per hour. Suppose that a chair can be produced using 4 hours of labor or machinery in any combination. If the firm is currently using 3 hours of labor for each hour of machine time, is it minimizing its costs of production? If so, why? If not, how can it improve the situation? If the firm can produce one chair with either four hours of labor or four hours of capital, machinery, or any combination, then the isoquant is a straight line with a slope of -1 and intercept at K = 4 and L = 4, as depicted in Figure 7.5. 69 Chapter 7: The Cost of Production 22  0.2 when plotted with 110 TC TC capital on the vertical axis and has intercepts at K  and L  . The cost 110 22 minimizing point is a corner solution, where L = 4 and K = 0. At that point, total cost is $88. The isocost line, TC = 22L + 110K has a slope of  Capital 4 Isoquant for Q = 1 3 2 Isocost, slope = -0.20 Cost-minimizing corner solution 1 1 2 3 4 5 Labor Figure 7.5 6. Suppose the economy takes a downturn, and that labor costs fall by 50 percent and are expected to stay at that level for a long time. Show graphically how this change in the relative price of labor and capital affects the firm’s expansion path. Figure 7.6 shows a family of isoquants and two isocost curves. Units of capital are on the vertical axis and units of labor are on the horizontal axis. (Note: In drawing this figure we have assumed that the production function underlying the isoquants exhibits constant returns to scale, resulting in linear expansion paths. However, the results do not depend on this assumption.) If the price of labor decreases while the price of capital is constant, the isocost curve pivots outward around its intersection with the capital axis. Because the expansion path is the set of points where the MRTS is equal to the ratio of prices, as the isocost curves pivot outward, the expansion path pivots toward the labor axis. As the price of labor falls relative to capital, the firm uses more labor as output increases. 70 Chapter 7: The Cost of Production Capital Expansion path before wage fall 4 Expansion path after wage fall 3 2 1 1 2 3 4 5 Labor Figure 7.6 7. You are in charge of cost control in a large metropolitan transit district. A consultant you have hired comes to you with the following report: Our research has shown that the cost of running a bus for each trip down its line is $30, regardless of the number of passenger’s it carries. Each bus can carry 50 people. At rush hour, when the buses are full, the average cost per passenger is 60 cents. However, during off-peak hours, average ridership falls to 18 people and average cost soars to $1.67 per passenger. As a result, we should encourage more rush-hour business when costs are cheaper and discourage off-peak business when costs are higher. Do you follow the consultant’s advice? Discuss. The consultant does not understand the definition of average cost. Encouraging ridership always decreases average costs, peak or off-peak. If ridership falls to 10, costs climb to $3.00 per rider. Further, during rush hour, the buses are full. How could more people get on? Instead, encourage passengers to switch from peak to offpeak times, for example, by charging higher prices during peak periods. 8. An oil refinery consists of different pieces of processing equipment, each of which differs in its ability to break down heavy sulfurized crude oil into final products. The refinery process is such that the marginal cost of producing gasoline is constant up to a point as crude oil is put through a basic distilling unit. However, as the unit fills up, the firm finds that in the short run the amount of crude oil that can be processed is limited. The marginal cost of producing gasoline is also constant up to a capacity limit when crude oil is put through a more sophisticated hydrocracking unit. Graph the marginal cost of gasoline production when a basic distilling unit and a hydrocracker are used. The production of gasoline involves two steps: (1) distilling crude oil and (2) refining the distillate into gasoline. Because the marginal cost of production is constant up to the capacity constraint for both processes, the marginal cost curves are “mirror” L-shapes. 71 Chapter 7: The Cost of Production Marginal Cost MCT MC1 MC2 Q1 Q2 Quantity Figure 7.8 Total marginal cost, MCT, is the sum of the marginal costs of the two processes, i.e., MCT = MC1 + MC2, where MC1 is the marginal cost of distilling crude oil up to the capacity constraint, Q1, and MC2 is the marginal cost of refining distillate up to the capacity constraint, Q2. The shape of the total marginal cost curve is horizontal up to the lower capacity constraint. If the capacity constraint of the distilling unit is lower than that of the hydrocracking unit, MCT is vertical at Q1. If the capacity constraint of the hydrocracking unit is lower than that of the distilling unit, MCT is vertical at Q2. 9. You manage a plant that mass produces engines by teams of workers using assembly machines. The technology is summarized by the production function. Q = 4 KL where Q is the number of engines per week, K is the number of assembly machines, and L is the number of labor teams. Each assembly machine rents for r = $12,000 per week and each team costs w = $3,000 per week. Engine costs are given by the cost of labor teams and machines, plus $2,000 per engine for raw materials. Your plant has a fixed installation of 10 assembly machines as part of its design. a. What is the cost function for your plant — namely, how much would it cost to produce Q engines? What are average and marginal costs for producing Q engines? How do average costs vary with output? K is fixed at 10. The short-run production function then becomes Q = 40 L. 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: TC(Q) = rK + wL + 2000Q = (12,000)(10) + (3,000)(Q/40) + 2,000 Q = 120,000 + 2,075Q The average cost function is then given by: AC(Q) = TC(Q)/Q = 120,000/Q + 2,075 and the marginal cost function is given by:  TC(Q) /  Q = 2,075 Marginal costs are constant and average costs will decrease as quantity increases (due to the fixed cost of capital). 72 Chapter 7: The Cost of Production b. How many teams are required to producing 80 engines? What is the average cost per engine? To produce Q = 80 engines we need L = Q/40 labor teams or L = 2. Average costs are given by AC(Q) = 120,000/Q + 2,075 c. or AC = 3575 You are asked to make recommendations for the design of a new production facility. What would you suggest? In particular, what capital/labor (K/L) ratio should the new plant accommodate? If lower average cost were your only criterion, should you suggest that the new plant have more production capacity or less production capacity that the plant you currently manage? 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 level of output Q. The cost-minimization rule is given by MPK r = MPL w . To find the marginal product of capital, observe that increasing K by 1 unit increases Q by 4L, so MPK = 4L. Similarly, observe that increasing L by 1 unit increases Q by 4K, so MPL = 4K. (Mathematically, MPK = Q/K = 4L and MPL = Q/L = 4K.) Using these formulas in the cost-minimization rule, we obtain: 4L/r = 4K/w or K / L = w / r = 3,000 / 12,000 = 1/4 The new plant should accommodate a capital to labor ratio of 1 to 4. The firm’s capital-labor ratio is currently 10/2 or 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 and increase output. *10. A computer company’s cost function, which relates its average cost of production AC to its cumulative output in thousands of computers CQ and its plant size in terms of thousands of computers produced per year Q, within the production range of 10,000 to 50,000 computers is given by AC = 10 - 0.1CQ + 0.3Q. a. Is there a learning curve effect? The learning curve describes the relationship between the cumulative output and the inputs required to produce a unit of output. Average cost measures the input requirements per unit of output. Learning curve effects exist if average cost falls with increases in cumulative output. Here, average cost decreases as cumulative output, CQ, increases. Therefore, there are learning curve effects. b. Are there increasing or decreasing returns to scale? To measure scale economies, calculate the output, Q: TC EC  TC  Q Q elasticity of total cost, TC, with respect to TC MC Q  . TC AC Q If this elasticity is greater (less) than one, then there are decreasing (increasing) returns to scale, because total costs are rising faster (slower) than output. From average cost we can calculate total and marginal cost: 2 TC = Q(AC) = 10Q - (0.1)(CQ)(Q) + 0.3Q , therefore 73 Chapter 7: The Cost of Production MC  dTC  10  01 . CQ  0.6Q . dQ Because marginal cost is greater than average cost (because 0.6Q > 0.3Q), the elasticity, EC, is greater than one; there are decreasing returns to scale. The production process exhibits a learning effect and decreasing returns to scale. c. During its existence, the firm has produced a total of 40,000 computers and is producing 10,000 computers this year. Next year it plans to increase its production to 12,000 computers. Will its average cost of production increase or decrease? Explain. First, calculate average cost this year: AC1 = 10 - 0.1CQ + 0.3Q = 10 - (0.1)(40) + (0.3)(10) = 9. Second, calculate the average cost next year: AC2 = 10 - (0.1)(50) + (0.3)(12) = 8.6. (Note: Cumulative output has increased from 40,000 to 50,000.) The average cost will decrease because of the learning effect. 11. The short-run cost function of a company is given by the equation C = 190 + 53Q, where C is the total cost and Q is the total quantity of output, both measured in tens of thousands. a. What is the company’s fixed cost? When Q = 0, C = 190 (or $1,900,000). Therefore, fixed cost is equal to 190 (or $1,900,000). b. If the company produced 100,000 units of goods, what is its average variable cost? c. With 100,000 units, Q = 10. Variable cost is 53Q = (53)(10) = 530 (or $5,300,000). TVC $530 Average variable cost is   $53. Q 10 What is its marginal cost per unit produced? With constant average variable cost, marginal cost is equal to average variable cost, $53. d. What is its average fixed cost? At Q = 10, average fixed cost is e. TFC $190   $19 . Q 10 Suppose the company borrows money and expands its factory. Its fixed cost rises by $50,000, but its variable cost falls to $45,000 per 10,000 units. The cost of interest (I) also enters into the equation. Each one-point increase in the interest rate raises costs by $30,000. Write the new cost equation. Fixed cost changes from 190 to 195. Variable cost decreases from 53 to 45. Fixed cost also includes interest charges: 3I. The cost equation is C = 195 + 45Q + 3I. *12. Suppose the long-run total cost function for an industry is given by the cubic 2 3 equation TC = a + bQ + cQ + dQ . Show (using calculus) that this total cost function is consistent with a U-shaped average cost curve for at least some values of a, b, c, d. To show that the cubic cost equation implies a U-shaped average cost curve, we use algebra, calculus, and economic reasoning to place sign restrictions on the parameters of the equation. These techniques are illustrated by the example below. 74 Chapter 7: The Cost of Production First, if output is equal to zero, then TC = a, where a represents fixed costs. In the short run, fixed costs are positive, a > 0, but in the long run, where all inputs are variable a = 0. Therefore, we restrict a to be zero. Next, we know that average cost must be positive. Dividing TC by Q: 2 AC = b + cQ + dQ . This equation is simply a quadratic function. When graphed, it has two basic shapes: a U shape and a hill shape. We want the U shape, i.e., a curve with a minimum (minimum average cost), rather than a hill shape with a maximum. To the left of the minimum, the slope should be negative (downward sloping). At the minimum, the slope should be zero, and to the right of the minimum the slope should be positive (upward sloping). The first derivative of the average cost curve with respect to Q must be equal to zero at the minimum. For a U-shaped AC curve, the second derivative of the average cost curve must be positive. The first derivative is c + 2dQ; the second derivative is 2d. If the second derivative is to be positive, then d > 0. If the first derivative is equal to zero, then solving for c as a function of Q and d yields: c = -2dQ. If d and Q are both positive, then c must be negative: c < 0. To restrict b, we know that at its minimum, average cost must be positive. The minimum occurs when c + 2dQ = 0. We solve for Q as a function of c and d: c  0 . Next, substituting this value for Q into our expression for average Q 2d cost, and simplifying the equation: Fc I  dFc I , or bc H2d K H2d K 2 AC  b  cQ  dQ AC  b  implying b  c2 2d  c2 3d 2  b 3c 2 6d  2c 2 6d  b c2 6d  0. c2 2 . Because c and d > 0, b must be positive. 6d In summary, for U-shaped long-run average cost curves, a must be zero, b and d must 2 be positive, c must be negative, and 4db > c . However, the conditions do not insure that marginal cost is positive. To insure that marginal cost has a U shape and that its minimum is positive, using the same procedure, i.e., solving for Q at minimum marginal cost c / 3d , and substituting into the expression for marginal cost b + 2cQ + 2 2 3dQ , we find that c must be less than 3bd. Notice that parameter values that satisfy 2 this condition also satisfy 4db > c , but not the reverse. 75 Chapter 7: The Cost of Production Costs 2 MC AC 1 0.17 0.33 0.50 0.67 0.83 Quantity in Dozens 1.00 Figure 7.12 2 3 For example, let a = 0, b = 1, c = -1, d = 1. Total cost is Q - Q + Q ; average cost is 2 2 1 - Q + Q ; and marginal cost is 1 - 2Q + 3Q . Minimum average cost is Q = 1/2 and minimum marginal cost is 1/3 (think of Q as dozens of units, so no fractional units are produced). See Figure 7.12. *13. A computer company produces hardware and software using the same plant and labor. The total cost of producing computer processing units H and software programs S is given by TC = aH + bS - cHS, where a, b, and c are positive. Is this total cost function consistent with the presence of economies or diseconomies of scale? With economies or diseconomies of scope? There are two types of scale economies to consider: multiproduct economies of scale and product-specific returns to scale. From Section 7.5 we know that multiproduct economies of scale for the two-product case, SH,S, are SH , S  a f afb g afb g TC H, S H MCH  S MCS where MCH is the marginal cost of producing hardware and MCS is the marginal cost of producing software. The product-specific returns to scale are: SH  SS a f a f and afb g TC a H , S f TC a H ,0 f  S MC g afb TC H, S  TC 0, S H MCH S where TC(0,S) implies no hardware production and TC(H,0) implies no software production. We know that the marginal cost of an input is the slope of the total cost with respect to that input. Since b g b g TC  a  cS H  bS  aH  b  cH S, 76 Chapter 7: The Cost of Production we have MCH = a - cS and MCS = b - cH. Substituting these expressions into our formulas for SH,S, SH, and SS: SH ,S  aH  bS  cHS H a  cS  S b  cH SH ,S  aH  bS  cHS  1 , because cHS > 0. Also, Ha  Sb  2cHS b g b g or baH  bS  cHSg bS , or Hb a  cS g baH  cHSg ba  cSg 1 and similarly  Hb a  cS g b a  cS g SH  SH SS  aaH  bS  cHSf aH  1. Sa b  cH f There are multiproduct economies of scale, SH,S > 1, but constant product-specific returns to scale, SH = SC = 1. Economies of scope exist if SC > 0, where (from equation (7.8) in the text): af a f a f aH  bS  a aH  bS  cHS f S  , or TC a H, Sf cHS S   0. TC a H , Sf Sc  af TC H,0  TC 0, S  TC H , S , or, TC H , S c c Because cHS and TC are both positive, there are economies of scope. 77