Daniel l. Rubinfeld

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210 Part 2 Producers, Consumers, and Competitive Markets Chapter 7 The Cost of Production 2 average variable cost (AVC) Variable cost diyided bv' the level of output. ' In §6.3, we explain that diminishing marginal returns occurs when additional inputs result in a decrease in additions to output. The marginal product of labor is discussed in §6.3. Average variable cost (AVe) is \'ariable cost di\'ided by the level of output, VC/Q. The a\'erage \'ariable cost of producing 5 units of output is 526-that is, 5130/5. Table 7.1 shows that variable and total costs increase with output.. The rate at which these costs in.crease depends on the nature of the production process and, in particular, on the extent to which production im'oh'es diminishing returns to variable factors. Recall from Chapter 6 that diminishing returns to labor OCCurs "when the marginal product of labor is decreasing. If labor is the only input, "what happens as we increase the firm's output To produce more output, the firm must hire more labor. Then, if the marginal product of labor decreases as the amount of labor hired is increased (owing to diminishing rehlrns), succeSSively greater expenditures must be made to produce output at the higher rate. As a result, variable and total costs increase as the rate of output is increased. On the other hand, if the marginal product of labor decreases only slightly as the amount of labor is increased, costs will not rise so fast when the rate of output is increased 1 Let's look at the relationship between production and cost in more detail by concentrating on the costs of a firm that can hire as much labor as it wishes at a fixed wage lL'. Recall that marginal cost MC is the change in variable cost for a 1- Lmit change in output (i.e., ..l VC/ ..lQ). But the change in variable cost is the perunit cost of the extra labor w tinles the amount of extra labor needed to produce the extra output..lL Since ..lVC = w..lL, it fo11o\\'s that MC = ..lVC/..lQ = 'U..lL!..lQ Recall from Chapter 6 that the marginal product of labor MPL is the change in output resulting from a I-unit change in labor input, or ..lQ/..lL Therefore, the extra labor needed to obtain an extra unit of output is ..lL/..lQ = l/MPL· As a result, MC = ,ujivlP L (7.1) Equation (7.1) states that Inarginal cost is equal to the price of the input di\'ided by its marginal product. Suppose, for example, that the marginal product of labor is 3 and the \vage rate is 530 per hour. In that case, 1 hour of labor will increase output by 3 units, so that 1 unit of output will require 1/3 additional hour of labor and will cost 510. The marginal cost of producing that unit of output is 510, which is equal to the 'wage, 530, divided by the marginal product of labor, 3. A low marginal product of labor means that a large amount of additionallabor is needed to produce more output, a fact that leads, in turn, to a high marginal cost. Conversely, a high marginal product means that the labor requirement is low, as is the marginal cost. More generally, whenever the marginal produ,ct of labor decreases, the marginal cost of production increases, and vice \'ersa.- 1 We are implicitly assuming that because labor is hired in competiti\"e markets. the pa\'ment per unit of factor used is the same regardless of the firm's output 2 With two or more \'ariable inputs. the relationship is more complex The basic principle, hO\\'e\'er, still holds: The greater the producth'ity of factors, the less the \'ariable cost that the firm must incur to produce any given le\'el of output m ar gi!1al returns means. that the marginal product Diminishing declines as the quantity at labor employed mcreases. As a result, when there are diminishina marai;1al returns, marginal cost will increase as output increases. This can be s~en bv looking at the numbers for marginal cost in Table 7.L For output levels from 0 through 4, marginal cost is declining; for output le\'els from 4 throuah 11, however, marginal cost is increasing-a reflection of the presence of diminishina marginal returns. b The Shapes the Cost Curves Figure 7.1 illust:'ates how \'arious cost measures change as output changes. The top part of the flgure shows total cost and its two components, variable cost and .. Cost ,100 L (dollars per year) 300 173 100 Cost 100 (dollars per unit) 73 30 2 3 ,1 5 6 7 8 (a) I 6 7 8 (b) TC VC FC 9 10 11 12 13 Output (units per year) MC AVC 9 10 11 Output (units per year) In (a) total cost TC is the vertical sum of fixed cost FC and variable cost Ve. In (b) average total cost ATC is the sum of averaae variable cost AVC and averaae fixed cost A.Fe. Marginal cost MC crosses the a\'er~ge variable cost and averaae tgtal cost curves at their minimum b
$ 212 Part 2 Producers, Consumers, and Competitive Markets fixed cost; the bottom part shows marginal cost and average costs. These cost cun'es, vvhich are based on the information in Table 7.1, provide different kinds of information. Observe in Figure 7.1(a) that fixed cost FC does not vary with output-it is shown as a horizontal line at 550. Variable cost VC is zero when output is zero and then increases continuously as output increases. The total cost cun'e TC is determined by vertically adding the fixed cost curve to the variable cost curve. Because fixed cost is constant, the vertical distance beh\'een the h\'o curves is always $50. Figure 7.1(b) shows the corresponding set of marginal and average variable cost curves 3 Because total fixed cost is £50, the average fixed cost cun'e AFC falls continuously from £50 \'{hen output is 1, toward zero for large output. The shapes of the remaining curves are determined by the relationship between the marginal and average cost CLUTes. Whene\'er marginal cost lies below average cost, the average cost curve falls. Whenever marginal cost lies abo\'e average cost, the average cost curve rises. vVhen average cost is at a minimum, marginal cost equals average cost. Marginal and average costs are another example of the average-marginal relationship described in Chapter 6 (with respect to marginal and average product). At an output of 5 in Table 7.1, for example, the marginal cost of 18 is below the average variable cost of $26; thus the average is lowered in response to increases in output. But when marginal cost is 529, which is greater than average variable cost (5255), the average increases as output increases. Finally, when marginal cost (525) and average cost (525) are the same, average variable cost remains unchanged (at about 525). The ATC curve sho'ws the average total cost of production. Because a\'erage total cost is the sum of average variable cost and average fixed cost and the AFC curve declines everywhere, the vertical distance beh·veen the ATC and AVC curves decreases as output increases. TIle AVC cost cun'e reaches its minimum point at a lower output than the ATC curve. This follO\'\'s because MC = A VC at its minimum point and MC = ATC at its minimum point. Because ATC is always greater than AVC and the marginal cost curve MC is rising, the minimum point of the ATC curve must lie above and to the right of the minimum point of the AVC curve. Another way to see the relationship between the total cost cun-es and the a\'erage and marginal cost curves is to consider the line drawn frorn origin to point A in Figure 7,l(a). In that figure, the slope of the line measures a\'erage \'ariable cost (a total cost of 5175 divided by an output of 7, or a cost per unit of 525). Because the slope of the VC curve is the marginal cost (it measures the change in variable cost as output increases by 1 unit), the tangent to the VC CLUTe at A is the marginal cost of production when output is 7. At A, this marginal cost of $25 is equal to the average variable cost of 525, because a\'erage variable cost is minim.ized at this output. Note that the firm's output is measured as a flow: The firm produces a certain number of units per yenr. Thus its total cost is a flo'w-for example, some number of dollars per year. (Average and marginal costs, however, are measured in dollars per unit.) For simplicity, we will often drop the time reference, and refer to total cost in dollars and output in units. But you should remember that a firm's production of output and expenditure of cost occur over some time period. For simplicity, we will often use cost (C) to refer to total cost. Likewise, unless noted otherwise, we will use nI'ernge cost (AC) to refer to average total cost. 3 The cun'es do not exactlv match the numbers in Table 7.1. Because marginal cost represents the change in cost associated \;ith a change in output, we have plotted the Me cun'e for the first unit of output by setting output equal to~, for the second unit by setting output equal to It and so on Marginal and a\'erage cost are \'ery important concepts. As we will see in Chapter 8, they enter critically into the firm's choice of output leyel. KnowledGe of short-fur: costs is particularly important for firms that operate in an enviro~lrnent in whlCh demand conditions fluchlate considerably. If the firm is cunently producing at a l~\'el of O~ltput a~ which marginal cost is" sharply increasing, and if demand ma,Y mcrease m the tuture, management might ,vant to expand production capaClty to avoid higher costs, Al~lmin~ml is ~ ligh~~reight versatile lr:etal used in.a wide vari~ty. of applicahons, mcludmg all planes, automobIles, packagmg, and bmldmg materials. The production of aluminum begins with the mining of bauxite in such cow1tries as Australia, Brazil, Guinea, Jamaica, and Suriname. Bauxite is an ore that contains a relatively high concentration of alumina (aluminum oxide), whicl: is .separated frorn the bauxite through a chemical refining process, The alum~1a 1S then .converted to aluminum through a smelting process in which an electnc current IS used to separate the oxygen atoms from the aluminum oxide :nolecule~. It is this smelt~lg process-which is the most costly step in producmg alummum-that we tocus on here, All of the major aluminum producers, including Alcoa, Alcan, Reynolds, Alumax, an~ Kai~el~ operate smelting plants. A typical smelting plant ,v"ill have hvo productlOn Imes, each of which produces approximately 300 to -l00 tons of alul1-:inum per day. We will focus on the short-run cost of pl:oduction. Thus we cons1der the cost of operating an existing plant because there is insufficient time in the short run to build additional plants. (It takes about four years to plan, build, and fully equip an aluminum smelting plant.) , Although the cost of a smelting plant is substantial (over 51 billion), we will assum~ that the plant carmot be sold, and therefore the expendihlre is sunk and can be 19nored. Furthermore, because fixed costs, which are larGely for admin- · . 0 " lsh'ahve expenses, are relatively small, we 'will ignore them also. Thus we can focus entirely on short-run variable costs. Table 7.2 shows the averaGe of,erat- · f 0 mg costs or a t~pical aluminum smelter.~ The cost numbers apply to a plant that nm~ ~:'o sh1fts. per day to produce 600 tons of aluminum per day. If prices w~re sufflClently Illgh, the firm could choose to operate the plant on a threesh1~ts-per-day basis by asking 'workers to work overtime. However, wage and mamtenance costs would likely increase about 50 percent for this third shift because of the need to pay higher overtime wages. We have divided the cost components in Table 7.2 into two groups. The first group includes those costs that would remain the same at any output level, and the second includes costs that,would increase if output exceeded 600 tons per day. · Note that the largest cost components for an aluminum smelter are electric- 1ty an~ the cost of alumina; together they represent about 60 percent of total perating costs. Because elech"icity, alumina, and other raw materials are used m direct proportion to the amOLmt of aluminum produced, they represent variable costs that are constant with respect to the level of output. The costs of labor, maintenance, and freight are also proportional to the level of output, Sc Tl his examp~e is based on Kelmeth S. Corts, "The Aluminum Industn' in 1994," Han'ard Business 1001 Case N9-799-129, April 1999 ' Chapter 7 The Cost of Production 213
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PART 1 Introduction: Marl
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PART 1 surveys the scope of microec
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F PART 3 examines a broad range of
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Daniel l. Rubinfeld

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