A business that is expanding or contracting its operation needs to predict how costs will change as output changes. Estimates of future costs can be obtained from a cost function, which relates the cost of production to the level of output and other variables that the firm can control.

Suppose we wanted to characterize the short-run cost of production in the automobile industry. We could obtain data on the number of automobiles Q produced by each car company and relate this information to the variable cost of production VC. The use of variable cost, rather than total cost, avoids the problem of trying to allocate the fixed cost of a muj|iproduct firm's production process to the particular product being studied.

Figure 7.13 shows a typical pattern of cost and output data. Each point on the graph relates the output of an auto company to that company's variable cost of production. To predict cost accurately, we need to determine the underlying relationship between variable cost and output. Then, if a company expands its production, we can calculate what the associated cost is likely to be. The curve in the figure is drawn with this in mind-it provides a reasonably close fit to the cost data. (Typically, least-squares regression analysis would be used to fit the curve to the data.) But what shape of curve is the most appropriate, and how do we represent that shape algebraically? One cost function that might be chosen is

By interpreting the two coefficients in footnote 16 in light of the levels of the output and plant size variables, one can allocate about 15 percent of the cost reduction to increases in the average scale of plants, and 85 percent to increases in cumulative industry output. (Suppose plant scale doubled, while cumulative output increased by a factor of 5 during the study. Then costs would fall by 11 percent from the increased scale and by 62 percent from the increase in cumulative output.)

18 If an additional piece of equipment is needed as output increases, then annual rental cost of the equipment should be counted as a variable cost. If, however, the same machine can be used at all output levels, then its cost is fixed and should not be included.

Variable Cost

General Motors

Nissan

Honda

Volvo

Ford

Chrysler

FIGURE 7.13 Total Cost Curve for the Automobile Industry. An empirical estimate of the total cost curve can be obtained by using data for individual firms in an industry. The total cost curve for automobile production is obtained by determining statistically the curve that best fits the points that relate the output of each firm to the total cost of production.

This linear relationship between cost and output is easy to use but is applicable only if marginal cost is constant.19 For every unit increase in output, variable cost increases by 3, so marginal cost is constant and equal to 3- (a is also a component of variable cost but it varies with factors other than output.)

If we wish to allow for a U-shaped average cost curve and a marginal cost that is not constant, we must use a more complex cost function. One possibility, shown in Figure 7.14, is the quadratic cost function, which relates variable cost to output and output squared:

This implies a straight-line marginal cost curve of the form MC = 3 + 2-yQ.2n Marginal cost increases with output if y is positive, and decreases with output if 7 is negative. Average cost, given by AC = aJQ + 3 +-yQ, is U-shaped when "Y is positive.

19In statistical cost analyses, other variables might be added to the cost function to account for differences in input costs, production processes, product mix, etc., among firms.

Short-run marginal cost is given by ATVC/AQ = ยก3 + y\(Q2}/&Q. But A(Q2)/AQ = 2Q. Check this using calculus or by numerical example.) Therefore, (3 + 2yQ.

If the marginal cost curve is not linear, we might use a cubic cost function:

Figure 7.15 shows this cubic cost function. It implies U-shaped marginal as well as average cost curves.

Cost functions can be difficult to measure. First, output data often represent an aggregate of different types of products. Total automobiles produced by General Motors, for example, involves different models of cars. Second, cost data are often obtained directly from accounting information that fails to re-

Figure 7.15 shows this cubic cost function. It implies U-shaped marginal as well as average cost curves.

Cost functions can be difficult to measure. First, output data often represent an aggregate of different types of products. Total automobiles produced by General Motors, for example, involves different models of cars. Second, cost data are often obtained directly from accounting information that fails to re-

FIGURE 7.15 Cubic Cost Function. A cubic cost function implies that the average and the marginal cost curves are U-shaped.

fleet opportunity costs. Third, allocating maintenance and other plant costs to a particular product is difficult when the firm is a conglomerate that produces more than one product line.

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