Nonlinear Cost Functions

Learning Objective 6 Explain nonlinear cost functions in particular, those arising from learning-curve effects

• graph of cost function is not a straight line, for example, because of quantity discounts or costs changing in steps
• either cumulative average-time learning, where cumulative average time per unit declines by a constant percentage, as units produced double
• or incremental unit-time learning, in which incremental time to produce the last unit declines by a constant percentage, as units produced double

Cost functions are not always linear. A Nonlinear cost function is a cost function for which the graph of total costs (based on the level of a single activity) is not a straight line within the relevant range. To see what a nonlinear cost function looks like, return to Exhibit 10-2. The relevant range is currently set at 2,000 to 6,500 snowboards. But if we extend the relevant range to cover the region from 0 to 8,000 snowboards produced, it is evident that the cost function over this expanded range is graphically not straight.

Consider another example. Economies of scale may enable an advertising agency to produce double the number of advertisements for less than double the costs. Even direct materials costs are not always linear. As Panel A of Exhibit 10-9 shows, total direct materials costs rise as the units of direct materials purchased increase. But, because of quantity discounts, these costs rise more slowly (as indicated by the changing slope coefficient) as the units of direct materials purchased increase. This cost function has a slope coefficient $b=\mathrm{\$}25$ per unit for 1–1,000 units purchased, $b=\mathrm{\$}15$ per unit for 1,001–2,000 units purchased, and $b=\mathrm{\$}10$ per unit for 2,001–3,000 units purchased. The direct materials cost per unit falls with each price cut. The cost function is nonlinear over the relevant range from 1 to 3,000 units. Over a more narrow relevant range (for example, from 1 to 1,000 units), the cost function is linear.

Step cost functions are also nonlinear cost functions. A step cost function is a cost function in which the cost remains the same over various ranges of the level of activity but increases by discrete amounts—that is, increases in steps—as the level of activity increases from one range to the next. Panel B in Exhibit 10-9 shows a step variable-cost function, a step cost function in which cost remains the same over narrow ranges of the level of activity in each relevant range. Panel B shows the relationship between units of production and setup costs. The pattern is a step cost function because setup costs increase with each production batch started and then stay the same as each unit is produced. In the relevant range from 0 to 6,000 production units, the cost function is nonlinear. However, as shown by the green line in Panel B, managers often approximate step variable costs with a continuously variable cost function. This type of step cost pattern also occurs when production inputs such as materials-handling labor, supervision, and process engineering labor are acquired in discrete quantities but used in fractional quantities.

Panel C in Exhibit 10-9 shows a step fixed-cost function for Crofton Steel, a company that operates large heat-treatment furnaces to harden steel parts. Comparing Panels B and C, the main difference between a step variable-cost function and a step fixed-cost function is that in the latter the cost remains the same over wide ranges of the activity in each relevant range. The ranges indicate the number of furnaces being used (operating costs of each furnace are $300,000). The cost increases from one range to the next higher range when another furnace is added. Within the relevant range of 7,500–15,000 hours of furnace time, the company expects to operate with two furnaces at a fixed cost of $600,000. However, in the relevant range from 0 to 22,500 hours, the cost function is nonlinear: The graph in Panel C is not a single straight line; it is three broken lines.

Linear regression techniques can be adapted to estimate the non-linear relationships described in this section. For example, piece-wise linear regression could be used to estimate the different linear segments in Exhibit 10-9, Panel A. Different non-linear functional forms such as a quadratic relationship could be used to fit a curve to data. One of the most commonly used tools for building models in a world of “big data” is logistic regression. This technique is used when the dependent variable takes on a limited set of values, for example, a binary variable, such as whether or not a customer will leave a telecommunications company in the next 3 months or whether or not a borrower will default on a loan.

Consider a bank that is evaluating how it might use publicly available FICO sores that range from 300 to 850 based on an applicant’s past credit history to evaluate the risk of default on a loan. Exhibit 10-10, Panel A, shows data on past loans where the dependent variable is coded as a 1 if an applicant defaulted on a loan and a 0 if the applicant did not. The independent variable is the applicant’s FICO score. Exhibit 10-10, Panel B, shows a plot of the data. It also shows the regression line if a linear regression was fitted to the data. Although it is technically possible to fit a line to the data, the relationship between the independent variable (FICO score) and the independent variable (a 0 or a 1) is not linear. Fitting a linear regression to these data does not seem appropriate. Instead, as in Panel B, a logistic regression fits an S curve to the data using a technique called maximum likelihood estimation (MLE). Although the mathematical details are beyond the scope of this text, think of the MLE estimate (which ranges from 0 to 1) as a probability. The estimation technique fits an S curve with the objective of assigning a high probability (closer to 1) to defaulted loans and a low probability (closer to 0) to repaid loans. Of course, it cannot achieve this perfectly since there are loans (such as Loan 5) that have low FICO scores (similar to loans that usually default) but are in fact repaid and other loans (such as Loan 18) that have high FICO scores (similar to loans that usually repay) but are in fact in default. As we will see in the next chapter, the management accountant will work with management to determine a cutoff value trading off the interest the bank would earn from making the loan against the losses it would incur if the loan defaults. All borrowers with calculated values below the cutoff value (which correspond to relatively high FICO scores) will be given loans while all borrowers with calculated values above the cutoff value (which correspond to relatively low FICO scores) will not be given loans. We next turn our attention to another form of non-linear estimation—learning curves.

Learning Curves

Nonlinear cost functions also result from learning curves, a phenomenon first documented in aircraft assembly. A learning curve is a function that measures how labor-hours per unit decline as units of production increase because workers are learning and becoming better at their jobs. As workers become more familiar with their tasks, their efficiency improves. Managers learn how to more efficiently schedule work and operate the plant. Unit costs decrease as productivity increases, and the unit-cost function behaves nonlinearly. These nonlinearities must be considered when estimating and predicting unit costs.

The term experience curve describes a broader application of the learning curve—one that extends to other business functions in the value chain, such as marketing, distribution, and customer service. An experience curve measures the decline in the cost per unit of these various business functions as the amount of these activities increases. For companies such as Dell Computer, Walmart, and McDonald’s, learning curves and experience curves are key elements of their strategies to reduce costs and increase customer satisfaction, market share, and profitability.

We now describe two learning-curve models: the cumulative average-time learning model and the incremental unit-time learning model.