Estimating a Cost Function Using Quantitative Analysis

Learning Objective 4 Outline six steps in estimating a cost function using quantitative analysis

• the end result (Step 6) is to evaluate the cost driver of the estimated cost function

There are six steps in estimating a cost function using quantitative analysis of past data. We illustrate the steps using the Elegant Rugs example.

1. Step 1: CHOOSE THE DEPENDENT VARIABLE. Which dependent variable (the cost to be predicted and managed) managers choose will depend on the specific cost function being estimated. In the Elegant Rugs example, the dependent variable is indirect manufacturing labor costs.

1. Step 2: IDENTIFY THE INDEPENDENT VARIABLE, OR COST DRIVER. The independent variable (level of activity or cost driver) is the factor used to predict the dependent variable (costs). When the cost is an indirect cost, the independent variable is also called a cost-allocation base. Frequently, the management accountant, working with the management team, will cycle through the six steps several times, trying alternative economically plausible cost drivers to identify the one that best fits the data.
2. Recall that a cost driver should be measurable and have an economically plausible relationship with the dependent variable. Economic plausibility means that the relationship (describing how changes in the cost driver lead to changes in the costs being considered) is based on a physical relationship, a contract, or knowledge of operations and makes economic sense to the operating manager and the management accountant. As you learned in Chapter 5, all the individual items of costs included in the dependent variable should have the same cost driver; that is, the cost pool should be homogenous. When this is not the case, the management accountant should investigate the possibility of creating homogenous cost pools and estimating more than one cost function, one for each cost pool/cost driver pair.
3. As an example, consider several types of fringe benefits paid to employees and the cost drivers of the benefits:

1. The costs of health benefits and cafeteria meals can be combined into one homogenous cost pool because they have the same cost driver—the number of employees. Pension benefits and life insurance costs have a different cost driver—the salaries of employees—and, therefore, should not be combined with health benefits and cafeteria meals. Instead, they should be aggregated into a separate homogenous cost pool, which can be estimated using the salaries of employees receiving these benefits as the cost driver.
2. Step 3: COLLECT DATA ON THE DEPENDENT VARIABLE AND THE COST DRIVER. This is usually the most difficult step in cost analysis. Management accountants obtain data from company documents, from interviews with managers, and through special studies. These data may be time-series data or cross-sectional data.
3. Time-series data pertain to the same entity (such as an organization, plant, or activity) over successive past periods. Weekly observations of Elegant Rugs’ indirect manufacturing labor costs and number of machine-hours are examples of time-series data. The ideal time-series database would contain numerous observations for a company whose operations have not been affected by economic or technological change. A stable economy and stable technology ensure that data collected during the estimation period represent the same underlying relationship between the cost driver and the dependent variable.
4. Cross-sectional data pertain to different entities during the same period. For example, studies of loans processed and the related personnel costs at 50 individual, yet similar, branches of a bank during March 2020 would produce cross-sectional data for that month. The cross-sectional data should be drawn from entities that, within each entity, have a similar relationship between the cost driver and costs. Later in this chapter, we describe the problems that arise in data collection.
5. Step 4: PLOT THE DATA. The general relationship between the cost driver and costs can be readily seen by graphing a plot of the data. The plot provides insight into the relevant range of the cost function and reveals whether the relationship between the driver and costs is approximately linear. Moreover, the plot highlights extreme observations (observations outside the general pattern) that analysts should check. Was there an error in recording the data or an unusual event, such as a work stoppage, that makes these observations unrepresentative of the normal relationship between the cost driver and the costs?
6. Exhibit 10-4 is a plot of the weekly data from columns B and C in Exhibit 10-3. This graph provides strong visual evidence of a positive linear relationship between Elegant Rugs’ number of machine-hours and indirect manufacturing labor costs (when machine-hours go up, so do indirect manufacturing labor costs). There are no extreme observations in Exhibit 10-4. The relevant range is from 46 to 96 machine-hours per week (weeks 8 and 6, respectively).

1. Step 5: ESTIMATE THE COST FUNCTION. The two most common forms of quantitative analysis are the high-low method and regression analysis. We present these methods after Step 6.
2. Step 6: EVALUATE THE COST DRIVER OF THE ESTIMATED COST FUNCTION. In this step, managers and management accountants ask questions such as, is the cause-and effect relationship between the cost driver and cost plausible? How strong is the relationship? We next describe the high-low method.

High-Low Method

The simplest form of quantitative analysis to “fit” a line to data points is the high-low method. It uses only the highest and lowest observed values of the cost driver within the relevant range and their respective costs to estimate the slope coefficient and the constant of the cost function. It provides a quick first look at the relationship between a cost driver and costs. We illustrate the high-low method using data from Exhibit 10-3.

The slope coefficient, b, is calculated as follows:

To compute the constant, we can use either the highest or the lowest observation of the cost driver. Both calculations yield the same answer because the method solves two linear equations with two unknowns, the slope coefficient and the constant. The equation of a line is given by $y=a+bX$, so $a=y-bX$. 

At the highest observation of the cost driver, the constant, a, is

At the lowest observation of the cost driver, a is

Thus, the high-low estimate of the cost function, $y=a+bX$ is:

The blue line in Exhibit 10-5 shows the estimated cost function using the high-low method, formed by joining the observations with the highest and lowest values of the cost driver (number of machine-hours). This simple high-low line falls “in between” the data points; there are three observations on the line, four above it and five below it. The intercept $(a=\mathrm{\$}23.68)\mathrm{,}$ the point where the dashed extension of the blue line meets the y-axis, is the constant component of the equation that provides the best linear approximation of how a cost behaves within the relevant range of 46–96 machine-hours. Managers should not interpret the intercept as an estimate of the fixed costs if no machines were run. Why? Because running no machines—that is, using zero machine-hours—is outside the relevant range.

Suppose Elegant Rugs’ indirect manufacturing labor costs in week 6 were $1,280, instead of $1,456. In this case, the highest observation of the cost driver (96 machine-hours in week 6) will not coincide with the new highest observation of costs ($1,316 in week 9). How would this change affect our high-low calculation? Because the cause-and-effect relationship runs from the cost driver to costs, the high-low method still uses the highest and lowest observations of the cost driver and estimates the new cost function using data from weeks 6 (high) and 8 (low).

The high-low method is simple to compute and easy to understand. It gives Elegant Rugs’ managers quick initial insight into how the cost driver—the number of machine-hours—affects the firm’s indirect manufacturing labor costs. However, it is sometimes misleading for managers to rely on only two observations to estimate a cost function. Suppose a labor contract guarantees a minimum payment in week 8. Indirect manufacturing labor costs in week 8 are $1,000, instead of $710, when 46 machine-hours are used. The green line in Exhibit 10-5 shows the cost function estimated by the high-low method using this revised cost. Note that all of the data points lie on or below the line! In this case, choosing the highest and lowest observations for machine-hours results in an estimated cost function that poorly represents the underlying linear cost relationship between number of machine-hours and indirect manufacturing labor costs. In such a situation, managers modify the high-low method so that the two observations chosen to estimate the cost function are a representative high and a representative low. This adjustment avoids extreme observations arising from abnormal events influencing the estimate of the cost function. The modified cost function is more representative of the relationship between the cost driver and costs and, therefore, more useful for making decisions. Next, we describe the regression analysis method. Rather than just high and low values, it uses all available data to estimate the cost function.