10 Estimate a Variable and Fixed Cost Equation and Predict Future Costs

Sometimes, a business will need to use cost estimation techniques, particularly in the case of mixed costs, so that they can separate the fixed and variable components, since only the variable components change in the short run. Estimation is also useful for using current data to predict the effects of future changes in production on total costs. Three estimation techniques that can be used include the scatter graph, the high-low method, and regression analysis. Here we will demonstrate the scatter graph and the high-low methods (you will learn the regression analysis technique in advanced managerial accounting courses.

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### Functions of Cost Equations

The cost equation is a linear equation that takes into consideration total fixed costs, the fixed component of mixed costs, and variable cost per unit. Cost equations can use past data to determine patterns of past costs that can then project future costs, or they can use estimated or expected future data to estimate future costs. Recall the mixed cost equation: When using this approach, Eagle Electronics must be certain that it is only predicting costs for its relevant range. For example, if they must hire a second supervisor in order to produce 12,000 units, they must go back and adjust the total fixed costs used in the equation. Likewise, if variable costs per unit change, these must also be adjusted.

This same approach can be used to predict costs for service and merchandising firms, as shown by examining the costs incurred by J&L Accounting to prepare a corporate income tax return, shown in (Figure).

Cost Information for J&L AccountingCost IncurredFixed or VariableCost
Building rentFixed?1,000 per month
Direct labor (for CPAs)Variable?250 per tax return
Secretarial staffFixed?2,000 per month
Accounting clerksVariable?100 per return

J&L wants to predict their total costs if they complete 25 corporate tax returns in the month of February.

Determine total fixed costs: ?1,000 + ?2,000 = ?3,000Determine variable costs per tax return: ?250 + ?100 = ?350Complete the cost equation: Y = ?3,000 + ?350x

Using this equation, J&L can now predict its total costs (Y) for the month of February when they anticipate preparing 25 corporate tax returns:

$$\begin{array}{c}Y=\text{?}3,000+\left(\text{?}350\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}25\right)\hfill \\ Y=\text{?}3,000+\text{?}8,750\hfill \\ Y=\text{?}11,750\hfill \end{array}$$

J&L can now use this predicted total cost figure of ?11,750 to make decisions regarding how much to charge clients or how much cash they need to cover expenses. Again, J&L must be careful to try not to predict costs outside of the relevant range without adjusting the corresponding total cost components.

J&L can make predictions for their costs because they have the data they need, but what happens when a business wants to estimate total costs but has not collected data regarding per-unit costs? This is the case for the managers at the Beach Inn, a small hotel on the coast of South Carolina. They know what their costs were for June, but now they want to predict their costs for July. They have gathered the information in (Figure). ### Demonstration of the Scatter Graph Method to Calculate Future Costs at Varying Activity Levels

One of the assumptions that managers must make in order to use the cost equation is that the relationship between activity and costs is linear. In other words, costs rise in direct proportion to activity. A diagnostic tool that is used to verify this assumption is a scatter graph.

A scatter graph shows plots of points that represent actual costs incurred for various levels of activity. Once the scatter graph is constructed, we draw a line (often referred to as a trend line) that appears to best fit the pattern of dots. Because the trend line is somewhat subjective, the scatter graph is often used as a preliminary tool to explore the possibility that the relationship between cost and activity is generally a linear relationship. When interpreting a scatter graph, it is important to remember that different people would likely draw different lines, which would lead to different estimations of fixed and variable costs. No one person’s line and cost estimates would necessarily be right or wrong compared to another; they would just be different. After using a scatter graph to determine whether cost and activity have a linear relationship, managers often move on to more precise processes for cost estimation, such as the high-low method or least-squares regression analysis.

To demonstrate how a company would use a scatter graph, let’s turn to the data for Regent Airlines, which operates a fleet of regional jets serving the northeast United States. The Federal Aviation Administration establishes guidelines for routine aircraft maintenance based upon the number of flight hours. As a result, Regent finds that its maintenance costs vary from month to month with the number of flight hours, as depicted in (Figure). In scatter graphs, cost is considered the dependent variable because cost depends upon the level of activity. The activity is considered the independent variable since it is the cause of the variation in costs. Regent’s scatter graph shows a positive relationship between flight hours and maintenance costs because, as flight hours increase, maintenance costs also increase. This is referred to as a positive linear relationship or a linear cost behavior.

Will all cost and activity relationships be linear? Only when there is a relationship between the activity and that particular cost. What if, instead, the cost of snow removal for the runways is plotted against flight hours? Suppose the snow removal costs are as listed in (Figure).

Snow Removal CostsMonthActivity Level: Flight HoursSnow Removal Costs
January21,000?40,000
February23,00050,000
March14,0008,000
April17,0000
May10,0000
June19,0000 where Y2 is the total cost at the highest level of activity; Y1 is the total cost at the lowest level of activity; X2 is the number of units, labor hours, etc., at the highest level of activity; and X1 is the number of units, labor hours, etc., at the lowest level of activity.

Using the maintenance cost data from Regent Airlines shown in (Figure), we will examine how this method works in practice.

The first step in analyzing mixed costs with the high-low method is to identify the periods with the highest and lowest levels of activity. In this case, it would be February and May, as shown in (Figure). We always choose the highest and lowest activity and the costs that correspond with those levels of activity, even if they are not the highest and lowest costs.

We are now able to estimate the variable costs by dividing the difference between the costs of the high and the low periods by the change in activity using this formula: Using the high-low method, express the company’s maintenance costs as an equation where x represents the gallons of paint produced. Then estimate the fixed and variable costs.Predict the maintenance costs if 90,000 gallons of paint are produced.Predict the maintenance costs if 81,000 gallons of paint are produced.Using Excel, create a scatter graph of the cost data and explain the relationship between gallons of paint produced and equipment maintenance expenses.

(Figure)This cost data from Hickory Furniture is for the year 2017. Using the high-low method, express the company’s overtime wages as an equation where x represents number of invoices processed. Assume BC has monthly fixed costs of ?3,800.Predict the overtime wages if 9,000 invoices are processed.Predict the overtime wages if 6,500 invoices are processed.Using Excel, create a scatter graph of the cost data and explain the relationship between the number of invoices processed and overtime wage expense.

(Figure)This cost data from Hickory Furniture is for the year 2017. Prepare a scatter graph of the shipping data. Plot cost on the vertical axis and yachts shipped on the horizontal axis. Is the relationship between shipping costs and unit shipped approximately linear? Draw a straight line through the scatter graph.Using the high-low method, create the cost formula for Carolina Yachts’ shipping costs.The least-squares regression method was used and the analysis resulted in this cost equation: Y = 4,000 + 1,275x. Comment on the accuracy of your high-low method estimation.What would you estimate shipping costs to be if Carolina Yachts shipped 10 yachts in a single month? Use the cost formula you obtained in part B. Comment on how accurately this is reflected by the scatter graph you constructed.What factors other than number of yachts shipped do you think could affect Carolina Yachts’ shipping expense? Explain.

See more: Once Diminishing Returns Have Set In, As Output Increases, The Total Cost Curve:

(Figure)Gadell Farms produces venison sausage that is distributed to grocery stores throughout the Southeast. They have collected this shipping cost data: Prepare a scatter graph of the shipping data. Plot cost on the vertical axis and tons produced on the horizontal axis. Is the relationship between packaging costs and tons produced approximately linear? Draw a straight line through the scatter graph.Using the high-low method, estimate the cost formula for Gadell Farms’ packaging costs.The least-squares regression method was used and the analysis resulted in this cost equation: Y = 1650 + 78.57x. Comment on the accuracy of your high-low method estimation.What would you estimate packaging costs to be if Gadell Farms shipped 10 tons in a single month? Use the cost formula you obtained in part B. Comment on how accurately this is reflected by the scatter graph you constructed.What factors other than number of tons produced do you think could affect Gadell Farm’s packaging expense? Explain.

### Glossary

high-low methodtechnique for separating the fixed and variable cost components of mixed costsscatter graphplot of pairs of numerical data that represents actual costs incurred for various levels of activity, with one variable on each axis, used to determine whether there is a relationship between them