# 6.5 Functions of Cost Equations

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.

## 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:

where Y is the total mixed cost, a is the fixed cost, b is the variable cost per unit, and x is the level of activity.

Let’s take a more in-depth look at the cost equation by examining the costs incurred by Eagle Electronics in the manufacture of home security systems, as shown in Table 6.13.

Table 6.13 Cost Information for Eagle Electronics By: Rice University Openstax CC BY-NC-SA 4.0
Cost Incurred Fixed or Variable Cost
Lease on manufacturing equipment Fixed $50,000 per year Supervisor salary Fixed$75,000 per year
Direct materials Variable $50 per unit Direct labor Variable$20 per unit

By applying the cost equation, Eagle Electronics can predict its costs at any level of activity (x) as follows:

1. Determine total fixed costs: $50,000 +$75,000 = $125,000 2. Determine variable costs per unit:$50 + $20 =$70
3. Complete the cost equation: Y = $125,000 +$70x

Using this equation, Eagle Electronics can now predict its total costs (Y) for any given level of activity (x), as in Figure 6.44:

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 Table 6.14.

Table 6.14 Cost Information for J&L Accounting By: Rice University Openstax CC BY-NC-SA 4.0
Cost Incurred Fixed or Variable Cost
Building rent Fixed $1,000 per month Direct labor (for CPAs) Variable$250 per tax return
Secretarial staff Fixed $2,000 per month Accounting clerks Variable$100 per return

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

1. Determine total fixed costs: $1,000 +$2,000 = $3,000 2. Determine variable costs per tax return:$250 + $100 =$350
3. Complete 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 6.45. In June, they had an occupancy of 75 nights. For the Beach Inn, occupancy (rooms rented) is the cost driver. Since they know what is driving their costs, they can determine their per-unit variable costs in order to forecast future costs: $\begin{array}{ccc}\hfill \frac{\text{Front Desk Staff}}{\text{75 nights}}& =\hfill & \frac{3,800}{75}=50.67\phantom{\rule{0.2em}{0ex}}\text{variable front desk staff costs per night}\hfill \\ \\ \hfill \frac{\text{Cleaning Staff}}{\text{75 nights}}& =\hfill & \frac{2,500}{75}=33.33\phantom{\rule{0.2em}{0ex}}\text{variable cleaning staff costs per night}\hfill \\ \\ \hfill \frac{\text{Laundry Service}}{\text{75 nights}}& =\hfill & \frac{1,200}{75}=16.00\phantom{\rule{0.2em}{0ex}}\text{variable laundry service costs per night}\hfill \end{array}$ Now, the Beach Inn can apply the cost equation in order to forecast total costs for any number of nights, within the relevant range. 1. Determine total fixed costs:$700 + $2,500 =$3,200
2. Determine variable costs per night of occupancy: $50.67 +$33.33 + $16.00 =$100
3. Complete the cost equation: Y = $3,200 +$100x

Using this equation, the Beach Inn can now predict its total costs (Y) for the month of July, when they anticipate an occupancy of 93 nights.

$\begin{array}{c}Y=\text{}3,200+\left(\text{}100\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}93\right)\hfill \\ Y=\text{}3,200+\text{}9,300\hfill \\ Y=\text{}12,500\hfill \end{array}$

In all three examples, managers used cost data they have collected to forecast future costs at various activity levels.

### Waymaker Furniture

Waymaker Furniture has collected cost information from its production process and now wants to predict costs for various levels of activity. They plan to use the cost equation to formulate these predictions. Information gathered from March is presented in Table 6.15.

Table 6.15 March Cost Information for Waymaker Furniture By: Rice University Openstax CC BY-NC-SA 4.0
Cost Incurred Fixed or Variable March Cost
Plant supervisor salary Fixed $12,000 per month Lumber (direct materials) Variable$75,000 total
Production worker wages Variable $11.00 per hour Machine maintenance Variable$5.00 per unit produced
Lease on factory Fixed $15,000 per month In March, Waymaker produced 1,000 units and used 2,000 hours of production labor. Using this information and the cost equation, predict Waymaker’s total costs for the levels of production in Table 6.16. Table 6.16 March Cost Information for Waymaker Furniture By: Rice University Openstax CC BY-NC-SA 4.0 Month Activity Level April 1,500 units May 2,000 units June 2,500 units Table 6.16 Solution Total Fixed Cost =$12,000 + $15,000 =$27,000. Direct Materials per Unit = $75,000 ÷ 1,000 Units =$75 per unit. Direct Labor per Hour = $11.00. Machine Maintenance =$5.00 per unit. Total Variable Cost per Unit = $75 +$11 + $5 =$91 per unit.