The definition of variable cost assumes a linear relationship between the cost of an activity and its associated driver. For example, Reddy Heaters uses one 3-inch seg- ment of pipe in each insert heater. Each 3-inch segment costs $4. The total variable cost of 3-inch segments can be expressed as:
Total variable cost $4 Units produced
If 100 insert heaters are produced, the total cost of pipe segments is $400 ($4 100). If 200 insert heaters are produced, the total cost is $800 ($4 200). As pro- duction doubles, the cost of the 3-inch segments doubles. In other words, cost increases in direct proportion to the number of units produced. The linear relation- ship for the pipe-segments example is shown in Exhibit 3-7. How reasonable is this assumption that costs are linear? Do variable activity costs really increase in direct proportion to increases in the level of the activity driver? If not, then how closely does this assumed linear cost function approximate the underlying cost function?
Economists usually argue that variable costs increase at a decreasing rate up to a certain volume, at which point they increase at an increasing rate. This type of non- linear behavior is displayed in Exhibit 3-8. Here, variable costs increase as the num- ber of units increases, but not in direct proportion. For example, a power supplier that initially has ample capacity may set prices that decrease per kilowatt-hour to encourage consumption; yet once the power plant capacity has been met, any fur- ther demands may produce higher prices to ration a now-scarce resource among users. What if the nonlinear view more accurately portrays reality? What do we do
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C h a p t e r 3 / A c t i v i t y C o s t B e h a v i o r
Cost (dollars)
Volume (units)
VC $4X
Exhibit 3-7
Linearity of Variable Costs: 3-Inch Segments, Reddy Heatersthen? One possibility is to determine the actual cost function. But every activity could have a different cost function, and this approach could be very time consum- ing and expensive (if it can even be done). It is much simpler to assume a linear relationship.
If the linear relationship is assumed, then the main concern is how well this assumption approximates the underlying cost function. Exhibit 3-9 gives us some idea of the consequences of assuming a linear cost function. Recall that the relevant range is the range of output for which the assumed cost relationships are valid.
Here, validity refers to how closely the linear cost function approximates the under- lying cost function. Note that for units of the activity driver beyond X1the approxi- mation appears to break down.
84 P a r t 2 / A c t i v i t y - B a s e d A c c o u n t i n g
Cost (dollars)
Volume (units)
Exhibit 3-8
Nonlinearity of Variable CostsCost (dollars)
Volume (units)
Large Error Region
Error
Relevant Range
0 X1
Exhibit 3-9
Linear ApproximationThe equation for a straight line is
Total cost Fixed cost (Variable rate Output)
This equation is a cost formula. Let’s take a closer look at each term in the cost for- mula. The dependent variableis the cost we are trying to predict, or “Total cost.” In this equation, total cost depends on only one variable, “Output.” Output is the measure of activity; it is the independent variable.“Fixed cost” is the intercept parameter,and it is the fixed cost portion of total cost. Finally, “Variable rate” is the cost per unit of activity; it is also called the slope parameter.Exhibit 3-10 shows this graphically.
Thedependent variableis a variable whose value depends on the value of another variable. It is easy to see that we are trying to find the “Total cost”—and that its value depends on the values of the parameters and variable on the right-hand side of the equation. The independent variableis a variable that measures output and explains changes in the cost. It is an activity driver. The choice of an indepen- dent variable is related to its economic plausibility. That is, the manager will attempt to find an independent variable that causes or is closely associated with the depen- dent variable. The intercept parametercorresponds to fixed cost. Graphically, the intercept parameter is the point at which the mixed-cost line intercepts the cost (ver- tical) axis. The slope parametercorresponds to the variable cost per unit of activity.
Graphically, this represents the slope of the mixed-cost line.
Since accounting records reveal only the amount of activity output and the total cost, those values must be used to estimate the intercept and slope parameters (the fixed cost and the variable rate). With estimates of fixed cost and variable rate, the fixed and variable components can be estimated, and the behavior of the mixed cost can be predicted as activity usage changes.
Three methods will be described for estimating the fixed cost and the variable rate. These methods are the high-low method, the scatterplot method, and the method of least squares. The same data will be used with each method so that com- parisons among them can be made. The data have been accumulated for the setup activity of Reddy Heaters’ Newark, New Jersey, plant. The plant manager believes that setup hours are a good driver for the activity setting up the production line.
Assume that the accounting records of the plant disclose the following setup costs and setup hours for the past five months:
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C h a p t e r 3 / A c t i v i t y C o s t B e h a v i o r
Cost
Fixed Component Variable Component
Exhibit 3-10
Mixed-Cost BehaviorMonth Setup Costs Setup Hours
January $1,000 100
February 1,250 200
March 2,250 300
April 2,500 400
May 3,750 500