Linear Cost Functions

To understand three basic types of linear cost functions and to see the role of cost functions in business decisions, consider the negotiations between StoreBox, a technology startup, and Forest Web Services (FWS), which provides StoreBox with enterprise-class cloud computing and data analytic services.

• ALTERNATIVE 1: StoreBox pays $0.50 per CPU hour used. Total cost to StoreBox changes in proportion to the number of CPU hours used. The number of CPU hours used is the only factor whose change causes a change in total cost. 

Panel A in Exhibit 10-1 presents this variable cost for StoreBox. Under alternative 1, there is no fixed cost for cloud services. We write the cost function in Panel A of Exhibit 10-1 as

1. where X measures the number of CPU hours used (on the x-axis) and y measures the total cost of the CPU hours used (on the y-axis), calculated using the cost function. Panel A illustrates the $0.50 slope coefficient, the amount by which total cost changes when a 1-hour change occurs in CPU usage. Throughout the chapter, uppercase letters, such as X, refer to the actual observations, and lowercase letters, such as y, represent estimates or calculations made using a cost function.

• ALTERNATIVE 2: StoreBox pays a fixed amount of $1,000 per month. Total cost is fixed at $1,000 per month, regardless of the number of CPU hours used. (We use the same activity measure, number of CPU hours used, to compare cost-behavior patterns under the three alternatives.)

1. Panel B in Exhibit 10-1 shows the fixed-cost alternative for StoreBox. We write the cost function in Panel B as

1. The fixed cost of $1,000 is called a constant; it is the component of the total cost that does not vary with changes in the level of the activity. Graphically, the value of the constant is the point where it intersects the y-axis. Therefore, the constant is also called the intercept. In this example, the constant accounts for all the cost because there is no variable cost. Graphically, the slope coefficient of the cost function is zero.

• ALTERNATIVE 3: StoreBox pays a fixed amount of $300 per month plus $0.20 per CPU hour used. This is an example of a mixed cost. A mixed cost—also called a semivariable cost—is a cost that has both fixed and variable elements.

1. Panel C in Exhibit 10-1 plots the mixed-cost function for StoreBox. We write the cost function in Panel C of Exhibit 10-1 as

Unlike the graphs for alternatives 1 and 2, Panel C has both a constant value of $300 and a slope coefficient of $0.20. The total cost in the relevant range increases as the number of CPU hours used increases. However, the total cost does not vary strictly in proportion to the number of CPU hours used within the relevant range. For example, when 4,000 hours are used, the total cost equals $\$\text{1,100}\hspace{0.17em}[\$300+(0.20\hspace{0.17em}\text{per hour}\times \text{4,000} \mathrm{hours})],$ and when 8,000 hours are used, the total cost equals $\$\text{1,900}\hspace{0.17em}[\$300+(\mathrm{\$}0.20 \text{per hour}\times \text{8,000} \mathrm{hours})]\mathrm{.}$ Although the usage in terms of hours has doubled, the total cost has increased by only about 73% $\left[\right(\mathrm{\$}\text{1,900}-\mathrm{\$}\text{1,100})÷\text{1,100}]$ because some of the costs are fixed costs, which do not increase with the number of CPU hours.

StoreBox’s managers must understand the cost-behavior patterns in the three alternatives to choose the best deal. Suppose StoreBox expects to use at least 4,000 hours of CPU time each month. Its cost for 4,000 hours under the three alternatives are

• ALTERNATIVE 1: $\$\text{2,000} (\mathrm{\$}0.50 \text{per hour}\times \text{4,000} \text{hours})$
• ALTERNATIVE 2: $1,000
• ALTERNATIVE 3: $\$\text{1,100 }[\$300+(\text{\$}0.20 \text{per hour}\times 4\text{,}000 \text{hours})]$

Alternative 2 is the least costly. Moreover, if StoreBox expects to use more than 4,000 hours, alternatives 1 and 3 would be even more costly. StoreBox’s managers, therefore, should choose alternative 2.

Note that the graphs in Exhibit 10-1 are linear (straight lines). We simply need to know the constant, or intercept, amount (commonly designated a) and the slope coefficient (commonly designated b) to describe and graphically plot all values within the relevant range. The general form of this linear cost function is

Under alternative 1, $a=\mathrm{\$}0$ and $b=\mathrm{\$}0.50$ per CPU hour used; under alternative 2, $a=\mathrm{\$}\text{1,000}$ and $b=\mathrm{\$}0$ per hour used; and under alternative 3, $a=\mathrm{\$}300$ and $b=\mathrm{\$}0.20$ per hour used.