2 The Cost Function

The Theory

The cost function of a firm splits the firm’s typical costs into variable and fixed portions, and expresses those costs algebraically:

The variable cost per unit stays relatively constant over time, as does the fixed cost in total, so once a firm has estimated a cost function, they can plug in the expected volume of units and solve for projected total cost.

A cost function can be estimated using past costs. The firm gathers data on unit volume at several points in time, and the corresponding total costs at those times. These volume-cost pairs can be plotted on a scatterplot, and a line can be fitted to them using linear regression or the high-low method. The firm’s cost function is the equation for that line.

The cost function can only be used to predict costs within the relevant range, or the range of unit volumes that occurred in the past. The firm only has information about costs that occur at unit volumes from the lowest past volume to the highest past volume, so they have no way of knowing whether the cost function will hold true outside that range.

The linear regression method is a more accurate way to estimate a cost function. However, we will focus on the high-low method because it will allow you to better understand the principles underlying cost estimation.

The Method

The high-low method builds a cost function from only two data points.

First, find the volume-cost pairs with the highest volume and the lowest volume. These are the high and low points, respectively.

Next, find the variable cost per unit by using the slope formula:

Remember that “High Cost” is not the highest cost given in the data—it is the cost at the point where volume is highest. Likewise, “Low Cost” is the cost at the point where volume is lowest.

Next, find the total fixed cost by plugging the variable cost per unit and either the high or low point into the cost function formula and solving for fixed cost:

Next, build the firm’s cost function by replacing the variables for variable cost per unit and fixed cost with the calculated numbers. Total cost and units remain variables in the equation.

Next, determine whether the cost function can be used at the predicted volume by ascertaining whether the predicted volume falls within the relevant range of the cost function.

Finally, if the cost function can be used at the predicted volume, use the firm’s cost function to predict future costs at that volume.

Illustrative Example

Jaworski Company gathered information on sales volume and total costs for the first three years of the company’s operation:

Determine the firm’s cost function and predict costs for a year in which 275,000 units are sold.

• First, find the high and low points, based on volume:
  • The high volume is 300,000 units, so (300,000, $2,250,000) is the high point.
  • The low volume is 240,000 units, so (240,000, $1,950,000) is the low point.
• Next, find the variable cost per unit:
  • Variable cost per unit = ($2,250,000 – $1,950,000) / (300,000 – 240,000) = $5.00
• Next, find the total fixed cost:
  • Fixed cost = $2,250,000 – 300,000 × $5 = $750,000
  • Or fixed cost = $1,950,000 – 240,000 × $5 = $750,000
• Next, build the firm’s cost function:
  • Total cost = $5 × Units + $750,000
• Next, determine whether the cost function can be used at the predicted volume.
  • Is the predicted unit volume of 275,000, in the relevant range? Yes, it is higher than the low volume of 240,000 units and lower than the high volume of 300,000 units.
• Finally, use the firm’s cost function to predict future costs:
  • Total cost = $5 × 275,000 + $750,000 = $2,125,000

Lecture Example

Your firm has collected the following information on unit sales and total costs for the past six months:

1. Identify the high and low points.
2. Make a scatterplot of the points.
3. Calculate variable cost per unit.
4. Calculate fixed cost.
5. Formulate the firm’s cost function.
6. If the firm sells 25,000 units this month, what will total costs be?
   1. First, can the firm use the cost function at this volume?
   2. If so, what are total costs?

Bonus: How would you draw the firm’s cost function on the scatterplot? Draw the cost function.