### Analysis of Mixed Costs

**Analysis of Mixed Costs**

To make decisions, managers must be able to estimate how costs will change as a result of a specific decision, such as introducing a new product pr producing more units. While it is fairly easy to predict what will happen for variable costs and fixed costs, mixed costs are more difficult because they contain both variable and fixed components. It is important that one develop the skill in looking at mixed costs and breaking these costs down into their fixed and variable components for managerial decision making.

- Linear Approach to Analyzing Mixed Costs – The methods we use to analyze mixed costs are based on the linearity assumption, or the assumption that the relationship between total cost and activity can be approximated by a straight line: Y = A + B(X)
- Y is the total cost, which is shown on the vertical axis. It is called the dependent variable because we assume that Y is dependent on X.
- X is the activity that causes Y (total cost) to change. This variable is also called the cost driver or the independent variable.
- A is the amount of cost that will incurred regardless of activity level (X), or the total fixed cost. This term is also call the intercept term or the constant
- B indicates how much Y (total cost) will increase with each additional unit of A (activity or cost driver). In other words, B is the variable cost per unit of X and is represented by the slope of the line.

- Scattergraph- The first step in analyzing mixed costs is to prepare a visual representation of the relationship between total cost and activity. A scattergraph is a graph with total cost plotted on the vertical (Y) axis and a measure of activity, or cost driver, plotted on the horizontal (X) axis. A scattergraph is useful for getting a “feel” for the data and helps answer preliminary questions such as whether the linear assumption is reasonable and whether there are unusual patterns or outliers in the data.

Once a scattergraph has been created and we have confirmed that the relationship between total cost and activity is roughly linear, the next step is to fit the line through the data that will provide an estimate of total fixed cost (intercept) and variable cost per unit (slope). There are several ways to “fit the line”:

- Visual Fit – Involves “eye balling” the data on the scattergraph and drawing a line through the graph to capture the relationship between total cost and activity. The method is simple and intuitive but is very subjective and imprecise. If the line slopes upward, the total cost increases with the activity, indicating a variable cost. The steeper the slope, the higher the variable cost per activity, but it is difficult to determine the exact slope of the line by looking at the scattergraph.

- High-Low Method – use the two most extreme activity (X) observations to “fit the line.” This method uses only two data points to solve for variable cost per unit (slope or B) and total fixed cost (intercept). Although this method only uses two points, this approach may provide a reasonable estimate of the fixed and variable costs as long as the high and low data points represent the general trend in the data. Once the extreme data points are found, one first can find the slope of the line (variable cost per unit), then use the algebraic formula for a line (Y= A + B (X)) to find the Y intercept (fixed cost).

- Least-Squares Regression – is a statistical technique for finding the best fitting line based on all available data points. Although it is more complicated than the high-low method, a spreadsheet program such as Excel can be used to do the calculations. Least squares regression is a statistical technique that uses all of the available data to find the “best fitting” line. The best fitting line is the one that minimizes the sum of the squared errors, where error is the difference between the regression prediction and the actual data values. One should use at least six data points to get a reliable regression result. The regression method uses all available data to find the best fitting line or the one that minimizes the sum of the squared error around the regression line. The regression output also provides information about the “goodness of fit” of the model, or how well the regression line fits the data points. The most common measure of goodness of fit is the R square value. R square tells managers how much of the variability in the Y variable (total cost) is explained by the X variable (the activity or cost driver). The closer R square is to 1, the more reliable are the results.

- Contribution Margin Approach – Now that we have analyzed cost behavior and classified costs as either variable or fixed, we can prepare a new type of income statement called the contribution margin income statement. This particular statement is appropriate for internal use only. Instead of differentiating between manufacturing (product) and non-manufacturing (period) costs, a contribution margin income statement is based on cost behavior, or whether cost is variable or fixed. In a contribution margin format, variable costs are deducted from sales revenue to get contribution margin, and then fixed costs are subtracted to arrive a profit. Contribution Margin is the difference between sales revenue and variable costs: Contribution Margin = Sales Revenue – Variable Costs.

The contribution margin income statement is not used for external reporting (GAAP). Rather, it provides a tool for managers to do “what-if” analysis or to analyze what will happen to profit if something changes. To do so, managers focus on either the unit contribution margin or the contribution margin as a percent of sales.

- Unit Contribution Margin – Unit contribution margin tells us how much each additional unit sold contributes to the bottom line

- Contribution Margin Ratio – The contribution margin tells managers how much contribution margin is generated by every dollar of sales.