What are mixed costs?

Mixed costs are costs that have both a fixed component and a variable component; they are also known as semi-variable costs.

In practice, many costs are mixed, i.e. they have both a variable (i.e. proportional to the volume of activity) and a fixed (i.e. not proportional to the volume of activity) component. In addition, the variable component is not necessarily linear and the fixed component not necessarily constant. This means that you can observe the following pattern of costs:

Solution. The cost behaviors mixed in this case are:

  • committed fixed costs: you can recognize them at the fact that costs are incurred even when there is no activity;
  • variable costs: total costs increase with the volume of activity, so there is definitely a variable component. Moreover, you can observe that as volume increases, the slope decreases first (concave variable costs) and then increases (convex variable costs);
  • step (fixed) costs: you may notice that just after 1 unit, 500 units, and 1,000 units there is a sudden jump in total costs: this discontinuity is characteristic of step costs. They are step fixed costs because changes incur on relatively wide ranges of activity - and because we will use these discontinuities to define relevant ranges;
  • discretionary fixed costs: there is also some random variation in total costs. This apparent randomness (variations which are not proportional to the volume of activity) are characteristic of discretionary fixed costs.

One possibility is that the company rents machines under a very flexible contract (for such a clean pattern over several periods, it must be possible to stop renting quickly, otherwise the cost would be committed fixed). The capacity of each machine is capped at about 500 units. Moreover, the more the company produces, the more efficient it becomes at first (economies of scale) and maybe even benefits from greater discounts on raw materials. However, beyond 1,000 units it seems that coordination becomes too difficult, and it may have to pay extra-time to its employees (congestion costs). As for the discretionary component, it can be some expenses in employees’ training. It could also suggest that some of the costs included have a different cost driver (i.e. these costs are not homogeneous).

A single linear cost function (straight line in the following graph) would result in systematic errors (vertical distance between the straight line, our prediction, and the real cost, the observation):

You can see the pattern of these errors (i.e. vertical distances) in the following graph plotting the residuals of a linear regression:

These errors are not only large (from a cost under-estimation of 1,768.14 to a cost over-estimation if 1,998.88), but also clearly not random. This is a clear indication that we need to work a bit more so that we can use simple linear equations to model our costs.

The next subsection explain why management accountants use linear equations rather than more complex parametric models and discuss the preparatory work they do to make these linear equations good enough. This requires first disaggregating costs, identifying underlying cost driver(s) and then defining meaningful relevant ranges. For instance, in the following diagram we cut the previous observations in three ranges of activity where linear approximations seem to work:

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