# Why linear cost functions?

It is possible to model complex cost patterns with a single parametric cost function using exponents, logarithms, and interactions between variables. But such an equation is difficult to build, not very robust and difficult to understand.

First, parametric models on complex patterns require a lot of data to be estimated, and the amount of data available is often insufficient to yield reliable estimates (or even fit the model). Parametric models on just enough data also tend to “over-fit”: they model perfectly past costs, but are usually not reliable to predict future costs.

Second, and more importantly, the parameters of complex models are very difficult to interpret. For instance what a negative coefficient on a square term means can only be interpreted in light of the coefficient on the non-squared term… Even if complex parametric models resulted in accurate predictions, they would not provide good explanations for, and may even prevent building a sufficient understanding of, cost behaviors. This makes them inadequate for managerial purposes. Therefore, management accountants prefer simple linear cost equations which are easy to build, easy to communicate, express very clearly the relationship between cost drivers and costs, and are often surprisingly robust.

Now, linear cost equations are only valid if costs are indeed linear, i.e. have a relatively constant slope $$V_c^d$$ (estimate for the unit variable cost) and intercept $$FC_p$$ (estimate for the fixed cost). If this is the case, a linear cost equation $$TC_p = \sum_{d=1}^{l} Q_p^d \times V_c^d + FC_p$$ can predict and explain costs without much error. In other words, costs should ideally be linear variables. They may have a fixed component (committed or discretionary) and perhaps even a step variable component. But they should not be concave, convex, or step fixed.

The linearity assumption is that the relationship between two variables can be approximated by a straight line characterized by a constant slope ($$V_c^d$$) and intercept ($$FC_p$$).

It is useful to systematically plot a scatter graph to visually check whether a linear approximation of costs makes sense, at least over some range of activity. If it does not with a specific measure of activity, you may try with another one. If no single measure of activity alone provides something close to a straight-line, you would have to look for multiple underlying cost drivers and make more complex statistical tests of linearity. Finally, if the assumption of cost linearity cannot be supported, cost estimation should not even be attempted with the techniques I will introduce later. Fortunately, in practice, a linear cost equation can always be used by specifying the properly scope of the estimation. This is what we will see in the next few pages.

Linear costs

Non-linear costs