Cost Structure: Ideas for a Practical Calculation - Pt. 2

As already mentioned several times in the first part, and as is easy to imagine, in reality it is impossible to make an exact subdivision between Fixed Costs and Variable Costs.

However, taking as an example a certain reference cost center (for example a certain department), and eliminating all those costs that are clearly fixed (rent) or clearly variable (raw materials), it is possible to concretely identify the Fixed Costs and the unit Variable Cost referred to that cost center, and we can do this with two different approaches .

Let's see everything with a concrete example, which we will develop throughout the rest of the discussion.

Let us imagine the “lamps” department of a company that produces lighting products, and suppose that we have a series of weekly observations available, relating to a certain cost driver (in this case the units produced) that we will use as the independent variable in our analysis.

The independent variable is the known one and for us it will therefore be the Quantity produced, which we will use to try to determine the dependent variable, that is the unknown one, namely the Total Cost referred to the lamps department.

Below is the table from which we will start our calculations, which reports the results of 20 surveys carried out at the end of each week:

Based on the findings listed in the table, it will be necessary to find a function that can explain the trend of the Total Cost of the department as accurately as possible:

CT = CF + CVu × Units produced

To estimate this cost function we will use, as already mentioned:

• an empirical approach
• a statistical approach

The empirical approach consists in identifying the week with the lowest number of pieces produced and the week with the highest number , so as to have a range of values ​​of the independent variable that is sufficiently significant. See the following figure in this regard:

Once the extremes of our sample of observations have been identified, the following practical reasoning is carried out: if, going from the 154 units produced in the 6th week to the 221 units of the 17th week, the Total Cost of the department goes from €3,386 to €4,606, this means that – by the very definition of Variable Cost – the increase in the Total Cost of €4,606 – €3,386 = €1,220 compared to an increase of 221 – 154 = 67 units produced is due exclusively to the Variable Costs, so the unit Variable Cost can easily be obtained in the following way:

To find the Fixed Costs, it will be sufficient to calculate the total Variable Costs at the minimum output level (or maximum, it makes no difference), and subtract them from the Total Cost of the department.

So, by calculating the minimum level, we will have:

CF = 3,386 - (154 × 18.20) = €582.97

Similarly, by carrying out the calculation on the maximum level, we will have:

CF = 4,606 - (221 × 18.20) = €582.97

Both calculations, obviously, lead to the same result, since all the reasoning done so far is based on the fact that the difference in total cost between the minimum and maximum output level is due exclusively (and this is quite plausible!) to the Variable Costs, so that in the 2 cases (minimum and maximum) the Fixed Costs are the same!

Below are the Excel formulas used to develop the calculations.

So here we have empirically found our cost function, to try to represent the behavior of the Total Costs of our lamp department:

CT = €582.97 + €18.20 × Units produced

Based on the quantities detected in the 20 weeks, and using the formula above, the following values ​​are obtained:

In column G there are the "theoretical" Total Costs, that is those determined through the empirical formula, while column H shows the square of the deviations between the Total Cost values ​​actually detected (referred to in the already known column D) and the theoretical Total Costs in column G. Column H represents in a certain sense a measure of the "goodness" of the cost function that we have just found, but we will go into the concept in more detail shortly.

This method was deliberately called empirical as it has no statistical value: simply 2 surveys were taken as a basis, the one with the smallest independent variable and the one with the largest independent variable, and a Total Cost function was obtained from them.

As can be seen, in correspondence with these 2 findings the deviation is equal to zero since – as it was conceived – the Total Costs line will necessarily pass through those 2 points, as shown in the following figure.

The dashed line is the straight line that we have just identified empirically and which, as can be seen, passes through the minimum (week 6) and maximum (week 17) output levels.

The method seen so far has no statistical value.

And if we instead used statistical calculations, could we find a function that was more in line with the actual behavior of the Total Cost of the department? A function, in other words, capable of minimizing the deviations between the observed Total Costs and the theoretical Total Costs?

(continued in part 3 )