What is a profit function?

For a single-product business

The profit function is nothing more the combination of two sub-functions: (1) a revenue function \(R_p = Q_p \times P + FR_p\), where \(R_p\) stands for total revenues and \(FR_p\) for some fixed revenues (like for instance subscriptions) over the period \(p\); and (2) a cost function \(TC_p = Q \times V_c + FC_p\). If we assume that there is no fixed committed revenues (\(FR_p = 0\)) and that there is a single product driving both revenues and costs we obtain:

\[ \begin{aligned} OI_p & = Q_p \times P && - (Q \times V_c &&& + FC_p) \\ & = Q_p \times P && - Q \times V_c &&& - FC_p \\ & = Q_p \times && (P - V_c) &&& - FC_p \\ \\ & = Q_p \times && UCM &&& - FC_p \end{aligned} \]

where \(OI_p\) is the Operating Income of the period \(p\), \(P\) is the unit price of the product, \(V_c\) is the unit variable cost of the product, and \(FC_p\) are the fixed costs of the period \(p\). The last form of the profit equation also reveals a very important estimate that we will keep using over and over again: \(UCM\), the unit contribution margin of the product.

\[ UCM = P - V_c \]

The unit contribution margin is the change in contribution margin, and therefore the change in profit (since fixed costs are fixed) for every additional unit sold.

In other words, the unit contribution margin is the contribution that each unit makes to first covering fixed costs and, then, once fixed costs are covered, generating a profit.

Quite often in practice, volumes are not available. In these circumstances, the analysis can be conducted using the contribution margin ratio, \(CMR\):

\[ \begin{aligned} CMR_p & = \frac{R_p-VC_p}{R_p} = \frac{Q_p \times P - Q_p \times V_c}{Q_p \times P} && = \frac{Q_p \times (P - V_c)}{Q_p \times P} \\ CMR & = \frac{CM}{R} && = \frac{UCM}{P} \end{aligned} \]

where \(VC_p\) is the total variable cost for the period \(p\) and \(CM_p\) the total contribution margin of the period \(p\) defined as the revenues minus the variable costs:

\[ CM_p = R_p - VC_p = R_p \times CMR \]

The contribution margin ratio is the increase in contribution margin and thus in profit for every additional euro of revenues. It is the fraction of each sales euro that is available first to cover fixed expenses and then, when this is done, to generate a profit.

\[ OI = R_p \times CMR - FC_p \]

For a multi-product business

The preceding profit function can of course be generalized to the case of multiple (m) products (or revenue and cost drivers):

\[ \begin{aligned} OI_p & = \sum_{d = 1}^{m} Q_p^d \times UCM^d - FC_p \\ & = \sum_{d = 1}^{m} R_p^d \times CMR^d - FC_p \end{aligned} \]

where \(UCM^d\) is the unit contribution margin of the product \(d\), \(R_p^d\) is the sales revenues generated by product \(d\) over the period \(p\), and \(CMR^d\) is the contribution margin ratio of the product \(d\).

Now, the beauty of CVP analysis is that it fortunately lends itself to a nice visual representation which is far easier to read and understand than this analytic model: the CVP graph that I introduce in the next sub-section.

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