Business Functions — Cost, Revenue and Profit

The whole point of studying functions in a Business Mathematics & Statistics syllabus is to use them to model real economic quantities. Three functions recur in almost every numerical problem of this kind.

Cost function, $C(x)$, gives the total cost of producing $x$ units of a good. It is usually written as

where $F$ is the fixed cost (rent, salaries, depreciation — cost incurred even if $x=0$, so $C(0)=F$) and $v$ is the variable cost per unit (raw material, labour directly tied to each unit produced). A cost function of this form is linear in $x$.

Revenue function, $R(x)$, gives the total money received from selling $x$ units. If every unit is sold at the same fixed price $p$ per unit, then

which is also linear, and passes through the origin (selling zero units brings in zero revenue).

Profit function, $P(x)$, is defined simply as revenue minus cost:

When both $R(x)$ and $C(x)$ are linear, $P(x)$ is linear too. The break-even point is the output level $x_0$ at which profit is exactly zero, i.e. $R(x_0)=C(x_0)$ — the firm neither gains nor loses money. Producing fewer units than $x_0$ results in a loss; producing more results in a profit, provided the cost and revenue functions stay linear over that range.

<!-- FIGURE-NEEDED: graph — a cost line C(x) = F + vx starting above the origin at height F, and a steeper revenue line R(x) = px starting at the origin, both plotted against quantity x, with their intersection point marked and labelled as the break-even point --> …