Cost function
A
cost function specifies the cost $C$ as a function of the number of items $x.$ Thus, $C(x)$ is the cost of $x$ items, and has the form
Cost = Variable Cost + Fixed Cost
where the variable cost is a function of $x$ and the fixed cost is a constant. A cost function of the form
$C(x) = mx + b$
is called a
linear cost function; the variable cost is $mx$ and the fixed cost is $b.$ The slope $m$ in a linear cost function is the
marginal cost, and measures the incremental cost per item.
Example
The daily cost to your donut service to prepare $x$ boxes of organic donuts is
$C(x) = 2x + 100\qquad$ \t
(which happens to be a linear function). The fixed cost is $\$100$, the variable cost is $2x,$ and the marginal cost is $2.$
Revenue function
The
revenue resulting from one or more business transactions is the total payment received, sometimes called the
gross proceeds. If $R(x)$ is the revenue from selling $x$ items at a price of $m$ each, then $R$ is the linear function $R(x) = mx$ and the selling price $m$ can also be called the
marginal revenue.
Example
Your donut service sells organic donuts for $\$4.50$ per box. Thus, the revenue from the sale of $x$ boxes is
$R(x) = 6.50x$ $\qquad$ \t
The marginal revenue is $m = \$4.50$ per box.
#[Profit Function][Función de ganancia]#
The
profit is the net proceeds, or what remains of the revenue when costs are subtracted. If the profit depends linearly on the number of items, the slope $m$ is called the
marginal profit. Profit, revenue, and cost are related by the following formula.
#[Profit][Ganancia]# \t ${}={}$ #[Revenue − Cost][Ingreso − Costo]#
\\ #[$P$][$G$]# \t #[${}= R - C$][${}= I - C$]#
If the profit is negative, say &minus\$500, we refer to a
loss (of \$500 in this case). To
break even means to make neither a profit nor a loss. Thus, break even occurs when $P = 0,$ or
#[$R = C \iff P = 0$][$I = C \iff G = 0$]# \t \t
The
break even point is the number of items $x$ at which break even occurs.
Continuing with the donut scenario: Given the above cost and revenue functions, the profit function is
#[$P(x){}$][$G(x){}$]# \t #[${}= R(x) - C(x)$][${}= I(x) - C(x)$]#
\\ \t ${}= 4.5x - (2x + 100)$
\\ \t ${}= 2.5x - 100$
For break even, we set $P = 0:$.
$P {}= 2.5x - 100 = 0$${}\implies x = \dfrac{100}{2.5} = 40$ #[boxes][cajas]#
#[So, to avoid a loss, you would need to sell at least 40 boxes of organic donuts.][Entonces, para evitar pérdidas, necesitarías vender 40 cajas de donas orgánicas.]#