Cost, Revenue, & Profit

Definitions

Definition

The total cost function, \(C(x)\), measures the costs incurred from operating a business and is defined by

\[C(x) = F(x) + V(x)\]

where \(F(x)\) denotes the fixed costs (i.e., costs that remain the same regardless of the level of production \(x\)) and \(V(x)\) denotes the variable costs (i.e., costs that vary depending on the level of production \(x\)) of operating a business.

Definition

The average cost function, \(\overline{C}(x)\), measures the average cost per unit produced and is defined by

\[\overline{C}(x) = \frac{C(x)}{x}.\]

Definition

The total revenue function, \(R(x)\), measures the amount of money received from the sale of \(x\) units and is defined by

\[R(x) = x\cdot p(x)\]

where \(x\) is the number of units demanded and \(p(x)\) is the unit price.

Definition

The average revenue function, \(\overline{R}(x)\), measures the average amount of money received per unit sold and is defined by

\[\overline{R}(x) = \frac{R(x)}{x}.\]

Definition

The total profit function, \(P(x)\), measures the difference between the total revenue and total cost functions and is defined by

\[P(x) = R(x) - C(x)\]

where \(R(x)\) is the total revenue function and \(C(x)\) is the total cost function.

Definition

The average profit function, \(\overline{P}(x)\), measures the average profit earned per unit produced and sold and is defined by

\[\overline{P}(x) = \frac{P(x)}{x}.\]

Example 1

Cost, revenue, profit and average profit

A manufacturer of Robot Tutors has a fixed monthly cost of $500 and a processing cost of $9 for each robot tutor produced. Assuming each robot sells for $19, compute the total profit and the average profit per robot when 250 robots are produced and sold.

Step 1:   Compute the total cost function,   \(C(x)\).

Here, \(x\) denotes the number of robots produced.

\[\begin{align*} C(x) &= F(x) + V(x) && \text{total cost equals fixed costs plus variable costs}\\ &= 500 + 9x && \hbox{\$500 of fixed costs and \$9 for each robot} \end{align*}\]

Step 2:   Compute the total revenue function,   \(R(x)\).

\[\begin{align*} R(x) &= x \cdot p(x) && \text{revenue equals number of units times price per unit}\\ &= 19x && \hbox{since each robot sells for \$19} \end{align*}\]

Step 3:   Compute the total profit function,   \(P(x)\).

\[\begin{align*} P(x) &= R(x) - C(x) && \text{profit equals revenue minus cost}\\ &= \left( 19x \right) - \left( 500 + 9x\right) && \text{using Steps 1 and 2}\\ &= 19x - 500 - 9x && \text{distribute minus sign}\\ &= 10x - 500 && \text{combine like terms} \end{align*}\]

Step 4:   Plug in   \(x=250\)   into the profit function.

Plug in \(x=250\) into the profit function to find the profit associated with the production and sale of 250 robots.

\[\begin{align*} P(250) &= 10(250) - 500 \\ &= 2500 - 500\\ &= 2000 \end{align*}\]

Step 5:   Plug in   \(x=250\)   into the average profit function.

Plug in \(x=250\) into the average profit function to find the average profit associated with each robot when 250 robots are produced and sold.

\[\begin{align*} \overline{P}(250) &= \frac{P(250)}{250} && \hbox{since $\overline{P}(x) = \frac{P(x)}{x}$}\\ &= \frac{2000}{250} && \hbox{using Step 4}\\ &= 8 \end{align*}\]

Therefore, each of the 250 robots produced and sold earns an average profit of $8.