Cost, Revenue, and 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

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

Definition

The break-even point refers to the value of \(x\) where the total revenue from the sale of \(x\) units equals the total cost of producing \(x\) units. In other words, the break-even point is the value of \(x\) such that

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

or equivalently,

\[P(x) = 0.\]

Example 1

Find cost, revenue, profit, and break-even point.

A manufacturer of wireless security cameras has fixed monthly costs of $10,000 and a processing cost of $49 for each camera produced. Assuming each camera sells for $99, compute the break-even point.

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

Here \(x\) denotes the number of cameras produced.

\[\begin{align*} C(x) &= F(x) + V(x) && \text{total cost equals fixed costs plus variable costs}\\ &= 10000 + 49x && \hbox{\$10,000 of fixed costs and \$49 for each camera} \end{align*}\]

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

\[\begin{align*} R(x) &= x \cdot p(x) && \text{revenue equals number of units times price per unit}\\ &= 99x && \hbox{since each camera sells for \$99} \end{align*}\]

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

\[\begin{align*} P(x) &= R(x) - C(x) && \text{profit equals revenue minus cost}\\ &= 99x - ( 10000 + 49x) && \text{using Steps 1 and 2}\\ &= 99x - 10000 - 49x \\ &= 50x - 10000 && \end{align*}\]

Step 4:   Compute the break-even point.

\[\begin{align*} P(x) = 0 &~~\rightarrow~~ 50x - 10000 = 0 \\ &~~\rightarrow~~ 50x = 10000 \\ &~~\rightarrow~~ x = \frac{10000}{50} = 200 \end{align*}\]

Therefore, the break-even point is 200 cameras.