Profit

Cost Function

Total cost (CT) is the sum of fixed costs (CF) and variable costs (CV):

Variable cost is a function of demand

Where, cV = variable cost per unit

Therefore,

Figure 2-4 shows the fixed cost, variable cost, and total cost as a function of units of demand.

Equation 2-5

Profit Function

Total profit is calculated by subtracting total cost (in Equation 2-6) from total revenue (in Equation 2-2).

Equation 2‑6

Figure 2-5 shows the graphical representation for the profit function in Equation 2-6 for a linear price–demand relationship.

Similar to the revenue function, this quadratic relationship shows that profit increases with demand (D) up to a certain point, after which it decreases due to the negative D² term. The maximum total profit occurs at a specific demand level (D*), which can be found using calculus by solving for the maximum of the function 

To make any profit, the following two conditions must be met:

1. The maximum price (intercept) that results in no demand must be greater than the variable cost per unit, ensuring non-negative demand.

2. Total Revenue (TR) > Total Cost (CT): Revenue must exceed cost.

If both conditions hold, the optimal demand (where maximum profit occurs) can be found by:

• Taking the first derivative of the profit equation with respect to demand, D,

• Setting it equal to zero and solving.

Equation 2‑7

To confirm that profit is maximized, the second derivative of the profit function must be negative, which is true since

The maximum profit can be obtained by substituting the optimum demand into the profit(loss) equation as given below. 

Equation 2-8