Revenue Function

Total revenue (TR) is calculated as the product of price per unit (p) and demand (D):

TR = p · D

If price is a function of demand, such as p = a - bD, then:

TR = (a - bD)D = aD - bD² for 0 ≤ D ≤  a/b and a > 0,b > 0

Equation 2 2

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

Taking the first derivative (the slope) and setting it equal to zero provides the point at which the slope is zero — the maximum (or minimum) point of the function.

d(TR)/dD=a-2bD=0

Equation 2 3

Therefore, the demand that maximizes total revenue can be obtained as:

D̂=a/2b

Equation 2 4

Therefore, the maximum revenue can be calculated by substituting the optimal demand, D̂, into the total revenue function.

TR_Maximum  = aD̂- bD̂²= a a/2b  - b(a/2b)²=a²/4b

Equation 2 5

Taking the second derivative of the total revenue function (TR) yields a negative value, indicating that the revenue is at a maximum.

d²/dD²(TR) =-2b

Nevertheless, maximizing revenue does not necessarily lead to maximum profits due to cost–volume relationships. Therefore, it is essential to consider both costs and revenues, as cost reduction is a major driver of engineering process improvements.