So far we have considered that a sale is a linear function of selling price. This means that although output increases, there is a change in selling price. Similarly, we have assumed that variable cost is a linear function of output. But this seldom happens. Often variable cost bears a non-linear relationship.
In a linear cost function, the cost equation is presented as follows:
TC = VC + FC
TC = Total Cost
VC = Variable Cost
FC = Fixed Cost
V = bX, where b is per unit variable cost is and X is no. of units.
So, TC or C = a + bX, given a is FC.
In case per unit variable cost is $ 5 and fixed cost is $ 100000, the linear cost function becomes
TC = 100000 + 5X
Given selling price of $ 10 per unit, total revenue (TR) is
TR = 10X
So, the BEP equation stands as
TR - TC
i.e., 10X = 100000 + 5X
or, 5X = 100000
X = 100000/5
X = 20000
It gives BEP 20000 units.
However , if the variable cost becomes non linear in the form of second degree polynomial, the BEP equation may appear as follows.
V = bX + c (X)(X)
Accordingly, TR – TC stands as follows
sX - ( a + bX + c(X)(X) ) = 0
i.e, - a + (s-b)X - c(X)(X) = 0