Question

Suppose a monopoly faces the total revenue schedule ?? = ???? − ??^? and the total cost schedule ?? = ???^? − ???^? + ??? + ??. Are there two output levels at which MC = MR?

Which is the profit-maximizing output? In a graph show the profit maximizing condition of the monopolist.

Answer 1

Answer : Given,

TR = 300q - 2q^2

MR = TR / q = 300 - 4q

TC = 12q^3 - 44q^2 + 60q + 30

MC = TC / q = 36q^2 - 88q + 60

For monopolist at profit maximizing output level MR = MC occur. So,

300 - 4q = 36q^2 - 88q + 60

=> 0 = 36q^2 - 88q + 4q + 60 - 300

=> 0 = 36q^2 - 84q - 240

=> 0 = 12 (3q^2 - 7q - 20)

=> 0 / 12 = 3q^2 - 7q - 20

=> 0 = 3q^2 - 7q - 20

=> 0 = 3q^2 - (12 - 5)q - 20

=> 0 = 3q^2 - 12q + 5q - 20

=> 0 = 3q (q - 4) + 5 (q - 4)

=> 0 = (q - 4) (3q + 5)

Either,

q - 4 = 0

=> q = 4

Nor,

3q + 5 = 0

=> 3q = - 5

=> q = - 5/3

As here q = 4 and q = - 5/3 occurs hence, yes, at MR = MC there occurs two output levels.

As profit maximizing output level can never be negative, hence here the profit maximizing output level is, q = 4.

The profit maximizing condition of monopolist is MR = MC. This is shown by the following picture's diagram.

In above picture's diagram Q is the profit maximizing output level of monopolist and P is the price.