Question

5. The demand and total cost equations for the output of a monopolist are
Qd = 90 - 2P, TC =Q3 - 8Q2 + 57Q+ 2w
i. Find the firm’s profit-maximizing output level.
ii. What is the profit at this output level?

Suppose further that the monopolist’s total cost of production function is given by the equation TC = 2Q2
Find the output level that will maximize profit.
i. Determine the monopolist’s profit at the profit-maximizing output level.
ii. What is the monopolist’s average revenue (AR) function?
iii. Determine the price per unit at the profit-maximizing output level.
iv. Suppose that the monopolist was a sales (total revenue) maximizer.
v. Compare the sales maximizing output level with the profit-maximizing output level.//
vi. Compare total revenue at the sales-maximizing and profit-maximizing output levels.

Answer 1

5. The demand and total cost equations for the output of a monopolist are

Qd = 90 - 2P, so P = 45 - 0.5Q, TC = Q3 - 8Q2 + 57Q+ 2w

i. The firm’s profit-maximizing output level is:

MR = MC,

MR = TR'(Q) = 45 - Q,

$MC = TC'(Q) = 3Q^2 - 16Q + 57,$

$3Q^2 - 16Q + 57 = 45 - Q,$

$3Q^2 - 15Q + 12 = 0,$

$Q^2 - 5Q + 4 = 0,$

Q1 = 4 units, Q2 = 1 unit.

ii. The profit at this output level is:

$TP = TR - TC = (45Q - 0.5Q^2) - (Q^3 - 8Q^2 + 57Q + 2w),$

$TP1 = (45*4 - 0.5*4^2) - (4^3 - 8*4^2 + 57*4 + 2w) = 8 + 2w,$

$TP2 = (45*1 - 0.5*1^2) - (1^3 - 8*1^2 + 57*1 + 2w) = -5.5 + 2w.$

Suppose further that the monopolist’s total cost of production function is given by the equation TC = 2Q2

The output level that will maximize profit is:

MR = MC,

MC = TC'(Q) = 4Q,

45 - Q = 4Q,

5Q = 45,

Q = 9 units.

i. The monopolist’s profit at the profit-maximizing output level is:

$TP = TR - TC = (45*9 - 0.5*9^2) - 2*9^2 = 202.5.$

ii. The monopolist’s average revenue (AR) function is:

AR = TR/Q = P = 45 - 0.5Q.

iii. Determine the price per unit at the profit-maximizing output level is:

P = 45 - 0.5*9 = 40.5.

iv. If the monopolist was a sales (total revenue) maximizer, then:

TR'(Q) = MR = 0,

45 - Q = 0,

Q = 45 units.

P = 45 - 0.5Q = 22.5.

v. The sales maximizing output level is much more higher than the profit-maximizing output level (45 > 9).

vi. Total revenue at the sales-maximizing output level is:

TR = 22.5*45 = 1,012.5.

Total revenue at the profit-maximizing output level is:

TR = 40.5*9 = 364.5.