Question

4. You are the manager of a monopoly, and your demand and cost functions are given by P = 300 - 3Q and C(Q) = 1,500 + 2Q^2 , respectively.

a. What price–quantity combination maximizes your firm’s profits?

b. Calculate the maximum profits.

c. Is demand elastic, inelastic, or unit elastic at the profit-maximizing price–quantity combination?

d. What price–quantity combination maximizes revenue?

e. Calculate the maximum revenues.

f. Is demand elastic, inelastic, or unit elastic at the revenue-maximizing price–quantity combination?

Answer 1

Answer : Given,

P = 300 - 3Q

TR (Total Revenue) = P * Q

=> TR = (300 - 3Q) * Q

=> TR = 300Q - 3Q^2

MR (Marginal Revenue) = TR / Q

=> MR = 300 - 6Q

Given, C(Q) = 1500 + 2Q^2

MC (Marginal Cost) = C(Q) / Q

=> MC = 4Q

At monopoly equilibrium, MR = MC.

=> 300 - 6Q = 4Q

=> 300 = 4Q + 6Q

=> 300 = 10Q

=> Q = 300 / 10

=> Q = 30

Now, P = 300 - (3 * 30)

=> P = 210

Therefore, the monopoly firm maximizes it's profit when price is $210 and quantity is, Q = 30.

b) TR = P * Q = 210 * 30

=> TR = 6300

TC = 1500 + 2 * (30)^2

=> TC = 3300

Profit = TR - TC

=> Profit = 6300 - 3300

=> Profit = 3000

Therefore, the maximum profit for monopolist is $3000.

c) Given,

P = 300 - 3Q

=> 3Q = 300 - P

=> Q = (300 - P) / 3 = (300 / 3) - (P / 3)

=> Q = 100 - (1/3 * P)

Elasticity of demand (Ed) = (Q / P) * (P/Q)

=> Ed = (- 1/3) * (210 / 30)

=> Ed = - 0.33 * 7

=> Ed = - 2.31

Here Ed = - 2.31 < 1. As here Ed is less than 1, hence the demand is inelastic.

d) Revenue becomes maximum when, MR = 0.

=> 300 - 6Q = 0

=> 300 = 6Q

=> Q = 300 / 6

=> Q = 50

Now, P = 300 - (3 * 50)

=> P = 150

Therefore, the revenue becomes maximum when P = $150 and Q = 50.

e) Maximum Revenue = P * Q = 150 * 50

=> Maximum Revenue = 7500

Therefore, maximum revenue for the monopolist is $7500.