Question

4. The total cost (TC) and demand equations for a monopolist is

TC=100+5Q²

P=200-5Q

a)     What is the profit-maximizing quantity?

b)     What is the profit-maximizing price?

Answer 1

$TC=100+5Q^2$

 $MC = \frac{dTC}{dQ}$

$= 0+10Q$

$Monopolistic maximizing condition$$MR = MC$

$P = 200-5Q$

$TR = PQ$

$(200-5Q)Q$

$= 200Q-5Q^2$

$MR = \frac{dTR}{dQ}$

$MR = 200-10Q$

$MR = MC$

$200-10Q = 10Q$

$200 = 10Q +10Q$

$200= 20Q$

$Q = \frac{200}{20}$

$Q = 10$

$Substituting into P equation$

$P = 200-5Q$

$P = 200-5(10)$

$P = 200-50$

$Profit-Maximizing price = 150$$b) Profit-Maximizing quantity$$P = MC$

$200-5Q = 10Q$

$200 = 10Q + 5Q$

$200 = 15Q$

$Q = \frac{200}{15}$

$Profit-Maximizing Quantity = 13.3$