Question

5. Suppose a monopolist has TC = 100 + 10Q + 2Q2, and the demand curve it faces is P = 90 - 2Q. What will be the price, quantity, and profit for this firm?

Answer 1

$TC = 100 + 10 Q + 2Q^2 \\ MC = 10 + 4Q \\ P = 90 - 2Q \\ TR = PQ \\ = 90Q - 2Q^2 \\ MR = 90 - 4Q$

At equilibrium MR = MC

$90 - 4Q = 10 + 4Q \\ 8Q = 80 \\ Q = \frac{80 }{ 8} = 10$

(equilibrium quantity)

$Price = 90 - 2Q \\ = 90 - 2 \times 10 \\ = 90 - 20 \\ = 70$

(equilibrium price)

$Profit = TR - TC \\ = P \times Q - (100 + 10 \times 10 + 2\times 10^2 ) \\ = (70 \times 10 ) - (100 + 10 \times 10 + 2 \times 10^2 ) \\ = 700 - ( 100 + 100 + 2 \times 100 ) \\ = 700 - ( 100 + 100 + 200 ) \\ = 700 - 400 = 300$

Answer: Price = 70 , Quantity = 10 , Profit = 300