Question

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

P = 90-2Q ------------- (1)

Multiply eq. 1 with Q and differentiation w.r.t. Q, will give marginal revenue (MR).

R = Q*P = 90Q – 2Q^2

MR = dR/dQ = 90-4Q ----------------- (2)

TC = 100+10Q + 2Q^2 -------- (3)

Differentiation of eq. 3 w.r.t. Q will give Marginal cost (MC).

MC = d(TC)/dQ = 10+4Q ----------- (4)

For profit maximizing output for a monopolist firm,

MR = MC

90-4Q = 10+4Q

Or,

Q = 80/8 = 10

P = 90-2*10 = $70

Profit = revenue – cost = 70*10 – (100+10*10 + 2*10^2)

Profit = $300

So for a monopolist firm:

Quantity of output = 10

Price = $70

Profit = $300