Question

A monopolist faces the demand function Q = 20 – 2P. Its cost function is TC(Q) =0.5Q. Solve for the monopolist’s profit-maximizing price and output and calculate its profit as well as the consumer surplus and deadweight loss.

Answer 1

Answer : Demand :  Q = 20 – 2P

=> 2P = 20 - Q

=> P = (20 - Q) / 2

=> P = 10 - 0.5Q

TR (Total Revenue) = P * Q = (10 - 0.5Q) * Q

=> TR = 10Q - 0.5Q^2

MR (Marginal Revenue) = TR / Q

=> MR = 10 - Q

TC = 0.5Q

MC (Marginal Cost) = TC / Q

=> MC = 0.5

At monopoly equilibrium, MR = MC .

=> 10 - Q = 0.5

=> 10 - 0.5 = Q

=> Q = 9.5

Now, P = 10 - 0.5Q = 10 - (0.5 * 9.5)

=> P = 10 - 4.8

=> P = 5.2

Therefore, for monopolist the profit maximizing price level is $5.2 and output level is 9.5 units.

TR = P * Q = 5.2 * 9.5

=> TR = 49.4

TC = 0.5 * 9.5

=> TC = 4.8

Profit = TR - TC = 49.4 - 4.8

=> Profit = 44.6

Therefore, the profit level of monopolist is $44.6 .

Consumer surplus (C.S.) = P_d(Q) * dQ - (P * Q)

=> C.S. = (10 - 0.5Q) * dQ - (P * Q)

=> C.S. = 10Q - (0.5Q^2 / 2) - (5.2 * 9.5)

=> C.S. = (10 * 9.5) - [{0.5(9.5)^2} / 2] - (5.2 * 9.5)

=> C.S. = 95 - (45.1 / 2) - 49.4

=> C.S. = 95 - 22.6 - 49.4

=> C.S. = 23

Therefore, here the consumer surplus is $23 .

For perfectly competitive firm at equilibrium condition, P = MC.

=> 10 - 0.5Q = 0.5

=> 10 - 0.5 = 0.5Q

=> 9.5 = 0.5Q

=> Q = 9.5 / 0.5

=> Q = 19

P = 10 - (0.5 * 19)

=> P = 10 - 9.5

=> P = 0.5

Deadweight loss = 0.5 * (Pm - Pc) * (Qc - Qm)

Here Pm = Monopoly price

Pc = Competitive price

Qc = Competitive output

Qm = Monopoly output

=> Deadweight loss = 0.5 * (5.2 - 0.5) * (19 - 9.5)

=> Deadweight loss = 0.5 * 4.7 * 9.5

=> Deadweight loss = 22.3

Therefore, here the deadweight loss is $22.3 .