Question

. A natural monopolist has the total cost function C(Q) = 500 + 5Q and faces the inverse demand curve P = 100 – Q

a) Find the monopolist’s price, quantity, profits, consumer surplus and deadweight loss if the monopolist is not constrained by a regulator (you probably want to draw a picture to help you answer CS and DWL)

P = _________ Q = _________ π = _________ CS = ¬_________ DWL = _________

b) A regulator constrains the monopolist to marginal cost pricing and will subsidize the firm to stay in business and earn 0 profit. Find the monopolist’s price, quantity, subsidy, consumer surplus and deadweight loss (ignore welfare losses associated with funding the subsidy).

P = _________ Q = _________ subsidy = _________ CS = ¬_________ DWL = _________  

c) A regulator constrains the monopolist to average cost pricing. Find the monopolist’s price, quantity, consumer surplus and deadweight loss.

P = _________ Q = _________ CS = ¬_________ DWL = _________

Answer 1

MC = dC/dQ = 5

In efficient outcome, P = MC

100 - Q = 5

Q = 95

P = MC = 5

(a)

Monopolist will set MR = MC.

TR = PQ = 100Q - Q2

MR = dTR/dQ = 100 - 2Q

100 - 2Q = 5

2Q = 95

Q = 47.5

P = 100 - 47.5 = 52.5

TR = 52.5 x 47.5 = 2493.75

TC = 500 + 5 x 47.5 = 500 + 237.5 = 737.5

Profit = TR - TC = 1756.25

When Q = 0, P = 100

CS = (1/2) x (100 - 52.5) x 47.5 = (1/2) x 47.5 x 47.5 = 1128.125

DWL = (1/2) x (Monopoly P - Efficient P) x (Efficient Q - Monopoly Q) = (1/2) x (52.5 - 5) x (95 - 47.5)

= (1/2) x 47.5 x 47.5 = 1128.125

(b)

When P = MC,

P = 5

Q = 95

TR = 5 x 95 = 475

TC = 500 + 5 x 95 = 500 + 475 = 975

Subsidy = TC - TR = 975 - 475 = 500

CS = (1/2) x (100 - 5) x 95 = (1/2) x 95 x 95 = 4512.5

DWL = 0 [Since efficiency is maximized with P = MC]

(c)

AC = C/Q = (500/Q) + 5

Setting P = AC,

100 - Q = (500/Q) + 5

100Q - Q2 = 500 + 5Q

Q2 - 95Q + 500 = 0

Q2 - 95Q + 500 = 0

Solving this quadratic equation using online solver,

Q = 89.41 or Q = 5.59

It is assumed that the monopolist will choose higher output. So

Q = 89.41

P = 100 - 89.41 = 10.59

CS = (1/2) x (100 - 10.59) x 89.41 = (1/2) x 89.41 x 89.41 = 3997.07

DWL = (1/2) x (10.59 - 5) x (95 - 89.41) = (1/2) x 5.59 x 5.59 = 15.62