1 Assume that a monopolist faces the following demand curve

1) Assume that a monopolist faces the following demand curve: P=250-2Q

Also assume that the total cost function is given as: TC=1/2 Q2

a) What would be the monopolist price and quantity? Calculate the profit for the monopolist and the DWL .

b) Calculate the learner index for this monopolist.

c) Now assume that the government imposes a specific tax of $1 per unit. What will happen to price and quantity? What will happen to the profit of the monopolist?

d) The government decides to impose a price ceiling on the monopolist to increase total welfare. What should be the optimal price ceiling?

Solution

a) Given that total cost is 1/2Q² Marginal cost is the derivative of total cost so MC = Q. Applying the rule MR = MC, we get

Q = 250 - 4Q

5Q = 250

Q = 50, P = 250 - 2*50 = $150

Profit = TR - TC

= P*Q - 1/2Q²

= Q(P - 1/2Q)

= 50(150 - 25)

= $6250

DWL for the monopolist = 1/2*(150 - 50)*(83.33 - 50) = $1667

b) Lerner’s index is a measure of the difference between price and marginal cost as a fraction of the product’s price. It is given by

L = (P - MC)/P

= 150 - 50/150

= 0.66

A value greater than 0.5 ensures that firms in this market do not compete rigorously for consumers through price competition.

c) With a tax of $1 per unit, MC becomes Q + 1

MR = MC

250 - 4Q = Q + 1

249 = 5Q

Q = 49.8, P = $150.4

Profit = TR - TC

= 49.8*150.4 - 1/2(49.8)² - 49.8

= $7489.92 - 1289.82

= $6200

Monopoly profits fall after tax.