4. Suppose a monopolist faces the following demand curve: P = 596 – 6Q. Marginal

Question

4. Suppose a monopolist faces the following demand curve:

P = 596 – 6Q. Marginal cost of production is constant and equal to $20, and there are no fixed costs.

a) What is the monopolist’s profit maximizing level of output?
b) What price will the profit maximizing monopolist produce?
c) How much profit will the monopolist make if she maximizes her profit?
d) What would be the value of consumer surplus if the market were perfectly competitive?
e) What is the value of the deadweight loss when the market is a monopoly?

Explanation / Answer

P = 596 – 6Q , MC = 20 a) What is the monopolist’s profit maximizing level of output? Set the derivative of profit with respect to quantity equal to zero. We'll call V profit and MV marginal profit. V = (P-MC)*Q V = (596 - 6Q - 20)*Q V = (576 - 6Q)*Q MV = 576 - 12Q = 0 Q* = 576/12 Q* = 48 b) What price will the profit maximizing monopolist produce? P* = 596 – 6Q* P* = 596 – 6*48 P* = 308 c) How much profit will the monopolist make if she maximizes her profit? V* = (P*-MC)*Q* V* = (308-20)*48 V* = 13,824 d) What would be the value of consumer surplus if the market were perfectly competitive? First, we have to find the quantity produced by a perfectly competitive market. Set price equal to marginal cost. P = MC 596 – 6Q = 20 Q = 576/6 Q = 96 P = 20 Now, take the integral of the demand curve minus price between 0 and Q = 96. CS = int[0,96] {P(Q) - mc}dQ CS = int[0,96] {596 – 6Q - 20}dQ CS = int[0,96] {576 – 6Q}dQ CS = {576Q - 3Q^2} | [0,96] CS = 576*96 - 3*(96^2) CS = 27,648 e) What is the value of the deadweight loss when the market is a monopoly? This is the integral of the demand curve minus the marginal cost from the monopoly quantity to the perfectly competitive quantity. That is, this is the welfare that exists in a perfectly competitive market but not in a monopoly. CS = int[48,96] {P(Q) - mc}dQ CS = int[48,96] {596 – 6Q - 20}dQ CS = int[48,96] {576 – 6Q}dQ CS = {576Q - 3Q^2} | [48,96] CS = (576*96 - 3*(96^2)) - (576*48 - 3*(48^2)) CS = 6,912

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