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Problem :

Assume a monopolist has MC = 10 and no fixed costs. The monopolist faces a demand curve of P = 100 - 3Q. Find the equilibrium quantity and price.

Revenue = P·Q = (100 - 3Q)Q = 100Q - 3Q2
Marginal Revenue = 100 - 6Q

Setting MC = MR:
10 = 100 - 6Q
=> Q = 15

Problem :

Assume a monopolist has MC = 10 and no fixed costs. The monopolist faces a demand curve of P = 100 - 2Q. The government imposes a tax of 10 dollars for every unit sold. Find the equilibrium quantity and price.

To find the equilibrium quantity, we can simply assume the consumer absorbs the tax (the equilibrium quantity is the same whether the tax is shouldered by the firm or the consumer). The demand curve is then:

P + T = 100 - 2Q
=> P = 90 - 2Q

Revenue = P·Q = (90 - 2Q)Q = 90Q - 2Q2
Marginal Revenue = 90 - 4Q

Setting MC = MR:

10 = 90 - 4Q
=> Q = 20
=> P + T = 60
=> P = 50

Problem :

True or False: A monopolist who faces a monotonically decreasing demand curve will always make profits.

False. The monopolist sets its quantity by the profit maximizing rule, MR = MC. With no fixed costs, assume it extracts profits of some positive amount. If we imagine an alternate scenario where the fixed costs are equal to these extracted profits, the monopolist would make no profits. Its profit maximizing quantity would not be affected by a change in fixed costs.

Problem :

Assume a monopolist has MC = 20 and no fixed costs. The monopolist faces a demand curve of P = 100 - 4Q. Calculate the deadweight loss.

Revenue = P·Q = (100 - 4Q)Q = 100Q - 4Q2
Marginal Revenue = 100 - 8Q
Setting MC = MR:
20 = 100 - 8Q
=> Q = 10

To find Q*, find the intersection of P and MC.

100 - 4Q* = 20
=> Q* = 20

The deadweight loss is simply the area between the demand curve and the marginal cost curve over the quantities 10 to 20. The deadweight loss is thus 200.

Problem :

For a non-Giffen good (a good with a non-increasing demand curve), show that price can never be less than marginal revenue.

Rather than give a rigorous proof, we'll just state the reasoning behind this principle. In a perfectly competitive market, marginal revenue is simply the price. In a monopoly, in order to sell the next unit, the firm must cut its price by some non-negative amount (non-Giffen good). The marginal revenue is thus the price received for the next unit minus the revenue lost by the price cut over all other units sold. Because the price cut is non-negative, the revenue lost by the price cut is also non-negative. Therefore, the marginal revenue must be no greater than the price.

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