**Problem : **

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

Revenue =

*P*·*Q* = (100 - 3*Q*)*Q* = 100*Q* - 3*Q*^{2}
Marginal Revenue =

100 - 6*Q*
Setting

*MC* = *MR*:

10 = 100 - 6*Q*

=> *Q* = 15

**Problem : **

Assume a monopolist has *MC* = 10 and no fixed costs. The monopolist faces a
demand curve of *P* = 100 - 2*Q*. 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 - 2*Q*

=> *P* = 90 - 2*Q*

Revenue = *P*·*Q* = (90 - 2*Q*)*Q* = 90*Q* - 2*Q*^{2}

Marginal Revenue = 90 - 4*Q*

Setting *MC* = *MR*:

10 = 90 - 4*Q*

=> *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 - 4*Q*. Calculate the deadweight loss.

Revenue = *P*·*Q* = (100 - 4*Q*)*Q* = 100*Q* - 4*Q*^{2}

Marginal Revenue = 100 - 8*Q*

Setting

*MC* = *MR*:

20 = 100 - 8*Q*

=> *Q* = 10

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

100 - 4*Q*^{*} = 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.