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# the monopolist's optimal quantity and price

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1. Suppose a monopolist faces the market demand function P = a - bQ. Its marginal cost is given by MC = c + eQ. Assume that a > c and 2b + e > O.
a) Derive an expression for the monopolist's optimal quantity and price in terms of a, b, c, and e.
b) Show that an increase in c (which corresponds to an upward parallel shift in marginal cost) or a decrease in a (which corresponds to a leftward parallel shift in demand) must decrease the equilibrium quantity of output.
c) Show that when e 2:: 0, an increase in a must increase the equilibrium price.

2. A monopolist serves a market in which the de¬mand is P = 120 - 2Q. It has a fixed cost of 300. Its marginal cost is 10 for the first 15 units (MC = 10 when 0 &#8804; Q &#8804; 15). If it wants to produce more than 15 units, it must pay overtime wages to its workers, and its mar¬ginal cost is then 20. What is the maximum amount of profit the firm can earn?

3. Suppose that demand and supply curves in the market for corn are Qd = 20,000 - SOP and QS = 30P. Suppose that the government would like to see the price at \$300 per unit and is prepared to artificially increase demand by initiating a government purchase program. How much would the government need to spend to achieve this? What is the total deadweight loss if the government is successful in its objective?
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#### Solution Preview

1. Suppose a monopolist faces the market demand function P = a - bQ. Its marginal cost is given by MC = c + eQ. Assume that a > c and 2b + e > O.
a) Derive an expression for the monopolist's optimal quantity and price in terms of a, b, c, and e.
If a demand curve has the equation P = a - bQ, then marginal revenue has the equation MR = a - 2bQ. That is, when the demand curve is linear, the marginal revenue curve has the same intercept, but is twice as steep.
To maximize the total profit, firm will produce when MR = MC, i.e.,
a - 2bQ= c + eQ
a -c = (2b+e)Q
Q* = (a -c) / (2b+e)
From the demand curve,
P* = a - bQ*= a - b(a -c) / (2b+e)
b) Show that an increase in c (which corresponds to an upward parallel shift in marginal cost) or a decrease in a (which ...

#### Solution Summary

Derive the price for the monopolist's optimal quantity and price and other functions.

\$2.19