Solution to problem set 7 - NYU Stern School of Business

Microeconomics
Prof. N. Economides
Solutions to problem set 7
1.a. As the average cost is constant at $6, the marginal cost is also constant at $6. MR =
d(revenue)/dQ = d(11Q - Q 2 )/dQ = 11 - 2Q. When profit is maximized, MR = MC = 11 - 2Q
= 6. Therefore, Q = 2.5 i.e., 2,500 units and P = 11 - 2.5 = 8.5. Profit, Π = Revenue - total
cost = (revenue per unit - average cost)*unit sold = (8.5 - 6)*2.5 = 6.25 i.e., $6,250.
b. When the price is regulated at $7, quantity demanded = (11 - 7)*1,000 = 4,000. Profits, Π =
revenues - cost = (11Q - Q 2 ) - 6Q = (5Q - Q 2 ) = 4 ($4,000) when Q = 4.
c. If the price is set below $6, the monopoly will go out of business as it cannot cover the
cost. At p = 6, output for a competitive market would be 5,000 so a monopolist would
produce less than 5,000 units at any price above $6. Therefore, the price ceiling should
be set at $6.

2a. Revenue = price*quantity. Since price = 120 - y, revenues = p*y = (120 - y)*y = 120y - y 2 .
Marginal revenue = change in revenue/change in quantity = d(120y - y 2 )/dy = 120 - 2y. To
maximize profits, monopolist produces where MR = MC ⇔120 - 2y = 2y ⇔ y = 30. Price =
120 - y, so p = 90. Profits = R - C = 30*90 - 30 2 = 2700 - 900 = $1,800.
b. Revenues remain unchanged, but c(y) = y 2 + 100. MC = d(y 2 + 100)/dy = 2y. Since this is the
same as in a), the monopolist would not change his level of output from y = 30. Profits are
still positive at $1,700, so the monopolist would not shut down.
c. Surplus is maximized at the level of output reached in competitive equilibrium, when the
supply and demand curves intersect. This occurs where MC = price consumers are willing to
pay. Thus, 2y = 120 - y, and y =40. Price = 120 - y = 80.
d. y = 40, as shown in part c.
e. Now c(y) = y 2 + 20y, so MC' = 2y + 20. Setting MR = MC', 120 - 2y = 2y + 20 or y = 25, p =
95.

3. Referring to the graph, MC 2 intersects with the MR curve but MC 1 doesn’t. Therefore, only
MC 2 is relevant. As the demand curve is P = 20 - 3Q, then, MR = 20 - 6Q. At profit maximization,
MR = MC ⇔ 20 - 6Q = 10 + 5Q ⇔ 11Q = 10 and Q = 10/11 and the price = 20 - 3(10/11) = 190/11
= 17.3.