INTERMEDIATE MICROECONOMIC THEORY
ECON301
Fall 2010
Third Problem Set (Answers)
1.
Suppose a monopolist produces its output subject to the following total cost
equation:
C = 100 + 10Q + 2Q
2
, and the demand curve it faces is p = 90  2Q.
Determine the price,
quantity, and profit for this firm.
Answer: Set marginal cost equal to marginal revenue => MC = 10 + 4Q = MR = 90 – 4Q
=> 8Q = 80 and Q* = 10. P* = 90 – 2Q = 90 – 20 = 70. Profit equals revenue minus cost
=> PQ – C = 70(10) – (100 + 10(10) + 2(10)
2
) = 700 – (100 + 100 + 200) = 700 – 400 =
300.
2.
Assume that the inverse demand equation a monopoly faces is P = 40  Q, and
marginal cost is constant at $5 per unit.
a.
Draw a graph showing the demand curve, marginal revenue curve and marginal
cost curve.
b.
What are the monopolist’s profitmaximizing output and price?
c.
Calculate the consumer surplus, producer surplus and social welfare at the
monopoly output and price.
d.
What is the value of Lerner’s index at the profitmaximizing price and quantity?
Show your work.
Answers: The demand curve has a Pintercept of 40 and a Qintercept of 40.
The marginal
revenue equation is MR = 40 – 2Q.
The MR curve has a MRintercept of 40 and a Qintercept of
40.
Marginal cost is a horizontal line at MC = 5.
The firm maximizes profits by setting MR
equal to MR => 40 – 2Q = 5 => Q* = 35/2 = 17.5.
The monopolist’s price is P* = 40 – 17.5 =
22.5.
Consumer surplus equals (40 – 22.5)(17.5)/2 = 153.13. Producer surplus is (22.5 – 5)(17.5)
= 306.25. The Lerner’s index equals (P – MC)/P = (22.5 – 5)/22.5 = 17.5/22.5 = 0.778.
Consumer surplus equals (40 – 22.5)(17.5)/2.
Producer surplus equals (22.5 – 5)(17.5) = (17.5)
2
.
1
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3.
Assume that a firm has a monopoly.
Its demand curve is given by the equation
P = 60 – Q.
It produces its output subject to the following shortrun cost equation: C =
Q
2
+ 20.
a.
Draw a graph of the monopolist’s demand curve, shortrun marginal cost curve
and marginal revenue curve and determine the profitmaximizing price and
quantity.
b.
Now assume that there is a $4 per unit specific tax. Determine the new profit
maximizing price and quantity.
c.
Determine what effect the $4 specific tax has had on consumer surplus, producer
surplus, tax revenues, and social welfare.
Be sure to calculate the dollar amounts.
Answers: The marginal revenue equation is MR = 60 – 2Q.
Since MR equals MC at the
profitmaximizing output level, 60 – 2Q = 2Q => Q* = 60/4 = 15. P* = 60 – Q = 60 – 15 =
45.
If there is a $4 specific tax, SMC = 2Q + 4.
In this case, MR = MC => 60 – 2Q = 2Q + 4
=> 4Q = 60 – 4 => Q* = 56/4 = 14.
Consumers pay a price equal to P* = 60 – Q = 60 – 14 =
46.
The monopolist receives a net price of 46 – 4 = 42.
The government collects a revenue
of 4(14) = 56.
Both consumer and producer surplus decrease due to the tax. Without the tax
CS = (60 – 45)(15) /2 = 121.5. PS = 15(45) – (15)
2
= 450. With the tax CS = (60 – 46)(14)/2
= 98. PS =
14(46) – (14(14) + 14(4)) = 392.
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 Fall '08
 Arnold
 Consumer Surplus, Supply And Demand, monopolist, marginal revenue curve

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