1. A firm operating in a monopolistically competitive market faces demand and marginal
revenue curves as given below:
P = 10 - 0.1Q
MR = 10 - 0.2Q
The firm's total and marginal cost curves are:
TC = - 10Q + 0.0333Q3 + 130
MC = -10 + 0.0999Q2,
where P is in dollars per unit, output rate Q is in units per time period, and total cost C is in
dollars.
a. Determine the price and output rate that will allow the firm to maximize profit or
minimize losses.
b. Compute a Lerner index.
Solution:
a.
Calculate MR and equate it to MC.
MC = MR
- 10 + 0.10Q2 = 10 - 0.2Q
0.1Q2 + 0.2Q - 20 = 0
The quadratic formula yields:
Q1 = 13.17
Q2 = -15.15.
Use Q1 since negative quantities are not meaningful.
At Q1 = 13.17
P = 10 - 0.1(13.17) = 8.68
b.
Computation of monopoly power. The Lerner index is computed below:
L=
P − MC
P
At Q = 13.17, P = 8.68, and MC = 7.34
L = (8.68 – 7.34)/8.68 = 0.154
L=
8.68 − 7.34
= 015
.
8.68
SD&GY_2011-2012/4
2. Bartels and Jaymes are two individuals who one day discover a stream that flows wine
cooler instead of water. Bartels and Jaymes decide to bottle the wine cooler and sell it. The
marginal cost of bottling wine cooler and the fixed cost to bottle wine cooler are both zero.
The market demand for bottled wine cooler is given as:
P = 90 - 0.25Q
where Q is the total quantity of bottled wine cooler produced and P is the market price of
bottled wine cooler.
a. What is the economically efficient price of bottled wine cooler?
b. What is the economically efficient quantity of bottled wine cooler produced?
c. If Bartels and Jaymes were to collude with one another and produce the profitmaximizing monopoly quantity of bottled wine cooler, how much bottled wine cooler will
they produce?
d. Given the output level in (c), what price will Bartels and Jaymes charge for bottled wine
cooler?
e. At the output level in (c), what is the welfare loss?
f. Suppose that Bartels and Jaymes act as Cournot duopolists, what are the reaction
functions for Bartels and for Jaymes?
g. In the long run, what level of output will Bartels produce if Bartels and Jaymes act as
Cournot duopolists?
h. In the long run, what will be the price of wine coolers be if Bartels and Jaymes act as
Cournot duopolists?
Suppose that after Bartels and Jaymes have arrived at their long run equilibrium as Cournot
duopolists, another individual, Paul Mason, discovers the streams. Paul Mason, who will sell
no wine cooler before its time, decides to bottle wine coolers. There are now three Cournot
firms producing at once. In the long run, what level of output will Bartels produce?
Solution:
a.
The economically efficient level of price is found where price equals marginal
cost. The marginal cost is zero. Therefore, the efficient price is zero.
b.
At a price of zero, Q = 360.
c.
The profit maximizing level of output is found where
MR = MC. The MR curve has the same price intercept as the demand curve
and is twice as steep. Thus, a monopolist will produce half as much as the
competitive level (this is only true because marginal cost is constant). The
competitive level of output is 360. Therefore, the monopoly level of output is
180.
Mathematically, the marginal revenue curve is:
MR = 90 - 0.5Q
Equating MR to MC yields:
90 - 0.5Q = 0
Q = 180
SD&GY_2011-2012/4
d.
When Q = 180 we have P = 90 - 0.25(180) = 45.
e.
The welfare loss is the value of the output that would have been produced
under the conditions of economic efficiency, but is not produced due to the
monopoly. This is the area of the triangle from Q = 180 to Q = 360, under the
demand curve. The base of the triangle is 180, the height of the triangle is 45,
and therefore the welfare loss is (1/2)(180)(45) = 4,050.
f.
The Cournot equilibrium is found by using the reaction curves of the two firms
to solve for levels of output. The reaction curve for firm 1 (Bartels) is found as
follows:
R1 = PQ1 = (90 - 0.25Q)Q1
= 90Q1 - 0.25(Q1 + Q2)Q1
= 90Q1 - 0.25Q12- 0.25Q2Q1
The firm's marginal revenue MR1 is just the incremental revenue dR1 resulting
from an incremental change in output dQ1:
MR1 = dR1/dQ1 = 90 - 0.5Q1 - 0.25Q2
Setting MR1 equal to zero (the firm's marginal cost) and solving for Q1 yields
the reaction curve for Q1:
Firm 1's Reaction Curve: Q1 = 180 - (1/2)Q2
Going through the same calculations for firm 2 yields:
Firm 2's Reaction Curve: Q2 = 180 - (1/2)Q1
g.
Solving the reaction curves simultaneously for Q1 and Q2 yields: Q1 = Q2 =
120.
h.
The total output is 240 (120 by each firm). Therefore:
P = 90 - 0.25(240) = 30.
i.
The Cournot equilibrium for three firms is found by solving the reaction curves
of the three firms simultaneously for levels of output. The reaction curve for
firm 1 (Bartels) is found as follows:
R1 = PQ1 = (90 - 0.25Q)Q1
= 90Q1 - 0.25(Q1 + Q2 + Q3)Q1
= 90Q1 - 0.25Q12 - 0.25Q2Q1 - 0.25Q3Q1
The firm's marginal revenue MR1 is just the incremental revenue dR1 resulting
from an incremental change in output dQ1:
MR1 = dR1/dQ1 = 90 - 0.5Q1 - 0.25Q2 - 0.25Q3
Setting MR1 equal to zero (the firm's marginal cost) and solving for Q1 yields
the reaction curve for Q1:
Firm 1's Reaction Curve: Q1 = 180 - 0.5Q2 - 0.5Q3
Going through the same calculations for firm 2 yields:
SD&GY_2011-2012/4
Firm 2's Reaction Curve: Q2 = 180 - 0.5Q1 - 0.5Q3
Going through the same calculations for firm 3 yields:
Firm 3's Reaction Curve: Q3 = 180 - 0.5Q2 - 0.5Q3
Solving the reaction curves simultaneously for Q1, Q2 and Q3 yields: Q1 = Q2 =
Q3 = 90.
3. Lambert-Rogers Company is a manufacturer of petrochemical products. The firm's research
efforts have resulted in the development of a new auto fuel injector cleaner that is
considerably more effective than other products on the market. Another firm, G.H. Squires
Company, independently developed a very similar product that is as effective as the LambertRogers formula. To avoid a lengthy court battle over conflicting patent claims, the two firms
have decided to cross-license each other's patents and proceed with production. It is unlikely
that other petrochemical companies will be able to duplicate the product, making the market a
duopoly for the foreseeable future. Lambert-Rogers estimates the demand curve given below
for the new cleaner. Marginal cost is estimated to be a constant $2 per bottle.
Q = 300,000 - 25,000P.
where P = dollars per bottle and Q = monthly sales in bottles.
a. Lambert-Rogers and G.H. Squires have very similar operating strategies. Consequently,
the management of Lambert-Rogers believes that the Cournot model is appropriate for
analyzing the market, provided that both firms enter at the same time. Calculate LambertRogers' profit-maximizing output and price according to this model.
b. Lambert-Rogers' productive capacity and technical expertise could allow them to enter
the market several months before Squires. Choose an appropriate model and analyze the
impact of Lambert Rogers being first into the market. Should Lambert-Rogers hurry to enter
first?
Solution:
Denote Lambert-Rogers price and quantity as PL, QL and Squires as PS, QS.
Demand function is given as:
Q = 300,000 - 25,000P
Solve for P:
Q - 300,000 = -25,000P
P = 12 - 0.00004Q
SD&GY_2011-2012/4
Outcome under Cournot model:
a.
TRL = PL · QL
TRL = (12 - 0.00004Q)QL
Q = QL + QS
TRL = [12 - 0.00004(QL + QS)]QL
TRL = 12QL - 0.00004QL2 - 0.00004QLQS
MRL = 12 - 0.00008QL - 0.00004QS
Set MRL = MC
12 - 0.00008QL - 0.00004QS = 2
-0.00008QL - 0.00004QS = -10
QL = 125,000 - 0.5QS
So, QS = 125,000 - 0.5QL
Substitute for QS:
QL = 62,500 + 0.25QL
QL =
62,500
= 83,333
0.75
Q = QL + QS
Q = 83,333 + 83,333 = 166,666
P = 12 - .00004(166,666)
P = 12 - 6.67 = $5.33
P = $5.33 per bottle
166,666 bottles sold per month
b.
The Stackelberg model is appropriate when one firm enters first.
Lambert-Rogers determines its output, which Squires then takes as given.
Lambert's total revenue function is given as:
TRL = 12QL - 0.00004QL2 - 0.00004QLQS
SD&GY_2011-2012/4
Squires reaction function QS = 125,000 - 0.5QL can be substituted into TRL,
since Squires will take Lambert's output as given.
TRL = 12QL - 0.00004QL2 - 0.00004QL(125,000 - .5QL)
TRL = 12QL - 0.00004QL2 - 5QL + 0.00002QL2
TRL = 7QL - 0.00002QL2
MRL = 7 - 0.00004QL
Set MRL = MC
7 - 0.00004QL = 2
-0.00004QL = -5
QL = 125,000
To find QS substitute QL into S reaction function
QS = 125,000 - 0.5QL
QS = 125,000 - 0.5(125,000)
QS = 62,500
Q = QL + QS
Q = 125,000 + 62,500
Q = 187,500
P = 12 - 0.0004(187,500)
P = 12 - 7.5 = $4.50
Lambert-Rogers gets a much larger share of the market by entering first. It
should advance its schedule in order to enter first.
SD&GY_2011-2012/4
4. The two leading U.S. manufacturers of high performance radial tires must set their
advertising strategies for the coming year. Each firm has two strategies available: maintain
current advertising or increase advertising by 15%. The strategies available to the two firms,
G and B, are presented in the payoff matrix below.
Firm G
Increase Adv.
Maintain Adv.
Firm B
Increase Adv.
Maintain Adv.
27, 27
50, 12
12, 50
45, 45
The entries in the individual cells are profits measured in millions of dollars. Firm G's
outcome is listed before the comma, and Firm B's outcome is listed after the comma.
a. Which oligopoly model is best suited for analyzing this decision? Why? (Remember it is
illegal to collude in the United States.)
b. Carefully explain the strategy that should be used by each firm. Support your choice by
including numbers.
Solution:
a.
The prisoner's dilemma model is most appropriate for analyzing this situation.
We can conclude that the prisoner's dilemma is most appropriate because each
firm must set its advertising strategy without knowledge of the rival's strategy.
b.
Increasing the advertising level is the dominant strategy, since the firm is better
off increasing regardless of the rival's action. For example, if Firm B
increases, Firm G earns 27 if it increases and 12 if it does not increase. G is
better off increasing. If Firm B doesn't increase, Firm G earns 45 by not
increasing and 50 by increasing. Again, Firm G is better off to increase. It is
obvious that no matter what B does, G is better off to increase. Firm B faces
the same situation.
5. The market for an industrial chemical has a single dominant firm and a competitive fringe
comprised of many firms that behave as price takers. The dominant firm has recently begun
behaving as a price leader, setting price while the competitive fringe follows. The market
demand curve and competitive fringe supply curve are given below. Marginal cost for the
dominant firm is $0.75 per gallon.
QM = 140,000 - 32,000P
QF = 60,000 + 8,000P,
where QM = market quantity demanded, and QF = the supply of the competitive fringe.
Quantities are measured in gallons per week, and price is measured as a price per gallon.
a. Determine the price and output that would prevail in the market under the conditions
described above. Identify output for the dominant firm as well as the competitive fringe.
b. Assume that demand curve shifts rightward by 40,000 units. Show that the dominant
firm is indeed a price leader. What output (leader and follower) and market price will prevail
after the change in demand?
SD&GY_2011-2012/4
Solution:
a.
QM = 140,000 - 32,000P
QF = 60,000 + 8,000P
Denote dominant firm demand curve as QD.
QD = QM - QF
QD = 140,000 - 32,000P - (60,000 + 8,000P)
QD = 80,000 - 40,000P
Solve for P
QD - 80,000 = -40,000P
P = 2 - 0.000025QD
MRD = 2 - 0.00005QD
Marginal cost for the dominant firm is $0.75. Equate MRD to MCD
2 - 0.00005QD = 0.75
-0.00005QD = -1.25
QD = 25,000
P = 2 - 0.000025(25,000)
P = 2 - 0.625 = 1.375 per gallon
Fringe takes dominant firm price as given
QF = 60,000 + 8,000(1.375)
QF = 71,000
QT = 25,000 + 71,000 = 96,000
b.
A 40,000 increase in demand curve to:
QM = 180,000 - 32,000P
QF = 60,000 + 8,000P
QD = 180,000 - 72,000P - (60,000 + 8000P)
QD = 120,000 - 40,000P
Solve for P
QD - 120,000 = -40,000P
PD = 3 - 0.000025QD
MRD = 3 - 0.00005QD
setting MRD = MCD
3 - 0.00005QD = 0.75
-0.00005QD = -2.25
QD = 45,000
PD = 3 - 0.000025(45,000)
PD = 3 - 1.125 = $1.875
Fringe again follows
QF = 60,000 + 8,000(1.875)
QF = 75,000
SD&GY_2011-2012/4
QT = 45,000 + 75,000 = 120,000
We can see that when demand changed, the dominant firm raised price. The
competitive fringe took the new price as given and adjusted output accordingly.
6. Consider two identical firms (no. 1 and no. 2) that face a linear market demand curve.
Each firm has a marginal cost of zero and the two firms together face demand:
P = 50 - 0.5Q, where Q = Q1 + Q2.
a. Find the Cournot equilibrium Q and P for each firm.
b. Find the equilibrium Q and P for each firm assuming that the firms collude and share the
profit equally.
c. Contrast the efficiencies of the markets in (a) and (b) above.
Solution:
a.
Determine the reaction curve for no. 1. Equate MR1 to MC1.
R1 = P1Q1 = (50 - 0.5Q)Q1 = 50Q1 - 0.5QQ1
= 50Q1 - 0.5(Q1 + Q2)Q1 = 50Q1 - 0.5Q12 - 0.5Q1Q2
MR1 = 50 - 1Q1 - 0.5Q2.
Since MC1 = 0, then
50 - 1Q1 - 0.5Q2 = 0
Q1 = 50 - 0.5Q2
The reaction curve for firm no. 2 is calculated in the same way as that for firm
no. 1.
Q2 = 50 - 0.5Q1
At the intersection of two reaction curves, we find the equilibrium Q1 and Q2.
By substitution:
Q2 = 50 - 0.5(50 - 0.5Q2)
Q2 = 50 - 25 + 0.25Q2
0.75Q2 = 25
Q2 = 33.33
SD&GY_2011-2012/4
Now solve for Q1:
Q1 = 50 - 0.5Q2 = 50 - 0.5(33.33) = 33.33
The total quantity produced Q = Q1 + Q2 = 66.67
The market equilibrium price is:
P = 50 - 5Q = 50 - 0.5(66.67) = $16.67/unit.
Each firm is maximizing its own profit, given its competitor's production rate.
b.
The total revenue for the two firms is:
R = PQ - (50 - 0.5Q)Q = 50Q - 0.5Q2, and thus
MR = 50 - Q
Set MR equal to MC = 0 to find Q that maximizes profit.
50 - Q = 0
Q = 50
If profit is shared equally, then Q1 = Q2 = 25.
The collusive price is P = 50 - 0.5(50) = 25.
c.
When competing, each firm produces 33.33 at a price of 16.67; and when
colluding, each produces 25 at a price of 25. The market is more efficient
when the firms compete, because in this situation selling price is more nearly
equal to marginal cost.
7. The demand for on-line brokerage services is: Q D = 6, 500 − 100 P ⇔ P = 65 − 0.01Q D . If the online brokerage firms collude, the collusive marginal revenue function is: MR ( Q ) = 65 − 0.02Q.
The brokerage firm specific marginal cost functions are:
 MC1 ( q1 ) = 1.5q1

 MC2 ( q2 ) = 2.0q2
.

 MC3 ( q3 ) = 2.5q3
 MC4 ( q4 ) = 3.0q4

Calculate the
collusive output level and market price. If the brokerage firms behaved competitively and
each firm set its own marginal cost equal to price, what would be the output level and market
price?
SD&GY_2011-2012/4
Solution:
 MC1 ( q1 ) = 1.5q1

 MC2 ( q2 ) = 2.0q2

 MC3 ( q3 ) = 2.5q3
 MC4 ( q4 ) = 3.0q4

2

 q1 = 3 MC

 q = 1 MC
 2 2
.
⇔
 q = 2 MC
 3 5

1
 q4 = MC

3
This implies the collusive supply sets
10
 2 1 2 1  19
Q S = MC  + + +  = MC. The collusive marginal cost function is: MC = Q. The
3
2
5
3
10
19


collusive profit maximization level sets marginal cost equal to marginal revenue. This is:
65 − 0.02Q =
10
Q ⇔ Q = 118.979.
19
The resulting price is $63.81. If brokerage firms behave
competitively, each firm sets marginal cost equal to price.
2

 q1 = 3 P

 MC1 ( q1 ) = 1.5q1
q = 1 P

2

MC
q
=
2.0
q
( )
2
2
. This implies competitive supply is
Thus,  2 2
⇔
2.5
2
MC
q
q
=
(
)
3
3
3

q = P
 MC4 ( q4 ) = 3.0q4
 3 5


1
 q4 = P

3
 2 1 2 1  19
S
Q = P  + + +  = P. Setting supply equal to demand and solving for market price gives
 3 2 5 3  10
us the competitive solution. This is: 19 P = 6500 − 0.01P ⇔ P = 63.78. The competitive market
10
output level is 121.182.
SD&GY_2011-2012/4