Industrial Organization
ECNS 406
Fall 2013
Exam #: 2
Thursday November 7, 2013
Name:
You must answer all of the following questions. Each question is worth the same amount.
You have the class period to complete the exam.
Answer each question clearly and concisely. You must show your work to receive credit.
This exam is given under the rules of the Montana State University. By printing your
name above you acknowledge the University’s Honor Code and agree to comply with the
provisions of the Honor Code. You may not use notes or receive any assistance. There is to
be no talking during the exam. You may use a calculator, but are never allowed to use device
allowing you to take photographs or transmit over a network. No notes, no assistance,
no talking, no cell phones, but you can use a calculator.
Clearly print your name above, in the space provided on the next page and in your blue
book(s). You must turn in your blue book(s).
ECNS 406
Exam #: 2
11/7/2013
1. Given an inverse demand function of P = 100 − 5Q and costs for firm i of C(qi ) = 25qi ,
answer the following questions:
(a) What is the Cournot duopoly equilibrium firm quantity, market quantity and price?
Solution: Setup one firm’s problem, find their best response and impose symmetry to find the equilibrium
max πA = (100 − 5qA − 5qB )qA − 25qA
qA
dπA
= 100 − 10qA − 5qB − 25
dqA
0 = 75 − 10qA − 5qB
10qA = 75 − 5qB
Now impose symmetry.
10q ∗ = 75 − 5q ∗
15q ∗ = 75
q∗ = 5
The Cournot duopoly equilibrium firm quantity is q ∗ = 5, market quantity is
Q = 10, and the price is P = 100 − 5(10) = 50.
(b) Now instead of Cournot competition, suppose the two firms choose to differentiate
their products. Each consumer’s baseline valuation is 100 and consumers are uniformly distributed around a circular preference space. The circular preference space
has a circumference equal to the Cournot duopoly equilibrium market quantity in
part a. Firm costs remain the same and each consumer must incur a transportation
cost of 2 times the distance traveled. What is the equilibrium price with product
differentiation?
Solution: The circumference of the circle is 10. Let firm A be at the top of the
circle (0 and 10) and firm B be at the bottom of the circle (5).
First consider the utility maximization problem to find the indifferent consumer
located at x on the left part of the circle indifferent to buying from A or B.
uA (x) = 100 − pA − 2(x), uB (x) = 100 − pB − 2(5 − x) and set uA (x) = uB (x)
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ECNS 406
Exam #: 2
11/7/2013
and solve for x.
uA (x) = uB (x)
100 − pA − 2(x) = 100 − pB − 10 + 2x
4x = pB − pA + 10
pB − pA 5
+
x=
4
2
Now we use the indifferent consumer to find demand. We know that qA = 2x =
pB −pA
+ 5.
2
With demand the firms profit maximization problem is as follows.
pB − pA
+5
max πA = pA qA − 25qA = (pA − 25)qA = (pA − 25)
pA
2
pB − pA
1
dπA
=
+ 5 − (pA − 25)
dpA
2
2
pB − pA
1
0=
+ 5 − (pA − 25)
2
2
pB − pA
1
(pA − 25) =
+5
2
2
pA − 25 = pB − pA + 10
2pA = pB + 35
Now impose symmetry to find the equilibrium.
2p∗ = p∗ + 35
p∗ = 35
The equilibrium price with product differentiation is p∗ = 35.
(c) Using the above scenarios, do firms want to produce standardized or differentiated
products?
Solution: Firm profits from the Cournot duopoly equilibrium are π = P ∗ q ∗ −
25q ∗ = 50(5) − 25(5) = 125.
With the product differentiation example above, p∗ = 25, q ∗ = 5, and π =
p∗ q ∗ − 25q ∗ = 35(5) − 25(5) = 50.
Firms would rather produce standardized products because their profits are
higher.
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ECNS 406
Exam #: 2
11/7/2013
(d) Using the above scenarios, do consumers want the firms to produce standardized or
differentiated products?
Solution: Consumer surplus
h from the
i10Cournot duopoly equilibrium is CS =
R 10
2
(100 − 5Q∗ − 50)dQ∗ = 50x − 5x2
= 250.
0
0
With product differentiation, the consumer surplus is as follows.
Z
5
2
CS = 4
(100 − p∗ − 2x)dx
0
Z
CS = 4
5
2
(65 − 2x)dx
0
5
CS = 4 65x − x2 02
5 25
CS = 4 65 −
2
4
625
CS = 4
4
CS = 625
Consumer’s want the firms to produce differentiated products because their
consumer surplus is higher.
2. The inverse demand function is P = 120 − 2Q. Two firms compete in quantities and
each firm has a cost of C(qi ) = 2qi2 . The interest rate is 25%.
(a) If a competitive equilibrium is maintained over time, what is the discounted sum
of profits?
Solution: Firm A’s problem is as follows.
max πA = (120 − 2qA − 2qB )qA − 2qA2
qA
The first order condition is
dπA
= 120 − 2qB − 8qA
dqA
and setting this equal to zero and solving for qA yields the following.
qA = 15 −
qB
4
The above is a firm’s best response which will be used later.
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ECNS 406
Exam #: 2
11/7/2013
Imposing symmetry conditions and solving for the equilibrium quantity yields
q ∗ = 12, Q∗ = 24, and P ∗ = 72. The per period competitive profits are
π C = 72(12) − 2(12)2 = 576
The discounted sum of profits are
1 C
π = 4(576) = 2304.
r
(b) If firms can perfectly collude and share the profits evenly, what would be the per
period profits of each firm?
Solution: For a collusive equilibrium, Q = 2q and a firms profit is
max π = (120 − 2(2q))q − 2q 2
q
The first order condition is
dπ
= 120 − 12q
dq
and setting this equal to zero and solving for q yields the following.
q ∗ = 10
Q∗ = 20
P ∗ = 120 − 2(20) = 80
The per period collusive profits are
π T C = 80(10) − 2(10)2 = 600
(c) Consider a trigger strategy where if a firm deviates from a collusive equilibrium,
the other firm behaves competitively forever after. What is the discounted sum
of profits obtained by deviating from a collusive equilibrium and is it possible to
maintain a collusive equilibrium if the other firm plays the trigger strategy?
Solution: Assume firm A deviates from the collusive equilibrium. We first
need to find firm A’s optimal deviation. In a collusive equilibrium, q ∗ = 10, so
assume qB = 10 and firm A plays their best response. Firm A’s best response
is qA = 15 − q4B = 25
. The market quantity is then Q = 45
and the price is
2
2
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ECNS 406
Exam #: 2
11/7/2013
= 75. Firm A’s profits from the deviation are
P = 120 − 2 45
2
πAD
= 75
25
2
−2
25
2
2
= 625
The discounted sum of profits would then be as follows.
1
1 1 C
πAD +
π
1+r
1+rr
4
4
11716
(625) + (4)(576) =
5
5
5
It’s possible to maintain a collusive equilibrium if the following condition holds.
2400 >
11716
5
which is does, so yes it is possible to maintain a collusive equilibrium if the other
firm plays the trigger strategy.
(d) Consider another type of trigger strategy where one firm tries to establish a collusive equilibrium from the competitive equilibrium. If one firm deviates from the
competitive quantity to the collusive quantity, then the other firm will play along
forever after (unless one firm deviates from the collusive equilibrium). What is the
discounted sum of profits obtained by deviating from a competitive equilibrium and
is it possible to switch from a competitive equilibrium to a collusive one?
Solution: Assume firm A deviates from the competitive equilibrium. From the
question, we know that firm B plays the competitive outcome of qB = 12 and
firm A plays the collusive outcome of qA = 10. The market quantity is then
Q = 22 and the price is P = 120 − 2(22) = 76. Firm A’s profits from the
deviation are
πAD = 76 (10) − 2 (10)2 = 560
Note that this is not a profitable deviation for firm A, but if they can then get
firm B to play along, the discounted sum of profits for firm A is as follows.
1 1 TC
1
πAD +
π
1+r
1+rr
4
4
(560) + (4)(600) = 2368
5
5
It’s possible to switch from a competitive equilibrium to a collusive one if 2368 >
2304, so yes it is possible.
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ECNS 406
Exam #: 2
11/7/2013
3. Given an inverse demand function of P = 100 − 2Q and costs for Firm i of C(qi ) = 20qi ,
find the Stackelberg duopoly equilibrium:
(a) Firm Quantities
Solution: Let the two firms be A and B. Assume firm A sets their quantity first
and then firm B follows. Using backwards induction, consider firm B’s PMP
and find their best response.
max πB = (100 − 2qA − 2qB )qB − 20qB
qB
dπB
= 100 − 2qA − 4qB − 20
dqB
0 = 100 − 2qA − 4qB − 20
4qB = 80 − 2qA
1
qB = 20 − qA
2
Substitute firm B’s best response into firm A’s PMP.
1
max πA = (100 − 2qA − 2 20 − qA )qA − 20qA
qA
2
max πA = (100 − 2qA − 40 + qA )qA − 20qA
qA
dπA
= 100 − 4qA − 40 + 2qA − 20
dqA
0 = 100 − 4qA − 40 + 2qA − 20
2qA = 40
qA = 20
qB = 20 − 12 qA = 10
(b) Market Price
Solution: Q = qA + qB = 30
P = 100 − 2Q = 100 − 60 = 40.
(c) Firm Profits
Solution:
πA = 40(20) − (20)20 = 400
πB = 40(10) − (20)10 = 200
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ECNS 406
Exam #: 2
11/7/2013
(d) Consumer Surplus
Solution: The market quantity is Q = qA + qB = 30. The consumer surplus is the area below the demand curve and above the price for all the goods
bought/sold.
Z 30
CS =
(100 − 2x − 40)dx
0
CS = [60x − x2 ]30
0
CS = (60)(30) − (30)2
CS = 900
(e) How do your answers compare with the Cournot Duopoly equilibrium?
, Q = 80
, P = 140
and
Solution: The Cournot Duopoly equilibrium has q = 40
3
3
3
6400
CS = 9 .
Compared to the Cournot duopoly equilibrium, firm A produces more, firm B
produces less, overall output is higher, the price is lower and consumer surplus
is greater.
4. A durable goods monopolist faces an inverse demand function of P = 20 − Q and has
a constant marginal cost of production equal to 4. The monopolist can choose two
prices before the good becomes obsolete: a price today (P1 ) and a price tomorrow (P2 ).
Tomorrow’s profits are discounted at an interest rate of 25%.
(a) What is the quantity demanded in the second period taking into consideration the
consumers who purchased the durable product in the first period?
Solution: Rearrange the inverse demand to get the demand: Q = 20 − P . In
the second period, the quantity demanded is Q2 = 20 − P2 , but Q1 = 20 − P1 of
those consumers bought in the first period and will not buy again. This leaves
Q2 − Q1 consumers in the second period. Q2 − Q1 = 20 − P2 − 20 + P1 = P1 − P2 .
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(b) Find the firm’s period 2 price expressed as a best response to their first period price.
Solution: The firm’s profit maximization problem in the second period is as
follows.
max π2 = (P2 − 4)(P1 − P2 )
P2
∂π2
= P1 − P 2 − P2 + 4 = 0
∂P2
2P2 = P1 + 4
P1
P2 =
+2
2
(c) Setup the firm’s first period profit maximization problem.
Solution:
4
max π = (P1 − 4)(20 − P1 ) + (P2 − 4)(P1 − P2 )
P1
5
where P2 = P21 + 2.
1
Note that (1+r)
= 45 . Also note that P1 − P2 = P1 − P21 − 2 = P21 − 2 and
P2 − 4 = P21 + 2 − 4 = P21 − 2. Substituting this into the profit maximization
problem yields the following.
2
4 P1
−2
max π = (P1 − 4)(20 − P1 ) +
P1
5 2
(d) What are the equilibrium prices in each period?
Solution:
2
4 P1
max π = (P1 − 4)(20 − P1 ) +
−2
P1
5 2
∂π
4
1
P1
= 20 − P1 − P1 + 4 +
2
−2 =0
∂P1
5
2
2
200 − 20P1 + 40 + 4P1 − 16 = 0
224 = 16P1
P1∗ = 14
P2∗ = 7 + 2 = 9
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