ECOS3005 Practice exam solutions
1. Two firms produce an identical product in a linear city of length 1 unit. Consumers are uniformly
located along the city. Suppose that consumer i’s utility derived from buying firm j’s product is
given by
ui j = 20− t(xi− y j)2− p j,
where j = 1,2 indicate the two firms, t is the per unit cost of travelling along the city, xi is the
location of consumer i, y j is the location of firm j, and p j is the price of product j. Assume that
the two firms are located at each end of the city. That is y1 = 0 and y2 = 1. Each firm has constant
marginal costs of c per unit and no fixed costs.
(a) Suppose the firms compete by simultaneously choosing prices in a single period.
i. Find the demand for each firm’s product in terms of the market prices set by each firm.
[3 marks]
ANS: To find the demand for each firm, we need to find the fraction of consumers buying
from each firm. So, we need to work out the location of the consumer who is indifferent
between each firm’s product. Call the location of this consumer x. It must therefore be
that
20− tx2− p1 = 20− t(x−1)2− p2
p1 + tx2 = t(x−1)2 + p2
t(x2− (x−1)2) = p2− p1
2tx = p2− p1 + t
x =
p2− p1 + t
2t
Hence, the demand for each firm is given by
q1 = x =
p2− p1 + t
2t
q2 = 1− x = p1− p2 + t2t
Hence the quantity demanded for each firm decreases with its own price and increases
with its rival’s price.
ii. Find the reaction function for each firm. [3 marks]
ANS: To find the market prices for each firm, we need to first work out firm profits,
then solve for the profit maximising problem for each firm. This will give us reaction
functions for each firm, which we can solve to find the equilibrium market prices. First,
profits for Firm 1 are given by
pi1 = q1(p1− c)
= (p1− c) p2− p1 + t2t
The first order conditions for firm 1’s profit maximising problem are then
p2− p1 + t
2t
− p1− c
2t
= 0
p2−2p1 + t+ c = 0
p1 =
p2 + t+ c
2
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ECOS3005 Practice exam solutions
This is Firm 1’s reaction function.
Similarly, solving for Firm 2’s reaction function we obtain
p2 =
p1 + t+ c
2
iii. Find the Nash equilibrium price and profit for each firm. [3 marks]
ANS: Solving these two reaction functions together,
p1 =
t+ c
2
+
p1 + t+ c
4
3
4
p1 =
3t+3c
4
p1 = t+ c
p2 = t+ c
Hence, the market price of each firm is positively related to both marginal costs, and
transport costs. Notice that both firms have the same equilibrium market price. Substi-
tuting into the demand functions, we obtain q1 = q2 = 1/2. Profits are then given by
pii = qi(pi− c) = t2 , i= 1,2
(b) Suppose the firms play an infinitely repeated game. Each period the firms simultaneously
choose prices. We want to examine whether a cartel is sustainable. Consider the following
grim-trigger strategies:
p=
{
pC if it is the first period or both firms set p= pC in every prior period
pN otherwise
where pN is the Nash equilibrium price you obtained in part 1(a)iii and pC = 3t+c is the cartel
price. Under these strategies, both firms set the cartel price unless one of them deviated from
this in a previous period. If either firm deviates, both firms punish for the rest of the game by
setting p = pN . Firms discount the future using the discount factor δ, where 0 < δ < 1. For
what values of δ is the cartel sustainable using the above grim-trigger strategies? Explain.
[6 marks]
ANS: We want to find out if the grim-trigger strategies form a subgame perfect Nash equi-
librium (SPNE). We therefore need to establish that there is no incentive to deviate from this
strategy if your rival is playing this strategy.
First, note that if there is ever a deviation, the grim-trigger strategy calls on players to set the
price PN which we identified as the Nash equilibrium price to the one-period game. Therefore,
there is no incentive to deviate (again) once punishment begins.
Next, we need to check whether there is an incentive to deviate at the start of the game (or
if no players have deviated yet). To do this, we compare the payoffs to cooperating with the
payoffs to deviation. Suppose Firm 2 plays the grim-trigger strategy and consider Firm 1’s
incentives. If Firm 1 cooperates, she receives payoffs:
VC =
piC
1−δ ,
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ECOS3005 Practice exam solutions
where piC are the profits if both firms cooperate by setting p= pC. If she deviates, she receives
payoffs:
VD = piD+δ
piN
1−δ ,
where piD are the profits she obtains by choosing an optimal deviation and piN = t/2 are the
non-cooperative profits we identified earlier.
So, collusion is sustainable if
VD =
piC
1−δ ≥V
D = piD+δ
piN
1−δ
piC ≥ (1−δ)piD+δpiN
δ(piD−piN)≥ piD−piC
δ≥ pi
D−piC
piD−piN
Next, note that if both firms set pC, then q1 = q2 = 1/2 and
piC = q(p− c) = 1
2
3t =
3t
2
.
If firm 1 deviates, the optimal price to set is given by their reaction function. So,
p1 = (t+ c)/2+ p2/2 = (t+ c)/2+(3t+ c)/2 = 2t+ c.
Demand for Firm 1 is then q1 = (p2− p1 + t)/2t = 2t/2t = 1. Hence,
piD = q1(p1− c) = 1×2t = 2t.
Hence, the grim-trigger strategies form an SPNE and collusion is sustainable if
δ≥ pi
D−piC
piD−piN
=
2t−3t/2
2t− t/2 =
1
3
.
If the firms are sufficiently patient, they will be able to resist the temptation to earn higher
short-term profits because of the prospect of collusive profits in the future.
2. Two firms engage in quantity competition in a single period. They play a three stage game. In stage
1, the Leader chooses an output, q1. In stage 2, the Follower decides whether to enter. In stage 3,
if the follower entered, it chooses an output, q2. Market demand is given by
P= 70−0.5Q,
where Q= q1 +q2 is market output, and P is the market price. Each firm has the cost function
C(q) = 10q+400.
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ECOS3005 Practice exam solutions
(a) Assume that the Follower decides to enter.
i. Solve for the reaction function of the follower. [3 marks]
ANS: The Follower chooses output to maximise profits
pi2 = q2(70−0.5(q1 +q2))−10q2−400.
This leads to FOCs:
70−0.5q1−q2−10 = 0
q2 = 60−0.5q2.
This is the reaction function for Firm 2.
ii. Solve for the Stackelberg solution to the game. Show that the Leader chooses an output
of 60. What are the profits of each firm? [4 marks]
ANS: The leader earns profits of
pi1 = q1(70−0.5q1−0.5q2)−10q1−400
= q1(70−0.5q1−30+0.25q1)−10q1−400
= q1(30−0.25q1)−400.
Maximising profits by choosing p1 leads to FOCs:
30−0.5q1 = 0
q1 = 60.
The leader chooses an output of 60. Substituting into the follower’s reaction function,
we obtain
q2 = 30.
The market price is then
P= 70−0.5Q= 25.
Profits are given by
pi1 = q1P−10q1−400 = 60.25−60.10−400 = 500
pi2 = 30.25−30.10−400 = 50.
(b) Is the Stackelberg solution also a subgame perfect Nash equilibrium to the full game? Explain.
[Hint: Does the leader have an incentive to increase output to, say, 64?] [4 marks]
ANS: To check whether the Stackelberg solution above is a subgame perfect NE, we need to
check whether either firm has an incentive to deviate in any of the subgames. In particular,
does the leader have an incentive to deter entry by producing a larger output?
Deterring entry involves ensuring that pi2 ≤ 0. Letting F = 400 be the fixed cost,
pi2 = q2(70−0.5(q1 +q2))−10q2−F
= (60−0.5q1)(60−0.5q1−0.5(60−0.5q1))−F
= 0.5(60−0.5q1)2−F.
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ECOS3005 Practice exam solutions
Setting pi2 = 0, we have
(60−0.5q1)2 = 2F
60−0.5q1 =
√
2F
q∗1 = 120−2
√
2F ≈ 63.43.
Any q1 > q∗1 will be sufficient to deter entry. For example, suppose q1 = 64. Would this be a
profitable deviation for Firm 1?
pi1 = q1(70−0.5q1)−10q1−F
= 64(60−32)−400 = 1392.
This gives Firm 1 a higher profit than the Stackelberg solution. Hence, Firm 1 has an incentive
to deter entry by setting q1 > q∗1. The Stackelberg solution cannot therefore constitute a
subgame perfect NE.
(c) Suppose stages 1 and 2 were reversed (that is, the Follower’s entry decision takes place before
the Leader’s output decision). How would this change your analysis of the game. Explain
briefly. [4 marks]
ANS: If the Follower is able to commit to entry before the Leader chooses output, this will
have a major impact on the game. We know from above that the Stackelberg solution involves
positive economic profits for both Leader and Follower. Consider then the following strate-
gies. The Follower enters in stage 1; the Leader chooses the Stackelberg output in stage 2
if the Follower enters (and chooses the monopoly output otherwise); the Follower adopts its
reaction function from above in Stage 3. With these strategies, neither player has an incentive
to deviate. The Follower’s reaction function was derived as a best response to the output of
the Leader; the Leader cannot deter entry, so the Stackelberg output is a profit-maximising
output; and the Follower has an incentive to enter in Stage 1 because it earns positive profits.
In this case, the Follower’s ability to commit to entry before the leader chooses output is very
important. This removes the possibility of entry deterrence for the Leader and ensures the
Stackelberg outcome results.
3. Firm 1 is a monopoly manufacturer of Bam. Bam is a homogeneous product with market demand
given by Q(p) = 110−P, where P is the final retail price. Production of Bam involves constant
marginal costs of 20 and no fixed costs. Distribution of Bam incurs a constant marginal cost of 10
and no fixed costs. The following two stage game is played once. In stage 1, Firm 1 chooses a
price PU to sell its product to distributors. In stage 2, distributors choose a retail price to charge to
customers.
(a) Suppose that Firm 2 is the sole distributor of Bam.
i. Find the reaction function for Firm 2 in terms of the upstream price PU . [3 marks]
ANS: Firm 2 has profits of
piD = Q(PD)(PD−PU −10) = (110−PD)(PD−PU −10)
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ECOS3005 Practice exam solutions
Maximising profits leads to the first order conditions:
−PU +PU +10+110−PD = 0
2PD = 120+PU
PD = 60+
PU
2
This is Firm 2’s reaction function.
ii. Solve for the subgame perfect Nash equilibrium to the game. What are the upstream and
downstream prices? How many units of Bam are sold? [3 marks]
ANS: We solve by backward induction. In stage 2, Firm 2 adopts the above reaction
function. In stage 1, Firm 1 maximises profits, taking Firm 2’s reaction function into
account. Firm 1 profits are
piU = Q(PD(PU))(PU −20) = (110−60− P
U
2
)(PU −20)
= (50− P
U
2
)(Pu−20)
Maximising profits leads to the first order conditions:
50− P
U
2
− 1
2
(PU −20) = 0
PU = 60
Substituting into Firm 2’s reaction function,
PD = 60+
PU
2
= 90.
So, in the SPNE, Firm 1 sets a price of PU = 60 and Firm 2 uses the above reaction
function, leading to a downstream price of PD = 90 in equilibrium. Output is given by
Q= 110−PD = 20.
iii. Firm 1 is considering the idea of taking over Firm 2. How much would Firm 1 be willing
to pay for Firm 2? Explain. [3 marks]
ANS: Firm 1 currently earns profits of
piU = Q(PU −20) = 20×40 = 800.
Suppose the firms were to integrate. Then the integrated firm would maximise profits
given by
piI = (110−P)(P−30)
This leads to first order conditions
110−P−P+30 = 0
P= 70
The integrated firm would set a price of 70 and earn profits of piI = 40× 40 = 1600.
Hence, Firm 1 would be willing to pay up to piI−piU = 1600−800 = 800 for Firm 2.
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ECOS3005 Practice exam solutions
iv. Would Firm 2 be willing to accept a take-over by Firm 1? Explain. [2 marks]
ANS: Firm 2 would have to be compensated for current profits of
piD = Q(PD−PU −10) = 20(90−60−10) = 400.
Therefore, Firm 2 should be happy to accept any offer of at least 400.
(b) Suppose that Firms 2 and 3 are both distributors of Bam. In stage 2, they compete for final
customers by simultaneously choosing prices. [Assume that Firm 1 must charge the same
price PU to both firms.] Find the subgame perfect Nash equilibrium to the game. What price
does each firm charge? Explain. [4 marks]
ANS: Because Bam is a homogeneous product, Firms 2 and 3 engage in Bertrand competition.
The only Nash equilibrium to this subgame involves both firms setting price equal to their
marginal cost. Thus, P2 = P3 = PU + 10. We could think of this as the reaction function of
Firm 2 and Firm 3. Firm 1 then maximises profits:
pi1 = (110−PD)(PU −20) = (100−PU)(PU −20)
Solving first order conditions yields
100−PU −PU +20 = 0
PU = 60.
Hence, the SPNE involves Firm 1 setting a price of 60 and Firms 2 and 3 following the above
reaction function. In equilibrium Firms 2 and 3 will set a price of 70.
Intuitively, because Firm 2 and Firm 3 behave competitively in stage 2, there is no double
marginalisation problem and we have the same outcome as the integrated monopoly problem.
4. Good A is a homogeneous product. Entry into the market for Good A is free. Each producer of
Good A has an identical U-shaped average cost curve given by C(q) = q
2
4 + 5q+ 25. L = 1000
consumers are interested in buying Good A. Each consumer is willing to buy up to one unit of
Good A for a price up to pu = 12. A fraction α of consumers have perfect information about
prices, where 0 ≤ α ≤ 1. The remaining consumers know the distribution of prices, but do not
know the price set by any individual store. These uninformed consumers arrive at one store at
random and learn the price of that store. If they wish to go to another store, they incur a search cost
of c = 3 to find the store and learn its price. Let pc be the price that would prevail in the market
for Good A in the long run if it were a perfectly competitive market (that is, if all consumers were
perfectly informed about prices).
(a) Find the competitive price pc. [3 marks]
ANS: In the long run competitive equilibrium, all firms set a price equal to marginal cost.
Free entry ensures that firms earn zero economic profits in the long run. Therefore, we must
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ECOS3005 Practice exam solutions
have marginal cost equal to average cost in equilitbium. Therefore,
MC = 5+
q
2
= AC = 5+
q
4
+
25
q
q
4
=
25
q
q2 = 100
q= 10
Each firm produces an output of 10. At this quantity, marginal costs are given by 5+1/2= 10.
Therefore, pc = 10.
(b) Suppose that α= 0. That is, all consumers have imperfect information about prices.
i. Is there a Nash equilibrium in which all firms set price p= pc? Explain.
[3 marks]
ANS: No. If each store were to set a price equal to p = pc, firms have an incentive to
raise price to pu. At this price, consumers have no incentive to visit another store (notice
that pc+ c > pu). Thus, the firm would not lose any sales while maximising its profit
margin.
ii. Is there a Nash equilibrium in which all firms set a price p= pu? Explain.
[3 marks]
ANS: Yes. Suppose that each store set price p= pu, and examine the incentives of one of
those stores. There is no incentive to undercut by a small amount (less than c), because
this does not encourage consumers to search. Firms may be able to encourage consumers
to search only if p < pu− c. However, pu− c < pc and hence firms would make a loss
trying to attract customers.
(c) Suppose that α= 0.1.
i. Is there a Nash equilibrium in which all firms set price p= pu? Explain.
[2 marks]
ANS: No. If all firms set a price of pu, there would be an incentive to undercut slightly
to attract all of the perfectly informed consumers.
ii. Is there a long run equilibrium in which all firms set a price p= pc? Explain.
[4 marks]
ANS: Recall from part 4a that pc = 10. At this price, firms are setting price equal to
marginal cost and earning zero economic profits. Each firm produces an output of q= 10.
Let us consider the incentive to deviate. First, note that it is not worth undercutting this
price. This will lead to losses. If a firm sets a price above pc, they will lose all of the
informed customers, but will keep their share of the uninformed customers as long as
they do not have an incentive to search. Notice that pc + c > pu and the uninformed
consumers would not have an incentive to search if they arrived at a store charging pu.
We just have to check whether it is profitable to deviate to pu. It will be profitable if
pu > AC(qu), where qu is the output of a firm setting price pu. If all other firms set a
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ECOS3005 Practice exam solutions
price of pc and produce q = 10 units, then there will be n = 100 firms. The deviating
firm would obtain an output of
qu =
1000(1−α)
n
= 10(1−α) = 9.
Average costs are then
AC(qu) = 5+
q
4
+
25
9
= 5+
9
4
+
25
9
= 5+
81+100
36
= 10+
1
36
< 12.
Therefore, it is profitable to deviate to a price of pu and there can be no long run equilib-
rium in which all firms set a price of pc.
END OF EXAMINATION
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