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Managerial Economics, SS14
Second Homework Solution
(1) Bundling
Kebap price
Coke price
P
e10
e5
e7
e0
e2
e2.5
e10
e7
e9.5
e3
e1
e4
A
B
C
MC
Optimal separate prices1
Kebaps:
• PK = 10 → πK (10) = 10 · (10 − 3) = 70
• PK = 7 → πK (7) = 20 · (7 − 3) = 80
• PK = 5 → πK (5) = 30 · (5 − 3) = 60
Cokes:
• PC = 2.5 → πC (2.5) = 10 · (2.5 − 1) = 15
• PC = 2 → πC (2) = 20 · (2 − 1) = 20
• PC = 0 → πC (0) = 30 · (0 − 1) = −30
max
=⇒ πK
= 80
=⇒ πCmax = 20
Given the WTP’s of the three costumer segments, the highest possible profit selling Kebaps and
max
Cokes separately is e100 (= πK
+ πCmax ).
Optimal pure bundle prices
• PK,C = 10 → 10 · (10 − 4) = 60
• PK,C = 9.5 → 20 · (9.5 − 4) = 110
• PK,C = 7 → 30 · (7 − 4) = 90
max
=⇒ πK,C
= 110
The optimal pure bundle price is e110.
(2) Price leadership
a) The total supply curve for the follower firms is obtained by horizontally aggregating their
marginal cost curves. First we rewrite the M C curves as follows:
M C(qi ) = 20 + 5qi = P
5qi = −20 + P
1
qi = −4 + P
5
The total supply curve is thus
5
X
1
Qf =
qi = 5 · −4 + P
5
i=1
= −20 + P
b) The dominant firm’s residual demand curve is given by the difference between the market
demand curve and the total supply curve of the follower firms:
Qd = 400 − 2P − (−20 + P ) = 420 − 3P
1
Note that potential profits are calculated by multiplying the profit margin P −M C by the number of costumers
who are willing to buy the product at the respective price (i.e. whose willingness-to-pay exceeds the price).
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Managerial Economics, SS14
Second Homework Solution
c) The dominant firm will set marginal revenue equal to marginal cost. First we have to rewrite
the dominant firm’s demand curve as P = 140− 31 Q, marginal revenue is thus M R = 140− 32 Q.
Hence,
2
140 − Q = 20,
3
Q∗d = 180,
Pd∗ = 140 −
1
· 180 = 80
3
d) Each follower firm will charge the same price as the dominant firm, Pf = Pd = 80. The total
output produced by the follower firms is Qf (80) = −20 + 80 = 60. Since they are identical,
each firm will therefore produce 12 (= 60/5) units of output.
(3) Oligopoly I
a) Firm a faces the following maximization problem:
πa = T Ra − T Ca = 300qa − 3qa2 − 3qa qb − (30qa + 1.5qa2 )
max
q
(1)
a
The first-order condition for qa yields firm a’s reaction function:
∂πa
= 270 − 9qa − 3qb = 0
∂qa
1
qa (qb ) = 30 − qb
3
(2)
By symmetry, the reaction function of firm b is
1
qb (qa ) = 30 − qa
3
(3)
b) If firm a produces 9 units, firm b will produce qb (9) = 30 − 13 · 9 = 27 units.
c) Solving the reaction functions simultaneously by substituting (2) into (3) yields
qb = 30 −
1
1
1
30 − qb = 20 + qb
3
3
9
∗
qb = 22.5
qa∗ = 22.5
Since qa = qb , the profit function in (1) can be rewritten as
πi = 270qi − 7.5qi2
∀i ∈ {a, b}
Thus, each firm’s profit is πi = 270 · 22.5 − 7.5 · 22.52 = $2,278.13.
d) If firms decide to collude, they maximize joint (monopoly) profits. Because both firms are
identical, each produces Q/2 units. The cartel’s profit function therefore reads
Π(Q) = P (Q)Q − 2C
Q
2
2
" Q
= 300Q − 3Q − 2 30
2
= 270Q − 3.75Q
2
+ 1.5
Q 2 #
2
(4)
2
Managerial Economics, SS14
Second Homework Solution
Thus, the first-order condition for joint profit-maximization is
∂Π(Q)
= 270 − 7.5Q = 0
∂Q
QC = 36
Substituting QC = 36 into the profit-function given by (4) yields a joint-profit of $4,860,
of which half is earned by the single firms (πaC = πbC = $2,430). Each firm can therefore
improve its profit by $151.87 through colluding.
e) If firms engage in price competition, each price above marginal cost will be underbid by the
rival. Profits and quantity produced will be lower, whereas the price will be higher compared
to the Cournot-Nash equilibrium.
Using the demand and cost functions given in this example, the Bertrand equilibrium is
found where qaB = qbB = 30 and P B = 120. The profit of each firm is πiB = 1,350.
(4) Oligopoly II
P = 10 − qx − qy
ACx = 5
ACy = 2qy
−→
−→
T Cx = ACx qx = 5qx
T Cy = ACy qy = 2qy2
−→
−→
M Cx = ∆ACx /∆qx = 5
M Cy = ∆ACy /∆qy = 4qy
Under Bertrand competition, the market equilibrium is found where price equals marginal cost,
because every price above will be underbid by the rival:
P = M Cx
10 − qx − qy = 5
qx = 5 − q y
P = M Cy
10 − qx − qy = 4qy
qy = 2 − 0.2qx
(5)
(6)
Solving the system of reaction functions by plugging (5) into (6) yields
qy = 2 − 0.2 · (5 − qy )
qy = 1.25 =⇒ qx = 5 − 1.25 = 3.75
The price will end up at P = 10 − 1.25 − 3.75 = $5. The firms’ profits are therefore
πy = P qy − T Cy = 5 · 1.25 − 2 · 1.252
πy = $3.125
πx = P qx − T Cx = 5 · 3.75 − 5 · 3.75
πx = $0
If firm y was able to push its rival out of the market entirely, e.g. through rent-seeking activities
such as bribery, it would serve the market as a monopolist. The monopoly equilibrium is found
where M R = M C, resulting in a quantity of QM = 35 and a price of P M = 25
. Firm y’s profit
3
would then be (coincidentally)
π
M
2
25 5
5
=
· −2·
3 3
3
=
25
3
Compared to the Bertrand equilibrium, profits could be improved by approximately $5.21.
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Managerial Economics, SS14
Second Homework Solution
(5) Game theory I
a) A dominant strategy is a strategy that earns a player a larger pay-off than any other strategy,
regardless of the strategies played by other players. In this example, neither player has a
dominant strategy.
b) A Nash equilibrium is a situation where no player has an incentive to unilaterally change his
strategy. In this example, there are two Nash equilibria; one where firm A enters but firm
B doesn’t – {10,0} –, and one where firm B enters but firm A doesn’t – {0,5}.
c) It is not obvious at first sight what the outcome of this game will be. However, we see that
firm A has higher potential pay-offs from entering than firm B, so the equilibrium is most
likely {10,0} .
(6) Game theory II
Suppose each decision node is a day of the week. On the third day (the decision node on the right),
firm K decides between abiding (a) and cheating (c). K will choose c, because it yields a higher
pay-off compared to a (3 > 2). On day 2, firm L therefore expects a pay-off of 3 if it abides (α)
by the agreement, because it anticipates firm K to cheat the day after. Hence, firm L will cheat
also on day 2, since 4 > 3. On day 1, firm K’s pay-off is 0 when it expects firm L to cheat on day
2, thus it will choose to cheat on day 1 as well (1 > 0). The outcome of this game is therefore
that firm K will already cheat on day 1 if it knows that it will cheat sometime in the future – the
cartel will therefore break down.
Two crucial assumptions have been made here: (1) players act rationally, and (2) both players
correctly anticipate all actions of the other player.
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