Problem set on Collusion
1. Collusion between two °rms is possible no matter what their marginal cost di/erence
is.
True or False. Explain.
Answer. False. If °rms±costs are very asymmetric it may not be posible to allocate
shares so that °rms don±t want to cheat. The problem is that a very low cost °rm will
need a very large share which leaves too little for the other °rms.
2. Explain why a long detection lag (the time it takes for a °rm to detect a deviation by
other °rms) may limit the possibilities of collusion. Relate your answer to the model
of collusion discussed in class.
Answer. A long detection lag results in a long time between a °rm±s deviation and the
punishment. This will increase the time the °rm enjoys the higher pro°ts from deviation
so it increases the gains from deviation and reduces the cost of the punishment. This
is like raising the discount factor
R
.
3. Collusion is more likely in industries where individual sales are large and take place
ocassionaly than in industries where individual sales are small but frequent.
True or
False. Explain.
Answer. False. If sales are large and take place ocassionaly, it is like in the previous
question where the immediate gains from deviating are large and the punishment takes
time to come. This makes collusion harder.
4.
r
= 1%
; c
1
= 5
; c
2
= 0
;
demand function
Q
= 10
°
p:
It is not possible to have collusion
where °rm 1 gets a positive share.
True or False. Explain.
Answer. No, it is not possible. The monopoly price for °rm 2 is 5 and at this price
the other °rm is out of the market.
So in the Bertrand equilibrium °rm 2 gets the
maximum it could ever get under collusion  it obviously will not be willing to share
this with °rm 1.
5. The demand function is given by
p
= 100
°
Q
and the constant marginal cost of produc
tion is denoted by
c
= 20
:
Assume °rms engage in Bertrand competition. There are 16
°rms in the industry. If these °rms could collude, they would set the monopoly price
and equally share the monopoly output. By deviating from this collusive agreement, a
°rm undercuts the rest by a tiny bit and gets all the market. Assume that in this case
the °rm gets during the period of undercutting the full monopoly output at essentially
the monopoly price (this corresponds to an in°nitesimally small price reduction) and
thus the total pro°ts of a monopolist. The °rms will confront each other for in°nite
time and the interest rate is
r:
(a) If the rate of interest in this market is 10%, could you support cooperation at the
monopoly level with trigger strategies?
Answer. It can easily be checked that the monopoly price is 60 and total output
is 40.
Each °rm gets a share 1/N of this market, giving pro°ts (40/N)
±
40 =
1600
=N
= 100
:
A °rm that deviates takes away the share of the remaininig 15
°rms, so the gain from deviating is
1500
:
The cost of deviating is
100
=r
= 1000
<
1500
;
so °rms will deviate.
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 Spring '07
 Hopenhayn
 Economics, Game Theory, Oligopoly, Collusion, Bertrand competition

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