Microeconomics: Collusion & Cartel Policy (Answers included! Just need answer explanation help)

TUTORIALS - SOLUTIONS

Section 3 - Collusion and Cartel Policy

Problem 3.1

a) The monopolist’s problem is

max(p1,p2) ⇧ M

= p1(1 � p1 + 1

2

p2) + p2(1 � p2 + 1

2

p1)

The first partial derivatives are

@⇧

M

@p1 = 1 � p1 +

1

2

p2 � p1 + 1

2

p2 = 0

@⇧

M

@p2 = 1 � p2 +

1

2

p1 � p2 + 1

2

p1 = 0

Solving simultaneously gives p

M 1 = p

M 2 = 1

b) Consider the following trigger strategies for each firm:

Play pMi in each period if the history in this period only contains prices p M i from

the rival; otherwise, play pCi

This strategy gives rise to two distinct subgames, those with a deviation and those without deviation in the history. In a deviation subgame, (pCi , p

C i ) is a NE because no firm can do

better given the other is playing pCi for the rest of the game. In the collusion subgame, playing pMi in the current period gives a profit of

⇧

M i + �⇧

M i + �

2 ⇧

M i + ... =

⇧

M i

1 � � .

The best deviation gives

⇧

D i + �⇧

C i + �

2 ⇧

C i + ... = ⇧

D i + �

⇧

C i

1 � � .

To calculate ⇧Di , we need to find a firm’s best deviation given the rival prices at p M

= 1. Plugging pM = 1 in the reaction function calculated is A.5b), gives the best deviation price

p

D i =

1

2

+

1

4

p

M =

3

4

.

With this price the deviator has a demand of 3/4 and deviation profits are

⇧

D i = 9/16.

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TUTORIALS - SOLUTIONS

Collusion is profitable if ⇧

M i

1 � � � ⇧Di + �

⇧

C i

1 � � .

Substituting all values gives

1

2(1 � �) �

9

16

+ �

4

9(1 � �)

or � �

9

17

⇡ 0.529

If this condition holds, then (pMi , p M i ) is a NE equilibrium of the collusion subgame and the

defined trigger strategies are a subgame perfect equilibrium of the repeated game.

Problem 3.2

a) Firms are symmetric and have constant marginal cost. Thus, the industry-profit maxi- mizing total quantity coincides with the monopoly solution. Therefore, they maximize

⇧(Q) = (60 � Q/2 � 12)Q

which yields 48 � Q = 0 () QM = 48

i.e. each firm produces qCi = 16 and the market price is p = 60 � 48/2 = 36 and profits are ⇧Ci = (36 � 12) ⇤ 16 = 384. In the static Nash equilibrium each firm maximises its own profits

⇧i(qi, Q�i) = (60 � (qi, Q�i)/2 � 12)qi

where Q�i is the sum of production of firm i’s rivals. Maximising wrt to qi yields

48 � Q�i/2 � qi () Ri = 48 � Q�i/2

Imposing symmetry qi = qj one gets qNEi = 24 with an equilibrium price of p = 60 � 3 ⇤ 24/2 = 24 and profits of ⇧NEi = (24 � 12) ⇤ 24 = 288. Finally, if a firm deviates from the collusive quantity qCi = 16 it optimally deviates to Ri = 48 � 2qCi /2 = 32 which implies a price of (60 � 64/2) = 28 and a profit of

⇧

D i = (28 � 12) ⇤ 32 = 512

To make sure firms stick to their collusive quantities it has to hold that

⇧

C i

1 � � � ⇧Di +

�⇧

NE i

1 � �

10

TUTORIALS - SOLUTIONS

or � �

⇧

D i � ⇧Ci

⇧

D i � ⇧NEi

=

512 � 384 512 � 288

=

128

224

=

4

7

' 0.5714

b) A one period lag means firms can deviate from the collusive quantity without being detected in the current period and being punished in the next period. This means firms can effectively deviate in two consecutive periods. Firms can use the same trigger strategies as above. Collusion is sustainable in the collusion subgame if

⇧

C i

1 � � � ⇧Di (1 + �) +

�

2 ⇧

NE i

1 � �

or

� �

s ⇧

D i � ⇧Ci

⇧

D i � ⇧NEi

=

r 4

7

' 0.756

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