The monopoly price in both states of demand
TUTORIALS – SOLUTIONS
Problem 3.3
a) The monopoly price is identical in both states of demand, i.e. p = 10. The Nash equilibrium of the static price game is pF = pM = 0. The following trigger strategy for each firm can induce collusion:
Charge p = 10 in the current period as long as the history up to this period only contains prices p = 10, otherwise charge p = 0.
Note first, that since pF = pM = 0 is the NE of the stage game, these prices will also form a NE in the deviation subgame. In the collusion subgame we have to distinguish between periods with high demand and with low demand. First calculate the expected per-period profits under collusion before firms observe in which demand state they are in,
⇧
e i =
1
4
1200
2
+
3
4
2000
2
= 900
In a high-demand state/subgame, playing p = 10 yields
2000
2
+ �⇧
e i + �
2 ⇧
e i + … = 1000 +
900�
1 � � ,
deviation to a slightly lower price than 10 would yield 2000 but trigger the Nash equilibrium afterwards,
2000 + �0 + �
2 0 + … = 2000.
Therefore, collusion is sustainable in the high-demand state if
1000 +
900�
1 � � � 2000
or � �
10
19
‘ 0.526
In a low-demand state/subgame, playing p = 10 yields
1200
2
+ �⇧
e i + �
2 ⇧
e i + .. = 600 +
900�
1 � � ,
deviation to a slightly lower price than 10 would yield
1200 + �0 + �
2 0 + … = 1200.
Therefore, collusion is sustainable in the low-demand state if
600 +
900�
1 � � � 1200
12
TUTORIALS – SOLUTIONS
or � �
2
5
Hence, collusion is easier to sustain in the low-demand state. Putting the two conditions together yields that the specified trigger strategy equilibrium holds if
� � 10
19
b) The following trigger strategy for each firm induces collusion in both markets
Charge pE = 10 in the electricity market and pG = 6 in the gas as long as the history up to the current only contains prices of pE = 10 and pG = 6 in the respective market, otherwise charge pG = pE = 0 in both markets.
With this strategy, any deviation in one of the two markets will be punished by the static Nash equilibrium in both markets. The optimal deviation for each firm is to deviate in both markets at a time. This leads to a pooling of the incentive conditions for collusion. In a high-demand period in the electricity market, playing pE = 10 and pG = 6 yields
2000
2
+
900�
1 � � +
600
2
+
300�
1 � �
deviation to a slightly lower price in both markets would yield
2000 + 600 = 2600
Therefore, collusion is sustainable in the high-demand state if
1300 +
1200�
1 � � � 2600
or � �
13
25
= 0.52
In a low-demand period in the electricity market, playing pE = 10 and pG = 6 yields
1200
2
+
900�
1 � � +
600
2
+
300�
1 � �
deviation to a slightly lower price in both markets would yield
1200 + 600 = 1800
Therefore, collusion is sustainable in the low-demand state if
900 +
1200�
1 � � � 1800
13
TUTORIALS – SOLUTIONS
or � �
3
7
‘ 0.428
Therefore, collusion is sustainable in both states of demand if
� � 13
25
= 0.52
which is a lower threshold than the one in a). Hence, multi-market contact increases the sustainability of collusion. The reason for this is that firms can pool their collusion and deviation profits of the two markets. This helps if the two markets are different. In the gas market alone firms could have colluded for all � > 1/2 (check or see example in lecture). In the electricity market, collusion in the high-demand state was possible for all � > 0.526. This means for 0.5 < � < 0.526 firms earn more than enough collusion profits to sustain collusion in the gas market but they are unable to sustain collusion in the electricity market. When firms are present in both markets, they add up the incentive constraints in the gas and electricity market. This allows them to use some of the excessive collusion rent they earn in the gas market to prevent deviation in the electricity and gas market. Or in other words, by pooling the incentive conditions, deviation in the electricity market becomes less attractive because it loses valuable collusion in the gas market.
14
TUTORIALS – SOLUTIONS
Problem 3.4
a) The monopoly price is identical in both states of demand, i.e. p = 110. The Nash equilibrium of the static price game is pi = 10, 8i = 1, 2, 3. The following trigger strategy for each firm can induce collusion:
Charge p = 110 in the current period as long as the history up to this period
only contains prices p = 110, otherwise charge p = 10.
Since pi = 10 is the NE of the stage game, these prices form a NE in the deviation subgame. In the collusion subgame one has to distinguish between periods with high demand and with low demand. First calculate the expected per-period profits under collusion in the future. Since firms don’t know the demand level in advance they expect
⇧
e i =
1
2
60(110 � 10) 3
+
1
2
30(110 � 10) 3
= 1500
In a high-demand collusion subgame, setting p = 110 yields
60(110 � 10) 3
+
1500�
1 � � = 2000 +
1500�
1 � � ,
deviation to a slightly lower price than 110 would yield 60(110�10) = 6000 but trigger the static Nash equilibrium afterwards. Therefore, collusion is sustainable in the high-demand state if
2000 +
1500�
1 � � � 6000
or � �
8
11
‘ 0.727
Collusion is sustainable in a low-demand collusion subgame if
1000 +
1500�
1 � � � 3000
or � �
4
7
‘ 0.57
The specified trigger strategy equilibrium holds if
� � 8
11
b) The following trigger strategy for each firm is considered:
As long as the history up to this period only contains prices of 110 in low
demand periods and prices p in high demand periods, charge p = 110 if the
current demand is low or charge p if current demand is high; otherwise charge
p = 10.
15
TUTORIALS – SOLUTIONS
The expected per-period profits of future collusion are
⇧
e i =
1
2
60(p � 10) 3
+
1
2
30(110 � 10) 3
= 10(p � 10) + 500
In a high-demand collusion subgame, setting price p yields
60(p � 10) 3
+ [10(p � 10) + 500] 2 3
1 � 2 3
= 20(p � 10) + 20(p � 10) + 1000 = 40(p � 10) + 1000.
Deviation to a slightly lower price than p would yield approximately 60(p � 10) but trigger the static Nash equilibrium afterwards. Therefore, collusion is sustainable in the high- demand state if
40(p � 10) + 1000 � 60(p � 10).
or p 60.
16
TUTORIALS – SOLUTIONS
c) The trigger strategy here is the same as in (a). First consider the high demand collusion subgame. The expected continuation profit after setting the monopoly price p = 110 is
V
H = �
1000
1 � �2 + �
2 2000
1 � �2
=
5
6
1000
1 � (5 6 )
2 + (
5
6
)
2 2000
1 � (5 6 )
2
=
5
6
36
11
1000 +
25
36
36
11
2000
=
30000
11
+
50000
11
=
80000
11
‘ 7272.72
Hence, collusion is sustainable since
2000 +
80000
11
� 6000
Now consider the low demand collusion subgame. The expected continuation profit after setting the monopoly price p = 110 is
V
L = �
2 1000
1 � �2 + �
2000
1 � �2
=
25
36
1000
1 � (5 6 )
2 +
5
6
2000
1 � (5 6 )
2
=
25
36
36
11
1000 +
5
6
36
11
2000
=
25000
11
+
60000
11
=
85000
11
‘ 7727.27
Hence, collusion is sustainable since
1000 +
85000
11
� 3000
Overall, the trigger strategies are sustainable in this problem.
17