## 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.

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