Collusion and Cartels

1
Collusion and Cartels
2
Collusion and Cartels

• What is a cartel?
• attempt to enforce market discipline and reduce
  competition between a group of suppliers
• cartel members agree to coordinate their actions
• prices
• market shares
• exclusive territories
• prevent excessive competition between the cartel
  members

3
Collusion and Cartels

• Cartels have always been with us
• electrical conspiracy of the 1950s
• Alitalia
• Some are explicit and difficult to prevent
• OPEC
• De Beers

4
Collusion and Cartels

• Other less explicit attempts to control
  competition
• formation of producer associations
• publication of price sheets
• peer pressure
• violence
• Cartel laws make cartels illegal in the US and
  Europe
• Authorities continually search for cartels
• Have been successful in recent years
• Nearly 1 billion in fines in 1999 (USA)

5
Recent cartel violations
6
Recent cartel violations 2

• Fines steadily grew during the 1990s

7
Collusion and Cartels

• What constrains cartel formation?
• they are generally illegal
• per se violation of anti-trust law in US and EU
• substantial penalties if prosecuted
• cannot be enforced by legally binding contracts
• the cartel has to be covert
• enforced by non-legally binding threats or
  self-interest
• cartels tend to be unstable
• there is an incentive to cheat on the cartel
  agreement
• MC gt MR for each member
• cartel members have the incentive to increase
  output
• OPEC until very recently

8
The Incentive to Collude

• Is there a real incentive to belong to a cartel?
• Is cheating so endemic that cartels fail?
• If so, why worry about cartels?
• Simple reason
• without cartel laws legally enforceable contracts
  could be written by cartel members
• De Beers is tacitly supported by the South
  African government
• gives force to the threats that support this
  cartel
• not to supply any company that deviates from the
  cartel
• Without contracts the temptation to cheat can be
  strong

9
The Incentive to Cheat

• Take a simple example
• two identical Cournot firms making identical
  products
• for each firm MC 30
• market demand is P 150 Q where Q is in
  thousands
• Q q1 q2

Price
150
Demand
30
MC
Quantity
150
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The Incentive to Cheat
Profit for firm 1 is p1 q1(P - c)
q1(150 - q1 - q2 - 30)
q1(120 - q1 - q2)
Solve this for q1
To maximize, differentiate with respect to q1
?p1/?q1
120
- 2q1
- q2
0
q1 60 - q2/2
This is the best response function for firm 1
The best response function for firm 2 is then
q2 60 - q1/2
11
The Incentive to Cheat
These best response functions are easily
illustrated
q2
q1 60 - q2/2
q2 60 - q1/2
120
Solving these gives the Cournot-Nash outputs
R1
qC1 qC2 40 (thousand)
60
The market price is
C
40
PC 150 - 80 70
R2
Profit to each firm is
q1
p1 p2 (70 - 30)x40 1.6 million
120
60
40
12
The Incentive to Cheat (cont.)
What if the two firms agree to collude?
They will agree on the monopoly output
q2
This gives a total output of 60 thousand
120
Each firm produces 30 thousand
Price is PM (150 - 60) 90
R1
Profit for each firm is
60
p1 p2 (90 - 30)x30 1.8 million
C
40
30
R2
q1
120
60
40
30
13
The Incentive to Cheat (cont.)
Both firms have an incentive to cheat on their
agreement
If firm 1 believes that firm 2 will produce 30
units then firm 1 should produce more than 30
units
q2
Cheating pays!!
120
Firm 1s best response is
qD1 60 - qM2/2 45 thousand
R1
Total output is 45 25 70 thousand
Price is PD 150 - 75 75
60
C
Profit of firm 1 is (75 - 30)x45 2.025
million
40
30
R2
Profit for firm 2 is (75 - 30)x25 1.35
million
q1
40
30
120
60
45
14
The Incentive to Cheat (cont.)
Both firms have the incentive to cheat on their
agreement
Firm 2 can make the same calculations!
This gives the following pay-off matrix
Firm 1
Cooperate (M)
Deviate (D)
This is the Nash equilibrium
Cooperate (M)
(1.8, 1.8)
(1.35, 2.025)
Firm 2
(1.6, 1.6)
Deviate (D)
(2.035, 1.35)
(1.6, 1.6)
15
The incentive to cheat (cont.)

• This is a prisoners dilemma game
• mutual interest in cooperating
• but cooperation is unsustainable
• However, cartels do form
• So there must be more to the story

16
Cartel Stability

• The cartel in our example is unstable
• This instability is quite general
• Can we find mechanisms that give stable cartels?
• violence in one possibility!
• are there others?
• must take away the temptation to cheat
• staying in the cartel must be in a firms
  self-interest
• Suppose that the firms interact over time
• Then it might be possible to sustain the cartel
• Make cheating unprofitable
• Reward good behavior
• Punish bad behavior

17
Repeated Games

• Formalizing these ideas leads to repeated games
• a firms strategy is conditional on previous
  strategies played by the firm and its rivals
• In the example cheating gives 2.025 million
  once
• But then the cartel fails, giving profits of 1.6
  million per period
• Without cheating profits would have been 1.8
  million per period
• So cheating might not actually pay
• Repeated games can become very complex
• strategies are needed for every possible history
• But some rules of the game reduce this
  complexity
• Nash equilibrium reduces the strategy space
  considerably
• Consider two examples

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Example 1 Cournot duopoly
The pay-off matrix from the simple Cournot game
Firm 1
Cooperate (M)
Deviate (D)
Cooperate (M)
(1.8, 1.8)
(1.35, 2.025)
Firm 2
(1.6, 1.6)
Deviate (D)
(2.025, 1.35)
(1.6, 1.6)
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Example 2 A Bertrand Game
Firm 1
105
130
160
(7.3125, 7.3125)
(7.3125, 7.3125)
(8.25, 7.25)
(9.375, 5.525)
105
(8.5, 8.5)
130
(8.5, 8.5)
(7.25, 8.25)
(10, 7.15)
Firm 2
(5.525, 9.375)
160
(7.15, 10)
(9.1, 9.1)
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Repeated Games (cont.)

• Time matters in a repeated game
• is the game finite? T is known in advance
• Exhaustible resource
• Patent
• Managerial context
• or infinite?
• this is an analog for T not being known each
  time the game is played there is a chance that it
  will be played again

21
Repeated Games (cont.)

• Take a finite game Example 1 played twice
• A potential strategy is
• I will cooperate in period 1
• In period 2 I will cooperate so long as you
  cooperated in period 1
• Otherwise I will defect from our agreement
• This strategy lacks credibility
• neither firm can credibly commit to cooperation
  in period 2
• so the promise is worthless
• The only equilibrium is to deviate in both
  periods

22
Repeated Games (cont.)

• What if T is large but finite and known?
• suppose that the game has a unique Nash
  equilibrium
• the only credible outcome in the final period is
  this equilibrium
• but then the second last period is effectively
  the last period
• the Nash equilibrium will be played then
• but then the third last period is effectively the
  last period
• the Nash equilibrium will be played then
• and so on
• The possibility of cooperation disappears
• The Selten Theorem If a game with a unique Nash
  equilibrium is played finitely many times, its
  solution is that Nash equilibrium played every
  time.
• Example 1 is such a case

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Repeated Games (cont.)

• How to resolve this? Two restrictions
• Uniqueness of the Nash equilibrium
• Finite play
• What if the equilibrium is not unique?
• Example 2
• A good Nash equilibrium (130, 130)
• A bad Nash equilibrium (105, 105)
• Both firms would like (160, 160)
• Now there is a possibility of rewarding good
  behavior
• If you cooperate in the early periods then I
  shall ensure that we break to the Nash
  equilibrium that you like
• If you break our agreement then I shall ensure
  that we break to the Nash equilibrium that you do
  not like

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Example 2 / 1

• Consider the following pricing game

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Example 2 /2

• There are two Nash equilibria

Both agree that this is bad
Both agree that this is good
(7.31, 7.31)
Both agree that this is best
(8.5, 8.5)
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A finitely repeated game

• Assume that the discount rate is zero (for
  simplicity)
• Assume also that the firms interact twice
• Suggest a cartel in the first period and good
  Nash in the second
• Set price of 160 in period 1 and 130 in period
  2
• Present value of profit from this behavior is
• PV2(p1) 9.1 8.5 17.6 million
• PV2(p2) 9.1 8.5 17.6 million
• What credible strategy supports this equilibrium?
• First period set a price of 160
• Second period If history from period 1 is (160,
  160) set price of 130, otherwise set price
  of 105.

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A finitely repeated game

• These strategies reflect historical dependence
• each firms second period action depends on the
  history of play
• Is this really a Nash subgame perfect
  equilibrium?
• show that the strategy is a best response for
  each player

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A finitely repeated game

• This is obvious in the final period
• the strategy combination is a Nash equilibrium
• neither firm can improve on this
• What about the first period?
• why doesnt one firm, say firm 2, try to improve
  its profits by setting a price of 130 in the
  first period?

Defection does not pay in this case!

• Consider the impact
• History into period 2 is (160, 130)
• Firm 1 then sets price 105
• Firm 2s best response is also 105 Nash
  equilibrium
• Profit therefore is PV2(p1) 10 7.3125
  17.3125 million
• This is less than profit from cooperating in
  period 1

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Example 2/3

• Best response to a price of 160 is 130
• This gives profit of 10
• So deviation is period 1 gives the profit stream
  10 7.31 17.31
• Cooperation in period 1 gives the profit stream
  9.1 8.5 17.6
• Undercutting does not pay

30
A finitely repeated game

• Defection does not pay!
• The same applies to firm 1
• So we have credible strategies that partially
  support the cartel
• Extensions
• More than two periods
• Same argument shows that the cartel can be
  sustained for all but the final period strategy
• In period t lt T set price of 160 if history
  through t 1 has been (160, 160) otherwise set
  price 105 in this and all subsequent periods
• In period T set price of 130 if the history
  through T 1 has been (160, 160) otherwise set
  price 105
• Discounting

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A finitely repeated game

• Suppose that the discount factor R lt 1
• Reward to good behavior is reduced
• PVc(p1) 9.1 8.5R
• Profit from undercutting in period 1 is
• PVd(p1) 10 7.3125R
• For the cartel to hold in period 1 we require R gt
  0.756 (discount rate of less than 32 percent)
• Discount factors less than 1 impose constraints
  on cartel stability
• But these constraints are weaker if there are
  more periods in which the firms interact

32
A finitely repeated game

• Suppose that R lt 0.756 but that the firms
  interact over three periods.
• Consider the strategy
• First period set price 160
• Second and third periods set price of 130 if
  the history from the first period is (160,
  160), otherwise set price of 105
• Cartel lasts only one period but this is better
  than nothing if sustainable
• Is the cartel sustainable?

33
A finitely repeated game

• Profit from the agreement
• PVc(p1) 9.1 8.5R 8.5R2
• Profit from cheating in period 1
• PVd(p1) 10 7.3125R 7.3125R2
• The cartel is stable in period 1 if R gt 0.504
  (discount rate of less than 98.5 percent)

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Cartel Stability (cont.)

• The intuition is simple enough
• suppose the Nash equilibrium is not unique
• some equilibria will be good and some bad for
  the firms
• with a finite future the cartel will inevitably
  break down
• but there is the possibility of credibly
  rewarding good behavior and credibly punishing
  bad behavior
• make a credible commitment to the good
  equilibrium if rivals have cooperated
• to the bad equilibrium if they have not.

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Cartel Stability (cont.)

• Cartel stability is possible even if cooperation
  is over a finite period of time
• if there is a credible reward system
• which requires that the Nash equilibrium is not
  unique
• This is a limited scenario
• What happens if we remove the finiteness
  property?
• Suppose the cartel expects to last indefinitely
• equivalent to assuming that the last period is
  unknown
• in every period there is a finite probability
  that competition will continue
• now there is no definite end period
• so it is possible that the cartel can be
  sustained indefinitely

36
A Digression The Discount Factor

• How do we evaluate a profit stream over an
  indefinite time?
• Suppose that profits are expected to be p0 today,
  p1 in period 1, p2 in period 2 pt in period t
• Suppose that in each period there is a
  probability r that the market will last into the
  next period
• probability of reaching period 1 is r, period 2
  is r2, period 3 is r3, , period t is rt
• Then expected profit from period t is rtpt
• Assume that the discount factor is R. Then
  expected profit is
• PV(pt) p0 Rrp1 R2r2p2 R3r3p3 Rtrtpt
  
• The effective discount factor is the
  probability-adjusted discount factor G rR.

37
Cartel Stability (cont.)

• Analysis of infinitely or indefinitely repeated
  games is less complex than it seems
• Cartel can be sustained by a trigger strategy
• I will stick by our agreement in the current
  period so long as you have always stuck by our
  agreement
• If you have ever deviated from our agreement I
  will play a Nash equilibrium strategy forever

38
Cartel Stability (cont.)

• Take example 1 but suppose that there is a
  probability r in each period that the market will
  continue
• Cooperation has each firm producing 30 thousand
• Nash equilibrium has each firm producing 40
  thousand
• So the trigger strategy is
• I will produce 30 thousand in the current period
  if you have produced 30 thousand in every
  previous period
• if you have ever produced more than 30 thousand
  then I will produce 40 thousand in every period
  after your deviation
• This is a trigger strategy because punishment
  is triggered by deviation of the partner
• Does it work?

39
Example 1 Cournot duopoly
The pay-off matrix from the simple Cournot game
Firm 1
Cooperate (M)
Deviate (D)
Cooperate (M)
(1.8, 1.8)
(1.35, 2.025)
Firm 2
(1.6, 1.6)
Deviate (D)
(2.025, 1.35)
(1.6, 1.6)
40
Cartel Stability (cont.)
A cartel is more likely to be stable the greater
the probability that the market will continue and
the lower is the interest rate

• Profit from sticking to the agreement is
• PVC 1.8 1.8R 1.8R2

1.8/(1 - G)

• Profit from deviating from the agreement is
• PVD 2.025 1.6 G 1.6 G 2

2.025 1.6 G /(1 - G)

• Sticking to the agreement is better if
• PVC gt PVD

1.8
1.6 G
gt 2.025
this requires
which requires G rRgt 0.592
1 - G
1 - G
if r 1 we need r lt 86 if r 0.6 we need r lt
13.4
41
Cartel Stability (cont.)

• This is an example of a more general result
• Suppose that in each period
• profits to a firm from a collusive agreement are
  pM
• profits from deviating from the agreement are pD
• profits in the Nash equilibrium are pN
• we expect that pD gt pM gt pN
• Cheating on the cartel does not pay so long as

There is always a value of G lt 1 for which this
equation is satisfied
This is the short-run gain from cheating on the
cartel
This is the long-run loss from cheating on the
cartel
pD - pM
G gt
pD - pN

• The cartel is stable
• if short-term gains from cheating are low
  relative to long-run losses
• if cartel members value future profits (high
  probability-adjusted discount factor)

42
Cartel Stability (cont.)

• What about Example 2?
• two possible trigger strategies
• price forever at 130 in the event of a deviation
  from 160
• price forever at 105 in the event of a deviation
  from 160
• Which?
• there are probability-adjusted discount factors
  for which the first strategy fails but the second
  works
• Simply put, the more severe the punishment the
  easier it is to sustain a cartel

43
Trigger strategies

• Any cartel can be sustained by means of a trigger
  strategy
• prevents destructive competition
• But there are some limitations
• assumes that punishment can be implemented
  quickly
• deviation noticed quickly
• non-deviators agree on punishment
• sometimes deviation is difficult to detect
• punishment may take time
• but then rewards to deviation are increased
• The main principle remains
• if the discount rate is low enough then a cartel
  will be stable provided that punishment occurs
  within some reasonable time

44
Trigger strategies (cont.)

• Another objection a trigger strategy is
• harsh
• unforgiving
• Important if there is any uncertainty in the
  market
• suppose that demand is uncertain

A firm in this cartel does not know if a
decline in sales is natural or caused by
cheating
Suppose that the agreed price is PC
Price
There is a possibility that demand may be low
Actual sales vary between QL and QH
And a possibility that demand may be high
This is the expected market demand
Expected sales are QE
PC
DH
DL
DE
Quantity
QE
QL
QH
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Trigger strategies (cont.)

• These objections can be overcome
• limit punishment phase to a finite period
• take action only if sales fall outside an agreed
  range
• Makes agreement more complex but still feasible
• Further limitation
• approach is too effective
• result of the Folk Theorem

Suppose that an infinitely repeated game has a
set of pay-offs that exceed the one-shot Nash
equilibrium pay-offs for each and every firm.
Then any set of feasible pay-offs that are
preferred by all firms to the Nash equilibrium
pay-offs can be supported as subgame perfect
equilibria for the repeated game for some
discount factor sufficiently close to unity.
46
The Folk Theorem

• Take example 1. The feasible pay-offs describe
  the following possibilities

p2
1.8 million to each firm may not be sustainable
but something less will be
Collusion on monopoly gives each firm 1.8 million
The Folk Theorem states that any point in
this triangle is a potential equilibrium for
the repeated game
2.1
If the firms collude perfectly they share 3.6
million
2.0
If the firms compete they each earn 1.6 million
1.8
1.6
p1
1.8
2.1
1.5
1.6
2.0
47
Stable cartels (cont.)

• A collusive agreement must balance the temptation
  to cheat
• In some cases the monopoly outcome may not be
  sustainable
• too strong a temptation to cheat
• But the folk theorem indicates that collusion is
  still feasible
• there will be a collusive agreement
• that is better than competition
• that is not subject to the temptation to cheat

48
Cartel Formation

• What factors are most conducive to cartel
  formation?
• sufficient profit motive
• means by which agreement can be reached and
  enforced
• The potential for monopoly profit
• collusion must deliver an increase in profits
  this implies
• demand is relatively inelastic
• restricting output increases prices and profits
• entry is restricted
• high profits encourage new entry
• but new entry dissipates profits (OPEC)
• new entry undermines the collusive agreement

49
Cartel formation (cont.)

• So there must be means to deter entry
• common marketing agency to channel output
• consumers must be persuaded of the advantages of
  the agency
• lower search costs
• greater security of supply
• wider access to sellers
• denied access if buy outside the agency (De
  Beers)
• trade association
• persuade consumers that the association is in
  their best interests

50
Cartel formation (cont.)

• Costs of reaching a cooperative agreement
• even if the potential for additional profits
  exists, forming a cartel is time-consuming and
  costly
• has to be negotiated
• has to be hidden
• has to be monitored
• There are factors that reduce the costs of cartel
  formation
• small number of firms (recall Selten)
• high industry concentration
• makes negotiation, monitoring and punishment (if
  necessary) easier
• similarity in production costs
• lack of significant product differentiation
• Similarity in costs

51
Cartel formation (cont.)

• Lack of product differentiation
• if products are very different then negotiations
  are complex
• need agreed price/output/market share for each
  product
• monitoring is more complex
• Most cartels are found in relatively homogeneous
  product markets
• Or firms have to adopt mechanisms that ease
  monitoring
• basing point pricing

52
Cartel formation (cont.)

• Low costs of maintaining a cartel agreement
• it is easier to maintain a cartel agreement when
  there is frequent market interaction between the
  firms
• over time
• over spatially separated markets
• relates to the discussion of repeated games
• less frequent interaction leads to an extended
  time between cheating, detection and punishment
• makes the cartel harder to sustain

53
Cartel formation (cont.)

• Stable market conditions
• accurate information is essential to maintaining
  a cartel
• makes monitoring easier
• unstable markets lead to confused signals
• makes collusion near to monopoly difficult
• uncertainty can be mitigated
• trade association
• common marketing agency
• controls distribution and improves market
  information
• Other conditions make cartel formation easier
• detection and punishment should be simple and
  timely
• geographic separation through market sharing is
  one popular mechanism

54
Cartel formation (cont.)

• Other tactics encourage firms to stick by
  price-fixing agreements
• most-favored customer clauses
• reduces the temptation to offer lower prices to
  new customers
• meet-the competition clauses
• makes detection of cheating very effective

55
Meet-the-competition clause
? the one-shot Nash equilibrium is (Low, Low)
? meet-the-competition clause removes the
off-diagonal entries
? now (High, High) is easier to sustain
Firm 2
High Price
Low Price
High Price
12, 12
5, 14
5, 14
Firm 1
Low Price
14, 5
6, 6
14, 5
56
Example 2 A Bertrand Game