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**Price-Fixing and Repeated Games**

Chapter 14: Price-fixing and Repeated Games

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**Chapter 14: Price-fixing and Repeated Games**

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 Chapter 14: Price-fixing and Repeated Games

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**Chapter 14: Price-fixing and Repeated Games**

Collusion and cartels 2 Cartels have always been with us; generally hidden electrical conspiracy of the 1950s garbage disposal in New York Archer, Daniels, Midland the vitamin conspiracy But some are explicit and difficult to prevent OPEC De Beers shipping conferences Chapter 14: Price-fixing and Repeated Games

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**Chapter 14: Price-fixing and Repeated Games**

Recent events The 1990s saw record-breaking fines being imposed on firms found guilty of being in cartels illegal conspiracies to fix prices and/or market shares Archer-Daniels-Midland $100 million in 1996 UCAR $110 million in 1998 Hoffman-LaRoche $500 million in 1999 Chapter 14: Price-fixing and Repeated Games

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**Recent cartel violations**

F. Hoffman-LaRoche Ltd. Vitamins 1999 $500 International BASF AG (1999) $225 SGL Carbon AG Graphite Electrodes $135 UCAR International Inc. 1998 $110 Archer Daniels Midland co. Lysine and Citric Acid 1997 $100 Haarman & Reimer Corp. Citric Acid $50 HeereMac v.o.f. Marine Construction $49 Hoechst AG Sorbates $36 Showa Denko Carbon Inc. $32.5 Fujisawa Pharmaceuticals Co. Sodium Gluconate $20 Dockwise N.V. Marine Transportation $15 Dyno Nobel Explosives 1996 Domestic $14 Eastman Chemical Co. $11 Jungblunzlauer International Lonza AG $10.5 Akzo Nobel Chemicals BV & Glucona BV $10 Chapter 14: Price-fixing and Repeated Games

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**Recent cartel violations 2**

Fines steadily grew during the 1990s Chapter 14: Price-fixing and Repeated Games

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**Chapter 14: Price-fixing and Repeated Games**

Cartels Two implications cartels happen generally illegal and yet firms deliberately break the law Why? pursuit of profits But how can cartels be sustained? cannot be enforced by legal means so must resist the temptation to cheat on the cartel Chapter 14: Price-fixing and Repeated Games

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**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 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 Investigate incentive to form cartels incentive to cheat ability to detect cartels Chapter 14: Price-fixing and Repeated Games

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**Chapter 14: Price-fixing and Repeated Games**

An example Take a simple example two identical Cournot firms making identical products for each firm MC = $30 market demand is P = Q Q = q1 + q2 Price 150 Demand 30 MC 150 Quantity Chapter 14: Price-fixing and Repeated Games

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**The incentive to collude**

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 q*1 = 60 - q2/2 This is the best response function for firm 1 The best response function for firm 2 is then: q*2 = 60 - q1/2 Chapter 14: Price-fixing and Repeated Games

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**The incentive to collude 2**

Nash equilibrium quantities are q*1 = q*2 = 40 Equilibrium price is P* = $70 Profit to each firm is (70 – 30)x40 = $1,600. Suppose that the firms cooperate to act as a monopoly joint output of 60 shared equally at 30 units each price of $90 profit to each firm is $1,800 But there is an incentive to cheat firm 1’s output of 30 is not a best response to firm 2’s output of 30 Chapter 14: Price-fixing and Repeated Games

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**Chapter 14: Price-fixing and Repeated Games**

The incentive to cheat Suppose that firm 2 is expected to produce 30 units Then firm 1 will produce qd1 = 60 – q2/2 = 45 units total output is 75 units price is $75 profit to firm 1 is $2,025 and to firm 2 is $1,350 Of course firm 2 can make the same calculations! We can summarize this in the pay-off matrix: Chapter 14: Price-fixing and Repeated Games

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**Chapter 14: Price-fixing and Repeated Games**

The Incentive to cheat 2 Both firms have the incentive to cheat on their agreement Firm 1 Cooperate (M) Deviate (D) This is the Nash equilibrium Cooperate (M) (1800, 1800) (2250, 1250) Firm 2 (1600, 1600) Deviate (D) (1250, 2250) (1600, 1600) Chapter 14: Price-fixing and Repeated Games

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**Chapter 14: Price-fixing and Repeated Games**

The incentive to cheat 3 This is a prisoners’ dilemma game mutual interest in cooperating but cooperation is unsustainable However, cartels to form So there must be more to the story consider a dynamic context firms compete over time potential to punish “bad” behavior and reward “good” this is a repeated game framework Chapter 14: Price-fixing and Repeated Games

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**Finitely repeated games**

Suppose that interactions between the firms are repeated a finite number of times know to each firm in advance opens potential for a reward/punishment strategy “If you cooperate this period I will cooperate next period” “If you deviate then I shall deviate.” once again use the Nash equilibrium concept Why might the game be finite? non-renewable resource proprietary knowledge protected by a finite patent finitely-lived management team Chapter 14: Price-fixing and Repeated Games

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**Finitely repeated games 2**

Original game but repeated twice Consider the strategy for firm 1 first play: cooperate second play: cooperate if firm 2 cooperated in the first play, otherwise choose deviate Firm 1 Firm 2 Cooperate (M) Deviate (D) (1800, 1800) (2250, 1250) (1250, 2250) (1600, 1600) Chapter 14: Price-fixing and Repeated Games

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**Finitely repeated games 3**

This strategy is unsustainable the promise is not credible at end of period 1 firm 2 has a promise of cooperation from firm 1 in period 2 but period 2 is the last period dominant strategy for firm 1 in period 2 is to deviate Firm 1 Firm 2 Cooperate (M) Deviate (D) (1800, 1800) (2250, 1250) (1250, 2250) (1600, 1600) Chapter 14: Price-fixing and Repeated Games

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**Finitely repeated games 4**

Cooperation in the second period is based on a worthless promise but suppose that there are more than two periods with finite repetition over T periods the same problem arises in period T any promise to cooperate is worthless so deviate in period T but then period T – 1 is effectively the “last” period so deviate in T – 1 and so on Selten’s Theorem “If a game with a unique equilibrium is played finitely many times its solution is that equilibrium played each and every time. Finitely repeated play of a unique Nash equilibrium is the equilibrium of the repeated game.” Chapter 14: Price-fixing and Repeated Games

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**Finitely repeated games 5**

Selten’s theorem applies if there is a unique equilibrium to the game What if this condition is not satisfied? suppose that there is more than one Nash equilibrium both firms agree that one equilibrium is “good” and the other “bad” both firms benefit from cooperation Then the possibility of cooperation is opened up reward early cooperation by moving to the “good” equilibrium in later periods punish early deviation by moving to the “bad” equilibrium Chapter 14: Price-fixing and Repeated Games

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**Chapter 14: Price-fixing and Repeated Games**

An example Consider the following pricing game Firm 2 Firm 1 $105 $130 $160 (7.31, 7.31) (8.25, 7.25) (7.25, 8.25) (9.38, 5.53) (5.53, 9.38) (8.5, 8.5) (10, 7.15) (7.15, 10) (9.1, 9.1) Chapter 14: Price-fixing and Repeated Games

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**An example 2 There are two Nash equilibria Firm 2 Firm 1 $105 $130**

$160 (7.31, 7.31) (8.25, 7.25) (7.25, 8.25) (9.38, 5.53) (5.53, 9.38) (8.5, 8.5) (10, 7.15) (7.15, 10) (9.1, 9.1) 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) Chapter 14: Price-fixing and Repeated Games

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**Chapter 14: Price-fixing and Repeated Games**

An example 3 Suppose that the game is played twice and that second period profits are not discounted Consider the strategy for both firms First period: Set a price of $160 Second period: If history from the first period is ($160, $160) then set price of $130, otherwise set price of $105 Depends on history of play equilibrium path is ($160, $160) then ($130, $130) why? second period is easy ($130, $130) is a Nash equilibrium for the second period subgame what about the first period? Chapter 14: Price-fixing and Repeated Games

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**Chapter 14: Price-fixing and Repeated Games**

An example 4 Best response to a price of $160 is $130 This gives profit of 10 So deviation is period 1 gives the profit stream = $17.31 Cooperation in period 1 gives the profit stream = $17.6 Undercutting does not pay Firm 2 Firm 1 $105 $130 $160 (7.31, 7.31) (8.25, 7.25) (7.25, 8.25) (9.38, 5.53) (5.53, 9.38) (8.5, 8.5) (10, 7.15) (7.15, 10) (9.1, 9.1) Chapter 14: Price-fixing and Repeated Games

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**Chapter 14: Price-fixing and Repeated Games**

Extensions Two extensions more than two periods: say T periods no discounting play ($160, $160) for T – 1 periods and ($130, $130) in period T cooperation can be sustained in all but the last period discounting: suppose that discount factor is R two periods cooperation gives R deviation gives R cooperation is profitable if and only if R > 0.756 three periods cooperation gives R + 8.5R2 deviation gives R R2 cooperation is sustainable if and only if R > 0.398 Chapter 14: Price-fixing and Repeated Games

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**Lessons from finite games**

If Nash is unique cooperation cannot be sustained in a finite game If Nash is not unique cooperation can be sustained by a credible strategy part of the time depending upon discount factor and number of repetitions by a credible threat to punish deviation Chapter 14: Price-fixing and Repeated Games

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**Repeated games with an infinite horizon**

With finite games the cartel breaks down in the “last” period assumes that we know when the game ends what if we do not? some probability in each period that the game will continue indefinite end period then the cartel might be able to continue indefinitely in each period there is a likelihood that there will be a next period so good behavior can be rewarded credibly and bad behavior can be punished credibly Chapter 14: Price-fixing and Repeated Games

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**Valuing indefinite profit streams**

Suppose that in each period net profit is pt Discount factor is R Probability of continuation into the next period is r Then the present value of profit is: PV(pt) = p0 + Rrp1 + R2r2p2 +…+ Rtrtpt + … valued at “probability adjusted discount factor” Rr product of discount factor and probability of continuation Chapter 14: Price-fixing and Repeated Games

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**Chapter 14: Price-fixing and Repeated Games**

Trigger strategies Consider an indefinitely continued game potentially infinite time horizon Strategy to ensure compliance based on a trigger strategy cooperate in the current period so long as all have cooperated in every previous period deviate if there has ever been a deviation Take our first example period 1: produce cooperative output of 30 period t: produce 30 so long as history of every previous period has been (30, 30); otherwise produce 40 in this and every subsequent period Punishment triggered by deviation Chapter 14: Price-fixing and Repeated Games

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**Chapter 14: Price-fixing and Repeated Games**

Cartel stability Profit from sticking to the agreement is: PVM = Rr R2r2 + … = 1800/(1 - Rr) Profit from deviating from the agreement is PVD = Rr R2r2 + … = Rr/(1 - Rr) Sticking to the agreement is better if PVM > PVD this requires 1800/(1 - Rr) > Rr/(1 - Rr) or Rr > (2.025 – 1.8)/(2.025 – 1.6) = 0.529 if r = 1 this requires that the discount rate is less than 89% if r = 0.6 this requires that the discount rate is less than 14.4% Chapter 14: Price-fixing and Repeated Games

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**value of R < 1 for which**

Cartel stability 2 There is always a value of R < 1 for which this equation is satisfied This is an example of a more general result Suppose that in each period profits to a firm from a collusive agreement are pC profits from deviating from the agreement are pD profits in the Nash equilibrium are pN we expect that pD > pC > pN Cheating on the cartel does not pay so long as: This is the short-run gain from cheating on the cartel This is the long-run loss from cheating on the cartel pD - pC Rr > 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 (low discount rate) Chapter 14: Price-fixing and Repeated Games

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**Chapter 14: Price-fixing and Repeated Games**

Cartel stability 3 Consider the price game with two Nash equilibria two possible strategies to sustain cooperation trigger with deviation to the “good” Nash trigger with deviation to the “bad” Nash the latter entails a harsher punishment so is sustainable with a lower probability adjusted discount factor Firm 2 Firm 1 $105 $130 $160 (7.31, 7.31) (8.25, 7.25) (7.25, 8.25) (9.38, 5.53) (5.53, 9.38) (8.5, 8.5) (10, 7.15) (7.15, 10) (9.1, 9.1) Chapter 14: Price-fixing and Repeated Games

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**Chapter 14: Price-fixing and Repeated Games**

Trigger strategies With infinitely repeated games cooperation is sustainable through self-interest But there are some caveats examples assume speedy reaction to deviation what if there is a delay in punishment? trigger strategies will still work but the discount factor will have to be higher harsh and unforgiving particularly relevant if demand is uncertain decline is sales might be a result of a “bad draw” rather than cheating on agreed quotas so need agreed bounds on variation within which there is no retaliation or agree that punishment lasts for a finite period of time Chapter 14: Price-fixing and Repeated Games

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**Chapter 14: Price-fixing and Repeated Games**

The Folk theorem Have assumed that cooperation is on the monopoly outcome this need not be the case there are many potential agreements that can be made and sustained – 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. Chapter 14: Price-fixing and Repeated Games

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**$1800 to each firm may not be sustainable but something less will be**

The Folk theorem 2 Take example 1. The feasible pay-offs describe the following possibilities p2 The Folk Theorem states that any point in this triangle is a potential equilibrium for the repeated game $1800 to each firm may not be sustainable but something less will be $2100 Collusion on monopoly gives each firm $1800 $2000 If the firms collude perfectly they share $3,600 If the firms compete they each earn $1600 $1800 $1600 p1 $1500 $1600 $1800 $2000 $2100 Chapter 14: Price-fixing and Repeated Games

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**Chapter 14: Price-fixing and Repeated Games**

Balancing temptation 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 Chapter 14: Price-fixing and Repeated Games

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**Chapter 14: Price-fixing and Repeated Games**

Summary Infinite or indefinite repetition introduces real possibility of cooperation stable cartels sustained by credible threats so long as discount rate is not too high and probability of continuation is not too low There are good reasons for the Justice department and the Competition Directorate to be concerned about cartels but they have limited resources so where should they look? how might they detect cartels? Chapter 14: Price-fixing and Repeated Games

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