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Price-Fixing and Repeated Games

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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 Price-Fixing and – cartel members agree to coordinate their actions • prices Repeated Games • market shares • exclusive territories – prevent excessive competition between the cartel members Chapter 14: Price-fixing and 1 Chapter 14: Price-fixing and 2 Repeated Games Repeated Games Collusion and cartels 2 Recent events • Recent years have seen record-breaking fines being • Cartels have always been with us; generally hidden imposed on firms found guilty of being in cartels. For – electrical conspiracy of the 1950s example – garbage disposal in New York – illegal conspiracies to fix prices and/or market shares – Archer, Daniels, Midland – €479 million imposed on Thyssen for elevator conspiracy in 2007 – the vitamin conspiracy – €396.5 million imposed on Siemens for switchgear cartel in 2007 • But some are explicit and difficult to prevent – $300 million on Samsung for DRAM cartel in 2005 –OPEC – Hoffman-LaRoche $500 million in 1999 – De Beers – UCAR $110 million in 1998 – shipping conferences – Archer-Daniels-Midland $100 million in 1996 Chapter 14: Price-fixing and 3 Chapter 14: Price-fixing and 4 Repeated Games Repeated Games Recent cartel violations 2 Cartels • Justice Department Cartel Fines grew steadily since 2002 • 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 5 Chapter 14: Price-fixing and 6 Repeated Games Repeated Games 1 The incentive to collude An example • Is there a real incentive to belong to a cartel? • Is cheating so endemic that cartels fail? • Take a simple example • If so, why worry about cartels? – two identical Cournot firms making identical products – for each firm MC = $30 • Simple reason – market demand is P = 150 - Q – without cartel laws legally enforceable contracts could be written –Q = q + q De Beers is tacitly supported by the South African government 1 2 Price • gives force to the threats that support this cartel 150 – not to supply any company that deviates from the cartel • Investigate Demand – incentive to form cartels – incentive to cheat – ability to detect cartels 30 MC 150 Quantity Chapter 14: Price-fixing and 7 Chapter 14: Price-fixing and 8 Repeated Games Repeated Games The incentive to collude The incentive to collude 2 Profit for firm 1 is: = q (P - c) • Nash equilibrium quantities are q*1 = q*2 = 40 1 1 • Equilibrium price is P* = $70 = q (150 - q -q - 30) 1 1 2 • Profit to each firm is (70 – 30)x40 = $1,600. = q1(120 - q1 -q2) • Suppose that the firms cooperate to act as a monopoly – joint output of 60 shared equally at 30 units each To maximize, differentiate with respect to q1: – price of $90 1/q1 = 120 - 2q1 -q2 = 0 – profit to each firm is $1,800 q* = 60 - q /2 •But 1 2 – there is an incentive to cheat The best response function for firm 2 is then: • firm 1’s output of 30 is not a best response to firm 2’s output of 30 q*2 = 60 - q1/2 Chapter 14: Price-fixing and 9 Chapter 14: Price-fixing and 10 Repeated Games Repeated Games The incentive to cheat The Incentive to cheat 2 • Suppose that firm 2 is expected to produce 30 units d • Then firm 1 will produce q 1 = 60 – q2/2 = 45 units – total output is 75 units – price is $75 Firm 2 – profit to firm 1 is $2,025 and to firm 2 is $1,350 Cooperate (M) Deviate (D) • Of course firm 2 can make the same calculations! • We can summarize this in the pay-off matrix: Cooperate (M) (1800, 1800) (1350, 2025) Firm 1 Deviate (D) (2025, 1350) (1600,(1600, 1600)1600) Chapter 14: Price-fixing and 11 Chapter 14: Price-fixing and 12 Repeated Games Repeated Games 2 The incentive to cheat 3 Finitely repeated games • This is a prisoners’ dilemma game • Suppose that interactions between the firms are repeated a – mutual interest in cooperating finite number of times known to each firm in advance – but cooperation is unsustainable – opens potential for a reward/punishment strategy • However, cartels to form •“If you cooperate this period I will cooperate next period” • So there must be more to the story • “If you deviate then I shall deviate.” – once again use the Nash equilibrium concept – consider a dynamic context • firms compete over time • Why might the game be finite? • potential to punish “bad” behavior and reward “good” – non-renewable resource –this is a repeated game framework – proprietary knowledge protected by a finite patent – finitely-lived management team Chapter 14: Price-fixing and 13 Chapter 14: Price-fixing and 14 Repeated Games Repeated Games Finitely repeated games 2 Finitely repeated games 3 • Original game but repeated twice • This strategy is unsustainable • Consider the strategy for firm 1 – the promise is not credible – first play: cooperate • at end of period 1 firm 2 has a promise of cooperation from – second play: cooperate if firm 2 cooperated in the first play, firm 1 in period 2 otherwise choose deviate • but period 2 is the last period • dominant strategy for firm 1 in period 2 is to deviate Firm 2 Firm 2 Cooperate (M) Deviate (D) Cooperate (M) Deviate (D) Cooperate (1800, 1800) (1350, 2025) Cooperate (1800, 1800) (1350, 2025) (M) (M) Firm 1 Firm 1 Firm Deviate (D) (2025, 1350) (1600, 1600) Deviate (D) (2025, 1350) (1600, 1600) Chapter 14: Price-fixing and 15 Chapter 14: Price-fixing and 16 Repeated Games Repeated Games Finitely repeated games 4 Finitely repeated games 5 • Promise to cooperate in the second period is not credible • Selten’s theorem applies under two conditions – but suppose that there are more than two periods – The one-period equilibrium for the game is unique • with finite repetition for T periods the same problem arises – in period T any promise to cooperate is worthless – The game is repeated a finite number of times – so deviate in period T • Relaxing either of these two constraints leads to the – but then period T – 1 is effectively the “last” period possibility of a more cooperative equilibrium as an – so deviate in T . and so on alternative to simple repetition of the one-shot equilibrium • Selten’s Theorem – “If a game with a unique equilibrium is played finitely many times • Here, we focus on relaxing the second constraint and its solution is that equilibrium played each and every time. Finitely consider how matters change when the game is played repeated play of a unique Nash equilibrium is the equilibrium of the repeated game.” over an infinite or indefinite horizon Chapter 14: Price-fixing and 17 Chapter 14: Price-fixing and 18 Repeated Games Repeated Games 3 Repeated games with an infinite horizon Valuing indefinite profit streams • With finite games the cartel breaks down in the “last” period • Suppose that in each period net profit is t – assumes that we know when the game ends • Discount factor is R – what if we do not? • some probability in each period that the game will • Probability of continuation into the next period is p continue • Define “probability adjusted discount factor” Rp • indefinite end period • Then the present value of profit is: • then the cartel might be able to continue indefinitely 2 t t – in each period there is a likelihood that there will be a next –PV(t) = Rp R p +…+ R p t + … t period –PV(t) = +…+ t + … – so good behavior can be rewarded credibly – product of discount factor and probability of continuation – and bad behavior can be punished credibly Chapter 14: Price-fixing and 19 Chapter 14: Price-fixing and 20 Repeated Games Repeated Games Trigger strategies Cartel stability • Consider an indefinitely continued game • Expected profit from sticking to the agreement is: – potentially infinite time horizon – PVM = 1800 + 1800 + 1800 + … = 1800/(1 - ) • Strategy to ensure compliance based on a trigger strategy • Expected profit from deviating from the agreement is – cooperate in the current period so long as all have cooperated in every previous period – PVD = 2025 + 1600 + 1600 + … = 2025 + 1600/(1 - ) – deviate if there has ever been a deviation • Sticking to the agreement is better if PVM > PVD • Take our earlier example – this requires 1800/(1 - ) > 2025 + 1600/(1 - ) – period 1: produce cooperative output of 30 –or = Rp (2025 – 1800)/(2025 – 1600) = 0.529 – period t: produce 30 so long as history of every previous period has been (30, 30); otherwise produce 40 in this and every subsequent – Note that R=1/(1+r) period •if p = 1 this requires that the interest rate (r) is less than 89% • Punishment triggered by deviation •if p = 0.6 this requires that the interest rate (r) is less than 13.4% Chapter 14: Price-fixing and 21 Chapter 14: Price-fixing and 22 Repeated Games Repeated Games Cartel stability 2 Trigger Strategy Issues • With infinitely repeated games • This is an example of a more general result – cooperation is sustainable through self-interest • Suppose that in each period • But there are some caveats C – profits to a firm from a collusive agreement are – examples assume speedy reaction to deviation D – profits from deviating from the agreement are • what if there is a delay in punishment? N – profits in the Nash equilibrium are – trigger strategies will still work but the discount factor will – we expect that D > C > N have to be higher • Cheating on the cartel does not pay so long as: – harsh and unforgiving D - C • particularly relevant if demand is uncertain Rp> D - N – decline in sales might be a result of a “bad draw” rather than cheating on agreed quotas • The cartel is stable – so need agreed bounds on variation within which there is no – if short-term gains from cheating are low relative to long-run losses retaliation – or agree that punishment lasts for a finite period of time – if cartel members value future profits (high discount rate or low interest rate ) Chapter 14: Price-fixing and 23 Chapter 14: Price-fixing and 24 Repeated Games Repeated Games 4 The Folk theorem The Folk theorem 2 • Have assumed that cooperation is on the monopoly • Take example 1.
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