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Lecture 14 Cartels: Price-fixing and repeated games (I)

Lecture 14 Cartels: Price-fixing and repeated games (I). ECON 4100: Industrial Organization. Introduction. Collusion Incentives to cheat in a cartel Repeated Games and Cartel stability. Introduction.

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Lecture 14 Cartels: Price-fixing and repeated games (I)

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  1. Lecture 14Cartels: Price-fixing and repeated games (I) ECON 4100: Industrial Organization

  2. Introduction • Collusion • Incentives to cheat in a cartel • Repeated Games and Cartel stability

  3. Introduction • People of the same trade seldom meet together, even for merriment and diversion, but the conversation ends in a conspiracy against the public, or in some contrivance to raise prices. Adam Smith

  4. Collusion and Cartels • A cartel is an 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

  5. Collusion is there… • Cartels have always been with us • electrical conspiracy of the 1950s (using the phases of the moon trick) • Some are explicit and difficult to prevent • OPEC • De Beers (they are trying to get their problem solved, but never manage!)

  6. Collusion and Cartels • Other less explicit attempts to control competition • formation of producer associations • publication of price sheets • peer pressure (NASDAQ?) • violence • Cartel laws make cartels illegal in the US and Europe

  7. Collusion is difficult • What constrains cartel formation? • they are generally illegal • per se violation of anti-trust law in US • 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 cheaton the cartel agreement

  8. 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 We can use Cournot analysis to see that colluding is the Pareto efficient equilibrium and go directly to the associated payoff matrix 150 Demand 30 MC Quantity 150

  9. 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 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

  10. The Incentive to Cheat These best response functions are easily illustrated q2 q*1 = 60 - q2/2 q*2 = 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 40 60 120

  11. 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 30 40 60 120

  12. 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 1’s 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 30 40 60 120 45

  13. The Incentive to Cheat (cont.) Both firms have the incentive to cheat on their agreement 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)

  14. Cartel Stability • This is a typical “prisoners’ dilemma” game, this cartel is unstable, as it often happens • Can we find mechanisms that give stable cartels? • violence in one possibility! • are there others? • 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

  15. Repeated Games • Formalizing these ideas leads to repeated games • a firm’s 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

  16. Repeated Games • 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

  17. 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)

  18. Example 2: A Bertrand Game, with two Nash-equilibria 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 (7.25, 8.25) (8.5, 8.5) (10, 7.15) Firm 2 $160 (5.525, 9.375) (7.15, 10) (9.1, 9.1)

  19. Repeated Games (cont.) • Time “matters” in a repeated game • is the game finite? T is known in advance (Exhaustible resource, Patent, etc.) • 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

  20. 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

  21. 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, by backward induction • The possibility of cooperation disappears

  22. Repeated Games (cont.) • 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 Even with repetition, if it is finite, cooperation will not succeed!!!

  23. Repeated Games (cont.) • How to resolve this? Consider our 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

  24. 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.

  25. A finitely repeated game • These strategies reflect historical dependence • each firm’s 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

  26. 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 doesn’t 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 2’s 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

  27. A finitely repeated game • Defection does not pay! It triggers a backlash that makes it unprofitable! • The same applies to firm 1 • So we have credible strategies that partially support the cartel

  28. A finitely repeated game • Extensions • More than two periods • Same argument shows that the cartel can be sustained for all but the final period: strategy • In period t < 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

  29. A finitely repeated game • Suppose that the discount factor R < 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 > 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

  30. A finitely repeated game • Suppose that R < 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?

  31. 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 ifR > 0.504 (discount rate of less than 98.5 percent)

  32. Cartel Stability (cont.) • The intuition is simple: • 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 commitmentto the good equilibrium if rivals have cooperated • to the bad equilibrium if they have not.

  33. 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 • 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 • so it is possible that the cartel can be sustained indefinitely

  34. 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.

  35. Next • More on the stability of cartels • Read Ch. 15

  36. Firm 1 $105 $130 $160 (7.3125, 7.3125) (8.25, 7.25) (9.375, 5.525) $105 $130 (7.25, 8.25) (8.5, 8.5) (10, 7.15) Firm 2 $160 (5.525, 9.375) (7.15, 10) (9.1, 9.1) Example 2: A Bertrand Game

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