Puerto Rico, US, Dec 2013: 5-year sentence for pricefixing

Transcription

1 Dynaic oligopoly theory Collusion price coordination Illegal in ost countries - Explicit collusion not feasible - Legal exeptions Recent EU cases - Banking approx..7 billion Euros in fines (03) - Cathodic ray tubes.5 billion Euros (0) - Gas approx.. billion Euros in fines (009) - Car glass approx..4 billion Euros (008) Puerto Rico, US, Dec 03: 5-year sentence for pricefixing Tacit collusion Hard to detect not any cases. Repeated interaction Theory of repeated gaes Deviation fro an agreeent to set high prices has - a short-ter gain: increased profit today - a long-ter loss: deviation by the others later on Tacit collusion occurs when long-ter loss > short-ter gain Tore Nilssen Strategic Copetition Thee Slide

2 Model Two firs, hoogeneous good, C(q) = cq Prices in period t: (p t, p t ) Profits in period t: (p t, p t ), (p t, p t ) History at tie t: H t = (p 0, p 0,, p, t, p, t ) A fir s strategy is a rule that assigns a price to every possible history. A subgae-perfect equilibriu is a pair of strategies that are in equilibriu after every possible history: Given one fir s strategy, for each possible history, the other fir s strategy axiizes the net present value of profits fro then on. T nuber of periods T finite: a unique equilibriu period T: p T = p T = c, irrespective of H T. period T : the sae and so on Tore Nilssen Strategic Copetition Thee Slide

3 T infinite (or indefinite) At period, fir i axiizes t t i p t pt,, r The best response to (c, ) is (c, ). But do we have other equilibria? Can p > c be sustained in equilibriu? Trigger strategies: If a fir deviates in period t, then both firs set p = c fro period t + until infinity. [Optial punishent schees? Renegotiation-proofness?] Monopoly price: p = arg ax (p c)d(p) Monopoly profit: = (p c)d(p ) A trigger strategy for fir : Set p 0 = p in period 0 In the periods thereafter, p t (H t ) = p, if H t = (p, p,, p, p ) p t (H t ) = c, otherwise Tore Nilssen Strategic Copetition Thee Slide 3

4 If a fir collaborates, it sets p = p and earns / in every period. The optiu deviation: p, yielding for one period. An equilibriu in trigger strategies exists if: ( ) The sae arguent applies to collusion on any price p (c, p ]. Infinitely any equilibria. The Folk Theore. Tore Nilssen Strategic Copetition Thee Slide 4

5 Collusion when deand varies Deand stochastic. Periodic deand is low: D (p) with probability ½ high: D (p) with probability ½ D (p) < D (p), p. The deand shocks are i.i.d. Each fir sets its price after having observed deand. What are the best collusive strategies for the two firs? Trigger strategies: A deviation is followed by p = c forever. What are the best collusive prices? One price in lowdeand periods and one in high-deand periods: p and p. s (p) total industry profit in state s when both firs set p. With prices p and p in the two states, each fir s expected net present value is: t D p D p p t 0 c p c = [D (p )(p c) + D (p )(p c)] 4 p p = 4 Tore Nilssen Strategic Copetition Thee Slide 5

6 The best possible collusive price in state s is: p s = arg ax (p c)d s (p), s =,. s = (p s c)d s (p s ), s =,. If the firs can collude on these prices, then: 4 A deviation in state s receives a gain equal to: s For (p, p ) to be equilibriu prices, we ust have: s ½ s + s The difficulty is state (high-deand), since <. The equilibriu condition becoes: < < < 0 < 3 Tore Nilssen Strategic Copetition Thee Slide 6

7 But what if [, 0 )? Can we still find prices at which the firs can collude? The proble is again state. We need to set p so that p p 4 3 p < 3 3 So: prices below onopoly price in high-deand state during boo. Could even be that p < p. But is this a price war? More realistic deand conditions: Autocorrelation business cycle. Collusion ost difficult to sustain just as the downturn starts. Haltiwanger & Harrington, RAND J Econ 99 Kandori, Rev Econ Stud 99 Bagwell & Staiger, RAND J Econ 997 [Exercise 6.4] Tore Nilssen Strategic Copetition Thee Slide 7

8 Epirical studies of collusion the railroad cartel - Porter Bell J Econ Ellison RAND J Econ 994 collusion aong petrol stations - Slade Rev Econ Stud 99 collusion in the soft-drink arket: prices and advertising - Gasi, et al., J Econ & Manag Strat 99 collusion in procureent auctions - Porter & Zona J Pol Econ 993 (road construction) - Pesendorfer Rev Econ Stud 000 (school ilk) Infrequent interaction Suppose the period length doubles. Collusion feasible if: 0.7 Tore Nilssen Strategic Copetition Thee Slide 8

9 Multiarket contact Market A: Frequent interaction, period length. Collusion if ½. Market B: Infrequent interaction, period length. Collusion if ½. (How could frequency vary across arkets?) What if both firs operate in both arkets? Can the firs obtain collusion in both arkets even in cases where < ½ <? A deviation is ost profitable when both arkets are open. Deviation yields: Collusion yields: [ /] every period, plus [ /] every second period (starting today) Collusion can be sustained if: [ ] + [ ] Tore Nilssen Strategic Copetition Thee Slide 9

10 Secret price cuts, or: Price coordination when supervising the partners is difficult Own deand observable Market deand not observable Other firs prices not observable When own deand is low, is it because arket deand is low, or because partners default? Punishent (p = c) is necessary. But punishent forever? Can firs coordinate prices without being able to observe each other s prices? Punishent starts when one observes low deand. Punishent phase lasts for a finite nuber of periods. Even colluding firs have periods of price wars. Model: Two firs; hoogeneous products; MC = c. In each period: firs set prices; consuers choose the fir with the lowest price. Market deand is either: D = 0, with probability ; D = D(p), with probability ( - ). Tore Nilssen Strategic Copetition Thee Slide 0

11 Both firs know it if at least one fir has zero profit in a period. Either: - arket deand is zero and both firs have zero profit, or - one fir has cut its price and knows that the other fir has zero profit Strategy: Start with p = p. Set p = p until (at least) one fir has zero profit. If this happens, then set p = c for T periods. After T periods, return to p = p until (at least) one fir has zero profit. And so on. Is there an equilibriu in which each fir plays this strategy? T ust be deterined. Tore Nilssen Strategic Copetition Thee Slide

12 Tore Nilssen Strategic Copetition Thee Slide Two phases: Colluding phase Punishent phase + = net present value of a fir in the colluding phase = net present value of a fir at the start of the punishent phase = T + Equilibriu condition: + ( )( + ) + = ( ) + T

13 Tore Nilssen Strategic Copetition Thee Slide 3 T T T T ( ) + ( ) T + The best equilibriu has the highest possible +. The firs proble: ax T +, such that: ( ) + ( ) T + But: d + /dt < 0. So we restate the proble. in T, such that: ( ) + ( ) T +

14 T = 0 is too low there has to be soe punishent, even under collusion: ( ) + ( ) = < And the lefthand side ust be increasing in T: d dt T ln T 0 0 If ½, then collusion is ipossible: The probability of zero arket deand is too large. If < ½, then < 0. But ( ) T + 0 as T. Equilibriu condition satisfied for soe T if also ( ) All in all: Collusion can occur in equilibriu if: < ½ T is chosen as the lowest integer that satisfies: ( ) + ( ) T + Exaple: = ¾, = ¼. Condition: (¾) T + ¼ T* = 4. But often T* is saller: = 0.9, = 0. T* =. Tore Nilssen Strategic Copetition Thee Slide 4

15 Price rigidities Menu costs Price reactions not punishents, but attepts to regain arket share Suppose - a price is fixed for two periods - firs alternate at setting price Duopoly with alternating price setting A discrete price grid Markov strategies: strategies based only on directly payoff-relevant inforation Exaple: A trigger strategy is not Markov; no price fro the past has a direct effect on a fir s profit today, only an indirect effect, because other firs use trigger strategies. A restriction to Markov strategies would be too strong when oves are siultaneous. Here, oves are alternating. Model: duopoly; each fir s price fixed for two periods; fir sets price in odd-nubered periods ( 3 5 ), fir in even-nubered periods ( 4 6 ). Tore Nilssen Strategic Copetition Thee Slide 5

16 Markov reaction functions: Let p it be the price set by fir i in period t. Fir s reaction function: p, k + = R (p, k ), k = 0,,, Fir s reaction function: p, k + = R (p, k + ), k = 0,,, Markov perfect equilibriu: An equilibriu in Markov reaction functions. At the start of each subgae, the fir that akes the ove chooses an optiu strategy, given the restriction only to pay attention to payoff-relevant inforation, and given the other fir s equilibriu strategy. The two firs at any point in tie: the active and the other Consider the active fir s decision today. Suppose the other fir set the price p h last period; this is also its price today. We are in state h. h the active fir s net present value in state h. W h the other fir s net present value in state h. Toorrow, the roles are changed. Tore Nilssen Strategic Copetition Thee Slide 6

17 Profit per period: (own price, the other s price) ax p, p h k k h W A syetric equilibriu: R () = R () = R() k Mixed strategy: A fir ay be indifferent between one or ore prices, and in equilibriu, the other fir has beliefs about which of these prices will be chosen. These beliefs will then constitute the fir s ixed strategy. hk the probability (according to the other fir s beliefs) that a fir in state h chooses price p k. Note: k hk A syetric equilibriu can be described by a transition atrix: Suppose there are H possible prices. fro state h... H H to state k... HH = A Tore Nilssen Strategic Copetition Thee Slide 7

18 Equilibriu conditions h W k p, p W k hk p, p l kl k k h l l k These are the values of h and W k that follow fro the transition atrix A. [ h (p k, p h ) W k ] hk = 0, h, k. h (p k, p h ) + W k, h, k. Copleentary slackness: If hk > 0, it ust be because h = (p k, p l ) + W k, that is, because p k axiizes the fir s net present value in state h. k hk, h hk 0, h, k. Tore Nilssen Strategic Copetition Thee Slide 8

19 Exaple: D(p) = p; c = 0 The price grid: p h = h, h = 0,, 6. 6 Copetitive price: p 0 = 0. Monopoly price: p = p 3 = ½. Two (syetric Markov perfect) equilibria (at least):. Kinked deand curve : The other fir does not follow you if you increase the price but undercuts you if you decrease the price. R() = R( 65 ) = R( 3 ) = R( ) = R(0) = ; R( 3 ) = 6 ; R( 6 ) { 6, }. Either the gae starts in state 3 and stays there, or it ends there sooner or later (absorbing state). A ixed strategy in state a waiting gae ( war of attrition ): Each fir is indifferent between eeting p with p, and aking a short-ter sacrifice in order to get the onopoly price fro next period on. The equilibriu is sustainable only if each fir is able to supply the whole arket deand at p = 6 : D( 6 ) = 6 5. In the absorbing state 3, each fir sells D(p 3 ) = 4 but needs to keep an excess capacity of = 7. Tore Nilssen Strategic Copetition Thee Slide 9

20 . Price war: The firs undercut each other. R() = R( 65 ) = 3 ; R( 3 ) = ; R( ) = 3 ; R( 3 ) = 6 ; R( 6 ) = 0; R(0) {0, 65 }. Unstable prices: no absorbing state. Edgeworth cycle. Again a waiting gae. But now the price jups beyond the onopoly price. * Multiple equilibria, even when we restrict attention to Markov strategies. Fewer equilibria than in an ordinary repeated gae. p = c is no longer an equilibriu; there is always soe price collusion in equilibriu. Tore Nilssen Strategic Copetition Thee Slide 0