3.3.3 Illustration: Infinitely repeated Cournot duopoly.

Transcription

1 will begin next period less effective in deterring a deviation this period. Nonetheless, players can do better than just repeat the Nash equilibrium of the constituent game Illustration: Infinitely repeated Cournot duopoly. The setup of the stage game is the following. There are 2 firms, denoted Firm and Firm 2. Each firm i chooses its output q i, which is produced at marginal cost c 0. Total inverse demand is P (Q) = Q, Q = q + q 2. The Cournot outcome of the stage game: firm i s best-response function is q i (q j ) = q j, and thus the Nash equilibrium is (qne 2, q2 NE ) = (, ), yielding 3 3 an equilibrium price p NE =, and payoffs πne 3 i =. 9 The collusive outcome, obtained when firms form a cartel that maximizes joint profit, is such that Q c = arg max Q P (Q)Q and q c i = 2 Qc. Here, Q c = 2, q c = q c 2 = 4, the cartel price is pc = 2, and firms profits amount to πc i = 8. If a firm i had to unilaterally deviate from the collusive agreement, it would choose to produce an output qi d produces qj c =, that is 4 that maximizes its profit, given that firm j q d i = arg max q i π d i (q i, q c j) where π d i (q i, q c j) = ( q i q c j)q i = ( 3 4 q i)q i πi d (q i, qj) c q q d i = 0 3 i 4 2qd i = 0 qi d = 3 8 > qc i. The instantaneous profit from deviating is πi d = ( ) >. 8 Now we can analyze the following questions:. For which values of the discount facto δ is perfect collusion sustainable, given that firms follow trigger strategies? 2. For a given value of δ smaller than 9/7, what output could be sustained by a trigger strategy? 3. Carrot-and-stick strategies (Gibbons, p ). 6

2 - For which values of the discount facto δ is perfect collusion sustainable, given that firms follow trigger strategies? Each firm i follows the trigger strategy Produce the collusive output q c i in period. In period t >, produce qi c if the outcome (q, c q2) c was observed in all previous periods. If, in at least one previous period, the outcome was different than (q c, q c 2), then produce the non-cooperative output q NE i. We must show that such a strategy profile constitute a SPNE. Consider any given period, and suppose that firm j is following the trigger strategy. strategy. We must show that firm i does not want to deviate from the trigger In period, there are two possible categories of subgames:. The subgames that start after the outcome (q c, q c 2) in every ( ) previous period; 2. The subgames that start after at least one outcome was different than (q c, q c 2). In the first category of subgames, firm i knows that firm j will produce q c j. Firm i can either produce q c i or deviate (and then deviate to its best possible deviation) to produce q d i. By deviating to q d i, firm i obtains π d i instead of π c i in the current period, but this deviation triggers firm j to a revert to the non-cooperative outcome qj NE in all following periods, to which firm i s best-response is qi NE. Therefore, the present value (at time ) of the flow of profits when firm i deviates is V d π d i + δ s=+ δ s ( ) π NE i = ( ) δ 8 δ 9. Conversely, if firm i does not deviate and produces qi c in period, it obtains a profit πi c = at the current period. Next period, firm i will have the same decision 8 to make between deviating and cooperating. Denote V the discounted present value of the infinite sequence of payoffs firm i receives when it is optimal for firm i to collude: V c πi c = δ 8. 62

3 At period, firm i chooses to collude if V c V d δ 8 ( ) δ 8 δ 9 δ Hence, the trigger strategy constitute a Nash equilibrium of the first-category subgames if δ 9 7. In the second category of subgames, firm i knows that firm j will produce q NE j, in the current period and in every following one. Firm i best response is to produce qi NE. Therefore, the trigger strategy forms a Nash equilibrium of the second-category subgames. It remains to show that firm i chooses to produce qi c in period. By choosing to collude in the first period, firm i s discounted flow of profits is V c =. By δ 8 deviating, it is V d = ( ) δ. Again, firm i agrees to collude in the first period δ 9 as long as δ > For a given value of the discount factor δ < 9, what level of output 7 would be sustainable, given that firms follow trigger strategies? Denote q the output on which firms agree to cooperate on. Because δ < 9 7, we know that q < qi c. We look for q (qi c, qi NE ) that is sustainable when each firm i follows a trigger strategy Produce the output q in period. In period t >, produce q if the outcome (q, q ) was observed in all previous periods. If, in at least one previous period, the outcome was different than (q, q ), then produce the non-cooperative output q NE i. When both firms cooperate on q, each one obtains an instantaneous payoff of π = ( 2q )q. If one of them wants to deviate, it will choose the best possible deviation, q d (q ), that is the one that maximizes its profit, given that its competitor is currently producing q : q d (q ) = arg max q ( q q)q. It follows that q d (q ) = 2 ( q ). The current profit of the deviating firm is written π d i (q d ; q ) = ( q d q )q d = 4 ( q ) 2. For the trigger strategy to be an equilibrium, given δ < 9, the present value of 7 cooperating on q should be at least as high as the present value of deviating and 63

4 then reverting to the Nash outcome forever after: V c = δ ( 2q )q V d = 4 ( q ) 2 + δ δ 9. (3.2) The condition expressed in (3.2) can be written as (q ) 2 ( δ 4 [ and is satisfied for any q ) ( ) δ + 2 q + + δ , 9 5δ 3(9 δ), 3 ]. The highest cooperative profit is obtained when firms produce the corresponding lowest cooperative output q = instance, if δ = 0.4, q = 9 2 3(9 0.4) 9 5δ. For 3(9 δ) This level of output is of course larger than the perfect collusive output q c i = 0.25, but smaller than the Nash equilibrium output q NE i = Carrot-and-Stick strategy. Now, we consider a strategy different than a trigger strategy, in which a deviation is followed by a punishment phase that lasts a finite number of periods. Basically, firms cooperate on an output q, allowing them to obtain profits higher than the Cournot profits. The punishment that follows a deviation consists in a level of output q p > q, that yields an individual profit π p i (qp, q p ), and a number of periods T during which the punishment is carried out. Cooperation can be sustained with carrot-and-stick strategies if these strategies are feasible and credible. A carrot-and-stick strategy is be feasible if firms actually want to cooperate on the level of output q, and it will be credible if it is indeed optimal for each of them to implement the punishment phase after one of them has deviated. The cooperative profit is π π (q, q ) = ( 2q )q. The best deviation from q is q d (q ) = 2 ( q ), yielding a profit π d π d (q d (q ); q ) = 4 ( q ) 2. During the punishment phase, both firms are supposed to produce q p, and should obtain a profit π p π p (q p, q p ) = ( 2q p )q p. But each firm has always the option to deviate from the punishment output. If it does so, it would choose the best possible deviation. i.e. q dp = arg max q ( q p q)q. The best deviation from the punishment output is thus q dp (q p ) = 2 ( qp ), yielding an instantaneous profit π dp π dp (q dp ; q p ) = 4 ( qp ) 2. Now, consider the following strategy for firm i: 64

5 Produce q in period. In period t >, produce q if (q, q ) or (q p, q p ) was the outcome observed in period (t ). Otherwise, produce q p. This two-phase strategy involves a one-period punishment phase in which each firm produces q p. This punishment output is produced in two possible situations, i.e. after either firm deviated from the cooperative output q, and after either firm deviated from the punishment output during the punishment period. If both firms adopt this carrot-and-stick strategy, the subgames in the infinitely repeated game can be grouped in two categories:. The cooperative subgames, in which the outcome of the previous period was either (q, q ) or (q p, q p ); 2. The punishment subgames, in which the outcome of the previous period was neither (q, q ) nor (q p, q p ). To be a SPNE, the strategy profile (consisting of both firms adopting the carrotand-stick strategy) must be a Nash equilibrium in each category of subgames. (.) In cooperative subgames, the strategy profile is a Nash equilibrium if neither firm wants to unilaterally deviate, that is if ( δ π π d + δ π p + δ ) δ π δ (π π p ) π d π (3.3) Inequality (3.3) can be read as follows: The gain from deviating in the current period (the difference in the RHS) must not exceed the discounted value of the loss due to the next period s punishment phase (the difference in the LHS). (2.) In punishment subgames, each firm must prefer to administer the punishment rather than to deviate (that is, punishment must be credible). This condition can be written as π p + δ ( δ π π dp + δ π p + δ ) δ π δ (π π p ) π dp π p. (3.4) Inequality (3.4) means that the gain from deviating from the punishment this period (the RHS) must not exceed the discounted loss of another period of punishment (the LHS). 65

6 We need both inequalities (3.3) and (3.4) to hold simultaneously, which implies that we must solve the following system δ [( 2q )q ( 2q p )q p ] 4 ( q ) 2 ( 2q )q δ [( 2q )q ( 2q p )q p ] 4 ( qp ) 2 ( 2q p )q p (3.5) Note that there are for the moment 3 unknown in this system: δ, q and q p. To make things simple, suppose δ = and that firms wish to cooperate on the perfectly 2 collusive output, q =. We can thus find the optimal punishment 4 qp. With these values, the system (3.5) becomes 2 2 [ ( 8 2qp )q p] ( ) [ ( 8 2qp )q p] ( 4 qp ) 2 ( 2q p )q p q p [ ] [ 0, 8 3, ] 8 q p [ 3, 0 2]. 2(q p ) 2 q p (qp ) 2 2q p Finally, the system (3.5) is satisfied for q p [ 3 8, 2]. The two-phase (or carrotand-stick ) strategy, allowing firms to cooperate on the collusive output q = 4, and including a one-period punishment in case one of them deviates, constitutes a subgame perfect Nash equilibrium as long as the punishment output q p is in the interval [ 3 8, 2]. Note that the smallest punishment, corresponding to q p = 3 8, is harsher than a reversion to the Nash equilibrium of the stage game, q NE = 3, as π p = 3 32 < πne = 9. 66