6.6 Secret price cuts

Transcription

1 Joe Chen Secret price cuts As stated earlier, afirm weights two opposite incentives when it ponders price cutting: future losses and current gains. The highest level of collusion (monopoly price) is sustainable with the severest level of punishment (eternal reversion to pricing at marginal cost). When price choices are perfectly observable, it makes sense to resort to extreme punishments because such punishments never occur in equilibrium and therefore are costless to firms (they are just threats). In some situations, maximal punishments need not be optimal. Suppose firms initially coordinate on the monopoly-price perfect equilibrium, and some firm deviates by undercutting the price in the first period. By the trigger strategy with maximal punishment, firms charge the marginal cost forever after the deviation. But the firms, who expect no profits from period 2 on, have every incentive to renegotiate to avoid the punishment phase and reach another equilibrium anew. The possibility of renegotiating undermines the strength of punishments and, therefore, adds incentive to undercutting. This opens the door for the discussion of renegotiation equilibrium in supergames. We will not get into that here. Instead, we consider a situation where the demand is stochastic and firms cannot tell a secret price undercutting from a slump in the demand. Under such an uncertainty, mistakes are unavoidable and maximal punishments need not be optimal. Under imperfect information, the fully collusive outcome is not sustainable. The collusive outcome could be sustained only if firms kept on colluding (charging the monopoly price) even when making small profits, because even under collusion small profits can occur as a result of low demand. However, a firm that is confident that its rivals will continue cooperating even when their profits are low has every incentive to undercut secretly. Thus, fully collusive outcome is inconsistent with preventing (deterring) secret price undercutting.

2 Joe Chen Setup price game Porter (1983) and Green and Porter (1984) propose a supergame model that formulizes the issue of secret price cutting. In their model, Green and Porter assume quantity competition. Here, we will go through a version of the model where firms compete in prices. The essence is the same. Consider a market of two firms choosing prices in every period. The goods are perfect substitutes and are produced at constant marginal cost c. Consumers all buy from the low price firm, and the demand is split in halves if both firms charge the same price. In each period, there are two possible states of nature. With probability α, there is no demand (the low-demand state ); with probability 1 α, the demand is D(p) > 0 (the high-demandstate ) Denote p m arg max p (p c)d(p), andπ m =(p m c)d(p m ). Notethatafirm that does not sell at time t is not able to observe whether the absence of demand is due to the realization of the low-demand state or to rival s secret price cut. Let s look for an equilibrium with the following trigger strategies: At the beginning, both firms charge p m ; Both firms continue to charging p m until one or both of them makes zero profit and the game goes to the punishment phase. Call the phase that both firms charging p m the collusive phase; In the punishment phase, both firms charge the marginal cost c for exactly T periods (T can, a priori, be finite or infinite), and revert to the collusive phase. Note that at least one firm makes zero profit, whenever it happens, is common knowledge, even though both firms does not observe its rival s profits. Note also that the punishment phase is unavoidable in equilibrium; in other words, the punishment phase is on the equilibrium path.

3 Joe Chen Trigger strategy equilibrium To look for an equilibrium given the above trigger strategies is to look for a T such that the expected present discounted value of profits of each firm is maximal subject to the constraint that the associated strategies form an equilibrium. Let V + denote the expected present discounted value of a firm s profit fromt on, given that at t 1, the game is in the collusive phase. Let V denote the payoff of a firm s profit from t on, given that at t, the game starts the punishment phase. Then: V + =(1 α)( Πm 2 + δv + )+αδv V = δ T V +. Or, And, the incentive constraint is: V + = V = (1 α)π m 2[1 (1 α)δ αδ T +1 ] (*) (1 α)δ T Π m 2[1 (1 α)δ αδ T +1 ]. V + (1 α)(π m + δv )+αδv ; or (using the definition of V + ), δ(v + V ) Π m /2. (**) Note that, on the one hand, V must be sufficiently lower than V + to prevent undercutting. This implies T has to be long enough. On the other hand, because punishments are costly and occur with positive probability, T should be chosen as small as possible given that the constraint is satisfied. Substituting the equations of (*) to equation (**) yields: 2(1 α)δ +(2α 1)δ T +1 1.

4 Joe Chen 78 When the game starts in the collusive phase, the highest profit for the firms is obtained by solving the following problem: max T V + = (1 α)π m 2 1 (1 α)δ αδ T +1 s.t. 2(1 α)δ +(2α 1)δ T Observe first, V + is decreasing in T (again, this implies we need to find T as small as possible). Following conclusions can be drawn: For all T, 2(1 α)δ +(2α 1)δ T +1 < 1, ifα [1/2, 1]. Thisisbecausewhenα =1/2, 2(1 α)δ +(2α 1)δ T +1 = δ<1, and2(1 α)δ +(2α 1)δ T +1 is decreasing in α. When α is high, no equilibrium with the above trigger strategies is sustainable; Assume 2(1 α)δ 1 (1 α)δ 1/2, the above trigger strategy equilibrium is guaranteed using maximal punishments (T = ). Note that this generalizes the result for the deterministic demand case which corresponds to α =0. To maximize V + subject to the incentive constraint, given α and δ, itsuffices to choose the smallest T (finite) that satisfied the incentive constraint. As an example, if α =1/4, the incentive constraint requires 3δ δ T This is possible only when δ 2/3. Ifδ =0.7, thesmallestt is around Now we have price wars that are involuntary in that they are triggered, not by a price cut, but by an unobservable slump in demand. Note also that price wars are triggered by recessions, contrary to the Rotemberg-Saloner model.

5 Joe Chen Price rigidities Inthesupergameframework,weassumethatfirms always choose prices simultaneously (the synchronicity assumption). Two features of this setup are important: First, a firm s current profit isnotaffected by its rival s previous price choices when it chooses its own price (the game is a repeated game, not a fullly fledged dynamic one); second, the only reason a firm conditions its pricing behavior on previous price choices is that the other firms do so. Hence, the firms strategies are bootstrap strategies in that: The achievement of collusion stems from a subtle self-fulfilling expectation. There are no real business strategies, such as trying to regain losing market shares. In reality, past price choices affect current profits in many scenarios. This raises the discussions that price reactions are not bootstrap reactions but are real attempts to regain market shares. Past price choices may affect current profits through: Price rigidity: the existence of menu cost; On the demand side, consumers may face costs of product learning, or switching costs, or both; On the supply side, past prices affect current workload, if orders take time to be filled. In this subsection, we introduce the concept of Markov perfect equilibrium (MPE) through the examination of a model of asynchronous pricing. Let s consider two firms producing perfect substitutes. At odd (respectively, even) periods, firm 1 (respectively 2) chooses its price. At any period t, apricep it chosen by firm i lasts for two periods: p it = p i,t+1. In period t +2, firm i chooses a new price, which again will be locked in for another two periods. The assumption that firm 1 (respectively, firm 2) chooses a price at odd (respectively, even) periods and the assumtpion that the prices are lock-in for two periods, are not important; the key is the lock-in of prices for some periods. We look for an equilibrium in which the firms price choices are simple in that they depend only on the payoff-relevant information. More precisely, at date 2k +1, firm 2

6 Joe Chen 80 is still committed to the price p 2k itchoseatperiod2k. Notethatthispriceaffects firm 1 s profit atperiod2k +1and therefore, it is payoff-relevant. Consider a strategy of the form: p 1,2k+1 = M 1 (p 2,2k ).Thatis,firm 1 s strategy is conditioned on as little information as is consistent with profit maximizing. Similarly for firm 2, p 2,2k+2 = M 2 (p 1,2k+1 ). M i ( ) is called a Markov reaction function. A Markov perfect equilibrium is a perfect equilibrium in which firms use Markov strategies. Forpricep 2,2k at time 2k +1, firm 1 s reaction must maximize its objective function given that the firms will react according to M 1 ( ) and M 2 ( ) in the future. Mathematically, at period 2k +1,denotep 2,2k as p 2,andp 1,2k+1 as p 1, firm 1 s profit fromperiod2k +1on is: V 1 (p 2 )=max p 1 Π 1 (p 1,p 2 )+δπ 1 (p 1,M 2 (p 1 )) + δ 2 Π 1 (M 1 (M 2 (p 1 )),M 2 (p 1 )) + In equilibrium, p 1 = M 1 (p 2 ) must maximize the profit expression for all p 2.Firm2behaves similarly The kinked-demand story revisited Let D(p) =1 p, andfirms are producing at marginal cost c =0. The price grid is discrete: p h = h/6, whereh =0, 1,...,6. Note that p 0 =0is the competitive price, and p 3 =1/2 is the monopoly price. Consider a symmetric reaction function M 1 ( ) =M 2 ( ) =M( ) as follows: p Π(p) =36p(1 p) M(p) p 6 =1 0 p 3 p 5 =5/6 5 p 3 p 4 =2/3 8 p 3 p 3 =1/2 9 p 3 p 2 =1/3 8 p 1 p 1 =1/6 5 p 1 p 3 with prob. α with prob. 1 α p 0 =0 0 p 3 According to M( ), starting from p 3,ifafirm raises its price, its rival does not follow suit. If a firm undercuts to p 2, its rival reacts with a price war. At p 1,thefirms engage in a

7 Joe Chen 81 war of attrition. Bothfirms want the price to go back to p 3 ; however, each of them wants the other firm to move first, because the relenting firm loses market share in the short run. The outcome is a typical mixed strategy behavior in which firms raise price with positive probability. The pair of strategies (M( ),M( )) forms an MPE when the discount factor is close enough to one. To check, we need to verify that no firm would deviate from M( ): No price cutting at p 3.Atp 3, undercutting to p 2 results in: 8+δ 0+δ 2 0+δ 3 V (p 3 ); pricing at p 3 results in: V (p 3 )=4.5(1 + δ + δ ) =4.5/(1 δ). So, as long as δ is close enough to one, undercutting is not profitable; At p 2, continue the price war. Charging p 1 results in: 5+δ W (p 1 ),wherew (p 1 ) is the payoff when: a firm chose p 1 in the previous period, it is now the other firm s term to choose price, and firms use (M( ),M( )). Notethatatp 1, firms are indifferent between staying p 1 and raising the price to p 3. Hence, 2.5 +δw(p 1 )=δv (p 3 ) or, W (p 1 )=V (p 3 ) 2.5/δ. So,thepayoff of charging p 1 is: 2.5+δV (p 3 ).Thisislarger than the pay off of charging p 3 which is δv (p 3 ). At p 1, firmsplayamixedstrategywiththeprobabilityofplayingp 1 defined as: or, δ 4.5 =2.5+δ {α[2.5+δv (p 1 )] + (1 α)[5 + δv (p 3 )]}; 1 {z δ {z } } continuing p 1 raising price to p 3 Note that V (p 1 )=δv (p 3 ). α = 5+δ 5δ +9δ 2. Let p 3 be the focal price, this equilibrium is the same as the static kinked-demand-curve story, but now the reactions are real and fully rational.

8 Joe Chen Edgeworth cycle There also exist equilibria in which the price never settles. following strategy: For instance, consider the p Π(p) =36p(1 p) M(p) p 6 =1 0 p 4 p 5 =5/6 5 p 4 p 4 =2/3 8 p 3 p 3 =1/2 9 p 2 p 2 =1/3 8 p 1 p 1 =1/6 5 p 0 p 0 =0 0 p 0 p 5 with prob. β with prob. 1 β

9 Joe Chen Some final thoughts Despite multiple equilibria, it can be shown that in every MPE, profits are always bounded away from the competitive profit (whichis0). Both supergames and Markov games suggest some collusion is always possible as long as the discount factor is close enough to one. In Markov games, unlike supergames, the current profit is determined by pervious actions. When firms compete in prices, the response to a rival s previous action would be to regain its losing market share. Based on this reasoning, suppose one introduces demand fluctuation into the model, one would expect price adjustments more sluggish during booms than during recessions (more Green-Porter alike). Some sort of collusion is possible because of the fear of retaliation (triggering a price war). However, the motives for retaliation are very different with the two different setups. In supergames, the price war is a purely self-fulling phenomenon. A firm charges a low price because it expects the other firms to do so (bootstrapping). In Markov games with price rigidities, the reaction of one firm to a price cut by another firm is motivated by its desire to regain a market share that has been and continues to be eroded by its rivals aggressive pricing strategy. In an intertemporal setup, it is also possible that firms can sustain collusion through nonphysical factors such as reputation. We stop here.