On the Effects of Selective Below-Cost Pricing in a Vertical Differentiation Model

Colombo, Stefano Working Paper On the Effects of Selective Below-Cost Pricing in a Vertical Differentiation Model Economics Discussion Papers, No. 2009-25 Provided in Cooperation with: Kiel Institute for the World Economy – Leibniz Center for Research on Global Economic Challenges Suggested Citation: Colombo, Stefano (2009) : On the Effects of Selective Below-Cost Pricing in a Vertical Differentiation Model, Economics Discussion Papers, No. 2009-25, Kiel Institute for the World Economy (IfW), Kiel This Version is available at: https://hdl.handle.net/10419/27506 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by-nc/2.0/de/deed.en Discussion Paper Nr. 2009-25 | May 6, 2009 | http://www.economics-ejournal.org/economics/discussionpapers/2009-25 On the Effects of Selective Below-Cost Pricing in a Vertical Differentiation Model Stefano Colombo Catholic University of Milan Abstract We analyse the effects of predation in a vertical differentiation model, where the highquality incumbent is able to price discriminate while the low-quality entrant sets a uniform price. The incumbent may act as a predator, that is, it may price below its marginal costs on a subset of consumers to induce the rival’s exit. We show that the entrant may adopt an aggressive attitude to make predation unprofitable for the incumbent. In this case predation does not occur and the equilibrium prices are lower than the equilibrium prices which would emerge in a contest of explicitly forbidden predation. Moreover, we show that when the incumbent may choose whether to price discriminate or not before the game starts, if the quality cost function is sufficiently convex, there always exists a parameter space on which the incumbent prefers to commit not to price discriminate. JEL: D43, L12, L41 Keywords: Vertical differentiation; selective below-cost pricing; predation; price discrimination Correspondence Stefano Colombo, Largo A. Gemelli 1, I-20123, Catholic University of Milan, Milan, Italy; e-mail: [email protected] The author is indebted to Michele Grillo for his suggestions. All remaining errors are his own. © Author(s) 2009. Licensed under a Creative Commons License - Attribution-NonCommercial 2.0 Germany 2 1. Introduction Predatory prices are said to occur when a firm sets prices at a level which implies the sacrifice of short-run profits in order to reduce competition and obtain higher long-run profits (Motta, 2004). Moving from theory to practice, predatory prices are usually defined as prices which are below the marginal costs (Areeda and Turner, 1975), the average variable costs (Areeda and Turner, 1975), the average total costs (Joskow and Klevoric, 1979), the average avoidable costs (Baumol, 1996), and the average incremental costs (Bolton et al., 2000). Following the influential article by McGee (1958), the mere existence of predatory pricing has been debated for a long time. Nowadays, several theories explaining the rationale of predatory pricing have been developed, and economists are well convinced that predation may emerge as a complete rational choice of firms (Motta, 2004). This paper is not about the rationale of predation, but concerns the effects of predation. Notwithstanding the importance of this issue, quite surprisingly the literature about the effects of predatory pricing is scarce. To be convinced about this, one may look at the analysis of predatory pricing – presumably, the most complete one – by Bolton et al. (2000), which is very extensive about the rationale of predation, but is totally lacking in considering the effects of predation. Similarly, one may look at three recent surveys about price discrimination (Armstrong, 2006 and 2008, and Stole, 2007), where no theory concerning the effects of predatory selective price cuts is mentioned. On the same line is Spector (2005). Taking for granted predation rationality, our paper investigates on the effects of predation within a very simple vertical differentiation framework, where an incumbent firm faces the threat of the entrance by another firm. The incumbent is assumed to be able to price discriminate between consumers, while the entrant (if enters) has to set a uniform price to all consumers. This assumption can be rationalised noticing that in order to price discriminate a firm must have a quite deep knowledge of the market in which it operates. In this sense, it appears reasonable to assume that the incumbent has a better knowledge of the market than the entrant, due to the fact that it is in the market when the game starts while the entrant is outside the market 1 . Moreover, the incumbent may act as a predator in the sense of Areeda and Turner (1975), i.e. it may set belowmarginal cost prices on a subset of consumers. We show that there exists a range of parameters over which the threat of predation induces an aggressive attitude by the entrant which ultimately determines no predation and lower equilibrium prices with respect to the case in which predation is a priori impossible. Moreover, we show that when the incumbent may choose whether to price discriminate or not before the game starts, if the quality cost function is sufficiently convex, there always exist conditions on which the incumbent prefers to commit not to price discriminate in order to assure the entrant that predation will not be tempted in case of entrance. Finally, in a T-periods model, conditions are derived for the equilibrium prices to increase over time until they stabilizes at the level that would result in absence of predation. This paper is largely indebted with the fast-growing literature on price discrimination (see Liu and Serfes, 2005, Choudary et al. 2005, Encaoua and Hollander, 2007, for 1 “Adoption of discriminatory pricing by the incumbent reveals information about buyers’ reservation prices that the entrant cannot possess. Then entrant is more likely – at least initially – to set a uniform price, or divide consumers into fewer classes for pricing purposes than the incumbent” (Encaoua and Hollander, 2007, p.15). 3 recent contributions on vertical price discrimination, as well as the surveys we mentioned above). However, we want to stress that the focus of this paper is not on price discrimination, but on the predatory use of price discrimination, an issue which has been largely neglected by theory. An exception is represented by a recent paper by Karlinger and Motta (2007). The authors develop a horizontal differentiation model in which an incumbent and an entrant compete by offering a network good to asymmetric buyers. They compare the exclusionary impact of three different pricing schemes (uniform pricing, second-degree price discrimination and third-degree price discrimination), and conclude that the scheme inducing the lower equilibrium prices has also the highest exclusionary power. Our paper differs in many aspects from the work by Karlinger and Motta (2007). Just the mention the most relevant ones, we adopt a vertical differentiation setup and firms’ asymmetry instead of consumers’ asymmetry. Moreover, second-degree price discrimination is left aside. The paper proceeds as follows. In section 2 we describe the model. In section 3 we solve the model and we illustrate the main result. In section 4 we consider the price policy choice by the incumbent. In section 5 the main result is generalized to a T- periods framework. Section 6 concludes. 2. The model The framework we adopt is inspired by Tirole (1988). There is a continuum of consumers, differing in their tastes, described by the parameter ϑ which is assumed to be uniformly distributed on the interval ]1,0[ with density 1. Suppose to have two firms, H (the incumbent) and L (the entrant). Firm H produces a good of quality H s , while firm L, if it enters, produces a good of quality L s . Assume: 01 ≥>≥ LH ss : that is, firm H is the high-quality firm, while firm L is the low-quality firm 2 . Firm H is able to price discriminate between the consumers, while firm L is not able. Define with H p the price schedule set by firm H. The term “price schedule” has the same meaning as in Encaoua and Hollander (2007): it refers to a positive valued function (.) H p defined on ]1,0[ that specifies the price )( ϑ H p at which firm H is willing to sell one unit to consumer ϑ . Define with L p the uniform price set by firm L. Each consumer buys at most one unit of the good. The utility of a consumer ϑ when he buys from firm H is given by: H H psvu −+= ϑ , while his utility when he buys from firm L is given by: L L psvu −+= ϑ . Assume variable costs of quality improvement, represented by )( j sc , where LHj , = , with 0(.)' ≥ c and 0(.)" > c 3 . In what follows, we use the simplified notation H c for )( H sc and L c for )( L sc . We make the following assumptions on the parameters of the model: 2 This assumption is rooted in Lehmann-Grube (1997) article, where the author shows that in a sequential game where a leader chooses quality, then a follower chooses quality, and finally firms simultaneously set prices, the leader chooses the higher quality: that is, there is an incentive for the firm that enters first in the market to be the high-quality firm since this allows to obtain higher profits. 3 Variable costs of quality improvement arise when quality improvement depends on more skilled labour or more expensive materials (see for example Gal-Or, 1983; Motta, 1993; Crampes and Hollander, 1995; Encaoua and Hollander, 2007). 4 Assumption 1: )(2 LHHL sscc −−> Assumption 2: 2)( LH ccv +> Assumption 1 guarantees firm H has positive profits in the non-predatory duopoly, while assumption 2 guarantees that in equilibrium market is covered (see footnote 7). The timing of the game is the following. At time 0 firm L decides whether to enter the market or stay out. There are no entrance costs. If firm L enters, firms compete for two periods, period 1 and period 2. At the end of period 1 firm L leaves the market if it obtains non-positive profits, while firm H has no such financial constraint 4 . In period 2, firms compete if firm L is still in the market, otherwise firm H acts as a monopolist. Following the traditional approach in price discrimination literature with asymmetric firms, we assume that in each period first firm L (if it is present) sets its uniform price, and then firm H sets its price schedule 5 . The sub-game Nash equilibrium concept is used in solving the game. 3. Solution of the model We start from period 2. First, consider the case in which firm L is still in the market (duopoly). In this case, predation by firm H is not a relevant issue: firm H has no incentive to prey, since there are no periods left to take advantage from the monopolistic position deriving from predation. Let define L p 2 as the price set by firm L in period 2. The best price schedule firm H can set is such to serve as many consumers as possible without pricing below marginal costs. That is, the price schedule of firm H is obtained by solving: L L HD H psvpsv 2 , 2 −+=−+ ϑϑ and imposing H HD cp ≥ , 2 6 . It follows: H L LH HD cpssp ≥+−= 2 , 2 )( ϑ (1) Solving for ϑ we get the consumer which is indifferent between the two firms: LH L H ss pc − − = 2 ˆ ϑ (2) 4 Things do not change if firm L has a (limited) access to credit. What matters is that firm H has better access to credit than firm L. There are many possible explanations for this asymmetry. For example, banks have a better knowledge of firm H than firm L (firm H has entered the market first), and therefore they are more prompt to give credit to firm H than to firm L. Alternatively, given that in the nonpredatory equilibrium firm H obtains larger profits than firm L, firm H has larger collateral than firm L. For more about the credit issue in predation models, see Motta (2004). 5 See, among the others, Thisse and Vives (1988), De Fraja and Norman (1993) and Tabuchi (1999). As Tabuchi (1999) argues: “such a leader-follower relationship may be justified by the flexibility of the price schedule used by the discriminatory pricing firm since it could easily cut the price at each location in secret if it were profitable” (p.619) 6 The superscript D indicates that firm H is acting as a non-predator duopolist. Similarly, in what follows the superscripts M and P indicate respectively that firm H is acting as a monopolist and as a predator. 5 The demand of firm H is ϑ ˆ 1− , while the demand of firm L is ϑ ˆ . The profit functions of the two firms are respectively: )(2 )( )( 2 2 1 ˆ , 2 , 2 LH H L LH H HDHD ss cpss dxcp − −+− =−=Π ∫ ϑ (3) LH L HL L L LL ss pccp cp − −− =−=Π ))(( ˆ )( 22 22 ϑ (4) Consider now firm L. It chooses L p 2 in order to maximize L 2 Π. The equilibrium uniform price is: 2 * 2 LH L cc p + = (5) Substituting (5) into (1) we get the equilibrium discriminatory price schedule of firm H: 2 )(* , 2 LH LH HD cc ssp + +−= ϑ (6) Substituting (5) into (2) we get the equilibrium indifferent consumer 7 : )(2 * ˆ LH LH ss cc − − = ϑ (7) Substituting (6) and (7) into equation (3) we get firm H’ equilibrium duopolistic nonpredatory profits: )(8 )22( * 2 , 2 LH LHLH HD ss ccss − +−− =Π (8) Similarly, substituting (5) and (7) into equation (4) we get firm L’ equilibrium duopolistic profits in case of no predation: )(4 )( * 2 2 LH LH L ss cc − − =Π (9) Now, consider the case in which firm L left the market at the end of period 1. Firm H is a monopolist and it is able to extract the whole consumer surplus by setting the appropriate price schedule, which is given by: 7 Firm H’ demand is positive when 1* ˆ< ϑ , which amounts to require )(2 LHHL sscc −−> (Assumption 1). Moreover, the market is covered when the consumer with the lowest taste for quality buys the good. This requires that 0* 2 >− L pv , or 2)( LH ccv +> (Assumption 2). 6 H HM svp ϑ +=* , 2 (10) Equilibrium monopolistic profits of firm H follow from equation (10). We get: H H H HMHM c s vdcp −+=−=Π ∫ 2 )*(* 1 0 , 2 , 2 ϑ (11) By using equation (8) and equation (11) we can calculate the future gains from predation. They are simply the difference between the monopolistic profits and the duopolistic profits. Therefore: ) )(8 )22( 2 (** 2 , 2 , 2 LH LHLH H H HDHM ss ccss c s vB − +−− −−+=Π−Π≡ δ (12) where )1,0( ∈ δ is the discount factor. We move now to period 1. Consider firm H. Given the price set by the rival in the first period, L p 1 , firm H has two possibilities: on one hand it can price aggressively, in order to induce firm L’ exit at the end of the period; on the other hand, it can maximize profits in period 1. Suppose first that firm H acts in a predatory way. Firm H has to push firm L’ demand (given L p 1 ) to zero: in this way firm L obtains zero profits and leaves the market. The equilibrium aggressive price schedule is obtained through the indifference condition: L L HP H psvpsv 1 , 1 −+=−+ ϑϑ , from which it follows: L LH HP pssp 1 , 1 )( +−= ϑ (13) Note that firm H may want to price below marginal costs, while this is excluded in a non-predatory situation (compare equation 13 with equation 1). Predatory profits of firm H are therefore: 2 22 )( 1 1 0 , 1 , 1 H L LH H HPHP cpss dxcp −+− =−=Π ∫ (14) Suppose now that firm H does not prey firm L. The equilibrium prices are never lower than the marginal costs and they coincide with the prices defined in equation (1): H L LH HD cpssp ≥+−= 1 , 1 )( ϑ (15) The non-predatory profits correspond to equation (3): 7 )(2 )( 2 1 , 1 LH H L LH HD ss cpss − −+− =Π (16) Using equation (14) and (16) we can calculate the losses from predation, which amount to the reduction of current profits induced by the adoption of a sub-optimal discriminatory price schedule. Therefore: )(8 )](48[)2(4)( 11 2 , 1 , 1 LH LH LL LHHLH HPHD ss ccpbpccscc Y− +−+−++− =Π−Π≡ (17) Equation (12) and equation (17) provide the necessary and sufficient condition for predation to occurs (given L p 1 ). Since predation occurs when future gains outweigh current losses, the following inequality must be satisfied in order to observe predation: Y B > → )(8 4)]2()[(4)1)((4 2 1 LH LHLHLHLHLHL L ss ssccvccssccs p− −−−−+−++−+ ≡Γ> δδδδ (18) We state the following result: Result 1: 1) When L L cp >>Γ * 1 , at the profit-maximizing uniform price * 1 L p predation is not convenient for firm H. Therefore, the equilibrium prices are 2)(* 1LH L ccp += and *)(* 1 , 1 LHD pp , and no predation occurs. At time 0 firm L enters. 2) When Γ>> L L cp * 1 , the only prices which induce no predation are below the marginal costs of firm L. Therefore, predation occurs if firm L enters. At time 0 firm L stays out, and in equilibrium firm H sets the monopolistic price schedule H HMHM svpp ϑ +== ** , 2 , 1 in both periods. 3) When L L cp >Γ>* 1 , firm L can avoid predation. Since by avoiding predation firm L obtains positive profits in both periods, it has the incentive to avoid predation. It sets the highest uniform price which induces no predation by firm H 8 . Therefore the equilibrium prices of firm L and firm H are respectively ],max[* ˆ 1HLH L sscp −+Γ= 9 and *) ˆ (* ˆ 1 , 1 LHD pp , and predation does not occur. At time 0 firm L enters. The most interesting case is case 3). Firm L is aggressive (it sets a low price) in order to reduce the aggressiveness of firm H (firm H does not set predatory prices). Let call this strategy by firm L as a fight-to-survive strategy. The most striking consequence of the adoption of this strategy concerns the level of the equilibrium prices. By comparing the equilibrium prices under this strategy with respect to the non-predation case, we observe 8 It is immediate to note that for any price lower than * 1 L p the profits of firm L are increasing in price. 9 From equation (2) follows that for firm L’ prices lower than HLH ssc −+ the demand of firm H is zero. Therefore, firm L has never the incentive to decrease the price below HLH ssc −+ . 8 that the adoption of the fight-to-survive strategy lowers the prices for all consumers. This is due to the fact that firm L increases competition in order to reduce the incentive to prey by firm H. Note that when the threat of predation is absent, there is no need for a fight-to-survive strategy, and all the equilibrium prices would be higher. In this sense, the possibility to predation unambiguously improves the consumer surplus through the increase of competition it generates, provided that firm L is able to resist to predation: if firm L is too weak (or if the gains from predation are too high), predation occurs and consumer welfare decreases. To gain insight, in what follows we investigate on the determinants of the fight-to- survive strategy. Assume that the cost function takes the following form: 2 )( jj kssc = , with LHj , = . By taking derivates of Γ with respect to v and δ , it can be easily verified that Γ is decreasing in v and δ . Since the marginal costs of firm L are invariant in v and δ , it follows that the higher is the size of the market or the discount factor, the more stringent is the condition for the emerging of the fight-to-survive strategy. Therefore, predation is more likely to occur. The intuition is straightforward. The future gain from predation depends on the expected monopolistic profits, which in turn are affected positively by the dimension of the market. At the same time, whatever is the difference between monopolistic and duopolistic profits, such difference is more valued by firm H when the discount factor is high. It turns out that firm H is more prone to predation and the set of firm L’ marginal costs allowing for the fight-to-survive strategy shrinks. On the contrary, the derivative of Γ with respect to H s is positive. It follows that the condition for the emerging of the fight-to-survive strategy is less stringent, i.e. predation is less likely to occur. The reason is that the gains from predation (equation 12) decrease with the level of quality while the losses from predation (equation 17) increase with the level of quality. It follows that the higher is the quality of the high-quality firm the less firm H is induced to prey, and the fight-to- survive strategy is more likely to occur. Consider now parameter k (the degree of convexity of the cost function). It can be shown that when k increases, function Γ increases too. However, the marginal costs of the low-quality firm increase with k as well. Therefore, it is not obvious whether higher convexity implies more or less opportunity for predation. However, it can be proved that 2 L sk >∂Γ∂ , which implies that Γ increases with respect to k faster that L c . Therefore, higher convexity of the cost function makes predation less sustainable, all else being equal 10,11 . 4. Selecting the price policy An interesting implication of the analysis developed in section 3 is that under the fight- to-survive strategy there is actually no predation in equilibrium. However, the 10 The sign of the derivatives and the comparison between k∂Γ∂ and 2 L s have been calculated using the software Mathematica. 11 The impact of L s (the quality level of the low-quality firm) is instead ambiguous. In fact, function Γ initially decreases with L s , then increases, and finally it decreases again, while the marginal costs of firm L are obviously increasing in L s . Therefore, an unambiguous relationship between L s and the likelihood of predation cannot be found. Please note: You are most sincerely encouraged to participate in the open assessment of this discussion paper. You can do so by posting your comments. Please go to: http://www.economics-ejournal.org/economics/discussionpapers/2009-25 The Editor © Author(s) 2009. Licensed under a Creative Commons License - Attribution-NonCommercial 2.0 Germany