Competition in Two-Part Taris Between Asymmetric Firms

Transcription

1 Competition in Two-Part Taris Between Asymmetric Firms Jorge Andres Tamayo and Guofu Tan April 1, 2016 Abstract We study competitive two-part taris (2PTs) oered by duopoly rms when consumers have private information regarding horizontal brand preferences and/or vertical quality preferences over the products. We consider a general model in which the rms have asymmetric marginal costs and asymmetric demands (or oer dierentiated goods). We provide necessary and sucient conditions under which marginal cost pricing arise in equilibrium under horizontal dierentiated consumers with homogeneous and heterogeneous taste parameter for quality. We restrict our general model to studying the equilibrium outcomes of each asymmetry separately. First, we consider a model in which both rms have symmetric demands but asymmetric marginal costs. We show that the optimal strategy for the rm with the higher marginal cost is to set its marginal price below its own marginal cost and compensate this loss with the xed fee, while the optimal strategy for the ecient rm is to set its marginal price above its own marginal cost but below its rival's price. The second model assumes that both rms have symmetric marginal costs but asymmetric demands. Similar to our rst model, we show that the optimal strategy for the rm with the vertically inferior product is to set its own price below its marginal cost, while in this case the optimal strategy for the superior rm is to set the marginal price above its rival's price. For each of these models, the disadvantaged rm has incentives to use cross-subsidies between the taris (xed fee and marginal price). However, the optimal strategy of the advantaged rm depends on the nature of each model. Finally, we provide conditions for uniqueness and comparative statics properties of the equilibrium of each model. Keywords: Oligopoly, two-part taris, marginal cost pricing, cross-subsidies JEL code: L11, L1, L15. jta- Department of Economics, University of Southern California and Central Bank of Colombia. mayo8@gmail.com. Department of Economics, University of Southern California. guofutan@usc.edu. 1

2 1 Introduction In this paper, we study competitive two-part taris (2PTs) oered by duopoly rms when consumers have private information regarding horizontal brand preferences and/or vertical quality preferences over the products. 1 We consider a general model in which consumers have continuous demands and rms have asymmetric marginal costs and/or asymmetric product demands. We analyze equilibrium outcomes of these two versions of our general model and determine comparative static properties of the equilibrium. Traditionally, 2PTs was viewed as a monopoly price discrimination tool. 2 Most of the articles that have studied competition in 2PTs assume unit demands, horizontal dierentiated consumers with homogeneous taste parameter for quality and rms with symmetric marginal costs and product demands. However, the latter assumptions are too strong for many applications of interest and do not provide a complete and accurate description of the industries that actually use 2PTs. For instance, we may be interested in situations where rms use 2PTs and oer multiple products and where consumers may want to purchase multiple units of a given product. Common examples where this relaxed assumption (continuous demand) is relevant are health clubs (tennis and country clubs), ski resorts and bars. Similarly, we may be interested to considerer situations where consumers are heterogeneous and dier by unobservable characteristics (in addition to their diering tastes for the utilities oered by rms). In fact, implementing 2PTs is often subject to the vertical uncertainty regarding consumers' quality preferences. This may explain why too little price discrimination is observed compared to what theory suggests (Armstrong, 2006). Finally, the most salient assumption adopted by previous articles (that have studied 2PTs and other forms of price discrimination) is that rms are symmetric. However, we may want to consider competition in 2PTs between asymmetric rms which have dierent marginal costs and product demands. Some of the industries in which 2PTs are widely practiced have evolved from natural monopolies before the recent worldwide liberalization of their sectors - e.g. energy, transports and communication and have experienced competition from more ecient rms (lower marginal costs) with new products and dierent demands. Here we suppose that consumers have idiosyncratic horizontal preferences a la Hotelling for diering brands of products as well as heterogeneous vertical preferences for product quality. We assume that there are two rms, A and B, along the linear market between which 1 Examples of 2PTs include credit cards, telephone services, car rentals, club memberships, equipment leasing, royalty contracts for high-tech patents, amusement parks, TV program subscriptions and many others which charge an annual membership and a usage fees; bars and night clubs set cover fees and prices for drinks. Recently, Amazon has expanded its loyalty program, Amazon Prime, through dierent benets like: access to Amazon Instant Video, free cloud storage through Amazon Web Services and the possibility to shop Lightning Deals on Prime Days. 2 In fact, traditional theories viewed 2PTs as price discrimination devices, employed exclusively by rms with market power (Hayes, 1987). A seminal contribution is the study of Oi (1971). We will refer to homogeneous consumers when agents are horizontal dierentiated only. That is, consumers have private information about their location but are homogeneous. If consumers have private information about their tastes for the goods (and not just about their location) we will refer to heterogeneous consumers. That is consumers have heterogeneous horizontal preferences. 2

3 all consumers are located. We consider a general model in which both rms have asymmetric marginal costs as well as asymmetric demands (or oer dierent products) and use 2PTs. We provide necessary and sucient conditions under which marginal cost pricing arises in equilibrium under horizontal dierentiated consumers with homogeneous and heterogeneous taste parameters for quality. In order to understand the impact of the asymmetries considered here on equilibrium outcomes, we restrict our general model and consider two particular cases: Asymmetric marginal costs: we assume that the goods oered by each rm are symmetric (or both rms have symmetric demands) but have asymmetric marginal costs. Asymmetric demands: we assume that both rms have symmetric marginal costs, but oer dierent goods (or have asymmetric demands). Finally, we provide conditions for uniqueness and comparative statics properties of the equilibrium of each model. The rest of the paper can be summarized as follows: In Section 2 we introduce our framework and discuss the basic assumptions of the primitives of the model. In Section we assume that rms have asymmetric marginal costs and asymmetric demands. We start by considering a model in which consumers have private information about their horizontal preferences (location) only. That is, we assume that the taste parameter for quality is constant for all consumers while their location is private information. We show that there exists a unique equilibrium in which rms set marginal prices equal to marginal costs and provide properties of the game. We show that this result is intimately related with Mathewson and Winter's (1997) proposition for goods that are strongly complementary in demand. Finally, we provide necessary and sucient conditions for marginal cost pricing being a Nash equilibrium when consumers are heterogeneous. Section 4 presents a model of two-part tari competition which assumes that both rms have symmetric demands but asymmetric marginal costs (Asymmetric Costs model). From Section we know that, under the assumptions described in Section 2, if consumers are heterogeneous marginal cost pricing is not a Nash equilibrium. 4 Moreover, we show that the optimal strategy for the inecient rm (the rm with the highest marginal cost) is to set its marginal price below its own marginal cost and compensate for this loss with the xed fee. On the other hand, the optimal strategy for the ecient rm is to set its marginal price above its own marginal cost, but below of its rival. This result contrasts sharply with a model in which both rms use an LP schedule; because, as the number of tools for the rms increases (from one to two) there are incentives to establish cross-subsidies between the taris (xed fee and marginal price). This is not possible in the linear pricing model. Section 5 presents the Asymmetric Demands model in which we assume that both rms have symmetric marginal costs but asymmetric demands. Similar to our previous model, from Section we know that if consumers are heterogeneous marginal cost pricing is not a 4 While if consumers are homogeneous under the assumption of full market coverage in which all consumers buy at least from one rm and both rms sell strictly positive quantities, marginal cost pricing arise in equilibrium.

4 Nash equilibrium. We show that the optimal strategy for the rm that is disadvantage in demand (which products are vertical inferior than its rival) is to set its own price below the marginal cost, while in this case the optimal strategy for the superior rm is to set its marginal price above its rival's price (and above the marginal cost). This result contrasts with our previous model, in which the rm that is disadvantage in marginal cost sets its price below its own marginal cost, but above its rival's marginal price. One reason that explain these results is that in the rst model (Asymmetric Costs) the ecient rm sets a marginal price below its rival but above its own marginal cost, while in the second model (Asymmetric Demands) by setting a price below the common marginal cost, rm A (advantage in demand) has to compensate this loss increasing the xed fee and decreasing the demand for its products. Hence, in both models the disadvantaged rm use crosssubsidies between the taris. However, the equilibrium outcome of the ecient rm depends on the nature of each model. Finally section 6 concludes and suggests directions for future research. Related literature on competitive price discrimination: A seminal contribution to the literature is Armstrong and Vickers (2001) who study competitive non-linear pricing when consumers are dierentiated á la Hoteling, have private product specic information about their tastes (vertical preferences) and purchase all products from a single rm (onestop shopping). Armstrong and Vickers (2001) suggest that in equilibrium, when market is fully covered and rms are symmetric, rms oer a simple two-part tari contract with marginal price equal to marginal cost. Note that, although Armstrong and Vickers's (2001) model is close to our approach, we cannot use their dual approach framework (modelling rms competing in the utility space) since they only considered symmetric equilibria. Rochet and Stole (2002) use a similar model and nd similar results. Particularly, Rochet and Stole (2002) interpreted the quantity in Armstrong and Vickers (2001) as quality (so consumers choose which price-quality pair to select) and show that if rms are symmetric and transportation cost is suciently low to guarantee full coverage, rms oer a costplus-fee pricing schedule. 5 As is remarked by Rochet and Stole (2002) this result implies that in equilibrium, prot margins are constant over qualities (quantities in the case of AV). However, the feasibility of this simple solution depends heavily on the assumption of symmetry, excluding from the analysis cases in which rms may oer asymmetric contracts; may have dierent marginal costs, or may oer dierent products. Armstrong and Vickers (2010) generalize Armstrong and Vickers (2001) model assuming that consumers are allowed to multi-shop (buy from both rms or from just one). Surprisingly, although the complexity of the model, Armstrong and Vickers (2010) nd that in equilibrium rms oer ecient two-part taris Note that if rms are symmetric and the market is competitive (all consumers buy at least from one rm) Armstrong and Vickers (2001) and Rochet and Stole (2002) result implies that there would be an ecient quantity or quality provision supported by the cost-based two part taris. 6 Hoernig and Valletti (2011) consider a simple version of Armstrong and Vickers' (2010) model where vertical and horizontal taste parameters are correlated. He show that neither two-part taris nor full exclusivity can arise in equilibrium 7 See also Thanassoulis (2007). 4

5 Competitive 2PTs: Yin (2004) considers a model of two-part tari competition with general horizontal preferences in which the transportation cost interacts with the quantity (transportation cost is a shipping cost). He shows that marginal prices are equal to marginal costs if and only if the demand of the marginal consumer (who is indierent to between buying the A-good or B-good in the full competition equilibrium) is equal to the average demand. So for instance, if the horizontal taste parameter is additively separate from the price (transportation cost is a shopping cost), marginal price is equal to the marginal cost in equilibrium. 8 We show that this result does not hold if we introduce vertical preferences and assume asymmetric marginal costs or asymmetric demands. In this case, the disadvantaged rm (the one with the highest marginal cost) sets prices below its marginal cost. In a context of vertically dierentiated duopoly, Herweg (2012) considers a non-cooperative three stage game: In the rst stage, two potential rms decide whether to enter the market. After observing entry decision, rms choose a quality level for its products. Finally, rms oer two part tari contracts conditional on the two previous choices. The author nds that product dierentiation arises in equilibrium if and only if two part taris are feasible. Interestingly, Herweg (2012) shows that marginal prices are greater than marginal cost due to similar reasons suggested by Yin (2004). Using a similar framework, Griva and Vettas (2012) consider a duopoly model with homogeneous goods with consumers dierentiated by their usage rate. When pricing take the form of two-part taris, the authors show how some elements of product dierentiation arise when one of the price components (the xed fee or the rate) is xed. Hoerning and Valleti (2007) model competition between two rms with consumers dierentiated á la Hotelling, and show how taris structure aects location decision, consumers and prots. The authors assume unit demands, allow for multihoming and quadratic transportation cost. Interestingly, the authors show that when both rms use two part taris, variable prices (marginal prices) are equal to marginal cost if and only if both rms are located at the same spot. Asymmetric Firms: As we mentioned before, the literature has studied dierent oligopoly pricing models with asymmetric marginal costs and demands (dierent quality between the products supplied by the rms). In the context of Bertrand and Cournot competition, Hackner (2000) extends Singh and Vives (1984) seminal work on oligopoly theory allowing for a large number of rms and vertical product dierentiation. He shows that Singh and Vives (1984) conclusions are sensitive to the duopoly assumptions. Particularly, these depend on whether the goods are substitutes or complements (as in Singh and Vives, 1984) and also on how large are quality dierences. Zanchettin (2006) extends Hackner (2000) model, allowing for asymmetric cost and demands. For the case of substitute goods, he shows that for high degree of asymmetry and low degree of product dierentiation the ecient rm's prots and industry prots are higher under Bertrand than Cournot compe- 8 Note that in our model all consumers with the same vertical parameter, θ, purchase the same quantity of a good if they decide to buy it, independently of his/her location. Thus, by construction the demand of the marginal consumer is equal to the average demand. 5

6 tition. 910 Cross-subsidization and loss-leading literature: Cross-subsidization is commonly observed in multiproduct rms who often price some products below marginal cost and subsidize the resulting loss by the prots from other products. The literature provides dierent explanations for competitive cross-subsidization. Pricing below marginal cost or loss-leading could be treated as a predatory strategy; a rm with market power pushes rivals out of the market by setting prices below marginal cost during a given period of time, raising the prices afterwards. 11 However, loss leading practices appear to be adopted consistently over time (Chen and Rey, 201), yet has been associated as a practiced commonly adopted by rms with market power. The literature has interpreted loss-leading as an advertising strategy adopted to attract consumers with imperfect information about prices (Ellison, 2005); optimal cross subsidization by multiproduct monopolists with heterogeneous elasticities across its products (Ambrus and Weistein, 2008); as an exploitative device by pricing below marginal cost the products on which the large rm competes with the smaller but more ecient rival, and rising the price on the other products (Chen and Rey, 2012); and as a device to screen consumers between one-stop shopping heterogeneous consumers (DeGraba, 2006) or between one-stop or multi-stop consumers (Chen and Rey, 201). This paper provides a dierent rationale for cross-subsidization. Here, the disadvantaged rm is the one that has incentives to use cross-subsidies between the taris (xed fee and marginal price) as an optimal strategy to extract surplus. 2 Model We assume that consumers have idiosyncratic horizontal preferences for diering brands of products and have private information about their tastes for the goods (services). 12 Particularly, we assume that there is a mass of consumers uniformly located on the unit interval [0, 1], parametrized by a distance parameter x [0, 1], with intensity for quality preferences represented by θ (θ 1,..., θ n ) Θ [ θ, θ ]n which is continuously distributed with cumulative distribution G ( ). We suppose that the distribution of θ is independent of the distribution of x. Finally, we assume that θ is associated. 1 We assume that there are two symmetrically placed rms, A and B, along the linear market between which all consumers are located. Without loss of generality we assume that 9 Notice that for all of these models there is no uncertainty, only Bertrand or Cournot competition is considered and most of them consider quadratic utility functions 10 Reverse order of Singh and Vives (1984). See also Ledvina and Sircar (2012) 11 See for example Bolton et al. (2000). 12 Tastes for the goods can be interpreted as quality preferences or as a vertical taste parameter. 1 A vector θ of random variables is associated if Cov[f (θ), g (θ)] 0 for all nondecreasing functions f and g for which E [f (θ)], E [g (θ)] and E [f (θ) g (θ)] exist. For a complete reference on association of random variables and its properties see Esary et al. (1967). See also Holmstrom and Milgrom (1994) and Milgrom and Weber (1982) for an economic application. 6

7 rms A and B are located at zero and 1, respectively. Consumers buy all products from one or the other rm, or they consume their outside option. 14 We assume that the deal oered by rm i A, B} gives utility u i R + if the consumers decides to buy from rm i, and u 0 if no purchase is made. A consumer located at x [0, 1] receives a net utility u A tx if she buys from A and a net utility u B t (1 x) if she buys from B, where t > 0 is the consumer transportation cost per unit of distance. To avoid expositional complications, we dene the set of feasible unit prices both rms can choose as P. 15 We suppose both rms oer dierent goods at a marginal (unit) price, p i, with lump-sum entry fee, F i, for i = A, B (2PT scheme). Given p i and F i, a consumer with vertical taste parameter θ Θ decides to buy q i : P Θ R + units from rm i A, B}, where, So the aggregate utility U i (p i, F i, θ) is, q i (p i, θ) = arg max q i R + u i (q i, θ) p i q i } U i (p i, F i, θ) v i (p i, θ) F i where v i (p i, θ) is the indirect utility oered by rm i, dened by, v i (p i, θ) max q i R + u i (q i, θ) p i q i } We start by considering a general model in the next section, in which both rms have asymmetric marginal cost i.e. c i c j and oer dierentiated goods. We present conditions for marginal cost pricing under the assumption of homogeneous and heterogeneous consumers. 16 We restrict our general model to studying the equilibrium outcomes of each asymmetry separately. First, we assume that the indirect utility provided by both rms are equal i.e. v i (p, θ) = v j (p, θ) = v(p, θ) for all p P and θ Θ where v (p, θ) satises (A1) but the marginal cost for rm A, the ecient rm, is lower than the marginal cost for rm B, i.e. c A < c B. The second model, assumes that both rms have symmetric marginal costs but oer dierentiated goods. In particular we assume that the product oered by rm A is vertically superior to the product oered by rm B, i.e. v A (p, θ) > v B (p, θ) for all p P and θ Θ. Before proceeding, we discuss some basic assumptions on the primitives of the model. We will focus on the case with E [v i (c i, θ)] > 0, where v i (c i, θ) is the maximum surplus oering a good at the marginal cost, c i > 0, by rm i A, B} for any θ Θ. The next assumption characterizes the set of utility functions considered here. 14 That is, we are assuming a one-stop shopping model. 15 For each model we dened the set of feasible unit prices, P. 16 Note that the term homogeneous and heterogeneous refer to the taste parameter θ. We will denote homogeneous consumers when θ is constant in the model and heterogeneous consumers when θ follows a distribution G ( ) independent of x. Note that in both cases consumers are horizontal dierentiated. 7

8 Assumption 1 The utility function u i (q i, θ) : R + Θ R + is twice continuously dierentiable, satises u i(q i,θ) q i > c, qi 2 u i (q i,θ) =0 qi 2 and j 1, 2,..., n}. < 0 θ Θ and 2 u i (q i,θ) q i θ j > 0 i A, B} Note that the indirect utility function, v i (p i, θ) satises q i (p i, θ) = v i( )/ p i by Roy's identity with 2 v i ( ) p i θ j < 0 for all i A, B} and j 1, 2,..., n}. Moreover, by continuity of the rst and second derivatived of v i (p i, θ) and by Roy's identity we know that v i (p i, θ) is submodular in (p i, θ). 17 From the properties of supermodular (submodular) functions we know that v i (p i, θ) satises increasing dierences property. That is, v i (p i,1, θ) v i (p i,2, θ) must be monotone nondecreasing in θ for all p i,1, p i,2 P and p i,1 p i,2 i A, B}. Moreover, (A1) implies that buyer's demand function and monopoly prot function, q i (p i, θ) and π i (p i, θ), respectively, are continuously dierentiable and q i (p i, θ) is strictly decreasing on p i, i A, B}. Furthermore, the expected value of the monopoly prot function, E [ π i (p i, θ) ] = E [q i (p i, θ)] (p i c) is single peak in p i under the assumption that µ i (p i ) < 1, i.e., where µ i (p i ) E[q i(p i,θ)] E[q i (p i,θ)]. E [ π i (p i, θ) ] = E [q i (p i, θ)] (p i c i ) µ i (p i )} Assumption 2 µ i (p i ) < 1, i A, B}. 18 Under (A2), there is a unique optimal monopoly price p m i P dened by E[πi (p i,θ)] p i = 0. Note that as p i p m i we have that µ i (p i ) (p m i c). In order to simplify our analysis we focus on the case of full market coverage in which all consumers buy at least from one rm i A, B} and both rms sell strictly positive quantities. This assumption implies a lower and an upper bound for t, which will depend on the model considered in each section. Finally, as we mentioned before, in this paper we don't consider situations where F i is negative. That is, we exclude the possibility that rms could subsidize consumers to go to theirs stores e.g. suppose shopping centers far from urban areas that subsidized transportation to their locations. All of the proofs are presented in Appendix A. > 0. From Topkis (1978) we know that v (p, θ) is supermodular in (p, θ). See also Milgrom and Shannon (1994), Theorem An identical assumption is used by Carrillo and Tan (2015) in a model of platform competition. Likewise, Armstrong and Vickers (2001) have a similar assumption for a model with homogeneous consumers and 17 Notice that, 2 v(p,θ) p θ i = q(p,θ) θ i symmetric rms. They assume σ (u) 0 where σ (p) = q (p) q(p) (p c) for u = v (p). The function σ (p) represents the elasticity of (expected) demand express in terms of the mark-up (p c) instead of the price p. Note also that as p c, σ (p) 0 and as p p m we have that σ (p) 1. Is straightforward to show that µ (p) < 1 implies that σ (u) 0. 8

9 Marginal Cost Pricing In this section we consider a general model and assume that rms oer dierentiated goods and have dierent marginal cost. We start by considering a model in which consumers are homogeneous i.e. we assume that the taste for quality is constant for all consumers while the location remains unknown to the rm. We show that there exists a unique equilibrium in which rms set marginal price equal to marginal costs and characterized the game. Next, we consider a model in which consumers are heterogeneous and provide necessary and sucient conditions for marginal cost pricing being a Nash equilibrium. We show that under specic conditions this equilibrium is unique. Homogeneous consumers We rst dene the set of feasible unit prices both rms can choose and the conditions for full market coverage in which all consumers buy at least from one rm and both rms sell strictly positive quantities. The set of feasible unit prices both rms can choose is, P = [c, p m ] where c min c A, c B } and p m = max p m A, pm B }. Similarly, we introduce conditions for t that guarantees full coverage and strictly positive prots for both rms in the following assumption. Without loss of generality we assume that v A (c A ) v B (c B ) > 0. Assumption a t T a where T a = } t R + v A(c A ) v B (c B ) < t < v A(c A )+v B (c B ). We will come back on the analysis of this assumption once we characterize the equilibrium. Now consider the choice of prices and xed fees by each rm. Due to our full market coverage assumption (Aa), the market share of consumers who decide to buy from rm i A, B} is, s i (p i, F i, p j, F j ) ϕ i (p i, p j ) F i + F j 2t (1) where ϕ i (p i, p j ) v i (p i ) v j (p j ). The market share for rm j i is s j (p j, F j, p i, F i ) = 1 s i (p i, F i, p j, F j ). The problem of rm i A, B} is, max p i,f i Π i = s i (p i, F i, p j, F j ) [ π i (p i ) + F i ] j i Proposition 1. Suppose the analogue of (A1) and (A2) for θ identical for all consumers as well as (Aa) are satised. Then, there exist a unique equilibrium in two-part taris characterized by p i = c i and Fi = t + ϕ i(c i,c j ) for i = A, B and i j If t < v A(c A ) v B (c B ) then the optimal strategy (corner equilibrium) for rm B is to set p B = c B and F B = 0 while rm A sets p A = c A and F A = t 2 + v A(c A ) v B (c B ) 2. For the rest of the paper we consider only interior equilibria. 9

10 Taking rst order conditions with respect to p i and F i for i = A, B and using the fact that xed fees are strictly positive, we show that rms set their prices equal to their marginal cost. Note that (Aa) guarantees that the xed fee of the most disadvantage rm is strictly positive in equilibrium. 20 Proposition 1 is close to Mathewson and Winter (1997) result for goods that are strongly complementary in demand. In our model of one-stop shopping and homogeneous consumer preferences, consider rm i's choices for i = A, B: we can interpret the permission to allow consumers to enter the shop as the rst product (product 1) and it's price to be equal to the xed fee F i, and treat the real product oered by rm i as product 2 with price equal to p i. The demand for product 1 is the market share function for rm i's product, s i (p i, F i, p j, F j ), and the demand for product 2 is the market share multiplied by the individual demand for such product s i (p i, F i, p j, F j ) q i (p i ). Note that the ratio is independent of the fee, F i. Hence the two products are strong complements. Using Proposition 2 in Mathewson and Winter we would be able to conclude that the prots are maximized for rm i at p i = c i. Hence, independently of what rm j i does, rm i will always charge the marginal cost of the second product, c i. 21 Note that if we modify our model to make it compatible with Yin (2004), Proposition 1 would be able to be derived from his Proposition 1. When the location parameter does not interact with quantity, the demand of the marginal consumer is equal to the average demand, satisfying the condition for marginal cost pricing. However, there are three important remarks: First, he assumes a general distribution for the consumers while we assume they are uniform distributed on [0, 1]. Second, although he considers the particular case in which consumers are uniform distributed, he assumes symmetric cost and symmetric demands for this case. Third, we cannot deduct important features like uniqueness (or the conditions needed) from his result. Corollary 1 If v i (c i ) > v j (c j ) then F i > F j, s i > s j and Π i > Π j for i j and i, j A, B}. From Corollary 1 we know the outcome for the asymmetric marginal costs model (indirect utilities are symmetric e.g. v i (p) = v j (p) for all p P and i j); if c i < c j then F i > F j, s i > s j and Π i > Π j. Similarly, we also know the outcome for the asymmetric demands model (marginal cost are symmetric e.g. c i = c j for i j); if v i (p) > v j (p) for all p P, then F i > F j, s i > s j and Π i > Π j. Finally, note that if both marginal costs and indirect utilities are symmetric, we get the standard 2PT symmetric result. 20 Note that we exclude from the analysis cases in which the xed fees oered by the two rms are equal to zero, otherwise we will end up considering LP contracts. If (Aa) holds, rms will have incentives to deviate and oer a 2PT scheme with positive xed fees. 21 Note that this game (and in general the set of games presented here) satises strategic complementarity on rivals' strategy (like in Bulow et al., 1985) but is not a supermodular game (like in Vives 1990 and Milgrom and Roberts 1990). The reason is that the product under consideration and access by each rm are complements to consumers, not substitutes. So, these two "products" are substitutes across the rms but complements within each rm. 10

11 Heterogeneous consumers. We shift our attention to the case where consumers dier both in their brand preferences (horizontal dierentiation) and in their quality preferences (vertical dierentiation). Under this scenario (heterogeneous consumers) we show that marginal cost pricing is an equilibrium only if certain conditions are satised. We start by modifying (A2a) to make it compatible when consumers are heterogeneous. Assumption b t T b, where T b = t R + E[v i(c i,θ) v j(c j,θ)] + ɛ < t < E[v i(c i,θ)+v j(c j,θ)] ɛ} and ɛ mine[q i(c,θ)],e[q j (c,θ)]} c i c j for i, j A, B} and j i. Due to our full market coverage assumption (Ab), the expected market share of consumers who decide to buy from rm i A, B} is, [ 1 E [s i (p i, F i, p j, F j ; θ)] E 2 + ϕ ] i (p i, p j, θ) F i + F j 2t where ϕ i (p i, p j, θ) v i (p i, θ) v j (p j, θ). The problem of each rm i A, B} is, max E s i (p i, F i, p j, F j ; θ) [ π i ]} (p i, θ) + F i for j i p i,f i First order conditions for rm i A, B} are, [ [p i ] : E q ] i (p i, θ) [ π i ] (p i, θ) + F i + π i (p i, θ) s i (p i, F i, p j, F j ; θ) = 0 (2) 2t [ [F i ] : E s i (p i, F i, p j, F j ; θ) 1 [ π i ] ] (p i, θ) + F i = 0 () 2t From (2), the marginal price for rm i in equilibrium satisfy, p i c i = 1 ( 1 ɛ i p i ɛ i s sf/p) i sq (4) where ɛ i sq E[s i(p i,f i,p j,f j ;θ)q i (p i,θ)] p i p i E[s i (p i,f i,p j,f j ;θ)q i (p i is the elasticity of the expected,θ)] total demand (the demand of the product oered times the market share of rm i) with respect to p i ; ɛ i s E[s i(p i,f i,p j,f j ;θ)] p i p i E[s i (p i,f i,p j,f j is the elasticity of the expected market ;θ)] share with respect to p i and s i F i E[s i (p i,f i,p j,f j ;θ)] F/p E[s i (p i,f i,p j,f j ;θ)π i (p i is the ratio of the expected total,θ)] income derived from the xed fee to the expected total revenue derived from the product oered by rm i for i A, B}. Observe that if F i = 0 (linear pricing competition) we get the standard representation of the monopolist's price-cost margin in terms of the elasticity of total demand. Nonetheless, if rm i uses 2PT then the optimal price (or price margin) depends on the inverse of the elasticity of the expected total demand adjusted by a proportion of the expected relative contribution of both instruments (xed fee and price) to the total revenue of the rm. Finally, note that if ɛ i sq,ɛ i s and ɛ i sq remain constant, an increase of the 11

12 xed fee relative to the expected revenue from product i, reduces the optimal price of rm i for i A, B}. Similarly, note that if marginal cost pricing is an equilibrium then 1 /ɛ i s = s i ; that is, the F/p inverse of the elasticity of the expected market share is equal to the ratio of the expected total income obtained from the xed fee to the expected total revenue obtained from product i. Likewise, the marginal price for rm i in equilibrium is above (below) the marginal cost if and only if 1 /ɛ i s > s i (<). F/p From (2) and () for rm i A, B}, we can establish general conditions under which marginal cost pricing is an equilibrium. Suppose marginal cost pricing is an equilibrium. Then, from () for rm i A, B} the xed fee is dened by, F i = t + E [ϕ i (c i, c j, θ)] (5) if, Using (5) for rm i A, B} in (2), marginal cost pricing is an equilibrium if and only Cov (q i (p i, θ), ϕ i (p i, p j, θ)) = 0 (6) for p i = c i for both i, j A, B} and i j. We summarize this result in the following proposition. Proposition 2 Suppose (A1), (A2) and (Ab) are satised. (i) For a given c i, c j P, marginal cost pricing is an equilibrium if and only if (6) holds for p i = c i for i, j A, B}. (ii) If for any p i, p j P (6) holds, marginal cost pricing is a unique equilibrium. We can use our previous example to explain under what conditions Mathewson and Winter's (1997) result holds when consumers are heterogeneous in our model; remember that we can interpret the permission to allow consumers to enter the shop as the rst product and treat the real product oered by rm i as product 2 with prices F i and p i, respectively. In this case the demand for product 1 is the expected market share for rm i's product, E [s i (p i, F i, p j, F j, θ)], and the demand for product 2 is the expected value of the market share multiplied by the individual demand for such product E [s i (p i, F i, p j, F j, θ) q i (p i, θ)]. Proposition 2(i), shows indirectly that if for i, j A, B} and i j, E [s i (c i, F i, c j, F j, θ) q i (c i, θ)] E [s i (c i, F i, c j, F j, θ)] = E [q i (c i, θ)] (7) marginal cost pricing is an equilibrium i.e., if (6) holds the ratio of the demands of the two products is independently of the xed fee, F i, for p i = c i. Hence, from Mathewson 12

13 and Winter's result we know that marginal cost pricing is an equilibrium. 22 Moreover, Proposition 2(i) shows that for a given c i, c j P this is a necessary and sucient condition. An implication of Proposition 2 is that if θ is associated, since q i (c i, θ) is monotone increasing, and if v i (c i, θ) v j (c j, θ) is monotone increasing or decreasing (depending on marginal cost and functional forms for v i ( ) i A, B}) with respect to θ, marginal cost pricing is not an equilibrium (6, is violated) except for the case when v i (c i, θ) v j (c j, θ) = k θ Θ where k is a constant. That is, note that Proposition 2 is a general result in the following sense: If marginal cost are asymmetric and the products of the two rms are symmetric e.g. v i (p, θ) = v j (p, θ) = v(p, θ) for all p P and θ Θ, and v (p, θ) satises (A1), then (6) does not hold. Likewise, if marginal cost are symmetric but v i (p, θ) v j (p, θ) is monotonic with respect to θ, then marginal cost pricing is not a Nash equilibrium. 2 An ilustrative example of the last case with symmetric marginal cost is the following: suppose for instance that for any p P and θ Θ the indirect utility oered by rm i is v (p, θ) (satises A1) and the indirect utility oer by rm j, j i, is α v (p, θ) for any α (0, 1). Then, marginal cost pricing is not an equilibrium. However, if marginal costs are asymmetric and v (p, θ), c i for i A, B} and α are such that v (c i, θ) αv (c j, θ) = 0 for all θ Θ we get an opposite result. 24 Finally, note that as c B c A 0 and v i (p, θ) = v j (p, θ) for all p P and θ Θ, we get the standard result of 2PTs; marginal cost pricing is (always) an equilibrium. 25 The reason for this remarkable dierence is related with the dependence of the xed fees and marginal prices on θ. If both marginal costs and indirect utilities (demand of the two goods) are symmetric, the most protable way for both rms to attract consumers and extract prots is to set its marginal prices equal to marginal cost and set xed charges equal to t. Note that this cost-based 2PT does not depend on θ, therefore this tari remains an equilibrium even when θ is unknown for both rms (Armstrong and Vickers, 2001; Armstrong, 2006b). However, if marginal costs or the products oered by the two rms are asymmetric, the marginal price and the xed fee would depend on θ, which makes previous result irrelevant. In sum, when we assume asymmetric marginal costs or asymmetric indirect utility functions, information about vertical taste parameters has a substantial eect on equilibrium outcomes. That is, vertical uncertainty varies the slope and the intercept of the implicit best response functions. 22 Note that the fact that (6) implies (7) for p i = c i is a special feature of the Hotelling's market share. That is, this result would not be true for a model with a general market share e.g., Armstrong and Vickers (2001). 2 By monotonic with respect to θ we refer to the following example: Let θ, θ Θ such that θ > θ (high and low type). Then if v i (p, θ) v j (p, θ) is monotonic with respect to θ, the sum of the indirect utilities oered by rm i and j to the high and low type, respectively, is higher than the sum of indirect utilities oered by rm i and j to the low and high type, respectively, e.g. v A (p, θ)+v B ( p, θ ) > v A ( p, θ ) +v B (p, θ). That is, product A is vertically superior to product B. 24 Marginal cost pricing is also an equilibrium if for example the indirect utilities oered by the two rms are such that v i (p, θ) v j (p, θ) is a constant for all p P and θ Θ. 25 See Armstrong and Vickers (2001) and Rochet and Stole (2002). 1

14 4 Asymmetric Costs In this section we suppose that indirect utilities oered by the two rms are symmetric (the goods oered by the two rms are the same) but marginal costs dier i.e., c i c j for i, j A, B and i j. Without loss of generality we assume that the marginal cost for rm A, the ecient rm, is lower that the marginal cost of rm B, the inecient rm. We assume in this section that consumers are heterogeneous i.e. dier both in their brand preferences (horizontal dierentiation) and in their quality preferences (vertical taste parameter). From Proposition 2 we know that marginal cost pricing is not a Nash equilibrium. We provide two lemmas that help us to characterize the solution. Finally we show existence and uniqueness of the game. To make notation analogous to previous sections we need to redene the set of feasible unit prices both rms can choose and the conditions for full market coverage in which all consumers buy at least from one rm and both rms sell strictly positive quantities. Note that as t tends to 0, the market share of rm B goes to 0. Thus, we need a lower bound for t so that both rms sell positive quantities and an upper bound so that full coverage condition is satised. The set of feasible unit prices both rms can choose is, ˆP = [c A, p m B ] where, c A > 0, is the marginal cost of rm A and p m B corresponds to the monopoly price for rm B. That is we restrict the set of feasible unit prices of the ecient rm to be always above its marginal cost, c A, however, prices for rm B are allowed to be lower than its own marginal cost, c B. Similarly, we need to modify (Ab) to make it compatible for the model with symmetric products. Assumption c t T c, where T c = (c B c A ). ɛ E[q(c B,θ)] The problem of each rm i A, B} is, t R + E[v(c A,θ) v(c B,θ)] + ɛ < t < E[v(c A,θ)+v(c B,θ)] ɛ} and max p i,f i ( 1 E 2 + v (p ) i, θ) v (p j, θ) F i + F j [π i ] } (p i, θ) + F i for j i 2t Taking rst order conditions with respect to p i and F i for each rm i = A, B and after some algebra, the two equations that dene the equilibrium prices, p A and p B, are, f A (p A, p B ) = ( ψ A (p A ) 1 ) ( t + E [v (p A, θ) v (p B, θ)] 2E [ π A (p A, θ) ] E [ π B (p B, θ) ] ) (8) E [v (p A, θ) v (p B, θ)] + E [ π A (p A, θ) ] + E [ π A (p A, θ) (v (p A, θ) v (p B, θ)) ] E [π A φ A (p A ) = 0 (p A, θ)] 14

15 f B (p A, p B ) ( ψ B (p B ) 1 ) ( t E [v (p A, θ) v (p B, θ)] E [ π A (p A, θ) ] 2E [ π B (p B, θ) ] ) (9) +E [v (p A, θ) v (p B, θ)] + E [ π B (p B, θ) ] E [ π B (p B, θ) (v (p A, θ) v (p B, θ)) ] E [π B φ B (p B ) = 0 (p B, θ)] where ψ i (p) E[q(p,θ)] and φ i (p) E[q(p,θ)2 ](p c i ). 26 Using the rst order conditions with E[π i (p,θ)] E[π i (p,θ)] respect to F i for i = A, B the xed fee in equilibrium is dened by, F i = t + E [v (p i, θ) v (p j, θ)] 2E [ π i (p i, θ) ] E [ π j (p j, θ) ] i j (10) Note that, as p A c A in (8) we have that p B c A and as p A c B is not true that p B c B. In fact, p B α A > c B. Similarly, from (9) as p B c B we have that p A c B while as p A c A, p B α B > c A. This analysis corroborates Proposition 2 i.e. in equilibrium rms will not set their unit prices equal to their marginal cost. We now show that the equilibrium is uniquely determined by p B < c B and p A < p B. We divide the proof in three parts: In the rst, we show that for every p A, p B ˆP that satisfy (8), it has to be true that p A < p B (Lemma 1). Next, in Lemma 2, we prove that for every p A,p B ˆP that satisfy (9) and p B > c B, it has to be true that p B < p A. Using Lemma 1 and Lemma 2 we conlude that no solution exists for p B > c B. Note that if (Ac) holds, Fi > 0 for i = A, B and (p A, p B ) [c A, c B ] 2. Next, in Lemma we show that the slopes of the implicit functions dened by (8) and (9), ξ i (p A ) : ˆP ˆP, are positive, where ξ i ( p A ) = p B is such that p A and p B satisfy (8) and (9) for i = A and i = B, respectively. Finally, we show that in equilibrium the slope of the implicit function dened by (8), ξa (p A ), is greater than the slope of the implicit function dened by (9), ξb (p A ) (Porposition 4). Lemma 1 Suppose (A1) (A2) and (Ac) are satised. If c A < c B and p A, p B ˆP satisfy (8), then it has to be true that p A < p B. To prove Lemma 1 we show that if we assume that p A p B then (8) will not hold, which implies that in equilibrium, (8) is satised if p A < p B. Note that this result holds for values of p B below and above c B. The above result is intuitive since the best response of the ecient rm, A, is always to set a price below its rival, p B, and extract a higher surplus from consumers by setting a higher xed fee, than rm B. The next lemma shows that rm B has incentives to follow a similar strategy by setting prices even below its own marginal cost. Lemma 2 Suppose (A1), (A2) and (Ac) are satised. If c A < c B, no solution exist for p B > c B. 26 From (A2) we can show that ψ i (p) > 0. Likewise, Carrillo and Tan (2015) show that (A2) implies that φ i (p) > 0. 15

16 Under the assumption that F B is nonnegative we can show that for values of p B strictly higher than c B, (9) will not hold for p B p A. 27 This result jointly with Lemma 1 conrms that there is no equilibrium for which p B is above c B. From the previous analysis we may conclude that if there is any equilibrium of the model studied in this section then it must be the case that rm B sets unit prices below its marginal cost, c B. Intuitively, the optimal strategy for rm B is to set very low unit prices (below c B ) to attract consumers and compensate this loss with a xed charge. To analyze the slope of the best response functions we need to introduce a new assumption that help us to characterize it. 28 First we introduce the following denition. Denition 1 v (p, θ) : ˆP Θ R + is separable if there exist functions v (p) : ˆP R + and h, l : Θ R where v ( ) is strictly decreasing and h ( ) is strictly increasing, such that for all (p, θ) ˆP Θ, v (p, θ) = v (p) h (θ) + l (θ) Assumption 4 v (p, θ) : ˆP Θ R + is separable. Lemma Suppose (A1), (A2), (Ac) and (A4) are satised. Then, the slope of the implicit functions dened by (8) and (9), ξi (p A ) for i = A and B, respectively, are positive for (p A, p B ) [c A, c B ] 2. This result will be useful to show existence of the solution in the following proposition. Proposition Suppose (A1), (A2), (Ac) and (A4) are satised. Then there exist a unique equilibrium in two-part taris in which p A, p B ˆP are determined by (8) and (9) and Fi for i = A, B satises (10). We rst show that the solution exists and then we prove that is unique. Intuitively, to prove existence, from Lemma 1 and 2 we know that no solution exist for p B > c B. Thus, we need to show that there is a unique solution in the set Ω (p A, p B ) where Ω (p A, p B ) = (pa, p B ) (p A, p B ) [c A, c B ] 2}. As we mentioned before, as p A c A in (8) we have that p B c A and as p A c B, p B α A > c B. Similarly, from (9) as p B c B we have that p A c B while as p A c A, p B α B > c A. Using Lemma is easy to show that α A > c B and α B (c A, c B ). This implies that the two implicit functions ξ A (p A ) and ξ B (p A ) derived from (8) and (9), respectively, cross at least once (See Figure 1). Next, from Lemma we know that the slopes of the implicit functions dened by (8) and (9), ξ A ( p A ) and ξ B ( p A ), respectively, are positive. To prove uniqueness we show that in equilibrium is true that ξ A (p A ) > pa ξb (p A ) =p A pa =pa. 27 Since rm B sets prices below its marginal cost, c B, F B cannot be negative given that under this scenario rm B will have negative prots. 28 Note that (8) and (9) implicitly dene quasi-best response functions for each rm in terms of the marginal prices p A and p B. 16

17 The intuition of the result presented in Proposition is the following: suppose that initially both rms sell products at the marginal cost and charge a positive xed fee. Then, the inecient rm, B, has incentives to decrease the marginal price below its own marginal cost and compensate this loss increasing the xed fee, keeping the market share for its products relatively constant. On the other hand, the ecient rm increases its marginal price, but keeps it below its rival's price, and decreases slightly its xed fee. Therefore, rm B is following the strategy suggested by the games with symmetric 2PTs: extract the largest share of the total income through the xed fee. While rm A is using its advantage over rm B. Figure 1: Equilibrium with Asymmetric Cost Corollary 2 In equilibrium for t T c, (i) c A < p A < p B < c B; (ii) As c B goes to c and c A goes to c, p i converges to c and F i to t, for i = A, B. Note that the rst part of Corollary 2 follows from Lemma 1 and 2. As was remarked before, the optimal strategy for rm B is to subsidized the price, setting p B < c B, and extract prots through the xed fee. The second part of Corollary 2 follows from Proposition. As the marginal cost for rm B and rm A converge to a common value c, marginal prices tend to marginal cost and xed fees to the transportation cost, t, which is the standard symmetric 2PTs result. 5 Asymmetric Demands This section presents the second model, which analyze the second type of asymmetry related with the goods oered (or equivalently with the demand) by the two rms. We consider a model in which both rms have the same marginal cost, c, but we assume that rms oer dierentiated goods. Without loss of generality, we assume that for any p P 17

18 and θ Θ the indirect utility oered by rm A is higher than the one oered by rm B i.e. v A (p, θ) v B (p, θ) > 0 for all θ Θ. We need to redene the set of feasible unit prices that both rms can choose and conditions for full market coverage in which all consumers buy at least from one rm and both rms sell strictly positive quantities. The set of feasible unit prices both rms can choose is, where γ B is such that, P = [ γ B, p m A ] E [ϕ (c, γ B, θ)] E [q A (c, θ)] = E [q A (c, θ) ϕ (c, γ B, θ)] where ϕ (p A, p B, θ) = v A (p, θ) v B (p, θ). Since ϕ (p, p, θ) > 0 for any p P and θ Θ, γ B is stricltly less than c (marginal cost of both rms). We restrict our analysis to the set of indirect utilities that satisfy (A1) such that γ B is strictly positive. This condition implies that the dierence between the two indirect utilities is bounded (equivalently, we can say that the dierence between the demands of the two products oered by the rms is bounded). In order to simplied the analysis we introduce the following assumption. Assumption 5 (i) Let v A (p, θ) = v (p, θ) and v B (p, θ) = αv (p, θ) for α (0, 1), where v (p, θ) satises (A1). (ii) v (p, θ) is separable. Intuitively, (A5i) implies that for any p P and for two indirect utility functions that satises (A5) e.g. v A (p, θ) v B (p, θ) > 0, we have that for θ, θ Θ such that θ > θ (high and low type), the sum of the indirect utilities oered by rm A and B to the high and low type, respectively, is higher than the sum of indirect utilities oered by rm A and B to the low and high type, respectively, i.e. v A (p, θ) + v B (p, θ ) > v A (p, θ ) + v B (p, θ). That is, product A is vertically superior to product B. A second implication of (A5i) is that cases in which the two indirect utilities dier by an additive constant are excluded from analysis. 29 We proceed to characterize the equilibrium of the game following a similar strategy used before. From Proposition 2 we know that marginal cost pricing is not a Nash equilibrium. We introduce two lemmas that help us to determine the solution. Finally we show existence and uniqueness of the game. We start by modifying (Ac) to make it compatible for this model. Assumption d t T d where T d = E[q(c,θ)] (c γ B ). t R + E[v(c,θ) αv(c,θ)] + ɛ < t < E[v(c,θ)+αv(c,θ)] ɛ} and ɛ In the case where both rms are allowed to oer a 2PT contract and oer dierent goods (have asymmetric demands), the problem of rm i A, B} is, 29 From Proposition 2 we know that marginal cost pricing would be an equilibrium for this case. 18