Competition in Two-Part Taris Between Asymmetric Firms

Transcription

1 Competition in Two-Part Taris Between Asymmetric Firms Jorge Andres Tamayo and Guofu Tan November 15, 2017 Abstract We study competitive two-part taris in a general model of asymmetric duopoly rms oering (both vertically and horizontally) dierentiated products. We provide a necessary and sucient condition for marginal-cost pricing to be in equilibrium, in both the Hotelling and general discrete choice approaches to horizontal dierentiation. In the Hotelling setting, when the rms face symmetric demands but have asymmetric marginal costs, we show that in equilibrium the inecient rm sets its marginal price below its own marginal cost and compensates this loss with the xed fee, while the ecient rm sets its marginal price above its own marginal cost but below its rival's price. When the rms have identical marginal costs but asymmetric demands, we show that in equilibrium the rm with the vertically inferior product sets its price below the marginal cost, while the superior rm sets its price above the marginal cost. In each case, the inferior rm cross-subsidies between xed fee and marginal price. When the market shares are determined by Logit, even in the symmetric model we show that the marginal-cost pricing is not an equilibrium. Keywords: Competition, two-part taris, marginal-cost pricing, cross-subsidies, product dierentiation JEL code: L11, L1, L15. We are grateful to to Odilon Camara, Keneth Chuk, Yilmaz Kocer, Dan Marino, and Simon Wilkie for their comments and seminar participants at the the 14th annual International Industrial Organization Conference, 91st Annual Conference - Western Economic Association International, 2016 Asian Meeting of the Econometric Society, LACEA-LAMES Anual Meeting 2016, Asia-Pacic Industrial Organisation Conference 2016, University of Southern California and Lingnan University. Department of Economics, University of Southern California. jtamayo8@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 duopolistic rms when consumers have elastic demands and private information regarding horizontal brand preferences and quality preferences over the products. We consider a general model in which rms have asymmetric marginal costs or face asymmetric vertically dierentiated demands. We analyze equilibrium outcomes of our models and determine comparative static properties of the equilibrium. The liberalization of the British electricity market at the end of the 1990s is a good example of competition in 2PTs between asymmetric rms. 1 The retail sector was separated in fourteen monopolized regional markets before it was opened to competition. In the years before and after the liberalization, rms predominantly oered single 2PTs. A stylized fact of these markets is the considerable variability of the taris oered to each region at each point in time, for the period following the liberalization. According to Davies et al. (2014) in two-thirds of all cases, the entrant oered a lower marginal price with a higher xed fee than the incumbent. 2 persistent over time. Moreover, they show that this asymmetry between the taris was 2PTs, traditionally, were viewed as a monopoly price discrimination tool. Most of the articles that have studied competition in 2PTs assume horizontally dierentiated consumers with homogeneous taste parameter for quality (or homogeneous demand), and rms with symmetric marginal costs and product demands. However, these assumptions are too strong for many applications of interest and do not provide a complete and accurate description of the industries that use 2PTs, like the British electricity market. 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 the same product. Typical examples where this assumption (elastic demand) is relevant are on-line grocery delivery services (e.g., Amazon Fresh and Instacart), membership-only retail stores (e.g., Costco and Sam's Club), health 1 Other examples of 2PTs include credit cards, telephone services, car rentals, club memberships, equipment leasing, amusement parks, TV program subscriptions and many others which charge an annual membership and usage fees; bars and nightclubs set cover fees and prices for drinks. Recently, Amazon has expanded its loyalty program, Amazon Prime, through dierent benets: access to Amazon Instant Video, free cloud storage through Amazon Web Services and the possibility to shop Lightning Deals on Prime Days. According to Morgan Stanley Research (2017) Amazon Prime penetration is expected to increse to 51% of the US households by the end of In this market, there are other types of asymmetry not considered in our paper. In particular, most of the electricity suppliers were also active in the gas market. Some of the rms were vertically integrated into the generation; National Grid provides transmission, and there is a monopoly distributor in each of these regions. For a complete description of the British electricity market see Davies et al. (2014). 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). 2

3 clubs (e.g., tennis and country clubs), and the retail electricity market (e.g., British electricity market). Similarly, we may be interested in considering situations where consumers are heterogeneous and dier by unobservable characteristics, e.g., taste parameter for quality, in addition to their brand (horizontal) preferences. For the British market, Davies et al. (2014) show that the levels of consumption (of electricity) vary signicantly across households. In fact, implementing 2PTs is often subject to the uncertainty regarding consumers' quality preferences, which may explain why too little price discrimination is observed compared to what theory suggests (Armstrong (2006)). The most salient assumption adopted by the previous literature that have studied 2PTs and other forms of price discrimination, is that rms are symmetric. However, 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 and communication and have experienced competition from more ecient rms (lower marginal costs) with new products and dierentiated demands. Thus, we may want to consider competition in 2PTs between asymmetric rms with dierent marginal costs, and oering asymmetrically dierentiated products. In this paper, we assume that consumers' types are described by a horizontal brand preference parameter as well as taste parameters for product quality, which are independent each other. We consider two dierent assumptions regarding the horizontal parameter. In the rst ve sections we study a model in which consumers have horizontal brand preferences à la Hotelling (uniformly distributed). In Section 6 we relax this assumption and consider a general discrete choice model of random utility. We assume that there are two rms with asymmetric marginal costs as well as asymmetric demands (or oer dierent products), both oering 2PTs. 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 where consumers have private information about their horizontal preferences (location) and homogeneous tastes for quality. We show that there exists a unique equilibrium in which rms set marginal prices equal to marginal costs. This result is intimately related with Mathewson and Winter's (1997) proposition for a multi-product rm selling goods that are strongly complementary in demand. Further, we provide necessary and sucient conditions for marginal-cost pricing being a Nash equilibrium when consumers have heterogeneous tastes for quality. In Section 4, we assume that both rms have symmetric demands but asymmetric marginal costs (Asymmetric Costs model). Our analysis in Section, implies that if consumers have heterogeneous tastes for quality, marginal-cost pricing is not a Nash equilib-

4 rium. 4 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 that of its rival. This result contrasts with a model in which both rms use linear pricing (LP): as the number of tools available to rms increases (from one to two), they have incentives to establish crosssubsidies across the tari instruments (xed fee and marginal price), which, of course, is not possible in the LP model. Section 5 presents the Asymmetric Demands model, in which both rms have symmetric marginal costs but asymmetric demands. Similar to our previous model (Asymmetric Costs), from Section we know that if consumers are heterogeneous in their tastes for quality, marginal-cost pricing is not a Nash equilibrium. In fact, the optimal strategy for the rm disadvantaged in demand (whose products are vertically inferior to those of its rival) is to set its own price below the marginal cost, whereas in this case the optimal strategy for the advantaged rm is to set its marginal price above its rival's price (and above the marginal cost). This result contrasts with our previous model, where the rm with higher marginal cost (disadvantaged rm) sets its price below its own marginal cost but above its rival's marginal price. This is because in the Asymmetric Costs model, the ecient rm sets a marginal price below its rival but above its own marginal cost, while in the Asymmetric Demands model, by setting a price below the common marginal cost, the rm with the advantage would have to compensate this loss increasing the xed fee, decreasing its market share. Hence, in both models, the disadvantaged rm uses cross-subsidies between the taris (e.g., marginal price below marginal cost and positive xed fee). However, the equilibrium outcome of the ecient rm depends on the nature of each model. Section 6 extends our analysis by considering a general market share (discrete choice model of random utility). We propose a model in which consumers have private information about their horizontal brand preferences and consider homogeneous as well as heterogeneous tastes for quality. In the rst case (homogeneous tastes for quality), we show that there exists a unique equilibrium in which rms set marginal prices equal to marginal costs and provide comparative statics properties of the equilibrium. In the second case, with heterogeneous tastes preferences, we provide a necessary and sucient condition for marginal-cost pricing to be an equilibrium. We further show that the results of Section 4 hold in a simplied discrete-type model. 4 Whereas if consumers are homogeneous in their tastes for quality, under the assumption of full market coverage (all consumers buy at least from one rm and both rms sell strictly positive quantities), marginalcost pricing is an equilibrium. 4

5 In Section 7, we present a second extension and study under what conditions cost-based 2PT is an equilibrium if both rms use nonlinear taris, instead of 2PTs. Armstrong and Vickers (2001) and Rochet and Stole (2002) showed that if the two rms are symmetric and under full market coverage, each rm oering a cost-based 2PT is an equilibrium. We extend this result and consider asymmetric rms and derive necessary and sucient conditions for cost-based 2PT to be an equilibrium. Finally, Section 8 concludes and suggests directions for future research. The proofs are presented in the Appendix. Related literature A seminal contribution to the literature on competitive price discrimination is Armstrong and Vickers (2001) who study competitive non-linear pricing when consumers are dierentiated á la Hotelling, have private information about their tastes for quality, and purchase all products from a single rm (one-stop shopping). They show that when the market is fully covered, and rms are symmetric, rms oer a simple 2PT contract with a marginal price equal to the marginal cost in equilibrium. Rochet and Stole (2002) interpreted the quantity in Armstrong and Vickers (2001) as quality (so consumers choose a price-quality pair) and show that if rms are symmetric and transportation cost is suciently low to guarantee full coverage, rms oer a cost-plus-fee pricing schedule in equilibrium. 5 However, this simple result strongly depends on the assumption of symmetry of the rms, excluding from the analysis cases in which rms may have dierent marginal costs, or may oer dierentiated products. 6 Our analysis extends Armstrong and Vickers' (2001) and Rochet and Stole's (2002) ndings in two ways. First, we provide necessary and sucient conditions under which marginal cost-based 2PTs is an equilibrium under horizontally dierentiated consumers with heterogeneous quality preferences. This condition allows us to identify environments in which marginal cost-based 2PTs are not an equilibrium when rms have smaller pricing spaces like 2PTs, and hence are not an equilibrium either when they are allowed to use a larger pricing space. Similarly, we show that even if rms are symmetric, marginal cost-based 2PTs may not be an equilibrium, like in the logit model with outside option. Second, we characterize 5 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's (2002) result implies that there would be an ecient quantity (or quality) provision supported by the cost-based 2PTs. 6 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 2PTs. Hoernig and Valletti (2011) consider a simple version of Armstrong and Vickers' (2010) model where vertical and horizontal taste parameters are correlated. They show that neither 2PTs nor full exclusivity can arise in equilibrium. See also Thanassoulis (2007). 5

6 the equilibrium outcome of the model when marginal cost-based 2PTs is not an equilibrium, and show for the case of the asymmetric marginal costs and asymmetric demands that the optimal solution involves cross-subsidization between the marginal price and the xed fee for the disadvantaged rm. Yin (2004) considers a model of 2PTs 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 between buying the i-good and the j-good for i j in the full competition equilibrium) is equal to the average demand. So for instance, if the horizontal taste parameter is additively separable from the price (transportation cost is a shopping cost), marginal price is equal to the marginal cost in equilibrium. 7 We show that this result does not hold in the presence of vertical preferences and asymmetric marginal costs or asymmetric demands. In this case, the disadvantaged rm (the one with the highest marginal cost) sets its prices below its own marginal cost. 8 Related to this article is the literature on cross-subsidization, which is commonly observed in multiproduct rms who often price some products below marginal cost and subsidize the resulting loss of the prots from other products. The literature provides dierent explanations for competitive cross-subsidization. DeGraba (2006) show that pricing below cost could serve as a strategy to screen the most protable consumers in a setting in which rms face heterogeneous consumers. Chen and Rey (2012) show that pricing below marginal cost could be treated as an exploitative device by pricing below marginal cost the products on which the large rm competes with the smaller rival and raising the price on the other products, allowing the large rm to discriminate between multi-shoppers from one-stop shoppers. Chen and Rey (2016) study multi-product rms with dierent comparative advantages, compet- 7 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. 8 Hoernig and Valletti (2007) consider a model where consumers are horizontally dierentiated á la Hotelling and mix goods oered by two rms, and show that taris structure aects location decision, consumers and prots. In particular, they assume unit demands (consumers can buy from only one rm or combine products from the two rms) and quadratic transportation cost. The authors show that when both rms use 2PTs, marginal prices are equal to the marginal cost if and only if both rms are located at the same spot. Griva and Vettas (2015) consider a duopoly model in which rms use 2PTs and oer homogeneous goods to a population of vertically dierentiated consumers (e.g., heterogeneous usage rate). The authors show that when one price of the components is xed for both rms, e.g., the xed fee or the marginal price, and the dierence between these xed components is large, the market is segmented, e.g., low usage consumers choose the low fee rm, and high usage consumers choose the low rate rm. Our analysis does not consider any of these cases e.g., interaction of the transportation cost with the quantity or location decisions. Thus, the reasons for marginal cost-based 2PTs (or not) are dierent from the previous models. Here, is related to the asymmetry of the rms and whether there is no correlation between each rm's ecient quantity and the dierence between the ecient consumers surpluses oered by the two rms. 6

7 ing for customers with heterogeneous transaction costs. The authors show that rms price strong products (on which they have a comparative advantage) above cost and weak products below cost. Our 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 consumer surplus. 2 Model There are two rms, A and B, oering dierentiated products to a population of heterogeneous consumers. There is a mass of consumers with types (x, θ) where x is uniformly distributed on the unit interval independently of the distribution of θ (θ 1,..., θ n ) Θ [ θ, θ] n, which is continuously distributed with cumulative distribution G ( ). 9 We adopt a one-stop Hotelling model with heterogeneous consumers with dierent tastes for quality, i.e., consumers buy all products from one or the other rm, or they consume their outside option. Consumers' preferences for the two dierentiated products can be represented by the utility function u A (q A, θ) tx if she buys from A and u B (q B, θ) (1 x) t if she buys from B, where x is the distance to rm A (and 1 x the distance to rm B), t > 0 is the consumer transportation cost per unit of distance, θ represents the preference for quality and q i is the number of units that consumer (x, θ) purchases from rm i. The next assumption characterizes the set of utility functions considered here. 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} We assume that both rms can produce their products at constant marginal costs, denoted by c A and c B, respectively. The rms use 2PTs, which include a marginal (unit) price, p i, and a lump-sum fee, F i, for i = A, B. To avoid expositional complications, we dene the set of feasible unit prices of both rms as P. 10 Given (p i, 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 net 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 } 9 In Section 6 we relax this assumption and assume that x follows a general distribution. 10 For each model, we dene the set of feasible unit prices, P. 7

8 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 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, by rm i A, B} for any θ Θ. 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 derivative 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). 11 From the properties of supermodular (submodular) functions we know that v i (p i, θ) satises increasing dierences property. 12 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}. 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. Moreover, (A1) implies that the buyer's demand function and the 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}. Assumption 2. q i (p i,θ) p i. 1 µ i (p i ) p i < 1, i A, B} where µ i (p i ) E[q i(p i,θ)] E[q i (p i,θ)] and q i (p i, θ) Under (A2), there is a unique optimal monopoly price p m i value of the monopoly prot function, P. Furthermore, the expected > 0. From Topkis (1978) we know that v (p, θ) is supermodular in (p, θ). See also Milgrom and Shannon (1994), Theorem See for example Milgrom and Shannon (1994). 1 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 11 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 demand express in terms of the mark-up (p c) instead of the price p. Is straightforward to show that µ (p) < 1 implies that σ (u) 0. 8

9 E [π i (p i, θ)] = E [q i (p i, θ)] (p i c) is single-peaked in p i under (A2). Due to our full market coverage assumption, the share of θ-consumers who decide to buy from rm i A, B} is, 14 s i (p i, F i, p j, F j ; θ) v i (p i, θ) v j (p j, θ) F i + F j 2t (1) and share for rm j i is s j (p j, F j, p i, F i ; θ) = 1 s i (p i, F i, p j, F 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 ]} (2) p i,f i for j i. We present conditions for marginal-cost pricing under the assumption of homogeneous and heterogeneous taste preferences for consumers. 15 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. 14 The full market coverage assumption requires a lower bound for t that guarantees that both rms sell strictly positive quantities and an upper bound such that all consumers buy at least from one rm. Note for each model we need dierent bounds. In Section we dene the lower and upper bound for t. For the rest of the models, the bounds are similar, so we exclude them from the analysis. 15 Note that the term homogeneous and heterogeneous refer to the taste parameter θ. We will denote homogeneous preferences when θ is constant in the model and heterogeneous preferences when θ follows a distribution G ( ) independent of x. Note that in both cases consumers are horizontally dierentiated. 9

10 Marginal Cost Pricing In this section, we assume that rms oer dierentiated products and have dierent marginal costs. We start by considering a model in which consumers are homogeneous in their tastes for quality, whereas their horizontal brand preferences remain unknown to the rm. We show that there exists a unique equilibrium in which rms set their marginal prices equal to their marginal costs. Next, we consider a model in which consumers have heterogeneous tastes preferences 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 Preferences. The set of feasible unit prices is, P = [c, p] where c min c A, c B } and p = max p m A, pm B }. Now consider the choice of prices and xed fees by each rm. Due to our full market coverage assumption, the market share of consumers, s i (p i, F i, p j, F j ), who decide to buy from rm i A, B} is dened by the analogue of (1) for θ identical for all consumers. 16 The problem of rm i A, B} is, max p i,f i Π i = max p i,f i s i (p i, F i, p j, F j ) [ π i (p i ) + F i ] for j i. Proposition 1. Suppose the analogue of (A1) and (A2) for θ identical for all consumers are satised. Then, marginal cost-based 2PT is a unique equilibrium where Fi = t + v i (c i ) v j (c j ) for i A, B} and j i. 17 Proposition 1 shows that if consumers are homogeneous in their tastes for quality, under the assumption of full market coverage, the optimal strategy for each rm is to set its prices equal to their marginal costs and extract surplus through the xed fee. Note that in this } 16 v The full market coverage assumption requires t t R + ; A (c A ) v B (c B ) < t < v A(c A )+v B (c B ). For the rest of the paper, we omit the conditions for t. 17 If t < v A(c A ) v B (c B ) then there exists an corner equilibrium in which rm B sets 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. 10

11 model the marginal costs of the two rms may be dierent, which implies that the marginal prices (and xed fees) may also be dierent. 18 Proposition 1 is close to Mathewson and Winter's (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 its 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 of 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 xed 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 rm j's actions, rm i j always charges the marginal cost of the second product, c i. 19 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 uniformly distributed on [0, 1]. Second, although he considers the particular case in which consumers are uniformly distributed, rms have symmetric costs and demands for this case. Third, we cannot deduct uniqueness (or the conditions needed) from his result. From Proposition 1 we know that in equilibrium both rms set their prices equal to their marginal costs, and that in equilibrium, the rm that provides the highest surplus (at its own marginal cost) has the highest xed fee, market share and total prots. Similarly, in 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 in equilibrium F i > F j, s i > s j and Π i > Π j. Likewise, note that for the asymmetric demands model (marginal cost are symmetric e.g. c i = c j for j i); if v i (p) > v j (p) for all p P, then in equilibrium F i > F j, s i > s j and Π i > Π j. Finally, 18 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 full market coverage assumption is satised, rms will have incentives to deviate and oer a 2PT scheme with positive xed fees. 19 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). However, neither is a game with strategic complementarities like in Vives (1990) nor a supermodular game like in Milgrom and Roberts (1994). 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. Thus, we cannot use the results derived for these set of games e.g. Nash equilibria exist and have a certain order structure. 11

12 note that if both marginal costs and indirect utilities are symmetric, we get the standard 2PTs symmetric result. Heterogeneous Preferences. We shift our attention to the case where consumers dier both in their brand preferences (horizontal dierentiation) and in their quality preferences (vertical taste parameter), and provide necessary and sucient conditions under which marginal-cost pricing is an equilibrium. Due to our full market coverage assumption, the market share and the problem of rm i A, B} is dened by (1) and (2), respectively. First order conditions for rm i A, B} are, [p i ] : E [q i (p i, θ) π i (p i, θ)] E [q i (p i, θ)] F i + 2t E [π i (p i, θ) s i (p i, F i, p j, F j ; θ)] = 0 () [F i ] : 2t E [s i (p i, F i, p j, F j ; θ)] E [π i (p i, θ)] F i = 0 (4) We can establish general conditions under which marginal-cost pricing is an equilibrium. From () and (4) we get the following condition, Cov (v i (p i, θ) v j (p j, θ), q i (p i, θ)) = 0 (5) for p i = c i for i = A, B and i j. We summarize this result in the following proposition. Proposition 2. (i) For a given c i, c j P, marginal cost-based 2PT is an equilibrium if and only if (5) holds for p i = c i for i, j A, B}. (ii) If for any p i, p j P, (5) holds, marginal cost-based 2PT is a unique equilibrium. Note that if (5) holds for p i = c i, the covariance of the demand, q i (p i, θ), and the market share, s i (p i, F i, p j, F j ; θ), is also zero. The reason is that the market share is linear with respect to dierences of the two indirect utilities, v i (p i, θ) v j (p j, θ). Thus marginal-cost pricing is an equilibrium if and only if the covariance of the demand and the market share for both rms, evaluated at the marginal cost, is zero. Now, if the demand is independent of 12

13 the market share for rm i A, B} for all feasible prices, 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 have heterogeneous tastes for quality 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)] (6) marginal-cost pricing is an equilibrium i.e., if (5) 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 and Winter's result we know that marginal-cost pricing is an equilibrium. 20 Moreover, Proposition 2(i) shows that for a given c i, c j P this is a necessary and sucient condition. Note that this condition is always satisfy for the symmetric case, as we discuss in the following corollary. Corollary 1. If c i = c j = c and v i (p, θ) = v j (p, θ) for all p P, θ Θ and j i, marginal cost-based 2PT is a Nash equilibrium. Corollary 1 is related to the standard result of 2PTs (e.g., Armstrong and Vickers (2001), and Rochet and Stole (2002)), that is, marginal cost pricing is an equilibrium for the symmetric case. An implication of Proposition 2 is that if θ is associated, 21 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 20 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 like the model in Section We assume that θ is associated for the rest of the paper. 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 economic applications. 1

14 pricing is not an equilibrium (5 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. Corollary 2. (i) If c i c j and v i (p, θ) = v j (p, θ) = v(p, θ) for all p P and θ Θ, marginal cost-based 2PT is not a Nash equilibrium; (ii) if c i = c j and v i (p, θ) v j (p, θ) is monotonic with respect to θ, then marginal cost-based 2PT is not a Nash equilibrium. Corollary 2(i) shows that if marginal costs are asymmetric and the products of the two rms are symmetric then (5) does not hold. Likewise, if marginal costs are symmetric but v i (p, θ) v j (p, θ) is monotonic with respect to θ, then (5) does not hold from Corollary 2(ii). 22 An illustrative 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 i (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. 2 The reason for this dierence -between Corollary 1 and 2- 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 the xed fee 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 (2006)). 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. In the next section we show that as the dispersion of θ increases, the quasi-best response function in terms of (p i, p j ) rotate clockwise for rm i, and counterclockwise for rm j, for i j. This implies that rms would have incentives to deviate from marginal cost-bases 2PTs. In sum, when rms have asymmetric marginal costs or asymmetric demands, information 22 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. 2 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. 14

15 about vertical taste parameters has a substantial eect on equilibrium outcomes. That is, vertical uncertainty aects the slope of the implicit best response functions regarding the marginal prices. We will further investigate this issue in the next sections. 4 Asymmetric Costs In this section we suppose that indirect utilities oered by the two rms are symmetric but marginal costs dier. Without loss of generality we assume that the marginal cost of rm A, the ecient rm, is lower than the marginal cost of rm B, the inecient rm, i.e., c B > c A 0. We assume that consumers are heterogeneous in their taste for quality, thus from Proposition 2 we know that marginal-cost pricing is not a Nash equilibrium. We rst show in Proposition that no equilibrium exists for p B c B. Indeed, we show that there exists a pure-strategy Nash equilibrium, in which (p A, p B ) (c A, c B ) 2. Next, we show that the quasi-best response functions of the two rms are increasing, 24 and that the equilibrium is unique. Finally, we provide some comparative statics with respect to θ. To make notation compatible with previous sections we need to redene the set of feasible unit prices both rms can choose, ˆP = [c A, p m B ] where p m B corresponds to the monopoly price of rm B. 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. 25 The problem of each rm i A, B} is, max p i,f i E ( v (p ) } i, θ) v (p j, θ) F i + F j [π i (p i, θ) + F i ] 2t From the rst order conditions with respect to p i, for rm i, (p i c i ) E [ 2t q (p i, θ) s i q (p i, θ) 2] + E [q (p i, θ) (t + v (p i, θ) v (p j, θ))] (7) 24 By quasi-best response functions we refer to the best response functions only in terms of p A and p B (substituting both xed fees). 25 We discuss later that this assumption is without loss of generality. 15

16 +E [q (p i, θ)] (F j 2F i ) = 0 where s i is the market share of rm i. When both rms use LP, the rst and the second term on the left-hand side of (7) characterized the best response functions of each rm i. 26 Thus, when rms use 2PT there are two eects compare to LP: rst, there is a direct eect that moves the curve of quasi-best response function of rm i to left, in the p i, p j plane, for j i. 27 This implies that rm i reacts more aggressively with its marginal price for each value of p j. Second, there is an indirect eect, since F j, also makes rm j react more aggressively, decreasing p j for each value of p i, increasing F j. The intuition for these two eects is the following: when rms are allowed to use 2PT, they can extract surplus also through the xed fee, which does not depend directly on the curvature of the demand. Thus the best strategy is to set a low price to attract consumers and extract surplus through the xed fee. We next show that in equilibrium both rms decrease their marginal prices compare to the case when both rms use LP. Now, from the rst order conditions with respect to F i for each rm i = A, B the xed fee in equilibrium is dened by, F i + E [π i (p i, θ)] = t + T S i (p i ) T S j (p j ) (8) where T S i (p i ) E [v (p i, θ)] + E [π i (p i, θ)] is the expected total surplus for rm i A, B} for j i. From (8) we know that the total prot per-consumer depends on the transportation cost and the dierence between the surplus oered by both rms. Similarly, from (7) and (8), p i is dened by, (p i c i ) 2t E [q (p i, θ) s i ] Var [q (p i, θ)]} + Cov (v (p i, θ) v (p j, θ), q (p i, θ)) = 0 (9) where s i 1 + v i v j +T S i T S j and v 2 2t i v (p i, θ) E [v (p i, θ)] for i = A, B and j i. Note that if the second term on the left-hand side is zero, marginal cost-based 2PT is an 26 If both rms use LP, the problem of rm i is, max pi E 27 See Figure 1. ( ) } v(pi,θ) v(pj,θ) 2t π i (p i, θ). 16

17 equilibrium (Proposition 2) i.e., the rst term on the left-hand side is zero if and only if p i = c i. If p i > c i, the rst term of the left-hand side of (9) is negative. This implies that the second term must be positive; p i > p j. If p i < c i, the rst term is positive, which implies that in this case p i < p j. Thus, no equilibrium exists for p B > c B. We formalize this idea in the following proposition. Proposition. There exists a pure-strategy Nash equilibrium. In any equilibrium, the following hold: (i) c A < p A < p B < c B; (ii) the expected market share, per customer prots and total revenue per consumer is greater for rm A than rm B. For existence, we rst show that no equilibrium exists for values of p B c B, as we mentioned before. Thus, we show that the two curves dened by (9) for i = A, B cross each other at least once in the set (c A, c B ) 2 - see Figure Note that (ii) follows from (i) and (8), and the fact that the expected market share is a linear function of the dierence in the expected surplus. That is, from (i) we know that in any equilibrium p A < p B, then the expected total surplus and then the market share for rm A is greater than for rm B. Similarly, note that in any pure-strategy Nash equilibrium in which, c A < p A < p B < c B, we have that, E [π B (p B, θ)] < 0 < E [π A (p A, θ)], and thus the expected revenue per consumer is greater for rm A than rm B. The intuition of this result 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. Although rm B sets its price below its own marginal cost, in any equilibrium, rm A sets a price below its rival's price, which guarantees it a greater market share. Thus, rm A does not need to set its price below its own marginal cost to get a higher market share than rm B. In sum, in any equilibrium, the expected market share, prots, total revenue per consumer 28 Particularly, note that as p A c A in (9) for i = A, 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) for i = B, as p B c B we have that p A c B while as p A c A, p B α B > c A. 17

18 and total revenue is greater for rm A than for rm B. In particular, note that the marginal price is lower and the xed fee is higher for the ecient rm than for the inecient one. These results may explain the empirical regularities observed in the British electricity market and highlighted by Davies et al. (2014); if the entrant rms are more ecient than the incumbent, we should expect lower marginal prices and higher xed fees for the entrant than the incumbent. 29 We next show in Lemma 1 that the slope of the implicit functions dened by (9) for i = A, B, R i (p A ) : ˆP ˆP, is positive, where R i ( p A ) = p B is such that p A and p B satisfy (9) for i = A, B. We show that there exists a unique equilibrium in 2PTs. Particularly, we show that in equilibrium the slope of the implicit function dened by (9) for i = A, RA (p A ) p A, is greater than the slope of the implicit function dened by (9) for i = B, RB (p A ) p A (Proposition 4). Finally, we illustrate some comparative statics properties of the equilibrium with respect to θ. To analyze the slope of the quasi-best response functions we need to introduce a new assumption that helps us to characterize it. 0 First we introduce the following denition. Denition 1. v (p, θ) : ˆP Θ R + is separable if there exist functions v : ˆP R +, h : Θ R and m : ˆP R + where v ( ) and m ( ) are strictly decreasing and h ( ) is strictly increasing, such that for all (p, θ) ˆP Θ, v (p, θ) = v (p) h (θ) + l (θ) + m (p) Assumption. v (p, θ) : ˆP Θ R + is separable. An example of the class of indirect utilities that satisfy (A) are the power functions (or constant elasticity demand) e.g., suppose that u (q, θ) = θ q then v (p, θ) = θ2 ; log function, 4p e.g., u (q, θ) = θ log q then v (p, θ) = θ (log θ 1) θ log p; and linear demand-type function, e.g., u (q, θ) = αq θq2 2 then v (p, θ) = (α p)2 2θ. 29 As we mentioned before, there are other types of asymmetries that are important in the British electricity market. Some of the rms were integrated upstream into generation, and some of them were active in the gas market. Although Davies et al. (2014) suggest small cost asymmetries between rms, we need to assume that the other type of asymmetries (vertical integration and gas oer) can be projected into the marginal cost of the rms. This may result in rms with asymmetric marginal cost. 0 Note that (9) for i = A, B implicitly dene quasi-best response functions for each rm in terms of the marginal prices p A and p B. 18

19 Lemma 1. Suppose (A) is satised. Then, the slope of the implicit functions dened by (9), Ri (p A ) p A for i = A, B, is positive for (p A, p B ) [c A, c B ] 2. Note that although both rms are using xed fees to extract surplus, the quasi-best response functions with respect to the prices of the two rms are increasing, as in the standard LP game. This result will be useful to show uniqueness of the game in the following proposition. Proposition 4. Suppose (A) is satised. Then there exists a unique equilibrium in 2PTs in which p i ˆP is determined by (9) and F i satises (8), for i = A, B. From Proposition we know that the two implicit functions R A (p A ) and R B (p A ) derived from (9) for i = A, B, cross at least once (see Figure 1). Next, from Lemma 1 we know that the slope of the implicit functions R A (p A ) and R B (p A ) is positive. To prove uniqueness we show that in equilibrium the slope of R A (p A ) is greater than the slope of R B (p A ). Figure 1: Equilibrium with Asymmetric Cost Corollary. In equilibrium, 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 Corollary follows from Proposition 4 and Proposition 2. As the marginal cost for rm B and rm A converge to a common value c, both marginal prices tend to the marginal cost, and both xed fees to the transportation cost, t, which is the symmetric 2PTs result like in Armstrong and Vickers (2001). 19

20 Finally, (A4) allows us to express (9) as a function of σ Var [h (θ)] Corollary 4. In equilibrium, (i) as σ 0, p A c A and p B c B ; (ii) as σ, p A p A and p B p B where c A < p A < p B < c B. Corollary 4(i) follows from Proposition 1 and the monotonicity of the quasi-best response functions with respect to the prices for both rms. Note that when σ = 0, the quasi-best response function for rm A is a vertical line at p A = c A in the p A, p B plane. Similarly, for rm B is a horizontal line at p B = c B. From dierent numerical simulations, we nd that as σ increases p A increases and p B decreases, i.e., as σ increases the quasi-best response function rotates to the right around (c A, c A ). Similarly, for rm B; as σ increases the quasi-best response function rotates to the left (counterclockwise) around (c B, c B ). Thus, for p i > c i, as σ increases rm i reacts less aggressively (sets a higher price) to each p j, for j i. However, for p i < c i, as σ increases rm i reacts more aggressively (sets a lower price) to each p j, for j i. This explains why when consumers are heterogeneous in their tastes marginal-cost pricing is not a Nash equilibrium. In particular, it explains why the optimal strategy for the inecient rm (B) 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 (A) is to set its marginal price above its own marginal cost but below that of its rival. Finally, from Corollary 4(ii), note that as σ increases, the marginal increase of p A is decreasing, and the marginal decrease of p B is also decreasing. 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 products. Without loss of generality, we assume that for any p P 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. In order to simplify the analysis we introduce the following assumption. Assumption 4. Let v A (p, θ) = θv (p) and v B (p, θ) = αθv (p) for α (0, 1), where v (p, θ) satises (A1) and v : ˆP R + is strictly decreasing. 20