Two-part tariff competition between two-sided platforms

European Economic Review 68 (2014) 168–180

Contents lists available at ScienceDirect

European Economic Review journal homepage: www.elsevier.com/locate/eer

Two-part tariff competition between two-sided platforms Markus Reisinger n Department of Economics, WHU - Otto Beisheim School of Management, Burgplatz 2, 56179 Vallendar, Germany

a r t i c l e in f o

abstract

Article history: Received 28 September 2012 Accepted 14 March 2014 Available online 22 March 2014

Two-sided market models in which platforms compete in two-part tariffs, i.e., a subscription and a per-transaction fee, are often plagued by a continuum of equilibria. This paper incorporates heterogeneous trading behavior of agents into the existing framework. We show that this natural and realistic extension yields a unique equilibrium that has several reasonable properties. The equilibrium stays unique as the heterogeneity vanishes, thereby selecting a unique equilibrium from the continuum that exists under homogeneous trading behavior. We show that this equilibrium differs from equilibria obtained through other selection criteria. The analysis also provides novel empirical predictions. & 2014 Elsevier B.V. All rights reserved.

JEL classification: D43 C72 L14 Keywords: Two-sided markets Equilibrium uniqueness Heterogeneous trading behavior Two-part tariffs

1. Introduction Many industries are organized around platforms on which two groups of agents trade with each other. Prominent examples are shopping malls that facilitate transactions between shoppers and retailers, real estate agents, who act as intermediaries between house buyers and sellers, or credit card companies that allow payment by credit card between buyers and merchants. In many such two-sided markets, platforms charge two-part tariffs that consist of a subscription (or fixed) fee and a per-transaction fee to at least one side. For example, in the credit card industry merchants pay pertransaction fees each time a buyer pays via credit card and in addition are charged fixed fees e.g., for opening an account with the credit card company. Buyers usually just pay an annual fee but are not charged per transaction. The frequent use of two-part tariffs naturally begs the question what the implications of this form of price discrimination are for the profits of competing platforms and for the surpluses of the two sides. However, as pointed out by Armstrong (2006a), when platforms compete using two-part tariffs, a continuum of equilibria exists, each one with a different profit and surplus for both sides. This presents major problems for the predictive power of such models. Armstrong (2006a) assumes, like most papers in the literature,1 that agents are homogeneous in their trading behavior. Given this assumption, a platform receives the same profit via different combinations of fixed and per-transaction fee and therefore has a continuum of best responses. Only the total amount that an agent pays is important but not how this amount is divided between the two fees. Since this applies to all platforms, an equilibrium multiplicity emerges. In this paper, we show that heterogeneous trading behavior uniquely ties down both fees of the two-part tariff, eliminating the multiplicity problem. n

Tel.: þ49 261 6509 290; fax: þ49 261 6509 289. E-mail address: [email protected] 1 See, for example, Caillaud and Jullien (2001, 2003) or Rochet and Tirole (2003, 2006).

http://dx.doi.org/10.1016/j.euroecorev.2014.03.005 0014-2921/& 2014 Elsevier B.V. All rights reserved.

M. Reisinger / European Economic Review 68 (2014) 168–180

169

Heterogeneous trading behavior is a realistic assumption in most platform markets because some buyers are more active than others, and some sellers offer a larger variety of products, and therefore enjoy a larger trading volume than other sellers. We consider such heterogeneity of agents in a simple way. We suppose that there are two types of agents in each group who differ in their trading volume when interacting with an agent from the other group.2 Ex ante, platforms cannot distinguish between these two types and therefore charge the same prices to them. We show that this simple form of heterogeneity is powerful enough to reduce the set of equilibria from a continuum that prevails under homogeneity to a unique one. The intuition behind the uniqueness is that the different types react differently to a change in the tariff combination. For example, if a platform raises the fixed fee and lowers the per-transaction fee in such a way that the type with the lower trading volume is indifferent, the other type is strictly better off because he trades more often. As a consequence, the exact value of both fees matters and not only the total payment. This implies that each platform has a unique best response, resulting in a unique equilibrium. We show that this equilibrium stays unique in the limit as the heterogeneity vanishes, thereby selecting a unique equilibrium from the continuum that exists under homogeneity. The selected equilibrium is different from the profitdominant one, implying that profit-dominant equilibrium selection is not robust to the introduction of a very small degree of heterogeneity. We demonstrate in a simple linear model that the equilibrium two-part tariff with heterogeneous trading behavior has many reasonable properties, and that our analysis provides novel empirical predictions: First, the per-transaction fees are determined by the trading benefits and costs but are independent of fixed benefits. This is an appealing property since trading benefits and costs only accrue via transaction and not via subscription. Second, the per-transaction fee for side k is low if side k benefits a lot from an additional member on side k. This result complements the finding of previous literature that platforms price aggressively to the side that exerts the larger externality. This paper shows that this is still true under two-part tariffs but the lower payment is only reflected in the per-transaction fee and not in the subscription fee. Third, the per-transaction fee to a group is high if an agent of this group enjoys a large benefit from interacting with an agent from the other group. This finding is new in the literature on platform competition. It shows that when platforms compete using twopart tariffs, they couple low prices to the group who exerts a high externality with high prices to the group that receives a large benefit from interacting. Finally, we demonstrate that an alternative way to obtain equilibrium uniqueness is to consider a model in which agents are homogeneous ex ante but are uncertain about their transaction benefits at the time they decide to join a platform. This uncertainty is resolved after the participation decision, implying that agents are heterogeneous ex post. In that case, each agent makes two sequential decisions—first, to join a platform or not, and, second, how much to trade. Via the two-part tariff, the platform then influences not only consumers' participation decisions (extensive margin) but also their transaction decisions, conditional on the participation choice (intensive margin). A platform then needs both pricing instruments to optimally control both margins, implying that the tariff is tied down uniquely. The seminal papers on two-sided markets by Rochet and Tirole (2003) and Armstrong (2006a) focus on either pure pertransaction or pure subscription fees. Whereas Rochet and Tirole (2003) consider mainly per-transaction fees, Armstrong (2006a) analyzes subscription charges.3 However, for the case in which agents on both sides single-home, Armstrong (2006a) also considers two-part tariffs and demonstrates that a continuum of not payoff-equivalent equilibria exists.4 Rochet and Tirole (2006) and Bedre-Defolie and Calvano (2013) allow for two-part tariffs but focus on a monopoly platform. Since there is no competition, equilibrium multiplicity does not emerge.5 In an extension, Bedre-Defolie and Calvano (2013) consider (imperfect) platform competition. In their model, consumers learn their transaction benefits only after the decision to join a platform, which, as discussed above, eliminates the multiplicity problem. Liu and Serfes (2013) analyze perfect price discrimination in a spatial model, that is, platforms can charge different prices to consumers at different locations. With this form of price discrimination the problem of equilibrium multiplicity does not emerge. We compare our findings with the ones of Liu and Serfes (2013) in more detail in Section 3.2. As the above-mentioned papers, the present paper does not deal with the coordination problem of consumers that can occur in markets with (indirect) externalities. This problem leads to a different kind of equilibrium multiplicity than the continuum-of-tariffs problem, i.e., the equilibrium depends on consumers' expectations of other consumers joining one or the other (or both) platforms. Consistent with e.g., Armstrong (2006a), we avoid this problem by assuming that platforms are sufficiently differentiated, so that both are active in any equilibrium.6

2

This can also be interpreted as one type interacting with a smaller probability than the other type. Both papers work out several important insights of pricing in two-sided markets that apply to a variety of market structures. For a summary of the results appearing in industries with two-sided platforms and its implications for antitrust policy, see Rysman (2009). 4 Rochet and Tirole (2003) in the Appendix also analyze two-part tariffs but suppose that platforms compete just in the sum of the two charges. Since, as pointed out by Armstrong (2006a), this is not equivalent to offering two-part tariffs, they obtain a unique equilibrium. 5 Roger and Vasconcelos (forthcoming) also study two-part tariffs by a monopoly platform, but focus on reputational concerns, and find that two-part tariffs can overcome the sellers' moral hazard problem to supply low quality. 6 Papers actively dealing with the consumer-coordination problem include Caillaud and Jullien (2001, 2003) and Ambrus (2006). Caillaud and Jullien (2001, 2003) assume that consumers have favorable beliefs for a particular platform, implying that the equilibrium with the highest participation on the favorable platform is selected. This concept is used e.g., in Hagiu (2006) and Jullien (2011). Ambrus (2006) develops the concept of coalitional rationaliziability, which takes into account not only single-player deviations but also deviations by coalitions, thereby reducing the set of Nash equilibria. This concept is applied to two-sided markets in Ambrus and Argenziano (2009). 3

170

M. Reisinger / European Economic Review 68 (2014) 168–180

Weyl (2010) allows for general tariffs and considers a monopoly platform. He develops the concept of insulated equilibrium that helps to overcome the consumer-coordination problem and derives the profit and welfare maximizing price structure.7 White and Weyl (2012) develop this concept further and apply it to platform competition. They show that using the concept leads to a unique equilibrium, i.e., it resolves the consumer-coordination problem and the multiplicity-oftariffs problem. They find that the pricing structure in the insulated equilibrium generalizes the pricing structures obtained in previous papers, e.g., in Armstrong (2006a) or Anderson and Coate (2005). In contrast to their paper, we focus on twopart tariffs and show that a simple form of heterogeneity is sufficient to guarantee equilibrium uniqueness. The rest of the paper is organized as follows: Section 2 presents the model with homogeneous trading behavior. Section 3 introduces heterogeneous trading behavior, shows that the resulting equilibrium is unique, and analyzes the properties of the equilibrium. Section 4 discusses the robustness of the results, and Section 5 concludes.

2. A competitive bottleneck model with homogeneous trading behavior In the main part of the paper, we focus on the competitive bottleneck model, in which agents of group b—the buyers— deal only with one platform (single-home), while agents of group s—the sellers—wish to deal with each platform (multihome).8 There are several markets that fit well with the competitive bottleneck structure, e.g., the credit card market, where merchants accept many cards while customers often possess only one card,9 or Internet trading platforms, where sellers post offers on many platforms while buyers often use just one. There are two platforms denoted by i¼ 1,2 that enable interaction between two groups of agents denoted by k ¼ b; s. Each i platform i can set two different prices to each group. The first is a fixed or subscription fee denoted by pk that an agent of i group k pays for joining platform i. The second is a per-transaction charge, denoted by γk, that an agent of group k has to pay each time she interacts with a member of the other group on platform i. So overall each platform charges four prices. To make the point in the simplest possible way, we focus on a set-up with linear demand and cost functions, which i allows us to solve the model in closed form.10 Denote the number of agents of group k who join platform i by nk. We start with the buyers. To capture competition for buyers in the most tractable way we model it in Hotelling fashion, as e.g., in Anderson and Coate (2005), Armstrong (2006a) or Peitz and Valletti (2008). There is a mass 1 of buyers that is uniformly distributed on a line of length 1, platform 1 is located at point 0 while platform 2 is located at point 1. Transportation costs are denoted by t. Each buyer obtains a benefit of αb from interacting with a seller. The utility of a buyer (gross of transport costs) when joining platform i is given by U ib ¼ ðαb γ ib Þnis  pib . Therefore, the number of buyers on platform i is j j j 1 ðαb  γ ib Þnis  pib  ðαb γ b Þns þ pb nib ¼ þ ; 2 2t

i a j; i; j ¼ 1; 2:

ð1Þ

An implicit assumption in the set-up is that each buyer who joins platform i trades with each seller on platform i, i.e., buyers are homogeneous with respect to their trading behavior. We now turn to the sellers. The utility of a seller who joins platform i is given by U is ¼ ðαs  γ is Þnib pis , where αs measures the benefit that a seller enjoys from interacting with a buyer.11 Sellers differ with respect to αs. Let αs be uniformly distributed between 0 and α s . Normalizing a seller's outside utility to zero, a seller joins platform i if αs Zpis =nib þγ is . Assuming that there is a unit mass of sellers, we obtain ! pis γ is i ns ¼ 1  þ : ð2Þ α s nib α s i

Thus, ns does not depend directly on the fees charged by platform  i because sellers multi-home. In line with the literature, i we assume that frictionless payment between the two sides is not possible, that is, if γs increases, a seller cannot raise the trading price when interacting with a buyer. This is the case e.g., in the credit card industry due to the no-surcharge rule.12 Each platform incurs costs c per transaction between any pair of agents, e.g., due to installing devices to monitor these i i transactions.13 The profit of platform i is then given by Π i ¼ pib nib þ pis nis þðγ ib þ γ is  cÞnib nis , where nb and ns are defined in (1) and (2), respectively. 7 The idea of insulated equilibrium is that the platform chooses an allocation—a participation rate for both sides—directly instead of choosing a price pair. To avoid failure of the implementation of the desired allocation, the platform changes the price to side k if less or more than expected agents of side  k participate. Thereby, it insulates the participation of side k from the participation of side  k. 8 We consider the case of two-sided single-homing in Section 4.3. 9 To focus on our main point of interest, we abstract from other features of the credit card industry, e.g., the interaction between issuer and acquirer bank. For in-depth studies of the credit card industry, see, for example, Rochet and Tirole (2002) or Wright (2003, 2012). 10 However, the main insights remain valid in a more general setting allowing for non-linearities. This analysis is presented in an online appendix available at http://ssrn.com/abstract=2314134. 11 Consistent with most of the literature, we assume that there are no direct externalities within agents of one group. For papers that consider intragroup externalities between sellers, see, for example, Nocke et al. (2007), Belleflamme and Toulemonde (2009), Galeotti and Moraga-González (2009), and Hagiu (2009). 12 For a detailed discussion on the implications of this assumption, see Wright (2012). 13 For simplicity, we abstract from fixed costs per seller or buyer.

M. Reisinger / European Economic Review 68 (2014) 168–180

171

To ensure that the profit function of a platform is concave, we assume that the differentiation parameter t is large enough. This assumption is necessary and sufficient for market-sharing equilibria to exist.14 To simplify the exposition we restrict attention to the case in which an interior mass of sellers participates in equilibrium. This is the case if αb þ α s 4 c 4 αb  α s . We also assume γ ib r αb , that is, the buyer per-transaction fee must be lower than the per-transaction benefit, otherwise buyers will never find it optimal to trade. Finally, since each platform has four optimization variables, there may exist asymmetric equilibria. Consistent with Armstrong (2006a), we will focus on symmetric equilibria. Maximizing platform i's profit function with respect to the four strategy variables, we obtain that the first-order i i conditions for pk and γk are exactly the same. Therefore, two first-order conditions are redundant, and we have a system of two equations for four unknowns. As shown in the Appendix, a possible solution to this system, which specifies the fixed fees but leaves the per-transactions fees undetermined, is15 pb ¼ t 

2α s αb þ ðα s Þ2  3ðαb Þ2 þ c2  2cðα s  αb Þ þ4γ b ðαb þ γ s Þ 4αb γ s 4α s

ð3Þ

and ps ¼

α s þ c αb  2γ s : 4

ð4Þ

Inserting the equilibrium fees into the profit function of a platform we obtain that the equilibrium profit is t ðα  γ b Þð2γ s þαb  α s cÞ Π¼ þ b ; 2 4α s

ð5Þ

leading to the following result: Remark 1. With homogeneous trading behavior there is a continuum of symmetric equilibria. In these equilibria, the total payment of group k ¼ b; s is given by pk þγ k n  k , where pb and ps are given by (3) and (4), respectively, while γb and γs are free parameters. The profit of each platform is given by (5). Proof. See Appendix. The analysis shows that Armstrong's (2006a) result that there exists a continuum of equilibria in a model with two-sided single-homing also holds for competitive bottlenecks. The multiplicity of equilibria is due to the fact that a platform has a continuum of best responses to the rival's tariff. This is because only the total price that an agent pays is relevant for profit but not the composition between fixed and pertransaction fee.16 Thus, there is a continuum of combinations of fixed and per-transaction fee that give the same surplus to buyers and sellers, and hence also the same profit to a platform. In addition, this continuum of best responses does not only hold for the equilibrium tariff of the rival but also for any tariff, resulting in a continuum of equilibria. In this respect, the game differs from e.g., matching pennies, which has a unique (mixed strategy) equilibrium. In such a game, a player has a continuum of best responses only to the equilibrium strategy of the rival player. It is evident from (5) that the different equilibria are not payoff-equivalent, that is, platforms' profits depend on the exact value of the per-transaction fee. The reason is as follows: Charging a different per-transaction fee to one side, while keeping the total price constant, affects the cross-group externality that agents of this side enjoy. This changes the demand elasticity of agents on the other side because attracting e.g., an additional buyer leads to an increase in the seller demand, the extent of which depends on the per-transaction fee to sellers. Since the demand elasticity has changed, the rival platform optimally reacts with a different tariff, which leads to an equilibrium with a different profit. Determining the best equilibrium for the platforms, it is easy to see that the equilibrium profit is increasing in γs since αb Zγ b . Hence, it is optimal for the platform to set γs as high as possible. The second bracket in the numerator of (5) is then positive, leading to an optimal γb that is as low as possible. Therefore, the best equilibrium for the platforms involves γs as large as possible and γb as small as possible.17 The intuition is as follows: Since platforms compete for buyers, reducing the cross-group externality from buyers to sellers by increasing the seller per-transaction fee γs reduces the competitive pressure in the buyer market. By contrast, platforms do not compete for sellers. Therefore, they benefit from setting γb as low as possible and raising the fixed fee pb to extract more buyer surplus. Since equilibrium profits depend on the selected equilibrium, the utility of at least one of the two groups also depends on the selected equilibrium. This is the case for the buyers while the utility of the sellers is independent of the selected equilibrium. Since nb ¼ 1=2, the total payment of a seller is ps þγ s =2. As is evident from (4), γs then cancels out in the total 14

The precise condition on t is provided in the Appendix. Since only two of the four unknowns are specified, an alternative way to present the solution is e.g., to solve (3) and (4) for γb and γs and leave pb and ps free. i 16 Consider for example the fees to the sellers. Since in a symmetric equilibrium nb ¼ 1=2, an increase in γs by one unit can exactly be compensated by a i decrease in ps by half a unit. 17 The model as formulated does not include an upper ceiling on γs or a lower ceiling on γb. To focus on the main point, we sidestep this issue here. However, such ceilings can be incorporated in a more elaborate model, thereby restricting platforms to increase γs beyond a particular level and reduce γb below a particular level. 15

172

M. Reisinger / European Economic Review 68 (2014) 168–180

payment. Because the equilibrium profit and the utility of the buyers depend on the selected equilibrium, the continuum of equilibria causes major problems for the predictive power of models with two-part tariffs.

3. Heterogeneous trading behavior Suppose now that there are two different types of agents on each side. These two types are heterogeneous with respect to their trading behavior. On the buyer side, there is a mass 1  qb of “small” buyers, denoted as β-type buyers, while the remaining mass qb is of normal type. Similarly, on the seller side there is mass 1  qs of “small” sellers, denoted as λ-type sellers, while the remaining mass qs is of normal type. If a buyer and a seller of normal type meet, their transaction volume (or transaction probability) equals one, as in the previous section. However, when a small buyer meets a seller of normal type, the transaction volume is now only β o 1. Similarly, when a small seller meets a normal buyer, the transaction volume is λ o 1. Finally when a small buyer and a small seller meet, the transaction volume equals βλ oβ; λ.18 A natural interpretation of these two types is that there are some buyers who buy only a small number of products from each seller while others shop a large amount of goods. Similarly, some sellers offer only a small variety of goods, implying that they sell a smaller amount than sellers with a big variety. In addition to the difference in the trading volume, we allow the small buyers to differ from the normal types also in their trading benefit. In particular, instead of obtaining a benefit of αb per transaction as the normal types do, they obtain a benefit of αbβ , which can be larger or smaller than αb.19 i i As before, platform i sets four prices, i.e., a subscription fee pk and a per-transaction fee γk, k ¼ b; s, to each side. This implies that we rule out price discrimination both directly and indirectly through menus of contracts. Although such an analysis is of interest, it would shift the focus away from the main goal of the paper, which is the determination of a unique equilibrium.20 We can now determine the number of agents on both sides that choose to trade on platform i. The number of buyers of different types can be written as j j i j 1 ðαb  γ ib Þn s  pib  ðαb  γ b Þn s þpb nib ¼ þ 2 2t

ð6Þ

j j i j 1 βðαbβ  γ ib Þn s  pib βðαbβ  γ b Þn s þ pb ; nibβ ¼ þ 2 2t

ð7Þ

and

with n is  ð1  qs Þλnisλ þ qs nis . Thus, n is represents the trading volume of a buyer of normal type who is active on platform i. We can write the number of sellers of different types as ! ! pis γ is pis γ is i i ns ¼ 1  þ þ and nsλ ¼ 1  ; ð8Þ α s n ib α s α s λn ib α s with n ib  ð1  qb Þβnibβ þqb nib , i.e., n ib represents the trading volume of a seller of normal type who is active on platform i. The profit function of platform i is then given by Π i ¼ pib ðð1  qb Þnibβ þ qb nib Þ þ pis ðð1 qs Þnisλ þ qs nis Þ þ ðγ ib þγ is  cÞn ib n is . 3.1. Equilibrium uniqueness Solving for the equilibrium, we obtain that with heterogeneous trading behavior none of the four first-order conditions is redundant, and the equilibrium fees are given by21 pb ¼ t;

ps ¼ 0

γb ¼

and

αbβ βð1  qb Þ þαb qb α s  c ;  2 2ðβð1  qb Þ þ qb Þ γs ¼

α s þ c αbβ βð1 qb Þ þ αb qb  : 2 2ðβð1  qb Þ þqb Þ

ð9Þ

ð10Þ

18 As will become evident below, it is not important if β or λ are larger or smaller than 1 but just that they differ from 1. Therefore, introducing “large” buyers and sellers, i.e., β 41 and λ 41, does not alter the main result. 19 To simplify the exposition, we assume that the λ-type sellers have the same benefit distribution as the normal types, i.e., αs is uniformly distributed between 0 and α s . In the online appendix, we also allow for different benefit distributions of the two seller types. 20 For an analysis of screening contracts in the presence of externalities between agents, see, among others, Csorba (2008) and Gomes and Pavan (2013). 21 The reason why the equilibrium fees are independent of qs and λ is, first, the linearity of the model and, second, that sellers differ only with respect to their trading behavior but not with respect to the benefit distribution. In the more general model, the fees are more complicated and depend on all parameters.

M. Reisinger / European Economic Review 68 (2014) 168–180

173

Inserting these fees into the platforms' profits, we obtain Π ¼ t=2, leading to the following result: Proposition 1. With heterogeneous trading behavior there is a unique symmetric equilibrium, in which fees are given by (9) and (10), and profits are Π ¼ t=2. Proof. See Appendix. The intuition behind the equilibrium uniqueness is that the two types in each group react differently to a change in the per-transaction fee. Because a seller of type λ trades less often than a seller of normal type, an increase in the per-transaction fee together with a reduction of the subscription fee, which keeps the utility of a seller of type λ constant, makes a seller of normal type strictly worse off.22 This implies that different tariffs lead to different profits. A platform then has a unique optimal best response to any tariff of its rival, resulting in a unique equilibrium. It is also instructive to understand why the particular form of heterogeneity helps to overcome the multiplicity while other approaches do not work. In the model, we allowed the two buyer types to differ not only in the trading volume but also in the trading benefit. Although this difference is realistic as well, it alone does not help to get rid of the continuum of equilibria. The reason is that if trading behavior were the same for the two types, i.e., β¼1, the effect of a change in the per-transaction fee on the utilities of the two types would still be the same. This is because they are only heterogeneous in a dimension that is not affected by the per-transaction fee. We finally note that the assumption of only two types is not restrictive, i.e., the main result holds for any number of types, as long as at least two of them differ in their trading behavior. To make the point in the simplest way, we focus on the two-type case. The limiting case: A case of particular interest is the one in which the heterogeneity between the two types vanishes. Analyzing this case allows us to determine if introducing heterogeneous trading behavior also selects a unique equilibrium from the continuum that prevails under homogeneity. As the heterogeneity becomes negligible, we have qb -1 and qs -1. Applying this to the equilibrium prices given by (9) and (10) yields pb ¼ t;

γb ¼

c þ αb  α s ; 2

ps ¼ 0

γs ¼

and

c þ α s αb : 2

ð11Þ

We therefore obtain the following result: Proposition 2. In the limit as the heterogeneity vanishes, equilibrium fees are uniquely determined by (11). Fig. 1 depicts the equilibrium fees (left-hand side) and equilibrium profits (right-hand side) with heterogeneous and homogeneous trading behavior. In the left-hand side of the figure, the solid lines including the dots denoted by pb, ps, γb and γs represent the equilibrium fees for the model with heterogeneous trading behavior, while the dashed vertical lines pb ðγ b ; γ s Þ and ps ðγ b ; γ s Þ represent the set of equilibrium fees with homogeneous trading behavior. Similarly, for the equilibrium profit in the right-hand side of the figure. The figure illustrates that a continuum of equilibria exists when analyzing the model in the limit with the small types not being present. By contrast, approaching the limit by letting the mass of the small types go to zero results in a unique equilibrium that is in the set of equilibria of the limit case. As a consequence, the equilibrium with this simple form of heterogeneity is a natural candidate for the “correct” equilibrium in the limit with homogeneity. It is important to note that the selected equilibrium in the limiting case is not the profit dominant one. As shown in the last section, the profit-dominant equilibrium involves γs as high as possible and γb as small as possible. As is evident from (11), introducing heterogeneity leads to a different equilibrium. In case of homogeneous trading behavior, one might be tempted to use profit dominance as a natural selection criterion. Our analysis shows that introducing a tiny amount of heterogeneity then involves a discrete jump in the equilibrium outcome, implying that profit dominance is not robust to a very small degree of heterogeneity. Alternatively, one could argue that platforms may coordinate on an equilibrium in which both either use pure subscription or pure per-transaction fees. The model shows that this selection criterion is also not robust to the introduction of heterogeneous trading behavior. Relation to one-sided models: A natural question is why the equilibrium multiplicity is particularly problematic in a twosided market model as compared to a one-sided one. To analyze this relationship, let us now consider a simple one-sided market. Suppose that two firms compete in Hotelling fashion only for buyers and that the number of sellers is fixed for both platforms and is equal to n^ s . The demand function of firm i is then 1 ðαb γ ib Þn^ s pib  ðαb  γ b Þn^ s þ pb nib ¼ þ 2 2t j

j

i

i

and the profit function is Π i ¼ pib nib þðγ ib  cÞnib n^ s . Maximizing with respect to pb and γb delivers again that one of the firstorder conditions is redundant. The equilibrium prices can be written as pb ¼ t þ ðc  γ b Þn^ s . However, inserting pb into the profit function yields a unique equilibrium profit of Π ¼ t=2 and a unique buyer utility (gross of transport costs) of v  t þðαb  cÞn^ s . Therefore, although the equilibrium fees are also not uniquely determined, the allocation is unique and the model has full predictive power. Heterogeneity can pin down the equilibrium fees uniquely leading to pb ¼ t and γ b ¼ c. 22 In graphical terms, drawing the indifference curves of the two types of side k in a plane with the two fees on the axes, the only intersection point between the indifference curves is the point at which the per-transaction fee is equal to zero.

174

M. Reisinger / European Economic Review 68 (2014) 168–180

Fig. 1. Equilibrium fees (left-hand side) and equilibrium profit (right-hand side).

This analysis shows that the problem of multiple equilibrium allocations is inherent in two-sided markets. The reason why there is a unique allocation in one-sided markets is as follows: Although a firm has a continuum of best responses, the exact price combination it plays does not change the demand elasticity since there is only one consumer group. Therefore, the equilibrium profit is unique. By contrast, in a two-sided market, playing a different best response in the tariff to one side affects the cross-group externality of this side and thereby changes the demand elasticity on the other side. This in turn leads to a different best response of the rival, leading to continuum of equilibria with different profits. 3.2. Properties of the unique equilibrium After having derived the equilibrium fees we can study their properties in more detail. We note that the simplicity of the fees is partly due to the linearity of the model. Allowing for non-linearities and/or agents to differ in multiple dimensions complicates the expressions. In this respect, the following analysis is meant to provide an illustration of what can be done with the approach of heterogeneous trading behavior and to determine predictions on prices for the linear case, which do not necessarily carry over in this simplicity to more general settings. A closer look at the equilibrium fees given by (9) and (10) reveals that they have natural properties that make them intuitively appealing. In particular, the per-transaction fees on both sides are determined by parameters that are only related to the costs and benefits of a transaction, but are independent of the transportation cost. The transportation cost is a component of the fixed benefit and is therefore reflected only in the fixed fee to the buyers. Also, the fixed fee to the sellers is zero, which is natural since there are no fixed costs or benefits per seller. An implication is that platforms do not necessarily use both parts of the tariff even if they have the possibility to do so. The equilibrium uniqueness also allows us to perform comparative-static analyses to gain insights into the model's predictions regarding tariffs and payoffs. We start with the equilibrium fees: Proposition 3. (i) The buyer per-transaction fee γb increases in αb and αbβ and decreases in α s . (ii) The seller per-transaction fee γs decreases in αb and αbβ and increases in α s . (iii) The sum of γb and γs is equal to the per-transaction cost c. Proof. See Appendix. The first two statements of Proposition 3 are intuitive. The per-transaction fee of group k falls if group  k obtains a higher per-transaction benefit. This is because via attracting an additional agent of group k a platform becomes more attractive for group k. This result is well-known in two-sided markets. Our analysis complements this result by showing that with two-part tariffs this effect is only reflected in the per-transaction fee but not in the fixed fee. In addition, the pertransaction fee of a group rises if agents of this group benefit to a greater extent from a transaction with a member of the other group. Although intuitive, this result does not occur in models that only consider fixed fees, see e.g., Armstrong (2006a). In that case, the equilibrium fixed fee to group k only depends on the benefit of group  k. The present result shows that when allowing for two-part tariffs, the per-transaction fees of both groups depend on the benefits of both groups. Platforms combine lower prices to the group who exerts large indirect externalities with higher prices to the group who enjoys large benefits.23 The third statement of Proposition 3 shows that in the simple linear model, the effects just described in sum exactly balance out, and platforms in equilibrium only recoup per-transaction costs via per-transaction fees. 23 We note that this result also obtains in the more general model considered in the online appendix, which allows for non-linearities. Therefore, it does not hinge on the linear structure.

M. Reisinger / European Economic Review 68 (2014) 168–180

175

It is instructive to compare these results with the ones obtained by Liu and Serfes (2013), who consider perfect price discrimination in a spatial model. As the present paper, they find that the discriminatory prices depend on both cross-group externalities. However, in their model cross-group externalities unambiguously make the market more competitive, that is, prices to both groups fall in the indirect externality of each group. By contrast, in our model the per-transaction fee to a group increases in the externality that this group enjoys but falls in the externality enjoyed by the other group. Therefore, the indirect externalities have opposing effects in our model, while they unambiguously reduce prices in Liu and Serfes (2013). In the online appendix, we compare platforms' profits and agents' surpluses in the selected equilibrium with the ones for the case in which platforms can set either only per-transaction fees or only subscription fees. We confine this analysis to the limiting case. We find that platforms benefit from the possibility to charge two-part tariffs.24 The reason is that pertransaction fees mitigate the indirect externalities that make two-sided markets particularly competitive, while the fixed fees allow platforms to extract more of agents' surpluses. The increase in platforms' profits comes at the expense of the single-homing agents while the surplus of multi-homing agents is not affected by the possibility to price discriminate. 4. Discussion 4.1. Two-part Tariffs on only one side Our analysis focused on the case in which platforms can charge two-part tariffs on both groups of agents. However, sometimes platforms have the possibility to charge two-part tariffs only on one side.25 We now briefly explain that in this case heterogeneity on only one side—the side for which two-part tariffs are possible—yields uniqueness. In particular, if trading behavior is homogeneous, by the same line of reasoning as in Section 2, a continuum of nonpayoff equivalent equilibria emerges. The fees on the side at which platforms can charge a two-part tariff are not pinned down uniquely. However, there is now only one free parameter and not two as in case with two-part tariffs on both sides. Therefore, to resolve the problem, it is sufficient to introduce heterogeneity only on the side with the two-part tariff. We then obtain a unique equilibrium for the same reasons as laid out in Section 3.1. Instead, if we considered heterogeneity on the other side, the continuum of equilibria would remain. The reason is that what is needed to obtain uniqueness is that the group facing a two-part tariff consists of agents who react differently to changes in the per-transaction fee. 4.2. Intensive margin In our analysis we assumed that agents do not choose how many units to trade on a platform, but only which platform to join. An alternative way to obtain equilibrium uniqueness is to consider a model in which an agent not only decides on which platform to be active but also chooses his trading volume, conditional on being active on a platform. In such a model there is an extensive margin—the participation of agents—and an intensive margin—the trading volume of each agent. In such a model, the transaction benefit of each agent is stochastic, and the agent only learns its realization after the decision which platform to join. Therefore, an agent makes two sequential decisions. First, he decides to be active on platform or not. This participation decision depends on the fixed and per-transaction fee set by the respective platform. Second, conditional on being active on a platform, the agent decides about his trading volume, which depends on the realized transaction benefit and the per-transaction fee.26 In this scenario, the platform influences the individual demand of each agent with the per-transaction fee, even if the agent has already decided to be active on the platform, implying that none of the two fees is redundant. 4.3. Two-sided single-homing Finally, we analyze the case of two-sided single-homing, thereby showing that the insights of the analysis of the competitive bottleneck model remain valid when agents of both sides can join only one platform. To approach this problem consider the model of two-sided single-homing developed by Armstrong (2006a).27 Both buyers and sellers are uniformly distributed on a Hotelling line of length 1, the platforms are located at the endpoints of the line, with platform 1 located at point 0 and platform 2 located at point 1. The transport costs are tb for the buyers and ts for the sellers. Each buyer (seller) obtains a benefit of αb (αs) when interacting with a seller (buyer). To simplify the exposition we assume that the platforms can only charge two-part tariffs to the buyers and a fixed fee to the sellers. Assuming full 24 Although this result is intuitive, several studies on one-sided markets have shown that the effect of price discrimination on profits under imperfect competition is ambiguous because an additional pricing instrument can intensify competition, see e.g., Armstrong (2006b) and Stole (2007). 25 This is because e.g., it is technically not feasible to collect fixed fees from one side or because agents of one group would not carry out transactions if they were charged by the platform. For example, in the credit card industry consumers may refuse to pay using credit cards if this implies that they have to pay a higher price than when paying by cash. 26 See Bedre-Defolie and Calvano (2013) for a formulation of such a model applied to the credit card industry. 27 This model is commonly used to capture two-sided single-homing, see e.g., Kaiser and Wright (2006) or Belleflamme and Peitz (2010).

176

M. Reisinger / European Economic Review 68 (2014) 168–180

market coverage, the demand functions in case of homogeneous trading behavior can be written as j j j 1 ðαb  γ ib Þnis  pib  ðαb γ b Þns þ pb nib ¼ þ 2 2t b

and

njb ¼ 1 nib

and j j 1 αs nib  pis αs nb þ ps nis ¼ þ 2 2t s

and

njs ¼ 1  nis :

The profit function of platform i is Π i ¼ pib nib þ pis nis þ ðγ ib  cÞnib nis .28 To guarantee an interior solution we suppose that 16t b t s 4 ð2ðαb þαs Þ  c γ b Þ2 , that is, tb and ts must be sufficiently large relative to αb and αs to ensure that there is a marketsharing equilibrium.29 i i Solving the model for the equilibrium fees yields that the two first-order conditions for pb and γb are equivalent. So there is again a continuum of equilibria, in which the equilibrium fees can be written as pb ¼ t b  αs þ ðc  γ b Þ=2 and ps ¼ t s  αb þ ðc þ γ b Þ=2 leading to an equilibrium profit of Π ¼ ðt b þ t s  αb αs Þ=2 þðc þ γ b Þ=4, which depends on γb, implying that the equilibria are not payoff-equivalent. Consider now the model with trading heterogeneity at the buyer side. Introducing this heterogeneity in the same way as above, i.e., there is mass 1 qb of small buyers whose trading volume with each seller is β o 1, while the remaining mass qb have a trading volume of 1, and solving for the equilibrium then yields a unique equilibrium, with fees given by pb ¼ t b ;

γ b ¼ c  2αs

and

ps ¼ t s  ðαb þ αs  cÞðβð1  qb Þ þ qb Þ

and an equilibrium profit of Π¼

t b t s ð2αs þ αb  cÞðβð1  qb Þ þ qb Þ : þ  2 2 2

The equilibrium stays unique when considering the limiting case qb -1, leading to a profit of Π ¼ ðt b þt s  ð2αs þαb  cÞÞ=2. It is evident that the selected equilibrium is again not the profit-dominant one. Hence, selecting the equilibrium according to profit dominance is not robust to the introduction of a tiny amount of heterogeneity, thereby sharing the same feature as the competitive bottleneck model.

5. Conclusion This paper provided a two-sided market model with heterogeneous trading behavior of agents in both groups. It shows that this heterogeneity leads to equilibrium uniqueness, whereas a model with homogeneous trading behavior is plagued with a continuum of equilibria. The equilibrium stays unique in the limiting case when the heterogeneity vanishes, implying that the model selects a unique equilibrium from the continuum. The equilibrium tariff has many reasonable properties and differs from the one obtained by other selection criteria, such as profit dominance. The idea of obtaining equilibrium uniqueness through trading heterogeneity is related to Klemperer and Meyer (1989). They analyze a model of supply function competition and show that when demand is deterministic, a continuum of equilibria exists. If instead there is aggregate demand uncertainty with a large enough support, a unique equilibrium obtains.30 In contrast to the approach by Klemperer and Meyer (1989), our model uses individual heterogeneity to pin down a unique best response, that is, platforms know the demand with certainty but cannot distinguish between the different types.31 Since the model obtains clear predictions on equilibrium prices, it lends itself naturally to explore different questions on the effects of two-part tariffs in two-sided markets. For example, an interesting question is how sellers' investment incentives change when platforms charge two-part tariffs as compared to linear fees. Another direction for further research could be to endogenize the market structure i.e., competitive bottleneck or two-sided single-homing, which is exogenous in our analysis. This could be done e.g., along the lines of Armstrong and Wright (2007). Such an analysis could give insights into the conditions for price discrimination to be beneficial for platforms or consumers, which are derived from the market's primitives. 28

In contrast to Armstrong (2006a), we abstract from fixed costs per agent but consider transaction costs. For the derivation of this condition and the calculations to solve the model, see the online appendix. 30 See Green and Newberry (1992) for an application of this approach to the electricity market, or Hendricks and McAfee (2010) for an application to vertical mergers in the gasoline industry. 31 An equilibrium selection approach that also relies on individual heterogeneity is global games, see e.g., the survey by Morris and Shin (2003). There, each agent observes a signal about the true state of the world but cannot observe the signals of other agents. However, an agent's own signal is informative about the other signals. By contrast, in our case agents in fact differ in their trading behavior, independent of the trading volume of other agents. 29

M. Reisinger / European Economic Review 68 (2014) 168–180

177

Acknowledgments I would like to thank an associate editor and three anonymous referees for very helpful comments and suggestions that helped me improve the paper substantially. I also thank Mark Armstrong, Özlem Bedre-Defolie, Helmut Bester, Jan Eeckhout, Paul Heidhues, Simon Loertscher, Benny Moldovanu, Martin Peitz, Sven Rady, Roland Strausz, Dezsö Szalay and Piers Trepper as well as seminar participants at the Free University of Berlin, the University of Bonn, the University of Munich, the conference on ”Platform Markets” in Mannheim and the EEA Meeting in Glasgow for many very helpful comments and suggestions. Financial support by the German Science Foundation through SFB/TR-15 is gratefully acknowledged. Appendix A Proof of Remark 1. Differentiating Πi with respect to the four strategic variables we obtain first-order conditions of ! i i i i   ∂Π i i i dnb i dns i i i dnb i dns ¼ n þ p þ p þ γ þ γ  c n þ n ¼ 0; s s s k b b b i i i i ∂pik dpk dpk dpk dpk and   ∂Π i dn dn dn dn ¼ pib ib þpis is þ nib nis þ γ ib þ γ is c nis ib þnib is i ∂γ k dγ k dγ k dγ k dγ k i

i

i

i

! ¼ 0; i

i

i

i

with k ¼ b; s. To obtain the solutions we first need to determine the derivatives of nb and ns with respect pk and γk, i respectively. Totally differentiating nb yields32 i

i

dnb ¼

j

i

ðαb  γ b Þdns  ðαb  γ b Þdns  dpb  nis dγ ib : 2t i

Similarly, totally differentiating ns yields that, in equilibrium, i

i

dns ¼ 

i

2dps dγ is 4ps dnb  þ αs αs αs

i

j

dns ¼ 

and

4ps dnb : αs

Here we used that in a symmetric equilibrium prices are the same on both platforms, i.e., pik ¼ pjk ¼ pk and γ ik ¼ γ jk ¼ γ k , which implies market sharing at the buyer side, nib ¼ njb ¼ 1=2, and an equal number of sellers on each platform, nis ¼ njs ¼ ns . i i We can now use these equations to determine the derivatives of the number of buyers and sellers with respect to pb and γb to get i

dnb i dpb

¼

αs ; 2ρ

i

dnb α s ns ; ¼ 2ρ dγ ib

i

dns i dpb

¼

2ps ; ρ

i

and

dns 2p ns ¼ s ; ρ dγ ib i

i

where ρ  ðtα s  4ps ðαb  γ b ÞÞ. In the same way we can determine the derivatives with respect to ps and γs. Here we obtain i

dnb i dps

¼

αb  γ b ; ρ

i

i

dnb α  γb ; ¼ b 2ρ dγ is

dns i dps

¼

2ψ ; αsρ

i

and

dns ψ ; ¼ αsρ dγ is

with ψ  tα s  2ps ðαb  γ b Þ. i i i i i Inserting the derivatives for dnk =dpb and dnk =dγ ib into the first-order conditions for pb and γb and rearranging, we obtain t 4αb ps þγ b ð4ps þγ s α s Þ  α s pb  4ðps Þ2 þðc  γ s Þðα s  γ s Þ ¼ 0

ð12Þ

for both equations. As a consequence, there exists a continuum of combinations of pb and γb that fulfill both first-order conditions. i i i i i Inserting the respective values for dnk =dps and dnk =dγ is into the first-order conditions for ps and γs yields that both firstorder conditions are the same and given by tα s ðα s þc  4ps  γ b  2γ s Þ þ ðαb  γ b Þðcðα s  4ps γ s Þ þ ðγ s Þ2  γ s ðα s  γ b  8ps Þ  α s ðpb þ γ b Þ þ 12ðps Þ2  4ps ðα s γ b ÞÞ ¼ 0:

ð13Þ

Thus, there also exists a continuum of ps–γs-combinations that fulfill both first-order conditions. Solving (12) and (13), we obtain that pb and ps are implicitly defined by (3) and (4). We now turn to the second-order conditions. Although each platform has four strategic variables, one can easily reduce their number to two, as shown by Armstrong (2006a). This is because, given the prices of its rival, platform i's profit can be i i written as a function that depends only on the utilities ub and us that it offers to the two sides. We can define i i i i i i i i i i i i i i ub  ðαb  γ b Þns  pb and us  ps  γ s nb . After replacing pb, ps, γb and γs by ub and us in the equations determining the number of buyers and sellers joining platform i, these numbers can be written as nib ¼ 1=2 þðuib  ðαb γ jb Þnjs þ pjb Þ=ð2tÞ and 32

j

i

Note that because njb ¼ 1 nib , we have dnb ¼  dnb .

178

M. Reisinger / European Economic Review 68 (2014) 168–180

nis ¼ 1 þ uis =ðα s nib Þ. The profit function of platform i is given by Π i ¼ ðαb  cÞnib nis  nib uib  nis uis : To show that Πi is concave in the utilities, we have to verify (i) that ∂2 Π i =∂ðuik Þ2 o 0, k ¼ b; s, and (ii) that the determinant of the matrix of second derivatives of Πi is positive. From the equations determining the number of agents we can derive the first and second derivatives of the demand i functions with respect to uk. We can use these derivatives to determine ∂2 Π i =∂ðuik Þ2 . We can then substitute the respective expressions for the utilities into ∂2 Π i =∂ðuik Þ2 , use that in a symmetric equilibrium pik ¼ pjk ¼ pk and γ ik ¼ γ jk ¼ γ k with k ¼ b; s, and insert the solutions for pb and ps given by (3) and (4). After tedious but routine calculations we obtain that ∂2 Π i =∂ðuib Þ2 o0 if ð2tα s þc2 þcðα s  3αb þ γ b Þ þ 2γ s ðαb  γ b Þ  ðα s  αb Þð2αb γ b ÞÞ o 0 while ∂2 Π i =∂ðuis Þ2 o 0 if  4=α s o 0, which is always fulfilled. By proceeding in the same way we obtain that condition (ii) for concavity, i.e., ð∂2 Π i =∂ðuib Þ2 Þ  ð∂2 Π i =∂ðuis Þ2 Þ ð∂2 Π i =∂uib ∂uis Þ2 4 0, holds if 16tα s  c2 þ cð2α s 6αb þ8γ b Þ þ 16γ s ðαb  γ b Þ þ7ðαb Þ2 þ 8γ b ðα s αb Þ  α s ð10αb þα s Þ 40: Π i =∂ðuib Þ2 Þ

Π i =∂ðuis Þ2 Þ  ð∂2 Π i =∂uib ∂uis Þ2 4 0

It is easy to check that ð∂  ð∂ implies ∂ function is concave if (14) holds, which is the case if t is sufficiently large, i.e., 2

t4

2

2

Π i =∂ðuib Þ2 o 0.

c2  cð2α s  6αb þ 8γ b Þ  16γ s ðαb γ b Þ  7ðαb Þ2  8γ b ðα s  αb Þ þ α s ð10αb þ α s Þ : 16α s i

ð14Þ As a consequence, the profit

□

i

Proof of Proposition 1. Differentiating Πi with respect to pk and γk we obtain first-order conditions of ! i i i   dn i ∂Π i dn^ dn^ i dn s b i ^ ik þ pib b þ pis s þ γ ib þγ is c ¼ n n þ n ¼0 s b i i i i ∂pik dpk dpk dpk dpk and i i   dn i ∂Π i dn^ dn^ dn i b i ¼ pib ib þ pis is þn ib n is þ γ ib þγ is c n s þ n ib is i i ∂γ k dγ k dγ k dγ k dγ k

! ¼ 0;

i i with n^ b ¼ qb nib þ ð1  qb Þnibβ and n^ s ¼ qs nis þð1  qs Þnisλ . i

i

i

i

As above, we need to determine dnm =dpk and dnm =dγ ik , where now m ¼ b; bβ; s; sλ and k ¼ b; s. Totally differentiating nb and nibβ given by (6) and (7), and using the fact that in a symmetric equilibrium pik ¼ pjk ¼ pk and γ ik ¼ γ jk ¼ γ k , which implies that nib ¼ nibβ ¼ 1=2, nis ¼ njs ¼ ns and nisλ ¼ njsλ ¼ nsλ , yields i

i

dnb ¼

ðαb γ b Þdn is  ðαb γ b Þdn js  dpb  n s dγ ib 2t

and i

i

dnbβ ¼

ðαbβ  γ b Þβdn is ðαbβ  γ b Þβdn js  dpb βn s dγ ib ; 2t i

i

where dn is ¼ qs dns þ ð1  qs Þλdnsλ . i Totally differentiating ns and nisλ given by (8) yields i

dns ¼ 

2 dγ i 4ps i dps  s þ dn i α s ðqb þβð1  qb ÞÞ α s α s ðqb þβð1  qb ÞÞ2 b

and i

dnsλ ¼ 

2 dγ i 4ps i dp  s þ dn i ; α s λðqb þβð1  qb ÞÞ s α s α s λðqb þ βð1 qb ÞÞ2 b i

j

i

i

where dn ib ¼ qb dnb þð1  qb Þβdnbβ . Finally, differentiating ns and njsλ with respect to nb and nibβ yields j

dns ¼ 

4ps dn ib 2

α s ðqb þβð1  qb ÞÞ

and

j

dnsλ ¼ 

4ps dn ib α s λðqb þ βð1  qb ÞÞ2 i

i

:

i

Equipped with these equations we can determine dnm =dpk and dnm =dγ ik , m ¼ b; bβ; s; sλ and k ¼ b; s. Inserting the respective values into the first-order conditions we obtain that none of the four first-order conditions is redundant. Solving the first-order conditions for the four prices yields that in a symmetric equilibrium the tariffs are implicitly defined by (9) and (10).

M. Reisinger / European Economic Review 68 (2014) 168–180

179

We now turn to the second-order conditions. We need to determine under which conditions the Hessian (4  4 matrix) is negative definite. This is a tedious matter since there are overall 14 conditions. However, the calculations are standard ones. i i As an example, consider the derivatives ∂2 Π i =∂ðpib Þ2 and ∂2 Π i =∂ðγ is Þ2 .33 Building the second derivatives of dnm =dpb , i m ¼ b; bβ; s; sλ, with respect to pb we obtain that they are all zero. We therefore have ! ! ! i i i   dnibβ  dnibβ  dnisλ ∂2 Π i dnb  dnb  dns  i i ¼ 2 q þ 1  q þ γ  c q þ 1  q þ 1  q β λ þ γ q : b b b b s s s b i i i i i i ∂p2i dpb dpb dpb dpb dpb dpb i

i

Inserting the respective values for dnm =dpb and the equilibrium prices we obtain that the expression simplifies to ∂2 Π i =∂pi2 ¼  1=t o0. i i Similarly, the second derivatives of dnm =dγ is with respect to γs are also zero, and so the second derivative of the profit i function with respect to γs can be written as ! i i   dnbβ    ∂2 Π i dnb  qs nis þ 1 qs λnisλ ¼ 2 qb i þ 1  qb β i i ∂γ s dγ s dγ s ! i      dni dn  þ 2 qb ni1 þ 1  qb βnibβ qs is þ 1 qs λ sλ dγ s dγ is ! ! i i    dnibβ  dnisλ dnb  dns  i i þ 2 γ b þ γ s  c qb i þ 1  qb β i qs i þ 1  qs λ i : dγ s dγ s dγ s dγ s i

Inserting the respective values for dnm =dγ is and the equilibrium prices yields that ∂2 Π i =∂γ is o 0 if and only if t4

ðqs þ λð1  qs ÞÞðqb ðαb þ α s Þ þ βð1  qb Þðαbβ þα s Þ  cðqb þ βð1 qb ÞÞÞ α s ðqb þ βð1  qb ÞÞ3

ððα s cÞðqb þ βð1  qb ÞÞðqb þβ2 ð1 qb ÞÞ þ αb qb ðqb þβð1  qb Þð2  βÞÞ þ βαbβ ð1  qb Þðqb ð2β 1Þ þ β2 ð1  qb ÞÞÞ: Proceeding in the same way for all other conditions yields that they are satisfied if the transportation cost parameter t is sufficiently large relative to αb, αbβ , and α s . □ Proof of Proposition 3. Differentiating γb with respect αb and αbβ yields ∂γ b qb 40 ¼ ∂αb 2ðβð1  qb Þ þ qb Þ

and

∂γ b βð1  qb Þ 40 ¼ ∂αbβ 2ðβð1  qb Þ þ qb Þ

and differentiating γb with respect α s yields ∂γ b =∂α s ¼  1=2 o 0. This establishes part (i). Part (ii) can be shown in exactly the same way by differentiating γs. Finally, to show part (iii) simply summing up γb and γs yields c. □ References Ambrus, A., 2006. Coalitional rationalizability. Q. J. Econ. 121, 903–930. Ambrus, A., Argenziano, R., 2009. Asymmetric networks in two-sided markets. Am. Econ. J.: Microecon. 1, 17–52. Anderson, S.P., Coate, S., 2005. Market provision of broadcasting: a welfare analysis. Rev. Econ. Stud. 72, 947–972. Armstrong, M., 2006a. Competition in two-sided markets. RAND J. Econ. 37, 668–691. Armstrong, M., 2006b. Recent developments in the economics of price discrimination. In: Blundell, R., Newey, W., Persson, T. (Eds.), Advances in Economics and Econometrics: Theory and Applications, Ninth World Congress of the Econometric Society, vol. 2. , Cambridge University Press, Cambridge, pp. 97–141. Armstrong, M., Wright, J., 2007. Two-sided markets, competitive bottlenecks and exclusive contracts. Econ. Theory 32, 353–380. Belleflamme, P., Peitz, M., 2010. Platform competition and seller investment incentives. Eur. Econ. Rev. 54, 1059–1076. Belleflamme, P., Toulemonde, E., 2009. Negative intra-group externalities in two-sided markets. Int. Econ. Rev. 50, 245–272. Bedre-Defolie, Ö., Calvano, E., 2013. Pricing payment cards. Am. Econ. J.: Microecon. 5, 206–231. Caillaud, B., Jullien, B., 2001. Competing cybermediaries. Eur. Econ. Rev. 45, 797–808. Caillaud, B., Jullien, B., 2003. Chicken and Egg: competition among intermediation service providers. RAND J. Econ. 34, 309–328. Csorba, G., 2008. Screening contracts in the presence of positive network effects. Int. J. Ind. Organ. 26, 213–226. Galeotti, A., Moraga-González, J.L., 2009. Platform intermediation in a market for differentiated products. Eur. Econ. Rev. 53, 417–428. Gomes, R., Pavan, A., 2013. Cross-subsidization and Matching Design. Working Paper, Toulouse School of Economics and Northwestern University. Green, R., Newberry, D.M., 1992. Competition in the British electricity spot market. J. Polit. Econ. 100, 929–953. Hagiu, A., 2006. Pricing and commitment by two-sided platforms. RAND J. Econ. 37, 720–737. Hagiu, A., 2009. Two-sided platforms: product variety and pricing structures. J. Econ. Manag. Strategy 18, 1011–1043. Hendricks, K., McAfee, R.P., 2010. A theory of bilateral oligopoly. Econ. Inq. 48, 391–414. Jullien, B., 2011. Competition in multi-sided markets: divide and conquer. Am. Econ. J.: Microecon. 3, 186–219. Kaiser, U., Wright, J., 2006. Price structure in two-sided markets: evidence from the magazine industry. Int. J. Ind. Organ. 24, 1–28. Klemperer, P.D., Meyer, M.A., 1989. Supply function equilibria in oligopoly under uncertainty. Econometrica 57, 1243–1277. Liu, Q., Serfes, K., 2013. Price discrimination in two-sided markets. J. Econ. Manag. Strategy 22, 768-786.

33 As in the proof of Proposition 1, we could work with the utilities instead of the fees. However, since there are now four different types of agents, we need four different utilities, implying that using the utilities does not reduce the dimensionality.

180

M. Reisinger / European Economic Review 68 (2014) 168–180

Morris, S., Shin, H.S., 2003. Global games: theory and applications. In: Dewatripont, M., Hansen, L.P., Turnovsky, S.J. (Eds.), Advances in Economics and Econometrics: Theory and Applications, Ninth World Congress of the Econometric Society, vol. 1. , Cambridge University Press, Cambridge, pp. 56–114. Nocke, V., Peitz, M., Stahl, K., 2007. Platform ownership. J. Eur. Econ. Assoc. 5, 1130–1160. Peitz, M., Valletti, T., 2008. Content and advertising in the media: pay-TV versus free-to-air. Int. J. Ind. Organ. 26, 949–965. Rochet, J.-C., Tirole, J., 2002. Cooperation among competitors: some economics of payment card associations. RAND J. Econ. 33, 549–570. Rochet, J.-C., Tirole, J., 2003. Platform competition in two-sided markets. J. Eur. Econ. Assoc. 1, 990–1029. Rochet, J.-C., Tirole, J., 2006. Two-sided markets: a progress report. RAND J. Econ. 37, 645–667. Roger, G., Vasconcelos, L. Platform pricing structure and moral hazard. J. Econ. Manag. Strategy, forthcoming. Rysman, M., 2009. The economics of two-sided markets. J. Econ. Perspect. 23, 125–143. Stole, L.A., 2007. Price discrimination and competition. In: Armstrong, M., Porter, R. (Eds.), Handbook of Industrial Organization, vol. 3. , Elsevier, Amsterdam, North-Holland, pp. 2221–2299. Weyl, E.G., 2010. A price theory of multi-sided platforms. Am. Econ. Rev. 100, 1642–1672. White, A., Weyl, E.G., 2012. Insulated Platform Competition. Working Paper, University of Chicago. Wright, J., 2003. Optimal card payment systems. Eur. Econ. Rev. 47, 587–612. Wright, J., 2012. Why payment card fees are biased against retailers. RAND J. Econ. 43, 761–780.