Tying and freebies in two-sided markets

International Journal of Industrial Organization 30 (2012) 436–446

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International Journal of Industrial Organization journal homepage: www.elsevier.com/locate/ijio

Tying and freebies in two-sided markets Andrea Amelio a, Bruno Jullien b,⁎ a b

Toulouse School of Economics (GREMAQ), France Toulouse School of Economics (GREMAQ&IDEI), France

a r t i c l e

i n f o

Article history: Received 15 September 2007 Received in revised form 2 March 2012 Accepted 7 March 2012 Available online 31 March 2012 JEL codes: L11 L13 L42 L81 L86 D43

a b s t r a c t In two-sided markets where platforms are constrained to set non-negative prices, tying can be deployed by platforms as a tool to introduce implicit subsidies. For a monopoly, this raises participation and benefits consumers on both sides. In a duopoly, tying on one side makes a platform more or less competitive on the other side depending on externalities. Tying may not be ex-ante optimal while the competing platform may benefit from it. The impact on consumers' surplus depends on whether competition is softened or intensified on the profitable side. Moreover tying increases total welfare if network effects are strong. © 2012 Elsevier B.V. All rights reserved.

Keywords: Tying Two-sided market Platform competition

1. Introduction Two-sided platforms need to coordinate consumers to reach an efficient allocation. Based on the initial work of Armstrong (2006), Caillaud and Jullien (2003) and Rochet and Tirole (2003), the literature has emphasized the role of the price structure in solving coordination problems. Armstrong (2006) and Caillaud and Jullien (2003) show that platforms may set “negative prices” on one side in order to enhance participation. However, a direct implementation of negative prices is not always possible, due to adverse selection or opportunistic behaviors. As emphasized in Jullien (2005), when a platform is constrained to set non-negative prices and perceives a monetary transfer as too risky, one alternative is to rely on tying. In the paper, we develop a model where a platform constrained to set non-negative prices ties the sales of another good with the access to the platform as a way to relax the non-negativity constraint. By giving away the bundle for free or at a discounted price, conditional on participation from one side, the platform eventually implements implicit negative subscription prices in a context where monetary subsidy would be inefficient or could generate opportunistic behaviors. The concept of two-sided markets refers to a specific instance of networks where the services are used by two distinct groups of ⁎ Corresponding author. Tel.: + 33 5 61 12 85 61; fax: + 33 5 61 12 86 37. E-mail addresses: [email protected] (A. Amelio), [email protected] (B. Jullien). 0167-7187/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.ijindorg.2012.03.002

customers who seek to interact with each other. This includes, among other activities, intermediation, credit card services, media, computer operating systems, video games, shopping malls and yellow pages. 1 In these markets, tying is a widespread phenomenon and may take several forms. One form is the practice of offering gifts along with the service, such as a magazine coming with a free DVD. Another form, illustrated by the case of the Windows Media Player, consists of bundling a monopoly good with a complementary competitive twosided good. 2 Last but not least, a widespread practice among web portals like Yahoo! or Google is offering for free a large bundle of services to one side of the market. From a competition policy perspective, assessing situations in which the bundle is offered for free to consumers can be difficult. This paper offers a framework for that analysis. In the paper, the rationale for tying differs from entry deterrence purposes emphasized by Whinston (1990), and from price discrimination motives as developed in Adams and Yellen (1976) or Schmalensee (1984). The only purpose for tying is to stimulate demand on one side in order to increase the membership value and profits on the other. The advantage is that the platform may avoid attracting undesirable customers by tying the platform service with a good of particular interest for the targeted side. Clearly for such a strategy to be effective, there must be some advantage of payments in 1 2

See Evans (2003a, 2003b) and Rochet and Tirole (2003, 2006) for more examples. See Choi (2010).

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kind over money subsidies. Thus, the best candidate is a complementary good. Free parking at a shopping mall is only interesting for customers intending to shop, as Google desktop-bar is only interesting for web-surfers. However, since the purpose is to give subsidies to some participants, even an independent good can be used provided that its demand is related to the demand of two-sided services. For instance, a magazine offering a CD or a DVD can choose to target a particular population. To this extent, the theory developed in the paper illustrates an instance of second-degree price discrimination implemented through tying which confirms the more general idea that price discrimination may help a network to coordinate the customers' participation and thereby be welfare improving (see Jullien (2011)). The tying strategy is studied in the context of a monopolistic and a duopolistic two-sided market, similar to Armstrong (2006), where a consumer participating in one side derives a positive externality from participation in the other side. In the duopoly case, platforms are horizontally differentiated on both sides, and agents register with one platform only (single-homing). We allow one platform to deploy mixed bundling (which reduces to tying in our model) when the service is free due to a non-negativity constraint on prices, and analyze their impact on the allocation, as well as on the consumer surplus, the platforms' profits and the total welfare. We then discuss the results obtained with pure bundling, multi-homing and demand expansion. In a monopolistic context, when the platform has some market power on the tied good market, it uses a tying strategy and sells the good unbundled and bundled. The only reason to sell a bundle is to subsidize participation. The bundle (that includes the subscription) is thus offered for free or at a discounted price. We show that all consumers benefit from this practice, since the enhanced participation of one side raises the perceived quality of the service on the other side. In a duopolistic context, we account for strategic effects as the use of bundles affects the competitor's behavior. In a one-sided context, Whinston (1990) shows that mixed bundling is neutral while pure bundling reduces the equilibrium profit of all firms and thus may result in entry deterrence. Since it implies a lower price of the platform service on the targeted side, tying is not neutral to profit in our model. The key point is that tying occurs when the platform is subsidizing participation, thus when one side is a loss leader. Standard intuition may then be misleading until one realizes that the subsidized size should be considered only as the “input” sold to the profitable side. Increasing the subsidy through tying has two effects. On one hand, enhanced participation of one side gives to the tying platform a “quality” advantage on the other side. On the other hand, as a byproduct, it also increases the opportunity cost of selling on the profitable side. The reason is the feedback effect, usual in a two-sided market framework. More sales on the profitable side implies more sales, in our case at a loss, on the subsidized side. The opportunity cost of selling on the profitable side then accounts for the loss generated on the subsidized side, and increases with tying. Thus tying on one side may also reduce the intensity of competition on the other side. Therefore, the impact of tying on the behaviour of the platform on the profitable side has to be analyzed as the combined effects of increasing the value and increasing the (opportunity) cost. There is a demand shifting effect associated with higher quality, and a competition softening effect associated with higher cost. The impact on equilibrium profits is then ambiguous and due to softened competition, the competitor's profit may increase when one platform uses tying. The impact on consumers is also ambiguous. Consumers on the subsidized side always benefit, but consumers on the profitable side may benefit or not depending on which of the two effects dominates. The main conclusions are: i) when the two sides evaluate the participation of each other in a symmetric way, total consumer surplus and total welfare decrease with tying; ii) when the subsidy is given to consumers who do

437

not value the participation of the other side, then consumer surplus increases on both sides. In any case, some consumers, who may be clients of either platform, are hurt. Overall, total welfare increases if competition is intense and/or network externalities are large. Despite the importance of tying in two-sided markets, there has been little contribution to this issue. Rochet and Tirole (2006) analyze the practice of tying credit and debit cards on the merchant side of the payment cards market, and show that this results in a more efficient setting of interchange-fees. They share with us the conclusion that tying may enhance efficiency by inducing better coordination between the various sides. Farhi and Hagiu (2008) analyze the impact on profits of a reduction of the marginal cost of one platform on one side. A possible interpretation is that this reduction results from a pure bundling strategy, as in Whinston (1990). Although the key mechanisms are different, our paper shares the conclusion that bundling may increase competitors' profits. Moreover they confirm the insight that this effect is related to the existence of a negative margin on some side (the opposite side in their case, the bundling side in our case). Choi (2010) analyzes tying in a situation where one or both sides multi-home. In his model, the tied good is essential to the participation to any platform and as a result tying has an exclusionary effect. He shows that even if foreclosure arises the welfare implications are ambiguous. His analysis differs from ours in that we rule out exclusionary effects of tying to focus on the subsidy dimension, while that dimension is absent in his model. Hence the two analyses are complementary. The paper is organized as follows. In Section 2 we set up a twosided monopolistic framework, constraining prices to be nonnegative. When it is binding, we study the impact of relaxing the non-negativity constraint by means of tying. Section 3 presents the duopolistic framework with tying and the results when both platforms use tying. Section 4 discusses the effect of tying by a single platform on profits and welfare. Section 5 presents some extensions. First it compares pure bundling and tying from a strategic perspective. Then it discusses the implication of multi-homing and demand expansion effects. Section 6 draws conclusions. 2. A monopoly platform Consider a platform serving two sides, denoted by 1 and 2. Each side i consists of a unit mass of consumers and the platform incurs a cost fi for each consumer subscribing to side i. Every consumer of each group cares about the total number of consumers in the other group. The numbers of consumers buying the service on each side are denoted n1 and n2 and are given by the demand system: n1 ¼ D1 ðp1 −α 1 n2 Þ and n2 ¼ D2 ðp2 −α 2 n1 Þ;

ð1Þ

where αi ≥ 0 is the benefit accruing to consumers in side i from interacting with every consumer belonging to the other group. We assume that the function D1 and D2 are decreasing, continuously differentiable on a convex support. 3 The demand on each side at monopoly prices is assumed to be positive. The platform's problem is then to maximize the profit Π = (p1 − f1) n1 + (p2 − f2)n2. The optimal prices when negative prices are allowed are denoted by pi∗ and are solutions of the system: 



0

pi −f i þ α −i n−i ¼ −Di =Di ; f or i ¼ 1; 2; where D′i is the slope of the demand function at fixed participation of the other side. The price on side i is the monopoly price for an opportunity cost fi − α− in− i, reflecting the fact that one additional customer on side 3 Notice that Eq. (1) defines a unique mapping from quantities to prices but this mapping may not be invertible. For conciseness, we assume that the monopoly is able to implement any pairs of prices and of quantities that satisfy Eq. (1).

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i allows raising the price by α− i for the n− i consumers of the other side.4 When this opportunity cost is negative, the optimal price may also be negative. Suppose that p1∗ b 0 and that negative prices are not feasible. 5 Then the platform offers the service for free on side 1. To show how tying helps to relax the zero price constraint, we assume that the platform can sell some good that is demanded only by side 1 consumers. 6 Let θ be the willingness to pay for the good of a side 1 consumer, assumed to be homogeneous within side 1. Let the platform's cost of the good be c ≤ θ. 7 Given that all subscribers on side 1 are willing to buy the good, it is straightforward to see that an optimal strategy for the platform consists of selling the good alone at price θ and a bundle including the platform service and the good at price p~ ≤θ. 8 Notice that setting p~ bθ is tantamount to a subsidy side 1 given to subscribers. The utility of a consumer on side 1 buying the bundle is the same as if she were buying the service at price p~ −θ and the good at the price θ. Since the consumers always buy the good, the implicit price of the service in the bundle is p1 ¼ p~ −θb0. The demand for the bundle is then n1 = D1(p1 − α1n2) while the demand for the good alone is 1 − n1. As long as p~ is non-negative, only consumers on side 1 consider buying the bundle. The non-negativity constraint thus writes as p1 + θ ≥ 0. The profit under tying is then ðp~ −f 1 −cÞn1 þ ðp2 −f 2 Þn2 þ ðθ−cÞð1−n1 Þ and the maximal profit is obtained by solving: ΠT ¼

max

p1 ;p2 ;n1 ;n2

ðp1 −f 1 Þn1 þ ðp2 −f 2 Þn2 þ θ−c

s:t: p1 ≥−θ and ð1Þ: Clearly, since θ > 0, the platform strictly prefers tying when the zero price constraint binds, as it allows subsidizing the participation to the platform without selling the good to non-interested consumers. Notice that the platform is price constrained when it would benefit from raising demand on side 1 so as to propose a larger value to side 2. Hence it is no surprise that tying raises demand on both sides: Proposition 1. Suppose p1∗ b 0. If the platform deploys tying, participation is higher on both sides than under no bundling, and it is nondecreasing with θ, maximal for θ ≥ − p1∗ . If the profit function is quasi-concave in prices, then the price of the
 bundle is p~ ¼ max p1 þ θ; 0 . Proof. Using the demand equation, we rewrite the program as:     −1 Π θ ¼ max D−1 1 ðn1 Þ−f 1 n1 þ D2 ðn2 Þ−f 2 n2 þ ðα 1 þ α 2 Þn1 n2 þ θ−c n1 ;n2

s:t:

D−1 1 ðn1 Þ þ α 1 n2 ≥−θ

with solution ni (θ). The first part of the results follows from monotone comparative results for supermodular programs. Indeed, theorem 6 of Milgrom and Shannon (1994) shows that the objective is supermodular in (n1,n2) as the cross-derivative is positive. From theorem 5 of Milgrom and Shannon (1994), (n1(θ), n2(θ)) is non-decreasing in θ, in the sense of the strong order set, since relaxing θ expands the feasible sets.

4

See Armstrong (2006) or Jullien (2005). We do not consider the constraint on p2 since it will be positive in equilibrium if the constraint binds on p1. 6 In all that follows, the term “service” refers to the platform and the term “good” stands for the good bundled with the service. 7 The case c b θ correspond to a situation with market power. The case c > θ is briefly discussed at the end of the section. 8 Let pu be the price of the good, setting p~ ¼ pu would amount to giving the service for free, so this is weakly optimal. Morever pu = θ as increasing both pu and p~ below θ doesn't affect sales on side 1. 5

When p1∗ + θ > 0, the solution coincides with the unconstrained solution and p1 = p1∗ . When p1∗ + θ b 0, quasi-concavity implies that the constraint binds, thus that p1 = − θ. □ When the value of the good tied is small, the bundle is free. The implicit subsidy is equal to θ and p1 = − θ. However, when the value θ is above the optimal subsidy that an unconstrained platform would set for side 1, the price of the bundle is positive at p1∗ + θ and reflects any increase in θ. In this latter case, the platform implements the unconstrained monopoly solution. An immediate consequence of the result is that since tying combines a subsidy and increased participation, it raises the surplus of all consumers. Proposition 2. Consumer surplus is higher on both sides under tying than under no bundling. v

Proof. The consumer surplus on side i writes as ∫Di −i 1(ni) Di(x)dx and it is increasing with ni. From Proposition 1 participation increases with tying on both sides, thus consumer surplus increases. □ To complete the discussion of the monopoly case, pure bundling is also discussed. Clearly selling only the bundle would be less profitable if θ > c. But it can be profitable if the value of the good is below its cost, c > θ > 0. In this case, the platform may still benefit from subsidizing participation through bundling, but the good will not be sold unbundled as this would involve a loss. Defining the implicit price of the service as above, p1 ¼ p~ −θ; the platform's profit under pure bundling writes as: Π ¼ ðp1 −f 1 þ θ−cÞn1 þ ðp2 −f 2 Þn2 ; where the demands are still given by Eq. (1). The behavior of the platform is thus the same as the behavior of a platform using tying with a marginal cost f1 − (θ − c) on side 1. A similar argument, such as for Proposition 1, shows that the participation levels are nondecreasing with θ and non-increasing with c on both sides. Given that tying and pure bundling coincide when θ = c, pure bundling would result in higher participation on both sides if θ > c and lower participation if θ b c. 3. The duopoly model In this section, we extend the model to the duopolistic case. As in the monopoly case, using tying to subsidize the participation of one side expands the demand on this side, but this may be achieved either by attracting new consumers to the market, or by attracting consumers from the competing platform. Clearly tying should have a more favorable effect on welfare when it expands the market demand than when it steals business from competitors. In the paper, we focus on the business stealing effect by assuming a fixed market size. We then discuss market expansion in a separate section. We base the analysis on the model developed by Armstrong (2006) and Armstrong and Wright (2007). There are two sides, denoted 1 and 2. Consumers on each side are distributed uniformly on the unit line. There are two platforms, A and B, located at the two extremes of the unit interval of side i, xiA = 0, xiB = 1. They compete for consumers' participation on both sides and consumers can join only one platform, i.e. they single-home. If a consumer on side i located at xi ∈ [0, 1] joins platform j = A, B he enjoys a utility:     j j j ui ¼ vi −t i xi −xij  þ α i n−i −pi ; where pij is the subscription fee of joining platform j in side i, vi is the j intrinsic valuation assumed to be platform independent, and n− i is the mass of consumers on the other side joining platform j.

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We maintain throughout the section that v1 and v2 are large enough so that both platforms are active on both sides and the market is covered: 0 b niB = 1 − niA b 1. Then the participations on both sides of the market are related by the quasi-demand system:   8 A B B A > α 1 n2 −n2 þ p1 −p1 > 1 A > > þ ¼ n < 1 2 2t1  > > α 2 nA1 −nB1 þ pB2 −pA2 > 1 > : nA2 ¼ þ : 2 2t 2

ð2Þ

To define a stable demand system, we assume the same condition as in Armstrong (2006). Assumption 1. 4t1t2 − (α1 + α2) 2 > 0. The implied demands of platform A on each side are:     8 B A B A > α 1 p2 −p2 þ t 2 p1 −p1 > 1 1 A > > < n1 ¼ þ 2 2  t1 t 2 −α 1 α 2  B A B A > > α p −p 1 þ t 1 p2 −p2 > A 1 1 2 1 > : n2 ¼ þ : 2 2 t 1 t 2 −α 1 α 2

ð3Þ

We denote the numerator by: γ ¼ t 1 t 2 −α 1 α 2 :

3.1. Tying and best reply Let us now turn to tying. In the duopoly context, we have to consider two possibilities depending on whether one or both firms rely on tying. The case where θ = c corresponds to a competitive supply of the good so it is natural to assume that both firms can offer the good. We shall thus examine both cases. Before we do that, it is worth examining the effect of tying on the best reply of one firm. Suppose that platform A ties on side 1 its the good to the service. The stand-alone price of the good is θ and, as in the monopoly case, all consumers on side 1 buy the good unbundled if they do not join the platform. The price of the bundle is p~ A , and we denote by pA1 ¼ p~ A −θ the implicit subscription price in the bundle. Given implicit or explicit subscription prices pij, the subscription levels are given by the same system of demand Eq. (3) as above. Tying relaxes the non-negativity price constraint and platform's A problem becomes:     max pA1 −f 1 nA1 þ pA2 −f 2 nA2 þ ðθ−cÞ pA1 ;pA2

s:t: pA1 ≥−θ and ð3Þ: As long as the non-negativity constraint is binding, platform A sells the bundles at a zero price on side 1 (p1A = − θ) and optimally sets its price on side 2 according to the following response function: A

p2 ¼

Then t1/γ and t2/γ are the slopes of the demand functions with respect to the price differential. 9 Each platform has costs f1 and f2 to serve both sides. The profit of platform j writes as follows:     j j j j p1 −f 1 n1 þ p2 −f 2 n2 : The equilibrium with no constraint on prices is characterized in Armstrong (2006). Unconstrained symmetric equilibrium prices are then: UU

439

 α 1 B 1 CC α 2  B A A p2 þ p2 þ p1 −p1 − 1 p1 : 2 2 2t 1 2t 1

The best response functions show the standard complementarity between the prices of platforms A and B in side 2. However, the platform adjusts the price in side 2 by taking into account the discount on the good offered by platform A to participants on side 1. To interpret the channels through which the constraint on the price p1A affects the best-response, we can rely on the quasi-demand and demand equations. The first-order condition on side 2 writes as:   ∂nj   ∂nj j j 2 1 2 ¼ p −f þ p −f þ nj : 1 2 1 2 ∂pj2 ∂pj2 ∂pj2

∂Π j

UU

p1 ¼ f 1 þ t 1 −α 2 and p2 ¼ f 2 þ t 2 −α 1 ;

From Eqs. (2) and (3), we have for i ≠ j

and the profit of each platform is Π UU ¼ 12 ðt 1 −α 2 þ t 2 −α 1 Þ, strictly positive. As in the monopoly case, we constrain the prices on side 1 to be non-negative and assume that the constraint is binding.

A ∂nAi α ∂nj α ¼ i A¼− i: A t i ∂pj 2γ ∂pj

Assumption 2. p1UU = f1 + t1 − α2 b 0.

Using linearity of the demand Eq. (3), we can separate the impact of p1A on the derivative of profits Π A and Π B as follows:

Under Assumption 2, platforms set prices equal to zero on side 1 and adjust prices on side 2 consequently. As shown in the Appendix, the constrained symmetric equilibrium prices are:

∂ ∂pA1

∂Π A ∂pA2

∂ ∂pA1

∂Π B ∂pB2

CC p1

¼ 0 and

CC p2

¼

UU p2

α UU UU þ 1 p1 bp2 : t1

When the externality α2 is large, consumers of side 2 attach a high value to the participation of consumers on side 1 and platforms have an incentive to subsidize side 1 participation. Under Assumption 2, unconstrained platforms would operate with a negative price on side 1 and a positive margin on side 2. Imposing a non-negativity constraint on prices then results in higher prices on side 1 but the prices on side 2 are reduced due to a lower opportunity cost of servicing side 2 consumers. 9

Notice that Assumption 2 implies that γ is strictly positive.

! ¼

∂nA1 ∂nA2 α α þ ¼− 1− 2; 2γ 2γ ∂pA2 ∂pA1

¼

α2 : 2γ

!

ð4Þ

The impact on the first-order conditions shows that there are two effects associated with the fact that the implicit subscription fee p1A is now negative. 3.1.1. Demand shifting The terms proportional to α2 capture the fact that a fall of p1A below zero shifts the demand on side 1 toward platform A. As a result, platform A becomes more attractive on side 2 (n2A increases at given prices). The slope of profit and thus the best reply of platform A shifts upward. Similarly platform B becomes less attractive (n2B decreases) and its best-reply shifts downward. This effect increases with α2

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which determines the sensitivity of side 2 demand to the quantity on side 1. The demand shifting effect is at the heart of the motive of the platform for setting prices below zero as it is precisely this increase in side 2 demand that makes the subsidy profitable.

Proof. See Appendix. ■

3.1.2. Competition softening The term in α1 reflects the fact that platform A incurs a higher loss per customer on side 1 when p1A decreases. As a consequence, it has less to gain by cutting prices on side 2 and raising demand. In order to see that effect, it is worth reminding that any additional customer on side 2 implies α1/t1 additional customers on side 1 (from Eq. (2) and n2B = 1 − n2A). These customers are costly since p1A b 0. As a consequence, platform A faces a higher opportunity cost on side 2 which shifts its best reply further upwards. This effect is key to the analysis that follows as, unlike the preceding one, the competition softening effect tends to increase both equilibrium prices (because prices are strategic complements on side 2). This suggests that to some extent, tying can be used by platform A as a commitment to set higher prices. Thus this practice may aim to enforce some form of collusive prices on the other side of the market, rather than achieving some exclusion of rivals. While the demand shifting effect is clearly detrimental to platform's B profit, on the contrary, the competition softening effect is beneficial to it. The balance between these two effects then determines the global impact on profit and welfare.

4. Tying by a single platform

Proposition 4. Under tying, platform A sets a price for the bundle p~ A ¼ p1A þ θ, where p1A = max{−θ, p1UC}. Platform B sets a zero price on side 1.

3.2. Tying by both platforms

Proof. See Appendix. □

Let us start with the case where both platforms tie the service to the good. Clearly if θ is large, θ ≥ − p1UU, the equilibrium coincides with the unconstrained equilibrium, pij = piUU and the bundle is sold at UU price p~ ¼ pUU 1 þ θ. Whenever θ b − p1 , the platforms offer the bundle A B for free on side 1 : p1 = p1 = − θ. The price on side 2 is then given by: A

B

CC

p2 ¼ p2 ¼ p2 þ

α1 θ: t1

In this symmetric constrained equilibrium, the two platforms share the market equally and make the following equal profit (for θ ≤ − p1UU): Π¼Π

CC

  α1 θ : þ −1 2 t1

When both firms use tying, the demand shifting effect cancels out as prices decrease for both firms on side 1 leaving quantities unchanged. Therefore, only the competition softening effect affects equilibrium prices. The result is a reduction θ of prices on side 1 and an increase α1/t1θ of prices on side 2. Given that quantities are the same on both sides, the profit is larger with tying than without if the prices increase more on side 2 than they decrease on side 1, hence if α1/t1 is larger than 1. Clearly consumers on side 1 benefit from tying while consumers on side 2 are harmed. The total (summing the two sides) consumer surplus in the constrained case is TCS ¼ TCS

UU

   α UU − 1 −1 p1 þ θ t1

where TCS UU is the total consumers surplus in the unconstrained case (θ > − p1UU) and is given in the Appendix. Again tying can be beneficial or harmful for consumers depending on the sign of α1 − t1. Proposition 3. When both platforms use tying on side 1, the profits are higher for both firms and total consumer surplus is lower if the externality is large on side 1 (α1 > t1). The reverse holds otherwise.

The total welfare is not affected: market shares remain unchanged and platforms re-adjust prices without welfare losses. Thus the impact on profits and the impact on consumers have opposite signs.

Let us now consider the asymmetric case where only platform A can tie. The situation is similar to before in that for low values of θ the platform sells the bundle at p~ ¼ 0, while for large values of θ, the nonnegativity constraint is not binding and the allocation becomes independent of θ. The critical level of θ corresponds to the price that platform A would set at the equilibrium of a game where only platform B is constrained to set a non-negative price on side 1. Due to the strategic interaction this price differs from p1UU and is computed in the Appendix to be p1UC, where 10: UC



p1 ¼

 3γ UU p1 : 2 6γ−ðα 2 −α 1 Þ

ð5Þ

We thus obtain:

When the value of the good is low, platform A always gives the bundle for free. The (implicit) negative subscription fee is compensated by the value of the bundled product. If platform A disposes of a good that has high value (θ big enough), the price of the bundle is positive since the value of the tied good is larger than the optimal subsidy. 4.1. Strategic effects and equilibrium allocation Given price p1B = 0 and p1A b 0, platforms set their prices on side 2 according to the following response functions:   1 1 CC α þ α 2 A B p1 pA2 pB2 ¼ p2 þ p2 − 1 2 2 2t 1   1 1 CC α A A pB2 pA2 ¼ p2 þ p2 þ 2 p1 2 2 2t 1 Since only platform A is able to subsidize, the demand shifting effect induces platform B to reduce its price on side 2 and platform A to increase its price. The equilibrium prices on side 2 are then: 2α 1 þ α 2 A p1 3t 1 α 2 −α 1 A þ p1 : 3t 1

pA2 ¼ pCC 2 − pB2 ¼ pCC 2

ð6Þ

An increase in absolute terms of the subscription fee p1A results in a higher price p2A as both the competition softening effect and the demand shifting effect shift the reaction curve of platform A upward. Depending on the relative magnitude of externalities, the equilibrium reaction of platform B in side 2 may be to decrease or increase its price. If α2 > α1 (see Fig. 1.1) the competition softening effect is not strong enough to offset the shift in demand and platform B therefore reduces its price. On the contrary (see Fig. 1.2), when the externality is higher on side 1, α2 b α1, the competition softening effect dominates and all prices increase on side 2. 10

Recall that γ = t1t2 − α1α2.

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pB 2.0

441

pB 2.0

A's best reply

A's best reply B's best reply

1.5

no tying

1.0

B's best reply

1.5

1.0

no tying

tying

tying 0.5

0.5

0.0 0.0

0.5

1.0

1.5

2.0

0.0 0.0

0.5

1.0

pA

1.5

2.0

pA

Fig. 1. Platforms' best-reply on side 2 and equilibrium.

Notice that platform B always sets a lower price than platform A on side 2. The equilibrium platform market shares are then asymmetric and are given by:   1 α ðα −α 1 Þ pA1 − 3þ 1 2 2 γ 6t 1   A 1 α 2 −α 1 p1 A n2 ¼ − 2 γ 6 A

n1 ¼

4.2. Profits and welfare

The next graph (Fig. 2) plots platform A's market shares as a function of α1 for α2 = 2/3, t1 = 1/2, t2 = 2 and p1A = − 0.1. The striking feature is that despite the subsidy of platform A on side 1, its market share may not increase on side 2. Indeed n2A decreases with tying under circumstances leading the competitor to increase its price, i.e. when α2 b α1. The reason is that platform A opts for a higher mark up but lower sales on side 2 at given prices of the competitor. This rather counter-intuitive result is explained by the competition softening effect that reduces the incentive to sell on side 2. In a duopoly context this combines with the reaction of the competitor, which is why this can occur. Notice also that the market share on side 1 declines with α1 once it is larger than α2. When the competition softening effect is very strong, the tying platform A raises its price on side 2 substantially more than a monopoly would because the price increase is magnified by the reaction of platform B. In this range, the equilibrium effect on side 2 prices of an increase in the externality parameter α1 is strong

A's demands

enough for α1n2A to decrease, thereby reducing the attractiveness of platform A on side 1 and offsetting the effect of the subsidy. As we can see, the competition softening effect may induce some counter-intuitive effects. As another illustration, the simulation shows that for α1 > 0.185, the utility obtained by a consumer on side 2 decreases on both platforms, which contrasts with the monopoly case.

0.7

Due to the strategic interaction and the competition softening effect the impact of tying on the platforms' profit is not obvious. It depends on the relative levels of externalities on each side of the market, and in particular on the equilibrium reaction of the competitor. Before we proceed with the general case let us examine the situation where only the demand shifting effect is present. 4.2.1. No competition softening effect (α1 = 0) Suppose that only side 2 cares about the participation of the other side: α1 = 0. The fact that consumers on side 1 do not benefit from participation neutralizes the competition softening effect. In this case we have γ = t1t2 and p1UC/p1UU = 3γ/(6γ − α22), which varies from 1/2 to 3/2 when α2 varies from 0 to 2γ. Thus for large θ, the subsidy by platform A on side 1 can be smaller or higher than in the case where both platforms subsidize. As in the monopoly case, tying induces platform A to increase its market shares unambiguously on both sides. Platform A expands its market share on side 1 due to the subsidy. It becomes more attractive on side 2 and it can charge higher prices while expanding its market share also on side 2. Platform B reacts by setting lower prices on side 2, p2B b p2CC. This would hurt platform B for large network effects. Proposition 5. When only platform A uses tying and α1 ≃ 0:

side 1 A's demand 0.6

side 2 A's demand 0.5

i) Consumer surplus increases on each side; ii) For α2 large, platform A's profit increases, platform B's profit decreases and total welfare increases; iii) For α2 small (close to f1 + t1), platform A's profit decreases, total welfare decreases if t1t2 is large and platform's B profit increases if t1 b f1/2.

0.4 Proof. See Appendix. □

0.3 0.0

0.2

0.4

0.6

0.8

1.0

alpha_1 Fig. 2. Platform A's markets shares.

Tying is beneficial for consumer surplus when it induces only a demand shifting effect. Consumers on side 1 benefit from the subsidy and consumers on side 2 on average pay lower prices due to the platform B pricing strategy.

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The striking feature is that for small network externalities, tying benefits the competing platform at the expense of the tying platform. The fact that platform A profit decreases is due to the reaction of the competitor, who reduces its price on side 2: this offsets the benefits of tying to a point where the subsidy becomes costly. In other words, subsidizing is ex-post optimal at any prices of platform B but due to a negative strategic effect, it is ex-ante suboptimal for platform A. In this case one may expect the platform A to commit to not engage in tying strategies. The fact that platform B may benefit from tying pursued by its rival is more surprising. This occurs when the subsidy is small, f1 + t1 − α2 is negative and close to zero, and side 1 is competitive, α 2 −f 1 ≃t 1 b f21 : It is related to the fact that platforms have negative margins on the subsidized size. As a result, platform B benefits from lower losses on side 1 due to lower sales. Combined with the ability to mitigate the demand shifting effect by adjusting p2B, this is sufficient to allow platform B to obtain higher profits.

4.2.2. General case Let us now consider the effect of a marginal subsidy in the case where 0 b α1 ≠ α2. The main conclusion that we will discuss at the end of the section is that tying increases total welfare only if γ = t1t2 − α1α2 is small enough. Beforehand, we provide some results on profits and consumer surplus. Proposition 6. There exist ϕ A and ϕ B such that platform A profit is higher under tying if and only if p1A > ϕ A, while platform B profit is higher under tying if and only if p1A b ϕ B. Proof. See Appendix. □ The thresholds ϕ A and ϕ B may be positive or negative, and above or below p1UC. For instance in Proposition 1, ϕ A is positive when α2 is small. On the contrary ϕ A and ϕ B are below p1UC when α2 is large. Both platforms benefit from tying if ϕ B > − θ > ϕ A. It is shown in the Appendix that ϕ B > 0 > p1UC > ϕ A if α1 ≥ α2. Thus when tying occurs on the side perceiving the largest externality, the profits of both platforms increase. This simple conclusion contrasts with the one-sided case where tying would hurt the competitor. Whenever the thresholds lie between p1UC and 0, platform A benefits from tying if θ is not too large while platform B benefits if θ is large enough. The conclusion for A is due to the reaction of platform B. While tying improves A's position on side 2, p2B decreases when α1 b α2 which hurts platform A and offsets the gains from tying when the subsidy θ is large. The result for platform B reflects the fact that it benefits from tying only if the competition softening effect is strong enough, which occurs when the subsidy θ is large. Concerning consumers, tying generates distributional effects within sides, and some types of consumers will be negatively affected by tying. Indeed, we could not identify a case with a Pareto improvement on the consumer side. Aggregating consumers by side yields: Proposition 7. The consumer surplus is always larger on side 1 with tying than without. It is larger on side 2 if and only if the subsidy is large enough. Proof. See Appendix. □ The demand shift in side 1 has the effect of raising the perceived quality for platform A in side 2 and reducing it for platform B, while the competition softening effect reduces utility on both platforms. The overall impact is therefore ambiguous and depends on the level of prices, and thus on the intensity of the competition softening effect.

To conclude, let us consider total welfare. We show in the Appendix that it increases for α1 ≠ α2 if: 2

2

ðα 2 −α 1 Þðα 1 þ 5α 2 Þγ þ α 1 ðα 2 −α 1 Þ ðα 1 þ α 2 Þ−9γ > 0: which is violated when γ is large. More precisely ^ ðα 1 ; α 2 Þ > ðα 2 −α 1 Þ2 =4 such Proposition 8. If α1 ≠ α2, there exists γ ^ ðα 1 ; α 2 Þ. Moreover that tying is welfare improving if and only if γbγ ^ ðα 1 ; α 2 Þ tends to zero when α2 − α1 tends to zero. γ Proof. See Appendix. □ Hence tying raises total welfare if γ is small (recall that γ = t1t2 − α1α2), which corresponds to a situation with large externalities and strong competition. Otherwise total welfare is reduced. To understand these results, notice that in the Hotelling model, the change in welfare is the sum of two effects (transfers are neutral): variation of the total transport cost and of the overall size of network externalities. At the symmetric constrained equilibrium, transport costs are minimized. Hence tying can only increase the total transport cost. But on the other hand, network effects would be maximized with all consumers on the same platform. By shifting demand on both sides toward platform A, tying raises the value of network effects. The term γ = t1t2 − α1α2 summarizes the difference between these two effects. Indeed transport costs matter more if t1t2 is large, while network effects matter more if α1α2 is large. Hence when γ is small, the effect on network externalities offsets the effect on the transport cost and total welfare increases. As an illustration consider the case where α1 = α2. Then from above we obtain: Corollary 1. When only platform A uses tying and α2 = α1: i) Total consumer surplus and total welfare decrease; ii) Profit increases for both platforms. Proof. See Appendix. □ When α1 = α2, platform A expands its market share on side 1, but the demand shifting effect and the competition softening effect cancel each other out so that the market shares on side 2 are left unchanged, n2A = n2B = 1/2. Moreover platform B also maintains its price unchanged: p2B = p2CC. Clearly this situation is favorable for both platforms as B sells less on side 1, while A moves closer to its best-response. In this case tying hurts consumers on side 2 who obtain a lower externality at platform B and a higher price on platform A. This negative effect is sufficient to offset the other effects so that overall the total consumer surplus is lower under tying. 5. Extensions 5.1. Pure bundling So far we have considered the case where the platform introduces on side 1 tying by selling the good both independently and bundled with the service. An alternative strategy would be to sell only the bundle, referred to as pure bundling. As already mentioned, pure bundling arises if the value of the good is below its cost so that selling the good alone is not profitable. It may also arise if it helps the firm to foreclose the market in situations where mixed bundling would not (Whinston (1990)). Let us define the margin on the good to be m = θ − c. In the case of pure bundling, the margin can be positive or negative. As discussed in the monopoly section, the strategy of platform A under pure bundling is the same as under tying but with a marginal cost on side 1 equal to f1 − m instead of f1. The comparison between pure bundling and

A. Amelio, B. Jullien / Int. J. Ind. Organ. 30 (2012) 436–446

mixed bundling thus reduces to the comparison of equilibria when the platform 1 has cost f1, and when it has cost f1 − m. The first paper to analyze the effect of the marginal costs in a duopoly unconstrained model is Farhi and Hagiu (2008). When m is far away from zero, the analysis becomes quite involved. The effect of the marginal cost of platform A on prices and profits may be quite complex. It depends on the feedback effects and the levels of the margin of the firms on both sides of the market. In particular it may happen that although platform B sets p1B = 0 with no bundling, the equilibrium response of platform B when platform A sells a pure bundle is to raise its price so that p1B is positive. As our concern is mostly with free services, we refer the reader to Farhi and Hagiu (2008) and to our working paper (Amelio and Jullien (2007)) for the detailed analysis of pure bundling when m is large and only discuss here some of the differences with tying for the case where m is close to zero. Notice that when m = 0, the equilibrium prices and profits are the same under pure bundling and tying. As a result, when the margin m is not too far away from zero, the analysis of the equilibrium is qualitatively similar to the case of tying. The price p1B is zero, and there exists a critical price p1P such that the price of the bundle is max{0, p1P + θ}. The price p1P depends on m, P p1

¼

þ α 1 ðα 2 −α 1 Þ m 6γ−ðα 1 −α 2 Þ2

3γ UC p1 −

and decreases with m if 3γ + α1(α2 − α1) > 0. Under pure bundling, the prices on side 2 are then given as a function of the prices on side 1 by: 2α 1 þ α 2 A 2α 1 p1 − m 3t 1 3t 1 α −α 1 A α 1 þ 2 p1 − m 3t 1 3t 1

pA2 ¼ pCC 2 − pB2 ¼ pCC 2

Thus comparing with the equilibrium under tying (Eq. (6)), we see that pure bundling reduces the prices on side 2 if the margin m is positive. This is due to the change in perceived margin for side 1 consumers. So compared with tying, pure bundling generates a weaker (resp. stronger) competition softening effect when the margin m is positive (resp. negative). Increasing m from m = 0 (tying) makes platform A more aggressive on side 2. This hurts platform B on side 2 and leads B to reduce both the price p2B and the quantity n2B. On the other side the quantity n1Bdecreases which mitigates the negative impact on the profit ΠB = − f1n1B + (p2B − f2)n2B. When evaluating the implication for platform B's profit, one would expect the first effect to dominate. This is the case only if α1 is small or the subsidy θ is small. Proposition 9. Suppose that θ b − p1UC and platform A pursues a pure bundling strategy, then in the vicinity of m = 0, the profit of platform B decreases with m if (α2 − α1)θ b 3γ. Proof. See Appendix. □ The proof also shows that the differential profit between bundling and tying of platform A decreases with m in the vicinity of m = 0 for θ small. Thus, for a small subsidy θ by platform A, both platforms would prefer tying when m is positive and pure bundling when m is negative. In this case, pure bundling (as opposed to mixed bundling) has an exclusionary effect only if the margin m is positive. 5.2. Multi-homing The analysis that precedes emphasizes the strategic implications for side 2 competition of tying on side 1. These strategic effects are only at work if there is effective competition on side 2, which is the case in our model because consumers are bound to choose only one

443

platform, i.e. they single-home. By contrast, these effects would evaporate if all consumers were multi-homing on side 2. To see that, let us follow Armstrong and Wright (2007) by assuming that on side 2 all consumers join the two platforms. This is the case if on this side, multi-homing is possible and t2 = 0. Then, the utility of a consumer on side 2 when participating on both platforms is:     A A B B v2 þ α 2 n1 −p2 þ v2 þ α 2 n1 −p2 ; where we assume that the stand-alone utility is additive. Since n2A = n2B = 1, the value of network effects at any platform for a consumer on side 1 is α1 and the demand is j

n1 ¼

−j j 1 1 p1 −p1 þ : 2 2 t1

For a consumer on side 2, the participation in one platform doesn't depend on the offer of the other platform but only on the extent of network effect and the price level. Thus, the platforms do not compete on side 2 and charge monopoly prices p 2j = v2 + α2n1j . The profit is then:   j j j p1 −f 1 n1 þ v2 þ α 2 n1 : The model is similar to the Hotelling model with a cost f1 − α2. The unconstrained equilibrium price is then p1UU = f1 + t1 − α2. Again this price may be negative in which case firms will offer free services on side 1. Tying may then be used to relax the constraint. A first immediate result is that when both firms use tying, firms face a prisoner dilemma type of game. Indeed, in equilibrium they reduce their prices on side 1, with no change on sales and prices on side 2. Thus profits decreases. Consider now the effect of tying by a single platform. Without tying, profits (net of the profit θ − c on the good) are (α2 − f1)/2 + v2 with price p1A = p1B = 0. Conversely, with tying the (implicit) prices are p1A b 0 = p1B and profits are: pA1 −pUU 1 t1

!

Π TA ¼

1 ðα −f Þ þ v2 þ 2 2 1

Π TB ¼

1 pA ðα 2 −f 1 Þ þ v2 þ 1 ðα 2 −f 1 Þ 2 2t 1

−pA1 2

!

The optimal price for platform A is such that 0 > p1A > p1UU. Thus tying benefits platform A and hurts platform B. (Recall that α2 b f1 when p1UU b 0.) Clearly consumers' surplus on side 1 increases also. However in the Hotelling model, the total welfare would decrease because tying doesn't affect network effects in the case of multi-homing but it does raise the average transport cost on side 1. Overall, the extent to which consumers multi-home has the effect of progressively neutralizing the strategic effects of a tying strategy on the competition in side 2, which are the relevant aspects in our model. Thus, in our model, multi-homing has the consequence of reducing the strategic interaction to a more standard “one-sided” market. 5.3. Demand expansion The model focuses on the strategic interaction between the two platforms and for this purpose ignores the possibility that the subsidy raises total demand. We discuss here the extension to elastic aggregate demand on side 1. First, remark that, in an equilibrium with zero prices on side 1, an elastic demand would reduce the intensity of competition on side 2 because the impact of increased side 2 participation on side 1 loss would be stronger. Thus the price p2CC would increase as demand on side 1 becomes more elastic (preserving quantities). In other words

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the pricing strategies of the two platforms would be more affected by the competition softening effect. Moreover, starting from the constrained equilibrium, decreasing p1A below zero would induce not only a transfer of demand from platform B to platform A but also an inflow of new consumers on platform A. Then for given prices the demand of platform A would increase faster on both sides if demand were elastic. This would raise the advantage of platform A on both sides. Notice that this should make tying more attractive for platform A insofar as it allows her to obtain the same demand differential n1A − n1B with a lower subsidy. For the analysis of strategic effects, besides the previously explained demand shifting effect and the competition softening effect, we need to consider an additional effect corresponding to the demand expansion. Demand expansion would be concentrated on platform A as long as n2B decreases (since p1B is unchanged) and would act in a similar way to the demand shifting effect: it enhances the shift of demand from platform B to platform A on side 2. Moreover, for platform B, this shift is not compensated by a reduction in the loss on side 1 since the expansion is due to new consumers. According to this logic, demand expansion would tend to make tying by platform A more detrimental to platform B. This however should be tempered by our first remark that an elastic demand weakens competition on side 2. Indeed one may surmise that the competition softening effect would be stronger with an elastic demand. Overall while this suggests that demand expansion would make tying more attractive to platform A and more detrimental to platform B, there may still be instances where the reduction in competition is sufficient to offset the impact of demand expansion in such a manner that platform B profit increases when aggregated demand becomes more elastic. 6. Conclusion In this paper we show that under two-sided externalities a platform constrained to set non-negative prices ties the sales of another good with the access to the platform as a way to relax the non-negativity constraint. By giving away the bundle for free or at a discounted price, conditional on participation to one side, the platform eventually implements implicit negative subscription prices in a context where monetary subsidy would be inefficient or could generate opportunistic behaviors. Such a tying strategy doesn't involve the firm leveraging market power from one market to another. The aim is to stimulate participation on one side and to raise the profit on the other side by offering a higher membership value. We thus corroborate the idea that tying in two-sided markets is a way to enhance efficiency by inducing better coordination between various sides. This use of tying is also implicitly related to the literature on price discrimination. By tying the platform service to a good of particular interest for the targeted side, the platform may avoid attracting undesirable customers. To this extent, the theory developed in this paper illustrates an instance of second-degree price discrimination implemented through tying which confirms the more general idea that price discrimination may help a network to coordinate the customers' participation and calls for further research in this field. Acknowledgments The authors thank Bernard Caillaud and two anonymous referees for the valuable comments. We are also indebted to Simon Anderson, Mark Armstrong, David Evans, Carole Haritchabalet, Michael Katz, Massimo Motta, Patrick Rey, and Jean-Charles Rochet for the insightful discussions, as well as participants to the IDEI Conference on Software and Internet Industries, and to the Network of Industrial Economists (University of Bristol).

Appendix A. Tying by both platforms Proof of Proposition 3. Denote by λ A the multiplier associated to the non-negativity constraint. Taking derivatives of the Lagrangians we obtain the following first order conditions for platform A:         B A B A α 2 pA2 −f 2 t 2 pA1 −f 1 1 α 1 p2 −p2 þ t 2 p1 −p1 A 0¼ þ − − þλ 2 ð t −α α Þ ð t −α α Þ ð t −α α Þ 2 t 2 t 2 t 1 2 1 2 1 2 1 2 1 2 1 2         B A B A t 1 pA2 −f 2 α 1 pA1 −f 1 1 α 2 p1 −p1 þ t 1 p2 −p2 0¼ þ − − 2 2ðt 1 t 2 −α 1 α 2 Þ 2ðt 1 t 2 −α 1 α 2 Þ 2ðt 1 t 2 −α 1 α 2 Þ

and the symmetric for B. If p1UU b − θ, the problems of the two platforms are constrained. Therefore, setting p1A = p1B = − θ and λA = λB > 0, the constrained symmetric equilibrium prices are: p1 ¼ −θ: p2 ¼ f 2 þ t 2 −α 1 þ

α1 ðf þ t 1 −α 2 þ θÞ: t1 1

Platforms share the market equally and the equilibrium profits are Π

CC

¼Π

UU

þ

    α 1 f 1 þ t 1 −α 2 þ θ f þ t 1 −α 2 þ θ − 1 : 2 2 t1

Moreover the unconstrained total consumers surplus is UU

TCS

3 5 ¼ 2v−f 1 −f 2 þ ðα 1 þ α 2 Þ− ðt 1 þ t 2 Þ: 2 4

□

Appendix B. Duopoly with one platform tying Proof of Proposition 4. When platform A uses mixed bundles, first order conditions yield:         B A B A A A α 2 p2 −f 2 t 2 p1 −f 1 1 α 1 p2 −p2 þ t 2 p1 −p1 A − − þλ ; 0¼ þ 2 t 2 −α 1 α2 Þ 2ðt 1t 2 −α 1 α2 Þ 2 −α 1 α  2ðt 1 t 2Þ  2ðt 1 B A B A α 2 pB2 −f 2 t 2 pB1 −f 1 1 α 1 p2 −p2 þ t 2 p1 −p1 B 0¼ − − − þλ ; 2 2ðt 1 t 2 −α 1 α 2 Þ 2ðt 1 t 2 −α 1 α 2 Þ 2ðt 1 t 2 −α 1 α 2 Þ

on side 1, and         B A B A t 1 pA2 −f 2 α 1 pA1 −f 1 1 α 2 p1 −p1 þ t 1 p2 −p2 − − ; 0¼ þ 2 t 2 −α 1 α2 Þ 2 −α 1 α  2ðt 1 t 2Þ  2ðt 1t 2 −α 1 α2 Þ 2ðt 1 B A B A t 1 pB2 −f 2 α 1 pB1 −f 1 1 α 2 p1 −p1 þ t 1 p2 −p2 − − ; 0¼ − 2 2ðt 1 t 2 −α 1 α 2 Þ 2ðt 1 t 2 −α 1 α 2 Þ 2ðt 1 t 2 −α 1 α 2 Þ on side 2. First, for given prices p1A and p1B on side 1, the optimal response functions on side 2 are:   1 1 CC α þ α 2 A α 2 B B pA2 pB2 ¼ p2 þ p2 − 1 p1 þ p ; 2 2 2t 1 2t 1 1   1 1 α þ α α A CC A 2 B pB2 pA2 ¼ p2 þ p2 − 1 p1 þ 2 p1 : 2 2 2t 1 2t 1 Solving for these prices we obtain pA1 pB þ ðα 2 −α 1 Þ 1 3t 1 3t 1 pB1 pA and pB2 ¼ pCC þ ðα 2 −α 1 Þ 1 : 2 −ðα 2 þ 2α 1 Þ 3t 1 3t 1 pA2 ¼ pCC 2 −ðα 2 þ 2α 1 Þ

A. Amelio, B. Jullien / Int. J. Ind. Organ. 30 (2012) 436–446

The first order conditions on market 1 then write: A

    6t 1 t 2 −ðα 1 þ α 2 Þ2 −2α 1 α 2 pA1 − 3t 1 t 2 −ðα 1 þ α 2 Þ2 þ α 1 α 2 pB1

B

6t 1 ðt 1 t 2 −α   1 α2 Þ  6t 1 t 2 −ðα 1 þ α 2 Þ2 −2α 1 α 2 pB1 − 3t 1 t 2 −ðα 1 þ α 2 Þ2 þ α 1 α 2 pA1

λ ¼ λ ¼

6t 1 ðt 1 t 2 −α 1 α 2 Þ

Total welfare: Total welfare, denoted TW, changes according to the following expression: −

ðf 1 þ t 1 −α 2 Þ 2t 1

−

ðf 1 þ t 1 −α 2 Þ 2t 1

First notice that it is not possible that no constraint binds. Now suppose that both constraints are still binding. Therefore the prices are p1A = − θ and p1B = 0. Observe that λ A b λ B which implies that this is the solution if λ A > 0 or A p1

¼ −θ >

UC p1

445

3ðt 1 t 2 −α 1 α 2 Þ ¼ ðf 1 þ t 1 −α 2 Þ : 4t 1 t 2 −ðα 1 þ α 2 Þ2 þ 2ðt 1 t 2 −α 1 α 2 Þ

 2  pA1 36t 1 γ

2

2

2

2

2

ð3γ þ α 1 ðα 2 −α 1 ÞÞ þ t 1 t 2 ðα 2 −α 1 Þ −18γ þ 4ðα 2 −α 1 Þ γ



or  2  pA1 36t 1 γ

2

2

ðα 2 −α 1 Þðα 1 þ 5α 2 Þγ þ α 1 ðα 2 −α 1 Þ ðα 1 þ α 2 Þ−9γ

2

 :

ð12Þ

Now suppose θ > − p1UC. Setting λ A = 0 and λ B > 0, the equilibrium prices are p1A = p1UC and p1B = 0. We verify that:

□

9t 1 t 2 −2ðα 1 þ α 2 Þ2 −α 1 α 2

ðf 1 þ t 1 −α 2 Þ > 0: λ ¼− 2t 1 6t 1 t 2 −ðα 1 þ α 2 Þ2 −2α 1 α 2

Proof of proposition 5. In the case where α1 = 0, the feasible range pffiffiffiffi

of α2 is f 1 þ t 1 ; 2 γ where γ = t1t2 > (f1 + t1) 2/4. Consumers' surplus increases on both sides. Platform A profit increases if

B

□

2α 2 þ

Derivation of profits and surplus Profits: The function ΠTA − (Π CC + (θ − c)) has value pA − 1 6t 1

The LHS is increasing with α2. When α2 is small, it must be the case that p1UU >−α2 is close to zero and thus p1A. Thus for α2 close to f1 +t1, the LHS is −f1 −t1 b 0 and the condition is violated. At the upper bound the pffiffiffiffi value is 4 γ þ 53 pA1 −3ðf 1 þ t 1 Þ which is positive for γ large. B ΠT > Π C if

! ! ðα −α 1 Þ2 A p1 : 3− 2 3γ

2α 2 þ α 1 −3t 1 −3f 1 þ

This is positive if 2α 2 þ α 1 þ

! ðα 2 −α 1 Þ2 A 3− p1 −3ðf 1 þ t 1 Þ > 0 3γ

ð7Þ

!

ðα −α 1 Þ2 A p1 2ðα 1 −α 2 Þ þ 3f 1 − 2 3γ

α2 þ

α 22 A 3 p − f b0: 6γ 1 2 1

At the lower bound this reduces to (again with p1A close to zero) t1 − f1/2 b 0. Total welfare increases if − 9γ + 5α22 > 0, and the values at the lower bound is − 9γ + 5(f1 + t1) 2 which is positive if γ is small, while the value at the upper bound is 11γ > 0. ■

We also have ΠTB > Π C if −pA1 6t 1

! α 22 A 3− p −3f 1 −3t 1 > 0: 3γ 1

! >0

which holds if

Proof of proposition 6. Condition (7) write as 2

ðα 2 −α 1 Þ A 3 p1 − f 1 b0: 2 6γ

α 2 −α 1 þ

ð8Þ

A

Consumer surplus: The consumer surplus on side i writes as:  CSi ¼ vi þ

B B B α i n−i −t i ni −pi

þ ti

nAi

2

 2 þ nBi 2

¼

pA − 1 2t 1

A

B

p1 bϕ ¼

:

!    3γ þ α 1 ðα 2 −α 1 Þ 2  A −p1 : t1 þ 2 6γ

A

¼

−p1 2t 1

! −α 1 þ 2t 1 t 2

   α 2 −α 1 2  A −p1 : 6γ

ð9Þ

ð10Þ

A

ϕ

−pUC 1

pA1 2t 1

!

!  ð3γ þ α 1 ðα 2 −α 1 ÞÞ2 þ t 1 t 2 ðα 2 −α 1 Þ2  A −p −α þ t 1 1 1 : 18γ 2

  UU 3γ 3p1 þ α 2 −α 1

 3γ UU p1 2 6γ−ðα 2 −α 9γ−ðα 2 −α 1 Þ2 !1 Þ ! 3γ 9γ−2ðα 2 −α 1 Þ2 UU ¼ p1 þ α 2 −α 1 6γ−ðα 2 −α 1 Þ2 9γ−ðα 2 −α 1 Þ2 ¼

 −

is negative if α1 > α2. □ Proof of proposition 7. The result for side 1 is immediate given Eq. (9). For side 2 the condition is

The total consumer surplus changes by

−

  6γ 3 f −α þ α : 1 2 1 ðα 2 −α 1 Þ2 2

Clearly ϕ B > 0 if α1 > α2. Moreover

On side 2 we obtain the change in surplus: T C CS2 −CS2

3γ ð3ðf 1 þ t 1 Þ−2α 2 −α 1 Þ: 9γ−ðα 2 −α 1 Þ2

The condition (8) writes as

The total change in consumer surplus on side 1 is: T C CS1 −CS1

A

p1 > ϕ ¼

ð11Þ

2t 1 t 2

   α 2 −α 1 2  A −p1 > α 1 : 6γ

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A. Amelio, B. Jullien / Int. J. Ind. Organ. 30 (2012) 436–446

Proof of proposition 8. Assumption 1 implies that γ > (α2 − α1) 2/4. When γ reaches this minimal value, the total welfare change is positive (Eq. (12)): ðα 2 −α 1 Þðα 1 þ 5α 2 Þγ þ α 1 ðα 2 −α 1 Þ2 ðα 1 þ α 2 Þ−9γ2  α −α 2  1 ≃ 2 11α 22 þ 3α 21 þ 18α 2 α 1 > 0 4 The condition is concave in γ and positive at the lower bound if ^ α2 ≠ α1 which implies that it becomes negative above some level γ which is finite since it is negative for large γ. More precisely we have

^¼ γ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðα 2 −α 1 Þðα 1 þ 5α 2 Þ þ jα 2 −α 1 j ðα 1 þ 5α 2 Þ2 þ 9α 1 ðα 1 þ α 2 Þ 18

which tends to zero when α2 − α1 tends to zero. □ Proof of corollary 1. The result on total welfare is immediate. Then ϕ A = p1UU and ϕ B = + ∞ so that profits increase for both platforms. Finally since profits increases, the fact that total welfare decreases implies that consumer surplus decreases. □ Proof of proposition 9. When p1B = 0: 2α 1 þ α 2 A 2α 1 p1 − m 3t 1 3t 1 α −α 1 A α 1 þ 2 p1 − m 3t 1 3t 1

pA2 ¼ pCC 2 − B

CC

p2 ¼ p2 and

   A 1 α −α 1 p1 α2 þ 3 þ α1 2 − 1 m 2 γ 6t 1 6γt 1   1 α 2 −α 1 pA1 α 1 B − m n2 ¼ þ 2 γ 6 6γ

nB1 ¼

The profit of platform B is ΠB = − f1n1B + (p2B − f2)n2B. The slope when p1A = − θ is 2 α ∂Π B α α B  B 1 ¼ f 1 1 − 1 n2 − p2 −f 2 6γt 1 3t 6γ ∂m 1  α α α 2 −α 1 α α θþ 1 1m ¼− 1 þ 1 3t 1 9t 1 γ 9t 1 6γ

This is negative at m = 0 if (α2 − α1)θ b 3γ.

The profit with pure bundling of platform A is ΠA = (m + p1A − f1) n1A + (p2A − f2)n2A while it is (p1A − f1)n1A + (p2A − f2)n2A + m with tying. The change in the profit differential between pure bundling and tying is then   α2 α ∂Π A 2α A  A A A 1 1 − 1 n2 þ p2 −f 2 −1 ¼¼ n1 þ m þ p1 −f 1 6γt 1 3t 1 6γ ∂m After substitution of the values, we obtain   ∂Π A 1 α 2α ðα −α 1 Þ θ α2 þ 1 m −1 ¼ − − 1 þ 3 þ 1 2 2 6t 1 γ 6t 1 9γt 1 3 ∂m which is negative at m = 0 for θ close to zero. □ References Adams, W., Yellen, J., 1976. Commodity bundling and the burden of monopoly. Quarterly Journal of Economics 90 (3), 475–498. Amelio, A., Jullien, B., 2007. Tying and Freebies in Two-Sided Markets. IDEI, Working Paper, 445. Armstrong, M., 2006. Competition in two-sided markets. The Rand Journal of Economics 37 (3), 668–691. Armstrong, M., Wright, J., 2007. Two-sided markets with multi-homing and exclusive dealing. Economic Theory 32 (2), 353–380. Caillaud, B., Jullien, B., 2003. Chicken & egg: competition among intermediation service providers. The Rand Journal of Economics 34 (2), 309–328. Choi, J.P., 2010. Tying in two-sided markets with multi-homing. The Journal of Industrial Economics 58 (3), 607–626. Evans, D., 2003a. Some empirical aspects of multi-sided platform industries. Review of Network Economics 2 (3), 191–209. Evans, D., 2003b. The antitrust economics of two-sided markets. The Yale Journal on Regulation 20 (2), 325–381. Farhi, E., Hagiu, A., 2008. Strategic Interactions in Two-Sided Market Oligopolies. Working Paper. Harvard University. Jullien, B., 2005. Two-sided markets and electronic intermediaries. CESifo Economic Studies 51 (2–3), 233–260. Jullien, B., 2011. Competition in multi-sided markets: divide-and-conquer. American Economic Journal: Microeconomics 3, 1–35. Milgrom, P., Shannon, C., 1994. Monotone comparative statics. Econometrica 62 (1), 157–180. Rochet, J., Tirole, J., 2003. An economic analysis of the determination of interchange fees in payment card systems. Review of Network Economics 2 (2), 69–79. Rochet, J., Tirole, J., 2006. Two-sided markets: a progress report. The Rand Journal of Economics 37 (3), 645–667. Schmalensee, R., 1984. Gaussian demand and commodity bundling. Journal of Business 57 (1), 211–230. Whinston, M., 1990. Tying, foreclosure, and exclusion. The American Economic Review 80 (4), 837–859.