Homogeneous Platform Competition with Heterogeneous Consumers

Transcription

1 Homogeneous Platform Competition with Heterogeneous Consumers Thomas D. Jeitschko and Mark J. Tremblay Prepared for IIOC 2014: Not Intended for Circulation March 27, 2014 Abstract In this paper we investigate a two-sided market platform model. We develop a theoretical model for platforms with two sides, buyers or consumers and sellers or producers, who buy and sell through the platform. This occurs in the market for smartphones with consumers, app producers, and smartphone operating systems; similarly, it occurs in the video game market with consumers, video game producers, and video game consoles. Only consumers who purchase the platform can purchase content from the producers. In these markets consumers are heterogeneous in their gains from the producer side. Some consumers purchase many games while others only buy a few. In developing platform competition, instead of using the Hotelling model we develop a model of competition between two homogeneous platforms that allows consumers and firms to optimize with respect to how they home, i.e. we allow both individual consumers and individual producers to multi-home or single-home depending on whether it is optimal based on their type. This leads to multiple equilibrium allocations of consumers and firms all of which are seen in existing markets. 1 Introduction Many transactions among multiple parties are conducted through an intermediary or platform. For many consumers, purchases are made online through platforms like ebay or Amazon. Some buyers actually consume their purchases on the platform or network: Google users obtain searches on Google, Facebook users visit Facebook pages of their friends, smartphone users use apps on their phones, and video game users play games on their gaming console. Today, platforms provide many commodities in our society and their market structures vary considerably. For example, Google and Facebook are near monopolists who provide their services for free; however smartphones and video game consoles are generally priced well above zero. 1

2 The early development of the literature began with markets where network externalities were prominent, Katz and Shapiro (1985) and Economides (1996a). Caillaud and Jullien (2003) investigate competition further but assume that coordination favors the incumbent platform; otherwise platforms may fail to gain a critical mass, or fail to launch. There has been a considerable amount of focus on two-sided markets: Evans (2003); Rochet and Tirole (2003); Armstrong (2006); Rochet and Tirole (2006). Many have attempted to resolve the issues surrounding the failure to launch and the multiple equilibrium aspects which occur in multi-sided market models, see Ellison and Fudenberg (2003) and Hagiu (2006). Weyl (2010) generalizes many of the previous pricing models including those of Armstrong and Rochet and Tirole for the monopoly case using what he calls insulating tariffs. With an insulating tariff, the price that a platform charges on one side of the market depends on the number of agents on the other side. An insulating tariff makes price a function of the number of users on the other side and in doing so resolves the failure to launch and multiple equilibrium issues which occur in other models. Weyl and White (2013) extend this model with insulating tariffs to the case where there exist multiple competing platforms. Both models are very general, allowing for rich heterogeneity. However, they also find that more heterogeneity of agents can lead to issues with the platform s profit maximization platform; Deltas and Jeitschko (2007) come to a similar conclusion. One of the main empirical contributions is Lee (2013b). He investigates the video game market when Xbox entered the market from He finds that exclusive contracts with game developers allowed Microsoft to enter the video game market with Xbox. The market with video game consoles and smartphones as platforms are the examples of platforms that this paper aims to focus on. In the majority of the previous models on platforms, agents marginal utilities for the number of agents on the other side of the platform are homogeneous. In considering a two-sided market with consumers and firms then this assumption is not reasonable since consumers vary in their value for the other side. For example, in the market for video games, Lee 2013 finds there exists a large variance in the number of video games purchased across consumers. This suggests that there are many types of consumers whose marginal utilities for video games differ. To investigate markets with consumers and firms where consumers marginal utilities are heterogeneous we develop a model where consumer marginal utilities depend on the consumer type. This leads to equilibria where consumers who join the platform consume different amounts of video games or apps. This differs from much of the previous literature yet it closely resembles what is seen in reality. Furthermore, 2

3 the monopoly model we develop generalizes naturally to a model of competition between two homogeneous platforms which provides new insights regarding multiple equilibrium and equilibrium allocations of consumers and firms. The rest of the paper is organized as follows. Section 2 constructs the base model for a monopoly platform. In Section 3, we generalize this model to include two competing platforms. We determine the equilibrium prices the platforms charge and determine how consumers and firm allocate in equilibrium. Unlike other models we allow both consumers and firms to optimize over participation, i.e. they choose whether or not to single-home, multi-home, or not participate on a platform. Section 4 concludes. 2 Base Model Consider two groups of agents who would benefit from interaction but are unable to do so without an intermediary. This intermediary platform charges agents in each group a price to participate on the platform. We will call this price the membership fee. The utility of one group depends on the number of agents of the other group who join the platform. Agents on both sides of the platform are described by continuous variables. Agents on side 1, the consumers or buyers, are indexed by τ [0, N 1 ] and agents on side 2, the firms or producers, are indexed by θ [0, ). The number of consumers that join the platform is denoted by n 1 [0, N 1 ] and the number of firms that join the platform is denoted by n 2 [0, ). The utility for a consumer of type τ is given by: u 1 (τ) = v + α 1 (τ) n 2 p 1. (1) Here, v is the membership value every consumer receives from joining the platform. This is the stand alone utility of being a member of the platform when no firms join the platform. Note that it is possible for v = 0 but for smartphones and video game consoles v > 0. For smartphones v is the utility from using a smartphone as a phone and for video game consoles v is the utility from using the console to watch DVDs. Also notice that consumers are homogeneous in their membership benefit to the platform; v does not depend on consumer type τ. The price or membership fee that consumers pay the platform is given by p 1. The network effect or the marginal benefit to a consumer for an additional firm on the platform is given by α 1 (τ). Thus, consumers are heterogeneous in their marginal benefit for firms. We focus on the case when network effects are positive so we assume α 1 (τ) 0 for all τ. We assume α 1 ( ) is a decreasing continuous and twice continuously differentiable function. 3

4 Since α 1 (τ) is decreasing, it orders consumers by their marginal benefits. Consumers whose type is τ close to zero have marginal benefits that are high relative to those consumers whose type is located far from zero. The platform knows v and α 1 (τ) but cannot distinguish the individual values for each consumer τ. Thus, it cannot price discriminate between consumers. On the other side of the platform, side 2, are the firms. Their utility or profits are given by: π 2 (θ) = α 2 n 1 cθ p 2, (2) Here c > 0 is the synchronization cost parameter for the firms. We assume firms must pay to synchronize their product to the platform s operating system and this cost depends on firm type. The price or membership fee the firms pay to the platform is given by p 2. The marginal benefit firms receive from an additional consumer on the platform is given by α 2 which does not depend on firm type. The platform knows the firm s profit structures but cannot identify firms individually; hence, it cannot price discriminate between firms. Firms differ from consumers in a few aspects. First, notice that the marginal benefit of firms for an additional consumer is homogeneous across firm type. The logic here is that an additional consumer will shift each firms demand curve up in the same way. The assumption we are making here is that each consumer sees firm products as homogeneous, but consumers differ in their preferences for the number of products they wish to consume. Second, firms are heterogeneous in their synchronization cost. Firms with type θ close to zero pay lower synchronization costs relative to those with higher type θ. Lastly we introduce the platform. The platform s profits are given by: Π M = n 1 (p 1 f 1 ) + n 2 (p 2 f 2 ), (3) The platform faces no fixed costs. The marginal cost the platform faces for an additional agent i to join the platform is given by f i. The platform is a price setter so it maximizes profits with respect to p 1 and p 2. This model for platforms is similar to much of the previous literature. One key difference of our model is that consumers preferences for the seller side is heterogeneous. This leads to consumers purchasing different amounts of products on the platform. In other models consumers are homogeneous with respect to their marginal benefit for firms and thus all purchase the same products once on the platform. 4

5 2.1 Pricing with a Monopoly Platform In this section we solve the platform s problem and determine the optimal prices it charges consumers and firms. Given prices p 1 and p 2 the last agent on each side to join the platform has utility or profit equal to zero. The last consumer to join the platform is denoted by τ [0, N 1 ] and the last firm to join the platform by θ [0, ). We assume consumers and firms are distributed uniformly. Furthermore, since α 1 ( ) is decreasing in τ, the last consumer to join, τ, gives the number of consumers who join: n 1 = τ. For firms, the relationship is similar since firm synchronization costs are increasing in firm type, hence firm profits will decrease in θ. Thus θ = n 2. Since u 1 (τ = τ) = 0, π 2 (θ = θ) = 0, θ = n 2, and τ = n 2 we can use equations (1) and (2) to solve for prices as a function of the number on consumers and firms that join the platform: p 1 (n 1, n 2 ) = v + α 1 (n 1 ) n 2, (4) p 2 (n 1, n 2 ) = α 2 n 1 cn 2. (5) By substituting these into the platform s profit function, equation (3), the platform maximizes its profits with respect to n 1 and n 2 to determine the optimal prices. This gives two first-order conditions and three second-order conditions that must be satisfied: [v + α 1 (n 1 )n 2 f 1 ] + n 1 [α 1(n 1 )n 2 ] + α 2 n 2 = 0, (6) n 1 α 1 (n 1 ) + [α 2 n 1 cn 2 f 2 ] cn 2 = 0, (7) 2n 2 α 1(n 1 ) + n 1 n 2 α 1(n 1 ) < 0, (8) 2c < 0, (9) 2c[2n 2 ( α 1(n 1 )) n 1 n 2 α 1(n 1 )] [α 1 (n 1 ) + n 1 α 1(n 1 ) + α 2 ] 2 > 0. (10) The first two equations, (6), and (7), implicitly define the optimal participation levels n 1 and n 2. Substituting these levels into the price equations, (4) and (5), determines the 5

6 optimal prices the monopoly platform charges to guarantee the optimal participation levels. The second-order conditions, equations (8), (9), and (10), essentially imply α 1( ) cannot be too negative. Note, the second-order conditions hold for all linear α 1 ( ). The first-order conditions, equations (6) and (7), give the optimal prices the platform charges: p 1 = f 1 α 2 n 2 + [ α 1(n 1 )]n 1 n 2, (11) p 2 = f 2 α 1 (n 1 )n 1 + cn 2. (12) For the price consumers face, f 1 is the marginal cost to the platform for an additional consumer, n 2 α 2 is the marginal benefit to the firm side for an additional consumer, and [ α 1(n 1 )]n 1 n 2 > 0 is the monopoly markup. The consumer price is the marginal cost minus their benefit to the firm side plus the monopoly markup. For the firm price, f 2 is the marginal cost to the platform for an additional firm, α 1 (n 1 )n 1 is the benefit to the marginal consumer for an additional firm, and cn 2 is the monopoly markup. The firm price is the marginal cost minus their benefit to the consumer side plus the monopoly markup. α 1 (τ). Using this base model we derive closed form solutions assuming a specific functional form 2.2 Closed Form Solutions and Welfare In this subsection we investigate closed form solutions and welfare for the case when α 1 ( ) is linear. Assume α 1 (τ) = a bτ. This occurs when consumer α 1 s are distributed uniformly over [0, a] and the number of potential consumers present on the platform is N 1 = a. To b simplify calculations, we further assume v = f 1 and f 2 = 0. 1 These assumptions do not affect the firm s profit function but consumer utility is now given by: u 1 (τ) = v + [a bτ] n 2 p 1. (13) 1 These assumptions are not extremely critical in the analysis and they make computations simple. This case, with v = f 1 0 and f 2 = 0 fits with the smartphone and video game examples we have discussed. The marginal cost of firms, or app producers, for the platform is zero. Firms pay synchronization costs that drive marginal cost to zero for the platform. Consumers receive membership benefits from having a smartphone or a video game console, they can make phone calls or watch DVDs, and producing each piece of hardware for an individual consumer has a positive marginal cost for the platform. This assumption means these values are equal. 6

7 Since α 1 (τ) = a bτ, we have α 1(τ) = b for all τ [0, a b ]. conditions, (6) and (7), yields: 2 Using the first order n 1 = 1 2b [a + α 2], (14) n 2 = n 1 2c [a bn 1 + α 2 ]. (15) Since it must be that n 1 [0, a ], equation (14) implies a corner solution occurs when b α 2 > a. When α 2 > a, the marginal gain from the other side of the platform is greater for firms; firms have a higher marginal gain then every consumer. To extract the most surplus, the platform captures as many consumers as possible with a low consumer price; this will generate a much greater surplus on the firm side which the monopoly platform will extract from. When α 2 a we compare the platform profits between the corner solution and the interior solution and determine which parameter values result in each type of solution. For the interior solution, the assumptions α 2 < a, v = f 1 0, and f 2 = 0 and equations (11), (12), (14), and (15) imply: n 1 = 1 2b [a + α 2], (16) p 1 = v bc [a + α 2] 2 (a α 2 ) > 0, (17) n 2 = 1 8bc [a + α 2] 2, (18) p 2 = 1 8b [a + α 2](3α 2 a). (19) There are two key things to notice in this equilibrium. First, recall the usual monopoly problem with inverse demand, p = a bq and marginal cost equal to zero. The equilibrium number of consumers who purchase from the monopoly is given by Q = a < 2b n 1. So in equilibrium, a monopoly platform will have more consumers than a traditional monopolist. This is since additional consumers generate additional surplus on the platform so a monopoly platform will want to have more consumers then in the traditional monopoly model so that it 2 The second-order conditions hold for this example. This can be seen by substituting the linear α 1 ( ) into the second-order conditions, equations (8), (9), and (10). 7

8 can generate more surplus on the firm side which it will then extract through the firm price. Second, firm price can be negative. Firms are subsidized to join the platform when a > 3α 2. Intuitively this means that if adding firms generates a significantly larger amount of surplus for consumers than consumers generate for firms, then the total surplus on the firm side is less important. The platform will subsidize firms allowing for a greater extraction of surplus on the consumer side. Given equilibrium prices, the number of consumers, and the number of firms, we calculate platform profits. The superscript i denotes the interior solution. Π i = n 1 (p 1 f 1 ) + n 2 (p 2 f 2 ) = 1 64b 2 c [a + α 2] 4. (20) Looking at the comparative statics we see that Πi, Πi < 0. This implies that an increase in c b c, the synchronizing cost firms face, will decrease platform profits, and an increase in b, the rate at which α 1 decreases across consumers, will also decrease platform profits. We also have Π i a, Πi α 2 > 0. This implies an increase in a, or an upward shift in α 1 for all consumers, will increase platform profits, and an increase in the gain firms receive from additional consumers, α 2, will increase platform profits. These are the comparative static results one would expect and they give insight as to how the platform would like to affect these parameters if they were made endogenous through some form of technological improvement. We can also derive consumer, firm, and total surplus given the platform s optimal pricing scheme: CS i n 1 0 [v + α 1 (τ)n 2 p 1]dτ = (a + α 2) 4 64b 2 c = Π M, (21) F S i n 2 0 [α 2 n 1 cθ p 2]dθ = (a + α 2) 4 128b 2 c = (1/2)Π M, (22) W i = T S i = 5(a + α 2) 4. (23) 128b 2 c Notice, consumer surplus is equal to platform profits which are twice as large as firm surplus. Thus, comparative statics for the types of surplus are the same as those for the platform s profit. This is not surprising since the comparative statics are all in relation to generating or reducing some kind of surplus which affects all agents through the platform in this model. We now investigate the corner solution which always occurs when α 2 a, and once this is 8

9 complete we investigate for which cases it occurs when α 2 < a. In the corner solution we know all consumers participate on the platform so for the consumer side of the market we have: n 1 = a b = ˆN 1, (24) p 1 = v = f 1 0. (25) where p 1 = v = f 1 is the highest consumer price the platform can charge that will induce all consumers to join the platform. Given this, the platform maximizes profits with respect to n 2 with p 2 = p 2 (n 1 = a/b, n 2 ) = α 2 a b cn 2. This gives the firm side equilibrium for the corner solution: n 2 = aα 2 2bc, (26) p 2 = aα 2 2b. (27) Given the corner solution, we calculate welfare. The superscript c denotes the corner solution results. Π c = n 1 (p 1 f 1 ) + n 2 (p 2 f 2 ) = n 1 (0) + n 2 p 2 = α2 2a 2 F S c CS c n 2 0 n 1 0 4b 2 c, (28) [v + α 1 (τ)n 2 p 1 ]dτ = a3 α 2 4b 2 c, (29) [α 2 n 1 cθ p 2 ]dθ = α2 2a 2 8b 2 c = (1/2)Πc, (30) W c = T S c = α 2a 2 8b 2 c [3α 2 + 2a]. (31) The comparative statics are what we expect. Surpluses, firm price and firm participation are increasing in a and α 2, which are the parameters that generate surplus, and are decreasing in b and c, which are the parameters that reduce surplus. By comparing platform profits for when α 2 < a, Π c > Π i if and only if a 4 + 4a 3 α 2 10a 2 α aα α 4 2 < 0. This inequality never holds for α 2, a 0; so when α 2 < a we 9

10 have the interior solution and when α 2 a we have the corner solution. The relationship between a and α 2 will play a key role throughout the paper in determining welfare. In the next section we generalize the base model to include platform competition and we determine the equilibria and welfare. 3 Homogeneous Platform Competition 3.1 The Model Setup Assume there exists two platforms, A and B, that play a static pricing game. With multiple platforms consumers and firms can either join a single platform (single-home) or join multiple platforms (multi-home). Previous work has made restricting assumptions on how some of the agents can participate; for example, they assume that on one side of the platform agents can only multi-home and on the other side agents can only single-home. In this model, the allocation of consumers and firms is part of the equilibrium. We assume the synchronization cost for a firm to join the platform is the same for platforms A and B. So the cost to joining one platform for firm θ is cθ and 2cθ to join both platforms. This means synchronization to one platform has no affect on the cost to synchronize to the other platform, there are no economies of scale for synchronization. On the consumer side, if a consumer participates on two platforms his benefit from membership to the second platform diminishes by δ [0, 1], so that his total membership benefit from the two platforms is (1 + δ)v. If δ = 0, then there is no additional membership benefit from joining the second platform. If δ = 1, then the membership benefit is unaffected by being a member of another platform. We assume the platforms are homogeneous: Definition 1. Platforms A and B are homogeneous if α 1,A ( ) = α 1,B ( ) α 1 ( ) for all τ. This implies that all consumers have the same marginal utilities for the firm side across platforms. Let N 1 (N 2 ) denote the total number of consumers (firms) who participate in at least one platform and let n A 1 (n B 1 ) denote the total number of consumers who participate on platform A (B). Finally, let n m 1 be the number of consumers who multi-home; hence, N 1 = n A 1 + n B 1 n m 1. Similarly, we have n A 2, n B 2, and n m 2 for the firm side. If firm θ single-homes on platform i, its profit is given by π2(θ); i if it multi-homes its profit is given by π2 AB (θ). These functions are given by: 10

11 π i 2(θ) = α 2 n i 1 cθ p i 2, (32) π AB 2 (θ) = α 2 (n A 1 + n B 1 n m 1 ) 2cθ p A 2 p B 2. (33) If consumer τ single-homes on platform i his utility is given by u i 1(τ); if he multi-homes his utility is given by u AB 1 (τ) where, u i 1(τ) = v + α 1,i (τ)n i 2 p i 1, (34) u AB 1 (τ) = (1 + δ)v + α 1 (τ)(n A 2 + n B 2 n m 2 ) p A 1 p B 1. (35) The sequence of play is as follows: first the platforms simultaneously choose consumer and firm prices, p i 1 and p i 2 for i = A, B; then consumers and firms simultaneously choose which platforms to join. This form of differentiation describes many types of two-sided markets better than much of the literature which use the Hotelling line on both sides of the market to differentiate two competing platforms. For example, the synchronization costs app producers incur rarely varies across platforms; this cannot be modelled with the Hotelling line. Furthermore, in the Hotelling model we cannot allow for the rich choice participation sets that agents have. Thus in our model of competition, differentiation occurs through the consumer side of the market; then based on the unrestrictive participation decisions allowed of the consumers and firms, firms must decide whether it is worth joining a platform based on the additional fixed cost and the additional price they would incur. To the best of our knowledge, this is the first paper to introduce platform differentiation in this way. In the following subsection we determine which consumers and firms single-home or multihome in equilibrium. The equilibrium allocations we get closely resemble what is seen in several two-sided markets like smartphones and video game platforms. Thus demonstrating the importance of allowing both sides of the market to determine their homing choice in a static game. 3.2 Equilibrium Configurations We investigate the equilibrium configurations that occur when the two homogeneous platforms, A and B, compete in prices. If platforms charge the same consumer prices, p A 1 = p B 1, 11

12 then consumers will join the platform with more firms. If consumer prices and the number of firms are equal across platforms then consumers are indifferent between the two platforms; in this case, consumers will either not participate, multi-home, or single-home on platform A or B with equal probability. Assuming that consumers and firms cannot coordinate with any other agents, then if prices are equal on both sides of the platform then in expectation the number of consumers will be the same on each platform and the number of firms will be the same on each platform, E(n A 1 ) = E(n B 1 ) and E(n A 2 ) = E(n B 2 ). Once prices are set, consumers and firms simultaneously choose which platforms to join. This can lead to failure to launch issues. Specifically, how will consumers and firms allocate if p i 1 > p j 1 and p i 2 < p j 2? When platforms price both sides equally, then, in expectation, the number of consumers and firms on each platform are equal. However, when prices on each side are not equal and neither platform sets both the low prices then how consumers and firms allocate themselves will depend on the beliefs and the beliefs of beliefs and so on. To resolve this we assume that if p i 1 p j 1 > p j 2 p i 2 then platform i fails to launch and agents who join a platform only join platform j. 3 As we will see, in many cases the homogeneous platforms will set their prices equal to each other for both the consumer and firm prices so that p A 1 = p B 1 = p 1 and p A 2 = p B 2 = p 2. Before investigating how, and for which parameter sets this occurs, we first investigate how firms and consumers allocate on the platforms when they face symmetric prices. Given the timing of the game and the lack of coordination between and within both sides of the platform, if a consumer or firm decides to single-home on a platform then it must be indifferent between which platform it chooses. We assume it then joins platform A or B with equal probability; hence, it must be true that in expectation n A 1 = n B 1 and n A 2 = n B 2. In the following subsections we investigate how firms and consumers allocate by either not participating, single-homing, or multi-homing on the platforms Allocation of Firms with Symmetric Pricing Firms who join the platform can be separated into two disjoint groups. Without loss of generality we compare multi-homing to the case when a firm single-homes on platform A; the group who single-home and those who multi-home are given by: 3 Then analysis in what follows still holds for more belief systems. For example, if instead the inequality is with respect to percent makeup then the results are the same. We also require p A 1, p B 1 > δv, but this is not important at this time. The additional requirement, p A 1, p B 1 > δv, will become clear in the analysis below; but notice that when consumer prices fall below δv then all consumers will join regardless of what the firms do. So this requirement is important. 12

13 Group 1: The θ such that π AB 2 (θ) > π A 2 (θ) and π AB 2 (θ) > 0 are the multi-homing firms. Group 2: The θ such that π A 2 (θ) > π AB 2 (θ) and π A 2 (θ) > 0 are the single-homing firms. Group 1 is the set firms such that multi-homing is more profitable than single-homing when multi-homing is better than not participating. Equation (33) implies this occurs when 0 < α 2 (n B 1 n m 1 ) cθ p 2 and 0 < α 2 (n A 1 +n B 1 n m 1 ) 2cθ 2p 2. However, for all θ 0 and since in expectation n A 1 = n B 1, we have α 2 (n B 1 n m 1 ) cθ p 2 α 2 (n A 1 +n B 1 n m 1 ) 2cθ 2p 2, i.e. the first inequality implies the second. Thus the firms who multi-home are the θ such that θ α 2 (n B 1 nm 1 ) p 2. Note, if the firm price, p c 2, is large enough, then this set is empty and no firms multi-home. Alternatively, if n m 1, the number of consumers who multi-home, is large enough, then the set is again empty and no firm will multi-home. Thus for multi-homing firms to exist we must have firm price and the number of multi-homing consumers to be sufficiently low. Assuming this is true, i.e. the inequality 0 < α 2 (n B 1 n m 1 ) cθ p 2 holds, then the set of firms who multi-home are: θ [ 0, α ) 2 (n B 1 n m 1 ) p 2 = [0, n m 2 ). (36) c Group 2 is the set of firms such that single-homing is more profitable than multi-homing when single-homing is better than not participating. Equation (32) implies this occurs when 0 > α 2 (n B 1 n m 1 ) cθ p 2 and 0 < α 2 n A 1 cθ p 2. For this set to be nonempty we need α 2 n A 1 cθ p 2 > α 2 (n B 1 n m 1 ) cθ p 2 which is true since n m 1 0. Thus, so long as prices are sufficiently low for the market to exist, the set of firms who single-home is always non-empty when the competing platforms set prices equal. By solving for θ we can determine the total number of firms that single-home on platforms A or B: [ α2 (n B 1 n m 1 ) p 2 θ, α ] 2n A 1 p 2 = [n m 2, N 2 ]. (37) c c When prices are such that both sets are nonempty there are two key points worth making here. First, the number of firms who will join each platform is n A 2 = n B 2 = n m 2 + (1/2)(N 2 n m 2 ) = (1/2)(N 2 + n m 2 ) where (1/2)(N 2 n m 2 ) is the expected number of single-homing firms to join each platform. Second, the firms that choose to multi-home instead of singlehome are the firms with the sufficiently low synchronization costs. For firms with higher synchronization costs it becomes to costly to join more than one platform. Thus, for given prices, a firm is more likely to multi-home if it faces a lower synchronization cost to join a platform. 13

14 3.2.2 Allocation of Consumers with Symmetric Pricing In a similar fashion we can separate the consumers into those who multi-home and those who single-home. Group 1: All τ such that u AB 1 (τ) > u A 1 (τ) and u AB 1 (τ) > 0 multi-home. Group 2: All τ such that u A 1 (τ) > u AB 1 (τ) and u A 1 (τ) > 0 single-home. Group 1 is the set of consumers such that multi-homing provides more utility than singlehoming when multi-homing is better than not participating. Equation (35) implies this occurs when 0 < δv + α 1 (τ)(n B 2 n m 2 ) p 1 and 0 < v + α 1 (τ)n 2 p 1. However, for all τ 0 we have δv+α 1 (τ)(n B 2 n m 2 ) p 1 v+α 1 (τ)n 2 p 1. This means that the first equation implies the second. Thus the consumers who multi-home are those with 0 < δv+α 1 (τ)(n B 2 n m 2 ) p 1. Note, if the consumer price, p 1, is large enough, then this set is empty and no consumer will multi-home. If n m 2, the number of firms who multi-home, is large enough, then the set may again be empty and no consumer will multi-home. For consumers, unlike firms, there always exists an additional membership benefit, δv, to join the additional platform. Thus if δv p 1, then all consumers will multi-home regardless of the firms decisions. Assuming 0 < δv + α 1 (τ)(n B 2 n m 2 ) p 1 holds, then the set of consumers who multi-home is given by: τ [ ( 0, α1 1 p1 δv n B 2 n m 2 )] = [0, n m 1 ]. (38) Group 2 is the set of consumers such that single-homing provides more utility than multi-homing when single-homing is better than not participating. Equation (34) implies this occurs when 0 > δv + α 1 (τ)(n B 2 n m 2 ) p 1 and 0 < v + α 1 (τ)n A 2 p 1. For both of these to hold we need v + α 1 (τ)n A 2 p 1 δv + α 1 (τ)(n B 2 n m 2 ) p 1 to hold for all τ, which it does. Since α 1 ( ) is increasing in τ, this set of single-homing consumers are the consumers with smaller τ relative to those τ who multi-home. Furthermore, the above inequalities imply that if p 1 > δv then for all δ < 1 the set of single-homing consumers is nonempty since v + α 1 (τ)n A 2 > δv + α 1 (τ)(n B 2 n m 2 ). It is reasonable to assume that p 1 > δv will hold in most practical cases. For most products the membership benefit will depreciate almost to zero when a consumer multi-homes. In the video game case, the membership benefit would be the ability to play music and watch movies from the video game console. Once someone has one console they can do this and adding an additional console is of almost no added benefit. This implies a δ close to zero. Thus for p 1 > δv, the set of single-homing consumers 14

15 is given by: τ [ ( α1 1 p1 δv n B 2 n m 2 ), α 1 1 ( )] p1 v = [n m 1, N 1 ]. (39) Assuming the sets of multi-homing and single-homing consumers are nonempty, the consumers who multi-home are those with τ closer to zero, i.e. those with the higher marginal utility for producers. This makes sense. In the case for video games, for example, this implies that consumers who purchase multiple gaming consoles are those with the highest marginal utility for the number of games available on these consoles. The consumers who single-home are those who have lower marginal utilities for the number of games available and those with the lowest marginal utilities, τ large, do not enter the market. Furthermore, we have n A 1 = n B 1 = n m 1 + (1/2)(N 1 n m 1 ) = (1/2)(N 1 + n m 1 ) where (1/2)(N 1 n m 1 ) is the number of single-homing consumers to join each platform. Investigating the allocation decisions of the consumers and firms provides insight into two-sided markets where both sides can either single-home or multi-home if it is optimal to do so. In reality this is how many two-sided markets work. Given the consumer and firm allocation rules for equal prices across platforms, we can determine the Nash Equilibria that occur in this model. n A Equilibria In determining the equilibrium configurations, the main parameter inequality that affects > both how consumers and firms allocate and how platforms choose to price is f 1 δv. For < many products the membership benefit will depreciate almost to zero when a consumer multi-homes. This implies the marginal cost for an additional consumer on the platform is greater than the additional membership benefit from joining an additional homogeneous platform, f 1 δv. In the video game case, the membership benefit would be the ability to play music and watch movies from the video game console. Once someone has one console they can do this, but adding an additional console is of almost no added benefit since they can already watch movies and play music on the first console. This would imply a δ close to zero and the additional benefit would not overcome the cost of producing the additional console. Consider first the case when f 1 δv. In this case, competition between the two homogeneous platforms results in the Bertrand Paradox: Proposition 1. If f 1 δv then p A 1 = p B 1 = f 1 and p A 2 = p B 2 = f 2 are the unique symmetric 15

16 prices in equilibrium and Π A = Π B = 0. Given these prices, and using the allocation rules above we can determine the Nash Equilibrium for the case when p 1 = f 1 δv. We first investigate the case when there exists both consumers who single-home and consumers who multi-home. Theorem 1. When f 1 δv we have unique Nash Equilibrium prices charged by the two platforms, p A 1 = p B 1 = f 1 and p A 2 = p B 2 = f 2, and at least one, potentially four, allocations of firms and consumers that can be present in Nash Equilibrium. 1. All consumers single-home and all firms multi-home. This is always a Nash Equilibrium. 2. There exists consumers that multi-home and single-home and firms that multi-home and single-home; here, existence depends on model parameters. 3. All firms single-home and many, potentially all, consumers multi-home. This has strong restrictions when p 2 > 0, and when p 2 = 0 it requires v = 0. Case 1, which always exists, is an important equilibrium because it very closely resembles the two-sided market for smartphones where almost all consumers single-home, they own one phone, and almost all firms multi-home, every app is available on all types of smartphones. In case 2, allocations resemble current allocations we see in many two-sided markets. For video game platforms, there exists consumers who multi-home and single-home and there exists game designers who multi-home and single-home. In case 3, v = 0 is needed for such an equilibria to exist and this equilibria is seen with antique malls. An antique mall is the platform with many firms, antique shops, and consumers. Antique shops are only in one mall and all consumers shop at the many malls to find what they are looking for. There is no membership benefit for being a mall so v = 0, all consumers multi-home and all firms single-home. We now turn to the case when δv > f 1. In this case a platform can charge a consumer price of p i 1 = δv and guarantee itself profit since consumers will either single-home on platform i or if a consumer is already on platform j i then they will be indifferent to multi-homing. Hence, if p i 1 = δv then consumers will join platform i regardless of the firms decisions. Thus for this case with δv > f 1, both platforms are guaranteed profits and all consumers τ [0, N 1 ] are guaranteed to join at least one platform. Furthermore, in this case we do not have failure to launch issues since both platforms are completely able to establish themselves 16

17 on the consumer side of the market. This implies we can investigate and determine the Nash Equilibrium allocations for the cases when p i 1 > p j 1 and p j 2 > p i 2. This also leads to potentially uncountably many Nash Equilibrium. Theorem 2. When δv > f 1, there exists at least one and potentially uncountably many Nash Equilibria. 1. There exists a unique symmetric Nash Equilibrium where p A 1 = p B 1 = δv, p A 2 = p B 2 = f 2, all consumers τ [0, 1] multi-home, and the firms who join a platform do so by singlehoming on platform A or B with equal probability; this Nash Equilibrium always exists. 2. The uncountably many asymmetric Nash Equilibria that can potentially exist are characterized by p i 2 = f 2, p j 2(ɛ) = f 2 ɛ, p i 1 = δv, p j 1 = p j 1(ɛ) δv, and p i 2 = f 2, for ɛ 0 with p j 1(ɛ = 0) = δv. If there exists an ɛ > 0 where certain conditions hold, see proof, then the resulting Nash Equilibrium allocations will have all consumers multi-homing and all firms who decide to participate do so by joining platform j. Furthermore, in all NE platforms receive positive profits: Π A = Π B = N 1 (δv f 1 ) > 0. Thus, if a platform has significantly high retained membership benefit for consumers when they multi-home, then competing homogeneous platforms can avoid the Bertrand Paradox on the consumer side of the market and make positive profits. This result obviously generalizes to n homogeneous platforms who compete in prices. The Nash Equilibrium outcomes provide insight into the decision of an agent to single-home or multi-home. We next investigate the theorems by looking at closed form solutions. 3.3 Closed Form Solutions and Welfare for Static Nash Equilibrium In this subsection we investigate the same closed form solution that we determined above in the case for a monopoly platform. We calculate welfare and compare it with the welfare that occurs with a monopoly platform. Note, there exists two effects on welfare between having a monopoly platform and two competing platforms. Competition will result in lower prices; however, it also can destroy network surplus or create more synchronization costs from firms who choose to multi-home. Depending on which effect dominates, welfare may increase or decrease with increased competition. 17

18 As before we assume α 1 (τ) = a bτ so that the maximum number of consumers that can join a platform is N 1 = a/b. Furthermore, we assume v = f 1 0 and f 2 = 0. This implies f 1 = v δv so Proposition 1 and Theorem 1 hold but Theorem 2 does not. Proposition 1 implies p A 1 = p B 1 = f 1 = v 0 and p A 2 = p B 2 = f 2 = 0. Given this, we investigate allocation 1 in Theorem 1 where all consumers single-home and all firms multi-home. We know this equilibrium exists and is relevant for real world applications such as smartphones and video game consoles. Given that all consumers single-home and all firms multi-home we have n m 1 = 0 and N 2 = n A 2 = n B 2 = n m 2. Furthermore, allocations (36) and (39) imply: n m 2 = (1/c)(α 2 n A 1 p 2 ) = α 2n A 1 c, n A 1 = a 2b. These two equations and two unknowns determine the equilibrium: n A 1 = n B 1 = (1/2)N 1 = (1/2) N 1 and n A 2 = n B 2 = n m 2 = aα 2 with p 2bc 1 = f 1 = v, and p 2 = f 2 = 0. Notice, all consumers join a platform i.e., we have full participation. Given this equilibrium, consider welfare. The superscript 1 represents allocation 1 in Theorem 1. Π 1 A = Π 1 B = 0, (40) CS 1 N1 0 [v + α 1 (τ)n m 2 p 1 ]dτ = F S 1 n m [(a bτ) ( aα 2 2bc )]dτ = a3 α 2 4b 2 c = CSc, (41) [α 2 N 1 2cθ 2p 2 ]dθ = a2 α 2 2 4b 2 c = 2F Sc, (42) W 1 = T S 1 = a2 α 2 4b 2 c [α 2 + a] = a2 α 2 8b 2 c [2α 2 + 2a] < a2 α 2 8b 2 c [3α 2 + 2a] = W c. (43) Recall from the monopoly model at the end of section 2.1 that the corner solution occurs where the monopolist captures all consumers when α 2 a, otherwise the interior solution occurs. Notice how consumer surplus is the same as in the monopoly model when a corner solution occurs. This is because consumer prices are the same in both cases and all consumers are single-homing in this case with two platforms. Firm surplus is greater here than in the monopoly corner solution. 18

19 The most striking result is that welfare with perfect competition is strictly less than welfare with a monopoly platform that implements the corner solution and captures all consumers. Comparing welfare between the case here and the case when the monopolist implements an interior solution (which requires α 2 < a) we see that W 1 < W i if and only if 0 < 5a 3 17a 2 α aα α2. 3 This inequality never holds for α 2, a 0. Thus we have the following theorem. Theorem 3. For v = f 1 0, f 2 = 0, α 1 (τ) = a bτ. When α 2 a the monopoly corner solution that occurs has higher welfare than the perfectly competitive equilibrium between two homogeneous platforms when all consumers single-home and all firms multi-home. When α 2 < a the monopoly interior solution that occurs has lower welfare than the equilibrium between two homogeneous platforms when all consumers single-home and all firms multihome. This means that when the two-sided market is driven by the firms, α 2 a, then the monopoly will capture all consumers and this will generate more welfare than the case where we have two homogeneous platforms who compete perfectly competitively with all consumers single-homing and all firms multi-homing. When the monopoly platform is already capturing all consumers then in an sense it is already pricing low enough for competition to not be enough to increase surplus. Otherwise, the tradeoff for lower prices is sufficient to increase welfare. There are three more important points to be made here. First, we are assuming that there do not exist decreasing marginal returns for the other side of the platform. In many instances consumers have decreasing marginal returns from the number of apps available on their smartphone. Overall this decreases surplus for both levels of competition but it would hurt the monopolist more since now its prices would change and destroy more surplus. Second, we assumed v = f 1 0. For v > f 1 similar arguments follow and welfare results will closely resemble what we have here. However, for v < f 1 it will be less likely for the corner solution to occur since in that case it will be costly for the monopoly platform to capture all consumers. This makes competition welfare improving in more cases. Finally, we assumed homogeneous platforms. If platforms are quite different more cases with high price competition but strong network benefits will occur. This would make competition welfare enhancing. Even with these pitfalls, this result is quite interesting and one can imagine a case where a monopoly is welfare improving. For example, when the first iphone came out it was essentially a monopolist in terms of pure smartphones. At this time, if a homogeneous competitor 19

20 was also present, welfare arguably would suffer by destroying and splitting network surpluses between these two platforms even if we assume no coordination issues. 4 Conclusion This paper makes three contributions to the existing platform and two-sided market literature. It establishes a base model for two-sided markets, with consumers, firms, and a monopoly platform, that is intuitive and general while allowing for extensions of competition and well-behaved closed form solutions. Second, it introduces competition between two homogeneous platforms where both sides of the market, consumers and firms, may either single-home or multi-home. Third, the model admits welfare comparisons across levels of competition; a monopoly platform and two competing homogeneous platforms. The framework of this model allows for more extensions. For instance, this model will be extended to allow for entry. The results of the stage game will depend on the amount of differentiation between platforms. For example, given a situation where an incumbent platform has established a presence in the market, a homogeneous potential entrant will have no way of gaining momentum to enter the market. This could lead to many interesting results about entry and exit in two-sided markets which will be left for future research. 5 Appendix 5.1 Proofs Proof of Proposition 1 Proof. Clearly, symmetric prices that are less than their respective marginal costs is worse for each platform and so both platforms deviate. Thus symmetric prices below the respective marginal cost is not feasible. If the symmetric firm price is greater than f 2 then both platforms have the incentive to undercut the price by ɛ and capturing all firms and thus all consumers. Thus we must have p 2 = f 2. If p 1 > f 1 then both platforms have an incentive to undercut the price by ɛ and capture the whole market. This undercutting only captures all consumers when p 1 δv which we assume to be true in Proposition 1. Otherwise all consumers will multi-home because price is less than the additional membership gain from joining a platform. Thus p 1 = f 1. This implies Π i = n i 1(p 1 f 1 )+n i 2(p 2 f 2 ) = n i 1(0)+n i 2(0) = 0 for i = A, B. 20

21 Proof of Theorem 1 Proof. Here we show the equilibrium allocations for general prices p 1 and p 2. Equilibrium prices are then given by Proposition 1, this completes the Nash Equilibrium. Allocation 1: For all firms to multi-home, notice that allocation (37) implies all firms multi-home only when all consumers single-home. Furthermore, since n m 2 = n A 1 = n B 1, allocation (38) implies no consumer multi-homes. Hence, the allocation where all firms multi-home and all consumers single-home is a Nash Equilibrium when prices are equal and p 1 δv. Allocation 2: Since p 1 δv, allocation (38) implies the set of multi-homing consumers is non-empty only when the number of multi-homing firms is not to large. In what proceeds we determine a Nash Equilibria that occurs under certain parameter restrictions. Let x [0, 1] be the percent of consumers who multi-home of those N 1 who join a platform so that in expectation n m 1 = xn A 1 = xn B 1. Note, this implies N 1 = n A 1 + n B 1 n m 1 and since in expectation n A 1 = n B 1 we have N 1 = (2 x)n A 1. Given this, using allocation (36) we can find the cutoff x = x m such that above x m no firm will multi-home. Allocation (36) implies 0 = α 2 (1 x m )n B 1 p 2. Thus, x m = 1 p 2. (44) α 2 n B 1 And for all x > x m no firm multi-homes. Note, we are assuming p 2 < α 2 n B 1 the market collapses, hence x m (0, 1). since otherwise Now if we let 0 < x < x m then some firms will single-home and some firms will multihome. Allocation (36) implies n m 2 = α 2(1 x)n B 1 p 2 c and allocation (37) implies n B 2 = (1/2)(N 2 + n m 2 ) = (1/2c)[α 2 (2 x)n A 1 2p 2 ]. Similarly, allocation (38) defines the number of multihoming consumers: 0 = δv + α 1 (n m 1 )(n B 2 n m 2 ); using this equation and the equations for n m 2, n B 2, and n m 1 = xn A 1 = xn B 1 we can characterize x by: 0 = δv+α 1 (xn A 1 )(1/2c)[α 2 (2 x)n A 1 2p 2 2α 2 (1 x)n B 1 +2p 2 )] = δv+α 1 (xn A 1 )(1/2c)[α 2 xn A 1 ], (45) Furthermore, allocation (39) defines N 1, the number of consumers on platform A: 0 = v + α 1 (N 1 )n A 2 p 1. Thus we have: 21

22 0 = v + α 1 (N 1 )n A 2 p 1 = v + α 1 ((2 x)n A 1 )(1/2c)(α 2 (2 x)n A 1 2p 2 ) p 1. (46) Thus, we have two equations (45) and (46) and two unknowns, x and n A 1. 4 If the resulting value of x is such that x (0, x m ) then we have a Nash Equilibrium for the whole static game where there is a complete mix of single-homing and multi-homing consumers and firms. Note this equilibrium may not exist for sets of parameters. If the x characterized in equations (45) and (46) is not in [0, x m ) then this equilibrium does not exist. Allocation 3: For all firms to single-home, allocation (37) implies all firms single-home only if the number of multi-homing consumers is large relative to the number of single-homing consumers, n B 1 n m 1 + p 2 /α 2. If p 2 = 0, then this holds only when all consumers multi-home. By allocation (39), this will only be an equilibrium when v = 0. If p 2 > 0, then allocation (39) implies there exists an equilibrium where all firms single-home and a large portion of consumers multi-home given prices such that N 1 n m 1 = α 1 ) α 1 ) 2p n A 2 n A 2 2 α 2. This potential equilibrium requires relatively restrictive assumption. In reality we do not see equilibrium 1 ( p 1 v 1 ( p 1 δv of this form where all firms single-home and many or all consumers multi-home. Proof of Theorem 2 Proof. We first prove the case for ɛ = 0. We must show there does not exist a non-epsilon deviation that makes the platform worse off and the allocation stated above is indeed Nash. Given prices are p A 1 = p B 1 = δv and p A 2 = p B 2 = f 2, consumers are indifferent between single-homing on either platform and multi-homing so we assume they multi-home. If p 1 lowered by ɛ then this would occur. Thus every consumer available in the market joins both platforms A and B. Given all consumers multi-home and firms face equal prices between platforms A and B, for all firms θ we have π A 2 (θ) = π B 2 (θ) > π AB 2 (θ). Thus the firms who join a platform will single-home to platform A or B with equal probability and the number of single-homing firms who join a platform is given by π A 2 (θ = N 2 ) = 0. Using equation (32) this implies N 2 = (1/c)[α 2 f 2 ]. To see why the prices are NE we investigate deviations. If a platform were to increase its consumer price from δv, then it will lose consumers and thus lose all firms and thus lose all consumers driving profits to zero. If a platform lowers consumer price below δv, then the allocation is unchanged and they receive less payment; 4 These equations and prices equal to marginal costs completely determine the Nash Equilibrium. 22