Price Competition in Multi-sided Markets *

Transcription

1 Prce Competton n Mult-sded Markets * Guofu Tan Junje Zhou July 27, 207 Abstract Ths paper studes a general model of prce competton among platforms offerng dfferentated servces n mult-sded markets. We ncorporate a general form of both wthn-sde and cross-sde externaltes nto a dscrete choce model of random utlty maxmzaton by consumers on each sde of the markets. We consder a two-stage game n whch the platforms choose prces (or user fees) smultaneously n the frst stage, followed by consumers on all sdes smultaneously decdng whch platform to jon (sngle-homng) n the second stage. We show that n a symmetrc settng wth full market coverage, there exsts a symmetrc equlbrum n prces and the equlbrum prce on each sde follows a smple rule: The prce equals the cost, plus a mark-up due to product dfferentaton, mnus a subsdy due to cross-sde externaltes. The subsdy to each sde accounts for the degree of the aggregate margnal externaltes of that sde mposed on all the other sdes. As competton among platforms ncreases, both the product dfferentaton effect and the crosssubsdy are shown to decrease. As such, the prce on one sde can decrease whle the prce on another sde may ncrease wth the number of platforms. We also dscuss the ncentves for platforms to merge and the extent of excessve free entry of platforms nto the markets as compared to the socal optmum. We further compare unform prcng rule wth dscrmnatory prcng across dfferent sdes of the markets and fnd that the average prce across sdes under the dscrmnatory prcng s hgher than the unform prce when the externaltes are small or when the number of platforms s large. The mpacts of consumers outsde optons on the equlbrum prces are also studed. *We are grateful for comments from Alessandro Pavan, Andrew Rhodes, Adam Szedl, Alex Whte, Adam Wong, Julan Wrght, Jdong Zhou and semnar partcpants at CUHK, HKU, HKUST, Melbourne, Monash, NUS, Shangha Jaotong Unversty, Shangha Mcroeconomcs Workshop, SHUFE, SoCal NEGT Symposum, Wuhan Unversty, and Zhejang IO Workshop. Department of Economcs, Unversty of Southern Calforna. guofutan@usc.edu Department of Economcs, Natonal Unversty of Sngapore. ecszjj@nus.edu.sg Electronc copy avalable at:

2 JEL classfcaton: L3, L4. Keywords: mult-sded markets, platform prce competton, cross subsdzaton, dscrete choce models, free entry, merger ncentves, dscrmnatory prcng, unform prcng, outsde opton. Introducton In ths paper, we study a general model of prce competton among platforms provdng dfferentated servces to customers n mult-sded markets. We ncorporate a general form of cross-sde externaltes nto a dscrete choce model of random utlty maxmzaton by customers on each sde of the markets and study the mpacts of platform competton on the equlbrum prces and customer partcpaton rates. There are numerous examples of platforms provdng servces to allow multple groups (or sdes) of customers to nteract, where one group s beneft from jonng a platform depends on the szes of the other groups that also jon the platform (.e., crosssde externaltes). Examples of ths nclude Internet companes such as Albaba, ebay, Facebook, Uber. Albaba and ebay facltate e-commerce transactons between buyers and sellers; Uber connects rders and drvers through ts rde-sharng platform; Facebook can be vewed as a three-sded platform wth users, advertsers, and publshers (or content provders). The presence of cross-group externaltes provdes a ratonale for the platforms to charge dfferental prces across dfferent sdes, often n the form of subsdzng some groups of customers. The huge success of these platform busness models nevtably nvte competton from entrants. Dd Chuxng and Uber Chna fercely competed for several years to nduce passengers and tax drvers to adopt ther own platforms; Uber and Lyft have been aggressvely competng wth local cab companes n the U.S. markets. The popularty of onlne advertsement exchange platforms (Ad Exchanges) has attracted major competng operators ncludng Google s DoubleClck, Mcrosoft s AdECN, Yahoo s RghtMeda and Facebook s FBX, through whch users, advertsers, and publshers (or content provders) nteract wth each other. How do platforms compete for multple groups or sdes of potental customers? Would an ncrease n platform competton reduce cross-subsdzaton and ncrease the prces (or user fees) and welfare? How would the presence of dstnct multple sdes of customers affect the equlbrum outcome n terms of both dscrmnatory and unform prcng across sdes? There s now a large lterature on platform provsons n mult- 2 Electronc copy avalable at:

3 sded markets. Wth some exceptons, most of the papers n ths lterature focus on two sdes, duopolstc platforms, and a Hotellng specfcaton of product dfferentaton. To answer the above questons and partcularly study the mpacts of platform competton on the prces and welfare n many applcatons, we need to allow an arbtrary number of platforms, a more general form of product dfferentaton, and possbly more than two sdes. The purpose of ths paper s to provde a general model ncorporatng both own-sde and cross-sde externaltes nto a dscrete choce model of random utlty maxmzaton by customers on each sde of the markets. We follow closely the framework of Armstrong (2006) and consder a two-stage game n whch n 2 platforms choose prces (or user fees) smultaneously n the frst stage, followed by s groups of customers smultaneously decdng whch platform to jon (sngle-homng or one-stop shoppng) n the second stage. The decson by a customer on each sde of whch platform to jon s formulated as a dscrete choce maxmzaton of random utltes. In the presence of own-sde and crosssde externaltes, the demand system n our settng s nterdependent across platforms and s mplctly determned by a system of nonlnear equatons. Whle the exstence of partcpaton equlbrum (PE) follows from Brouwer s fxed pont theorem, multple equlbra may exst due to the externaltes across sdes. We provde suffcent condtons to ensure the unqueness of PE for any prce profle, and such condtons hold as long as the extent of externaltes s not so large, relatve to the degree of product dfferentaton among platforms. To show the exstence of the prce equlbrum n the frst stage nvolves two techncal challenges. Frst, as mentoned above, the demand system does not have an explct expresson, whch mples that the changes n a platform s demand respect to hs own prces (.e., the Jacoban matrx) and the platform s frst-order condtons for proft maxmzaton cannot be explctly determned. Second, the condtons to deter global devatons n prces by each platform (.e., the second-order condtons) are not easy to specfy. 2 We provde an approach of transformng the proft maxmzaton problem n The semnar work on two-sded markets starts wth Rochet and Trole (2003, 2006), Callaud and Jullen (2003) and Armstrong (2006), whch provdes a basc framework for studyng prcng schemes under both monopoly and duopoly structures. See Rysman (2009) for a survey of the early lterature on the economcs of two-sded markets. Weyl (200) provdes a theory of mult-sded markets under monopoly usng nonlnear nsulatng tarffs. Jullen (20) studes Stackelberg prce competton n a mult-sded market. For examples of more recent studes and references, see Whte and Weyl (206) and Jullen and Pavan (206). 2 Wthout externaltes, each platform s proft maxmzaton problem n prces s separable across sdes 3

4 prces nto one n quanttes, and ths allows us to crcumvent the two techncal challenges mentoned above. In the symmetrc settng of platforms wth full market coverage, n Theorem of our paper, we prove the exstence of a symmetrc equlbrum n whch the prce on each sde follows a smple, decomposable formula: The prce equals the cost (c), plus a mark-up (M (n)) due to market power assocated wth product dfferentaton, mnus a subsdy (η (n)) due to cross-sde externaltes,.e., p (n) = c + M (n) η (n), where both the producton dfferentaton effect and subsdy effect depend on the number of platforms n. To support ths symmetrc equlbrum, we check the ncentve of a platform, say platform, to devate from the proposed prces. Instead of wrtng the devatng proft as a functon of prces, whch does not have an explct expresson, we rewrte t as a functon of quanttes by consderng hs nverse demand system condtonng on other platforms prces. Intutvely, gven the other platforms prces at the symmetrc equlbrum, for platform to mplement demands q = (q, q s), the addtonal prce he could charge on each sde s equal to the nverse demand wthout externaltes, H ( q ), adjusted by the dfferences n the externaltes that a customer on each sde enjoys between platform and any other platform who shares the the remanng market wth the n platforms equally. The devatng proft as a functon of quanttes, s separable n terms of the mpacts of the dstrbutons and the externaltes, and separate assumptons on these terms can be easly made to guarantee ts concavty. The characterzaton of the equlbrum prces allows us to study the mpacts of platform competton on prces, profts, and welfare. We rase the followng three questons. The frst s how prces change wth respect to competton. Based on the explct prcng formulae, we show that the market power mark-up due to product dfferentaton, M, s monotoncally decreasng n n under the log-concavty assumpton on the dstrbutons, whch s satsfed for many commonly used dstrbutons. The subsdy to each sde, η, accounts for dfferent degrees of total externaltes of a group on all the other groups, whch s shown to be decreasng n the number of platforms under farly weak condtons. The net effect of competton on prces depends on the degree of product of the markets, and for each sde the demand functon, explctly derved from the dscrete choce model, can be shown to be log-concave n ts own prce under certan assumptons on the dstrbutons of the random utltes (e.g., log-concavty of the dstrbutons, see Capln and Nalebuff (99)), and hence the proft on each sde s log-concave n prce and the second-order condton s satsfed. 4

5 dfferentaton and the sze of externaltes. To llustrate the net effect we consder a class of dstrbutons (.e., Gumbel dstrbutons) and lnear forms of externaltes and fnd that as competton ncreases, the prce on one sde can decrease, but the prce on another sde may ncrease. In other words, due to cross subsdzaton, the prces charged to certan sdes can actually ncrease as competton among platforms goes up. Second, how would equlbrum profts and total surplus change wth respect to platform competton, and n partcular, would free entry lead to excessve entry as compared to the socal optmum when there s a fxed cost of entry? In the presence of cross-sde externaltes, the equlbrum proft per platform may not necessarly decrease wth n. Except for the prce effects, the customers prefer varetes, so the expected consumer surplus may ncrease or decrease wth n. The total surplus may not necessarly ncrease wth the number of platforms snce the magntudes of the externaltes decrease even though the prce effects do not affect the total surplus. We provde suffcent condtons for excessve entry, whch rely only on the dstrbutons, but not on the externalty functons. Under Gumbel dstrbutons, we show that there s always excessve entry nto the markets, regardless of the degree of externaltes, although both the socally optmal number of platforms and the equlbrum number of platforms under free entry are affected by cross-subsdzaton. A thrd queston we ask s whether platforms have ncentves to rase prces through a merger. Utlzng the equlbrum condtons for platform proft maxmzaton, we are able to demonstrate that the mergng platforms have ncentves to ncrease ther prces margnally on every sde of the markets. Furthermore, we fnd the magntudes of cost reductons on the mergng platforms that make the equlbrum prces unchanged before and after the merger are proportonal to the pre-merger prce-cost margns. Moreover, we compare unform prcng wth dscrmnatory prcng across sdes. The prce-cost mark-up under the symmetrc unform prce s determned by the famlar demand sem-elastcty, evaluated aggregately over all sdes. The product dfferentaton effect and subsdy effect under the unform prcng cannot be solated as n the case of dscrmnatory prcng. Snce the total welfare remans unchanged under the two prcng rules, platforms and customers have opposte preferences, dependng on the relatve magntudes of the average dscrmnatory prces across sdes and the unform prce. We show that the average dscrmnatory prces across sdes s greater than the unform prce and the platforms prefer dscrmnatory prcng over unform prcng rule when the number of platforms s large or when the degree of externaltes s small. Fnally, we explctly study the mpacts of outsde optons, manly focusng on two- 5

6 sded markets wth Gumbel dstrbutons. In the presence of explct outsde optons, a full characterzaton of the equlbrum s avalable, but the equlbrum prces do not have the decomposable three-term formulae as n the case of no outsde optons. Nevertheless, we show that the mpacts of outsde optons s small when there s suffcent competton among platforms. The remander of ths paper s organzed as follows. Secton 2 ntroduces the model and assumptons. Secton 3 characterzes the equlbrum outcome of the two-stage game. The effects of platform competton on prces and welfare are presented n Secton 4, and some nsghts nto merger analyss are dscussed n Secton 5. Secton 6 compares the equlbrum outcomes under dscrmnatory prcng and unform prcng rules. Secton 7 studes the mpacts of outsde optons, and Secton 8 concludes. All the proofs are presented n Appendx. 2 Model There are n platforms competng for customers from s sdes by chargng prces (or user fees), n 2 and s. Let p k = (p k,, pk s) denote the prces charged by platform k. On each sde, there s a contnuum of customers wth measure. The utlty functon of a customer on sde S := {, 2,, s} from jonng platform k N := {, 2,, n} s u k = v k + φ (x k ) p k + ɛk. () Here v k denotes the ntrnsc valuaton. Snce we focus on ex ante symmetrc platforms, we set v k = v for any k. Let x k = (x k,, xk s) denote the demand profle of platform k. The functon φ s a mappng from [0, ] s to R, and φ (x k ) captures the externaltes, enjoyed by customer on sde from all other sdes (possbly sde tself) n platform k. The term ɛ k s random utlty representng customer preference characterstcs, product characterstcs, functonal form msspecfcaton, and so on. 3 In our specfcaton (), we focus on the prce as user fee that s not condtonal on the partcpaton of customers on any sde. 4 3 See Perloff and Salop (985), Capln and Nalebuff (99) and Anderson et al. (992) for examples of nterpretatons of the random utlty term. 4 Armstrong (2006) consders a platform s two-part tarff on each sde condtonal on the partcpaton of the other sde on the same platform. Whte and Weyl (206) allow general nonlnear tarffs that are condtonal on the partcpatons of customers on all platforms. 6

7 Let P = (p,, p n ) denote the prce profle and p k denote all the prces except of platform k. Gven P and demand profle X = (x,, x n ), platform k s proft s Π k (p k, p k ; X) = (p k c )x k, (2) S where c s the margnal cost of servng a customer on sde, whch s assumed to be dentcal across platforms. We further make the followng assumptons: Frst, our model of customer behavor s based on a dscrete choce model wth random utlty. Each customer on sde has a unt demand, she jons at most one platform (sngle homng). Moreover, we focus on full market coverage by assumng away outsde optons. As such, each customer on sde partcpates n one and only one platform. An alternatve modelng assumpton s that there s ndeed an outsde opton whch yelds utlty level u 0 = v 0 + ɛ 0 on each sde, but the ntrnsc value v s suffcently large as compared wth v 0 so that customers always opt out the outsde opton. Later n Secton 7, we explctly model the mpact of non-trval outsde optons on equlbrum prces. Second, for our man result n the next secton, we make two assumptons on random utltes. We assume that across dfferent sdes = j S, random varables {ɛ, ɛ2,, ɛn } are ndependent from {ɛ j, ɛ2 j,, ɛn j }.5 Wthn each sde, we assume that ɛ, ɛ2,, ɛn are symmetrcally dstrbuted,.e., ther jont dstrbuton s nvarant under any permutaton of the order of these n random varables. 6 For certan comparatve statcs results n Secton 4, we consder the case of ndependent and dentcally dstrbuted (IID) shocks and condtonally ndependent and dentcally dstrbuted (CIID) shocks, respectvely. For the case of IID, we assume that ɛ k IID F ( ), k N (sde-specfc). Moreover, we assume that F s contnuous and dfferentable on the support [a, ā ] wth contnuous probablty densty functon (PDF) f. Beyond these, we do not mpose any functonal form restrctons on F, although the followng examples are frequently adopted for llustratons n our paper: Type I Extreme value dstrbuton (double exponental or Gumbel dstrbuton); Normal dstrbuton; 5 See Jullen and Pavan (206) for a model of platform competton n two-sded markets wth correlated customer preferences between the two sdes. Assumng full market coverage and symmetry between two platforms, they study the mpacts of correlaton between customers nformaton on the equlbrum prces and partcpaton rates. 6 Our assumpton of wthn-sde symmetry could accommodate some spatal models of prce dfferentaton such as the Hotellng specfcaton wth unform dstrbuton of consumer locaton (for n = 2) and the Spokes model (for any n 2, see Chen and Rordan (2007)). 7

8 Unform; Exponental dstrbuton. For the case of CIID, we assume that condtonal on τ, ɛ k IID F ( τ ), k N, where τ s dstrbuted accordng to G ( ). When G s degenerate, the model reduces to the IID case. Thrd, we assume that φ (x) s contnuously dfferentable n x wth φ (0) = 0 (normalzaton). The most common specfcaton of φ s lnear,.e., φ (x, x 2,, x s ) = j S γ j x j. The parameters γ j s not necessarly nonnegatve,.e., negatve externaltes (congeston effect) are also allowed. Our specfcaton could ncorporate wthn-sde φ x network externaltes: = 0, whch s natural n some applcatons. Moreover, we are not lmted to lnear functonal forms, and ncreasng (decreasng) return to scale of externaltes are also permtted, for example, φ (x, x 2,, x s ) = j S γ j ρ(x j ) wth ρ > 0, ρ 0(ρ < 0); or φ (x, x 2,, x s ) = µ( j S γ j x j ), wth µ 0, µ > 0(µ < 0). We study the subgame perfect equlbrum of the followng two-stage game: Platform provders smultaneously choose ther prces frst, followed by customers smultaneously decdng whch platform to partcpate. We use the followng notaton: Let Ψ j (x) = φ (x) x, and j Ψ(x) = (Ψ j (x)),j s denote the margnal externaltes matrx at x. When φ takes the lnear form,.e., φ (x) = j S γ j x j, we let Γ = (γ j ) s s be the lnear externalty matrx,.e., Ψ(x) = Γ, x. For gven vector w = (w,, w n ), we let Dag(w) = Dag(w,, w n ) denote the n by n dagonal matrx wth w on ts -entry. For a matrx M, ts transpose s denoted as M. The n by n dentty matrx s denoted as I n, whle 0 denotes the zero matrx wth sutable dmenson. The column vectors of s of length s s denoted as s = (,,, ). 3 Equlbrum analyss In ths secton we determne the subgame perfect equlbrum of the two-stage game. By backward nducton, we frst study the equlbrum n the second stage, characterzng customers smultaneously partcpaton decsons. For each fxed prce profle P, we call demand profle X(P) a partcpaton equlbrum (PE) assocated wth P f t s a pure strategy Nash Equlbrum n the second stage. Gven P, each customer on sde jons the platform that yelds the hghest utlty, as specfed n (). Clearly, X(P) s a PE assocated wth P f and only f the followng equatons smultaneously hold: x k (P) = Pr(v + φ (x k (P)) p k + ɛk (3) max t =k {v + φ (x t (P)) p t + ɛt }) 8

9 for any S, k N. In other words, the demand system X(P) s nterdependent across platforms and mplctly determned by the above system of equatons. Applyng Brouwer s fxed pont theorem, we see that the system (3) has at least one soluton. Proposton. For any prce profle P, there exsts at least one partcpaton equlbrum. Due to cross-group externaltes, the unqueness of PE s not guaranteed. 7 Nevertheless, suffcent condtons can be derved to ensure unqueness. Intutvely, when the degree of externaltes s small, relatvely to the dsperson of customer preferences for dfferentated products (or the nose dstrbutons), there exsts a unque PE. To llustrate, we focus on the lnear form of externaltes. For n = 2 and general dstrbutons, we apply the contracton-mappng theorem and obtan the followng suffcent condton for unqueness: 8 2{max h (θ ; θ 2)}{ γ j } <, S, j S where h s defned n the next paragraph. 9 When n 2 and ɛ t s IID accordng to Gumbel dstrbuton wth parameter β, the followng condton s suffcent: max { γ j } < 2 mn(β,, β s ). S j S Note that ths condton does not rely on n. We now characterze the prce equlbrum n the frst stage. Let H (θ; n) and h (θ; n) denote the CDF and PDF of the random varable ɛ max(ɛ 2,, ɛn ), respectvely.0 7 See Brock and Durlauf (200, 2002) for dscussons on possble multple equlbra n a sngle-sded market wth externaltes. Multple equlbra also arse n coordnaton games wth strong complementarty, see Carlsson and Van Damme (993); Morrs and Shn (998, 2003); Hellwg (2002); Vves (2005). 8 When γ j 0, alternatve condton s gven by: 2λ max (Γ) max S {max θ h (θ ; 2)} <, where λ max (Γ) s the largest egenvalue of nonnegatve matrx Γ, whch also equals the spectral radus of Γ by Perron-Frobenus Theorem. For n 2, smlar condtons can be derved but are more nvolved. 9 See Appendx B for detals. 0 The exact formula of H and h depends on the specfcaton of random shocks. For Gumbel dstrbuton, H s a logstcs functon, see (24). For Hotellng specfcaton wth unform dstrbuton of consumer locaton (n = 2), H s lnear (see Example on page ). For presentng our man result we mpose mld restrctons on H. For comparatve statc analyss n the next secton, we wll focus on the case of CIID and several examples of the IID case. 9

10 For a gven vector a R s, defne the followng mult-varable functon R : [0, ] s R { R(z; a) := z H ( z ; n) + z φ (z) φ ( } s z n ) + z a. (4) S S We assume the followng regularty condton on R. Assumpton. For any a R s, R(z; a) s concave n z [0, ] s. Notce that the concavty of R(z; a) n z s clearly not affected by a, as the thrd term z a n (4) s lnear n z. An equvalent statement of Assumpton s that for some a, R(z; a) s concave n z [0, ] s. Later we show that the R functon, for approprately chosen a, becomes the proft functon of a platform, f the platform uses quanttes {z,, z s } as hs strategy varables, nstead of prces. Essentally, Assumpton requres that platform s revenue functon, hence proft functon, be concave n quanttes. Ths assumpton can be easly verfed once the dstrbutons of random utltes and the functons of externaltes are gven. To state our man result on the equlbrum prces, we defne M (n) := H (0; n), and η h (0; n) (n) := n Ψ j (x) x= n s, j S S where M represents a sem-elastcty of prce and measures the degree of product dfferentaton among platforms on sde, whch s completely determned by the dstrbutons of random utltes, and η measures the mpact of externaltes that sde generates to all other sdes when the markets are equally dvded among platforms. Theorem. Under full market coverage and Assumpton, there exsts a subgame perfect Nash equlbrum wth the outcome that all the platforms charge the same prces p = (p,, p s ) n the frst stage and the equlbrum market demand s x = n s for each platform n the second stage, where p (n) = c + M (n) η (n), S (5) Theorem shows that n our general settng, the equlbrum prces follow a smple formula. The prce charged to each sde conssts of three addtvely separable terms: the cost, a mark-up due to product dfferentaton, and a subsdy due to externaltes. Each term n the prcng formula represents a separate economc factor and s explctly Under the suffcent condton for the unqueness of PE, the symmetrc prcng equlbrum s unque, and takes the form gven by (5). 0

11 determned by the prmtves of the model. 2 The term M (n) s the standard measure of market power of the olgopolstc frms offerng dfferentated products and s determned by the dstrbutons of random utltes (see Perloff and Salop (985)), whch depends explctly on the number of platforms n the case of symmetry. What s nterestng here s the determnaton of the subsdy term. Note that the partcpaton by sde customer generates externaltes to all sdes (ncludng her own sde). When a platform, say platform, attracts addtonal users on sde, the externaltes enjoyed by any other sde j from jonng platform s enhanced by Ψ j, whle the ex- ternaltes from jonng any other competng platform s reduced by Ψ j n, snce each of the remanng n platforms would lose n on sde due to a busness stealng effect. The dfference n wllngness to pay by each sde j customer then equals Ψ j ( + n ). The addtonal proft that the platform could capture from attractng addtonal user s thus equal to the sum of the margnal externaltes multpled by the equlbrum market share x j that sde mposed on all sdes. In other words, to encourage customer from sde to partcpate, n equlbrum platform needs to lower hs prce by an amount equal to the aggregate margnal profts: Ψ j ( + n )x j, j S whch s the subsdy n Theorem, where x j platform. = /n s the equlbrum market share of Example. To llustrate Theorem, we consder a specal case of our model. In hs model of platform competton (n = 2) n two-sded markets (s = 2), Armstrong (2006) uses a Hotellng specfcaton wth unform dstrbuton of consumer locaton. The equlbrum prces n hs settng are gven by p = c + t α 2, p2 = c 2 + t 2 α, where t and t 2 are the product dfferentaton (or transport cost) parameters, and α and α 2 are the degrees of cross-group externaltes enjoyed by sde and 2, respectvely (see equaton (2) n Proposton 2 n hs paper). Ths prcng formula s the same as 2 The smplcty of the prcng formula s a jont product of symmetry, one-stop shoppng, full market coverage, and the addtve separablty of the consumer payoff, and does not rely on the functonal forms of externaltes and the dstrbutons of random utltes. We wll show n Secton 7 that n the presence of explct outsde optons, the markets are not fully covered, and the equlbrum prces cannot easly decomposed nto the three separate effects.

12 ours n (5). Indeed, n ths Hotellng specfcaton of product dfferentaton, the random utltes can be represented by ɛ = t ɛ, ɛ 2 = t ( ɛ ) wth ɛ beng unformly dstrbuted on the unt nterval [0, ]. It follows that H (θ; 2) = /2 + 2t θ, θ [ t, t ], and hence ( H (0; 2))/h (0; 2) = t [, =, 2. Moreover, ] n ths settng, R s a quadratc 2t α + α 2 functon of z wth Hessan matrx 2, and Assumpton means that α + α 2 2t 2 ths Hessan matrx s negatve defnte, or equvalently 4t t 2 > (α + α 2 ) 2, whch s what Armstrong (2006) assumes (see equaton (8) n hs paper). What s mplctly assumed n the Hotellng duopoly model n Armstrong (2006) s that the ntrnsc value v s relatvely large so that customers always jon at least one of the two platforms, and hence the market are always fully covered. Ths argument s applcable to our general settng wth an outsde opton mposed on each sde, and hence the symmetrc prcng n Theorem remans to be an equlbrum f () ntrnsc values are suffcently large, relatvely to the values of the outsde optons, and () the supports of the dstrbutons of random utltes are bounded. Before further analyzng the above prcng formula, we provde a sketch of the proof of Theorem. We need to show that there s no ncentve for any platform to devate unlaterally from the proposed equlbrum prce vector p n (5). To check the ncentve of a platform, say platform, to devate from p to p, we need to pn down hs devatng proft as a functon of p. One challenge s that due to the cross-sde externaltes, the demand system for platform s servces s mplctly defned by (3), and hence the devatng proft functon does not have an explct expresson. Instead, we rewrte platform s devatng proft as a functon of quanttes q by consderng hs nverse demand system. Intutvely, gven other platforms choces of p, platform behaves as a monopolst, and choosng prces p by the monopolst s equvalent to choosng quanttes q, subject to the demand system condtonal on other platforms prces. Assume that the n platforms have the same demand when chargng the same prce p. Ths together wth full market coverage assumpton mples that whatever quanttes q platform offers, the remanng market s shared equally among the n platforms, wth each takng s q n. It follows that the (condtonal) nverse demand system that platform faces has an explct expresson: The extra wllngness to pay by each sde, p p, equals H ( q ) (the nverse demand wthout externalty), adjusted by φ (q ) φ ( s q n ) (the dfference n the externaltes that customer enjoys between platform and any other platform). We can then reformulate the devatng proft as R(q ; p c) n (4). Devatng s not proftable f R(q ; p c) s maxmzed 2

13 at the equlbrum demand profle, q = n s. The frst-order condtons for ths proft maxmzaton problem hold when p satsfes (5). Moreover, gven that R s concave n q by Assumpton, the frst-order condtons are necessary and suffcent for proft maxmzaton. Therefore, there s no proftable devaton by platform. Moreover, to support the equlbrum outcome gven n Theorem, we need to specfy partcpaton equlbrum (PE) for each prce profle P. As dscussed prevously, there are suffcent condtons under whch for each prce profle P, there s a unque PE. However, n general there mght be multple partcpaton equlbra. We have made the followng reasonable selecton: on the equlbrum path as every platform offers p, we pck PE such that markets are shared equally among these n platforms; off the equlbrum path when only one platform devates to a dfferent prce p whle remanng n platforms charge p, we focus on the sem-symmetrc PE such that the remanng n platforms charge the same prce receve the same demand. Such sem-symmetrc PE always exsts and can be characterzed by fxed-pont condtons, smlar to, but smpler than (3). When more than 2 platforms devate from p, smply pck any PE. One advantage wth transformng a platform s proft maxmzaton n prces nto the one n quanttes s that suffcent condtons for maxmzaton can be easly specfed and checked. 3 Indeed, snce R functon n (4) s separable n terms of the mpacts of the dstrbutons and externaltes, separate assumptons on these functons can be made to guarantee Assumpton. Ths s not possble when consderng proft maxmzaton n prces, snce there s no explct demand system n ths general settng of mult-sded markets. Assumpton can be replaced by smpler but stronger assumptons. Suppose H (θ ; n) s log-concave n θ. Then z H ( z ; n) s concave n z (as shown n Appendx C). As a consequence, the frst term n R(z, a), z H ( z ; n), s concave n z [0, ] s. In other words, wthout cross-sde externaltes, the log-concavty of each H s suffcent to mply Assumpton. In general, the cross-sde externalty term φ non-trvally affects the Hessan matrx of R, and Assumpton roughly means that the degree of externaltes s relatvely small, as compared wth the product dfferentaton effect. Under the lnear form of externaltes, we can nstead mpose the followng alternatve assumptons: 3 See Nocke and Schutz (206) for a model of mult-product olgopolstc competton usng technques from aggregate games. Duet to the mult-sded features and cross-group externaltes, the demand system n our model s not explct and the products wthn each platform are complements nstead of substtutes. Therefore, ther approach does not apply to our model. 3

14 Assumpton 2. () For each S, H (θ ; n) s log-concave n θ ; () the matrx Dag(w,, w s ) s postve defnte, where /w = max θ h (θ ; n) > 0. n n (Γ + Γ ) In Appendx C, we formally show that Assumpton 2 mples Assumpton. Note that for IID random varables, Assumpton (2-) holds f for each, the PDF f ( ) s log-concave, whch holds for many commonly used dstrbutons ncludng normal, unform, exponental, and Gumbel dstrbutons. Assumpton (2-) only requres postve defnteness of a sngle matrx, whch s smpler to check. For Gumbel dstrbutons, (2-) s equvalent to 4Dag(β,, β s ) n n (Γ + Γ ) beng postve defnte, whch s true f the maxmal egenvalue of matrx (Γ + Γ ) s less than 4 mn{β }(n )/n. 4 The effect of platform competton on prces and welfare Theorem reveals decomposable prcng formula for symmetrc equlbrum prces: cost, product dfferentaton effect and cross subsdy, where the product dfferentaton effect s M (Perloff and Salop 985) and the cross subsdy equals η, whch summarzes the externaltes that group offers to all other groups. Both product dfferentaton effect and cross subsdy vary wth the degree of competton (n), and the net effect depends on the degree of product dfferentaton and the sze of externaltes. To study the mpact of platform competton on the equlbrum prces and welfare, we frst consder the case of CIID random varables. In ths case, we have ( ) H (θ; n) = Pr(ɛ max {t=2,,n} ɛt θ) = F (θ + ξ τ )df n (ξ τ ) dg (τ ) and h (θ; n) = ( ) f (θ + ξ τ )df n (ξ τ ) dg (τ ). Therefore, H (0; n) = (n )/n and h (0; n) = ( ) f (ξ τ )df n (ξ τ ) dg (τ ). Proposton 2. Suppose under the CIID, for each τ, F (θ τ ) s log-concave n θ. Then the product dfferentaton effect M (n) s decreasng n n. 4

15 For the case of IID random varables, the effect of competton on market power s known n the lterature. For nstance, Zhou (207) shows that under the log-concavty of F, M (n) s monotoncally decreasng n n, and approaches zero as n goes to nfnty, f addtonally lm θ f (θ )/( F (θ )) = +. 4 Proposton 2 extends the monotoncty of product dfferentaton effect to the case of CIID. On the other hand, n the lterature on mult-sded markets, not much attenton has been pad to the mpact of platform competton on the extent of cross-subsdes. The followng Proposton provdes some nsghts. Proposton 3. Suppose φ (x) = S γ j ρ(x j ) wth ρ > 0. 5 Then the equlbrum crosssubsdy equals η (n) = ρ (/n) n γ, where γ := j S γ j. Assume xρ (x) + ρ (x) > 0 and lm x 0 xρ (x) = 0. 6 Then η decreases wth n f and only f γ > 0. Moreover, η converges to 0 as n. Proposton 3 shows that for a general class of externaltes wth weakly ncreasng returns to scale, the magntude of subsdy to each sde decreases wth n. As competton ncreases, the equlbrum number of users on each sde decrease, and the loop effect s weaker, moreover the margnal externaltes s also weaker gven weakly ncreasng returns to scale, thus the cross-subsdy term decreases wth n. Here the loop effect s equal to + n as each addtonal user on platform mples /(n ) loss for each of the other platforms. Moreover, note that for any two sdes, and j, η η j = ρ (/n) n ( γ γ j ). The relatve degree of subsdes to the two sdes depends on the relatve margnal externaltes that each sde mposes on all the other sdes. Suppose c = c j and M = M j, then γ < γ j mples p > p j. That s, each platform subsdzes customers on sde j by chargng more on sde. As competton ncreases, the extent of cross-subsdes decreases snce ρ (/n) n decreases wth n. In other words, an ncreases n competton erodes the relatve degree of cross subsdes. Propostons 2 and 3 show the monotoncty of the product dfferentaton effect and cross-subsdy effect wth respect to the degree of competton. To llustrate the net effect 4 When the support of θ has a upper bound θ + and f (θ + ) > 0, lm θ θ + f (θ )/( F (θ )) s nfnte. When θ s unbounded from above, the lmt can be nfnte (e.g, normal dstrbutons) or fnte (e.g., Gumbel and exponental dstrbutons). See Proposton 2 and 3 and related dscussons n Perloff and Salop (985). 5 The case wth φ (x) = µ( S γ j x j ) wth ncreasng µ s smlar. 6 The condtons xρ (x) + ρ (x) > 0 and lm x 0 xρ (x) = 0 are satsfed when ρ s convex, or when ρ s a power functon (ρ(x) = x r, 0 < r < ) or log functon (ρ(x) = log( + tx), t > 0). 5

16 of competton on prces, we consder the followng example. Example 2. Suppose ɛ t s IID wth exponental dstrbuton F (θ) = e λ θ and λ > 0. It follows that the product dfferentaton effect M (n) = /λ, whch s constant n n (see Perloff and Salop 985). Wth lnear form of externaltes, the equlbrum prces are gven by p = c + λ n γ, S. Therefore, we have an ntrgung observaton that the equlbrum prce on each sde s ncreasng n the number of platforms n whenever there are postve aggregate externaltes ( γ > 0). For welfare analyss, we defne δ (n) = E[max k=,2, n ɛ k ] as the expected value of the maxmum of n random varables. The followng proposton presents the equlbrum proft for each platform, the expected consumer surplus for each sde, and the total surplus (the sum of total consumer surplus and total profts). Proposton 4. In the symmetrc equlbrum characterzed by Theorem, the followng hold. Each platform k N earns the same proft The expected consumer surplus for sde S equals Π(n) = n (M η ) ; (6) S CS (n) = v c M + δ + η + φ ( n s); (7) The total surplus equals ( TS(n) = v c + δ + φ ( ) n s). (8) S Note that δ reflects customer s preference over varety and ncreases wth n for the case of CIID. Wthout externaltes, Propostons 2 and 4 ndcate standard results that the equlbrum proft for each frm decreases wth n whle both the expected consumer surplus and total surplus ncrease wth n. In the presence of cross-group externaltes, as shown n Proposton 3, platforms have ncentves to subsdze across dfferent sdes, whch may cause the equlbrum prces on some sdes to ncrease wth n. Moreover, for monotoncally ncreasng functon of externaltes, φ ( n s) decreases wth n. As a result 6

17 of the mpact of cross-group externaltes, the equlbrum proft per platform may not necessarly decrease wth n, and the expected consumer surplus and total surplus may not ncrease wth n ether. One mplcaton of Proposton 4 s that there mght be excessve entry nto the multsded markets even though there are cross-group externaltes. To llustrate we frst note that the margnal beneft of an addtonal platform on the total externaltes enjoyed by all sdes s negatvely related to the total cross-subsdes: { S φ ( n s)} n =,j S n 2 Ψ j( n s) = n n ( 2 η ), S whch holds for any functonal form of externaltes. When the aggregate externaltes are postve, the total subsdes are also postve, whch mples that the total benefts from externaltes decreases wth the number of platforms. Second, by Proposton 4, as n ncreases, the change n the equlbrum total surplus can be related to the per-platform proft as follows: TS n n n Π = ( δ n n S n 2 M ), where the rght-hand sde s ndependent of externaltes, and depends only on the dstrbutons of random utltes. 7 Suppose there s a fxed cost K > 0 assocated wth buldng up a platform. There s excessve entry f the equlbrum free-entry number of platforms determned by Π(n) K = 0 exceeds the socally optmal number of platforms that maxmzes the total socal surplus TS(n) nk. The above dscussons mply a suffcent condton for excessve entry as stated n the followng Corollary. Corollary. Suppose that TS(n) nk s quas-concave n n, and that for every sde, the dstrbuton of random utltes satsfes the condton that δ n n M n 2 0. Then there s excessve entry. The condtons n Corollary are mposed only on the dstrbutons of random utltes, not on externalty functons. To see how competton affects the prces and welfare through the trade-off between the product dfferentaton effect and cross-sde subsdy, we consder the followng class of dstrbutons and the lnear form of externaltes. 7 The rght-hand sde s always 0 for Gumbel dstrbutons, and negatve for unform dstrbutons. 7

18 Proposton 5. Suppose ɛ t s IID accordng to the Gumbel dstrbuton wth parameter β, and lnear forms of externaltes. Let β = S β and γ = S γ. Then the followng hold. () For each S, the equlbrum prce s p (n) = c + whch decreases wth n f and only f β > γ. () Each platform s equlbrum proft s whch decreases wth n f and only f γ/ β < n 2 /(2n ); () The total surplus s n n β n γ, (9) Π(n) = n β γ n(n ), (0) TS(n) = (v c ) + (ln(n) + κ) β + γ, () n S whch ncreases wth n. Here κ s the Euler-Marcheron constant. (v) There s always excessve entry. Part () In Proposton 5 provdes a smple formula for the equlbrum prces, whch mples that t s feasble to have the prce on one sde (say ) decrease wth n (when β > γ ) whle the prce on another sde (say ) ncrease wth n (when β < γ ). The latter nequaltes can hold for a number of sdes wthout contradctng wth the condtons for unqueness of partcpaton equlbrum and suffcency for prce equlbrum. In equlbrum, each platform balances out all sdes and the proft per platform and total surplus depend only on the aggregate parameter values, β and γ, as ndcated n parts () and () of Proposton 5. As competton ncreases, platform proft decreases when γ/ β < n 2 /(2n ), and total surplus ncreases when γ/ β < n. The latter condton s equvalent to postve proft for each platform. Both condtons are easly satsfed gven the requrement for suffcency of the prce equlbrum. Therefore, n ths settng the presence of cross-group externaltes does not affect the qualtatve mpact of competton on the aggregate welfare. 8 8 The platform proft ncreases wth the product dfferentaton parameter β, but decreases wth the degree of ndvdual externaltes γ j. The total surplus ncreases wth both β and γ j. It can be easly checked that the expected consumer surplus ncreases wth γ j, but decreases wth β when n = 2, and ncreases wth β when n 3. 8

19 The strength of externaltes and the curvature of φ, whether φ s concave, convex or lnear, clearly affect both the equlbrum number of platforms wth free entry and the socally optmal one, but do not affect the comparson between these two numbers under Gumbel assumptons. 9 5 Merger analyss One applcaton of our equlbrum analyss s to study the mpact of a merger between platforms n mult-sded markets. For nstance, would a merged entty have ncentves to rase the prces gven that all platforms engage n cross-subsdzaton both before and after the merger? Whle our analyss n the prevous secton reles on symmetry, t stll offers some nsghts nto ths queston. In general, a key to the merger analyss s to examne the changes n demands wth respect to prces both wthn and across platforms, whch determne the own-prce and cross-prce elastctes as well as the dverson matrces. Gven that platform k charges prces p k whle other platforms choose symmetrc prces p, assume the demands for other platforms servces are symmetrc. Then, the full market coverage mples a smple relaton between the demand of platform x k and that of other platforms,.e., x k = s x k n, k = k. Ths further mples that xk = p k n xk. As a consequence, the dverson p k matrx from k to k s smply ( xk p k ) ( xk p k ) = n I s, (2) whch s ndependent of p k, p, and x k. The standard measure of upward prcng pressure (UPP) at the equlbrum prces s then n (p c), whch s not nformatve for llustratng ncentves to rase or reduce prces, snce some of the margns can be negatve n ths settng. To evaluate the ncentves for the merged entty to rase prces or not, we need to compute precsely the matrx of the cross-prce partal dervatves. The followng Lemma presents these partal dervatves when all the prces are symmetrc across platforms. 9 When γ = 0, the number of frms under free entry mnus the socally optmal number equals exactly (see Anderson et al. (992), Table 7. on page 226, for ths observaton). In the presence of cross-sde externaltes wth γ > 0, the dfference s strctly larger than. 9

20 Lemma. Wth full market coverage, at any symmetrc prce p wth symmetrc allocaton x = s n, the followng hold: For any k, k N and k = k, x k p k = E and x k p k = E, (3) n where E = Dag{ h (0; n),, h s (0; n) } n n Ψ(x) x= n. (4) s Note that these partal dervatves n Lemma are surprsngly smple and do not depend on the symmetrc p. In Appendx A, we prove a general verson of Lemma, whch presents the matrces of partal dervatves when platform k charges prces p k whle other platforms choose symmetrc prces p. As expected, these matrces depend on (p k, p, x k ). When p k = p, x k = s n and the matrces greatly smplfy to (3) and (4) n Lemma. We now use Lemma to llustrate the margnal ncentve for the merged platforms to rase or reduce prces n mult-sded markets. For notatonal smplcty we consder platforms and 2 to merge and maxmze the jont profts by coordnatng on ther prces. The followng Proposton shows that the merged entty has ncentves to rase locally the prces across all sdes. Proposton 6. At the symmetrc equlbrum prce p, both mergng platform and 2 have strct ncentve to ncrease prce on every sde of the markets n the sense that for l =, 2, {Π + Π 2 } p l p k =p, k N = n(n ) s 0. Wthout cross-group externaltes, the above result s well known for prce competton wth substtute goods, as the mergng frms would nternalze some of the beneft from ncreasng prces. However, n our settng t s possble that the prce-cost margn p c s negatve on some sde before the merger, snce platforms engage n cross subsdzaton. It s not clear at the frst glance that the margnal beneft to platform 2 s postve or not. However, snce both platforms have the same pre-merger margns whch satsfes platform s frst-order condtons p c = ( x p ) n I s. 20

21 Elmnatng the margns, we have a smple expresson for the margnal beneft of rasng prces at the pre-merger symmetrc equlbrum prces as follows: { x2 p } (p c) = { x2 p } ( x p ) n s, whch s equal to n(n ) s by Lemma. Ths mples postve ncentves to ncrease prces on every sde locally. 20 Second, we dentty the magntudes of the cost reductons, due to post-merger synerges, to make the equlbrum prces unaffected. Ths result can be useful when all the pre-merger prces are postve. Proposton 7. Suppose n > 2. There exsts a new cost vector ĉ = (ĉ,, ĉ s ) for the mergng platforms and 2 such that the equlbrum prces after the merger under the new cost structure (ĉ, ĉ, c,, c) stay the same as n Theorem wth orgnal symmetrc cost structure (c, c, c,, c). More precsely, we have ĉ = c n 2 (p c) where the symmetrc prcng p s gven by Theorem. The percentage of proft ncrease by each mergng platform from the above cost savng equals /(n 2). Proposton 7 suggests that to keep the equlbrum prces unchanged, the cost reducton on each sde by the merger needs to be proportonal to the pre-merger margn (when the margn s postve), and ths proporton depends on n but s not affected by sde. Moreover, post-merger equlbrum prces and equlbrum market allocatons stay the same as before, each customer obtans the same expected surplus. the beneft to each mergng platform from the above cost savng s proportonal to equlbrum proft as the cost reducton needed s proportonal to the pre-merger margn. 6 Dscrmnatory vs unform prcng The equlbrum prces n Theorem n general exhbt some degree of prce dscrmnaton across dfferent sdes. Now suppose government regulaton prohbts ths type ( ) ( ) 20 Note that f UPP vector s defned by dg x x 2 p p (p 2 c) as n Affeldt et al. (203), where dga denotes the dagonal matrx whch conssts of the dagonal elements n A, then the above argument suggests that n our settng UPP= n(n ) (dge) s > 0, and hence ths modfed measure of UPP provdes an ndcaton of the merged platforms to rase ther prces locally. 2

22 of prcng behavor by forcng each platform to charge a unform prce on all sdes. How does ths polcy affect the equlbrum prces, profts of platforms, and consumer welfare? The followng theorem determnes the symmetrc equlbrum unform prce: Theorem 2. Suppose c = c, S and all the platforms adopt the unform prcng rule. Then there exsts a symmetrc equlbrum unform prce p u wth where E s gven by (4). p u = c + s/n,j E j, (5) Here E s the sem-elastcty matrx x / p at the equlbrum prces. The prcng formula n (5) follows from standard optmal prcng rule, where s/n s the aggregate demand n equlbrum, and,j E j s the aggregate margnal demand ncrement when a platform rases hs prces unformly on all sdes. Notce that there should be stll product dfferentaton effect and subsdy effect, but the two effects cannot be solated as n the case of dscrmnatory prcng. Nevertheless, the mark-up n (5) s explctly determned by the prmtves of the model. Under both prcng rules, the demand profles are dentcal, hence the total surplus remans unchanged. However, a ban on prce dscrmnaton makes some group(s) better off and other group(s) worse off, and t also affects the platform profts. To compare platform profts and consumer surplus between the two prcng rules, t suffces to compare the average prces across sdes, as shown n the followng Proposton. Proposton 8. The equlbrum outcomes under dscrmnatory prcng and unform prcng characterzed n Theorem and 2 have the followng propertes: () Π d > Π u f and only f p /s > pu ; () For S, CS d > CS u f and only f p < p u ; () TS d = TS u. Snce the equlbrum prces n Theorem and Theorem 2 have explct expressons, the sgn of p /s pu can be determned. Let us frst consder the benchmark case wthout externaltes. Proposton 9. Suppose that at least two of M, S dffer. Wthout externaltes, p /s > p u and the platforms strctly prefer dscrmnatory prcng over unform prcng. 22

23 The logc for the superorty of dscrmnatory prcng over unform prcng s as follows: Wthout externaltes, we have p u c = [ ] s/n S h (0; n) = s M, S whch s the harmonc mean of the product dfferentaton effects M, S. Meanwhle, (p c)/s = s M S S whch s the arthmetc mean. Snce the arthmetc mean s greater than or equal to the harmonc mean, p u S p /s, where the nequalty must be strct by the assumpton on M, S. By Proposton 8, Π d > Π u,.e., platforms obtan strctly hgher profts under dscrmnatory prcng. By contnuty, the above results hold when the degree of externaltes s small. The followng Proposton provdes a comparson between the average prce under dscrmnaton and the unform prce n the presence of cross-sde externaltes when n s large. Let M ( ) = lm n + M (n) denote the product dfferentaton effect n the lmt. 2 Proposton 0. Suppose that the assumptons n Proposton 2 and 3 hold, and that at least two M ( ), S, dffer. Then lm n + S p /s > lm n + p u. In the presence of externaltes across sdes, f the externalty functons satsfy the assumptons n Proposton 3 then the mpact of cross subsdes η, S decreases as competton ncreases and vanshes n the lmt. The prce comparson s then largely determned by the product dfferentaton effects, and hence n the lmt we have smlar results to that n absence of externaltes. Proposton 0 suggests that there exsts a cutoff n such that platforms prefer prce dscrmnaton over unform prcng as long as n > n. Moreover, f at least one M ( ) s zero, lm n p u equals c; f at least one M ( ) s nonzero, lm n S p /s s strctly above c. The above result, although applcable only for large n, s vald for any s. For twosded markets, we have the followng smpler characterzaton for any fnte n. 2 Under the assumptons specfed by Proposton 2, the product dfferentaton term M (n) s monotoncally decreasng n n, and hence lm n + M must exst. 23