Strategic Interactions in Two-Sided Market Oligopolies

Transcription

1 Strategc Interactons n Two-Sded Market Olgooles Emmanuel Farh and ndre Hagu rl 0th, 008 bstract Strategc nteractons between two-sded latforms deend not only on whether ther decson varables are strategc comlements or substtutes as for one-sded frms, but also -and crucally so- on whether or not the latforms subsdze one sde of the market n equlbrum. For examle, wth rces beng strategc comlements across latforms, we show that a cost-reducng nvestment by one frm may have a ostve effect on ts rval s rofts and a negatve effect on ts own rofts when one sde s subsdzed n equlbrum. y contrast, f latforms make ostve margns on both sdes, the same nvestment has the regular, exected effects. Our analyss mles that the strategy sace and the logc of comettve advantage are fundamentally dfferent n two-sded markets relatve to one-sded markets. Keywords: Two-Sded Markets, Two-Sded Platforms, Strategc Comlements, Strategc Substtutes, Comettve dvantage. JEL Classfcatons: L, L, L4, L8 Introducton Fudenberg Trole(984)(hereafterFT) andulow Geanakolos and Klemerer (985) (hereafter GK) have roosed a tyology of strategc nteractons n one-sded markets olgooles. They analyze a cometton game receded by an nvestment stage. Three factors are shown to determne whether an ncumbent wll over-nvest or under-nvest: whether the objectve of the ncumbent s to accomodate or to deter entry, whether actons n the cometton game are strategc comletements or substtutes and whether nvestment ncreases or decreases rvals rofts (cf. Trole 988). In ths aer, we show that the ossblty of subsdzaton of one sde n a two-sded market can lead to fundamentally new (and somewhat surrsng) strategc confguratons n olgooly. For nstance, we show that a cost-reducng nvestment by a two-sded latform may be a successful entry accomodaton strategy and at the same tme rase the rofts of ts rval (n a one-sded market, cost reductons by one frm unambguously hurt ts comettors). The ntuton behnd ths result s as follows. cost-reducton by one of the latforms gves t a comettve advantage relatve to ts rval, so that n the new equlbrum ts rces on both sdes wll be lower than ts rval s and t wll steal customers from ts rval on both sdes of the market relatve to the ntal Harvard Unversty, efarh@fas.harvard.edu Harvard usness School, ahagu@hbs.edu

2 equlbrum. However, the rato between the number of customers stolen from the subsdzed sde and the number of customers stolen from the other (roftable) sde may be suffcently hgh so that the rval s hay to get rd of both, and suffcently low so that the more cost-effectve latform can serve them roftably at the new equlbrum rces. In other words, the cost reducton allows a "rce-rebalancng" act by both latforms, whch may end u beng benefcal to both of them - n stark contrast wth the one-sded case. n mortant alcaton of ths result concerns tyng n the srt of Whnston (990): a twosded latform also has a monooly ower over another roduct M whch s homogenously valued by all customers on one sde of. Tyng M and the urchase of the latfom on ths sde of then acts as a commtment to rce agressvely by rasng the oortunty cost of a foregone sale. In the rcng game that follows, t has the same effect as a reducton of the margnal cost of dstrbuton of on the sde of the market whch buys M, relatve to rval two-sded latforms. In a one-sded market wth rce cometton and homogenous valuatons of the tyng good, tyng s always a "to dog" strategy: t decreases rvals rofts whle ncreasng one s own. y contrast, the result mentoned above mles that n a two-sded market, tyng can be art of a "fat cat" strategy: a roftable way to accomodate entry whle at the same tme beng "soft" (.e. benefttng rvals as well). More broadly, our analyss reveals that the set of strategc confguratons n a two-sded market s strctly larger than n a one-sded market - not n terms of the nature of the strateges but n terms of the condtons under whch they emerge. Ths s due to the fact that n two-sded markets, the sgn of the strategc effect that determnes whether the ncumbent wll over- or under-nvest can no longer be entrely determned by the effect on rvals rofts as n one-sded markets. Ths concluson lends sgnfcant substance and suort to the contenton that two-sded markets have new mlcatons, both from a strategy and from a ublc olcy ersectve. Indeed, the recent and burgeonng lterature on two-sded markets s bult on ths remse, but the man argument ut forward u to now s that two-sded latforms are dfferent because they mght end u subsdzng one sde of the market n order to recou on the other. Ths argument has receved sgnfcant attenton from anttrust scholars (e.g. Evans (003), Wrght (004)), whch have onted out several mlcatons, such as for nstance that below-cost rcng n a market may not be ndcatve of redaton, but of a two-sded rcng strategy, whch can be roftable regardless of the resence of cometton. Our aer takes a ste beyond exhbtng ths tye of skewed two-sded rcng structure by dervng deeer mlcatons of ths feature for strategc nteractons among latforms and a systematc categorzaton of strateges n two-sded markets olgooles (whch we show can be very dfferent from one-sded markets). Relaton to the lterature. Ths aer contrbutes to the recent two-sded markets lterature, ntated by rmstrong (006), Callaud and Jullen (003) and Rochet and Trole (003). Most of ths lterature has analyzed how latforms mght solve the chcken and egg roblem assocated Cf. Rochet and Trole (006).

3 wth two sded-markets and focused almost exclusvely on the condtons whch determne whch (fany)sdessubsdzedandhowmuch.westudythemlcatonsofthetwo-sdedrcnggame between two latforms on strategc nvestment choces that must be made ror to the rcng game. To llustrate the conclusons of our general analyss, we borrow from rmstrong (006) s two-sded Hotellng cometton settng, but we extend hs model by allowng for "hnterlands" - ths extenson s absolutely necessary n order to exhbt results whch are dfferent from the ones occurrng n one-sded markets. There are three aers n the two-sded market lterature that focus on tyng. Rochet and Trole (003b) rovde an economc analyss of the tyng ractce ntated by ayments card assocatons Vsa and MasterCard n whch merchants who accet ther credt cards were forced also to accet ther debt cards. They show that n the absence of tyng, the nterchange fee between the merchant s and the cardholder s banks on debt s too low and tends to be too hgh on credt comared to the socal otmum. Tyng s shown to be a mechansm to rebalance the nterchange fee structure and rase socal welfare. Cho (007) analyzes the welfare effects of tyng n a model of cometton between two-sded latforms (connectng consumers and content rovders), when one or both sdes can multhome. In hs model, tyng smly allows one of the latforms to reach all consumers by bundlng the latform roduct n queston wth another roduct that all consumers need (the motvatng examle s the tyng of Wndows Meda Player to the Wndows Oeratng System, whch every PC user has). The mact of tyng on socal welfare deends on whether consumers can multhome or not, but n all cases, tyng unambguously hurts the rval latform. melo and Jullen (007) consder a settng n whch two-sded latforms would lke to set rces below zero on one sde of the market n order to solve the demand coordnaton roblem, but are constraned to set non-negatve rces. Tyng can then serve as a mechansm to ntroduce mlct subsdes on one sde of the market n order to solve the aforementoned coordnaton falure. saresult,tyngcan raseartcaton onbothsdesandcanbeneft consumers n the case of monooly latform. In a duooly context tyng also has a strategc effect on cometton. Contrary to the monooly case, tyng may not be ex-ost and/or ex-ante otmal for a contested latform. Moreover, the cometng latform benefts from t f the equlbrum mlct subsdy s large enough. We also obtan ths result, although as a artcular case of a broader settng. The remander of the aer s organzed as follows. In Secton, webrefly revew the tyology ntroduced by FT for one sded-markets. In Secton 3, we lay out a very general two-sded market olgooly settng, derve the corresondng characterzaton of the strategc sace and exlan the key dfferences relatve to the one-sded envronment. In Secton 4, we rovde secfc examles and analyze the mlcatons for tyng. We conclude n Secton 5. 3

4 One-sded markets Ths secton rovdes a bref summary of the framework and anmal strategy nomenclature ntroduced by FT; t draws heavly on Trole (988), to whch the reader s referred for more detals. There are two frms, an ncumbent,, and an entrant,. The tmng of the duooly game comrses three stages. In stage, frm chooses a varable K for examle a caacty, an nvestment n roduct qualty or n a cost reducng technology. In stage, frm observes K and decdes whether or not to enter. In stage 3, the stage game deends on wether or not has entered at stage. If has not entered at stage, chooses x whch could be a rce or a quantty to maxmze rofts as a monoolst. If has entered then and smultaneously choose x and x. In stage 3, fbothfrms are actve, frm s rofts are denoted by Π (K,x,x ). Denote by (x (K ),x (K )) the Nash equlbrum n stage 3 gven s choce of K n stage. There are two cases to consder. If entry deterrence s the most roftable strategy for then ts choce of K s drven by Π.Frmwll choose K so as to just deter entry by frm : Π (K,x (K ),x (K )) = 0 () y contrast, f entry accomodaton s the most roftable strategy for frm, then ts choce of K s drven by Π.Inthscase,K s chosen so as to maxmze Π (K,x (K ),x (K )) : dπ = Π + Π dx =0 K {z x } dk {z } drect effect strategc effect In both cases, the nfluence of K on frm s rofts s gven by: ssumng by symmetry that sgn dπ = Π + Π dx K {z x } dk {z } drect effect strategc effect ³ ³ Π Π x = sgn x, wehave: µ µ µ Π dx Π x dx sgn = sgn sgn x x x () (3) where x > 0 (< 0) fthevarables(x x,x ) are strategc comlements (substtutes). The focus of the analyss s on the strategc effects. Drect effects would exst even f frm dd not observe K (oen loo soluton). Therefore, the sgn of the strategc effect - under both entry deterrence and accomodaton - determnes whether frm should over- or under-nvest relatve to the level of nvestment whch would reval n the oen loo equlbrum. FT adot the conventon t the stage 3 Nash equlbrum values (x (K ),x (K )) we have: Π x = Π x =0. 4

5 that nvestment makes frm "tough" ("soft") f the strategc effect of K on frm s rofts s negatve (ostve). We wll mantan ths conventon throughout our aer. They then defne the followng strateges: () "to dog", be bg (.e. overnvest) n order to look tough (aggressve); ()"lean and hungry look", stay small (.e. undernvest) n order to look tough (aggressve), () "uy dog": stay small n order to look soft (noffensve) and (v) "fat cat": be bg n order to look soft (noffensve). These four strateges turn out to be suffcent for fuly descrbng frm s desred behavor n all cases, as shown n the followng table: Table where stands for entry accomodaton and D for entry deterrence. We now turn to a smle alcaton, to whch we wll refer n the next sectons of the aer. ssume that the actons x are rces - x for =, -an s an nvestment whch reduces the margnal cost c of frm. lso, assume that rces are strategc comlements (ths s the case wth lnear demand models): > 0 for {, }. Denotng by n (, ) the demand for frm wth the roerty that n dπ = d d > 0 > n,for {, } wehave: d (( c ) n )=( n d dc c ) < 0 {z } {z} dc {z} dk {z } where the frst term s ostve because frm s rofts have to be ostve n equlbrum n order for frm to be actve. The thrd term s ostve as a result of the frst order condton for roft maxmzaton for frm 3. Thus, a cost-reducng nvestment by frm n a one-sded market can only be tough. Ths s qute ntutve: a margnal cost reducton by one frm can only hurt ts rval n a one-sded context, rresectve of whether cometton s n rces or quanttes. s we wll see n the next secton, thngs change radcally n a two-sded context. 3 ssumng the second order condtons are satsfed,.e. the second dervatve of s rofts wth resect to s negatve. >0 >0 >0 <0 5

6 3 Two-sded markets We now move to a two-sded context, n whch the two frms and are cometng not for just one tye of customers, but for two nterrelated grous of customers and. Each latform {, } chooses actons and, whch corresond to sdes and resectvely. lthough n rncle these could be any strategc actons, we wll focus on the case n whch they corresond to rces that the latforms have to set on both sdes of the market. 4 Two-sdedness s catured by assumng the demand a latform faces on each sde s decreasng n the rce t charges to that sde, ncreasng n the rce charged by the rval latform on the same sde, ncreasng n the realzed demand on the other sde of the same latform and decreasng n the realzed demand on the other sde of the rval latform. In short, the demand for latform on sde j s gven by: Nj =ˆn j j, j,n j,n j (4) ˆn j wth ˆn j < 0 < ˆn j ; > 0 > ˆn j. j N j j N j The tmng of the game s the same as n the revous secton. In stage, frm chooses nvestment K.Instage, frm observes K and decdes whether or not to enter. In stage 3: f has not entered at stage, then chooses and as a monoolst; f has entered then and smultaneously choose and, {, }. Throughout the rest of the aer we wll make the followng four assumtons n order smlfy the analyss (the substance of our conclusons s not affected). ssumton (non-sngularty) The functons n j are well-behaved so that for any quartet of rces { j}, the four equatons (4), determne a unque, stable, confguraton of demands: Nj = n j { k l } where n j j < 0 < n j j and n j j < 0 < n j. j The rofts of latform can then be wrtten as: Π = c n + c n where c j s the margnal costs of latform on sde j. ssumton (concavty) Each latform s rofts are concave n both of ts rces holdng the rval latform s rces constant and there always exsts a unque Nash equlbrum n the stage 3 rcng game (whch deends on the value of K chosen n stage and on latforms costs c j), denoted by ª j wth {, }, j {, }. 4 In fact, we are unaware of any model of two-sded markets n whch latforms comete n varables other than rces. 6

7 ssumton 3 (strategc comlementarty) Prces are strategc comlements: j / l > 0 and equlbrum rces for each latform are ncreasng n ts own costs: j c j 0 > 0 for all {, }, j, j 0 {, } (5) Note that ths assumton mles that d = d + d < 0 for =,. In a one-sded market, the corresondng condton to (5) s automatcally verfed and can be seen at the most general level as an alcaton of the monotone comaratve statcs results n Mlgrom and Shannon (994). These rncles, however, cannot be nvoked n the two-sded context analyzed n ths secton. ssumton 4 (symmetry) For K =0and the corresondng Nash equlbrum, we have: Π j { k l } = Π { k l } j Indeed, nstead of solvng the last two stages of the game for all values of K, we wll only erform a local comaratve statcs analyss n K around K =0,whchssuffcent for our uroses. Fnally, from now on we wll focus on the case n whch K s a cost-reducng nvestment on sde for latform : dc < 0 and dc = dc = dc =0 Equatons (5) mly then: d j = d j dc < 0 dc for {, }, j {, }. 3. Strategc nteractons wth varable rces on both sdes s n the one-sded context, we are nterested n determnng the strategc effects of K on the rval latform () s rofts as well as on latform s own rofts. The noton of under/over-nvestment s relatve to a stuaton n whch only the drect effects exsts. Hence our nterest les n sgnng and dπ Π K. Usng the enveloe theorem, the sgn of the strategc effect on latform s rofts s: dπ Π K ½ ¾ ½ dπ sgn Π Π = sgn K X = sgn j {,} µ c n j d + Π + c n j ¾ d d j (6) 7

8 The sgn of the strategc effect of K on frm s own rofts s: ½ ¾ ½ dπ sgn Π Π = sgn K X = sgn j {,} d + Π µ c n j ¾ ½ d Π = sgn + c n j d j d + Π ¾ d These two exressons consttute the core of our analyss. Note that even when rces are strategc comlements (whch we wll assume below), the resectve sgns of the strategc effects for frms and are dsconnected. Ths s a crucal dfference wth the case of one sded markets. Indeed, n GK and FT, whch analyze the same knd of strategc nteractons, but n one-sded olgooles, all the acton s n the terms d whether the "actons", j acton comes from the sgn of the terms Π "actons", j and d j, the sgns of whch deend crucally on are strategc comlements or substtutes. y contrast, here the, whch can be ostve or negatve even when the are n fact rces and therefore startegc comlements. In GK and FT these terms are ostve f the actons are rces (and negatve f the actons are quanttes). The set of strategc confguratons n a two-sded market s strctly larger than n a onesded market - not n terms of the nature of the strateges but n terms of the condtons under whch they emerge. Indeed, even wth rcng beng strategc comlements (cf. nassumtono 3), equatons (6) and (7) mly that all four confguratons are ossble 5 dπ : () sgn Π K > n o dπ Π K n o dπ Π K n o dπ Π K 0 and sgn > 0; () sgn > 0 and sgn < 0; () n o n o n o n o dπ sgn Π dπ K < 0 and sgn Π dπ K > 0;(v)sgn Π dπ K < 0 and sgn Π K < 0. Under entry accomodaton, these cases lead to the followng strateges for latform : () fat cat; () to dog; () uy dog; (v) lean and hungry look. y contrast, n a one-sded market wth strategc varables beng strategc comlements, the otmal strategy for entry accomodaton by can only take two forms: fat cat and uy dog (cf. Table ). Ths s because n a one-sded market wth strategc comlement actons, the two strategc effects of K necessarly have the same sgn. Consequently, table above needs to be adjusted n order to allow for the full range of strategc scenaros n a two-sded market. We wll say that nvestment K s self-servng (self-harmng) for frm f the strategc effect on s ostve (negatve). We then have the followng table corresondng to the case where the actons are strategc comlements and fnds t otmal to accomodate entry by : (7) 5 nd whle stll mantanng the condton that s rofts are ostve. 8

9 Table Frst, note that t s ossble that (6) s ostve,.e. K has a ostve effect on latform s rofts so that K s "soft" wthout volatng the condton that s rofts are ostve n equlbrum: c n + c n > 0 The ossblty for a cost reducton by one latform to ncrease the rofts of ts rval s novel. Ths stuaton never occurs n one-sded markets as we showed n the revous secton. In a two-sded market settng, t s ossble that n equlbrum the latforms subsdze one sde of the market say sde and recou the losses on the other sde: c < 0 and c > 0. Note that for K to be soft n ths case t s necessary that: n / j n > n / j n (8) for at least one j {, }. Ths condton has an ntutve nterretaton. Softness s more lkely to occur when the "busness stealng" effect on the subsdzed sde n n / j s hgher relatve to the busness stealng effect on the roftable sde n n / j. Equaton (8) holds the key to nterretng ths surrsng result. y the enveloe theorem, a small change n K macts Π only through s equlbrum rces. Gven that K reduces c,we know that s rces wll be lower n the new equlbrum on both sdes ( d, d < 0), whch means that steals customers from on both sdes of the market. However, f the roorton of customers stolen by from on the loss-makng sde s suffcently hgh relatve to the roorton of customers stolen on the roft-makng sde, ths mght ncrease s rofts. Gettng new customers on both sdes n these roortons can be roftable for because t has now ganed some comettve advantage relatve to ( dc < 0). Insectng (7), t should be clear however that the strategc effect of K on s own rofts can be negatve. Tyng. Tyng can be analyzed as a reducton n margnal costs on one sde of the market. Confguraton 4 above s then a case n whch tyng makes the ncumbent soft and s roftable at 9

10 thesametme. Inotherwords,tyngbecomesafatcatstrategy. Wehavethereforeuncoveredthe ossblty that tyng can be a roftable entry-accomodaton strategy, n stark contrast wth the case of one sded markets. To ut ths result n ersectve, t s worth revewng the logc of Whnston (990) s argument. Whnston consders a model wth two frms and. Frm roduces two dfferent goods and. Producton of good s monoolzed by frm. Goods and good are merfect substtutes: the demand for good s gven by x (P,P ) wth x j 0 f 6= j and x j < 0 f = j. Frm however, needs to ay an entry cost K n order to oerate. In addton to ts rcng decsons, frm can offer bundles of ts roducts. Two cases have to be dstngushed. The commtment case occurs when frm commtstoasecfc bundle of roducts before entry and rcng decsons are made. The no-commtment case occurswhenbundlng decsonsaremadeatthesametmeas rcng decsons. Whnston s man result corresonds to the case that s closest to our setu, where valuaton for the tyng good s homogenous. Whnston shows that tyng s useless n the no-commtment case: frm makes sure that all customers urchase good and the equlbrum s equvalent to the equlbrum of the ndeendent rcng game where bundlng s rohbted. In the commtment case tyng has the same effect on best resonse functons as a reducton n margnal cost for frm : every sale of good comes wth the extra beneft of a sale of good, thus lowerng the effectve margnal cost of the sale. Tyng therefore acts as a commtment by frm to be more aggressve. If frm remans actve, ths reduces rofts for both frms. Tyng s then a self-deafeatng strategy to accomodate entry. On the other hand however, tyng can result n foreclosure whereby frm decdes not to ay the fxed cost of entry. Frm then monoolzes themarketforgood. The benft forfrm s reduced cometton for good. The otental loss comes from the fact that frm wll be a monoolst who can only offer a bundle. Thus, the resence of a large number of consumers who dslke oduct may make a commtment to bundlng unroftable, even when t leads to excluson. The concluson s that n ths context, tyng s a to dog strategy that s always unroftable to accomodate entry, and can sometmes be roftable to deter entry. 6 6 Whnston then nvestgates the consequences of relaxng the assumton that the valuaton for good s homogenous across consumers. Theresultsaremodfed as follows. Frst and most related to ths aer, a commtment to tyng need not always result n foreclosure: t can be the case that frm s rofts ncrease as a result of a commtment to tyng by frm. The reason s twofold. Frst, enough consumers may fnd unattractve so that the margn on every bundle sale s lower than the margn on an ndeendent sale of good (t s always hgher n the homogenous case) the monooly ower of frm s thus too weak. Second, the elastcty of bundle sales to the rce of the bundle s no longer dentcal to that arsng n market : ndeed, when and are nearly homogenous, tyng essentally transforms the nearly homogenous market ntoavertcallydfferentated market, otentally rasng frm s rofts. Second, tyng can be a roftable strategy even n the absence of an ablty to commt, and when t s, t may lower frm s roftablty. 0

11 3. Strategc nteractons wth rces fxed on one sde It s nterestng to brefly comare the results above wth a stuaton n whch latforms margns ononesdeofthemarket,saysde,areexogenouslyfxed,.e. c = c π.thsmay be nterreted as a context n whch regulaton, common ractce or other nsttutons ut rces on one sde of the market beyond the control of the two latforms. Then each latform only has onevarabletochoose- - and the demand functons can be wrtten as Nj = n j {, }. The strategc effect of K on latform s: dπ = Π d = c n + π n d whereas the strategc effect on latform s own rofts s: Π d = Π d d d = c n + π n d d d = dπ d d The frst exresson above mles that the ossblty of K beng soft survves. If for examle π < 0 and n n / > n n /, then we can have smultaneously dπ > 0 ³ and Π d > 0 n equlbrum. However, gven that sgn dπ = sgn sgn,f d d ³ Π ³ d d > 0 whch s always the case wth lnear demands and rces fxed on one sde then the two strategc effects have the same sgn, just lke n a one-sded market. Hence, by fxng rces on one sde, the set of ossble otmal strateges for entry accomodaton for latform s reduced back to {uy dog, fat cat}. 4 Examles In ths secton we use secfc models to rove that the results antcated by the general analyss above are ndeed ossble. In artcular, for each examle, our goal s to exhbt the ossblty of a small cost reducton by latform beng "soft" (.e., ncreasng latform s rofts), whle rces are strategc comlements across latforms. n addtonal beneft of gong through these examles s that they allow us to gan further ntuton regardng the two-sded strategc nteractons we are llustratng. 4. Fxed rces on one sde We begn wth the smler case, n whch latform rces or markus are fxedononesdeof the market. In ths case, we can wrte wthout loss of generalty the rofts of latform {, } as: Π = c n, j + π f n,n j

12 where n, j s decreasng n and ncreasng n j ; f n,n j s ncreasng n n and decreasng n n j. We want to show that the strategc effect of a decrease n c on Π can be ostve,.e., < 0 7.Thssequvalentto 8 : Π " c + π f n + f n n µ n # < 0 (9) The frst order condton for yelds: " c + π f n + f n n µ n # = n µ n > 0 (0) Thus, n order for our examle to work, we need to have f n π > 0 and n n < n < 0 and ether: n () or: π < 0 and n n > n n () Stablty of the demand system n, n requres n > n n, hence we must have π < 0 9. Most models found n the two-sded market lterature u to now rmstrong (006) n artcular have a symmetc lnear strucuture whch mles n = n = n = n. Hence they cannot satsfy our condtons () or (). We therefore need to deart from the symmetry assumton and look to obtan the followng tye of exresson: n, j = N + γ j wth 0 <γ<. In other words, we need the elastcty of a latform s demand n ts own rce to be strctly hgher n absolute value than the elastcty n the rval latform s rce. Ths can be acheved by addng "hnterlands" on sde to the two-sded Hotellng model consdered n rmstrong (006). In the aendx we rovde the detaled mcro-foundatons of ths model, whch allows us to 7 Recall that, wth fxedrcesononesde, d dc 8 ll terms are evaluated at. 9 Note that the requrement, f n > 0 whenever Π s concave n. < 0 rules out multhomng by all sde agents, snce n that case demands for each latform on sde would be ndeendent of each other (assumng away economes of scale n multhomng) and hence f would only deend on n.

13 obtan: n = N + γ j n = N + N α n t n j Π = c n + π n (3) where N > 0, N > 0, t > 0, α > 0, π < 0 (necessary as we have seen above) and γ ]0, [ are all constants. The lnearty of demand n rces mles that latform s rofts are concave n,whchn turn mles d d > 0, and that latform rces are strategc comlements,.e. dc > 0 for all d j 6= j {, } (so that assumtons -4 are satsfed). In the aendx we rove: Prooston necessaryandsuffcent condton for a cost reducton by latform to be roft enhancng for latform (9) above s: N c ( γ) π N α ( γ) ( + γ) t γ < 0 (4) The condton n the text of Prooston can be comared to the condton for n to be ostve n equlbrum 0 : N c ( γ) π N α ( + γ)( γ) > 0 (5) t and to the condton for latform rofts to be ostve n equlbrum: h h N c ( γ)+ π N α t ( + γ) N c ( γ) π N α t ( + γ)( γ) ( γ) > π N (6) It s then easly verfed that the last three condtons can hold smultaneously for a range of arameter values. Note that γ<s absolutely necessary n makng t ossble that (4) and (5) hold at the same tme. To gan some ntuton, note that (4), (5) (6) are clearly satsfed when γ =0and N s suffcently large relatve to N. In ths case, (3) yelds n = N and so there s no cometton between latforms on sde. Meanwhle, on sde latforms are "forced" to lose money and each gans unroftable sde customers n roorton to ts market share advantage relatve to ts rval on sde. Consequently, laform s always hay to lose market share on sde (snce that 0 We use the equlbrum exressons of and n rovded n the roof of Prooston n the aendx. 3

14 entals no loss n market share on sde ) and ths haens whenever latform s margnal cost c decreases because t leads to lower,hghern and hgher n. The condton that N s suffcently large relatve to N smly ensures that the market of unroftablecustomers(sde)s suffcently small relatve to the market of roftable customers (sde ) so that the latforms can make ostve rofts n equlbrum. Theresultsthusstraghtforward whenthetwolatforms aresuffcently dfferentated so that they don t comete on sde. When cometton among them on sde becomes more ntense (.e. γ ncreases), t becomes less and less lkely that a cost reducton by one latform on sde wll beneft ts rval. Ths s because each latform s now ambvalent about losng customers on sde : although that rds t of unroftable customers, t also decreases ts market share of roftable sde customers. One can confrm ths ntuton by notng that the left hand sde of (4) s ncreasng n γ and strctly ostve for γ =. Therefore, for γ hgh enough, a cost reducton by one latform always hurts ts rval, as standard ntuton suggests. 4. Fully two-sded rcng Let us now turn to the general case, when rces on both sdes of the market are flexble. n nvestment that reduces latform s margnal cost c s soft f and only f (recall 6): c n + c n d dc + c n + c n d dc < 0 whch means we want at least one of the two terms n-between square brackets to be negatve. ssume then that: c n + c n < 0 or, equvalently µ c + c n n < 0 (7) Just lke n the revous examle, the man ssue s also satsfyng the frst order condtons. In artcular, the frst order condton n s: c + c n µ n = n µ n > 0 (8) To smlfy thngs somehwat, assume that ˆn N = ˆn N > 0. Then: n µ n = ˆn N Ã + n µ n! and n µ n = ˆn N Ã + n µ n! Ths holds for examle when latforms comete a la Hotellng wth no hnterlands on sde. 4

15 Stablty requres we must have: n n n n c < 0 and, therefore, n order to satsfy (7) and (8) smultaneously, n µ n < n µ n gan, ths condton cannot be satsfed n the rmstrong (006) model. Just lke n the revous examle, we need to have: n = n = γ n = γ n wth 0 <γ<. In ths case: n µ n = ˆn N µ + > ˆn γ N To obtan ths, we ost the followng demand functons: ( + γ) = n µ n n = + u u j + x V + u t t " # n = N + u u j t (9) (0) wth: Note that we have: u = α n and u = α n () n + n = N and n + n = Relegatng the tedous calculatons n the aendx, we have: Prooston Thereexstsarangeofarameters(α,α,t,t,x,V,N ) for whch an nvestment that slghtly decreases latform s margnal cost c s soft,.e. ncreases the rofts of latform j 6=,, j {, }. To get some more ntuton for ths result, let us magne that the ntal stuaton s a symmetrc equlbrum where rces are gven by and and market shares are gven by n and n, wth c > 0 and c < 0. Then dπ dc n = d n + d n +( c )dn +( c )dn Note that under our assumtons, d < 0 and d < 0 so that In the rmstrong (006) model, γ =. d n + d n < 0 for all. 5

16 Hence a necessary condton for dπ dc n to be ostve for all, and hence for tyng to be a fat cat strategy s that ( c )dn +( c )dn > 0 for all. () Clearly, f the total market sze s fxedoneachsde,thendn j = dn j for all, j so that ( c )dn +( c )dn = ( c )dn ( c )dn and() cannotbeverfed. Tyng beng a fat cat strategy requres an exanson n market szes. Indeed, suose that on sde, total market sze s not fxed. Denote by dn the total exanson n market sze. Then we have ( c )dn +( c )dn = ( c )dn ( c )dn +( c )dn wth ( c )dn > 0. Therefore, f the market sze exanson s strong enough, () can be verfed. 5 Concluson We have shown that strategc nteractons n two-sded market duooles are fundamentally dfferent from those n one-sded markets. The ossblty of cross-subsdzaton between the two sdes leads to novel and counterntutve results. In artcular, a cost-reducng nvestment by one frm can both ncrease the rofts of ts rvals and be desrable for the frm undertakng the nvestment. Thus, ncreasng comettve advantage through cost advantage for one latform may end u beneftng both latforms (n a one sded market, cost reductons by one frm always hurt ts rvals). The fundamental reason s that n the event one sde of the market s heavly subsdzed, the latform ganng a cost advantage hels the rval latform by stealng customers on both sdes n roortons that t was unroftable for the latter to serve. Ths means that a cost-reducng nvestment can be art of a fat cat strategy under entry accomodaton, whch s never the case n a one-sded market, as shown by FT. romnent alcaton s tyng, whch n a one-sded market s unambguously tough. y contrast, our analyss mles that t can actually be soft n a two-sded market, whch means that tyng may n fact beneft boththetyngfrm and ts rval. Ths can for examle mean that when Mcrosoft s tyng Wndows Meda Player nto Wndows (whch can smly be nterreted as leveragng a dstrbuton cost advantage on the consumer sde over Real Networks and other rvals), everyone could ossbly beneft. Thus, even before worryng about socal welfare, an anttrust tyng clam may be nvaldated on grounds of lack of harm to the cometton. 6

17 References [] melo,. and. Jullen (007) Tyng and Freebe n Two-Sded Markets, unublshed manuscrt, June 007. [] rmstrong, M. (006) "Cometton n Two-Sded Markets," Rand Journal of Economcs, Vol. 37 (3), 006. [3] ulow, J. I., J. D. Geanakolos, and P. D. Klemerer (985) "Multmarket Olgooly: Strategc Substtutes and Comlements," Journal of Poltcal Economy, 93(3), [4] Cho, J. P. (007) "Tyng n Two-Sded Markets wth Mult-Homng," unublshed manuscrt, May 007. [5] Evans, D. S. (003) The nttrust Economcs of Mult-Sded Platform Markets, Yale Journal on Regulaton, 0(), [6] Fudenberg, D., and J. Trole (984) "The Fat-Cat Effect, the Puy-Dog Ploy, and the Lean and Hungry Look," mercan Economc Revew, 74(), [7] Hagu,. (006a) "Prcng and Commtment by Two-Sded Platforms," Rand Journal of Economcs, Vol. 37 (3), 006. [8] Mlgrom, P. and C. Shannon (994) "Monotone Comaratve Statcs," Econometrca, Vol. 6(), 57-80, 994. [9] Rochet, J.-C., and J. Trole (003a) Platform Cometton n Two-Sded Markets, Journal of the Euroean Economc ssocaton, (4), [0] Rochet, J.-C. and J. Trole (003b) Tyng n Two-Sded Markets and the Imact of the Honor ll Cards Rule, unublshed manuscrt, Setember 003. [] Rochet, J.-C., and J. Trole (006) Two-Sded Markets: Where We Stand, Rand Journal of Economcs, Vol. 37 (3), 006. [] Trole, J. (988) "The Theory of Industral Organzaton," MIT Press. [3] Whnston, M. D. (990) Tyng, Foreclosure, and Excluson, mercan Economc Revew, Setember 990, 80, [4] Wrght, J. (004) "One-Sded Logc n Two-Sded Markets," Revew of Network Economcs, 004, 3,

18 6 endx Mcro-foundatons leadng to the demand system (3) We assume that on sde there s a Hotellng segment of consumers wth lnear transortaton costs t, standalone valuaton for ether latform V and addtonal utlty α er customer on sde j of the same latform, wth 6= j {.}. We assume the total mass of these customers s on sde and M on sde. Furthermore,onsde, each latform faces a downward-slong demand of "loyal" customers (.e. who are never nterested n the rval latform), wth standalone valuaton V, utlty er sde customer on the same latform β and transortaton costs t. Ths yelds x the followng exresson of for latform s demands: n = + α n n j + j + x V + β t t n à n = M + α! n n j t where α, α, x, V, β, t, t > 0. Solvng ths demand system for (n,n ) asfunctonsofrcesonly,weobtan: wth: M = t M + xv + β t θ = λ = n = M θ + λ j ³ ( + x) M α α +M α xβ 4t t + M α α +M α xβ 4t t ³ 4t M α α +M α xβ 4t t ³ M α α +M α xβ 4t t +(+x) M α α +M α xβ 4t t ³ 4t M α α +M α xβ 4t t We need to mose: whch not surrsngly mles: > M α α + M α xβ 4t t 0 <λ<θ To smlfy thngs, we let then γ = λ < and N θ = M ; N θ = M, snce we can factor all rofts θ by θ wthout changng anythng. 8

19 So everythng s as f we had: n = N + γ j n = N + N α n t n j wth γ<. Proof. (Prooston ) The frst order condton ((0) n the text) at the symmetrc equlbrum ( = = and n = n = n )canbewrttenas: t the symmetrc equlbrum: c + π N α t ( + γ) =n > 0 n = N ( γ) Pluggng n the frst order condton above, we obtan: = c γ + N N π α t ( + γ) γ The condton for a cost reducton by latform to be roft enhancng for latform ((9)above)sthen: N α γ + c + π < 0 t γ Relacng wth ts exresson as a functon of the model arameters, ths condton s equvalent to (4) 3. Proof. (Prooston ) Solvng (9), (0) and () for n and n, =,, we obtan: n = N + α ( + x ) N +t 4t t α α ( + x ) N n = N + α ( + x ) N +t 4t t α α ( + x ) N n = t + V x + α N + α ( + x ) t 4t t α α ( + x ) N + 4t t + α α x ( + x ) N 4t t α α ( + x ) N t 4t t ( + x ) α α x ( + x ) N 4t t α α ( + x ) N t 3 Recall that π < 0 so that π = π > 0. 9

20 n = t + V x + α N + α ( + x ) t 4t t α α ( + x ) N + 4t t + α α x ( + x ) N 4t t α α ( + x ) N t 4t t ( + x ) α α x ( + x ) N 4t t α α ( + x ) N t In order to smlfy notaton for the calculaton of the rcng equlbrum, let: γ δ α ( + x ) 4t t α α ( + x ) N ; γ α ( + x ) N 4t t α α ( + x ) N t 4t t α α ( + x ) N ; ε 4t t + α α x ( + x ) N t [4t t α α ( + x ) N ] N t + V x + α N ; +u 4t t ( + x ) α α x ( + x ) N t 4t t + α α x ( + x ) N Clearly u>0 snce we must have 4t t α α ( + x ) N > 0. Wecannowwrte: Recall that latform rofts are: n = N + γ + ε ε ( + u) n = N + γ + ε ε ( + u) n = N + γ + δ n = N + γ + δ Π = c n + c n Π = c n + c n We start from a symmetrc stuaton wth c = c = c and c = c = c and consder a slght decrease n c. Usng the demand exressons above, the frst order condtons n rces are: c ε ( + u)+ c γ = N + γ + ε ε ( + u) (3) c γ + c N δ = + γ + δ (4) c ε ( + u)+ c γ = N + γ + ε ε ( + u) (5) c γ + c N δ = + γ + δ (6) We can use (5) and (6) to determne latform s best resonse rces, as a functon of latform 0

21 s rces, : = = 4δε ( + u) (γ + γ ) 4δε ( + u) (γ + γ ) δ N (γ + γ ) N +[δε ( + u) (γ + γ ) γ ] c + δ (γ γ ) c +[δε (γ + γ ) γ ] + δ (γ γ ) ε ( + u) N (γ + γ ) N +ε ( + u)(γ γ ) c +[εδ ( + u) (γ + γ ) γ ] c +ε [γ ( + u) γ ] +[εδ ( + u) γ (γ + γ )] Therefore, rces are strategc comlements across latforms ( k l j {, }) f and only f the followng 5 condtons hold: 4δε ( + u) (γ + γ ) > 0 δε (γ + γ ) γ > 0 γ γ > 0 γ ( + u) γ > 0 εδ ( + u) γ (γ + γ ) > 0 > 0 for all, j {, } and k 6= l Takng the dervatve n c of both sdes of the equalty n (3), (4), (5) and (6) above, we obtan the followng system of 4 equatons wth 4 unknowns ( d, d dc, d dc, d dc ): dc (7) d dc ε ( + u)+ d dc (γ + γ ) d dc ε d dc γ = ε ( + u) d dc (γ + γ )+ d dc δ d dc γ d dc δ = γ d dc ε ( + u)+ d dc (γ + γ ) d dc ε d dc γ =0 d dc (γ + γ )+ d dc δ d dc γ d dc δ =0 Takng the dfference between the frst and thrd equaltes, then between the second and fourth, we obtan: µ d dc µ d d ε (3 + u)+ dc dc µ d dc µ d d (γ dc +γ )+ dc d (γ dc + γ )=ε( + u) d 3δ = γ dc Smlarly, summng frst and thrd, then second and fourth, we also have: µ d dc µ + d d ε ( + u)+ dc dc + d γ dc = ε ( + u)

22 µ d dc µ + d d γ dc + dc Solvng the two systems above, we fnally obtan: + d δ = γ dc d dc d dc = 3δε ( + u) γ (γ + γ ) 3δε (3 + u) (γ +γ )(γ + γ ) d dc d dc = γ ε (3 + u) (γ +γ ) ε ( + u) 3δε (3 + u) (γ +γ )(γ + γ ) = ε [γ ( + u) γ ( + u)] 3δε (3 + u) (γ +γ )(γ + γ ) d dc + d dc d dc + d dc = δε ( + u) γ γ δε ( + u) γ γ = γ ε ( + u) γ ε ( + u) δε ( + u) γ γ = δε ( + u) γ γ We then mmedately get the two terms we are nterested n: d = 3δε ( + u) γ (γ + γ ) dc 3δε (3 + u) (γ +γ )(γ + γ ) + δε ( + u) γ γ (8) δε ( + u) γ γ d = ε [γ ( + u) γ ( + u)] dc 3δε (3 + u) (γ +γ )(γ + γ ) + γ εu (9) δε ( + u) γ γ We want both of these terms to be ostve. Fnally, we also have to determne the symmetrc equlbrum = = and n = n = n for =, and c = c = c.itsuffces to use the four frst order condtons above (3, 4, 5, 6) n order to obtan: ε ( + u)+ γ = N + c ε ( + u)+c γ whch we can mmedately solve to get: c = c = γ + δ = N + c γ + c δ γ εu δεu N δ N γ c + δε ( + u) γ γ [δε ( + u) γ γ ] γ εu δε ( + u) γ γ c + N ε ( + u) N γ [δε ( + u) γ γ ] We can now wrte all the condtons that need to be satsfed n order for an equlbrum n whch a slght reducton n c ncreases s rofts whle rces are strategc comlements across latforms to exst. Usng the exressons of n k ( {, }, k {, }) asfunctonsof,,,, the softness condton s: [ε ( c )+γ ( c )] d dc +[γ ( c )+δ ( c )] d dc < 0

23 whch can be rewrtten as: ( c ) µ ε d dc µ d d + γ +( dc c ) γ dc + δ d < 0 (30) dc t the same tme, just lke n the revous examle, we need to make sure that the equlbrum n s ostve: n = + x µ N V + α t > 0 (3) Fnally, latforms must make ostve rofts n equlbrum: ( c ) + x µ N V + α t +( c ) N > 0 (3) Together wth (7), (8) and (9), (30), (3) and (3) comlete the set of necessary condtons. Tedous calculatons reveal that all of these condtons can be satsfed for a range of arameter values (ε, δ, u, γ, γ, N, N ) or of the rmtve arameters (α, α, t, t, x, V, N ). 3