RESEARCH DISCUSSION PAPER

Transcription

1 Reserve Bank of Australa RESEARCH DISCUSSION PAPER Competton Between Payment Systems George Gardner and Andrew Stone RDP

2 COMPETITION BETWEEN PAYMENT SYSTEMS George Gardner and Andrew Stone Research Dscusson Paper Aprl 2009 Payments Polcy Department Reserve Bank of Australa We are grateful to Mchele Bullock, Chrstopher Kent, Phlp Lowe and colleagues at the Reserve Bank for helpful comments. The vews expressed n ths paper are those of the authors and are not necessarly those of the Reserve Bank of Australa. Authors: gardnerg or stonea at doman rba.gov.au Economc Publcatons: ecpubs@rba.gov.au

3 Abstract Ths paper s the frst of two companon peces examnng competton between payment systems. Here we develop a model of competng platforms whch generalses that consdered by Chakravort and Roson (2006). In partcular, our model allows for fully endogenous mult-homng on both the merchant and consumer sdes of the market. We develop geometrc frameworks for understandng the aggregate decsons of consumers to hold, and merchants to accept, dfferent payment nstruments, and how these decsons wll be nfluenced by the prcng choces of the platforms. We also llustrate a new potental source of non-unqueness n the aggregate behavour of consumers and merchants whch s dstnct from the well-known chcken and egg phenomenon and ndeed can only arse n the context of multple competng platforms. Fnally, we brefly dscuss how ths new source of non-unqueness may nevertheless shed lght on the chcken and egg debate n relaton to the development of new payment systems. JEL Classfcaton Numbers: D40, E42, L14 Keywords: payments polcy, two-sded markets, nterchange fees

4 Table of Contents 1. Introducton 1 2. A Model of Competng Payment Systems The General Model Notaton Platforms Consumers Merchants Possble Applcatons of the Model Understandng Merchants and Consumers Card Choces Merchants Card Acceptance Decsons Consumers Card Choces The Case of the Chakravort and Roson Model Potental Non-unqueness of Market Equlbra The Extent and Nature of Non-unqueness The Modellng and Other Implcatons of Non-unqueness Conclusons 32 Appendx A: Dervaton of the Geometrc Frameworks n Secton 3 34 A.1 The Framework for Merchants Card Choces 34 A.2 The Framework for Consumers Card Choces 35 Appendx B: Platforms Prcng Incentves 38 B.1 The Effects of an Increase n a Platform s Consumer Fee 38 B.2 The Impact on a Platform s Profts and Incentves 42 References 45

5 COMPETITION BETWEEN PAYMENT SYSTEMS George Gardner and Andrew Stone 1. Introducton Over the past decade, analyss of the prcng strateges of payment system operators has been an area of growng nterest. Much of ths analyss has, however, been conducted n the context of a sngle payments platform, competng wth the default alternatve of cash. Only recently has a lterature developed examnng the more complex, but more realstc, stuaton of competng payment systems. Important contrbutons n ths latter area nclude Rochet and Trole (2005), Armstrong (2006) and Guthre and Wrght (2007). In these and other papers, consderable progress has been made n dentfyng key ssues nfluencng platform prcng. However, the complexty of modellng competton between payment systems has typcally requred such papers to adopt a range of smplfyng assumptons, some of whch lmt the applcablty of any fndngs. One such assumpton s that payment platforms levy purely per-transacton fees on both consumers and merchants. Adoptng ths assumpton has the analytcal advantage of ensurng that every consumer (merchant) faces the same charge for each transacton on a gven platform as every other consumer (merchant). However, n many payments markets competng schemes tend to use annual, rather than per-transacton, fees n ther prcng to consumers. 1 In such markets, the effectve per-transacton fee faced by consumers, rather than beng dentcal for all consumers, vares dependng on ther propensty to use a gven network. A second assumpton commonly adopted relates to the degree to whch consumers (merchants) tend to hold (accept) multple payment nstruments known as multhomng. In recent years the relatve tendency of consumers and merchants to mult-home has emerged as an mportant potental determnant of platforms 1 An example s the credt card market n Australa. Of course many credt cardholders n Australa also receve reward ponts per dollar spent equvalent to a negatve per-transacton charge. However, they must usually pay an addtonal annual fee for membershp of a rewards program.

6 2 allocaton of ther fees between the two groups. However, allowng for multhomng on both sdes of payments markets sgnfcantly complcates analyss of aggregate consumer and merchant behavour, and hence of platforms prcng ncentves. It has therefore been common an example s Chakravort and Roson (2006), dscussed n greater detal below to assume that mult-homng s allowed only on one sde of the market, wth partcpants on the other sde permtted to subscrbe to at most one platform (sngle-home). Fnally, three further smplfyng assumptons often adopted are: that competng platforms provde dentcal bundles of payment servces to both consumers and merchants; that they face dentcal costs n provdng these servces; and that consumers or merchants are homogeneous n the values that they place on the benefts they receve from transactng wth each platform. These assumptons consderably ease the task of understandng the mechancs of competton between platforms. They also allow consumers or merchants to be treated as dentcal n the transactonal benefts they receve not only across platforms but across ndvduals. Ths dramatcally (albet unrealstcally) smplfes modellng of the balancng task each platform faces n tryng to get both sdes on board, so as to maxmse proft. These assumptons do, however, lmt the scope for such analyss to nform our understandng of competton between, say, dfferent types of payment nstruments, such as debt versus credt cards. Aganst ths background, the central goal of ths paper s to construct a model of competton between payment systems whch relaxes as many as possble of these assumptons. We would also lke the model to be mplementable, so as to allow the use of smulaton analyss to study the prcng mplcatons of such competton. Much of the lterature to date on such competton has tended to be purely analytcal focusng, for example, on dervng the margnal condtons that wll be satsfed at a proft-maxmsng equlbrum (n terms of sutably defned elastctes of consumer and merchant demand) and how these condtons wll be affected by underlyng features of the two sdes of the market. Such analyss s both llumnatng and mportant. However, t can also be valuable at tmes to be able to study the full soluton to a model of any system. Such global solutons n the current settng comprsng each platform s ultmate prcng choces, ther profts, and the fnal take-up of each platform s servces by both consumers and merchants can help not only to draw out nterestng features of the system

7 3 beng modelled, but also llustrate how these features may respond as underlyng parameters of the system are vared. Wth these goals n mnd, the model we develop s an extenson of Chakravort and Roson (2006). Ther paper consders the case of two payment platforms competng wth each other, along wth the default payment opton of cash. A desrable feature of the Chakravort and Roson (CR) model s that each platform, whle chargng merchants on a per-transacton bass, leves a fxed fee on consumers for onng the platform. Ther model thus avods the frst (and arguably most restrctve) of the common lmtng assumptons descrbed above. It also explctly allows for: heterogenety among both consumers and merchants n the values they place on the transactonal benefts provded by each platform; and, n prncple at least, varaton between the platforms n both the payment servces they provde and the costs they ncur n dong so. 2 Fnally, an addtonal strength of the CR model s that t ncorporates the derved demand aspect of payments that many generc models of two-sded markets fal to capture. Ths s the property that payment transactons occur only as a by-product of the desre to undertake some other transacton namely the exchange of a good or servce rather than beng suppled or demanded for ther own sake. All of these features contrbute to makng Chakravort and Roson s framework a good startng pont for modellng competton between payment systems. 3 The CR model does, however, assume that whle merchants may choose to accept payments from nether, one or both platforms, consumers may at most 2 In practce, however, t should be noted that for the CR model the analyss of competton n the case of non-symmetrc platforms s very complex, and general analytcal results cannot be readly obtaned (Chakravort and Roson 2006, p 135). Ths practcal lmtaton carres over to the model we develop n ths paper. 3 The CR model does not, however, allow for busness stealng consderatons. Ths s the phenomenon analogous to the well-known prsoner s dlemma whereby each ndvdual merchant may feel compelled to accept payments from a platform, even f they would prefer (say) to be pad n cash, for fear that f they do not then some consumers who wsh to use that platform mght transfer ther busness to a compettor. In developng our extenson of the CR model we also do not attempt to allow for busness stealng consderatons. Ths s not because we regard them as unmportant, but smply because analyss of them s not a partcular goal of ths paper and omttng them smplfes the model, wthout obscurng those aspects of payments system competton whch we do wsh to nvestgate.

8 4 subscrbe to one platform. It thus ncorporates the second of the common lmtng assumptons lsted earler. The key extenson we make s to remove ths restrcton on consumers, so as to be able to study the mplcatons of fully endogenous mult-homng on both sdes of the market for the prcng strateges of competng payment platforms. Ths turns out to have substantal ramfcatons for the behavour of both consumers and merchants, and hence for platforms prcng strateges towards each group. The remander of ths paper s devoted to descrbng our extenson of the CR model henceforth referred to as our ECR model (for Extended Chakravort and Roson ) and explorng ts mplcatons, n theoretcal terms, for the behavour of consumers, merchants and platforms. Secton 2 of the paper sets out the detals of our ECR model, as well as ntroducng essental notaton. It also dscusses possble applcatons of the model, ncludng to competton between dfferent types of payment nstruments. Secton 3 then focuses on establshng geometrc frameworks for understandng the aggregate decsons of consumers to hold and merchants to accept the competng payment nstruments n the model, and how these wll be nfluenced by the prcng choces of the platforms. In Secton 4 we analyse an nterestng new potental source of non-unqueness n the behavour of consumers and merchants n our model, and explore ts possble mplcatons for the chcken and egg debate n relaton to payment systems (and two-sded markets more generally). Numercal smulaton of the model s deferred to the sequel to ths paper, where (nter ala) the results of such smulatons are used to further nvestgate the lkely effects of competton on platforms prcng strateges. 4 Conclusons are drawn n Secton 5. 4 See Gardner and Stone (2009a). After developng our ECR model we became aware that a somewhat smlar study of the mpact of allowng endogenous mult-homng on both sdes of the market had already been undertaken n Roson (2005). Indeed, that paper allows, n prncple, for addtonal features whch our ECR model does not, such as mult-part (flat and per-transacton) prcng by platforms to both sdes of the market, as well as both mult-part costs to platforms and mult-part benefts to consumers and merchants. However, t does not develop the geometrc frameworks for understandng consumers and merchants card choces developed here, nor does t appear to explctly address the non-unqueness ssues, canvassed n Secton 4 of ths paper, whch can arse n such a model. Our own smulaton analyss also suggests that there may be startng-pont dependency ssues assocated wth the teratve approach used there as descrbed n Footnote 8 of Roson (2005, p 14) to generate numercal smulaton results.

9 2. A Model of Competng Payment Systems 5 In ths secton we set out the detals of our ECR model of two competng payment systems. To fx deas, and to smplfy the exposton, we take these to be card payment networks. However, there s nothng nherently specal about cards. Hence, the analyss whch follows apples ust as well n prncple to non-card payment systems. 2.1 The General Model The model contans three types of agents: a set of C consumers, denoted Ω c, a set of M merchants, denoted Ω m, and the operators of two card payment platforms, and. The platforms offer card payment servces to consumers and merchants, n competton wth the baselne payment opton of cash. We focus prmarly on the case where the two platforms are rvals. However, for comparson purposes, we also consder the case where both platforms are operated by a monopoly provder of card payment servces. Every consumer s assumed to make precsely one transacton wth each merchant, usng ether cash or one of the platforms cards. By fxng the number of transactons at each merchant, ndependent of the prcng decsons of the platforms, ths assumpton s consstent wth the derved demand aspect of payments dscussed earler. However, as noted, t also explctly rules out busness stealng consderatons from the model (see Footnote 3). For a transacton to be made usng a partcular payment type, two condtons must be satsfed. Frst, both the consumer and merchant must have access to that nstrument; for example, for a transacton to occur on platform the consumer must hold a card from platform and the merchant must accept platform s cards. All consumers and merchants are assumed to hold/accept cash, so cash s always a payment opton. Second, the decson must be made to select that payment method n preference to all other feasble optons. Consstent wth most treatments of payment systems, ths choce at the moment of sale s assumed to fall to the consumer. Each consumer makes ths choce to maxmse the net beneft he or she wll accrue n makng that partcular payment transacton.

10 6 For smplcty, both consumers and merchants are assumed to receve zero utlty f cash s used to make a payment. By contrast, as dscussed further below, all consumers are assumed to receve non-negatve utlty from payng by card so that consumers who hold ether platform s card wll always prefer to pay by card rather than by cash, f possble. We now descrbe the ncentves facng: platforms n ther choce of prcng strateges; merchants n ther decsons whether or not to accept the cards of each platform; and consumers n ther decsons whether or not to hold the cards of each platform, and to prefer one or the other f they hold both when makng any gven payment. Frst, however, t s helpful to ntroduce some further notaton. 2.2 Notaton Let Ω c denote the subset of consumers who choose to hold the card of platform, and let Ω c, denote the further subset of these consumers who choose not to hold the card of platform. Let Ω c and Ω c, be defned correspondngly, and let Ω c, denote the subset of consumers who choose to hold the cards of both platforms and. Fnally, let Ω c 0 denote the subset of consumers who choose to hold no cards and use only cash. Clearly we then have that and Ω c = Ω c, Ω c,, Ω c = Ω c, Ω c, (1) Ω c = Ω c 0 Ω c, Ω c, Ω c,. (2) Next, among those consumers holdng both cards t wll be necessary to dstngush also between those who would prefer to use card over card, n the event that a merchant accepts the cards of both platforms, and those who would nstead prefer to use card over card. Let Ω c, ; and Ω c, ; denote these two subsets respectvely, so that we then also have, n turn: 5 Ω c, = Ω c, ; Ω c, ;. (3) 5 For smplcty, and wthout mpact on the model, we assume that any consumers who hold both cards, and who would be ndfferent between usng the two f a merchant accepted both, are ncluded n the set Ω c, ;.

11 7 Fnally, defne D c 0 to be the fracton of consumers who choose to hold no cards, and smlarly for D c, D c, D c,, D c,, D c,, D c, ; and D c, ;. 6 Correspondng to Equatons (1) to (3) above we then also have that D c = D c, + D c,, D c = D c, + D c, (4) 1 = D c 0 + D c, + D c, + D c, (5) and D c, = D c, ; + D c, ;. (6) Turnng to the merchant sde, let Ω m 0, Ω m, Ω m, Ω m,, Ω m, and Ω m, denote the analogous subsets of Ω m (so, for example, Ω m, s the subset of merchants who choose to accept the cards of platform but not of platform ). 7 Then, as for the consumer sde, we have the followng obvous relatonshps Ω m = Ω m, Ω m,, Ω m = Ω m, Ω m, (7) and Ω m = Ω m 0 Ω m, Ω m, Ω m,. (8) If we agan defne D m 0, D m, D m, D m,, D m, and D m, analogously to ther consumer counterparts, we then also have the correspondng denttes: D m = D m, + D m,, D m = D m, + D m, (9) and 1 = D m 0 + D m, + D m, + D m,. (10) For ease of reference, ths notaton and that ntroduced n Sectons 2.3 to 2.5 below are summarsed n Table 1. 6 Thus, for example, D c 0 Ω c 0 / Ω c Ω c 0 /C, where s used to denote the sze of a set. 7 Note that there s no need to defne analogues of the subsets Ω c, ; and Ω c, ;, snce the choce of payment nstrument at the moment of sale s assumed to fall to the consumer, not the merchant. Ths makes descrpton of the merchant sde somewhat smpler than that of the consumer sde.

12 8 Varable Descrpton Table 1: Lst of Model Notaton Consumer market segments (fractons) Ω c Set of all consumers Ω c 0 (D c 0) Subset (fracton) of consumers who choose not to hold any cards Ω c (D c ) Subset (fracton) of consumers who choose to hold card Ω c, (D c, ) Subset (fracton) of consumers who choose to hold card but not card Ω c, (D c, ) Subset (fracton) of consumers who choose to hold both cards and Ω c, ; (D c, ;) Subset (fracton) of consumers who choose to hold both cards and who prefer to use card over card whenever merchants accept both Merchant market segments (fractons) Ω m Set of all merchants Ω m 0 (D m 0 ) Subset (fracton) of merchants that choose not to accept any cards Ω m (D m ) Subset (fracton) of merchants that choose to accept card Ω m, (D m, ) Subset (fracton) of merchants that choose to accept card but not card Ω m, (D m, ) Subset (fracton) of merchants that choose to accept both cards and Platform fees f c f c, f c, f m Flat fee charged to consumers to subscrbe to card The flat fee f c converted to per-transacton terms for a consumer n Ω c, or Ω c, ; (that s, the quantty f c /MD m ) The flat fee f c converted to per-transacton terms for a consumer n Ω c, ; (that s, the quantty f c /MD m, ) Per-transacton fee charged to merchants by platform Platform costs c Cost ncurred by platform for each transacton processed over the platform g Flat cost to platform of sgnng up each consumer g The quantty g /MD m (representng the flat cost g converted to per-transacton terms for subscrbers n Ω c, or Ω c, ;) Other C (M) Total number of consumers (merchants) τ Maxmum per-transacton beneft receved by any consumer on platform µ Maxmum per-transacton beneft receved by any merchant on platform h c Per-transacton beneft receved by a gven consumer on platform h m Per-transacton beneft receved by a gven merchant on platform Total proft earned by platform Π Notes: For smplcty, where there s analogous notaton for both platforms only that for platform s shown. Consumer (merchant) market fractons represent the proporton of all consumers (merchants) that are members of the correspondng set.

13 9 2.3 Platforms The two platforms are assumed to be proft-maxmsng and to face per-transacton costs of c for platform and c for platform. In addton, they ncur fxed costs g and g respectvely for each consumer that they sgn up. In terms of prcng, we assume that the platforms charge flat fees, f c and f c, to each consumer, but do not levy per-transacton fees on consumers (nor do they offer per-transacton rewards). Conversely, platforms do not mpose flat, up-front fees on merchants, but do charge per-transacton fees f m and f m to merchants for the use of ther cards. Thus, for platform, each consumer that subscrbes generates drect revenue f c and cost g, whle each transacton generates revenue of f m from the relevant merchant and ncurs a processng cost of c ; and smlarly for platform. For each platform, profts wll be determned by both the number of consumers whom they manage to attract and the volume of transactons subsequently undertaken on the platform. When n competton wth one another each platform must, n makng ts prcng decsons, take nto account the expected effects of any fee ncreases on both consumers and merchants. These effects nclude causng some consumers (merchants) to abandon the platform n favour of holdng (acceptng) only cash or the card of the other platform ether as a drect result of the fee mpact, or ndrectly by reducng the number of merchants prepared to accept the card (consumers wshng to use the card). Platform s proft functon can be wrtten explctly as: Π = CD c ( f c g ) +CM(D c, + D c, ;)D m ( f m c ) +CMD c, ; D m, ( f m c ) = CD c ( f c g ) +CM [ D c D m D c, ; D m ], ( f m c ). (11) The three rght-hand-sde terms n the top equalty of Equaton (11) are, respectvely: profts from subscrptons; profts from transactons made by cardholders who ether only hold card, or hold both and prefer t over card ; and profts from transactons made by cardholders who prefer card over card but hold both.

14 10 If we then ntroduce the further notaton dscussed n greater detal below that f c, f c /MD m and g g /MD m (12) t s readly checked that Equaton (11) may alternatvely be wrtten { [( Π = CM f c, g ) + ( f m c ) ] D c D m ( f m c )D c, ; D m, }. (13) Platform s proft functon s then correspondngly gven by { [( Π = CM f c, g ) + ( f m c ) ] } D c D m ( f m c )D c, ;D m, (14) where f c, f c /MD m and g g /MD m. Sectons 3.1 and 3.2 below descrbe, n greater detal, geometrc frameworks for understandng the ncentves facng proft-maxmsng platforms n ther fee choces, n terms of the mpact of these fee choces on consumers and merchants card holdng and acceptance decsons Consumers Consumers make ther payment choces so as to maxmse ther utlty. They are assumed to receve a per-transacton beneft for payng by non-cash means, equal to h c for payments made over network and h c for payments made over network. Consumers are heterogeneous n ther benefts, whch are randomly (though not necessarly ndependently) drawn from dstrbutons over the ntervals [0,τ ] for platform and [0,τ ] for platform. 9 In Chakravort and Roson (2006), and n the sequel to ths paper (Gardner and Stone 2009a), these dstrbutons are taken to be unform, as ths represents an nterestng case and one whch sgnfcantly smplfes analyss of the model. Consumers draws of benefts for each platform are also assumed to be ndependent so that the beneft any ndvdual consumer receves from makng a payment over network s unrelated to the beneft they receve from makng a 8 The mplcatons of these ncentves for platforms prcng are then explored n Appendx B. 9 The pars of quanttes {h c,h c } thus typcally dffer from consumer to consumer but, for each consumer, are the same for every transacton.

15 11 payment over network. Whle t may help the reader to adopt these assumptons mentally n what follows, t s mportant to note that they are not necessary for the model and, other than n Secton 4, we do not requre them for the remander of ths paper. Indeed, at the end of Secton 2.6 we brefly descrbe a natural settng n whch an alternatve ont dstrbutonal assumpton for consumers per-transacton benefts mght be approprate. As noted above, a consumer makng a payment over network or faces no pertransacton fee. Snce each consumer s per-transacton beneft from usng ether platform s always non-negatve, consumers who hold cards wll therefore always prefer to pay by card rather than by cash, whenever ths s possble. Unlke n Chakravort and Roson (2006), t s assumed that each consumer can choose to hold no cards, one card or both cards; and, n the event that they sgn up to both platforms, can choose to use card n preference to card, or vce versa, where a merchant accepts both. In assessng ther expected utlty, each consumer s assumed to have a good understandng of the fracton of merchants who wll sgn up to each platform, for gven platform fees { f c, f m } and { f c, f m }. The equatons whch descrbe the utlty a consumer wth per-transacton benefts {h c,h c } wll obtan from each of ther possble card holdng/use optons are thus as follows: U0 c = 0 (15) = Mh c D m f c = M { h c D m f c /M } (16) U, c = Mh c D m f c = M { h c D m f c /M } (17) U c, U, c ; = M { h c D m, + h c D m, + h c D m, = M { h c D m + h c D m, f c U c, ; = M { h c D m, + h c D m, + h c D m, } f c f c /M f c /M } (18) } f c f c = M { h c D m + h c D m, f c /M f c /M }. (19) Here, consstent wth prevous notaton, the quanttes U0, c U, c and U, c denote the utlty the consumer would receve, respectvely, from choosng to hold nether platform s cards, the card of platform only, or the card of platform only. Smlarly, U, c ; and U, c ; denote the utlty the consumer would receve from choosng to hold both platforms cards and then choosng, respectvely, to use card over card, or vce versa, whenever a merchant accepts both.

16 12 Fnally, before turnng to the merchant sde, t s useful to compute what the effectve charge s, n per-transacton terms, for dfferent consumers who elect to hold each platform s card. Focusng wthout loss of generalty on the cards of platform, we see that consumers n the subsets Ω c, and Ω c, ; of Ω c wll each expect to make MD m transactons on ther platform cards. Hence, such consumers face an effectve per-transacton charge for these payments of ( f c /MD m ) f c, Ths provdes the ntutve nterpretaton for the quanttes f c, n Secton 2.3. and f c,. ntroduced On the other hand, a consumer n the subset Ω c, ; of Ω c wll expect to make only MD m, transactons on ther platform card, snce they wll use t only when a merchant accepts card and does not accept ther preferred card. Hence, these consumers face a hgher effectve per-transacton charge for payments on ther platform cards than consumers n Ω c, and Ω c, ;. Ths effectve charge s f c, f c /MD m, f c,. (20) Smlarly, consumers n the subset Ω c, ; face a correspondng effectve pertransacton cost for payments on ther platform card of f c,, where f c, f c /MD m, f c,. As we shall see, the quanttes f c,, f c,, f c, and f c, wll play an mportant role n the geometrc framework descrbed n Secton 3.2 for understandng consumers card holdng decsons. 2.5 Merchants Each merchant can choose to sgn up to both networks, one network, or nether network, based on the net beneft t wll receve from dong so. Lke consumers, merchants are assumed to receve a per-transacton beneft for acceptng noncash payments, equal to h m for those receved on network and h m for those receved on network. Merchants are also heterogeneous n ther benefts, whch are randomly (but not necessarly ndependently) drawn from sutable dstrbutons over the ntervals [0, µ ] and [0, µ ] for platforms and. 10 If a merchant accepts a payment over network t s charged a per-transacton fee of f m, and smlarly for 10 As on the consumer sde, to fx deas t may help to focus on the case of unform and ndependent dstrbutons throughout the remander of ths paper. Nevertheless, t should agan be noted that there s no reason n prncple why non-unform and/or correlated dstrbutons could not be used here.

17 13 platform. However, merchants do not face any fxed costs n choosng to accept ether platform s cards. In assessng the beneft t wll receve from sgnng up to one or more platforms, each merchant s once agan assumed to have a good knowledge of both: the fracton of consumers who wll sgn up to each platform, for gven platform fees { f c, f m } and { f c, f m }; and the fractons of those choosng to hold both cards who wll then prefer to use a partcular card at the moment of sale. Gven ths, the equatons whch descrbe the net beneft whch a merchant, wth per-transacton benefts {h m,h m }, wll obtan from each of ts possble card acceptance optons are as follows: U0 m = 0 (21) U, m = C(h m f m )D c (22) U, m = C(h m f m )D c (23) U, m = C(h m f m ) { D c, + D c, ;} +C(h m f m ) { D c, + D c, ; }. (24) Once agan, the quanttes U m 0, U m,, U m, and U m, denote the net beneft the merchant would receve, respectvely, from choosng to accept nether platform s cards, the card of platform only, the card of platform only, or those of both platforms. Note that these equatons also rest upon the feature of the model, dscussed earler, that consumers wll always prefer to pay by one or other card rather than by cash, f possble. 2.6 Possble Applcatons of the Model Havng specfed our ECR model, the next step s to derve descrptons of the aggregate card holdng and acceptance behavour of consumers and merchants n t. Before dong so, however, t s worth brefly addressng the queston: to what real-world stuatons mght the model potentally apply? Snce platforms n the model nteract drectly wth partcpants on both sdes of the market, the obvous applcaton s to competton between rval three-party card schemes, such as Amercan Express and Dners Club. The absence of separate ssuers and acqurers n the model makes t approprate to such a settng. Despte the absence of dstnct ssuers and acqurers (and consequent lack of explct nterchange fees), the model could arguably stll be used to shed lght

18 14 on some features of competton between four-party credt card schemes (such as MasterCard and Vsa). Whle clearly less well adapted to ths stuaton, the model nevertheless accurately captures many features of the dynamcs of the consumer and merchant sdes of the market whch would arse n ths settng. It mght also offer some nsghts nto competng four-party platforms lkely prcng behavour, wth the tltng of platforms prce structures n favour of consumers or merchants potentally ndcatve of ther lkely nterchange fee choces n ths settng. That sad, cauton would need to be exercsed before usng our ECR model to draw any frm conclusons about the case of competton between four-party schemes. For example, n the event that the ssung sde were domnated by a small number of large nsttutons, the model s applcablty to ths case would be lmted, gven ts lack of a proper treatment of olgopsony effects (n relaton to platforms prcng behavour towards such ssuers). 11 Our ECR model potentally also allows us to draw some nferences about the case of competton between dfferent types of payment nstrument, such as debt versus credt cards (or cheques versus ether of these). Chakravort and Roson (2006) emphassed the scope for ther model to be used to study such competton stressng, n ths regard, ts capacty to handle stuatons where platforms provde dfferent maxmum per-transacton benefts to consumers and/or merchants (so τ τ and/or µ µ ). Our ECR model also offers scope for such dfferentaton between platforms based on the maxmum per-transacton benefts they provde. However, ths s not somethng whch we pursue n the smulaton analyss n the sequel to ths paper. Rather, there s a more fundamental reason why we beleve that our ECR model, lke Chakravort and Roson s earler one, plausbly covers the case of competton between dfferent types of nstrument namely, that t allows for heterogeneous benefts, to both consumers and merchants, across the two competng platforms. 11 Another mportant dstncton n the case of competng four-party credt card schemes would be that annual credt card fees are pad by consumers to ssuers, rather than to the schemes. We mght expect ths dstncton to have mplcatons for the transferablty of any model results regardng how the use of flat fees to consumers, rather than per-transacton fees, would affect platforms prcng. The mportance of ths dstncton n practce, however, would depend on the extent to whch schemes mght be able to extract some or all of these flat consumer fees from ssuers say through the use of scheme fees to ssuers based on subscrber numbers.

19 15 Even where consumers (merchants ) benefts from transactng on each platform are unformly dstrbuted, as long as they are not perfectly correlated then some consumers (merchants) wll value usng platform more hghly than platform, and vce versa. Ths s consstent wth the fact that n the real world dfferent agents wll, for example, place dfferent ntrnsc values on usng debt and credt cards. Some consumers, for nstance, may be partcularly averse to takng on debt, and so apprecate the budgetng dscplne provded by a debt card. Others, by contrast, may value the flexblty afforded by a credt card relatve to a debt card n managng ntra-month cash flow constrants. Fnally, by allowng for non-unform and/or correlated dstrbutons of consumer and merchant per-transacton benefts across platforms, our ECR model potentally even apples to the case of competton between a premum credt or charge card brand and a non-premum one. To the extent that some consumers mght value the prestge assocated wth holdng a certan exclusve payment card, ths could generate an ncentve for one platform to target ths market segment hopng to charge hgher fees to cardholders and, f possble, to merchants compared to a rval platform focused nstead on ncreasng proft by maxmsng ts subscrber numbers and transacton volumes. The attracton of such a targeted busness approach would be greater, the stronger the concentraton of consumers placng an asymmetrcally hgh prestge value on transactng wth a premum rather than run-of-the-mll payment nstrument. Ths s somethng whch our ECR model could, n prncple, ncorporate va usng a sutable non-unform dstrbuton for consumers per-transacton benefts Understandng Merchants and Consumers Card Choces Havng descrbed the general model, the next step s to try to understand what fractons of merchants and consumers wll choose to sgn up to none, one or both platforms, and what factors wll nfluence these proportons. In Secton 3.1 we frst descrbe a geometrc framework for thnkng about merchants card acceptance decsons, and for understandng how these decsons wll be affected by platforms prcng strateges. The more complex geometrc framework for 12 In ths way our ECR model mght represent an approprate vehcle for nvestgatng both the presence of premum credt cards n the marketplace alongsde more prevalent ordnary credt cards, and the market dynamcs of competton between the two.

20 16 understandng the analogous decsons on the consumer sde s then set out n Secton Detals of the dervatons of the geometrc frameworks descrbed n Sectons 3.1 and 3.2 are provded n Appendx A. Fnally, n Secton 3.3 we show how these frameworks smplfy to the versons derved and analysed n Chakravort and Roson (2006), for the specal case consdered there. 3.1 Merchants Card Acceptance Decsons Each merchant has some draw of per-transacton benefts {h m,h m } for acceptng a payment on platform or. It s possble therefore to represent each merchant as a pont n h m,h m -space, wth the populaton of all merchants, Ω m, then beng dstrbuted across the rectangle bounded by the ponts (0,0), (µ,0), (0, µ ) and (µ, µ ). 14 Recall here that µ and µ denote the maxmum per-transacton benefts whch any merchant wll receve from processng a payment on network or respectvely. Usng Equatons (21) to (24), t s then possble to subdvde ths rectangle nto four mutually exclusve regons, correspondng to the subsets Ω m 0, Ω m,, Ω m, and Ω m, (see Appendx A for detals). Dong so yelds that Ω m may be convenently represented geometrcally as shown n Fgure The fact that Lne 1 of Fgure 1 may be non-horzontal reflects that there may be some merchants who wll choose not to accept platform s cards, despte the per-transacton beneft they receve from takng a payment on platform, h m, exceedng the per-transacton charge they would face for dong so. Ths s 13 Havng establshed frameworks for understandng agents behavour n aggregate on both the merchant and consumer sdes, these frameworks are then used n Appendx B to dscuss, from a theoretcal perspectve, the ncentves facng platforms n ther prcng choces. 14 In the event that merchants draws of per-transacton benefts are from unform and ndependent dstrbutons then the populaton of merchants wll (on average) be evenly dstrbuted across ths rectangle, wth concentraton M/µ µ. 15 Wthout loss of generalty, Fgure 1 has been drawn wth f m < µ and f m < µ. Ths reflects that f ether platform were to set ts per-transacton fees above these levels t would attract no merchants to accept ts cards and so would make no proft.

21 17 the phenomenon known as steerng dscussed, for example, n Rochet and Trole (2003). 16 For these merchants, although the net beneft they would receve from processng a payment on platform, h m f m, s postve, the net beneft they would receve from processng a payment on platform, h m f m, s greater agan. If the dfference between these net benefts s suffcently large then, provded enough consumers who hold card also hold card, t may be worthwhle not to accept platform s cards. Ths would see some payments shft to cash (namely, those by consumers n Ω c, ), at some loss to the merchant. However, t would also steer those consumers holdng both cards to pay usng the cards of the merchant s preferred platform, platform generatng a gan suffcent, for some merchants, to ustfy declnng the cards of platform. 16 The fact that Lne 2 may also be non-vertcal reflects correspondng steerng of consumers by some merchants from platform to platform. Note that here we use the term steerng n the sense n whch t s generally used n the theoretcal lterature on payment systems; that s, the refusal by a merchant to accept a platform s card, so as to force those consumers who mult-home to use a dfferent card preferred by the merchant. Ths s n contrast to the colloqual sense n whch the term s sometmes used, of a merchant tryng to nfluence consumers choces through mlder means such as sgns or verbal suggestons about preferred payment optons.

22 18 Fgure 1: A Geometrc Representaton of the Populaton of Merchants µ h m f m Ω m, Lne 2 Ω m, Lne 1 Ω m 0 Ω m, f m h m µ Notes: The fgure shows a representaton of the populaton of all merchants n h m,h m -space, subdvded nto the four subsets Ω m 0, Ω m,, Ω m, and Ω m,. Lne 1 passes through the pont ( f m, f m ) and has slope D c, ; /(D c, +D c, ; ). Lne 2 also passes through the pont ( f m, f m ) and has slope (D c, + D c, ;)/D c, ;. 3.2 Consumers Card Choces Turnng to the consumer sde, each consumer also has some draw of pertransacton benefts {h c,h c } for makng a payment on platform or. Hence, we may also represent the populaton of all consumers, Ω c, as beng dstrbuted across

23 19 a rectangle ths tme n h c,h c -space, bounded by the ponts (0,0), (τ,0), (0,τ ) and (τ,τ ). 17 Usng Equatons (15) to (19), ths rectangle can be subdvded nto mutually exclusve regons now correspondng to the subsets Ω c 0, Ω c,, Ω c,, Ω c, ; and Ω c, ; (see Appendx A for the detals). Dong so yelds that the populaton of all consumers may be represented geometrcally n h c,h c -space as shown n Fgures 2 and 3. The added twst here, unlke on the merchant sde, s that the breakdown of Ω c turns out to dffer dependng on whether or not f c, f c,. Hence, our representaton of Ω c conssts of two separate dagrams, wth Fgure 2 depctng the stuaton f f c, f c, and Fgure 3 the stuaton f the reverse nequalty holds. Note also that, n these fgures, the further notaton f c, and f c, s used to denote the quanttes f c, f c, + ( D m, D m ) f c, and f c, f c, + ( D m, D m ) f c,. (25) Regardng the ntuton for the subdvsons shown n Fgures 2 and 3, the boundares of the regon Ω c 0 are straghtforward, gven the defntons of the quanttes f c, and f c, (see Secton 2.3). For consumers n ths regon, the fxed cost of holdng ether card exceeds the total beneft they would accrue from that card, even f they used t at every merchant that would accept t. Then Lne 1, whch passes through the pont ( f c,, f c, ) and has slope D m /D m, smply represents the boundary between those consumers not n Ω c 0 who would opt to hold card, f they could hold only one platform s card, and those who would opt to hold card. More nterestng are the structures of the Regons Ω c, ; and Ω c, ;, representng consumers who choose to hold both platforms cards. Focusng wthout loss of generalty on the Regon Ω c, ; n Fgure 2, ths conssts of those consumers for 17 Recall here that τ and τ represent the maxmum per-transacton benefts whch any consumer wll receve from makng a payment on network or respectvely, as set out n Secton 2.4. Also, as on the merchant sde, n the event that consumers draws of per-transacton benefts, h c and h c, are from unform and ndependent dstrbutons then the populaton of all consumers wll (on average) be evenly dstrbuted across ths rectangle, wth concentraton C/τ τ.

24 20 Fgure 2: A Geometrc Representaton of the Populaton of Consumers The case of f c, f c, τ f c, f c, f c, c h Ω c, Lne 1 Lne 2 Ω c, ; Ω c, ; Lne 3 Ω c 0 Ω c, f c, f c, f c, h c τ Notes: The fgure shows a representaton of the populaton of all consumers n h c,h c -space, for the case f c, f c,, subdvded nto the fve subsets Ω c 0, Ω c,, Ω c,, Ω c, ; and Ω c, ;. Lne 1 passes through the pont ( f c,, f c, ) and has slope D m /D m. Lne 2 passes through the pont ( f c,, f c, ) and has slope D m /D m,. Lne 3 passes through the pont ( f c,, f c, ) and has slope 1.

25 21 Fgure 3: A Geometrc Representaton of the Populaton of Consumers The case of f c, f c, τ f c, f c, f c, c h Ω c, Lne 1 Ω c, ; Ω c, ; Lne 2 Lne 3 Ω c 0 Ω c, f c, f c, f c, h c τ Notes: The fgure shows a representaton of the populaton of all consumers n h c,h c -space, for the case f c, f c,, subdvded nto the fve subsets Ω c 0, Ω c,, Ω c,, Ω c, ; and Ω c, ;. Lne 1 passes through the pont ( f c,, f c, ) and has slope D m /D m. Lne 2 passes through the pont ( f c,, f c, ) and has slope D m, /D m. Lne 3 passes through the pont ( f c,, f c, ) and has slope 1.

26 22 whom the followng three nequaltes all hold: h c h c (26) h c f c, (27) and ( h c D m D m, ) (h c f c, ). (28) The ntuton underlyng Inequalty (26) s obvous. Snce consumers face no pertransacton fees, ths smply represents the dvson between those consumers who, f they hold both cards, would prefer to use card, and those who would prefer to use card. The nterpretaton of Inequalty (27) s also straghtforward. Recall that, by defnton, f c, s the (relatvely hgh) effectve per-transacton prce that a consumer who holds both cards, but who would prefer to use card over card, faces for card transactons. Hence, Inequalty (27) s smply capturng that a consumer who would prefer to use card over card wll only wsh to hold both cards, rather than ust card, f the per-transacton beneft they would receve from usng card exceeds ths prce. The nterpretaton of Inequalty (28), however, s more nterestng. Ths condton captures that, even for a consumer who would prefer to use card over card, t may be that he or she would stll choose to sgn up to platform ahead of platform (f, say, far more merchants wll accept platform s cards than platform s). Hence, to wsh to hold both platforms cards, t s also necessary that the addtonal utlty for such a consumer from sgnng up to platform, f already holdng platform s card, should exceed the cost of dong so; otherwse such a consumer would be better off holdng only the card of platform. Ths translates drectly to the requrement that MD m, h c + MD m, (h c h c ) f c (29) where: the frst term on the left-hand sde equates to the extra utlty a consumer would obtan from beng able to pay by card, rather than by cash, at those merchants whch accept card but not card ; and the second term corresponds to the ncremental addtonal utlty that a consumer wth h c h c would gan from

27 23 beng able to swtch hs or her card payments from platform to platform, at those merchants whch accept both cards. Ths condton s then easly confrmed to be equvalent to Inequalty (28) above. 18 A further complcaton on the consumer sde Fnally, the need to allow for two possble cases rather than one s, unfortunately, not the only added complcaton wth the geometrc framework descrbed above for the consumer sde, relatve to the merchant sde. A further one relates to the magntudes of τ and τ. As on the merchant sde, we may assume wthout loss of generalty that f c, τ and f c, τ. Ths reflects that, f ether platform were to set ts fees so that one or other of these nequaltes faled, then that platform would attract no consumers, and hence would make no proft. However, although for smplcty we have also drawn Fgures 2 and 3 wth τ max{ f c,, f c, } and τ max{ f c,, f c, }, there s nothng whch ensures that proft-maxmsng platforms wll necessarly set ther fees so that ths wll be so. Hence, for example, Fgure 2 ought really to allow for the possblty that f c, τ < f c,, f c, τ < f c, or f c, τ ; and also ndependently for the possblty that f c, τ < f c,, f c, τ < f c, or f c, τ (and smlarly for Fgure 3). Fgures 2 and 3 really, therefore, each break nto nne sub-cases, of whch only those correspondng to the stuatons where τ and τ each exceed max{ f c,, f c, } have been shown. These addtonal possbltes do not fundamentally alter the structure of Fgures 2 and 3, snce they essentally ust alter where the lnes h c = τ and h c = τ st n relaton to the other parts of each dagram. However, they do sgnfcantly complcate the task of wrtng down equatons for the fractons D c 0, D c,, D c,, 18 An alternatve dervaton of Inequalty (28), workng drectly from the utlty formulae gven by Equatons (15) to (19), s provded n Appendx A.

28 24 D c, ; and D c, ; even n the specal case where consumers per-transacton benefts are drawn from unform and ndependent dstrbutons The Case of the Chakravort and Roson Model As noted n Secton 1, our model s a generalsaton of that ntroduced by Chakravort and Roson (2006). In ther model, the restrcton s mposed that consumers may only subscrbe to at most one card platform. Wth ths restrcton, the geometrc frameworks ust derved turn out to smplfy dramatcally. Consderng frst the merchant sde, n the event that D c, ; = D c, ; = 0 t s easly checked that Lnes 1 and 2 n Fgure 1 become (respectvely) horzontal and vertcal. Hence, Fgure 1 smplfes to the stuaton where a merchant wll choose to accept the cards of platform f and only f h m accept those of platform f and only f h m f m. Ths s consstent wth the fact that, when consumers hold the card of at most one platform, merchants have no scope to steer consumers n ther choce of payment card at the moment of sale, n the manner dscussed n Secton 3.1. f m, and (ndependently) wll Smlarly, on the consumer sde, when consumers are prohbted by fat from holdng the cards of more than one platform then the subsets Ω c, ; and Ω c, ; must vansh. Hence, Fgures 2 and 3 reduce to the sngle, far smpler representaton gven by Fgure 4 below. 20 Note that ths dramatc smplfcaton already hnts at how far-reachng the consequences can be, for the modellng of competton between payment systems, of a no mult-homng assumpton on ether sde of the market. Ths s a pont to whch we return n greater detal n the sequel to ths paper (see Secton 2.2 and Appendx B of Gardner and Stone 2009a). 19 Of course, as on the merchant sde, for the general stuaton of non-unform and/or correlated dstrbutons these equatons wll be even more complex, wth each possble case nvolvng the double ntegral over the relevant area of an approprate (non-constant) densty functon. For the specal case where consumers and merchants per-transacton benefts are drawn from unform and ndependent dstrbutons, detals of the equatons for the quanttes D m 0,...,D m, and D c 0,...,D c, ; are provded n a separate techncal annex: see Gardner and Stone (2009b), avalable on request. 20 Fgure 4 s the exact analogue of Fgure 1 n Chakravort and Roson (2006), except that everythng n Fgure 4 s shown n per-transacton terms. By contrast, n Fgure 1 of Chakravort and Roson consumers are represented n terms of ther aggregate net potental benefts (summed across all ther transactons) from usng the cards of platform or platform.

29 25 Fgure 4: A Geometrc Representaton of the Populaton of Consumers The specal case of the Chakravort and Roson model c h τ f c, Ω c, Lne 1 Ω c, Ω c 0 f c, h c τ Notes: The fgure shows a representaton of the populaton of all consumers n h c,h c -space, subdvded nto the three subsets Ω c 0, Ω c, and Ω c,, for the specal case where consumers are prohbted from holdng the cards of more than one platform. Lne 1 passes through the pont ( f c,, f c, ) and has slope D m /D m. Fnally, one further observaton s n order regardng our ECR model and the CR model. Ths s that both may actually be vewed as representng specal cases of a stll more general model of payment system competton, obtaned by ncorporatng an addtonal parameter, κ, nto our ECR model. Ths parameter represents the dsutlty to a consumer from holdng more than one card say, due to the clutterng of ther wallet, or the hassle of havng to check two perodc

30 26 transacton statements rather than one. Formally, ts ncorporaton s accomplshed smply by subtractng κ from the rght-hand sdes of both Equatons (18) and (19) n Secton 2.4 (the effects of whch would, of course, then flow through to alter the frameworks set out n Secton 3.2). For ths more general model, our ECR model would correspond to the specal case κ = 0, whle the CR model would correspond to any κ value greater than some threshold, κ, suffcent to deter even the most enthusastc of consumers from holdng more than one platform s card. Whle we do not pursue ths dea further here, we do take t up n a dfferent but closely related context n the sequel to ths paper (see Secton 5 of Gardner and Stone 2009a). 4. Potental Non-unqueness of Market Equlbra Sectons 3.1 and 3.2 appear to provde complete descrptons of the behavour of merchants and consumers n our ECR model, for gven platform fees. In prncple, ths should allow us to analyse from a theoretcal perspectve the factors nfluencng platforms n ther prcng strateges. 21 However, before such an analyss can be undertaken, t turns out that there s a sgnfcant ssue remanng to be clarfed about the geometrc frameworks ust descrbed. Ths ssue concerns whether or not, for gven fee choces by platforms and, the resultng merchant and consumer market outcomes are necessarly unquely determned. Interestngly, the short answer to ths at least for some fee choces s no! Moreover, non-unqueness can arse even for the case where merchant and consumer per-transacton benefts on each platform are unformly and ndependently dstrbuted, and where the platforms are dentcal n ther fee choces and the maxmum per-transacton benefts they provde to both merchants and consumers. For ths symmetrc case t s possble to establsh the exstence of such nonunqueness, as well as dentfy ts extent and the condtons under whch t wll arse, purely analytcally. 22 However, rather than go nto the detals here we content ourselves wth provdng a concrete llustraton. Ths s gven by Fgure 5, 21 One such analyss, focusng on the ncentves facng a platform contemplatng an ncrease n ts flat fee to consumers, s presented n Appendx B. 22 Detals are provded n a separate techncal annex: see Gardner and Stone (2009b).