A Puzzle of Card Payment Pricing: Why Are Merchants Still Accepting Card Payments? Fumiko Hayashi 1. December 28, 2004

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1 Puzzle o Card Payen Priing: Why re Merhans Sill eing Card Payens? Fuiko Hayashi eeber 8, 4 Payens Syse Researh, Federal Reserve Bank o Kansas Ciy Working Paer WP4- Preliinary: Please do no ie or quoe wihou auhor s erission. bsra This aer resens odels ha exlain why erhans ae ayen ards even when he ees hey ae exeed he ransaional beneis hey reeive ro a ard ransaion. Suh erhan behaviors an be exlained by oeiion aong erhans and/or he eeiveness o he erhan s ard aeane in shiing ardholders deand or goods uward. The revalen assuion used in ayen ard lieraure erhans ae ards only when heir ransaional beneis are higher han he ees hey ay holds only or a onooly erhan who aes an inelasi onsuer deand. ard nework ha wans all erhans in a given indusry o ae ards ses a lower erhan ee iniially and hen gradually inreases i o he highes ossible level, whih ay be higher han he su o he erhan s ransaional benei and he erhan s iniial argin wihou ards. Suh erhan ees oenially reae inequaliy beween ardholders and non-ardholders. Fuiko Hayashi is a senior eonois in he Payens Syse Researh earen, Federal Reserve Bank o Kansas Ciy, 95 Grand Boulevard, Kansas Ciy, MO, 6498, e-ail:uiko-hayashi@k.rb.org, Phone: The views exressed in his arile are hose o he auhor and do no neessarily rele hose o he Federal Reserve Bank o Kansas Ciy or he Federal Reserve Syse.

2 . Inroduion Credi and debi ard ayens are exeriening raid growh. This raid growh in ard ayens is araing onroversy and anirus sruiny in any ounries. Reenly, he Euroean Coission and he Reserve Bank o usralia have issued heir deisions on riing o ard ayens. Regulaory auhoriies in oher ounries, suh as Mexio and he Unied Kingdo, are evaluaing oenial uure regulaions. he ener o hose oliyakers aenion are inerhange ees, whih are aid by he bank alled he aquirer ha roesses he ard ransaion or he erhan o he bank alled he issuer ha has issued he ayen ard o he onsuer. Inerhange ees are yially se by he ard nework or by is eber inanial insiuions olleively and in any insanes hey are onsidered by regulaors o be oo high. Char shows he redi ard inerhange raes in usralia and he Euroean Union EU, beore and aer he reduions o inerhange raes were ored. In hese regions, he regulaor s involveens on riing o ard ayens lowered he redi ard inerhange raes signiianly. The har also inludes he UK redi ard inerhange raes. In he Unied Kingdo, he anirus auhoriy has no deided wheher i will regulae he redi ard inerhange raes. Bu i i does, hen soe indusry observers aniiae he raes will be u o.7 eren or below. 4 Curren U.S. redi ard inerhange raes are also inluded in he har. s seen in he har, he U.S. inerhange raes are even higher han he raes in he hree regions beore he inerhange raes were lowered. While he Reserve Bank o usralia has regulaed he redi ard inerhange raes, he Euroean Coission has issued is deision o exe Visa s inerhange ees under Euroean oeiion laws. Visa has volunarily redued is inerhange raes or ross-boarder ransaions wihin erain Euroean Union ounries. ording o The Herald, Noveber, 4, a sokesan o he UK anirus auhoriy said we would hoe o issue a deision by he suer o nex year. 4 Indusry observers exe he inerhange raes in he Unied Kingdo o be beween.5 o.7 eren. See, or exale, The Ties, May 7, 4 and Noveber, 4, and Finanial Ties, Noveber, 4.

3 Char : Credi Card Inerhange Raes in Seleed Counries.6%.4%.%.%.8%.6% Beore er Curren.4%.%.% usralia EU UK U.S. Soures: Reserve Bank o usralia, Visa Euroe, MaserCard Inernaional, and erian Banker Noes: Beore = beore he rae was ored o be lowered; er = aer he rae was lowered; Curren = as o Noveber, 4. In usralia, he regulaion is eeive or boh Visa and MaserCard. The beore and aer raes are he average o Visa and MaserCard raes. In he EU, he Euroean Coission ade is deision on he Visa rae or ross-boarder ransaions only. The beore rae is no ublily available, bu he rae was esiaed a abou eren aording o he reor Credi Card Servies by he UK Monoolies and Mergers Coission and ohers. In he Unied Kingdo, he anirus auhoriy has no ade a inal deision on he redi ard inerhange raes as o Noveber 4. Thereore, he aer rae is an exeaion by indusry observers. They redi he regulaed inerhange rae will be beween.5 o.7 eren. See oonoe 4. The U.S. rae is he average o Visa and MaserCard raes. Char : ebi Card Inerhange Fees or a $5 Transaion in Seleed Counries $.8 $.6 $.4 $. Visa/MC oline Visa/MC online oesi online $. -$. usralia EU UK U.S. Soures: Reserve Bank o usralia, Visa Euroe, MaserCard Inernaional, and TM & ebi News Noes: In usralia, here are no Visa/MaserCard online debi rodus. The doesi debi nework s inerhange ees are aid by issuers o aquirers. For he EU, only ross-boarder debi rodus are lised. In he Unied Kingdo, he doesi debi nework and MaserCard agreed in ha he doesi rodus will be igraed ino Maesro, he MaserCard online debi rodu by 7. The raes shown were he 998 raes. In he Unied Saes, here exis ore han en doesi online debi neworks. The rae shown as doesi debi nework is he weighed average o he o hree doesi online debi inerhange ees. To onver he urrenies, he average exhange raes or he irs hree quarers o 4 are used.

4 Char shows he debi ard inerhange ees in seleed ounries. Inerhange ees or oline debi are higher han hose or online debi. 5 In usralia, he online debi inerhange ees go in he oosie direion aid by he issuer o he aquirer. lhough he har does no inlude he, any ounries have doesi debi shees wih zero inerhange ees. 6 Siilar o redi ard inerhange ees, debi ard inerhange ees in he Unied Saes are aong he highes in he world. In he Unied Saes, inerhange ees or boh redi and debi ard ransaions are aong he highes in he world. Moreover, hey have been inreasing very raidly or he as several years. Char shows he various inerhange ees or a $5 ransaion a a reail sore in he Unied Saes. Inerhange ees or redi and online debi ransaions have been inreasing. lhough he oline debi inerhange raes were redued aer he seleen o a lawsui by a grou o erhans agains he wo oline debi neworks in ugus, hey have inreased sine 4. Char : Inerhange Fees or a $5 Transaion a Non-Suerarke: $. $.8 $.6 $.4 $. MaserCard redi Visa redi MaserCard oline Visa oline Sar online NYCE online Pulse online Inerlink online $ Year Soures: erian Banker and TM & ebi News 5 See Hayashi, Sullivan, and Weiner or he dierene beween online and oline debi. 6 Those inlude Belgiu, Canada, enark, Gerany, Neherlands, and Swizerland. 4

5 Sine inerhange ees are a oonen o erhan ees harged by aquirers, inreases in inerhange ees resul in inreases in erhan ees. More and ore erhans have exressed heir onerns wih he ees and soe even argue ha he urren ee level exeeds he beneis hey reeive in aeing ards. However, ew erhans have soed aeing ard ayens. Two ineresing quesions arise ro he exeriene o he U.S. ayens indusry. Firs, why have inerhange ees been inreasing desie he a ha he ayens indusry exeriened ehnologial advanes and inensiied oeiion? variey o soures reored ha ehnologial advanes have redued he oss o roessing ard ransaions. 7 Sine one o he raionales or he inerhange ee is o over he issuer s oss or roviding ard servies, he reduion o soe o heir oss ay redue he ees. Inensiied oeiion has been observed aong ard neworks, issuers, aquirers, and roessors, alhough soe o he have greaer arke ower han he ohers. 8 Tradiional eonoi heories sugges ha ehnologial advanes and oeiion will redue ries. Sine he ayen indusry is haraerized as a wo-sided arke, soe o he radiional heories ha analyze one-sided arkes are no aliable. 9 Neverheless, revious lieraure on ayen ard neworks redis ha nework oeiion will redue inerhange ees and/or erhan ees. 7 See, or exale, Evans and Shalensee, Paying wih Plasi, he MIT Press, 999,., TM & ebi News, February, 4, and ril, 4, and Murky Fuure or Inerhange Euroean Card Review, Seeber/Oober 4. 8 In he ruling in he earen o Jusie s anirus ase agains Visa and MaserCard, he our ound ha hese wo ard neworks have arke ower in ayen ard arkes. Unied Saes Cour o eals or he Seond Cirui, Unied Saes o eria v. Visa US, Visa Inernaional Cor, and MaserCard, ugus Ter,. 9 See, or exale, Evans or he dierene beween one-sided and wo-sided arkes. Sudies ha analyze oeiion aong ard neworks inlude Maneni and Soa, Bol, Guhrie and Wrigh, and Rohe and Tirole. 5

6 The seond quesion is why erhans are aeing ard ayens even when erhan beneis in aeing ards are no higher han he ees hey ae. Merhans exlain ha he oeiive ressure does no allow he o reje ard ayens and i hey do so, hey lose sales. Card neworks, on he oher hand, laied ha he beneis erhans reeive oday are greaer han hose in he as sine ard neworks rovide useul usoer inoraion ha erhans an use o inrease heir rois. Thereore, he neworks reason, erhans reeive ore beneis han ees even hough he ees oday are higher han in he as, and i no hey an dro heir ard aeane. The neworks also laied ha beause o nework exernaliies, ayen ards generae ore beneis sine he nuber o erhans ha ae ayen ards has inreased. The lieraure on ayen ards has been growing, bu only a ew sudies have analyzed erhan ard aeane. Mos sudies sily assue eiher ha erhans ae ards when heir ransaional beneis ro ards exeed he erhan ees or ha erhans ae ards regardless o he level o erhan ees. Thereore, one anno use hese odels o answer he seond quesion above. Soe exeions are Rohe and Tirole, Chakravori and To, and Wrigh b. Rohe and Tirole and Wrigh b have ound ha i erhans oee agains eah oher, hey ae ards as long as he erhan ees do no exeed he su o he erhan s ransaional beneis and he ardholder s average ne ransaional beneis ro ards. Chakravori and To ous on he redi ard s revolving union and exlain ha oeing erhans ae redi ards beause i allows he o ake sales o illiquid usoers oday raher han o wai or unerain sales oorrow. Their Models wih he assuion ha erhans only ae ards when heir ransaional beneis ro ards exeed he ees inlude Baxer 98, Shalensee, Bol and Tiean, and Wrigh a, 4. In onras, Frankel 998, Gans and King, Kaz, and Shwarz and Vinen assue ha erhans ae ards regardless o he level o erhan ees. 6

7 odel also onludes ha alhough eah erhan hooses o ae ards, sine all erhans ae ards in equilibriu eah erhan s roi is lower. Chakravori and To exlain he erhans redi ard aeane bu do no exlain heir debi ard aeane. The odels by Rohe and Tirole and Wrigh b exlain boh redi and debi ard aeane by erhans even when erhan ees are higher han heir ransaional beneis. In Wrigh b, erhans an inrease heir roi argins by aeing ards, whih onradis wha soe o he erhans are saying. In onras, in Rohe and Tirole, erhans anno inrease heir argins by aeing ards. Their odel, however, exes ha he axiu erhan ee dereases as oeiion aong issuers inensiies, whih sees onrary o he exeriene in he Unied Saes. Undersanding erhan ard aeane behavior ay be a key in answering he irs quesion. s enioned, no revious lieraure is suessul in exlaining ha nework oeiion raises he inerhange ees. This is so, arly beause os o he sudies assue ha erhans ae ards only when he erhan ees do no exeed heir ransaional beneis, and arly beause os o he do no odel issuers behavior when he issuers deide whih neworks hey will join. This aer analyzes he erhan inenives o ae ayen ards boh redi and debi ards. Four dieren arkes ha are haraerized by erhan oeiiveness onooly or a Hoelling odel o oeiion and by rie elasiiy o he arke aggregae onsuer deand inelasi or elasi are onsidered in urn. The resuls sugges ha only onooly erhans who are aing an inelasi onsuer deand do no ae ards i he While he o en redi ard issuers arke share in ers o anaged reeivable is high and has inreased ro 79.4 eren in o 85.7 eren in, he o en debi ard issuers arke share is low and seady. ording o EFT aa Book various years, boh he online and oline debi issuers arke shares eiher in ers o nuber o ards issued or in ers o ransaion volue have been around o 5 eren sine. 7

8 ees exeed he erhan s ransaional beneis. In he oher hree arkes, erhans ae ards even when he ees exeed heir ransaional beneis. s revious sudies ound, oeing erhans ae ards or sraegi reasons. Merhans iniially hoe ha heir ard aeane an lure usoers away ro heir rivals, bu laer on hey ae ards o kee heir urren usoers. Even onooly erhans ae ards when heir ransaional beneis are lower han he ees hey ay i hey ae an elasi onsuer deand. They do so no beause hey have a sraegi reason bu beause ard aeane shis heir ardholder usoers deand uward. In he odel, here is one ard nework ha deerines boh he erhan ees and ardholder ee. The nework is assued o be onservaive in he sense ha i ses a erhan ee so ha all o he erhans in a given indusry ae ards. I he erhans rie seing is oleely lexible, he nework will se he highes ossible erhan ees ro he irs eriod in he arkes wih a onooly erhan, while he nework will se lower erhan ees in he irs eriod and will raise he o he highes ossible erhan ees in he seond eriod in he arkes o he Hoelling oeiion. In realiy, however, erhans are unwilling o raise heir ries by a large aoun over a shor eriod. 4 In suh ases, he nework will se a erhan ee lower in earlier eriods o ard aeane in a given indusry, and i will gradually inrease he ee as ore eriods have assed aer he indusry irs ados ard ayens. The aer also analyzes he welare o ardholders, non-ardholders, and erhans searaely beause oliyakers ay wan o evaluae he ard riing ro boh an eiieny and equiy oin o view. To rele realy, he odel assues ha erhans se he sae rie In os neworks, erhan ees are deerined by eah individual aquirer and ardholder ees are deerined by eah individual issuer. However, he ees do no vary very uh by aquirer or issuer. 4 See, or exale, Roeberg 4. 8

9 or ard users and ash users. 5 Under his riing, in oarison wih an equilibriu wihou ards, i he nework harges he highes erhan ee hen ardholders are beer o or a leas indieren, non-ardholders are worse o, and erhans are eiher beer o or indieren. In he long run, however, he oal o he onsuers and erhans surlus will gradually derease as he nework raises he erhan ees. In ariular, in arkes where aggregae onsuer deands are inelasi, he oal o he onsuers and erhans surlus will onverge o he oal o he onsuers and erhans surlus wihou ards a all. The res o he aer is organized as ollows. Seion develos he odel. Seion resens resuls o our dieren ases in urn, and seion 4 onludes.. The odel For siliiy, only wo ayen insruens are available in he odel. They are ard and ash. Eah indusry is sall enough so ha he ard aeane by he erhans in he indusry does no ae he onsuer s ardholding deision and hereore a ixed eren o he usoers or he indusry hold a ard. ard ransaion brings ransaional beneis o boh he ard users and he erhans who ae hose ards. One o he ransaional beneis o ards is reduing ransaional oss assoiaed wih ash ransaions or boh onsuers and erhans. When onsuers ay wih ash, he onsuers inur soe ransaional oss besides he rie o goods or servies hey urhase and he erhans also inur soe ransaional oss in addiion o he oss o selling goods or servies. Generally, ransaional oss wih ash or onsuers inlude oss o obaining ash suh as TM ees and ie o go o he bank and risks assoiaed wih ash 5 ue o no-surharge rules iosed by ard neworks. 9

10 suh as he; hose or erhans inlude handling ash ransaions suh as oss o labor, ared ar, and bank ees. Soe ransaions require ore oss o onsuers and erhans i hey are aid by ash. Think abou he ransaions ha require reservaions, suh as a hoel reservaion. lhough he reservaion ay no be iossible wih ash, ash is inonvenien i a usoer wans a lae arrival guaranee as well as an abiliy o anel he reservaion wihou aying ees. The hoel ay wan soe advane ayens ro he ash usoer o kee he roo unil he or she arrives, bu i ay hesiae o do so sine i needs o send he ash bak on he hane ha i he usoer anels beore he dae ha a anellaion ee is assessed. So, eiher he ash usoer anno obain a lae arrival guaranee or he hoel ay loose business should he ash usoer no show u. Card ayen an diinish suh inonveniene or loss. he ie o reservaion, a ard usoer gives he hoel his or her ard nuber. I he ard usoer anels he reservaion earlier, he hoel does no harge anyhing o he usoer s ard aoun. The hoel an kee he roo unil he or she arrives, bu in ase he usoer did no show u wihou aneling he reservaion, he hoel an sily harge a anellaion ee o he aoun. lhough here ay exis soe oher ransaional beneis o ards, he odel assues ha ard ransaions redue boh ard users and erhans ransaional oss o zero bu reae no oher ransaional beneis o eiher erhans or ard users. Card ransaions require er ransaion ees o ardholders,, i rewards insead o ees, hen is negaive and o erhans,. For onsuers, he rue os o urhasing a good or servie is wih ash and wih a ard, where is he rodu rie harged by he erhan and is he ransaional os or ash users. For erhans, he rue os o selling a good or servie is d wih ash and d wih a ard, where d is he os o selling a rodu regardless o

11 he ayen ehods used or he ransaion and is he erhan s ransaional oss wih ash. ssue ha ransaional oss wih ash or onsuers and erhans do no vary by eah individual onsuer or erhan bu vary by indusry. Thus, onsuers who urhase goods ro he sae indusry inur he sae ransaional oss wih ash, and erhans who are in he sae indusry inur he sae ransaional oss wih ash. Soe eirial evidene an jusiy his assuion. The onsuer ayen sudy by ove Consuling showed ha he onsuers ayen ix varies by he ye o sore. Hayashi and Klee also ound ha a onsuer s ayen hoie deends on he ransaion haraerisis, suh as average ransaion value and hysial haraerisis a he oins o urhase. lhough larger erhans in a given indusry ay ay lower erhan ees han heir saller ounerars, yially in he U.S. erhan ees vary by a ye o indusries. 6 This assuion is also adoed by soe o he revious sudies. Wrigh 4 assues ha ard beneis vary by indusry. Wrigh and Kaz assue ha wihou ards, soe o he ransaions exra he onsuer s enire surlus. Suh ransaions ay haen only in ariular indusries. Merhans are assued o se he sae rie or ash users and ard users. The nework ioses he no-surharge rule, whih rohibis erhans ro harging a higher rie o ard users. lhough he rule does no rohibi erhans ro giving ash disouns, erhans likely hoose o se he sae ries sine seing dieren ries ay be osly. 7 In soe 6 Boh redi ard neworks and debi ard neworks in he Unied Saes se dieren inerhange ees aording o indusry yes. Tyially, debi ard neworks have wo ees: one or suerarkes and he oher or nonsuerarkes. Visa and MaserCard redi ard neworks se dieren ees or dieren aegories o indusry, suh as suerarkes, general reail arkes, eerging reail arkes, hoel and ar renals, auoaed uel disenser, and so on. 7 Seing dieren ries ay inrease soe adinisraion oss or usoer dissaisaion.

12 ounries, a surharge has been allowed, bu only a sall erenage o erhans aually se dieren ries or ard users and he res o he onsuers. 8 Four dieren ases are analyzed and hese our are haraerized by onsuer deand or goods eiher elasi or inelasi and erhan arke oeiiveness eiher onooly or oeiion aording o he Hoelling odel. Eah indusry aes eiher an elasi or inelasi aggregae onsuer deand or goods. In he indusry ha aes an elasi deand, he onsuer s quaniy deanded deends on he onsuer s rue oss o urhasing goods or servies raher han he rie isel. Le us hink abou he hoel indusry again. s exlained above, aking reservaions wih ash will likely be ore osly or boh usoers and hoels han ha wih ards. Sine ards redue suh ransaional oss or usoers, even when he rie o he hoel say does no hange, here will eiher be ore ravelers or he urren ravelers will ravel ore. Indusries ha ae inelasi deand, on he oher hand, anno shi indusry aggregae deand by sily aeing ayen ards. The onsuer hoe alianes indusry is an exale o his. Eah household needs a os a erain nuber o rerigeraors or washers and dryers. Sine hese are large-dollar ies, aeing ards redues he onsuer s risk or inonveniene assoiaed wih ash ransaions. However, he onsuers do no buy ore rerigeraors or washers and dryers beause he erhans ae ard ayens. To siliy he analysis, eah individual ardholder s and eah individual non-ardholder s deand unions are assued o be idenial. 9 In eah indusry, here is eiher a onooly erhan or wo erhans oeing aording o he Hoelling odel. Merhans deide no only heir ard aeane sraegy bu also heir ries. Merhans are assued o ae soe resriions when hey adjus heir ries 8 Those ounries inlude usralia, Neherlands, Sweden, and he Unied Kingdo.

13 uward, sine hey ay be araid o anirus sruiny or usoers dissaisaion oward heir rie inreases. In he odel, erhans an inrease heir ries wih a os he growh rae o i in he eriod. i is an indusry seii and is a rando draw ro he range o [, i ] eah eriod. i is realized aer he nework ses he erhan ee and beore he erhans se heir ries. The odel assues ha here exiss only one ard nework, whih ses boh erhan ees and he ardholder ee. The nework ses a unique erhan ee in a given indusry and a universal ardholder ee. lhough a variey o nework objeives, suh as axiizing he oal ebers rois, balaning he rois or oss beween aquirers and issuers, or axiizing he oal ransaion volue, are suggesed, sine he aer ouses on he erhan ard aeane behavior, he nework is assued o have an inenive o se erhan ees as high as ossible. The nework is also assued o be onservaive when i ses erhan ees. I ses a erhan ee so ha all o he erhans in a given indusry ae ards. When he nework ses he erhan ee, however, i anno observe he realized axiu rie growh rae, so i onservaively redis ha he growh rae o he rie is zero. I is assued ha onsuers are eiher ardholders or non-ardholders. Non-ardholders need o ay wih ash all he ie. On he oher hand, ardholders an hoose heir ayen ehod eiher ash or a ard. Consuers observe all he deisions he erhans ade ard aeane and new rie beore hey deerine heir ayen ehods, he erhans ro whih hey ake urhases, and quaniies i deand is elasi. 9 This assuion ay no be realisi. The average inoe o redi ard holders is higher han he average inoe o non-redi ard holders. Thus deand unions are likely dieren. In he Unied Saes, he reen growh o inerhange ees o redi and ajor debi ard neworks has ouaed he growh o onsuer ries in soe indusries. For exale, Consuer Prie Indies CPI or household urnishing and aarels have delined or he las ive years, and he growh rae o CPI or ubli ransoraion or or rereaion is lower han he growh rae o inerhange ees.

14 Table : Tiing o he Gae Period Sage N ions Neiher he onooly erhan nor he duooly erhans ae ards. They se he rodu rie, so as o axiize heir rois. Given, he nework ses he ardholder ee,, and he indusry-seii erhan ee,. The erhan deides wheher o ae ards or no, and deerines he new rie,, i i hooses o ae ards. The onsuer deides ro whih erhan he or she akes urhases i duooly, whih ayen ehod he or she uses i a ardholder, and how uh he or she urhases i deand is elasi. N Given, reea he gae in eriod. The iing o he gae is desribed in Table. We onsider a uli-eriod gae and in eah eriod, here are hree sages: a sage he nework ses a erhan ee and a ardholder ee, a sage he erhans deide wheher o ae ards and heir ries, and a sage he onsuers deide ro whih erhan hey ake urhases, whih ayen ehod hey use, and how uh hey urhase i deand is elasi.. Merhan aeane and he highes ossible erhan ees. Monooly erhan... Inelasi deand In eriod, or whaever reason he onooly erhan does no ae ards. ll o is usoers ass ay wih ash. ssue ha eah usoer akes a ixed nuber o ransaions, n. The erhan roi in eriod is: π = d n, Closed syse neworks as well as oen syse neworks an inluene boh erhan ees and ardholder ees. See, or exale, Shwarz and Vinen and Wrigh a. 4

15 where is he rodu rie he onooly erhan harges in eriod. ssue ha eah onsuer reeives gross benei by urhasing one uni o he rodu, υ. Sine he onooly erhan exras he onsuer s enire surlus, i ses he rodu rie so ha: = υ. Saring wih sage o eriod, a ardholder will use a ard i >, and i he erhan aes ards sine he erhan ses a unique rodu rie or boh ash users and ard users. Tha is, by using a ard, he ardholder an save > er ransaion as oosed o aying wih ash. sage o eriod, given he erhan ee,, harged by he nework, he erhan deides wheher o ae ards or no and deerines he rodu rie. The erhan s roi i i does no ae ards is he sae as he roi in eriod. Three dieren levels o he erhan s roi ro aeing ards are ossible. One o he is he roi wihou hanging he rodu rie ro. Sine he erhan has already exraed he ash users enire surlus, raising he rie ro ilies ha he erhan losses all ash usoers. noher level o he erhan s roi is ha o when he erhan exras he ard users enire surlus. By doing so, he erhan raises he rie o: * = υ =. his rie, he erhan exras he ard users enire surlus bu loses he enire roi ro ash users. This level o roi, however, ay no be aainable beause he erhan anno raise he rie by a large aoun in a single eriod. Sine he realized axiu rie growh rae is i in eriod, he rie in eriod,, anno be higher han i. I 5

16 i < *, hen he erhan anno exra he ard users enire surlus. The oher level o he roi, hereore, is ha when he erhan ses = i. Given, he erhan s axiu roi in eriod is deined as: π = d n d n, i d / in{, i }, = i d n, i > i > d /, 4 = d n, i i > d /. Fro equaions and 4, π π always holds as long as. This ilies ha a any o he hree levels o he erhan s roi, he erhan aes ards i he erhan ee is lower han he erhan ransaional oss wih ash. Fro equaion 4, he erhan ay ae ards even when he erhan ee exeeds he erhan ransaional oss wih ash, i raising he rie inreases he erhan s roi. sage o eriod, he nework ses a ardholder ee,, whih should be lower han he ransaional oss wih ash or onsuers,. Sine, by assuion, he nework onservaively redis he indusry s realized axiu rie growh rae is zero, he nework ses a erhan ee,, as high as. Thereore, he highes ossible erhan ee in eriod, =. 5 Proosiion Prie and Welare: Suose he nework ses < and. onooly erhan who aes an inelasi onsuer deand aes ards. Coared o he equilibriu wihou he ards, a The erhan raises he rodu rie i he ardholder s er ransaion ee is low relaive o his or her ransaion beneis and/or he ardholder base is high enough; 6

17 b Non-ardholders surlus is lower i he erhan raises he rie, oherwise i is unhanged; Cardholders surlus is higher or a leas he sae; d ggregae onsuer surlus is lower i he erhan raises he rie o is roi axiizing level; i he erhan raises he rie o he resried level, i is eiher higher or lower; i he erhan does no hange he rie i is higher; e The erhan s surlus is higher or a leas he sae. In eriod, an he nework raise he erhan ee higher han he highes ossible erhan ee in eriod? Obviously, i he erhan did no raise he rie in eriod, he nework anno raise he erhan ee, beause he gae in eriod is exaly he sae as he gae in eriod. In his ase, he highes ossible erhan ee in eriod,, is he sae as. I, on he oher hand, he erhan raised he rie in eriod, he nework an raise he erhan ee. Sine he erhan an always go bak o he original sraegy o no ae ards and o se he original rie,, he new erhan ee needs o guaranee a leas he sae roi as he erhan s roi wihou aeing ards. I he erhan has already exraed he ard users enire surlus in eriod, he highes ossible erhan ee in eriod is: = d. 6 I he erhan did no exra he ard users enire surlus in eriod due o he riing resriion, is lower han he level o equaion 6. I a onsuer has no disuiliy ro no urhasing any goods, non-ardholders surlus is unhanged. Oherwise i is lower. 7

18 In he long run, i he erhans hose o serve ard users only, he erhan ee evenually onverges o he level o equaion 6 and he rodu rie also onverges o he level o equaion. Oherwise, he erhan ee does no hange ro eriod and he rie does no hange ro he iniial rie,. The long-run equilibriu erhan ee and rie are desribed in equaions 7 and 8. N = =, d i d, oherwise. /, 7 N =, i d / =, oherwise., 8 Noie ha i he erhan s rie seing is oleely lexible and he nework knows ha lexibiliy, he long-run equilibriu erhan ee will be se ro he irs eriod he erhan aes ards even when he erhan hooses o serve ard users only, and he long-run equilibriu rie will be harged by he erhan ro eriod... Elasi deand In eriod, he onooly erhan does no ae ards. ll o is usoers ass ay wih ash. ssue ha eah individual usoer s deand union is idenial regardless o his or her ardholding. The erhan s roi union in eriod is: n π = d. 9 The onooly erhan ses he rodu rie so as o axiize is roi. The rie in eriod is deined as: 8

19 9 argax n d = = π Saring wih sage o eriod, a ardholder will use he ard i > and i he erhan aes ards. sage o eriod, he erhan deides wheher o ae ards; i i deides o ae, i adjuss he rodu rie. The erhan s roi i i does no ae ards is he sae as he roi in eriod. Given a erhan ee,, he erhan s roi union i i aes ards is deined as: ; d d = π. The erhan aes ards i and only i ; n π π. lhough, by assuion, he rodu rie growh rae is liied, sine he erhan sill has he reedo o se he rodu rie dieren ro, ; n π π always holds i ; n π π holds. Given, he erhan s roi axiizing rie, *, is deined as: * * * * * * d =. Sine he rodu rie growh rae is liied o i, he rie in eriod will be: oherwise.,, * i *, i i = = sage o eriod, he nework ses a ardholder ee,, whih should be lower han. Sine he nework onservaively redis he indusry s axiu rie growh rae is zero, i will se he erhan ee so ha he erhan s roi by aeing ards under is a leas

20 n as high as he roi wihou ards. Tha is π ; π. The highes erhan ee in eriod se by he nework is, hereore: = d. 4 Equaion 4 yields he ollowing roosiion. Proosiion Merhan ee: The onooly erhan who aes an elasi onsuer deand aes ards even i he erhan ee is higher han he erhan ransaional benei ro ards. Proo: Clearly, he seond er o equaion 4 is osiive. Thus, >. Wrigh b analyzes erhan ard aeane in he indusry where aggregae onsuer deand is elasi and erhans are oeing aording o a Courno odel. The odel also assues ha eah onsuer buys a os one uni in he indusry. In suh an indusry, erhans will ae ards even when he erhan ee is higher han heir ransaional benei ro ards. In y odel, eah onsuer has an elasi deand and buys as any unis as he or she wans ro he sae erhan. lhough a onooly erhan does no have o oee, he erhan will ae ards even when he erhan ee is higher han heir ransaional benei. This is beause he erhan an ake he ardholder s deand urve shi uward by aeing ards. Sine general resuls are no available or he ase o elasi onsuer deand, he res o his subseion resris is aenion o linear deand. Proosiion oares he rodu rie wih and wihou he ards and Proosiion 4 suaries he onsuers and erhan s welare. Proosiion Prie: onooly erhan who aes linear onsuer deand will adjus he rodu rie i i aes ards.

21 i For, he rodu rie wih ards is higher i > i. ii For < : i <, he rodu rie wih ards is higher when i > ; i, or > he rodu rie wih ards is higher i i > and or < he rodu rie wih ards is lower. Proo: Fro equaions and, * = { }. 5 By assuion >. Obviously, * > when. When <, i is greaer han, * >, oherwise *. Proosiion 4 Welare: Suose he nework ses < and. onooly erhan who aes linear deand aes ards. Coared o he equilibriu wihou ards: a Non-ardholders surlus is higher lower i he erhan ses a lower higher rie aer aeing ards; b Cardholders surlus is higher; ggregae onsuer surlus is higher; d The erhan s surlus is higher or a leas he sae. Proo: The onsuers surlus is easured by using he Marshallian deand urve. Nonardholders surlus deends only on he rodu rie. The higher he rie, he lower he nonardholders surlus is. Thereore, i he rodu rie wih ards is higher han ha wihou ards, non-ardholders are worse o and vise versa. Cardholders surlus deends on he rodu rie, he ardholder s er ransaion ee, and he onsuer s ransaional oss wih ash. The oal oss or a ardholder when he or she

22 uses a ard,, is always lower han he oal oss or he ardholder when he erhan does no ae ards,. To show his, suose ha he nework ses he highes erhan ee so ha he erhan aes ards wih he good rie,. Suose also ha he erhan an harge he roi axiizing rie in eriod, whih is *. The roi axiizing rodu rie is also he highes, beause * / > ro equaion. his highes rie,, he ollowing equaion holds. = [ { } ] 6 Sine { } <, and >, equaion 6 is negaive. This ilies ha is always lower han, and hereore ardholders are always beer o. enoe v is a onsuer s surlus. ggregae onsuers surlus wihou ards, v, is lower han aggregae onsuers surlus when he erhan aes ards and ses he rodu rie a he highes level, v v. v v v = b { } { { 4 } } > By assuion, he nework ses he erhan ee a os as high as he level ha akes he erhan s roi wih ards equal o he roi wihou ards, under he rie o. Sine he erhan will adjus he rie aer deiding o ae ards, he erhan s roi should be a leas as high as he roi wihou ards. Fro equaion, > when =, unless i =, he erhan is beer o.

23 Proosiions and 4 ily ha he erhan s ard aeane ias he welare o non-ardholders, ardholders, and he erhan dierenly. While ardholders and he onooly erhan are always beer o, non-ardholders are likely worse o. Even when he nework ses he erhan ee lower han he erhan ransaional benei, non-ardholders are likely worse o. I ardholders ne ransaional benei ro ards,, is greaer han he erhan s ransaional benei ro ards,, non-ardholders are always worse o, unless i =. This resul suors he indings o Shwarz and Vinen. They oare he equilibriu wih and wihou he no-surharge rules and show ha when a onooly nework harges a roi axiizing erhan ee ro a onooly erhan, ash users non-ardholders surlus is lower under no-surharge han under surharge. Sine ash users surlus a he equilibriu wihou ards and ha a he equilibriu wih ards under surharge are he sae, y resuls sugges ha even when he nework harges a erhan ee ha is lower han he roiaxiizing level, ash users surlus is lower under no-surharge han under surharge. erhan ee, Now, le us onsider he gae in eriod. Suose in eriod he nework ses he. Can he nework raise he erhan ee higher han, wihou hanging he ardholder ee? I he realized axiu rie growh rae in eriod were greaer han zero, he erhan raised he rie o inrease his roi under n π ; > π ; = π. The nework an se. Thus, by deiniion,, suh ha erhan s roi wih ards equal o he axiu roi wihou ards: π ; = π n >, o ake he. 7

24 I he realized axiu rie growh rae in eriod was zero, he rodu rie in eriod was n he sae as he rie in eriod. Sine π ; = π ; = π, he nework anno raise he erhan ee higher han. The erhan will oninue o ae ards and adjus he rie o axiize roi wih he erhan ee,. n π ; π ; = π 8 Sine, he erhan s roi in eriod is equal o or lower han ha in eriod : π ; π ; 9 In he long run, he highes erhan ee and he rodu rie will gradually inrease: N N K, N N K, and he erhan s roi onverges o he axiu roi wihou ards: π N ; N N n π. Proosiion 5 Long-run equilibriu: Suose ha he nework will se he onservaive highes erhan disoun ee every eriod. Coared o he equilibriu wihou he ards, a he long run equilibriu: a Non-ardholders are worse o; b Cardholders are beer o; ggregae onsuer surlus is higher; d The erhan is indieren. Noie ha i he erhan s rie seing is oleely lexible and he nework knows ha lexibiliy, he longrun equilibriu deined above an be reahed ro he irs eriod ha he erhan aes ards. 4

25 Proo: a is obvious sine N is higher han. d is obvious ro equaion. The roo o b and are in endix. In he long run, in onooly arkes wih inelasi onsuer deands, all ariians are a os indieren wih and wihou ards. On he oher hand, in onooly arkes wih elasi onsuer deands, alhough non-ardholders and he erhan are a os indieren wih and wihou ards, ardholders are beer o. Moreover, ardholders welare gain surasses non-ardholders welare loss ro ard aeane, and hereore he su o onsuers and he erhans surlus is higher wih ards han wihou ards.. uooly erhans Hoelling oeiion ddiional assuions are ade or his seion. There are wo erhans, Merhan and Merhan B, who are oeing wih eah oher aording o he Hoelling odel. Consuers ass are uniorly disribued on he inerval o [, ], whih is indeenden o heir ardholding. Merhan is loaed a oin and Merhan B is loaed a oin. For he onsuer loaed a oin x, where x, he ransoraion os o Merhan is x, and he ransoraion os o Merhan B is x. Only ure sraegy Nash equilibria are onsidered in his seion... Inelasi deand Beore onsidering he equilibriu wih ards, irs we desribe he equilibriu wihou ards equilibriu in eriod. s is usual in he syeri Hoelling odel, he equilibriu ries,,, are he sae or boh erhans and are equal o he erhan s arginal os, B d, lus he ransoraion os, : = B = = d. 5

26 Eah erhan s roi is equal o he argin ies he arke share: π = π B = π =. Now, le us onsider he equilibriu in eriod. sage o eriod, a non-ardholder hooses he erhan based on he ries. I >, a ardholder hooses he erhan based on he erhans ard aeane and heir ries. I he ardholder hooses he erhan who does no ae ards he or she uses ash, oherwise he ardholder ays wih a ard. I he ardholder as as i he or she is a non-ardholder., Suose >. sage o eriod he erhans deide wheher o ae ards and deerine he rodu ries. Firs, le us desribe he ases where boh erhans ake he sae ard aeane sraegy. Suose ha he erhans ae ards. Given he oher erhan s rie j, erhan l s roi union is deined as: π ; l l j j l = l d l d j l, or l=, B Sine he erhans anno raise he rodu ries by ore han he rie growh rae, i, in eriod, heir riing and rois are deenden on i. Merhan l s rie in eriod is: l l : ae; j : ae = d, i i = i, oherwise., 4 Merhan l s roi is: π l l : ae; j : ae =, = { i }, i i oherwise., 5 6

27 Suose insead ha he erhans do no ae ards. In his ase, he equilibriu is he sae as he equilibriu wihou ards. Nex, le us desribe he ases where eah erhan akes a dieren ard aeane sraegy ro is rival s. Suose ha Merhan aes ards and Merhan B does no. Eah erhan s roi union deends on and. I he raio o a ardholder s ne benei ro a ard ransaion o he ransoraion oss, /, is large, he ardholder s erhan hoie does no deend on he rie dierene bu deends on he dierene o erhan ard aeane. The analysis below ouses on /. 4 Merhan s roi union is deined as: π ; B B = d d and Merhan B s roi union is: B,6 π n B B ; B = B d { B }. 7 The rodu rie and roi or eah o he erhans are deenden on he indusry s realized axiu rie growh rae: ae; reje = d { }, i i { }, 8 = i, oherwise. B reje;ae = d = { { i }, i i { }, oherwise. }, 9 4 The oher ases igh be ore ioran. In he Unied Saes, any o he ard issuers give rewards o heir ardholders, hus negaive is very oular. In addiion, soe o he erhan arkes are very oeiive, hus ay be very sall in suh arkes. 7

28 π ae; reje = [{ = [ i i { } i i { } oherwise. i { ], }, }], π reje; ae = B = { i { i }{ }, i i { }, oherwise. }, Consider Merhan s ard aeane behavior. Suose ha Merhan B aes ards. Merhan will ae ards i and only i π ae;ae π reje; ae. This is equivalen o he erhan ee being lower han, where is deined as: =, i i { }, =, i { } > i, = [ i i { i }{ }], oherwise. Suose insead ha Merhan B does no ae ards. Merhan will ae ards i and only i π ae;reje π reje; reje. This is equivalen o, where is deined as: = = { i i i, { i } } i i { oherwise. 5 },, Fro equaions and, > always holds. 8

29 Sine Merhans and B are idenial, Merhan s ard aeane behavior desribed above is also aliable o Merhan B. Thus, ure sraegy Nash equilibria are deined as ollows: When he erhan ee is higher han, here is one Nash equilibriu boh erhans do no ae ards; when he erhan ee is beween and, here are wo Nash equilibria boh erhans do no ae ards, or boh ae ards; when he erhan ee is lower han, here is one Nash equilibriu boh erhans ae ards. sage o eriod, he nework ses a ardholder ee,, suh ha <. I also ses he erhan ee,, so ha boh erhans ae ards. Sine he nework anno observe he indusry s realized axiu rie growh rae, i, when i ses he erhan ee, i redis i o be zero. Fro equaions and, when i = : { = }, 4 and =. 5 In order o have boh o he erhans ae ards, he nework will se he erhan ee lower han eiher or. While a unique Nash equilibriu in whih boh o he erhans ae ards exiss when he nework ses he erhan ee lower han, wo Nash equilibria eiher boh ae ards or boh do no ae ards exis when he nework ses he erhan ee lower han bu higher han. Sine neiher erhan reviously aeed ards, eah erhan hardly believes ha his rival erhan will ae ards when boh do no ae ards an be an equilibriu. Suh erhan s belie akes he onservaive nework se as he highes erhan ee in eriod. The highes erhan ee in eriod,, is hereore: 9

30 =. 6 Proosiion 6 Merhan ee: In he indusry whose aggregae onsuer deand is inelasi, he duooly erhans oeing aording o he Hoelling odel ae ards even i he erhan ee is higher han he erhan ransaional benei ro ards. Proosiion 6 ilies ha he erhans sraegi oive o ae ards enables he nework o raise he erhan ee higher han he ee or a onooly erhan. While he nework harges he erhan ee o a onooly erhan who aes an inelasi deand as high as in eriod, he ee i harges duooly erhans who ae he sae aggregae onsuer deand as he onooly erhan is higher han in eriod. Sine boh erhans ae ards or any, he new rodu ries, B are given in equaion 4. Clearly, i >, he erhans raise he rodu ries unless i =, and i < he erhans lower he ries even when i >. Proosiion 7 Welare: Suose he nework ses < and. In he indusry whose aggregae onsuer deand is inelasi, he duooly erhans who are oeing in he Hoelling odel ae ards. Coared o he equilibriu wihou ards: a Non-ardholders surlus is lower higher i he erhans se higher lower rodu ries; b Cardholders surlus is higher; ggregae onsuers surlus is higher; d The erhans surlus is lower or a os he sae.

31 Proo: Sine onsuer deand is elasi, assue ha eah onsuer reeives gross benei by urhasing one uni o he rodu, υ. The ne uiliy is, hereore, υ or a ash ransaion and υ or a ard ransaion. Fro equaion 4, i i >, he new ries are higher lower when he erhan ee is higher lower han he ransaional oss wih ash. Sine he non-ardholders surlus deends only on he rodu rie, i he rodu rie wih ards is higher han ha wihou ards, nonardholders are worse o and vise versa. The oal os or a ardholder when he or she uses a ard,, is always lower han he oal os or he ardholder when he erhans do no ae ards,. The erhan will se he rie o a os = d, under he highes erhan ee,. = < ggregae onsuers surlus wihou ards is υ, and ha wih ards and rodu rie υ υ. is { υ υ } υ = { } >. Fro equaions and 5, i he erhans an adjus heir ries o heir roi axiizing level eiher, or > and i, he erhans rois are unhanged, oherwise heir rois are lower. Le us urn o he gae in eriod. Can he ard nework raise he erhan ee higher han he highes erhan ee in eriod? In onras o onooly erhans, duooly

32 erhans anno go bak o he original sraegy easily. Given he a ha he oher erhan is now aeing ards, oninue o ae ards is he beer sraegy han reje ards, as long as he erhan ee is lower han he level deailed below. Suose ha boh erhans aeed ards and raised heir ries in eriod. Consider he ase where one o he erhans rejes ards and he oher oninues o ae ards in eriod. The roi or he erhan who oninues o ae ards is given as: π l ae; reje = [{ = [ i { { } i i i i { } }], ], oherwise. Even i he realized axiu rie growh rae in eriod is zero, as long as he erhan ee in eriod is lower ha, where is deined as: }, 7 = { } /, 8 π l ae; reje π l reje; reje =. This ilies ha when he nework ses he erhan ee lower han, boh erhans oninue o ae ards. Sine >, is higher han he highes erhan ee in eriod. I he nework wans o have a unique Nash equilibriu where boh erhans ae ards, hen he highes erhan ee he nework an harge is. Thus, i boh o he erhans raised heir ries in eriod, he nework an raise he erhan ee in eriod.

33 Suose, insead, ha boh erhans aeed ards bu did no raise heir ries in eriod i.e., he realized axiu rie growh rae in eriod was zero. In his ase, i he nework wans o have a unique equilibriu where boh erhans ae ards, hen he highes erhan ee he nework an harge is he sae as. Neverheless, he nework ay raise he erhan ee in eriod beause he nework ay no wan o have suh a unique equilibriu. In eriod, sine neiher erhan had reviously aeed ards, eah erhan hardly believed ha his rival erhan would ae ards when boh do no ae ards an be an equilibriu. In eriod, however, sine boh erhans have aeed ards in eriod, eah erhan ay believe ha his rival will oninue o ae ards when boh erhans oninue o ae ards an be an equilibriu. The erhans hange in ereion enables he nework o harge a erhan ee as high as, whih is deined in equaion 4. Noie ha >. Thus, even when he erhans did no raise heir rodu ries, i eah o he erhans srongly believes ha is rival will ae ards in eriod, he nework an raise he erhan ee in eriod. In eriod, boh erhans will oninue o ae ards and raise heir ries aording o he erhan ee inrease unless he realized axiu rie growh rae in he eriod is zero. The inrease in rodu ries enables he nework o raise he erhan ee urher in he sueeding eriods. In his way, he erhan ee and rodu ries gradually inrease. In he long run, boh he erhan ee and rodu ries onverge o he oin where hey anno inrease anyore. The long run equilibriu erhan ee and rodu ries are deined as: N =, 9 N N N and = = = d. 4

34 In he long run, eah o he erhans believes ore srongly ha is rival will oninue o ae ards when boh oninue o ae ards an be an equilibriu. Thereore, he nework an se he erhan ee in he range where here exis ulile Nash equilibria ha inlude an equilibriu in whih boh erhans ae ards. In addiion, in he long run, as he rodu ries inrease, he erhans rie resriions beoe less resriive so ha he erhans an se heir roi axiizing ries. Noie ha he long-run equilibriu erhan ee is he sae as wha Rohe and Tirole ound. In heir odel, he erhans rie seing is oleely lexible. In y odel, however, even i he erhans rie seing is oleely lexible, he nework ay no se he long-run equilibriu erhan ee in he irs eriod. This is beause he onservaive nework wans o ake sure ha boh o he erhans ae ards in eriod, and by doing so i avoids seing a erhan ee whih ay lead o he equilibriu where boh do no ae ards. Proosiion 8 Long-run equilibriu: Suose ha he nework will se he onservaive highes erhan ee in every eriod. Coared o he equilibriu wihou ards, a he long run equilibriu: a Non-ardholders are worse o; b Cardholders are beer o; ggregae onsuers surlus is he sae; d The erhan is indieren. Proo: a I is obvious sine N N >. b Fro equaions and 4, <. { υ N υ N } υ = { N } =. d I is obvious ro equaions and 5. 4

35 Merhan oeiion allows he nework o se erhan ees higher. In indusries whose aggregae onsuer deand is inelasi, he nework an harge onooly erhans he erhan ees as high as he erhans ransaional beneis ro ards or in soe irusanes i an harge he ees higher han he erhans ransaional beneis. I he sae indusries are ore oeiive, he nework an always harge he erhan ees higher han he erhans ransaional beneis. nd in he long run i harges he ees ha are equal o he erhans ransaional beneis lus ard users ne ransaional beneis. Moreover, in a oeiive arke where eah erhan s iniial roi argin,, is saller han he ard user s ne benei,, he long-run erhan ee exeeds he su o he erhans ransaional benei and he erhans iniial argin. In eiher onooly or oeiive arkes, in he long run, he su o onsuers and erhans surluses wih ards onverges o he su o heir surluses wihou ards... Elasi deand In order o ake he odel analyially solvable, we odiy he Hoelling odel in he ollowing way. We assue ha a onsuer buys a leas one uni o rodu er ri. The onsuer hooses a erhan based on he oal os he su o he rodu rie, he ransaional os wih he ayen insruen he or she uses, and ransoraion os. er he onsuer deides ro whih erhan o urhase he rodus, he onsuer deerines he quaniy based on he rodu rie and he ransaional os. Beore onsidering he equilibriu wih ards, irs we desribe he equilibriu wihou ards. This odiied Hoelling odel gives he equilibriu ries,,, whih are he sae or boh erhans. B 5

36 = B = = d. 4 Eah erhan s roi is: π = π B = π =. 4 { } Now onsider equilibriu wih ards. In order o ake he odel analyially solvable, he erhan s deision-aking is broken ino wo ses. Firs, he erhans deide wheher o ae ards beore deerining heir ries. Seond, aer observing eah oher s ard aeane sraegy, hey deerine heir ries. This assuion ay ake he highes erhan ee he nework will harge slighly lower in eah eriod, oared wih he odel where he erhans deide heir ard aeane and ries a he sae ie. Suose >. sage o eriod, a ardholder hooses he erhan based on he erhans ard aeane and ries. I he ardholder hooses he erhan who does no ae ard, he or she ays wih ash, oherwise he or she ays wih a ard. Siilar o he ase where onsuer deand is inelasi, erhan ard aeane and riing behavior deend on he araeer values. Here, we analyze equilibriu only or /. sage o eriod, irs le us suose boh erhans ake he sae ard aeane sraegy. Given, eah erhan s roi when boh erhans ae ards under rie is: π l ae; ae = d d, or l =, B,4 and when boh erhans reje he ards under rie is: π l reje; reje = d. 44 6

37 Suose insead a erhan akes a dieren ard aeane sraegy ro is rival s. Consider he ase where Merhan aes ards and Merhan B does no. Merhan s roi is: π ae;reje = d d,45 and Merhan B s roi is: π B reje; ae = d. 46 Suose ha Merhan B aes ards. Merhan aes ards i and only i: d { }. 47 Suose insead Merhan B does no ae ards. Merhan aes ards i and only i: d { }. 48 eine and ha saisy he equaliy in equaions 47 and 48. Clearly, is greaer han. When he erhan ee is higher han, here is one Nash equilibriu; boh erhans do no ae ards. When he erhan ee is beween and, here are wo Nash equilibria; boh erhans do no ae ards, or boh ae ards. When he erhan ee is lower han here is one Nash equilibriu; boh erhans ae ards. Sine he nework wans as any erhans o ae ards as ossible, i will se a erhan ee lower han. The highes erhan ee in eriod,, is hereore: = d { }. Proosiion 9 Merhan ee: In he indusry whose aggregae onsuer deand is elasi, he duooly erhans oeing in he odiied Hoelling odel ae ards even i he 7

38 erhan ee is higher han he erhan ransaional benei ro ards. I all araeer values are he sae,,,, and he deand union, he raio o he irs-eriod highes erhan ee o he iniial equilibriu rie,, in he duooly arke wih elasi onsuer deand is higher han ha in he onooly arke wih elasi onsuer deand or in he duooly arke wih inelasi onsuer deand. For any o he ard ransaions, erhan ees are se as a rae o he ransaion value, whih is equivalen o soe erenage o he rie. Proosiion 9 ilies ha in he arke whose aggregae deand is elasi, he ore oeiive he arke, he higher he erhan ee rae is and ha in he oeiive arke, he ore elasi he onsuer deand, he higher he erhan ee rae is. er deiding ard aeane, boh erhans se heir ries. The roi axiizing ries are deined as: * = * = B * = d { } { }, 49 { } { } where x = * x and x = * x. Sine he rodu rie growh rae is liied o i, he rie in eriod will be: = *, i * < i, = i, oherwise. Sine general resuls are no available or he ase o elasi onsuer deand, he res o he subseion resris is aenion o linear deand. Fro equaions 4 and 49, we obain equaion 5. b B * = b { * b}, 5 B 8