Discriminatory prices, endogenous locations and the Prisoner Dilemma problem

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1 Disriminory ries, endogenous loions nd he Prisoner Dilemm rolem Sefno Colomo* sr In he Hoelling frmework, he equilirium firs-degree disriminory ries re ll lower hn he equilirium unorm rie. When firms loions re fied, rie disriminion emerges s he unique equilirium in gme in whih every firm my ommi no o disrimine efore seing he rie shedule. This er ssumes endogenous loions nd shows h unorm riing emerges s he unique equilirium in gme in whih every firm my ommi no o disrimine efore hoosing where o loe in he mrke. Prie disriminion sill is he unique equilirium ouome when firms my ommi only fer he loion hoie. JEL odes: D43; L11 Keywords: Prie disriminion; Commimen; Loion. *DEFP, Universià Coli del Sro Cuore Milno, Lrgo. Gemelli 1, I-013, Milno. E-mil: sefno.olomo@uni.i. I m indeed o Mihele Grillo for his guidne, suggesions nd enourgemen. ll remining errors re my own. 1

2 1. Inroduion Prie disriminion is widely used usiness rie. However, in oligooly i my e ossile h he equilirium disriminory ries re ll lower hn he equilirium unorm rie. This henomenon is lled ll-ou omeiion 1. When ll-ou omeiion ours, equilirium rofis under he disriminory rie regime re lower hn he equilirium rofis under he unorm rie regime. ll-ou omeiion yilly emerges in he Hoelling 199 frmework wih liner or qudri rnsorion oss. Thisse nd Vives 1988 hve sudied he se of firs-degree rie disriminion wihin he Hoelling s 199 model ssuming h he firms re eogenously loed he endoins of he mrke. Firs, hey show h when firms n erfely rie disrimine nd simulneously hoose he rie shedule, unorm riing is never n equilirium. Then, Thisse nd Vives 1988 ssume wosge gme, where in he firs sge eh firm hs he ossiiliy o nnoune h i will no rie disrimine, while in he seond sge he rie shedules re effeively se. For emle, firm my nnoune in he firs sge h i would no hold sles or would no issue ouons. Of ourse, suh nnounemens hve o e redile. There re lo of ries h mke hese nnounemens redile: he mos-fvoured nion luse 3 is one of hese ries. Thisse nd Vives 1988 show h even when every firm my redile ommi no o disrimine efore seing he rie shedule, he disriminory ries sill rise in equilirium, sine no-ommimen is he dominn sregy for eh firm in he firs sge of he gme ondiioned on he equilirium h in he seond sge of he gme. This siuion gives rise o yil Prisoner Dilemm: oh firms would e eer off seing unorm ries, u he dominn sregy of eh firm indues he disriminory equilirium, h in urn yields lower rofis. The im of his er is o es wheher Thisse nd Vives 1988 resul sill holds when he loions of he firms re endogenous. Firs of ll, llowing firms o hoose where o loe in he mrke, we oin h he loion-rie equilirium wih erfe rie disriminion is hrerized y less hn miml dfereniion degree: in 1 Cors Ulh nd Vulkn The mos-fvoured nion luse engges firm o offer onsumer he sme rie s is oher onsumers: he luse is no reseed, he firm mus y k he onsumer he dferene eween he rie he effeively id nd he lowes rie fied y he firm.

3 f, firms loe 1/4 nd 3/4. ll-ou omeiion ours. seond se of he nlysis onsiss in suosing hree-sge gme. In he firs sge of he gme, firms simulneously hoose where o loe in he mrke. In he seond sge of he gme eh firm deides wheher o ommi no o rie disrimine or no o ommi: firm hs ommied, i is oliged o se he sme rie for ll onsumers when he omeiion in rie rises; firm hs no ommied, i omees wih unresried rie shedules. In he hird sge he firms se simulneously he rie shedules. We show h here eiss unique su-gme erfe equilirium, whih is hrerized y rie disriminion. Therefore, even wih endogenous loions, disriminory ries rise in equilirium. Finlly, we suose hnge in he iming of he hree-sge gme. Th is, in he firs sge of he gme, eh firm ommis or does no ommi; in he seond sge he firms simulneously hoose he loion in he mrke; in he hird sge he firms se simulneously he rie shedules. Ineresingly, in his se he unique su-gme erfe equilirium is hrerized y unorm riing: oh firms ommi in he firs sge, mimlly dferenie in he seond sge nd se unorm ries in he hird sge. No Prisoner Dilemm is resen. The inuiion ehind his resul is he following. When he ommimen deision is ken for given loions of he firms firs iming, he only effe of ommimen is o redue he firm s fleiiliy on seing ries. Therefore, he dominn sregy for eh firm is no-ommimen, nd he Thisse nd Vives 1988 resul sill is vlid. Insed, when he ommimen deision ffes no only he rie deision u lso he loion deision seond iming, eh firm niies h is deision o ommi will indue higher equilirium dfereniion degree, nd his mkes rofile for eh firm o ommi, even he fleiiliy in seing rie is redued: in his se, he Thisse nd Vives 1988 resul does no hold. This er is orgnized s follows. In Seion we desrie he model nd we riefly rell he well-known loion-rie equilirium under he hyohesis of unorm rie regime. In seion 3 we nlyse he loion-rie equilirium when he firms n erfely rie disrimine. In seion 4 we nlyze he hree-sge gme wih he wo dferen imings. Seion 5 onludes. 3

4 . Unorm rie ssume liner mrke of lengh 1. Consumers re unormly disriued long he segmen. Define wih [0,1] he loion of eh onsumer. Eh oin in he liner mrke reresens erin vriey of given good. For onsumer osiioned given oin, he referred vriey is reresened y he oin in whih he onsumer is loed: he more he vriey is fr from he oin in whih he onsumer is loed, he less i is reied y he onsumer. Eh onsumer onsumes no more hn 1 uni of he good. Define wih v he mimum rie h onsumer is willing o y for uying his referred vriey. Suose h v is equl for ll onsumers. Suose furher h v is lrge enough o gurnee h eh onsumer lwys uys he good. There re wo firms, nd, omeing in he mrke. oh firms hve idenil onsn mrginl oss,, nd zero fied oss. The firms deision onerning where o loe oinides wih he deision of whih vriey o rodue. Define wih he loion hosen y firm nd wih he loion hosen y firm. Wihou loss of generliy, ssume: 0 1. Define wih he unorm rie se y firm nd wih he unorm rie se y firm 4. The uiliy of onsumer deends on v, on he rie se y he firm from whih he uys, nd on he disne eween his referred vriey nd he vriey rodued y he firm. We ssume qudri rnsorion oss. Define wih, equl for ll onsumers, he imorne riued y he onsumer o he disne eween his referred vriey nd he vriey offered y he firm. The uiliy of onsumer loed when he uys from firm is given y: u v, while he uiliy of onsumer loed when he uys from firm is given y: u v. Define wih * he onsumer whih is indferen eween uying from firm or from firm for given 4 Given he inerreion of he liner mrke h we re doing, he rnsorion oss re neessrily susined y he onsumers: herefore, ries re f.o... However, he liner mrke n lso hve sil inerreion: in his se eh oin of he segmen reresens oin in he hysil se. Sine he disne eween onsumer nd he firm imlies now effeive rnsorion oss, wo riing mehods re ossile: f.o.. ries, when he rnsorion oss re susined y he onsumer whih goes nd kes u he rodu he firm s mill, nd delivered ries, when he rnsorion oss re susined y he firm h rries he rodu from he mill o he onsumer. Thisse nd Vives 1988 do sil inerreion of he mrke nd ssume delivered ries. 4

5 oule of loions, nd, nd for given oule of unorm ries, nd. Equing he uiliy in he wo ses nd solving for i follows: * Given he unorm disriuion of he onsumers, * is he demnd funion of firm nd 1 * is he demnd funion of firm. I is well known h in wo-sge gme in whih firms firs hoose loions nd hen hoose he unorm rie, he unique su-gme erfe equilirium imlies miml dfereniion, s he following roosiion indies: Proosiion 1 D sremon e l. 1979: in wo-sge gme in whih he firms firs simulneously deide where o loe nd hen simulneously deide he [unorm] rie, here is unique su-gme erfe equilirium, defined y * 0 nd * 1, nd * *. Given he equilirium loions nd he equilirium ries, he equilirium rofis for eh firm re: Π Π. 3. Disriminory ries We sudy now he loion-rie equilirium when oh firms n erfely rie disrimine eween onsumers. We suose wo-sge gme, in whih he firms firs deide where o loe nd hen omee on ries. efore o sr, noe h he f h he firms hve he ossiiliy o rie disrimine does no imly h he firms effeively rie disrimine: firm my deide o rie unormly even i n rie disrimine. In he following we show h when firms n rie disrimine, hey do i. Consider onsumer loed in. Define wih J he rie hrged y firm J, o he onsumer. The uiliy of h onsumer when he uys from firm is 5

6 6 given y: v u, while his uiliy when he uys from firm is given y: v u. Oviously, he onsumer uys from he firm whih gives him he higher uiliy. If he uiliy of he onsumer is he sme when he uys from firm nd when he uys from firm, we suose h he uys from he nerer firm 5. Suose h onsumer is nerer o firm hn o firm. For given oule of firms loions nd for given rie se y firm, he es hing firm n do is seing rie h gives he onsumer he sme uiliy he reeives from firm : his is he highes ossile rie h gurnees h onsumer uys from. Suose insed h he onsumer is nerer o firm. For given oule of firms loions nd for given rie se y firm, in order o serve onsumer he es hing firm n do is giving him slighly higher uiliy hn he uiliy rovided o him y firm. Of ourse, n nlogous resoning holds for firm. Therefore, defining wih ε osiive nd infiniely smll numer nd relling h rie lower hn he mrginl os enils loss, he esrely funions of firm nd firm re reseively: ] ; m[ ] ; m[ ] ; m[ ] ; m[ r r ε ε 1 Noe h he es-rely rie shedule of eh firm deends on. Therefore, he dominn sregy of eh firm enils rie disriminion. This mens h when firm n rie disrimine, i rie disrimines. The following roosiion defines he equilirium rie shedule for ny oule of loions. Proosiion : when he firms n erfely rie disrimine eween he onsumers, he equilirium ries in he seond sge of he gme re he following: 5 This ssumion is ommon in sil models, nd i is neessry o void he ehniliy of ε- equilirium ones when oh firms rie disrimine.

7 *, nd *, *, nd *, Proof. Suose h is ner o, h is, <. Consider firm. Firs, we show h > nno e n equilirium. When >, he es-rely of firm onsiss in seing:. I follows h he oiml rie for firm is: ' ε, h is lwys higher hn due o he f h is higher hn y hyohesis nd ε is n infiniely smll nd osiive numer y definiion. Therefore, > nno e n equilirium, euse firm would oin higher rofis seing: ' ε. We show now h is n equilirium. The es-rely > of firm is:. Wih suh rie firm oins zero rofis from onsumer, whih uys from firm, u i hs no inenive o hnge he rie, euse inresing he rie i would oninue o oin zero rofis, nd seing rie lower hn he mrginl oss would enil loss. Sine firm is seing he rie indied y he es-rely funion, i is seing he oiml rie y definiion. I follows h nd reresens he unique rie equilirium. The roof for > is symmeri o he roof for <. Finlly, when he onsumer is he sme disne from he wo firms, h is, he sndrd errnd s resul holds: he unique rie equilirium when wo undferenied firms omee on rie is reresened y oh firms seing rie equl o he mrginl os. The equilirium loions in he firs sge of he gme re defined in he ne roosiion: Proosiion 3: in he firs sge of he gme he unique Nsh equilirium is given y * 1 4 nd *

8 Proof. Using Proosiion, he firms rofis n e wrien direly s funions of nd. Then: Π, 4 1 Π, 4 Mimizing hem wih rese o nd i follows: Π Π Consider equion 3 s funion of. This equion hs wo soluions: 3 nd. The seond soluion is imossile, sine neiher or n e negive, nd 0 does no solve equion 4. Therefore i mus e: 3. Susiuing i in equion 4 nd solving wih rese o we oin wo soluions: 1 4 nd 1. The seond soluion is imossile, sine we hve 3 3 > 1, whih is imossile. Therefore, he only dmissile vlues whih solve he sysem defined y equions 3 nd 4 re * 1 4 nd * 3 4. The following roosiion omres he loion-rie equilirium when erfe rie disriminion is ossile wih he loion-rie equilirium under he unorm rie regime: Proosiion 4: ll ries re lower under erfe rie disriminion hn under unorm rie. Therefore, rofis re lower under erfe rie disriminion hn under unorm rie. The surlus of eh onsumer is higher under erfe rie disriminion hn under unorm rie, nd he more he onsumer is loed ner o he middle he higher is he dferene. The equilirium loions under erfe rie disriminion mimize ol welfre 6. 6 Lederer nd Hurer 1986 oin he sme resul ssuming delivered insed of f.o.. ries. 8

9 Proof. Susiuing 1 4 nd 3 4 ino he equilirium disriminory rie shedules, i follows: * 1, [0,1 ] nd * 1, [1,1]. Then, * * > 1 *, [0,1 ] nd * * > 1 *, [1,1]. Under rie disriminion ol rofis re: Π D 4, while under unorm rie hey re: Π U ΔΠ Π D Π U 3 4 < 0.. Then: Under rie disriminion, he surlus of onsumer loed [0,1 ] is given y: CS D D v * * v 1 1 4, while he surlus D D of onsumer loed [1,1] is given y: CS v * * v Under unorm rie, he surlus of onsumer loed U U [0,1 ] is: CS v * * v, while he surlus of U U onsumer loed [1,1] is: CS v * * v 1. Define: Δ CS. I follows h: ΔCS > 0, [0,1 ], nd D U CS CS ΔCS > 0, [1,1]. Moreover, ΔCS > 0 [0,1 ] nd ΔCS < 0 [1,1]. Sine he ouu is he sme under he unorm rie regime nd he disriminory rie regime nd he ries hve only redisriuive effe, ol welfre deends only on rnsorion oss, whih in urn re deermined y he equilirium loions. Define wih â nd ˆ he oiml loions from he ol welfre oin of view. They re simly: ˆ, ˆ minct 1 rg rg min{ z dz z dz} 0, where he rkeed eression indies he ol rnsorion oss. The roof hs wo ses: firs we lule he oiml shring of onsumers, nd hen we lule he oiml vlues of nd. CT 1 1 { z dz z dz} 0 0 ^ 9

10 CT, d d CT CT Sine equions 5 nd 6 oinide reseively wih equions 3 nd 4, he oiml loions â nd ˆ oinide wih he equilirium loions * 1 4 nd * 3 4. The hrerisis of he loion-rie equilirium under he wo riing regimes re summrized in he following figure: Figure 1: Illusrion of Proosiion 4 The hin nd sloed lines in he oom r of he grh reresen he equilirium ries se y he firms o eh onsumer under erfe rie disriminion, while he old nd fl line reresens he equilirium ries under unorm rie. I is immedie o see h ll equilirium disriminory ries ly elow he rie line under he unorm rie regime ll-ou omeiion, nd h he disriminory ries derese moving from onsumers loed he endoins o onsumers loed he middle. From he onsumers oin of view, he surlus deends on he rie id nd on he rnsorion oss susined. The urves in he uer r of he grh desrie he 10

11 surlus, gross of he rie, of eh onsumer: he old urve refers o he unorm rie regime while he hin urve refers o he disriminory rie regime. Under he disriminory rie regime, he gross onsumer surlus is mimum for onsumers loed 1 4 nd 3 4, where he firms re loed, nd dereses he more he onsumers re disn from hese oins. The minimum gross onsumer surlus is oins 0, 1 nd 1. Under he unorm rie regime he gross onsumer surlus is mimum oins 0 nd 1 sine firms re loed he endoins of he segmen, nd i is minimum 1. The ne onsumer surlus is given y he dferene eween he uer urves nd he rie lines. In Proosiion 4 we se h he surlus of eh onsumer is higher under rie disriminion hn under he unorm rie regime. For onsumers loed eween 1 8 nd 7 8 his is immedie, sine oh he rnsorion oss nd he ries derese ssing from he unorm rie regime o he disriminory rie regime. For he oher onsumers we oserve wo oosie effes: he rnsorion oss inrese under rie disriminion sine he firms now re frher from hese onsumers u he equilirium ries derese. In order o rove h even for hese onsumers he surlus is higher under he disriminory rie regime hn under he unorm rie regime i is suffiien o omre he surlus of he mos eernl onsumers in he wo ses, sine he onsumers loed oin 0 nd 1 re he es-osiioned onsumers under he unorm rie regime nd he wors-osiioned onsumers under he disriminory rie regime. Under unorm riing, he surlus of he onsumers loed oins 0 nd 1 is equl o v ; under erfe rie disriminion, he sme onsumers oin surlus whih is equl o v Sine he surlus of hese onsumers inreses ssing from he unorm rie regime o he disriminory rie regime, he sme mus e rue for ll oher onsumers. Finlly, in Proosiion 4 we se h ol welfre is mimized under rie disriminion. Sine he ol ouu is he sme under he unorm rie regime nd he disriminory rie regime nd sine ries hve only redisriuive funion, ol welfre deends only on he equilirium loions whih deermine he ol rnsorion oss susined y he onsumers: he equilirium loions under rie disriminion, 1 4 nd 3 4, minimize he ol rnsorion oss nd herefore mimize ol welfre. 11

12 4. hree-sge model In seion 3 we hve shown h erfe rie disriminion yields lower rofis hen unorm riing. Now, suose h eh firm n nnoune is inenion no o rie disrimine efore seing he rie shedule. This ommimen n e mde redile y he doion of usiness ries like he mos-fvoured nion luse. When firms loions re fied, Thisse nd Vives 1988 show h he ossiiliy o ommi efore omeing on ries does no ler he fundmenl resul: unorm riing does no emerge in equilirium. In his seion we sk wheher his resul is sill vlid when loions re endogenous insed of eogenous. Therefore, we need o move from wo-sge model o hreesge model. Two imings re ossile. Unil now we hve ssumed h he finl deision of he firms regrds he rie shedule o e lied, nd we minin his hyohesis. However, he deision regrding he ommimen o unorm riing my reede he deision on rie nd ome fer he deision on loion, or i my reede oh he deision on rie nd he deision on loion: hese wo lernives genere wo dferen imings of he gme. In wh follows we solve he gme in oh ses. We show h when he deision on ommimen is ken fer he deision on loion here eiss unique su-gme erfe equilirium in whih oh firms rie disrimine, while when he deision on ommimen is ken efore he deision on loion here eiss unique su-gme erfe equilirium in whih oh firms se unorm rie. Gme 1 Timing: ime 1, oh firms simulneously hoose he loion long he mrke; ime oh firms simulneously deide wheher o ommi U or no D; ime 3 oh firms simulneously hoose he rie shedule. We solve he gme y kwrd induion. Consider he hird sge of he gme. We need o lule he equilirium ries when one firm hs ommied while he oher hs no. Suose h firm hs ommied sge while firm hs no ommied. To oin he es-resonse funion of firm we susiue wih ino 1, sine 1

13 now he rie se y firm mus e he sme for ll onsumers. For ske of lriy we wrie elow he es-rely funion of firm when he rivl hs ommied no o rie disrimine: r m[ m[ ε; ] ; ] Consider generi onsumer. If firm ses unorm rie suh h >, firm n lwys serve he onsumer seing he esresonse rie defined y equion : herefore onsumer will lwys uy from firm nd firm will oin zero rofis. In order o hve osiive demnd, firm mus se unorm rie of his ye:, whih nno e underu y firm wihou seing rie lower hn he mrginl os, whih would enil loss. Therefore, he highes unorm rie h firm nno underu is given y:. Solving for, we oin he mos he righ onsumer served y firm, *', whih resuls o e: *'. I follows h he demnd of firm is given y *', while he demnd of firm is given y 1 *'. We se he following roosiion: Proosiion 5: firm hs ommied nd firm hs no ommied, he equilirium ries in he hird sge of he gme re he following 7 : *, 7 Unforunely, ε -equilirium nno e voided for suse of onsumers when one firm ses unorm rie nd he oher n erfely rie disrimine. 13

14 14 4 4, * ε If firm hs no ommied nd firm hs ommied, he equilirium ries in he hird sge of he gme re he following: , * ε, * Proof. Consider Π *' ] [. Mimize i wih rese o he rie. I follows: 0 Π. Therefore, *. Using his resul ino equion, * follows immediely. The demonsrion of he seond r of Proosiion 5 roeeds in he sme wy. We n wrie he firms rofis direly s funions of nd in he four ossile ses: U,U, U,D, D,U nd D,D 8. We do i in he following le: Tle 1 П П U D U 18 ; ; 16 4 D 16 ; 8 4 ; 4 8 The rofi funions in D,D re simly he funions 1 nd ; he rofi funions in U,D nd D,U ome from Proosiion 5 disregrding he ε s; he rofi funions in U,U n e oined y sndrd lulions see, for emle, Tirole, 1988, g. 81.

15 I is immedie o see h, for ny oule of loions, he dominn sregy of eh firm is D. Given h he seond sge oh firms do no ommi, in he hird sge hey rie disrimine nd he equilirium loions re given y Proosiion 3. The following roosiion summrizes nd defines he unique su-gme erfe equilirium: Proosiion 6: in gme 1, he unique su-gme erfe equilirium is given y * 1 4 nd * 3 4, D,D, * 1 nd * for 1, nd * nd * 1 for 1. Proof. Consider Tle 1. If firm hooses U, hen firm hooses D for ny nd, sine 1 16 > When firm hooses D, firm hooses D for ny nd, sine 1 4 > 1 8. Then, D is he dominn sregy for firm. The sme is rue for firm. I follows h in he seond sge of he gme he equilirium is given y oh firms hoosing D. The res of he Proosiion follows from Proosiions nd 3. Proosiion 6 shows h he Prisoner Dilemm is resen in gme 1, sine oh firms do no ommi no o rie disrimine even his sregy is onduive o lower equilirium rofis. Th is, ssuming endogenous hoie of he loions efore he ommimen deision does no ler he Thisse nd Vives 1988 resul: firms rie disrimine in equilirium. Gme Timing: ime 1 oh firms simulneously deide wheher o ommi or no; ime oh firms simulneously hoose he loion long he mrke; ime 3 oh firms simulneously hoose he rie shedule. s usul, in order o solve he gme we sr from he ls sge. We lredy hve he equilirium ries nd loions when oh firms se unorm rie Proosiion 1 nd when oh rie disrimine Proosiions nd 3. Moreover, we lredy know he equilirium ries when one firm hs ommied nd he oher hs no Proosiion 5. 15

16 Therefore, i remins o lule he equilirium loions in he su-gme h rises when only one firm hs ommied in he firs sge. Equilirium loions in his sugme re defined y he following roosiion: Proosiion 7: he firs sge firm hs hosen U nd firm hs hosen D, he equilirium loions he seond sge re given y * 1 3 nd * 1; he firs sge firm hs hosen D nd firm hs hosen U, he equilirium loions he seond sge re given y * 0 nd * 3. Proof. Mimize he rofi funions in U,D of Tle 1. I follows: Π 3 8 nd Π Consider he ler equion. Sine i is lwys osiive, firm loes he righ eremiy of he mrke: h is, 1. Susiue i ino he firs equion nd solve. There re wo soluions: 1 3 nd 1. Sine he ler soluion is imossile, he equilirium loions re * 1 3 nd * 1. The seond r of Proosiion is demonsred in he sme wy. Mimize he rofi funions in D,U of Tle 1. I follows: Π 3 16 nd Π 3 8. The firs equion is lwys negive: herefore, firm hs lwys he inenive o move o he lef, h is, 0. Susiue ino he seond equion nd solve. There re wo soluions: 3 nd. Sine he seond soluion is imossile nno e higher hn 1 he unique equilirium loions re * 0 nd * 3. Sine we hve he equilirium ries hird sge nd he equilirium loions seond sge in ll ossile ses, we n wrie he equilirium rofis of eh firm direly s funions of he deision wheher o ommi or no ken he firs sge of he gme. The equilirium rofis re summrised in he following le: 16

17 П Tle П U D U ; 4 7 ; 8 7 D 8 7 ; 4 7 8; 8 The ne roosiion follows direly from Tle : Proosiion 8: in gme, he unique su-gme erfe equilirium is given y U,U, * 0 nd * 1, nd * *. Perhs surrisingly, he loion deision is ken one he deisions regrding he ommimen hve een lredy ken, here eiss unique su-gme erfe equilirium in whih oh firms ommi. On he onrry, when he deision wheher o ommi or no is ken fer he loion deision, he equilirium is hrerized y rie disriminion y oh firms nd, onsequenly, y lower rofis. However, one kes ino oun he dferen fores working in he wo gmes, suh resul hs n inuiive elnion. The min dferene eween ommimen sregy nd non-ommimen sregy is h he former redues he fleiiliy of firm in seing ries: when firm hs ommied, i n hoose is rie shedule only from suse of he omlee rie shedules se nmely, he suse omosed y he unorm rie shedules. Therefore, here is no reson for firm o hoose o ommi he only onsequene of he ommimen is o redue is own fleiiliy in seing ries. This is ely wh hens in gme 1. The deision wheher o ommi or no ffes only he deision regrding he ries o e se. When eh firm nnounes h in he fuure i will no rie disrimine, he loions hve een lredy fied, nd herefore hey nno e modied y ny ommimen deision. The only onsequene for firm h deides o ommi is o redue is own iliy o underu he rivl for eh onsumer: i is ovious h no firm would find i onvenien, nd he dominn sregy for eh firm neessrily is no-ommimen. This is wh hs een oined lso y Thisse nd Vives 1988 in heir wo-sge model wih eogenous loions: here is rous endeny for firm o hoose he disriminory oliy sine i is more fleile nd does 17

18 eer gins ny generi sregy of he rivl, lhough firms my end u worse off hn hey hoose o rie unormly g Why his does no our in gme? The iming is dferen: in his se he loion is se fer he deision regrding he ommimen. This imlies h he deision regrding wheher o ommi or no hs n im on he loions hosen y he firms. The loions of he firms in urn deermine he equilirium ries nd he equilirium rofis. The more he firms re dferenied, he higher is heir mrke ower, nd, onsequenly, he higher is he equilirium rie. I is reisely he im of he ommimen deision on he equilirium degree of dfereniion h mkes he ommimen deision more rofile for eh firm, even i redues he fleiiliy in seing ries. Eh firm niies h is own ommimen no o rie disrimine indues n higher equilirium degree of dfereniion in ny se: when he rivl hooses o ommi, deiding o ommi oo llows o oin he mimum degree of dfereniion; he rivl hooses no o ommi, deiding o ommi llows o oin n higher degree of dfereniion hn in he siuion in whih oh firms do no ommi 9. Induing higher dfereniion is rofile for oh firms, even suh higher dfereniion is oined he os of losing he fleiiliy in seing ries gurneed y he no-ommimen sregy. In his model he enefis from he higher dfereniion ouweigh he oss from he redued fleiiliy, nd herefore ommimen is onvenien for eh firm nd for ny ossile deision y he rivl. Th is, he dominn sregy of eh firm is ommiing o unorm riing. This in urn indues unorm rie equilirium, whih is hrerized y higher rofis. Summing u, in gme he deision wheher o ommi or no deermines he equilirium dfereniion eween he firms. Tking no ommimen indues lower dfereniion, whih in urn dmges oh firms hrough lower rofis. niiing his f, eh firm hs he inenive o ommi in he firs sge of he gme, nd he equilirium is hrerized y no disriminion. On he onrry, when he ommimen is deided fer he loion sge, he degree of dfereniion eween he firms is given. Therefore, he inenive o ommi disers, while is sill resen he 9 When oh firms ommi, he equilirium disne eween he firms is 1 Proosiion 1; one firm ommis no o rie disrimine while he oher does no, he equilirium disne is 3 Proosiion 7; when oh firms do no ommi, he equilirium disne is 1 Proosiion 3. 18

19 inenive o no ommi, linked o he omeiion on ries he ls sge of he gme: no ommimen nd rie disriminion y oh firms follow. 5. Conlusions Using he Hoelling s model 199 wih endogenous loions, we sudy he loion-rie equilirium when firms n erfely rie disrimine. If firms nno ommi no o rie disrimine efore omeing on rie, rie disriminion emerges s he unique su-gme erfe equilirium nd firms loe reseively 1/4 nd 3/4 Proosiions nd 3. Equilirium firs-degree disriminory ries re ll lower hn he equilirium unorm rie of wo-sge loion-rie gme where rie disriminion is imossile Proosiion 4. If firms n ommi no o rie disrimine efore omeing on rie u fer loing in he mrke, he unique equilirium is hrerized y rie disriminion Proosiion 6. On he onrry, firms n ommi no o rie disrimine efore omeing on rie nd efore loing in he mrke, he unique equilirium is hrerized y unorm riing Proosiion 8. Referenes Cors, K. 1998, Third-Degree Prie Disriminion in Oligooly: ll-ou Comeiion nd Sregi Commimen, RND Journl of Eonomis, 9, D sremon, C., J.J. Gszewiz, nd J.F. Thisse 1979, On Hoelling s Siliy in Comeiion, Eonomeri, 47, Hoelling, H. 199, Siliy in Comeiion, Eonomi Journl, 39, Lederer, P.J. nd.p. Hurher 1986, Comeiion of Firms: Disriminory Priing nd Loions, Eonomeri, 54, Thisse, J. F. nd X. Vives 1988, On he Sregi Choie of Sil Prie Poliy, merin Eonomi Review, 78, Tirole, J The Theory of Indusril Orgnizion. Cmridge, Mss.: M.I.T. Press. Ulh, D. nd N. Vulkn 000, Eleroni Commere nd Comeiive Firs-Degree Prie Disriminion, w, Eonomi nd Soil Reserh Counil ESRC Cenre, London. 19