Bertrand Edgeworth Competition in Differentiated Markets: Hotelling revisited

Transcription

1 Bertrad Edgeworth Competto Dfferetated Markets: Hotellg revsted Ncolas Boccard * & Xaver Wauthy ** March 000 A BSTRACT Ths paper deals wth stuatos where frms commt to capactes ad compete prces the market for a dfferetated product. Frst, we show that capacty-costraed prcg games wth product dfferetato, there s a fte umber of mxed strateges equlbra wth fte support. Next, wth the caocal model of Hotellg, we characterse subgame perfect equlbra (SPE) a two-stage game wth capacty commtmet followed by prce competto. Ether equlbrum capacty choces cover the market exactly ad there s o room for prce competto ( whch case the equlbrum outcomes replcate Courot equlbra), or SPE volve excess capactes ad lmted prce competto the secod perod exhbtg Edgeworth cycles. These two types of SPE exst whe the costs for capacty stallato s eglgble; however, f ths cost s large eough, oly SPE exhbtg Courot outcomes survve. Keywords: Hotellg, Capacty, Prce Competto JEL Classfcato: D43, F3, L3 The authors thak Jea Gabszewcz, Isabel Grlo ad Jea-Fraços Mertes for commets ad advses. * CORE & Uversté de Lège. Facg by Commuauté Fraçase de Belgque, DRS, ARC 98/03. Address: CORE, 34 voe du roma pays, 348 LLN, Belgum. Emal: boccard@core.ucl.ac.be ** CEREC, Facultés Uverstares Sat Lous, Brussels ad CORE, xwauthy@fusl.ac.be

2 ) Itroducto Bertrad's [883] crtque of Courot [838] s probably oe of the most famous story udergraduate Mcroecoomc textbooks. Accordg to Bertrad, two frms are eough for compettve outcomes to emerge, provded these frms set prces rather tha quattes. The paradox s so uplausble from a emprcal pot of vew that t essetally rase questos, frst as to why exactly t occurs ad secod, about how frms maage to avod t. Both questos have geerated a huge amout of research recet years. O the theoretcal groud the result of Bertrad rests o very fragle assumptos, amely costat returs to scale, product homogeety ad a statc settg. Buldg o ths fraglty, ecoomc scholars who foud ther way out of the paradox very early by relaxg oe or several of these assumptos. I partcular, Edgeworth [5] shows that decreasg returs to scale esure postve profts uder prce competto. Hotellg [9] puts forward product dfferetato order to escape the paradox. More recetly, collusve outcomes have bee show to emerge from repetto of the prcg games. All all, models of mperfect competto avod to fall to the Bertrad paradox by elargg the prcg game may drectos. As such, they study the dfferet meas through whch frms relax prce competto. As s wdely uderstood owadays, swtchg from a Bertrad model to a Courot oe volves more tha a smple chage the strategc varable. Buldg o the observato that most cases frms set prces ad quattes, may papers tred to recocle the two approaches. Kreps & Schekma [83] (KS hereafter) offers the most spectacular result ths respect. I ther model, Courot outcomes obta as the uque subgame perfect equlbrum outcome of a stage game volvg capacty commtmet ad prce competto. I other words, the Courot model ca be vewed as the reduced form of a elarged game whch frms ultmately do set prces uder a extreme form of decreasg returs to scale. The KS result s at least as famous as t s fragle, partcular to the specfcato of the ratog rule (see Davdso & Deeckere [86]). Stll, t pots the rght drecto: troducg decreasg returs to scale drves prce competto towards courota outcomes the sese that equlbrum market outcomes are maly depedet o frms' output possbltes. Eve though t s clear today that recoclg Bertrad ad Courot uder geeral codtos s a hopeless task, ther qualtatve mplcatos have bee made somewhat compatble. Gve these results, t s qute surprsg that the study of capacty-costraed prcg games remaed cofed to markets for homogeeous goods. After all, almost all dustres products are dfferetated. The fact that product dfferetato by tself relaxes

3 prce competto (ad thereby avods the Bertrad paradox) may expla why the vrtues of capacty costrats have ot bee vestgated markets for dfferetated products. Stll, very lttle s kow about the ature of prce competto uder decreasg returs to scale such markets. Ths s clearly damagg sce t meas fact that the relevace of the lterature o prce competto dfferetated markets s formally cofed to dustres exhbtg costat returs to scale. Beyod the fact that product dfferetato s ot suffcet to restore the exstece of pure strategy equlbra the presece of capacty costrats (see Beassy [89] or Fredma [88]), very lttle s kow o the ature of mxed strategy equlbra uder product dfferetato. Secodly, gve that quattatve costrats ad product dfferetato are, separately, powerful relaxg prce competto, t s mportat to kow to whch extet they are substtutes or complemet ths respect. Our motvato ths paper reflects these cosderatos. We pursue deed two ams: frst, we wsh to provde a charactersato of prce equlbrum dfferetated markets uder capacty costrats ad, secod, usg ths result, we shall study the extet to whch Courot ad Bertrad ca be recocled uder product dfferetato. Several recet papers have deed tred to recocle Courot ad Bertrad by cosderg prcg games wth capacty commtmet whle eglectg the ratog ssues whch where the heart of Edgeworth's argumet. For stace, Dastdar [95], [97] shows that forbddg ratog s suffcet to restore pure strategy equlbra whe products are homogeeous. Magg [96] adds product dfferetato to the pcture ad ot oly restores pure strategy equlbra but also esures uqueess. Whe ratog s forbdde ad products are dfferetated, equlbrum outcomes of a prcg game volvg capacty costrats have a strog courota flavour. It seems therefore mportat to kow to whch extet Magg's coveet shortcut ca be provded a more sold foudato tha by smply assumg ratog away. To ths ed, t s atural to start wth a "true" capacty costraed prcg game.e., oe that allows ratog. We make two specfc cotrbutos. Frst, we clarfy the ature of equlbra capacty-costraed prcg games (more geerally prcg games wth creasg margal costs) wth product dfferetato. We show that pure strategy equlbra are preserved oly to the extet that quattatve costrats are loose eough. Whe a pure strategy equlbrum does ot exst, frms use mxed strateges equlbrum. Because of product dfferetato, the equlbrum mxed strateges has a fte support, thus volves o destes. Furthermore, there s a fte umber of equlbra ( mxed strateges) ad o uqueess. Frms are "forced" to ame prces the rage that correspods to the sales of both capactes-quattes. 3

4 Thus these equlbra qualtatvely dffer from the oes prevalg for homogeous products where, accordg to the exstg lterature, destes ad uqueess are the rule. Secod we relate the Courot ad Bertrad results through capacty commtmet the stadard Hotellg model of dfferetated products. To ths ed, we replcate the KS aalyss wth the stadard Hotellg model. I a subgame perfect equlbrum (hereafter SPE), capacty commtmet softes prce competto, as KS, but more drastcally: I most of the SPE, the capacty choces exactly cover the market ad there s o room for prce competto at all. Other SPE volve excess capactes ad a lmted prce competto the secod perod. The, we show that SPE volvg exact market coverage are formally equvalet to Courot equlbra. Ths exteds to horzotally dfferetated dustres the KS result accordg to whch capacty precommtmet followed by prce competto leads to Courot outcomes. It should be also metoed that all the prevously stated results are depedet of the costs for capacty stallato; f the capacty cost s large eough, the oly SPE exhbtg Courot outcomes exst. The paper s orgased as follows. We start the ext secto by charactersg the ature of a prce equlbrum a duopoly market where products are dfferetated ad frms face creasg margal costs. The we tur secto 3 to the aalyss of the Hotellg model uder capacty pre-commtmet. We apply there the results of secto to the charactersato of prce equlbra the Hotellg model. I secto 4 we characterse frms' capacty choces before showg how our SPE ca be related to Courot equlbrum outcomes. Secto 5 cocludes. ) Equlbrum Capacty-costraed Prcg games wth Dfferetated products I order to overcome the Bertrad paradox, Edgeworth [5] shows that capacty costrats preclude the exstece of pure strategy equlbra prcg models. The argumet rests o a very smple dea: a frm may beeft from spllovers whe ts oppoet s ether ot wllg or ot able to serve full demad at prevalg prces. Ideed, the cosumers who are ratoed by the "low prce" frm may report ther purchase to the "hgh prce" frm. Whe products are homogeeous, these spllovers are spectacular because a hgh prce frm's sales jump from zero to some strctly postve level. However, oly the dscotuty of the spllover s specfc to the case of homogeeous goods. Whe rasg ts prce agast that of a oppoet whch sells a dfferetated product, a frm wll, smoothly, crease the oppoet's demad up to a pot where capacty becomes bdg. Beyod that pot, spllovers accrue, smoothly, to the "hgh prce" frm, as the homogeeous products case. 4

5 Therefore, the reaso why a equlbrum fals to exst the aalyss of Edgeworth s stll preset uder product dfferetato. Note that Edgeworth hmself does ot restrct the possblty of "cycles" to the case of homogeeous goods. I hs ow words, "It wll be readly uderstood that the extet of determatess dmshes wth the dmuto of the degree of correlato betwee the artcles" (Edgeworth [5], p.). O the other had, Hotellg [9] thought that product dfferetato would solve the Edgeworth problem of cycles completely, as he wrote "The assumpto, mplct ther work [Courot, Amoroso ad Edgeworth] that all buyers deal wth the cheapest seller leads to a type of stablty whch dsappears whe the quatty sold s cosdered as a cotuous fucto of the dffereces prces" (Hotellg [9] p 47, bracket added). Although Hotellg was rght argug that cotuous demad would solve the Bertrad paradox, he was wrog o the Edgeworth's frot. Shapley & Shubk [69] ad McLeod [85] provde a formal treatmet of the role of "correlato" Edgeworth's tuto: product dfferetato s ot suffcet to restore the exstece of a pure strategy equlbrum a prcg game wth creasg margal costs because profts fuctos typcally rema o quas-cocave. However, t s of some help the sese that product dfferetato teds to elarge the set of capacty levels for whch a pure strategy equlbrum s preserved. If the o-exstece of pure strategy equlbra the presece of capacty costrats, eve uder product dfferetato, s a (farly) well-documeted ssue, very lttle s kow about the ature of a mxed strategy equlbrum such settgs. Notceable exceptos are Krsha (989) ad Furth ad Koveock (99) who provde some partal charactersatos. Accordgly, our frst task wll cosst clarfyg the ature of prce equlbra whe both decreasg returs to scale ad product dfferetato are preset. To ths ed, we cosder the market for a dfferetated product. The demad addressed to frm s D(p,p ) whle that of frm s the symmetrc D(p,p ). The fucto D(p,p ) s assumed cotuously dfferetable of order ad satsfes the followg assumptos: A) D(0,.) > 0 Dp (, p ) Dp (, p ) A) > 0 A) assumes that a frm's demad s postve whe ts prce s zero whereas A) meas that ow prce effect o a frm's demad domate crossed oes. Cosder a complete formato stage game Γ. At stage oe, frms vests to techologes yeldg creasg Ths pot s studed by Fredma [88], Beassy [89], Caoy [96] ad Wauthy [96]. These papers share the dea that the more dfferetated the products, the more lkely a pure strategy equlbrum wll exst. 5

6 ad covex cost fuctos C ad C. At stage two, they set prces. Lastly stage three, frms perform ratog wheever they wsh to ad some of the ratoed cosumers tur to the other frm. Before cosderg ths last stage t s useful to descrbe the case where ratog s forbdde.e., Bertrad competto. 3 The proft fucto s the B Π p p j p D p p j C D p p j (, ) (, ) ( (, )). I order to guaratee uqueess of the Bertrad equlbrum rrespectve of the cost fuctos chose at stage oe we assume the followg cotracto property: D D D D j j A3) 0 p + p { } Let the ϕ ( p ) argmax qp C( q) be the compettve supply of frm (t s equal q 0 to C C ( p) f C s strctly covex) ad Π ( p) pϕ( p) C( ϕ ( p) ) be the compettve proft. It s creasg covex sce C Π ( p) =ϕ ( p) ad ϕ ( p ) s tself weakly creasg. Let us the allow frms to rato cosumers wheever t s proftable for them to do so. If (p,p j ) s such that D( p, p ) >ϕ ( p ) the stage three, frm j ratos some cosumers ad j j j frm obtas a fracto λ(p j,p ) of the resdual demad, ths s the spllover effect. The mportace of the spllover depeds o the degree of dfferetato of the products, the prefereces of the cosumers ad the ratog rule used by frms; we assume that t s cotuously dfferetable ad satsfes: A4) Postve spllovers decreasg wth respect to ow prce: λ(p,p j ) > 0, λ 0 ad λ λ + p 0 Gve these four assumptos we aalyse a prce subgame Γ(C,C ) ad look for subgame perfect equlbra (SPE). Note that whe ratog s forbdde (o thrd stage Γ) the Bertrad payoff apples over the whole rage of prces. If ratog s permtted, a Bertrad-Edgeworth aalyss s called for because frms' sales may dffer from demads. It s well-kow that such crcumstaces a pure strategy equlbrum ofte fals to exst (see for stace Beassy [89]). The followg theorem provdes a geeral result about the ature of the equlbrum mxed strateges for such games. 3 Magg (996) provdes a recet case where such a vew of prce competto uder capacty costrats s edorsed. 6

7 THEOREM Cosder ay prce subgame Γ(C,C ) that satsfes A-A4. Uder Bertrad competto, there exsts a uque pure strategy SPE. Uder Bertrad-Edgeworth competto, a SPE always exsts, the support of a mxed strategy prce SPE s fte ad prces are larger tha those of the Bertrad equlbrum. The detaled proof has bee relegated to the appedx, however the tutve argumet s relatvely easy to summarse. Whe ratog s ot allowed, A ad A esure that a frm's payoff s cocave. Assumpto A3 eables to derve the Bertrad equlbrum as the uque fxed pot of the best reply operator. To show that ths Bertrad equlbrum s also a lower boud to prces played equlbrum of the Bertrad-Edgeworth competto we use the fact that Π B ad Π C are both creasg over the doma where frm wshes to rato. The, order to prove that frms do ot use destes equlbrum, oe shows that, f frm uses a desty aroud some prce p, the frm j must usg a desty aroud the prce p j that makes t wllg to rato.e., such that D( pj, p) =ϕ j( pj). Symmetry the mples that frm uses a desty aroud ˆp such that D(ˆ p, p ) =ϕ (ˆ p ). Ths process leads to lower ad lower prces j precsely because goods are dfferetated. We reach a cotradcto because frms do ot put mass below the Bertrad prces equlbrum. If frms do ot use destes the support of equlbrum dstrbutos must be fte. The equlbra charactersed Theorem are qute dfferet from those prevalg market for homogeeous goods where frms use destes uder stadard assumptos o demad. Note also that the argumet developed above does ot help to prove uqueess ad deed multplcty of equlbra ofte obtas. Ths wll partcular be the case for the model of capacty pre-commtmet the Hotellg market we cosder hereafter. 3) Prce equlbrum the Hotellg model wth capacty commtmet. I what follows, we adapt the stage-game proposed Kreps & Schekma [83] to the Hotellg model of dfferetato: Frms choose capacty levels ad the compete prce a horzotally dfferetated market. After presetg a smplfed verso of the Hotellg model, we defe the full game as well as the assumptos uder whch our aalyss s coducted. The we characterse equlbra the prcg games. The aalyss of prcg games s rather log ad volved. We have chose to cocetrate o tutve argumets the core of the paper. Almost all techcal proofs have bee relegated to the appedx. 7

8 3.) THE SET U P Two shops are located at the boudares 4 of the [0;] segmet alog whch cosumers are uformly dstrbuted. Each cosumer s detfed by ts address x [0;] ad has a commo reservato prce S. A aget buys at most oe ut of the good ad bears a trasportato cost, whch s lear the dstace to the shop. Wthout loss of geeralty, we set the trasportato cost betwee the two shops to. Therefore, the utlty derved by a cosumer located at x s thus S x p f he buys the product at frm (located 0) ad S ( x) p f he buys at frm (located ). Refrag from cosumg ay of the two products yelds a l level of utlty 5. Frms ame prces o-cooperatvely. The essece of the Hotellg model s best summarsed as follows. Whe frms ame low prces, the market s covered (.e. all cosumers purchase oe of the good ). Frms' market shares are defed by the address of the dfferet cosumer deoted by x(.). By defto, p p t s the soluto of S x p = S ( x) p.e., xp (, p) +. Cosumers [ 0, x( p, p )] buy at frm whose demad s xp (, p ) as cosumers are uformly dstrbuted o [0;]. Demad addressed to frm s x ( p, p ). If prces are too large the market s ot covered. I such cases, frm 6 s a local moopoly; ts demad s m { S, p }. Ths happes f S x ( p, pj) p < 0 p > S pj. The demad fucto of frm s thus defed by equato p + p p S pj D( p, pj) = f. m {, S p} f p > S pj Pluggg ths demad the proft fucto, we may detfy the respectve argumet + p maxmsers Hp ( j j) for the frst brach ad the moopoly prce p m S, for { } m S the secod. Agast a low p j frm plays H(p j ) whle agast a large p j t plays p m ; for prces the mddle rage, the optmal respose s to cover the market wth S p j. Sce D (p,p j ) s pecewse lear ad decreasg p j, proft s cocave p j, thus the best reply to a mxed strategy s the best reply to ts expectato.e., a pure strategy. Therefore, a Nash equlbrum of ths prcg game s pure. The best reply tersect at the ut prce for both frms wheever S > 3/. I ths case, frms face o capacty costrats the uque Nash equlbrum of the prcg game s (,) ad the market s covered. Otherwse 4 We choose maxmum dfferetato to relax prce competto as much as possble. If frms fd t proftable to further relax prce competto through capacty precommtmet, t s lkely that they would face eve greater cetves f they were less dfferetated o the horzotal dmeso. 5 I Hotellg's orgal model, ths possblty s ot cosdered, formally, ths correspod to a fte S. 6 I the remader of the text, stads for ether of the frms ad j for ts compettor. 8

9 whe S < 3/, there s a cotuum of equlbra o the froter (p + p j = S + ) whch etal o "real" prce competto. I order to troduce capacty commtmet, we ow add a prelmary step where frms choose smultaeously sales capacty k ad k before they smultaeously choose prces p ad p kowg the chose capacty of ther compettor. The ut cost of capacty s assumed equal to ε >0. Ths two stage-game of complete formato s deoted G. The subgame after the choces of k ad k s deoted G(k,k ) ad called the prcg game. Formally, G s thus detcal to the game cosdered by KS. Gve that frms may act as local moopolsts provded prces are hgh eough, game G s really terestg f oly frms are lead to choose capactes whose sum exceeds the market sze. Oly ths case wll they eter to a prce competto at the secod stage. Obvously ths caot happe for very large capacty costs. Proposto clarfes ths pot. PROPOSITION If the ut cost of capacty stallato ε s larger tha S, the uque SPE etals moopoly prcg by both frms. If ε < S, the market must be covered a SPE. Proof Suppose that frm s a moopoly over the market ad has stalled a capacty k, ts demad s f m k, S p. The secod perod proft p ( S p ) s maxmum for p = { } { } ( ) m S k S,, thus the frst perod proft s k S k ε f k S < ad S 4 ε k f k S. Sce the latter s decreasg wth k, oly the frst matters for the optmal capacty choce whch s k m m, S ε. Now, t clear that ε > S mples k m < / ; Beg located { } respectvely at 0 ad, both frms are able to acheve ther equlbrum moopoly proft wthout teractg whch meas that k m s a domat strategy ad thus characterse the uque SPE allocato. It s clear that whe ε < S, capacty choces such that k + k < are ot stable sce oe of the frm (may be both) has a cetve to choose at least a capacty complemetary to ts compettor. Yet, exact market coverage capactes (k + k =) subject to the costrat max { k, k } k m are caddates SPE of the whole game. I the sequel of the paper, we cocetrate o cases where ut capacty costs are eglgble relatve to S. The presece of lmted capactes affects frms' cetves the prcg game two ways. Frst, a lmted capacty may decrease the cetve of a frm to reply to the other's prce wth a low prce sce the market share a frm s wllg to serve caot exceed ts stalled capacty. A secod observato duced by lmted capactes s that some cosumers mght be ratoed at the prevalg prces. Ths possblty s the corerstoe of the prce competto aalyss as t may reverse frms' cetves the prce 9

10 game. More precsely, oe frm could fd t proftable to quote a hgh prce, atcpatg the fact that some cosumers wll be ratoed by the other frm ad could therefore be wllg to report ther purchase to t. Ths was the orgal tuto of Edgeworth. The cetves for that behavour bascally deped o the wllgess of cosumers to swtch to the hgh prce frm case of ratog. I a model wth ut demad ad heterogeeous cosumers the extet to whch ratoed cosumers wll be recovered by ths frm drectly depeds o who the ratoed cosumers are. We follow KS assumg that the effcet ratog rule s at work the market. Uder ths rule, ratoed cosumers are those exhbtg the lowest reservato prce for the good. Cosder the example depcted o Fgure below. 0 x(p,p) k Fgure x [ ] wat to buy at frm but some wll be ratoed. All cosumers located 0; xp (, p) Uder effcet ratog, those are located [ k; x ( p, p) ] ad are precsely the most cled to swtch to frm. Despte a potetally low demad for frm, the fact that frm s costraed by ts capacty k, ca gve frm a effectve demad of k. It s the case f p S + k whch s the et reservato prce of the cosumer located k. Ths feature of the market allocato rule also lowers frm 's cetves to eter a prce competto "à la Bertrad" sce ts demad s locally depedet of ts ow prce. Note fally that the Hotellg model wth maxmal dfferetato, effcet ratog defes the largest resdual demad for frm, so that, cotrarly to KS, the cetves to use ratog strategcally are maxmsed. Ths pheomeo wll have a strog feedback o the choces of capactes. The ext secto studes the prcg game whe the choce of capactes exceed the market sze. 3.) EQUILIBRIA IN PRICE SUBGAMES As a frst step we derve the effectve sales of the frms the prcg game usg cosumer demads ad the ratog rule. The, we characterse the best reply fuctos ad detfy the support of the equlbrum mxed strateges. Three types of equlbra are charactersed: the pure strategy equlbrum that prevals uder stadard Hotellg competto, equlbra whch oe frm mxes over two prces whereas the other plays a pure strategy (we refer to these as sem-mxed equlbra) ad last, equlbra where both frms use o-degeerated mxed strateges. I all of the possble prcg games G(k,k), at least oe of these equlbra prevals but multple equlbra ofte preval. I the mxed strategy equlbra, frms use oly atoms. 0

11 The shape of frms' sales ca be formally captured by referrg to Fgure. p D = S p ρ(k ) D = k D = k D = k c D = S p D = S p δ(k ) b D = x(p, p ) D = x(p, p ) a D = k D = k D = k D = S p p δ(k ) ρ(k ) Fgure Assume frms quote smlar (low) prces correspodg to a pot the area delmted by les a, b ad c. I ths rego, the classcal Hotellg demads preval. Now, f p creases, frm looses sales utl frm s costraed by her capacty.e., we reach the upper tragle. From that pot o, f p creases further, D remas costat at k utl p s so large that the market s ot covered aymore. From the o, frm moves alog ts moopoly demad fucto. Whe both prces are hgh, the market s ot covered ad, obvously, o frm s capacty costraed. We derve ow each frm's sales fucto formally. Sce the ratog rule that we use s the effcet oe, f demad addressed to frm exceeds k, t serves the segmet [0;k ]. Thus, f we let D be the sales of frm, the demad addressed to frm j s bouded by D. O the other had, D s bouded by the capacty k ad by the moopoly sales S p. Lettg f m { k, S p}, we thus have { } { j} D m S p, k, Dj = m f, D { } ad Observe that there s a equvalece betwee xp (, p) < f, xp (, p) < f { D = x ( p, p), D = x ( p, p) } because demads addressed to frms ca be served by both whle the reverse mplcato s true as oe ca see from the defto of D. Let us deote by (E) the equato xp (, p) < f ad by (E) the equato x ( p, p) < f. We vestgate whe they hold: p + p - p S k f = k ad ( E) become s x = k p a( k, p ) p + k

12 - p > S k f = S p ad ( E) read s x = S p p c( p ) S p p + p p + p - p S k f = k ad (E) reads x = k p b( k, p ) p + k - p > S k f = S p ad (E) becomes x = S p p c( p ) p + p Thus, the classcal Hotellg demads D= x ( p, p) ad D = ( x p, p) preval f ak (, p) p m { bk (, p), cp ( ) }. The maxmum prce compatble wth sales of k s δ( k ) S k whle the prce guarateeg frm a demad of k j s ρ( k ) S + k. We call ths prce the "securty" prce of frm. We choose k > k wthout loss of geeralty so that k >. As show o Fgure above, k + k > mples a(k,.) < b(k,.) ad δ(k j ) < ρ(k ). Lastly, a(k,.) = c(.) = δ(k ) for p = ρ(k ) ad b(k,.) = c(.) = ρ(k ) for p = δ(k ). The area delmted by the fuctos a, b ad c wll be called the bad. Observe that k > postve (as o Fgure ) or egatve. mples a(k,0) < 0 whle b(k,0) ca be We ow derve the best reply of frm to a prce charged by frm. We already oted that S > + ε was a ecessary codto for frm to egage to a meagful competto. We go a step further 7 by assumg S > to create a ferce prce competto betwee the duopolsts. Uder ths assumpto a moopolst located at oe ed of the market would choose to cover the market. Techcally, t mples S < δ(k ) ad S < ρ(k ) for =,. j j LEMMA Frm =, ever charge prces above ρ( k j ); the best reples are dscotuous ad defed by BR( pj) = ρ( kj) f pj γ( k, kj) + pj f γ( k, kj) < pj < α( k) pj + k f max { ( k), ( k, kj) }< p α γ j Proof Let F be the cumulatve dstrbuto fucto of the mxed strategy used by frm equlbrum. Recall that p, p ρ ( k ), D ( p, p ) = S p. Sce the moopoly prce S s less tha ρ(k ), Π (p,.) must be decreasg over [ρ(k );+ [ ad the same s true for Π( F,.) = Π( p,.) df( p). Therefore F,beg a best reply to F, has o mass above ρ(k ) ad symmetrcally the support of F s cluded [ 0; ρ( k) ]. 7 A close look at the proofs shows that t s uecessary but t smplfes the exposto by removg subcases. Notce that the stadard aalyss of the Hotellg model, t s geerally assumed that S large eough to esure market-coverg equlbrum.

13 We ca ow restrct the study of the best reply of frm to a prce p lesser tha ρ(k ). Frm ca act a classcal fasho "à la Hotellg" wth a aggressve prce order to ga market shares. It ca also hde behd frm 's capacty by servg that part of market that s out of reach for frm.e., the [0; k ] terval. Over ths resdual market, frm 's payoff s gve by ts moopoly payoff fucto (ths s the key feature of the Hotellg framework) ad the optmal prce s ρ(k ) ; we call t the securty strategy. The, we dstgush four areas o Fgure above: the bad, the lower tragle, the upper tragle ad the doma of moopoly demad above ρ(k ) ad c. For the latter we have just show that the best choce s the lower froter of the doma.e., ρ(k ) for p < δ(k ) ad c(p ) beyod. Observe that c(p ) s tself domated by the best reply wth the bad. I the upper ad lower tragles demad s costat so that proft s creasg ad the best choces are respectvely ρ(k ) ad a(k,p ). Ths latter value s domated by the optmum wth the bad. I the bad the best reply s ether H(.) or oe of the froters a(k,.) ad b(k,.). Solvg for b(k,p ) > H(p ) > a(k,p ), the frst equalty leads to p > β(k ) 3 4k (t s always satsfed f k > 3/4) whle the secod leads to p < α(k ) 4k. Sce b(k,p ) also belogs to the upper tragle, t s domated by ρ(k ). Ths observato eables wthout loss of geeralty to take Max{ H(.), a( k,.) } as be the best choce the bad because we wll the choose betwee ths caddate ad the securty strategy ρ(k ). Summg up, the best reply s ether ϕ securty prce ρ(k ). f p < α( k) ( p ) = p + k fα( k) < p S or the Comparg the assocated profts, we derve a cut-off prce γ(k,k ) such that whe the prce p s low, frm reples wth a hgh prce to beeft from the resultg ratog at frm. Agast a hgh prce p, frm fghts for market shares. The dervato of γ(k,k ) ca be foud Lemma A. of the appedx. Formally, we obta: BR ( p ) = ρ( k) f p γ( k, k). ϕ( p) f p > γ( k, k) The prevous aalyss eables us to defe the support of a equlbrum mxed strategy. PROPOSITION I equlbrum, the support of the mxed strategy F s cluded max ( )( ), kj S + kj ; ( ) ρ k k j 3

14 Proof As a corollary of Theorem each frm ames prces larger tha the Hotellg ut prce. Now observe that Frm ca guaratee tself the demad k by playg ρ(k ), thus ts s equlbrum proft s larger tha Π ( k )( S + k ). Now, as sales are bouded by the s capacty k, frm must ame a prce above p Π equlbrum. A symmetrcal result k holds for frm. The statemet o the upper boud was proved lemma. We ow characterse the equlbra of the prce subgame proposto 3 ad provde a sketch of the proofs. Aalytcal developmets have bee relegated to the appedx. PROPOSITION 3 Three o exclusve types of prce equlbra exst: - A) both frms quote the pure strategy Hotellg prce; - B) oe frm plays a pure strategy ad the other mxes over two atoms; - C) both frms use a mxed strategy volvg the same umber of atoms. Sketch of the Proof: A) Equlbra volvg oly pure strateges If both capactes are arbtrarly close to, the stadard Hotellg equlbrum of proposto s preserved. Ideed f γ(k,k ) < ad γ(k,k ) <, the the best reply curves tersects at (,) meag that the pure strategy prce equlbrum (,) exsts. Those equaltes defe two sets A ad A the capacty space whch are symmetrc wth respect to k =k. Ther tersecto s a square area Φ; Φ; S S. B) Equlbra volvg a pure strategy ad a mxed oe [ ] [ ] where Φ + ( ) The set A \A correspods to a "large" k ad a "smaller" k ; the pure strategy equlbrum ceases to exsts ths set because γ( k, k) >. To uderstad the charactersato of type B equlbrum, t s useful to gve the tuto of ths result by presetg the Edgeworth cycle a market for dfferetated products. To ths ed, we use the Fgure 3 below. 4

15 Startg wth p =, frm uses the fact p that k s ot very large to ejoy the market share k at the securty prce ρ(k ) rather ρ(k ) tha fghtg agast p = wth ts "Hotellg" best reply H() =. If frm sets p = ρ(k ), there s o competto ad the best reacto of frm s to crease ts prce to δ(k ), the maxmum prce compatble wth sales of k. δ(k ) q γ(k,k ) Now, both prces are at ther peak ad the oly way to crease proft s to capture ew market δ(k ) shares by udercuttg oe's oppoet prce. q γ(k,k ) α(k ) ρ(k ) Fgure 3 The ext best move of frm s p = H(p ) (above q ), followed by a low value p = H(p ) (slghtly above q ); at ths momet we are back to the begg of the story: t s better for frm to retreat over ts protected share k wth a hgh prce. Accordg to the Nash defto ths cotext, the equlbrum sees frm playg the pure strategy γ(k,k ) whle frm mxes betwee ρ(k ) ad the lower prce q as descrbed o Fgure 3 above. Ths kd of equlbrum was detfed frst by Krsha [89]. Note that the symmetrc vector of strateges s ot a equlbrum. Ideed to make frm dfferet betwee ρ(k ) ad q, frm would have to play the pure strategy γ(k,k ) whch s strctly less tha. Ths cotradcts the fact that equlbrum prces are larger tha as establshed lemma. As k s "large", the default opto appears to be ever relevat for frm because t volves a almost zero resdual demad ad thus zero profts. The aalyss of area A \A s etrely symmetrc. I the complemet of A A, both γ(k,k ) ad γ(k,k ) are greater tha uty so that both type B equlbrum ca coexst. C) Equlbra volvg completely mxed strateges All A ad B type equlbra prevously metoed exst areas that shrks as S gets larger. Whe these equlbra do ot exst, completely mxed strategy equlbra must exst. The pecewse learty of the demad fuctos mples that frms do ot use destes equlbrum (cf. theorem ). I lemma A. of the appedx we show that whe S creases, the umber of atoms ecessary to buld a equlbrum creases ad s the same for each frm. I order to characterse a -atom equlbrum we proceed as follows. Whe a frm uses atoms, t has to solve codtos of local optmalty ad proft equaltes usg the prces of hs ow support ad the m probabltes over hs oppoet's prces. If m < the ths p 5

16 problem has geercally o soluto whle f m > t has a fty. It s therefore atural to obta m = order to be able to derve each player's best reples prces p k fucto of the vector of prces played by the other p j k ( ) as a k = ( ). The operator thus obtaed may k = the have a fxed pot whch wll be a equlbrum caddate. A atom equlbrum s a quadruple p m, p m, µ m, µ m m where µ m s the weght put by frm o her m th m atom p ( ) (prces are raked by creasg order). To derve a atom equlbrum of the prcg game G(k,k ), we cosder a grd of capacty couples over the [0,]x[0,] area ad solve umercally 8 a system of polyomal equatos varables. The we check two codtos o the vector of prces derved from the system relato to k, k ad S.e., we elmate some capacty pots whose assocated caddate equlbra volate oe of those codtos. The symmetry eables us to lmt ourselves to the case where k > k. The frst m m ecessary codto (cf. lemma A. the appedx) states that k > p p > k for every atom m ; t dsqualfes capacty pots (k,k ) exhbtg a too large dfferetal. The reduced form of the codto reads k > g (k ) where each g s a creasg ad covex fucto. As creases, more equaltes have to be satsfed, more capacty pots are elmated ad the area where atomc equlbrum exst shrks ; hece those fuctos satsfy g < g 3 < g 4 < g 5... The secod ecessary codto s related to the upper boud o prces ; t lks the upper prces of the dstrbutos to the reservato prce by p + p < S. Sce the equlbrum prces do ot crtcally deped o the capacty dfferetal but o the total capacty, we study ths codto o the dagoal. For a gve symmetrc capacty choce m m (k,k), we compute the symmetrc caddate equlbrum ( p ( k) ) ad the mmal p k reservato prce for whch the codto s satsfed.e., Sm ( k ) ( ) +. The verse of ths fucto gves us the maxmal capacty Kmax ( S ) such that pots (k,k ) wth k +k. K ( S) have a atoms prce equlbrum at the gve S. Those fuctos wll be useful max the subsequet secto. As oe could expect, the larger the capactes, the larger the prces a caddate equlbrum. I fact, our computatos show that the upper prces of a caddate equlbrum ted to fty as capactes ted to (,). Now, as Proposto showed that prces are 8 It s deed ecessary as for a 5 atoms equlbrum, a system of 8 equatos volvg polyomals of degree 7 wth more tha 500 moomals has to be solved wth the Mathematca software. 6

17 bouded by ρ( k ) S + k, capacty choces aroud (,) have o atomc prce equlbra ; j j for that reaso our umerc computatos ca safely stop at k = It s mportat at ths step to ote the reaso why type B equlbrum may fal exst. The pure strategy γ(k,k ) s optmal for frm oly f the pot (γ(k,k ),q ) of Fgure 3 above les strctly the "bad" because otherwse the demad addressed to frm at γ(k,k ) s k whe facg ρ(k ) ad k whe facg q. Ths meas that frm has a strct cetve to rase ts prce utl δ(k ). So our sem-mxed caddate equlbrum s ot a vald caddate. I fact, whe γ(k,k ) > α(k ) (whch happes whe capactes are smlar) there exsts a completely mxed equlbrum (called type B') where both frms play two prces ad where frm who has the largest capacty plays the securty prce ρ(k ). Ths equlbrum shares wth the sem-mxed oe a very ce property: that of yeldg a payoff whch s depedet of capacty. Ideed, both cases frm plays the securty prce wth a strctly postve s probablty. Evaluatg the payoff at ths atom yelds Π ( k )( S + k ). Ths equlbrum s formally derved lemma A.4 of the appedx. Note that the descrpto of a Edgeworth cycle part B) of the precedg argumet should ot be crtcsed for ts dyamc presetato of the statc cocept of Nash equlbrum. Beyod showg why there s o equlbrum pure strateges, t helps to uderstad the ature of the ew equlbrum. I ths equlbrum where frm s playg the pure strategy γ(k,k ), f frm perceves a slghtly larger prce, t reples aggressvely for sure whle f t perceves a slghtly lower prce, t plays the securty prce for sure. We follow here the purfcato argumet of Harsay [73]. Accordg to ths terpretato our mxed strategy equlbrum satsfes the o-regret property for the frm ad therefore escape the stadard crtcsm of ths equlbrum cocept. Moreover, the expermetal study of Brow-Kruse & al. [94] suggest that dsequlbrum adjustmet process (called Edgeworth cycle ther paper) or mxed strategy equlbra are the most robust theoretcal explaato of the observed prcg patter a Bertrad-Edgeworth olgopoly game. Buldg o our charactersato of prce equlbra we may state Corollary whch s strumetal for the resoluto of the capacty game. By dervg a explct formula for the secod perod proft of oe frm, we wll be able to costruct our focal SPE. It s a equvalet to Proposto KS whch states that the hgh capacty frm s pad accordg to ts Stackelberg follower payoff.e., as a fucto of the small capacty frm. COROLLARY Whe pure strategy equlbra do ot exst, there always exsts a mxed strategy equlbrum whch frm, the large capacty frm, s pad Π = ( k j )(S + k j ). 7

18 Proof Assume frm s the large capacty frm ad cosder frst frm 's proft a type B equlbrum. It s computed from ether prce the support of ts equlbrum strategy. At the atom ρ( k) S + k, frm 's demad s k so that Π = ( k )(S + k ). I prcg games where type B equlbra do ot exst, we ca buld a completely mxed strategy equlbrum where frm plays ts securty strategy wth a postve probablty. Lemma A.4 provdes a explct dervato of the two atom equlbrum. Usg Lemma A.3 ad the procedure descrbed Lemma A.4, equlbrum volvg more tha two atoms ad oe frm amg ts securty prce ca be buld as well. 4) Capacty commtmet ad Courot outcomes Gog backward s dffcult the game G because G(k,k ) has ofte several prce equlbra as show proposto 3. The focal SPE of our model volves symmetrc choces by the frms. The correspodg equlbrum outcomes replcate those of a moopoly owg the two frms: the market s shared evely, there s o excess of capacty, global surplus ad frms' jot profts are maxmsed, ad cosumer surplus s mmsed. PROPOSITION 4 Commttg to the Hotellg equlbrum quattes / ad amg the moopoly prce S / s a Subgame Perfect Equlbrum. Proof A dowward devato caot be proftable sce we assumed that the moopoly proft of a costraed frm ( S ε k) k s creasg up to S ε >. To deter a upward devato, we defe the cotuato prce equlbrum of G(k,/) to be of type B or B' (cf. proposto 3). It yelds a proft of ( S ) ε k for the devat. Ths s ot proftable because of the supplemetary capacty cost. Theorem cofrms the tuto accordg to whch prces are a sese "too low" the stadard Hotellg equlbrum,.e. there s room for relaxg competto. Capacty precommtmet allows frms to sell exactly ther Hotellg demads, but at a much hgher prce. At ths equlbrum, frms are o ther local moopoly proft curve so that, cotrary to the stadard equlbrum result, prces drectly deped o S. Notce that ths prce s the hghest oe that esures full market coverage sce t leaves the margal cosumer located / wth zero surplus. Sce a moopoly ower of both frms would mplemet precsely ths outcome, the ssue of ths o-cooperatve competto seems to be collusve. However, ths subgame perfect equlbrum s ot uque as the followg proposto shows. 8

19 PROPOSITION 5 Whe the capacty cost s eglgble, two kds of SPE coexst ) Complemetary capactes equlbra (CCE) where the total capacty exactly covers the market ad each frm ejoys a mmum share; furthermore, the prce equlbrum s pure strateges. ) Overlappg capactes equlbra (OCE) where the total capacty exceed the market sze (overlappg capactes), the dfferece the capacty choces s lmted ad the prce equlbrum s mxed strateges. Proof ) Complemetary capactes equlbra Wthout loss of geeralty, cosder capactes choces (m, m) wth m /. The proft fucto of frm ad o the tervals [ 0;m] ad [ 0; m ] respectvely s (S ε k)k. We have show whe provg proposto that ths fucto s creasg up to S ε, whch s therefore a upper lmt to m order to deter dowward devatos (for a large S, ths lmt s ot bdg). A upward devato k > m by frm ca oly lead to a prce equlbrum of type B or C. There s o possblty for a type A equlbrum because t requres both capactes to be large ad sce m /, the capacty choce of frm s smaller tha the requred lmt Φ( S, / ) (as derved Lemma A. appedx). To deter a upward devato of frm, we defe the cotuato prce equlbrum of G(k, m) to be of type B or B' whch yelds a et proft for frm of (S m)m εk ; ths s a o proftable devato because of the supplemetary cost of capacty. 9 If m < Φ( S, / ), a upward devato k > m by frm ca oly be followed by a type B or C equlbrum ad we apply the same trck as for frm to deter ths upward devato. Wheever m Φ( S, / ) frm 's payoff s almost l, hece t has a cetve to devate to the large capacty Φ( S, / ) order to play the Hotellg prce equlbrum ad ear a et proft of / ε Φ( S, / ). Wheever ε s less tha half (the relevat codto for the orgal Hotellg model), the soluto of the equato / εφ( S, / ) = [ S ε + m] ( m ) gve us a boud o m whch s obvously less tha oe. We may coclude that ay par (k, k ) such that k, +k = ca be sustaed as part of a SPE as log as each frm obtas at least ts "Hotellg proft". ) Overlappg Capactes Equlbra 9 As mxed strateges eable large prces, type C equlbra provde too large payoffs ad caot be used to susta our caddate SPE. 9

20 Cosder ow a caddate SPE outcome (m,m ) such that m +m >. To prove that t s ot a SPE, we must cosder devato to (m,k ) or (k,m ) ad look at the worst prce equlbrum for the devat ; f the devato s stll proftable the (m,m ) caot be sustaed as a SPE. Clam If (m,m ) s such that o type C equlbra exsts G(m,m ), the ths choce s ot part of a SPE. If the prce equlbrum G(m,m ) s of type A, frms ear a proft depedet of ther capacty choces. Therefore, each has a cetve to reduce capacty sce the cost ε s postve (almost l s exactly what s eeded). If the prce equlbrum s of type B or B', the payoff of oe frm, say, the prcg game s Π d ( m j) ; by choosg k = m j, frm sets tself a o overlappg stuato ad acheves (S + m j )( m j ) = Π d ( ) wth a lower cost of capacty stallato, thus t wll devate. Ths artefact s our ma strumet to rule out "uwated" equlbra ; we also obta a frst result: overlappg capactes are sustaable as SPE choces oly f capactes are ot too dssmlar. The pot (m,m ) must satsfy m > g (m ),.e. there exsts type C prce equlbra. We ow buld a SPE wth capacty choces (m,m ) such that the equlbrum of G(m,m ) s a atom oe. Ths couple must satsfy m > g (m ) (by symmetry of G, we ca always assume m < m ) ad m + m < Kmax ( S) ; those codtos take together defe a upper boud o capactes. We ow defe the strateges out of the equlbrum path: at (k,m j ), we defe prcg strateges to be the pure strategy γ(k,m j ) for frm j whle frm +γ( k, m ) mxes betwee ρ(m j ) ad the lower prce j (type B equlbrum). Frm obtas Π d (m j ) ad to deter the devato k, t must be less tha Π (m,m ), the proft accrug to frm at the atom equlbrum. Ths latter fucto mostly depeds o the total capacty, therefore we may study the prevous codto o the dagoal. We thus solve Π ( kk, ) εk [ S ( k) ε] ( k) kk k the varable S to get S Smax ( k, ) Π (, ) ε ε + k + ε. The umercal computato s k performed for ε = 0 as we are studyg the case of almost l capacty cost. The, we ca vert Smax ( k, 0 ) to obta a lower boud Km ( S) o capactes whch s compatble wth the upper boud Kmax ( S) derved proposto 3 (the above smplfcato s therefore vald up to small umercal errors). Cotrarly to type ) SPE, the capacty combatos that appear as SPE of type ) are fuctos of S. Fgure 4 below summarses our result: the varous Km ( S ) ad K max ( S) fuctos are plotted for =, 3, 4 ad 5. Cosder for example S = 4. There exsts a m j 0

21 symmetrcal 0 SPE wth a atoms prce equlbrum f the capacty s betwee.8 ad.85 ad SPE wth a 3 atoms prce equlbrum f the commo capacty s betwee.6 ad.6. For S = 6, there exsts SPE wth, 3, 4 or 5 atoms prce cotuato equlbra. The larger S, the larger the umber of mxed strategy equlbra the prcg games ad therefore the larger the umber of possble capacty overlappg SPE equlbra. k 5 K max (S) 5 K m (S) S Fgure 4 Obvously, the exstece of our overlappg capacty SPE s eased by the fact the capacty cost s eglgble. The followg theorem studes the robustess of our equlbra to the level of the capacty cost. PROPOSITION 6 If the capacty cost s larger tha /4 oly CCE subgame perfect equlbra exst. Proof We have show that a type ) SPE exsts for the symmetrc capacty k oly f S ( k) S S ( k, ε ). As the latter fucto s decreasg ε, S ( k) = S ( k, ε ) has a m max soluto ε (k) ad for ay ε > ε (k), the caddate SPE s removed. The equato to solve s kk k Sm ( k ) Π (, ) ε = + k + ε ε Π ( kk, ) Sm( k) ( k) ( k) ( k). k k m [ ] max 0 Wheever a symmetrcal -atom equlbrum exsts, there also exsts asymmetrcal oes for all capacty choces wth the same mea ad satsfyg m > g (mj).

22 The ε fuctos satsfy ε > ε 3 > ε 4 > ε 5 ad are cocave, decreasg, reachg zero at 4. It s clear that for every ε > /4, o type ) equlbra remas. The followg commets are order. Frst, ote that the SPE's volvg exact market coverage, both frms are o ther moopolst's proft curve. Ths perfectly llustrates why frms may beeft from capacty precommtmet. Ideed, the ma feature of the Hotellg model s that frms may ejoy local moopoles aroud ther locatos. However, the absece of capacty costrats, they caot prevet prce competto to take place because ther moopolst's atural markets overlap. Although postve mark-ups are preserved equlbrum, prce competto s damagg to the frms. Ths s clearly see by observg that the Hotellg equlbrum, prces do ot deped o S. I other words, frms fal to capture a large part of the cosumers' surplus. The ma vrtue of capacty precommtmet s precsely to avod ths falure. Ideed, through capacty precommtmet, frms are ow able to capture the greatest part of the cosumers' surplus. I partcular, ther equlbrum payoffs deped postvely o S. Secod, the exstece of equlbra volvg excess capactes s maly due to the exstece of multple equlbra the prce subgame where frms fght for market shares. However, t remas true that these equlbra, prces are always above the Hotellg prces ad are postvely related to S. The correspodg payoffs are also postvely lked to the reservato prce. Thus, the qualtatve cocluso remas: through capacty commtmet, frms systematcally susta hgher prces. Summg up, whatever the subgame perfect equlbrum cosdered, we are led to coclude that capacty precommtmet eables frms to take full advatage of the local moopoly structure whch s heret to the Hotellg model. I the case of a homogeous product, KS show that frms ted to avod destructve prce competto through capacty commtmet. I Proposto 5 ad 6, we have exteded ther result to the case of horzotal dfferetato. The cest feature of the KS result s that t provdes a theoretcal foudato for Courot competto that allows for a explct prce mechasm. We ow show that a smlar result obtas our model of horzotal dfferetato. PROPOSITION 7 The equlbrum quattes of a Courot game the Hotellg model correspod to the Complemetary Capactes SPE of the capacty precommtmet game.

23 Proof I the Courot game, frms supply quattes q ad q to a otherwse compettve market:.e. prces must clear the market (see Grlo & Mertes [99] for a foudato). If the proposed quattes q ad q do ot cover the market, there s excess demad ad the prces crease utl supply equals demad o each sde of the market.e., q = S p for =,. Ths stuato s ustable for S>, sce at least oe frm has a cetve to crease ts quatty above the complemet of the other. If ow the proposed quattes q ad q exceed the market sze, there s excess aggregate supply ad at least oe of the prce, say p, must be l o ths compettve market. Therefore frm has a proftable devato by offerg a quatty slghtly less tha q to be o ts moopoly proft curve. The caddates for a Courot equlbrum are (q, q) wth q /. As a frm ca supply q ε to guaratee the prce S q + ε, the market-clearg prces have to be S q ad S + q a SPE. Wthout loss of geeralty, we assume that frm offers q, thus sells less tha / equlbrum. Hece, p caot be l because t would attract at least oe half of the cosumers, thereby mplyg a excess demad. Frm caot proftably devate to a larger quatty tha q because t would face a zero prce (oe prce s l ad by the precedg argumet, t must be ts prce). Frm may proftably devate to some Q larger tha q but stll less tha /. Sce there s excess supply, p s l, thus frm sells all of Q ad the cosumer located at x = Q must be dfferet equlbrum so that p = Q. The proft Q( Q ε) reaches a maxmum of ( ) ε at ε 4 to be compared wth q(s q ε). Sce q /, the oly relevat 8 ε ( ε) ( ε) 4 root s q * S S > 0. Thus, the Courot equlbra must feature exact market coverage (q, q) wth q larger tha ths lower boud q *. 5) Commets ad Cocluso I ths paper, we made a frst step toward recoclg the two les of research tated by the Bertrad paradox.e., models of capacty commtmet ad product dfferetato. We have show that the ature of equlbra capacty-costraed prcg games wth product dfferetato sgfcatly dffer from the case of homogeeous products. Multple equlbra preval ad frms ever use destes o the support of a mxed strategy equlbrum. Usg the Hotellg model, we have show that product dfferetato does ot preclude the use of capacty commtmet as a devce to relax prce competto. O the cotrary, frms always lmt producto capactes equlbrum. The ma terest of such strateges s to susta Ths lower boud s dfferet from that derved theorem 3 but both are small so that our equvalece apples for the most lkely sharg of the market. 3

24 hgher prces order to approprate a larger part of the cosumer surplus. The mechasm at work replcates that of the market for a homogeeous product. However, the specfcty of dfferetated markets s to allow for may equlbra the prcg game. Therefore, besde Courot-lke outcomes, there exst equlbra exhbtg excess capactes ad completely mxed prcg equlbra. These result has bee establshed a partcular framework that calls for dscusso. For stace, t s well kow that the ature of the ratog rule plays a cetral role prcg models wth capacty costrats. Davdso & Deeckere [86] show that the KS result etrely rests upo ther assumpto of effcet ratog. I the preset aalyss also, the partcular rule of ratog s strumetal achevg clear-cut results. Ay alteratve to the effcet rule would result a lower resdual demad addressed to the "hgh" prce frm. However, the local moopoly structure of the model does ot crucally deped o the ratog rule. Therefore, t s tutve that equlbra would have the same qualtatve features, t would oly take a capacty cost larger tha /4 to elmate Overlappg capactes SPE. Moreover, our settg, the effcet ratog rule may be vewed as a rather atural oe whe terpretg the Hotellg model as a spatal model. I ths case deed, the rule bascally orgases ratog o a "frst arrved-frst served" bass. We cosder a market whch the locato of frms are exogeeously fxed at the extremtes of the market. As metoed the troducto ths assumpto s motvated by ts mplcatos for prce competto: as product dfferetato s maxmsed, ths s the case where the frms have the lowest cetves to further relax prce competto. Havg prove the stregth of the cetves to use capacty precommtmet, we ca expect that the same forces would apply whe frms are located sde the market. If frms are ot located sde the frst ad thrd quartles (see ext pot), the each has a protected market so that the cetves to play the prcg game as a local moopolst are reforced. All demads fuctos charactersed sub-secto 3. have a added elastc moopolst demad term of the form m{d,s-p } where d s the locato of frm sde the market. More geerally though, the robustess of our result to alteratve locato patters s ot easy to trace. Ideed, eve wthout capacty costrats, pure strategy equlbrum may fal to exst the lear Hotellg model whe frms are located sde the frst ad thrd quartles (see Osbore & Ptchck [87] for a charactersato of mxed strateges equlbra). No doubt ths possblty severely complcates the charactersato of equlbra prcg subgames. Two remarks are order ths respect. Frst, t should be oted that the presece of capacty costrats may help to restore the Hotellg equlbrum for locatos sde the frst ad thrd quartles (see Wauthy [96]). At the same tme, sde locatos wll ted to make upward devatos less proftable, 4