International Journal of Industrial Organization

Transcription

1 Internatonal Journal of Industral Organzaton 7 (009) Contents lsts avalable at ScenceDrect Internatonal Journal of Industral Organzaton journal homeage: Endogenous locaton leadersh Sergo Meza a, Mhkel Tombak a,b,c, a Unversty of Toronto, 05 St. George St., Toronto, ON, Canada M5S 3E6 b CIRANO, Montreal, Canada c ESMT, erln, Germany artcle nfo abstract Artcle hstory: Receved 6 June 006 Receved n revsed form March 009 Acceted March 009 Avalable onlne Arl 009 JEL classfcaton: L D43 Keywords: Asymmetres Horzontal dfferentaton Stackelberg Games of tmng We analyze a game of tmng where Sellers, whch have margnal roducton cost asymmetres, can delay entry and a commtment to a locaton n a Hotellng tye settng. When cost dfferences are large enough the game becomes a war of attrton that yelds Stackelberg behavor where the hgh cost frm wll delay choosng a locaton untl the low cost frm commts to ts oston. We fnd nteracton effects between tmng and the degree of roduct dfferentaton and comute tmng/locaton and mxed strategy equlbra through a range of margnal cost dfferences. The frms maxmally dfferentate wth moderate cost dfferences; wth somewhat greater cost dfferences there s ntermedate dfferentaton, and; wth large cost dfferences there s a blockadng monooly. The low cost frm always commts to entry mmedately whereas the hgh cost frm ether enters mmedately, shortly after the low cost leader, or never, deendng on the cost dfferences. Fnally, we fnd that n equlbrum the duooly s sustaned for a larger range of cost dfferentals and that dfferentaton s greater than the socal otmum. 009 Elsever.V. All rghts reserved.. Introducton Frms are not all created equal. Indeed, whole ndustres and nsttutons of hgher learnng are dedcated to creatng and sustanng dfferences between frms. There s an abundant management strategy lterature where such dfferences are characterzed as comettve advantages, but rgorous studes and the recse mlcatons of such advantages have been lackng n ths area. One such comettve advantage s havng lower costs of roducton. Cost dfferences between frms can arse from a myrad of causes rangng from dfferng technologcal oortuntes, labor condtons, and management qualtes. Frms make sgnfcant nvestments to try to The authors would lke to thank semnar artcants at: EARIE, Toulouse; Purdue Unversty; Unversté de Montreal; Unversty of Toronto, IIOC, Washngton, and; the Worksho n Stablty n Cometton at UCL, Louvan-la-Neuve, L, and esecally an Assocate Edtor and two referees for ther comments and suggestons on an earler verson of ths aer. Meza would lke to thank Unversty of Toronto for fnancal suort from the Connaught Fund. Corresondng author. Unversty of Toronto, 05 St. George St., Toronto, ON, Canada M5S 3E6. E-mal addresses: Sergo.meza@rotman.utoronto.ca (S. Meza), mhkel.tombak@rotman.utoronto.ca (M. Tombak). For a survey of the varous exlanatons for ersstent dfferences between cometng frms and for an examnaton of when frm costs would dverge, see Röller and Snclar- Desgagné (996). For emrcal evdence of such cost asymmetres see Syverson (004). obtan lower costs wth varyng degrees of success. When these cost dfferences exst managers of hgher cost frms must fnd ways to reman comettve,.e., they must dfferentate ther roduct. A low cost frm may choose closer locatons to cature most (or sometmes all) of the market. The hgher cost frm then may delay entry and commtment to a roduct ostonng n order to be able to choose a oston dfferentated from ts lower cost rval. Alternatvely, a frm may wsh to reemt ts rval to secure the best oston n the market. In ths aer we study how cost dfferences of frms affect the strategc tmng of market entry and ts nteracton wth market locaton. What t means to enter and commt to a certan market s catured well n ertrand models of horzontal dfferentaton (Hotellng, 99). However to ensure rce equlbra and equlbrum n the locaton game (when frms are symmetrc), for ths aer we use the varaton by D'Asremont et al. (979), (hereafter D'AGT) wth quadratc cost of transortaton. Smlar to oyer et al. (003) and Zss (993) we allow for asymmetrc frms by modelng dfferent cost of roducton. We defne a game of tmng of market entry where the roduct market subgame s the dfferentated ertrand model of D'AGT wth asymmetrc sellers to see how cost asymmetres result n the Substantal long term nvestments are made by frms n: ersonnel tranng; new technologes; n seekng new and cheaer sources of raw materals, these nvestment oortuntes may not be avalable to all frms to the same extent or be effectve n varyng degrees. We treat the cost dfferences here as gven, but the reercussons of any cost advantage obtaned on how frms comete would be an mortant justfcaton of those nvestments /$ see front matter 009 Elsever.V. All rghts reserved. do:0.06/j.jndorg

2 6 S. Meza, M. Tombak / Int. J. Ind. Organ. 7 (009) moston of a certan order of entry. Snce frms may not be able to change margnal cost n the short run, ths varable s treated as exogenous n our study. ut because frms may be strategc n decdng the tme of entrance to the market, to comlete ths framework, our study endogenzes tme of entry. Deendng on ther relatve costs the equlbrum tmng could then defne whch roles are taken by frms. That s, low cost frms may be leaders whle hgh cost frms may be followers. y allowng the delay of entry 3 and locaton choce, we fnd that Stackelberg behavor naturally arses under certan crcumstances and yelds unque locaton equlbra. In our analyss, beyond a certan level of cost dfferences, the low cost frm takes the role of a Stackelberg leader and chooses a oston antcatng that the follower maxmally dfferentates. In ths tmng/locaton game, we fnd the followng tyes of equlbra: () wth lower cost dfferences frms enter mmedately, smultaneously and maxmally dfferentate, () wth somewhat hgher cost dfferences the low cost frm moves frst wth the hgher cost follower movng as soon as ossble and stll the frms maxmally dfferentate, () wth hgher cost dfferences stll the low cost frm leads and chooses an nteror ont whle the hgh cost follower chooses the boundary ont farthest away from ts rval, (v) wth even hgher cost dfferences, there are no ure strategy equlbrum n the game of tmng, and two equlbrum locatons n mxed strateges emerge; and (v) fnally wth very hgh cost dfferences, the low cost leader locates n the mddle of the market and blockades entry. Interestngly we fnd that the Stackelberg leader can never obtan ts frst-best locaton and that the weakest frm (the one wth hgher margnal costs) wns what turns out to be a war of attrton. We subsequently comute the welfare otmzng behavor and fnd that cost asymmetres can be welfare mrovng (as Salant and Shaffer, 999, fnd wth the asymmetrc Cournot model) and that duooly erssts for much larger cost dfferences than s socally otmal... The related lterature The bulk of Industral Organzaton theory assumes symmetrc frms but some recent studes have reexamned the semnal theory to study how the devaton from the usual assumton of symmetrc frms affects the revously derved results and yeld new nsghts. 4 There are two strands n the lterature closely related to ths study. In the frst related trbutary of studes the tmng of the dfferentated ertrand game s fxed and locaton and rce are endogenzed (.e. Zss, 993, for smultaneous locaton; and oyer et al., 994, 003 and Tyag, 000 for sequental locaton). Snce our game embeds a smlar locaton game n the game of tmng, for a range of low cost dfferences the results of our model concde wth some of the revous results (e.g., for low cost dfferences frms locate smultaneously and as far as ossble from each other). When margnal cost dfferences ncrease, however, our results dffer from those revously obtaned. For nstance, beyond some value of cost dfferences a locaton equlbrum does not always exst n the tradtonal locaton game (Zss, 993). Deste the hgh cost frm wantng to kee maxmum dfferences, t could be otmal for the low cost frm to st on to of the hgh cost frm and thus monoolze the market. Prevous studes avoded the roblem of a lack of locaton equlbrum by changng the choce set 3 We allow for the ossblty that a seller may delay entry untl a certan tme or, that a seller may adot a wat-and-see strategy. There may be a tme lag for a rval's move to be confrmed and for the seller to react, for the uroses of ths analyss, we assume ths tme lag to be fnancally neglgble. 4 For examle, the mact of frm cost asymmetres has been exlored: n a Cournot settng (by Leahy and Neary, 005; Röller and Snclar-Desgagné, 996; Röller et al., 007, Salant and Shaffer, 999, and Tombak, 00); n a ertrand settng (Marquez, 997; Thomas, 00); n models of horzontal dfferentaton (oyer et al., 003; Vogel, 007; Zss, 993) and; n vertcally dfferentated models (Sutton, 96; Anderson and DePalma, 00). of locatons. Tyag (000) broadened the sectrum of ossble locatons of sellers to beyond the locatons of the consumers as er Gabszewcz and Thsse (96) and Tabuch and Thsse (995). oyer et al. (003) lmted the set of locatons of each frm wthn ther half of the lne. Dfferently, we allow the locatons where the frms can lay to be anywhere on the Hotellng lne. Also, we solve for mxed strateges where equlbrum dd not exst for smultaneous game. Fnally, the nature of the game of tmng allows us to fnd erfect equlbrum for most of that regon wthout any further restrcton. Another dfference wth revous aers s that n our study the sequence of entrance s not mosed lke n Tabuch and Thsse (995), Tyag (000), or oyer et al. (003) but nstead t emerges naturally from the endogenous choce of tme of entrance of the frms. To our knowledge, ths s the frst study to endogenze; tmng, locaton, and rce. In the second related strand of lterature the tmng of settng a certan strategc varable s endogenzed. Ths strand s under the rubrc of endogenous leadersh (Amr and Steanova, 006; rad, 96; van Damme and Hurkens, 004, when the strategc varable s rce; Amr and Grlo, 00; Hamlton and Slutsky, 990; van Damme and Hurkens, 999, where frms set quanttes). In much of ths lterature (wth the exceton of Hendrcks et al., 9) the model used s one of two dscrete tme erods. Ths facltates tractablty but could n some cases sacrfce generalzaton. Instead, we allow tme to be contnuous (from zero to nfnty) and we also allow frms to choose whether they would lay an oen-loo (frms commt to a tmng and locaton at tme zero) or feedback (frms emloy strateges that are contngent on the hstory of the rval's moves) games. Snce the varable frms choose n sequence s locaton and not rce or quantty, there s a second way n whch our aer dfferentated from ths strand. In ths lterature of endogenous leadersh the models rely on whether the strategc varable s a strategc substtute or a strategc comlement to determne the nature of the tmng equlbra. van Damme and Hurkens (004), n ther rce settng game, fnd that although both frms refer to follow, the low cost frm wll emerge as the rce leader. Amr and Steanova (006) generalze these results to stuatons where the rce could be strategc substtutes and/or comlements and fnd unque equlbra wth suffcent cost dfferences where agan the low cost frm s the rce leader. Generally, n these models, the hgh cost frm has a second mover advantage. As our strategc varable s locaton, where the sloe of the reacton functon can change sgn deendng on cost dfferences, t s not ossble to nfer the tmng from the results of ths lterature. 5 Instead, we need to solve for the erfect equlbrum by analyzng the three stages n sequence and for dfferent ranges of cost dfferences. In the followng secton we ntroduce our model; we descrbe our horzontally dfferentated ertrand game and how we model margnal cost dfferences and; we exlan our game of tmng. After comutng the rce equlbra, we examne equlbrum locaton choces and fnd the regons of cost dfferentals where those equlbra n ure strateges exst. A mxed strategy equlbrum n locaton choces are comuted n ths secton for the entre sectrum of cost asymmetres. We fnd and characterze three ossble outcomes maxmal dfferentaton, ntermedate dfferentaton, and a blockadng monooly for each of the ossble relatve tmngs. In Secton 3 we comute the tmng equlbra. In Secton 4 a welfare analyss s conducted. In ths secton we fnd the frst-best soluton of the socal lanner (.e., where the lanner determnes the locaton and rce) and comare that to the equlbrum outcomes of the revous sectons. In ths fnal secton, we summarze and dscuss our fndngs. 5 Anderson (97) fnds equlbra for the orgnal lnear transortaton cost Hotellng model by mosng Stackelberg behavor for symmetrc (zero) roducton cost frms. He fnds a roft advantage for the rce leader whch mles that there would be a cometton for the frst move n that model.

3 S. Meza, M. Tombak / Int. J. Ind. Organ. 7 (009) The model In ths secton we develo a model of two frms whch may be asymmetrc n ther margnal costs layng a three stage game of erfect nformaton. In the frst stage frms decde on a tme of entry. The relatve tmngs chosen would then determne whch frm s the leader and whch frm s the follower or f the tmng s smultaneous. In the second stage frms enter the market and choose a locaton. If the tmng n the frst stage dffered then the follower would be nformed of the locaton of the leader when makng ts locaton decson. In the thrd stage of the game frms smultaneously choose rces gven the locatons as er D'AGT. In the frst stage two frms decde the lanned tme of entrance (T) from a contnuum of ossbltes from 0 to nfnty. A strategy for layer A () s a tme T A [0, ](T [0, ]) at whch frm A () lans to enter gven that ts rval has not entered before that tme. If rval frm (A) enters at a tme ror to T A (T ) frm A () may react by adjustng ts tme of entry to τ A (τ ). The actual tmes of entry (τ A, τ ), however, may devate from T deendng on the otmal strategy of the rval. Gven a entry tmng ar (τ A, τ ) [0, ] [0, ] a ayoff to layer A could then be defned as ether a Leader's ayoff f τ A bτ ; a Follower's ayoff f τ A Nτ, or; a smultaneous ayoff. We wll further defne these ayoffs subsequently but they would be the dscounted equlbrum ayoffs from the locaton and rcng subgames gven ther relatve tmng. In our comlete nformaton settng, the frms each have full knowledge of the hstory, n other words, we have a nosy game of tmng. That s, the frms nstantly learn when a rval has entered. A frm may adot a wat-and-see strategy and thus we seek feedback equlbra to the tmng game,.e., the strategy ar becomes (τ A (H τ ), τ (H τ )), where H τ s the hstory of entry and locaton behavor untl tme τ. However, f no frm has entered the market when tme T A (T ) has elased, Seller A (Seller ) wll mmedately enter at tme τ A =T A (τ =T ). Also, f a frm wants to reemt ts rval, the tme cannot be below 0 and f mmedate entry s chosen by both frms then they would enter smultaneously. Furthermore, we assume that frms wll not enter unroftable markets (.e., there s a cost of entry). The order of entry tme defnes the sequence of entrance, so that we can have sequental or smultaneous games. We assume that entry and resonse to entry take lace n contnuous tme. If a frm wshes to be a follower as soon as ossble after the leader has moved there would be a certan tme lag to confrm that the rval has entered and commtted to a locaton but we assume that lag to be fnancally neglgble. In ths stage the strategy set s the resectve lanned tme of entrance (T A and T )as well as the strategy (τ A (H τ ), τ (H τ )) to follow f the rval frm has entered before the lanned tme. In the second stage the sellers enter at entry tmng (τ A, τ ) and smultaneous to entry they lay a locaton game on a Hotellng lne. The sellers are located on a lne of length l at resectve dstances a (for Seller A) and b (for Seller ) from the ooste ends of ths lne (a+b ; a, b 0). At the tme of entry τ A (τ ) they choose locatons a (b). The locaton game s contngent on the two ossble relatve tmngs of entrance: sequental or smultaneous. When the relatve tmng s sequental the locaton game layed s Stackelberg where the follower has nformaton of the locaton of the leader before enterng. If frms enter smultaneously we assume that f both frms want to maxmally dfferentate they wll always lay n ooste locatons. When a ure equlbrum doesn't exst, and frms choose the same tme of entrance we solve for mxed strategy equlbrum. Once layers choose a locaton, t remans fxed for the remnder of the game. At ths stage the strategy set s the resectve locatons a and b n the case of ure strategy equlbra or the dstrbuton of robabltes for a and b n the case of mxed strateges. In the thrd stage the frms lay a ertrand game gven the degree of horzontal dfferentaton. Here frms have to smultaneously decde mll rces ( A and, where agan, the subscrts denote the Sellers) gven ther resectve locatons. Frms decde rces by maxmzng current rofts. Consumers are unformly dstrbuted across a lne of unt length l. They contnuously urchase and consume one unt of a roduct er unt of tme from any of the two Sellers A and that roduce the homogeneous roduct. Consequently, n what follows whenever we refer to roduct market rofts π t s to be understood as the rofts n a tme erod τ. A customer wll buy from the seller who quotes the least delvered rce and there are zero costs of swtchng sulers. For the Hotellng model, transortaton costs are lnear (tx, where t s the transortaton cost, x s the dstance from a consumer to the locaton of the seller). For D'AGT and our subsequent model, the transortaton cost wll be quadratc tx n order to guarantee the exstence of rce equlbra. Furthermore, as s common n such analyses, we assume that rces and locatons are such that there s full market coverage,.e., that the reservaton values of consumers are hgh enough and that the lmt rces are low enough to attract even the most dstant consumer. We also assume low barrers to entry. Ths means that a frm whch forces ts rval to ext (or revents t from enterng n the monoolstc case) must be cognzant of that frm otentally enterng. In other words, the frm whch fnds t otmal to become a monoolst must engage n lmt rcng. Ths lmt rce s a functon of the locaton (onanno, 97) and so blockadng can be done va a combnaton of the locaton and rce. We wll henceforth refer to such a monoolst as a blockadng monoolst. Our modfcaton to the revous horzontal dfferentaton models, analogous to Zss (993) and oyer et al. (003), wll be to nclude constant margnal costs of roducton that are dfferent for A and. Somewhat dfferently from Zss (993), Seller A has a margnal cost of roducton of ( (r/))c, whle Seller has margnal costs of (+(r/))c. Thus cr reresents the dfference n margnal costs, and we can nduce a mean reservng sread n those costs by ncreasng r. Note that r can take on any real value and consequently there s no loss of generalty. 6 We ntroduce ths cost structure whch dffers from Zss and oyer et al. as t allows us to change the dfference n margnal costs wthout changng the average ndustry costs and allows us to comare our results to those of Salant and Shaffer (999) and ergstrom and Varan (95) whch examne asymmetrc Cournot models wth a mean reservng sread n costs. We seek subgame erfect equlbra and to solve ths game we roceed n the reverse order of stages as s usually done n these multstage roblems. We frst solve n Aendx A the thrd stage rcng game. Prce equlbra are found for all locatons a and b and the resultng rce equlbra rofts are comuted. Then we move n the next secton to the second stage and solve all ossble locatons n three ossble games: the two sequental games (Seller A beng the leader, beng the follower; beng the leader and A beng the follower); and, the smultaneous game. For the smultaneous game we solve for ure strategy equlbra of the locaton game and we solve for the mxed strategy equlbra. The dervaton of the mxed strategy equlbrum allows us to: () comare the result wth that under ure strateges, and () fnd a soluton for the comlete range of cost dfferentals. Fnally, we solve the frst stage n whch the feedback equlbra for the tmng of market entrance are derved. 6 Our results translate recsely nto those of oyer et. al. for the relevant tmng scenaros. Our cr s the same as ther Δ/k (ther dfference n roducton costs dvded by ther transortaton rate), whch oyer et. al. use throughout ther analyss.

4 690 S. Meza, M. Tombak / Int. J. Ind. Organ. 7 (009) The locaton stage Wth the equlbrum results of the rcng game (see Aendx A) we now roceed to examnng the second stage of the game where frms choose locatons gven a certan sequence of moves. Ths tmng of moves could be sequental or smultaneous. We assume throughout that once a locaton s chosen then the frm s commtted to that locaton. The nformaton sets dffer accordng to ther relatve tmng, under sequental moves the follower knows the leader's locaton. As we have no ure strategy locaton equlbra n a certan range of cost dfferentals under smultaneous moves we comute a mxed strategy equlbrum n those cases. The outcomes wll then be comared n the subsequent secton to derve tmng equlbra.... Locaton wth smultaneous moves Movng forward to the locaton stage of the game, one can characterze the equlbrum locaton choces by takng the dervatves of the equlbrum rofts (Eq. (A)) wth resect to the resectve frm locaton choces, and comarng the resultng otmum wth that obtaned from the monoolstc regon defned by Eq. (A5). Prooston. (Zss, 993; oyer et al., 003; A.3, A.5) In duooly, the ncentves to change locaton are gven by: Aπ A Aa Aπ Ab Δt 3+a b = ½ ð Þ + cr Š ½ Δt ð +3a + bþ cr Š ; and Δ t ½Δt 3 +a b = ð Þ + cr Š ½ Δt ð +a +3b Þ + crš : Δ t ðþ where Δ= a b, Δ, a, b 0. When r=0, the results of D'AGT are obtaned; that s, the dervatves are both negatve and Sellers A and both have an ncentve to maxmally dfferentate. Wth margnal cost dfferences, however, the sgns of the dervatves may change. Substtutng condton (A3) nto Aπ n Eq. (), one can see that for all a, b and cr that satsfy ths condton, the dervatve s always negatve. Ab Therefore, t s otmal for Seller to locate as far as ossble from A (b=0 f a /, b= f a /). Knowng that Seller locates always as far as ossble from Seller A, we can contnue our analyss assumng b=0. 7 Substtutng b=0 nto Aπ A Aa of Eq. (), one obtans the condton, cr=t b ð aþð+3aþ ðþ for ths dervatve to be negatve. Such a condton always holds for cr/tb for all a that satsfy condton (A3). Therefore, f both frms artcate n the market, they would refer to maxmally dfferentate when cr/tb. Ths means that the hgh cost frm always fears rce cometton and seeks to avod t through roduct dfferentaton, but beyond a certan cost dfferental the low cost frm s no longer as afrad of rce cometton and starts to decrease the degree to whch ts roduct s dfferentated from ts rval's roduct. Recall, however, that Seller A can choose locatons outsde the regon defned by condton (A3). Locatons n ths regon would mly the ext of Seller and monoolst rofts for Seller A. The maxmum monooly rent can be easly derved from Eq. (A5) and would be obtaned when Seller A stuates on to of Seller (a= b). In that case, the rofts of Seller A would be equal to cr. Ths roft would be greater than the duoolstc roft π A (a,0) when, cr t zð aþ ð6 aþ 3 3 a : ð3þ Notethat the R.H.S. of condton (3) s decreasng n a n the relevant range of ths varable. When a s zero, then the R.H.S. reduces to cr = t N 3 3 ð 0:04Þ. Thus, when 0 V cr = t b 3 3, then both frms refer to maxmally dfferentate (a, b )=(0, 0) and obtan duooly rents. Thslocaton choce s a Nash equlbrum snce Seller A, f ts comettor locates at b=0, would fnd t otmal to locate at a =0 so long as cr = t b 3 3. Furthermore, for these values of a and cr/t, the dervatve of Seller 's rofts s negatve so t wll not devate from b=0. In ths case, the equlbrum rofts are: π A = ð 3t + cr t Þ and; π = ð 3t cr t Þ : ð4þ The above analyss shows when ure strategy equlbrum n locatons exsts for smultaneous move games. It also mles, as Zss (993) had found, that there s no duooly ure strategy locaton equlbrum for the smultaneous game when cr = t N 3 3. That s, Seller (the hgh cost seller) contnues to have a tendency to maxmally dfferentate, whle Seller A (the low cost seller) wshes to mnmally dfferentate. Furthermore, margnal costs dfferences can become large enough (satsfyng condton (3)), that Seller A refers to become a monoolst rather than obtan the duooly rofts of Eq. (A), regardless of the locaton choce a. Seller A becomng the monoolst mles that t mnmally dfferentates and undercuts Seller to drve out of the market regardless of the choce of b. 7 There s an analogous roblem assumng that Seller chooses locaton b=. Zss' condton for the nonexstence of locaton equlbra was cl where c s the transortaton cost and L was the length of the Hotellng lne whch s analogous to the value we obtan here.

5 S. Meza, M. Tombak / Int. J. Ind. Organ. 7 (009) As margnal cost dfferences become large enough (cr/tn5/4) the equlbrum n the smultaneous game reemerges. It s easy to show that by locatng at the mddle of the market (a=0.5) the low cost seller becomes a monoolst realzng rofts of cr t/4. Seller wll not enter the market as t wll not be able to make ostve rofts.... A mxed strategy soluton In order to attan locaton equlbra n the duooly throughout the arameter range, we solve for a mxed strategy equlbra. 9 Prooston. When 0 b cr = t b 3 3 there exsts a mxed strategy equlbrum of the smultaneous locaton subgame such that the exected ayoffs are equal to the frms' ayoffs when they choose to maxmally dfferentate at a=0 and b=0. Proof. See Aendx. The above rooston mles that the equlbrum exected roft from layng mxed strateges n ths arameter regon yelds the same roft as the equlbrum n ure strateges. An equal weghtng of robablty mass extends to a regon of greater cost asymmetry as shown n the next rooston. Prooston 3. When 3 3 b cr = t b there exsts a mxed strategy equlbrum of the smultaneous locaton subgame such that each frm chooses locaton 0 or l wth the robablty /. Proof. Ths follows from Prooston 4 and Prooston 5 n Aendx. The equlbrum robabltes yeld the exected ayoffs for smultaneous locaton of h E π MS A = h crt + 3t + cr 36t ð Þ h and E π MS = 3t cr 36t ð Þ : ð5þ In other words, n ths regon of cost dfferences the hgh cost frm and the low cost frm would collocate and the hgh cost frm would obtan zero rents f the low cost frm correctly antcates the locaton of the hgh cost frm. The hgh cost frm, however, can stll earn rofts by ntroducng nose nto ts decson makng rocess n the hoes that the low cost frm would antcate ncorrectly the locaton of the hgh cost frm and locate at the ooste extreme. Prooston 4. When bcr/tb.77 there exsts a mxed strategy equlbrum of the smultaneous locaton subgame such that Seller chooses b=0 or b= each wth the robabltes of /, and Seller A chooses a=a or a= ã each wth the robabltes of /. Proof. See Aendx. Lemma. When cr/tn.77 then Seller A refers to monoolze at a=/. Proof. See Aendx. To summarze, when the moves by the frms are smultaneous then: when 0 V cr = t b 3 3 the frms maxmally dfferentate and earn the duooly rofts n Eq. (a,b); when 3 3 V cr = t b then the frms lay mxed strateges over the extreme onts n the sectrum and frms earn the exected rofts n Eq. (5); when cr/tb.77 then the frms lay mxed strateges where there s ntermedate dfferentaton by the low cost frm, 0 and; when cr/t s greater than.77 the low cost frm wll locate n the center of the sectrum and monoolze the ndustry obtanng a roft ofcr t/ Locaton wth sequental moves In our settng there are two otental sequental games: one n whch the hgh cost frm s the leader and the low cost frm locates afterwards, and another game where the low cost frm takes the lead and the hgh cost frm locates afterwards. Once frms have chosen locatons they are commtted to those locales. The follower s able to observe the locaton of the leader before decdng ts entry/locaton decson. In ths secton we solve these two sequental games for the whole range of values of cr. oyer et al. (003) also solved the sequental game of locaton choce but frms are restrcted to locate on ther half of the lne. Ths constrant of where frms could locate negated the ossblty of frms leafroggng one another whch hels n ther obtanng locaton equlbra. As already seen n Secton.., for the regon of low cost dfferences when 0 V cr = t b 3 3, both frms refer to maxmally dfferentate (a, b ) =(0, 0) and obtan duooly rents. Therefore even f the two frms lay sequentally the frm layng second n ths regon would always choose a locaton that maxmzes roduct dfferentaton. Consequently, order of entry does not have a materal mact on locaton decsons when frms are so symmetrc. Tmng, however (as we shall see n Secton 3), starts to lay an mortant role when cost asymmetres are larger. 9 Mxed strategy equlbra wth symmetrc cost frms have been comuted: for rce subgames n the orgnal lnear transortaton cost model of Hotellng (Ptchk and Osborne, 97) and, for locatons n the model of D'AGT (ester et.al., 996) whereas we consder locaton subgames of a broader game. 0 In Aendx we show that these mxed strategy equlbra have hgher exected ayoffs than any other ossble mxed strategy equlbra and as such we use them as the bass for comarson.

6 69 S. Meza, M. Tombak / Int. J. Ind. Organ. 7 (009) For hgher cost dfferences (cr = t N 3 3 ), as can be seen n Sectons.. and.., there s no ure strategy equlbrum n the smultaneous game because whle the hgh cost frm refers to maxmally dfferentate, the low frm nstead refers to locate on to of the hgh cost frm and thus monoolze the market. For the sequental case n whch the hgh cost frm lays frst the results are straghtforward. After the hgh cost frm has chosen a locaton, the low cost frm enters and locates at the same lace as ts rval. ecause of ts cost advantage, the low cost frm has only to rce slghtly below the margnal cost of the hgh cost frm of c(+(/)r). The hgh cost frm cannot reduce ts rce and wthout any dfferentaton wll not sell any roduct and wll realze a roft of zero. The low cost frm would cature the whole market realzng the monoolstc roft ofcr. For the sequental case n whch the low cost frm lays frst the results are a lttle more comlex. The low cost frm A s gong to enter frst and let's assume that t chooses a locaton a, a 0,0.5. As we know from Eq. () that for all values of a Aπ s always negatve, therefore the hgh cost Ab frm s gong to follow choosng the locaton that maxmzes dfferences at b=0. If frm A chooses nstead a locaton n the range (0.5, ], the frm would choose the ooste locaton at b=. ecause ths later roblem s symmetrc to the former one, we wll focus only n the frst roblem where a 0,0.5 and b=0. The roft functon that the low cost frm has to maxmze s that from Eq. (A) knowng that layer s gong to choose b=0, and constraned to the values of a n the range a 0,0.5. Also, gven the cost advantages of frm A, monoolzaton s ossble and the resultng rofts are obtaned by substtutng b=0 n Eq. (A5). Monoolzaton would be obtaned when the rofts comuted ffff from Eq. (A5) n ths manner are hgher than the rofts from the duoolstc game,.e., π Mono A (a)nπ L tt+ ð crþ A (α,0) ths condton holds when a N t. In summary, the rofts obtaned by Seller A for any choces of locaton a are: ðð aþð3+aþt + crþ π L ð aþt Aða; 0Þ = cr tð aþ ff tt+ ð crþ f 0 V a V t ff tt+ ð crþ f b a V t ð6þ Solvng for the frst order condtons for the maxmzaton of Eq. (6) wth resect to a and comarng those results wth the border condtons results n three soluton regons for the otmal value of a. In the regon 3 3 b cr = t b Seller A locates at a =0, that ffff s Seller A fnds t otmal to maxmally dfferentate. When cost asymmetres are such that b cr = t b cr =t, where cr =t = :674 Seller A fnds t otmal to move to a == 3 ffff 4 3cr. That s, the Stackelberg leader chooses ntermedate dfferentaton and can engage n some degree of rce cometton and steal some busness from the hgh cost seller. Fnally when cr = t z cr =t, Seller A fnds t otmal to monoolze and locates at a =/. Seller wll then not enter. Summarzng the rofts realzed for the low cost roducer beng the Stackelberg leader for the varous cost asymmetry and locaton outcomes π A ð 3t + cr Þ a ; b = ð0; 0Þ f 0 V cr V t a ; b 4t +3cr + ffff ð4t 3crÞt ff! = 43 t + ð ff a ; b ð4t 3crÞt = = 3 ; 0 f b cr b cr =t 4t 3crÞt 3t t cr t a ; b cr = ð= ; NEÞ f cr =t V 4 t : Smlarly, for Seller (the hgh cost roducer) beng the Stackelberg follower obtans π ð 3t cr Þ a ; b = ð0; 0Þ f 0 V cr V t a ; b 6cr 5 t + ð ffff 4t 3crÞt ff! = 43 t + ff a ; b ð4t 3crÞt = = 3 ; 0 f b cr b cr =t ð4t 3crÞt 3t t 0 a ; b = ð= ; NEÞ f cr cr =t V t : These ayoffs are llustrated as functons of ther locatons n the followng fgures (Fgs. and ). Ths nteror locaton s determned by settng b=0 n π A of Eq. (A4) and takng the dervatve wth resect to a. Ths frst order condton yelds four ossble solutons for a, 3t t F ff ff ð4t + crþt ð4t 3crÞt and F t. Only the soluton a = 3t t ff ð ðcr +Þt 4t 3crÞt satsfes second order condtons and the requrements that a be less than a = t and greater than zero for the relevant values of cr/t.

7 S. Meza, M. Tombak / Int. J. Ind. Organ. 7 (009) Fg.. The ayoff for Seller A for varous cost dfferentals (cr/t) gven b =0... Summary The followng table (Table ) summarzes the ossbltes of locaton equlbra as a functon of the cost dfferental and the sequence of entry. 3. The game of tmng Wth the equlbrum locaton solutons for the three ossble sequences of entry games as defned n Table we roceed to solve for the equlbrum n the game of tmng (T A,T ). Equlbra can be determned for a gven level of cost asymmetry f both frms fnd t otmal to locate n the same cell of relatve tmng structure n Table. For examle, f both frms refer smultaneous moves then that would be the equlbrum behavor. When, however, frms dffer n ther reference of relatve tmng (.e. both frms want to be leaders or followers) then ether a reemton game or a war of attrton would arse and a feedback equlbrum may solve that game. efore we solve for each of the regons of cost dfferentals, we start by examnng the actual tme of entry (τ A, τ ) gven the relatve tmng structure. We fnd that at low cost asymmetres the tmng would be nstantaneous and smultaneous. For hgher cost asymmetres where the smultaneous move locaton equlbra n ure strateges dd not exst we fnd that the low cost frm would be the Stackelberg leader and the hgh cost frm would be the follower. At the most extreme cost dfferences the low cost frm would agan be the Stackelberg leader but the hgh cost frm would not enter. 3.. A model for the tme of entry (τ A, τ ) If Seller A enters at tme τ A at locaton a and f ths tmng s earler than ts rval's entry tme, then t obtans the leader's ayoff, Z τ Z L A ðτ A ; τ Þ = π m A ða; A Þe ρτ dτ + π L A a; b; A ; e ρτ dτ Φ; τ A τ Fg.. The ayoff for Seller for varous cost dfferentals (cr/t) gven a=0.

8 694 S. Meza, M. Tombak / Int. J. Ind. Organ. 7 (009) Table Tyes of equlbra for each sequence of entry game. Range of cr/t Smultaneous game Low cost frm s Stackelberg leader Hgh cost frm s Stackelberg leader 0to3 3 Duooly/maxmal dfferentaton Duooly/maxmal dfferentaton Duooly/maxmal dfferentaton 3 3 to Duooly/mxed strateges Duooly/maxmal dfferentaton Colocaton/ext/monooly to cr =t Duooly/mxed strateges Duooly/ntermedate dfferentaton Colocaton/ext/monooly cr =t to :77 Duooly/mxed strateges Monooly a=/ Colocaton/ext/monooly.77 to Monooly/a=/ Monooly Colocaton/ext/monooly where π A m (a, A ) are the rate of monooly rents of Seller A when t charges the monooly rce A, π A L (a, b, A, ) are the duooly rent flows of Seller A when t chooses locatons frst and locates after A, ρ s the contnuous dscount rate, and Φ s the fxed cost of entry. Seller A's ayoff f t enters the market as the follower (τ A Nτ ) s then, Z F A ðτ ; τ A Þ = π F A a; b; A ; e ρτ dτ Φ: τ A Fnally, f the Sellers enter and locate smultaneously the ayoff wll be Z S A ðτ S Þ = π S A a; b; A ; e ρτ dτ Φ: τ S Note that the ayoff for smultaneous moves has the same structure as the follower's ayoff but the ntegrand s not determnstc for the smultaneous game when the locaton equlbrum n ure strateges does not exst. In that case, the exected ayoffs are used. Analogously, there are leader's and follower's ayoffs for Seller nterchangng the A and subscrts n the above ayoffs. The structure of these ayoffs lead to the followng. Lemma. The follower, f t wshes to enter at all, would want to enter as soon as ossble,.e., τ F τ L. Proof. UsngLebntz'srule, we can dfferentate the follower's ayoff wth resect to ts entry tme and obtan the frst order condton, AF = πf Aτ a; b; A ; e ρτ, and AF A Aτ A = π F A a; b; A ; e ρτ A, whch are both negatve. Takng the second dervatve of the above follower's ayoffs we can fnd that they are ostve,.e., the ayoffs are convex. The above lemma mles that there s no ossblty for ether frm to obtan monooly rents for some erod of tme f ts rval wshes to enter. Thus the leaders' ayoffs smlfy to, and Z L A ðτ A ; τ Þ = π L A a; b; A ; e ρτ dτ Φ; τ A Z L ðτ ; τ A Þ = π L a; b; A ; e ρτ dτ Φ: τ Smlarly, for the leader's smultaneous ayoffs we fnd, Lemma 3. The ayoffs are all dmnshng and convex wth tme. Proof. As n Lemma. A Aτ = π a; b; A ; e ρτ, and A A Aτ A = π A a; b; A ; e ρτ A, for =L, F, S whch are both negatve. Agan the second dervatves of the above are ostve mlyng that the ayoffs, regardless of the role of the frm, are convex n tme. These Lemmas () and (3) mly; Corollary. The leader, f t wshes to enter at all, would want to enter as soon as ossble,.e., τ L =0, and, Corollary. Smultaneously enterng frms, f they wsh to enter at all, would want to enter as soon as ossble,.e., τ S =0, and that the follower, f t enters, would enter at tme τ F 0, or τf =0 +. The above s an mlcaton that there are roduct market rofts to be had and so long as the dscounted sum of those rofts exceeds Φ, the fxed cost of entry, then frms are caturng the maxmum surlus by movng as radly as ossble. Thus domnatng tmng of entry strateges (τ A, τ ) are: (0,0), (0, 0 + ), (0 +, 0), (0, ), (,0), (, ). In the followng subsectons we derve the tmng equlbra deendng on the cost asymmetres.

9 S. Meza, M. Tombak / Int. J. Ind. Organ. 7 (009) Tmng under low cost dfferences (cr = t b 3 3 ) 695 In ths secton we determne the equlbrum tmng when there are small dfferences n cost. From the revous secton (Secton 3) we found that for ths range of cost dfferental the equlbrum locaton regardless of the tmng s maxmal dfferentaton. As we can see from Table the ossble tmngs are: (0,0) for the smultaneous game, (0, 0 + )whenthefrm A s leader, and (0 +,0)whenfrm s leader. Prooston 5. For small margnal cost dfferences (cr = t b 3 3 ) the equlbrum lanned and entry tmng s smultaneous at (T A,T )= (τ A, τ )=(0, 0). Proof. From Zss, when cr = t b 3 3, the unque locaton equlbrum entals maxmal dfferentaton and then π S (a, b, A, )=π F (a, b, A, ) and π S A (a, b, A, )=π F A (a, b, A, ) as the Sellers wll locate such that a=b=0 and Lemma stll ales. The ayoffs S A (0)NF A (0, 0 + ) and S (0)NF (0, 0 + ). In ths case the unque equlbrum tmng s smultaneous and mmedate. In other words, wth such slght cost dfferentals there s no dfference n frm behavor wth resect to locaton (whch s maxmal dfferentaton) and rcng whether the tmng of entry s smultaneous and mmedate or f any frm wats. As watng entals a loss of roduct market rofts durng the delay there s no ncentve to delay and the orgnal results of Zss hold The game of tmng when cost dfferences become sgnfcant (3 3 b cr = t b cr=t) For the arameter regon cr = t N 3 3 our model has the ayoff structure of a war of attrton. Such games are often lagued by multle equlbra (Hendrcks et al., 9). As Smth (974) onted out, An ntal asymmetry n the condtons of a contest can be used to settle t and he descrbes some examles from the anmal kngdom. Ths suggests that the asymmetres between frms that we descrbe above may hel to sharen the rognostc ower of these models. From Table we can see that n ths regon of cost dfferentals the ossble scenaros are: smultaneous entry wth mxed strategy equlbra n locatons; the low cost frm acts as the Stackelberg leader and there s maxmal dfferentaton, or; the hgh cost frm s the Stackelberg leader and there s collocaton wth the hgh cost frm forced to ext. Clearly ths last scenaro s domnated for the hgh cost frm by the other crcumstances where t obtans ostve exected rofts. Also, the low cost frm wll always make ostve rofts by enterng the game and so the strategy of never enterng s domnated. Ths structure of ayoffs leads to the next two roostons. Lemma 4. When the tmng/locaton game s a war of attrton the hgh cost frm, f t enters, wll always follow (T = ). Proof. Say Seller followed the strategy to enter after Seller A has entered untl a maxmum of T =τ at whch tme t wll enter. The dscounted ayoffs to Seller enterng on or before τ (bτ + ) for all ossble tmngs of entry by Seller A are: L (τ A, τ )fτ A Nτ + ; L (τ A, τ )fτ A =τ + ; S (τ A, τ )fτ A =τ ; and, F (τ A, τ + A )fτ A bτ. If, however, Seller were to enter after Seller A has entered or wat untl tme τ + then ts dscounted ayoffs for all ossble entry tmngs of Seller A would be: L (τ A, τ + )fτ A Nτ + ; S (τ A, τ + )fτ A =τ + ; F (τ A, τ + A )fτ A =τ ; and, F (τ A, τ + A ) f τ A b τ. Ths later stream of ayoffs domnates the former as: L (τ A, τ )bl (τ A, τ + ); L (τ A, τ )fτ A = τ + = Φ b S (τ A, τ + )fτ A = τ + ;and, F (τ A, τ + A )NS (τ A, τ )fτ A = τ. Consequently, τ + τ τ + N τ τ =. Recall that T s the tme of entry f no frm has entered by that tme. The above rooston mles that at any tme τ the hgh cost frm wll always be better off to wat and see whether the low cost frm has entered. Ths logc extends to the lmt of watng forever (T = ). Ths means that f Seller A never enters then Seller wll also never enter. As the low cost frm (Seller A) can always roftably enter the market, the rooston above together wth Corollary mly that the game of tmng reduces to nclude only the followng tmng: Lemma 5. The equlbrum tmng of the low cost frm s to enter mmedately (T A =0). Proof. Gven that the hgh cost frm wll always follow (Lemma 4) or never enter, the low cost frm's otons are to ether be the leader or to never enter. As L A (τ A, τ + A )N0 τ A, leadng always domnates never enterng. The rooston then follows from Corollary. Thus the ossble ure strategy equlbrum tmngs are: (0,0), (0, 0 + ), (0, ). In what follows we endeavor to fnd the equlbrum tmng ars/ locatons for ths regon of sgnfcant cost dfferences. These equlbra dffer n the hgh cost frm's tmngs and locatons of both frms wth the extent of those cost dfferences and so we dvde ths regon of cost dfferences nto fner sub-regons Tmng under moderate cost dfferences (3 3 b cr = t b ) For ths subset of cost dfferences wthn that whch cause a war of attrton we fnd, Lemma 6. For moderate cost dfferences (3 3 b cr = t b ) the domnant tmng strategy for the hgh cost frm s to wat untl the low cost rval commts to a locaton and f that rval does not enter then t never enters (T = ). Proof. L (0 +,0)= Φ as the low cost frm wll collocate f allowed to move second (see Table ). F (0 +,0 ++ )N0 as ths nvolves maxmal dfferentaton and the roduct market roft flows n the ntegrand are those gven n Eq. (a,b). F (0, 0 + ) N0 and NS (0), F 0; 0 + R = 0 + ð3 crþ te ρt dτ N S ð0þ = R 0 36 ð3 crþ te ρt dt as R 0 + e ρt dt N 36ρ. In other words Lemma 6 states that the domnant strategy for the hgh cost frm s a feedback strategy whereby ths Seller wats untl the other Seller has commtted to a locaton. We are now n a oston to determne the equlbrum tmng.

10 696 S. Meza, M. Tombak / Int. J. Ind. Organ. 7 (009) Prooston 6. The equlbrum tmng for moderate cost dfferences (3 3 b cr = t b ) s where the low cost frm moves mmedately and the hgh cost frm follows mmedately thereafter and they maxmally dfferentate (τ A, τ )=(0, 0 + ). Proof. Gven Lemmas 5, 6 and that L A (0, 0 + )NL A (0 +,0 ++ ) as both ayoffs nvolve maxmal dfferentaton but the latter nvolves some delay to obtanng the roduct market rofts. Thus the result that the low cost frm takes the role of Stackelberg leader erssts for these games of locaton for moderate cost dfferentals. The smultaneous game nvolves mxed strateges whch yeld some robablty of zero or negatve rofts for the hgh cost frm whereas a slght delay by the hgh cost frm ensures that t wll obtan the hghest rofts ossble. As the cost dfferental grows larger stll, as we observed n Secton 3., the otmal locaton for the low cost frm changes and ths changes the rofts n the ntegrand of the ayoffs under varous tmngs Tmng under ntermedate cost dfferences ( b cr = t b cr =t) When cost asymmetres are such that b cr = t b cr =t a duooly has the tmng of a Stackelberg equlbrum where the low cost frm s the leader and there s ntermedate dfferentaton. As shown n Secton.., when cr/tn then dervatve of π A wth resect to a s no longer negatve for all values of a. In artcular, for a=0 the dervatve s ostve and consequently there s an ncentve for Seller A to move nto the nteror. From Secton..3 we fnd that for cr = t b cr =t, even for the cases where Seller A could monoolze, A would refer obtanng the duooly rents at some nteror value for a. The ayoffs for Seller A for the regon bcr/tb5/4 are llustrated n Fg.. So as n the revous cost dfference regon, the equlbrum tmng s where the low cost frm moves mmedately and the hgh cost frm follows mmedately thereafter (τ A, τ )=(0, 0 + ) Wth large cost dfferences ( cr =t b cr = t) Here we can fnd two sub-regons that may result n wth the same equlbrum outcome but they must be analyzed searately as the logc by whch they arrve at that outcome s dfferent Wth substantal cost dfferences ( cr =t b cr = t b :77) From Table we can see that f both frms lay mmedately and smultaneously than the equlbrum would nvolve a mxed strategy wth locatons for the low cost frm beng somewhere n the nteror. These ayoffs (shown n Aendx A), however, are lower n an exected value sense for the low cost frm than f t were to delay entry and become the Stackelberg follower (n whch case t would acheve a ayoff of cr). The hgh cost frm, however, would then devate from ts mmedate entry and mxed strategy n locatons to also delay as ts exected ayoff becomes negatve. The low cost frm, as t would acheve zero rents, would then want to devate to becomng the Stackelberg leader. From the mddle column n Table, the hgh cost frm would then want to devate to smultaneous entry and mxed strateges n locatons. Thus we have no ure strategy equlbrum n the game of tmng for ths range of cost dfferentals. Alternatvely we can try to solve the game n mxed strateges. In mxed strateges the low cost Seller A can enter at each erod wth robablty α. And the hgh cost seller can enter each erod wth robablty β. In equlbrum we must fnd β (and α) that makes Seller A (and Seller ) ndfferent between enterng ths erod τ a (and τ b ) or watng and enter some tme after at τ + a (and τ + b ). If Seller A decdes to enter at tme τ a gven that no seller has entered yet, the exected roft s: E½Π A = β S MS ðτ a ; τ a Þ + ð βþ L A ðτ a ; Þ ð7:aþ A If nstead Seller A decdes to wat and enter at tme τ + a the exected roft s: E½Π A + a = β F A τ + a ; τ a + ð βþ β S MS A τ + a ; τ + a + ð βþ L A τ + a ; ð7:bþ In a smlar way, f Seller decdes to enter at tme gven that no seller has entered yet, the exected roft s: E½Π = α S MS ðτ b ; τ b Þ ð αþ Φ e ρτ b ð:aþ If nstead Seller decdes to wat and enter at tme the exected roft s: E½Π + = ð αþ a S MS τ + b ; τ + b ð αþ Φ e ρτ+ b b ð:bþ Prooston 7. For substantal cost dfferences ( cr =t b cr = t b :77) and searate decsons for entrance and locaton, two ossble tmng equlbrums emerge:. One n whch for each erod Seller A and Seller can enter the market wth robabltes: β = Φ α = Φ + S MS ðτ b ; τ b Þ qffff ð d 4 F A ð+dþ S MS A ð dþ d +4 F A S MS A ð dþ 4d ð d 4FA ð dþþ 4 F A S MS A d 4 F A S MS A where d=e ρ(τ a + τ a) =e ρ(τ b + τ b) also, F and S are evaluated at τ a. One n whch Seller A enters at tmes and locates at τ a =/ Monoolzng the market, and Seller never enters.

11 S. Meza, M. Tombak / Int. J. Ind. Organ. 7 (009) Proof. For art () we make the substtuton L A τ + a ; = FA τ + a ; 4ρ e ρτ+ a nto Eqs. (7.a) and (7.b). We also make use of the roerty that Π(τ + a )=Π(τ a ) e ρ(τ a + τ a). Then α and β are obtaned as the unque solutons n the range (0,) that solves and E[Π A =E[Π A + and E[Π =E[Π + n For art () we relaced α n Eqs. (.a) or (.b) and we obtan E[Π =E[Π +=0. Ths mles that Seller s ndfferent to never enterng the market. Snce we are assumng that Seller s rsk neutral, never enterng the market leads the same ayoff as E[Π As a resonse to the never enterng soluton of Seller (or β =0 n Eqs. (7.a) or (7.b)), Seller A wll beneft for enterng as soon as ossble at tme τ a =0, to realze rofts L A by locatng at a=/. The hgh cost frm s ndfferent between the two ossble equlbra above t obtans zero exected economc rents n both cases. The mxed strategy game, however, s senstve to any erturbaton of rsk neutralty. The low cost frm, however, would refer the mxed strategy game as t obtans some ossblty of the ayoff from the smultaneous game of mxed strateges n locaton. The mxed strategy equlbrum, however, s unstable. Whle we do not model ths here, ths means that there may be an ncentve for the low cost frm to offer the hgh cost frm a sde ayment for commttng to lay the mxed strategy game. The exected tme of the low cost frm enterng the market s then τ A = ð αþlnð αþ whch then delays any ayoff to the low cost Seller Under extreme cost dfferences (.77bcr/t) For ths range of cost dfferentals the low cost frm wll monoolze regardless of the mxed strateges n locatons the hgh cost frm emloys (see Aendx ). The low cost monoolst would locate n the mddle of the Hotellng lne and engage n lmt rcng. The tmng of the low cost frm would thereby be mmedate whle the hgh cost frm would never enter,.e., (T A, T )=(τ A, τ )=(0, ). 4. Summary and dscusson In ths secton we summarze the results from our analyss of merfectly comettve markets n the revous sectons. We then comute the frst-best welfare outcomes and comare the results of rvately otmal behavor to the socally otmal. We then contrast our results wth the dfferentated ertrand model wth those of others usng the Cournot model. Summarzng the dscusson n the above secton, we have the followng rooston. Prooston. In a Stackelberg game of locaton choce, the equlbrum locatons of Sellers A and (a, b ) deend on the margnal cost dfferences as follows: ð0; 0Þð0; 0Þ f 0 V cr = t V 3 3 0; 0 + ð0; 0Þ f 3 3 V cr = t V ðτ A ; τ Þ a ; b = 0; 0 + = 3 ffff 4 3cr ; 0 f b cr = t b cr DE f cr b cr = t b :77 ð0; Þð = ; NEÞ f :77 V cr = t where NE denotes No Entry and DE Dual Equlbra whch are those defned n Prooston 7 one equlbrum beng where the low cost frm enters mmedately and monoolzes at locaton a =, the other equlbrum where the frms lay mxed strateges n tmng. Interestngly, note that the low cost frm always enters mmedately, whle the hgh cost frm could enter mmedately, delay entrance or never enter. Ths ssue has mortant mlcatons by enterng frst the low cost frm relnqushes ts otentally domnant oston and must accet ts second best rofts (for most of the cost dfferental range). The hgh cost frm, however, wns whenever the game s a war of attrton n the sense that t moves last n equlbrum. Havng determned the equlbrum behavor, we now comute the frst-best soluton for a total welfare maxmzng olcy maker and we then comare ths soluton to that derved above. In ths case the frst-best nvolves the socal lanner settng both locatons and rces for both the frms. The total welfare gven a reservaton value of consumers of v s and the ont of the ndfferent consumer beng x, Z x h S = v r Z h c tx a ð Þ dx + v + r c tx ð ð bþþ dx: 0 x That s, the total surlus s the reservaton values less the costs of roducton and the transortaton costs for all the consumers served. Unlke the symmetrc cost case where the rces dd not matter for total welfare comutatons, here the cost structure does affect total surlus through the allocaton of consumers to ether the hgh roducton cost frm or the low roducton cost frm. The socal lanner here must, on occason, tradeoff some transortaton cost wth roducton costs. Smultaneously solvng the above exresson for the mnmum cost values of a, b, and x yelds the socally otmal values, a = 4 + cr t ; b = 4 cr t ; x = + cr cr V t t 4 : eyond cr/t=/4 t s socally otmal to have the low cost frm monoolze the market at a locaton of a =/. The above show a very smle and ntutve relatonsh between the socally otmal locatons and the cost dfferental related to the socally otmal locatons found for symmetrc frms (a, b ) =(/4,/4) (see, for examle, Trole, 99, g. ). That s, cost dfferentals (of a mean reservng sread) do not ncrease the socally otmal degree of dfferentaton but shft both sellers and allocate more consumers to the lower cost frm. The other Wth the exceton of the regon where one of the ossble equlbrum s to enter mmedately wth robablty α but, here too the low cost frm commts to begn layng the mxed strategy game mmedately.