When Do Switching Costs Make Markets More or Less Competitive?

Transcription

1 When Do Swtchng Cot Make Market More or Le Compettve? Francco Ruz-Aleda December, 013 Abtract In a two-perod duopoly ettng n whch wtchng cot are the only reaon why product may be perceved a d erentated, we provde neceary and u cent condton for wtchng cot to lead to hgher prce n the rt perod a well a to hgher overall pro tablty. We how that th happen f and only f wtchng cot are not too large. A neceary condton for wtchng cot not to be "large" that econdperod pro tablty be nonmonotonc n rt-perod market hare. Becaue prevou lterature ha examned tuaton n whch wtchng cot are uch that econd-perod pro t grow wth rt-perod market hare, a un ed treatment of how wtchng cot (and only wtchng cot) a ect competton mng. We manage to provde uch a un ed framework for th analy by makng wtchng cot heterogeneou acro the populaton of conumer. In o dong, we llutrate the undered byproduct of aumng that product exhbt ubtantal horzontal d erentaton, drawng mplcaton for the clacal lterature on competton wth wtchng cot a well a the more recent one bult upon uch an aumpton. Key word: Swtchng Cot Heterogenety, Market Share Accumulaton, Conumer Foreght, Lock-n. JEL code: L13, L14. Ecole Polytechnque. E-mal: francco.ruz-aleda@polytechnque.edu

2 1 Introducton It by now very well documented that many conumpton decon nvolve makng unk nvetment that are product-pec c (or, more generally, eller-pec c). Th relatonhppec cty create an nerta toward the contnued ue of the ame product, even f there are eller of ubttute product wth very mlar functonal feature. Such cot of wtchng uppler of a product are n ndutre a dvere a computer, telecommuncaton, pad- TV, ketchup, credt card, cgarette, or retal bankng, and ther extent can range from beng relatvely mall to beng qute large, dependng on the ndutry. 1 The mot obvou e ect of wtchng cot that, once a conumer tart conumng a gven product, t degree of ubttutablty wth repect to competng product decreae, whch ha an mpact on how rm compete for conumer. In fact, rm that foreee that conumer wll get (partly) locked-n to ther product n the future wll alo factor thee element nto ther current decon-makng. Gven ther wdepread mportance, t hould come a no urpre that there ha been a lot of theoretcal work analyzng the mpact of wtchng cot on compettve outcome n olgopoly. The mot crucal nght from th lterature tem from Klemperer (17a, b) emnal work. In a two-perod ymmetrc duopoly model, he nd that the econd-perod pro t of a rm ncreang n t rt-perod market hare. Th lead to very aggreve competton before conumer are locked-n, wth competton greatly relaxng afterward. Th temporal evoluton for prce commonly known a the "bargan-then-rpo " pattern. Th receved wdom ha been very n uental n formng the conceptual apparatu for full- edged dynamc model that greatly extend the workhore two-perod model ntally condered. In the quet for a tractable framework, th tream of the ndutral organzaton lterature ha had to addre gn cant techncal challenge. Indeed, pure-trategy equlbra ealy fal to ext n multperod model wth wtchng cot owng to endogenouly formed dcontnute and aymmetre n demand functon. Th drawback ha been amended by makng product ubtantally d erentated from a horzontal pont of vew. 3 When a conumer locked-n by a rm, t may be really hard for the compettor to teal t away from 1 See Calem and Meter (15), Dubé et al. (010), Keane (17), Larkn (004), Shcherbakov (013), Shum (004), Shy (00) and Vard (007) for ome relevant emprcal tude. Excellent urvey of earler lterature can be found n Klemperer (15) and Farrell and Klemperer (007). 3 See Klemperer (17b) for the workhore model wdely ued n the lterature. A complementary way to deal wth equlbrum nextence to further aume that that mot conumer leave the market before they can pobly bear any wtchng cot, a alo done n Klemperer (17b). 1

3 the rm (unle tate are largely uncorrelated over tme). A a reult, xng a techncal problem create a fundamental artfact n the economc of the tuaton to be analyzed: competton once conumer are locked-n end up beng too oft, o t hould not be urprng that econd-perod pro t are found to be monotone ncreang n rt-perod market hare. The objectve of th paper to provde a un ed analy of the e ect of wtchng cot, and only wtchng cot, on compettve outcome. In our two-perod ymmetrc duopoly ettng, conumer are heterogeneou wth repect to ther wtchng cot: the only reaon why demand from locked-n conumer omewhat elatc becaue they bear d erent wtchng cot. Even though th eem a natural feature of many real world market n whch wtchng cot are preent, heterogenety n uch cot unuual n the lterature. 4 Characterzng the full equlbrum et for any value taken by the average wtchng cot qute traghtforward under wtchng cot heterogenety. We nd when (average) wtchng cot are relatvely large that econd-perod competton qute mld, o the receved wdom apple n thee cae, and econd-perod pro tablty grow wth a rm rt-perod market hare. Th reult n rm ntally competng ercely for market hare n the unque equlbrum of the game, n antcpaton of hgh future prce gven the oft competton n the econd perod. The man contrbuton of th paper to nally provde a framework for analyzng competton n the preence of wtchng cot that need not be that large. When uch cot are n fact relatvely mall, we nd that econd-perod pro t do not vary monotoncally wth rt-perod market hare anymore. Startng from a low market hare n the rt perod, a rm econd-perod pro t grow a t ell to more conumer becaue the rm ha a larger bae of (partly) captve conumer. However, accumulatng market hare come at the expene of the rval. So once the rm ha accumulated a large enough market hare, the rval wll be led to charge lower prce n the econd perod n face of t hrnkng cutomer bae. Therefore, the rm that develop a larger cutomer bae bene t from havng more conumer locked-n, but harmed by havng a more aggreve rval. The latter e ect domnate the former one once the rm capture at leat half of the market n the rt perod, o econd-perod pro tablty eventually declne a a rm attract more conumer n the rt perod. Becaue econd-perod pro t rt ncreae wth rt-perod market hare, peak at half, and then decreae, a trong dere to match the rval prce are, whch 4 Up to the bet of our knowledge, the only paper that make wtchng cot heterogeneou acro conumer are Klemperer (17a) and Bglaer et al. (013). The latter make a very d erent pont from our gven t focu on entry deterrence, wherea the former much more cloely related. However, t aume extence of an equlbrum n the rt perod of play, omethng that cannot be taken for granted.

4 create a multplcty of (ymmetrc) equlbra. Retrctng attenton to Pareto-domnant equlbra, we nd that an ncreae n (ntally low) wtchng cot lead to hgher prce and pro t n both the rt and the econd perod of the game. Thee reult pert when product are horzontally d erentated n the rt perod and when conumer are forward lookng: ndeed, rm charge hgher rt-perod prce and earn greater pro t a conumer care more about the future. We conclude our analy by payng attenton to wtchng cot that are nether hgh nor low. In thee cae, we nd that econd-perod pro t are tll nonmonotonc n rtperod market hare, whch agan reult n a contnuum of ymmetrc equlbra. However, the unque Pareto-domnant equlbrum uch that ncreang wtchng cot nten e competton n the rt perod n a way that harm overall pro tablty. Summarzng our reult, rt-perod prce and payo rt ncreae wth (average) wtchng cot and then decreae. Alo, econd-perod pro t rt ncreae wth a rm market hare and then decreae f wtchng cot are not too large; otherwe, econd-perod pro t alway ncreae wth market hare. In h poneerng contrbuton to the wtchng cot lterature, Klemperer (17a) doe note a a caveat to h nght on the "bargan-then-rpo " pattern that econd-perod pro t may decreae wth rt-perod market hare. He doe not acknowledge though that uch pro t may n fact vary nonmonotoncally wth market hare. In fact, he doe not even upport h clam by gvng a ngle example, let alone by gvng condton on prmtve that may lead to econd-perod pro t decreae wth rt-perod market hare. Indeed, the ubequent lterature ha treated th apect a a rarty, f not a pathology (for example, Farrell and Klemperer 007 do not even menton t a a poblty n ther extenve urvey). We how that the decreangne of future pro t on current market hare, even f t doe not hold everywhere, are naturally once we focu on ettng n whch the only ource of product d erentaton are wtchng cot, provded uch cot are low enough. The ntal lterature on the compettve mplcaton of wtchng cot worked wth mple two-perod ettng n order to hed lght on the man ue at play. Our paper contrbute to th clacal lterature, but t alo ha mplcaton for more recent work on the topc. Indeed, the early nght were ued to buld a currently well-developed lterature on n nte-horzon game that tudy competton wth wtchng cot and contant arrval of conumer. In thee ettng, a rm ha to conder ncentve to extract rent from locked-n conumer a well a from thoe not yet locked-n to any rm. When wtchng cot are large, the ncentve of a rm to mlk t cutomer bae puhe prce up, but the ncentve to grow the future cutomer bae create a tenon to lower prce. The common 3

5 wdom receved from pat lterature that the ncentve to rae prce domnate, o hgh wtchng cot often dynamc competton (ee e.g., Begg and Klemperer 1, Padlla 15, To 15 and Anderon et al. 004). Subequent work by Are and Greco (013), Cabral (013), Dubé et al. (00), Fabra and García (01), and Rhode (013) ha examned the tuaton that are f wtchng cot are not large, whch lead to reverng the prevou ndng (under ome condton). Agan, th ha requred makng extraneou aumpton that make product greatly d erentated from a horzontal tandpont, aumpton that are lkely to drve reult a they do n the mpler two-perod model. In a ettng n whch the only ource of d erentaton between product the extence of (low) wtchng cot, we contend that there no con ct between capture of newly arrved conumer and rent extracton from conumer who are locked-n: ncreang wtchng cot hould unambguouly often dynamc competton. Techncally peakng, our reult mply that the wdepread focu on lnear Markov Perfect Equlbra may have been hghly retrctve. Thu, monotoncty of equlbrum tratege hould be expected jut n the cae of hgh wtchng cot; f t are when uch cot are not hgh, our reult ugget that they mut be drven by other aumpton made. 5 The remander of the paper a follow. Secton ntroduce the game-theoretc model. Secton 3 olve t for the cae of mall wtchng cot. Secton 4 how how reult n the mall wtchng cot cae are robut to a number of relaxaton of aumpton. Secton 5 olve the game when wtchng cot are not mall and Secton 6 conclude. All proof not n the text can be found n the Appendx. The model We conder a two-perod game played by two rm labeled 1 and. Such rm produce homogeneou good at no cot and compete n prce at each perod of play. 6 There alo a unt ma of ex-ante homogeneou conumer. Each ha unt-demand at each perod and wllng to pay at mot v > 0 for one unt of the good. Upon purchang rm good at the ntal perod of play, a conumer uppoed to bear a random cot e f he wtche to rm rval ( = 1; ). In partcular, e unformly dtrbuted between 0 and > 0, o the fracton of conumer of rm that have a cot of wtchng greater than [0; ] equal R d = 1 ( = 1; ). Frm do not oberve wtchng cot of any conumer, 5 Other than a gn cant extent of horzontal d erentaton, another tandard aumpton that doyncratc tate toward competng product exhbt no pertence over tme. 6 Cotle producton wthout lo of generalty f margnal cot are contant (and not too large). Gven th normalzaton, prce n our model hould be nterpreted a markup. 4

6 but t common knowledge how they are dtrbuted acro the populaton of conumer. We aume that conumer have a dcount factor equal to zero (ee ubecton 4. for a relaxaton of th aumpton) and rm have a dcount factor equal to one. Our oluton concept for th game wll be ubgame perfect Nah equlbrum (n pure tratege). 3 Reoluton of the model when wtchng cot are low We wll aume throughout that =v o mall that all conumer alway purchae one of the good at any perod of play. 7 Becaue we eek to characterze the et of ubgame perfect Nah equlbra, we work backward, tartng wth the econd perod of play. 3.1 Second-perod competton We let [0; 1] denote rm cutomer bae, that, the meaure of conumer who purchaed rm good n the rt perod, wth [0; 1] for all f1; g and 1 + = 1. Among the cutomer bae of rm, we can ealy compute the meaure of conumer that prefer uch a rm when t charge p over t rval chargng p 3 p : t equal Pr(e p p 3 ) = ( + p 3 p )=. Clearly, all (partally) locked-n conumer prefer rm over t rval f p 3 > p, wherea the convere hold for p p 3 +. Therefore, the total demand from rm locked-n cutomer : >< f p p 3 0 Q own (p ; p 3 ) = ( + p 3 p ) f 0 p p 3 >: 0 f p p 3 7 A we hall prove below, th wll mply requre that =v 1.. 5

7 Smlarly, the demand from the rval locked-n conumer : >< Q rval (p ; p 3 ) = >: 3 f p p 3 3 ( p 3 p ) f p p f p p 3 0. A a reult, the econd-perod demand functon for rm + 3 f p p 3 >< + 3 ( p 3 p ) f p p 3 0 Q (p ; p 3 ) = ( + p 3 p ) f 0 p p 3 >: 0 f p p 3, and t pro t functon >< (p ; p 3 ) = >: p ( + 3 ) f p p 3 p ( + 3 ( p 3 p )) f p p 3 0 p ( + p 3 p ) f 0 p p 3 0 f p p 3. Wthout lo of generalty, let 3. It eay to prove that there can ext an equlbrum only n the regon n whch 0 p p 3, o let u proceed to characterze t. In uch a regon, Q (p ; p 3 ) = (+p 3 p )= (.e., rm make fewer ale than n the rt perod), o (p ; p 3 ) = p (+p 3 p )=. Smlarly, Q 3 (p 3 ; p ) = 3 + (p p 3 )= and 3 (p 3 ; p ) = p 3 [ 3 + (p p 3 )=]. The rt-order (neceary) condton for an (nteror) optmum are: and + p 3 p = 0 (1) 3 + ( p p 3 ) = 0. () Accountng for corner oluton, the repectve bet repone functon for rm and t Note that, condtonal on a conumer beng wllng to wtch to rm, t necearly hold that uch a conumer make a potve utlty upon wtchng, nce chargng a prce p 3 v domnated for rm 3 and hence p 3 p < v p. Therefore, Pr(e 3 p 3 p ) Pr( e 3 v p j e 3 p 3 p ) = (p 3 p )=. 6

8 rval are and b (p 3 ) = mn(v; + p 3 ) (3) b 3 (p ) = mn(v; ( 3 ) + p ), (4) nce no rm wll ever nd t optmal to charge a prce greater than v. Hence, the game exhbt trategc complementarte, a wa to be expected gven that rm compete n prce. Alo, an ncreae n rm cutomer bae can never make t rval prce le aggrevely. We eek for nteror equlbra, wth the analy of corner equlbra delayed untl Secton 5. A we prove n the propoton below, there ext a unque nteror equlbrum n whch the rm wth larget cutomer bae prce hgher than t compettor, o the latter teal ome conumer n the former cutomer bae. Further, greater wtchng cot often competton, but the rm wth larget cutomer bae ncreae prce more than t rval. Propoton 1 Gven that 3, t hold that p = 3 (+ 3 ) and p 3 = 3 (1+ 3 ) n the unque (nteror) equlbrum. Pro t for rm = ( + 3 ), wherea pro t for rm 3 3 = (1 + 3 ). It hold that p > p 3 and > 3 f > 3, wherea p = p 3 and = 3 f = 3. Proof. See Appendx. In order for all conumer to derve a nonnegatve utlty n an nteror equlbrum, we need that v p = 3 ( + 3 ) regardle of the value taken by 3, that, we need v. Gven th aumpton, Propoton 1 then apple and t qute mmedate to how the followng. Corollary If 3 = 1 and 1=, then both and 3 decreae wth, wth 3 decreang fater than. It eaer to llutrate the corollary by plottng a rm econd-perod pro t a a functon of t own market hare n the rt perod, under the aumpton that all conumer tranacted wth one of the rm. Denotng th pro t functon by (), we have hown that >< (1 ) () = >: Note that th aumpton yeld that v + p 3 nterect at nteror oluton, a we aumed. (1 + 1 ) f [0; 1=] (1 + 1 ) f [1=; 1] 7, > ( 3 ) + p, o bet repone functon do

9 whch can be vualzed n Fgure 1. Proft Fgure 1: () plotted for = gma Fgure 1 llutrate the man departure from prevou lterature, whch ha focued on the cae n whch econd-perod pro t monotoncally ncreae wth rt-perod market hare. A we have hown, the nonmonotoncty of econd-perod pro t may naturally be expected when wtchng cot are mall (recall our mantaned aumpton that v). Such nonmonotoncty wll be crtcal n howng that there can be a multplcty of ymmetrc equlbra. Fgure 1 llutrate that there a tendency toward chargng a prce that matche that of the rval o that = 1= and econd-perod pro t are not too mall: when rm have aymmetrc cutomer bae, the mall rm unhappy becaue t ha few conumer n t cutomer bae; even though the large rm ha more locked-n cutomer, the large rm unhappy too becaue t face a very aggreve rval. Frm bene t from matchng the rval prce, but there are many way to match t, o multplcty of equlbra become an ue, a we hall ee. The followng fact related to Fgure 1 wll prove ueful n undertandng our man ndng when wtchng cot are not too large. Remark are both potve for < 1=. Thu, ncreang a mall market hare n the rt perod at the expene of the rval enhance a rm econd-perod pro tablty, 10 and uch an enhancement greater the hgher ; market hare accumulaton valuable, omethng accentuated a wtchng cot ncreae. What mportant n th regard that a rm drven by th ncentve wll be more wllng to lower the rt-perod prce wth the am of ganng market hare, the wllngne beng 10 The reaon that the rm ncreae t cutomer bae and at the ame tme decreae the rval cutomer bae, thu makng capture of the compettor locked-n conumer le mportant.

10 greater the hgher. Th the uual tuaton, exempl ed for ntance by Klemperer (17b), but, nteretngly, qute the oppote hold when a rm market hare exceed half. Remark are both negatve for > 1=. In word, ncreang an already large market hare n the rt perod at the expene of the rval reduce a rm econd-perod pro tablty, 11 and uch a reducton greater the hgher ; market hare accumulaton not valuable, omethng accentuated a wtchng cot ncreae. A rm drven by th ncentve wll be more wllng to rae the rt-perod prce wth the am of avodng ganng market hare, the wllngne beng greater the hgher. Although th tuaton ha never been formally tuded n the lterature, Klemperer (17a) dd pont out that market hare accumulaton may actually harm econd-perod pro tablty. A rm that uppoed to charge the ame prce a t rval wll take Remark 3 nto account when contemplatng downward prce devaton, but t wll conder Remark 4 when contemplatng upward prce devaton. Th relevant becaue, a we explaned earler, there a trong preure toward chargng the ame prce, even f there are many way n whch th can be done. The next ubecton formalze thee nght more precely. 3. Frt-perod competton We now turn to the rt perod, keepng n mnd that rm antcpate how rt-perod acton n uence econd-perod competton. We wll lghtly abue notaton and denote prce wth the ame notaton ued for the econd perod, nce econd-perod prce are not ued anymore. Gven that product are homogeneou, t eay to ee baed on our prevou analy that rm 1 payo gven the prce p charged by t rval >< p 1 + 4= f p 1 < p 1 (p 1 ; p ) = (p 1 + )= f p 1 = p, >: = f p 1 > p nce (1) = 4=, (1=) = = and (0) = =. We rt examne ymmetrc equlbra n whch both rm charge prce p and each get half of the conumer. Th ymmetrc prcng wll conttute equlbrum behavor f there 11 The reaon that the rm compettor ha fewer conumer n t cutomer bae and therefore compete more aggrevely for the rm larger cutomer bae. Th negatve e ect more mportant for the rm pro t than the bene cal one of havng more locked-n cutomer.

11 are no ncentve for a rm to lghtly decreae or ncreae t prce, that, f the followng two condton hold at the ame tme: (p + )= p + 4= and (p + )= =. Therefore, there ext a contnuum of ymmetrc equlbra that can be Pareto-ranked: any par of prce (p ; p ) uch that p [ 7=; =] can be charged n equlbrum. Note that our aumpton that v mple that v p, o we ndeed have an equlbrum. Equlbrum pro t, denoted by, range from = to 5=, o the equlbrum preferred by rm the one n whch prce are hgher, namely, (p ; p ) = (=; =). To fully characterze the equlbrum et, we now rule out aymmetrc equlbra. Suppoe now that rm 1 charge p 1 and rm charge p, wth p 1 < p, o that rm 1 attract all conumer. In order for th to conttute an equlbrum, everal condton hould be multaneouly at ed: p 1 + 4= (p + )=, p 1 + 4= =, = (p 1 + )= and = p 1 +4=. Thu, rm 1 hould have no ncentve to ether match rm prce or prce hgher than rm, wherea rm hould have no ncentve to ether match rm 1 prce or lghtly undercut rm 1. Becaue p 1 + 4= = mple p 1 3= and = (p 1 + )= mple 7= p 1, t follow that an aymmetrc equlbrum cannot ext, and all equlbra mut nvolve market harng and ymmetrc prcng, wth prce growng over tme. Propoton 5 Suppoe that v. Then (p ; p ) conttute a ymmetrc equlbrum f and only f p [ 7=; =], wth pro t rangng from = to 5=. No other equlbrum ext. Underlyng any equlbrum there a trong force that nduce rm to behave ymmetrcally n a way that doe not dpate pro t. In fact, a rm that lghtly undercut the compettor dcontnuouly enhance ale n the hort run, but dcontnuouly nten e future compettve nteracton: when wtchng cot are not too large, the compettor can tll attract ome conumer and wll actvely try to do o followng the undercuttng. In ome ene, wtchng cot allow the rval of a rm to credbly commt to punhng the rm f t undercut the rval prce. It alo worth notng that, depte the multplcty of ymmetrc equlbra, the nature of the boundare of the equlbrum et very d erent, each beng determned by dtnct economc force. The lower bound on the ymmetrc equlbrum et are from rulng out devaton that nvolve hgher prce and hence accumulatng a maller market hare, o Remark 3 apple. In turn, the upper bound are from rulng out devaton that nvolve accumulatng a larger market hare n the rt perod, o Remark 4 apple. Even though there ext n ntely many equlbra, t eem reaonable from a generc pont of vew to expect that rm wll play the Pareto-domnant ymmetrc equlbrum, a 10

12 tandard equlbrum re nement. 1 Ung th re nement to ngle out a unque ubgame perfect Nah equlbrum, we obtan the followng reult drectly from Propoton 5. Corollary 6 Gven that v, uppoe that rm alway coordnate on playng ther preferred equlbrum. Then rt-perod competton oftened a grow, o payo ncreae. Larger alway often econd-perod competton, a we howed n Propoton 1. When the Pareto-domnant equlbrum played, larger alo often rt-perod competton. The pont that rt-perod prce cannot be too hgh n order to duade a rm from focung on hort-run pro t garnerng, even f uch hort-term focu aocated wth lower econdperod pro tablty by Remark 4. Becaue the harm n econd-perod pro t greater the hgher wtchng cot are, t follow that hgher rt-perod prce can be utaned a grow. 4 Extenon We now examne the robutne of our reult when wtchng cot are mall, wth the am of better undertandng what drve our reult. 4.1 Horzontally d erentated product One may legtmately upect that the dcontnute mpled by product homogenety drve the multplcty of (ymmetrc) equlbra. We proceed to how that th upcon not well founded, the drver beng the ngle-peakne property of econd-perod pro t llutrated by Fgure 1. To th end, let u generalze the rt perod of the model by placng each rm on a d erent end of a Hotellng egment of unt length. Aume that the unt ma of conumer unformly dtrbuted along the egment and denote each conumer unt tranportaton cot by t to how that Lettng (p 1 ; p ) denote rm 1 demand n the rt perod, t tandard 1 f p >< 1 p t t + p (p 1 ; p ) = p 1 f t p 1 p t t >: 0 f p 1 p t 1 We note that the et of ymmetrc equlbra contan the ymmetrc equlbrum that would obtan n the abence of wtchng cot. Th hold too n all the extenon we conder later on n Secton The cae we have jut analyzed correpond to t =

13 Droppng the argument of (p 1 ; p ) for mplcty, we have that rm 1 payo functon : 1 (p 1 ; p ) = p 1 + (). Recallng that >< () = >: (1 + 1 ) f p 1 p 0 (1 ) (1 + 1 ) f p 1 p 0 t hold that 1 (; p ) contnuou, but t not d erentable at p 1 = p. So let u look for a ymmetrc equlbrum n whch both rm charge p. A neceary condton that 1 (p 1 ; p 1 ; p 1 1(p 1 ; p ) p1 1. p1 #p 1; p ) p1 1 = p1 #p and lm p1 "p (p 1; p ) = lm p1 #p (p 1; p ) = 1=, th condton can be rewrtten a: 1 t, t t p t + 3 1t 0 t t p t 15 1t, o p [t 15=; t + 3=] and 1 (p ; p ) = (p + )=. Up to now, our analy wa local around the nond erentablty pont dplayed by pro t functon, o we dd not yet account for large prce devaton. Th mportant becaue rulng out ncentve to perform uch prce devaton may dcard ome prce p [t 15=; t + 3=], perhap all. The analy complex becaue pro t functon are hghly rregular and may well be nether qua-concave nor qua-convex. Even though we have been unable to how that the et of ymmetrc equlbra ha more than one element for all parameter value, we can prove that th the cae for a large number of parameter value. We can alo characterze the full et of ymmetrc equlbra for large value of =t a well a for low value of =t. Propoton 7 Suppoe that v + max(0; t =3). Then there ext a contnuum of ymmetrc equlbra whenever =t = [=; ). When =t =16, (p ; p ) conttute a ymmetrc equlbrum f and only f p [t 15=; t + 3=], wth pro t rangng from (3t )=6 to (3t + 4)=6. When =t 1, (p ; p ) conttute a ymmetrc equlbrum f and only f p [ 7=; t + =], wth pro t rangng from = to (1t + 10)=1. Proof. See Appendx. 1

14 Th propoton generalze Propoton 5, howng that the extence of horzontal dfferentaton mply moderate the mportance of wtchng cot n drvng prcng behavor. It alo how that havng a contnuum of ymmetrc equlbra that can be Pareto-ranked unrelated to demand not beng contnuou. Fgure llutrate rm 1 payo functon n one of thee equlbra. 4 payoff prce Fgure : Plot for 1 (p 1 ; 1) when = 7 and t = 1 A can be een n the Fgure, pro t functon dplay a knk at ther maxmum and need not be ether qua-concave or qua-convex. In th gure that repreent a ymmetrc equlbrum n whch both rm charge prce p = 1, rm 1 ha no ncentve to lower the prce, nce dong o ncreae rt-perod loe and decreae econd-perod pro t. Frm 1 ha no ncentve to ncreae the prce ether. Even though rt-perod loe are cut down, the advere e ect of ntenfyng econd-perod competton more powerful. Increang reult n greater econd-perod pro t, o greater (average) wtchng cot have a potve drect e ect on overall rm pro tablty. However, larger alo a ect rt-perod competton. Aumng that rm play ymmetrc equlbra that are Paretoundomnated, Propoton 7 mple that rt-perod competton often a ncreae (by Remark 4), o we have the followng corollary to the propoton. Corollary Suppoe that rm alway coordnate on playng ther preferred equlbrum. Gven that v + max(0; t =3), =t =16 and =t 1 mply that rt-perod competton oftened a grow, o payo ncreae. 4. Forward lookng conumer Returnng to the cae where t = 0, we now analyze how equlbrum prcng a ected by droppng the aumpton that conumer are myopc. A conumer become more forward 13

15 lookng, ther rt-perod demand become more nelatc, o the prce that can be utaned n a ymmetrc equlbrum hould be greater n prncple. We proceed to how that th partly true: the hghet prce that can be utaned n a ymmetrc equlbrum ncreae, but the lowet one doe not vary. A rt ponted out by Klemperer (17b), rm mght perfectly obtan greater pro t a conumer become better foreghted. Indeed, th wll be the cae f rm coordnate on playng the Pareto-domnant ymmetrc equlbrum. To formally demontrate all th, uppoe now that conumer dcount future utlty at rate [0; 1]. 14 In order to avod havng to entrely redo the analy of econd-perod play, uppoe alo that conumer do not learn ther wtchng cot untl they conume one product. Then the expected future net utlty when dealng wth rm f1; g f t market hare equal 1= can be computed a follow: Z p p 3 u () = (v p 3 ) d Z 0 + (v p ) d p p 3 = v p + (p p 3 ) = v 3 (1 + 1 ) + 1 ( 1 ). In turn, dealng wth rm 3 f t market hare equal 1 1= delver the followng future utlty to a conumer: u 3 (1 ) = v p 3 = v 3 ( 1). So the total utlty of a conumer who buy from rm f1; g when t charge p and attract 1= conumer equal: U () = v p + u (). It rval delver the followng utlty: U 3 (1 ) = v p 3 + u 3 (1 ). When conumer are forward lookng, a rm demand functon become contnuou, nce conumer a a whole do not repond to lower prce by mavely wtchng conumpton, n 14 The cae we analyzed earler correpond to conumer beng myopc (.e., = 0). 14

16 antcpaton of the compettve mbalance that would otherwe reult n the econd perod. Lemma It hold that rm 1 demand a follow: >< (p 1 ; p ) = >: 1 1 f p 1 < p 5=1 r 1+ 1(p p 1 ) 1(p p 1 ) r 1+ 1(p 1 p ) 1(p 1 p ) f p 5=1 < p 1 p f p p 1 < p + 5=1 0 f p 1 > p + 5=1, wth rm demand equal to 1 (p 1 ; p ). Proof. See Appendx. If one retrct attenton to ymmetrc equlbra, t can be readly hown that forward lookng behavor by conumer mply ncreae the larget prce that can be utaned n a ymmetrc equlbrum, a the followng propoton how. Propoton 10 Suppoe that v. Then t hold that (p ; p ) conttute a ymmetrc equlbrum f and only f p 7 (1 + 5) [ ; ], wth pro t rangng from 5( + ) to. 1 Proof. See Appendx. If we retrct attenton to ymmetrc equlbra that are not Pareto-domnated by other, then we obtan the followng corollary. Corollary 11 Suppoe that rm alway coordnate on playng ther preferred equlbrum. Gven that v, rt-perod competton oftened a or grow, o payo ncreae. Conumer foreght mply add an extra force that renforce the one that gve re to Corollary 6, namely, the nelatcty of rt-perod demand that generated by conumer realzng that prce cut n the rt perod are partly ntended to lock them n later on. 5 Reoluton of the model when wtchng cot are not mall Havng examned the equlbrum mplcaton of mall wtchng cot, we turn now to the cae n whch > v. Such cae reult n corner equlbra n the econd tage. Ung the bet repone functon n (3) and (4), we have the followng ueful reult. 15

17 Lemma 1 Suppoe that 3. Then: () v > 1 3 ( + 3 ) mple that p = 3 ( + 3 ) and p 3 = 3 (1 + 3 ), wth = ( + 3 ) and 3 = (1 + 3 ). () 3 < v 1 3 ( + 3 ) mple that p = v and p 3 = 1 [v + ( 3 )], wth 0 < p p 3 < a well a = v [( + 3 ) v] and 3 = 4 [v + ( 3 )]. () v 3 mple that p = v and p 3 = v, wth 0 = p p 3, a well a = v and 3 = 3 v. Proof. See Appendx. Lettng 3 = 1 and = n the prevou lemma, and rewrtng the condton that gve re to each of the three cae n term of, we can dtnguh two tuaton, dependng on whether wtchng cot are "large" or "ntermedate." 5.1 Large wtchng cot Let 3v=. Then the cae that hould be dealt wth baed on Lemma 1 are 0 1 v +, v v + v + and < 1, o econd-perod pro t for a rm a a v + functon of a follow: (1 ) [v + ( >< 4 1 )] f 0 v v + v () = v f v +. v + >: v [(1 + 1 ) v] f v + < 1 By contrat wth our ntal analy, econd-perod pro t can be ealy hown to be monotonc n rt-perod market hare when wtchng cot are large. The pont now that a mall rval that very aggreve wll not matter that much for a large rm wth a cutomer bae that "tcky" and hard to teal away. Proceedng to the reoluton of the full game when 3v=, rm 1 payo n the rt perod >< p 1 + v( v)=() f p 1 < p 1 (p 1 ; p ) = (p 1 + v)= f p 1 = p, >: v =(4) f p 1 > p o any ymmetrc equlbrum prce p mut atfy both (p + v)= p + v( v)=() and (p + v)= v =(4). Therefore, v( v)= p v( v)=, that, t hold that 16

18 p = v( v)= < 0 n the unque ymmetrc equlbrum, wth rm earnng v =() > 0. A wtchng cot ncreae, both prce and pro t decreae n a ymmetrc equlbrum, and ndeed pro t vanh a! 1, nce p! v n uch a cae. Regardng aymmetrc equlbra, the followng condton mut be at ed by an equlbrum (p 1; p ) n whch rm 1 capture all conumer: p 1 + v( v)=() (p + v)=, p 1 + v( v)=() v =(4), v =(4) (p 1 + v)= and v =(4) p 1 + v( v)=(). Becaue p 1 + v( v)=() v =(4) p 1 + v( v)=() mple that p 1 = v(3v 4)=(4), o we cannot pobly have v =(4) (p 1 + v)= = 3v =(). A n the cae n whch v, no aymmetrc equlbrum ext, o we have hown the followng. Propoton 13 Suppoe that 3v=. Then (p ; p ) conttute a ymmetrc equlbrum f and only f p = v( v)= < 0, wth pro t equal to v =(). No other equlbrum ext. The equlbrum played when wtchng cot are large exhbt the followng properte. Corollary 14 Gven that 3v=, the unque equlbrum of the game uch that rtperod competton toughen a grow, wth overall payo decreang. 5. Intermedate wtchng cot We conclude our analy by tudyng the cae n whch (v; 3v=). Snce 3 = 1 and =, Lemma 1 yeld the followng ubnterval for market hare: 0 1 3v, 3v 3v 1 v +, v v + v +, v + < 3v and 3v < 1. Accountng for the pro t made by a rm n each of thee ubnterval, we can ealy contruct t econd-perod pro t a a functon of rt-perod market hare: >< () = >: (1 ) (1 + 3v 1 ) f 0 3v (1 ) 3v [v + ( 4 1 )] f 3v v v + v v f v + v + v [(1 + 1 ) v] f v + < 3v (1 + 1 ) f 3v < 1 A n the mall wtchng cot cae, th functon agan nonmonotonc: t rt ncreae a grow, peak at 3v (1 ; 1) and then decreae a further ncreaed. 17.

19 A uual, t mmedate to contruct rm 1 payo n the rt perod wth the ad of (): >< p 1 + 4= f p 1 < p 1 (p 1 ; p ) = (p 1 + v)= f p 1 = p. >: = f p 1 > p Alo, t eay to ee that a ymmetrc equlbrum n whch rm charge p mut atfy the followng two condton: (p + v)= p + 4= and (p + v)= =. A a reult, any (p ; p ) uch that p [= v; v =] conttute a ymmetrc equlbrum for the rt tage, wth pro t rangng from = to v 4=. Wth repect to aymmetrc equlbra n whch rm 1 charge ome p 1 and rm charge ome p > p 1, they hould atfy the followng et of condton: p 1 + 4= (p + v)=, p 1 + 4= =, = (p 1 + v)= and = p 1 + 4=. Becaue p 1 + 4= = p 1 + 4= mple that p 1 = 3=, t mpoble to have = (p 1 +v)= = (v 3=)= (nce the cae we are examnng requre that < 3v=). Conequently, only ymmetrc equlbra ext, and we have therefore proved the followng reult. Propoton 15 Suppoe that (v; 3v=). Then (p ; p ) conttute a ymmetrc equlbrum f and only f p [= v; v =], wth pro t rangng from = to v 4=. No other equlbrum ext. Aumng that rm do not play Pareto-domnated equlbra ngle out a unque equlbrum wth the followng properte. Corollary 16 Suppoe that rm alway coordnate on playng ther preferred equlbrum. Gven that (v; 3v=), rt-perod competton toughen a grow, wth overall payo decreang. Unlke Corollary 6, rt-perod prce (whch are negatve) fall wth wtchng cot when at an ntermedate level. Prce are negatve becaue equally harng the market create a trong ncentve to become a large rm n the econd perod, and the only way to deter devaton to make conumer attracton cotly n the rt perod. Prce fall wth becaue the econd-perod pro t of a rm that ell to all conumer grow wth, o the payo to devatng grow wth, and the only manner to make uch devaton more cotly to make the prce charged n the rt perod even more negatve. 1

20 6 Dcuon of the reult Our model allow for a un ed treatment of how wtchng cot a ect competton and payo. We have een that the compettve mplcaton of wtchng cot crtcally depend on how the relatonhp between econd-perod pro tablty and rt-perod market hare a ected by. We can readly ee n Fgure 3 that t nonmonotonc relatonhp for low wtchng cot mtgated a grow, eventually becomng monotone ncreang. proft market hare Fgure 3: Second-perod pro t a a functon of rt-perod market hare for v = 1 a well a = 1= (dotted curve), = 5=4 (dahed curve) and = (old curve) The change n th relatonhp between econd-perod pro t and market hare n the rt perod due to the change n the demand elatcty of rm cutomer bae: ncreae, uch demand functon become more nelatc. Prevou lterature ha made techncal aumpton that have the undered byproduct of makng econd-perod demand become very nelatc, o they correpond to the large wtchng cot cae we have analyzed. Such lterature ha remaned lent on the e ect of wtchng cot on competton when they are not too large. A we have hown, demand can become o elatc that a large rm wll care negatvely about accumulatng more market hare becaue of the negatve trategc reacton elcted on the rval. It not nnocuou to aume that cutomer bae are qute "tcky" for reaon other than wtchng cot. For mall wtchng cot, we have hown (c.f. Propoton 1) that the mall rm charge a hgher prce a t gather more market hare, wherea the large rm doe the oppote. A a reult, prce are not monotone n market hare, whch ha ubtantve mplcaton for the current lterature on n ntehorzon model of competton wth wtchng cot and arrval of new conumer over tme. In partcular, the tandard focu on prce that monotoncally vary wth market hare hold 1 a

21 becaue product are gn cantly d erentated from a horzontal tandpont: low wtchng cot have d erent e ect from the one unveled o far. Havng a un ed treatment of how wtchng cot, and only wtchng cot, a ect compettve outcome that t allow u to tudy whether rm bene t from competng for conumer who bear cot of wtchng acro d erent product. Our reult, baed on electng out the Pareto-undomnated equlbrum a the unque one whenever there multplcty, unambguouly ndcate that greater wtchng cot ncreae rt-perod prce and overall pro t when wtchng cot are mall and reduce them otherwe. Th repreented n Fgure 4. payoff, prce wtchng cot Fgure 4: Frt-perod prce (dahed curve) and overall payo (old curve) a a functon of for v = 1 Swtchng cot ntenfy rt-perod competton f and only f they are u cently large. Th the tradtonal meage from the lterature, but agan t drven by the aumpton that cutomer bae are very tcky for reaon other than the wtchng cot themelve. When cutomer bae are eay to teal away, wtchng cot wll not ntenfy competton n the rt perod and wll not be detrmental for overall pro tablty. 7 Concluon In a two-perod game played by two ymmetrc rm and by conumer who d er n ther wtchng cot, we have analyzed how ncreang (average) wtchng cot a ect rt-perod competton and rm payo. When wtchng cot are low, a rm that ncreae t market hare n the rt perod need not enhance future pro tablty, a tuaton overlooked by pat 0

22 lterature. We have hown that the nonmonotonc relatonhp between rt-perod market hare and econd-perod pro t reult n a multplcty of ymmetrc equlbra. However, the unque Pareto-undomnated equlbrum uch that greater (average) wtchng cot lead to greater prce and pro t n both perod. When wtchng cot are hgh, greater (average) wtchng cot lead to a lower prce n the rt perod and a lower overall pro tablty. We have provded a un ed treatment of how wtchng cot a ect competton wthout makng any aumpton on the degree of horzontal d erentaton between product. Such an aumpton a key feature of prevou lterature on the topc, ncludng the more recent one ung n nte-horzon model. Indeed, there a currently ongong debate on whether hgher wtchng cot alway lead to hgher prce and pro t n dynamc ettng wth contnuou arrval of new conumer over tme. Our reult eem to ndcate that a potve relatonhp between prce and wtchng cot may be expected for mall wtchng cot, but the reult for larger wtchng cot n prncple ambguou. We hope our model can be ued a a buldng block for future analye that clear th ue. Another relevant extenon that would be worthwhle purung to omehow allow for market expanon e ect. In our model, no conumer left out of the market, o a change n wtchng cot mply redtrbute overall urplu created between rm and conumer. Our pont wa to analyze how change n wtchng cot hape competton and rm payo. Examnng ther e ect on ocal welfare an mportant ue that very challengng and left for future reearch on the topc. 1

23 APPENDIX Proof of Propoton 1. Solvng the ytem that cont of equaton (1) and () yeld p = 3 ( + 3 ) and p 3 = 3 (1 + 3 ), wth 0 p p 3 becaue of our workng aumpton that 3. Alo, (p ; p 3 ) = ( + 3 ) and 3 (p 3 ; p ) = (1 + 3 ), o 3, wth equalty f and only f = 3. It only reman to how that, keepng the rval prcng trategy xed, nether rm ha an ncentve to charge a ubtantally d erent prce from the one t uppoed to. We tart by rulng out pro table devaton by rm 3, takng nto account that p 3 ( 3 + ) f p 3 p >< p 3 [ 3 + ( p p 3 )] f p 3 p 3 (p 3 ; p 0 ) = p 3 3 ( + p p 3 ) f 0 p 3 p >: 0 f p 3 p a contnuou functon (even f t not d erentable at three pont). We have jut hown that the optmal prce to be charged by rm 3 wthn [p ; p ] p 3. Alo, t clear that chargng a prce below p cannot be pro table. 15 So the only devaton to be condered by rm 3 would nvolve (ubtantal) ncreae n the prce o a to focu on mlkng t cutomer bae, but not above p +, nce they would yeld a zero payo. Note that p 3 3 ( + p p 3 )= maxmzed at bp 3 ( + p )=, whch alway maller than p +. Alo, bp 3 p, o 3 (p 3 ; p ) ha a local maxmum at bp 3 [p ; p + ), and we need to how that 3 (p 3 ; p ) = (1 + 3 ) 3 (bp 3 ; p ) = 3 36 (5 + 3 ) n order to rule out a pro table devaton for rm 3. Th nequalty hold f and only f ( ) 3 (5 + 3 ), whch alway at ed for any 3 [0; 1] Gven the contnuty of the payo functon, th follow from the fact that 3 (p 3 ; p ) > 3 (p ; p ) and that 3 (p 3 ; p ) everywhere ncreang on the et [0; p ]. 16 To how th, let y 3 = and de ne (y) (+4y) y(5+y). Becaue (0) > 0 = (1), 0 (1) = 0 and 00 (y) = 6( y) > 0 for y [0; 1], t hold that (y) 0 for all y [0; 1].

24 We conclude by rulng out pro table devaton by rm, takng nto account that p ( + 3 ) f p p 3 >< p [ + 3 ( p 3 p )] f p p (p ; p ) = p ( + p 3 p ) f 0 p p 3 >: 0 f p p 3 a contnuou functon (even f t not d erentable at three pont). We have hown that the optmal prce to be charged by rm wthn [p 3 ; p + ] p. Alo, t clear that chargng a prce above p 3 + cannot be pro table. So the only devaton to be condered by rm would nvolve (ubtantal) drop n the prce o a to focu on capturng ome of the rval cutomer bae. Note that p [ + 3 ( p 3 p )] maxmzed at bp ( 3 +p 3 )=, where bp p Becaue bp > p 3, chargng a prce below p 3 cannot be pro table. Alo, bp p 3 mple that (p ; p 3 ) everywhere ncreang on the et [p 3 ; p 3 ], and hence (p ; p 3 ) ncreang for p < p, whch how that no pro table devaton ext for rm ether. Proof of Propoton 7. We conder ncentve by rm 1 to devate from chargng prce p 15 [t ; t + 3 ] gven that t rval ndeed chargng p. Suppoe rt that rm 1 contemplate chargng p 1 < p, o that t pro t become l 1(p 1 ; p ) = p 1 + (1 + 1 ). We wll work on the quantty pace, o let p 1 = t + p t be the prce that mplement ale equal to [0; 1] gven that the rval charge p 15 [t ; t + 3 ]. Then rm 1 chooe > 1= o a to maxmze 1 () (t + p t) + (1 + 1 ). Note that d 1 () = t + p 4t + d (1 1 ) and d 1 () = 4( t). Let K d 1 3 1=3 > 0. Then 1 () concave for > K and convex for < K. We conder three 1t cae: (a) Suppoe rt that t 36 16, o that K 1=. Then 1 () trctly concave for 17 To demontrate th, note that bp p 3 equvalent to ( 3 ) Let y 3 = [0; 1] and de ne (y) y + y 3. Becaue (0) < 0 = (1) and 00 (y) > 0, t hold that (y) 0 for all y [0; 1], a dered. 3

25 > 1=. Gven that d 1 () = p d #1= t 3 0 (nce p [t that 1 (1=) > 1 () for all > 1=. Therefore, any p [t devaton nvolvng large prce drop. (b) Suppoe now that ; t + 3 ]), t follow 15 ; t + 3 ] mmune to 16 < t < 1, o that 1= < K < 1. Then 1 () dplay an n ecton pont at = K, wth 1 () convex for (1=; K) and concave for (K; 1). If d 1 () < 0, then d 1 () mut be decreang for all > 1=. In turn, d 1 () 0 d =K =K mple that 1 () wll attan a local maxmum at ome (K; 1) f t alo hold that d 1 () < 0, wherea d 1 () wll be ncreang for (K; 1) f d 1 () 0. Note d "1 "1 rt that d 1 () < 0 f and only f d =K wherea d 1 () d p t < 4K 1 t (1 1 K ) = ( 1 t )1=3 < 0 f and only f p t "1 t 1, < 3. So we have when p t < (1 t )1=3 t 1 that rm 1 cannot mprove by ellng more. Snce < < 1 (the cae that gve re to (b)) t 15 mple that 1 t < (1 t )1=3 1 < 1+ 3 t t, t hold that any 15 p [t ; t(1 t )1=3 15 t] [t ; t + 3 ] mmune to devaton nvolvng large prce rae. If ntead p t (1 t )1=3 1, havng p 3 yeld that the optmal devaton would be to ell to t t all cutomer. 1 If p 3t, 1 (p ; p ) 1 (1) f and only f p t Snce 16 < t < mple that t + < 3t, there can be no p 3t uch that 1 (p ; p ) 1 (1) whenever t <. If ntead t < 1, then any p [3t; t + ] [t 15 ; t + 3 ] mmune to devaton nvolvng large prce drop. (c) Suppoe nally that t 1, o that K 1. Then 1 () convex, o the fact that 1 Note that 3 > ( 1 t )1=3 t 1 for t (36 p p ; 1), o 3 enure that 16 t t (1 t )1=3 t 1. 4

26 d 1 () 0 mple that the only reaon why rm 1 may want to devate by chargng d #1= a prce o that = 1. Snce 1 (p ; p ) = p + 1 (1) = p t + 4, we mut have p t+, where t+ 3 < t+ becaue we are analyzng the cae n whch =t 1. Snce t 15= < t + =, t hold that any p 15 [t ; t + ] mmune to devaton nvolvng large prce drop. Havng examned downward prce devaton, we now uppoe that rm 1 contemplate chargng p 1 > p, o that t pro t become r 1(p 1 ; p (1 ) ) = p 1 + (1 + 1 ). Agan, we wll work on the quantty pace, o let p 1 = t+p t be the prce that mplement ale equal to [0; 1] gven that the rval charge p 15 [t ; t + 3 ]. Then rm 1 chooe < 1= o a to maxmze b 1 () (t + p t) + (1 ) (1 + 1 ). Note that db 1 () = t + p (1 + )(3 ) 4t + and d 1 b () = 4( d (1 ) d (1 ) 3 t). Let 1=3 k 1. Then t b 1 () concave for k and convex for k. We dtnguh three cae: (a) Suppoe rt that t 16, o that k 1=. Then b 1 () trctly concave for < 1=. Gven that db 1 () = p + 15 t 0 (nce p 15 [t d ; t + 3 ]), t follow "1= that b 1 (1=) > b 1 () for all < 1=. Therefore, any p 15 [t ; t + 3 ] mmune to devaton nvolvng large prce rae. (b) Suppoe now that 16 < t < 7 16, o that 0 < k < 1=. Then b 1 () dplay an n ecton pont at = k, wth b 1 () concave for (0; k) and convex for (k; 1=). If d b 1 () > 0, then d b d 1 () mut be ncreang for all < 1=. In turn, b 1 () 0 d =k =k mple that b 1 () wll attan a local maxmum at ome (0; k) f t alo hold that d b 1 () > 0, whch alway the cae nce db 1 () = d d. Note that db 1 () > 0 d #0 #0 =k 5

27 f and only f p t > t + (1 + k)(3 k) p 4tk + = 3 + (1 k) t 1=3 6. t 1=3 6 that rm 1 cannot mprove by ellng le. Snce t So we have when p t > 3 + t 16 < t < 7 16 (the cae that gve re to (b)) mple that t, t hold that any p [3t + t to devaton nvolvng large prce rae. 6 t t < 3 + t 1=3 ; t + 3 ] [t 15 1=3 6 < t ; t + 3 ] mmune (c) Suppoe nally that t, o that k 0. Then t tandard to how that b 1 () convex, o the fact that db 1 () 0 mple that the only reaon why rm 1 may want d "1= to devate by chargng a prce o that = 0. Snce 1 (p ; p ) = p + b 1 (0) =, we mut have p 7, 7 > t 15 becaue we are analyzng the cae n whch 7 =t =. Snce < t + 3, t hold that any 7 p [ ; t + 3 ] mmune to devaton nvolvng large prce rae. Table 1 ummarze how prce devaton, both upward and downward, may re ne the et of canddate equlbrum prce: Set of parameter Downward Upward value for =t devaton devaton [0; 16 ] 15 p [t ; t + 3 ] 15 p [t ; t + 3 ] [ 16 ; ) 15 p [t ; t + 3 ] p [3t + 1=3 6 t ; t + 3 t ] [ ; ) 15 p [t ; t(1 t )1=3 t] p [3t + 1=3 6 t ; t + 3 t ] [ ; ) [; 1) p [3t; t + ] p [ [1; 1) p 15 [t ; t + ] p [ 7 ; t + 3 ] 7 ; t + 3 ] 6

28 Accountng for the retrcton arng from rulng out ncentve to devate, Table how equlbrum prce dependng on parameter value: Set of parameter Admble value for =t equlbrum prce [0; 16 ] 15 p [t ; t + 3 ] [ 16 ; ) p [3t + 1=3 6 t ; t + 3 t ] [ ; ) p [3t + 1=3 6 t ; t( 1 t t )1=3 [ ; ) [; 1) p [3t; t + ] [1; 1) p 7 [ ; t + ] t] Notce for the cae n whch = =t < that we have been unable to how the optmalty of ome of the canddate for ymmetrc equlbrum prce, reaon why the cell n both table are left empty. Note alo for Table that, except for the cae n whch =t =16 and =t 1, the et of ymmetrc equlbrum prce (wth hgh lkelhood n our vew) a uperet of the one gven. When =t =16 and =t 1, the et decrbed the actual et of equlbrum prce, a we have formally hown above. We conclude the proof by notng that we have mplctly aumed that all conumer were alway wllng to purchae one of the good, even f one of the rm devate and charge a lower prce, o that the rval ell to conumer farther away than the one n the mddle of the Hotellng egment. For all conumer to be wllng to conume any of the good, we need v t + p. A u cent (but not neceary) condton for th to hold for all p that t v t + 3, nce we alway have that p t + 3. v + t. Recallng the 3 aumpton that v, we can ummarze both by aumng that v + max(0; t 3 ). Proof of Lemma. Note that U 1 (1) > U (0) f and only f p 1 < p 5=1, o rm 1 demand 1 f p 1 < p 5=1, and t 0 f p 1 > p + 5=1. For p 1 [p 5=1; p ], rm 1 demand olve U 1 () = U (1 ), that, (4 + 1 )( 1 ) = 1(p p 1 ). (5) 7

29 Therefore, If p 1 [p ; p + 5=1], then = = 1 r 1(p p 1 ) (p p 1 ) r 1(p 1 p ) (p 1 p ) Proof of Propoton 10. Suppoe that rm 1 chooe p 1 to maxmze >< p (p 1 ; p ) = (1 + 1 ) f p 1 p 0 (1 ) >: p 1 + (1 +, 1 ) f p 1 p 0 where a hortcut for the demand functon n Lemma. Suppoe that rm charge ome prce p. In order for rm 1 to have ncentve to charge prce p a well, the followng hould hold: Takng nto account that 1 (p 1 ; p 1 ; p 1 th condton can be rewrtten a: 1(p 1 ; p ) p1 1. p1 #p 1; p ) p1 1 = p1 #p lm (p 1; p ) = lm (p p 1 "p p1 1; p ) = 1=, #p (p 3 ) (p + 15 ). Hence, any ymmetrc equlbrum prce p mut be uch that ( 5)=3 p (1+)=3. We conder now ncentve by rm 1 to devate from chargng prce p ( 5) (1 + ) [ ; ] 3 3 gven that t rval ndeed chargng p. Suppoe rt that rm 1 contemplate chargng p 1 < p, o that t pro t become l 1(p 1 ; p ) = p 1 + (1 + 1 ). A uual, we wll work

30 on the quantty pace, o ung expreon (5) yeld that p 1 = p 1 (4 + 1 )( 1 ) the prce that mplement ale equal to > 1= gven that the rval chargng p. Then rm 1 chooe > 1= o a to maxmze 0 1() (p 1 (4 + 1 )( 1 )) + (1 + 1 ). Becaue d 1 () ( + ) = > 0, t clear that the only devaton that rm 1 would d 3 conder f t lowered the prce would be accumulatng the maxmum poble market hare. Suppoe now that rm 1 contemplate chargng p 1 > p, o that t pro t become r 1(p 1 ; p (1 ) ) = p 1 + (1+ 1 ). A uual, we wll work on the quantty pace, o let p 1 = p 1 ( )( 1 ) be the prce that mplement ale equal to < 1=.1 1 Then rm 1 chooe < 1= o a to maxmze b 0 1() (p 1 ( )( 1 (1 ) )) + ( ). Becaue d b 0 1 () [(4 ) + (1 )] = > 0, t clear that the only devaton that rm d (1 ) 4 1 would conder f t raed the prce would be not accumulatng any market hare. We have hown that the only devaton worthwhle takng nvolve ether not capturng any market hare or capturng all of t. Therefore, we need both 0 1(1=) = p + 0 1(1) = p + ( 5) 1 and that, 7 p only prce p uch that p [ 0 1(1=) = p + b 0 1(0) =, (1 + 5). Becaue ( 5)= < 7= and (1 + 5)= < (1 + )=3, 7 (1 + 5) ; ] are (ymmetrc) equlbrum prce. Note that we have mplctly aumed throughout that all conumer would be wllng to trade wth rm when rm 1 devate from p and ncreae t prce o a to accumulate no market hare. Th requre that U (1) = v p + u (1) = v p 11 + (v 1 ) be 1 Th follow from replacng wth 1 n expreon (5) gven that p = p, nce uch an expreon wa derved under the preumpton that > 1=.