We re Number 1: Price Wars for Market Share Leadership

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1 MANAGEMENT SCIENCE Vol. 64, No. 5, May 28, pp ISSN (prnt), ISSN (onlne) We re Number : Prce Wars for Market Share Leadershp Luís Cabral a, b Downloaded from nforms.org by [ ] on 9 June 28, at 8:. For personal use only, all rghts reserved. a Stern School of Busness, New York Unversty, New York, New York 2; b Centre for Economc and Polcy Research, London ECV DX, Unted Kngdom Contact: lus.cabral@nyu.edu (LC) Receved: March 3, 25 Revsed: Aprl 28, 26 Accepted: September 2, 26 Publshed Onlne n Artcles n Advance: May 9, 27 Copyrght: 27 INFORMS Keywords: Abstract. I examne the dynamcs of olgopoles when frms derve subjectve value from market leadershp. In equlbrum, prces alternate n tandem between hgh levels and occasonal prce wars, whch take place when market leadershp s at stake. The statonary dstrbuton of market shares s typcally multmodal; that s, much of the tme, there s a stable market leader. Even though shareholders do not value market leadershp per se, a corporate culture that values market leadershp may ncrease shareholder value. Hstory: Accepted by Bruno Cassman, busness strategy. Supplemental Materal: The onlne appendx s avalable at dynamc olgopoly prce wars market shares ordnal rankngs. Introducton In many ndustres, the prevalng manageral atttude places a dsproportonate weght on beng number one the market share leader. For example, n May 22, Arbus accused Boeng Co. of tryng to start a prce war after the U.S. company pledged to work aggressvely to regan a 5% share of the market (Ostrower 22). A February 2 headlne announced that IBM reclams server market share revenue crown n Q4, addng that IBM and HP [Hewlett-Packard] wll contnue to duke t out (Dgnan 2). Accordng to CNN, GM [General Motors] held onto ts No. rank by cuttng prces on cars to the pont where they were unproftable (Isdore 22). And durng a 27 ntervew wth a group of bloggers, SAP chef executve offcer Hennng Kagermann stated, We are not arrogant, we are the market leader (Farber 27). In ths paper, I examne the mplcatons of ordnal comparsons, n partcular the number one bas, for market competton. Specfcally, I examne the behavor of managers who receve an extra utlty kck from beng market share leaders. 2 (I do not develop a theory to explan why managers derve utlty from beng market leaders, though I do dscuss some ratonal and behavoral reasons for ths pattern.) I develop a model wth two sellers and multple buyers, all of whom lve forever. Buyers reassess ther choce of seller at random ponts n tme. Buyers have preference for sellers and for money. Sellers have a preference for money and for beng number one. The paper makes two central ponts, one normatve and one postve. Frst, I show that a corporate culture that emphaszes the mportance of market leadershp may ncrease shareholder value even f shareholders do not care about market leadershp (or market shares) per se. Second, I propose a smple model that leads to a rch theory of prce wars and the evoluton of market shares. Specfcally, I show that a frm s utlty from begn market leader mples a prce drop when market shares are close to 5%, and thus a lot s at stake. Moreover, I provde condtons such that, fearful of enterng nto a prce war, competton s softened at states close to the prce war regon so much so that shareholder value ncreases wth respect to a stuaton where managers do not care about market share leadershp. The softenng of prce competton also mples that the statonary dstrbuton of market shares s bmodal; that s, most of the tme, one frm s larger than the other one and occasonally prce wars for market share take place. My paper also has mplcatons for a central queston n strategy and ndustral organzaton: the persstence of dfferences across frms. Typcally, these are explaned by prmtve dfferences across frms, such as unque resources (Wernerfelt 984, Barney 986, Derckx and Cool 989); endogenous dfference due to ncreasng returns, such as learnng curves or network effects (Cabral and Rordan 994, Cabral 2); or stckness n market shares due to swtchng costs or related effects (Beggs and Klemperer 992). My model features none of the above characterstcs and stll nduces a statonary dstrbuton of market shares that can be bmodal. In other words, for a long perod of tme, there s a large frm and a small frm (even though the model s symmetrc); the only barrer to moblty that stops the small frm from becomng large s the prce war t must go through to ncrease market share. In ths sense, my model also provdes a new perspectve on the concept of moblty barrers. In a semnar paper, Caves and Porter (977) proposed an extenson 23

2 Cabral: We re Number : Prce Wars for Market Share Leadershp 24 Management Scence, 28, vol. 64, no. 5, pp , 27 INFORMS Downloaded from nforms.org by [ ] on 9 June 28, at 8:. For personal use only, all rghts reserved. of the theory of entry barrers, one that goes beyond the movement of a frm from zero output to some postve level: for example, n some cases, establshed frms enter a new segment of a gven ndustry. Exogenous or endogenous mpedments to such segment entry are denoted moblty barrers. My theory of dynamc prce competton suggests an addtonal nstance of ntrandustry moblty: a frm that s a market share follower becomng a market share leader. To the extent that the statonary dstrbuton of market shares s multmodal (as I wll show s frequently the case), ths shft n relatve postons s suffcently dscontnuous that the analogy of moblty barrers s meanngful. The barrer I wll consder s endogenous and results from the market leader s aggressve prce behavor when the laggard s market share becomes threatenngly close to the leader s... Related Lterature and Contrbuton The paper makes several contrbutons to the strategy and ndustral organzaton lteratures. Frst, t studes the mplcatons of a farly pervasve phenomenon namely, frms desre to be market share leaders. Baumol (962) and others have developed models where frms follow objectves other than proft maxmzaton. However, to the best of my knowledge ths s the frst paper n the ndustral organzaton lterature explctly to consder prcng dynamcs when number one effects are n place. Second, I develop a realstc theory of prce wars. For all of the rchness of ndustral organzaton theory, the core theory of prce wars s stll connected almost exclusvely to colluson models. In Green and Porter (984), prce wars result from the breakdown of collusve equlbra durng perods of (unobservable) low demand. Rotemberg and Saloner (986) suggest that prce wars correspond to frms refranng from colluson durng perods of observable hgh demand. 3 By contrast, I assume that frms do not collude (they play Markov strateges). Instead of a repeated game, I assume frms play a dynamc game where the state s defned by each frm s market share. In ths context, prce wars emerge n states where a frms value functon s partcularly steep that s, durng perods when a frm s gan from ncreasng market share s partcularly hgh. In ths sense, the prcng equlbrum n my model bears some resemblance to models wth learnng or network effects (Cabral and Rordan 994, Besanko et al. 2, Cabral 2). However, the dynamcs n these papers are drven by ncreasng returns, whereas I consder a settng wth constant returns to scale. Thrd, I provde an nstance where corporate culture has a clear nfluence on the way frms compete. Specfcally, I provde condtons such that a devaton from proft maxmzaton may n effect lead to hgher frm value. Vckers (985), Fershtman and Judd (987), and Sklvas (987) have shown that proft-seekng shareholders may have an nterest n delegatng decsons to managers based on ncentve mechansms that dffer from proft maxmzaton. 4 Specfcally, f the frms decson varables are strategc complements (as s the case n my model), then equlbrum delegaton contracts ask managers to pay less mportance to profts than shareholders would: such contracts soften prce competton and lead to overall hgher profts than n the normal prce competton game. My approach to delegaton s very dfferent, and so are the results. Specfcally, number one effects ask frms to place more weght on market shares than shareholders would. Ths makes frms more, not less, aggressve..2. Roadmap The rest of ths paper s structured as follows. In Secton 2, as a prelude to the full-fledged framework, I develop a smple, two-stage model of foremarket and aftermarket competton. Although the model s rather stylzed, t conveys the ntuton that a corporate culture of strvng to be number one may nduce a credble commtment that s valuable to shareholders (who, by assumpton, do not share the manager s utlty from beng market share leaders). In Sectons 3 5, I develop an nfnte-perod, multstate model of prce competton for market share. Besdes confrmng some of the qualtatve features of Secton 2, I also propose a novel theory of prce wars for market shares. 5 Secton 6 dscusses robustness and extensons of the basc results. Secton 7 concludes the paper. 2. A Two-Stage Model Consder a two-stage model wth a foremarket and an aftermarket. For example, the foremarket may correspond to a hardware purchase and the aftermarket to some consumable. Each consumer frst chooses a seller n the foremarket, wllng to pay for at most one unt from ether seller s product. Upon purchasng the basc product, a consumer s wllng to pay µ n the aftermarket for at most one unt. However, f a consumer purchases n the aftermarket from a dfferent seller than n the foremarket, then the consumer must pay an addtonal swtchng cost. The value of s each consumer s prvate nformaton and has the followng dstrbuton: ( L wth probablty, H wth probablty, where H > L > and 2(, ). Suppose that two frms, a and b, have equal-szed consumer nstalled bases, n a n b n, where n > /µ. ()

3 Cabral: We re Number : Prce Wars for Market Share Leadershp Management Scence, 28, vol. 64, no. 5, pp , 27 INFORMS 25 Downloaded from nforms.org by [ ] on 9 June 28, at 8:. For personal use only, all rghts reserved. The tmng of the game s as follows. Frst, frms smultaneously set prces p ( a, b) for a new consumer comng on the market (foremarket). Next, the new consumer chooses to buy from frm a, frm b, or none. Next, frms smultaneously set prces q ( a, b) n the aftermarket. Fnally, all consumers choose whether to purchase n the aftermarket from the same frm they purchased before or rather from a dfferent frm (payng f they do so). I now come to the central feature of the model: the beneft from leadershp. I assume that frm a s corporate culture s such that ts manager enjoys an addtonal payoff when t s the leader as measured by the aftermarket nstalled base; that s, when n a > n b. 6 When >, I dstngush between frm a s shareholder value, whch corresponds to frm a s proft (from foremarket and aftermarket sales) from frm a s manager s utlty, whch ncludes, n addton, the beneft from market leadershp (f t apples). The man result s that although shareholders do not beneft from market leadershp, they may beneft from a culture that creates such a beneft n the manager s eyes. Proposton. There exsts such that f, then frm a s equlbrum shareholder value s greater when than wth. The proof of ths and subsequent results may be found n the appendx. The ntuton for Proposton s that, by creatng a corporate culture of market leadershp, frm a effectvely commts to becomng very aggressve should t lose ts market leadershp (whch happens when frm b makes a sale n the foremarket). Aggressve prcng by frm a s harmful to frm b. In fact, t mples that frm a poaches all of frm b s customers n the aftermarket. Fearng such aggressve prcng, frm b softens up ts foremarket behavor. Fnally, such softenng up by frm b ncreases frm a s foremarket profts and, along the equlbrum path, the event of aggressve prcng by frm a never takes place, so frm a s shareholders are better off than they would be f frm a s managers were straght value maxmzers. Proposton corresponds to the frst of the two man ponts n the paper: the normatve pont that a corporate culture of seekng to be number one nduces a credble threat of prce aggresson n states when market leadershp s at stake. Whle the executon of the threat destroys shareholder value, the strategc commtment tself ncreases shareholder value. If, along the equlbrum path, the threat does not need to be followed through, then only the postve effect remans. The dea that credble threats may mprove a frm s poston s hardly new (see, for example, the top dog strategy descrbed n Fudenberg and Trole 984). The novel dea s that the pervasve culture of market leadershp may do that job. In other words, one may thnk of corporate culture as a frst-stage nvestment that nfluences the outcome of a second-stage market game. The dea that contracts wrtten between shareholders and managers may have strategc value s also not novel n of tself. Vckers (985), Fershtman and Judd (987), and Sklvas (987) have shown that proft-seekng shareholders may have an nterest n delegatng decsons to managers based on ncentve mechansms that dffer from proft maxmzaton. Specfcally, f the frms decson varables are strategc complements (as s the case n my model), then equlbrum delegaton contracts ask managers to pay less mportance to profts than shareholders would: such contracts soften prce competton and lead to overall hgher profts than n the normal prce competton game. My approach s very dfferent, and so are the results, essentally because my approach s dynamc that s, the game I consder evolves over a seres of states whereas that of Vckers (985), Fershtman and Judd (987), and Sklvas (987) s essentally statc. Specfcally, number one effects ask frms to place more weght on market shares than shareholders would. Ths makes frms more, not less, aggressve. From a statc pont of vew, ths effect s bad news for shareholders, for excessvely aggressve prcng means lower equlbrum profts. However, the prce wars that follow from number one effects are rare (n the above example, they do not take place at all); the negatve effect of overly aggressve prcng s more than compensated for by the deterrence effect mpled by the threat of a prce war. 2.. Corporate Culture as Busness Strategy Proposton begs the queston of when a frm would want to have a number one culture. Formally, we can answer ths queston by consderng a corporate culture metagame (see Fgure ) where, n a frst stage, frms smultaneously choose the value of ( a, b); then, the frms havng observed the choces of values, n a second stage, both play the game consdered above. There are dfferent versons that the frst stage of ths game could take. Here, I assume that each frm must choose between not havng and havng a number Fgure. Corporate Culture Metagame Frm a à n å n å + ì Frm b n å n å n å à à n å + ì à

4 Cabral: We re Number : Prce Wars for Market Share Leadershp 26 Management Scence, 28, vol. 64, no. 5, pp , 27 INFORMS Downloaded from nforms.org by [ ] on 9 June 28, at 8:. For personal use only, all rghts reserved. one corporate culture, where the latter s defned by a specfc value of. Proposton 2. Suppose that each frm must choose 2 {, }. There exsts such that, f, then a the choces are strategc substtutes, and b there exst two asymmetrc equlbra n pure strateges (, ) and (,). In essence, Proposton 2 states that the corporate culture metagame s akn to a game of chcken: both players prefer an aggressve culture gven that the rval does not have one, but both players prefer not to have one gven that the rval has one. In other words, the choces of an aggressve corporate culture are strategc substtutes. Part of Proposton 2 s a corollary of Proposton : f my rval has no number culture, then I want to have one. The addtonal part of Proposton 2 s that frms prefer to avod a clash of aggressve number cultures, n the same way that, n a game of chcken, the worst possble outcome s for both players to choose not to swerve. In the present context, a subgame where both frms have a number one culture nduces a prce war n the aftermarket, resultng n a loss for shareholders of the order of. In fact, that s approxmately how far below cost managers are wllng to take prces so as to guarantee market share leadershp (from whch managers, but not shareholders, derve utlty). Proposton 2 states that there exst two asymmetrc equlbra n pure strateges. As often s the case n games of ths type, there also exsts a symmetrc equlbrum n mxed strateges: n the frst stage, each frm chooses a number one culture wth probablty p. In turn, ths mples that all four possble outcomes from no frm choosng a number one culture to both frms dong so takes place wth postve probablty Summary We may summarze ths secton by statng that a number one culture may be used as a value-enhancng commtment to be aggressve n prcng should a frm fall behnd ts rval. In a metagame where frms choose corporate culture before competng n the market, there exst asymmetrc equlbra where one frm (and not the other) chooses a number one culture. The specfc extensve form consdered n ths secton s hghly stylzed. My man purpose has been to llustrate one of the central ponts n the paper: the strategc value of commttng to a number one corporate culture. In the next two sectons, I propose a more general and realstc model, one that evolves over nfnte perods and features many possble market share states. The result of hgher shareholder value under aggressve number one management wll reappear and so wll an addtonal result: a postve theory of prce wars that s, the dea that prce wars may result from the aggressve behavor of managers for whom market shares, n partcular market leadershp, are partcularly mportant. Specfcally, n the next secton, I lay down the basc model, ncludng results that establsh key propertes of ts statonary state. In Secton 4, I show, by means of analytcal results and numercal smulatons, how the model leads to a natural theory of prce wars. In Secton 5, I return to the basc questons examned n the present secton namely, the extent to whch commtment to an aggressve corporate culture may beneft shareholders who do not drectly care about market share. The value added by Secton 5 s that t s based on a more realstc model than the one consdered n the present secton. In Secton 6, I consder a varety of robustness and extensons of the nfnte-perod model, ncludng the possblty of demand-sde number one effects and more than two competng frms. 3. An Infnte-Perod Model Consder a duopoly wth two frms, a and b. I wll use and j to desgnate a frm genercally; that s,, j a, b. Tme s dscrete and runs ndefntely: t, 2,...The total number of consumers s gven by. The model dynamcs are gven by the assumpton that agents make durable decsons nfrequently. Specfcally, at random moments n tme, a consumer s called to reassess ts decson regardng the frm t buys from. One way to thnk about ths s that each consumer s swtchng cost follows a stochastc process, alternatng between the values of nfnty (nactve consumer) and zero (actve consumer). Alternatvely, I may assume that consumers leave the market (death) and are replaced by new consumers n equal number (brth). 7 Untl later n the paper, I consder a symmetrc equlbrum of a symmetrc game. In partcular, I assume both frms have the same corporate culture and play the same prcng strategy. I do ths for several reasons. Frst, the man analytcal results are cleaner n the symmetrc case. Second, the symmetrc model better hghlghts the dstncton between model symmetry and outcome symmetry. Thrd, as mentoned n the prevous secton, the symmetrc equlbrum of the corporate culture game admts an outcome whereby both frms choose a number one culture. The tmng of the game, as well as the state transton process, are descrbed n Fgure 2. Each perod starts wth each frm havng a certan number of consumers, and j, attached to t (where + j ). Frms set prces p() and p(j). I constran prces to be a functon of the state (, j); that s, I restrct frms to playng Markov strateges. Snce the total number of consumers s constant, the state space s one-dmensonal and can be summarzed by. After frms set prces, nature chooses a partcular agent, whom I wll call the actve agent. Each agent

5 Cabral: We re Number : Prce Wars for Market Share Leadershp Management Scence, 28, vol. 64, no. 5, pp , 27 INFORMS 27 Fgure 2. Tmng á q q Downloaded from nforms.org by [ ] on 9 June 28, at 8:. For personal use only, all rghts reserved. j á becomes actve wth equal probablty. Then nature generates the actve agent s preferences: values a and b, correspondng to consumer specfc preference for each frm s product. I assume these values are ndependent and dentcally dstrbuted (..d.), drawn from a cumulatve dstrbuton functon (cdf) ( ), and that a b s dstrbuted accordng to cdf ( ). 8 The actve consumer then chooses one of the frms and perod payoffs from sales are pad: the sale prce to the frm that makes a sale and utlty mnus prce to the consumer who makes a purchase. In addton to sales revenues, I assume that frm receves an extra beneft f t s the market leader that s, f > j. To preserve model symmetry, I also assume that f < j, then frm receves an extra negatve beneft and that f j, then both frms receve zero extra beneft. Ths assumpton guarantees that regardless of the state, the frms jont payoff from market leadershp s zero. Market leadershp payoff may be summarzed by (), where () s an ndcator varable defned as follows: ()sgn( 8>< + f > j, j) f j, > : f < j, where j. Recall that ths term does not correspond to real value; rather, t s smply value perceved by frm s managers. 9 There are two sources of randomness n the model. One s that each perod consumer s selected by nature to be an actve consumer. Second, nature generates utlty shocks for the actve agent such that the dfference j s dstrbuted accordng to cdf ( ). Many of the results below requre relatvely mld assumptons regardng. q j q j Assumpton. () The cdf ( ) s contnuously dfferentable, () ( ) ( ), () ( ) >, 8, and (v) ( )/ ( ) s strctly ncreasng. 3.. A Note on Model Assumptons The model outlned above s farly parsmonous. As s often the case, ths begs the queston of whether t captures realty approprately. In partcular, (a) I measure market shares by addng up prevous purchase decsons, and (b) I assume there s one consumer only per perod. Regardng the measurement of market shares, I have n mnd the stuaton where consumers make both durable and nondurable purchases. For example, wreless consumers buy smartphones and commt to long-term plans occasonally (when they are actve agents) and then buy usage on a monthly bass. To the extent that average nondurable purchases are relatvely constant across frms, market shares n terms of nondurable purchases correspond to market shares n terms of my state space that s, /. In other words, a frm that s a market leader n a gven perod s a frm wth hgher value of. If varable profts from sellng nondurables are zero, then the payoff functon I consder s approprate, for all proft s derved from sellng durables. The case when nondurables produce varable proft s consdered n Secton 6. Consder now the assumpton of one sale per perod. As I mentoned earler, ths may be nterpreted as the reduced form of a contnuous tme model where each consumer becomes actve as the result of a Posson process. In that case, assumng the Markov state s gven by amounts to assumng that frms can adjust prces nstantly after each purchase. Ths s obvously a smplfyng assumpton. In the onlne appendx, I argue that the man qualtatve results are unlkely to change f I allow consumers to become actve each perod,

6 Cabral: We re Number : Prce Wars for Market Share Leadershp 28 Management Scence, 28, vol. 64, no. 5, pp , 27 INFORMS Downloaded from nforms.org by [ ] on 9 June 28, at 8:. For personal use only, all rghts reserved. whch corresponds to the case when frms must commt to a prce for a perod of tme. Note that f I allow for a fracton of the total number of consumers to become actve n each perod, then, as!, the model becomes determnstc, and all of the rch dynamcs developed n Secton 4 fall through. The reason s that, unless there are aggregate preference shocks, the law of large numbers kcks n. In ths sense, my assumpton that only one out of consumers s actve each perod may also be nterpreted as a reduced form of a world where a fracton / of all consumers became actve each perod and are subject to a common preference shock. In that case, f were the churn rate per calendar tme perod, we would have /, where s the number of model perods per perod of calendar tme Equlbrum I wll focus on symmetrc Markov equlbra, whch are characterzed by a prcng strategy p(), where s the number of lvng consumers who have purchased from frm. In the remander of the secton, I frst derve the determnants of consumer demand. Next, I derve the frm value functons and the resultng prcng strategy. Puttng together demand and prcng, I derve a master equaton that determnes the evoluton of market shares. The secton concludes wth two prelmnary results: one regardng equlbrum exstence and unqueness and another regardng the statonary dstrbuton of market shares Consumer Demand At state, an actve consumer chooses frm f and only f p() > j p(j), (2) or smply, where j > x(), x()p() p(j). (3) Frm s demand functon s smply gven by Notce that q() (x()). (5) 3.4. Prcng Suppose that frms costs are zero. Frm s value functon s then gven by v() q() p() + (q()( () + v()) + ( q())( ( ) + v( ))) + j (q()( ( + ) + v( + )) + ( q())( () + v())), (6) where,..., and j. The varous terms n (6) correspond to varous possbltes regardng consumer death and brth. Suppose, for example, that the actve consumer s a frm j consumer, somethng that happens wth probablty j/. Suppose, moreover, that ths consumer chooses frm, whch happens wth probablty q(). Then frm receves sales revenue q()p() (frst row), current extra payoff ( + ), and contnuaton payoff v( + ). Note that, wth some abuse of notaton, (6) corresponds both to frm s Bellman equaton and to the recursve system that determnes the value functon. As a Bellman equaton, the v( )on the rght-hand sde should be treated as v c () that s, contnuaton value. Ths s mportant when dervng frst-order condtons, to the extent that the terms on the rght-hand sde should be treated as constant n the frm s optmzaton problem. Defne w() ( ( + ) ()) + (v( + ) v()). (7) Put nto words, ths denotes frm s value from poachng a customer from frm j. Ths s dvded nto two dfferent components: the mmedate value n terms of market leadershp, ( +) f frm makes the sale and mnus () f t does not; and the dscounted future value, v( + ) f frm makes the sale and mnus v() f t does not. Usng (7), the frst-order condton for maxmzng the rght-hand sde of (6) wth respect to p() s gven by w( or smply, p() (x()) (x()) ) + w( ) j w(), (8) where I substtute (4) for q() and (5) If, then there are no number one effects: v() v( +), w( ) w(), and we have a standard statc product dfferentaton model. Specfcally, only the frst term on the rght-hand sde of (8) matters, where x() p() p(j). By contrast, f >, then w(),, and frms lower ther prce to the extent of what they have to gan from makng the next sale, whch s gven by (/)w( ) + ( j/)w(). From frm s perspectve, wth probablty /, the next sale s a battle for keepng one of ts customers; that s, t s the dfference between the contnuaton value of state and the contnuaton value of state. Wth probablty j/, the next sale s a battle for attractng a rval customer; that s, t s the dfference between the contnuaton value of state + and the contnuaton value of state.

7 Cabral: We re Number : Prce Wars for Market Share Leadershp Management Scence, 28, vol. 64, no. 5, pp , 27 INFORMS 29 Downloaded from nforms.org by [ ] on 9 June 28, at 8:. For personal use only, all rghts reserved. Pluggng ths back nto the value functon (6) yelds v() ( (x()))2 (x()) + ( ( ) + v( )) + j ( () + v()). (9) Under a statc olgopoly, we would only have the frst term on the rght-hand sde. The addtonal terms suggest that a frm s value corresponds to the value n case t loses the challenge for the next consumer: ether losng the battle for keepng one of ts consumers (a battle that takes place wth probablty /) or losng the battle for capturng one of the rval s consumers (a battle that takes place wth probablty j/). Ths s the ntuton underlyng the Bertrand paradox (also known as the Bertrand trap; see Cabral and Vllas-Boas 25): to the extent that frms lower ther prce by the value of wnnng a sale, ther expected value s the value correspondng to losng the sale (zero n the standard symmetrc Bertrand model and the frst term on the rght-hand sde f there s product dfferentaton). In other words, prce competton mples rent dsspaton; n the present case, the w() rent. System (9) can be solved sequentally: v() j ( (x())) 2 (x()) + ( ( ) + v( )) + j (). () Fnally, I am also nterested n dstngushng frm value (the functon that frm decson makers maxmze) from shareholder value (the frm s fnancal gan). The latter s gven by s() q() p() + (q() s() + ( q()) s( )) + j (q() s( + ) + ( q()) s()). () In other words, () corresponds to (6) wth the dfference that t excludes number one effects; that s, Market Shares Recallng that x() p() p( j) and subtractng (8) from the correspondng p(j) equaton, we get p() p(j) (x()) (x()) (x(j)) (x(j)) w( ) j w() + j w(j ) + w(j), (2) or smply, x() 2 (x()) (x()) (w( ) w(j)) j (w() w(j )), (3) where I use the fact that (x(j)) (x()). Equaton (3) s the master equaton determnng the evoluton of market shares (n expected value). Recall that q() (x()), so a hgher x() mples a lower probablty that frm makes the next sale. If, so that w() for all, then we have a standard statc product dfferentaton model: all terms on the rght-hand sde except the frst one are zero, and as a result x(), too: each frm makes a sale wth the same probablty. More generally, what factors nfluence the value of x()? Essentally, the dfference across frms s the value of wnnng the sale: as shown before, frms lower ther prces to the extent of ther ncremental value of wnnng a sale; the frm that has the most to wn wll be the most aggressve, thus ncreasng the lkelhood of a sale. The value of wnnng a sale may be decomposed nto (a) the mmedate beneft from an ncrement n market share, ( + ) () or () ( ) as the case may be; and (b) the dscounted future value from market share, v( + ) v() or v() v( ), as the case may be Equlbrum Equatons () and (3) defne a Markov equlbrum, where I note that w() s gven by (7). Gven the values of v() and x(), prces p() and sales probabltes q() are gven by (8) and (4), respectvely. Many of the results n the next sectons pertan to the lmt case when!. These results are based on the followng exstence and unqueness result. Lemma. There exsts a unque equlbrum n the neghborhood of. Moreover, equlbrum values are contnuous n Statonary Dstrbuton of Market Shares Gven the assumpton that ( ) has full support (part () of Assumpton ), q() 2(, ), 8 ; that s, there are no corner solutons n the prcng stage. It follows that the Markov process of market shares s ergodc, and I can compute the statonary dstrbuton over states. Ths s gven by the (transposed) vector m that solves mm m. Snce the process n queston s a brth-and-death process, whereby the state only moves to adjacent states, I can drectly compute the statonary dstrbuton of market shares.

8 Cabral: We re Number : Prce Wars for Market Share Leadershp 22 Management Scence, 28, vol. 64, no. 5, pp , 27 INFORMS Downloaded from nforms.org by [ ] on 9 June 28, at 8:. For personal use only, all rghts reserved. Lemma 2. The statonary dstrbuton m() s recursvely determned by where m() m() m() + X Y q( ) q() +, k Y k q( ) q() +. Lemmas and 2 allow for a partal analytcal characterzaton of equlbrum. I wll develop two types of analytcal results: one corresponds to takng lmts as! ; the second, to takng dervatves wth respect to at (that s, lnearzng the model). I complement these analytcal results wth numercal smulatons for hgher values of. These numercal smulatons confrm the analytcal results for small but also uncover addtonal features not present n the small case. 4. A Theory of Prce Wars I cannot fnd a general analytcal closed-form soluton for the model s equlbrum. However, I can characterze the equlbrum when, and, by Lemma, n the neghborhood of, the equlbrum values take on values close to the lmt case. In the followng results, I assume for smplcty that s even, and I denote the symmetrc state by /2. Proposton 3. There exsts a unque equlbrum n the neghborhood of. Moreover, lm! p() lm q()! 2 8>< lm! v() 2 () 8>< 2 () > : 2 () 4 () 4 () f, + f ±, otherwse, f apple, f, 4 () + f +, > : 4 () + f + 2,! lm m()!! ( )!2. The lmtng statonary dstrbuton s maxmal at. Put nto words, when frm market shares are close to each other, frms engage n a prce war for market leadershp, whereby both frms decrease prce by up to from the statc Hotellng prce level /(2 ()). Ths s smlar to the dea underlyng the Bertrand paradox: the potental gan from beng a market leader s competed away through prcng. Specfcally, I defne the prce war regon of the state space as the set {,, + }. Proposton 3 then states that, n the lmt as!, prces are set lower than /(2 ()) (prce war) when 2{,, + } and are equal to /(2 ()) (peace) when < {,, + }. Note that, n the lmt as!, p() p(j). As a result, the probablty of makng a sale s unform at. Ths 2 mples that market share dynamcs follow a straghtforward reverson to the mean process: smaller frms ncrease ther market share on average, whereas larger frms decrease ther market share on average. Ths s partcularly bad for profts because t mples a constant tendency to engage n a prce war. The dashed lnes n Fgure 3 llustrate ths stuaton. (In ths and n the remanng fgures n the paper, I assume, so that s both the state and frm s market share. 2 I also assume that s dstrbuted accordng to a standardzed normal. 3 ) The top left panel depcts the equlbrum prce functon, whereas the top rght panel shows the statonary dstrbuton of market shares. (Note that, snce the equlbrum s symmetrc, p() and m() are only a functon of the state, not of the frm s dentty.) The bottom panels show the value functons for frm managers (left) and shareholders (rght). Begnnng wth the prce mappng, we see that prces are set at a constant level (the statc equlbrum level) when the state s outsde the prce war regon. Insde the prce war regon, frm prces drop by up to, whch s the change n frm value from movng up one unt n the state space. Snce the prce mappng s symmetrc about, each frm s sale probablty s flat at. It follows that the statonary dstrbuton of market shares 2 s a smple multnomal centered around (that s, around 5% market share). The bottom rght panel shows that shareholder value drops sharply when s near that s, n the prce war regon. Ths follows form the fact that prces are lower near the symmetrc state and also the fact that shareholders do not receve any beneft from beng number one. In other words, snce shareholders do not care for market leadershp per se, number one effects are only bad news: they lead to prce wars, whch n turn destroy shareholder value. Wth respect to frm value, the bottom left panel ndcates that, n the lmt as!, v() s ncreasng n. In partcular, f >, then frm receves utlty n addton to expected revenues. Ths beneft from leadershp s balanced out by the negatve utlty suffered by the laggard.

9 Cabral: We re Number : Prce Wars for Market Share Leadershp Management Scence, 28, vol. 64, no. 5, pp , 27 INFORMS 22 Fgure 3. Equlbrum When and (Dashed Lnes) and 3 (Sold Lnes) 4.5. Downloaded from nforms.org by [ ] on 9 June 28, at 8:. For personal use only, all rghts reserved. p() ( Ñ)v() Fnally, although not obvous from Fgure 3, ndustry jont value, v() + v(j), at states near s actually lower when > than when. Ths follows from Proposton 3, as the next result attests. Corollary. In the lmt as!, jont ndustry value v() + v(j) s strctly decreasng n f 2{,, + } and constant otherwse. Ths s an mportant pont, one that warrants further elaboraton. The dea s akn to the Bertrand paradox. In a frst-prce aucton where the payoff from wnnng s gven by + and the payoff from losng s gven by, the greater the value of, the lower the equlbrum value by both bdders: the wnner gets from wnnng mnus 2, the equlbrum bd, whereas the loser gets. In the present context, an ncrease n ncreases the payoff from wnnng a sale and decreases the payoff of losng t. Although the total payoff from market leadershp s constant (specfcally, () + ( j) ), the equlbrum value receved by each frm s decreasng n : n equlbrum, each frm fares as well as when t loses the sale. 4 An addtonal mplcaton of Proposton 3, smlar to Corollary, s that ndustry jont value s hgher at asymmetrc states than at symmetrc states, so that, at symmetrc or near-symmetrc states, the leader has more to gan from ncreasng ts lead than the laggard has to lose from fallng farther behnd. 5 5 m() ( Ñ)s() Corollary 2. At, v() + v(j) s strctly ncreasng n j f j apple2. Moreover, v( + ) v( ) > v( ) v( ), v( + 2) v( + ) > v( ) v( 2). Put nto words, the second part of Corollary 2 states that, at state +, what the leader has to lose by movng down one step s more than what the laggard has to gan by movng up one step, and what the leader has to gan by movng up one step s more than what the laggard has to lose by movng down one step. Ths s the dynamc equvalent of the jont-proft effect of Glbert and Newbery (982). In ther paper, the effect results from the convexty of the proft functon; n my paper, t results from the convexty of the value functon. 5 Notce that the two parts of Corollary 2 are equvalent: both stem from the value functon beng convex. In fact, v( + ) v( ) > v( ) v( ) s equvalent to v( +)+ v( ) > v( )+ v( ), and v( +2) v( +) > v( ) v( 2) s equvalent to v( + 2) + v( 2) > v( ) + v( + ). In other words, f the value functon s convex, then ts slope s greater for the leader than for the laggard. Smlarly, by a dscrete analog of Jensen s nequalty, jont proft ncreases when the state becomes more asymmetrc.

10 Cabral: We re Number : Prce Wars for Market Share Leadershp 222 Management Scence, 28, vol. 64, no. 5, pp , 27 INFORMS Fgure 4. Market Leadershp Beneft (Left) and Value Functon (Rght) at Where Is the Symmetrc State for (Lght Lnes) and > (Dark Lnes), +à 2î() + à Downloaded from nforms.org by [ ] on 9 June 28, at 8:. For personal use only, all rghts reserved. à () à * 3 * 2 * * * + * + 2 * + 3 Fgure 4 llustrates Corollares and 2. The left-hand panel depcts the market leadershp mappng. As can be seen, the mappng s symmetrc about (, ), so that the sum () + (j) s equal to zero at every state. The same s not true, however, regardng value functons, as can be seen from the rght-hand panel. For example, at state, each frm s payoff when > s lower than t would be f (Corollary ). Moreover, v() s convex. At, ths corresponds to the fact that v( ) v( ) < v( + ) v( ); at, t corresponds to the addtonal fact that v( + 2) v( + ) > v( ) v( 2) (Corollary 2). Corollares and 2 have mportant mplcatons for system dynamcs n the neghborhood of, as I wll show next. 4.. Postve, Small Values of Proposton 3 consders the lmt when!. From Lemma, I know that the system s behavor s contnuous around ; that s, the lmt! s a good ndcaton of what happens for low values of. Addtonal nformaton can be obtaned by lnearzng the system around and thus determnng the drecton n whch equlbrum values change as moves away from zero. 6 Recall that n the lmt, as!, p() p(j) and q() q(j). My next result shows that, n the near symmetrc states and +, the market leader sets a low prce and sells wth hgher probablty. Moreover, the laggard s strctly worse off by ncreasng ts market share. Proposton 4. There exsts a > such that f < apple, then > mples p( + ) < p( ), q( + ) > q( ), v( ) < v( 2). (Notce that, gven the demand curve (4), the frst two nequaltes are equvalent.) v() Ñ = 2î() 2î() à * 3 * 2 * * * + * + 2 * + 3 As mentoned earler and as shown by (8) frm s frst-order condton ncludes the value of wnnng a sale, ether the value of keepng an exstng customer, w( ), or the value of poachng a rval s customer, w(). When, the value of wnnng a customer s based on the mappng (), as llustrated n the left panel of Fgure 4. Consder, for example, a frm wth customers. If ths frm gans one customer, ts payoff ncreases by, whereas ts rval, by movng from + to, decreases by. Conversely, f the frm at loses one customer, then ts leadershp payoff remans the same, whereas ts rval, by movng from + to + 2, also sees ts payoff reman constant. In sum, for, what the leader has to gan (respectvely, lose) from makng a sale s the same as the laggard has to lose (respectvely, gan). As a result, both frms apply the same subsdy to ther prce level, and q() /2 for all, as stated n Proposton 3. Consder now the case when s postve but nfntesmal. Gven that the actve consumer s a j consumer, frm s value from wnnng a sale s gven by w() ( +) ()+ v( +) v().at, as we have seen, the values of w() for leader and laggard balance out exactly. As we ncrease nfntesmally, the value of w() ncreases at the rate v( + ) v(), where the value functons are evaluated at. Proposton 4 explots the fact that, whle the values of () add up to a constant, so that leader and laggard have the same to wn or lose, the same s not true for v( + ) v(), as Corollary 2 states. Specfcally, consder the near-symmetrc state (, +). As Corollary 2 shows, the laggng frm has less to gan from movng up the value functon than the leader has to lose from losng to the laggard. Moreover, the laggard has less to lose from fallng farther behnd than the leader has to gan from movng further ahead. In other words, the value functon s convex. Gven the ntuton underlyng the frst-order condtons (8), ths mples that the leader prces more aggressvely, whch results n t makng a sale wth a hgher probablty than the laggard.

11 Cabral: We re Number : Prce Wars for Market Share Leadershp Management Scence, 28, vol. 64, no. 5, pp , 27 INFORMS 223 Downloaded from nforms.org by [ ] on 9 June 28, at 8:. For personal use only, all rghts reserved Hgher Values of For hgh values of, I cannot fnd a closed-form analytcal soluton or lnear expanson approxmaton. However, I can solve the model numercally. The dark lnes n Fgure 3 show the model s soluton for The soluton looks qualtatvely smlar to n varous respects namely, n the property that prces drop when frms market shares are close to each other. However, upon closer nspecton, mportant dfferences become apparent as well. Frst, as suggested by Proposton 4, when >, the prcng functon s no longer symmetrc around. In partcular, just outsde the prce war regon, the large frm s prce s lower, whereas the smaller frm s prce s hgher. Ths mples that the probablty of a sale by a leader ncreases when the leader s market share drops to close to. As the top rght panel n Fgure 3 shows, ths (perhaps) also mples that the statonary dstrbuton of market shares s bmodal. 8 That s, most of the tme, the system les at an asymmetrc state, where one frm s larger and the other frm smaller Prce and Market Share Dynamcs Proposton 3 shows that frms engage n prce wars when the state space s close to the symmetrc state, whereas Proposton 4 suggests that market shares tend to reman stable around asymmetrc outcomes. I now examne the mplcatons of these propertes. Fgure 5 llustrates the dynamcs of prce and market shares by showng the results of a model smulaton when 3 and (the parameter values correspondng to Fgure 3). 9 In the top panel, the sold lne 4 represents frm s prce and the dashed lne frm j s Fgure 5. Prce and Market Share Dynamcs ( 3 4, ) p(), p( j) /á t t p( j) p() 2 2 prce. A horzontal lne marks the equlbrum prce level when. 2 The bottom panel depcts frm a s market share. Accordng to my model, a prce war s a perod of sgnfcantly lower prces that takes place when the frms market shares are close enough (f, when j apple2). Fgure 5 shows some of these prce wars. The deepest prce war takes place rght from the start, whch s not surprsng snce I started the smulaton at. Mostly out of sheer luck, frm a makes most of the sales durng ths perod; that s, frm a wns the prce war. As a result, frm a s market share ncreases to over 5%, as can be seen from the bottom panel. Now that there s a clear market share leader, prces ncrease to a hgh level, at or slghtly above the statc equlbrum prce level. Whenever frm a s market shares decrease to close to 5%, a prce war begns agan. Notce that frm a, the market share leader, s also the prce leader gong nto a prce war. Ths s consstent wth the dea that prce wars are a defense aganst threats to market share leadershp. Fnally, just as prce wars are trggered by a drop n the leader s market share, so the end of a prce war s determned by an ncrease n the leader s market share. In other words, frm (the market leader n the partcular smulaton on Fgure 3) has more to lose from lowerng ts market share than frm j has to gan from ncreasng ts market share. Ths results n more aggressve prcng by the market share leader. Most of the tme (and n the frst 2 perods of the smulaton I consdered), the market share leader remans so; that s, my model mples persstence n leadershp. However, wth probablty, market leadershp changes n fnte tme. My paper s by no means the frst paper to feature symmetrc equlbra wth asymmetrc outcomes and prce wars near symmetry states. Besanko et al. (2), for example, show that learnng curves lead to trenchy prce equlbra whereby prces drop when compettors market shares are close to each other. 2 My model dffers from the prevous lterature n that t does not feature ncreasng returns to scale. In fact, by constructon, () + ( j) s equal to zero. Specfcally, f prces were set at a constant level, then my model would mply that ndustry jont value v() + v(j) s constant across states, whereas Besanko et al. (2) or Cabral (2), for example, would mply that v() + v( j) s ncreasng n j. Moreover, whle the statonary dstrbuton of market shares s multmodal, t stll places sgnfcant mass on symmetrc or near-symmetrc states. (If, the statonary dstrbuton of market shares s a bnomal centered around 5%.) As a result, prce wars are relatvely frequent, whereas n models wth ncreasng returns to scale they are rare: once one of the frms

12 Cabral: We re Number : Prce Wars for Market Share Leadershp 224 Management Scence, 28, vol. 64, no. 5, pp , 27 INFORMS Downloaded from nforms.org by [ ] on 9 June 28, at 8:. For personal use only, all rghts reserved. becomes domnant, t takes a long tme for tppng to take place. Ths s an mportant dstncton, one that warrants further dscusson. In dynamc market share models, there s a natural reverson-to-the-mean force: consumer death (a frm wth % of the market can only decrease ts market share). Aganst ths force pushng toward market share balance, there may be varous forces pushng the system away from symmetry. Increasng returns (learnng curves or network effects) represent one such force. In my model, the force that pushes away from symmetry s prce wars. However, to the extent that prce wars only kck n at states close to symmetry, the effect of prce wars s only felt at states close to symmetry. As a result, we have a statonary dstrbuton where much of the weght s on states close to the threshold of the prce war regon. Ths results n frequent movements nsde the prce war regon. In other words, unlke models wth ncreasng returns, prce wars are observed cyclcally along the equlbrum path. In Secton, I mentoned the server market as an example where market share leadershp s consdered mportant. For several years, IBM and HP fought for the number one poston (as measured by market share of dollar sales). In 24, IBM sold a consderable porton of ts server assets to Lenovo. Ths essentally ended a perod of fght for market share leadershp. My model would predct an ncrease n HP s prce to ensue, bascally, the result of a shft to a peace phase. The data, depcted n Fgure 6, are broadly consstent wth the model s predctons though admttedly, t s also consstent wth other models. In both cases, we observe a prce that s relatvely constant untl 23 and gradually ncreases as the market share gap between HP and IBM ncreases that s, as HP becomes a clear Fgure 6. Server Market: Prces and Market Shares market share leader. (Consderng that there may be many spurous factors affectng prce, I also plot the prce ndex: HP s prce dvded by ndustry average prce, measured n the rght scale of the upper panel. The pattern s smlar.) 5. Corporate Culture and Shareholder Value The bottom rght panel n Fgure 3 shows an mportant dfference between the equlbrum wth and the equlbrum wth 3. In the former case, number one 4 effects are unambguously detrmental to shareholder value. Ths s farly ntutve: number one effects lead frms (symmetrcally) to lower prces when n state 2 {,, + }. Lower prces lower shareholder value; moreover, number one effects accrue no shareholder utlty. All n all, wantng to be number one s bad for shareholders. However, f s suffcently hgh (e.g., 3 ), then 4 there are states when shareholder value s greater wth > than wth. To understand ths, t helps to notce that, as shown n Proposton 4, v() s decreasng for values of lower than, but close to,. In other words, a laggard becomes worse off as ts market share approaches the leader s. The reason s that the ncrease n market share nduces very aggressve prcng behavor by the leader, whch n turn reduces the laggard s value: the laggard receves no beneft from market leadershp but pays the cost of a leader eager to defend ts beneft from market leadershp. As seen earler, the frst-order condton for optmal prcng ncludes a subsdy n the amount of the expected contnuaton gan from makng a sale, ether the value of keepng an own contested consumer, w( ), or the value of poachng a consumer from the rval frm, w(). If the value functon s decreasng (and the payoff from market leadershp does not change), then a declnng v() mples a negatve w(), whch n turn mples that the prce subsdy becomes a tax. In other words, the threat of enterng a prce war wth the leader softens the laggard. Ths effect may be so strong as to ncrease the leader s shareholder value (n the states where the laggard softens up). In other words, even though shareholders do not care about market leadershp per se, shareholder value may ncrease when managers care for market leadershp. (I should stress the word may as ths s a possblty result, not a general analytcal result.) Although the Markov equlbrum I consder dffers greatly from a repeated game (where, by defnton, there s no state space such as market share), there s an nterestng smlarty between the above effect and the so-called topsy-turvy prncple n collusve repeated game equlbra (Shapro 989). Consder a repeated game where each perod consumer buys