UNINTENDED CONSEQUENCES OF PROMOTIONS: SHOULD MANAGERS WORRY ABOUT CONSUMER STOCKPILING? Manish Gangwar Nanda Kumar And Ram C.

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1 UNINTENDED CONSEQUENCES OF PROMOTIONS: SHOULD MANAGERS WORRY ABOUT CONSUMER STOCKPILING? Manh Gangwar Nanda Kumar And Ram C. Rao 1 Aprl 2012 Latet Veron: Aprl 2015 We have beneted rom eedback provded by everal colleague, revewer and edtor. We are grateul epecally to Maxm Sntyn and Dmtr Kukov or ther comment. Any remanng hortcomng are our reponblty. 1 Manh Gangwar Atant Proeor at Indan School o Bune. Nanda Kumar Aocate Proeor and Ram C. Rao Founder Proeor both at Naveen Jndal School o Management at UTD.

2 UNINTENDED CONSEQUENCES OF PROMOTIONS: SHOULD MANAGERS WORRY ABOUT CONSUMER STOCKPILING? ABSTRACT Increae n ale due to promoton could come at the expene o compettor; uch ale come rom conumer who have relatvely weak brand preerence. Increaed ale rom conumer wth trong brand preerence are lkely to be at the expene o the promoted brand. In other word, brand loyal conumer can take advantage o promoton to tockple or uture conumpton. Thu, loyal conumer who would be otherwe wllng to buy at hgh prce can trategcally tockple at low prce. What t mpact on rm prot? How hould rm adapt to conumer tockplng? To anwer thee queton we model a duopoly competng or loyal and wtchng conumer. In contrat to extant ndng that tockplng by wtchng conumer doe not aect rm protablty, we nd that tockplng by loyal conumer ndeed reduce rm long-run prot. We alo nd that even when tockplng may nduce hgher conumpton, t reduce but doe not elmnate loe. Furthermore, we etablh an upper bound on the lo due to loyal conumer tockplng. Surprngly, we nd that t amount to a relatvely mall percentage o prot. We alo obtan a novel ndng on mxed tratege that rm equlbrum prcng dtrbuton can have ma pont n the nteror o the upport. Our reult alo oer everal counter-ntutve nght o relevance to manager. Keyword: Game Theory, Conumer Stockplng, Promotonal Stratege, Loyal Conumer, Interor Ma Pont, Prcng Dtrbuton, Prcng 1 P age

3 1. Introducton One goal o promoton by packaged good manuacturer to appeal to prce entve conumer who mght buy a competng brand abent promoton. But promoton may have an unntended conequence that could have negatve allout. Conumer may tockple and purchae or uture conumpton when brand are promoted. I loyal conumer who hun competng brand tockple ther avorte brand opportuntcally or later conumpton, then uture ale, potentally at hgher prce, are cannbalzed. Thu, when promotng to attract prce entve (wtchng) conumer rm hould nternalze th poble downde. Gangwar, Kumar and Rao (2014) explored the eect o tockplng by prce entve conumer on rm prot and ound that t doe not reduce rm prot. Would that reult hold loyal conumer tockple? We could ak: t mportant to conder tockplng by loyal conumer? Recent emprcal evdence ugget that brand loyal conumer are more lkely to engage n tockplng n repone to promoton than wtchng conumer. The ntuton behnd th ndng that loyal conumer derve relatvely more benet rom tockplng durng a promoton becaue they do not ntend to aval o uture promoton by competng brand. In contrat, wtcher are not commtted to any partcular brand and hence can take advantage o promoton oered by any brand makng tockplng le attractve. For example, Coke lover are lkely to tockple Coke oer a lower prce but cola lover can take advantage o promoton not only by Coke but alo thoe oered by Pep and other competng brand o cola. Chan, Naramhan and Zhang (2008) ound that brand loyal conumer repond to promoton by tockplng whle wtcher merely change brand and do not tockple. Sun, Neln and Srnvaan (2003) have alo ound mlar reult. Thee recent tude, emphazng modelng nnovaton, have ndng that are content wth the earler evdence n Krhnamurth and Raj (1991) that loyal conumer, though le prce entve n the brand choce decon, are more entve to prce n the quantty decon. Neln, Henderon and Quelch (1985) ound loyal purchaer n the coee market have learned to change ther purchae pattern n order to take advantage o promoton or ther preerred brand. In ummary, emprcal reearcher have ound loyal conumer to be more lkely than wtcher to tockple due to promoton. I t conceded that a model o tockplng by loyal conumer worth explorng we can then ak what the eect o uch behavor on rm prcng decon and prot? On the one hand, tockplng by loyal conumer n a gven perod leave rm wth a maller tock o loyal conumer n the ollowng perod. Th n turn make rm compete or wtcher wthout needlely ubdzng too 2 Page

4 many loyal conumer. On the other hand, th reduced leakage reult only ater ncurrng the expene o nducng loyal conumer to tockple at reduced prce. Thereore, t not apparent whether tockplng by loyal conumer on promoton reult n lower or hgher prot compared to a tuaton o no tockplng. It could alo turn out that n a compettve ettng, rm promotonal tratege balance out thee two eect o that there no net eect on rm prot jut a n Gangwar, Kumar and Rao (2014). So, the rt queton we addre n th tudy : doe tockplng by loyal conumer lower, rae or leave unchanged rm equlbrum prot? Second, tockplng by loyal conumer doe aect rm protablty what the nature and extent o t? Thrd, how are rm equlbrum tratege aected by tockplng? Fourth, how hould manager addre practcal ue arng rom tockplng by loyal conumer? Why t that certan promotonal prce appear more requently n equlbrum? Fnally, how would our reult change the very act o tockplng lead to ncreae n conumpton? Thu, our work contrbute to extant undertandng o the eect o compettve promoton on rm tratege and prot, and n dong o alo oer manageral nght. 1.1 Lterature Revew Prce promoton a well-tuded topc n economc and marketng (Shlony 1977, Varan 1980, Naramhan 1988, Raju, Srnvaan and Lal 1990, Rao 1991). Th body o theoretcal work aume that ome conumer repond to promoton by wtchng brand, and the rm that oer the lowet prce ell to all wtchng conumer. The preence o uch wtchng conumer ntroduce dcontnuty n demand and that n turn lead to equlbrum prce n mxed tratege wth realzaton o low prce nterpreted a promoton. Early emprcal reearch (Chang 1991, Chntagunta 1993, Buckln, Gupta and Sddarth 1998, and Bell, Chang and Padmanabhan 1999) ha ound ubtantal brand wtchng among conumer. In partcular, reearcher have ound that a large racton (about 75%) o the demand expanon due to promoton can be attrbuted to brand wtchng and the ret to purchae acceleraton and ncreaed purchae quantty. Recent work (Pauwel, Hanen and Sddarth 2002, Heerde, Gupta and Wttnk 2003, and Steenburgh 2007) that ocue on ale volume nd that n many ntance only a thrd o the ncremental unt ale n the promotonal perod can be attrbuted to brand wtchng. Th ugget that there ubtantal purchae acceleraton and ncreae n purchae quantty, whch cannot be gnored n dervng equlbrum prcng tratege. Thereore, n th paper we develop a model that explctly account or both brand wtchng and conumer tockplng behavor n characterzng rm equlbrum prcng tratege. Our work hould be een n the context o pror exploraton o conumer tockplng. Salop and Stgltz (1982) have hown that conumer orward lookng behavor by tel may lead to promoton n 3 Page

5 the orm o mxed trategy even when conumer preerence are homogeneou. Bell, Iyer and Padmanabhan (2002) nnovate on Salop and Stgltz model o homogeneou conumer by makng conumpton tate dependent, and o rm may nduce tockplng to ncreae conumpton to get hgher ale. Dudne, Hendel and Lzzer (2006) provde an analy o the role o commtment n a monopoly market wth torable good. They how that n contrat to the lterature on Coae conjecture, abence o commtment lead to hgher prce or torable good. Hendel, Lzzer and Roketky (2014) provde an alternatve explanaton or cyclcal pattern n ale and prce and how that torablty mpoe a contrant on a monopolt' ablty to extract urplu under non-lnear prcng that lead to cyclcal pattern. In our paper the ocu on competton or prce entve brand wtchng conumer that lead to opportuntc buyng by brand loyal conumer thu orcng the rm to orego prot. The queton then what a rm equlbrum prcng trategy hould be to balance the orce o competng or wtchng conumer and takng nto account opportuntc buyng by loyal conumer. In th way we can oer manageral nght and gudelne. Our work cloer to Hong, McAee and Nayyar (2002) who ue Varan' (1980) model to tudy conumer tockplng by aumng that all wtchng (prce entve) conumer tockple at an exogenouly peced threhold. Our work der rom th n two mportant way. Frt, we explore tockplng by loyal conumer. Second, we treat conumer tockplng rule a endogenou. Th mportant nce conumer tockplng rule and rm prcng tratege are mutually dependent. For example, rm equlbrum prce do not contan requent deep promoton, conumer may nd t protable to tockple at relatvely hgher prce. On the other hand, deep promoton were requent, ncdence o tockplng at relatvely hgher prce would be le. Frm equlbrum prcng tratege wll depend on conumer tockplng rule, a conceptualzed n our work. Although there extant work that make conumer tockplng rule endogenou our work ocue on derent ue and o add to extng reearch. For example, Guo and Vlla-Boa (2007) conder a derentated market n whch conumer preerence or product are dtrbuted over a Hotellng lne. In ther model conumer wth trong brand preerence are more lkely to tockple. They etablh an mportant reult that when conumer have contant preerence over tme they do not tockple n equlbrum becaue o the trategc prcng behavor o rm. However, conumer preerence can change over tme rm have an ncentve to nduce tockplng by conumer who preer ther product n the rt perod to oet potental wtchng by them due to changed preerence n perod 2. The equlbrum prcng tratege n Guo and Vla-Boa are n pure tratege gven that they have employed the Hotellng model tructure. Our model, on the other hand, lead to promoton n the orm o mxed tratege becaue we ocu on 4 P age

6 conumer heterogenety n prce entvty. Some conumer are not prce entve but are brand loyal whle other are prce entve and wtch brand. Gangwar, Kumar and Rao (2014) analyze a model where ome prce entve conumer tockple by endogenouly determnng the tockplng rule, and how why rm equlbrum prot do not uer a a reult o conumer tockplng behavor relatve to the cae o no tockplng a long a rm recognze t n determnng ther equlbrum promotonal tratege. Intutvely th happen becaue all potental prot rom wtchng conume are competed away n equlbrum. Hence, tockplng by wtchng conumer doe not reduce rm prot n equlbrum. Alawad et. al. (2007) ound through mulaton that tockplng benet would be ubtantal tockplng alo ncreae conumpton that oet the negatve apect o tockplng by loyal conumer. In an extenon, we develop a theoretcal model o promoton that allow or conumer tockplng and tockplng nduced lexble conumpton to examne thee ue. In ummary, our paper add to pror work n everal way. Frt, we ncorporate tockplng behavor by brand loyal conumer n dervng rm equlbrum prcng tratege whle keepng the conumer tockplng rule endogenou; th poe techncal challenge, whch we addre ucceully. In partcular, not only do we nd that equlbrum prcng tratege can have a hole n the nteror o the upport a een n pat work, but we alo obtan a novel reult o the extence o a ma pont n the nteror o the upport. Second, we nd that unlke n pat work, long run tatonary equlbrum prot n our model are reduced becaue o conumer tockplng. What mportant that we obtan an upper bound on the loe due to tockplng by loyal conumer. Fnally, we how that even ncreaed conumpton nduced by tockplng doe not mprove prot or the rm compared to no tockplng cae. The ret o the paper organzed a ollow. In Secton 2, we decrbe the conumer model and rm technology and then dene the equlbrum we eek. In Secton 3, we tate conumer and rm decon problem and characterze conumer tockplng behavor. In ecton 4, we analyze rm equlbrum prcng tratege. We then derve cloed orm expreon or rm prot condtonal on conumer nventory and obtan both lower and upper bound on the prot. We compare th wth a benchmark cae where conumer do not engage n tockplng. We then extend our analy to allow lexble conumpton by tockplng conumer. In Secton 5, we dcu the relaton o our work to manageral concern and tratege. Fnally, we conclude n Secton 6. 5 P age

7 2. Model We tudy a market wth two rm ervng conumer over an nnte horzon. We rt decrbe our model o conumer ollowed by our model o rm. 2.1 Conumer In our model, there a contnuum o conumer. They each conume one unt o the product n every perod. A n Naramhan (1988) there are two type o conumer n our model, brand loyal and wtcher: a proporton α o conumer n the market loyal to each o the two brand. The remanng (1-2α) proporton o conumer cont o wtchng conumer who examne the prce o both brand and purchae the lower prced brand n each perod. All conumer are aumed to have a reervaton prce o r. In other word, conumer maxmum wllngne to pay r per unt o the product. Motvated by emprcal ndng that conumer who are brand-loyal are more lkely (than wtcher) to tockple n repone to prce promoton we nnovate wth repect to Naramhan (1988) by lettng a racton λ o loyal conumer to tockple or uture conumpton. Sad derently, even though loyal conumer n our model need one unt o the product per perod, they may buy more than one unt, or none, n ome perod. Thu, n our model a racton λ o the loyal conumer o each brand may tockple or the next perod, the prce o ther preerred brand ucently low. We alo aume that thee conumer can tockple or at mot one perod. 2 So nventory held by conumer can be 0 or 1. Fnally, utlty maxmzng conumer dcount the uture ung a dcount actor δ c. 2.2 Frm Frm n our model maxmze expected prot dcounted over an nnte horzon. Frm are aumed to dcount the uture ung a dcount actor δ. They chooe prce multaneouly n each perod. Wthout lo o generalty, we normalze margnal cot o both rm to zero, o per perod prot are mply the product o demand and prce. The wtchng conumer and the non-tockplng loyal conumer demand one unt each perod. However, the tockplng loyal conumer demand can vary. In partcular, tockplng loyal conumer have nventory, they demand zero unt at hgh prce and one unt at low prce and they do not have nventory, they demand one unt at hgh prce and two unt at low prce. Thereore, rm prcng tratege would depend on the tate o tockplng conumer nventory. 2 Nraj, Padmanabhan and Seetharaman (2008) and Alawad et. al. (2007) ound that mot o the purchae are concentrated at zero, one or two unt. Moreover, allowng tockplng or only one perod keep our expoton and analy mple. 6 P age

8 We hould note that all pay-o relevant pat normaton or rm n our model captured by the nventory held by tockplng conumer o each brand. Hence, we dene a tate varable { I, I3 } where I, or =1,2, nventory o tockplng conumer o brand. Snce conumer n our model can tockple or at mot one perod, I =0 or 1. Th lead to our poble tate: 00,01,10,11. 3 The tate =00 correpond to the cae n whch tockplng loyal conumer o nether rm have nventory. The tate =01 (=10) correpond to the cae where tockplng loyal conumer o the ocal (competng) rm do not have nventory but that o the competng (ocal) rm have nventory o 1 unt. Fnally, =11 correpond to the cae where tockplng loyal conumer o both rm have nventory. 2.3 Equlbrum We nvoke Markov Perect Equlbrum (MPE) to characterze the rm optmal prcng tratege that maxmze expected prot o the rm gven conumer tockplng decon. We ocu on the expected prot n the tatonary tate. In other word, we derve rm equlbrum prcng tratege condtonal on tate recognzng that contnuaton prot o rm depend on the equlbrum tratege n teady tate. Our equlbrum ub-game perect, rulng out non-markovan devaton by rm (Makn and Trole 1998 and 2001). Makn and Trole oered everal reaon or nvokng MPE a a derable equlbrum concept. Moreover, Markov tratege oten have more predctve power compared to tratege that rely on punhment becaue punhment tratege can have multple equlbra. MPE ha alo been ued n analyzng marketng tratege, or example, Vlla-Boa (1993), Anderon and Kumar (2007), Gangwar, Kumar and Rao (2014). Fnally, n our model the payorelevant tate varable low dmenonal and o the model reman tractable. 3. Conumer Rule and Frm Decon There are our agent n our model: rm, wtchng conumer, non-tockplng loyal conumer and tockplng loyal conumer. Frm MPE tratege are dened a ollow: rm take the current tate a gven and et prce multaneouly : 00,01,10,11. We know rom pror reearch that the preence o wtchng conumer reult n a mxed trategy equlbrum, and rm 3 In th paper, we ue the tate pace notaton ung the ocal rm' perpectve. In other word, we ollow the conventon that rt dgt o the tate pace correpond to current nventory poton o tockplng loyal conumer o the ocal rm and econd dgt correpond to that o the competng rm tockplng loyal conumer. We ue upercrpt to denote the tate and ubcrpt to denote the rm. 7 P age

9 charge prce over an nterval uch that expected prot are equal at all un-domnated prce. Thereore when we reer to rm prcng tratege we envage mxed tratege. Swtcher ue the prce charged by the two rm to decde whether or not to purchae and whch rm to purchae rom, : 2 {0,1},3-1 decde whether or not to purchae a unt ; non-tockplng loyal conumer ue the prce o the preerred rm to : 0,1 2. The tockplng loyal conumer purchae quantty decon n general may ue the prce o both rm and ther own nventory tate to decde whether to buy 0, 1 or 2 unt. It helpul to conder the tockplng decon rather than the quantty decon o the tockplng loyal conumer. The conumer may buy 0 or 1 unt when her nventory 1, and buy 0, 1 or 2 unt when her nventory zero. Thu, the quantty decon depend on her current nventory. The decon to buy or the uture (tockplng), on the other hand, doe not depend on the current nventory poton but on realzed current prce and expectaton o uture prce. We can thereore dene the trategy o tockplng loyal conumer a tockple or not 2 3 : Don't tockple,stockple. 3.1 Conumer Stockplng Rule How do tockplng loyal conumer decde to tockple? There are many way to model the conumer tockplng rule. Recall that rm need to know the nventory tate among other thng to enable them to ormulate ther trategy. One approach to aume that lke the rm n our model, conumer too have all normaton relevant to ther tockplng decon. For them the relevant normaton prce o both brand and the equlbrum tratege o both rm. Th can be thought o a a cae o perect normaton. We explctly develop and analyze the perect normaton model n the techncal appendx. But rt we take a derent route. Why? We thnk t ueul, and mportant, to conder other approache that endow conumer wth le normaton. 4 Th can be thought o a nvokng bounded ratonalty o conumer. For example, loyal conumer n our model derve no drect benet rom montorng the prce o the competng brand, whch they never buy. Thu, an approach that aume they know the prce o only the brand they buy may make ene under ome condton, dependng on the cot o montorng prce. Marketng Scholar have long recognzed that conumer knowledge o prce ar rom perect (Dckon and Sawyer, 1990) but neverthele conumer can recognze prce, meanng they can tell whether an oberved prce the one they have n mnd (Vanhuele and Dreze, 2002). Moreover, conumer eem to have better prce normaton when an tem 4 The prt o the need to thnk o careully modelng the conumer relected n the ollowng: "There are two knd o people n the world: Johnny Von Neumann and the ret o u." Attrbuted to Eugene Wgner, a Nobel Prze wnnng phyct. 8 P age

10 on promoton meanng they can recognze low prce. (Le Boutller, Le Boutller and Neln, 1994). Gven the lkelhood that conumer would have le prece normaton on prce o brand that they don t buy, aumng that conumer know the prcng trategy o ther brand but not that o the other brand ha ome appeal. In act, becaue rm tratege are tate dependent the cogntve cot o computng or nerrng the tratege o rm could be hgh. Any model o conumer decon makng hould take nto account thee conderaton. The challenge or u to ncorporate bounded ratonalty n an approprate way (Smon 1972, Ellon 2006). We approach th challenge n two tep. We ue a parmonou repreentaton o conumer decon makng that doe not aume conumer to act a powerul computer. Rather, we model them a recognzng the benet o tockplng when they encounter a low prce on the brand they are loyal to. Accordngly, we mply the loyal conumer tockplng decon baed only on the prce o the brand they are loyal to: 3 : Don't tockple,stockple. We urther model the normaton avalable to them to evaluate the benet o tockplng a tattc that they could develop baed only on obervng prce o the brand they buy (For a dcuon o human nerence ung heurtc, ee Gegerenzer and Goldten, 1996). We ue th conumer model to analyze MPE or rm prcng tratege. In the techncal appendx we analyze the perect normaton cae and very that our reult reman robut even to that vew o conumer decon makng. Recall that conumer know the prce o ther brand beore makng ther purchae. We aume that tockplng loyal conumer wth bounded ratonalty have a threhold n mnd uch that they tockple whenever the brand prce below the threhold and do not tockple when t above the threhold. Where doe the threhold come rom? We enure that our model cloed under ratonal expectaton thu provdng a condton to olve or the threhold. Speccally, we aume conumer tockplng rule a bet repone to the prcng trategy o ther preerred brand. We now turn to characterzng the threhold. Imagne a brand loyal tockplng conumer who encounter brand prce when he ha nventory, I=1 (or doe not have nventory, I=0). Fgure 1 depct th pctorally, or ether cae. The prce allow the conumer to make nerence about two (equlbrum) prce dtrbuton. And o, n our model we aume that loyal tockplng conumer know the prce dtrbuton condtonal on ther own nventory tate I=0 or 1. Denote the cumulatve dtrbuton uncton o prce condtonal on loyal 0 1 conumer nventory by G p and G p ; lkewe denote the correpondng probablty denty 9 P age

11 0 1 uncton by g p and g p. 5 How can conumer ue the knowledge o prce condtonal on ther nventory to characterze tockplng threhold, t? In propoton 1, we derve the tockplng rule that the conumer bet repone under bounded ratonalty. Snce conumer n general may ue derent threhold dependng on ther level o nventory, we do not retrct them to a ngle threhold acro tate. However, t turn out that or conumer wth bounded ratonalty n the way we have modeled the optmal rule ue a ngle threhold that ndependent o ther nventory tate. 6 ye ye ye ye ye ye ye Stockplng Decon Prce Threhold, t Tme Fgure 1: Prce oberved by conumer and ther tockplng decon Propoton 1: Stockplng loyal conumer tockple only p the tockplng threhold, t, ndependent o the tate and mut aty: Proo: See Appendx G t G t t and do not tockple p t, where t r r t G t 2 pg ( p) dp pg ( p) dp pg ( p) dp G t pg ( p) dp l 0 t t l 1 t c c Note that the threhold t ndependent o nventory tate. So, regardle o tate, the tockplng decon only depend on the current prce o the brand he loyal to and t. The equaton n propoton 1 contan an mplct oluton or t. I conumer ue threhold t, and they expect rm How G p and G 0 1 nventore o conumer o both rm, we can ntegrate out the rm mxed tratege utably to obtan G p and p related to rm prcng tratege? Snce rm tratege depend on the tate characterzed by the 10 P age G p. 6 Th reult alo due to the act that conumer n our model can only be at one o two level o nventory. In general, they could be at one o n level o nventory, they could ue n-1 threhold.

12 tratege wll be uch a to render ther choce to be optmal under bounded ratonalty, then ther expectaton wll ndeed be ullled. Thu, our model cloed under ratonal expectaton. Beore analyzng rm prcng decon, n the next ecton, we decrbe how the tate pace evolve baed on rm current prce and conumer tockplng decon. Table 1: State Tranton under Derent Condton (or all current tate) Focal rm, prce Focal rm, loyal conumer current perod nventory tate Focal rm, loyal conumer buyng decon Focal rm, loyal conumer next perod nventory tate Competng rm, 3- prce Next perod tate =(I, I 3- ) Hgh I = 0 I = 1 I = 0 I = 1 buy 1 unt or current perod do not buy, conume rom nventory buy 1 unt or current perod do not buy, conume rom nventory I = 0 Hgh =00 Low =01 Low I = 0 I = 1 I = 0 I = 1 buy 2 unt or current and next perod buy 1 unt or next perod buy 2 unt or current and next I = 1 Hgh =10 perod Low =11 buy 1 unt or next perod 3.2 Conumer Inventory / State Evoluton To ee how current perod prce and behavor o cutomer determne the next perod tate conder the cae when ocal rm prce hgh. In th cae they would not tockple, and o n the next perod they have no nventory, I =0. Note that th tranton ndependent o current nventory. Now conder the cae when the ocal rm prce low enough and o t loyal conumer decde to tockple. They wll end up wth an nventory o one unt, I =1, n the next perod. Agan, note that, rrepectve o ther ntal nventory poton when the rm oered a low prce t tockplng loyal conumer ended up wth nventory, I =1, n the next perod. Smlarly, competng rm prce wll 11 Page

13 dctate the nventory poton o t loyal conumer. All other conumer, the non-tockplng loyal and wtchng conumer buy only one unt every perod becaue they do not engage n tockplng. Table 1 ummarze how current perod prce and behavor o cutomer determne the next perod tate. In each tate, rm have two tratege: oer a low prce, t, that would nduce tockplng or oer a hgh prce, > t. Both rm can oer low prce, or both can oer hgh prce, or the ocal rm can oer a low prce and the competng rm can oer a hgh prce or vce vera. For example, rm are n tate =00 n the current perod they could end up n one o our poble tate n the next perod. I both rm charge hgh prce then the tockplng loyal conumer purchae only one unt and rm wll end up agan n tate =00 n the next perod. I the ocal rm charge a hgh prce and the competng rm oer a low prce, tockplng loyal conumer o the ocal rm wll buy only one unt whle the tockplng loyal conumer o the competng rm wll buy two unt and the next perod tate wll be =01. In contrat, the ocal rm current perod prce low and the competng rm prce hgh we end up n tate =10. Fnally, both the rm oer low prce, the next perod tate wll be = Demand Characterzaton Frt note that no conumer wllng to pay more than the reervaton prce, hence we only characterze the demand uncton or prce below the reervaton prce, r. Conder, or example, ocal rm' demand, D, n tate 00 when tockplng loyal conumer o the both rm have no nventory z 0 D ( p, pj, t). 1 ( t p). 1 ( pj p), where (ndcator uncton), 1 ( z) = 0 z 0 Note that nce the ocal rm tockplng conumer have no nventory, all loyal conumer o proporton wll buy at leat one unt. Further, the prce o the ocal rm t, all tockplng loyal conumer (a racton ) wll buy an addtonal unt. In addton, the ocal rm prce le than the compettor prce, wtcher o ze wll alo buy one unt rom the ocal rm. There are two cae, whch requre pecal conderaton: (a) when both rm charge the ame prce, and (b) when the ocal rm charge prce equal to the tockplng threhold. The rt cae are only both rm charge a certan prce wth potve probablty. In that cae, we aume that the wtchng egment buy rom the ocal rm wth probablty and they buy rom the competng rm 12 P age

14 wth probablty 1. 7 In the econd cae, p t, tockplng conumer o brand are nderent between tockplng and not tockplng. When that happen we uppoe that tockplng loyal egment o rm decde to tockple wth probablty, 0 1. Note that we treat a endogenou. A we wll ee n propoton 3, th turn out to be mportant, and t ha orce, n characterzng the equlbrum tratege. Equaton (1) below repreent the demand takng nto account all poblte ncludng p p and p t. j. 1 ( t p). 1 ( pj p) p pj, p t. 1 ( ), t p p pj p t D ( p, pj, t). 1 ( pj p) p pj, p t p pj t tate 00,01 1 z 0 where and (ndcator uncton). 1 ( z) = 1 tate 10,11 0 z 0 Notce that n equaton (1) the rt term,, repreent a guaranteed demand rom loyal conumer at any prce n the upport. In tate {10, 11} tockplng loyal conumer o the ocal rm already have tock and do not necearly have to buy or the current perod, hence guaranteed demand lower, note 1. Loyal conumer may buy one unt or uture the prce below the tockplng 1 threhold; the econd term,. ( t p ) n (1) repreent that at any prce p t, the ocal rm wll get 1 j one unt o addtonal demand rom t tockplng loyal conumer. Term. ( p p ) and n (1) account or the act that wtchng conumer buy the lower prced brand and n cae o a te p p buy rom the ocal rm wth probablty. Fnally term decrbe that when ocal rm oer a prce (p ) equal to the tockplng threhold, t t tockplng loyal conumer are nderent between buyng and not buyng and buy wth probablty. (1) j 3.4 Frm Decon Problem A noted earler, we eek a Markov Perect Equlbrum (MPE) n rm prcng tratege n each tate 00,01,10,11 that ymmetrc acro rm and perod. However, we want to hghlght that ymmetry acro rm and perod but not tate. In act, n tate =01 or example, the two rm 7 In other word, we aume that when prce charged by the two rm are dentcal, wtcher employ an exogenou cue to break te and purchae rom one o the two rm. Th te-breakng rule aumed n the prt that all wtchng conumer are homogenou and make mlar decon. It n the cla o unpot equlbra. 13 P age

15 wll have derent prcng tratege becaue o the derent nventory poton o ther tockplng conumer. Frm need to denty ther optmal prcng trategy gven conumer tockplng rule. Let F p denote the rm equlbrum cumulatve dtrbuton o prce correpondng to tate and let V denote the nnte perod expected payo n tate 00,01,10,11. One mportant conderaton rom equaton (1) that relevant or our analy that n our model, dcontnute n a rm' demand uncton are rom two derent ource. The rt dcontnuty n the rm' demand at t compettor' prce, reultng rom wtchng conumer. When the ocal rm' prce below the compettor' prce, the rm obtan the demand o wtchng conumer whle when the ocal rm' prce above the compettor' prce t doe not obtan that demand. The econd ource o dcontnuty n demand are becaue o tockplng by conumer o the rm; th dcontnuty occur at the tockplng threhold t. 8 Keepng th n mnd, let u rt conder the ymmetrc tate =00. In a mxed trategy Nah equlbrum, the competng rm 3- mxed trategy hould make ocal rm nderent over all un-domnated prce. Recognzng that at hgh prce conumer do not tockple and that lead to a derent uture tate compared to when the rm oer a low prce, we explctly wrte the ocal rm' equ-prot condton at prce above the conumer tockplng threhold. Frt, conder the cae 00 p p below and t. The expected prot V p t cont o the current perod prot and the dcounted contnuaton prot. The contnuaton prot depend on the tate n the next perod, whch aected by whether p 3 t or p 3 t. Keepng that n mnd and nvokng equaton (1) we can wrte 00 V by condtonng t on p relatve to t. V p t p[ 1 F3 ( p) ] 1 F3 ( t) V F3 ( t) V (2) The rt term n equaton (2) the current perod prot when t charge prce p. The econd term denote the dcounted expected contnuaton prot ater takng expectaton over compettor current perod prce. Thu, the relevant contnuaton prot 00 V wth probablty 1 F 00 3 ( t) when compettor prce above the tockplng threhold, and lkewe, the compettor prce below the threhold. 01 V wth probablty F 00 3 () t, when 8 Recall that the rt knd o dcontnuty alo occur n Varan (1980) and Naramhan (1988) whle the latter doe not. From ther analy, we know that the preence o wtchng conumer reult n a mxed trategy equlbrum over an nterval uch that expected prot are equal at all un-domnated prce. The mxng dtrbuton n ther model trctly ncreang and contnuou everywhere over the upport except pobly at r. The dcontnuty n the mxng dtrbuton (ma pont) at r are only when rm are aymmetrc. In our model another knd o dcontnuty n the mxng dtrbuton can occur due to tockplng. Speccally, tockplng could lead to a ma pont n the nteror o the upport at t, a eature that doe not occur n extant model.. A urther novel eature o our model that both rm mxng dtrbuton could have a ma pont at t. 14 P age

16 In the ame way, expected prot V 00 p t can be computed a hown n equaton (3). In th cae, there are two derence. Frt, the current perod prot account or tockplng n the demand. Second, the contnuaton prot now dened or uture tate 10,11. V p t p[(1 ) 1 F3 ( p) ] 1 F3 ( t) V F3 ( t) V (3) Smlarly, we can wrte the equ-prot condton or the ocal rm n all other tate. Equ-prot condton n all tate 00,01,10,11 are a ollow: V V V V p 1 F3 ( p) 1 F3 ( t) V F3 ( t) V t p p (1 ) 1 F3 ( p) 1 F3 ( t) V F3 ( t) V p t p 1 F3 ( p) 1 F3 ( t) V F3 ( t) V t p p (1 ) 1 F3 ( p) 1 F3 ( t) V F3 ( t) V p t p (1 ) 1 F3 ( p) 1 F3 ( t) V F3 ( t) V t p p 1 F3 ( p) 1 F3 ( t) V F3 ( t) V p t p (1 ) 1 F3 ( p) 1 F3 ( t) V F3 ( t) V t p p 1 F3 ( p) 1 F3 ( t) V F3 ( t) V p t (4) (5) (6) (7) Gven the equ-prot condton n (4)-(7), we are n a poton to determne the rm equlbrum trategy condtonal on the tockplng threhold t. 4. Equlbrum We wh to explore an equlbrum n whch conumer tockple. To that end, we derve the ucent condton n propoton 2 or conumer to tockple. We alo wh to ocu on a mult-tate equlbrum n whch all tate are vted wth potve probablty. A ucent condton or th to occur that the reervaton prce r hould be un-domnated n all tate. We wll derve the ucent condton or th n propoton 4 ater characterzng the equlbrum and the value uncton. Propoton 2: I * c tockplng conumer tockple wth potve probablty, where * (1 ) ln 15 P age

17 Proo: Pleae ee Appendx Propoton 2 gve the condton to rule out an equlbrum n whch the tate alway 00. In partcular, when conumer dcount actor ucently hgh then n =00 tate, rm wll charge prce below the tockplng threhold wth potve probablty o that conumer wll nd t n ther bet nteret to tockple. For the remander o the paper we aume that c hgh enough o that the condton n Propoton 2 ated. In 4.1 we characterze the mxed tratege and hghlght the nteretng apect o the ame wth ocu on the ma pont at t and r. We evaluate the value uncton and derve the lower and upper bound or prot n 4.2 and nvetgate the eect o lexble conumpton n Mxed Strategy Prce n Mult-tate Equlbrum Wherever applcable we nvoke pror reult due to Varan (1980), Naramhan (1988), and Makn and Dagupta (1986) to characterze the equlbrum mxed tratege. In our model, each rm ha to ormulate a mxed trategy correpondng to the our tate 00,01,10,11 condtonal on t. Note that becaue conumer tockple below t but not above t, the cd F ( p ) cont o two uncton: above below F ( p ) and F ( p ). In general, below F ( t ) above F ( t ). How can th be reolved? Recall that cd, F ( p) hould be a non-decreang uncton n p. Thereore, below F ( t ) above F ( t ), then prce jut above t domnated and F ( p) contant over an nterval above t. We wll reer to th a a hole below above n F ( p ). Alternatvely, the F ( t ) F ( t ) then, F ( p ) wll contan a ma pont, M () t, at t. Fgure 2 how th graphcally. Fgure 2: Behavor o Cumulatve Dtrbuton Functon around the Stockplng Threhold 16 P age

18 It ueul to note that n extant work, or ntance n Varan (1980) and Naramhan (1988), both rm cannot have a ma pont at ame prce n the upport o ther mxng dtrbuton. 9 That becaue when both rm have a ma at the ame prce they hare the wtchng egment at that prce. Hence, or each rm t become a domnant trategy to ht t ma pont to a prce lghtly below the compettor ma pont to capture the entre wtchng egment. However, n our model, or both rm F ( p ) can have a ma pont at t. Th poe a unque challenge n characterzng F ( p ). We how n propoton 3 how the ma pont at t utaned n equlbrum. Denote the ma pont at t by M ( t ), n 00,01,10,11. Propoton 3: When both rm have a ma pont at t, t can be utaned,.e., probablty that rm loyal egment tockple at t equal to the probablty that the wtchng egment buy brand when both brand are prced equally. Proo: Pleae ee Appendx When both rm charge the ame prce the wtchng egment buy brand (or 3-) wth probablty (or1 ). Careully conderng the opton or tockplng conumer when they are nderent between tockplng and not tockplng reolve the ma pont ue and p t rendered un-domnated even n the preence o a ma pont at t or both rm. Gven our ocu on a ymmetrc equlbrum we et 1/ 2, 1,2. Turnng to the cae when below F ( t) above F ( t), a n Gangwar, Kumar and Rao (2014), we how n lemma 1 n the Appendx that the mxng dtrbuton wll have a hole above the tockplng threhold. Next, we contemplate the poblty o ma pont at other prce n the mxng dtrbuton. It eay to rule out the poblty o ma pont at any other prce except r and t. Note that n the ymmetrc tate =00 and =11 there can be no ma pont at r. 10 Th leave u wth the aymmetrc cae =01 and =10. In Naramhan (1988) model, a ma pont at r occur n the mxed trategy o the rm wth the larger loyal egment. We cannot make drect ue o th reult becaue o the preence o contnuaton prot. How then can we denty whch rm ha the ma pont? We do that next n lemma 2A and 2B 9 Naramhan(1988) and Raju, Srnvaan and Lal(1990) have hown that n ome cae one rm can have a ma pont but only at the reervaton prce. 10 Th becaue both rm charge r, then one can devate to r-ε and capture the wtchng egment wthout aectng current prot and uture tate. Th reult occur alo n Varan (1980) and Naramhan (1988) where uture tate do not matter. 17 P age

19 and how that mxed trategy o the rm n tate =01 wll have a ma pont at the reervaton prce r. We denote, M ( p ) a the ma pont at prce p n the mxng dtrbuton o the ocal rm n tate. d F p F p ; moreover, dp Lemma 2A: At p l 01, t, ( ) ( ) F3 ( t ) F ( t ) /. Proo: Pleae ee Appendx Lemma 2B: At p tr, d F 3 p F p ; moreover, dp, 0 01 ( ) 01 ( ) 0 F ( r ) F ( r ) / Proo: Pleae ee Appendx Note that 0 F ( r ) F ( r ) mple that the ocal rm () wll have a ma pont at the reervaton prce (r) 01 0 M ( r) and nce both rm can not have a ma pont at (r) 01 M3 () r It turn out that n our cae alo (even wth contnuaton prot) the ntuton mlar to the aymmetrc cae n Naramhan (1988). 4.2 Contnuaton prot under mult-tate equlbrum We now derve the rm value uncton, the um o expected prot over the nnte horzon, n each tate, to evaluate the prot mplcaton o tockplng by loyal conumer. Toward th end, we olve or the expected prot o the rm n each tate n the mult-tate equlbrum n lemma 3-5. Speccally, we ue (4) (7) to derve cloed orm expreon or the value uncton by evaluatng the equ-prot condton n (4) (7) at p r. Lemma 3: V r V Proo: Pleae ee Appendx Lemma 4: V 11 r 1 r Proo: Pleae ee Appendx r Lemma 5: r V V and V V By ymmetry M ( r) 0 and 0 M ( r) P age

20 Proo: Pleae ee Appendx Note that the contnuaton prot are equal n tate 00,01. It alo nteretng to note that n thee tate V and V do not depend on. Th becaue loyal conumer o the ocal rm do not have nventory n ether tate. Thereore, all loyal conumer, rrepectve o, demand one unt at r. In contrat, n tate 10,11 V and V do depend on. Th occur becaue only the loyal conumer who do not have nventory purchae at r. The ze egment 1, whch a uncton o. Th n turn make both V and V le thanv 00. Furthermore, we can ee rom lemma 5 the prot are lower n 11relatve to 10. The ntuton or th that n 11wtchng conumer compre a larger racton o demand at r relatve to Unntended conequence o promoton Denote the probablty o tate under teady tate a. Then the expected value over all tate, V V. We now tate our man reult n propoton 4. Propoton 4: In a mult-tate equlbrum rm expected prot, V ate r r V 1. Proo: Follow drectly rom lemma 3-5. Gven that V a convex combnaton o the tate dependent value uncton the tatement n propoton 3 ollow drectly rom lemma 3-5. To undertand the bound on the prot preented n propoton 3, t helpul to compare t wth the prot rm would obtan when conumer do not tockple. We denote the no tockplng cae a the benchmark cae. In that cae rm prot, 19 P age bench V, obtaned by conderng a ymmetrc, nnte perod veron o Naramhan (1988), and o the rm expected prot n each perod r and the dcounted um o thee expected prot over an nnte bench r horzon yeld: V 1. Note that n our model the prot bounded rom above by r, whch 1 equal tov bench. Let u compare the lower bound, 11 V, o equlbrum prot n propoton 4 to bench V. Frm maxmum loe n a mult-tate equlbrum can be expreed n percentage term a bench 11 mult V V MAXL, do not exceed bench 1. Note that V tockplng egment but decreang n. I rm are extremely patent, 1 mult MAXL ncreang n the ze o the mult MAXL

21 approache zero. Suppoe 50% o the loyal conumer tockple, the purchae requency n the product mult category once every month, and the rm dcount actor 0.99 per month. Then MAXL wll be a mere 0.5%. Snce the maxmum abolute lo, r, repreent jut one perod ale (at a maxmum prce o r) to the tockplng conumer t become nntemally mall relatve to um o the dcounted prot. In other word, the lo n the abolute dollar term may be gncant, dependng on the category but a a proporton t relatvely mall. To undertand the loe better t ueul to compare the benchmark prot to the prot condtonal on tate. Note that the contnuaton prot benchmark cae. Alo note that the contnuaton prot 00 V and 01 V are dentcal to the prot n the 11 V lowet and le than the prot n the benchmark cae by r. The r repreent the orgone ale to tockplng conumer at the reervaton prce. Thu, the ntal tate =11, t a tockplng loyal conumer got a ree ample r o the product (opportunty cot o whch r to the rm) and the rm make, equal to the 1 benchmark prot thereater. 12 What nteretng here that equlbrum prcng ully take nto account uture tockplng behavor and o we are let only wth a one-tme lo captured by the ntal tate. 4.3 Extence o Equlbrum We how that the et o parameter where our propoed equlbrum n mxed tratege ext, non-empty. In lemma 6 we combne our earler reult n propoton 2 to delneate the parameter regon n whch our propoed equlbrum ext. Then, n lemma 7 we very that the rm mxed tratege n ecton 4.1 aty the neceary condton or an MPE. Fnally, we demontrate the extence o equlbrum by contructon wth a numercal example. Lemma 6: A ucent condton or r to be n the upport o rm' mxed trategy 1. Proo: Pleae ee Appendx I the condton n Lemma 6 volated then the ucent condton that enure that r n the * 1 2 upport o the mxed trategy wll be volated. Recall rom propoton 2 that c. 1 (1 ) ln c 12 Suppoe we allow conumer to tockple multple unt. Then, the maxmum lo wll be the one tme opportunty cot o ellng tockpled unt to tockplng loyal conumer. 20 P age

22 * 1 2 Thereore a ucent condton or r to be n the upport requre that (1 ). 1 (1 ) ln Th condton guarantee that the parameter regon, or the extence o equlbrum o the knd we are explorng, non-empty. Snce th a ucent condton, we hould note that our equlbrum could ext or other parameter value a well. Lemma 7: The mxed tratege aty the neceary condton or an MPE. Proo: Pleae ee Appendx To demontrate extence let u turn to a numercal example. Suppoe 0.25, 0.30, r 100, 0.95, 0.99 c Then recall rom lemma 3-5 that the upper bound o prot V V r and the lower bound o prot V 11 r r Our numercal analy conrm th. We nd ollowng reult V 2500, V 2500, V , 11 V and V. Note that the upper bound ndeed 00 V 2500 and the lower bound 11 V We next ocu on the mxed tratege. They are gven by: F 1.5p p 100 p, t , 1.65 p p p p M ph pl F t , , P age

23 1.35p p p F p, M r0.1004, M t , 1.5 p p p ph , pl , F t p p p 10 F p, M t , 1.65 p p p ph , pl , F t F p M ph pl F t Here 1.35p p 100 p, t , 1.5 p p p , , ph and pl repectvely are the expected prce above and below the threhold n each tate. Frt, we can very that the equ-prot condton hold at all prce wth potve probablty gven the 01 computed mxed tratege. Next, we note that at the lower lmt F l and 10 F l df 01. We alo note that, a mentoned n lemma 2A, F 01 p F 10 p dp p 2 le than the lope o df 10 dp p 2. nce the lope o Smlarly, we note that the lope o df 01 dp le than the lope o df10 p 2 dp 50. We alo, note 2 p that 01 M r le than / 0.15 a hown n lemma 2B. Moreover ma pont at t, M t le than M t Next, we ee rom , , wth potve probablty n equlbrum. Other quantte o nteret are noted below , that all tate are vted P age

24 Conumer contnuaton utlte are 0 1 U and U , whch lead to 0 1 t U - U Thu, we can ee the extence o the c propoed equlbrum. More example can be ound n the Appendx. How derent are the prce n a tockplng equlbrum rom thoe n a tuaton wthout tockplng? To undertand th, we computed the dtrbuton o prce n a randomly choen perod by ntegratng out the mxed tratege over the tate under tatonary condton, denotng t a the uncondtonal PDF wth tockplng. In gure 3 we dplay that and compare t to the dtrbuton o prce wthout tockplng. We can readly ee that manager move away rom requently promotng below the threhold. Moreover, there an ncreae n the probablty o chargng the reervaton prce. Thu, t mportant to undertand that the relatvely low loe due to tockplng accompany gncant change n prcng tratege. We wll elaborate on th urther n ecton 5. Fgure 3: Equlbrum Probablty Denty Functon o Prce wthout (wth) Conumer Stockplng 4.4 Flexble Conumpton Due to Stockplng Alawad, Gedenk, Lutzky and Neln (2007) dented two major eect o conumer tockplng whch can have a potve eect on rm prot: preemptve wtchng and ncreaed conumpton. The dea that conumer' nventory can ncreae conumpton alo nd upport n Wank (1996). The nteracton o ncreaed or lexble conumpton and tockplng ha been tuded by Bell, Iyer and Padmanabhan (2002) and Heerde, Leelang and Wttnk (2004) whle Jan (2012) account or ncreaed conumpton baed on nventory n modelng the eect o package ze on conumpton. How mght lexble conumpton due to tockplng aect rm prot n our model? Gangwar, Kumar and Rao (2014) how that when only wtcher tockple, even ncreaed conumpton due to tockplng 23 P age

25 doe not lead to hgher prot. The queton reman th would hold tockplng by loyal conumer lead to ncreaed conumpton. We addre that n th ecton. We uppoe that when loyal conumer tockple ther conumpton lexble. When conumer tockple they have two unt on hand. We aume that they ether conume both unt n the current perod wth probablty 1 or conume jut one unt n the current perod, wth probablty, where, 0,1. Gven that conumer may exhaut ther nventory earler than antcpated t can lead to a tuaton where proporton o tockplng conumer have nventory o I =1 whle the remanng 1 proporton end up wth no nventory, I =0, n the next perod. From the rm' perpectve, th mean, t charge a low prce n the next perod, proporton o tockplng conumer wll buy only 1 unt and 1 proporton wll buy 2 unt. A beore, th lead to our poble tate {00,0, 0, } n our model. It ntructve to note that n the lexble conumpton cae, tate correpond to the average nventory poton o tockplng conumer o the ocal rm. So 1 wll correpond to our bae model. The conumer decon problem the ame a n the bae cae. In Table 2, we dplay how conumer decon to tockple (or not) and average nventory poton or the ocal rm tockplng loyal conumer determne the tate tranton n the cae wth lexble conumpton. Table 2: Inventory Poton Evoluton and State Space Tranton under Flexble Conumpton rm current perod prce nventory poton n the current perod buyng decon rm tockplng conumer nventory poton beore conumpton conumpton decon nventory poton n the next perod rm tockplng loyal conumer nventory tate n next perod Hgh Prce (No Stockplng) I = 0 buy 1 I = 1 I = 1 not buy I = 1 conume one unt conume one unt I = 0 I = 0 All loyal conumer o brand have zero nventory Low Prce (Stockplng) I = 0 buy 2 I = 2 1 proporton conume both unt 1 proporton have I = 0 proporton o loyal conumer o brand j have 24 P age

26 I = 1 buy 1 I = 2 I = 2 proporton conume one unt 1 proporton conume both unt proporton have I = 1 1 proporton have I = 0 nventory o one unt and 1 proporton have zero nventory I = 2 proporton conume one unt proporton have I = 1 Note that the average nventory poton n the next perod ndependent o current nventory poton o tockplng conumer. A n our bae model, the next perod tate wll only depend on current perod prce. In the lexble conumpton cae, the tate 00 correpond to the tuaton n whch tockplng conumer o nether rm have nventory whle tate correpond to the tuaton where proporton o tockplng conumer o both rm have nventory and 1 proporton o both rm have zero nventory. The tate 0 ( 0 ) correpond to the cae where tockplng conumer o the ocal (competng) rm and 1 proporton o competng (ocal) rm have no nventory and a proporton o the competng (ocal) rm tockplng loyal conumer have an nventory o one unt. The demand dynamc and the oluton procedure are mlar to that n the bae model. Stockplng threhold wth lexble conumpton alo ndependent o tate. Conder, or example, ocal rm' demand, tockplng loyal conumer o the both rm have nventory. D, n tate when proporton o 1 z 0 D ( p, pj, t) 1. 1 ( t p). 1 ( pj p), where (ndcator uncton). 1 ( z) = 0 z 0 Frt, note that nce the proporton o tockplng conumer o ocal rm have nventory, they do not buy when prce are hgh. In other word, only non-tockplng conumer 1 and tockplng conumer wth no nventory 1, buy 1 unt at hgh prce. Thereore, guaranteed demand at all prce n tate Further, the prce o the ocal rm t, all tockplng loyal conumer (a racton ) wll buy an addtonal unt. In addton, the ocal rm 25 P age

27 prce le than the compettor prce, wtcher o ze wll alo buy one unt rom the ocal rm. Then, or any tate demand can be wrtten a 1 1. ( t p). ( pj p) p pj, p t. 1 ( ), t p p pj p t D ( p, pj, t) (8). 1 ( pj p) p pj, p t p pj t tate 00,0 1 z 0 where and (ndcator uncton). 1 ( z) = 1 tate 0, 0 z 0 Compared to the bae cae, note that n (8) the ocal rm get an extra demand o 1 rom tockplng conumer who end up conumng both unt n tate 0,. To characterze the equlbrum n the lexble conumpton cae, we ollow the ame procedure outlned earler n Secton 4.1. Ater determnng the equlbrum mxng dtrbuton n each o the our tate, we evaluate the equ-prot condton at p r to compute prot o the rm n each tate. In Propoton 5 below, we provde bound on the rm prot n a mult-tate equlbrum under lexble conumpton. Propoton 5: In a mult-tate equlbrum wth lexble conumpton rm expected prot, r bounded uch that r V 1 lex. 26 P age V lex, Proo: Once we recognze that the guaranteed demand under lexble conumpton 00,0 00,01, whle n our bae model, then the proo (1 ) 0, (1 ) 10,11 traghtorward and mlar to that n Propoton 4 When we compare the lower bound o prot n a mult-tate equlbrum wth and wthout lexble conumpton (Propoton 5 and 4), t eay to ee that the lower bound o rm' prot wth lexble conumpton hgher. Recall that n Propoton 4, the maxmum lo wa r, whch the orgone ale to tockplng conumer that the rm could have made conumer had no nventory. Wth lexble conumpton th lo r r, 0,1. Thu, relatve to our bae model, the maxmum lo wth lexble conumpton lower. In partcular, t lower by 1 r whch the maxmum prot that the rm can make rom tockplng conumer who tockple, conume both unt and re-enter the market wth no nventory. What nteretng that even wth ncreaed conumpton, the mult-tate equlbrum prot