THE IMPACT OF QUICK RESPONSE IN INVENTORY-BASED COMPETITION

Workng Paper WP no 722 November, 2007 THE IMPACT OF QUICK RESPONSE IN INVENTORY-BASED COMPETITION Felpe Caro Vícor Marínez de Albénz 2 Professor, UCLA Anderson School of Managemen 2 Professor, Operaons Managemen and Technolog, IESE IESE Busness School Unvers of Navarra Avda. Pearson, 2 08034 Barcelona, Span. Tel.: +34 93 253 42 00 Fax: +34 93 253 43 43 Camno del Cerro del Águla, 3 Cra. de Caslla, km 5,80 28023 Madrd, Span. Tel.: +34 9 357 08 09 Fax: +34 9 357 29 3 Coprgh 2007 IESE Busness School. IESE Busness School-Unvers of Navarra -

The Impac of Quck Response n Invenor-Based Compeon Felpe Caro Vcor Marínez-de-Albénz UCLA Anderson School of Managemen 0, Weswood Plaza, Sue B420 Los Angeles, CA 90095, USA IESE Busness School Unvers of Navarra Av. Pearson 2 08034 Barcelona, Span fcaro@anderson.ucla.edu valbenz@ese.edu Ocober 28, 2007 We propose a mul-perod exenson of he compeve newsvendor model of Lppman and McCardle 997 o nvesgae he mpac of quck response under compeon. For hs purpose, we consder wo realers ha compee n erms of nvenor: cusomers ha face a sockou a her frs-choce sore wll look for he produc a he oher sore. Consequenl, he oal demand ha each realer faces depends on he compeor s nvenor level. We allow for asmmerc reorderng capables, and we are parcularl neresed n he case when one of he frms has a lower orderng cos bu can onl produce a he begnnng of he sellng season, whereas he second frm has hgher coss bu can replensh sock n a quck response manner akng advanage of an ncremenal knowledge abou demand f s avalable. We vsualze hs problem as he compeon beween a radonal makeo-sock realer ha bulds up nvenor before he season sars versus a realer wh a responsve suppl chan ha can reac o earl demand nformaon. We provde condons for hs game o have a unque pure-sraeg subgame-perfec equlbrum, whch hen allows us o perform numercal comparave sacs. Our resuls confrm n a compeve seng he nuve fac ha quck response s more benefcal when demand unceran s hgher, or exhbs a hgher correlaon over me. Fnall, we fnd ha par of he compeve advanage from quck response arses from he asmmer n response capables.. Inroducon In recen ears, he apparel ndusr has seen he rse of wha has been called fas fashon realers. These are clohng companes ha are able o respond quckl o marke rends and nroduce new producs ver frequenl. Mos of hese producs have a lfe-ccle of no more han a few weeks, and b he me he ccle s over, he are prompl replaced b a more fashonable em. The hgh assormen roaon has become a dsncve feaure ha has ncreased he average number of vss o he sore, snce now cusomers are aware ha he can alwas fnd somehng new.

In Europe, where he concep began, fas fashon has been denomnaed a 2s-cenur realng phenomenon wh represenave companes such as H&M and he Index group, owner of Zara Davdson 2005. Besdes her naural alen o desgn on-rend clohes, a crucal par of her success s due o her flexble suppl chan and operaonal compeences Ghemawa and Nueno 2003. In parcular, fas fashon realers can make n-season replenshmens hanks o remarkabl low lead mes, n he order of weeks raher han monhs. The laer s acheved n mos cases hrough local producon or expedng, whch obvousl ranslaes no hgher un coss. In he case of Norh Amerca, fas fashon remans a nche ha represens no more han wo percen of he apparel busness Foroohar 2006. Large clohng realers lke The Gap seem oo bg and mgh no have he ncenves o resrucure her enre suppl chan o mmc her European compeors snce her cusomers have been hsorcall less fashonforward. The mgh however borrow a few elemens of fas fashon. For example, he can srengh he lnk wh her supplers, or he can move he producon of render ems o Mexco nsead of Asa n order o shoren lead mes. The exen o whch radonal realers should adop or conver o fas fashon remans an open queson and serves as par of he movaon for hs paper. Despe he ncpen bu growng success of fas fashon n Norh Amerca, he concep self bulds upon quck response QR, whch was an apparel manufacurng nave ha sared prmarl n he Uned Saes durng he md 980s Hammond and Kell 990. The man obecve of QR s o drascall reduce lead mes and seup coss o allow he posponemen of orderng decsons unl rgh before or durng he real sellng season when beer demand nformaon mgh be avalable. A successful mplemenaon of QR s pcall based on he effecve use of nformaon echnologes. Fas fashon has aken QR o a hgher level and has leveraged on he mnmal lead mes b nroducng new producs on a regular bass, and herefore enablng a dnamc assormen ha bascall fulflls he deal of provdng fashon on demand. The overall success of fas fashon s arbuable o a combnaon of mulple facors. The neracon beween all he elemens nvolved s a a prelmnar sage of beng undersood. There has been exensve qualave work ha descrbes he dfferen cases or examples of fas fashon companes. However, he academc leraure on hs opc remans scarce. In hs paper we am a undersandng he mpac of one specfc elemen of fas fashon. Namel, he QR componen ha s arguabl he bass for all he oher elemens ha laer come no pla. Therefore, we dsregard assormen, prcng or marke posonng decsons 2

relaed o fas fashon, and we focus on he essenal capabl of havng more flexbl n erms of nvenor replenshmen. We look a he problem n a compeve seng, snce from s ncepon, QR advocaes have clamed ha s he onl vable sraeg under he curren condons n he apparel marke, smlar o wha us-n-me manufacurng has mean o he auo ndusr Hammond and Kell 990. We consder a model wh wo realers sellng a subsuable produc over a fne horzon. The wo realers compee based on nvenor. When a sockou occurs a one realer, he unsasfed cusomer walks no he second realer, where she s served f sock s avalable. Thus, he nvenor decson of a gven realer depends on he level of nvenor a he compeor. Moreover, as me goes b, he realers can ncorporae an addonal demand nformaon no her sockng decsons. The model allows for asmmerc realers and we analze her compeve sraeges, consderng he equlbrum nvenor decsons. Our model can be seen as a mul-perod exenson of he compeve newsbo developed b Lppman and McCardle 997. As n her case, we are neresed n deermnng and characerzng he exsence of a unque pure-sraeg subgame-perfec equlbrum, whch hen allows us o undersand and compare he oucomes for each realer, and n parcular assess he poenal benefs of mplemenng QR. Our paper makes conrbuons o he operaons leraure from boh he mehodologcal and manageral sandpon. From a mehodologcal perspecve, we solve an asmmerc mul-perod nvenor-based compeon model, where he asmmer s n erms of reorderng and demand learnng capables. We are no aware of an oher paper ha sudes dnamc horzonal nvenor-based compeon wh demand correlaon over me. Several auhors have prevousl suded he nfne horzon case, bu under such condons ha reduces o a mopc sngle-perod problem. On he conrar, we formulae a dnamc fne horzon model, and we analze four dfferen cases. For each one of hem, we provde suffcen condons ha guaranee he exsence of a unque equlbrum. The condons are summarzed n Table he meanng of each assumpon s dscussed laer. Noe ha we pa specal aenon o he wo-perod case snce s he mos common approach ha has been used n he leraure o model QR see, for nsance, secon 0.4 n Cachon and Terwesch 2005. Also, our resuls exend Theorem 3 of Lppman and McCardle 997 n wo was: o asmmerc realers n he sngle-perod case, and o an arbrar number of perods wh nformaon updaes n he smmerc case under a lnear demand spl. We also allow for arbrar nal nvenor levels n boh cases. Asmmerc ne margns and demand splng funcons are also allowed. 3

From a manageral sandpon, hs paper s a frs aemp a rng o undersand wha he compeve advanage of fas fashon s compared o more radonal realng operaons. As menoned before, our analss s resrced o he benefs ha sem from QR. In parcular, we provde a dealed numercal sud where we compare he profs n a compeon of wo slow response SR frms, wh hose acheved n a compeon where one or boh realers have mplemened QR. Our resuls ndcae ha boh realers are beer off n he QR vs. SR compeon compared o he SR vs. SR case. In addon, we show ha par of he compeve advanage for a QR realer comes from he asmmer,.e., from beng faser han he compeor, and hese benefs are larger under hgher demand unceran or hgher correlaon over me. Table : Suffcen condons for a unque pure-sraeg subgame-perfec equlbrum. The remander of he paper has he followng srucure. In secon 2 we revew he exsng leraure, mosl on nvenor-based compeon models. Then, n 3 we develop our model, and n 4 we esablsh he exsence and unqueness of a pure-sraeg subgameperfec equlbrum for he four dfferen cases menoned n Table. In 5 we perform numercal comparave sacs n order o undersand how he equlbrum depends on he parameers of he model. Fnall, n he las secon 6 we conclude and dscuss fuure research drecons. All he proofs are avalable n he echncal appendx, unless oherwse noed. 2. Leraure Revew Fas fashon has been dscussed exensvel n he popular press, see for nsance The Economs 2005. In more academc erms, he leraure s mosl descrpve wh an emphass on he qualave aspecs of he realng sraeg. Man cases have been wren, n parcular for he Spansh compan Zara, e.g., Ghemawa and Nueno 2003, McAfee e 4

al 2004, and Ferdows e al 2004. From a quanave perspecve, durng he 990s sgnfcan progress was made n undersandng he mpac of QR n an solaed suppl chan, mosl n a wo-perod seng see Fsher and Raman 996, Ier and Bergen 997, and references heren. More recenl, Cachon and Swnne 2007 have looked a QR n he presence of sraegc cusomers. All hs work s relaed o ours because we focus on he QR aspec of fas fashon. However, as menoned before, fas fashon goes beond QR, n parcular b nroducng a large number of new producs durng he real sellng season. In ha respec, Caro and Gallen 2007 provde a closed-form polc for one of he dsncve operaonal challenges faced b fas fashon frms, namel he dnamc assormen problem To he bes of our knowledge, here has no been much analcal work ha res o denf he drvers of fas fashon s success n a compeve conex. Clearl, he answer s no smple snce here are man nerweaved facors ha come no pla. For ha reason, as a frs aemp o undersand he poenal compeve advanage, n hs paper we focus exclusvel on he QR capabl and we openl dsregard oher mporan elemens of fas fashon. Wh hs scope n mnd, we are lef wh an nvenor-based compeon problem for subsuable producs. Several models have been developed n he leraure for hs problem. In Table 2 we provde a non-comprehensve summar of precedng work. Our paper conrbues o hs sream of research b solvng a mul-perod model ha allows for asmmerc realers n erms of reorderng and demand learnng capables. In he sngle-perod case wh n realers, Lppman and McCardle 997 prove he exsence of a pure Nash equlbrum under he general assumpon ha he effecve demand faced b a parcular frm s sochascall decreasng n he nvenor levels of he oher frms, whch comes naurall n he case of subsuable producs, see Neessne and Zhang 2005. The exsence of a unque Nash equlbrum requres addonal assumpons as hose lsed n he las column of Table 2. In he nfne horzon case, several auhors have shown ha, under suable condons, here exss a Nash equlbrum n whch each realer follows a saonar base-sock polc. All hese resuls sem from he dnamc olgopol model b Krman and Sobel 974. In pracce, hs s equvalen o solvng a sngle-perod problem. Noe ha even f he laer has a unque Nash equlbrum, ha does no guaranee a unque subgame-perfec equlbrum n he mul-perod case. Several oher realng compeon models n whch nvenores pla an mporan role have been suded n he leraure. For nsance, n an nfne horzon seng, L 992 looks a delver-me compeon and shows ha when all realers are dencal he end 5

o make-o-sock. Anupnd and Bassok 999 consder nvenor-based compeon à la Parlar 988, and sud he mpac of marke search.e., he spll-over fracon on he manufacurer s prof. Bernsen and Federgruen 2004 examne he case of realers ha compee on prce and hen se her nvenor levels accordngl. Gaur and Park 2007 consder cusomers sensve o negave experences such as a sockou, and sud he compeon of realers on he bass of her servce levels. As before, gven he saonar model formulaon n hese papers, he soluon s mopc n he sense of Sobel 98 and he analss reduces o a sngle-perod problem. Table 2: Invenor-based compeon models for subsuable producs. Gven he exremel shor lfe-ccle of fashonable clohng, fne horzon models seem more approprae. In ha maer, he work b Hall and Poreus 2000 s concepuall close o ours because, despe he fac ha he consder compeon based on cusomer servce nsead of non-pershable nvenor and nformaon updaes are no allowed, he realers can onl ake acons o preven leakage of demand o he compeor raher han proacvel arac demand o self. Under hese condons ogeher wh a mulplcave demand model, he are able o show he exsence of a unque subgame-perfec equlbrum. Lu e al 2007 exend he resul o a more general demand model ha ncludes he addve case. Olsen and Parker 2006 provde an alernave exenson n whch he realers can 6

hold nvenor over me and can adverse o arac dssasfed cusomers from s compeor s marke. The exsence of a unque equlbrum s guaraneed b assumng a parcular salvage value funcon and low nal nvenor levels as n Avsar and Bakal-Gürso 2002. Ineresngl, n equlbrum, he game effecvel becomes wo parallel Markov decson processes where each frm can make sockng decson ndependen of he oher frm s choces. We oban a smlar resul for he case of wo smmerc realers, hough under a dfferen se of assumpons. 3. A Mul-Perod Invenor Compeon Model In hs secon we formulae he mul-perod nvenor compeon model ha wll be used laer o sud he benefs of QR. In 3. we presen he basc feaures and assumpons. Then n 3.2, we nroduce he dnamc aspec of he model and he soluon approach.e., sub-game perfecon. 3. Basc Feaures and Assumpons In wha follows, we presen each one of he assumpons ha lead o our mul-perod model. The assumpons appear n alcs and an explanaon or dscusson follows whenever approprae. An relaed noaon s nroduced as well. A There are onl wo frms ha sell subsue producs, and each frm maxmzes he oal expeced profs over a fne horzon of T perods. Le ndces and denoe realers and perods respecvel, wh =, 2 and = T,...,. Noe ha perods are couned backwards, and we consder a fne horzon o represen he shor produc lfe-ccle n he fashon apparel ndusr. The ndex s also reserved o denoe a realer, and hroughou he paper s undersood ha. Snce he realers sell perfec subsues, a cusomer ha canno fnd he produc a her preferred realer wll check f s avalable a he compeor. A2 The aggregae cusomer demand n perod denoed D s connuous, sochasc, and ma be correlaed across perods. Le f be he p.d.f. of he demand n perod, F s c.d.f., F F, and F nverse. When here s correlaon across me, we denoe he p.d.f. and c.d.f. of he demand 7 s

n perod as f I and F I respecvel, where I s he demand nformaon avalable a ha pon n me, whch s assumed o be common knowledge. Tpcall, he nformaon would be he vecor of pas demand realzaons ha s, D +, D +2,..., D T } and/or an daa or demand sgnal ha has become avalable. An mporan remark s ha we do no allow he nformaon vecor o be a funcon of he decsons n he curren perod. In oher words, nformaon s uncensored, us as n mos QR models. A3 In a gven perod, he effecve demand faced b realer s composed of wo pars: he orgnal demand, and he overflow demand. The orgnal demand s expressed as qd, where q s he nal allocaon funcon also referred o as he demand splng funcon whch s assumed o be srcl ncreasng, and we have ha D = q D +q 2 D. 2 The orgnal demand s made of cusomers ha naurall choose realer over he compeor, and he overflow demand s made of hose cusomers ha nall choose bu end up bung a because runs ou of sock. Ths overflow demand s equal o max0, q D }, where s frm s nvenor level afer replenshmen n perod. 3 Then he effecve realzed demand faced b realer s gven b R q D + max0, q D }. Snce hs s a ke assumpon n our model, several mporan observaons follow: The effecve demand R depends onl on he compeor s nvenor level. Therefore, compeon s based on he nvenor levels, bu realer can onl lm he cusomers loses raher han nfluence hose gans. In an auhenc fas fashon seng, a realer would pcall arac more demand b susanng a hgh assormen roaon. Our model does no consder such feaure snce we focus on undersandng he mpac of he QR capabl for one parcular produc. We also noe ha, excep for Neessne e al 2006, all he papers menoned n Table 2 consder compeve models n whch he realer can onl preven leakage raher han arac addonal demand. The same happens n Hall and Poreus 2000 and Lu e al 2007. 2 In he demand sgnal case suded n 4.2., snce here s no demand realzaon n he frs perod, we le q2 = 0, =, 2. 3 Our model can be drecl exended o he case of mperfec subsuon. Tha s, when onl a fracon δ of cusomer choose o subsue when he face a sockou. I suffces o mulpl Equaon b δ, and all he equlbrum resuls follow hrough. 8

In Parlar 988 and oher smlar papers, ndependen frm demands are aggregaed no ndusr demand. On he conrar, n Lppman and McCardle 997 and s successors, aggregae ndusr demand s allocaed across frms. If he allocaon s deermnsc, hen n each perod here s onl one source of unceran, namel, he oal demand D. We have followed he laer approach snce we beleve represens beer he case of fashonable ems n whch he man unceran s he sze of he marke.e., how well a produc wll sell raher han he nal allocaon across realers. I s worh nong ha our resuls for wo perods T = 2 can be exended o he case when he nal allocaon depends on he curren demand nformaon I. As n he proof of unqueness b Lppman and McCardle 997, we requre he nal allocaon funcon q o be srcl ncreasng n D. I mus be srcl monoone because we need he nverse q o be well defned, and mus be ncreasng because our analss requres ha once he sockng decsons have been made, he realer ha runs ou of sock frs s he same one under all possble demand scenaros n a gven perod. These condons mplcl mpose a posve correlaon beween he orgnal demands of frms and 2. Agan, hs s reasonable when he man source of unceran s he marke sze. Noe ha he correlaon can be anwhere beween 0 o. I s perfec equal o one for he lnear demand splng case, bu can be close o zero as well. 4 Some models n he leraure, e.g., Nagaraan and Raagopalan 2005, assume negave correlaon beween he orgnal demands of he wo frms, whch s an approprae assumpon when he realers are compeng for a fxed pool of cusomers. Anoher class of models assume ha he effecve demand allocaon s proporonal o he ndvdual sockng levels see Cachon 2003. In ha case, demands a he realers are perfecl correlaed, and a realer ha socks more wll ge more. Ths goes back o he prevous dscusson abou he realer beng able o nfluence s effecve demand, bu also has he nconvenence of makng he mul-perod analss unracable. A4 A he begnnng of perod boh realers decde smulaneousl he order-up-o levels, 2 based on he nal sock levels x, x 2 and he demand nformaon I. 4 Consder he followng example: q D = 0 for 0 D a and q 2 D = a when a D, and D s unform n [0, ]. Then Covq D, q 2 D = a2 a 2, V arq D = a3 + 3a and 4 2 V arq 2 D = a3 4 3a. Hence, Corrq D, q 2 3 a a D =, whch s close o 0 for 2 + 3a + 3 a a 0 or a. 9

The saus of boh realers a he begnnng of a gven perod s descrbed b he nal sock levels x, x 2, and he saus of he marke n whch he realers compee s descrbed b he curren demand nformaon I. Then, he sae of he ssem.e, he realers and he marke s gven b he vecor x, x 2, I and we assume ha he realers decde her acons,.e. he order-up-o levels, 2, conngen on he sae. In oher words, he realers pla Markovan sraeges see Fudenberg and Trole 99. A5 The un cos and prce for realer n perod are consan parameers denoed c and p respecvel. The realers are sad o be smmerc f he have he same cos and prce n all perods. We exclude prcng decsons from he model. Ths allows us o focus on he use of nvenor as a compeve lever. Ths assumpon s conssen wh he fac ha fas fashon realers rel less on markdowns, see Ghemawa and Nueno 2003. Noe ha f c p, hen realer wll no order n perod. Therefore, b choosng he approprae cos and prce parameers we can model he suaon n whch frm has a lower orderng cos bu s onl allowed o produce a he begnnng of he sellng season, whereas realer has hgher coss bu can replensh sock ever perod akng advanage of an ncremenal knowledge abou demand f s avalable. Ths represens he case of wo realers ha are asmmerc n erms of reorderng and demand learnng capables. A6 We gnore holdng and los sales penal coss, and here s no mnmum orderng quan. Overall, we am a formulang a parsmonous model. We neglec nvenor holdng coss, as he are less relevan for shor lfe-ccle producs. However, hese coss can also be ncorporaed n he model. The same holds for he los sales penal coss. 5 On he conrar, he QR problem wh mnmum orderng quanes les beond he scope of hs paper. A7 Lefover nvenor can be carred over o he nex perod, and s los a he end of he season. If boh realers sockou n a gven perod, he unsasfed demand s los as well. 5 As a maer of fac, f one wshed o ncorporae los sales penal coss, one would add a erm v ER } + = v ER } v E mn, R } o he revenue; o ncorporae he nvenor holdng cos, one would add a erm h E R } + = h h E mn, R }. Hence, we would replace he revenue mnus he purchasng coss p E mnr, } c x b p + v + h E mnr, } v ER } c + h c x. 0

A salvage value could be easl ncorporaed n he model. Smlarl, a sraghforward exenson allows for backlogged demand o be share beween he frms n a deermnsc wa. Fnall, under assumpons A-A7, he mmedae expeced prof of realer n perod s equal o expeced revenues p E mn, R }}, mnus purchasng coss c x, and he lefover nvenor s equal o R +. 3.2 Sub-Game Perfec Sraeges Snce he overflow demand depends on he nvenor level of he compeor, see Equaon, we mus use game-heorecal ools o analze he replenshmen decson. For exposonal purposes, we nall consder he case when he demand nformaon I s vod, and we begn wh he ermnal perod =. Throughou he paper, when s clear from he conex, we om he argumens of a gven funcon. Frs, le r be frm s unconsraned expeced prof, whch can be expressed as r, = E c + p mn, R }}. 2 Second, f frm knew frm s order-up-o level, hen frm s bes response would come from maxmzng expeced profs akng no accoun s nal sock x and he compeor s acon. In oher words, realer would solve: max x c x + r, 3 Snce he erm c x n Equaon 3 s consan, frm s bes response acuall comes from maxmzng r, subec o x. If he soluon o hs opmzaon problem s unque, hen we can defne he bes-response funcon b x, = argmax x r, } as he opmal sockng level n perod for frm n response o a level from frm, sarng wh a poson of x. Of course, n real, frm does no know wha level of frm wll selec. Thus, we use he noon of Nash equlbrum. In our seng, a Nash equlbrum, f exss, s gven b wo funcons e and e 2 ha mgh depend on he nal sock levels x, x 2, and are such ha b x, e 2 = e and b 2 x 2, e = e 2. Pu dfferenl, no plaer s beer off b unlaerall devang from he equlbrum. We can defne he equlbrum expeced prof π b replacng he equlbrum acons n he obecve funcon of Equaon 3 o oban π x, x = c x + r e x, x, e x, x. 4

If we now consder a mul-perod seng, he noon of Nash equlbrum exends o subgame-perfec equlbrum. An equlbrum s subgame-perfec f nduces a Nash equlbrum as defned above n each subgame of he orgnal game see Fudenberg and Trole 99. In our conex, a subgame corresponds o a game ha s smlar o he orgnal one bu wh one perod less o go.e., he las perod s a subgame of he wo-perod game, and n urn, he laer s a subgame of he hree-perod game, and so on and so forh Therefore, n perod we can consruc he bes response funcons us as we dd for he las perod, bu he onl cavea s ha now he expeced prof s he sum of he mmedae prof plus he fuure prof-o-go, and he laer mus be he equlbrum profs of he game wh one perod less. Formall, he unconsraned expeced prof n perod s gven b r, = E c + p mn, R } +, + +π R R, 5 where π s he equlbrum expeced prof of he subgame ha sars n perod. The bes response funcons are he same as before,.e., b x, = argmax x r, }, 6 and a Nash equlbrum n perod, f exss, s gven b wo funcons e and e 2 ha mgh depend on he nal sock levels x, x 2, and are such ha b x, e 2 = e and b 2 x 2, e = e 2. In order o close he loop, f we wan o verf he exsence, and hen compue a subgameperfec equlbrum n perod +, we would need he expeced equlbrum profs n perod, whch s gven b π x, x = c x + r e x, x, e x, x, 7 and he procedure repeas self b replacng wh + n Equaon 5. 6 The prevous defnons were gven for he case when he demand nformaon I s vod. The laer would be vald f he demand across perods were ndependen. However, f he realers can use curren or pas demand nformaon o predc fuure demand, hen her acons, and consequenl he compeve equlbrum, wll be conngen on he nformaon ha s acuall avalable. Therefore, n ha case, we replace he subndex wh I n r, b, e, and π, and all he expecaons are condonal on I. For nsance, nsead of r, we wre r I,, and he expecaon on he rgh hand sde of Equaon 5 s condonal on I. 6 In game heor, hs procedure s known as backwards nducon see Fudenberg and Trole 99. 2

4. Exsence and Unqueness of Equlbrum We now presen he srucural resuls of he paper. Our goal n hs secon s o prove he exsence of a unque pure-sraeg subgame-perfec equlbrum for he four cases menoned n Table. These resul have heorecal value, bu also allow us o undersand he benefs of QR b compung comparave sacs of he unque equlbrum see 5. We sar b consderng he sngle-perod problem n 4.. Then, n 4.2 we consder he wo-perod case and we sud he wo mos ced QR models n leraure. Namel, he demand sgnal and he md-season replenshmen models. 7 The more general case wh wo or more perods s suded n 4.3. There, we analze he suaon when one compeor s passve and when boh realers are smmerc. Noe ha n Table he md-season replenshmen and he passve compeor cases are grouped ogeher because he have smlar proofs. 4. Sngle-Perod Case T = The sngle-perod problem s an essenal buldng block n our model. To smplf he exposon we om he dependence on he demand nformaon vecor I, bu all he dscusson hroughou hs secon remans vald f we replace he subndex = wh wh I, and all he expecaons are condonal on I. We frs consder he unconsraned problem. Tha s, he verson of he problem n whch he nal nvenor levels are equal o zero. From Equaon 2, s clear ha r s concave n, for all. Thus, he opmal nvenor polc s a base-sock polc wh arge level s, whch can be obaned from he frs-order condons Pr R s = c /p. Solvng he laer elds s NV = when NV q NV 8 oherwse, where NV F c /p corresponds o he newsvendor sockng quan of frm when s a monopols,.e., when = 0. In Fgure we plo s and s. For smplc, we om he subndex =, and frm s such ha c /p < c /p. The base-sock funcons s and s nersec onl once, whch means ha n he unconsraned compeve game here exss a unque Nash equlbrum, whch we denoe E = E, E. The shaded regons I-IV are used n he proofs provded n he echncal appendx. 7 In Cachon and Terwesch 2005 hese wo models are defned n erms of he reacve capac, and are referred o as lmed and unlmed bu expensve respecvel. 3

If now we allow he nal nvenor levels o be non-zero.e., we consder he consraned problem, gven he convex of r, follows ha frm s bes response s b x, = maxx, s }. Therefore, n he consraned compeve game, a graph of he bes responses would look us as Fgure, excep ha he vercal and horzonal sreches would move rgh and down respecvel. Hence, a graphcal argumen s enough o prove our frs resul. Fgure : Unconsraned sngle-perod base-sock funcons. Theorem For all x, x 2, here exss a unque Nash equlbrum e x, x 2, e 2 x 2, x of he sockng game. In addon, we can characerze e, e 2 as follows. Whou loss of general, assume ha c e 2 x 2, x = max x 2, q 2 p c2 p 2 frm has a hgher gross margn. Then NV 2 } and e x, x 2 = max x, q NV, NV e 2 x 2, x }. We can see from Theorem ha he equlbrum sraeg of he lower-margn frm s ndependen of he compeor s nal nvenor level x. Indeed, when he nal nvenor levels are zero, hen he hgher cos drves he lower-margn frm o gnore he overflow from 4

he oher compan. The reverse s no rue: he hgher-margn equlbrum sraeg ma depend on x 2. Theorem 3 of Lppman and McCardle 997 proves ha he unconsraned compeve game wh smmerc realers has a unque Nash equlbrum. The resul requres he nal allocaon funcons o be deermnsc and srcl ncreasng, us as n our seng. Theorem n hs paper exends he resul b Lppman and McCardle n he sense ha we consder he consraned game.e., he nal nvenor levels can be non-zero and we allow for asmmerc realers.e., he can face dfferen coss and prces. From a echncal sandpon, he unque equlbrum n he sngle-perod problem follows from he fac ha he unconsraned expeced prof r s concave n frm s acon. In order o prove he exsence of a unque pure-sraeg subgame-perfec equlbrum n he wo-perod game, we wll need o show ha r2 s srcl quas-concave n 2. For ha, we frs need o show ha π, he equlbrum expeced prof n he sngle-perod problem, s concave n frm s nal sock level x. Noe ha we have o consder he equlbrum expeced prof because, b he defnon of subgame-perfec, he realers assume ha n he nex perod a Nash equlbrum wll be plaed, gven an nal sae x, x 2, I we refer he reader back o he dscusson a he end of 3.2. The followng proposon provdes he heorecal resul ha we wll need as a buldng block n he nex secon. Proposon For all I, he expeced equlbrum prof π I x, x s concave n x, for all x. 4.2 Two-Perod Case T = 2 We now look a he wo-perod case whch s arguabl he mos mporan one snce QR models pcall onl have wo perods. In fac, he wo models we consder seem o concenrae mos of he aenon n he leraure see, for nsance, Cachon and Terwesch 2005. Fgure 2 shows a schemac descrpon of boh models. 4.2. Demand Sgnal The frs QR model we consder s based on he one suded for a sngle frm n Ier and Bergen 997. Smlar sequence of evens have been used n several oher papers see, for nsance, Cachon and Swnne 2007. The plannng horzon s dvded n wo perods. The las one represens he real sellng season, whereas he frs one represens a perod durng whch a demand sgnal s revealed. The laer could smpl represen daa ha s colleced rgh before he season sars for example, n fashon shows, mock sores, focus groups, or 5

Fgure 2: QR models, demand sgnal op and md-season replenshmen boom. b consulng expers. We assume ha he demand sgnal s nformave, meanng ha s correlaed wh he acual demand durng he season. Oherwse, hs model reduces o he sngle-perod problem suded n he prevous secon. A QR realer can place orders before and afer observng he demand sgnal, whereas a radonal realer, due o longer lead mes, can onl place a sngle order before he addonal demand nformaon becomes avalable. The sequence of evens for a QR realer are depced n Fgure 2 op melne. Gven ha here s no demand realzaon n he nal perod R2 = R 2 = 0, he unconsraned expeced prof r2 reduces o r 2 2, 2 = c 2 2 + E π I 2, 2}, where he expecaon s wh respec o he a pror dsrbuon of I. Noe ha r 2 2, 2 s concave n 2, for an 2 from Proposon. Therefore, he exsence of a pure-sraeg sub-game perfec Nash equlbrum n he unconsraned compeve game s guaraneed b Theorem.2 n Fudenberg and Trole 99. 8 r 2, frm s bes response s a base-sock polc s 2 2. Moreover, also due o he concav of Therefore, us as n he sngleperod model, he bes-response funcon n he consraned game s equal o b 2x 2, 2 = argmax 2 x 2 r 2 2, 2 } = max x 2, s 2 2}. The followng Theorem provdes condons 8 The heorem acuall requres ha he sraeg space s compac. Snce he realers would never order an nfne sock, s alwas possble o resrc her acons o a compac se. 6

under whch he equlbrum s unque. 9 The proof s omed snce follows drecl from Proposon and he dervaves compued n Table 3. Afer sang and dscussng he heorem we provde an example ha shows an applcaon of hs resul. Theorem 2 If c 2 c for =, 2, hen he sockng game wh a demand sgnal has a unque subgame-perfec equlbrum. In ha equlbrum, boh realers pla pure-sraeges. Three echncal observaons abou Theorem 2 are worh nong. Frs, when c 2 = c, he equlbrum exss n fac, no orderng n he nal perod would be an equlbrum, bu n general mgh no be unque hough he profs acheved are he same under an equlbrum. 0 Second, he heorem rules ou he exsence of a mxed-sraeg equlbrum, snce n he unque equlbrum he realers pla pure-sraeges. Thrd, per Assumpon A4, he sraeges plaed b he realers mus be Markovan. Hence, wha we acuall prove s ha here exss a unque Markovan perfec equlbrum see 3 n Fudenberg and Trole 99. However, snce he sae x, x 2, I conans all he relevan nformaon from he pas.e., s suffcen, s no hard o see ha an sraeg ha nduces a sub-game perfec equlbrum mus be Markovan, or a leas equvalen o a Markovan sraeg. Ths observaon usfes he clam ha here s a unque subgame-perfec equlbrum. A smlar usfcaon s gven n Hall and Poreus 2000. Example Ier and Bergen 997: The wo-perod demand sgnal case allows us o model he compeon beween a radonal realer.e., one wh ver long lead mes and a QR realer ha s modeled as n Ier and Bergen 997. In ha paper, demand durng he real sellng season s assumed o be normall dsrbued,.e. D Nθ, σ 2. The varance σ 2 s assumed o be known, whereas he average sze of he marke θ s unceran. Informaon abou θ n he nal perod s modeled as a normal dsrbuon wh mean µ and varance τ 2. Thus, a me = 2, he predcon of season demand s normall dsrbued wh mean µ and varance σ 2 +τ 2. Then he demand sgnal d s realzed, and he QR realer performs a Baesan updae of s belef regardng θ. In oher words, we have ha I = d}, and D I N µd, σ 2 + /ρ, where µd = σ2 µ + τ 2 d σ 2 + τ 2 and ρ = σ 2 + τ 2. 9 When he equlbrum s no unque, s 22 and b 2x 2, 2 are acuall correspondences raher han funcons. 0 Noce ha, f c 2 > c, hen he equlbrum s o no order anhng n he nal perod. On he conrar, f c 2 < c, hen a posve amoun s ordered n = 2. To be precse, here s a Markovan sraeg ha plas he exac same acons han he non-markovan one, for all possble hsores. 7

Noe ha /ρ < τ 2. Therefore, afer he realzaon of he demand sgnal d s observed, he QR realer has a more accurae predcon of season demand. The radonal realer canno make use of he demand sgnal because of long lead mes. In oher words, canno place a second order afer he demand sgnal s realzed. Ths s ncorporaed n our model b leng c = p here we assume ha ndex represens he radonal realer. As long as s cheaper o order n he nal perod for boh realers, he condons of Theorem 2 hold, and he compeon beween he QR response and he radonal realer has a unque subgame-perfec equlbrum. 4.2.2 Md-Season Replenshmen In he prevous QR model, he demand sgnal does no deplee sock. Tha s, he acquson of addonal demand nformaon n he nal perod does no affec he nvenor ha s carred over o he fnal perod. Ths smplfes he analss and allows us o prove he exsence of a unque equlbrum under farl general condons see Table. In he curren secon, we consder a second QR model ha dffers from he prevous one n few suble bu fundamenal aspecs. Specfcall, he sellng season comprses boh perods, he acual sales ha occur n he nal perod pla he role ha he demand sgnal prevousl had, and he procuremen n he fnal perod comes o replensh he nvenor ha has been depleed hence he name for hs case. The respecve sequence of evens s depced n he boom of Fgure 2. An example ha would f hs QR model s he Spor Obermeer case Fsher and Raman 996, where 20% of nal sales provdes an excellen esmaon of he remanng 80%. As before, he unconsraned expeced prof r2 2, 2 n he nal perod = 2 s gven b Equaon 5. However, now he prof-o-go π I s evaluaed n he remanng nvenor, and r2 s no longer guaraneed o be concave. Forunael, we are able o show ha s quas-concave under ceran condons o be nroduced nex, and hence, he opmal polc s sll a base-sock polc dependen on 2. Proposon 2 Assume ha for =, 2, c c 2 and p p 2; D 2 has nfne suppor and a log-concave p.d.f.,.e., logf 2 d s concave n d; I = D 2 } and D I = kd 2 + ɛ, where k 0 and ɛ s a random varable ndependen of D 2 wh p.d.f. g, such ha, for all x, for =, 2, 8

v f k > 0, hen q d = α d for =, 2. max 0, f } } 2 q f 2 x max 0, g 2 g q x ; 9 Then, r 2 2, 2 s quas-concave for all 2, and he consraned bes response s b 2x 2, 2 = max x 2, s 2 2 }, where s 2 2 s he unconsraned base-sock level, whch s unque. The frs condon n Proposon 2 requres ha he margns do no ncrease over me. Ths would be he case f he prce s fxed and he md-season replenshmen s more expensve han he nal procuremen. The second condon requres demand D 2 o be log-concave wh an nfne suppor. 2 The former s needed o generae bounds n he proof, and he laer s used o guaranee a unque maxmum. The hrd condon specfes he dependenc beween D and D 2 ha s allowed. Noce ha he nequal 9 s sasfed f f 2 has a decreasng p.d.f., or f s no larger han g n he lkelhood rao order. The laer s pcall he case when he nal perod represens less han half of he oal season. Fnall, he las condon v requres for echncal reasons a lnear splng rule n he fnal perod whenever D and D 2 are no ndependen. The proof of Proposon 2 nvolves several clams ha are saed and proved n he echncal appendx, bu he cenral dea s o show ha he followng nequal holds 2 r2 2 < φ 2 2 q2 2 r 2 2, where φ 2 = max 0, f } 2. 0 f 2 Noe ha quas-concav follows drecl from 0, snce an crcal pon s a maxmum, and herefore, he funcon r 2 s unmodal. We can now sae he man resul of hs secon. Theorem 3 If he condons -v of Proposon 2 are sasfed, hen he sockng game wh md-season replenshmen has a unque pure-sraeg subgame-perfec equlbrum. As n Theorem 2 for he demand sgnal case, Theorem 3 shows exsence and unqueness of a pure-sraeg subgame-perfec equlbrum. However, here are a some dfferences. Frs, Theorem 3 requres a few more condons han Theorem 2 see Table for a comparson. Second, snce n he md-season replenshmen case r 2 s quas-concave raher han concave, we canno rule ou he exsence of a mxed equlbrum. Anoher consequence s ha he equlbrum prof π 2 defned n Equaon 7 s no concave eher, and herefore, we are no able o exend Theorem 3 o a larger number of perods. 2 To be precse, he suppor mus be eher he real lne or an nerval of he pe [a, +. 9

Despe he addonal condons requred n Theorem 3, here are several neresng cases for whch he hold. Two of hem are gven n he nex corollares. Corollar shows he smples applcaon of Theorem 3 b assumng..d. demand. On conrar, Corollar 2 shows an applcaon wh demand ha s correlaed across perods. We hen use he laer n an example ha resembles he QR model n he Spor Obermeer case Fsher and Raman 996. Corollar Independen demands Assume ha c c 2, p p 2, and q = q 2 for =, 2. If he demands D 2, D are..d. and D 2 s log-concave wh nfne suppor, hen he sockng game wh md-season replenshmen has a unque pure-sraeg subgame-perfec equlbrum. Corollar 2 Correlaed demands Assume ha c c 2, p p 2, and q d = q 2d = α d, for =, 2. Le ɛ 2, ɛ be wo ndependen random varables such ha D 2 = ɛ 2 and D I = kd 2 + ɛ, wh k > 0 hus, ρ CorrD 2, D = k V ard 2 /V ard > 0. Furhermore, le eher ɛ 2, ɛ follow normal dsrbuons wh parameers µ 2, σ 2 and µ, σ respecvel and µ 2 µ and σ 2 σ ; or ɛ 2, ɛ follow runcaed normal dsrbuons wh parameers µ 2, σ 2 and µ, σ respecvel and µ 2 µ and σ 2 µ 2 σ µ ; or ɛ 2, ɛ follow gamma dsrbuons wh parameers a 2, θ 2 and a, θ respecvel and θ 2 θ and a 2 a ; or ɛ 2, ɛ follow exponenal dsrbuons. In he four cases above, he sockng game wh md-season replenshmen has a unque pure-sraeg subgame-perfec equlbrum. Example 2 Fsher and Raman 996: Consder he case when he demand vecor D 2, D follows a mulvarae normal wh margnal dsrbuons D Nµ, σ and covarance CovD 2, D = ρσ 2 σ, wh ρ 0. Then we have ha, he demand n he las perod condonal on he demand realzaon n he nal perod, s gven b D I = kd 2 + ɛ wh k = ρσ /σ 2, and ɛ s normall dsrbued wh parameers µ kµ 2, σ ρ2 and s ndependen of D 2 = ɛ 2. Noce ha, f here s posve correlaon ρ > 0, hen V ard I = σ 2 ρ 2 < σ 2 = V ard. In oher words, as n Example, he updaed 20

forecas D I s more accurae han he uncondonal predcon of D. 3 I can be shown ha he condon on he normal dsrbuons n Corollar 2, n hs case, reduces o µ } 2 µ 2 < µ and σ 2 σ max ρ2, ρ. µ µ 2 Hence, f he laer holds, hen Theorem 3 apples. In parcular, hs would be he case f a QR realer can place an order afer observng a small fracon e.g., 20% of he oal season demand so ha µ 2 µ, and hese earl sales provde relevan nformaon abou wha should be expeced n he remander of he season.e., he correlaon ρ s hgh. 4.3 Two or More Perods T 2 Mos QR models n he leraure onl consder wo perods, whch also represens he acual suaon of a large number of realers n pracce, who have a mos wo procuremen opporunes n a gven season. Tha usfes dedcang an enre secon o sud he wo-perod case c.f. 4.2. Now we urn o he general case wh wo or more perods, whch agan s parl movaed b fas fashon realers lke Zara who have he abl o replensh sock several mes durng he sellng season. However, as was menoned before, we found ha exendng Theorem 3 o more han wo perods was no drecl feasble wh he mehodolog used o prove Proposon 2. 4 Indeed, he equlbrum prof π, ma no be concave nor quas-concave n for 2. Ths can be worked around f we can prove ha he compeor s bes response b s consan whn a ceran range. In hs secon we show wo cases when ha s possble. Namel, when one of he realers does no ac sraegcall, and when he are smmerc. 4.3. Invenor conrol when one realer s passve We consder here he specal suaon where one realer s passve and canno place orders durng he season: we assume ha realer 2 s such ha mnc 2 } maxp 2 } for all,.e., s oo expensve o acqure nvenor and hus b 2 x 2, = x 2. In oher words, compeon here does no reall ake place as a game, bu nsead realer smpl responds o he nvenor level of realer 2, who s passve and canno purchase nvenor besdes s nal quan x 2 T. Ths suaon models he exsng compeon beween a fashon-leader,.e., a realer ha desgns and produces earl, whou even consderng he compeon, and a fashon-follower,.e., a realer ha copes he fashon-leader s desgns and s able o 3 Under he condons of Proposon 2, alwas holds ha V ard = k 2 V ard 2 + V arɛ V arɛ = V ard I. 4 The demand sgnal case can be exended o have an arbrar number of perods before he season begns, bu essenall reduces o he wo-perod case. 2

produce he ems quckl hrough QR. Ths could represen he case of Zara, a ver quck fashon follower, accordng o Ghemawa and Nueno 2003, p.2. Theorem 4 Assume ha c T... c and p T... p, and mn c 2 } maxp 2 }; demand s ndependen across perods, and for = T,..., 2, D s log-concave and has nfne suppor; for all d, f q d max 0, f } q f f d. Then, he sockng game has a unque pure-sraeg subgame-perfec equlbrum where, n each perod, frm follows a base-sock polc wh order-up-o level s x 2, and frm 2 never orders. When a realer canno order durng he season and sars wh an exogenousl gven nvenor level, s compeor faces a mul-perod nvenor problem. The heorem shows ha hs non-rval problem as curren nvenor decsons nfluence fuure demands s well-behaved and ha a base-sock polc opmal. Of course, he base-sock level depends on he compeor s nvenor level. 4.3.2 Smmerc realers When addonal condons are placed on he realers and he demand srucure, sronger resuls can be derved. Ths secon provdes a general resul for an arbrar number of perods, when he realers are smmerc,.e., boh realers have dencal coss and prces n ever perod. Noce ha our defnon of smmer sll allows he realers o a have dfferen nal nvenores. Here, he generalzaon of he model o T 2 can be obaned wh sandard mehods, assumng ha he demand splng rule s lnear n all perods. Theorem 5 Assume ha, gven T, for all = T,...,, coss and prces are dencal for boh frms,.e., c = c 2 and p = p 2. In addon, assume ha qd I = α D I for all = T,...,. Then, he T -perod sockng game has a unque pure-sraeg subgame-perfec equlbrum, for all I T. In hs equlbrum, a perod, e I s characerzed b e I x, x = maxx, α s I }, 2 22

where s I s he monopol base-sock level,.e., he soluon o he nvenor problem for a sngle frm ha receves D I per perod. As saed n he heorem, he equlbrum level of realer does onl depend on s own nvenor level x and s ndependen of he compeor s nvenor level x. To be more precse, we show ha realer has a domnan sraeg ha nvolves he monopol sockng quan s I, defned as follows. Consder he followng dnamc program, wh c = c = c 2 and p = p = p 2 : U 0 I0 x 0 U I x = c x + max x c + p E mn, D I } + E U I D I + } 3 Snce hs s a sandard nvenor problem, s eas o see ha he opmal polc for s o se = maxx, s I }, where s I problem. Ths s he quan used n Theorem 5. s he order-up-o level of he respecve unconsraned The proof mehodolog s relavel sandard, and uses he same lnes of Lppman and McCardle 997. Essenall, we show b nducon ha r I, s concave n for all and submodular, whch elds ha he opmal replenshmen polc s a base-sock polc. Also, a ke par of he proof nvolves ha when, realer never receves α α overflows from, n perod or n fuure perods,...,. As a resul, he bes response b I s ndependen of, and equal o α s I, whch elds he equlbrum srucure presened n Theorem 5. Fgure 3 llusraes he shape of he bes-response funcons descrbed b he heorem. Ths plo s for T = 2, c = 0.6, p =, D unform [0,], for = 2, and =, 2, x 2 = x 2 2 = 0 and α = 30%. We observe ha, ndeed, b 2 s fla n 2, when 2 α s I. We hus compleel characerze he mul-perod equlbrum n he case of smmerc frms n erms of cos and prce, no necessarl n erms of marke share. Noe ha s allowed ha he frms sar wh an nal nvenor. Ths conrass wh he work b Avsar and Bakal-Gürso 2002, who requre ha he nal nvenor levels of realers are below her equlbrum saonar base-sock levels. Also noe ha s sll possble ha spll-overs beween realers occur, even hough he equlbrum quan of each realer s ndependen of he compeor s nvenor level, see Equaon 2. Noce ha, when boh frms sar wh zero nvenor a = T, he aggregae ndusr nvenor s equal o α s T IT + α 2 s T IT = s T IT, whch s also he ndusr nvenor arge level under a monopol,.e., when frms and 2 merge. Our resul hus exends Theorem 23

0.8 Sock of realer 2: 2 2 0.6 0.4 s 2 2 2 0.2 s 2 2 2 0 0 0.2 0.4 0.6 0.8 Sock of realer : 2 Fgure 3: Shape of he bes-response funcons b 2 n he case of smmerc frms. 3 of Lppman and McCardle 997 o a mul-perod seng wh demand learnng. Unforunael, when coss are asmmerc, urns ou ha he bes response b I s never ndependen of, whch prevens he use of some of he argumens of he proof. Ineresngl, he same sor of proper s used n Nagaraan and Raagopalan 2005 b requrng ha he newsbo rao s larger han 0.5. 5. Quck vs. Slow Response Compeon In hs secon, we provde an exensve numercal sud on he wo-perod T = 2 nvenor game. Whle smple, he wo-perod case capures mos of he neresng dnamcs arsng as one moves from a sngle-perod game o a mul-perod one. In addon, Theorems 2 and 3 provde he heorecal resuls ha guaranee he exsence and unqueness of equlbrum. In hs secon we compue such equlbrum and perform comparave sacs,.e. we sud how he equlbrum depends on he ke parameers of he model. We do hs numercall snce analcal comparave sacs n our model are no sraghforward. We are neresed n quanfng he advanages of a QR realer over a SR slow response compeor, and compare nvenor levels and profs of such compeve suaon when compared o he radonal compeon beween wo SR realers we assume ha boh sar wh zero nvenor. For hs purpose, we consder hree man suaons, all fallng whn our generc model presened before c.f. 3. The frs case, ha we ake as a benchmark, consders wo SR realers, n he sense ha he can onl place orders before he sar of he sellng season, a a cos c SR. Thus, he are no able o place an order a me =. In hs suaon, he compeon corresponds o he sngle perod equlbrum 24

Fgure 4: Three marke confguraons: SR vs. SR compeon op, QR vs. SR compeon mddle and QR vs. QR compeon boom. denfed n Lppman and McCardle 997. The second case, whch s he focus of he paper, consders he asmmerc compeon beween a SR and a QR realer. The SR realer s dencal o he ones of he frs case. On he conrar, he fas realer s characerzed b he possbl o place orders n boh perods, a a cos c QR. We allow o have he same or possbl hgher orderng cos han he slow realer.e., c QR c SR, as n real. Fnall, he hrd case consders wo smmerc QR realers, whch wll be used o evaluae how much of he advanage of beng QR arses from he asmmeres beween realers. In our smulaons, we use dencal prces n each perod p = p 2 =, for =, 2. Therefore, f frm s a SR realer, hen c 2 = c SR and c = p =, whereas f frm s a QR realer, hen c 2 = c = c QR. Fgure 4 summarzes he hree cases ha we analze hroughou our numercal sud. Our demand model follows he srucure presened n Proposon 2, namel, D I = kd 2 + ɛ. Ths allows realers o learn from he realzaon of D 2 and mprove he forecas of he las perod demand D. In he demand sgnal case, D 2 s he addonal marke nformaon ganed n he nal perod, whereas n he md-season replenshmen case, D 2 s he acual demand n = 2. The parameer k deermnes he correlaon beween D 2 and D. Specfcall, ρ = CorrD 2, D = k k2 +. 25

Noe ha, n he fgures below, we plo our resuls drecl as a funcon of he demand correlaon nsead of k. For smplc, and o avod negave demand, we consder ha D 2 and ɛ are dencall dsrbued, and follow gamma dsrbuons wh mean µ = and sandard devaon σ. Fnall, we use a lnear splng rule qd = q2d = α d wh a 50% marke share for each realer.e., α = α = 0.5, for =, 2. 5. Equlbrum Invenor and Spll-Overs We sar b comparng he nal equlbrum nvenor levels beween he benchmark SR vs. SR and he asmmerc QR vs. SR scenaros, when boh realers sar wh zero nvenor,.e., x 2 = x 2 2 = 0. Of course, when boh frms are SR realers, all he nvenor s ordered a he begnnng of he season a = 2. In he QR vs. SR case, however, he QR realer ma place an addonal order a =. Noce ha n hs secon we solel look a he md-season replenshmen case. We do no consder here he demand sgnal case because, snce c 2 = c, he equlbrum n ha game mgh no be unque see he dscusson afer Theorem 2. In Fgure 5 we show he bes-response funcon of each realer n he benchmark lef and n asmmerc case rgh, when here s no demand correlaon across perods. We observe, frs, ha he SR vs. SR case corresponds exacl o he sngle-perod model. The QR vs. SR case fgure llusraes a qualavel dfferen behavor: neresngl, he besresponse funcon of he QR realer s no necessarl decreasng n he nvenor level of he compeor, as s alwas he case n sngle-perod models. We found hs behavor n mos of he smulaons performed. Furhermore we observe ha he shape of boh realers s bes-response funcons s shfed down n he asmmerc case compared o he benchmark. Anoher neresng observaon s ha he bes-response of he QR realer s decreasng, hen ncreasng, and evenuall becomes consan. The laer s wha should be expeced. In fac, f he SR realer has an exremel large quan of nvenor, hen he bes-response of he QR realer s o gnore he compeor snce wll no face spll-over demand. The same occurs when he compeor s passve and has a large amoun of nal sock c.f. 4.3.. Nex, n Fgure 6 we sud how he equlbrum nvenor levels n he nal perod change when correlaon s nroduced and he cos of he QR realer c QR ncreases. The fgure shows he equlbrum values for wo suaons:..d. demand and hence no forecas updang, and correlaed demand wh ρ = 0.7 n boh suaon σ = 0.3. Noce ha he values have been normalzed b he oal expeced orgnal demand αed 2 + D }. We depc he SR vs. SR equlbrum levels damond and sar n he rgh boom, and 26

.6 Bes response funcon of SR realer Bes response funcon of SR realer 2.6 Bes response funcon of SR realer Bes response funcon of QR realer.4.4 Sock SR realer 2.2 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8.2.4.6 Sock SR realer Sock SR realer.2 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8.2.4.6 Sock QR realer Fgure 5: Bes-response funcons a = 2 for he benchmark case lef and he asmmerc case rgh. Here we use k = 0, σ = 0.3 and c QR = c SR = 0.5. he curves of he QR vs. SR equlbrum levels for a vare of parameer values c QR from 0.5 boom par of he curves o 0.9 op par of he curves. The cos c SR of he SR realers remans fxed a 0.5. Frs, we observe ha he oal nal nvenor placed n he asmmerc case QR vs. SR decreases wh respec o he benchmark case SR vs. SR, and he decrease s more pronounced when demand s correlaed and/or when he orderng cos of he QR realer s hgher. Inuvel, he ndusr nal nvenor,.e., he sum of boh realers nvenor, should be lower han n he SR vs. SR case because he QR realer can pospone par of s order, and wll do even more so when he value of performng forecas updaes s hgher.e., when ρ s larger. Second, we observe ha no onl he nvenor placed b he QR realer s alwas lower han n he benchmark case, bu he nvenor level of he SR realer s hgher, even hough he oal ndusr nvenor level decreases. Thus, he SR realer akes advanage of compeng wh a QR realer whch carres lower nvenor b placng hgher nvenor levels, and hence n he nal perod capures hgher sales n expecaon. Fnall, o conclude hs secon, we nvesgae he magnude of spll-overs beween a QR and a SR realer. As poned ou before, he SR realer benefs from he decrease n nvenor of he QR realer and receves a spll-over n he nal perod, = 2. However, as Fgure 7 llusraes, he spll-over n he fnal perod, = goes from he SR realer o he QR realer and s sgnfcanl larger. For hgher demand varabl and demand correlaon, he average spll-over ma be as hgh as 0% of oal demand sales. I s worh nong ha n he demand sgnal case, snce he demand realzaon and hence, compeon onl akes place n he las perod, we generall observed consderabl lower spll-overs. 27

40% c QR =0.9 Equlbrum nvenor of realer 2 SR, as % of oal orgnal demand 20% 00% QR vs. SR case,..d. demand SR vs. SR case,..d. demand QR vs. SR case, ρ=0.7 SR vs. SR case, ρ=0.7 c QR =0.5 80% 40% 60% 80% 00% Equlbrum nvenor of realer QR or SR, as % of oal orgnal demand Fgure 6: Inal equlbrum nvenor levels for he benchmark and asmmerc cases, wh and whou demand correlaon values normalzed b αed 2 + D }. 5.2 Prof Comparson Afer analzng he dfferences n nvenor levels n equlbrum, we sud he correspondng realers profs for he demand sgnal and md-season replenshmen cases. Fgure 8 llusraes he ncrease of he realers equlbrum expeced profs as a funcon of he demand correlaon across me. The fgure compares hese profs o he benchmark case, SR vs. SR. In each graph, hree curves appear: a he op, he profs of a QR realer compeng agans a SR realer; n he mddle, he profs of a QR realer compeng agans a QR realer; and a he boom, he profs of a SR realer compeng agans a QR realer. The wo graphs on op correspond o he demand sgnal case, and he wo n he boom correspond o he md-season replenshmen. Lkewse, he graphs a he lef have σ = 0.3 and he wo a he rgh have σ = 0.6. In erms of cos we use c QR = c SR = 0.5. 5 Several observaons can be made from lookng a an of he graphs n Fgure 8. Frs, snce he boom curve s posve, mples ha a SR realer prefers o have a QR compeor nuvel, he former gans from he spll-over demand ha occurs n he nal perod. Second, comparng he boom and he mddle curves shows ha a SR realer compeng agans a QR opponen would raher be QR self. Ths confrms ha flexbl pas off. Thrd, comparng he mddle and he op curves shows ha a QR realer would raher have a SR compeor. Thus, par of he compeve advanage of QR comes from he replenshmen agl asmmer. The prevous remarks are summarzed n he followng 5 Agan, he demand sgnal case mgh have mulple equlbra, bu he all acheve he same profs c.f. 4.2.. 28

2% Perod Perod 2 2% Perod Perod 2 Ne average spll over from SR o QR realer 8% 4% 0% σ=0.3 Ne average spll over from SR o QR realer 8% 4% 0% σ=0.6 4% 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Correlaon ρ 4% 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Correlaon ρ Fgure 7: Ne average spll-over from he SR realer o he QR realer values normalzed b αed 2 + D }, for σ = 0.3 and σ = 0.6 c QR = c SR = 0.5 n boh graphs. preference orderng, where represens a suaon ha s preferred b he frs realer: QR vs. SR QR vs. QR SR vs. QR SR vs. SR. 4 Fgure 8 also shows ha he equlbrum prof ncrease due o he QR capabl s larger when he correlaon across perods and/or he demand unceran s larger. In he mdseason replenshmen case, can generae up o 50% hgher expeced profs. In he demand sgnal case, he ncrease s lower, especall for low levels of demand correlaon. Ths s wha should be expeced snce here s sgnfcanl less demand spll-over see he commen a he end of 5., and ndcaes ha n he demand sgnal case he benef of QR s mosl explaned b reduced under and oversock coss raher han capurng addonal demand ha splls over from he compeor. The resuls of Fgure 8 analzed he ncrease n equlbrum prof when c QR = c SR. However, as c QR ncreases, he QR prof clearl decreases. We can hus compue he breakeven cos for whch a realer would be ndfferen beween beng QR or SR. Ths resul s depced n Fgure 9 as he percenage ncrease over c SR ha leaves a realer ndfferen whle compeng agans a SR opponen. We plo he break-even coss as a funcon of he demand correlaon, and for several values of c SR = 0.3, 0.5, 0.7. Agan, he op and boom graphs correspond o he demand sgnal and md-season replenshmen cases respecvel, and he lef and rgh graphs have σ = 0.3 and σ = 0.6 respecvel. Ineresngl, whle he SR realer alwas gans when s compeor moves from SR o QR, hs s no alwas rue for he QR realer. I all depends on he cos ncrease assocaed 29

Prof ncrease wh respec o SR vs. SR 50% 40% 30% 20% 0% QR vs. SR QR vs. QR SR vs. QR σ=0.3 0% 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Correlaon ρ Prof ncrease wh respec o SR vs. SR 50% 40% 30% 20% 0% QR vs. SR QR vs. QR SR vs. QR σ=0.6 0% 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Correlaon ρ Prof ncrease wh respec o SR vs. SR 50% 40% 30% 20% 0% QR vs. SR QR vs. QR SR vs. QR σ=0.3 0% 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Correlaon ρ Prof ncrease wh respec o SR vs. SR 50% 40% 30% 20% 0% QR vs. SR QR vs. QR SR vs. QR σ=0.6 0% 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Correlaon ρ Fgure 8: Increase n equlbrum prof compared o he benchmark scenaro for he demand sgnal op and he md-season replenshmen boom cases, wh c QR = c SR = 0.5. wh mplemenng QR. For fas fashon realers, he leraure esmaes hs cos ncrease o be 5-20% hgher han radonal SR frms producng n Asa see Ghemawa and Nueno 2003, p.. As Fgure 9 shows, hs s nsuffcen o usf QR for small demand varabl and/or small demand correlaon across me. For example, n he md-season replenshmen model boom lef, he break-even cos s below 5% for ρ < 0.5 and σ = 0.3. However, can be much hgher for ρ 0.9 and σ = 0.6. Thus, our model shows ha for small demand varabl and correlaons, a realer would raher prefer lower flexbl and lower cos,.e., beng SR. Ths apples for basc ems, e.g., whe T-shrs. On he oher hand, for hgh demand varabl and correlaons, a realer s beer off havng hgher producon cos bu a faser response. Ths maches fashon goods, as hose pcall found n a Zara sore. In oher words, n a compeve seng, our resuls confrm he fundamenal rule ha he suppl chan n parcular, s coss and flexbl should mach he pe of produc. Funconal producs, such as sandard garmens, should have an effcen.e, low cos, and usuall less flexble suppl chan, whereas nnovave produc, such as rend ems, should 30

Break even cos ncrease 50% 40% 30% 20% 0% c SR =0.3 c SR =0.5 c SR =0.7 σ=0.3 Break even cos ncrease 50% 40% 30% 20% 0% c SR =0.3 c SR =0.5 c SR =0.7 σ=0.6 0% 0 0.2 0.4 0.6 0.8 Correlaon ρ 0% 0 0.2 0.4 0.6 0.8 Correlaon ρ Break even cos ncrease 50% 40% 30% 20% 0% c SR =0.3 c SR =0.5 c SR =0.7 σ=0.3 Break even cos ncrease 50% 40% 30% 20% 0% c SR =0.3 c SR =0.5 c SR =0.7 σ=0.6 0% 0 0.2 0.4 0.6 0.8 Correlaon ρ 0% 0 0.2 0.4 0.6 0.8 Correlaon ρ Fgure 9: Break-even cos ha makes a realer wh a SR opponen ndfferen beween beng QR or SR, for he demand sgnal op and md-season replenshmen boom cases. have a suppl chan ha s responsve, whch pcall requres excess buffer capac, and herefore mples hgher operaonal coss see Fsher 997. 6. Conclusons and Fuure Research In hs paper, we formulaed a mul-perod fne horzon nvenor compeon model for wo realers sellng subsuable ems. The model can be used o analze he mpac of asmmerc producon coss and orderng flexbl on he compeve oucome, and specfcall on realer nvenor levels and profs. Tha s he case when one of he frms has a lower producon orderng cos bu can onl produce a he begnnng of he sellng season, whereas he second frm has hgher coss bu can replensh sock durng he plannng horzon, akng advanage of an addonal demand nformaon ha mgh become avalable. We vsualze he problem as he compeon beween a radonal SR realer ha makes-osock before he season sars, versus a QR frm ha has a flexble suppl chan and can order sock more han once. 3

For he smmerc case, we exended he exsence and unque equlbrum resul b Lppman and McCardle 997 o an arbrar number of perods wh demand learnng and possbl nonzero nal nvenor, where each frm follows a base-sock polc ha gnores he compeon. In oher words, realers adop he same polc as f he were alone n he marke. For asmmerc realers and wo perods T = 2, we provded condons ha guaranees he exsence of a unque pure-sraeg subgame-perfec equlbrum for he demand sgnal and he md-season replenshmen cases. In addon, we performed an exensve numercal sud o undersand he mpac of cos asmmeres, demand varabl, and correlaon across perods on he equlbrum nvenor levels and he correspondng profs. One of he srkng resuls of our model s ha a SR realer would raher compee agans a QR realer han a SR opponen. Also, a realer compeng agans a SR realer s beer off beng QR, f he orderng cos c QR s below a break-even value ha ncreases wh demand unceran and he correlaon across perods. Fnall, we show ha par of he compeve advanage for a QR realer comes from he asmmeres on suppl chan flexbl, snce he ncrease n profs s hgher n he QR vs. SR scenaro han n he QR vs. QR case. Several exensons of hs work are possble. Frs, n erms of nvenor compeon models, deal exensons nclude exsence and unque equlbrum resuls for he asmmerc case wh an arbrar number of perods, and possbl more general demand allocaon rules and a larger number of realers. However, he analss s presumabl no sraghforward. Second, n erms of undersandng he fas fashon phenomenon, here s sll plen o be done. In fac, n hs paper we have gnored oher dsncve aspecs such as he endogenous effec of hgher fll raes on marke share, smlar o Gaur and Park 2007. Incorporang hese elemens no our model s a challengng srand of fuure research. Acknowledgmens Boh auhors would lke o hank Gerard Cachon, an assocae edor, and hree anonmous referees for her helpful and consrucve commens on a prevous verson of hs paper. References Anupnd, R. and Y. Bassok. 999. Cenralzaon of Socks: Realers vs. Manufacurers. Managemen Scence, 452, pp. 78-9. 32

Avsar, Z.M. and M. Bakal-Gürso. 2002. Invenor Conrol Under Subsuable Demand: a Sochasc Game Applcaon. Naval Research Logscs, 494, pp. 359-375. Bernsen, F. and A. Federgruen. 2004. Dnamc Invenor and Prcng Models for Compeng Realers. Naval Research Logscs, 52, pp. 258-274. Cachon, G. 2003. Suppl Chan Coordnaon wh Conracs, n Handbooks n Operaons Research and Managemen Scence: Suppl Chan Managemen, eded b Seve Graves and Ton de Kok. Norh-Holland. Cachon, G. and R. Swnne. 2007. Purchasng, Prcng, and Quck Response n he Presence of Sraegc Cusomers. Workng paper. The Wharon School, Unvers of Pennslvana. Cachon, G. and C. Terwesch. 2005. Machng Suppl wh Demand: An Inroducon o Operaons Managemen. McGraw-Hll/Irwn. Caro, F. and J. Gallen. 2007. Dnamc Assormen wh Demand Learnng for Seasonal Consumer Goods. Managemen Scence, 532, pp. 276-292 Davdson, J. 2005. Chc Thrlls. Scoland on Sunda, Februar 3 ssue. Economs. 2005. The fuure of fas fashon. The Economs, June 6h 2005 ssue. Ferdows, K., M. Lews and J. A.D. Machuca. 2004. Rapd-Fre Fullflmen. Harvard Busness Revew, November ssue, Reprn R04G. Fsher, M. L. and A. Raman. 996. Reducng he Cos of Demand Unceran Through Accurae Response o Earl Sales. Operaons Research, 44 87-99. Fsher, M. L. 997. Wha Is he Rgh Suppl Chan for Your Produc? Harvard Busness Revew, March-Aprl ssue, Reprn 97205. Foroohar, R. 2006. A New Fashon Froner. Newsweek Inernaonal, March 20 ssue. Fudenberg, D. and J. Trole 99. Game Theor. MIT Press, Cambrdge, Massachuses. Ghemawa, P. and J. L. Nueno. 2003. ZARA: Fas Fashon. Harvard Busness School Mulmeda Case 9-703-46. Gaur, V. and Y. Park. 2007. Asmmerc Consumer Learnng and Invenor Compeon. Managemen Scence, 532, pp. 227-240. Hall, J. and E. Poreus. 2000. Cusomer Servce Compeon n Capacaed Ssems Source. Manufacurng and Servce Operaons Managemen, 22, pp. 44-65. Hammond, J.H. and M.G. Kell. 990. Quck Response n he Apparel Indusr. Harvard 33

Busness School Noe 9-690-038. Ier, A.V. and M.E. Bergen. 997. Quck Response n Manufacurer-Realer Channels, Managemen Scence, 434, pp. 559-570. Krman, A.P. and M.J. Sobel. 974. Dnamc Olgopol wh Invenores. Economerca, 422, pp. 279-287. L, L. 992. The Role of Invenor n Delver-Tme Compeon. Managemen Scence, 382, pp. 82-97. Lppman, S. A. and K. F. McCardle. 997. The Compeve Newsbo. Operaons Research, 45, pp. 54-65. Lu, L., W. Shang, and S. Wu. 2007. Dnamc Compeve Newsvendors wh Servce- Sensve Demands. Manufacurng and Servce Operaons Managemen, 9, pp. 84-93. Mahaan, S. and G. van Rzn. 200. Invenor Compeon Under Dnamc Consumer Choce. Operaons Research, 493, pp. 646-657. McAfee, A., V. Dessan and A. Söman. 2004. ZARA: IT for Fas Fashon. Harvard Busness School Case 9-604-08. Nagaraan, M. and S. Raagopalan. 2005. Invenor Models for Subsuable Producs: Monopol and Duopol Analss. Workng paper, Marshall School of Busness, U.S.C. Neessne, S. and N. Rud. 2003. Cenralzed and Compeve Invenor Models wh Demand Subsuon. Operaons Research, 52, pp. 329-335. Neessne, S. and F. Zhang. 2005. Posve vs. Negave Exernales n Invenor Managemen: Implcaons for Suppl Chan Desgn. Manufacurng and Servce Operaons Managemen, 7, pp. 58-73. Neessne, S., N. Rud and Y. Wang. 2006. Invenor Compeon and Incenves o Back-Order. IIE Transacons 38, pp. 883-902. Olsen, T.L. and R.P. Parker. 2006. Consumer Behavor n Invenor Managemen. Workng paper, Washngon Unvers n S Lous. Parlar, M. 988. Game Theorec Analss of he Subsuable Produc Invenor Problem wh Random Demands. Naval Research Logscs, 353, pp. 397-409. Sobel, M. 98. Mopc Soluons of Markov Decson Processes and Sochasc Games, Operaons Research, 295, pp. 995-009. 34

Techncal Appendx o The Impac of Quck Response n Invenor-Based Compeon Felpe Caro Vcor Marínez-de-Albénz UCLA Anderson School of Managemen 0, Weswood Plaza, Sue B420 Los Angeles, CA 90095, USA IESE Busness School Unvers of Navarra Av. Pearson 2 08034 Barcelona, Span fcaro@anderson.ucla.edu valbenz@ese.edu Ocober 28, 2007 General Remarks: The numberng of he equaons n he appendx connues he same sequence from he paper. Throughou he appendx, an nerchange of negraon and dfferenaon n defne negrals s usfed b Lebnz rule, whch holds whenever he negrand s connuousl dfferenable almos everwhere. Proof of Proposon Proof. We om I snce remans consan hroughou he proof. We can om he subndex = as well, snce all he funcons and varables n he proof refer o he ermnal perod. For each,, we use he followng noaon. Le κ = q and κ = q. Each of hese quanes represens he demand a whch each realer socks ou κ sands for knocked-ou. Noe ha f κ κ, hen here can onl be a spll-over of cusomers from o. In addon, le β, = P R } = κ κ β, = Pκ D + } = κ κ where A s he ndcaor funcon of even A. + fudu + κ <κ + κ κ fudu and fudu, 5 35

Followng Fgure, le E and E be he equlbrum sockng quanes n he unconsraned compeve game,.e, when boh realers sar whou an nvenor. Smlarl, le s and s be he unconsraned base-sock funcons. From he defnon of a Nash equlbrum, E = s E and E = s E, and we have ha e 0, 0 = E and e 0, 0 = E. We now compue he paral dervaves of π n each one of he four regons I-IV depced n Fgure. I When x E, x E, hen e = E and e = E. Hence, from Equaon 4, π s lnear n x, wh π p x = c p and ndependen of x. II When x E, x s x, hen e = x and e = s x. Hence, π = c x c s x + p E mn s x, R x }. Then, we have ha p π x = c p and p π x = P κ x D s x + x } = β e, e. where he laer follows from he envelope heorem. III When x E, x s x, hen e = x and e = s x. Hence, π = p E mn x, R s x }. I s ndependen of x and π } } p x = P x R s x ds d P κ s x D x + s x = β x, s x β x, s x ds d. Noce ha, from he mplc funcon heorem a = s, we have ha ds d = 2r 2 r 2 = κ <κ s. Tha s, when he dervave of he base-sock funcon s s no zero hen β, s s zero. In oher words, β, s ds d s alwas equal o zero. Hence, p π x = β e, e. IV When boh realers sar wh nvenores above her equlbrum quanes, hen e = x, e = x, and π = p E mn x, R x }. Then, π } p x = P x R x = β e, e 36

Regon π π p x p x c I p 0 II c p β s x, x III β x, s x 0 IV β x, x β x, x Table 3: Properes of π. and p π x = P κ x D x + x } = β e, e. We summarze hese fndngs n Table 3. Noe ha π s connuous n x, x, and s dfferenable almos everwhere. In addon, noe ha π x s connuous n x bu dsconnuous n x n he border of regons I and II. Ineresngl, π s connuous n boh argumens. In parcular, akng x fxed, and x ncreasng x, we move from eher regon I o II, or from regon I o II o IV, or from regon III o IV. In each one of borders, π x s connuous, as β = c n he border p II-IV and s x = x n he border III-IV. Smlarl, akng x fxed, and ncreasng x, we move from eher regon I o III, or from regon I o III o IV, or from regon II o IV. Agan, here s no ump n π as we move from regon o regon. Snce s x non-ncreasng as a funcon of x nsde he regons, π s concave n x. Proof of Proposon 2 Proof. As before, for smplc we om he subscrp D 2 and use subscrp nsead. The proof s que long, bu he man dea s raher smple. followng nequal holds 2 r2 2 < φ 2 2 q2 2 r 2 2 We wan o show ha he, where φ 2 = max 0, f } 2. 6 f 2 Ths s he same nequal as 0, and for exposonal purposes, frs assume ha acuall holds we wll prove shorl. Then, consder a crcal pon of r2, 2,.e., r2 2, 2 = 0. From nequal 6, he crcal pon s necessarl a src maxmum, 2 2 r2.e., 2 2 2, 2 < 0. Ths shows ha r2, 2 s frs ncreasng and hen decreasng,.e., 37

quas-concave. In addon, for 2 ver large, r 2 s evenuall decreasng. Hence, here exss a maxmzer of r 2, 2. Furhermore, hs maxmzer mus be unque. In fac, consder wo dsnc maxma. B 6, boh would have o be src. Ths would mpl ha n beween hem, here would have o be a mnmum, whch agan would be a conradcon wh 6. As a resul, denoe s 2 2 he unque maxmzer of r 2 2, 2. From quas-concav, he opmal unconsraned polc s base-sock, and from Equaon 6, he consraned bes response s b 2x 2, 2 = maxx 2, s 2 2}. Therefore, o prove Proposon 2 suffces o show ha he nequal 6 s vald. Tha s wha we do nex. We sar he proof of 6 usng he noaon and resuls from Table 3. We srucure he proof n hree seps: n clams and 2, we provde bounds on 2 π x, π 2 x n clam 3, we fnall prove he nequal 6. and π ; hen, x Clam Leng φ x = max 0, f x f }, 7 2 π we have ha x φ q x c 2 π. x We prove he clam for each one of he four regons n Fgure. As n he prevous proof, we use he noaon of Equaon 5 and le κ = q and κ = q. I II 2 π x = 0 c 2 π. 2 π x 2 = 0 x c π x. IV We leave case III for he end. Here, usng Equaon 5, we have β, = κ κ + f udu + κ <κ κ f udu and hus, nong ha dκ d = q κ, β = κ κ f + f κ κ <κ q κ Snce f s log-concave,.e., logf s concave, F s also log-concave, and for all v n he suppor of f, f v F v max 0, f } v, f v 38

see Bagnol and Bergsrom 2005 or Marínez-de-Albénz 2004. Thus, we have, usng ha q because q = q 0, β κ f κ + + κ <κ f κ κ κ max 0, f } + + f f udu 0 + κ <κ max 0, f } κ κ f f udu 0 = max 0, f +, f } κ f f κ κ + κ <κ + κ f udu f udu where o oban he max, we used ha f s non-ncreasng f max 0, f } q f c β. Ths can be rewren as 2 π x max 0, f } q 2 f x c π. x p III We have p 2 π x 2 = β + β ds d = β. Ths s rue because, when ds κ d 0, hen κ κ s. Ths mples ha β, s = f udu, and hence β = 0 a, s. Smlarl o case IV, 2 π x max 0, f } q 2 f x c π. x Ths complees he proof of he bound. We need a second bound on he frs dervaves. Clam 2 π x c ; for all x such ha F q x <, π 0, x > p c. x 39

The frs par of he clam, π x and π 0, x x = c. c, s a drec consequence of concav wh respec o x For he second par, we noce ha 0, x belongs o regons I or II. When π 0, x 0, we mus be n regon II. There, π 0, x = β s x, x. Recall ha we have, a x = s x unconsraned bes-response, x ds d = 2 r 2 r 2 = κ s κ x. Tha s, β s x, x s no zero f and onl f he bes-response s x = F from Theorem. As a resul, when π x 0, x 0, c p π x 0, x = β F p = κ x F = F κ c p c p x, x F κ F κ c p c p f udu c p x, Fnall, hs expresson s vald oherwse s zero whle we are n regon II: x mus be greaer han s s x. We know ha s has slope here, whch mples ha s mus be consan, snce a he nersecon a mos one of he bes-response curves has slope. In oher words, x q c. Hence, we can express p π F p x 0, x = c p F q x c p whch complees he second par of he clam. F q x c p 8 Wh hese wo clams we are read o prove he ke nequal 6. Clam 3 2 r2 2 < φ 2 2 q2 2 r 2 2, where φ 2 = max 0, f } 2. 9 f 2 40

For hs purpose, ake 2 as fxed. Smlarl as before, le κ 2 2 = q 2 2 and κ 22 = q2 2. Equaon 5 can be wren as c 2 r2 2, 2 2 + p 2 mn 2, R2 2 } = E +, + +π D 2 2 R2 2 2 R2 2 There are wo possble suaons gven 2, 2. Eher socks ou before, and hence here ma be a spll over from o, n whch case κ 2 κ 2; or vce-versa. The frs suaon occurs when κ 2 κ 2. Here, nong ha he negrand s connuousl dfferenable almos everwhere and hence dfferenaon and negraon can be nerchanged, r 2 2 = c 2 + p 2P } π 2 + D2 2 D 2 + E x π D2 +E 2 + 2 D 2, 0 κ 2 D 2 2 + 2 x 2 q2d 2, 2 q2d 2 κ 2 D 2 } Snce π D 2 s concave n x, for all D 2, r2 s concave n 2. In fac, he second dervave can be expressed as 2 r2 2 2 = π 2 p 2f 2 2 + 2 D + E 2 x 2 2 q2d 2, 2 q2d 2 } κ 2 D 2 2 π D +E 2 x 2 2 + 2 D 2, 0 } κ 2 D + π D 2 0, 0f 2 2 + 2 x 2 2 + 2 < E φ u q } π 2 q2d D 2 2 x 2 q2d 2, 2 q2d 2 c κ 2 D 2 } π +E φ u q 2 + D2 2 D 2 2 + 2 D 2, 0 c x x usng Clams and 2, and he nfne suppor of D 2 E φ 2 q π 2 2 D 2 x 2 q2d 2, 2 q2d 2 c +E φ 2 q π 2 2 D 2 2 + 2 D 2, 0 c } κ 2 D 2 2 + 2 κ 2 D 2 κ 2 D 2 2 + 2 usng ha φ u x = max0, g x ku} φ g 2x ku φ 2 x, from assumpon and ha φ 2 s non-ncreasng, assumpon φ 2 q 2 2 r2 c 2 c 2 p 2 c P } 2 + 2 D 2 φ 2 q 2 2 r2. 2 } } 4

o, Consder now he second suaon,.e., κ 2 < κ 2. Here, snce here s onl spll-over from r 2 2 = c 2 + p 2P } π κ D2 2 D 2 + E x π D2 +E x 0, 2 + 2 D 2 κ 2 D 2 2 + 2 2 q2d 2, 2 q2d 2 κ 2 D 2 } = c 2 + c + p 2 c f 2 udu 2 + 2 κ 2 π u + 0 x 2 q2u, 2 q2u c f 2 udu + 2 + 2 κ 2 p 2 c + π u x 0, 2 + 2 u f 2 udu. The second negral ma no be decreasng n 2, snce s varaon s relaed o he second dervave of he prof-o-go wh respec o x, whch ma no even exs. Usng Equaon 8, we can express as = 2 + 2 κ 2 2 + 2 κ 2 p 2 c + π u x 0, 2 + 2 u f 2 udu p 2 c p c p F u q 2 + 2 u c p } 20 F u q 2 + 2 u c f p 2 udu Le Gx := gd. We hus have F u q 2 + 2 u = G 2 + 2 u ku. α x When k > 0, consder he change of varables v = 2 + 2 u α ku, or equvalenl u = 2 + 2 αv + kα. We oban = 2 + 2 κ 2 2 + 2 κ 2 α kκ 2 0 p 2 c p c p F u q 2 + 2 u c p p 2 c p c p Gv c p Gv c p F u q 2 + 2 u c 2 + 2 α f v 2 + kα p f 2 udu α + kα dv 42

As a resul, he use of Lebnz rule for dfferenaon elds 2 + 2 p 2 2 c + π u κ x 0, 2 + 2 u f 2 udu 2 [ ] α + k f q 2 κ 2 κ α 2 2 + kα = p 2 c p 2 + 2 κ 2 c G p G 2 + 2 κ 2 α kκ 2 c α kκ 2 c p p 2 + 2 κ 2 α + kκ 2 p 2 c p c Gv c f 2 + 2 αv α 0 p Gv c p 2 p + kα + kα dv 2 Snce 0 α, k 0 and q 2 κ 2, he frs erm s non-posve. In addon, because of log-concav of D 2, f 2 s non-ncreasng and hence for 0 v 2 + 2 κ 2 kκ 2 f 2 f 2 2 + 2 αv f 2 + kα + kα f 2 κ f 2 2 + kα φ 2 q2 2. As a resul, 2 + 2 p 2 q2 2 2 c + π u x 0, 2 + 2 u f 2 udu φ 2 q2 2 2 + 2 p2 c + π u x 0, 2 + 2 u f 2 udu q 2 2 When k = 0,.e., D s ndependen from D 2, hen he same resul s obaned usng he change of varables v = 2 + 2 u, snce = 2 + 2 κ 2 2 + 2 κ 2 0 p 2 c p c p F u q 2 + 2 u c p p 2 c p c p Gq v c p α F u q 2 + 2 u c f p 2 udu Gq v c p Noe ha n hs case no assumpon on he lnear of q s made. f 2 2 + 2 v dv. 43

Thus, dfferenang Equaon 20 elds π 2 r2 κ 2 = 2 2 p 2 c f 2 2 + x 0, 2 + 2 q 2 κ 2 c f q2 κ 2 κ 2 2 κ 2 2 π u + 0 x 2 2 q2u, 2 q2u f 2 udu + 2 + 2 p 2 c + π u x 0, 2 + 2 u f 2 udu 2 κ 2 κ 2 < 2 π u 0 x 2 2 q2u, 2 q2u f 2 udu +φ 2 q 2 2 2 + 2 q 2 2 p 2 c + π u x 0, 2 + 2 u f 2 udu where we used π κ 2 0, x 2 q 2 κ 2 c 0 from Clam 2 and he nfne suppor. Usng Clam and assumpon of he proposon, we have ha 2 π u x 2 2 q2u, 2 q2u π φ u q 2 q2u u x φ 2 q π 2 2 q2u u x 2 q2u, 2 q2u c 2 q2u, 2 q2u c φ 2 q π 2 2 u x 2 q2u, 2 q2u c where we used ha f 2 s log-concave so ha φ 2 s non-ncreasng. Hence, 2 r2 q 2 < φ 2q2 2 2 2 π u 2 0 x 2 q2u, 2 q2u c f 2 udu 2 + 2 +φ 2 q2 2 p q2 2 2 c + π u x 0, 2 + 2 u f 2 udu = φ 2 q 2 2 r2 c 2 c 2 p 2 c f 2 udu 2 + 2 φ 2 q 2 2 r2. 2 Ths ends he proof of he hrd clam, and herefore, he proof of he proposon s also complee. Proof of Theorem 3 Proof. We use frs he followng clam. Clam 4 2 π x 2 2 π x x 0 almos everwhere. 44,

Noe ha hese quanes are well-defned almos everwhere because, as seen n he proof of Proposon, π s connuous, and he pons of non-dfferenabl are a he borders x 2 r of regons I-IV onl. The frs nequal s easl proved usng ha 2 r 2. The second par comes from he fac ha, n each regon, he cross-dervave s eher zero, n regons I-III, or non-posve n regon IV. Smlarl as n he prevous proofs, le κ 2 2 = q 2 2 and κ 2 2 = q 2 2. When κ 2 κ 2, and 2 r 2 2 2 2 r 2 2 2 2 r 2 2 2 = E [ ] 2 π D 2 x 2 π D 2 2 x x 2 q2d 2, 2 q2d 2 κ 2 D 2 = p 2 π D 2 2 π 0, 0 f x 2 2 + 2 D + E 2 x x 2 q2d 2, 2 q2d 2 } κ 2 D 2 2 π D +E 2 x 2 2 + 2 D 2, 0 } q 2 2 D 2 2 + 2 0. 22 In addon, when κ 2 < κ 2, 2 r 2 2 2 2 r 2 2 2 proof. because π D 2 x = p 2f 2 κ 2 q 2 κ 2 + f 2κ 2 q2 κ 2 +E [ π κ 2 x π κ 2 [ 2 π D 2 x 2 2 π D 2 x x < f 2κ 2 q 2 κ 2 p p 2 0 x ] 0, 2 + 2 κ 2 } 0 ] 2 q 2D 2, 2 q 2D 2 q 2 2 D 2 2 } 23 0, z π D 2 x 0, z < c + p c = p from Clam 2 n he prevous From he mplc funcon heorem, ds 2 d 2 we have ha ds 2 d 2 mus have eher ds 2 d 2 = 2 r 2 2 2 2 r 2 2 2. Thus, usng Equaons 2 and 23,. In addon, snce a each pon, eher κ 2 < κ 2 or he reverse, we 0 and ds 2 d 2 > ; or ds 2 d 2 45 > and ds 2 d 2 0. Ths mples ha n

each pon, ds 2 ds 2 d d 2 2 <. The wo bes-response funcons are connuous. s 22 s decreasng for small 2, whle q2 s 22 q2 2. As a resul, he wo bes-response funcons nersec n he regon [0, s 20] [0, s 20]: equlbrum exss. In addon, snce n an equlbrum pon ds 2 ds 2 <, equlbrum s unque, see Cachon and Neessne 2004. d2 d 2 Proof of Theorem 4 Proof. Snce realer s passve, s clear ha for all, s = 0. Thus, we need o show ha frm s polc s a base-sock polc ha ma depend on = x. In oher words, show ha b x, = maxx, s }. For hs purpose, we follow he lnes of proof from Proposon 2. Specfcall, we show b nducon on =,..., T, ha, for all, r, s quas-concave; hence a base-sock polc of level s s opmal; here s a unque equlbrum: e x, x = x and e x, x = maxx, s x }; π s concave n x, π x x, x c ; v Denong we have 2 π x 2 φ φ = max 0, f } f q x c π 0; x, 24 v 2 π x 2 2 π x x ; v π x 0, x p c. -v are rue for =, from he proofs of Proposon and Theorem 3. Assume ha he nducon properes are rue for. For, fx. Smlarl as n he prevous proofs, le κ = q and κ = q. There are wo possble suaons gven, : eher socks ou before, and hence here ma be a spll over from o,.e., κ κ ; or vce-versa. When κ κ, r = c + p P } π + D + E x qd, q D κ π +E + D, 0 } κ D + x 46 D }

Snce π s concave n x, r s concave n. In fac, he second dervave can be expressed as 2 r = 2 p f + 2 π + E x 2 qd, q D κ 2 π +E x 2 + D, 0 } κ D + E φ q qd π x π +E φ q + D o, usng par v of he nducon E φ q π x +E φ q π x x q D, q D D } q D, q D + D, 0 c + D, 0 c c c κ D κ D + } κ D κ D + usng of he proposon and ha φ s non-ncreasng, from of he proposon φ q r c c p c P } + D φ q r. 2 r Also, 2 r 2, usng par v of he nducon sep. Consder now he second suaon,.e., κ < κ. Here, snce here s onl spll-over from } } 25 } r = c + p P } π κ D + E x qd, q D κ } π +E x 0, + D κ D + = c + c + p c f udu + κ π + 0 x qu, q u c f udu + + q p c + π 0, + u x f udu. = c + c + p c f udu + κ π + 0 x qu, q u c f udu + + κ 0 p c + π x 0, v f + vdv, D } 47

where we made he change of varables v = + u n he las negral. Frs, observe 2 r ha 2 r 2. Moreover, dfferenaon elds 2 r 2 = f + p c + f κ π q κ x + q κ κ + + 0 + κ κ + 0 0, + κ c f κ p c + π x 0, + κ 2 π x 2 qu, q u f udu 0 + κ 0 p c + π x 0, v 2 π x 2 qu, q u f udu p c + π x 0, v f + vdv f + vdv, where we used par and v of he nducon sep,.e., π x p c + π x 0, 0. In addon, from he log-concav of D, f f and so s φ. Hence for v [0, + κ ], f f + v f f κ φ q. Fnall, from par v of he nducon, for u [0, q ], c 0, and s non-ncreasng 2 π x 2 qu, q u φ q qu π x qu, q u c φ q π qu, q u c x because he erm n parenheses s non-posve π φ q qu, q u c x from assumpon of he proposon 48

As a resul, κ π 2 r φ 2 q 0 x qu, q u c f udu + κ + p c + π 0 x 0, v f + vdv φ q r c c p c f udu + φ q r. Consder a crcal pon of r, r,.e.,, = 0. We have shown ha hs s 2 r necessarl a maxmum,.e., 2, 0. Ths proves ha r, s frs ncreasng and hen decreasng,.e., quas-concave. Snce, r s evenuall decreasng for large, here s a unque unconsraned maxmzer, s, and he opmal polc s base-sock: b x, = maxx, s }. Ths proves par of he nducon. Par s shown recallng ha realer does no order a all: e = x, and hus e = maxx, s x }. Hence, here are wo regons o consder: πx, x c = x + rs x, x for x s x c x + rx, x oherwse. In he frs regon replenshmen, frs lne n he expresson above, π = c x and hence 2 π x 0 = φ 2 q c π 2 π. Also, x x = 2 π 2 x x = 0. π In he second regon, = c x + r c, because x s x. In addon, from Equaons 25 and 26, 2 π x 2 = 2 r φ 2 q x r = φ q x c π. x 2 π Also, x 2 π 2 x x snce 2 r 2 r 2 shown before. Hence, π s concave n x, and pars -v are shown. Fnall, from he envelope heorem, 26 π x 0, x = r s x, x 49

0, Ineresngl, snce 2 r 2 2 r, r r s non-ncreasng n. Snce r s x, x = r 0, x r 0, x = p c r s x, x r Ths shows par v and complees he nducon. Proof of Theorem 5 s x, x = π x 0, x. Proof. In he proof, we use he dnamc program presened n Equaon 3. Ths s a sandard nvenor problem, and s eas o show ha for all, for all possble pas nformaon I, U I s concave. I follows ha he opmal polc s a base-sock polc wh level s I, where s I sasfes c = E p s I D + du } I s I D s I D dx I We show b nducon on =,..., T ha for all I whch we om as a subscrp below,.e., we wre subscrp nsead of I, for all, r, s concave and s s non-ncreasng n v when x α x α, hen π 2 r 2 and when s x, x = α U x α 2 r 0; α α, s = α s ; we have exsence and unqueness of equlbrum and for =, 2, e x, x = maxx, α s }; ; v 2 π x 2 2 π x x 0. For =, fx I and hence omed below. We know ha r s concave n for all. 2 r From Table 3, s clear ha 2 r 2 0, pon. Pon follows from Theorem : for α α, P R } = P α D }, and hence s = α s. Snce he bes-response funcons are non-ncreasng wh slope greaer han -, a Nash equlbrum exss Theorem agan; and snce eher s or s are consan dependng on α or no, hen he equlbrum s unque. In addon, s eas o check ha α E := e 0, 0 = α s and E := e 0, 0 = α s, and ha above he equlbrum level E, s e s consan equal o α s. Ths mples e = maxx, α s }, pon. 50

In parcular, f x x α α, hen e e. Ths mples ha here s no spll-over from α α realer s demand no s: mne, Re } = mne, α D }. Hence, x πx, x = c maxx, α s } x + p E mnmaxx, α s }, α D } = α U, α pon v. Fnall, snce s clear ha =. π x, x = c x + r maxx, α s }, maxx, α s } 2 π x 2 2 π x x 0, pon v. Thus he nducve proper s proved for Assumng ha s rue for, for all I, fx I omed as subscrp below. +, + r, = c + p E mn, R } + Eπ I R R and hence r = c + p P R } + E +, + π I R R. We have ha = +, + π I R R eher π I + D, 0 or π I 0, + D = α U I 0 or π I α D, α D or π I 0, 0 = α U I 0 Usng pon v of he nducon sep on π I, we have ha. 2 r 2 2 r 0, pon Ths mples ha he opmal polc s a base-sock polc wh base-sock level s, and from he mplc funcon heorem and pon, we have ha s s non-ncreasng. In addon, f α mn, α D } and. α, here s no spll-over from o, and hence mn, R } = +, π I R R from pon v of he nducon. Hence, when s 5 + = α U I α D + α α, α, s = α s. Ths shows

Agan, snce he bes-response funcons are non-ncreasng and wh slopes greaer han -, here exss a Nash equlbrum; as one of he bes-response funcons s consan, hs equlbrum s unque. Smlarl as before, e = maxx, α s } snce he bes-response funcon s s fla when α s. Ths shows. Agan, f x α x α, hen e α e α. We hen have ha mne, Re } = mne, α D }. Hence, πx, x = c maxx, α s } x + p E mnmaxx, α s }, α D } maxx +Eα U, α s } α D + I α x = α U, α pon v. Fnall, snce π x, x = c x + r maxx, α s }, maxx, α s }, 2 π s clear ha x 2 π 2 x x 0, usng par v of he nducon sep. Ths proves pon v for, for all I, complees he nducon and proves he heorem. References Bagnol, M. and T. Bergsrom. 2005. Log-Concave Probabl and Is Applcaons. Economc Theor, 26, pp. 445469. Cachon, G. and S. Neessne. 2004. Game Theor n Suppl Chan Analss. Suppl Chan Analss n he ebusness Era, eded b Smch-Lev D., S. D. Wu and Z.-J. Shen and publshed b Kluwer Academc Publshers. Marínez-de-Albénz, V. 2004. Porfolo Sraeges n Suppl Conracs. Ph.D. dsseraon, OR Cener, MIT. 52