MANUFACTURER-RETAILER CONTRACTING UNDER AN UNKNOWN DEMAND DISTRIBUTION

Transcription

1 MANUFACTURER-RETAILER CONTRACTING UNDER AN UNKNOWN DEMAND DISTRIBUTION Mrti A. Lriviere Fuqu School of Busiess Duke Uiversity Ev L. Porteus Grdute School of Busiess Stford Uiversity Drft December, 995

2 Abstrct We cosider mufcturer itroducig ew product ito distributio chel d exmie wht wholesle price should be chrged. The settig i my wys is simple. The chel is bbrevited with the mufcturer sellig directly to the retiler. The cotrct is lso simple, merely flt wholesle price. Demd is stochstic but idepedet d ideticlly distributed i ech period. Complictios rise from two dditiol ssumptios. First, either prty kows some prmeter of the demd distributio. The system evolves iformtiolly s the chel hs more experiece with, d iformtio bout, the product. Secod, we ssume umet demd is both lost d uobserved, so oly sles dt re vilble. The utoomous retiler s stockig level cosequetly dicttes the rte t which the chel cquires iformtio. The mufcturer s pricig policy, i tur, iflueces the retiler s ctios. We explore how the wholesle price evolves s beliefs re updted i Byesi fshio. Pricig is drive by the precisio of iformtio d ot the size of the mrket. I prticulr, we show tht the mufcturer chrges lower price followig stockout th fter exct observtio. Tht is, she prices more ggressively followig sigl of reltively wek demd (usold stock) th fter sigl of strog demd (empty shelves). The ppret omly is explied by reltig the precisio of iformtio to the umber of observed stockouts d the elsticity of retiler orders to the precisio of iformtio; stockouts re less iformtive, d ucerti retiler is reltively price sesitive.

3 I my mrkets, mufcturers do ot sell directly to cosumers but distribute products through decetrlized chels; wholeslers d distributors tke possessio of the goods before retilers offer them to customers. Whe the liks of the distributio chi re idepedet firms, mufcturer must budle her product with cotrct goverig chel trsctios. Cotrctig withi distributio chel hs received cosiderble ttetio withi ecoomics (e.g., Spegler, 950; Mthewso d Witer, 984; Tirole, 988, provides survey) d mrketig (e.g., Jeuld d Shug, 983; Moorthy, 987). Both fields emphsize the impct of the cotrctul form o totl chel profits d their divisio. Both geerlly presume kow determiistic demd curve. Issues of ivetory or stochstic demd re igored. I the opertios mgemet literture, reserchers hve exmied schemes tht lower supplier s orderig d ivetory costs i ecoomic order qutity settig. Moh (984) d Lee d Rosebltt (986) suggest qutity discouts to lter buyer s orderig ptters. Weg (995) shows tht qutity discouts loe will ot mximize chel profits. Psterck (985) cosiders both ivetory d stochstic demd; his retiler fces clssic, sigle period ewsvedor problem for kow demd distributio. He proves tht chel profits re mximized whe the mufcturer offers returs policy. Here, we study mufcturer who luches ew product i ucerti eviromet d the dymiclly djusts the wholesle price s iformtio is reveled. The mufcturer dels directly with self-iterested retiler who holds stock i ticiptio of demd d sells the product t fixed retil price. Demd is stochstic with ll reliztios idepedetly d ideticlly distributed, but either prty kows some prmeter of the demd distributio. Both gi iformtio bout tht prmeter by observig sles. Thus, despite sttiory demd, the demd distributio ppers differet i ech period becuse estimtes of the ukow prmeter chge. The system evolves iformtiolly s the chel gis experiece with the product. Further, we ssume tht umet demd is both lost d uobserved. Beliefs bout the demd distributio re updted bsed o observed sles. As sles dt provide oly lower boud o relized demd whe the retiler stocks out, the utoomous retiler s stockig decisio dicttes the rte t which the chel cquires iformtio. The mufcturer s pricig policy, i tur, iflueces the retiler s ctios.

4 We explore how the mufcturer s wholesle price schedule evolves over fiite horizo s beliefs bout the demd distributio re updted i Byesi fshio. I our model, the pricig decisio is drive by the precisio of iformtio d ot the size of the mrket. I prticulr, we show tht the mufcturer chrges lower price followig stockout th fter exct observtio. Tht is, she prices more ggressively followig sigl of reltively wek demd (usold stock) th fter sigl of strog demd (empty shelves). The ppret omly is explied by reltig the precisio of iformtio to the umber of observed stockouts d the elsticity of retiler orders to the precisio of iformtio; stockouts re less iformtive, d ucerti retiler is reltively price sesitive. The ext sectio gives the bsic ssumptios d structure of the model. We lso preset the retiler s optiml stockig policy for give price schedule. To miti the trctbility of the retiler s problem, we must restrict the mufcturer to limited set of dmissible price schedules. I sectio 2, we first show tht if the mufcturer uses such schedule her profits hve very simple form tht llows oe to fid the optiml dmissible schedule esily. We the rgue tht dmissible schemes re ll the mufcturer eed cosider; she my well prefer followig some ltertive schedule but is uble to commit to its use. Sectio 3 chrcterizes the behvior of the optiml price schedule for the specil cse of expoetil demd d perishble ivetory. It is here tht we discuss the reltio betwee the wholesle price d the stte of iformtio. Sectio 4 cosiders simple exmple tht llows us to exmie the behvior of prices over time d the impct of retiler ivetory.. MODEL BASICS AND THE RETAILER S PROBLEM Over fiite horizo of legth N, demd reliztios re idepedet d ideticlly distributed drws from some true distributio. The fmily of the demd distributio is kow, but some prmeter ω of its desity ψ( ω) is ot. There is prior o the ukow prmeter with desity g( ω ). Oe lso hs the predictive or mrgil desity, = g d. φξ ψξω ω ω φξ is the updted demd desity, the best estimte of the demd desity give the prior. The stdrd pproch is to ssume tht g( ω ) is from the cojugte fmily of the demd distributio (DeGroot, 970). The posterior of the ukow prmeter 2

5 (which is the prior for the ext period) would the come from the sme fmily s g( ω ), d the subsequet updted demd distributio would be from the sme fmily s φξ. Additiolly, oe would hve fixed-dimesiol sufficiet sttistic tht would completely chrcterize the prior. By updtig the sttistic followig observtio, the updted sufficiet sttistic would similrly fully chrcterize the subsequet posterior. Oe would ot eed to reti ech observtio explicitly. The difficulty is tht sles d ot demd re observed. Although demd reliztios re ideticlly distributed, sles observtios re ot. Sles dt coti both exct d cesored observtios, d cojugte prior does ot ecessrily exist eve if there is oe for fully observed demd. Brde d Freimer (99) chrcterize ewsboy distributios for which cojugte prior d fixed-dimesiol sufficiet sttistic exists eve with cesored observtios. The ewsboy fmily hs the form Ψξω e d with = ω ξ d ξ 0. The gmm distributio with shpe prmeter d scle prmeter S is the cojugte prior for ll ewsboy distributios. The updted gmm prmeters ( S ), re the sufficiet sttistics for period. Let m be the umber of exct observtios by the strt of period d s i the i-th sles observtio. ( S ), re clculted s: = + m S = S + d s i i= I updtig the scle prmeter, exct d cesored observtios re treted eqully: For either, d( ξ ) is evluted t the observed vlue d dded to the sum. The shpe prmeter, o the other hd, chges oly with exct observtios. It c (d will) be iterpreted s mesure of the precisio, or stte, of the system s iformtio. I wht follows, we restrict ψξω to be ewsboy distributio d ssume gmm prior. Additiolly, we wish to use the stte spce reductio techique of Azoury (985), which imposes further requiremets. Of the ewsboy distributios preseted i Brde d Freimer (99), oly the Weibull distributio stisfies the coditios of Azoury (985). We therefore ssume the true demd hs Weibull distributio with kow k d ukow ω, which yields: 3

6 k S k e g, S e (, S) k k ψξω = ωξ ωξ ω = ω ω φω = ξ Γ k ( S + ξ ) The Retiler s Problem S k S + We ow review the solutio to the retiler s stockig problem whe beliefs bout the ukow prmeter re updted i Byesi fshio d umet demd is lost d uobserved (herefter, we simply sy lost sles). A full lysis is preseted i Lriviere d Porteus (995). All costs d reveues re lier. The product is sold t fixed retil price r. The retiler is chrged h for ech uit tht remis usold d p for ech uit of ustisfied demd. Future profits re discouted by fctor α, d x deotes the retiler s ivetory level before orderig. The mufcturer follows lier price schedule, postig wholesle price for ech period d supplyig s much s the retiler wts t tht price. Further, we restrict the mufcturer to set of dmissible lier price schedules i which the posted wholesle price depeds o the period, the curret stte of iformtio, d the ormlized retiler ivetory. (The ivetory ormliztio is explied below.) A dmissible schedule is idepedet of the scle prmeter S or the exct history of sles. We deote the wholesle price for the curret period by w, suppressig its depedece o other prmeters for simplicity. The mufcturer cot cotrct o future prices lthough the retiler is ssumed to ticipte correctly the price for ech possible future stte. We retur to the importce of this ssumptio lter. Limitig the mufcturer to dmissible schedules is obviously restrictive. We rgue below tht such schemes re ll she eed cosider becuse they re the oly lier price schedules tht c be self-eforcig. Give the mufcturer s ibility to commit to future prices, dmissible schedule is the oly lier schedule whose use the retiler rtiolly ticiptes. Lriviere d Porteus (995) ssume fixed wholesle price is chrged over the horizo but poit out tht their Theorem (s stted below) geerlizes to the cse i which wholesle prices follow oe of our dmissible schedules. Thus, lettig f ( x S), deote the retiler s mximum expected discouted profit over periods, +,, N whe strtig period with x 4

7 =, be uits of stock before orderig d sufficiet sttistics d S, d lettig f( x) f( x ) the solutio of the ormlized problem with S =, we obti the followig result. Theorem. Assume tht the mufcturer uses dmissible schedule, tht the uderlyig demd distributio is Weibull with kow k d ukow ω, d tht the prior o ω is gmm with prmeters d S such tht k >. The lettig qs the followig hold: = S k, () The optiml vlues of the origil problem c be foud by sclig the ormlized x f x, S = q S f. vlues: qs (b) The optiml level, y ( x S),,, of ivetory fter orderig i period for the origil x problem c be foud by sclig the optiml level, y, qs, of ivetory fter orderig x i period for the ormlized problem: qs y x,, S = q S y, Heceforth, we ssume tht the coditios of the theorem hold uless otherwise stted. The orderig level for the origil problem depeds o the criticl umbers of the ormlized oe. It is strightforwrd to show tht the sme frctio of demd is met i either system. The optiml level of service depeds oly o the cost structure d the stte of iformtio s mesured by. The mout of stock eeded to provide tht service depeds o the size or scle of the mrket s mesured by qs. Whether the retiler expects profit or loss similrly depeds o the cost structure, ormlized ivetory, d the stte of iformtio; the mgitude of the profit or loss depeds o the scle of the mrket. The fuctio qs is turl mesure x of mrket size d is used to ormlize retiler ivetory (i.e., qs ) i pplyig dmissible price schedule. Ituitively, give ivetory level my be sigifict i smll mrket but trivil i lrge oe. 2. THE MANUFACTURER S PRICING PROBLEM We ow tur to the determitio of the optiml price schedule. We tckle the problem i stges, solvig dymic progrm by first limitig the policies tht my be used d the rguig tht the optiml policy must lie i the restricted set. We begi by cosiderig the 5

8 mufcturer s profits from usig dmissible schedule s defied bove. As these hve simple form, determiig the optiml dmissible schedule is strightforwrd. We the show tht the optiml dmissible schedule is the oly lier schedule to which the mufcturer my credibly commit. Chel Structure Before lyzig the pricig problem, we must formlize the mufcturer-retiler reltioship. The chel members re iformtiolly equivlet; both kow the fmily of the demd distributio d hve the sme prior o the ukow prmeter ω, d both observe retil sles. Admittedly, these re drmtic simplifictios. There is o compellig reso to believe tht the prties would hve ideticl ssessmet of ew product s prospects eve if they hd ccess to the sme iformtio, d mufcturer c rrely peer directly ito retiler s busiess. However, the ssumptios force updted estimtes of the ukow prmeter to coicide. Everyoe thus hs ideticl vlues of d S d derives the sme updted demd distributio. Further, ssumig tht retil sles re visible to the mufcturer prevets the retiler from mipultig iformtio to gi more fvorble price. The firms re differetited by their respective decisio rights. The retiler cotrols his shelf spce, orderig to mximize his ow profits give the posted price. He refuses the product if stockig it would be uprofitble. The mufcturer desigs d offers the terms of trde. Movig first, she ticiptes the stockig policy d cts s Stckelberg leder i settig the price. The optiml wholesle price is thus the oe tht is best for the mufcturer d ot the cotrct tht mximizes overll chel profits s cosidered by some (e.g., Jeuld d Shug, 983). The terms of trde re preseted s tke-it-or-leve-it offer with o opportuity for couteroffers or brgiig. The retiler ccepts y del offerig o-egtive retur. Expected Mufcturer Profits from Admissible Schedule The mufcturer hs costt mrgil cost of productio c d discouts future profits by fctor β per period. For ow, ssume the retiler ever drops the product. The mufcturer follows rbitrry dmissible schedule W, d the curret period s wholesle 6

9 price is deoted simply by w. The retiler correctly ticiptes the use of W d orders optimlly. For simplicity, we suppress the order qutity s depedece o y prmeter d write y y ( x,, S) d γ for y ( x ),. for Let ( x S W) π,, be the mufcturer s expected profits t the strt of period from followig the price schedule W for the remider of the horizo give curret shpe d scle prmeters d retiler ivetory. The oly ucertity the mufcturer fces is the evolutio of the system. Her curret period profits, ulike the retiler s, re determiistic; she kows exctly wht the retiler will order t y wholesle price. Expected profits re the: k (,, ) = ( )( ) + ( +, +, ) (, ) π x S W w c y x β π y ξ S ξ W φ ξ S dξ where ( x S W) π N k ( 0,, ) (, ) + β π S+ y Wφ ξsdξ y + y 0,, for ll x,, S, d W. As with the retiler s problem, we defie the ormlized cse s hvig S = d write π ( x, W) for π ( x,, W) φξ ( ) φξ =, be the ormlized updted demd distributio. Theorem 2. Lettig () = ( + ) Ut t k k, the followig hold:. Similrly, let γ t () () + U( t) + ( 0, ) ( ) π x, W = w c γ x + β U t π +, W φ t dt + β U γ π W φ ξ dt γ γ 0 () (b) The expected vlue of the origil problem c be foud by sclig the ormlized π x vlue: qs xsw,, = qsπ W,. The proof ppers (s do ll others) i the Appedix. From the first prt of the theorem, expected mufcturer profits from followig schedule W i the ormlized cse c be foud from simplified system i which future rewrds re expressed i terms of ormlized profits. 7

10 By the secod prt, the correspodig qutity for the origil system is determied simply by the pproprite sclig of the ormlized profits. As the coditios of Theorem hold, the x results re ot surprisig for sigle period problem; the retiler orders y x = q( S) qs γ. The mufcturer s sles, d hece her profits, re scled by the ormlizig fuctio. The propositio estblishes tht future profits re similrly scled whe the mufcturer uses dmissible schedule. The Optiml Admissible Price Schedule Becuse the expressio for ormlized mufcturer profits () is idepedet of the scle prmeter, price schedule tht mximizes expected ormlized profits must be dmissible schedule. Determiig the optiml dmissible schedule is the reltively strightforwrd. The process is further simplified by otig tht i both the origil d ormlized system, the mufcturer s curret period profits deped o wht she hs chrged erlier oly through pst retiler orders d pst demd reliztios. Wht hs bee ordered d wht is kow of pst demd, however, re completely cptured by the curret sufficiet sttistics ( S, ) d the retiler s ivetory x. The optiml dmissible schedule my cosequetly be developed recursively. Let ( x S) Π, be the expected mufcturer s profits from usig the optiml dmissible price schedule for the origil system from period to the ed of the horizo. For must stisfy the optimlity equtios: =,, N, it Π y x S w c y x β y ξ S ξ φ ξ S dξ w 0 0 k (, ) = mx ( )( ) + Π + ( +, + ) (, ) k ( 0, ) (, ) + β Π + S+ y φ ξsdξ y where Π N + ( x, S) 0 for ll x,, d S. Let Π ormlized system. x be the correspodig qutity for the 8

11 Corollry. Assume there is uique optiml wholesle price for ech period, iformtio stte, d ivetory level triplet i the ormlized cse, d deote the optiml dmissible schedule for tht cse s W. () W my be foud by solvig the followig optimlity equtios: Π γ t mx () Π + U ( t ) ( ) γ x = w c γ x + β U t + φ t dt w β U( γ ) Π + ( 0 ) φ( t) dt γ (2) for =,, N, where Π N + ( x) 0 for ll x, d. (b) The optiml vlues of the origil problem c be foud by sclig the ormlized Π x, S = q S Π. x vlues: q ( S ) (c) W is optiml for both systems. The proof follows directly from Theorem 2. Sice the mufcturer s profits from the full system re just sclr multiple of her profits from the ormlized system, the sme dmissible schedule is optiml for both. We heceforth ssume tht the coditios of the corollry hold d tht uique, optiml dmissible schedule exist. Giig Retiler Prticiptio To this poit, we hve ssumed tht the retiler ever drops the product lthough othig i the formultio of (2) gurtees the retiler stisfctory retur. The optiml dmissible schedule W my price the product out of the mrket, chrgig so much tht the retiler refuses to stock it. Recllig Theorem, the problem is esily resolved. Corollry 2. Solvig (2) subject to giig retiler prticiptio yields dmissible price schedule. 9

12 x The retiler s prticiptio costrit i period reduces to f q S 0. Thus, eve if the costrit bids, the resultig wholesle price will deped oly o the period, stte of iformtio, d ormlized ivetory. As the retiler c credibly threte to drop y product tht does ot llow him to brek eve, the mufcturer will lwys choose to ccommodte him d devite from the ucostried price schedule. The optiml dmissible price schedule should cosequetly be uderstood to be the optiml schedule subject to giig retiler prticiptio. Credible Price Schedules We hve so fr show tht the mufcturer s profits from dmissible schedules hve simple form d tht the structure of her profits my be exploited to fid the optiml dmissible schedule. The lysis hs bee predicted o restrictig the mufcturer to schedules drive by the curret period, iformtio stte, d ormlized retiler ivetory. Prices cot be fuctios of the sles level or the scle prmeter lthough the mufcturer could clerly do o worse with richer cotrcts. However, she will be uble to implemet more detiled schemes. She is limited by her ibility to commit to future prices. The mufcturer is ssumed boud oly by the price offered for the curret period, so she my post y price t the strt of the ext period regrdless of erlier promises. Oly price schedules tht form subgme perfect equilibri (Selte, 975) re the credible, d oly the optiml dmissible schedule meets tht criterio. Theorem 3. The optiml dmissible schedule determied subject to retiler prticiptio is the oly lier schedule tht forms subgme perfect equilibrium. The result follows from the uiqueess of the optiml wholesle price. For ech stte of iformtio d ormlized ivetory level, the mufcturer hs uique optiml price i the fil period. Ay prior clim to devite from those prices is ot credible. The mufcturer c cosequetly oly commit to the optiml dmissible schedule i the fil period. Give tht she will use the optiml dmissible schedule i the lst period, she hs uique optiml price for ech stte i the peultimte period, d similr rgumet the itertes. 0

13 Tble A Simple Idmissible Schedule The Optiml Admissible Schedule A Idmissible Schedule Period Period 2 (s < y) Period 2 (s = y) if s if s = 4 y is the retiler s Period order, d s observed Period sles. p = 0, h = 0., r = 0, c = 2, =,. S = 0 We illustrte the propositio by exmple. Tble presets the optiml dmissible schedule d simple idmissible schedule for two period horizo with expoetil demd d perishble ivetory. The idmissible schedule forms Nsh equilibrium but fils to be subgme perfect. Uder the optiml dmissible schedule, the mufcturer chrges 4.29 i the first period d either 4.59 (followig stockout) or 4.98 (followig exct observtio) i the secod. The retiler would order 2.7 uits i the first period. The idmissible schedule lso chrges 4.29 i the first period d 4.98 followig exct observtio but oly imposes 4.59 followig stockouts t or bove 4 uits. After lower stockouts, 9.50 is chrged. The itet is cler: To coerce lrger iitil order. The mufcturer would sell more i the first period but chrge her preferred prices i the secod. The retiler s secod period mrgi is imperiled uless he tkes 4 i the first. A order of 4 is thus prt of Nsh equilibrium. If the retiler believes the mufcturer will follow the ouced policy, he orders 4. Give tht the retiler orders 4, the mufcturer follows the posted schedule. Although Nsh, the equilibrium is ot subgme perfect. It requires the retiler to believe tht for some sttes of the world (i.e., stockouts below 4), the mufcturer will ot chrge her profit mximizig price. If the retiler were to cll her bluff (sy, by orderig 2.7), the

14 mufcturer would reege o the secod period prices d revert to the optiml dmissible schedule. Uless the mufcturer is edowed with some commitmet mechism, the retiler should igore y clim to follow other policy d expect the optiml dmissible schedule i the fil period. The logic crries over to loger horizos d more itricte pricig schemes. Uless the mufcturer costructs richer cotrct limitig future pricig decisios (which is beyod the scope of our model), the oly lier price schedule the retiler c rtiolly ticipte is the profit-mximizig dmissible price schedule. Ay other ouced pl lcks credibility. 3. BEHAVIOR OF THE OPTIMAL PRICE SCHEDULE We ow chrcterize the depedece of the optiml price schedule o model prmeters. We do this through simple exmple tht presumes true demd is expoetil d tht ll ivetory is perishble. With perishble ivetory, the retiler s problem becomes trivil series of ewsvedor problems if either the demd distributio is kow or Byesi updtig is doe uder fully observed demd. With Byesi updtig d lost sles, however, the retiler s problem remis chllegig becuse the stockig level still determies the rte t which iformtio is cquired. Lriviere d Porteus (995) show tht the optiml (ormlized) retiler order for period is: γ = ( + ( ) ρ+ ) r + p+ h+ αρ + w+ h where ρ ( ) f = 0 d ρ N + 0 for ll. Perishble ivetory lso simplifies the mufcturer s problem. From the retiler s orderig policy, the mufcturer fces determiistic demd curve. Whe the retiler crries ivetory ito the period, the mufcturer cofrots residul demd curve; the retiler buys oly the differece betwee his order-up-to qutity d his stock o hd. It is immedite to show tht the mufcturer s mrgil reveue flls, which fvors lower wholesle price. I the perishble cse, the mufcturer will ever be costried by retiler ivetory. The perishble ivetory logue of (2) is: 2

15 for γ Π = mx ( w c) γ + β U( t) Π + ( + ) ( t) dt+ U Π + ( t) dt w φ β 0 γ φ 0 γ =,, N, where ( ) Π N + N + stte of iformtio must stisfy: 0 for ll. The optiml wholesle price for period d w c w ( + h) βσ ( ( + ) ) + σ + w+ = δ δ where = ( ) Π d δ = r + p+ h+ α ρ ( + ) ρ σ h ( + + ). (3) The Iformtiol Dymics of Pricig i the Sigle Period Problem Clerly, there is geerlly o explicit solutio for (3). However, the correspodig coditio for the sigle period cse is substtilly simpler: w c w ( + h) w+ h = r+ p+ h (4) d llows us to exmie how the optiml price for the oe period horizo chges with vrious prmeters. Theorem 4. Cosider oe period problem, d ssume the retiler s prticiptio costrit does ot bid. () The optiml wholesle price w is icresig i the retil price r, the shortge pelty p, d the mrgil cost of productio c. (b) w r w = p <, d w w c c < if > c+ h. (c) The wholesle price lso icreses with the stte of iformtio. The first prt of the theorem is hrdly surprisig. It my seem odd to tke the wholesle price s fuctio of the retil price s covetiolly oe hs the reverse. I our cse, the retil price is fixed, mkig it piece of dt for the mufcturer s problem. To the mufcturer, 3

16 icreses i the retil price d the stockout pelty re equivlet becuse they hve ideticl impcts o retiler behvior (despite mrkedly differet impcts o retiler profits s discussed i Lriviere d Porteus, 995). The icrese i w followig icrese i either is less th oefor-oe; gis from higher retil price re split betwee the prties. Similrly, higher productio costs re pssed o to the retiler. If the stte of iformtio is sufficietly high, the icrese i the wholesle price is less th the icrese i mrgil cost. Otherwise, the wholesle price will icrese more th the uit cost. The more iterestig result is the fil prt of the theorem. The mufcturer chrges more s the precisio of chel iformtio improves. Cosequetly, the mufcturer posts higher price if there were exct observtio i the prior period th if ot. Stted differetly, the mufcturer chrges more followig sigl of wek demd (usold stock) th followig sigl of strog demd (empty shelves). The outcome cotrdicts stdrd ituitio; oe would geerlly expect higher price i lrger mrket. Here, the reverse holds, d the mufcturer demds higher price i smller mrket. The Elsticity of Retiler Orders d the Stte of Iformtio We idetify two fctors tht uderlie this ppret omly. First, by Corollry, the optiml price is idepedet of the mrket scle. Wht exct d cesored observtios ported for potetil mrket size is immteril. The type of observtio oly mtters to the extet tht it ffects the stte of iformtio. Secod, the retiler s orderig policy shifts systemticlly with the stte of iformtio. We ow show tht his sesitivity to price vritio depeds o the shpe prmeter d tht he becomes icresigly less sesitive to price chges s the chel hs more iformtio. Cosider the iverse demd curve the mufcturer fces for rbitrry demd distributio Fy: wy = p+ r+ h Fy h We ow substitute "w= w+ h d "r = r + h. The chge represets ltertive costig covetio. The retiler ow chrges himself for holdig the product t its cquisitio but rebtes the chrge whe sellig uit. As we hve implicitly ssumed tht costs re 4

17 deomited i strt-of-the-period uits, the lysis is uffected by the ccoutig chge. With the ew covetio, the iverse demd curve is wy " = ( p+ r" ) Fy. Lettig f( y ) be the desity of the demd distributio, the correspodig ow-price elsticity is: "ν y = Fy y f y (5) "ν ( y ) is the elsticity of demd the retiler plces o the mufcturer, ot the elsticity of cosumer demd fced by the retiler. We hve ot explicitly modeled the cosumer purchse decisio d hve ssumed costt retil price. O the other hd, by modelig the retiler s orderig policy, we hve defied iduced demd curve (d the equivlet iverse demd curve) tht the mufcturer fces. "ν ( y ) is the elsticity of retiler orders, mesurig the percetge chge i the qutity the retiler demds for percetge chge i the wholesle price. It is iversely relted to the hzrd rte of demd wheever the retiler fces sigle period ewsvedor problem. We defied "ν ( y ) for y distributio. If Fy is ow the pproprite updted demd distributio, we c relte the elsticity of retiler orders to the stte of iformtio. Theorem 5. If gmm prior, ow-price elsticity F y is the updted demd distributio for ewsboy distributio with "ν y decreses with the stte of iformtio if: ) the horizo is oe period d the true demd distributio is y ewsboy distributio, or b) the horizo is N d the true demd distributio is expoetil. Demd s perceived by the mufcturer becomes less elstic s the shpe prmeter icreses. Whe iformtio is imprecise, the icrese i volume fter price cut is lrge, d the mufcturer fvors lower price. Whe the chel hs more refied iformtio, the drop i sles followig price hike is less severe. The mufcturer exploits icresed certity by risig her price. Ideed, if demd were determiistic, the mufcturer would chrge rbitrrily close to the retil price d cpture ll chel profits, levig the retiler with the 5

18 smllest cceptble mrgi possible. She would succeed becuse retiler orders would be completely ielstic for y wholesle price below the retil price. At the other extreme, the retiler is very ucerti d hece extremely price sesitive. The mufcturer respods with reltively low levy tht icreses s iformtio becomes more precise. The determiistic exmple highlights the result s depedece o the ssumed power structure. I the determiistic cse, the mufcturer c exploit price isesitivity becuse she presets the terms of trde s tke-it-or-leve-it offer. We cot chrcterize the outcome of brgiig over price. The sme is true i our settig. The mufcturer s ledership positio llows her to tke dvtge of the growig ielsticity of retiler orders through higher price. 4. PRICES OVER TIME AND NON-PERISHABLE INVENTORY Our lysis of the optiml schedule hs so fr bee limited to chges withi give period. We hve exmied how the optiml price vries with the stte of iformtio holdig the remiig horizo fixed. Eqully iterestig is how the wholesle price chges with the remiig horizo holdig the stte of iformtio fixed. We would like to kow whether the mufcturer follows peetrtio (i.e., w w + ) or skimmig (i.e., w w > + ) strtegy. Also, our chrcteriztio to dte hs bee limited to perishble ivetory. It is ucler how the results of the previous sectio would chge if retiler ivetory ws ot perishble. We ddress these poits through exmple for which we hve explicit solutio for the oe period pricig problem. If ivetory is perishble, there is explicit solutio for loger horizos. We cotiue to ssume tht the true demd distributio is expoetil d work with the shifted costig covetio itroduced bove: "w= w+ h d "r = r + h. Additiolly, we hve "c= c+ h to mke mufcturer s costs comprble. Equtio (4) becomes: w" c" w" = w" r" + p (6) We ow ssume tht "c = 0. Usold product is slvged t its mrgil cost of productio. The product would be riskless for itegrted firm tht owed both the mufcturer d retiler. The itegrted chel would stock ifiite mout to serve 00% 6

19 of demd. While clerly urelistic, the exmple offers isights becuse the story is ot s simple i decetrlized chel. A profit mximizig mufcturer prices bove her mrgil cost, so the retiler will ot meet ll of demd. The outcome is vritio of the stdrd double mrgiliztio rgumet (Spegler, 950). A wholesle price bove mrgil cost does ot ffect the retil price but does iduce the retiler to hold less stock th the itegrted firm. Prices over Time Clerly, if "c = 0, the optiml wholesle price for the oe period problem with perishble ivetory is " = (" + )( ) w r p, which iduces order of γ = ; the retiler stocks to meet verge demd. It is ot coicidetl tht "ν ( ) =. As productio is costless, the mufcturer mximizes reveue i the lst period, d this is ccomplished where elsticity is oe. The correspodig mufcturer d retiler profits for oe period horizo re: σ ( r" p) ρ r" ( r" p)( ) ( ) = + = + + "+ r r p The rtio of decetrlized to cetrlized profits is " ( ), d the decetrlized chel uder-performs cetrlized oe. The reltive iefficiecy icreses with the stockout pelty. The itegrted firm ever icurs y shortge pelty becuse it stisfies ll demd. Both mufcturer d retiler profits icrese with the stte of iformtio. This goes beyod the results of Lriviere d Porteus (995). They show tht retiler profits re icresig i the stte of iformtio for fixed wholesle price. Here, retiler profits re icresig i eve s the mufcturer optimlly djust the wholesle price upwrds. To cosider two period problem, we first defie: σ 2 σ = ( + ) σ ( + ) ρ ρ ρ 2 = r" + p r" + p 2 2 Lriviere (995) shows tht either σ 2 ( ) or ρ depeds o the cost prmeters "r d p d tht this result c be exteded to rbitrry horizo. From (3), the first period wholesle price is: 7

20 2 w" (" = r + p) ( + 2 ) βσ αρ + + αρ2 Prices re costt cross the horizo oly if both the mufcturer d retiler totlly discout the future (i.e., α = β =0 ). If the mufcturer vlues the future t ll (i.e., β > 0 ), she prices below the level tht mximizes her curret profits. She scrifices curret erigs to iduce higher stockig level, icresig the likelihood of rechig more profitble future stte. Coversely, if the retiler vlues the future (i.e., α > 0 ), the mufcturer chrges more. Lriviere d Porteus (995) show tht forwrd-lookig retiler holds excess stock to icrese the rte of iformtio cquisitio. The retiler buys more t every wholesle price, expdig the mufcturer s mrket. Prt of tht growth is tke through higher wholesle price. As the first period price depeds o both the mufcturer d retiler discout fctors, the price pth over the horizo does s well. From mipultig the prices, we hve: Theorem 6. I two period problem with "c = 0, the mufcturer uses peetrtio w" w" ) if : pricig (i.e., 2 β β = 2 ( αρ2) αρ σ The criticl vlue β "w icreses with α while "w is icresig i the retiler s discout fctor α. The first period price 2 is uchged. Cosequetly, the more the retiler looks to the future, the greter the rge of β for which the mufcturer uses skimmig strtegy. The criticl vlue s depedece o the stte of iformtio is ot s simple. From Figure, oe sees tht β is ot mootoic i the shpe prmeter; the rge of mufcturer discout fctors i which skimmig strtegy is used t first icreses but the flls s more iformtio is vilble. The grphed exmple hs α equl to oe d is cosequetly upper boud o β becuse the criticl vlue is icresig i α : If the mufcturer would set w" w" 2 for α equl to oe, she would do the sme for y retiler discout fctor. As the mximum 8

21 0.0 Figure β for α = vlue o the upper boud for this specil cse is just greter th 0.0, the mufcturer would follow peetrtio strtegy for ll but extremely low mufcturer discout fctors. No-Perishble Ivetory We ow cosider how retiler ivetory iflueces the mufcturer s pricig decisio. As discussed bove, whe the retiler crries ivetory ito the period, the mufcturer fces residul demd curve mkig her mrgil reveue lower t every wholesle price. She cosequetly chrges less th she would if ivetory were perishble. For oe period problem i which the retiler hs ormlized ivetory x, the logue of (6) is: x + w" c" w" = w" r" + p (7) = r" + p If the retiler s ivetory is x ( c" ) hs uique solutio: w" c", the stockig level of the itegrted chel, (7) =. If the retiler hs ivetory bove x, the mufcturer would hve to price below cost to iduce the retiler to order more stock. For retiler ivetory below 9

22 the itegrted stockig level, the mufcturer c lwys fid price bove cost tht would iduce the retiler to tke more stock. The itegrted chel s stockig level is decresig i the stte of iformtio. Oe c therefore costruct exmples i which the optiml wholesle price flls s the shpe prmeter icreses for fixed level of ivetory. Suppose the retiler s ivetory is x ( ) +. If the stte of iformtio is, the mufcturer offers the product t some price bove mrgil cost. O the other hd, if the stte of iformtio icreses by oe, the price drops to "c, d the wholesle price would be decresig i the stte of iformtio cotrry to Theorem 4. Oe does ot hve to go to such extreme cse for the wholesle price to be decresig i the shpe prmeter. We retur to our exmple with "c = 0. The solutio to (7) is ow: w" = ( r" + p) x ( + ) As the itegrted firm would hold ifiite mout of stock, the mufcturer i the decetrlized chel c lwys iduce the retiler to order more t price bove mrgil cost. The price, however, is decresig i the retiler ivetory level. Figure 2 grphs the optiml price s fuctio of the shpe prmeter for three ivetory levels (0, 0.2, d 0.8). Oe sees tht the wholesle price is icresig for low sttes of iformtio but evetully flls with the stte of iformtio whe ivetory is positive. The shpe prmeter t which the price begis to declie decreses with the ivetory level. For this simple cse, we re ble to determie the ivetory level below which the wholesle price icreses with the stte of iformtio. 20

23 Figure 2 The Optiml Wholesle Price s Fuctio of the Shpe Prmeter for Three Ivetory Levels "r + p = x = 0 x=0.2 x=0.8 Theorem 7. If "c = 0, the wholesle price icreses with the stte of iformtio if: x ( ) e. Theorem 4 must the be qulified. If retiler ivetory is sufficietly low, the optiml wholesle price will be icresig i the stte of iformtio. If retiler ivetory is ot sufficietly low, the price will be decresig i the stte of iformtio. The cvet reflects two coutervilig forces tht ffect the reltioship betwee the wholesle price d the stte of iformtio. O the oe hd, the retiler becomes less price sesitive s the stte of iformtio icreses, d the mufcturer cosequetly wts to rise the wholesle price. O the other, the retiler s order-up-to level for fixed wholesle price flls with the stte of iformtio; he c stisfy greter frctio of his eeds with stock lredy o hd. The mufcturer compestes by cuttig the wholesle price. Which force previls depeds o the problem. 2

24 5. CONCLUSION We hve exmied mufcturer luchig ew product ito existig retil chel. At the time of itroductio, either the mufcturer or the retiler kows some prmeter of the demd distributio. Further, umet demd is both lost d uobserved; beliefs bout the demd distributio must be updted bsed o sles observtios. While the retiler s ivetory thus dicttes the rte t which the chel cquires iformtio, the retiler sets the stockig level to mximize his ow profits becuse he is idepedet of the mufcturer. Ideed, he my eve drop the product if he cot er stisfctory retur. The mufcturer must the cosider the retiler s stockig policy whe settig the wholesle price for ech period of the fiite horizo. She cts s Stckelberg leder, ticiptig the retiler s orderig policy d cosequetly fcig determiistic, iduced demd curve. Buildig o Lriviere d Porteus (995), we show tht the mufcturer s optiml lier schedule subject to giig retiler prticiptio is drive by the precisio of vilble iformtio, ormlized retiler ivetory, d the period. It is ot depedet o the size of the mrket or the exct history of sles. The optiml schedule c therefore be determied prior to luchig the product. For the cse of perishble ivetory with expoetil demd, we show tht the optiml wholesle price is icresig i the stte of iformtio. As cosequece, higher price is chrged followig sigl of wek demd (product left usold) th followig sigl of strog demd ( stockout). The result follows from the iduced demd curve the mufcturer fces becomig icresigly ielstic s the precisio of iformtio improves. As he hs better iformtio regrdig the ukow prmeter, the retiler becomes less price sesitive. The mufcturer exploits his growig isesitivity by risig the wholesle price. We lso show through simple exmple tht the mufcturer will follow peetrtio strtegy risig prices over the horizo if her discout rte is sufficietly high. Our results re subject to limittios. We demostrte, for exmple, tht the wholesle price my ot icrese with the stte of iformtio if ivetory is ot perishble. If retiler ivetory is lrge, the optiml wholesle price my fll with the stte of iformtio. I our exmple, the retiler s optiml stockig level (ll else beig equl) decreses with the stte of iformtio. The mufcturer s potetil mrket shriks puttig dowwrd pressure o the 22

25 wholesle price. Icresig retiler price isesitivity but decresig overll demd pull the optiml wholesle price i opposite directios. The optiml schedule s idepedece from the exct history of sles d the size of the mrket is due to our distributiol ssumptios d preclusio of loger term cotrcts. While we give exmple i which the mufcturer could do better with idmissible lier schedule if she could commit to its use, lyzig the retiler stockig policy for geerl idmissible schedule would be hopelessly complex. Similrly, the ssumptio of Weibull demd is bsed o the desire to use ewsboy distributio. As Brde d Friemer (99) ote, modeler delig with cesored dt but ustisfied with y of the distributio they suggest should expect serious lytic d computtiol problems. (p. 390) If we forgo ssumig tht umet demd is uobserved, greter rge of fmilies become lyticlly fesible. We cojecture tht the optiml lier price schedule with bcklogged sles would be idepedet of the scle of the mrket for the settigs cosidered by Azoury (985) becuse the retiler s orderig policy would be still give by sclig of ormlized problem. Filly, fixed retil price is problemtic. O the oe hd, the retiler my order i self-iterested mer, but o the other, he hs cot djust the retil price. Agi, trctbility ecessittes the ssumptio lthough it is more defesible th, sy, the ssumptio of Weibull demd. We hve implicitly ssumed tht the chel members hve some experiece i the product s ctegory o which to bse their priors. The product is ot completely ew d must fit ito existig mrket. The retil price is the set to trget prticulr segmet. The chel, however, is ucerti of the size of the segmet. 23

26 APPENDIX Proof of Theorem 2 We begi with lemm tht follows from simple lgebric mipultios: Lemm A. Give qs hold: () ( S) φξ, = φ ξ. qs qs qs+ ξ = qsu. (b) ( k ξ ) qs = S k, () = ( + ) Ut t k k, d φξ φξ ( ) =,, the followig We prove the theorem by bckwrd iductio o. It clerly holds for i suppose it is true for i Lemm A, d chge of vribles t = ξ qs. = N. We = + d verify tht it holds for i = usig the iductive hypothesis, k y ξ (,, ) = ( )( ) + ( + ) k +, + (, ) qs+ ξ π x S W w c y x β q S ξ π W φ ξ S dξ k + ( 0, ) (, ) + β qs+ y π Wφ ξsdξ y qs y ξ ( qs ) + ( 0, ) qs ξ y ξ ξ +, ξ ( qs ) = w c y x + β U π + W φ dξ U q S q S + β U π W φ dξ y y 0 y 0 γ x γ t = qs ( w c) ( γ ) + β Ut () π qs + +, Wφ tdt U t 0 + β U( γ ) π + ( 0, W) φ( t) dt γ = qs x π qs ( W, ) 24

27 Proof of Theorem 3 Defie the stte of the system i period s the previlig shpe prmeter d ormlized ivetory level. We mke the followig observtio: Lemm A2. It is optiml to use the optiml dmissible price schedule i the curret period for the previlig stte if the mufcturer pls o usig the optiml dmissible price schedule i ll possible future periods d sttes. The lemm follows immeditely from Corollry d the ssumptio of uique profit mximizig price: Give tht the optiml schedule will lwys be used i the future, there is uique, optiml price for the curret period d stte. Becuse the optiml schedule is determied recursively, tht uique price must be prt of the optiml schedule. Turig to the theorem itself, suppose there were lier pricig schedule W ~ tht formed subgme perfect equilibrium but devited from the optiml dmissible price schedule. As we re cosiderig idmissible schedules, the stte ow icludes ll vilble iformtio (e.g. the exct history of sles or retil orders) i dditio to the shpe prmeter d ormlized ivetory level. The ssumed order of ply hs the mufcturer postig price t the strt of ech period tht is bidig for tht period. A schedule for future sttes my be ouced but is ot bidig. Ech period-stte pir thus forms subgme. Let ~ be the ltest period for which ~ W devites from the optiml dmissible schedule for some stte. We hve two possibilities, ~ = N or ~ < N. For the former, the mufcturer hs uique profit mximizig price tht depeds o pst observtios oly through the shpe prmeter d is prt of the dmissible schedule. The proposed schedule, however, differs from the optiml dmissible schedule i period N for some stte. Thus, for t lest oe subgme, the mufcturer would hve icetive to uilterlly devite from W ~ ~. W cot be prt of subgme equilibrium. The secod cse is similr. By Lemm A2, there is uique optiml wholesle price for ech subgme tht begis i period ~ tht oly depeds o the shpe prmeter d ormlized ivetory level, but ~ W clls for differet price i t lest oe such subgme. Hece, ~ W cot be subgme perfect. 25

28 Proof of Theorem 4 We first show tht for oe period problem with expoetil demd, the mufcturer s profits re cocve i the wholesle price d thus y w solvig (4) i the text is thus the uique, optiml wholesle price. Lemm A3. Let π( w) = ( w c) γ π w is cocve i w.. Defie T r p h = + + w h. The γ = d: + T π ( w) w 2 π ( + h) ( w) w w c = T w = T w + c+ + 2h w h ( + ) Recllig from Theorem tht k > completes the proof. (4) follows settig π ( w) = 0. To prove the theorem, we pply the implicit fuctio theorem to the first order coditios to yield: T (( ) w h c) ( ) + ( + ) + 2 w = w = + + r p ( w c h) h ( w) w = + c ( w+ + c+ 2h) ( h w) ( ( w c) Log( T) ( w h) ( w c) ( ) w+ ( + ) c+ 2h ) w = + + As >, the first two expressios re clerly positive, provig the first prt of the propositio. The secod prt follows by otig tht T >. The fil prt will hold if: w c w h w c ( + ) ( ) > Log T which from the first order coditios is equivlet to T Log( T ) w > d thus lwys true. 26

29 Proof of Theorem 5 For (), the updted demd distributio for ewsboy distributio with gmm prior is S = S +. Deotig the first derivtive of dy d ( y ) s d Fy "ν y ( y) S + d y = y d y, (5) becomes: S which is decresig i the stte of iformtio. For expoetil demd, F( y) = ( S + y) α( ρ+ ( + ) ρ+ ) ( w+ h ) d y S r p h =. The iverse demd curve for the mufcturer i period is the: Its elsticity reduces to "ν ( y) ( ( + + ))( ) αρ ρ w" y = r" + p+ + F y = +, which is gi decresig i. S y y Proof of Theorem 7 Differetitig "w with respect to, oe hs: w" = + which is positive if Log ( x + ) ( r" p) x Log ( + ) x ( + ) or, equivletly, x e +. 27

30 REFERENCES Azoury, Kty S., Byes Solutio to Dymic Ivetory Models uder Ukow Demd Distributio, Mgemet Sciece, 3, 9 (September, 985), Brde, Dvid J. d Mrshll Freimer, Iformtiol Dymics of Cesored Observtios, Mgemet Sciece, 37, (November, 99), DeGroot, Morris H., Optiml Sttisticl Decisios, McGrw-Hill, New York, New York, 970. Jeuld, Abel P. d Steve M. Shug, Mgig Chel Profits, Mrketig Sciece, 2,3 (Summer, 983), Lriviere, Mrti A., Optiml Mufcturer-Retiler Cotrctig uder Ukow Demd Distributio, Upublished Doctorl Disserttio, Stford Uiversity, 995. Lriviere, Mrti A. d Ev L. Porteus, Stlkig Iformtio: Byesi Ivetory Mgemet with Uobserved Lost Sles, workig pper, Duke Uiversity. Lee, Hu L. d Meir J. Rosebltt, A Geerlized Qutity Discout Pricig Model to Icrese Supplier s Profits, Mgemet Sciece, 32, 9 (September, 986), Mthewso, G. F., d R. A. Witer, A Ecoomic Theory of Verticl Restrits, Rd Jourl of Ecoomics, 5 (Sprig, 984), Moh, Jmes P., A Qutity Discout Profit Model to Icrese Vedor Profits, Mgemet Sciece, 30, 6 (Jue, 984), Moorthy, K. Sridhr, Mgig Chel Profits: Commet, Mrketig Sciece, 6, 4 (Fll, 987), Psterck, Brry Al, Optiml Pricig d Retur Policies for Perishble Commodities, Mrketig Sciece, 4, 2 (Sprig, 985), Selte, R. Re-exmitio of the Perfectess Cocept for Equilibrium Poits i Extesive Gmes, Itertiol Jourl of Gme Theory, 4 (975), Spegler, Joseph J., Verticl Restrits d Atitrust Policy, Jourl of Politicl Ecoomy, 58 (August, 950),

31 Tirole, Je, The Theory of Idustril Orgiztio, The MIT Press, Cmbridge, Msschusetts, 988. Weg, Z. Kevi, Chel Coorditio d Qutity Discouts, Mgemet Sciece, 4, 9 (September, 995),