PRICING AND REPLENISHMENT STRATEGIES IN A DISTRIBUTION SYSTEM WITH COMPETING RETAILERS

Transcription

1 PRICING AND REPLENISHMENT STRATEGIES IN A DISTRIBUTION SYSTEM WITH COMPETING RETAILERS FERNANDO BERNSTEIN The Fuqua School of Busness, Duke Unversty, Durham, North Carolna 27708, fernando@duke.edu AWI FEDERGRUEN Graduate School of Busness, Columba Unversty, New York, New York 10027, af7@columba.edu We consder a two-echelon dstrbuton system n whch a suppler dstrbutes a product to N competng retalers. The demand rate of each retaler depends on all of the retalers prces, or alternatvely, the prce each retaler can charge for ts product depends on the sales volumes targeted by all of the retalers. The suppler replenshes hs nventory through orders purchases, producton runs) from an outsde source wth ample supply. From there, the goods are transferred to the retalers. Carryng costs are ncurred for all nventores, whle all suppler orders and transfers to the retalers ncur fxed and varable costs. We frst characterze the soluton to the centralzed system n whch all retaler prces, sales quanttes and the complete chan-wde replenshment strategy are determned by a sngle decson maker, e.g., the suppler. We then proceed wth the decentralzed system. Here, the suppler chooses a wholesale prcng scheme; the retalers respond to ths scheme by each choosng all of hs polcy varables. We dstngush systematcally between the case of Bertrand and Cournot competton. In the former, each retaler ndependently chooses hs retal prce as well as a replenshment strategy; n the latter, each of the retalers selects a sales target, agan n combnaton wth a replenshment strategy. Fnally, the suppler responds to the retalers choces by mplementng hs own cost-mnmzng replenshment strategy. We construct a perfect coordnaton mechansm. In the case of Cournot competton, the mechansm apples a dscount from a basc wholesale prce, based on the sum of three dscount components, whch are a functon of 1) annual sales volume, 2) order quantty, and 3) order frequency, respectvely. Receved December 1999; revson receved December 2001; accepted June Subject classfcatons: Games/group decsons, noncooperatve: prce and quantty competton among nondentcal retalers. Inventory/producton, mult-tem/echelon/stage: two-echelon supply chan wth procurement setup costs. Inventory/producton, polces, marketng/prcng: coordnatng mechansms va dscountng schemes. Area of revew: Manufacturng, Servce, and Supply-Chan Operatons. 1. INTRODUCTION In ther attempt to mprove or optmze aggregate performance, many supply chans ncreasngly nvestgate and compare ther performance under centralzed and decentralzed decson makng. In a decentralzed system, each chan member optmzes hs own proft functon. The challenge therefore conssts of structurng the costs and rewards of all of the chan members so as to algn ther objectves wth aggregate supply-chan-wde profts. Such a cost and reward structure s referred to as a coordnaton mechansm. If the decentralzed cost and reward structure results n chanwde profts that are equal to those acheved under a centralzed system, the coordnaton mechansm s called perfect. In ths paper, we address these questons for the followng prototype two-echelon dstrbuton system wth competng retalers. A suppler dstrbutes a sngle product or closely substtutable products to multple retalers, whch n turn sell these to the consumer. Each retaler s sales occur at a constant rate whch depends on the prces charged by hm as well as those charged by all other retalers, accordng to a gven retaler specfc demand functon. Alternatvely, the prce each retaler can charge for hs product depends on the sales volumes targeted by all of the retalers. The suppler replenshes her nventory through orders purchases, producton runs) from an outsde source wth ample supply. From there, the goods are transferred to the retalers. Carryng costs are ncurred for all nventores, whle all suppler orders and transfers to the retalers ncur fxed and varable costs, all wth faclty-specfc cost parameters. We consder one addtonal cost component: the suppler may ncur a specfc annual cost for managng each retaler s needs and transactons. We model the management costs assocated wth a retaler account by a concave functon of the retaler s annual sales volume, reflectng economes of scale. Ths cost component has been consdered by Chen et al. 2001); see there for a dscusson of how such account management costs arse n dfferent ndustres. All demand functons and cost parameters are statonary and common knowledge among all channel members. We frst characterze the soluton to the centralzed system n whch all retaler prces, sales quanttes, and the complete chanwde replenshment strategy are determned by a sngle decson maker, e.g., the suppler. The exact optmal centralzed) strategy s unknown and, n any case, of such complex structure as to preclude ts mplementablty, even f t could be computed n a reasonable amount of tme. Ths holds even for the far smpler case where all retaler prces and sales rates are exogenously gven. We are, however, able to derve effcently computable lower and upper bounds whch are shown to be tght, whenever the retalers gross proft margns = retal prce X/03/ $ electronc ISSN 409 Operatons Research 2003 INFORMS Vol. 51, No. 3, May June 2003, pp

2 410 / Bernsten and Federgruen wholesale prce)/wholesale prce] are not excessvely low say, at least 20%), and the annual holdng cost rate s not excessvely large say, less than 30%). The lower bound represents the proft of a strategy wth statonary retaler prces under the optmal power-of-two replenshment polcy to servce the correspondng sales rates at all retalers. Under a power-of-two polcy, all facltes replensh when ther nventory s down to zero and each uses constant replenshment ntervals, specfed as a faclty-specfc power-of-two multple of a gven base perod.) Smlarly, the upper bound represents the proft of a strategy wth statonary retal prces employng a lower bound for the mnmum setup and holdng costs ncurred to servce the correspondng sales at the retalers. We proceed wth the decentralzed system. Here, the suppler chooses a wholesale prcng scheme; the retalers respond to ths scheme by each choosng all of hs polcy varables. We dstngush systematcally between the case of Bertrand and Cournot competton. In the former, each retaler ndependently chooses hs retal prce as well as a replenshment strategy; n the latter, each of the retalers selects a sales target, agan n combnaton wth a replenshment strategy. Fnally, the suppler responds to the retalers choces by mplementng her own cost-mnmzng replenshment strategy. We focus ntally on lnear wholesale prcng schemes where each retaler pays a constant wholesale prce for each unt purchased. We show, under both Bertrand and Cournot competton that, whle a Nash equlbrum may fal to exst under completely general parameter combnatons, an equlbrum n pure strateges) s, n fact, guaranteed under a condton whch relates the retaler s prce elastcty of demand to the rato of hs annual sales and hs combned nventory and setup costs. Ths condton s, agan, shown to be comfortably satsfed n vrtually all product categores.) A related and slghtly stronger condton guarantees that the equlbrum s unque. We proceed wth a comparson between the equlbrum under Bertrand and Cournot competton. We next derve a perfect coordnaton mechansm. In the case of Cournot competton, the mechansm apples a dscount from a basc wholesale prce, based on the sum of three dscount components, whch are a functon of 1) annual sales volume, 2) order quantty, and 3) order frequency, respectvely. Under ths dscount scheme, the optmal centralzed soluton arses as a Nash equlbrum n the resultng retaler game. We derve condtons under whch all Nash equlbra n the retaler game acheve optmal supply-chan-wde profts, thus gvng rse to a strong form of perfect coordnaton. In the absence of retaler competton, dscounts based on the annual sales volume arse only n the presence of account management costs, as demonstrated n Chen et al. 2001). On the other hand, n the presence of retaler competton, such dscounts are requred even f no account management costs preval. The coordnaton mechansm thus provdes an economc ratonale, wthn the context of a model wth complete nformaton and symmetrc barganng power for all retalers, for wholesale prces to be dscounted on the bass of annual sales volumes, one of the most prevalent forms of prce dscount schemes see e.g., Brown and Medoff 1990, Sten and El-Ansary 1992, and Munson and Rosenblatt 1998). We show that for each retaler ths coordnatng wholesale prcng scheme s gven by the per unt ndrect costs ncurred for ths retaler, augmented by a markup, the magntude of whch ncreases wth the so-called compettve mpact, a measure for the degree of competton a retaler presents to the remander of the market. Ths measure was frst ntroduced n Bernsten et al ) If the retalers compete n prce space.e., face Bertrand competton), perfect coordnaton can be acheved wth a smlar, albet more complex, dscount scheme. We assess the value of perfect) coordnaton wthn a decentralzed system by analyzng settngs n whch a smple lnear wholesale prcng scheme s offered to the retalers and no other measures are taken to coordnate the channel members decsons. We analyze the performance of the system, assumng ether that the suppler has the market power to specfy the lnear wholesale prcng scheme, or that the constant wholesale prce s chosen so as to optmze the supply-chan-wde profts. In the frst case, the chan members are engaged n a Stackelberg game wth the suppler as the leader and the retalers followng by playng the noncooperatve retaler game descrbed above. The Stackelberg soluton often results n major losses n the supply-chan-wde profts. The marketng lterature on channel coordnaton focuses on prcng decsons. Jeuland and Shugan 1983) consder a smple channel wth one suppler and one retaler. Ther model does not consder any nventory replenshment decsons or resultng setup and nventory carryng costs. The authors found that a smple quantty dscount results n a perfect coordnaton mechansm. Because thers s a statc model, the quantty does not refer to the sze of a replenshment order but to the annual sales volume. As an alternatve, Moorthy 1987) showed that n ths sngle-retaler settng, perfect coordnaton can be acheved wth a smple two-part tarff,.e., by chargng the retaler the margnal cost plus a fxed franchse fee. McGure and Staeln 1983) consder the specal case of two dentcal retalers, competng n prce space under lnear procurement costs. These authors assume that the two retalers are suppled by two dfferent manufacturers whch are ether vertcally ntegrated wth ther retaler or not. See Moorthy 1987) for further observatons on ths model. Ingene and Parry 1995) generalze Jeuland and Shugan 1983), by allowng for two nondentcal retalers. The authors show that perfect coordnaton cannot be acheved by any constant wholesale prce whch s dentcal for both retalers. Instead, they derve a perfect coordnaton scheme by dscountng the wholesale prce as a lnear functon of the retalers purchase volumes. Whle attractve, the proposed scheme fals when the number of retalers s larger than two or when the procurement costs are nonlnear, as s clearly the case n our operatonal model wth nventory

3 and setup costs. Raju and Zhang 1999) analyze another varant of our model wth one domnant retaler capable of snglehandedly settng the retal prce whch s adopted by all other retalers n the market. Under a lnear cost structure, the authors show that wth a lnear wholesale prcng scheme, perfect coordnaton requres that double margnalzaton be avoded. A nonlnear prcng scheme s offered as an alternatve. Tyag 1999) addresses the case of an arbtrary number of competng retalers, albet that they are assumed to be dentcal and that, once agan, procurement costs are restrcted to be lnear. The marketng lterature has thus restrcted tself to the smplest of cost structures,.e., to the case of lnear costs. As summarzed above and demonstrated below, more complex, yet basc, operatonal cost structures such as those arsng under nventory carryng and fxed dstrbuton costs ntroduce addtonal and essental complextes to the challenge of desgnng approprate coordnaton mechansms. Ths pont has been brought out n an emergng stream of operatons management papers. The latter, on the other hand, restrct themselves to settngs where the demand processes are exogenously gven,.e., where the revenues cannot be controlled, or, n a few cases, to models n whch the retalers fal to compete wth each other n terms of ther retal prces and/or sales targets. We refer to Chen et al. 2001) for a revew of the lterature on models wth exogenously gven, determnstc demand processes. Ths stream of papers appears to have orgnated wth Crowther 1964, examnng quantty dscounts from both the buyer s and the seller s perspectve, and Lal and Staeln 1984 who deal wth a sngle retaler or multple but dentcal retalers.) Lee and Whang 1996), Chen 1999), and Cachon and Zpkn 1999) have developed perfect coordnaton schemes for a stochastc verson of our model wth a sngle retaler facng an exogenously gven demand process and n the absence of fxed costs for delveres from the suppler to the retalers. Ths work bulds on earler coordnaton results by Clark and Scarf 1960 and Federgruen and Zpkn 1984.) Weng 1995) s one of the frst attempts to treat the retalers demand rates as endogenous varables to be determned by a careful balancng of revenue as well as cost consderatons. Ths model consders the specal case of a sngle retaler or multple, but dentcal and noncompetng retalers. The author asserts that an order quantty dscount plus a perodc franchse fee suffce to acheve perfect coordnaton. Ths asserton, however, has not been substantated, as Boyac and Gallego 1997) pont out. Chen et al. 2001) address the centralzed and the decentralzed versons of our supply-chan model, n the absence of the retalers competng n prce or quantty space,.e., when each retaler s sales volume s a functon of hs own prce only. See Munson and Rosenblatt 1998), Boyac and Gallego 1997), Cachon 1999), Larvere 1999), and Tsay et al. 1999) for addtonal revews of the operatons management lterature related to channel coordnaton wth noncompetng retalers. Bernsten and Federgruen / 411 The exstence and desgn of perfect) coordnaton mechansms n vertcal supply chans s, n addton, a central topc n the ndustral organzaton economcs lterature; see, e.g., Trole 1988) and Katz 1989). See Bernsten et al. 2002) for a recent revew of ths part of the lterature. The latter paper addresses a varant of the model of ths paper n whch, contrary to our settng, the operatonal costs can be decomposed nto a part whch s determned only by the suppler and another part whch results exclusvely from the retalers replenshment strategy. The remander of ths paper s organzed as follows: 2 ntroduces the model and notaton. Secton 3 addresses the centralzed system. The analyss of the decentralzed system s gven n 4. Secton 5 develops the perfect coordnaton mechansm. Secton 6 reports on a numercal study, comparng the performance of the supply chan under centralzaton and varous forms of decentralzaton. We conclude the paper wth a conclusons secton. 2. MODEL AND NOTATION We consder a dstrbuton system wth a suppler dstrbutng a sngle product or closely substtutable products to N retalers. The retalers sell ther product to the fnal consumer. The suppler replenshes hs nventory from a source wth ample supply. All demands and all retaler orders must be satsfed wthout ncurrng any stockouts. We assume that all orders are receved nstantaneously upon placement. Postve but determnstc leadtmes can be handled by a smple shft n tme of all desred replenshment epochs. Thus, let p = retal prce charged by retaler, and q = consumer demand for retaler s product. The two sets of varables may be related to each other va the drect) demand functons q = d p 1 p N = 1 N or the nverse demand functons p = f q 1 q N = 1 N We assume that all demand functons are downward slopng, a property almost nvarably satsfed, wth the excepton of rare luxury, or Veblen goods: d p < 0 = 1 N 1) For all = 1 N, we use to denote the absolute prce elastcty of retaler s demand, measured from the drect demand functons, and ˆ measured from the nverse demand functons: = d p p p d p ˆ 1 f = q f q/q q Note that n the absence of retaler competton, = ˆ.) We assume that the demand volumes vary wthn a cube Q n the postve orthant of R N,.e., there exst numbers 0 q mn <q max such that q mn q q max for all = 1 N. Smlarly, the set of feasble prces for each retaler s a closed nterval p mn p max, where

4 412 / Bernsten and Federgruen p mn = f q max and p max = f q mn. The drect demand functons wll be used when the retalers compete n terms of ther announced retal prces,.e., under Bertrand competton. Lkewse, competton s best modeled va the nverse demand functons, when the retalers compete n terms of ther sales targets,.e., under Cournot competton. To smplfy some of the results, we shall confne ourselves to the case where all demand functons are lnear. In partcular, d p = a b p + j j p j wth a > 0b > 0 = 1 N 2) Because the retaler products are substtutes, we have, by a common defnton gong back to Samuelson 1947), that j 0 for all j 3) We assume n addton that the matrx B, wth B = b and B j = j for all j, s nonsngular, so that the nverse demand functons exst and are lnear as well,.e., f q =â ˆb q ˆ j q j = 1 N 4) j Moreover, we would lke the nverse demand functons to be downward slopng and products to be substtutes n terms of the nverse demand functons as well see Vves 2000): ˆb > 0 ˆ j 0 for all j = 1 N 5) Unfortunately, 5) s not necessarly mpled by the correspondng propertes 1) and 3) for the drect demand functons. The mplcaton holds, however, when D) b > j j for all = 1 N see Bernsten et al. 2002, Proposton 1). Ths domnant dagonal condton s hghly ntutve and s satsfed n most ndustres. It states that each retaler observes a decrease n hs sales volume f all retalers smultaneously ncrease ther prces by the same amount. Another equally ntutve suffcent condton to ensure that 1) and 3) mply 5) s b > j j for all = 1 N: It states that a prce ncrease by any one of the retalers results n a decrease of total sales n the market. See Bernsten et al. 2002, Proposton 1, for the most general, necessary and suffcent condton.) We henceforth assume D), throughout. We now turn to a descrpton of the cost structure. All delveres to and from the suppler ncur fxed and varable costs. In a decentralzed settng, t s useful to decompose the fxed cost assocated wth a delvery to a retaler nto a component ncurred by ths retaler and one ncurred by the suppler e.g., an order-processng cost). Inventory carryng costs are ncurred for each locaton s nventory and they are proportonal wth the prevalng nventory level. In addton, the suppler may ncur a specfc annual cost for managng each retaler s account. All cost parameters are statonary. For = 1 N, defne K 0 = fxed cost ncurred for each delvery to the suppler, K = fxed cost ncurred for each delvery to retaler = 1 N, K s = the component of K ncurred by the suppler n a decentralzed settng, K r = the component of K ncurred by retaler n a decentralzed settng, K = K s + Kr, h 0 = annual holdng cost per unt of nventory at the suppler, h = annual holdng cost per unt of nventory at retaler, h = h h 0, ncremental or echelon holdng cost at retaler, c 0 = cost per unt delvered to the suppler, c = transportaton cost per unt shpped from the suppler to retaler, d = annual cost ncurred for managng retaler s account, wth nondecreasng, concave, and 0 = 0. We assume that h 0 for all, whch means that the cost of carryng a unt at retaler s at least as large as the cost of carryng t n the suppler s warehouse. We also assume, wthout loss of generalty, that n a decentralzed settng the transportaton cost c s borne by the retaler. The above-specfed costs do not nclude any transfer payments between the suppler and the retalers. 3. THE CENTRALIZED SOLUTION In ths secton, we analyze the system, assumng that a central planner makes all decsons regardng retaler prces, sales volumes, and replenshment strateges so as to maxmze supply-chan-wde profts. The vector of retaler prces p unquely determnes the vector of sales volumes q and vce versa va the drect and nverse demand functons. Thus, n a centralzed settng, aggregate chanwde profts may be expressed as a functon of p or as a functon of q, and t s mmateral whch of the two functons s optmzed. Ths s n sharp contrast to the decentralzed system descrbed n 4 n whch retaler competton n prce space may result n equlbra qute dstnct from those acheved under competton n quantty space. The revenue component and the varable transportaton and account management costs can all easly be expressed n terms of the vector q: The revenue term s gven by N f qq, the varable order/transportaton costs by N c 0 + c q, and the account management costs are N q. Ths leaves us wth the specfcaton of the nventory and fxed delvery costs, the only components whch depend on the supply-chan-wde replenshment strategy. Even wth a gven vector of demand rates q, t s exceedngly dffcult to dentfy a replenshment strategy whch mnmzes these costs, let alone to express the optmal cost value as

5 a smple analytcal functon of a lmted set of decson varables. Untl Roundy s 1985) semnal paper, ths very problem remaned poorly understood. Roundy showed, however, that whle a fully optmal replenshment strategy s ntractable, a near-optmal soluton exsts wthn the class of so-called power-of-two polces. Under a power-of-two polcy, each faclty gets replenshed when ts nventory level s down to zero, replenshments come after constant ntervals and these ntervals are chosen as power-of-two multples of a gven base perod T b ;.e., they are part of the dscrete set 2 m T b m= A powerof-two polcy s thus fully characterzed by the vector of replenshment ntervals T = T 0 T 1 T N, wth T 0 the nterval used to replensh the suppler and T to make delveres to retaler, = 1 N. Roundy 1985) showed that under a power-of-two polcy, wth nterval vector T, the systemwde cost s gven by the followng relatvely smple analytcal expresson: { K 0 K T 0 T 2 h 0q maxt 0 T + 1 } 2 h q T 6) Thus, { K Cq=mn 0 + T 0 { K T h 0q maxt 0 T h q T } T =2 m T b m ZN represents the cost of the best power-of-two polcy. Roundy 1985) showed, n addton, that the unconstraned mnmzaton of 6 over all vectors T results n a lower bound for the cost of a fully optmal strategy: { { K Cq = mn 0 + T 0 K T h 0q maxt 0 T } 7) h q T } T>0 } 8) Moreover, Cq 106Cq. We conclude that, whle t s mpossble to compute the supply-chan-wde profts under a fully optmal replenshment) strategy, let alone to express the optmal value opt SC as an analytcal functon, opt SC can be approxmated very closely from below and above. Let SC q = f qq c 0 + c q SC q = q Cq 9) f qq c 0 + c q q Cq 10) Then, Bernsten and Federgruen / 413 max def SCq = SC opt SC def SC = max SC q 11) q Q q Q There exsts a vector q l q u whch acheves the maxmum to the left rght] of 11). Ths follows from the compactness of Q and the contnuty of SC and SC, a property whch s mmedate from the followng characterzaton of C and C: Lemma 1. a) C and C are jontly concave. b) CC s dfferentable, almost everywhere on Q,.e., whenever problem 7) 8)] has a unque mnmum T l qt u q, and for all = 1 N C = 1 q 2 h 0 maxt l 0 q T l q h T l q C = 1 q 2 h 0 maxt u 0 q T u q h T u q Proof. See the Appendx for the proof. We now show that the bounds SC and SC tend to be very close. Wrte opt SC = grosspropt cost opt. Here, the grosspr term s defned as the gross profts,.e., sales mnus varable costs mnus account management costs, and the cost term refers to setup and nventory holdng costs only.) Usng the optmalty gap results n Roundy 1985), t s easly verfed that SC opt grosspr SC /cost opt 1 12) SC opt grosspr SC /cost opt 1 13) Whle 12) and 13) do not result n an absolute worstcase gap for the two bounds, the gaps are very small for most product lnes wth reasonable gross proft margns. For example, we have computed a lower bound for the annual sales-to-nventory rato for a centralzed supply chan n 10 consumer goods categores, assumng the suppler s and the retalers ndvdual sales-to-nventory ratos are above the product category s lower quartles reported for the wholesale and retal sectors n Dun and Bradstreet ), respectvely. Ths lower bound vares between 1.7 and Ths mples a lower bound for the average rato grosspr opt /cost opt between 1.85 and 4.24, assumng an average gross proft margn of 32% 2 and an annual nventory carryng cost rate of no more than 30%. Thus, the rght-hand sde of 12) 13)] vares between ] and ]. Alternatvely, the U.S. Census Bureau reports on retaler gross margns for 30 retal sectors; see Table 7 n U.S. Census Bureau 2002) whch reports these values for the years The average across all retal sectors s close to 30% and only n 17 out of the 240 cases s the gross margn lower than 20%. These rare exceptons represent hgh-volume sectors e.g., the Warehouse Clubs and Superstores sector) where

6 414 / Bernsten and Federgruen Fgure 1. Effects of gross profts on bounds 12) and 13). a) Holdng Cost Rate = 20% b) Holdng Cost Rate = 25% S/I 1.7 S/I S/I 1.7 S/I Margn Margn sales-to-nventory ratos are hgh, offsettng the mpact of a relatvely low gross proft margn. Fgure 1 exhbts how the optmalty gap 006/ grosspr opt /cost opt 1 vares as a functon of the gross retal proft margns, based on the U.S. Census data taken to vary between 20% and 50%. Fgure 1a) assumes an nventory cost rate per sales dollar) of 20% and Fgure 1b) of 25%. Each fgure shows two curves, one for a salesto-nventory rato of 1.7 and one for a rato of 4. The optmalty gap becomes sgnfcant only when an exceptonally low proft margn arses n conjuncton wth a unusually low sales-to-nventory rato, whle, n practce, low margns tend to arse under hgh sales-to-nventory ratos. To compute SC and SC and the correspondng par of optmzng vectors q u T u and q l T l, frst note that the common term n SC q and SC q can be evaluated straghtforwardly for any q Q; Cq and Cq represent Roundy s 1985) proposed lower and upper bound for the one-warehouse multple-retaler model wth fxed demand rates. These can be evaluated n ON log N tme for any q Q, usng Roundy s algorthm a later refnement by Queyranne 1987) shows that C and C can, n fact, be evaluated n ON tme). Moreover, Lemma 1 shows that C and C are dfferentable almost everywhere, wth a gradent whch s easly computed n the process of calculatng C and C. These observatons allow for the effcent usage of a gradent-based, standard unconstraned optmzaton algorthm to compute SC and SC see, e.g., Denns and Schnabel 1989). 4. THE DECENTRALIZED SYSTEM In ths secton, we consder a decentralzed system n whch each retaler s responsble for hs own prce and sales volume decsons as well as hs own replenshment strategy, whle the suppler selects a wholesale prcng scheme as well as her replenshment polcy n response to the retaler orders. We start wth an analyss of the system under a smple lnear wholesale prcng scheme,.e., where retaler s charged a constant per-unt wholesale prce w The Retaler Game Under Lnear Wholesale Prcng Schemes Frst, assume that the retalers are engaged n prce or Bertrand competton. Under a lnear wholesale prcng scheme, t s clearly optmal for each retaler to replensh hs nventory when t drops to zero, and at constant ntervals of length T, say. Thus, assumng that all retalers smultaneously choose ther prces and replenshment strateges, ths gves rse to the followng proft functon for retaler : p T p w = p c w a b p + ) j p j j Kr T 1 2 d p h T 14) where p = p 1 p 1 p +1 p N. For the sake of notatonal smplcty, we ntally assume that h s ndependent of the wholesale prce w. In many settngs, holdng costs nclude captal costs, n whch case h should be modeled as an ncreasng functon of w : h w. As shown at the end of ths subsecton, all of the results for the decentralzed system contnue to apply n ths general settng. Note from 14) that whle retaler s prce p has an mpact on the profts acheved by all retalers, hs replenshment strategy affects only hs own profts. Ths observaton permts us to vew the noncooperatve retaler game as one n whch each retaler competes wth a sngle nstrument or decson varable,.e., hs retaler prce, and wth smplfed proft functons obtaned by replacng each varable T wth hs optmal EOQ value p p w = ) p c w a b p + ) j p j j 2d p h K r 15) In general, these proft functons fal to exhbt any known structural propertes to ensure that a Nash equlbrum exsts n the retaler game, let alone that ths equlbrum

7 be unque. We wll show, however, that a unque equlbrum can be guaranteed n most, f not all, realstc markets assumng sales-to-nventory ratos are not excessvely low and demand elastctes are not excessvely large n absolute value). More specfcally, we ntroduce the followng condton. Let INV = 2d p h K r = optmal total nventory and setup cost for retaler under the prce vector p, REV = p d p = total gross revenue for retaler under the prce vector p, C1) 8 REV INV Even f the annual nventory carryng cost s as large as 40% of the dollar value of the nventory a comfortable upper bound, n practce), the rato REV /INV s at least 2.5 tmes the annual) sales-to-nventory rato. For the sample of 10 consumer product lnes mentoned n 3, the average lower quartle of the retalers sales-to-nventory ratos vares between 2.8 and 6.7. Thus, for retalers n any of these sectors, wth a sales-to-nventory value above the lower quartles, the rght-hand sde of C1) s bounded from below by 56, and n some product lnes by 134. Compare these values wth estmated elastctes of demand, whch vary between 1.4 and 2.8. These estmates are obtaned from Tells 1988.) Condton C1) s equvalent to the nequalty d p 3/2 1 b 8 2 h K r def,ord p = 1 b 4 2 h K r 2/3. Thus, n vector notaton, C1) holds on the compact polyhedron P = p 0 a+ Bp. Ths polyhedron s a Leontef-type polyhedron, and has a largest element p,.e., for all p P, p B 1 a def = p. The last nequalty follows because B 1 = ˆb 0 and B 1 j = ˆ j 0, by 5. Because, as demonstrated above, C1) holds n equlbrum n almost all practcal settngs, we have that p>0 n all such cases. s con- We henceforth assume that the cube X N pmn taned n P. p max Theorem 1. Assume C1) apples. Then, the retaler game under Bertrand competton has a Nash equlbrum. Proof. In vew of Fredman 1977), t suffces to show that each of the proft functons p p w s concave n p. It s easly verfed that 2 p p w p 2 INV d 2 p 8 b = 2b + b2 4 2K r h d p d 2 p 0 = p b d p 8p d p = 8 REV INV INV Bernsten and Federgruen / 415 Several questons reman regardng the retaler game. Frst, we would lke to know whether the Nash equlbrum s unque, thus guaranteeng a fully predctable equlbrum) behavor by the retalers. In the presence of multple equlbra, t s, n general, dffcult to predct whch of the equlbra wll be adopted see, e.g., Harsany and Selten 1988). Second, we expect an ncrease n the wholesale prces to result n an ncrease of all equlbrum retaler prces. Fnally, we would lke to know whether an effectve procedure exsts to compute the Nash equlbrum or equlbra). All these questons can be answered n the affrmatve, f the retaler game can be shown to be supermodular see Topks 1998 and Mlgrom and Roberts 1990 for a precse defnton and detaled dscusson). In our context, the set of feasble decsons for each retaler s a smple p max closed nterval p mn and, hence, a compact set. Ths mples that the retaler game s supermodular f, for each retaler, the proft dfference p 1 p w p 2 p w s ncreasng n all p j j for any p 1 >p 2 16) Ths property s satsfed under the followng slght strengthenng of condton C1): C2) 4 REV INV As dscussed above, C2), lke C1), s satsfed n vrtually all realstc markets and the set of prces on whch C2) apples s agan a compact polyhedron P. We now assume that X N pmn p max P. It has been well known snce Topks 1998) that f the game s supermodular, the followng smple tatônnement scheme converges to a Nash equlbrum. In the kth teraton of ths scheme, each retaler determnes a prce p k whch maxmzes hs own proft functon p k 1 w assumng all other retalers mantan ther prces accordng to the current vector p k 1 obtaned n the k 1st teraton. Ths gves rse to a new vector p k. Theorem 2. Assume condton C2) apples. a) The retaler game s supermodular and has a unque Nash equlbrum p. b) The tatônnement algorthm converges to p for every startng pont. c) p s ncreasng n w. Proof. Part a): It suffces to show that 16 apples. Because the proft functons are twce dfferentable, 16 s equvalent to 2 /p j p 0 for all j. Note that 2 = p j p j b j INV 2K r h dp 4d 2 p 0 d 2 p 4 b = p b d p 2K r h d p p d p = 4 REV INV

8 416 / Bernsten and Federgruen Mlgrom and Roberts 1990), generalzng an earler condton by Fredman 1977), showed that a unque Nash equlbrum arses n a supermodular game f for all = 1 N, 2b b2 INV 4d 2 p = 2 p 2 > j = j j j 2 p p j b j INV 4d 2 p whch holds f and only f b + b ) j 1 b ) INV > 0 j 4d 2 p Observe now that b > 0, b j j > 0 by D), whle b INV 4d 2 p 1 because by C2), = b p d p 4REV INV = 4p d p INV Because the retaler game s supermodular, part b) s mmedate and part c) follows from 2 /p w = b 0 for all = 1 N. It s noteworthy that, whle the equlbrum retaler prces respond monotoncally to the wholesale prces), the same cannot be guaranteed for the sales volumes and profts, except n the specal case where retaler competton s absent,.e., where each retaler s demand functon depends on hs own prce only. Ths can be demonstrated wth a smple example see Bernsten et al. 2002). We next consder the case where the retalers are engaged n quantty or Cournot competton. As before, t s possble to vew the retaler game as one n whch each retaler competes wth hs sales volume q as the sngle nstrument or decson varable. The retalers profts, expressed as a functon of the vector q, are easly obtaned from 15, makng all approprate substtutons: C q q w = â ˆb q j 2q h K r 17) ˆ j q j c w ) q Once agan, n general these proft functons fal to exhbt any known structural propertes to ensure that a Nash equlbrum exsts n the retaler game. As n the case of Bertrand competton, an equlbrum can, however, be guaranteed under C1), and a unque equlbrum under C2),.e., n most markets wth realstc sales-to-nventory ratos and demand elastctes. Theorem 3. a) Assume C1) apples. Then the retaler game under Cournot competton has a Nash equlbrum. b) The Nash equlbrum q s unque under C2) and D ): ˆb > j ˆ j the counterpart of D)). Proof. In vew of Fredman 1977), t suffces agan to show that each of the proft functons C q q w s concave n q. It s easly verfed that 2 C q q w q 2 so that 2 C q q w q 2 f and only f 1 INV < 2ˆb 4 q 2 = 2ˆb + 1 INV 4 q 2 < 0 whch holds f and only f ˆ < 8 REV INV By Proposton 6.1 n Vves 2000), ˆ < < 8 REV INV where the second nequalty follows from C1). To prove part b), agan followng Fredman 1977), t s suffcent to show that 2ˆb 1 INV = 2 C q q w 4 q 2 q 2 > j or equvalently, that ˆb j ˆ j + ˆb 1 INV > 0 4 q 2 whch follows from ˆb > j 2 C q q w q q j ˆ j and ˆ < < 4 REV INV where the last nequalty follows from C2). = ˆ j j It should be noted that the Cournot game fals to be supermodular even under C2). As a consequence, and n contrast wth Theorem 2b) and c), we cannot guarantee that the smple tatônnement scheme converges to q or that the equlbrum sales volumes are monotone n the wholesale prce. Alternatve methods need to be nvoked to compute q as the unque soluton of the system of equatons C /q = 0= 1 N. Fnally, n the general model where h s an ncreasng functon of w, t s easly verfed that all of the above results contnue to apply. Ths s mmedate for all of Theorems 1 3 and Proposton 1, except for Theorem 2c). Assumng h s dfferentable wth dervatve h, the proof of Theorem 2c) generalzes, now wth 2 = b p w + b h 1 2 K r 2 h d > 0

9 4.2. A Comparson Between Prce and Quantty Competton We complete ths secton wth a bref comparson between the Bertrand and Cournot equlbra. Let p C = fq denote the prce vector under the Cournot equlbrum and q B = dp the demand volume vector under the Bertrand equlbrum. Proposton 1. Assume condton C2) apples. Then, p C p. Proof. By the proof of Theorem 2, the retaler game s supermodular under Bertrand competton and the proft functons p 1 p N are concave n p for all = 1 N. In vew of the proof of Proposton 6.2 n Vves 2000), t thus suffces to show that p C /p 0. Note that = d p p + p c w b + 2b h K r 2 INV p = d p b p c w + b INV p 2d p Substtutng q = d p, we obtan p C = q p b f q c w + b INV q 18) 2q Because q s the Nash equlbrum n the Cournot game, t satsfes the frst-order condton 0 = C q q w q = ˆb q + f q c w h K r INV q = ˆb q + f q c w INV q 19) 2q Substtutng 19 nto 18, we conclude that p C /p = q b ˆb q = 1 b ˆb q 0 because b ˆb > 1as1= BB 1 = b ˆb j j ˆ j. Thus, f the retalers compete n quantty space, each adopts a retal prce that s larger than ts equlbrum retal prce under prce competton. One mght conjecture that, smlarly q q B. A numercal example n 6 shows, however, that ths relatonshp may fal to hold. Larger sales volumes, under prce competton, can only be guaranteed n specal cases, e.g., when the retalers are dentcal so that the equlbra are symmetrc. In ths case, p = p and p C = p C for all = 1 N so that for all = 1 N, q B = a b j j p a b j j p C = q.) On the other hand, f q q B s satsfed, t follows that each retaler realzes lower profts under prce competton as compared to quantty competton; for all = 1 N: p = C q B = C q BqB C q Bq C q q = C q, where the frst nequalty follows f q q B gven that C /q j < 0 for all j, whle the second nequalty follows from the fact that q s a Nash equlbrum n the Cournot game. Bernsten and Federgruen / COORDINATION WITH THE SUPPLIER We now nvestgate the performance of the complete supply chan under a lnear wholesale prcng scheme, characterzed by a vector of wholesale prces w. Even f ths wholesale prce vector s chosen to optmze supply-chanwde profts, the resultng aggregate proft value s lkely to be dsappontng n the absence of any addtonal procedures to coordnate the suppler s replenshment actvtes wth those of the retalers. Recall from 4.1 that t s optmal for each retaler to replensh hs stock wth constant replenshment ntervals and, wthout any upfront restrctons, these ntervals wll be set accordng to the EOQ formula. In general, the resultng order stream for the suppler s hghly nonstatonary, fals to follow any smple, perodcally repeatng pattern and represents a dffcult manageral problem for the suppler. No satsfactory soluton s known for the correspondng nventory problem. Moreover, the suppler s costs are, n general, much hgher than f orders arrved accordng to a smple pattern, e.g., f all retalers were requred to choose ther replenshment ntervals from the dscrete set of power-of-two values 2 m T b m= Recall from 3 that even n a centralzed settng the proposed heurstc) strategy s based on all channel members restrctng ther consecutve replenshment ntervals from the set of power-of-two values. Wth ths restrcton, we showed that the best achevable strategy comes very close to an upper bound for the optmal systemwde profts; see 12) and 13) and the subsequent dscusson there. We therefore proceed to consder coordnaton mechansms, based on an upfront contract see, e.g., Trole 1988), specfyng that all channel members agree to choose each nterval between consecutve replenshments from the above dscrete set of power-of-two values. Whle ths restrcton nvolves major benefts for the suppler, t comes at mnmal addtonal expense to the retalers. See, e.g., Brown 1959, Roundy 1985, and the dscusson below.) As a consequence, the power-of-two value restrcton should be easly agreed upon by the channel members. In the worst case, the suppler may offer an annual rebate to the retalers equal to the modest ncrease n ther nventory and setup costs resultng from the nterval restrcton. Such rebates are most easly computed and clearly do not affect the supply-chan-wde profts.) Wth ths restrcton, we frst revst the retaler game that arses under an arbtrary lnear wholesale prcng scheme, specfed by a wholesale vector w. We consder ths restrcted game frst under Bertrand competton. As n the unrestrcted game, retalers choose prces and replenshment strateges smultaneously.) Note that the new proft functon ˆ p w for retaler s obtaned from 14) by replacng T by the power-of-two value 2 m Tb whch s closest, n the relatve sense, to the EOQ value. Clearly, ˆ = 1 N. In the restrcted retaler game, no condtons appear to preval to ensure that the game has any of the known structures guaranteeng exstence of an equlbrum: for example, the roundng procedure ntroduces volatons of both

10 418 / Bernsten and Federgruen concavty and supermodularty at varous dscrete ponts on the parameter spectrum. Under a lnear wholesale prcng scheme, we are therefore only able to guarantee a so-called -equlbrum, a concept ntroduced by Radner 1980) see also Fudenberg and Trole 1991, Def. 4.3). Defnton 1. In a Bertrand Cournot] retaler game, the vector p eq q eq ]sa-nash equlbrum for some >0, f no retaler = 1 N can mprove hs proft value by more than a -fracton through a unlateral) change of the retaler prce p eq quantty q eq ]. In Fudenberg and Trole 1998, refers to the absolute amount by whch a player s proft can be mproved by a unlateral devaton from the equlbrum.) Proposton 2. Fx a wholesale prce vector w. Assume condton C1) apples, so that a Nash equlbrum p q ] exsts for the contnuous game under Bertrand Cournot] competton. Then, p q ]sa-nash equlbrum n the correspondng restrcted retaler game, wth 006 = mn N grosspr /cost 106 where grosspr denotes retaler s annual gross revenues mnus varable purchase and transportaton costs, and cost hs annual setup and holdng costs n the contnuous game under p q ]. Proof. We gve the proof for the case of Bertrand competton. Fx = 1 N. It suffces to show that for all p -values ˆ p p w 1 + ˆ p p w. Let cost = K r/t m + 1 d 2 p h t m, the optmal annual setup and nventory holdng costs for retaler, under the power-of-two nterval restrcton, assumng the retalers adopt the prce vector p. It s well known that cost 106cost. Note that ˆ p p w p p w p p w = grosspr cost = grosspr cost 1 + cost ) cost grosspr cost ) ˆ p p w cost grosspr cost ˆ p p w 1 + The second nequalty follows from p beng a Nash equlbrum n the contnuous game.) Example 1. Consder a market wth N = 2 dentcal retalers, wth common demand functon d p = p + 4p 3 = 1 2. Let w +c = 16K = 800 h = 16= 1 2. Assume T b = 1p mn = 30, and p max = 40, so that 80 d p 290 because d p s maxmzed mnmzed) when p = p mn p max and p 3 = p max p mn. Condton C2) s equvalent to b 2 h K r 4 <d p 3/2 = 1 2 the rght-hand sde of whch s mnmzed when d p = 80. Thus, because ths nequalty s satsfed when d p = 80, C2) s satsfed throughout, ensurng concavty as well as supermodularty of the retalers proft functons n the contnuous game. To show that the retalers proft functon ˆ p 3 w fals to be concave, let p 3 = 35 and consder the proft functon on the nterval 32 35: ˆ 34 p 3 = 35w = < 1 3 ˆ 32 p 3 = 35w ˆ 35 p 3 = 35w = 1234 To show that the functon fals to be supermodular, note that ˆ 35 p 3 = 35w ˆ 32 p 3 = 35w = = 3 < ˆ 35 p 3 = 32w ˆ 32 p 3 = 32w = = 15. Observe that wth p 3 = 35 and p = 32 t s optmal to choose T = 2 1 whle for p 3 = 35 and p = 35, T = 1. It s ths dscrete jump n the optmal replenshment nterval whch causes the local volaton of concavty and supermodularty. Note that on the nterval of feasble demand rates d p , the correspondng optmal power-of-two nterval values are ether T = 05 ort = 1. A prce vector p1 p 2 s a Nash equlbrum of the restrcted game only f t s a Nash equlbrum n one of the four contnuous games whch arse when restrctng the vector T 1 T 2 to one of the four pars , and 1 1 and also the prces to four correspondng sets of ntervals). Each of these four contnuous games has a unque Nash equlbrum gvng rse to four prce vectors that are canddates for a Nash equlbrum of the restrcted game. It s therefore easly verfed that the restrcted game has exactly two Nash equlbra even though the proft functons fal to be concave or supermodular): p 1 = and p 2 = , the equlbra correspondng wth T 1 T 2 = 05 1 and T 1 T 2 = 1 05 and proft vector 1 2 = and 1 2 = , respectvely. It s clearly mpossble to predct whch of the two equlbra wll be adopted n the market place f ether). The contnuous game, on the other hand, has a unque symmetrc) Nash equlbrum p = snce satsfyng C2) see Theorem 2) wth a correspondng proft value of 1, for each retaler. Ths prce vector s a -Nash equlbrum n the restrcted game wth = 00095, a value obtaned by computng max p ˆ p p3 = 3358w. The fracton bears close smlarty to the optmalty gap n 12 and 13. As shown there, cannot be unformly bounded under completely general parameter values. However, t s clear that s very small for most product lnes and markets, wth reasonable gross proft margns. For example, assumng as before that the retalers gross profts represent at least 32% of the sales and that the annual nventory carryng cost rate s 30% or less of the dollar value of the nventory, a lower bound for the rato

11 grosspr /cost vares between 1.85 and 4.24 for the 10 product categores consdered n 3. Ths results n a value of varyng between 0.08 and In 6, we compare the performance of the supply chan n whch the suppler chooses a wholesale prce so as to maxmze hs own proft, wth one n whch a wholesale prce s chosen to optmze supply-chan-wde profts. The comparson s done under a Stackelberg game. Both settngs wll be compared wth the centralzed soluton. We show that even the best lnear wholesale prcng scheme fals to result n perfect coordnaton. Moreover, the optmzng wholesale prce vector s hard to compute, n partcular when the number of retalers s large. Fnally, we have shown that under a lnear wholesale prcng scheme, a full equlbrum for the retalers cannot be guaranteed unless the retalers are allowed to choose ther replenshment ntervals contnuously. On the other hand, the absence of an upfront restrcton of the replenshment ntervals to the set of power-of-two values comes at a severe expense for the suppler and hence for the chan as a whole Perfect Coordnaton va a Nonlnear Wholesale Prcng Scheme We now derve a nonlnear wholesale prcng scheme whch does result n a perfect coordnaton mechansm. We start wth the case where the retalers compete n quantty space. The desgn of the coordnatng prcng scheme s an applcaton of the Groves mechansm see Groves 1973 and Groves and Loeb 1979). The prncple behnd the mechansm s to algn, for each = 1 N, the margnal supply-chan-wde proft functon whch arses when all but retaler are commtted to the sales volumes and replenshment ntervals that optmze the supply-chan-wde profts, wth the proft functon retaler faces n the retaler game n the decentralzed chan. We wll show that, n our model, the Groves mechansm results n a farly smple prcng scheme wth three addtve dscounts off a constant base prce: Each of the dscount components depends on a sngle retaler characterstc. As motvated n the prevous subsecton, we contnue to requre that all chan members agree to choose ther replenshment ntervals from the dscrete set of power-oftwo values. Let q l T l denote a par of vectors whch acheve SC. Fx = 1 N, and let SC q T q l Tl = â ˆb q j ˆ j q l j ) q + â j ˆb j q l j j r j c 0 + c q + q j ˆ jr q l r ˆ j q ) q l j c0 + c j q l j + ql j ] K 0 T l 0 Bernsten and Federgruen / 419 K + 1 T 2 h 0q maxt l 0 T + 1 ] 2 h q T Kj + 1 j Tj l 2 h 0q l j maxt l 0 Tl j + 1 ] 2 h q l j T l j 20) denote the margnal supply-chan-wde proft functon whch arses when all but retaler are commtted to the sales volumes and replenshment ntervals n q l and T l respectvely, and observe that ths margnal proft functon s maxmzed at q = q l and T = T l. Moreover, ql Tl remans a maxmzer when ths functon s shfted n parallel by omttng terms that are constant n q T : SC q T q l Tl = â ˆb q ) ˆ j q l j q j ) ˆ j q l j q c 0 + c q + q j K + 1 T 2 h 0q maxt l 0 T h q T ] 21) Assume now that n the retaler game wth a possbly nonlnear wholesale prce w q T, all but retaler adopt sales volumes and replenshment ntervals n accordance wth the vectors q l and T l. The proft functon for retaler s then gven by C q T q l T l w = â ˆb q ) ˆ j q l j c w q j K r + 1 ] T 2 h q T 22) The proft functon C n the retaler game can thus be algned completely wth the functon SC by specfyng the wholesale prcng functon w q T as the sum of three components, one of whch s a decreasng functon of the retaler s order quantty, one a decreasng functon of hs replenshment nterval, and one a decreasng functon of the annual sales volume. More specfcally, w D q T = w 1 where T q + w 2 T + w 3 q 23) w 1 T q = c 0 + Ks 24) T q w 2 T = 1 2 h 0T l h 0 mnt l 0 T 25) w 3 q = q + q j = q q ˆ j q l j ) + Q l 1 ql Q l 26)

12 420 / Bernsten and Federgruen wth Q l = N j=1 ql j = total retaler sales n the centralzed soluton SC, and = j ˆ j q l j r q l r We refer to as the compettve mpact of retaler. It represents a measure for the compettve mpact retaler has on all other retalers, expressed as a weghted average of the margnal mpact of an ncrease of retaler s sales volume on the prces charged by all other retalers. The weghts are gven by the relatve magntudes, n sales volumes, of these retalers.) Theorem 4. Let q l T l denote a vector of sales volumes and a vector of replenshment ntervals under whch supply-chan-wde profts equal SC,.e., the optmal centralzed soluton under the power-of-two nterval restrcton. The par q l T l arses as a Nash equlbrum n the retaler game nduced by the wholesale prcng scheme 23. In other words, the prcng scheme 23 generates a perfect coordnaton mechansm. Proof. Note that for all = 1 N and all q T C q l Tl q l Tl wd ) = SC q l Tl q l Tl ) SC q T q l Tl = C q T q l Tl wd where the nequalty follows from the fact that q l T l acheves SC, so that q ltl maxmzes SC q l Tl and hence SC q l Tl because SC and SC dffer by a constant only. As mentoned, the frst component of the wholesale prcng scheme, w 1, provdes an ncentve for the retalers to ncrease ther order quanttes T q = 1 N. The second component w 2 offers a constant dscount, h 0, for each addtonal unt of tme the retaler s wllng to keep a unt of hs tem n stock, up to a cap of A = T0 l tme unts. Note that the thrd component n the scheme offers a drect ncentve to ncrease the sales volume. Note also that w D q T = c 0 + Ks + q + 1 T q q 2 h 0T l 0 1 ] 2 h 0mnT l 0 T ) +Q l 1 ql Q l 27) Here, the term wthn square brackets represents all cost components ntally) ncurred by the suppler that are related to retaler s sales. These are ) the varable procurement costs at unt rate c 0, ) the suppler s part of the fxed charges for delveres between the suppler and the retaler, ) the cost of carryng unts n the suppler s stock whch are sold va retaler, and v) the account management costs for retaler. We refer to these four cost components ) v) as retaler s ndrect costs. The term wthn square brackets s dentcal to the one requred n a settng wth noncompetng retalers,.e., one where all cross-elastctes n the demand functons are zero see Chen et al. 2001). The latter proved that all three dscount components are ndeed essental n ths smpler settng of noncompetng retalers. The authors showed, n partcular, that even n the absence of account management costs, no tradtonal dscount scheme based exclusvely on order quanttes s capable of achevng perfect coordnaton, regardless of ts shape or number of breakponts. Quantty dscounts based on the retaler s replenshment frequency and annual sales volume, as reflected by components w 1 and w 2 are prevalent n many ndustres. See Munson and Rosenblatt 1998 and Brown and Medoff See Chen et al for addtonal dscusson on ths ssue.) The second term n 27 represents a markup due to competton. Ths markup ncreases wth, retaler s compettve mpact. For a gven total sales volume Q l n the market, the suppler s markup for any gven retaler decreases wth ths retaler s market share. The perfect coordnaton scheme 27 thus provdes a ratonale for the wdely prevalent practce of offerng larger dscounts to larger retalers see Brown and Medoff 1990 and Munson and Rosenblatt 1998), beyond those that can be justfed by economes of scale n the costs ncurred. Observe that the wholesale prcng scheme tself dfferentates between the retalers. Dfferences between the retalers n the frst term of 27 are drectly justfed by dfferences n the costs ncurred to servce the retalers. Such quantty dscounts are permtted under 2a) of the Robnson-Patman Act, the prncpal federal act governng prce dscrmnaton. Dfferences n the markup,.e., the second term n 27, fal to be drectly related to cost dfferences. As far as ths component s concerned, complance wth federal trade regulatons s more questonable. On the other hand, the dfferences n the markup tend to vansh as the number of retalers becomes large see Corollary 3 n Bernsten et al for a more precse asymptotc analyss). One potental weakness of the coordnaton mechansm s the fact that whle the vector q l T l, whch optmzes supply-chan-wde profts, arses as a Nash equlbrum n the correspondng retaler game, exstence of alternatve equlbra wth suboptmal supply-chan-wde performance cannot be excluded, n general. The followng theorem shows, however, that supply-chan-wde optmalty s guaranteed for all equlbra, as long as the coeffcents ˆ j n the cross terms of the nverse demand functons are symmetrc,.e., ˆ j = f q j = f j q = ˆ j for all j. Theorem 5. Assume ˆ j = ˆ j for all j. Under the wholesale prcng scheme w D, all Nash equlbra n the retaler game result n supply-chan-wde optmal profts SC.

13 Proof. Let q l T l denote an optmal soluton of the centralzed system, wth the power-of-two nterval restrcton and let w D be the wholesale prcng scheme assocated wth ths soluton. Let q 0 denote an alternatve Nash equlbrum n the retaler game under w D, and T 0 a correspondng optmal vector of replenshment ntervals. Clearly, for all = 1 N, C q 0 T0 q 0 T0 wd C q l Tl q 0 T0 wd 28) Recall that for all q T C q T q 0 T0 wd = â ˆb q ) ˆ j q 0 j c w D q j K r + 2 h 1 ] q T 29) T Substtutng 23 nto 29, weget C q T q 0 T0 wd = â ˆb q ˆ j q 0 j c c 0 )q q j j ˆ j q l j q K T h 0q maxt l 0 T h q T ] 30) Next, substtutng 30 nto 28, we obtan for all = 1 N, ) â ˆb q 0 ˆ j q 0 j c c 0 q 0 q0 ˆ j q l j q0 j j K T h 0q 0 maxt l 0 T ] 2 h q 0 T 0 ) â ˆb q l ˆ j q 0 j c c 0 q l ql ˆ j q l j ql j j K T l + 1 ] 2 h 0q l maxt l 0 T l 1 2 h q l T l ) = â ˆb q l ˆ j q l j c c 0 q l ql ˆ j q 0 j ql j j K + 1 T l 2 h 0q l maxt l 0 T l + 1 ] 2 h q l T l where the equalty follows from ˆ j = ˆ j for all j. Addng all N nequaltes and because N j ˆ j qj lq0 = N j ˆ j qj 0ql whch also follows from ˆ j = ˆ j for all j), we obtan N f q 0 q 0 N c + c 0 q 0 q 0 K T h 0q 0 maxt l 0 T h q 0 T 0 ] Bernsten and Federgruen / 421 N f q l q l N c + c 0 q l q l K T l h 0q l maxt l 0 Tl + 1 ] 2 h q l T l Addng the term K 0 /T0 l to both sdes of the nequalty, we obtan SC q 0 T0 lt0 0 SCq l T l = SC, whch proves that q 0 nduces optmal supply-chan-wde profts. A smlar, Groves-based, perfect coordnaton mechansm can be acheved when the retalers compete n prce space. However, the structure of the resultng wholesale prcng scheme s more complex, a drect consequence of the fact that the choce of a retaler s retal prce has an mpact, not just on hs own sales volume, but on that of all other retalers and hence on the ndrect costs ncurred for them. Remark. The analyss above assumes that n the decentralzed system, the retalers holdng cost rates h are ndependent of the wholesale prces w. As dscussed n 4, t s often more realstc to assume that h s an ncreasng functon of w, e.g., h w = h 0 + w I, for some nterest rate I. It s easly verfed, along the lnes of Chen et al. 2001), that perfect coordnaton contnues to be achevable wth a slght modfcaton of the nonlnear prcng scheme 27): w D q T =c ) 1 2 IT j q l j +q /q +K s /q T j h 0 +Ic 0 T l 0 1 ] 2 h 0 +Ic 0 mnt l 0 T 6. NUMERICAL STUDY In ths secton we report on a numercal study comparng the performance of a supply chan under centralzed and decentralzed management. We also report on an example exhbtng nterestng dfferences n the equlbrum strateges adopted by the retalers when they compete n prce or quantty space. Our frst set of problem nstances s generated from the followng base scenaro, wth N = 5 dentcal retalers. Ther demand functon s gven by d p = a bp + j p j for all = 1 5, where a = 90 and b = 6. The cost parameters are as follows: c = 1q = 10 + q for q > 0c 0 = 10K 0 = 100h 0 = 5 h = 1.e., h = 6 K s = 4 and K r = 6.e., K = 10 for all = 1 5. In the base scenaro, the vector of prces p = corresponds to the vector of quanttes q = In addton to the base scenaro, we generated 9 addtonal nstances by rotatng the demand functons around the pont p q,.e., we augment a, the ntercept of the demand

14 422 / Bernsten and Federgruen functons, wth ncrements of 10.e., a k = k k = 0 1 9), and adjust the slope b upwards to ensure that d p = q. We have computed the optmal centralzed soluton for each of these nstances and compared t wth varous decentralzed systems. We frst compute the supply-chan-wde profts under the Stackelberg game wth the suppler as the leader and the retalers as the followers, competng n terms of ther prces quanttes]. In ths case, the suppler selects the wholesale prce that maxmzes her profts, antcpatng the prcng quantty] and replenshment strateges adopted by the retalers under ths wholesale prce. We also compute the best supply-chan profts that can be acheved under a lnear prcng scheme. We compute the best lnear prcng scheme, both when the retalers compete wth ther prces and when they compete wth ther sales target levels. We fnally compute, for the case of Cournot competton, the supply-chan profts arsng when the equlbrum wholesale prces from the nonlnear dscount scheme w D q l T l 3 are specfed as a lnear prcng scheme. Consder for example the base scenaro where w D q l T l = 2057; a rather dfferent equlbrum and assocated supply-chan-wde profts are acheved when each retaler s charged a constant per-unt cost of 2057.) Because computaton of the best lnear wholesale prcng scheme s rather tedous, we evaluate ths specfc choce of wholesale prces as a possble heurstc. In all the fgures, the horzontal axs descrbes each scenaro correspondng to a value of k = Fgure 2 exhbts the gaps vs-à-vs the optmal centralzed soluton of the Stackelberg game soluton and the soluton under the best lnear wholesale prcng scheme, n the cases of both Bertrand and Cournot competton among the retalers. We observe that the gaps of the Stackelberg game soluton average 13.8% and 16.0%, respectvely, and can be as large as 20.6%, demonstratng the extensve benefts whch a supply chan can accrue by mplementng an approprate coordnaton mechansm. The same set of nstances shows that the use of a nonlnear dscountng scheme, as opposed to the best lnear scheme, s mportant to nduce the proper equlbrum behavor, wth gaps as large as 4%. Recall that a larger scenaro ndex k s assocated wth a larger prce senstvty of the demand. Observe that the gap n supply-chan-wde profts ncurred under the best lnear wholesale prce decreases as the prce senstvty ncreases. Ths apples both to the case of Bertrand and that of Cournot competton. No pattern s apparent as far as the gaps of the Stackelberg solutons are concerned. Smlarly, for the Cournot case, the gaps between the centralzed soluton and the settng n whch the suppler charges w D q l T l, specfed as a lnear prcng scheme, average 3.1%. On the other hand, when the suppler charges each retaler = 1 5, ŵ D q ltl def = w D q ltl Q l 1 q l/ql 4 specfed as a lnear prcng scheme, the gaps vs-a-vs the centralzed soluton vary between 14% and 23%. Note that ŵ D q l T l only depends on retaler s own replenshment nterval T l and hs annual sales volume q l. Ths approxmaton of the coordnatng wholesale prcng scheme w D s therefore of a smpler structure. It s of nterest to gauge how closely the smplfed scheme wthout the externalty effect of compettors) approxmates the coordnatng scheme w D. Fgure 3 exhbts 1) the wholesale prce chosen by the suppler n the Stackelberg game soluton, 2) the best lnear wholesale prce value, 3) w D q l T l, and 4) ŵ D q l T l. We observe that n the Stackelberg soluton the suppler charges an excessvely large wholesale prce resultng n unnecessarly large retaler prces and suboptmal sales volumes. The best lnear wholesale prce s farly close to the coordnatng prce w D, although the former s somewhat hgher. Fnally, gnorng the term Q l 1 q l/ql, whch represents the externalty effect of competton, can result n large changes n the wholesale prces, of up to 44%. The coordnatng wholesale prce w D decreases as we move from left to rght,.e., as the prce senstvty of demand ncreases. The same monotoncty pattern fals to apply at least locally) for the other wholesale prcng schemes. Note also that w D ŵ D, the markup appled by the coordnatng wholesale prcng scheme, decreases as the prce senstvty of demand s ncreased. Ths s to be expected, because, the com- Fgure 2. Gaps wth centralzed soluton Bertrand and Cournot. Bertrand Cournot 20% 25% 15% 20% 10% 15% 10% 5% 5% 0% % Stackelberg Best Lnear Stackelberg Best Lnear

15 Fgure Wholesale prces. Stackelberg Best Lnear wd wd w/o Comp. Impact pettve mpact of each retaler, s a weghted average of the coeffcents ˆ of the cross terms n the nverse demand functons; the latter decrease rapdly wth b, e.g., when gong from scenaro k = 1 to scenaro k = 2, the coeffcents ˆ drop by 16%. Fnally, consder the followng example wth N = 3 competng retalers, wth drect demand functons gven by d 1 p = p 1 + p 2 + p 3, d 2 p = p 2 + p p 3, and d 3 p = p 3 + p p 2. The cost parameters are K 0 = 15 K s = 5 K r = 20 h 0 = 12, h = 04w c 0 = 3, and c = 2 for all = For varous values of w, we compare the resultng Bertrand and Cournot equlbrum strateges. For example, wth wholesale prces w = 252 = 1 2 3, we fnd that the Bertrand equlbrum prces are 37.8, 37.6, 37.6) wth correspondng quanttes 96.4, 112.8, 112.8). On the other hand, the Cournot equlbrum quanttes are gven by 106.5, 83.3, 83.3) and correspondng prces 40.0, 53.3, 53.3). Thus, prces are hgher n the case of Cournot competton, but quanttes are not necessarly lower. We also compute and compare the Stackelberg soluton for each settng. In the case n whch the retalers face Bertrand competton, the Stackelberg wholesale prce s w = 380, yeldng profts of $7136 for the suppler. In the case n whch the retalers engage n Cournot competton, the Stackelberg wholesale prce s w = 361, wth profts of $6061 for the suppler. 7. CONCLUSIONS In ths paper, we have compared the optmal performance of the centralzed supply chan wth that of varous decentralzed supply chans operatng under gven types of wholesale prcng schemes. Whle the exact optmal centralzed) strategy s unknown, we have derved an effcently computable lower bound and upper bound, and we have shown that these bounds are tght as long as the gross proft margns of the retalers are not excessvely low or the holdng cost rate excessvely large. The lower bound, for example, represents the profts of a strategy wth statonary retaler prces, under the optmal power-of-two replenshment polcy for the correspondng vector of sales rates Bernsten and Federgruen / 423 q l. Both q l and the vector T l of replenshment ntervals of the correspondng optmal power-of-two polcy are easly computable. When decson makng n the supply chan s decentralzed, t s easest to characterze the performance of the chan under a smple lnear wholesale prcng scheme, characterzed by an arbtrary vector of constant wholesale prces w. A Nash equlbrum of pure strateges) may fal to exst under completely arbtrary parameter combnatons, both when retalers engage n prce or n quantty competton. However, a Nash equlbrum s guaranteed under both types of competton) when condton C1) prevals: Based on emprcal data, we have shown that ths condton s comfortably satsfed n vrtually all retal ndustres. Under both Bertrand and Cournot competton, the equlbrum s n fact guaranteed to be unque under the related condton C2), whch contnues to be satsfed n vrtually all practcal settngs. There are, however, a number of mportant dfferences between the equlbrum behavor of the retalers under prce and quantty competton. If the retalers compete n quantty space, each adopts a retal prce that s larger than ts equlbrum prce under prce competton. One mght conjecture that, smlarly, the vector of equlbrum sales quanttes of the retalers under Cournot competton, q,s smaller than the vector of sales quanttes under Bertrand competton, q B, but a numercal example n 6 shows that ths relatonshp may fal to hold. Larger sales volumes, under prce competton, can only be guaranteed n specal cases, e.g., when the retalers are dentcal. On the other hand, f q q B s satsfed, t follows that each retaler realzes lower profts under prce competton than under quantty competton. In the case of prce competton, an equlbrum prce vector can be found under C1)) wth the help of the smple teratve tatônnement scheme; n the case of quantty competton, the equlbrum vector q can only be found by solvng the system of frst-order optmalty condtons for the N retalers proft functons. Under Bertrand competton all equlbrum retal prces ncrease as any of the wholesale prces ncrease; the same monotoncty fals to be guaranteed under Cournot competton. Unfortunately, perfect coordnaton cannot be acheved under any lnear wholesale prcng scheme. To acheve perfect coordnaton, a nonlnear wholesale prcng scheme s requred. We derve such a scheme, whch apples three addtve dscounts off a gven constant base prce: The frst dscount component represents a tradtonal dscount scheme, as t offers dscounts exclusvely as a functon of ndvdual order szes. The second dscount component offers a constant dscount for each addtonal unt of tme that the retaler s wllng to keep a unt of hs tem n stock, up to a gven cap of tme unts. The thrd and fnal dscount component offers a dscount exclusvely as a functon of the retaler s annual sales volume. Our coordnaton mechansm therefore provdes an economc ratonale, wthn the context of a model wth complete nformaton and symmetrc barganng power, for wholesale prces to be dscounted

16 424 / Bernsten and Federgruen on the bass of annual sales volumes. Ths type of dscount scheme s most prevalent n practce.) The wholesale prce charged to retaler under the coordnatng scheme equals the average cost per unt of sales) of all cost components ncurred by the suppler that are drectly related to retaler s sales, plus a markup. Ths markup ncreases wth, retaler s compettve mpact, a weghted average of the coeffcents of the crossterms n retaler s nverse demand functon. For a gven total sales volume n the market, the markup for a gven retaler decreases wth ths retaler s market share. If the coeffcents of the crossterms n the demand functons, and hence the cross prce elastctes of demand, are sgnfcantly large, the markups n the coordnatng wholesale scheme are essental. Ignorng these, may result n large gaps n the aggregate supply-chan-wde profts. Whle lnear wholesale prcng schemes fal to acheve perfect coordnaton, they appear to allow for modest gaps wth respect to the frst-best or centralzed soluton. The gaps are modest compared to those arsng under Stackelberg solutons.) Because t s computatonally tedous to dentfy the best lnear wholesale prcng scheme, the followng appears an effectve heurstc: Implement the coordnatng wholesale prces, under the vector of prces and replenshment strateges that are optmal for the centralzed system, as a lnear scheme, chargng each retaler a constant per-unt wholesale prce. More extensve numercal work s needed to compare the varous wholesale prcng schemes consdered n ths paper. APPENDIX Proof of Lemma 1. C s jontly concave as the mnmum of a countable number of affne functons n q. In addton, Q can be parttoned nto a fnte set of regons such that n the nteror of each regon a sngle vector T l q acheves the mnmum n Cq. Ths proves the lemma for C. As to C, ntroducng auxlary varables T 0 = maxt 0 T, 8) may be rewrtten as: P) K mn 1 + T T 0 =0 T 2 h 1 0q T h q T s.t. T 0 T 0 = 1 N 31) T 0 T = 1 N 32) T 0 = 0 1 N Ths s a convex program and, as such, has a strong) dual whch may be derved as follows. For = 1 N, let x and y denote the Lagrange multplers assocated wth constrants 31) and 32), respectvely. By strong dualty, we have that } Cq = max xy0 mn T 0T 0 K0 { K T 0 T 2 h 0q T h q T + x T 0 T 0 + ] y T T 0 = max N mn K 1 + =0 T 2 h 0q x y ) h N q + y T + xy0 T 0T 0 ) T 0 ) x T 0 ] 33) Note that for any par x y, f 1 h 2 0q x y 0 for some = 1 N, the nner mnmzaton problem s unbounded from below; such pars x y can therefore be excluded from the outer maxmzaton. Introducng auxlary varables v = 1 h 2 q + y = 1 N, and v 0 = N x, 33) can be rewrtten as )] K Cq 1 q N = max mn + v T 0 =0 T T = max 2 K v 34) =0 s.t. x + y = 1 2 h 0q = 1 N v 0 = x = 1 N v y = 1 2 h q x y 0 = 1 N = 1 N employng the well-known EOQ formula. The dual problem 34 conssts of the maxmzaton of a concave objectve subject to lnear constrants. Usng these propertes, one easly verfes that the optmum value s jontly concave n q, thus provng part a) for C. Assume now that for a gven vector q, the prmal problem P) has a unque mnmzer T0 uq TN uq T 01 u q Tu 0N q. It then follows from Rockafellar 1997) that Cq 1 q N q = T u 0 q 1 h 2 0q + T u q 1 h 2 q q q = 1 2 h 0 maxt u 0 q T u q h T u q Fnally, t s easly verfed from Roundy 1985) that the set of q-vectors for whch P) does not have a unque mnmum s of measure zero. ENDNOTES 1. The product categores are automobles, furnture, electrcal applances, sportng goods, statonery tems, books, men s clothng, women s clothng, footwear, and toys. If R 0 = N p d, I 0 = average nventory at the suppler, and I = average nventory at retaler, the sales-to-nventory rato for the centralzed supply chan s R 0 /I 0 + N I = 1/I 0 /R 0 + N I /R 0 1/I 0 /R 0 + maxi /p d, = 1 N= 1/1/R 0 /I 0 + 1/mnp d /I, = 1 N. A lower bound for the supply-chan-wde sales-to-nventory rato s thus obtaned

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