It is common for a retailer to sell products from competing manufacturers. How then should the firms manage

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1 MANAGEMENT SCIENCE Vol. 56, No. 3, March 2010, pp ssn essn nforms do /mnsc INFORMS Competng Manufacturers n a Retal Supply Chan: On Contractual Form and Coordnaton Gérard P. Cachon The Wharton School, Unversty of Pennsylvana, Phladelpha, Pennsylvana 19104, cachon@wharton.upenn.edu A. Gürhan Kök Fuqua School of Busness, Duke Unversty, Durham, North Carolna 27708, gurhan.kok@duke.edu It s common for a retaler to sell products from competng manufacturers. How then should the frms manage ther contract negotatons? The supply chan coordnaton lterature focuses ether on a sngle manufacturer sellng to a sngle retaler or one manufacturer sellng to many (possbly competng) retalers. We fnd that some key conclusons from those market structures do not apply n our settng, where multple manufacturers sell through a sngle retaler. We allow the manufacturers to compete for the retaler s busness usng one of three types of contracts: a wholesale-prce contract, a quantty-dscount contract, or a two-part tarff. It s well known that the latter two, more sophstcated contracts enable the manufacturer to coordnate the supply chan, thereby maxmzng the profts avalable to the frms. More mportantly, they allow the manufacturer to extract rents from the retaler, n theory allowng the manufacturer to leave the retaler wth only her reservaton proft. However, we show that n our market structure these two sophstcated contracts force the manufacturers to compete more aggressvely relatve to when they only offer wholesale-prce contracts, and ths may leave them worse off and the retaler substantally better off. In other words, although n a seral supply chan a retaler may have just cause to fear quantty dscounts and two-part tarffs, a retaler may actually prefer those contracts when offered by competng manufacturers. We conclude that the propertes a contractual form exhbts n a one-manufacturer supply chan may not carry over to the realstc settng n whch multple manufacturers must compete to sell ther goods through the same retaler. Key words: contractng; competton; retalng; wholesale-prce contract; quantty dscount; two-part tarff Hstory: Receved November 25, 2007; accepted November 6, 2009, by Ananth Iyer, operatons and supply chan management. Publshed onlne n Artcles n Advance January 29, Introducton The lterature on supply chan coordnaton has studed several contractual forms n settngs wth a sngle manufacturer and one or more retalers. One of the key results from ths lterature s that wholesaleprce contracts lead to suboptmal decsons for the supply chan (.e., double margnalzaton), and more sophstcated contracts (lke quantty dscounts or two-part tarffs) can be employed by a manufacturer to acheve both supply chan coordnaton (.e., maxmze the supply chan s proft) and rent extracton (.e., the ablty to allocate a hgh share of the profts to the manufacturer). Thus, they are vewed to work to the advantage of the manufacturer and to the possble dsadvantage of the retaler (coordnaton s good for the retaler but rent extracton s bad). Our objectve s to test these conclusons n a settng n whch multple manufacturers compete to sell ther products through a sngle retaler. In our model, two manufacturers smultaneously offer to the retaler one of three types of contracts: a wholesale-prce contract, a quantty-dscount contract (.e., a decreasng per-unt prce n the quantty purchased), or a two-part tarff (.e., a per-unt prce and a fxed fee). We refer to the latter two as sophstcated contracts. The retaler determnes the products demand rates (by choosng a prce for each of them) to maxmze her total proft gven the offered contracts and her nventory costs, whch may exhbt economes of scale. Ths model represents a common supply chan structure, as many manufacturers sell ther products through retalers that also sell smlar products from other manufacturers. A typcal example s Procter & Gamble s Crest toothpaste versus Colgate-Palmolve s Colgate toothpaste at a supermarket. To understand how competton among multple manufacturers nfluences contractng wth a retaler, consder an llustratve settng n whch manufacturers A and B sell ther products to a sngle retaler, as depcted n Fgure 1. Focus on the supply chan formed by manufacturer A and the retaler. Wthout 571

2 572 Management Scence 56(3), pp , 2010 INFORMS Fgure 1 Scenaro Product A s not n the assortment Retaler s maxmum proft wth A n the assortment An unacceptable allocaton of proft for the retaler An acceptable allocaton of proft for the retaler Four Proft Scenaros Holdng Manufacturer B s Contract Fxed Manufacturer A Retaler Manufacturer A B B Notes. Each box represents a unt of proft. Product A s ncremental value s 6 because that s the retaler s maxmum ncrease n her proft by addng product A to the assortment. Scenaro 3 s unacceptable to the retaler because the retaler prefers Scenaro 1n whch product A s not ncluded n the assortment. In Scenaro 4, manufacturer A earns hs ncremental proft, and the retaler s ndfferent between ncludng A n the assortment or not. product A n the assortment, say the retaler earns 12 unts of proft gven manufacturer B s contract (Scenaro 1). Hence, the retaler s reservaton proft s 12 when consderng A s contract offer; that s, the retaler accepts A s offer and add A s product to the assortment only f the retaler can earn a proft of at least 12. The supply chan s sad to be coordnated when ther combned proft s maxmzed. Ths occurs when A sells hs product at cost to the retaler n that case, the retaler earns all of the supply chan s proft and so chooses prces to maxmze t. We refer to ths proft as the maxmum proft, and n ths example, say, t s 18 (Scenaro 2). Fnally, defne product A s ncremental proft as the dfference between the supply chan s maxmum proft and the retaler s reservaton proft, whch n ths example s 6 (18 12). A product s ncremental proft s a useful construct because, among all possble contracts, t s the upper bound on how much a manufacturer can earn. To explan, f A were to offer a contract such that A earned more than hs ncremental proft, then t must be that the retaler s earnng less than her reservaton proft, and so she wll reject the contract (Scenaro 3). Thus, A s objectve s to try to earn as much of hs ncremental proft as possble. To earn all of the ncremental proft requres supply chan coordnaton and substantal rent extracton (.e., A earns 6 unts of proft, leavng the retaler wth her reservaton proft of 12), as n Scenaro 4. Ths s true whether A s a monopolst or s competng wth another manufacturer. For example, a wholesale-prce contract does not allow a monopolst manufacturer to earn hs ncremental proft (because t fals on coordnaton and has lmted rent extracton), but the manufacturer can acheve ths wth a properly desgned sophstcated contract, hence the advantage of sophstcated contracts for a monopolst manufacturer. In fact, the same s true even when the manufacturers must compete holdng the other frm s contract offer fxed, the sophstcated contracts allow a manufacturer to capture a large fracton (possbly all) of ts ncremental proft. In other words, holdng B s contract offer fxed, A prefers a well-desgned sophstcated contract over the wholesale-prce contract. So what changes n the contractng process when manufacturers compete? The answer s that a manufacturer s ncremental proft s exogenously determned when there s no competton and endogenously determned when there s competton. Returnng to our example, A s ncremental proft s 6 for a gven contract offered by B. If B were to offer a dfferent contract, then A s ncremental proft could be dfferent (possbly more, possbly less). Furthermore, we wll show that A s ncremental proft declnes as B uses a more sophstcated contract (and vce versa), and t declnes substantally when the products are close substtutes or f there are substantal economes of scale n nventory costs, or both. (Ths holds because addng product A to the retaler s assortment ncreases the retaler s proft more when manufacturer B offers a contract that does not maxmze the total proft from product B.) If they both use sophstcated contracts, they wll both be able to capture a large porton of ther ncremental profts (whch s why they prefer these contracts holdng the other frm s contract offer fxed), but f the products are close substtutes, ther ncremental profts wll be small. In contrast, f they both offer wholesale-prce contracts, then they wll both be able to capture only a modest porton of ther ncremental profts, but ther ncremental profts wll be large. In short, t can be better to have a pece of a large pe (wth wholesale-prce contracts) than to have all of a small pe (wth the sophstcated contracts). It follows that when the manufacturers sell close substtutes through a common retaler, sophstcated contracts can work to ther dsadvantage, whch s n sharp contrast to the results wth a sngle manufacturer. Put another way, although the exstng supply chan coordnaton lterature suggest that a retaler should be wary of a manufacturer offerng sophstcated contracts (because ths could lead the retaler to earn only her reservaton proft), our results wth competng manufacturers suggest that the retaler may prefer that they offer sophstcated contracts (because the supply chan s coordnated but the manufacturers have small ncremental values). The rest of ths paper s organzed as follows. Secton 2 revews the relevant lterature. Secton 3 descrbes the model. Secton 4 presents our analyss of the retaler s problem, 5 the analyss and comparson

3 Management Scence 56(3), pp , 2010 INFORMS 573 of the wholesale-prce, quantty-dscount, and twopart tarff games, and 6 the numercal study. Secton 7 concludes ths paper. All proofs are presented n the appendx. 2. Relaton to the Lterature The present paper s foremost a commentary on the supply chan coordnaton lterature. See Cachon (2003) for a revew of ths lterature. As already dscussed, ths lterature focuses on ether relatonshps wth blateral monopoly or models wth one manufacturer and multple retalers. Wholesale-prce contracts are nearly always found to be neffcent, and more sophstcated contracts can be used to elmnate that neffcency and reallocate rents arbtrarly between the partes n the supply chan. 1 There s an extensve lterature on supply chan coordnaton wth quanttydscount contracts and prce-dependent demand (e.g., Jeuland and Shugan 1983, Moorthy 1987, Ingene and Perry 1995) and on lot-sze coordnaton wth fxed demand (e.g., Monahan 1984, Corbett and de Groote 2000), but they consder only one manufacturer. Cho (1991), Trved (1998), Lee and Staeln (1997), and Martnez de Albenz and Roels (2007) study systems wth multple manufacturers and a common retaler, but they only consder wholesale-prce contracts. The lterature on common agency (Bernhem and Whnston 1985, 1986) and vertcal separaton n economcs s also relevant, because they study models wth multple manufacturers and retalers that could sell more than one product. Mathewson and Wnter (1987) explore whether or not exclusve dealng arrangements lead to foreclosure of rvals and the mplcatons for anttrust laws consderng only wholesale-prce contracts. Bernhem and Whnston (1998) and O Bren and Shaffer (1993, 1997) explore smlar ssues wth two-part tarffs. Other varatons of the basc model nclude manufacturers that nvest n the retalers to reduce the margnal sellng cost (Besanko and Perry 1994) and manufacturers that assgn exclusve terrtores to retalers to reduce competton (Rey and Stgltz 1995). The general result s that the manufacturers may prefer exclusve dealng due to reduced competton at the retaler level even though socetal welfare and ndustry profts may be hgher wth common agency. Ths s essentally a comparson of dfferent supply chan structures, whereas we are concerned wth the effect of dfferent contract 1 A wholesale-prce contract can maxmze profts n a system wth one manufacturer and multple quantty competng retalers. However, t provdes only one allocaton of the system s rents, and t s not even the manufacturer s optmal wholesale prce (see Cachon and Larvere 2005). If retalers compete on prce and quantty, then the wholesale prce no longer guarantees coordnaton (see Bernsten and Federgruen 2003, 2005). types n a gven supply chan structure. Also related to two-part tarffs, there s a lterature on slottng fees (whch are essentally two-part tarffs wth negatve payments to the manufacturer; see, for example, Marx and Shaffer 2010). Kuksov and Pazgal (2007) show that slottng fees do not occur n a settng wth smultaneous manufacturer competton and a sngle retaler, and the same result apples n our model. The lterature on strategc decentralzaton n marketng s hghly relevant. McGure and Staeln (1983) study two competng supply chans under two structural forms: n each supply chan ether the manufacturer sells to a dedcated retaler va a wholesale-prce contract or the manufacturer vertcally ntegrates nto retalng. In ether structure, the products of the two manufacturers are sold to consumers from dfferent frms, whereas n our model the manufacturers products are sold through a sngle ndependent retaler. Nevertheless, there are some smlartes n our results. McGure and Staeln (1983) fnd that the manufacturers may prefer to sell va wholesale-prce contracts, despte the fact that they do not coordnate the channel nor allow the manufacturer to extract all rents, because they dampen retal competton between the two products relatve to the vertcally ntegrated structure. In our model, competton to consumers s held constant, because we have a sngle retaler, so what changes s that the retaler s reservaton proft now depends on the contract offers by the manufacturers, enablng the retaler to gan more of the supply chan profts due to the manufacturers competton. Coughlan (1985) confrms the McGure and Staeln (1983) fndngs n an emprcal study of the nternatonal semconductor ndustry. In the context of the McGure and Staeln (1983) model, Moorthy (1988) provdes condtons on the characterstcs of the game and the demand functon for decentralzaton to be an equlbrum strategy, and Bonanno and Vckers (1988) show that decentralzaton s always the equlbrum when both manufacturers employ two-part tarffs. In a dfferent context wth strategc consumers tmng ther purchase of a product, Desa et al. (2004) and Arya and Mttendorf (2006) consder a monopolst manufacturer and show that decentralzaton through wholesale-prce contracts can yeld hgher profts for the channel. 3. The Model There are two products n the market suppled by two dfferent manufacturers. The products are partal substtutes and are sold through a common retaler. In the frst stage of the game, the manufacturers smultaneously announce the payment schemes for ther products,.e., ther contract offers. In the second stage, the retaler chooses prces, whch determne the products

4 574 Management Scence 56(3), pp , 2010 INFORMS demand rates, to maxmze her proft. In addton to the payments to the manufacturer, the retaler ncurs operatng costs that depend on the average volume sold of each product. The manufacturers ncur constant margnal producton costs. The retaler faces prce senstve customers. The revenue from product s R d p dd where d s the demand rate of product, d s the par of demand rates, and the nverse demand functon s p d d d j and > j > 0 for all j We elect to work wth the nverse demand functon for expostonal smplcty. The formulaton wth the demand functon s equvalent to the above. Let G d be the retaler s nventory related operatonal costs of product, G d K d K 0 0 <<1 where K and are exogenous constants. Ths functonal form, whch exhbts economes of scale, s a general representaton of the nventory costs that arse n common nventory replenshment models such as a base-stock model 2 or an economc order quantty (EOQ) model. 3 Let denote the retaler s proft from product and + j, the retaler s total proft. It follows that R d G d T d and T d s the payment made to the manufacturer based on the retaler s demand rate and ther agreedupon contract. 4 Manufacturer s proft s T d c d (1) where c s the manufacturer s cost per unt. 2 In a perodc revew model where demand follows a Normal dstrbuton wth mean d and standard devaton d, the total nventory related costs wth the optmal base-stock level s gven by b + hz d, where b s the backlog penalty per unt and h s the nventory holdng cost per perod. Defnng K b + hz leads to the G functon. 3 In the EOQ model, the retaler ncurs a fxed cost k per order and a holdng cost h per unt of nventory held for one perod. The well-known EOQ formula suggests orderng every 2k /h d perods. The resultng optmal total nventory and orderng cost s gven by 2k d h. Defnng K 2k h leads to the G functon wth 1/2. 4 The retaler s payment T d to a manufacturer can be nterpreted as a yearly (average) payment based on yearly (average) volume. Many manufacturer retaler purchasng contracts are based on the yearly volume rather than on the volume of ndvdual shpments. We focus on three dfferent types of contracts: wholesale-prce contracts, quantty-dscount contracts, and two-part tarffs. In the frst stage, the manufacturers strategy set ncludes one or more of these contractual forms. For expostonal clarty, we begn wth a lmted strategy space (e.g., just wholesaleprce contracts, or just quantty-dscount contracts, etc.) and then later consder a broader strategy space. Wth a wholesale-prce contract, the payment functon s T d w d where w s the wholesale prce chosen by manufacturer. Wholesale-prce contracts are common n practce and serve as a theoretcal benchmark. Quantty dscounts are common n many ndustres (see Munson and Rosenblatt 1998 for a feld study). We consder the followng famly of quantty dscounts: w d v d 2/2 f d w c /v T d T w c /v + c d w c /v (2) otherwse, where w and v are parameters, and v 0 v and v 2 + j. Note that wholesale-prce contracts are a subset of these quantty dscounts a wholesale-prce contract can be obtaned by settng v 0. Although we only consder a subset of possble quantty dscounts, ths s not overly restrctve. Our quantty dscounts are contnuous, dfferentable, concave, and the manufacturer does not sell even the margnal unt for less than ts producton cost (whch s the reason for the breakpont at w c /v. The upper bound on v mples that the quantty dscount s not too aggressve n the sense that the margnal prce pad does not fall too rapdly as the purchase volume ncreases,.e., T d v. In fact, t can be shown that our quantty dscounts are optmal for the manufacturer (holdng the other manufacturer s contract offer fxed) among all concave and ncreasng payment functons gven the T d v constrant (see Proposton 2 n the appendx). Furthermore, T d v ensures that the retaler s proft functon excludng the operatonal costs s strctly concave n d 1 d 2. Ths naturally rases the queston of whether the manufacturer could do better by offerng an even more aggressve quantty dscount. In a sngle-product envronment, the answer s no : as v approaches v, the supply chan s proft s maxmzed and the manufacturer earns all of that proft. In a twoproduct envronment, the answer s not clear because retal profts are no longer strctly concave wth v v. However, the next contract we descrbe can be nterpreted as the most aggressve quantty dscount, and we do have results for those contracts.

5 Management Scence 56(3), pp , 2010 INFORMS 575 Our thrd contract type s the two-part tarff, whch s characterzed by a fxed fee F and a margnal cost w : T d F 1 d >0 + w d (3) where ndcator functon 1 d >0 1 fd > 0 0 otherwse. The two-part tarff s an aggressve quantty-dscount contract because the margnal cost of the frst unt s F, whereas the margnal cost of all subsequent unts s only w (whch s generally much smaller than F. In a symmetrc game across manufacturers, the data for the two products are dentcal,.e., c,,,, K, and v are the same for any. The subscrpt wll be dropped n those cases. In a symmetrc soluton, the decsons (d at the retal level, w or T at the manufacturer level) are dentcal across products. In the followng sectons, we solve the problem usng backward nducton. We analyze the retaler s decson frst and then the game between the manufacturers. 4. The Retaler s Decson Ths secton studes the retaler s quantty decsons gven the two contract offers from the manufacturers. We present the analyss assumng the manufacturers offer the retaler quantty dscounts, whch s suffcent to understand the retaler s decson under any of the three contract forms we consder: wholesaleprce contracts are quantty dscounts wth v 0, and, from the perspectve of choosng demand rates for the products, two-part tarffs are dentcal to wholesale-prce contracts (because the fxed fees do not matter for the retaler s prcng decsons gven an assortment). Defne d d j arg max d d j (4) d d d j arg maxd d j d d j > 0 (5) d d j d d j (6) where d d j s the retalers optmal demand for product gven a demand choce for product j, d d j s the par of optmal demands condtonal that each product s ncluded n the assortment (wth some postve sales), and s the retaler s resultng proft. The retaler s optmzaton problem can now be wrtten as max 1 d d 2 0 (7) Hence, the optmal soluton to the retaler s problem s d 1 d 2 d 1 d 2 d d 2 0 For expostonal smplcty, we assume that the retaler breaks tes n favor of carryng a full product lne over a sngle product. Consder the problem of maxmzng the system s proft (.e., the total proft across the three frms). It s equvalent to the retaler s problem f the manufacturers charge only ther producton cost, T d c d for 1 2. Defne 1 1 d 1 0 0, 2 2 0d 2 0, and 12 d 1 d 2 under those contracts. These proft levels are, respectvely, the maxmum proft the system would earn f t were to carry only product 1, only product 2, and both products. We assume 12 > > 0 for 1 2 (8) whch mples that t s always optmal for the system to carry both products. Holdng manufacturer 2 s contract offer fxed, the retaler s reservaton proft for acceptng manufacturer 1 s offer s 2 0d 2 0,.e., the retaler accepts manufacturer 1 s offer only f addng product 1 to the assortment ncreases her proft over what she could earn wthout t n the assortment. To evaluate manufacturer 1 s ncremental proft, assume manufacturer 1 charges only hs producton costs, T 1 d 1 c 1 d 1 whle holdng manufacturer 2 s offer fxed. In that case, manufacturer 1 s ncremental proft s max 1 d d 2 0 where the frst term s the most the product 1 supply chan can earn (condtonal on product 2 s contract), and the second term s the retaler s reservaton proft. Now let us consder the retaler s demand decson wth ndependent manufacturers. Let H denote the frst dervatve of wth respect to d : H /d 2 d + j d j G d T d j (9) Consder the case wth no economes of scale (.e., K 1 K 2 0). The retaler s proft functon s jontly concave n d 1 d 2, so the unque soluton to H s the unque optmal soluton. If an nteror soluton does not exst, then the optmal soluton s ether d or 0d 2 0, where d 0 s the unque soluton to H 0 wth d j 0. Because s well behaved, the optmal soluton d1 d 2 can be easly characterzed, and t s a contnuous, dfferentable functon of the problem nputs such as the parameters of the manufacturers contracts. The case wth economes of scale, however, s rather complcated. Observe that 2 /d 2 2 G d T d s postve at d 0 and then decreasng n d Thus, s convex-concave n d for fxed d j. There are up to two solutons to H 0 and the larger of the two solutons s a local maxmum. The corner soluton,

6 576 Management Scence 56(3), pp , 2010 INFORMS d 0, s ether the larger soluton or zero. Evdently, s not jontly concave n d d j. As a result, there may be multple solutons to H 0 1 2, and we do not know whch one could be the nteror optmal soluton. Furthermore, the global optmal soluton may be at one of the boundares d 0. A fnal techncal note s that the proft functon s not necessarly unmodal (n one or two dmensons). The next theorem shows that there can be at most one nteror local maxmum and that the optmal soluton s ether that nteror soluton or at one of the boundary lnes; that s, there are at most three canddate optmal solutons and each s characterzed by a set of frst-order condtons. Furthermore, n a symmetrc problem, the unque nteror maxmum s a symmetrc soluton. Theorem 1. The retaler s optmal soluton s d1 d 2 d 1 d 2 d d 2 0, where d 1 d 2 s the unque nteror optmal soluton to H 0 1 2, and d 0 s gven by the larger of the two solutons to H 0 st d j 0. In a symmetrc problem, d 1 d 2 d d, where d s the larger of the two solutons to 2 + d G d T d 0 In summary, there are at most three local maxma for a retaler s problem: one nteror soluton n whch the retaler carres both products, and two boundary solutons n whch the retaler carres only one product. Although the retaler s proft functon s generally complex n the presence of economes of scale, the followng condtons ensure that t s jontly concave n d 1 d 2. We state and prove ths result n Lemma 1 n the appendx (10) T d v where 0 v < v (11) 2R /G for all where j j j p d (12) Condton (10) requres the own- and cross-prce coeffcents to be symmetrc. Ths s not a very restrctve assumpton, because we allow nondentcal, whch mples dfferent demand rates and prce elastctes for the products. Condton (11) s strcter than our earler assumpton: t requres the quantty dscount to be less concave to guarantee the concavty of the retaler s proft functon. Condton (12) stpulates that the own-prce elastcty ( s less than two tmes the revenues to average nventory costs rato of the product. (It s smlar to the condtons Bernsten and Federgruen (2003) developed for decentralzed retalers, and as they pont out, the condtons hold for most retalers based on the ndustry data n Dun and Bradstreet (2006) and Tells (1988).) 5. Competton Between the Manufacturers In ths secton, we study the game between the manufacturers n contract offers. We begn wth some observatons that apply no matter what type of contract s offered. In 5.1, we characterze the game s equlbra when both manufacturers choose quantty-dscount contracts, and 5.2 provdes the analogous analyss when the manufacturers choose two-part tarffs. Secton 5.3 expands the manufacturers strategy set to nclude all three contract types that we consder. Secton 5.4 compares the equlbra under dfferent contract types for the more analytcally tractable case of symmetrc products and no economes of scale n nventory costs. In the game between the manufacturers, each one chooses hs own best response T T j gven the other manufacturer s contract offer T j : T T j arg max d for all T d where d arg max (13) and t s understood that T d s chosen from a defned strategy space (e.g., quantty-dscount contracts wth a fxed v An equlbrum of the game s a par of contracts T T j such that nether manufacturer has an ncentve to offer a dfferent contract. The followng remark demonstrates how the contractng problem wth multple manufacturers s dfferent from that wth a sngle manufacturer. Remark 1. For any fxed contract offered by manufacturer 2such that 2 0d 2 0 > 0: 1. Consder the set of contracts such that the retaler s payment to manufacturer 1 s a nondecreasng functon of d 1. There does not exst a contract n ths set such that the manufacturer can extract all of the proft from hs product (.e., t s not possble to have 1 > 0 and 1 0). 2. The retaler accepts manufacturer 1 s contract offer and stocks both products only f 1 d d j + 2 d d j 2 0d 2 0 (14) Unlke n a supply chan wth a sngle manufacturer, the frst statement mples that a manufacturer must leave the retaler wth some proft to nduce the retaler to carry the manufacturer s product (see the appendx for a detaled proof). However, ths does not mean that a sngle reservaton proft exsts. As descrbed n 1, the rght-hand sde of (14) s the retaler s reservaton proft wth respect to manufacturer 1 s contract offer, and t depends on the partcular contract offered by manufacturer 2. Hence, there does not exst a sngle reservaton proft the reservaton proft s endogenously determned by the actons

7 Management Scence 56(3), pp , 2010 INFORMS 577 of the other manufacturer. Furthermore, t s not possble to replcate the dynamcs of ths model wth a carefully desgned seral supply chan and a fxed reservaton proft. To explan, va a smple manpulaton of (14), the retaler s reservaton proft for product 1 alone s the rght-hand sde of 1 d d j 2 0d d d j whch depends on manufacturer 1 s contract offer (even f the other contract s held fxed), and therefore cannot be represented by a sngle value. Before consderng the equlbrum contract offers, we add an observaton that rules out equlbra n whch the retaler carres only one product n symmetrc games. Remark 2. In a symmetrc game between the manufacturers, there does not exst an equlbrum where d 0 for some. The result s due to (8), whch guarantees that the ncluson of a manufacturer strctly ncreases system proft. Suppose there were an equlbrum n whch manufacturer s excluded. Regardless of the fracton of j the retaler earns, manufacturer can offer to sell to the retaler at c + for an arbtrarly small, and then the retaler s proft ncreases f t carres product. Ifj s excluded as a result, t wll react smlarly and get ts product ncluded Wholesale-Prce and Quantty-Dscount Contracts In ths secton, we consder the game between the manufacturers n whch they make quantty-dscount contract offers wth a gven v v j. Hence, the two manufacturers smultaneously announce ther w parameter to the retaler, then the retaler decdes how much to sell of each product. Recall that wholesale-prce contracts have v 1 v 2 0, and they are a specal case of ths analyss. In the presence of economes of scale (.e., f K > 0 for any, we assume that condtons (10) and (11) hold, and we restrct our attenton to a regon defned by (12). Defne w w j maxw dj w w j 0, the maxmum w that makes the retaler exclude product j, and w w j mnw d w w j 0, the mnmum wholesale prce of that makes the retaler exclude product. Note that w w j may not exst for every w j. In that case, set w w j c. Defne w w j as the best response of manufacturer, whch can be found va a lne search between w w. The followng theorem characterzes the unque symmetrc equlbrum of the contract offer game. Theorem 2. Consder a symmetrc game n whch the manufacturers offer quantty dscounts. There exsts a unque symmetrc soluton to d w 1w 2 + w v c d w 0 for all denoted w1 w 2, whch s the unque canddate to be a symmetrc equlbrum. As can be seen n the proof Theorem 2, showng the unmodalty of a manufacturer s proft n w requres the use of the second-order propertes of the retaler s optmal soluton d1 d 2. In the presence of economes of scale, we have shown that the retaler s decson problem s complex (e.g., not concave, not unmodal), and ths creates sgnfcant challenges to the analyss of the game between the manufactures, whch s bult on top of the retaler s decson problem. These challenges are not present n other competton papers n the lterature. We cannot guarantee that the canddate pont descrbed n Theorem 2 s an equlbrum. We show that at w1 w 2, w s a local optmum for manufacturer, and s concave for w >w. However, the optmal soluton w wj may be dfferent than w n the range w w. If w wj w, then w 1 w 2 s ndeed an equlbrum pont. If not, then there exsts no symmetrc equlbrum. Now consder the stuaton n whch there are no economes of scale. Ths substantally smplfes the analyss of both the retaler s demand decsons and the manufacturers contract offer problem. For any v 1 v 2 and asymmetrc products, we can now guarantee jont concavty of the retaler s proft and the exstence and the unqueness of the equlbrum wthout the symmetry assumptons. The next theorem provdes the closed form solutons for the demand rates and the contract parameters by solvng the frstorder condtons gven n Theorem 2. Theorem 3. Wth no economes of scale (.e., K 0), there exsts a unque equlbrum of the game n whch the manufacturers offer quantty dscounts. It s characterzed by the followng reacton functons and the optmal demand rates: d 2 j v j w + j j w j 2 v 2 j v j + j 2 w w j j + j j w j +v j +c j j 2+v j where j 2 j v j and 2 v 2 j v j + j Two-Part Tarffs In ths secton, we consder the game between manufacturers who offer two-part tarff contracts. Theorem 4. The equlbrum of the two-part tarff game s F 12 j and w c for 1 2, j 3. Hence, 12 j and

8 578 Management Scence 56(3), pp , 2010 INFORMS In the two-part tarff equlbrum, each manufacturer charges w c so that the retaler maxmzes ther combned profts. The manufacturer then extracts hs full ncremental proft va ts fxed fee, and the retaler earns her reservaton proft on each product. However, ths does not mean that the manufacturers earn a large proft or even more than the retaler. If the two products are close substtutes, then each manufacturer s ncremental proft, 12 j, can be qute small. To explan, magne the products were perfect substtutes. In that case, the retaler can earn as much from sellng just one as she can earn from sellng both,.e., , and each manufacturer s ncremental proft s then zero, leavng the manufacturers wth zero proft and the retaler wth 1 > 0. Ths logc extends to the case of economes of scale. As economes of scale become stronger, 12 decreases relatve to 1 or 2 : sellng both products s relatvely less attractve than sellng just one product when economes of scale ncrease because t becomes costly to fragment demand across multple products. Hence, an ncrease n economes of scale should decrease each manufacturer s ncremental proft, thereby resultng n lower profts for them and hgher profts (n a relatve sense) for the retaler Contract Choce In the prevous sectons, we assumed that the manufacturers offered contracts from a lmted set: wholesale-prce contracts, quantty dscounts wth a fxed v parameter, or two-part tarffs. Ths secton consders the manufacturers equlbrum contract choce when the set of contracts avalable to them s expanded. Suppose the manufacturers could choose any contract type to offer the retaler,.e., the unrestrcted contract offer game. Accordng to the next corollary (whose proof s actually part of the proof of Theorem 4), restrctng the frms to the set of two-part tarffs does not actually change the outcome of the game. Corollary 1. Consder the contract choce game n whch the manufacturers are free to choose any contract type and parameters. It s optmal for each frm to offer a two-part tarff wth w c, no matter what contract offer the other frm makes. Furthermore, the equlbrum of the two-part tarff game descrbed n Theorem 4 s also an equlbrum of the unrestrcted contract choce game. Now suppose that two-part tarffs are not n the contract space and the manufacturers smultaneously offer a quantty dscount w v where they are free to choose any v 0 ˆv for some ˆv < v. In other words, they can choose to offer a wholesale-prce contract (v 0 or a quantty dscount (v > 0 The next proposton ndcates, as n supply chans wth a sngle manufacturer, that a manufacturer prefers to offer a quantty-dscount contract and, n partcular, prefers more aggressve quantty dscounts. Quantty dscounts allow the manufacturer to mprove supply chan coordnaton (.e., reduce double margnalzaton) and to extract rents, so they are the preferred contract when the other manufacturer s contract s held fxed even when the retaler can adjust her demand allocatons between the two products n response. Proposton 1. In the quantty-dscount game where the manufacturers can choose from among the set of quantty dscounts, manufacturer s proft strctly ncreases wth v at the optmal wholesale prce w ; that s, f a manufacturer s gven the opton to choose between three contractual forms wth v 0ab such that 0 <a<b then w v 0< w v a < w v b. The mmedate mplcaton of ths proposton s that f the manufacturers are able to choose the quadratc parameter of ther quantty dscount as well as the startng wholesale prce, each manufacturer s best response s to choose the most aggressve contract n our consderaton set. The next corollary states that any equlbrum of ths game would then have the most aggressve contracts by both manufacturers. Corollary 2. In the contract choce game n whch the manufacturers smultaneously offer w v and they are free to choose any v 0 ˆv for some ˆv< v the equlbrum contracts are such that v ˆv for both manufacturers. To summarze, n equlbrum, the manufacturers choose the most aggressve contract n ther strategy set, be t a two-part tarff or a quantty dscount Comparson of Equlbra The prevous secton establshed that f the manufacturers are free to choose any contract they want to offer the retaler, n equlbrum they choose to offer two-part tarffs. We also characterzed the equlbrum contract offers when the manufacturers strategy set s restrcted to only wholesale-prce contracts or only quantty-dscount contracts. In ths secton, we compare these varous outcomes. To provde analytcal tractablty, we consder a system wth symmetrc products (.e., dentcal,,, c, and v and no economes of scale at the retaler.e., K 0 Furthermore, among the set of quantty dscounts, we provde results n the lmt as v goes to 2, the upper bound. (Recall that Corollary 2 ndcates that the manufacturers would choose those quantty dscounts f they are allowed to do so.) Table 1 presents the equlbrum proft terms under the three contract types. The dervaton of the proft expressons are presented under Theorem 7 n the appendx. As ndcated earler, two-part tarffs maxmze the system s combned profts, and Table 1 reveals that

9 Management Scence 56(3), pp , 2010 INFORMS 579 Table 1 Equlbrum Profts Under Wholesale-Prce, Quantty-Dscount, and Two-Part Tarff Contracts n the Case wth Symmetrc Products and No Economes of Scale Manufacturer Retaler Supply chan Contract Wholesale prce Quantty dscount as v 2 Two-part tarff c c c c c c c c c our chosen quantty dscounts do as well (ths result does not generalze to other quantty-dscount contracts). Nevertheless, the manufacturers are not able to extract ther full ncremental proft wth quantty dscounts (they earn less wth those contracts than the two-part tarff), whch mples the retalers s better off wth quantty dscounts than wth two-part tarffs. Furthermore, t s straghtforward to show that the wholesale-prce contract fals to maxmze the system s combned profts. Suppose the products are perfect substtutes,.e.,. In ths case, the ncremental proft of each manufacturer s zero under all contract types. Therefore, at equlbrum, the manufacturers offer margnal cost prcng, resultng n system optmal profts, and the retaler captures all the profts; that s, all contracts are equvalent for all players. When products are ndependent,.e., 0, the system s equvalent to two ndependent supply chans, and each manufacturer s able to acheve supply chan optmal proft and extract all of t ether wth quantty dscounts or two-part tarffs, but not wth wholesale-prce contracts. Ths case replcates the common assumpton n the supply chan contractng lterature wth a monopolst manufacturer. If s reduced further, <0 then the products are complements. Now, < 12 (.e., sellng both products together earns more than the sum of sellng them ndvdually). It follows that the retaler has no leverage over the manufacturers, and she earns zero proft, just as n the 0 case. Theorem 9 n the appendx provdes further detals for the equlbrum wth complementary products. Now we turn attenton to the nterestng stuaton wth mperfect substtutes, 0. The next theorem compares the manufacturers profts wth wholesale-prce contracts to the other two types. Theorem 5 (Manufacturers Perspectve). () The manufacturers profts are hgher under quantty dscounts than wholesale-prce contracts f and only f < () The manufacturers profts are hgher under twopart tarffs than wholesale-prce contracts f and only f < Theorem 5 ndcates that the manufacturers are better off wth the sophstcated contracts only f the products are not close substtutes, and ths effect s stronger for the quantty dscounts (n the sense that quantty dscounts are preferred over a narrower parameter range). Note, holdng the other manufacturer s contract offer fxed, a manufacturer always prefers a quantty-dscount contract over the wholesale-prce contract and a two-part tarff over the quantty dscount. Hence, although they would choose two-part tarffs f they are not restrcted n ther contract choce, they would be better off had they been restrcted to offer only wholesale-prce contracts. In effect, the contract choce game s lke the classc prsoner s dlemma. The manufacturers may be worse off n equlbrum wth the sophstcated contracts because the degree of product substtutablty,, nfluences both a contract s ncremental proft as well as the share of that proft that the manufacturer can extract from the retaler. To llustrate ths explctly, the followng are the ncremental proft of a manufacturer under the three contract types gven the equlbrum contract offer from the compettor (dervatons are presented n Theorem 8 n the appendx): IP W c IP Q IP T c2 4 + where the superscrpts W, Q, and T denote the wholesale-prce, quantty-dscount, and two-part tarff contracts, respectvely. It s straghtforward to show that the wholesale-prce contract has the hghest ncremental proft: IP W >IP Q for all 0. Ths holds because addng product A to the retaler s assortment ncreases the retaler s proft more when manufacturer B offers a contract that does not maxmze the total proft from product B (a wholesaleprce contract) relatve to when manufacturer B offers a contract that does maxmze product B s proft (the two sophstcated contracts). It s also worth notng that the wholesale-prce contract s relatve advantage n ncremental proft ncreases as the products become more substtutable ( ncreases): IP W IP Q 42 2 > 1 2 The drawback of the wholesale-prce contract s that t captures only a fracton of ts ncremental proft, one-half of t to be precse, for all. Even the

10 580 Management Scence 56(3), pp , 2010 INFORMS quantty-dscount contract extracts only a fracton of ts ncremental proft, / +, but t s greater than one-half n the range of nterest (though, t s also decreasng n ). Puttng these results together, f products are close substtutes, half of a large ncremental proft (under wholesale-prce contracts) may be larger than a larger share of a smaller ncremental proft (under sophstcated contracts). Now, let us consder the problem from the retaler s perspectve. The next theorem compares the retaler s equlbrum profts under the three contract types. Theorem 6 (Retaler s Perspectve). () The retaler s proft s hgher under quantty dscounts than wholesale-prce contracts f and only f > () The retaler s proft s hgher under two-part tarffs than wholesale-prce contracts f and only f > Wth a sngle manufacturer, the retaler s worse off wth quantty dscounts and two-part tarffs because they enable the manufacturer to extract rents from the retaler, leavng her wth her reservaton proft (possbly zero). However, when the manufacturers compete, the outcome s qute dfferent. Now the retaler s better off wth the more sophstcated contracts whenever the products are suffcently close substtutes. These more sophstcated contracts ncrease the system s total proft and as long as the products are close substtutes; the lon s share of that proft goes to the retaler (because the manufacturers ncremental profts are small). Fgure 2 llustrates under whch contract equlbrum the manufacturers and the retaler are better off for dfferent levels of substtutablty. As already dscussed, for hgh values of, the retaler prefers the sophstcated contracts because they lead to low ncremental profts for the manufacturers, and the manufacturers have the opposte preference although Fgure 2 Equlbrum Preferences of the Frms for Dfferent Levels of Substtutablty Manufacturers Retaler Manufacturers Retaler Wholesale-prce vs. quantty-dscount contracts < (0.354) > (0.354) Quantty dscount Wholesale prce < (0.177) Wholesale-prce contracts vs. two-part tarffs γ < (0.586) Wholesale prce Two-part tarff total rents n the system are lower wth wholesaleprce contracts and those contracts cannot extract rents as effcently (they earn only a fracton of ther ncremental proft), when s hgh they prefer the wholesale-prce contract. When s low, the manufacturers prefer the rent extractng ablty of the sophstcated contracts because ther ncremental profts reman reasonably hgh. The retaler s worse off wth the sophstcated contracts because the manufacturer s extract a large chunk of the system s proft. Hence, for low and hgh levels of product substtutablty, the manufacturers preference s the opposte of the retaler s preference. However, for ntermedate values of, they both prefer the sophstcated contracts. As wth all, the sophstcated contracts ncrease the system s total proft (.e., coordnates the system), but now both the manufacturers and the retaler are better off because the manufacturers ncremental profts are nether too hgh (as wth a low ) nor too low (as wth a hgh ). As mentoned, the manufacturers face the classc prsoner s dlemma when the products are close substtutes. In a sngle-perod game context, they always offer the more sophstcated contracts at equlbrum, but they prefer the system n whch they both offer wholesale-prce contracts. In a repeated game settng, the Folk Theorem suggests that they could theoretcally coordnate (on offerng wholesale-prce contracts) usng trgger-type punshment strateges. (Ths requres the ablty to observe the compettor s contract type, whch may be theoretcally nferred from the retal prces.) In ths case, the retaler may offer sde payments to brng the manufacturers to the system-effcent outcome (sophstcated contracts). When products are not so close substtutes, the equlbrum under sophstcated contracts s preferred by the manufacturers, but not by the retaler. In practce, the barganng powers of the frms n the negotaton > (0.177) Wholesale prce Quantty dscount Two-part tarff > (0.586) Wholesale prce < (0.382) > (0.382)

11 Management Scence 56(3), pp , 2010 INFORMS 581 process determne whch of these outcomes wll be observed. Although our fndngs bear a resemblance to those n McGure and Staeln (1983), there are some mportant dstnctons. In McGure and Staeln (1983), two manufacturers sell through dedcated retalers (.e., the retalers only carry one of the manufacturers products). As n our model, the manufacturers may prefer the equlbrum wth wholesale-prce contracts relatve to the equlbrum under vertcal ntegraton, whch can be acheved wth two-part tarffs or quantty dscounts. But the mechansm for ths result s dfferent n ther model, wholesale-prce contracts dampen competton between the retalers, whch ndrectly dampens competton between the manufacturers, whereas n our model wholesale-prce contracts drectly nfluence the competton between the manufacturers whle leavng constant the ndrect competton they face wth consumers. In addton, n ther model, the retalers always prefer the wholesale-prce equlbrum because two-part tarffs always leave them wth no proft, whereas n our model the retaler may prefer the two-part tarff equlbrum. 6. Numercal Study Ths secton presents a numercal study that compares the equlbrum solutons when the manufacturers offer wholesale-prce, quantty-dscount, and two-part tarff contracts. Our prmary goals are to study scenaros wth economes of scale (K >0) and to measure the magntude of the effects dscussed earler. Table 2 There are 108 scenaros formed by all combnatons of the followng parameters: 20 40, 1 2 4, , c 1 3, K 0 1 3, and 05. Wth each scenaro, we searched for an equlbrum under wholesale-prce, quantty-dscount, and two-part tarff contracts. Wth the K 0 scenaros, we ncluded v , and wth the K>0 scenaros, we ncluded v In total there are 360 scenaro/contract combnatons. In 8 of the 108 scenaros, we were unable to fnd an equlbrum for at least one of the contracts (ncludng fve scenaros wth wholesale-prce contracts). The best response functons n those scenaros reveal that the effect of economes of scale s very strong at the retaler, and the manufacturers cycle between undercuttng prces to get the retaler to exclude the other manufacturer and beng undercut. In those scenaros, there also does not exst an asymmetrc equlbrum. We report results for the remanng 100 scenaros. As a valdty check, we compared the average cost per unt that the retaler pays to the manufacturers, T d/d wth wholesale-prce and quantty-dscount contracts: the average cost s 5% lower n the quantty-dscount equlbrum than the wholesale-prce equlbrum when v 05, and 11% lower when v 095. These results ndcate that n these scenaros the quantty dscounts n equlbrum are modest. Table 2 compares the frms profts under wholesale-prce contracts to the other contracts. The results n the table expand upon our ntuton developed analytcally n the prevous secton wth K 0 and the most aggressve quantty dscount (v at ts Average Percentage Change n the System, Retaler, and Manufacturer Profts n Equlbrum Under Quantty-Dscount and Two-Part Tarff Contracts Relatve to the Equlbrum Under Wholesale-Prce Contracts K 0 K 1 K 3 Quantty dscount (%) Quantty dscount (%) Quantty dscount (%) Two-part Two-part Two-part Change n proft tarff (%) tarff (%) tarff (%) 025 System Retaler Manufacturers Scenaros manu. lose (%) 05 System Retaler Manufacturers Scenaros manu. lose (%) 075 System Retaler Manufacturers Scenaros manu. lose (%) Note. manu., Manufacturers.