PRODUCTION AND OPERATIONS MANAGEMENT POMS Vol. 20, No. 2, March–April 2011, pp. 181–197 DOI 10.3401/poms.1080.01173 ISSN 1059-1478|EISSN 1937-5956|11|2002|0181 r 2010 Production and Operations Management Society

Group Buying of Competing Retailers

Rachel R. Chen Graduate School of Management, University of California at Davis, Davis, California 95616, USA, [email protected]

Paolo Roma Department of Manufacturing Technology, Production and Management Engineering, University of Palermo, Palermo, Italy, [email protected]

nder group buying, quantity discounts are offered based on the buyers’ aggregated purchasing quantity, instead of U individual quantities. As the price decreases with the total quantity, buyers receive lower prices than they otherwise would be able to obtain individually. Previous studies on group buying focus on the benefit buyers receive in reduced acquisition costs or enhanced bargaining power. In this paper, we show that buyers can instead get hurt from such co- operation. Specifically, we consider a two-level distribution channel with a single manufacturer and two retailers who compete for end customers. We show that, under linear demand curves, group buying is always preferable for symmetric (i.e., identical) retailers. For asymmetric retailers (i.e., differing in market base and/or efficiency), group buying is beneficial to the smaller (or less efficient) player. However, it can be detrimental to the larger (or more efficient) one. Despite the lower wholesale price under group buying, the manufacturer can receive a higher revenue. Interestingly, group buying is more likely to form when retailers are competitive in different dimensions. These insights are shown to be robust under general nonlinear demand curves, except for constant elastic demand with low demand elasticity. Key words: group buying; competition; distribution channel; quantity discounts; retailing History: Received: January 2008; Accepted: December 2009, after 2 revisions.

bargaining power (Dana 2006). In this paper, we in- 1. Introduction vestigate situations where GB can be detrimental to Under group buying (GB), quantity discounts are buyers. Specifically, we examine the impact of com- offered based on buyers’ aggregated purchasing petition on the formation of group purchase. Buying quantity, instead of individual purchasing (IP) quan- groups are often concerned with the potential conflict tities. As the price decreases with the total quantity, of interest when their members compete. In his survey there exist positive externalities among buyers that al- of more than 100 firms, Hendrick (1997) reports that a low them to obtain lower prices than they otherwise majority (60%) of the purchasing consortia have mem- would be able to obtain individually. GB is widely berships consisting of noncompetitors only, whereas used for both business-to-business (B2B) and busi- 10% are exclusively made up of direct competitors. ness-to-consumer (B2C) transactions. In a B2B context, Some firms consider joining a competitor not the right co-operatives of independent grocers, convenience choice because ‘‘[they] do not want to help compet- stores, or retail hardware stores have long existed in itors.’’ In fact, mistrust among competitors can be a the United States as well as in Europe.1 Purchasing major hurdle for some (online) buying consortia (Gray consortia formed by independent companies are also 2003). common in procuring computers, office furniture, or Various mechanisms have been introduced to utilities such as electricity and gas (Anand and Aron reduce competition between members, e.g., requiring 2003). In some industries, such as food services and that members operate in tightly defined individual health care, group purchasing organizations have a territories (IGD Retailing Factsheets 2007). According dominant presence (Mitchell 2002). Sometimes buying to the Canadian Garden Centre Co-op, benefits from groups are also formed to meet minimum order cooperation are possible because its members are not quantities imposed by sellers. Recently, the rapid de- immediate competitors. Nevertheless, some rival velopment of information technology has enabled companies do cooperate. PhotoFair Stores, a buying cooperation and transactions among geographically group with members throughout New Jersey and distributed firms, which open up opportunities for New York, was founded on the belief that it is ben- demand aggregation over a larger scale. eficial to be helpful to competitors. In 2006, two of the Most studies on B2B GB focus on the benefit buyers main street supermarket rivals in United Kingdom, receive in reduced acquisition cost or enhanced The Co-operative Group and Spar UK, joined forces in 181 Chen and Roma: Group Buying of Competing Retailers 182 Production and Operations Management 20(2), pp. 181–197, r 2010 Production and Operations Management Society a buying alliance to pool volumes in targeted areas of for the retail sector (Zentes and Swoboda 2000). In own label products. Both firms stressed that they Italy, the top five buying groups account for almost would remain independent and collaborate only in 84% of the food retail market (Kuipers 2005). In buying, while maintaining ‘‘healthy and friendly United States and Canada, with the majority of fur- competition’’ (Grocer 2006). Thus, it remains unclear niture coming from overseas, independent retailers whether two competing firms should ever cooperate who do not get on board with a buying group often in purchasing. find themselves having a hard time surviving (Heist In this paper, we consider a two-level distribution 2008). Similarly, dealers of electronics and appliances channel consisting of a single seller (manufacturer) start purchasing together as they realize that they who offers a quantity discount schedule and two re- cannot continue to purchase products via traditional tailers who compete in the final market by selling to distribution and remain competitive (Knott 2009). end customers. The market starts with an equilibrium Most buying groups claim that they can generate where the retailers act individually in deciding their significant savings for their members. The appliance purchase quantities (i.e., IP). We examine the retailers’ retailer Filco receives rebates of 4–6% of total pur- decision of whether to purchase together (i.e., GB) in chases through its affiliation with the Selective order to obtain lower wholesale prices. Consolidated Dealers Co-Op. Members of Star furni- Under GB, two effects influence retailers’ profits: ture, a buying group based in Salem, Virginia, receive the cooperation effect through demand aggregation, and rebates on orders averaging 2–5%, with some rebates the competition effect in the final market. Depending on reaching 10% (Rauen 1992). However, not all retailers whether retailers are symmetric (i.e., identical) or engage in GB. While this can be explained by coor- asymmetric (i.e., differing in market base and/or effi- dination efforts or purchase timing required for GB, ciency), these effects influence retailers’ profits we show that competition could also deter coopera- differently. Using linear demand curves, we discover tion. Thus, our study offers another possible a number of insights concerning the preference of re- explanation of why some retailers may not join group tailers on their purchasing strategies. For instance, purchase. both retailers prefer GB under no competition. This is To the best of our knowledge, this is the first paper consistent with the observation that cooperation that studies GB in a distribution channel. In particular, within buying organizations is becoming increasingly we fully characterize the competition between two re- popular in the public sector in particular, where there tailers, with and without GB. Our study generalizes the is almost no competition (Schotanus 2007). Under existing knowledge on GB by considering a number of competition, symmetric retailers always prefer GB. structure features: price-sensitive market demand, di- While mild competition is clearly preferable under IP, rect competition in the marketplace, and general symmetric retailers might enjoy higher profits under quantity discount schedules. Results of this paper help more intense competition when they group purchase. managers to better understand the dynamics of coop- Also, a deeper discount from the manufacturer is not eration in purchasing and provide guidelines for their always beneficial for buyers who purchase individu- decisions in a competitive environment. Although our ally. When retailers mainly differ in one dimension model concerns the retailers, the framework developed (i.e., market base or efficiency), GB is always attractive here can be extended to other settings, such as GB of for the smaller (or less efficient) retailer. However, it competing manufacturers. can be detrimental to the larger (or more efficient) one. The outline of the paper is as follows. A review of Also, the manufacturer can receive a higher revenue, the relevant literature is presented in section 2. We despite the lower wholesale price under GB. Interest- introduce the model and describe the equilibrium so- ingly, GB is more likely to form when retailers are lutions in section 3, and present our results with competitive in different dimensions. These insights discussions in section 4. Section 5 considers two ex- are shown to be robust under general nonlinear de- tensions: nonlinear demand curves and multiple mand curves, except for constant elastic demand with retailers. Section 6 concludes. low demand elasticity. We also briefly examine the case of three retailers to shed light on the dynamics of GB of multiple competing retailers. 2. Literature Review In practice, retailers often face the opportunity of Our paper is related to several research streams, including group purchase. World Wide Retailer Exchange quantity discounts, competition in distribution channels, (WWRE.com), the premier integrated worldwide ex- group buying, and inventory centralization games. change community, promotes the idea of demand Quantity discounts have been extensively studied aggregation among its members, who are mostly large from one of three viewpoints: price discrimination retailers (e.g., Kroger, Safeway, CVS, Walgreens, etc.). (Buchanen 1953, Oi 1971), channel coordination (Jeu- In Europe, purchasing consortia are well established land and Shugan 1983, Ingene and Parry 1995, 1998, Chen and Roma: Group Buying of Competing Retailers Production and Operations Management 20(2), pp. 181–197, r 2010 Production and Operations Management Society 183

Raju and Zhang 2005), and operational efficiency literature that concerns B2C GB (Chen et al. 2002, (Crowther 1964, Dada and Srikanth 1987, Erhun et al. Kauffman and Wang 2002), in which demand uncer- 2008, Monahan 1984). Some studies combine the last tainty plays an important role. We consider retailers’ two streams by considering channel coordination routine purchase from the manufacturer, which in- with price-sensitive demand and operating costs volves little uncertainty. (Chen et al. 2001, Weng 1995). Interested readers are Finally, our work is related to the literature on in- referred to the reviews in Dolan (1987) and Weng ventory centralization in a distribution system, in (1995). Our work is close to the marketing literature which multiple independent retailers take advan- on channel coordination. Jeuland and Shugan (1983) tage of risk pooling in joint ordering under stochastic first discuss how quantity discounts could be used to demand (Anupindi et al. 2001, Chen 2009, Hartman maximize channel profits. Ingene and Parry (1995) et al. 2000, Slikker et al. 2005). This stream of literature introduce competing retailers to the quantity discount does not consider retailer competition, which is our literature. Raju and Zhang (2005) study channel co- focus. ordination in the presence of a dominant retailer. Similar to Ingene and Parry (1995), we consider a 3. Model Set-up and Analysis single manufacturer offering quantity discounts to Consider a two-level distribution channel where one competing retailers. However, our work focuses on manufacturer sells a homogeneous good to two inde- the retailers’ decision of whether to group purchase, pendent retailers, who compete in the final market by whereas most papers in this area examine channel selling to end customers. The manufacturer’s unit coordination. Moreover, we allow the discount to be wholesale price is decreasing in the purchase quantity based on the total quantity, instead of on the IP quan- (i.e., quantity discount). In the initial market equilib- tities as in most quantity discount studies. rium, each retailer acts individually in purchasing In terms of competition in distribution channels, from the manufacturer (i.e., IP). The retailers then our work is closely related to Tsay and Agrawal consider the option of purchasing together to obtain a (2000), who explore channel coordination when re- lower wholesale price (i.e., GB). GB is possible only tailers compete in both price and service. They point when both retailers agree to cooperate. out that, due to model complexity, typically deter- The retailers are indexed by iA 1, 2 and j 5 3 i. ministic formulations are found in most existing f g À Retailer i’s demand, q , decreases with its own price p , multi-echelon analyses that incorporate demand com- i i and increases with the opponent’s price p . We first petition. Such models are common in economics (cf j consider a linear demand function Katz 1989, Tirole 1988 for reviews) and marketing lit- erature (e.g., Choi 1991, Coughlan and Wernerfelt q A p y p p ; y 0; 1 1989, Ingene and Parry 1995, 1998, McGuire and Stae- i ¼ i À i þ ð j À iÞ  ð Þ lin 1983, Moorthy 1987, Padmanabhan and Png 1997, Trivedi 1998). Our model also considers deterministic where Ai is the demand for retailer i when both prices demands and price competition, but we focus on the are set at zero and y measures substitutability between impact of competition on retailers’ GB decisions, the retailers. The above demand function can be de- which has not been studied before. rived from the aggregate consumer utility function Many theoretical and empirical studies on buyer (Lee and Staelin 2000, Shapley and Shubik 1969), and is groups have emphasized enhanced buying power common in economic and marketing literature (see that justifies the formation of buyer co-operatives, Ingene and Parry 1995, McGuire and Staelin 1983, Tsay buyer alliances, and horizontal mergers (e.g., Chipty and Agrawal 2000). We extend our model to several 1995, Dana 2006, Horn and Wolinsky 1988, Inderst common nonlinear demand functions in section 5. and Wey 2003, Marvel and Yang 2008). Recently, Chen Ai reflects the comparative advantage of retailer i in et al. (2008) explore buyers’ bidding strategy under terms of access to customers, due to factors such as different GB mechanisms in B2B exchanges. None of location, loyalty, brand, or service. While Ai captures the work considers competition among buyers. retailers’ differentiation in ‘‘market base,’’ y reflects While buying groups usually involve a long-term their competition intensity. There is no competition relationship among buyers, recently developed web- when y 5 0. We also assume that retailer i incurs an based GB allows demand to dynamically aggregate operational cost ci in procuring and selling each unit online, so that price is set based on market-wide of the product. ci reflects the operational efficiency of demand. With demand uncertainty, GB (similar to retailer i. Thus, our study focuses on the difference in auctions) is a mechanism for price discovery. Anand retailers’ access to customers and operational effi- and Aron (2003) provide a survey of web-based prac- ciency, two common factors in the literature that tices of GB, and compare it with posted pricing under examines competing retailers (e.g., Ingene and Parry uncertain demand. There is also a growing body of 1995, Tsay and Agrawal 2000). Chen and Roma: Group Buying of Competing Retailers 184 Production and Operations Management 20(2), pp. 181–197, r 2010 Production and Operations Management Society

3.1. Quantity Discount Function (QDF) Figure 1 Range of Discount Scale and Steepness of the Quantity Discount According to the Robinson–Patman Act, a common Function price menu must be offered to each buyer to preclude d sellers ‘‘from giving different terms to different re- High wholesale price sellers in the same reseller class.’’ We assume that the Deep discount same price menu is offered under either individual or Infeasible Two-part group purchase. This is a reasonable assumption if the tariff manufacturer is selling to many retailers, and has no Flat discount knowledge of, or any control over the retailers’ deci- –1 sion regarding individual or group purchase. 0 1 e Quantity discount schedules come in all shapes and Flat discount sizes. In a recent study, Schotanus et al. (2009) derive a Linear general continuous quantity discount function (QDF) discount Infeasible that describes the underlying function of different Deep discount discount schedules. Under the QDF, the unit whole- Low wholesale price sale price w(q) is When e > 0, a higher d means a deeper discount but also a higher wholesale price. w q a d=qe; 2 When e < 0, a lower d means a deeper discount and a lower wholesale price. ð Þ¼ þ ð Þ where a 0 is the base price, d is the discount scale, and discount. Figure 1 depicts the possible range of dis-  e is the steepness. They show that this QDF fits well count scale and steepness of the general QDF. with 66 discount schedules found in practice, with e We assume that each retailer pays the same unit varying from 1.00 to 1.60. wholesale price determined by the total quantity under À Note that de40 is required to ensure that the unit group purchase, referred to as uniform cost allocation wholesale price w(q) decreases with q. The higher the (Chen and Yin 2008). Such a payoff scheme is easy to discount scale d, the higher the wholesale price. When calculate, achieves budget balance (i.e., the total pay- e40, a 5 w( ) represents the minimum price, ments equal the total cost TC(q)), and hence is appealing 1 whereas d 5 w(1) w( )40 reflects the price spread to the practitioners. For example, eWinWin.com, a major À 1 of the schedule. When eo0, we have do0, and online GB facilitator, uses this rule to allocate the total a 5 w(0) is the maximum price. The absolute value cost after demand is aggregated through its online plat- of the discount scale, |d|, reflects how fast w(q) de- form. To focus on the effects of competition, we make the creases with q and is referred to as the discount level. common assumption that all model parameters are de- The higher the |d|, the more effective the demand terministic and common knowledge (e.g., Choi 1991, aggregation by the retailers. Ingene and Parry 1995, 1998, Jeuland and Shugan 1983, In general, quantity discounts are deep for the initial McGuire and Staelin 1983, Tsay and Agrawal 2000). For quantities and become flat as q gets large. Thus, we as- ease of exposition, we assume zero coordination cost sume w(q)isconvex,whichrequirese 1. Also for under group purchase (which might involve adminis- À total cost, TC q aq dq1 e,tobeconcaveinq,weneed trative efforts such as phone calls or faxes). Such a (fixed) ð Þ¼ þ À e 1. Hence, we restrict to eA[ 1, 1], a range consistent coordination cost is likely to result in a threshold policy,  À with the empirical findings (Schotanus et al. 2009). Note so the assumption can be easily relaxed without affecting that TC(q)isnondecreasinginq if e40. For eo0, we fol- our results qualitatively. low Schotanus et al. (2009) by assuming q 1 e 1=e  ððÀ þ Þ d=a so that TC(q)isincreasingandw(q)remainsnon- 3.2. Individual Purchasing Þ negative. We also assume that retailers do not engage in We start with the initial market equilibrium where, any resale of the manufacturer’s product. Then under an given the discount schedule, each retailer acts indi- increasing and concave total cost function, no retailer will vidually in deciding its own retail price and purchase purchase more than what he can sell in the final market. quantity. Under IP, retailer i’s wholesale price, w(qi), When the steepness e 5 1, w(q) 5 a1d/q and the depends on its own purchasing quantity qi. With a per price schedule is in fact a two-part tariff, where the unit operating cost ci, retailer i’s profit function is retailers share a lump-sum fee d and incur a unit price Pi pi ci w qi qi: 3 a. When e 5 1, w(q) 5 a1dq with do0, and the unit ¼ð À À ð ÞÞ ð Þ À wholesale price linearly decreases with quantity q. Taking the first-order derivative with respect to pi Schotanus et al. (2009) report that five out of the 66 and setting it to 0 yield schedules collected have a steepness of 1. This dis- À @Pi count format is also common in the literature (Ingene 1 1 y w0 qi qi pi ci w qi 1 y and Parry 1995, 1998, Rosenblatt and Lee 1985, Tsai @pi ¼ ½Šþð þ Þ ð Þ Àð À À ð ÞÞð þ Þ 2007). We assume e 0, as e 5 0 is the trivial case of no 0: 4 6¼ ¼ ð Þ Chen and Roma: Group Buying of Competing Retailers Production and Operations Management 20(2), pp. 181–197, r 2010 Production and Operations Management Society 185

Table 1 Equilibrium Solutions under Individual Purchasing (e = 1 and 1) À Linear schedule (e 5 1) Two-part tariff (e 5 1) À IP l AX a c AX c IP t A a c 1 y DA Dc 1 y Retailer price p D D p þðþ Þð þ Þ þ ð þ Þ i Xþ 1þ 1 2yþX 1 i 2 y 2 3y ¼ þ Æ ð þ Þ þ ¼ þ Æ þ IP l A c a DA Dc 1 2y IP t A c a 1 y DA 1 y Dc 1 2y 1 y Retailer quantity q À ð þ Þ q ð À À Þð þ Þ ð þ ÞÀ ð þ Þð þ Þ i ÀX À1 1 2y X 1 i 2 y 2 3y ¼ þ Æ ð þ Þ þ ¼ þ Æ þ IP l A c a IP t A c a 1 y Total quantity Q 2 Q 2ð À À Þð þ Þ ÀX À1 2 y ¼ þ ¼ þ 2 qIP t 2 Retailer profit IP l X d qIP l IP t ð i Þ d Pi i Pi 1 y ¼ð À Þ ¼ þ À : Sign 1( ) is associated with retailer 1(2). X 1 2d ÀÁ 1 y : À ¼ þ þ

Maximizing (3) with respect to pi and jointly solving restricted to be positive. The superscripts l and t denote IP for both retail prices yield the equilibrium price pi , linear schedules and two-part tariffs, respectively. IP i 5 1, 2. Closed-form solutions for pi exist for e 5 2, 1, For each retailer, while a larger market base A is 0, 1, and 2 only. As we focus on eA[ 1, 1], we always beneficial, a higher operational cost c or base À À À obtain the solution for e 5 1 and 1 (see Table 1), and price a will be detrimental. Thus, A c a reflects the À À À explore the system behavior for other values of e environment in which retailers operate, and remains through numerical studies. positive throughout the paper. As seen in Table 1, a To focus on the impact of competition, we impose a higher A c a leads to higher quantity and profit À À IP number of conditions on the parameters to ensure a rea- under IP. Note that Q does not depend on DA or Dc sonable model with duopoly equilibrium under both IP under either linear discounts or two-part tariffs. Later and GB (e.g., nonnegativity of prices, quantities, and we will show that, for the general QDF, the total quan- profits for both retailers).2 For the analysis of e 5 1and tity under IP decreases with the level of asymmetry. À 1, these conditions are explicitly described in (30)–(34) and (36)–(38) in the supporting information Appendix 3.3. Group Buying S1. For general steepness of e,werestrictournumerical We now consider the option of cooperation in pur- analysis to the range where these conditions hold. chasing between the retailers. Under GB, each retailer IP i’s wholesale price, w(Q), depends on the total pur- Substituting pi into (1) yields the equilibrium IP IP IP IP chasing quantity, Q 5 qi1qj. The profit function of quantity qi . The total quantity is Q 5 q1 1q2 , and the manufacturer’s revenue under IP is retailer i is P p c w Q q : 7 RIP w qIP qIP w qIP qIP: 5 i ¼ð i À i À ð ÞÞ i ð Þ ¼ ð 1 Þ 1 þ ð 2 Þ 2 ð Þ Taking the first-order derivative with respect to p By (3) and (4), retailer i’s equilibrium profit under i IP and setting it to 0 yield IP, Pi , can be written as @Pi 1 de 1 w0 Q qi pi ci w Q 1 y IP IP 2 @pi ¼ð þ ð ÞÞ Àð À À ð ÞÞð þ Þ Pi e 1 qi ; i 1; 2: 6 ¼ "#1 y À qIP þ ð Þ ¼ ð Þ 0: 8 þ ð i Þ ¼ ð Þ : A1 A2 : A1 A2 Define A þ2 and DA À2 such that A1 5 Maximizing (7) with respect to pi and jointly solving ¼ ¼ : c1 c2 GB A1DA and A2 5 A DA. Similarly, let c þ2 and Dc for both retail prices yield the equilibrium price pi , : c c À ¼ 1À 2 such that c 5 c1Dc and c 5 c Dc. DA and Dc i 5 1, 2. Again, closed-form solutions exist for e 5 2, 1, ¼ 2 1 2 À reflect the level of asymmetry between the retailers, 0, 1, and 2, but we will obtain the results for e 5 1 À À with which we can write the equilibrium solutions for and 1 only (see Table 2). We then obtain the cor- À GB GB e 5 1and 1 as in Table 1. Note that DA and Dc are not responding quantity q by substituting p into (1). À i i

Table 2 Equilibrium Solutions Under Group Buying (e = 1 and 1) À Linear schedule (e 5 1) Two-part tariff (e 5 1) À GB t 2 GB l AY a c DA Y 2d Dc GB t QGB t DA Dc 1 y Q dDA Retailer price p ð À Þþ p A ½Šþ ð þ Þ À i Yþ 1þ 1 2y Y 2d 1 i 2 GB t 2 ðÞ ¼ þ Æ ð þ Þð À Þþ ¼ À Æ Q 2 3y d 1 2y ðÞð þ ÞÀ ð þ Þ GB t 2 GB l A c a DA Dc 1 2y GB t QGB t DA Dc 1 2y 1 y Q Retailer quantity q À ð þ Þ q ½ À ð þ ފð þ Þ i ÀY À1 1 2y Y 2d 1 i 2 GB t 2 ðÞ ¼ þ Æ ð þ Þð À Þþ ¼ Æ Q 2 3y d 1 2y ðÞð þ ÞÀ ð þ Þ 2 1=2 GB l A c a GB t A c a 1 y A c a 1 y d 1 2y Total quantity Q 2 Q ð À À ÞðÞþ ð À À ÞðÞþ ð þ Þ ÀY À1 2 y 2 y 2 y ¼ þ ¼ þ þ þ À þ 2  2 qGB t 2 Retailer profit GB l Y 2d qGB l GB t i 1 dQGB t À Pi i Pi ðÞ1 y ¼ð À Þ ¼ þ À : ÀÁ hiÀÁ Sign1( ) is associated with retailer 1(2). Y 1 d 2d 1þy : À ¼ þ þ Chen and Roma: Group Buying of Competing Retailers 186 Production and Operations Management 20(2), pp. 181–197, r 2010 Production and Operations Management Society

The manufacturer’s revenue under GB is given by GB retailer profit will increase with y when A c a e42qGB y e ey = 1 2y . That is, more RGB w QGB qGB qGB : 9 ðÞÀ À i ð þ þ Þ ðÞþ ¼ ð Þ 1 þ 2 ð Þ intense competition can result in higher retailer profit un- From (7) and (8), we canÀÁ write retailer i’s equilib- der GB. rium profit under GB as See the supporting information Appendix S1 for GB 1 de GB 2 proofs of all the propositions. Pi 1 e 1 qi ; ¼ 1 y À QGB þ ! 10 þ ðÞ ð Þ i 1; 2: ÀÁ Results on how retail price and demand quantity ¼ change with y are intuitive. Basically, competition Table 2 summarizes the equilibrium solutions for exerts downward pressure on retail prices, so that e 5 1 and 1 under GB. À demand quantities increase with y. Conventional wis- dom suggests that firms prefer less intense compe- 4. Comparison of IP and GB tition, as confirmed by the result that retailer profit In this section, we compare the two models (IP and decreases with y under IP. However, under GB retailer GB) for each retailer in order to find, if they exist, the profit will increase with y when A c a e4 GB ðÞÀ À conditions making cooperation in the form of GB 2qi y e ey = 1 2y . For e 5 1 (two-part tariffs), ð þ þ Þ ðÞþ 2 suitable in a competitive environment. We first ana- this condition can be reduced to d4 A c a y= ð À À Þ lyze the case of symmetric retailers (i.e., Ai 5 Aj, 1 2y , so retailer profit increases with competition ð þ Þ ci 5 cj). This is an important case because many buy- for deep discount (high d) and low competition (small ing groups consist of members of similar sizes, while y). For e 5 1 (linear discount), we have do0 and À it is unclear how competition affects their purchasing the condition is reduced to |d| 1/2 or y d =  j j and pricing decisions. For differentiated retailers, we 1 2 d . For general steepness e, numerical investi- ð À j jÞ focus on their differences in access to customers and gation confirms that this result holds for deep operational efficiency represented by market base and discount (high |d|) and mild competition (low y). unit operational cost, respectively. We first explore Figure 2 depicts how retailer profit under GB changes GB retailers’ decisions under asymmetry in the market with y for positive steepness of e, where Pi increases base, assuming that they have the same operational with y for y small. The result for negative steepness is costs (i.e., DA 0, Dc 5 0). For the case of asymmetry in similar. 6¼ efficiency (i.e., DA 5 0, Dc 0), the results do not differ Retailer competition in general leads to higher de- 6¼ qualitatively. The general case with asymmetry in mand in the final market, and consequently lower both dimensions (i.e., DA 0, Dc 0) is discussed at the unit wholesale price under the manufacturer’s quan- 6¼ 6¼ end. tity discount. Under deep discount and mild competition, this effect is strengthened through group 4.1. Symmetric Retailers For symmetric retailers, at the equilibrium we have pi 5 pj under either IP or GB. Under the general QDF, Figure 2 Group Buying Profit Changes With Competition Level h (A = 1, we have w q a d=qe, with which we can simplify ð iÞ¼ þ i c = 0.1, a = 0.3, d = 0.1, e40) the first-order condition (FOC) under IP (4) as 0.065 Π_GB 2 y d e = 0.1 A c a þ qi e 1 e ; i 1; 2: 11 e = 0.5 À À ¼ 1 y þ qi ð À Þ ¼ ð Þ þ e = 0.9 0.06 Similarly, under GB we have w Q a d=Qe, so ð Þ¼ þ that the FOC (8) becomes

2 y d e 0.055 A c a þ qi 2 ; À À ¼ 1 y þ 2e 1qe À 1 y 12 þ þ i þ ð Þ i 1; 2: ¼ Although closed-form solutions do not exist for 0.05 general steepness e, we obtain several results for sym- metric retailers based on Equations (11) and (12). 0.045 PROPOSITION 1 (Effect of competition intensity). When the retailers are symmetric, a higher competition level y leads to lower retail price and higher demand under both IP 0.04 and GB, as well as lower profit under IP. However, under 0 0.2 0.4 0.6 0.8 1 Chen and Roma: Group Buying of Competing Retailers Production and Operations Management 20(2), pp. 181–197, r 2010 Production and Operations Management Society 187 purchase so that each retailer’s profit increases with Figure 3 Individual Purchasing Profit Changes With Discount Scale d competition level y. However, when the discount is not (A = 1, c = 0.1, a = 0.3, h = 2, eo0) significant or competition is already fierce, a more in- 0.12 Π_IP tense competition will further depress retail prices and lower profits, passing most benefits on to consumers. 0.11 This possibly explains the present situation of German grocery buying groups. In the food retail sector in Eu- 0.1 rope, buying groups are able to obtain significant discounts from suppliers. However, Germany is 0.09 known for its fierce competition in food retailing, and such discounts are passed on to final consumers in 0.08 lower prices. By contrast, competition among grocery retailers is much less intense in France, United King- 0.07 dom, and Spain (Dobson Consulting 1999).

0.06 e = −0.1 ROPOSITION P 2 (Effect of discount scale). When retailers e = −0.5 are symmetric, a higher discount scale d leads to higher e = 0.9 0.05 − retail price and lower demand under both IP and GB, e = −1 as well as lower profit under GB. However, under IP re- tailer profit will increase with d when A c a e 0.04 ðÞÀ À −0.22 −0.18 −0.14 −0.1 −0.06 −0.02 d 4 2qIP 1 e ey = 1 y . This implies that retailers i ð þ þ Þ ðÞþ can get hurt from a lower wholesale price with deeper dis- count under IP. PROPOSITION 3 (Comparison between IP and GB). With symmetric retailers, retail quantity (price) will be higher e 1 As shown in Figure 1, the higher the discount scale (lower) under GB if yoy, with y e= 2 2 þ 1 e 1. ¼ ½ À ð À ފ À d, the higher the unit wholesale price w(q). This in Retailer profit is always higher under GB with a lower unit general leads to higher retail price and lower retail wholesale price. However,b it is possibleb for the manufacturer demand/profit, as is the case under GB. Under IP, to have higher revenue under GB. retailer profit also decreases with d under two-part tariffs (@PIP t=@d 1o0 for e 5 1). However, for Although retailers are able to get a lower unit i ¼À linear discount schedules (e 5 1), a lower discount wholesale price under GB, they do not necessarily À scale d, or equivalently, a deeper discount level |d|, purchase more. As a result, the total quantity is not can result in a lower retailer profit. In fact, for e 5 1, always higher under GB. When y4y, the fierce com- IP À the condition A c a e o 2qi 1 e ey = 1 y petition pushes each retailer to over purchase in order ðÞÀ Ày 2 ð þ þ Þ ðÞþ can be reduced to d À , which will hold when d to qualify for a deeper discount underb IP. Under GB,  2 1 y is sufficiently low. Thisð þ isÞ because a fairly deep since the discount is based on aggregated demand, discount can introduce very low retail prices that each retailer might scale back and buy less, resulting essentially intensify retailers’ competition. in higher retail prices in the final market. That is, re- For general values of e, numerical studies show that tailers’ cooperation might hurt end consumers.3 In retailer profit under IP decreases with d for e40. For particular, when e 5 1, we have y 1=2, so demand ¼À negative e close to 1, retailer profit can increase with quantity is always lower under GB (qGB qIP) for two- À i  i d, as illustrated in Figure 3. For negative e close to 0, part tariffs. However for e 5 1,b we have y , so À ¼1 retailer profit again decreases with d under mild quantity is always higher under GB (qGB qIP) for i  i competition, although the result might reverse when linear discounts. b competition is sufficiently high. In reality, the discount For profit comparison, the message is more consis- level varies across industries. For the optical buying tent. GB avails a deeper discount based on the total group ADO, the discount percentage for various quantity, and effectively improves each retailer’s eyewear models goes from 3% up to 30%. Proposition profit. Thus, symmetric retailers who face fierce com- 2 suggests that retailers may not always prefer deeper petition might consider cooperation as a mechanism discounts under IP. Interestingly, this never happens to soften competition and improve profits. This is under GB, probably because the cooperation in consistent with the observation that, while buying purchasing helps soften the competition when groups operate in various industries, they are most discounts are deep. In fact, for symmetric retailers, active in competitive retailing businesses such as the cooperation effect will dominate the competition appliances, jewelry, and consumer electronics (Rauen effect and GB is always preferable, as shown in the 1992). Table 3 summarizes the comparison for sym- next proposition. metric retailers. Chen and Roma: Group Buying of Competing Retailers 188 Production and Operations Management 20(2), pp. 181–197, r 2010 Production and Operations Management Society

Table 3 Comparison between Individual Purchasing and Group Buying support for such cooperation even in the presence of (Symmetric Retailers) competition. Some buying groups, e.g., FurnitureFirst, Retailers Manufacturer have requirements on the geographical location to prevent members from competing with each other Retailer price pIP 4pGB if yoy Unit wholesale price w qIP 4w 2qGB i i ð i Þ ð i Þ (Heist 2008). The cleaning supplies buying group, Retailer quantity qIP oqGB if yoy Total quantity QIP oQGB if yoy i i b DPA, typically has one distributor in a given market. Retailer profit PGB PIP Revenue R IP xR GB i  i b b On the other hand, buying group AFFLINK reasons that competition can be a benefit (Mollenkamp 2005). Proposition 3 shows that for symmetric retailers com- In general, GB leads to a lower unit wholesale price petition may not be a concern if they want to and consequently a lower revenue for the manufac- cooperate in buying. Moreover, under certain condi- turer. In fact, this is always the case under two-part tions, the more intense the competition, the higher the tariffs. However, for linear quantity discount sched- profit under GB as suggested in Proposition 1. ules, if the base price a is above certain threshold a0 (see proof in the supporting information Appendix S1 4.2. Asymmetric Market Base for details), the manufacturer will receive a higher revenue when retailers cooperate. To see this, recall In the previous section, we show that GB helps soften IP IP IP GB GB GB IP GB the competition between symmetric retailers. In prac- that R 2qi w qi , R 2qi w 2qi , qi oqi , IP¼ ðGB Þ ¼ ð Þ tice, group members can also differ from each other. and w qi 4w 2qi under linear discounts. When ð Þ ð Þ GB IP The industrial supplies buying group IDC include base price a increases, both qi and qi are decreasing, which leads to higher wholesale prices. However, this some members as small as US$2 million a year and GB a few as large as over US$50 million a year. Furni- impact is reinforced under GB because w(2qi ) lin- GB tureFirst, a major US furniture buying group, has early decreases with the total quantity 2qi , resulting IP GB members ranging from US$2 million up to US$30 in a reduced gap between w(qi ) and w(2qi ). When a is high enough, the higher quantity under GB will million annually. In 2002, the French supermarket overcome the lower wholesale price, yielding a higher group Intermarche, with sales totaling 37.2 billion revenue under GB. If a higher revenue is preferable Euros, joined forces with its competitor, Spanish for the manufacturer, GB can create win–win for all Eroski, whose sales were much lower at 4.6 billion the players. Numerical analysis confirms that this re- Euros (Eurofood 2002). In this section, we examine re- tailers’ difference in their access to customers while sult holds for general steepness e not close to 1, as assuming they are equally efficient (i.e., A A , c 5 c ). shown in Figure 4. i6¼ j i j In practice, we often observe co-ops or buying Without loss of generality, we consider positive DA groups of similar-size stores (e.g., Ace Hardware). only. That is, retailer 1 has a larger market base than While this can be explained by their common back- retailer 2. We first study whether retailers’ asymmetry ground and interests, our analysis provides further level affects the total quantity and the manufacturer’s revenue.

Figure 4 Manufacturer’s Revenue Comparison (Symmetric Retailers) PROPOSITION 4. When retailers are asymmetric in market (A = 1, c = 0.1, d = 0.01, h = 0.1) base, 0.8% R_GB- R_IP R_GB (a) under IP the total quantity (QIP) and the manufac- 0.6% turer’s revenue (RIP) are nonincreasing in DA. (b) under GB, the total quantity (QGB) and the manu- 0.4% facturer’s revenue (RGB) remain constant with DA, GB GB GB whereas Dq q1 q2 is proportional to DA. ¼ ÀGB 0.2% The total quantity (Q ) will increase with compe- tition level y and decrease with discount scale d. 0.0% (c) the manufacturer’s revenue under GB can be higher 0.3 0.4 0.5 0.6 0.7 0.8a 0.9 or lower than that under IP. −0.2% For IP, the total quantity QIP decreases or remains

e = −0.1 constant (e.g., under linear quantity discounts and −0.4% e = −0.2 two-part tariffs) with the asymmetry level DA, while e = −0.5 IP e = −0.9 the manufacturer’s revenue R generally decreases −0.6% e = 0.1 with DA. Thus, if a higher quantity and revenue are e = 0.2 desirable, the manufacturer would prefer symmetric −0.8% to asymmetric retailers. Chen and Roma: Group Buying of Competing Retailers Production and Operations Management 20(2), pp. 181–197, r 2010 Production and Operations Management Society 189

By contrast, under group purchase the total quan- Our next comparison shows that the small retailer tity QGB and the manufacturer’s revenue RGB do not will always prefer group purchasing, while the big depend on DA. That is, the manufacturer is indifferent retailer might get hurt from such cooperation. whether retailers are alike or different when they co- operate in purchasing. The difference in retailers’ PROPOSITION 6. When retailers are asymmetric in market quantity, DqGB, is proportional to retailers’ level of base and discount is of linear format, the total quantity asymmetry. Intuitively, retailers buy more under in- under IP does not depend on DA, and it is always below tense competition (higher y), whereas a higher that under GB (QGB l4QIP l). GB leads to lower price, discount scale d leads to a higher wholesale price, higher quantity, and higher profit for retailer 2 (with smaller and thus lower quantity. Consistent with Proposition market base) than IP. However, retailer 1 (with larger mar- 3, the manufacturer’s revenue is not always lower ket base) may not always prefer GB. Specifically, under GB for asymmetric retailers. As the model is not tractable for general steepness e, 4y3 12y2 6y (a) when d 3 þ 2 þ , there exists DAq DA 8y 20y 12y 2 we will first present the analytical results for two spe- À þ þ þ  such that for DA DA , retailer 1’s quantity is cial cases: linear quantity discounts (e 5 1) and two-  q À ; part tariffs (e 5 1), and explore the system behavior higher under IP (b) when d y , there exists DA DA such that under general steepness using a numerical approach. À1 2y p  for DA DAþ, retailer 1’s price is lower under IP;  p (c) when k(y, d) 0, there exists DA DA such that  P  4.2.1. Linear Quantity Discounts (e 5 1). When for DA DAP, retailer 1’s profit is higher under IP, À  e 5 1, the unit wholesale price is w(q) 5 a1dq, with where À do0. The equilibrium solutions under linear discount k y; d schedules are given in Tables 1 and 2. The conditions ð Þ 1 1 1 d 1 d 1 1 on the parameters that ensure nonnegativity of all 1 y 2d 1 2y 1þy 1þy 1 2y 1 y d 1 þ þ þ þ þ À þ þ þ þ þ variables are described in (30)–(34) in the supporting 2 qffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 ¼ þ À 1 2d 1 Á1 d 1 Àd 2d 1Á 1 d information Appendix S1. Note that when the 1 y 1þy 1þy 1 y 6 þ þ þ þ À þ þ þ þ þ 7 asymmetry level is too high, the small retailer may 4À1 d Áqffiffiffiffiffiffi À Áqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 1þy 2d 1 not make any positive profit. Define DA A c a þ þ þ : 1 2y X 1 ¼ð À À Þ Â 1 d 1 ð þ Þ þ "#1þy 1 2y X 1 as the largest gap in market base under which þ þ þ retailerþ 2 remains alive, and we will consider DA  DA only. We first compare the price, quantity, and Moreover, DA DA , DA DA . That is, if retailer profit of the two different retailers. q p q P 1’s profit (price) is higher (lower) under IP, then its quan- tity is higher under IP. PROPOSITION 5. When retailers are asymmetric in market base and discount is of linear format, under IP retailer 1 Proposition 6 suggests that the discount scale, (with a larger market base) may or may not set a higher competition intensity, and asymmetry level jointly retail price than retailer 2, although his quantity and affect retailers’ GB decisions. Basically, GB is favor- profit are always higher than retailer 2. Under GB the able for both retailers when k(y, d) 0. When  retail price, quantity, and profit for retailer 1 are always k(y, d) 0, the asymmetry between the retailers  higher than retailer 2. plays an important role. Specifically, GB is always preferable when the retailers are similar, which is Under IP, retailer 1 naturally enjoys a competitive consistent with Proposition 3. If the two retailers are advantage from its larger market base with a not alike (i.e., DA DA ), GB will hurt retailer 1.  P higher quantity and profit, and the retail price is in Although it is not possible to solve k(y, d) 5 0 by general higher than that of retailer 2. However, when explicitly expressing d as a function of y (or vice d 1/[2(11y)], or equivalently, |d| 1/[2(11y)], versa), we characterize the region numerically. As theÀ deep discount from the manufacturer will mo- shown in Figure 5, the duopoly model is feasible tivate retailer 1 to set a lower price than retailer 2. when d and y are below the solid curve (above which This result differs from Tsay and Agrawal (2000), in the equilibrium solution is invalid with negative which an asymmetry in market base always leads a quantities or profits).4 The feasible region is then di- higher retail price by the larger retailer. This is be- vided by the dashed curve k(y, d) 5 0 into two: cause the unit wholesale price is fixed in Tsay and Region I where GB is always advantageous, and Re- Agrawal (2000), rather than decreasing in quantity as gion II where GB is preferable when asymmetry in our study. Interestingly, GB removes this incentive level is below the threshold (DA DA ). Intuitively,  P so that retailer 1 always charges a higher price than GB is preferable for any value of d when y 5 0. When retailer 2. competition is intense (e.g., y 2), the dashed curve  Chen and Roma: Group Buying of Competing Retailers 190 Production and Operations Management 20(2), pp. 181–197, r 2010 Production and Operations Management Society

Figure 5 Group Buying and Individual Purchase Region (Linear Quantity for shipping and handling (e.g., transportation), this Discounts) price schedule would simply be a two-part tariff. 0.7 The total cost under a two-part tariff is TC(q) 5 aq1d for q40 and TC(0) 5 0. By (6) and (10), retailer 0.6 i’s profit under IP and GB can be written as Duopoly equilibrium does not exist: IP t 2 0.5 Small retailer is out of IP t qi market due to the ; Region I Pi ð Þ d and 13 competition ¼ 1 y À ð Þ 0.4 Always þ Group Purchase GB t 2 GB t 2 0.3 q d q PGB t i i ; 14 k( , d) = 0 i ¼ 1 y À 1 y QGB t ð Þ 0.2 ÀÁþ þ  Region II Individual Purchase if the asymmetry is respectively. In the initial equilibrium, both retailers 0.1 sufficiently large, otherwise Group Purchase make nonnegative profits by individually purchasing from the manufacturer. They now consider the option 0 01234of cooperation in buying. The equilibrium IP and GB solutions for two-part tariffs are given in Tables 1 and k(y, d) 5 0 is approaching the solid curve, so that the 2. The conditions that ensure a sensible model are asymmetry level becomes critical to the GB decision. given in (36)–(38) in the supporting information When competition is mild (e.g., y 2), the cooper- Appendix S1. The comparisons between the retailers ation effect will dominate the competition effect and the impact of discount scale and competition in- under deep discount (e.g., |d| 0.28) and GB is tensity are summarized below.  preferable. Otherwise for flat discounts (low |d|), PROPOSITION 7. When retailers are asymmetric in market again the asymmetry level determines whether base and the discount is a two-part tariff, retailers will cooperate. Recall that for the general QDF, the total quantity (a) retailer 1 (with larger market base) will set a higher and the manufacturer’s revenue under GB do not de- price, and enjoy a higher retail quantity and profit pend on DA,asshowninProposition4.Forlinear under either IP or GB compared to retailer 2; discount schedules, the total quantity under IP is also (b) under IP retail price and quantity do not depend on independent of DA,whereastherevenueunderIPde- d, while retail profit is linearly decreasing in d. Also creases with the level of asymmetry. Similar to the more intense competition leads to higher retail quan- symmetric case, the revenue comparison depends on tity for both retailers and lower retail price for the base price a as follows (see proof of Proposition 4(c) retailer 1 under IP. in the supporting information Appendix S1 for details): Under IP, the marginal cost remains constant at a after the fixed fee d is incurred, although a two-part tariff, de facto, corresponds to a quantity discount (a) if a a0, GB leads to higher revenue for sym- with decreasing unit wholesale price. Thus, the fixed  metric retailers and RGB l RIP l for DA DA; fee d behaves as a sunk cost and will not affect each   (b) if a1 aoa0, there exists a unique value retailer’s pricing or quantity decision. Note that d  GB l IP l DAo DA, where R R for DA DA0; cannot be too high for both retailers to have positive    2 (c) if aoa1, revenue under IP decreases but remains A c a GB l IP l profits (e.g., d 1 y 2À Ày by condition (37) in higher than that under GB, and R oR for Appendix S1). WeðÞþ now compareþ IP and GB under DA DA.   two-part tariffs.

We next present the analytical result under the PROPOSITION 8. When retailers are asymmetric in market special case of two-part tariffs. base and the discount is a two-part tariff, (a) GB leads to higher retail price and lower retail quan- 4.2.2. Two-Part Tariffs (e 5 1). Under a two-part tity for both retailers; tariff, the unit wholesale price is given by w(q) 5 a1d/q (b) both retailers are better off under GB. The profit with d40. Two-part tariffs are commonly used in advantage of group purchase is monotonically de- practice due to their simple structure (see Tirole 1988 creasing for the large retailer (@ PGB PIP = ð 1 À 1 Þ and the discussion therein), and have been well studied @DA 0) and monotonically increasing for the  in the literature (e.g., Dolan 1987, Ingene and Parry 1995, small retailer (@ PGB PIP =@DA 0); ð 2 À 2 Þ  Raju and Zhang 2005). If each retailer is required to pay (c) the manufacturer’s total quantity and revenue are the manufacturer a constant unit price plus a fixed fee higher under IP than under GB. Chen and Roma: Group Buying of Competing Retailers Production and Operations Management 20(2), pp. 181–197, r 2010 Production and Operations Management Society 191

Table 4 Comparison between IP and GB (Asymmetric Retailers)

Linear schedule (e 5 1) Two-part tariff (e 5 1) À Retailer 1 (larger market base) Retailer 2 Retailer i, i 5 1, 2

Retailer price pGB l wpIP l depending on d, y, DApGB l pIP l pGB t pIP t 1 1 2  2 i  i Retailer quantity qGB l wqIP l depending on d, y, DAqGB l qIP l qGB t qIP t 1 1 2  2 i  i Retailer profit PGB l wPIP l depending on d, y, DA PGB l pIP l PGB t PIP t 1 1 2  2 i  i Total quantity QGB l QIP l QGB t QIP t   Manufacturer revenue R GB l xR IP l R GB t R IP t 

Conventional wisdom suggests that when buyers group purchase. Numerical investigation shows that purchase together, they are likely to acquire a higher for eA[ 1, 1], the results obtained under linear dis- À quantity at a lower unit wholesale price. However counts are quite robust. As long as the steepness e is under two-part tariffs, buyers will actually buy less not close to 1, in a competitive and sufficiently asym- under group purchase, regardless of the presence of metric environment, GB can be detrimental for the competition. As discussed before, under IP the retailer with a larger market base, whereas the smaller fixed fee d does not affect each retailer’s quantity retailer always benefits from GB. Figure 6 depicts decision. Under GB, retailer i’s payment to the man- retailer 1’s relative profit difference under GB and IP, ufacturer is PGB PIP =PIP for e 5 0.1, 0.2, 0.5, and ð 1 À 1 Þ 1 Æ Æ Æ 0.9. Retailer 1’s profit difference for e 5 0.9 is al- qGB t À GB t i : ways positive and thus is omitted. aqi GB t d þ Q Under GB, retailer 1’s competitive advantage from That is, the fixed fee d is shared by retailers based its larger market base is weakened because now both GB t GB t retailers have the same unit wholesale price. On the on the ratio qi =Q . This serves as an incentive for each retailer to scale back and buy less under GB. other hand, the extra demand from retailer 2 helps The profit comparison suggests that if the two retailer 1 obtain a better discount from the manu- competing retailers currently purchase individually facturer. When the benefit from demand aggregation under a two-part tariff, they will be better off by is sufficient to cover its loss in competitive advan- switching to GB regardless of their asymmetry level tage (e.g., when asymmetry level is low), retailer 1 or competition intensity. Moreover, their cooperation will join group purchase. Otherwise if retailer 2 be- will always lead to a lower revenue for the manu- comes much more competitive through GB, retailer 1 facturer. These results are not surprising given the will receive a lower profit, and naturally, choose not special structure of two-part tariffs. Table 4 summa- to group buy. As a result, while the small retailer is rizes the comparison between IP and GB for both linear schedules and two-part tariffs. Figure 6 Profit Comparison for Retailer 1 (Large Market Base) (A = 1, 4.2.3. General Steepness e. As the results for the c = 0.3, a = 0.3, h = 3, d = 0.1 for eo0, d = 0.01 for e40) À two special cases differ, it would be interesting to 65% Π_GB- Π_IP investigate the system of asymmetric retailers under Π_IP general steepness of e. Not surprisingly, for e close to 55% 1 prices and quantities behave similarly as under À linear schedules, whereas for e close to 1 they behave 45% similarly as under two-part tariffs. In this section, we e=−0.1 e=−0.2 focus on the comparison of retailer profits and 35% manufacturer’s revenue between IP and GB. In all e=−0.5 the numerical studies, we restrict to the parameter 25% e=−0.9 range that ensures a sensible model with positive e=0.1 prices, quantities, and profits. 15% e=0.2 e=0.5 4.2.3.1. Retailer’s profit: GB Vs. IP. We have shown 5% that, under linear discount schedules, it is possible to 00.20.40.60.81∆A have the big retailer not willing to cooperate, whereas −5% the small player always benefits from GB. On the other hand, under two-part tariffs both retailers prefer −15% Chen and Roma: Group Buying of Competing Retailers 192 Production and Operations Management 20(2), pp. 181–197, r 2010 Production and Operations Management Society always interested in forming a group with the large steepness, numerical investigation confirms that re- retailer, such cooperation may never materialize. tailer 1 will always prefer GB, while retailer 2 may not Discussions from online GB forums show that always cooperate when efficiency gap Dc is large and people have very different opinions regarding local discount level |d| is low. buying groups. While some agree that such coopera- tion is more manageable because logistics are easier 4.4. Asymmetry in Both Market Base and Efficiency and there are no trust problems, others believe that We have shown that either dimension of asymmetry cooperation among local members will not be advan- has a similar impact on retailers’ GB decisions. For the tageous because they are competing in the same general case where firms differ in both dimensions, market. This study shows that both concerns are valid, there are two possible cases: (1) DADco0, i.e., the and a careful evaluation is needed for the large retailer. retailer with a larger market base also has a lower unit Our results add a caveat to the general belief that operating cost; and (2) DADc40, i.e., the retailer with GB always benefits buyers. Some buying groups a small market base is more efficient. Usually a high consist of local retailers who might compete with market base is correlated with low operating costs due each other. For example, Stagbuyinggroup.com, a to economies of scale. Nevertheless we sometimes company that specializes in sport, outdoor, and lei- do observe small retailers who excel in operational sure sectors, has over 400 independent retailers all efficiency. over the United Kingdom, with some outlets in For case (1), all the results obtained for asymmetry proximity to each other. Before engaging in any pur- in one dimension still hold. In fact, when DADco0, chasing cooperation, retailers should be aware of the the asymmetry between retailers is strengthened in possible outcomes depending on the competition the sense that the ‘‘strong’’ retailer becomes even and their differences. stronger. For case (2), results depend on which asym- metry dominates. In particular, if DA Dc(112y),  4.2.3.2. Manufacturer’s revenue: GB vs. IP. We have the asymmetry in market base outweighs the ampli- shown that under linear schedules, manufacturer’s fied gap in operational efficiency, and the retailer with a revenue can be higher under GB if the base price a larger market base is the ‘‘strong’’ one. Otherwise if is sufficiently high, whereas under two-part tariffs, DAoDc(112y), the efficient retailer is the strong player. revenue is always lower under GB. Numerical inves- Similar as before, the weak retailer will always prefer tigation suggests that GB can lead to higher revenue GB, while the strong one may or may not cooperate. for steepness e not close to 1 and sufficiently high Note that the asymmetry in operational efficiency (Dc) asymmetry. When e is close to 1, GB always results in is reinforced by competition through the multiplier DA Dc a lower revenue. 112y. For the special case y À , the two ¼ 2Dc asymmetries will cancel out, and these two retailers 4.3. Asymmetric Efficiency are virtually symmetric. This implies that when each We now study another form of asymmetry created by retailer is good at one dimension but not both, GB is the efficiency gap between the two retailers (Dc 0) more likely to materialize. This is an interesting result, 6¼ while assuming DA 5 0. Without loss of generality, we which suggests that retailers of different strengths are consider positive Dc only, i.e., retailer 2 is more effi- more likely to cooperate. cient. For comparisons between IP and GB, the results do not differ qualitatively from those under market 5. Extension base asymmetry. Hence, we briefly comment on the insights specific for the case of asymmetric efficiency 5.1. Nonlinear Demand Functions (the solutions are available from the authors). In many competitive equilibrium analyses, linear For two-part tariffs, the comparisons are similar to demand functions are often used because of their Propositions 7–8. For linear discounts, we find that tractability in providing analytical results. Yet we retailer 2 will always set a lower price, have higher expect demand nonlinearity in many real problems. In quantity and higher profit than (the less efficient) re- the investigation below, we supplement the preceding tailer 1, under both GB and IP. This result slightly discussion by analyzing three nonlinear demand differs from that under asymmetric market base. functions commonly used in the literature (Choi Parallel to the case of asymmetric market base, the 1991, Kadiyali et al. 2000, Lee and Staelin 1997, conditions identified in Proposition 6 will lead to cut- Moorthy 1988, Vives 1999) off points Dcp, Dcq, and DcP such that the price, m n quantity, and profit are higher in IP for the more effi- a q A pÀ p ; A 0; m41; 0 n m; ð Þ i ¼ i i j i    cient retailer 2 when Dc is beyond the cutoff points, A mp np b q e iÀ iþ j ; A 0; 0 n m; and respectively. Concerning the manufacturer, we have a ð Þ i ¼ i    c q A m ln p n ln p ; A 0; 0 n m: similar conclusion as in Proposition 4. For general ð Þ i ¼ i À i þ j i    Chen and Roma: Group Buying of Competing Retailers Production and Operations Management 20(2), pp. 181–197, r 2010 Production and Operations Management Society 193 n is a substitutability parameter; when it is zero, each than under GB, at almost no risk in losing demand. retailer’s demand is independent of the other re- When competition is intense (e.g., n m), GB is again  tailer’s price. m n 0 is required to ensure that advantageous because it helps soften the competition.   retailer i’s demand is more sensitive to its own price Note that GB is again preferable for symmetric re- than to the competitor’s and the total demand does tailers under higher values of m, where the higher not increase when both retail prices increase. (a) is demand elasticity depresses the retail price under IP, known as the constant elasticity demand function, and GB will again be preferable regardless of the where m and n are retailer i’s demand elasticities to competition level. Overall our numerical analysis its own price and to the competitor’s price, respec- suggests that results obtained under the linear de- tively. In the absence of quantity discounts and group mand function are robust, except for the constant purchase, this class of functions is known to result in elastic demand with low demand elasticity and mild price strategies that are independent of the compet- competition. For parameters in this range, retailers itor’s strategy (Choi 1991, Moorthy 1988). Such might continue to be worse off under GB when they property no longer holds under general quantity dis- become slightly different, so that even the weak counts. (b) and (c), referred to as semi-log demand player may prefer IP. functions, are commonly used in empirical studies, where (b) is sometimes called exponential demand func- 5.2. Three or More Competing Retailers tion (Lee and Staelin 1997). In the investigation of how competition affects retail- Owing to the functional complexity, analytical so- ers’ cooperation decision, we follow the literature lutions are elusive for these nonlinear demand (e.g., Ingene and Parry 1995, Tsay and Agrawal 2000) functions. For each demand function, we derive the by considering duopoly with two retailers. In reality, first-order conditions under IP and GB, based on some groups consist of multiple retailers who vary in which we find the equilibrium numerically. We ex- size. For example, members of NetPlus include some periment with a wide range of parameters by distributors with sales at US$50 million, two with normalizing A to 1 and c to 0.1, and varying a be- sales in US$40 million and a half dozen or so with tween 0.2 and 0.8, e between 1 and 1, and d across sales more than US$20 million. In this section, we fo- À feasible values. All of the numerical studies are cus on market base asymmetry and consider the carried within the range that ensures individual special case of two symmetric small retailers and one rationality and nonnegativity. Our analysis suggests big retailer, which sheds light on the analysis of three that results obtained under the linear demand func- or more competing retailers. tion (1) are robust. Specifically, cooperation may not Let retailers 1 and 2 be the two symmetric retailers materialize in presence of sufficient asymmetry be- with the same market base, and retailer 3 be the big retailer (A 5 A A ). Similar as before, we obtain cause the stronger retailer can get hurt from joint 1 2  3 purchasing. Also, the manufacturer’s revenue can be retailer i’s demand function from Shapley and Shubik higher under GB. (1969) as One result that differs concerns the constant elas- 1 n ticity demand. Proposition 3 shows that symmetric qi Ai pi y pj pi ; y 0: ¼ À þ À  0n j 1 1 retailers always prefer group purchase under the lin- X¼ ear demand function, a result that also holds for (b) @ A Define DA 1 A A so that A 1 A A and (c). However for (a), symmetric retailers may get ¼ 3ð 3 À 1Þ ¼ 3ð 1 þ 2þ A , A A A DA, A A 2DA. Again, DA hurt from cooperation when m is low, n is medium, 3Þ 1 ¼ 2 ¼ À 3 ¼ þ measures the level of asymmetry between the big re- and steepness e is not close to 1 (e.g., e 0.3). o tailer and the small retailers. The analysis is similar to To understand this result, note that, in general, de- the duopoly case, and is omitted here. We explore the mand elasticity increases with price (e.g., linear and coalition formation for general steepness numerically, semi-log demand functions). That is, consumers be- and report one representative example in Table 5. Our come more price sensitive when the product becomes major findings are as follows: more expensive. However, under constant elasticity demand, especially when elasticity m is low, a signifi- 1. If the asymmetry level is high, the big retailer cant change in price may not alter demand as much. will not cooperate with either small retailer, a As n varies from 0 to m, a more intense competition result consistent with Proposition 6. In this case, leads to lower retail price and higher quantity. When the two small retailers will group buy. This pos- competition is fairly weak (i.e., n 0), the cooperation sibly explains that in 2003, Mexico’s three largest  effect dominates and GB is preferable. When compe- retailers, Grupo Gigante, Organization Soriana, tition becomes mild for medium values of n, IP is and Comercial Mexicana, joined forces in a buy- preferable because the low demand elasticity m allows ing and operations alliance in order to compete each retailer to charge a much higher price under IP against the Wal-Mart–owned Walmex. Also, Chen and Roma: Group Buying of Competing Retailers 194 Production and Operations Management 20(2), pp. 181–197, r 2010 Production and Operations Management Society

Table 5 Profit Comparison for Three RetailersÃ

Retailer 1 profit (small retailer) Retailer 3 profit (large retailer)

DA GB(1, 2, 3) GB(1, 2) GB(1, 3) IP GB(2, 3) GB(1, 2, 3) GB(1, 2) GB(1, 3) IP 0 0.07835 0.07775 0.07775 0.07670 0.07653 0.07835 0.07653 0.07775 0.07670 0.1 0.06313 0.06239 0.06269 0.06150 0.06135 0.11370 0.11192 0.11313 0.11210 0.2 0.04955 0.04872 0.04925 0.04798 0.04783 0.15563 0.15398 0.15512 0.15419 0.3 0.03762 0.03674 0.03743 0.03614 0.03599 0.20412 0.20273 0.20373 0.20295 0.4 0.02733 0.02643 0.02723 0.02596 0.02582 0.25918 0.25816 0.25895 0.25839 0.5 0.01868 0.01781 0.01865 0.01747 0.01733 0.32080 0.32027 0.32080 0.32050 0.6 0.01167 0.01088 0.01169 0.01064 0.01052 0.38899 0.38909 0.38927 0.38931 0.7 0.00630 0.00564 0.00635 0.00549 0.00539 0.46374 0.46462 0.46438 0.46483 0.8 0.00257 0.00210 0.00263 0.00203 0.00195 0.54507 0.54693 0.54615 0.54711 0.9 0.00049 0.00026 0.00053 0.00025 0.00022 0.63295 0.63616 0.63466 0.63627

ÃGroup binding (GB) (1, 2, 3) means all three retailers group purchase. GB (1, 2) means only the two small retailers group purchase. GB (1, 3) means only retailer 1 and retailer 3 group purchase, while leaving retailer 2 alone. IP means all three retailers individual purchase. GB (2, 3) means retailer 2 and 3 group purchase, while leaving retailer 1 alone. (A 5 1; c 5 0.1; a 5 0.3; d 5 0.01; y 5 2; e 5 0.5). À À

large grocery buying groups in France typically advantageous in the absence of competition. Under refuse members with too low a turnover, while competition, symmetric retailers are always better off small retailers sometimes form their own groups under GB. Interestingly, when discount is deep and (Dobson Consulting 1999). competition is mild, symmetric retailers might prefer 2. For medium or low asymmetry levels, for a wide more intense competition when they cooperate. On range of parameters, all three players would the other hand, a deeper discount is not always ben- form one group. However, sometimes the big re- eficial to retailers under IP. tailer will be interested in cooperating with one, When retailers are asymmetric (i.e., differ in market but not both small retailers. Basically after the big base or operational efficiency), GB clearly helps the retailer and one small retailer form a subcoali- ‘‘weak’’ player (e.g., the retailer with smaller market tion, they might find it unattractive to further base or higher operational cost), while the ‘‘strong’’ include the third player. To some extent, the player may or may not prefer cooperation, depending formation of the subcoalition enlarges the asym- on the competition level, the discount schedule, and metry gap so that no further cooperation takes the level of asymmetry. In fact, GB in general weakens place. the competitive advantage of the strong player, who at the same time enjoys a lower wholesale price that be- Our limited analysis shows that ultimately compe- comes available because of its rival’s demand. tition together with asymmetry drives the decision of Especially when the asymmetry gap is large, the loss whether to group purchase. However, the general in competitive advantage will outweigh the gain from problem of three or more competing retailers can be joint purchasing so that GB may not materialize. In- quite complex because players can form subcoalitions terestingly, GB is more likely to form between retailers that enlarge or shrink the asymmetry gap. The full who are competitive in different, but not all dimen- investigation of this problem, although beyond the sions. Our investigation with nonlinear demand scope of this paper, is a promising direction for future curves suggests that results obtained under linear de- research. mand curves are robust, except for some minor differences under the constant elasticity demand with 6. Discussion and Conclusion low elasticity. Although GB has been widely used in a variety of Our results are based on several assumptions. First, markets and contexts, there is a lack of decision anal- we assume that the quantity discount schedule is ysis for firms who are involved. In this paper, we exogenously given, which is reasonable under the construct a general model of competing retailers with Robinson–Patman Act. However, sometimes the two possible strategies: IP or GB, given a quantity quantities of buying groups are beyond the price discount schedule from the manufacturer. Based on menu specified by the manufacturer, and they need linear demand curves, we offer a number of insights to negotiate the price with the seller. In the current on retailers’ GB decisions. Intuitively, GB is always study, we examine the retailers’ perspective and only Chen and Roma: Group Buying of Competing Retailers Production and Operations Management 20(2), pp. 181–197, r 2010 Production and Operations Management Society 195 consider the manufacturer’s revenue. An opportunity cost is needed for retailers to choose their strategies. to build upon this work is to consider the manufac- An important direction is to examine firms’ GB deci- turer’s optimal strategy when retailers purchase sion under incomplete information. together. Secondly, we have shown that under the commonly Acknowledgments used uniform allocation rule, the retailers pay the The authors would like to thank the editors, the area editor, and the two anonymous reviewers for their valuable and same unit price in group purchase and the strong re- insightful comments, which helped improve the paper tailer may not want to participate. In this case, there significantly. might exist other allocation rules that enable the co- operation and result in a win–win outcome. In Notes practice, buying groups operate on egalitarian princi- ples, whereas others arrange better deals for large 1Examples include the well-known grocery chain Indepen- members because they contribute more to the aggre- dent Grocers Association (or IGA), and Ace Hardware, gate purchase (Lindsay 2009). The majority literature TruServ, and Do It Best in retail hardware (Dana 2006). on cost-sharing mechanisms has focused on the 2 normative properties (e.g., fairness), without consid- Under certain conditions (e.g., very intense competition), it ering competition (Hougaard and Thorlund-Petersen is possible for one or both retailers to exit the market to avoid negative profits. As we study GB formation in a com- 2000, Moulin 1996). One important direction is to petitive environment, we restrict to the range of parameters examine alternative allocation schemes when buyers where the duopoly equilibrium exists under both IP and GB. compete. The issue of market exit or entry under GB is beyond the Our paper makes the intuitive assumption that the scope of this paper. total cost increases with purchasing quantity. Thus, it is never advantageous for retailers to hold back any 3Here we follow the standard economic argument that, inventory (i.e., purchase more than they can sell). when supply of a good expands, the price falls (assuming However, discount schedules in practice often include the demand curve is downward sloping) and consumer discrete breakpoints, under which the total cost might surplus increases. remain the same or even fall if more units are pur- 4This curve is generated by conditions (31), (32), and (34) chased over certain ranges. The area of the price that ensure nonnegative prices, quantities, and profits for schedule where this occurs is called a ‘‘window’’ asymmetric retailers. 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