Reputation and Multiproduct-firm Behavior: Product line and Price Rivalry Among Retailers

Transcription

1 Reputation and Multiproduct-firm Behavior: Product line and Price Rivalry Among Retailer Shaoyan Sun and Henry An Department of Reource Economic and Environmental Sociology, Univerity of Alberta, Canada Selected Paper prepared for preentation at the Agricultural & Applied Economic Aociation 2013 AAEA & CAES Joint Annual Meeting, Wahington, DC, Augut 4-6, Copyright 2013 by Shaoyan Sun and Henry An. All right reerved. Reader may make verbatim copie of thi document for non-commercial purpoe by any mean, provided that thi copyright notice appear on all uch copie.

2 Reputation and Multiproduct-firm Behavior: Product line and Price Rivalry Among Retailer Shaoyan Sun and Henry An 1. Introduction The current conumer demand for healthful and afe food ha become a driving force in the food retail market; for example the continued reitance to genetically modified or genetically engineered food and the growth of organic food. A coniderable number of tudie uing tated preference method have hown that conumer have a high willingne-to-pay (WTP) to healthful and high-quality food (i.e. Wang et al., 1997). In thi ituation, when making policy, retailer hould alo adjut their trategie to meet the changing nature of conumer preference. However, conumer tated demand are not alway an accurate reflection of their actual demand. Several tudie on conumer behavior have found that conumer tated preference do not alway match their revealed preference (Olen and Smith, 2001; Homburg et al., 2005; Laroche et al., 2001). An example from the fluid milk ector how that although many tudie find that conumer have a high tated WTP for milk that i not from cow treated with recombinant bovine omatropin (henceforth referred to a rbst-free milk for hort), actual ale data how that the market hare of rbst-free milk declined during the period 1997 to 2002 (Dhar, 2005). In thee cae, it may be in the bet interet of the retailer to follow what conumer actually do a oppoed to what they ay to determine what to provide. 1

3 However, there are good reaon to liten to what conumer ay, and not necearily to what they actually do. Thi i epecially true in the cae of product that receive high level of public crutiny, uch a genetically modified good. Another example from fluid milk i grocery retailer uch a Safeway, Walmart and Dean Food implemented a ban on elling milk from cow treated with rbst in 2007/2008. What i the incentive for them to pecialize in a niche market? A imilar quetion i why organic tore, uch a whole food, chooe to retrict their product line rather than to proliferate it. Our motivation i to invetigate the iue of retriction or pecialization of firm product line relative to proliferation of their product line. In our view, retailer that decide not to ell any controverial product can generate a lot of poitive publicity. It can end a poitive ignal to the public if a firm willingly ban a product becaue it i concerned with the public health. In addition, thi type of action can erve to differentiate the retailer from it competitor and further etablih their own reputation a a purveyor of high quality product to attract conumer who favor more healthful food. Our tudy examine competition among multiple-product retailer in a vertically and horizontally differentiated indutry where each retailer can chooe a niche or full product line trategy. We model a firm deciion with repect to product line offering in the preence of reputation effect. The objective of our work i to explore the role of reputation in product line competition and the condition under which retailer adjut their product offering. We ak and anwer thee quetion: (1) Can a retailer benefit from a niche product (high quality) trategy relative to a full product line when there are reputational benefit to be gained? (3) Can the 2

4 reputational benefit gained due to product line choice affect competition among multiple- product retailer? When modeling competition among firm, mot tudie impoe the aumption that each firm produce a ingle product due to the analytical convenience. However, multiproduct firm are ubiquitou in many indutrie. Studie are alo developed to addre iue aociated with competition among multiproduct firm. Among the literature modeling multiproduct firm competition, ome analyzed competition in term of product line choice. Katz (1984) analyzed multiproduct firm competition in market where the product produced by each firm have interaction on demand ide. He claimed that the degree of competition of one market can be affected by that of a related market. The conequence i that firm may chooe not to provide full product line. Gilbert and Matute (1993) ued a two-dimenional model to addre the problem that commitment can affect a multiproduct firm choice of product line. Their concluion i that in the preence of commitment the maller the degree of brand-pecific differentiation, the more likely the firm pecialize. In the abence of commitment, firm produce a full product line. Canoy and Peitz (1997) modeled a differentiation triangle where the low quality good i not ubject to horizontal differentiation. They alo addreed the quetion concerning when and why firm produce a full product line or follow a niche trategy. They modeled competition among firm in the differentiation triangle with equential entry. Their main finding i that incumbent have different motivation to develop a niche or full line trategy. Incumbent 1 who ha a firt-mover advantage chooe product for the purpoe of profit. Incumbent 2 ha trategic incentive to determine product line, either for the purpoe of 3

5 deterring entry, or accommodating entry. Different from their tudie, our work i the firt to endogenize reputation explicitly into a duopoly model where good are differentiated in two dimenion. Katz (1984) introduced firm-pecific reputation into hi model, yet he treated it a one of the two dimenion which ditinguih product manufactured by variou multiproduct firm. Many tudie have examined the iue of product line proliferation. Cheng, Peng and Tubuchi (2010) invetigate a two-tage competition in a vertically differentiated market and how that each firm ha an incentive to produce a range of qualitie under the aumption of an increaing and quadratic marginal cot of quality. To the bet of our knowledge, very few tudie have attempted to addre why firm chooe to retrict their product line or pecialize in a niche market. Our tudy i the firt to examine thi iue in the context of both vertically and horizontally differentiation. The tructure of thi paper i a follow. Section 2 preent the theoretical model of competition among multiproduct retailer. We decribe the imultaneou game i introduced and olve for the Nah equilibrium. Concluion are dicued in Section The model We preent a two-dimenional product differentiation model to depict firm behavior under a duopolitic market tructure. We aume that the product are differentiated both vertically (quality) and horizontally (location). Conumer have preference for quality that are a function 4

6 of their income (or omething imilar) and we aume that the ditribution i uniform. Retailer are located at two end of a linear city and can offer the ame product line or not. Our model i baed on Gilbert and Matute (1993) etup, and we build on their model by introducing reputation a an endogenou attribute. The key aumption that drive our reult i that conumer care about reputation; that i, firm reputation enter into their utility function. General aumption and notation Retailer Two retailer A and B, indexed by, compete in a duopolitic market of a generic good. The good i differentiated according to vertical characteritic (quality) and horizontal characteritic (location). We ue to index the vertically-differentiated variant of the good, where (high-quality variant) and (low-quality variant). The two retailer can chooe a full product line by providing both variant, or they produce only one variant and focu on a pecialized product line. We aume the high quality variant generate higher reputation for thoe who provide it. The good i alo differentiated by location, for the two retailer lie at two oppoite end of a linear city of length 1 or and. The retailer optimization problem i ( ) where i profit, denote the price, i the marginal cot, and i the market hare of the repective good by the repective retailer. 5

7 Conumer On the conumer ide, we aume the following: (a) conumer are uniformly ditributed along a linear city of length ; (b) each conumer located at which i uniformly ditributed over the interval ha a preference for quality denoted by (d) each conumer purchae one and only one good 1. The utility of a repreentative conumer buying product from firm i. In thi utility function, denote the quality of product and ; i tranportation cot; repreent the location of the two retail tore. Our contribution to the literature i to incorporate reputation into the utility function. Specializing in high-quality variant can generate higher reputation for retialer. If a retailer ha high reputation, i.e., the utility function become. If a retailer purchae from a retailer with low reputation, i.e., the conumer utility i. In our model, reputation enter into the utility function by effectively offetting the diutility caued by tranportation cot, it i eentially, though not actually, a reimburement of travel expene for any conumer who purchae from a retailer with higher reputation. We can eaily ee that the greater i, the greater the gain a conumer receive purchaing from a tore with high reputation. The duopoly game The competition between the two retailer in term of product line choice can be decribed by a imultaneou game. In the imultaneou game, retailer A and B chooe product line and 6

8 price imultaneouly. While deciding the product line, retailer A face the trategy et { }, where repreent retailer A offering the high quality product, repreent retailer A offering the low quality product, and repreent the null et (or no product). Similarly, retailer B trategy et i { }. Combining thee trategie yield four cenario: { } { }, { } and { }. Since our hypothei i that pecializing in high quality product can lead to high reputation, retailer have no incentive to pecialize in low quality and we do not conider the ituation where only the low quality good i provided 2. We alo ignore the cae where one or both retailer offer no product, but it i eay to verify that thee are never optimal trategie. Therefore, there are three cenario to conider in the duopoly game. The firt cenario i { }. In thi cenario, both retailer engage in ymmetric competition in which both of them chooe to provide a full line. The econd cenario i { }. Thi i alo a ymmetric competition, but both of them chooe to pecialize in high quality product. The third one i { }, which i an aymmetric competition in which retailer A provide only high quality and retailer B chooe a full line. We can prove that { } and { } have the ame reult. Thu we only look at two cenario in our tudy: { } and { }. We treat { } a the baeline cenario where both retailer have the ame reputation, in other word, reputation doe influence retailer deciion of product line trategy. In cenario 2, retailer A focue on the high quality good and gain higher reputation than the retailer who offer a full line. To find the Nah Equilibrium of the duopoly game, we need to 7

9 know their payoff which i imply their profit from each cenario. Thi i to be accomplihed in the following ubection. Baeline cenario: a ymmetric competition The baeline duopoly model draw on Gilbert and Matute (1993). In their model, each retailer chooe to provide a full product line. They do not conider the effect of retailer reputation. We firt take a look at the conumer deciion under horizontal and vertical differentiation, repectively. Horizontal differentiation We conider the model under horizontal and vertical differentiation eparately. In the etting of horizontal differentiation conumer are indifferent between buying good from retailer A or B if.thi yield If a conumer i located in the interval [0,, he buy product from retailer A. If he i located between the range of ], he prefer to purchae from retailer B. At the conumer i indifferent between buying from retailer A or B. which indicate the price differential between the two retailer cannot exceed tranportation cot. A neceary condition for thi to be true i that if, retailer B ha no demand for good, while retailer A capture the entire market of good. If,, retailer A and B plit the market 8

10 Vertical differentiation We next look at the conumer deciion between chooing a high quality or low quality good. The conumer i indifferent between buying good and good from a given retailer if. It yield If a conumer preference lie between [0,, he buy a low-quality good from a retailer. If her preference are located between ], he will chooe a high-quality good from the retailer. At he i indifferent a long a and. Thee two condition enure between and. If, there i no demand for the high quality product. In thi cae, we need to retrict the price differential to be equal to or lower than the quality differential to enure both product are purchaed. At, retailer charge the ame price for both good and drive the demand for the low- quality good to zero. Horizontal and vertical differentiation In the baeline cae, we want to look at how retailer A and B compete with each other when they both chooe to product a ymmetric full product line. The product they provide are both horizontally and vertically differentiated. We can think of the pace that depict conumer and firm when there i both vertical and horizontal differentiation a being a unit quare. Let denote that a conumer prefer 9

11 product from retailer to product from retailer. A conumer i indifferent between buying good from retailer and buying good from retailer if. For example, for a conumer who i indifferent with buying and, her phyical location and preference for quality have the following relationhip: ( ) Then we plug equation and into yielding From equation (5), we can ee that the conumer choice between and i determined by the following factor: (1) price differential; (2) tranportation cot ; and (3) quality differential. The effect of price differential occur at two level: within-retailer but acro product and acro retailer but within product. Therefore, the conumer choice between and can be decribe by a two-tep proce, he firt chooe between and, which i the quality deciion in a given retailer, if exceed, he would rather to chooe. Next he compare and, If i greater than, he will purchae. Thu a conumer purchae rather than if her preference ranking i. A general form of equation (4) can be written a ( ) 10

12 Equation (5) decribe the relationhip between and when, which i the conumer prefer good from retailer over good from retailer. i where the conumer doe not care about which retailer to go given a good. i where the conumer feel indifferent about the quality given a retailer. So far, we have decribed how conumer chooe between and (ee equation ), and (ee equation ) and and (ee equation ). Next we would like to ee under what circumtance can be choen by a conumer located in with quality tate. To ummarize the condition we talked previouly, a conumer located at with tate parameter for quality will purchae if and only if (i) require (ii) require ( ) (iii) require, where i where conumer are indifferent between and, and are where conumer are indifferent between and, and and, repectively. Under condition (i), retailer A win out in a competition with retailer B for ale of the lowquality good. In (ii), the rivalry i between different retailer producing different good, win. In cae (iii), the competition i between different good from the ame retailer and the low quality good win. Therefore the three condition are both neceary and ufficient condition of. In other word, a conumer located at with tate parameter will purchae if and only if 11

13 ; [ ( ) ] Given and, the ale region of i demontrated in Figure 1 Similarly, the conumer will purchae the if and only if (i) require (ii) require ( ) require (iii) Which can alo be written a ; [ ( ) ] Given and, the ale region of i demontrated in Figure 1. Likewie, we can draw ale region for retailer B in the ame unit quare. Then we obtain the ale region for all the product, which i illutrated in figure1. 12

14 Figure 1. Sale region of all product 1 A B Market hare We now can calculate the market hare for each of the good baed on the market egmentation above. They are the area of different region in figure 1. We have ( ) ( )( ) and ( ) 13

15 ( )( ) ( )( ) where, and are market hare for good,, and. Profit-maximization problem and olution We aume that the two retailer engage in Bertrand Nah competition in each ubmarket. Here we aume the retailer have the ame cot of providing a good and. We alo aume unit tranportation cot 3. Retailer A profit-maximizing problem i written a ( ) ( )( ) Retailer B problem i ( ) ( )( ) The equilibrium price we calculated are and. The market hare are and the profit of each firm i, given, where i a meaure of difference between marginal cot of offering high quality and low quality product and we ue to repreent it in the ret of the paper. 14

16 i conumer perception of quality differential between high and low quality. We ue and to denote marginal cot differential and quality differential in the remainder of the paper. and play eential role in the cenario of an aymmetric competition. We will give more detail about thee two parameter in the following decription of the aymmetric cenario. The computational detail are hown in Appendix 1. From the equilibrium we can ee that the price- cot markup i the ame for each good. Thi i the ame a in the ingle-product completion where each retailer only provide one quality. Our concluion i the ame a Gilbert and Matute (1993): price- cot markup are independent of the number of good provided and thu profit i independent of the number of variant provided, which mean a multi-product trategy cannot bring more profit for retailer relative to a ingle- product one. Furthermore, we can alo reach the concluion that the margin for a given quality i alo independent of conumer tate for quality attribute. The Nah equilibrium tell u that both retailer offer the full product line, although the retailer are not any better off by offering the product line vi-à-vi a ingle product. Thi i becaue if one firm i producing a ingle variant, it could alway generate additional ale without loing profit by introducing a econd variant with the ame price-cot margin. The additional profit made by the ingle-product retailer ha everal ource: (1) from the conumer that witch from the old variant to the new one without witching retailer; and (2) from the conumer who ued to buy the new variant from the other retailer and now witche. Scenario 2: an aymmetric competition 15

17 In cenario 2, retailer A pecialize in the high quality ubmarket. Here he gain higher reputation through the pecialization. In contrat, retailer B retain hi full product line and ha a low reputation. Market hare A conumer who purchae from retailer A gain extra utility due to the tore reputation, and the utility can offet her travel cot. The utility function of thi conumer i, a. If he hop from retailer B, her utility function i, a. When the conumer i indifferent between buying high-quality good from retailer A and B, the conumer obtain equal utility from and, which i Thu conumer are phyically located at When the conumer i indifferent between and, he obtain the ame utility from and, which i Conumer preference for quality are located at When the conumer i indifferent between and, we have 16

18 The relationhip between conumer location and their preference for quality can be repreented by In figure 8, we can ee how the entire market i egmented by, and. We need to note that the location of the difference curve between and depend on the relationhip between price differential and tranportation cot. When, we have, and AO i the indifference curve between and. Similarly, when, AC i the indifference curve; and when, AD i the indifference curve. Therefore, there are three cae of market egmentation. We can alo prove that the conumer i indifferent among, and at ( ). 17

19 Figure 2. Market egmentation if retailer A pecialize in high quality C O A B D Market hare under each cae can be computed a follow: The market hare in cae 1 are and ( )( ) ( ) 18

20 Where, and are the hare of, and. and are the total market hare captured by retailer A and B. Subcript denote cae. The market hare in cae 2 are and ( )( ) ( ) Latly, in cae 3, we have the following market hare: and ( )( ) ( ) Retailer problem and olution After having each product market hare under variou cae, we next need to olve retailer profit-maximization problem and compute price equilibrium. Their problem are ummarized in table 1. 19

21 Table 1. Retailer profit-maximization problem under variou cae Retailer A Retailer B Cae 1 Cae 2 Cae3 Note: The profit-maximization problem are ubject to everal contraint which are preented in Appendix 2. Here we only preent the reult of cae 2. We alo tried the other two cae. Cae 1 doe not make ene when impoing contraint. Cae 3 ha imilar reult a cae 2. Solution to the profit-maximizing problem under cae 2 can be found in table 2. Appendix 2 outline detail of how the problem i olved. The reult are ummarized in table 1. The reult are baed on different value of following parameter: marginal cot differential quality differential and tranportation cot ( i a meaure of the difference between the marginal cot of producing the high and low quality 20

22 good. The higher i, the more cotly it i to produce a high quality product relative to a low quality product. The factor that affect could be ize of the retailer and technology-related factor which become obtacle to economie of cope, etc. In our tudy, we let it vary between 0 and 10 4, where 0 mean it make no extra cot to improve quality and 10 mean it i expenive for retailer to offer high quality. meaure conumer perception of quality difference between high quality and low quality. It varie between 0 and 10 5 in our tudy, which mean conumer either perceive no difference between the two good (0) or a very large difference between high quality and low quality (10). The magnitude of affect how likely conumer are willing to commute to a tore. In our example, we chooe value of 2, 5 and 10 6 repreent low, moderate and high tranportation cot. We can infer the following reult from table 2. 1) Tranportation cot play an important role in affecting conumer and retailer behavior. Both retailer A and B chooe product line ubject to the magnitude of tranportation cot. For retailer A, he can be better off by pecializing in the high quality good when tranportation cot reache a certain threhold (which i conditional on the other parameter). Thi i becaue the higher tranportation cot i, the more conumer gain by purchaing from the high reputation retailer. We can eaily ee that retailer A benefit from an increae in tranportation cot through higher profit, a larger market hare and greater market power (i.e., markup). For retailer B, in mot cae, he alo gain in the aymmetric competition in term of greater profit and markup. However, her market hare of each product i declining with an increae in tranportation cot. 21

23 To ummarize, we can ay both retailer benefit from aymmetric competition, but retailer A i a winner relative to retailer B, given increaing tranportation cot. 2) The cot differential and quality differential ha a ignificant influence on the retailer choice of product line. We can ee from the table, retailer A market hare and markup for are greater at the point when conumer perceive a big difference between the high quality good and low quality good and when providing high quality good i cheap than the point when conumer perceive a mall difference between the high quality good and low quality good and providing high quality good i cotly. For retailer B, If conumer perceive the high quality good to be far uperior to the low quality good and difference in marginal cot between the two i mall, the retailer B would provide more high quality product relative to low quality product. We can ee that the market hare of BH i greater than BL. If conumer perceive the high quality good to be far uperior to the low quality good, but the marginal cot of producing the high quality good i alo much higher than that of the low quality good, then retailer B will capture almot none of the high quality good market. 22

24 Table 2. Reult of the aymmetric competition given certain parameter k t Markup Mkt Share Profit(A,B) Equilibrium of the game Baed on the payoff of the ymmetric and aymmetric cenario, we can find the Nah equilibrium of the imultaneou game. The game i preented in table 2 and 3 where we let,, and conider both low and high tranportation cot, repectively. 23

25 Table 3. Payoff matrix with low Retailer B Retailer A 0.5, , , , 0.5 Table 4. Payoff matrix with high Retailer B Retailer A 0.5, , , , 0.5 From table 3, we can ee that the Nah equilibrium i { When t i high, a table 4 how, there are two Nah equilibria { }, when t i low. } and { }. Thi indicate that the game end up with a ymmetric competition in which both retailer provide a full product line when tranportation cot i low. When the tranportation cot i high, the game end up with an aymmetric competition in which one retailer pecialize in high quality and the other one offer a full product line. 24

26 3. Concluion In thi tudy we analyzed competition among multiproduct firm, pecifically retailer, in term of product line choice and price. Our objective i to invetigate the condition under which retailer would chooe to retrict their product line. In contrat to firm that offer a very wide product line to price dicrimination, we how that that if reputation i a function of product line, concentrating on a high quality niche market can be an optimal trategy for retailer. We modeled a imultaneou game in a vertically differentiated market where two retailer decide their product offering price imultaneouly. We examined a cae of ymmetric competition in which both retailer provide a full product line and an aymmetric cae in which one retailer decide to pecialize in a high quality product. We found that retailer can be better off from pecialization if there i a reputation effect. The way we have modeled how reputation take effect i to aume that it reimbure conumer by offetting the travel cot they incur from hopping. In our model, travel cot i the only factor which generate diutility other than price. Therefore, baed on the aumption of our model, we alo found tranportation cot to be an important factor, the higher the tranportation cot, the greater the reimburement conumer can gain, and the more likely they will chooe to purchae from a retailer with higher reputation. In thi cae, the retailer incentive to pecialize in high quality good can be increaed. Judd (1985) and Gilbert and Matute (1993) both concluded that pecialization depend on the extent of brand-pecific differentiation. It occur if the degree of brand-pecific differentiation i mall. In contrat with their finding, our tudy further invetigate beyond brand-pecific 25

27 differentiation of product and conider reputation a a differentiating feature among firm. We explore the quetion what may affect pecialization even though the brand-pecific differentiation i contant. Our model hed light on endogenou reputation formation and it effect on multiproduct firm choice of product line. Reference Canoy, M., and M. Peitz The Differentiation Triangle. The Journal of Indutrial Economic 45 (3): Cheng, Y.L., S.K. Peng, and T. Tabuchi Multiproduct Duopoly with Vertical Differentiation. Dicuion Paper. Dhar, T., and J.D. Foltz Milk by Any Other Name... Conumer Benefit from Labeled Milk. American Journal of Agricultural Economic 87 (1): Gilbert, R., and C. Matute Product line rivalry with brand differentiation, Journal of Indutrial Organization 41 (3): Homburg, C., N. Kochate, and W.D. Hoyer Do Satified Cutomer Really Pay More? A Study of the Relationhip between Cutomer Satifaction and Willingne to Pay. Journal of Marketing 69 (2): Judd, K Credible Spatial Preemption. Rand Journal of Economic 16 (2): Katz, M Firm- pecific Dfferentiation and Competition Among Multiproduct Firm. The Journal of Buine 57 (1): Laroche, M. et al Targeting conumer who are willing to pay more for environmentally friendly product. Journal of Conumer Marketing 18 (6):

28 Olen, J.A., and R.D. Smith Theory Veru Practice: A Review of Willingne-to-pay in Health and Health Care. Health Economic 10: Wang, Q.B., C. Halbrendt, J. Kolodinky, and F. Schmidt Willingne to Pay for rbst-free Milk: a Two-limit Tobit Model Analyi. Applied Economic Letter 4: Thi aumption impoed on conumer i for the purpoe of implifying the analyi. 2 In thi tudy, we are not interet in the cae where the retailer would chooe pecialize in low quality. Specialization in low quality doe not have anything to do with reputation effect which i our main interet. We alo checked and it i not an optimal trategy. 3 We have checked that in the baeline ymmetric cenario, the magnitude of tranportation cot doe not affect the reult. 4 We ue number 0 and 10 to denote, repectively, low and high difference in conumer perception of high quality and low quality. can be an arbitrary number. We chooe thee two number becaue we want to ee the difference from a wide range of. 5 i alo arbitrary, jut a The reaon we make it vary between 0 and 10 i the ame a. We want want to ee the difference in a wide range. 6 We have tried variou number between 0.01 and 20. Note that if tranportation cot are o high that conumer would not patronize either retailer. So we think hould be retricted to enure conumer have a chance to witch between the two retailer. Ideally, we want to decribe in term of percent of price of each good, however, we have no further information of the magnitude of marginal cot of each good. 27

29 Appendix 1: olving retailer problem in Baeline cenario In thi appendix, we olve retailer A and B problem and derive the equalibrium price, market hare and pro t. We aume unit tranportation cot. According to the market hare egmentation in gure 1 and b h = p Bh p Ah + 1 ; 2 b l = p Bl p Al + 1 ; 2 A b p Ah p Al = ; B b p Bh = ; where A and B repreent retailer A and B. h and l refer to high quality and low quality. p i price. So p Ah mean price of high quality manufactured by retailer A. Retailer A problem i p Bl Max A = (p Al c l ) b A l b + (pah c h ) b A h (1 b ) = (p Al c l ) p Bl p Al + 1 p Ah p Al 2 +(p Ah c h ) p Bh p Ah + 1 p Ah (1 2 The rt order condition with repect to p Al i (p Al p Ah ) (p Al c l ) The reaction function i: 2 (p Al c l ) (1 p Al + p Bl ) 1 2 (p Al p Ah ) (1 p Al + p Bl ) 1 2 (c h p Ah ) (1 p Ah + p Bh ) = 0: p Al p Al = c l v 3c 1u h p Bh 3c h p Ah c l p Ah c l p Bl + 3c h c l + c 2 l t 3p Ah p Bh p Ah p Bl 3 4p Ah + 2p Bl + 4p 2 Ah + p2 Bl p Ah p Bl ) 28

30 The rt order condition with repect to p Ah i (p Ah p Al ) 1 (p Ah c h ) = 0; The reaction function i: (p Ah p Al ) 1 (1 p Ah + p Bh ) 1 2 (p Ah c h ) (1 p Ah + p Bh ) 1 2 (c l p Al ) (1 p Al + p Bl ) p Ah = c h p Bh p Al v 3c 1 l c h c h p Bh c h p Al 3c l p Al + 3c l p Bl c u h + c 2 h + 2p Bh 4p Al p Bh p Al 3t 3p Al p Bl p Bh + 2p Al +p 2 Bh + 4p2 Al : Retailer B problem i Max B = (p Bl c l )(1 b l ) b B + (p Bh c h )(1 b h )(1 B b ) = (p Bl c l )(1 p Bl p Al + ) p Bh p Bl 2 h l +(p Bh c h )(1 p Bh p Ah + p Bh p Bl )(1 ): 2 The rt order condition with repect to p Bl i = 0: The reaction function i: (p Bl + 1 p Al ) 1 (p Bl c l ) + 1 (p Bl p Bh ) 1 2 (p Bl + 1 p Al ) (1 p Ah + p Bh ) 1 (c h p Bh ) (p Bl p Bh ) (p Bl c l ) p Bl = c l v 1 3c u h p Ah 3c h p Bh c l p Bh c l p Al + 3c h t c l + c 2 l 3p Ah p Bh p Bh p Al 4p Bh 3 +2p Al + 4p 2 Bh + p2 Al p Bh p Al: The rt order condition with repect to p Bh i 29

31 2 6 4 = 0: 1 (p Bh p Bl ) (p Bh + 1 p Ah ) (p Bh p Bl ) 1 (p Bh c h ) 2 (p Bh + 1 p Ah ) 1 (p Bh c h ) 2 (1 p Al + p Bl ) 1 (c l p Bl ) The reaction function i: p Bh = c h p Ah p Bl v 1 3c u l c h c h p Ah c h p Bl 3c l p Bl t +3c l p Al c h + c 2 h 3 + 2p Ah 4p Bl p Ah p Bl : 3p Bl p Al p Ah + 2p Bl + p 2 Ah + 4p2 Bl The equation ytem of p Al ; p Ah ; p Bl ; p Bh i p Al = c l p Ah p Bl v 1 3c u h p Ah 3c h p Bh c l p Bh c l p Al + 3c h c l t +c 2 l 3p Ah p Bh 3 p Bh p Al 4p Bh + 2p Al + 4p 2 Bh + p2 Al + 1 p Ah = c h p Bh p Al v 1 3c u l c h c h p Bh c h p Al 3c l p Al t +3c l p Bl c h + c 2 h 3 + 2p Bh 4p Al p Bh p Al 3p Al p Bl p Bh + 2p Al + p 2 Bh + 4p2 Al p Bl = c l p Bh p Al v 1 3c u h p Ah 3c h p Bh c l p Bh c l p Al + 3c h c l t +c 2 l 3p Ah p Bh 3 p Bh p Al 4p Bh + 2p Al + 4p 2 Bh + p2 Al + 1 p Bh = 1 3 z c h h v 1 u t l p Ah p Bl 3c l c h c h p Ah c h p Bl 3c l p Bl +3c l p Al c h + c 2 h + 2p Ah 4p Bl p Ah p Bl 3p Bl p Al p Ah + 2p Bl + p 2 Ah + 4p2 Bl From (1) and (3) we can ee that the two equation are ymmetric, which implie p Al = p Bl : Similarly, from (2) and (4), we know that p Ah = p Bh : If we imploe p Al = p Bl and p Ah = p Bh ;the equation ytem become 30

32 p Al = c l p Ah p Al 1 p 2 Ah + p2 Al p Ah p Al c l p Ah 3 4p Ah c l p Al + 2p Al + 3c h + c 2 l c l + 1 p Ah = c h p Ah p Al v 1 p u 2 Ah + p2 Al p Ah p Al c h p Al 4p Al t c h p Ah + 2p Ah + 3c l + c 2 h c h 3 p Ah + 2p Al c h We can ee that equation (1) and (3) are ymmetric. Likewie, equation (2) and (4) are ymmetric. So we implify the equation ytem above by impoing ymmetry and yield 2p Al = 1 + c l + p Ah p 2 Ah + p2 Al p Ah p Al c l p Ah 4p Ah c l p Al + 2p Al + 3c h + c 2 l c l + 1 2p Ah = c h + p Al v p u 2 Al + p2 Ah p Al p Ah c h p Al 4p Al t c h p Ah + 2p Ah + 3c l + c 2 h c h. +1 p Ah + 2p Al c h + 2 The equation ytem ha a olution if p Al = c l + 1 and p Ah = c h + 1 Proof: 2p Al = 1 + c l + p Ah p 2 Ah + p2 Al p Ah p Al c l p Ah 4p Ah c l p Al + 2p Al + 3c h + c 2 l c l + 1 : if we rearrange it, it yield 2p Al 1 c l p Ah p = 2 Ah + p2 Al p Ah p Al c l p Ah 4p Ah c l p Al +2p Al + 3c h + c 2 l c l + 1 : Then plug p Al = c l + 1 and p Ah = c h + 1 into (1) 0 and yield 31

33 2(c l + 1) 1 c l (c h + 1) (c = h + 1) 2 + (c l + 1) 2 (c h + 1)(c l + 1) c l (c h + 1) 4(c h + 1 c l (c l + 1) + 2(c l + 1) + 3c h + c 2 l c l + 1 : For the righ hand ide = = (c h + 1) 2 + (c l + 1) 2 (c h + 1)(c l + 1) c l (c h + 1) 4(c h + 1) c l (c l + 1) + 2(c l + 1) + 3c h + c 2 l c l + 1 c l c h c l (c h + 1) c l (c l + 1) + c 2 l (c h + 1) (c l + 1) + (c h + 1) 2 + (c l + 1) 2 1 q (c h c l ) 2 : For the left hand ide Then we have 2(c l + 1) 1 c l (c h + 1) = c l c h : c l q c h = (c h c l ) 2 c l c h = q (c h c l ) 2 Similarly, we can proove that the econd equation alo hold at p Al = c l + 1 and p Ah = c h + 1. To ummary, p Al = c l + 1 and p Ah = c h + 1 i the olution to the equation ytem (1) 0 and (2) 0 : We cannot guarantee it i the unique olution for thi equation ytem. It guarantee only a local maximum. Similarly, we get p Bl = c l + 1 and p Bh = c h + 1 a the olution to retailer B problem. Therefore, the olution to the entire equation ytem i p Al = p Bl = c l + 1 and p Ah = p Bh = c h + 1:Then we go back the the pro t function and plug the olution into the pro t function. We have and A = (p Al c l ) b A l b + (pah c h ) b A h (1 b ) = (p Al c l ) p Bl p Al + 1 p Ah p Al 2 +(p Ah c h ) p Bh p Ah + 1 p Ah (1 2 = 1 2 ; p Al ) 32

34 B = (p Bl c l )(1 b l ) b B + (p Bh c h )(1 b h )(1 B b ) = (p Bl c l )(1 p Bl p Al + ) p Bh p Bl 2 h l +(p Bh c h )(1 p Bh p Ah + p Bh p Bl )(1 ) 2 = 1 2 : We can ee that the two retailer earn the ame pro t if they both provide the full product line. The pro t they make in thi ituation i the ame a they both reduce to the ame ingle quality and the mark-up keep unchanged. For example, if retailer A and B both produce high quality, and A = (p Ah c h ) b h 1 = (p Ah c h ) = 1 2 ; B = (p Bh c h )(1 b h ) 1 = (p Bh c h ) (p Bh c h )(1 b h ) = 1 2 : 33

35 Appendix 2: olving retailer problem in cenario 2 According to the market egmentation in gure 2, we have e h = e h = p bh p ah + t t = (p bh p ah ) 1 t + 1; and B e p bh = p bl ; when e = 0; when e = 0; When p ah e = p ah p bl t(1 e ) e p ah p bl = 1 : t e p ah p bl t = : p bl < t = 2; market hare ' ah = e h (e h e ) e B ; and ' bh = (1 e h )(1 e B ); Retailer A problem i ' bl = ( 1 2 (e h e ) e B + (1 e h ) e B ): A = A h = (p ah c h )( e h (e h e ) e B ) = (p ah c h )(((p bh p ah ) 1 t + 1) (((p bh p ah ) 1 t + 1) (1 p ah p bl )) p bh p bl ): t Firt order condition with repect to p ah i, given tranportation cot t = 2. We alo try other value of t;e.g. t = 5 and 10: p ah = 1 4 p 2 bh + 2p bh p bl + 2p bh p 2 bl c h : Retailer B problem i: 34

36 B = B h + B l = (p bh c h )(1 e h )(1 e B ) +(p bl c l )( 1 2 (e h e ) e B + (1 e h ) e B ) = (p bh c h )(1 ((p bh p ah ) 1 t + 1))(1 p bh p bl ) +(p bl c l )( 1 2 (((p bh p ah ) 1 t + 1) (1 p ah t p bl )) p bh p bl +(1 ((p bh p ah ) 1 t + 1))p bh p bl ): The rt order condition with repect to p bh given tranportation cot t = 2 i p bh = c 1 h 6 c l p ah p bl v 4 1u 2 4c h 4c l 4p ah + 12p bl + 4c 2 h t 4c h c l 4c h p ah + c 2 l 6 + 8c lp ah 6c l p bl + 4p 2 ah 12p ah p bl + 9p 2 bl Firt order condition with repect to p bl i, given di erent tranportation cot t = 2; p bl = 1 3 c l p ah + 1 c 2 l 2c l p ah + 4p 2 ah 12p ah p bh 3 +6c h p ah + 9p 2 : bh 6c h p bh The equation ytem of the rt order condition i when t=2, p ah = 1 p 2 bh + 2p bh p bl + 2p bh p 2 bl c h 4 p bh = c 1 h 6 c l p ah p bl v 4 1u 2 4c h 4c l 4p ah + 12p bl + 4c 2 h t 4c h c l 4c h p ah + c 2 l 6 + 8c lp ah 6c l p bl + 4p 2 ah 12p ah p bl + 9p 2 bl p bl = 1 3 c l p ah + 1 c 2 l 2c l p ah + 4p 2 ah 12p ah p bh 3 +6c h p ah + 9p 2 bh 6c h p bh 35

37 Let make p ah = c h + a; p bh = c h + b and p bl = c l + c, where a,b, and c are markup of high-quality product from retailer A, high-quality product from retailer B and low-quality product from retailer B repectively. We alo make k = c h c l and = h l, the equation ytem become given t = 2 (b c) 2 + 4a + (2k 6)b 2kc + k 2 4 = 0 3b 2 2ab + 2ac 3bc + ( k)a (2 2k)b = 0 3b 2 4ab + 4ac 3c 2 2ka + 4kc k 2 = 0: Similarly, we can derive the equaiton ytem given other value of tranportation cot. Here we alo have when t=5, the equation ytem i (b c) 2 + 4a + (2k 6)b 2kc + k 2 10 = 0 3b 2 2ab + 2ac 3bc + ( k)a (2 2k)b = 0 3b 2 4ab + 4ac 3c 2 2ka + 4kc k 2 = 0: When t=10, the equation ytem i (b c) 2 + 4a + (2k 6)b 2kc + k 2 20 = 0 3b 2 2ab + 2ac 3bc + ( k)a (2 2k)b = 0 3b 2 4ab + 4ac 3c 2 2ka + 4kc k 2 = 0: Here we have a few contrain to impoe on the equation ytem. Contraint 1 0 < e h = p bh p ah + 1 < 1;which mean conumer are located along the city between 0 and 1. From thi contraint, we have 0 < b a + 1 < 1: Contraint 2 0 < e B = p bh p bl < 1;which mean conumer preference for quality i located between 0 and 1. From thi contraint, we have 0 < k + b c < : Contraint 3 0 < p ah p bl < t = 2;which mean AD i the indi erence curve in gure 2. We preent only cae 2. From thi contraint, we have 0 < k + a c < 1: Contraint 4 e e < 0;which we can ee from gure 2. There i no analytical olution for thee equation ytem.we then olve the ytem through Matlab uing numerical method. 36