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					                  Channel Bargaining with Retailer Asymmetry


Anthony Dukes1                     Esther Gal-Or2                     Kannan Srinivasan3




1
  School of Economics & Management, University of Aarhus, Århus Denmark, adukes@econ.au.dk
2
  Glenn Stinson Chair in Competitiveness and Professor of Business Administration and of Economics,
Katz Graduate School of Business, University of Pittsburgh, Pittsburgh PA 15260, Phone (412) 648-1722,
esther@katz.pitt.edu
3
  H.J. Heinz II Professor of Management, Marketing and Information Systems, Graduate School of
Industrial Administration, Carnegie Mellon University, Pittsburgh PA 15213, Phone (412) 268-8840,
k annans@andrew.cmu.edu
                 Channel Bargaining with Retailer Asymmetry


                                         ABSTRACT

Manufacturers of consumer products often complain of lower profits in light of growing
channel dominance of “power retailers”. This complaint might not be valid. An analytical
model of competing manufacturers and multi-product retailers, might see manufacturers’
profits increase when a retailer gains a cost advantage over its rival. Efficiencies are
generated when the lower-cost retailer does a larger share of a manufacturer’s sales. If
manufacturers sell their products at a lower price to a low-cost retailer than to the high-
cost retailer, they transfer market share to the more efficient retailer, thus increasing
channel profits. In a bargaining relationship between manufacturer and retailer, some of
these enhanced efficiencies are transferred to the manufacturers. We examine the effect
of better product information on the distribution of channel profits and show that, by
informing consumers about product attributes, manufacturers improve their bargaining
position vis-à-vis retailers.




                                               2
                                           1 Introduction
Wal-Mart’s remarkable operational efficiencies give it an advantage over competing
chains such as Kmart and Target. This advantage enables Wal-Mart to offer famously
lower retail prices and generate higher sales volumes of many products. In fact, Wal-Mart
is the principle buyer for many manufacturers including, for example, Disney, Proctor &
Gamble, and Revlon. 4 With such high volumes, Wal-Mart is able to command
concessions from its suppliers through reduced prices or quantity discounts. This
provokes some suppliers to complain that Wal-Mart is using its buying advantage to reap
a higher share of channel profits at their expense. 5 But have manufacturers necessarily
suffered in terms of profits as a result of Wal-Mart’s growing channel power? We
illustrate that this complaint might be unsupported. Specifically, we argue that suppliers
can be, indeed, more profitable when an advantaged retailer gains buying advantage
relative to its retail competitors.
           For a fixed amount of products sold, there are additional efficiencies generated
when the lower cost retailer does a larger share of the sale. If manufacturers sell their
products at a lower price to a low-cost retailer than to the high cost retailer, they transfer
market share to the more efficient retailer, thus increasing channel profits. In a bargaining
relationship between a manufacturer and retailer, some of these enhanced efficiencies are
transferred to the manufacturers.
           Some manufacturers, such as Newell Rubbermaid, have recognized that obliging
a dominant retailer might be more profitable than fighting for channel power.
Rubbermaid, among others, has devoted a large portion of its marketing resources to
serve Wal-Mart. For example, Rubbermaid consults with Wal-Mart for all new product
designs for which Wal-Mart gets the best wholesale price, suggesting that other retailers
don’t.
           Lower wholesale prices might mean lower margins for Rubbermaid, but,
combined with Wal-Mart’s low retailing costs, this means a shift in the distribution of
Rubbermaid’s products from traditional retail channels to more efficient ones, which



4
    See Fortune, March 3, 2003.
5
    See, for example, “Mexico’s War of the Megastore,” Business Week, September 16, 2002.


                                                    3
makes other channels become relatively less important. Newell Rubbermaid can, in
principle, therefore, negotiate for better terms with other retailers.
         The previous example illustrates the intuition of our first of two main results,
which is illustrated in a model with two retailers who sell one or two differentiated
products sold by competing suppliers (e.g. manufacturers). We model the transaction
between a retailer and a manufacturer as a bilateral relationship, which takes place
through bargaining. This bargaining relationship is cruc ial for our result because it
endogenously captures the market power within the channel as dictated by the
competitive setting at each vertical level (e.g. retail competition and product
competition). Moreover, our negotiations-based framework captures bargaining power
asymmetries between retailers vis-à-vis a given manufacturer, under retailing cost
asymmetries. We further use our model to suggest a strategic response for manufacturers,
which can tilt the bargaining power in their favor.
         For a given product class, most retailers typically carry brands from several
different manufacturers. Such multi-product retailers often rely on consumers who visit
their store knowing what general product they wish to buy but not the particular model or
brand. For examp le, consider a consumer who wishes to buy a small household appliance
but needs to visit the store and inspect the features of the models carried by this retailer
before making a decision. Given sufficiently high search costs, this consumer might buy
one of these models rather than visit another retailer in hopes of learning about a more
suitable model. The retailer in this case has captured informational rents associated with
the consumer’s incomplete information. If, on the other hand, the consumer is informed a
priori about product attributes, she will visit the retailer who carries her most desired
model.
         In negotiations between a retailer and a manufacturer, the manufacturer is in a
better bargaining position when consumers are informed relative to being uninformed. In
the event that negotiations break down, this retailer will not carry this product and an
informed consumer who prefers this product will first visit another retailer instead, to buy
the manufacturer’s product.
         This improvement in bargaining position gives an incentive to the manufacturer to
endow the consumer with product information, by using advertising for example, before



                                               4
she makes her retailer choice. This intuition, which is based on our second main result,
indicates that manufacturers have marketing strategies at their disposal, which can
combat the concentration of market power toward a dominant retailer.
          Continuing with the previous example, Rubbermaid’s recent marketing strategy
illustrates this idea. Even though Wal-Mart plays a significant role in developing
Rubbermaid’s new products, Rubbermaid claims sole responsibility for advertising them.
When consumers are informed of Rubbermaid’s new products, Wal-Mart has a bigger
incentive to carry them. Otherwise, interested consumers will seek competing retailers,
which implies that Wal-Mart misses out on the sale of other products. Hence, by
advertising, Rubbermaid reclaims some leverage in its channel relationship with Wal-
Mart. 6
          Summarizing, the emergence of dominant retailers with significant cost
advantages, like those allegedly possessed by the “power retailers”, might not necessarily
leave manufacturers worse off, despite the fact that these retailers wield increased
channel power. Moreover, manufacturers of certain products can reclaim some of the
channel power by communicating directly with customers.
          The effects that may improve manufacturer’s profits, retailing advantages and
customer communication, are both augmented by an additional second order benefit.
Specifically, strategic complementarity in retail prices, which is present for competing
retailers offering similar products, gives retailers an opportunity for price coordination
(tacit collusion). In the presence of retailing cost advantages, manufacturers reap higher
wholesale prices with the weaker retailer. This induces the weak retailer to raise its retail
price, which in turn, allows the advantaged retailer to keep its price a bit higher than it
would have had its rival not raised price. Similarly, when a manufacturer improves its
bargaining position vis-à-vis retailers, via informed consumers, both retailers face higher
wholesale prices, which correspondingly gives retailers an incentive to raise retail prices.
But an incentive of one retailer to raise prices gives its rival an additional incentive (of




6
  To strengthen this effect, Rubbermaid has even tied local retailers to some of its advertising messages.
(See AdWeek, January 18, 1999). Geylani et al (2003) explores this issue in more detail.


                                                      5
second order) to raise retail price, which leads to higher margins. Manufacturers claim a
portion of this added surplus in negotiations. 7
         The present paper addresses three streams of research within the marketing
literature on channel management. The first stream regards studies that identify the
claimant of channel power. For example, Choi (1991) and Lee & Staelin (1997) use
theoretical models to illustrate that the timing of pricing decisions and the competitive
structure at manufacturer and retail levels can drastically alter the distribution of profits
within a channel. Asymmetry in costs between two competing multi-product retailers,
which is crucial for our result, has not been addressed in the existing theoretical literature.
Though Choi (1991) does consider a multi-product retailer, it does not address
competition between two such retailers. And competing multi-product retailers in Lee &
Staelin (1997) are symmetric and, thus, their paper is unable to identify how asymmetry
among retailers affects manufacturer profits. 8
         Lal & Narasimhan (1996) examine the implication of advertising on the
distribution of channel profits with a single, multi-product retailer carrying a branded
good and an unbranded good. Their paper illustrates that the manufacturer of the branded
good can improve its share of the channel surplus by increasing demand through
advertising. Similarly in our paper, product information, in the form of advertising for
example, can improve the manufacturer’s profit. However, because we consider
competing retailers, we are able to additionally capture the effect that manufacturer
advertising has on a consumer’s choice of retailers, as well as, choice of products.
         Also recent studies such as Kidyali et al (2000) and Sudhir (2001) use structural
models to estimate the distribution of channel power for several consumer products.
Again, the focus of these works is on situations with a single multi-product retailer, rather
than competing ones.
         The second stream of research regards the implication of bargaining on the
distribution and level of channel profits. Like this paper, Inderst & Wey (2002), Iyer &
Villas-Boas (2003), Shaffer (2001), consider bargaining contracts rather than take-or-


7
  This strategic effect, which arises in competing channels with intermediaries, has been well-established in
the marketing literature (McGuire & Staelin, 1983, Coughlin, 1985, Moorthy, 1988).
8
  Tyagi (2001) has previously pointed out the incentive of suppliers to price differently to suppliers
depending on their size.


                                                     6
leave it price offers, which is done in previous models of channel interaction. The focus
of Shaffer (2001) is to assess the ability of a channel to maximize profits when multi-
product retailers bargain bilaterally with manufacturers. Similarly, Iyer & Villas-Boas
(2003) are also interested in measuring how the bargaining relationship affects channel
coordination using a model of a single manufacturer and single retailer. Finally, Inderst &
Wey (2002) examine how growing bargaining power of a retailer (or downstream buyer)
affects the incentives of a manufacturer (upstream supplier) to strategically engage in
production or cost improvements. Given the focus of these three papers, they do not
explicitly consider how the strategic effects from competition between retailers affect the
distribution of surpluses within each channel, as is done in this paper.
        The third stream of research our paper addresses is the effect of information on
the distribution of channel profits. Chu & Messinger (1997) point out that, in the presence
of asymmetric information, the share of channel profits improves for the informed
channel member. (See also Messinger & Narasimhan, 1995.) Naraj & Narasimhan
(2002) examine information sharing strategies and the consequences on surplus
distribution with incomplete and imperfect information. In both of these studies, the
source of uncertainty is with respect to demand, while in the current paper, uncertainty is
with respect to consumers’ product information.
        On the other hand, Shaffer & Zettelmeyer (2002) examine the effect of consumer
information on the division of channel profits. In their paper, as well as here, certain
forms of third party consumer information can improve channel profits. However, in the
model of Shaffer & Zettelmeyer (2002), there is one retailer who acts as common agent
who can coordinate competing retail prices. Our result is achieved in a different manner,
namely with competing retailers, and relies crucially on consumers’ decisions regarding
which retailer to visit.
        In the next section, we describe the basic model and establish that, in equilibrium,
retailers obtain different wholesale prices as a result of retailer cost asymmetries (referred
to as retailer advantages). In section 3, we illustrate how retailing advantages shift the
distribution of surpluses throughout the channel. A modified version of the model is
presented in section 4, which illustrates the shift in channel power as a result of consumer




                                              7
information. We offer managerial implications and concluding comments in section 5.
The Appendix contains all technical details omitted from the main text.

                                               2 Model
Our model is described as a sequential game played by two manufacturers, 9 denoted M1

and M 2 , each of whom offer a product to two competing retailers, denoted W and K .
Retailers sell one or both of the products to a set of consumers with differentiated tastes
across products and across retailers. We assume that consumers’ preferences across
products are distributed independently from their preferences across retailers.
        In the first stage each manufacturer and each retailer negotiate over a wholesale
price. All negotiations are bilateral and simultaneous. Figure 1 illustrates all four
negotiations taking place in stage 1.
[INSERT FIGURE 1]
        At the end of stage 1, retailers carry the products of the manufacturers with whom
they have reached agreement in negotiations. We assume that retailers know the outcome
of their own negotiations but not those of their competitor. Then in stage 2, retailers set
retail prices for the products they carry. After stage 2, consumers decide which store to
visit and finally, what product to buy.
        We start the analysis with consumers and work backward to derive the
equilibrium pricing decisions of retailers in stage 2 given the outcomes of the
negotiations in stage 1.
        A consumer’s preference over retailers is formally represented by a location x on
the interval [0,1]. Assume that consumers are uniformly distributed on this interval and
retailers are located at opposite endpoints. This spatial distribution of consumers
represents heterogeneous preferences across retailers and may represent any factor in
which consumers view retailers differently – location, service, appearance, etc.
        A consumer’s preference over products is similarly represented on another
interval [0,1] by a location y , which is independent of x . Products are located at
opposite endpoints. Figure 2 clarifies this distribution of preferences.


9
 The term “manufacturer” is used in order to fix the context. These agents can be considered wholesalers,
distributors, or any supplier directly upstream from retailers.


                                                    8
[INSERT FIGURE 2]


        Consumers first decide which of the retailers to visit. After arriving at one of the
retailers, a consumer buys one product. Initially, a consumer is aware of her location x ,
what products are carried by each retailer and their corresponding retail prices ri j ,
i = 1,2 , j = W , K . The consumer is unaware, however, of her location y . After visiting a
retailer, she examines the available products and informs herself of her location y .
        One interpretation of this set-up is that both retailers send out flyers that inform
consumers of the available products and prices. To discover her location y , a consumer
must visit a retailer in order to physically examine the available products. An implication
of this set- up is that the model is most appropriate for durable and semi-durable goods,
such as small appliances, for which cons umers typically inspect physically before buying.
        To analyze the purchase decision, suppose a consumer visits retailer j and learns

that her location is y . Given prices ri j , i = 1,2 , a consumer receives utility v p − ty − r1 j

from brand 1 and v p − t (1 − y ) − r2j from brand 2, where v p denotes the utility of

consumption of a product which is an exact match of her preferences and t denotes the
degree of differentiation between the two products. If retailer j carries both brands, then

the consumer buys the brand giving the higher utility. We assume that v p is sufficiently

large that any consumer who visits a retailer who carries only one brand, buys a product.
        When deciding which retailer to visit, a consumer must anticipate her benefit
from purchasing a product at either store given that her location y is unknown at that
time. The anticipated benefit from visiting retailer j , if carrying both brands, is the

expected utility of purchase over all possible values of y given known prices r1 j and r2j :
                             yj
                 E ( j ) = ∫ ( v p − ty − r1 j ) dy + ∫ j ( v p − t (1 − y ) − r2j ) dy ,
                                                         1
(1)
                            0                            y


where

                                                      1 r2j − r1j
                                               yj =     +         .
                                                      2     2t




                                                       9
The value y j denotes the location of a consumer who is indifferent between the two
brands at retailer j . The first and second integral in (1) express the event that the

consumer located at y buys brand 1 (i.e. y < y j ) and buys brand 2 (i.e. y > y j ),
respectively.
        If retailer j carries only one product, then all consumers expect to incur the

average transportation costs in product space, or t / 2 . Hence, the expected benefit of
                                                       ~
visiting retailer j , when carrying only brand i is E ( j ) = v p − t / 2 − ri j .

        In order to capture differentiation among retailers, we assume that consumers
travel to a retailer and incur a costs of t r per unit traveled. Therefore, a consumer located

at x receives utility E (W ) − t r x when visiting W and E ( K ) − t r (1 − x ) when visiting K .

        Denote the demand for brand i at retailer j by Di j . Lemma 1 specifies the
product demand faced by each retailer, when both retailers carry two products or when
exactly one retailer carries both brands. As we show later, both retailers carry both brands
in equilibrium and no unilateral deviation from the equilibrium can result in any other
outcome than the cases presented in the lemma.




Lemma 1
    (i) If retailers W and K carry both brands and charge prices ri j , j = W , K ; i = 1,2

        then retailers’ product demands are:
                            1        I [ j =W ] δ  1               r j − r2j 
                       Di =  + ( −1)
                          j
                            2                         + ( −1) 2− i 1         ,
                                                2t r  2
                                                                        2t   
        where I [ j = W ] is an indicator function taking the value 1 if j = W and 0
        otherwise, and
                     r W + r2W r1K + r2K         ( r1W − r2W ) 2 (r1K − r2K ) 2   
                 δ = 1
                              −                −
                                                                 −                 .
                                                                                     
                         2         2                   4t             4t          




                                                  10
   (ii) If retailer j carries only brand m and charges price ~ j , and retailer l ≠ j carries
                                                             r

         both brands and charges ril , i = 1,2 , then demands are:

                ~j 1 1         ~ j r1l + r2l (r1l − r2l ) 2 t  ~ j
                Dm = −         r − 2 +                     +  , Di = 0 , m ≠ i ,
                    2 2t r                        2t        4

                        ~  1 r l − r2l                       ~ j  1 r l − r2l 
              ~
                    (      )
              D1l = 1 − Dmj  − 1
                            2            
                                             and
                                                      ~l
                                                            (
                                                      D2 = 1 − Dm  + 1
                                                                   2)     2t 
                                                                                 .
                                 2t                                           


         Lemma 1 expresses retailer j ’s demand for brand i as expected store traffic

times the conditional demand for brand i . The expression δ is the difference between a
consumers expected benefit E (K ) and E (W ) , when both retailers are carrying both

brands. This difference, δ , is a function of the mean and variance of product prices for
each retailer. This reflects that the consumer evaluates the risk associated with the
uncertainty of the location in product space when making her store choice.
         This uncertainty implies that, with positive probability, some consumers will, ex
post, buy the product they prefer least. For example, if retailer j carries only brand m ,
some consumers would have been better off visiting retailer l ≠ j and buying product
i ≠ m.
         Each retailer faces a constant marginal cost of retailing, which may include
inventory, distribution, or handling costs. Denote retailer j ’s marginal retailing cost by

c j . We assume that the retailing advantage comes in the form of lower marginal retail
costs for retailer W . Denote the cost advantage by ∆ c = c K − c W ≥ 0 . Manufacturers face

symmetric and constant marginal costs of production, which are normalized to 0.
         Before retail prices are specified, a given manufacturer negotiates with each
retailer a unit wholesale price for the manufacturer’s product. We designate by p ij the

unit price negotiated between manufacturer i and retailer j . Accordingly, if negotiations

between manufacturer i and retailer j result in agreement, retailer j pays p ij D ij to

manufacturer i . If negotiations result in disagreement, the retailer pays nothing to the
manufacturer and does not carry the brand.




                                               11
        To model the negotiations between a given manufacturer-retailer pair we utilize
the Nash bargaining solution. This cooperative solution concept implies that the parties to
the negotiation agree to split evenly the surplus generated in the trade between them. If
Vi and Π j designate the payoffs that accrue to manufacturer i and retailer j ,

respectively, in case they can reach an agreement and Vi− j and Π −j i designate their

respective payoffs when they are unable to reach an agreement (their “outside options”),
then the gain from trade is equal to (Vi + Π j ) − (Vi− j + Π −i ) . The distribution of these
                                                              j


gains between manufacturer i and retailer j , as defined by the Nash bargaining solution,

is determined by the wholesale price pij that yields (Vi − Vi− j ) = ( Π j − Π −i ) .
                                                                               j


        In spite of being a cooperative solution concept, the Nash bargaining solution
does not prevent us from capturing the competitive pressures that exist among the parties
in the wholesale and retail markets. Since the outside options Vi− j and Π −j i depend upon

the nature of competition among manufacturers and retailers, this competition is reflected
in the outcome of the negotiations. 10
        Another modeling issue, which arises as a result of the multi-party bargaining
arrangement, is the fact that each agent in our model can negotiate with multiple parties,
thus yielding complex interdependencies across negotiation outcomes. Any equilibrium
bargaining outcome in such a setting should have the property that no manufacturer-
retailer pair would want to renegotiate after learning the negotiated outcome between any
other negotiating pair.
        Using the expressions for market shares derived in Lemma 1, we can state the
agreement payoffs relevant to the negotiations between manufacturer i and retailer j as
follows:
(2)                                 Vi = ∑ j =W , K pij Dij , for i = 1,2

(3)                        Π j = ∑i =1, 2 ( ri j − p ij − c j ) Di j , for j = W , K .


10
   It is possible to consider other solutions to our multi-party bargaining problem. For example, a
cooperative solution using Shapley values could be used in the context of coalitional bargaining. However,
in our institutional setting, anti-trust law may constrain bargaining to be bilateral, between one
manufacturer and one retailer. For coalitions limited to size two, the Shapley value is equivalent to the
Nash solution used here.


                                                      12
The payoff to manufacturer i is the total sales from both retailers at wholesale prices pW
                                                                                         i


and piK . The payoff to retailer j is the total sales to consumers less marginal retail costs
and wholesale costs.
        Under this scenario, each retailer j = W , K chooses prices ri j , i = 1,2 in order to

maximize Π j as expressed by (3), which implies the following first order condition:

        ∂Π j                                    ∂Dij                        ∂D j
(4)              = Dij + ( ri j − pij − c j )         + ( rl j − plj − c j ) lj = 0, i = 1,2; l ≠ i ; j = W , K .
         ∂ri j                                  ∂ri j                       ∂ri

Symmetry across manufacturers implies that, in any equilibrium, p1j = p 2j = p j and that

each retailer j = W , K sets r1 j = r2j = r j . Apply symmetry to (4) in order to yield the
retailers’ optimal second stage pricing rules when all stage 1 negotiations result in
agreement:
(5)                 r j = t r + 2 ( p j + c j ) + 1 ( p l + c l ),
                                3                 3
                                                                         j, l = W , K ; l ≠ j .

Note from (5) that for any fixed wholesale prices p W ≤ p K , retailer W obtains higher

profit margins r W − p W − c W > r K − p K − c K despite lower prices r W < r K . It is this
advantage that makes W a potentially more profitable channel for manufacturers than K .
        In order to ensure that both retailers are earning positive margins, we invoke the
following assumption:
        A1: 3t r > ∆ c .
Assumption A1, guarantees that the spatial aspect of retailer competition is sufficiently
lower than the cost difference. If the cost difference were larger than 3t r , then the weaker
retailer is unable to achieve any market share at positive margins.
        Now suppose one of the negotiations result in disagreement. Specifically, suppose
that if manufacturer i and retailer j cannot reach an agreement, retailer j carries only
brand m ≠ i and manufacturer i sells only to retailer l ≠ j . The disagreement payoffs in
this case are:
                                                    ~
(6)                                     Vi− j = pil Dil , for l ≠ j ,
                                                           ~
(7)                           Π −j i = ( ~ j − p m − c j ) Dmj , for m ≠ i .
                                         rm      j




                                                            13
       Recall that retailer l ≠ j cannot observe that negotiations between i and j
resulted in disagreement and therefore does not modify its optimal stage 2 pricing rule
given in (5). Retailer j , however, revises its pricing rule from (5) in accordance to the

maximization of Π −j i with respect to ~mj . In particular, because consumers prefer variety,
                                       r

when retailer j carries only one product, consumers bare the risk that their location y is

far from product l and must be compensated by a lower retail price in order to visit the
retailer. Retailer j’s optimal stage 2 pricing rule under disagreement with manufacturer i
reflects this compensation:
(8)                                        ~j =rj − t .
                                           r        8

       Consumers’ incomplete information and heterogeneous preferences over products
imply that retailers are able to attract consumers by offering a variety. But when retailer
j is carrying only one product, it must compensate consumers for its competitive
disadvantage by reducing its price. The result says, in fact, that the greater the
differentiation between products the greater this compensation since this increases the
cost of consumers whose location y is far from brand l .
       We make the following additional assumption to guarantee that, upon
disagreement, the weak retailer maintains a positive market share:
       A2: t r / t > 1 / 8 .
Assumption A2 says that consumer loyalty to retailers is sufficiently high so that product
        ~
demand Di j for the retailer carrying only one brand is positive. If A2 fails to hold, it

might be possible that a retailer in disagreement with a manufacturer is unable to attract
any consumers at all positive price margins.
       Given retailers’ optimal pricing rules as functions of possible stage 1 outcomes,
we express the agreement and disagreement payoffs using equations (2), (3) and (5)-(8)
along with symmetry across manufacturers:
                                                                   2
                               t     ( p l − p j ) + (c l − c j ) 
(9)                        Πj = r   1 +                            ,
                                2                3t r             




                                                  14
                                         1 1  ( p l − p j ) + (c l − c j ) t 
                           Π −j i = t r  +                                 − 
                                         2 2t r            3                  8
(10)
                                          t  ( p l − p j ) + (c l − c j ) 
                                        8t  +
                                  × 1 −                                   ,
                                           r              3t r            

                                   pW     1 ( p K − p W ) + (c K − c W ) 
                           Vi =           +                              
                                    2    2              6t r             
(11)
                                      p K  1 ( p K − p W ) + (c K − cW ) 
                               +           −                             ,
                                       2 2               6t r            
                                       pl 1   1     ( p l − p j ) + (c l − c j ) 
(12)                       Vi − j =         +     −                              ,
                                       2  2 16t r               6t r             
where i = 1,2 ; j = W , K ; l ≠ j . Equilibrium wholesale prices are determined by
invoking the following condition,
(13)                       (Vi − Vi− j ) = ( Π j − Π −i ) for j = W , K ,
                                                     j


which is implied by the Nash bargaining solution, using (9)-(12), 11 and assumes a
splitting rule of equal shares. 12 The condition in (13) says that the equilibrium wholesale
price, denoted p j , is that price which ensures that each party receives half of the net

surplus derived from the bilateral agreement: ( Π j + Vi ) − (Π −j i + Vi − j ) .

        Note that the expressions (9)-(12) depend on cost difference ∆ c but not on

absolute costs c W , c K . Hence, ∆ c can be useful to examine the distribution of

bargaining power across retailers vis-à-vis manufacturer i when there exists a retailer
advantage of the form ∆ c > 0 . Retailer j ’s agreement payoff, expressed by (9), is

increasing in ∆ c for j = W and decreasing for j = K . Because of its cost advantage,

retailer W contributes more to the channel relationship with manufacturer i than K
does. Therefore, an increase in ∆ c also benefits manufacturer i upon agreement with
 j = W because the Nash bargaining solution implies that all bilateral surpluses are split
between the two parties.


11
  See Appendix for derivation of the specific equilibrium wholesale price conditions.
12
  The assumption of equal shares is made for convenience and the main results hold for arbitrary splitting
rules.


                                                       15
         Furthermore, an increasing cost advantage improves retailer W ’s disagreement
payoff while decreasing K ’s. (See (10).) Upon disagreement with i , retailer W enjoys
bigger margins than K would and, therefore, does not suffer as much in the absence of
product variety.
         From a manufacturer’s perspective, the retailing advantage ∆ c hurts its

bargaining position vis-à-vis retailer W , but enhances it relative to K . Note from the
disagreement payoff of manufacturer i , as expressed in (12), that Vi −W is decreasing and

Vi − K is increasing in ∆ c , for any fixed pair of prices pW and p K . As ∆ c increases, W

becomes a relatively more efficient retailing channel and can sell more of manufacturer
i ’s product than K . This says that manufacturer i ’s position upon disagreement with W
declines with the retailing advantage and suggests that the bargaining position of
manufacturer i vis-à-vis retailer W worsens with higher ∆ c . Note, however, that the
manufacturer profits are not necessarily harmed as a result, as is shown in the next
section, because they claim a portion of the channel profits enhanced through efficiency
gains, which is reflected in the sum Π j + Vi .

         The above intuition, which states that retailer W is in a better negotiating position
than retailer K , suggests that the outcome of the negotiations, as defined by (13), yields a
lower unit wholesale price for retailer W than for retailer K . The following proposition
confirms the intuition. 13


     Proposition 1
     Under assumption A2, if ∆ c > 0 then p W < p K .


         This proposition states that when retailing cost advantages exist, the advantaged
retailer, W in this case, obtains an additional advantage though its bargaining
relationship with manufacturers.
         It is instructive to note the role manufacturer competition plays in this result. It
can be shown, for instance, that the equilibrium wholesale price ordering is opposite (i.e.

13
  We are unable to obtain explicit expressions for p W and p K from the conditions in (13). Nevertheless,
we are able to obtain an order relation across the negotiated wholesale prices, as described in Proposition 1.


                                                     16
p W > p K ) when there is one, rather than two, manufacturers. In this case, retailers have
no outside option upon disagreement with the manufacturer, which claims half of each
channel’s profits through negotiations. And since channel profits are higher with the low
cost retailer, p W > p K .
        However, with competing manufacturers, a retailer improves its bargaining
position vis-à-vis manufacturers since, upon disagreement with a manufacturer, it is still
able to sell the competing brand. Moreover, this improvement is greater for the low cost
retailer since it is more profitable when selling only one brand.
        Note in addition that this outcome resembles an outcome in which a
manufacturer, with some degree of market power, engages in third degree price
discrimination by selling its product at different prices. However, the bargaining set up in
our model endogenizes the distribution of market power within the channel in accordance
with the relative bargaining positions of the retailer and manufacturer. Hence the price
discrimination in this case is a result of the shift in market power of the buyer, retailer
W , rather than from strategic pricing on the part of the seller.

                               3 Distribution of Surplus
In this section we use the model described in section 2 to characterize the distribution of
surpluses across manufacturers, the two retailers, and consumers. First we illustrate how
the retailing advantage, in terms of ∆ c , and market structure parameters t , t r , affect the
distribution of profits across firms. Next, we consider the effect of retailer advantages on
consumer surplus.
        The condition specified by (13), which defines the equilibrium wholesale prices
p W and p K does not permit an explicit solution. As such, the results of this section are

based on numerical simulations in which we solve (13) for a range of parameters ∆ c ≥ 0 ,

t , and t r that satisfy assumptions A1, A2, and the equilibrium wholesale price condition
specified in the Appendix. We graphically present the numerical results for
representative points in the parameter space. Furthermore, all quantities are computed
using the cost difference ∆ c = c K − c W with c K > 0 constant for all ∆ c ≥ 0 in order to
capture the effect of W’s asymmetric improvements in retailing costs.


                                               17
        Despite the fact that retailer W is in an improved bargaining position vis-à-vis a
given manufacturer, we find that manufacturer’s profits can increase with retailer
advantage. As Figure 3 suggests, the profits of manufacturers increase in ∆ c . There are
two sources of this profitability: efficient channeling and price coordination (collusion).
        The first of these sources comes from the fact that a retailing cost advantage
induces a greater portion of each manufacturer’s sales to be channeled through the more
efficient retailer. For any given retailing cost advantage, ∆ c > 0 , W obtains lower
wholesale prices than retailer K, as implied by Proposition 1, which means that retailer W
enjoys higher margins than K. (See equation (5), and the subsequent discussion.) As a
result, retail channel W is marginally more profitable (i.e. higher margins) than K. In
addition, the cost advantages p K − p W > 0 and ∆ c > 0 imply that retail prices are lower
for W than for K (see again equation (5)), which means that each manufacturer sells more
of its product through retailer W than through K.


[INSERT FIGURE 3]

        Hence, retailing advantages induces each manufacturer to channel its product
through the more efficient retailer, retailer W in this case. The manufacturer is entitled to
a portion of efficiency gains, as guaranteed by the Nash bargaining solution. On the other
hand, note that a portion of all efficiency losses associated with the retailer K is also
incurred by the manufacturers. However, in net, each manufacturer gains from the
retailing advantage because a greater share of its product is sold to W.
        In addition to the profitability source discussed above, a strategic effect induced
by the retailer advantage results in a collusive outcome with respect to the pricing
behavior retailers. As the disadvantaged retailer is forced to raise its price r K , retailer W
has a second order, strategic incentive to raise it price. (See (5)). To be sure, r W is
decreasing in ∆ c , as indicated in Figure 5. However, this decrease is smaller than the sum

of the decreases in c W and p W . On the other hand, retailer K has a strategic incentive to
lower its price in response to a lower price by its rival, W. But since market shares are
higher for retailer W, the net gain in extracted consumer surplus from the strategic effect



                                              18
is positive. (This fact is illustrated in Figure 6 by the fact that that the rate of increase in
consumer surplus with respect to ∆ c is less than 1.)                  14
                                                                            And, as in the case above, a portion
of this additional surplus is shared with manufacturers, as guaranteed by the Nash
bargaining solution. This last point is illustrated in Figure 4 by the fact that wholesale
prices are changing at a decreasing rate, with respect to ∆ c .


[INSERT FIGURE 4]



            To summarize, manufacturers are more profitable as a result of retailer advantages
because (i) the improved bargaining position of the advantaged retailer serves to shift the
distribution of products toward through the more profitable channel; and (ii) net strategic
effects of retail prices yields additional extracted consumer surplus accruing, in part, to
manufacturers.
            Finally, we examine the comparative statics with respect to brand and product
differentiation t and t r , respectively, in order to evaluate how the competitive structure
at each vertical level affects the distribution of profits within the channel. In general,
more differentiation, either at the brand or retailer level, raises channel profits as more
surpluses are extracted through higher retail prices. In order to determine the distribution
of these additional surpluses within the channel, one must examine how the respective
bargaining position respond to changes in differentiation parameters t and t r .


[INSERT FIGURE 5]


            First, consider changes in brand differentiation. The numerical results, as
presented in Figure 6, suggest that more brand differentiation shifts surpluses directly
from consumers to manufacturers, bypassing retailers. Recall that when retailer j does

not reach an agreement with manufacturer i , it must compensate consumers for not
carrying a variety and this compensation is increasing in t since consumers bear a higher

14
     Total ex ante consumer surplus is expressed CS = v − r − [(t + tr ) + ( r K − r W ) 2 / tr ] / 4 , where
r = r W ( D1 + DW ) + r K ( D1 + D2 ) is a market share weighted average price index.
           W
                2
                             K    K




                                                            19
cost when arriving at the retailer to learn that she does not carry the preferred brand.
Hence, more brand differentiation makes retailers less profitable upon disagreement and
weakens its bargaining position vis-à-vis the manufacturer. That is, its disagreement
payoff Π −j i is lower for higher levels of t , which results in higher wholesale prices

allowing the manufacturer to capture most of the additional surpluses generated by an
increase in brand differentiation.
           As illustrated in Figure 6, the effects of higher wholesale prices in this case are
not borne by retailers, but by consumers. Since increases in t raise wholesale prices
(approximately) symmetrically, it is possible for retailers to pass this additional cost
entirely on to consumers. 15 Hence, the effects of retailer advantages are relatively
invariant to the degree of brand differentiation.
           On the other hand, higher degrees of differentiation between retailers, as
measured by t r , brings additional profits to retailers at the cost of, not only consumers,

but also of manufacturers. (See Figure 3.) On the one hand, an increase in t r raises the
extracted surplus directly through higher retailer prices, as reflected in the optimal retailer
pricing rules in (5). This contributes to higher channel profits, potentially benefiting both
manufacturer and retailer. However, in the bargaining relationship with a given
manufacturer, each retailer’s position is improved equally upon agreement and upon
disagreement as a result of higher t . To see this, note that r j − ~ j is invariant to t . This
                                                 r                  r                                    r

implies that any marginal increase in retailer differentiation is not claimed by the
manufacturer.


[INSERT FIGURE 6]


           The main conclusion of this section is that a retailer cost advantage does not
necessarily harm manufacturers. In fact, a larger cost advantage increases channel profits
in the more efficient retail channel and benefits this retailer and both manufactures.



15
     Formally, assuming dpW / dt ≈ dp K / dt and differentiating retailer i’s optimal pricing rule in (5) with
respect to t imply that dr j / dt = dpi / dt .


                                                        20
         4 When Consumers are Informed of Product Preferences
The basic model discussed in the previous sections assumes that consumers are a priori
uninformed about their product preferences and must visit a retailer to know their
location in product space. In this section we compare that case to the case in which
consumers know their product preferences before making their retailer choice. This
distinction has an important implication with regard to the distribution of surpluses
       Recall from section 2, some consumers buy the brand they prefer least, ex post,
net of price whenever the retailer they visit carries only one brand. However, if informed,
a consumer would visit the retailer who carries her preferred brand. Thus, in a setting
with informed consumers, a manufacturer who sells to only one retailer obtains a higher
demand for its product than when consumers are uninformed, all else equal. And even
though this outcome never occurs in equilibrium, the payoffs determined in this outcome
affect the bargaining positions of the negotiating parties. In particular, since consumer
information raises demand for the manufacturer selling through only one channel, the
manufacturer’s bargaining position is improved, relative to the uninformed case.
Consequently, a larger share of the channel surplus accrues to the manufacturer.
       This last point illustrates an incentive for the manufacturer to advertise directly to
consumers. For example, a manufacturer who informs consumers about product attributes
in order to help them learn their preferences over products, can improve its negotiating
position vis-à-vis retailers. We formalize this intuition by illustrating that when
consumers are informed of their location in product space, manufacturer profits increase.

4.1 Model with Informed Consumers
Reconsider the game from section 2 in which consumers know their location y before
making their retailer choice. The remainder of the game is as in the original model.
Bilateral negotiations between manufacturer and retailer over wholesale price occur in
stage 1. Retailers set retail prices in stage 2 followed by consumers’ retail store choices
and product choices.
       We maintain the assumption from section 2 that all consumers are also informed
about prices and product availability at each retail store. As a result, a consumer’s store
choice involves no uncertainty since she knows what product she will buy before visiting



                                              21
the retailer. Formally, the consumer located at ( x, y ) facing retail prices ri j , i = 1,2 and

 j = W , K , chooses a retailer j and product i to maximize utility:

                                       v p   − t r x − ty − r1W               i = 1, j = W
                                       
                                       v p   − t r x − t (1 − y) − r W
                                                                               i = 2, j = W
                          U (i , j ) = 
                                                                     2

                                              − t r (1 − x) − ty − r           i = 1, j = K
                                                                      K
                                       v p                         1
                                       v p   − t r (1 − x) − t (1 − y) − r
                                                                           K
                                                                               i = 2, j = K .
                                                                         2


Note that if stage 1 negotiations between retailer j and manufacturer i end in

disagreement, then set ri j = ∞ .
           The main analytical distinction between the informed case and the uninformed
case of section 2 is in determining market shares to each retailer and manufacturer.
Moreover, it is the difference in market share between the two cases that is central to the
main result. We focus, therefore, on the market share analysis and relegate to the
Appendix the remaining details, which follow the same analytical logic as the uniformed
case of section 2.
           Market shares to each retailer and manufacturer are defined by areas in a partition
of the unit cube, [ 0,1] 2 , specified by a system of inequalities formed from the consumer’s
maximization problem. Of the many possible partitions allowable by the loosest
restrictions on the parameter space, we focus only on a special class of partitions in which
manufacturers earn higher profits in this modified game. In particular, we consider
outcomes that have positive market shares for both retailers and both products when all
negotiations result in agreement. (See Figure 7.) We also restrict attention to the case
when retailers are sufficiently differentiated relative to products. Specifically, we replace
assumption A2 with a stronger assumption:
           A2' t r / t > 7 / 8 .
           This assumption is sufficient to guarantee that all consumers “located” at the
retailers location will shop at that retailer even when it carries only one product. 16
           To illustrate the distribution of market shares when both retailers carry both
products, suppose that r2W − r1W ≥ r2K − r1K and let x * denote the location of a consumer
                                                       i


indifferent between buying product i from retailers W and K. Under this condition,
16
     It is not claimed that this assumption is necessary for our result.


                                                             22
retailer W is more attractive to consumers who prefer product 1 and retailer K to those
who prefer product 2. Because consumers know their location y before visiting the
retailer, they visit the store that offers a better value for their preferred brand. Figure 7
illustrates this by the fact that x1 > x * . This effect is not present in the uninformed case
                                   *
                                         2

in which a consumer’s store decision is based only on expected utility of product
consumption over all possible values of y . In the context of Figure 7, with uninformed

consumers yields x1 = x * .
                  *
                        2




[INSERT FIGURE 7]


[INSERT FIGURE 8]


        Despite this distinction in market shares, the marginal changes in market share
with respect to retail price ri j remain unchanged with informed consumers. As is shown

in the appendix, given wholesale prices p ij , i = 1,2 and j = W , K , the optimal retail

pricing rules is given by (5). Consequently, the agreement payoffs in the informed case,

denoted Π j and Vi , remain expressed by equations (9) and (11). As a result, if product
        ˆ        ˆ

information causes any difference in negotiated wholesale prices, then it must be
reflected in the disagreement payoffs.
        Now consider the market shares when a retailer, say W, and a manufacturer, say 2,
fail to reach an agreement in stage 2 negotiations. In the uninformed case, there is a set of
consumers who visit retailer W but would have been better off, ex post, shopping at
retailer K . With product information, however, these consumers always make the best
decision ex post. This set of consumers is represented in Figure 8 by the triangular region
defined by points ABC.
        Compared to the uninformed case, manufacturer 2’s disagreement position is
improved since product information has caused this set of consumers to switch stores in
order to obtain its product. As a result, its negotiating position vis-à-vis retailer W is
improved, leading to a share of the channel surplus in the form of higher negotiated
wholesale price.


                                               23
        Furthermore, as in the basic model, there exists a second order strategic effect that
works to further increase wholesale prices beyond the first order effect of the improved
bargaining position. In particular, each retailer faces higher wholesale prices, which
induces it to raise its price. Consequently, each retailer strategically reacts to its rival’s
price increase by raising its price further. (Recall the discussion in section 3.) As a result,
there is additional extraction of consumer surplus to which the manufacturers receive a
portion through the negotiations. These two effects are combined and generalized in the
following proposition, where p j denotes the equilibrium wholesale prices in the
                             ˆ
equilibrium with informed consumers.


    Proposition 2
    Under Assumption A2', p j > p j for j = W , K .
                          ˆ


        The proposition states when the ratio t r / t is sufficiently large, the manufacturer
obtains a larger wholesale price as a result of informed consumers. And, as we illustrate
numerically in the next section, this can improve manufacturers’ profits.
        Since our intent is to illustrate how consumer information might possibly improve
the bargaining position of the manufacturer, we have not fully characterized the
wholesale pricing outcome under other conditions. In particular, the question of whether
consumers’ product information can reduce wholesale prices when A2' does not hold
remains unanswered.

4.2 Distribution of Surplus with Informed Consumers

In this section, we establish that informing consumers can improve manufacturer profits.
In particular, we illustrate cases where V > V . Table 1 presents a sample of such results
                                          ˆ
from numerical simulations. Note that manufacturer profits increase in the presence
information.


[INSERT TABLE 1]




                                               24
           The previous section illustrated that consumer information about products can
raise the marginal cost of retailers by p j − p j . It is not necessarily the case, however,
                                        ˆ
that retailers suffer a loss in profits. In fact, because retailers retain a portion of the
additional (marginal) extracted surplus generated from the coordinated increase in retail
price, the advantaged retailer will benefit for all ∆ c ≥ 0 .17
           What is particularly noteworthy about this last result is that even though
consumers become better informed, they can be worse off. This counter-intuitive result
stems from the collusive effect discussed above. Consequently, information does not
necessarily always lead to more competitive outcomes.

                      5. Managerial Implications & Conclusions
The results from the discussions in sections 2-4 have several implications that should be
of interest to manufacturers when thinking about their relationship with retailers.
           First, the main result of section 2 states that equilibrium wholesale prices, as
dictated in the bargaining relationship between manufacturer and retailer, are not equal.
Specifically, the low-cost retailer is in a better bargaining position vis-à-vis the
manufacturer and is thus able to get a better price than the weak retailer. This suggests
that a manufacturer, when faced with a strongly positioned retailer, reevaluate its
bargaining position with its other, weaker retailers, and expect to receive higher
wholesale prices. 18
           Second, the results of section 3 imply that manufacturers’ profits might actually
be improved as a result of improved channel position of a retailer, when this
improvement is a result of efficiency gains. As such, manufacturers need not always fear
the retailer with the relatively strong channel presence. Rather, the efficiency gains of the
dominant retailer and the corresponding channel power it gains can serve to aid market
forces toward shifting the distribution of a manufacturer’s goods toward more efficient
outlets.


17
   Cabral and Villas-Boas (2002) refer to the outcome as a Bertrand Supertrap. They illustrate that in multi-
product Bertrand competition, when the strategic effect of a symmetric increase in marginal costs exceeds
the direct effect, competitors can be better off. We have illustrated this same idea in an asymmetric setting.
18
   There are delicate issues concerning the legality of this practice, as dictated by the Robinson-Patman Act
of 1934. Managers should be aware of these issues and of how to set wholesale prices in accordance with
the law.


                                                     25
       Third, the results of section 4 suggest that informing consumers about product
attributes (e.g. through manufacturer advertising) might be a way to shift channel power
(vis-à-vis a retailer) toward manufacturers of certain products. In particular, our model is
best suited for products (e.g. small appliances, semi-durable goods), where consumers
make their final brand choice after visiting a retailer. In the absence of full consumer
information, a retailer often can consummate a sale to a consumer who, ex post, would
have preferred a competing brand carried only by a competing retailer. By informing
consumers, via manufacturer advertising for example, before consumers make their
retailer choice, retailers are more inclined to reach an agreement with the manufacturer to
carry their product, thus improving the bargaining position of the manufacturer.
       This paper challenges the notion that the emergence of dominant “power
retailers” such Wal-Mart, Home Depot, Best Buy and others is necessarily bad for
manufacturers. In addition, manufacturers of products for which consumers inform
themselves upon visiting a retailer, can regain channel power by communicating product
information directly to consumers.
       The results of this highly stylized model simply serve to offer a new way of
thinking about channel relationships with a dominant retailer. Obviously, our analysis
leaves out many other aspects about actual channel relationships, which concern
practitioners. For example, issues such a trade promotions, slotting fees, or exclusive
dealing complicate the channel relationship relative to our simplified setting. As such, our
results should be taken in the context of highlighting some incentives for manufacturers
when facing a dominant channel member.




                                             26
                                                   Table 1
               Distribution of Profits & Surplus over Product Information Regimes
                        ( t = 1.00 , t r = 1.25 , c K = 2 , ∆ c = c K − c W )


  ∆c         pK        pW         rK      rW       ΠK       ΠW         CS           Vi
  0.0       0.528     0.528      3.778   3.778    0.625     0.625     0.35     0.2639
  0.0*      0.559     0.559      3.809   3.809    0.625     0.625     0.32     0.2794
  0.2       0.529     0.527      3.711   3.644    0.559     0.694     0.45     0.2639
  0.2*      0.561     0.557      3.743   3.675    0.559     0.695     0.42     0.2794
  0.4       0.530     0.526      3.645   3.511    0.498     0.767     0.55     0.2639
  0.4*      0.563     0.555      3.677   3.541    0.496     0.768     0.52     0.2795
  0.6       0.531     0.525      3.579   3.377    0.439     0.843     0.65     0.2639
  0.6*      0.566     0.554      3.612   3.408    0.438     0.846     0.62     0.2795
  0.8       0.532     0.525      3.513   3.244    0.385     0.923     0.75     0.2639
  0.8*      0.569     0.553      3.547   3.275    0.383     0.927     0.71     0.2796
  1.0       0.534     0.524      3.447   3.111    0.334     1.007     0.85     0.2640
  1.0*      0.572     0.552      3.482   3.142    0.331     1.011     0.81     0.2797
  1.2       0.536     0.524      3.382   2.978    0.286     1.094     0.95     0.2640
  1.2*      0.576     0.551      3.418   3.009    0.283     1.100     0.91     0.2798
  1.4       0.538     0.524      3.316   2.845    0.242     1.185     1.04     0.2640
  1.4*      0.580     0.551      3.354   2.877    0.239     1.192     1.01     0.2799
  1.6       0.540     0.523      3.251   2.712    0.202     1.280     1.14     0.2641
  1.6*      0.585     0.550      3.290   2.745    0.199     1.289     1.11     0.2801
  1.8       0.543     0.523      3.186   2.580    0.166     1.379     1.24     0.2642
  1.8*      0.591     0.550      3.228   2.614    0.162     1.390     1.20     0.2803

* Denotes Consumer Information Regime




                                           27
                                Figure 1
Bilateral Negotiations between Retailers W , K and Manufacturers M1 , M 2




                          M1                M2




                          W                 K




                                   28
                                         Figure 2
             Distribution of Consumer Preferences for Retailers and Products




        x                                                      y




Retailer W                       Retailer K        Product 1                   Product 2




                                              29
                      Figure 3
Manufacturers’ and Retailers’ Profits with respect to ∆ c
             ( t = 1, c K = 2 , ∆ c = c K − cW )




                             30
                  Figure 4
Manufacturer Wholesale Prices with respect to ∆ c
         ( t = 1, c K = 2 , ∆ c = c K − cW )




                         31
            Figure 5
Retail Prices with respect to ∆ c
( t = 1, c K = 2 , ∆ c = c K − cW )




                32
                       Figure 6
Effects from Changes in Brand Differentiation
        ( tr = 1 , c K = 2 , ∆ c = c K − c W )




                     33
                          Figure 7
           Market Shares when r2W − r1W ≥ r2K − r1K




Retailer
   K
                   Store K
                  Product 1             Store K
   x*                                  Product 2
    1




                    Store W
   x*
    2              Product 1
                                               Store W
                                              Product 2
Retailer
  W
                               y*
                                K
                                          *
                                         yW
        Product                                       Product
           1                                             2




                               34
                                       Figure 8
             Market Shares when Retailer W Carries only Product 1




Retailer
   K
                   Store K
                  Product 1
   ~*
   x1                                     Store K
                                         Product 2       B
                              A



                         Store W
                        Product 1        t
                                        tr               C
Retailer
  W                               ~
                                  yK
        Product                                      Product
           1                  x*                        2




                                             35
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                                           36
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                                           37
Appendix A

This appendix contains the proofs of Lemma 1 and Propositions 1 and 2 in addition to the
equilibrium wholesale price conditions.

Proof of Lemma 1

 (i) The demand for product i at retailer j is the market share of the retailer times the

 demand for brand i , conditional on visiting the retailer. Conditional on visiting
 retailer j , product demands are given by y j = 1 + ( r2j − r1j ) /( 2t ) for brand i = 1 and
                                                 2


  1 − y j = 1 − ( r2j − r1j ) /( 2t ) for brand i = 2 , which is compactly written as the second
            2


 multiplicative expression of Di j .


 To determine the market share of retailer j , consider a consumer located at x , who is
 indifferent between the two stores. Then
                                    E (W ) − tr x = E ( K ) − tr (1 − x )
 implies
                                             1 E (W ) − E ( K )
                                        x=     −                .
                                             2      2t r
 By evaluating the integrals in (1) for j = W , K , it can be verified that
  E (W ) − E( K ) = δ , as expressed in the statement of the lemma. All consumers x < x

 visit retailer W , implying market share for W of (1 − δ / t r ) / 2 and for K ,

  (1 + δ / tr ) / 2 .


 (ii) Suppose retailer W carries only brand m and k ≠ j carries both. The consumer
 decides which retailer to visit by solving
                                      ~
                             max{ E (W ) − t r x , E ( K ) − t r (1 − x)} .
 The consumer at ~ is indifferent between retailers when
                 x




                                                   38
                                        ~
                    ~ = 1 − E ( K ) − E(W )
                    x
                        2          2t r
                       1 1      
                                       1 r1K − r2K    K t r1K − r2K     
                      = −       v p −  −
                                                        r1 + −
                                                                           
                                                                             
                       2 2t r   
                                       2     2t           4   4         
                        1 r K − r2K    K t r1K − r2K           t           
                                                                                  
                      + + 1
                       2               r2 + +
                                                        −  v p − − ~W
                                                                       r        
                              2t           4   4               2           
                                                                                  
                       1 1       ~W r1K + r2K ( r1K − r2K ) 2 t 
                      = −       r −          +               + .
                       2 2t r           2           2t        4

 Since all consumers who visit W buy product m , expected demand for product m at
               ~                         ~W
 retailer W is DW = ~ . The remaining 1− Dm consumers shop at retailer K . Hence, the
                 m  x
 individual product demands at retailer K are given by

                  ~ K  1 r1K − r2K                                ~ K  1 r1K − r2K    
         ~K
                (
         D1 = 1 − Dm  −
                      2  )              
                                                and
                                                           ~K
                                                                 (       )
                                                           D2 = 1 − Dm  +
                                                                        2               .
                                                                                         
                             2t                                              2t       
 Similarly suppose that retailer K carries only brand m and W carries both to the
 deduce the general expression as given in the statement of the lemma.                        Q.E.D.

Equilibrium Price Conditions

 At the Nash bargaining solution, p j = arg max( Π j − Π −i )(Vi − Vi − j ) , j = W , K . First
                                                         j


 order conditions imply that − Dij (Vi − Vi − j ) + (Π j − Π −i ) Dij = 0 , which gives the
                                                             j


 condition in (13) for Dij ≠ 0 . Using this condition for j = W , K , equilibrium prices

  p W and p K must satisfy

                t     ∆    p K − pW   t             ∆    p K − pW          t K
 (A.1)            tr + c +          −   − pW  t r + c +                  + p =0
                2
                       3       3     16       
                                                       3       3             8
                                                                             

                t     ∆    p K − pW   t           ∆    p K − pW            t W
 (A.2)            tr − c −          −  − p K t r − c −                    + p =0
                2
                       3       3     16 
                                         
                                               
                                                     3       3               8
                                                                             
 For the proof of Proposition 1, note that these conditions imply that
                       t               ∆ ( p K − pW ) 2                     t
 (A.3)          t tr −  + ( p K − p W ) c +             − ( p K + pW ) t r −  = 0
                      16                3       3                           8




                                               39
                 t∆ c          W        11t  ( p K − pW )( p K + p W )               ∆
 (A.4)                + ( p − p ) t r +
                           K
                                             −                          − ( p K − pW ) c = 0 ,
                  3                     24              3                             3

 which are obtained from adding (A.1) and (A.2) and subtracting (A.2) from (A.1),
 respectively.


 The solution of (A.1) and (A.2) solves the maximization problem of the Nash
 bargaining solution under the second order conditions,
                           ∂ 2G j           1     2pj t                       ∆   pl 
(A.5)                                  =          
                                                   3 − − t r + (−1) I [ j =W ] c −  < 0 ,
                          ∂( p j ) 2       4t r       6                        3  3 
 for j = W , K and l ≠ j , where G j ( p j ) ≡ ( Π j − Π −j i )(Vi − Vi− j ) .


Before proving Proposition 1, we first state and prove an intermediate result in the form
of Lemma A1.

Lemma A1

 Under assumption A1, ∆ c + p K − p W > 0 .

Proof of Lemma A1

 If p K ≥ pW then the conclusion holds immediately. On the other hand, consider the

 case when p K < pW and suppose, by contradiction, that ∆ c + p K − p W ≤ 0 . Then (A.3)

 implies
                                                        t (t r − 16 )
                                                                  t
                                       p K + pW >                     .
                                                           tr − 8t


 Using this fact, it can be shown via algebra that the LHS expression of (A.4) must be
 less than
                              t 2∆c               W  t r − 192 t 
                                                        2     7 2
                          −                   K
                                                    
                                         + ( p − p )             ,
                            48(t r − 8 )
                                     t
                                                         tr − 8 
                                                               t
                                                                  
 which is non-positive under the assumption that p K < pW and assumption A1.
 Therefore, the LHS expression of (A.4) is negative – a contradic tion in equilibrium.
 Hence the conclusion holds.


                                                          40
Proof of Proposition 1

 Define x ≡ ∆ c + p K − p W . Then solving (A.1) and (A.2) for p W and p K yields

                                                         t               t             t 
                                              t r + x − 16  (t r − x ) + 8  t r − x − 16  
                                     t
 (A.6)                   pW =
                                2 D(t r + x)                                             

                                                          t              t             t 
                                               t r − x − 16 (t r + x ) + 8  t r + x − 16   ,
                                     t
 (A.7)                   pK =
                                2D( t r + x )                                            
 where
                                                        t2
                                  D = (t r − x) −                .
                                                    64(t r + x )

 Using the above expressions (A.6) and (A.7), it can be shown that p K > p W holds if
 and only if
                                            −8 x
                                             t
                                                 < 0.
                                            t+x
 This latter condition holds since x > 0 by Lemma A1.                                                Q.E.D.


In order to prove Proposition 2, we state and prove a lemma, which characterizes the
demand of each product at each retailer when consumers are informed of the location y
in product space.

Lemma A2
 Assume consumers are informed of their location y .

 (i)           If retailers W and K carry both brands and r2W − r1W ≥ r2K − r1K . Then product
                                      demands are given by

     1 r2K − r1K  1 r K − rW   r2W − r W r2K − r1K   1 ( r1K + r2K ) − ( r1W + r2W ) 
 D = +
   W
                   + 1       1
                                  +      1
                                               −              +
   12
           2t  2
                         2t r   2t
                                                2t   2
                                                                         4t r             
                                                                                            
                                1 r − r2  1 r2 − r2 
                                     W   W         K    W
                       DW =  + 1
                               2            +            
                         2
                                      2t  2
                                                   2tr   




                                                    41
                                      1 r K − rK       1 r1W − r K 
                               D1K =  + 2
                                     2
                                               1
                                                        +
                                                        2
                                                                   1
                                                                      
                                           2t               2t r  
         1 r W − r2W     1 r2W − r2K   r2W − rW r2K − r1K   1 ( rW + r2W ) − ( r K + r2K ) 
  D2K =  + 1             +           +         1
                                                     −              + 1               1
        2
               2t        2
                               2tr   2t
                                                       2t   2
                                                                               4t r             .
                                                                                                  
 (ii)           If retailers W and K carry both brands and r2W − r1W < r2K − r1K . Then product
                                        demands are given by

                                      1 r W − r1W  1 r1K − r1W 
                             D1W =  + 2
                                     2              +          
                                            2t  2      2tr  
       1       r − r2  1 r2 − r2   r2 − r
                 K    K        K      W      K      K
                                                       r − r1   1 ( r K + r2K ) − (r1W + r2W ) 
                                                        W    W
 DW = 
   2   2     + 1        +
                         2            +
                                        
                                                  1
                                                      − 2       2 +
                                                                      1
                                                                                                 
                  2t          2t r   2t              2t                   4t r            
        1 rW − r1W      1 r1W − r1K   r2K − r K r2W − rW   1 (r1W + r2W ) − ( r K + r2K ) 
 D1K =  + 2             +           +        1
                                                     −       1
                                                                    +                1
       2
             2t         2
                              2tr   2t
                                                       2t   2
                                                                              4t r             .
                                                                                                 
                                     1 r − r2  1 r2 − r2 
                                           K    K         W     K
                             D2K =  + 1
                                    2              +           
                                            2t  2
                                                          2tr  
 (iii)        Under Assumption A2', if retailer W carries only brand 1 and charges price
         ~W , and retailer K carries both brands and charges r K and r K − ~ W > 1 , then
         r                                                                 r
                                                              1       2          8

         product demands at retailer W are

         ~      1 r K − ~W
                          r    1 r2K − r1K   1 r1K + r2K − 2~ W
                                                                r      t  1 r1K − r2K 
         D1W =  + 1
               2              +
                               2            + +                 −       +         ;
                     2t r           2t   2
                                                        4t r        4t r  2
                                                                                2t   

         ~                  ~ K  1 r2K − r1K  1 ~W − r1K 
                                                   r
         DW = 0 ;           D1 =  +
                                  2           + 1
                                               2          ;
           2
                                       2t         2t r  
              ~K        ~     ~
              D 2 = 1 − D1W − D1K .

Proof of Lemma A2

 (i) Define x* as the location of the a consumer who is indifferent between buying
             i


 product i from retailer W or from retailer K. Then
                                      1 ri K − riW
                                   x = +
                                    *
                                    i              ,        i = 1, 2 .
                                      2     2t r

 Define y * by the location of a consumer indifferent between buying product 1 and
          j


 product 2 at retailer j. Then


                                                  42
                                              1 r2j − r1 j
                                       y* =
                                        j       +          ,                j = W,K .
                                              2     2t
If r2W − r1W = r2K − r1K then product demands are x1* yW , (1 − x1 ) yW , x1 (1 − yW ) ,
                                                       *         *    *    *       *



(1 − x1 )(1 − yW ) , respectively. Otherwise, r2W − r1W > r2K − r1K implies x1* > x* and
      *        *
                                                                                   2


yW > y * . A consumer at ( x, y ) buys product 1 at retailer K if and only if x > x1 and
 *
       K
                                                                                   *



y < y * . Therefore D1K = (1 − x1 ) y * . A consumer at ( x, y ) buys product 2 at retailer W
      K
                                *
                                      K


if and only if x < x* and y > yW . Therefore DW = x1* (1 − yW ) . A consumer at ( x, y ) in
                    2
                               *
                                              2
                                                            *



( x1 , x* ) × ( y * , yW ) such that
   *
        2         K
                       *



                                            r2K − r1W t + t r t
                                       x=            +       − y
                                               2tr     2t r   tr
is indifferent between product 1 at retailer W and product 2 at retailer K. Therefore,
                                      *        *
                                                y*
                                                 K

                                                yW
                                                     (
                           D1W = x1* yW + ∫ * x2 + r2 2−rr1 + t2ttrr − ttr y dy
                                                       t
                                                            K
                                                               + W
                                                                                          )
and

                                  2 K
                                               *
                                                y*
                           D2K = x* y * + ∫ * x1 +
                                               yW
                                                 K
                                                     (    r2K − r1W
                                                             2t r
                                                                      +   t +t r
                                                                           2t r
                                                                                         )
                                                                                   − ttr y dy ,

which yield the expressions stated in part (i) of the lemma.
(ii) Derived similarly as in above.
(iii) Define ~ * as the location of a consumer indifferent between buying product 1 and
             y   K


product 2 from retailer K and ~1* as the location of a consumer indifferent between
                              x
buying product 1 from either retailer. Then
                                                                                                        ~W
              ~ * = 1 + r2 − r1                                                       ~ * = 1 + r1 − r1 .
                          K    K                                                                  K
              yK                                         and                          x1
                    2       2t                                                              2       2t r

A consumer at ( x, y ) ∈ [ 0,1 − ~1* ] × [0, ~K ] buys product 1 from retailer K. Therefore,
                                 x           y*
~
D1K = ~K (1 − ~1* ) , which yields the expression stated in the lemma. To derive the
      y*      x
demand of retailer W, who sells only product 1, consider a consumer
( x, y ) ∉ [ 0,1 − ~1* ] × [0, ~K ] , such that x < ~1* or y > ~K , and break the demand into two
                   x           y*                   x          y*

parts. Part one consists of those consumers with x < ~1* and y < ~K , which has a
                                                     x           y*




                                                         43
 measure equal to ~1* ~K . For those consumers ( x, y ) ∉ [ 0,1 − ~1* ] × [0, ~K ] such that
                  x y*                                            x           y*

  y > ~K , they will buy (product 1) from retailer W if and only if
      y*

                                    r2K − ~ W + t + t r
                                          r                        t
                               x≤                             −      y ≡ h( y ) .
                                             2t r                 tr

 Note that under the assumption A2' and r2K − ~ W > 1 , h (1) ≥ 0 . The second part of the
                                              r     8


 retailer W’s demand is the area “under” h ( y) from ~K to 1. Thus, total demand is
                                                     y*
                                    ~
                                    D1W = ~1* ~K + ∫~ * h( y) dy ,
                                                    1
                                          x y*
                                                         yK


 which, upon evaluation, yields the expression given in the statement of the lemma.
                                                                          ~K
 Finally, since we assume that all consumers make a purchase, demand D 2 can be
 determined by computing the remaining area left over from the two demands computed
 above.                                                                                   Q.E.D.

Equilibrium Price Conditions with Informed Consumers
 In this section we derive equilibrium conditions for symmetric wholesale prices
  pW , p K . The derivation here is parallel to that of the basic model, which involves, first,
  ˆ ˆ
 determining the optimal stage 2 pricing behavior of the retailers given wholesale prices
 determined in stage 1 negotiations.


 Using Lemma A2, the payoff to retailer j when carrying both produc ts is

                                 Π j = ∑i=1, 2 ( ri j − pij − c j ) Dij ,
                                 ˆ

 where Di j are expressed in Lemma A2. Setting dΠ j / dri j = 0 and invoking symmetry
                                                ˆ

 across products gives the optimal second stage pricing rules when retailer j:
                              r j = tr + 2 ( p j + c j ) + 1 ( p j + c j ) ,
                                         3                 3

 which is analogous to (5). When negotiations between retailer j and manufacturer i end
 in disagreement, leaving j to sell only product m ≠ i in stage 2, it sets retail price ~ j in
                                                                                        r
 order to maximize
                                                                 ~j
                                     Π −j i = (~ j − p j − c j ) Dm .
                                     ˆ         r




                                                    44
     Setting dΠ −j i / d~ j = 0 and invoking symmetry, p j = pij = p2j , gives the optimal second
                        r

     stage pricing rule when retailer j sells only product m: ~ j = r j − 1 . Note that retailer
                                                              r           8

     j ’s optimal pricing rules with informed consumers, mimics those of the original
     model. 19


     The payoffs relevant for stage 1 negotiations, given optimal retailer behavior, in stage

     2, are agreement payoffs Π j , Vi and disagreement payoffs Π −j i , Vi− j i = 1,2 ;
                              ˆ      ˆ                          ˆ         ˆ

     j = W , K ; l ≠ j . For the model with informed consumers, the expressions for the
     agreement payoffs for both manufacturer and retailer remain as in equations (9) and
     (11). The disagreement payoff for retailer i also remains as before in (11). The
     important distinction between the two models occurs in the disagreement payoff of the
     manufacturer, which in the case of informed consumers is

                                       p j  1 3t    ( p l − p j ) + (c l − c j ) 
                              Vi − j =
                               ˆ              +    +
                                       2  2 16t r
                                                                6t r             .
                                                                                  
     The Nash bargaining solution is used to determine the outcome of the stage 1

     negotiations. Specifically, p j = arg max( Π j − Π −i )(Vi − Vi − j ) , j = W , K . First order
                                 ˆ              ˆ     ˆ j ˆ ˆ

     conditions for this maximization problem imply that prices pW , p K in an equilibrium
                                                                ˆ ˆ
     with informed consumers, must satisfy the following system:
                    t      ∆c p K − pW
                                ˆ    ˆ    t   W      ∆ c p K − pW  3t K
                                                            ˆ    ˆ
     (A.8)            tr +
                             +         −  − p t r +
                                             ˆ          +         + p =0
                                                                     8 ˆ
                    2      3      3     16          3       3    

                    t     ∆    pK − pW
                                ˆ    ˆ    t           ∆    p K − pW  3t W
                                                             ˆ     ˆ
     (A.9)            tr − c −
                                       −  − p K  tr − c −
                                             ˆ                      + p = 0.
                                                                       8 ˆ
                    2      3      3     16            3       3    
     The second order necessary condition for this maximization is identical to that of the
     original model, which is given in (A.5).




19
   Assumption A2' is sufficient, but not necessary, for this to be the case. If Assumption A2' is relaxed, it is
possible that these pricing rules change. In particular, by relaxing A2', the product demands of Lemma A2
(iii) are not guaranteed to hold, which alters the retailer’s first order condition with respect to ~ j .
                                                                                                    r


                                                      45
Proof of Proposition 2

 Define the LHS expressions of (A.8) and (A.9) as functions H W ( pW ) and H K ( p K ) ,
 which are decreasing under the second order condition for the maximization defined by
 the Nash bargaining solution. Substituting the (uninformed) equilibrium prices p W , p K
 gives
                                             t K
                              HW ( p W ) =     p > 0 = HW ( p W )
                                                            ˆ
                                             4
                                             t W
                              HK (p K ) =      p > 0 = H K ( pK ) ,
                                                             ˆ
                                             4
 since p W , p K solve (A.1) and (A.2) and p W , p K solve (A.8) and (A.9). The decreasing
                                           ˆ ˆ

 property of H j implies the result.                                                  Q.E.D.


Appendix B
This appendix provides more detailed results from the numerical calculations used in
Figures 3-6 in Section 3.
   ∆c         tr         pK       pW            rK        rW          ΠK      ΠW      CS        Vi
   0.0       1.00     0.536      0.536         3.536     3.536        0.500   0.500   0.71     0.2679
   0.0       1.25     0.528      0.528         3.778     3.778        0.625   0.625   0.35     0.2639
   0.2       1.00     0.537      0.534         3.470     3.402        0.435   0.570   0.81     0.2679
   0.2       1.25     0.529      0.527         3.711     3.644        0.559   0.694   0.45     0.2639
   0.4       1.00     0.539      0.533         3.404     3.269        0.374   0.645   0.91     0.2679
   0.4       1.25     0.530      0.526         3.645     3.511        0.498   0.767   0.55     0.2639
   0.6       1.00     0.541      0.533         3.338     3.135        0.318   0.723   1.01     0.2679
   0.6       1.25     0.531      0.525         3.579     3.377        0.439   0.843   0.65     0.2639
   0.8       1.00     0.543      0.532         3.273     3.002        0.266   0.807   1.11     0.2680
   0.8       1.25     0.532      0.525         3.513     3.244        0.385   0.923   0.75     0.2639
   1.0       1.00     0.546      0.531         3.208     2.869        0.219   0.896   1.21     0.2680
   1.0       1.25     0.534      0.524         3.447     3.111        0.334   1.007   0.85     0.2640
   1.2       1.00     0.549      0.531         3.143     2.737        0.176   0.988   1.31     0.2681
   1.2       1.25     0.536      0.524         3.382     2.978        0.286   1.094   0.95     0.2640
   1.4       1.00     0.553      0.531         3.079     2.605        0.138   1.086   1.41     0.2682
   1.4       1.25     0.538      0.524         3.316     2.845        0.242   1.185   1.04     0.2640
   1.6       1.00     0.557      0.531         3.015     2.473        0.105   1.189   1.51     0.2684
   1.6       1.25     0.540      0.523         3.251     2.712        0.202   1.280   1.14     0.2641
   1.8       1.00     0.563      0.531         2.952     2.342        0.076   1.297   1.60     0.2686
   1.8       1.25     0.543      0.523         3.186     2.580        0.166   1.379   1.24     0.2642
                       Table B1 Distribution of Profits & Surplus
                            ( t = 1, c K = 2 , ∆ c = c K − cW )


                                                46
∆c     t         pK       pW        rK       rW       ΠK        ΠW     CS     Vi
0.0   1.00      0.536    0.536     3.536    3.536     0.500    0.500   0.7   0.2679
0.0   1.25      0.683    0.683     3.683    3.683     0.500    0.500   0.5   0.3414
0.2   1.00      0.537    0.534     3.47     3.402     0.435    0.570   0.8   0.2679
0.2   1.25      0.685    0.681     3.617    3.549     0.434    0.570   0.6   0.3415
0.4   1.00      0.539    0.533     3.404    3.269     0.374    0.645   0.9   0.2679
0.4   1.25      0.688    0.68      3.552    3.416     0.373    0.645   0.7   0.3415
0.6   1.00      0.541    0.533     3.338    3.135     0.318    0.723   1.0   0.2679
0.6   1.25      0.69     0.678     3.486    3.282     0.317    0.725   0.8   0.3416
0.8   1.00      0.543    0.532     3.273    3.002     0.266    0.807   1.1   0.2680
0.8   1.25      0.694    0.677     3.422    3.149     0.265    0.810   0.9   0.3416
1.0   1.00      0.546    0.531     3.208    2.869     0.219    0.896   1.2   0.2680
1.0   1.25      0.698    0.677     3.357    3.017     0.217    0.898   1.0   0.3418
1.2   1.00      0.549    0.531     3.143    2.737     0.176    0.988   1.3   0.2681
1.2   1.25      0.702    0.676     3.294    2.885     0.175    0.993   1.1   0.3419
1.4   1.00      0.553    0.531     3.079    2.605     0.138    1.086   1.4   0.2682
1.4   1.25      0.708    0.676     3.231    2.753     0.137    1.092   1.2   0.3421
1.6   1.00      0.557    0.531     3.015    2.473     0.105    1.189   1.5   0.2684
1.6   1.25      0.715    0.676     3.168    2.622     0.103    1.195   1.3   0.3424
1.8   1.00      0.563    0.531     2.952    2.342     0.076    1.297   1.6   0.2686
1.8   1.25      0.723    0.677     3.107    2.492     0.074    1.304   1.4   0.3428
           Table B2 Effects from Changes in Brand Differentiation
                      ( tr = 1 , c K = 2 , ∆ c = c K − c W )




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