Vertical integration, exclusive dealing andproduct line differentiation in retailing

E. Avenel∗& S. Caprice†

June 25, 2003

Abstract

We analyze vertical integration and exclusive dealing in a modelof vertical differentiation in which retailers compete in product lines.Depending on quality and cost differentials, the equilibrium is charac-terized either by integration and partial product lines differentiationor by exclusive dealing and complete product lines differentiation. Themodel leads us to recommendations for antitrust policy makers: exclu-sive dealing should be banned per se, while a rule of reason approachof vertical integration should be adopted.Key-words : Vertical integration, exclusive dealing, product line

differentiation, antitrust policy.JEL Classification : L13, L22, L42.

1 Introduction

Retailing is characterized by the frequency of both exclusive dealing con-

tracts and vertical integration between the retailers and their suppliers. This

is true for a wide variety of products, from food and clothing1 to cable and

satellite TV. In a recent empirical analysis of the role of vertical integration

∗GREMAQ (UMR CNRS 5604, Université Toulouse I) & GAEL (UMR INRA, Univer-sité Grenoble 2). E-mail : eric.avenel@upmf-grenoble.fr

†INRA-ESR, Toulouse. E-mail : caprice@toulouse.inra.fr1Prominent examples are the British chain of department stores Marks & Spencer, that

relies heavily on exclusive dealing, and the Swiss chain of supermarkets Migros, that makesextensive use of vertical integration. See also Fearne (1998) and Hughes and Merton (1996)for a presentation of the vertical links between British food retailers and their suppliers.

1

in the US cable television industry, Chipty (2001) shows that vertically inte-

grated cable systems operators tend to exclude rival program services, which

reduces the welfare of consumers located in the areas where these operators

are in a monopoly position. Indeed, the frequency of vertical integration and

exclusive dealing in the retailing sector rises the two questions of (i) the in-

centives that manufacturers and retailers have to enter in such relations and

(ii) the impact of these relations on welfare.

There is an important theoretical literature on each of these topics, but

it doesn’t take into account a critical characteristic of this sector: retailers

offer several varieties of each product and face a strategic choice as regards

the range of varieties they offer and the differentiation of their product line

from the product lines of other retailers.

Absent differentiation between products, the qualitative effects of exclu-

sive dealing and vertical integration are identical: there is the same number

of firms competing on the final market. Assuming that there are two suppli-

ers at the upstream level2, neither exclusive dealing nor vertical integration

succeed in preventing entry or inducing exit at the downstream level : in

the case of a downstream duopoly, both downstream firms have access to the

intermediate good and, as a consequence, are present on the final market.

In the retail case, it means that both retailers offer the product. However,

despite this similarity of the qualitative effects, the quantitative effects of

exclusive dealing and vertical integration are different: both the price paid

by the downstream firm linked to one producer and the price paid by its

2There is an important literature on the use of exclusive dealing and vertical integrationas a barrier to entry at the upstream level that we don’t discuss here, since it is out of thescope of our analysis. See Bernheim and Whinston (1998) for a recent and comprehensivepresentation.

2

rival are different in the exclusive dealing case and in the vertical integration

case. In other words, the rising rivals’ costs effect has a different strength

in each case. It is in fact not clear what type of vertical relation leads to

the largest difference between the price paid by each of the two downstream

firms. Indeed, whereas exclusive dealing just means foreclosure, a vertically

integrated firm may either sell intermediate good on the market (as in Hart

and Tirole (1990)) or foreclose (as in Salinger (1988) and Ordover, Saloner

and Salop (1990)) or even strategically purchase this good (as in Gaudet and

Van Long (1996) and Avenel and Barlet (2000)). Furthermore, whereas the

internal transfer price within the integrated firm is just the marginal cost of

producing the intermediate good, it may for strategic reasons differ from the

marginal cost in an exclusive dealing relation, either toward higher prices

or toward lower prices (see Rey and Stiglitz (1988) and Caillaud and Rey

(1995)). In this literature, exclusive dealing and vertical integration differ in

their quantitative aspects, but each and every firm proposes the same prod-

uct as its competitors. For the reasons previously exposed, this is not best

suited to the analysis of the retailing sector.

Introducing competition in product lines in the analysis creates a qualita-

tive difference between the respective effects of exclusive dealing and vertical

integration on downstream competition: the number of firms offering each va-

riety of the product in equilibrium is different. Evidence from the European

satellite television industry (collected by the authors) illustrates this point.

Exclusivity trivially leads to complete product lines differentiation. When

BSkyB paid £1.1 billion for the exclusivity of the (UK) Premier League,

ITV Digital was left with lower level football matches. In contrast, vertical

integration doesn’t always imply foreclosure. For example, the two French

3

integrated groups supply each other with news channels: CanalSatellite pro-

poses the news channel LCI produced by TF1, a major shareholder of the

rival operator TPS, and TPS proposes the news channel Itelevision produced

by Canal +, the parent establishment of CanalSatellite.

Our objective in this paper is twofold. First, it is to contribute to the

strategic management literature by providing an analysis of both the product

line differentiation between firms linked to their suppliers through vertical

relations and the benefits and drawbacks of vertical integration as compared

to exclusive dealing in the retailing sector. Secondly, it is to provide guidance

for policy makers by asserting the effect of vertical integration and exclusive

dealing on welfare in the retailing sector. This question was raised recently

in the giant vertical merger between Vivendi and Universal reviewed by the

merger task force of the European Commission. We discuss this case, as well

as a French case about exclusive sports rights, in the conclusion.

We consider a model of vertically differentiated products, with two qual-

ities. The high quality is produced by a monopolist and the low quality by

a perfectly competitive industry. The pattern of vertical relations between

manufacturers and retailers is the result of a multi-stage game in which re-

tailers can propose exclusive dealing contracts or vertical integration to the

high quality manufacturer. We determine the vertical relationships and the

resulting product lines of retailers in equilibrium, and, contrary to Gilbert

and Matutes (1993) and De Fraja (1996), don’t obtain a “head-to-head” equi-

librium in which firms match exactly each other’s product line3. We then

3Gilbert and Matutes (1993) considers a variation of the basic model in which firms canoffer different product lines in equilibrium, but this is due to a Stackelberg assumption thatis not present in our paper. Champsaur and Rochet (1989) also find differentiated productlines in equilibrium, but in a model of Bertrand competition in which firms offer intervalsof qualities, whereas we consider a finite number of varieties and Cournot competition.

4

determine the optimal policy that a competition authority should implement

towards vertical integration and exclusive dealing. Whereas, as could be ex-

pected, exclusive dealing should be banned per se, the treatment of vertical

integration should be based on a rule of reason approach, not because of

efficiency gains, that we assume away, but because of the impact of vertical

integration on product lines and competition.

The plan of the paper is as follows: we present the model in section 2 ;

in section 3, we determine the equilibrium on the wholesale and the retail

market when exclusive dealing and vertical integration are ruled out and ex-

amine retailers’ incentives to differentiate their product line from the rival’s

one through exclusive dealing agreements ; in section 4, we determine the

equilibrium of the complete game in which retailers can propose exclusive

dealing and vertical integration to the high quality manufacturer ; section

5 presents the welfare analysis and policy recommendations. Section 6 dis-

cusses some evidence from the satellite TV industry and concludes.

2 The model

We consider an industry with two identical downstream retailers, D1 and

D2, who buy from an upstream producer, A, and a competitive fringe, B.

A produces a high quality product at marginal cost c while the competitive

fringe produces a low quality product at marginal cost 0. The retailing costs

are 0. Let qH = 1 be the high quality and qL < 1 be the low quality. There

is a unit mass of consumers, each of whom is interested in buying at most

one unit. The utility of consumers is of the Mussa-Rosen (1978) type: each

consumer obtains a utility θqi − pi if he buys one unit of good i, where θ isdistributed uniformly on the interval [0, 1], and a utility 0 if he does not buy

5

at all.

Competition on the final market is of the Cournot type. We denote re-

spectively by yiH and yiL the quantity of the high quality and the low quality

product offered by Di. Given consumers’ preferences, inverse demand func-

tions for the two qualities are as follows:

½pL (y

1H , y

2H , y

1L, y

2L) = (1− y1H − y2H − y1L − y2L) qL

pH (y1H , y

2H , y

1L, y

2L) = 1− y1H − y2H − (y1L + y2L) qL (1)

The strategic interactions between upstream and downstream firms are

represented by a four stage game:

Stage 1: Retailers simultaneously propose exclusive dealing contracts or

takeover offers to producer A. Producer A either accepts one of the offers or

rejects both.

Stage 2: If A rejects both offers, A proposes to D1 and D2 an identical

two-part tariff, T (yiH) = wyiH + F, i = 1, 2. If A has vertically integrated

with one of the retailers, then it offers a two-part tariff to the other retailer.

Stage 3: Retailers simultaneously accept or refuse the contract offered by

the manufacturer (if any).

Stage 4: Retailers (integrated or not) simultaneously choose (yiH , yiL)i=1,2.

Note that the retailers can buy the low quality of the good from the

competitive fringe at a price equal to 0. Information in this game is both

complete and perfect. In particular, contracts are public and cannot be

secretly renegotiated. We solve the game by backward induction. In the

next section, we show that retailers make acceptable offers in stage 1. The

solution of the game is presented in section 4.

6

3 Retailers’ incentives for product line differ-entiation

To prove that retailers make acceptable offers in stage 1, we show that re-

tailers would be better off offering an acceptable ED contract in stage 1 than

making no acceptable offer. Doing this requires us to first characterize the

equilibrium of the ”arm’s length relationships” subgame beginning in stage

2 when A rejects both offers in stage 1.

3.1 Arm’s length relationships

Assuming hereafter that c is not too large4, A supplies at least one retailer

in equilibrium. Wether it supplies one or both depends on the franchise fee.

In both cases, A saturates retailers’ participation constraints. Denote by

πi (wi, wj) the gross profit5 of retailer i when he pays wi and retailer j pays

wj for each unit of the high quality good (wi = +∞ means that Di is not

supplied by A). With F = πi (w,w)− πi (+∞, w), A supplies both retailers.Rising the franchise fee to πi (w,+∞)− πi (+∞,+∞), it supplies only one.In appendix A, we solve the downstream competition game (stage 4) and

establish the expressions of πi, yiH and yiL as a function of wholesale prices. We

provide the expressions of outputs in the following lemma.

Lemma 1 Given (wi, wj), retailers offer the following quantities of each

quality on the final market:

(i) If (wi, wj) = (w,+∞) and 0 ≤ w ≤ 1−qL3,©yiH , y

iL, y

jH , y

jL

ª=n2−qL−2w4−qL , 0, 0, 1+w

4−qL

o.

(ii) If (wi, wj) = (w,+∞) and w ≥ 1−qL3,©yiH , y

iL, y

jH , y

jL

ª=n(1−qL)−w2(1−qL) ,

3w−(1−qL)6(1−qL) , 0,

13

o.

(iii) If (wi, wj) = (w,w),©yiH , y

iL, y

jH , y

jL

ª=n(1−qL)−w3(1−qL) ,

w3(1−qL) ,

(1−qL)−w3(1−qL) ,

w3(1−qL)

o.

4More precisely, the high quality good is sold in equilibrium iff c < 1− qL.5πi includes the profits realized on both qualities.

7

In (i), the differentiation of product lines is complete, while, in (ii), it

is partial, since both firms sell the low quality. The difference between

(i) and (ii) is that, in (i), the wholesale price of the high quality is suffi-

ciently low for the supplied retailer to decide not to sell the low quality.

Corollary If (wi, wj) = (w,+∞) and w ≥ 1−qL3, yiH + y

iL = y

jL =

13and

pL =13qL.

In (ii), the impact of w is limited to interbrand competition within retailer

i.

We denote by wD and wDD the wholesale prices offered in equilibrium in

stage 2 by A respectively when there is one and two retailers. Appendix B

provides the expressions of wD and wDD.

Lemma 2 wD ≤ c, with a strict inequality for c < ca.Proof. See appendix B and note that 2c(4−qL)−(2−qL)qL

4(2−qL) < c since c < 1−qL.By charging a low wD, the manufacturer induces retailer i to be more

aggressive, which is profitable, since the outputs of retailer i (high quality)

and retailer j (low quality) are strategic substitutes. See Caillaud and Rey

(1995) for a discussion of these precommitment effects.

The corresponding profits are

Φ (wD,+∞) =Maxw(w − c) yiH (w,+∞) + πi (w,+∞)− πi (+∞,+∞)

Φ (wDD, wDD) =Maxw

2 (w − c) yiH (w,w) + 2πi (w,w)− 2πi (+∞, w)(2)

Introducing the joint profits of manufacturerA and retailer i for wholesale

prices (wi, wj), Π (wi, wj), we rewrite (2) as

Φ (wD,+∞) = Π (wD,+∞)− πi (+∞,+∞)Φ (wDD, wDD) = Π (wDD, wDD)− πi (+∞, wDD) (3)

8

WhenA supplies only one retailer, the rent left to this retailer is πi (+∞,+∞).Since this rent is constant, when A maximizes Φ, it also maximizes Π. When

A supplies both retailers, the rent left to retailer i is an increasing function of

the wholesale price on£0; 1−qL

3

¤and a constant on

£1−qL3; 1− qL

£(see corol-

lary of lemma 1). Let us define w̃ = ArgMaxwΠ (w,w). If wDD ≥ 1−qL

3,

Π (wDD, wDD) = Π (w̃, w̃) and πi (+∞, wDD) = πi (+∞,+∞). If wDD <

1−qL3, Π (wDD, wDD) < Π (w̃, w̃) and πi (+∞, wDD) < πi (+∞,+∞): switch-

ing from one to two retailers allows A to reduce the rent left to each supplied

retailer. This is what we call the ”rent effect”. Of course, the joint profits

are no longer maximized. However, the relevant comparison is not between

Π (wDD, wDD) andΠ (w̃, w̃), but betweenΠ (wDD, wDD) andΠ (wD,+∞). Assoon as Π (wDD, wDD) < Π (wD,+∞) and πi (+∞, wDD) < πi (+∞,+∞), itis not clear what retailing structure is optimal for the manufacturer. Indeed,

it faces a trade-off between sharing a large cake with one retailer, but leaving

it a high rent and sharing a smaller cake with two retailers, but leaving each

of them a smaller rent. There is a ”joint profits effect” opposite to the rent

effect. The manufacturer charges a franchise fee selecting only one retailer

iff

Φ (wD,+∞) > Φ (wDD, wDD) (4)

or, equivalently,

Π (wD,+∞)− Π (wDD, wDD) > πi (+∞,+∞)− πi (+∞, wDD) (5)

9

Proposition 1 The equilibrium in the retailing market is as follows:

(i) If c ≥ ca, the manufacturer is indifferent between supplying one retaileror two.

(ii) If cb ≤ c < ca, the manufacturer supplies only one retailer. This

retailer provides only the high quality while the other retailer provides only

the low quality.

(iii) If c < cb, the manufacturer supplies both retailers. The two retailers

either provide only the high quality or both high and low qualities.

Proof: We provide the expression of cb and a graphical representation of

ca and cb, as well as a detailed characterization of the equilibrium in appendix

C.

When c ≥ ca, choosing a price wDD below 1−qL3

is prohibitively costly

for the manufacturer, so that the rent effect vanishes in equilibrium and the

profits of the retailers are the same when one retailer is supplied and when

two retailers are supplied. As a consequence and because we assume public

contracts, A is indifferent to the number of retailers that it supplies: in both

cases, it maximizes the profits of the industry and receives the same part

of this profit. For intermediate values of c, it is still very costly for the

manufacturer in terms of joint-profits to reduce the rent by manipulating the

wholesale price, while in the “one retailer” case, it becomes easier to increase

the joint-profits by lowering the wholesale price. This optimal price ensures

that the retailer offering the high quality no longer offers the low quality.

Note that, although the low quality good is free, it is not profitable for the

vertical chain that the retailer offers this quality, because of the interbrand

competition that this would create. For low values of c, the rent effect is

strong enough for the manufacturer to decide to serve the two retailers. There

10

is then no differentiation in product lines between retailers.

3.2 Exclusive dealing vs. arm’s length relationships

Since exclusive dealing is not an issue for c ≥ ca, we assume hereafter thatc < ca.

If c < cb, the manufacturer supplies both retailers. It doesn’t maximize

its joint-profits with any of these two retailers since it distorts the wholesale

price in order to reduce the rent. There is thus a possibility for one of the

retailers to increase its profit by offering the manufacturer an acceptable

exclusivity contract in the first stage.

If c ≥ cb, each retailer knows that with probability 1/2, the exclusive

retailer of the high quality good will be the other retailer and thus has an

incentive to make sure in stage 1 that it will be the exclusive retailer by

signing a contract with the manufacturer.

Proposition 2 The manufacturer receives at least one acceptable offer in

the first stage.

Proof: See above.

Vertical restraints are always present in equilibrium and, as will be seen

in section 4, they play a key role in the determination of product lines differ-

entiation. To illustrate this point, we consider here a variation of the game in

which retailers can propose exclusivity contracts, but no takeover offer, to the

manufacturer in the first stage.

Corollary If exclusive contracts have been offered by the retailers at the

first stage, there is a complete differentiation.

The exclusivity contract accepted by the manufacturer stipulates the

wholesale price wD and the franchise fee Π (wD,+∞) − πi (+∞, wD) such

11

that the retailer is indifferent between signing the contract and renouncing

to the high quality good. Because of the competition between retailers in the

first stage, the opportunity to propose exclusive dealing in this stage leads

to lower profits for the retailers.

4 Equilibrium structure and product line dif-ferentiation

The retailer that obtains an exclusive supply of high quality good has monopoly

power on this quality, but the other retailer has free access to the low quality

good. In this section, we examine whether a retailer can further strengthen

its position in the retail market by acquiring the manufacturer of the high

quality good. In other words, we analyze the difference between exclusive

dealing arrangements and vertical integration, thus solving the first stage of

the game. The first step is to analyze the strategy of the vertically integrated

retailer on the intermediate market.

4.1 The integrated retailer’s strategy

The integrated retailer receives the profit of the whole industry less the reser-

vation profit of the nonintegrated retailer. This reservation profit is indepen-

dent from the wholesale price charged by the integrated retailer. As a conse-

quence, the integrated retailer maximizes the profits of the whole industry.

To do that, it can either supply or foreclose the independent retailer.

Proposition 3 The integrated retailer forecloses its non-integrated rival iff

c ≥ 1−qL3.

Proof: See appendix D.

12

For c ≥ 1−qL3, the integrated retailer offers both qualities of the good.

Supplying Dj induces Di to offer less high quality good (because Dj offers

more of this quality) and more low quality good (becauseDj offers less of this

quality). Because the slope of reaction functions is smaller than 1, intrabrand

competition increases for both qualities and interbrand competition increases

within both retailers. This reduces the industry’s profits.

For c < 1−qL3, the integrated retailer doesn’t offer the low quality good,

so that there is no intrabrand competition for the low quality and no in-

terbrand competition within retailer i. Vertical foreclosure is not optimal

in this case. The nonintegrated retailer sells the high quality good, which

reinforces intrabrand competition on this quality, but he reduces its offer of

low quality good, which reduces interbrand competition. The vertically inte-

grated retailer gives the non-integrated retailer an (restricted) access to the

high quality good (at wholesale price wI).

4.2 The trade-off between vertical integration and ex-clusive dealing

From the previous section, we know that the manufacturer is in equilibrium

necessarily bound with one retailer by an exclusive dealing contract rather

than an arm’s length contract. However, if integration leads to higher joint

profits, it emerges in equilibrium instead of exclusive dealing. The condition

for the emergence of vertical integration in equilibrium is thus:

Π (c, wI) > Π (wD,+∞) (6)

If A and retailer i merge, Di has access to the high quality good at an

internal transfer price equal to the marginal cost of production c, whereas

13

in the exclusivity contract, it pays wD ≤ c (lemma 2). In this sense, there

is a cost of integration: the retailer can no longer commit to an aggressive

behavior. There is also a benefit of integration, since the integrated firm

supplies Dj at a price wI that is not necessarily prohibitive.

Proposition 4 For any value of qL, there is a value cc such that 0 ≤ cc ≤ caand

(i) If c < cc, vertical integration emerges in equilibrium. There is no fore-

closure and product lines are partially differentiated (the integrated retailer

doesn’t offer the low quality of the good).

(ii) If ca ≥ c ≥ cc, there is exclusive dealing in equilibrium. Product

lines differentiation is complete, each retailer offering a different quality of

the good.

Proof: See appendix E.

Figure 1 summarizes the equilibrium structure of the industry.

0.2 0.4 0.6 0.8 1

0.05

0.1

0.15

0.2

0.25

0.3

c

Lq

cc

ac

No impact

Exclusive Dealing

Vertical Integration

3/)1( Lq−

Figure 1

14

If the high quality good has a sufficiently low cost of production, vertical

integration is profitable and is the equilibrium structure of the industry, be-

cause it induces an equilibrium of partial product lines differentiation. With

exclusive dealing, competition between high and low quality is tough. Di ben-

efits from the supply of its rival with high quality good because it reduces

this interbrand competition without creating too much intrabrand competi-

tion on high quality. In other words, vertical integration is an equilibrium

because it leads to an increase in the profits of the industry that is not fully

captured by the non-integrated retailer.

5 Welfare analysis and policy implications

In this section, we move from positive to normative analysis. The optimal

policy for a social welfare maximizing antitrust authority (AA) is to forbid

any subset of strategies such that the surplus realized in equilibrium without

these strategies in the set of strategies is higher than the surplus realized

in equilibrium with these strategies in the set of strategies. Proposition 5

describes the optimal policy for the framework considered in this model.

Proposition 5 Exclusive dealing should be considered as per se illegal, while

the following rule should be used in the treatment of vertical integration : the

AA should block a submitted vertical merger project if, previous to the merger,

the two retailers sell the high quality good. If, previous to the merger, only

one retailer proposes the high quality good, the merger should be blocked if the

degree of vertical differentiation is low and it should be allowed if the degree

of vertical differentiation is high.

Proof : See appendix F.

15

Figure 2 summarizes the optimal policy.

0.2 0.4 0.6 0.8 1

0.05

0.1

0.15

0.2

0.25

0.3

c

Lqbc

ac

M ergerallow ed

3/)1( Lq−

Figure 2

It is optimal to ban exclusive dealing because it has either no effect or

a negative effect on the social surplus due to the elimination of intrabrand

competition on the high quality.

Vertical integration reduces intrabrand competition on the high quality,

and thus social welfare, if both retailers are supplied under arm’s length

relations. On the contrary, it can be optimal to allow vertical integration

if under arm’s length relations only one retailer is supplied with the high

quality. Indeed, the integrated firmwill supply both retailers. This is however

not a sufficient condition, because, whereas under arm’s length relations the

supplied retailer is supplied at a wholesale price lower than the marginal cost,

vertical integration implies an internal transfer price equal to marginal cost,

which reduces the welfare. There is thus a trade-off that is solved in favor of

vertical integration when the quality differential is strong enough.

16

We conclude the analysis of welfare by a discussion of the robustness of

proposition 5 if we relax the assumption that contracts are nondiscriminatory.

Assume that A receives no acceptable offer in stage 1. If it can discriminate

between retailers in stage 2, it offers contracts (w1, F1) and (w2, F2), with

(w1, w2) in general different from (wD,+∞) and (wDD, wDD). This does notimply that we will not observe ED or VI at the equilibrium of the whole game.

Indeed, it may be possible for one of the retailers to make an acceptable

exclusivity offer to the manufacturer in stage 1. To keep things simple,

assume that retailer 1 offers an exclusivity contract with w = wD and retailer

2 makes no offer. If A refuses this offer, retailer 1 will make a profit equal

to π1 (+∞, w2). It is thus profitable for retailer 1 to make in stage 1 anacceptable offer that leaves him a profit at least equal to π1 (+∞, w2). Notethat, if A proposes exclusivity to retailer 1 in stage 2, it must leave him a

profit at least equal to π1 (+∞,+∞) > π1 (+∞, w2). Retailer 1 makes instage 1 an offer that he would refuse in stage 2. This is why exclusivity

may appear in equilibrium. A prefers ((w1, F1) , (w2, F2)) to an exclusivity

contract with w = wD leaving retailer 1 a profit equal to π1 (+∞,+∞), but itmay accept in stage 1 an exclusivity contract with w = wD that leaves retailer

1 a lower profit. The same argument holds for takeover offers. It turns out

that, in equilibrium, A accepts in stage 1 either an exclusive dealing contract

or a takeover offer and that proposition 4 holds. This does not imply that

proposition 5 also holds, since the arm’s length equilibrium is different. What

still holds is that exclusive dealing should be banned per se and that a rule

of reason approach should be adopted toward vertical integration because it

is socially optimal for a set of values of c and qL.

17

6 Conclusion

In this paper, we analyze product line differentiation between retailers com-

peting on a vertically differentiated market. The specificity of our model is

that we include in the analysis the vertical relations between retailers and

manufacturers. As far as we know, this was not done in any previous work.

This feature proves to be crucial for the determination of product line dif-

ferentiation. In particular, we find an equilibrium in which product lines are

partially differentiated, which is a new result. Our analysis suggests that

exclusive dealing should be banned per se, while a rule of reason approach

should be adopted toward vertical integration. This is related to the fact

that integration doesn’t necessarily imply foreclosure.

Our model sheds some light on competition policy issues in the satellite

TV sector. Consumers have the choice between bundles of channels, but de-

mand is driven mainly by sport and movies. Assuming that each consumer

values either sport or films, our model describes the competition between

satellite TV operators for the demand of either of these two groups of con-

sumers. Exclusivity of sport events, in particular football matches, is an im-

portant and intensely advertised element of competition between European

broadcasters. In October 2002, the French professional football league invited

bids for the rights on Ligue 1 football matches for 2004-2007. CanalSatellite’s

offer included an exclusivity premium of 290 millions euros per year. This

offer was accepted in November. Immediately, TPS, the rival operator, re-

ferred matter to the Competition Council which, in January, broke the deal6.

6See ”Décision n◦ 03-MC-01 du 23 janvier 2003 relative à la saisine et à lademande de mesures conservatoires présentées par la société TPS”, available onhttp://www.finances.gouv.fr/conseilconcurrence/.

18

Our results support the Council’s decision, since they suggest that if Canal-

Satellite had kept exclusive rights on Ligue 1 football matches, the surplus

would have been reduced. Between November 2002 and January 2003, the

estimated number of clients lost by TPS because of the exclusivity granted

to CanalSatellite is between 15000 and 20000. Exclusivity is also central in

the relations between major film studios and broadcasters. However, in 2000,

the European Commission examined a merger between Vivendi, Canal+ and

Seagram that had important implications in the sector7. Universal studios

were part of Seagram and Canal+ controlled CanalSatellite. In its apprecia-

tion of the case, the Commission considered that, after the merger, Universal

would not be willing to supply rival broadcasters. This view is not supported

by our analysis. Indeed, we show that vertical integration emerges only if

the integrated firm supplies its downstream rival (although at a potentially

high price). We would thus mitigate the Commission’s view on this point8.

Our model is quite general and can be useful to analyze the retailing of a

wide variety of products. Among the issues left open by the present analysis,

the most promising avenue for future work is to endogenize quality levels by

introducing in the model negotiations between manufacturers and retailers

on the characteristics of the products. Indeed, there is a growing evidence

that retailers, especially in the food sector, are more and more involved in

the definition of products (quality, packaging, ...).

7Case n◦ COMP/M.2050 - Vivendi / Canal+ / Seagram, available athttp://europa.eu.int/comm/competition/mergers/cases/decisions/m2050_en.pdf

8But we would not reject it, since it is also based on another argument, namely that”Universal’s acquisition by Vivendi is likely to have a significant impact on the abilityof Canal+ both to renew its existing contracts with the majors and to enter into newcontracts with those majors who have not signed with it yet”. This second point is out ofthe scope of our paper in which the high quality producer is a monopolist.

19

7 Appendices

7.1 Appendix A : Equilibrium values of outputs andprofits (stage 4)

Inverse demand functions are given by:

½pL (y

1H , y

2H , y

1L, y

2L) = (1− y1H − y2H − y1L − y2L) qL

pH (y1H , y

2H , y

1L, y

2L) = 1− y1H − y2H − (y1L + y2L) qL

We determine the equilibrium strategies of retailers in stage 4. We con-

sider a general case in which discrimination can happen in order to get results

that apply when one of the retailers is vertically integrated. Without loss of

generality, we assume that wi ≤ wj. Franchise fees are not relevant at thisstage of the game.

The equilibrium strategic profile©yiH (wi, wj) , y

iL (wi, wj) , y

jH (wj , wi) , y

jL (wj, wi)

ªsatisfies the conditions defining a Nash equilibrium:

(yiH , yiL) = ArgMax

xH ,xL

£pL¡xH , xL, y

jH , y

jL

¢¤xL +

£pH¡xH , xL, y

jH , y

jL

¢− wi¤xH¡yjH , y

jL

¢= ArgMax

xH ,xL[pL (y

iH , y

iL, xH , xL)]xL + [pH (y

iH , y

iL, xH , xL)− wj ] xH

The interior solution to this program (head-to-head competition on both

qualities) is©yiH (wi, wj) , y

iL (wi, wj) , y

jH (wj, wi) , y

jL (wj , wi)

ª=n(1−qL)−2wi+wj

3(1−qL) ,2wi−wj3(1−qL) ,

(1−qL)−2wj+wi3(1−qL) ,

2wj−wi3(1−qL)

oThe program admits an interior solution iff

¡wj ≤ 2wi and wi ≤ 1−qL

3

¢or¡

wj ≤ wi+1−qL2

and 1− qL ≥ wi ≥ 1−qL3

¢. We characterize below the corner

solutions.

• Partial product line differentiation (on low quality)

20

©yiH (wi, wj) , y

iL (wi, wj) , y

jH (wj, wi) , y

jL (wj , wi)

ª=n1+wj−2wi

3, 0,

2(1−qL)−wj(4−qL)+2wi(1−qL)6(1−qL) ,

wj2(1−qL)

oCondition of emergence: 2wi ≤ wj ≤ 2(1−qL)(1+wi)

4−qL . This condition re-

quires wi ≤ 1−qL3.

• Complete product line differentiation©yiH (wi, wj) , y

iL (wi, wj) , y

jH (wj, wi) , y

jL (wj , wi)

ª=n2−qL−2wi4−qL , 0, 0, 1+wi

4−qL

oCondition of emergence: wj ≥ 2(1−qL)(1+wi)

4−qL and 0 ≤ wi ≤ 1−qL3

• Partial product line differentiation (on high quality)©yiH (wi, wj) , y

iL (wi, wj) , y

jH (wj, wi) , y

jL (wj , wi)

ª=n(1−qL)−wi2(1−qL) ,

3wi−(1−qL)6(1−qL) , 0,

13

oCondition of emergence: wj ≥ wi+1−qL

2and 1− qL ≥ wi ≥ 1−qL

3

• Head-to-head competition on low quality©yiH (wi, wj) , y

iL (wi, wj) , y

jH (wj, wi) , y

jL (wj , wi)

ª=©0, 1

3, 0, 1

3

ªCondition of emergence: wi ≥ 1− qL.We now come back to the ”no discrimination” framework and consider

the two possible cases.

First case: (wi, wj) = (w,+∞)There is either product line differentiation on high quality or complete

product line differentiation or head to head competition on low quality, de-

pending on w.

21

If 0 ≤ w ≤ 1−qL3,©yiH , y

iL, y

jH , y

jL

ª=n2−qL−2w4−qL , 0, 0, 1+w

4−qL

oProfits (gross of franchise fees) are given by:

(πi (w,+∞) , πj (+∞, w)) =µ³

2−qL−2w4−qL

´2,³1+w4−qL

´2qL

¶If 1− qL ≥ w ≥ 1−qL

3,©yiH , y

iL, y

jH , y

jL

ª=n(1−qL)−w2(1−qL) ,

3w−(1−qL)6(1−qL) , 0,

13

oProfits (gross of franchise fees) are given by:

(πi (w,+∞) , πj (+∞, w)) =³5q2L−qL(14−18w)+9(1−w)2

36(1−qL) , qL9

´If w ≥ 1− qL,

©yiH , y

iL, y

jH , y

jL

ª=©0, 1

3, 0, 1

3

ªπi (w,+∞) = πj (+∞, w) = qL

9.

Second case: (wi, wj) = (w,w)

There is head-to-head competition, either on both qualities or on the low

quality, depending on w.

If 0 ≤ w ≤ 1− qL,©yiH , y

iL, y

jH , y

jL

ª=n(1−qL)−w3(1−qL) ,

w3(1−qL) ,

(1−qL)−w3(1−qL) ,

w3(1−qL)

oProfits (gross of franchise fees) are given by:

πi (w,w) = πj (w,w) = (1−w)2−qL(1−2w)9(1−qL) .

If w ≥ 1− qL,©yiH , y

iL, y

jH , y

jL

ª=©0, 1

3, 0, 1

3

ªπi (w,w) = πj (w,w) = qL

9.

22

7.2 Appendix B : Equilibrium values of wholesale prices

7.2.1 The “one retailer” case

wD is the solution of firm A’s profit maximization program:

wD = ArgMaxwΦ (w,+∞)

with Φ (w,+∞) = (w − c) yiH (w,+∞) + πi (w,+∞)− πi (+∞,+∞).The expression of Φ (w,+∞) depends on w as follows (see appendix A) :For w ∈ £0; 1−qL

3

¤, Φ (w,+∞) = (w − c)

³2−qL−2w4−qL

´+³2−qL−2w4−qL

´2− qL

9.

Forw ∈ £1−qL3; 1− qL

¤, Φ (w,+∞) = (w − c)

³1−qL−w2(1−qL)

´+5q2L−qL(14−18w)+9(1−w)2

36(1−qL) −qL9.

Φ is strictly concave on each interval and continuous at w = 1−qL3. We

solve the program on each interval and compare the solutions to determine

wD.

The solution to the producer’s program in the one retailer case is:

wD =

0 for (qL ≤ eq and c ≤ c0 (qL)) or (qL ≥ eq and c ≤ c2 (qL))

2c(4−qL)−(2−qL)qL4(2−qL) for (qL ≤ eq and c0 (qL) ≤ c ≤ c1 (qL))

c for (qL ≤ eq and c > c1 (qL)) or (qL ≥ eq and c > c2 (qL))with eq ' 0.721, c0 (qL) =

(2−qL)qL2(4−qL) , c1 (qL) =

√2−3qL+q2L3√2

and c2 (qL) =(1−qL)√qL(3√qL+2

√2+qL)

3(4−qL) (see figure below). Hereafter, we define ca by

ca (qL)=½c1 (qL) for qL ≤ eqc2 (qL) for qL ≥ eq .

23

0.2 0.4 0.6 0.8 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Lq

c

( )Lqc 0( )Lqc 2

( )Lqc1

q~

Lq−1

3/)1( Lq−

Equilibrium wholesale price with one retailer

7.2.2 The “two retailers” case

wDD is defined by

wDD = ArgMaxwΦ (w,w)

with

Φ (w,w) = [(w − c) yiH (w,w) + πi (w,w)− πi (+∞, w)]+£(w − c) yjH (w,w) + πj (w,w)− πj (+∞, w)

¤Because yiH (w,w) = y

jH (w,w), π

i (w,w) = πj (w,w) and πi (+∞, w) =πj (+∞, w), this simplifies to :

Φ (w,w) = 2£(w − c) yiH (w,w) + πi (w,w)− πi (+∞, w)

¤24

Since πi (+∞, w) has a different expression on £0; 1−qL3

¤and on

£1−qL3; 1− qL

¤,

this is also the case for Φ (w,w).

Φ (w,w) =

2

·(w − c) (1−qL)−w

3(1−qL) +(1−w)2−qL(1−2w)

9(1−qL) −³1+w4−qL

´2qL

¸for w ∈ £0; 1−qL

3

¤2h(w − c) (1−qL)−w

3(1−qL) +(1−w)2−qL(1−2w)

9(1−qL) − qL9

ifor w ∈ £1−qL

3; 1− qL

¤

Φ (w,w) is strictly concave on each interval, continuous at w = 1−qL3. We

solve the program on each interval and compare the solutions to determine

wDD.

Finally, the solution to the producer’s program with two retailers is :

wDD =

0 for (0 ≤ c ≤ c3 (qL) and qL ∈ [bq; 1])

−16−3c(−4+qL)2+42qL−27q2L+q3L2(−32+7qL+7q2L)

for(0 ≤ c ≤ c4 (qL) and qL ∈ [0; bq])

or (c3 (qL) ≤ c ≤ c4 (qL) and qL ∈ [bq; 1])14(1 + 3c− qL) for c4 (qL) < c ≤ 1− qL

with bq = 13 − 3√17 ' 0.63, c3 (qL) =(−16+26qL−q2L)(1−qL)

3(4−qL)2 and c4 (qL) =

19

³−15 + 3qL + 2

√2p32− 7qL − 7q2L

´(see figure below).

25

0.2 0.4 0.6 0.8 1

0.1

0.2

0.3

0.4

0.5

c

Lq−1

( )Lqc3 Lq

q̂

( )Lqc 4

3/)1( Lq−

Equilibrium wholesale price with two retailers

7.3 Appendix C: Proof of proposition 1

Defining cb by

cb (qL)=

−90+153qL−72q2L+9q3L+

√3√2560−9520qL+13112q2L−7290q3L+277q4L+1162q5L−301q6L

−42−60qL+48q2Lon

£0; q¤

(1−qL)√qL³√

qL(−138+51qL+6q2L)+√3√−640+3980qL−2140q2L−557q3L+322q4L+7q5L

´3(−4+qL)3 on

£q; 1¤ ,

with q ' 0.389, the comparison ofΠ (wD,+∞)−Π (wDD, wDD)with πi (+∞,+∞)−πi (+∞, wDD) shows that the number of supplied retailers, the wholesaleprice and the resulting product lines are as follows :

For (bq ≤ qL and 0 ≤ c ≤ c3 (qL)), the manufacturer supplies both retailersand charges wDD = 0. The retailers distribute only the high quality.

For (qL ≤ bq and 0 ≤ c ≤ cb (qL)) or (bq ≤ qL and c3 (qL) ≤ c ≤ cb (qL)), themanufacturer supplies both retailers and charges wDD =

−16−3c(−4+qL)2+42qL−27q2L+q3L2(−32+7qL+7q2L)

.

The retailers distribute both qualities.

26

For (cb (qL) < c ≤ ca (qL)), the manufacturer supplies only one retailerand charges wD = 0 for c ≤ c0 (qL), wD = 2c(4−qL)−(2−qL)qL

4(2−qL) for c ≥ c0 (qL). Inboth cases, product line differentiation is complete, i.e. one retailer proposes

the high quality and the other proposes the low quality, since both 0 and2c(4−qL)−(2−qL)qL

4(2−qL) are less than 1−qL3for the values of c and qL considered.

For ca (qL) < c, the manufacturer is indifferent between supplying one

retailer, with wD = c, or two retailers, with wDD =1+3c−qL

4.

0.2 0.4 0.6 0.8 1

0.1

0.2

0.3

0.4Lq−1

Lq

( )La qc

c

( )Lb qc

( )Lqc 3

( )Lqc 0

q q̂ q~

3/)1( Lq−

Arm’s length relationships-subgame equilibrium

7.4 Appendix D: Proof of proposition 3

Appendix A provides the expressions that we need in this appendix. Simply

replace (wi, wj) by (c, wj).

The profit maximization program for the integrated firm is as follows:

Maxwj ,Fj

πi (c, wj) + (wj − c) yjH (wj, c) + Fjst πj (wj, c)− Fj ≥ πj (+∞, c)

27

The constraint is binding and the program simplifies to:

Maxwj

πi (c, wj) + (wj − c) yjH (wj , c) + πj (wj , c)− πj (+∞, c)

The solution to this program is:

wI =

2(1−qL)+c(2+7qL)

4+5qLfor c ≤ 2(1−qL)

3(2+qL)

2c for 2(1−qL)3(2+qL)

≤ c ≤ 1−qL3

+∞ for c ≥ 1−qL3

with F I = πj¡wI , c

¢− πj (+∞, c) for c < 1−qL3and F I = 0 otherwise.

7.5 Appendix E: Proof of proposition 4

We show that there exists a function cc defined by:

cc (qL) =

2q3L−62q2L+172qL−112+(−q2L+2qL+8)

√16−4qL−22q2L+10q3L

2(−120+34qL+4q2L+q3L)on [0; 0.436]

2(−1+qL)qL³−8−22qL+3q2L+

√−192+288qL+500q2L−196q3L+5q4L

´64+16qL−16q2L+16q3L+q4L

on [0.436; 1]

such that, for c < cc (qL), Π (c, wI) − Π (wD,+∞) > 0 and vertical inte-gration is an equilibrium, while for c > cc (qL), Π (c, wI) − Π (wD,+∞) < 0and exclusivity is an equilibrium.

7.6 Appendix F : Proof of proposition 5

The first step of the proof is to compare the value of the social surplus for

each of the possible cases. Let us denote AL for arm’s length relationships,

ED for exclusive dealing and V I for vertical integration. We show that :

¨AL º ED for any (c, qL)

28

¨The function c5 defined by

c5 (qL) =

(−2+qL)³16−17qL+q2L−

√64−64qL+208q2L−276q3L+53q4L+15q5L

´2(−24−9qL+5q2L+q3L)

for qL ≤ 0.3562(−1+qL)

³8+18qL+q

2L−√64+96qL+84q

2L+52q

3L+q

4L

´(4−qL)(−12−16qL+q2L)

for 0.915 ≥ qL ≥ 0.35624−30qL+6q2L−(4−5qL+q2L)

√16−qL

3(4−qL)2 for qL ≥ 0.915is such that

ED Â V I for c ∈ [c5 (qL) ; ca (qL)]while ED ≺ V I otherwise¨AL ≺ V I for (c ∈ [cb (qL) ; c5 (qL)] and qL < 0.344)while AL Â V I otherwise.Thus, except for values of c and qL such that (c ∈ [cb (qL) ; c5 (qL)] and qL < 0.344),

AL dominates both ED and V I from the social point of view. Forbidding

both ED and V I is thus an optimal policy.

For (c ∈ [cb (qL) ; c5 (qL)] and qL < 0.344), AL and ED are equivalent andAL ≺ V I. Furthermore, for these values of c and qL, V I emerges in equi-

librium in the game in which it is allowed and ED is forbidden. It is thus

optimal to forbid ED and allow V I. Putting together these two results leads

to the proposition.

29

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31