Suppliers’merger and consumers’welfare Eric Avenel Université de Rennes I et CREM (UMR CNRS 6211) June 1, 2010

Abstract This article explores the consequences of a suppliers’ merger on consumers’ welfare when the product is sold to consumers by independent retailers competing à la Cournot. The literature shows that under the standard assumptions of private contracting and passive beliefs, there is no impact at all. I show that this unintuitive result strongly depends on the implicit assumption that suppliers have in…nite capacities of production. Indeed, assuming that suppliers face a capacity constraint and that retailers hold out-of-equilibrium beliefs compatible with this constraint, I show that the merger raises the price on the …nal market and reduces consumers’welfare.

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Introduction

Obtaining clearance from the antitrust authorities for a merger to monopoly is certainly a very di¢ cult task for …rms operating in countries where strict antitrust rules are rigorously enforced, as they are in the US and the European Union. Such a merger would inevitably lead to an increase in price and, thus, a reduction in consumers’welfare, unless spectacular (and improbable) e¢ ciency gains can be implemented. A ban can thus be expected, unless remedies are negociated between the …rms and the antitrust authority. Divestiture stands as an obvious candidate as remedies in such a case. However, the literature on vertical contracting suggests an alternative, more e¢ cient type of remedies. The solution is to forbid the monopolist to directly sell its product to consumers and to force it to sell through independent retailers. This leads to a complete loss of market power for the monopolist. More precisely, the monopolist captures industry pro…ts, but level of price on the market (and thus of industry pro…ts) is determined solely by the intensity of competition between retailers. Antitrust authority have accumulated a huge experience in dealing with competition issues at the retailing level. That these remedies are not adopted in practice and, as far as I know, are not even discussed, suggests that antitrust Address: 7 place Hoche, F-35000 Rennes, FRANCE. Email: [email protected]

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authorities are skeptical about the theoretical prediction of standard models of vertical contracting like McAfee and Schwartz (1994). Another, closely related, implication of previous results on vertical contracting to which the antitrust authorities would …nd di¢ cult to adhere is that a merger between duopolistic suppliers has no impact on the …nal price. The skepticism of antitrust authorities may …nd a justi…cation in the caveats that qualify these results. Economists are of course aware of these caveats. In fact, they were …rst to point at them, like Rey and Tirole (2007). Strangely enough, however, this did not lead to the emergence of models relying on more appealing assumptions and leading to results that could …nd their way into the practice of antitrust authorities and thus exert an in‡uence on antitrust case law. This article is an attempt to contribute to such an evolution by exploring the consequences of the fact that suppliers don’t enjoy an in…nite production capacity. This sounds fairly obvious, but the results quoted above rely on the opposite (implicit) assumption of in…nite production capacity. I show that this is far from neutral. Indeed, assuming that production costs are null, I show that a merger between capacity constrained duopolistic suppliers strictly increases the price for consumers. This result is obtained by comparing the equilibria of a game with duopolistic capacity constrained suppliers, presented and solved here for the …rst time, with the equilibria of a game with a monopolistic capacity constrained supplier, presented in a previous paper (Avenel (2010)). In section 2, I present the theoretical framework of this article. In section 3, I treat the case of in…nite production capacity and replicate previous results from the literature by showing that, under the usual assumption of passive beliefs, a merger to monopoly has no impact on the price paid by consumers. In section 4, I consider capacity constrained suppliers. I present in details the resolution of the game with duopolistic suppliers and show that a merger strictly increases the price to consumers. Section 5 concludes.

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A private bilateral contracting game with constrained suppliers

The industry is composed of two suppliers (S1 and S2 ) and two retailers (R1 and R2 ). Suppliers compete on the intermediate market to supply retailers. Retailers compete à la Cournot on the …nal market. Consumers’demand is assumed to be linear, with P (X) = 1 X. Suppliers’marginal cost of production is taken equal to zero as long as the output is less than or equal to the production capacity. I denote by Qi the production capacity of Si . It is exogenous and public knowledge. I will consider both the case where production capacities are in…nite and the case where they are …nite. Retailing costs are assumed to be null. The strategic interaction between …rms is modelled with the two-stage game (Q1 ; Q2 ) in which the timing of moves is: Stage 1 (o¤ers): S1 and S2 simultaneously make privately observed o¤ers

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to retailers. Rk is o¤ered (qk1 ; tk1 ) by S1 and (qk2 ; tk2 ) by S2 , where qki is the quantity supplied by Si to Rk and tki is the transfer from Rk to Si . Stage 2 (acceptances and competition): Each retailer decides whether to accept or reject the o¤er it received from each supplier. If Rk accepts the contract (qki ; tki ), it pays tki and receives qki . The quantity xk put on the market by Rk must be less than or equal to the total quantity of product received from suppliers. It is useful to introduce here some notations for later reference. Let Qm and m denote the output and the pro…t of a vertically integrated monopolist facing the inverse demand function P (:). Let q C and C denote the output and the pro…t of a vertically integrated duopolist and QC = 2q C be the total agregate output of the duopoly. Finally, let BR(:) denotes the best reply of an integrated duopolist to its rival’s output, i.e. BR(x) = arg maxP (x + y) y. y

Since Rl , l 6= k, does not observe (qk1 ; tk1 ) and (qk2 ; tk2 ), (Q1 ; Q2 ) is an extensive form game of imperfect information. Solving thus amounts to determining its (weak) perfect Bayesian equilibria (Mas-Colell, Whinston and Green (1995)). This requires that players’ strategies are sequentially rational given beliefs and beliefs are consistent with strategies. A strategy for retailer Rk is a triplet (zk1 ; zk2 ; xk ), where (i) zki (qk1 ; tk1 ; qk2 ; tk2 ) = 1 if Rk accepts (qki ; tki ) and 0 otherwise and (ii) xk is a function from 0; Q1 R+ 0; Q2 R+ into [0; zk1 qk1 + zk2 qk2 ]. Beliefs for retailer Rk are a quadruplet of functions (e ql1 ; e tl1 ; qel2 ; e tl2 ).1 Since a deviation by Sj does not convey information on Si ’s strategy (it is not even observed by Si ), qeli and e tli are functions of (q ; t ). Finally, a strategy for S is a quadruplet (q ; t i 1i 1i ; q2i ; t2i ) 2 Si := n ki ki o + 4 (q1i ; t1i ; q2i ; t2i ) 2 (R ) : q1i + q2i Qi . De…nition 1 An assessment

qki ; tki ; qeki ; e tki

i=1;2;k=1;2

; (zk1 ; zk2 ; xk )k=1;2 is

a (weak) perfect Bayesian equilibrium of (Q1 ; Q2 ) if and only if: (i) For any (qk1 ; tk1 ; qk2 ; tk2 ); (zk1 ; zk2 ) =

arg max (z1 ;z2 )2f0;1g2

and xk (qk1 ; tk1 ; qk2 ; tk2 ) =

max

x2[0;z1 qk1 +z2 qk2 ]

arg max

P (x + xl (e ql1 ; e tl1 ; qel2 ; e tl2 )

x2[0;zk1 qk1 +zk2 qk2 ]

P (x + xl (e ql1 ; e tl1 ; qel2 ; e tl2 )

x

z1 tk1 x

z2 tk2

zk1 tk1

zk2 tk2 .

(ii) Beliefs are derived from strategies through Bayes’rule whenever possible. (iii) (q1i ; t1i ; q2i ; t2i ) = arg max fz1i (q11 ; t11 ; q12 ; t12 ) t1i + z2i (q21 ; t21 ; q22 ; t22 ) t2i g (q1i ;t1i ;q2i ;t2i )2Si

While beliefs along the equilibrium pass are derived from the strategies using Bayes’ rule, the equilibrium concept imposes no a priori restriction on out-ofequilibrium beliefs. It is however necessary to specify these beliefs, because of the multiplicity of equilibria. Most contributions in the literature assume that players hold passive beliefs. 1 As usual in the literature on vertical contracting, I assume that R holds beliefs that put k probability one on one strategy of Si .

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De…nition 2 Under passive beliefs, when Rk receives the o¤ er (qki ; tki ) from Si , it believes that Si o¤ ers to Rl the equilibrium o¤ er (qli ; tli ). Under the common assumptions of a monopolist supplier with in…nite production capacity, assuming passive beliefs is reasonable. It is indeed optimal for the supplier to deal with retailers as if they were operating on di¤erent markets. Furthermore, in the type of game we consider, passive beliefs are wary beliefs, which reinforces their legitimacy (Rey and Vergé (2004)). Passive beliefs remain reasonable in a model with two unconstrained suppliers (Hart and Tirole (1990)). I assume passive beliefs in game 1 = (1; 1). As soon as a supplier has a …nite production capacity, passive beliefs have to be abandoned. I assume that Si cannot o¤er more than it is able to produce, Qi . When R1 receives an out-of-equilibrium o¤er q1i > namely q1i + q2i Qi q2i , where q2i is the quantity o¤ered by Si to R2 in equilibrium, R1 cannot believe that Si still o¤ers q2i to R2 . Beliefs must be compatible with suppliers’ strategy sets. A modi…cation of the assumption on out-of-equilibrium beliefs is thus necessary when suppliers are capacity constrained. In Avenel (2010) I de…ne full capacity beliefs in the case of a monopolist supplier. Here I complete the de…nition of full capacity beliefs to take into account the existence of a second supplier. It is useful to …rst de…ne precisely what an acceptable o¤er is. De…nition 3 An o¤ er (qli ; tli ) from Si to Rl is acceptable given (qlj ; tlj ), the o¤ er 8from Sj to Rl , if and only if: max F (qli ; qlj ; tli ; tlj ; x) > tli + tlj > x2[0;qli +qlj ] > < tli max F (qli ; qlj ; tli ; tlj ; x) max F (qli ; qlj ; tli ; tlj ; x) x2[0;qli +qlj ] x2[0;qlj ] > > > max F (qli ; qlj ; tli ; tlj ; x) max F (qli ; qlj ; tli ; tlj ; x) : tlj x2[0;qli +qlj ] x2[0;qli ] or 8 max F (qli ; qlj ; tli ; tlj ; x) > > tli x2[0;q > li ] < tlj > max F (qli ; qlj ; tli ; tlj ; x) max F (qli ; qlj ; tli ; tlj ; x) x2[0;qli +qlj ] x2[0;qli ] > > > max F (qli ; qlj ; tli ; tlj ; x) : tlj tli > max F (qli ; qlj ; tli ; tlj ; x) x2[0;qlj ]

x.

x2[0;qli ]

where F (qli ; qlj ; tli ; tlj ; x) = P (x+xk (e qki (qli ; tli ); e tki (qli ; tli ); qekj (qlj ; tlj ); e tkj (qlj ; tlj )))

When o¤ered (qli ; tli ) acceptable given (qlj ; tlj ), Rl may accept both contracts, which requires that this leaves it with non-negative pro…ts and that accepting both contracts leads to pro…ts not smaller than accepting only one contract. Alternatively, Rl may accept only (qli ; tli ). It is the best thing to do if this leads to a non-negative pro…t and to a strictly larger pro…t than either accepting both contracts or accepting (qlj ; tlj ) only. In this case, (qli ; tli ) could still be de…ned as acceptable when Rl is indi¤erent between (qli ; tli ) and (qlj ; tlj ). However, de…nition (3) doesn’t allow for this, for reasons explained below. In this sense, acceptability as described here is strict acceptability. Note that for any (qlj ; tlj ), there is an acceptable (qli ; tli ). In fact, any contract with tli = 0 is acceptable. 4

De…nition 4 Under full capacity beliefs, when Rk receives the o¤ er (qki ; tki ) from Si , it believes that Si o¤ ers (e qli (qki ; tki ); e tli (qki ; tki )) to Rl , where qeli (qki ; tki ) = Qi qki and e tli (qki ; tki ) is such that (e qli (qki ; tki ); e tli (qki ; tki )) is acceptable given that Rl receives from Sj the equilibrium o¤ er (qlj ; tlj ).

The acceptability of (qli ; tli ) depends on (qlj ; tlj ). Rk assumes that Si makes an acceptable o¤er to Rl . What this exactly means depends on Si ’s beliefs on (qlj ; tlj ). Indeed, (qli ; tli ) is considered by Si as acceptable, given Si ’s beliefs on (qlj ; tlj ). Consistency with the sequential rationality of Si requires that Si believes that Sj plays its equilibrium strategy. The previous de…nitions imply that Rk assumes that Si is not willing to o¤er Rl a contract such that Rl is indi¤erent between accepting only (qli ; tli ) and accepting only (qlj ; tlj ). In this situation, Rl will reject (qli ; tli ) with some probability, a risk that Si can eliminate by reducing tli by an in…nitely small amount. Rk assumes that this is indeed what Si does. Another implication of the previous de…nitions is that full capacity beliefs do not depend on the transfer o¤ered to the retailer. We thus drop transfers as arguments of the beliefs functions from now on. A merger between S1 and S2 will transform the game (Q1 ; Q2 ) into the game G Q1 + Q2 in which a monopolist supplier with production capacity Q1 + Q2 supplies retailers. In fact, G Q is simply (Q; 0). However, for clarity, it is better to introduce speci…c notations. In G Q , there is only one supplier, S, supplying R1 and R2 . Contract o¤ers are privately observed by retailers. I denote by (qk ; tk ) the contract o¤ered to Rk in stage 1. A strategy for retailer Rk is a pair (zk ; xk ), where (i) zk (qk ; tk ) = 1 if Rk accepts (qk ; tk ) and 0 otherwise and (ii) xk is a function from 0; Q R+ into [0; zk qk ]. Beliefs for retailer Rk are a pair (e ql ; e tl ) of functions of (qk ; tk ). Finally, a strategyofor n 4 S is a quadruplet (q1 ; t1 ; q2 ; t2 ) 2 S := (q1 ; t1 ; q2 ; t2 ) 2 (R+ ) : q1 + q2 Q . De…nition 5 An assessment

qk ; tk ; qek ; e tk

k=1;2

; (zk ; xk )k=1;2

is a (weak)

perfect Bayesian equilibrium of G(Q) if and only if: (i) For any (qk1 ; tk1 ; qk2 ; tk2 ); zk = arg max z2f0;1g2

and xk (qk ; tk ) = arg max x2[0;zk qk ]

P (x + xl (e ql ; e tl )

max

x2[0;z qk ]

x

zk

P (x + xl (e ql ; e tl )

x

tk .

(ii) Beliefs are derived from strategies through Bayes’rule whenever possible. (iii) (q1 ; t1 ; q2 ; t2 ) = arg max fz1 (q1 ; t1 ) t1 + z2 (q2 ; t2 ) t2 g. (q1 ;t1 ;q2 ;t2 )2S

The de…nition of passive beliefs is unchanged, up to notations. De…nition 6 Under passive beliefs, when Rk receives the o¤ er (qk ; tk ) from S, it believes that S o¤ ers to Rl the equilibrium o¤ er (ql ; tl ). With an upstream monopolist, it is easier to de…ne full capacity beliefs. Rl is willing to accept a contract if it leaves it a non-negative pro…t given its beliefs. 5

z tk

De…nition 7 An o¤ er (ql ; tl ) from S to Rl is acceptable if and only if: tl max P (x + xk (e qk (ql ; tl ); e tk (ql ; tl ))) x. x2[0;ql ]

De…nition 8 Under full capacity beliefs, when Rk receives the o¤ er (qk ; tk ) from S, it believes that S o¤ ers (e ql (qk ; tk ); e tl (qk ; tk )) to Rl , where qel (qk ; tk ) = Q qk and e tl (qk ; tk ) is such that (e ql (qk ; tk ); e tl (qk ; tk )) is acceptable.2

Given the previous de…nitions, asserting the impact of a suppliers’ merger on consumers’welfare amounts to comparing the perfect Bayesian equilibria of (Q1 ; Q2 ) and G Q1 + Q2 . In section 3, I do this comparison in the special case where suppliers are not capacity constrained, i.e. Q1 = Q2 = +1. In section 4, I consider the case where both …rms are constrained, but still have a rather large production capacity, that is QC < Qi < +1 for i = 1; 2. Each supplier is able to produce strictly more than the output of a Cournot duopoly of vertically integrated retailers.

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Unconstrained suppliers’merger and consumer welfare

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is similar to a game considered in Hart and Tirole (1990). There are however some di¤erences. In particular, Hart and Tirole assume that the upstream marginal cost is high, while I assume costless production. Their assumption ensures that retailers put on the market all the product received from suppliers. With costless production, retailers may retain part of the product. I deal with this issue in the resolution of 1 . I start with a characterization of the equilibrium on the …nal market in a perfect Bayesian equilibrium with passive beliefs of 1 . I also characterize pro…t-sharing between suppliers and retailers in equilibrium. Lemma 1 In a perfect Bayesian equilibrium with passive beliefs of and tki = 0 for k = 1; 2 and i = 1; 2.

1

, xk = q C

Proof. Assume that (x1 ; x2 ) 6= q C ; q C . Necessarily, for some k 2 f1; 2g, xk < q C . Furthermore, xl BR(xk ), with l 6= k. Rk would increase its pro…ts by increasing xk . If (qki ; tki ) is rejected, then it is pro…table for Si to deviate by reducing tki so that the contract becomes acceptable for Rk . If (qk1 ; tk1 ) and (qk2 ; tk2 ) are accepted and qk1 + qk2 < q C , then S1 can increase pro…ts by o¤ering (qk1 + "; tk1 + ), for some small, positive " and . Thus, in equilibrium, (x1 ; x2 ) = q C ; q C . Assume that tki > 0 for some i and some k. If (qki ; tki ) is accepted by Rk , then Sj , j 6= i, can increase its pro…ts by deviating from (qkj ; tkj ) to 2 This de…nition of full capacity beliefs di¤ers from the de…nition in Avenel (2010) in which e tl (qk ; tk ) is the highest acceptable transfer. Indeed, this aspect of the de…nition doesn’t change the results and its extension to the duopoly case raises many technical problems.

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(qkj + qki ; tkj + tki "). If (qki ; tki ) is rejected by Rk , then either Sj can increase pro…ts by increasing slightly tkj (for tkj = 0) or Si can increase pro…ts by reducing tki below tkj so that (qki ; tki ) becomes acceptable (for tkj > 0). The outcome of the strategic interaction between …rms in the industry is fairly competitive. In fact, suppliers are engaged in a very though competition that drives transfers to zero and makes it impossible for them to restrict retailers’output on the …nal market. The equilibrium on the …nal market is identical to the equilibrium that would result from competition between vertically integrated retailers. The comparison is all the more relevant as retailers keep all of their pro…ts for themselves. While lemma (1) characterizes equilibria, it doesn’t establish the existence of an equilibrium. I now turn to this point and deal with it by presenting an assessment and showing that it is an equilibrium. 2

De…nition 9 The assessment E 1 is characterized by (i) for any (k; i) 2 f1; 2g ; (qki ; tki ) = q C ; 0 , (ii) for any (qki ; tki ), qeli (qki ; tki ) ; e tli (qki ; tki ) = q C ; 0 and (iii) for any (qki ; qkj ), i 6= j, (zk1 ; zk2 ; xk ) is de…ned as follows: (a) (qki ; qkj ) 2 (qi ; qj ) 2 [0; q C )2 : qi + qj q C : (zki ; zkj ; xk ) = 1; 1; q C C C C for tki + tkj , tkj P (q C + qki )qki and tki P (q C + qkj )qkj . (zki ; zkj ; xk ) = (1; 0; qki ) for tki P (q C + qki )qki , tkj > C P (q C + qki )qki and tki tkj P (q C + qki )qki P (q C + qkj )qkj .3 (zki ; zkj ; xk ) = (0; 0; 0) for tki > P (q C + qki )qki , tkj > P (q C + qkj )qkj and tki + tkj > C . (b) (qki ; qkj ) 2 (qi ; qj ) 2 [0; q C )2 : qi + qj < q C : (zki ; zkj ; xk ) = (1; 1; qki + qkj ) for tki + tkj P (qki + qkj + q C )(qki + qkj ), tki P (qki + qkj + q C )(qki + qkj ) P (qkj + q C )qkj and tkj P (qki + qkj + q C )(qki + qkj ) P (qki + q C )qki . (zki ; zkj ; xk ) = (1; 0; qki ) for tki P (qki + q C )qki , tkj > P (qki + qkj + q C )(qki + qkj ) P (qki +q C )qki and tki tkj P (qki +q C )qki P (qkj +q C )qkj . (zki ; zkj ; xk ) = (0; 0; 0) for tki + tkj > P (qki + qkj + q C )(qki + qkj ), tki > P (qki + q C )qki and tkj > P (qkj + q C )qkj . (c) (qki ; qkj ) 2 (qi ; qj ) 2 [q C ; +1) [0; q C ) : (zki ; zkj ; xk ) = 1; 1; q C for C C tki + tkj , tkj 0 and tki P (q C + qkj )qkj . (zki ; zkj ; xk ) = 1; 0; q C C C for tki , tkj > 0 and tki tkj P (q C + qkj )qkj . (zki ; zkj ; xk ) = C C (0; 1; qkj ) for tkj P (q + qkj )qkj , tki tkj P (q C + qkj )qkj and tki > C C P (q + qkj )qkj . (zki ; zkj ; xk ) = (0; 0; 0) for tki > C , tkj > P (q C + qkj )qkj and tki + tkj > C . (d) (qki ; qkj ) 2 (qi ; qj ) 2 [q C ; +1)2 : (zki ; zkj ; xk ) = 1; 1; q C for tki + C C tkj , tkj 0 and tki 0. (zki ; zkj ; xk ) = 1; 0; q C for tki , tkj > 0 C C and tkj tki . (zki ; zkj ; xk ) = (0; 0; 0) for tki +tkj > , tki > and tkj > C . Lemma (1) implies that in equilibrium each retailer receives at least q C . In E 1 , each retailer receives exactly q C from each supplier. Both o¤ers are accepted and the retailer puts q C on the …nal market. Thus, the retailer covers its entire need for product with what it receives from only one supplier, while 3 In case of equality in this last condition, R is indi¤erent between accepting only the k contract o¤ered by Ri and accepting only the contract o¤ered by Rj . In this case, we assume the usual tie-breaking rule that with probability 1/2 it chooses one and with probability 1/2 it chooses the other. This plays no role in the proof of the following lemma.

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the product received from the other supplier is kept out of the market. It can be either stored or destroyed or returned to the supplier, but in this last case the assumption is that it is returned after the …nal market is closed. The clothing industry provides examples of such a mechanism. In this article, I don’t specify what happens with the product that retailers don’t use. They are assumed to get rid of it without a cost. Lemma 2 E 1 is a perfect Bayesian equilibrium with passive beliefs of

1

.

Proof. Whatever o¤ers it receives from the suppliers, Rk believes that Rl is o¤ered q C ; 0 by both suppliers and, thus, that (zl1 ; zl2 ; xl ) = 1; 1; q C . Sequential rationality for Rk thus requires that (zk1 ; zk2 ; xk ) is a best reply to (zl1 ; zl2 ; xl ) = 1; 1; q C . This leads for Rk to the strategy described in E 1 . As regards suppliers’rationality, given that Si o¤ers q C ; 0 to both retailers, there is no pro…table deviation for Sj . Lemma (2)is essentially an existence result. It proves that the set of equilibria characterized in lemma (1) is not empty. I brie‡y discuss why E 1 is an equilibrium. Each supplier supplies free of charge both retailers with a quantity of product that covers their entire needs. The other supplier is consequently unable to extract any transfer from retailers. It is thus indi¤erent between supplying both retailers with q C and supplying them with any other quantity, since production costs are null. This is why suppliers’strategies in E 1 are rational. As regards retailers, their rationality along the equilibrium path is easy to deal with because they can only accept contracts that require them no paiement at all. Once they accepts the contract, they are essentially competing à la Counot with a capacity constraint that is not binding. Of course, retailers’rationality has to be examined not only along the equilibrium path, but also out of equilibrium. Each retailer essentially believes that its competitor will put q C on the market and makes its decision regarding acceptance of contracts and output on the …nal market based on this belief. E 1 is not the unique equilibrium of 1 . A supplier may o¤er more than q C to a retailer, keeping the transfer at zero. However, in any equilibrium, the outcome on the …nal market is the same, as well as upstream and downstream pro…ts. I now turn to the case of a monopolist supplier. G1 is similar to a game considered in Rey and Tirole (2007) with, again, some di¤erences, in particular on the value of the upstream marginal cost. I proceed as for the duopoly case, starting with a characterization of the outcome on the …nal market and of pro…tsharing between suppliers and retailers. Lemma 3 In a perfect Bayesian equilibrium with passive beliefs of G1 , xk = q C and tk = P (QC )q C for k = 1; 2. Proof. Assume that (x1 ; x2 ) 6= q C ; q C . Necessarily, for some k 2 f1; 2g, xk < q C . Furthermore, xl BR(xk ), with l 6= k. Rk would increase its pro…ts by increasing xk . If (qk ; tk ) is rejected, then it is pro…table for S to deviate by reducing tk so that the contract becomes acceptable for Rk . If (qk ; tk ) is 8

accepted and qk < q C , then S can increase pro…ts by o¤ering (qk + "; tk + ), for some small, positive " and . Thus, in equilibrium, (x1 ; x2 ) = q C ; q C . A consequence of the above analysis is that in equilibrium S o¤ers q C to each retailers and the contracts are accepted. Given this, each retailer expects his competitor to put q C on the market. It is thus willing to pay up to P (QC )q C for the quantity q C . In equilibrium, S raises transfers up to this limit. Lemma (3) formulates the standard result that under passive beliefs a monopolist retailer cannot achieve the monopolization of the …nal market when it sells its product to consumers through competing retailers. More precisely, that the upstream industry is a monopoly or a duopoly is irrelevant if we consider only the output and the price on the …nal market. The di¤erence between the two situations appears when one considers pro…t-sharing between suppliers and retailers. A monopolist supplier is not able to increase industry pro…ts, but it is able to capture them entirely. I now deal with the issue of the existence of an equilibrium. De…nition 10 The assessment E1 is characterized by (i) (q1 ; t1 ; q2 ; t2 ) = q C ; P (QC )q C ; q C ; P (QC )q C , (ii) for any (qk ; tk ), ( q~l (qk ); t~l (qk ) = q C ; P (QC )q C and (iii) For 0 qk < q C , (zk ; xk ) = (1; qk ) for tk P (qk + q C )qk , (zk ; xk ) = (0; 0) otherwise. For q C qk , (zk ; xk ) = (1; q C ) for tk P (QC )q C , (zk ; xk ) = (0; 0) otherwise. Lemma 4 E1 is a perfect Bayesian equilibrium with passive beliefs of G1 . Proof. Given retailers’beliefs, it is rational for them to accept q C ; P (QC )q C and put q C on the market. The supplier cannot extract a larger transfer than P (QC )q C because this is max P (q + q C )q and each retailer believes that its q

competitor puts q C on the market. Lemmas (3) and (4) provide a full characterization of what happens on the …nal market when a monopolist supplier contracts with competing retailers under passive beliefs as it is the case in G1 . This concludes the analysis of the strategic interaction between suppliers and retailers both with an upstream monopoly and with an upstream duopoly. This makes it possible to conclude on the impact of an horizontal merger between suppliers. Proposition 1 When both suppliers have an in…nite production capacity and retailers hold passive beliefs, a merger to monopoly between suppliers has no impact on consumers’ welfare. Proof. The proposition is a straightforward consequence of lemmas (1) to (4). The impact of a merger between the two suppliers is limited to retailers. While industry pro…ts are captured by retailers before the merger, they are captured by the supplier after the merger. However, the monopolist supplier is not able to induce the monopoly outcome on the …nal market or at least to induce a price above the price observed on the …nal market before the merger.

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4

Constrained suppliers’ merger and consumer welfare

In this section, I characterize the equilibrium outcome on the …nal market for an upstream duopoly and an upstream monopoly. The comparison of both cases leads to a characterization of the impact of a merger on consumers’welfare. Lemma 5 In a perfect Bayesian equilibrium with full capacity beliefs of with Q1 > QC and Q2 > QC , x1 + x2 > Qm .

Q1 ; Q2 ,

Proof. Full capacity beliefs imply that in equilibrium all contracts are accepted. Indeed, Rk believes that S1 o¤ers Rl a contract that is acceptable given S2 ’s equilibrium o¤er and that S2 o¤ers Rl a contract that is acceptable given S1 ’s equilibrium o¤er. Rk thus believes that when Rl receives equilibrium o¤ers from both suppliers, it accepts both. This beliefs has to be correct in equilibrium. As a consequence, the total quantity transfered to retailers is Q1 + Q2 . It is easy to check that for any x 2 0; Q1 + Q2 , BR(x) Q1 +Q2 x and x+BR(x) Qm . In equilibrium, x2 = min (BR(x1 ); q21 + q22 ). x2 = BR(x1 ))x1 + x2 = x1 + BR(x1 ) Qm . x2 = q21 + q22 )x1 = min(BR(q21 + q22 ); Q1 + Q2 q21 q22 ) = BR(q21 + q22 ) and x1 + x2 = q21 + q22 + BR(q21 + q22 ) Qm . In a PBE with full capacity beliefs, x1 + x2 Qm . I now show that x1 + x2 6= Qm . The sequential rationality of retailers allows for x1 + x2 = Qm only if one of the retailers (say R1 ) receives q11 + q12 = Q1 + Q2 and the other (R2 ) receives q21 + q22 = 0. Suppliers’ strategies should then be (q11 ; t11 ; q21 ; t21 ) = Q1 ; 0; 0; 0 and (q12 ; t12 ; q22 ; t22 ) = Q2 ; 0; 0; 0 . Note that t11 = t12 = 0 because R1 can reject the o¤er of one of the suppliers without any impact on its market output. 0 0 Consider the following deviation for S2 : 02 = (q12 ; t012 ; q22 ; t022 ) = 0; 0; Q2 ; " , where " > 0 is in…nitely small. The deviation is pro…table for S2 if and only if it is accepted by R2 . Full capacity beliefs for R2 are (e q11 (q21 ); e t11 (q21 )) = e Q1 ; 0 and (e q12 (q22 ); t12 (q22 )) = (0; 0). (z21 ; z22 ; x2 ) must be a best reply to 0 ; t022 ), rational sequentiality implies x2 = x02 , x1 (Q1 ; 0; 0; 0). If R2 accepts (q22 0 with: x2 = arg max P (x2 + x1 (Q1 ; 0; 0; 0)))x2 . Obviously, z11 (Q1 ; 0; 0; 0) = x2 Q2

z12 (Q1 ; 0; 0; 0) = 1. R1 believes that S2 o¤ers R2 the quantity Q2 and a transfer that is acceptable given (q21 ; t21 ) = (0; 0). R1 chooses x1 (Q1 ; 0; 0; 0) so as to maximize P (x2 0; 0; Q2 ; + x1 )x1 subject to x1 Q1 . Note that x2 0; 0; Q2 ; is precisely x02 , so that x1 (Q1 ; 0; 0; 0) = arg max P (x02 + x1 )x1 x1 Q1

0 and x02 = x1 (Q1 ; 0; 0; 0) = q C . Accepting (q22 ; t022 ), R2 thus makes a pro…t equal to P (QC )q C " > 0 for " in…nitely small. 02 is thus a pro…table deviation for S2 and there is no equilibrium such that x1 + x2 = Qm . In equilibrium, suppliers produce at full capacity and o¤er their total output to retailers through contracts that are accepted. As a consequence, the retailing sector is a Cournot duopoly in which one …rm has a production capacity of q and the other has a production capacity of Q1 +Q2 q. With Q1 and Q2 strictly larger than the output of an unconstrained Cournot duopoly, this cannot lead to

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an output strictly lower than the monopoly output. It can lead to the monopoly output if both suppliers o¤er their total output to the same retailer. However, this is not an equilibrium because suppliers would deviate from this strategy pro…le. Indeed, the supplied retailer pays no transfer to the suppliers, while the other retailer is willing to pay a positive transfer to get access to the product. Q1 ; Q2 exists, it is characterized by a If a PBE with full capacity beliefs of price paid by consumers that is strictly below the monopoly price. I now address the question of the existence of an equilibrium. The …rst step is to de…ne an assessment that will afterwards be proved to be an equilibrium. De…nition 11 The assessment E Q1 ; Q2 is characterized by (i) (q11 ; t11 ; q21 ; t21 ) = q C ; 0; Q1 q C ; 0 and (q12 ; t12 ; q22 ; t22 ) = Q2 q C ; 0; q C ; 0 , (ii) for any (qki ; qkj ), i 6= j, qeli ; e tli ; qelj ; e tlj = Qi qki ; 0; Qj qkj ; 0 and (iii) for any (qki ; qkj ), i 6= j, (zk1 ; zk2 ; xk )nis de…ned as follows: o 2 (a) (qki ; qkj ) 2 (qi ; qj ) 2 0; q C : qi + qj > q C : (zki ; zkj ; xk ) = 1; 1; q C

C C C for tki + tkj , tkj P (q C + qki )qki and tki P (q C + qkj )qkj . C C (zki ; zkj ; xk ) = (1; 0; qki ) for tki P (q + qki )qki , tkj > P (q C + qki )qki C C and tki tkj P (q + qki )qki P (q + qkj )qkj . (zki ; zkj ; xk ) = (0; 0; 0) for C C tki > P (q C + qki )qn . ki , tkj > P (q + qkj )qkj and tki + o tkj > 2

(b) (qki ; qkj ) 2 (qi ; qj ) 2 0; q C : qi + qj q C : (zki ; zkj ; xk ) = (1; 1; qki + qkj ) for tki + tkj P (qki + qkj + BR(qki + qkj ))(qki + qkj ), tki P (qki + qkj + BR(qki + qkj ))(qki + qkj ) P (qkj + BR(qki + qkj ))qkj and tkj P (qki + qkj + BR(qki + qkj ))(qki + qkj ) P (qki + BR(qki + qkj ))qki . (zki ; zkj ; xk ) = (1; 0; qki ) for tki P (qki + BR(qki + qkj ))qki , tkj > P (qki + qkj + BR(qki + qkj ))(qki + qkj ) P (qki + BR(qki + qkj ))qki and tki tkj P (qki + BR(qki + qkj ))qki P (qkj + BR(qki + qkj ))qkj . (zki ; zkj ; xk ) = (0; 0; 0) for tki + tkj > P (qki + qkj + BR(qki + qkj ))(qki + qkj ), tki > P (qki + BR(qki + qkj ))qki and tkj > P (qkj + BR(qki + qkj ))qkj . 0; q C : qi + qj Q1 + Q2 q C : (zki ; zkj ; xk ) = (c) (qki ; qkj ) 2 (qi ; qj ) 2 (q C ; Qi ] C C C 1; 1; q for tki +tkj , tkj 0 and tki P (q C +qkj )qkj . (zki ; zkj ; xk ) = C C C 1; 0; q for tki , tkj > 0 and tki tkj P (q C +qkj )qkj . (zki ; zkj ; xk ) = C C (0; 1; qkj ) for tkj P (q + qkj )qkj , tki tkj P (q C + qkj )qkj and tki > C C P (q + qkj )qkj . (zki ; zkj ; xk ) = (0; 0; 0) for tki > C , tkj > P (q C + qkj )qkj and tki + tkj > C . (d) (qki ; qkj ) 2 (qi ; qj ) 2 (q C ; Qi ]2 : qi + qj Q1 + Q2 q C : (zki ; zkj ; xk ) = C 1; 1; q C for tki + tkj , tkj 0 and tki 0. (zki ; zkj ; xk ) = 1; 0; q C for C tki , tkj > 0 and tkj tki . (zki ; zkj ; xk ) = (0; 0; 0) for tki + tkj > C , C C tki > and tkj > . (e) (qki ; qkj ) 2 (qi ; qj ) 2 (q C ; Qi ]2 : qi + qj > Q1 + Q2 q C : (zki ; zkj ; xk ) = 1; 1; BR(Q1 + Q2 qki qkj ) for tki +tkj P (Q1 +Q2 qki qkj +BR(Q1 + Q2 qki qkj ))BR(Q1 + Q2 qki qkj ), tkj 0 and tki 0. (zki ; zkj ; xk ) = 1; 0; BR(Q1 + Q2 qki qkj ) for tki +tkj P (Q1 +Q2 qki qkj +BR(Q1 + Q2 qki qkj ))BR(Q1 + Q2 qki qkj ), tkj > 0 and tkj tki . (zki ; zkj ; xk ) = (0; 0; 0) for tki +tkj > P (Q1 +Q2 qki qkj +BR(Q1 +Q2 qki qkj ))BR(Q1 + 11

Q2 qki qkj ), tki > P (Q1 + Q2 qki qkj + BR(Q1 + Q2 qki qkj ))BR(Q1 + Q2 qki qkj ) and tkj > P (Q1 +Q2 qki qkj +BR(Q1 +Q2 qki qkj ))BR(Q1 + Q2 qki qkj ). Although the full description of retailers’ strategies necessitates a rather long de…nition of any assessment, what happens along the equilibrium path in E Q1 ; Q2 is simple. Each retailer is o¤ered by each supplier a quantity at least equal to its output in the unconstrained Cournot equilibrium. Retailers accept these o¤ers. The equilibrium on the …nal market is similar to that resulting from unconstrained Cournot competition. I now show that this assessment is a Q1 ; Q2 . perfect Bayesian equilibrium with full capacity beliefs of Lemma 6 E Q1 ; Q2 is a perfect Bayesian equilibrium with full capacity beliefs Q1 ; Q2 . of Proof. Full capacity beliefs For any (qk1 ; qk2 ), qel1 ; e tl1 ; qel2 ; e tl2 = Q1 qk1 ; 0; Q2 qk2 ; 0 . This corresponds to full capacity beliefs because S1 ’s equilibrium o¤er to Rl is either q C ; 0 or Q1 q C ; 0 , with Q1 q C > q C . In both cases, an o¤er from S2 to Rl is acceptable if and only if tl2 = 0. Similarly, an o¤er from S1 to Rl is acceptable if and only if tl1 = 0. Consistency of Rk ’s beliefs Full capacity beliefs are consistent with the suppliers’equilibrium strategies. Sequential rationality for suppliers Consider S1 .4 Given that (q12 ; t12 ; q22 ; t22 ) = Q2 q C ; 0; q C ; 0 , x1 = x2 = q C for any (q11 ; t11 ; q21 ; t21 ) and neither R1 nor R2 is willing to pay a strictly positive transfer to S1 . There is thus no pro…table deviation for S1 . n o 2 Sequential rationality for Rk (a) (qki ; qkj ) 2 (qi ; qj ) 2 0; q C : qi + qj > q C )xl = q C ) k (zki ; zkj ; xk ) = P (q C +xk )xk zki tki zkj tkj . (zki ; zkj ) = (1; 1))xk = q C and k = C tki tkj . (zki ; zkj ) = (1; 0))xk = qki and k = P (q C + qki )qki tki . Comparing the values of k proves that Rk ’s strategy in E Q1 ; Q2 is sequentially rational for these values of (qki ; qkj ).o n 2

(b) (qki ; qkj ) 2 (qi ; qj ) 2 0; q C : qi + qj q C )xl = BR(qki +qkj )) k (zki ; zkj ; xk ) = P (BR(qki + qkj ) + xk )xk zki tki zkj tkj . (c) (qki ; qkj ) 2 (qi ; qj ) 2 (q C ; Qi ] 0; q C : qi + qj Q1 + Q2 q C )xl = q C ) k (zki ; zkj ; xk ) = P (q C + xk )xk zki tki zkj tkj . (d) (qki ; qkj ) 2 (qi ; qj ) 2 (q C ; Qi ]2 : qi + qj Q1 + Q2 q C )xl = q C ) k (zki ; zkj ; xk ) = P (q C + xk )xk zki tki zkj tkj . (e) (qki ; qkj ) 2 (qi ; qj ) 2 (q C ; Qi ]2 : qi + qj > Q1 + Q2 q C )xl = Q1 + Q2 qki qkj ) k (zki ; zkj ; xk ) = P (Q1 +Q2 qki qkj +xk )xk zki tki zkj tkj . Given that suppliers charge transfers equal to zero, it is rational for retailers to accept the contracts. Then, retailers are competing on the …nal market with each of them able to produce the unconstrained Cournot equilibrium output. 4 The

proof is similar for S2 .

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They actually choose this level of output. No supplier can pro…tably deviate because its rival is o¤ering (for free) enough product to each retailer for retailers not to need its product. This is for the equilibrium path. However, proving that E Q1 ; Q2 is a perfect Bayesian equilibrium requires considering sequential rationality of retailers out of the equilibrium path. If for example R1 receives quantities of product corresponding to most of the production capacity of both suppliers, it believes that R2 was o¤ered (and accepted) a small quantity of product corresponding to the remaining production capacities. Upon reception of this small quantity, R2 believes that R1 receives the large quantity that it indeed received. Crossing best replies for retailers leads to R2 putting on the market all the product received from suppliers and R1 playing its best reply to this quantity, keeping out of the market part of the product. Proving that E Q1 ; Q2 is an equilibrium also requires checking that retailers’ beliefs are consistent and correspond to our de…nition of full capacity beliefs. Here, full capacity beliefs are very simple because the only transfer that is acceptable for a retailer given a suppliers’equilibrium o¤er, regardless of the quantity of product o¤ered, is zero. E Q1 ; Q2 is not the unique PBE with full capacity beliefs of Q1 ; Q2 . An assessment identical to E Q1 ; Q2 except for the fact that (q11 ; t11 ; q21 ; t21 ) = C qC q C + 1 ; 0; Q1 q C 1 ; 0 and (q12 ; t12 ; q22 ; t22 ) = Q2 2 ; 0; q + 2; 0 , C C with 1 2 (0; Q1 Q ] and 2 2 (0; Q2 Q ] is also an equilibrium of Q1 ; Q2 , leading to the same outcome on the …nal market. A more di¢ cult question is whether there are other equilibria leading to di¤erent market outcomes. It is not considered here. The two previous lemma allow me to claim that equilibria exist and that in any equilibrium, the price on the …nal market is strictly lower than the monopoly price. Let us now consider the game resulting from a merger between suppliers. G Q1 + Q2 is similar to a game considered in Avenel (2010). Lemma 7 In a perfect Bayesian equilibrium with full capacity beliefs of G Q1 + Q2 , x1 + x2 = Qm . Proof. Full capacity beliefs imply that in equilibrium all contracts are accepted. As a consequence, the total quantity transfered to retailers is Q1 + Q2 . Since for any q 2 0; Q1 + Q2 , BR(q) Q1 + Q2 q and q + BR(q) Qm , in a PBE with full capacity beliefs, x1 + x2 Qm . Now consider an assessment in which x1 + x2 > Qm . Rk correctly anticipates a gross pro…t equal to P (x1 + x2 )xk and pays tk to S. The supplier’s pro…t is thus P (x1 + x2 )(x1 + x2 ). S can pro…tably deviate from its strategy in this assessement by o¤ering (q1 ; t1 ; q2 ; t2 ) = (Q1 + Q2 ; m ; 0; 0). Indeed, R1 believes that it is in a monopoly position and is thus willing to pay a transfer up to the monopoly pro…t. This deviation is pro…table for S because m > P (x1 + x2 )(x1 + x2 ). As in the upstream duopoly case, and for the same reason, the output on the …nal market cannot be strictly less than the monopoly output. However, with an upstream monopolist, it also cannot be strictly more than the monopoly output. The upstream monopolist can avoid this by o¤ering all the product to 13

one retailer. While this is a pro…table deviation from equilibria including too large an output on the …nal market, it is still unproved that this is an equilibria strategy. I show it below. De…nition 12 The assessment E Q1 + Q2 is characterized by (i) (q1 ; t1 ; q2 ; t2 ) = Q1 + Q2 ; m ; 0; 0 , (ii) For 0 qk q C , (zk ; xk ) = (1; qk ) for tk P (qk + BR(qk ))qk , (zk ; xk ) = (0; 0) otherwise and ( q~l (qk ); t~l (qk ) = Q1 + Q2 qk ; P (qk + BR(qk ))BR(qk ) : For q C qk Q1 + Q2 q C , (zk ; xk ) = (1; q C ) for tk P (QC )q C , (zk ; xk ) = (0; 0) otherwise and ( q~l (qk ); t~l (qk ) = Q1 + Q2 qk ; P (QC )q C : For Q1 + Q2 q C qk Q1 + Q2 , (zk ; xk ) = (1; BR(Q1 + Q2 qk )) for tk P (BR(Q1 + Q2 qk ) + Q1 + Q2 qk )BR(Q1 + Q2 qk ), (zk ; xk ) = (0; 0) otherwise and ( q~l (qk ); t~l (qk ) = Q1 + Q2 qk ; P (Q1 + Q2 qk + BR(Q1 + Q2 qk ))(Q1 + Q2 qk ) : In E Q1 + Q2 , S o¤ers all the product to R1 . R1 pays the monopoly pro…t to S, puts the monopoly output on the …nal market and breaks even. R1 keeps part of the product received from S out of the market. Lemma 8 E Q1 + Q2 is a perfect Bayesian equilibrium with full capacity beliefs of G Q1 + Q2 . Proof. Full capacity beliefs The issue here is the acceptability of q~l (qk ); t~l (qk ) . For qk 2 0; q C , q~l (qk ) = Q1 + Q2 qk Q1 + Q2 q C ) q~k (~ ql (qk )) = qk and t~k t~l (qk ) = P (qk +BR(qk ))qk . xk (~ qk (~ ql (qk )); t~k t~l (qk ) ) = qk . t~l (qk ) = P (qk + BR(qk ))qk max P (x + qk )x = P (qk + BR(qk ))qk . q~l (qk ); t~l (qk ) is x2[0;Q1 +Q2 qk ] thus acceptable. The proof is similar for qk 2 (q C ; Q1 + Q2 q C ] [ (Q1 + Q2 q C ; Q1 + Q2 ]. Consistency of beliefs Full capacity beliefs are consistent with the supplier’s equilibrium strategy. In particular, t1 = m is acceptable for R1 . Sequential rationality for Rk For 0 qk q C , q~l Q1 + Q2 q C . (zl ; xl ) = (1; BR(qk )) and (zk ; xk ) = (1; min(BR(BR(qk )); qk )) = (1; qk ) () tk P (qk + Q1 + Q2 qC , qC q~l Q1 + Q2 qC . BR(qk ))qk . For q C qk C C C C (zl ; xl ) = 1; q and (zk ; xk ) = 1; q () tk P (Q )q . For Q1 + Q2 qC qk Q1 + Q2 , 0 q~l q C . (zl ; xl ) = Q1 + Q2 qk and (zk ; xk ) = 1; min(BR(Q1 + Q2 qk ); qk ) = 1; BR(Q1 + Q2 qk ) () tk P (Q1 + Q2 qk + BR(Q1 + Q2 qk ))BR(Q1 + Q2 qk ). Sequential rationality for S I consider all the possible deviations for S and check that none of those is pro…table. Without loss of generality, I consider only acceptable o¤ers. If S o¤ers q1 2 0; q C , it can o¤er any q2 up to Q1 + Q2 q1 > Q q C . For q2 2 0; q C , S gets transfers equal to t1 + t2 = P (q1 + BR(q1 ))q1 + P (q2 + BR(q2 ))q2 . This is obviously maximal for q1 = q2 = q C , but P (QC )q C < m . For q2 2 q C ; Q1 + Q2 q C , t1 + t2 = P (q1 + BR(q1 ))q1 + P (QC )q C = 21 (1 q1 ) q1 + 1 QC q C . arg max(t1 + t2 ) = q C , but t1 (q C ) + t2 (q C ) < m . For q2 2 Q1 + Q2 q C ; Q1 + Q2 q1 , t1 + t2 = P (q1 + BR(q1 ))q1 + P (BR(Q1 + Q2 q2 ) + Q1 + Q2 q2 )BR(Q1 + Q2 q2 ) = 14

2

1 2

(1 q1 ) q1 + 14 1 Q1 Q2 + q2 . S would set q2 = Q1 + Q2 q1 and t1 + t2 = P (q1 + BR(q1 ))q1 + P (BR(q1 ) + q1 )BR(q1 ) = 14 1 q12 ) (q1 ; q2 ) = (0; Q1 + Q2 ) and S = m . If S o¤ers q1 2 q C ; Q1 + Q2 q C , it can o¤er any q2 up to Q1 +Q2 q1 2 q C ; Q q C . The case q2 2 0; q C was examined above (by symmetry). For q2 2 q C ; Q1 + Q2 q1 . t1 + t2 = P (QC )QC < m . If S o¤ers q1 2 Q1 + Q2 q C ; Q1 + Q2 , it can o¤er any q2 up to Q1 +Q2 q1 q C . t1 + t2 =

1 4

1

Q1 + Q2 + q1

1 4

2

+ 12 (1 q2 )q2 . S would choose q2 = Q1 + Q2

q1

2

and t1 +t2 = 1 Q1 + Q2 q1 . This is maximal for q1 = Q1 +Q2 , which leads to t1 + t2 = m . Transfering all the product to one retailer is optimal for the supplier because this retailer is willing to pay the monopoly pro…t to the supplier. There is no way the supplier can extract from retailers more than the monopoly pro…t. In particular, o¤ering a positive quantity to both retailers would lead to an increase of the output on the …nal market and a decrease in pro…ts. Because the supplier has a …nite production capacity it is able to commit to the supply of only one retailer. In Avenel (2010), I discuss this situation under more general assumptions on marginal cost and production capacity. In the case we consider here, the two previous lemma show that equilibria exist and that in any equilibrium, consumers pay the monopoly price on the …nal market. The comparison with the upstream duopoly case leads to the following proposition: Proposition 2 When both suppliers have a large …nite production capacity and retailers hold full capacity beliefs, a merger to monopoly between suppliers reduces consumers’ welfare. Proof. The proposition is a straightforward consequence of lemmas (5) to (8). Before the merger, the two suppliers face a coordination failure. They could induce the monopoly outcome on the …nal market, but they are not able to do so in an equilibrium of the non-cooperative game . While I prove that there exists a highly competitive equilibrium in which retailers put on the …nal market the unconstrained Cournot output, I do not present a full characterization of the set of perfect Bayesian equilibria with full capacity beliefs, even if it is reasonable to suspect that they are quite competitive. It is thus di¢ cult to measure the impact of the merger on the price and thus on consumers’welfare. What I show is that there is an impact and that it is strictly negative. This is a key aspect in the evaluation of a merger by antitrust authorities and on this aspect the di¤erence with the absence of impact of a merger on consumers’welfare when production capacities are in…nite and passive beliefs are assumed is striking. As soon as production capacities are …nite, there is something to see for antitrust authorities in a merger between suppliers.

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5

Conclusion

Central to the analysis of the impact of a merger between suppliers is the ability of a monopolist supplier to eliminate competition between retailers. This is achieved by committing to supply only one retailer. The ability to commit to such a behavior …nds its roots in the fact that the supplier’s production capacity is …nite. It is thus able to o¤er all of its production to one retailer and, receiving this o¤er, the retailer knows that the other retailer will not be supplied at all. While this can be done regardless of the marginal cost of production, the assumption that production is costless ensures that this is indeed an equilibrium strategy for the supplier. Duopolists, even if facing a capacity constraint, cannot induce the same equilibrium on the …nal market because they cannot coordinate on supplying only one retailer. Quite reasonably, I …nd that a merger solves coordination problems between suppliers and reduces competition. Relaxing the assumption of costless production is certainly a promising avenue for future research. The ability for a monopolist supplier to implement the monopoly outcome on the …nal market remains with a positive, small marginal cost of production. The di¢ culties are more in the duopolistic suppliers’ game. Our resolution of this game relies on the fact that transfers are null in equilibrium, but positive marginal costs imply positive transfers in equilibrium. Solving the game with positive equilibrium transfers is more involved and left for future work.

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References

Avenel, E., 2010, Upstream capacity constraint and the preservation of monopoly power in bilateral vertical contracting, manuscript submitted for publication in the Journal of Industrial economics. Hart, O. and J. Tirole, 1990, Vertical integration and market foreclosure, Brookings papers on Economic Activity (Microeconomics), 1990, 205-285. Mas-Colell, A., M. Whinston and J. Green, 1995, Microeconomic theory, Oxford University Press. McAfee, R.P. and M. Schwartz, 1994, Opportunism in multilateral vertical contracting: Nondiscrimination, exclusivity, and uniformity, American Economic Review, 84(1), 210-230. Rey, P. and J. Tirole, 2007, Chapter 33: A primer on foreclosure, in Handbook of Industrial Organization, Vol. 3, 2145-2220. Rey, P. and T. Vergé, 2004, Bilateral control with vertical contracts, RAND Journal of Economics, 35(4), 728-746.

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