Vertical integration and foreclosure strategies in bilateral oligopoly.

Transcription

1 Vertical integration an foreclosre strategies in bilateral oligopoly. Lorenzo Clementi [JOB MARKET PAPER] This version: 31 st of October 011 Abstract The prpose of this paper is to sty the implications of vertical integration in centralize markets with the presence of oble marginalization of the price. To this extent I bil p a simple bilateral oligopoly moel with pstream an ownstream Cornot competition, where procers face inivial an exogenos costs. The moel fits both forwar an backwar vertical integration. I provie a simple argment that shows that a policy that consiers vertical integrations between big firms anti-competitive per se, an that forbis them in favor of other types of vertical mergers, may have etrimental implications for consmers. I show that the integration ecision epens both on pstream an ownstream costs, while the foreclosre strategy playe by the integrate firm epens on the pstream an ownstream post-merger markps. I state the general conitions ner which consmer welfare increases after the integration, both when the inpt foreclosre strategy is playe, or not. Finally, I sty what type of vertical integration can take place whenever a merger between big firms is forbien. Keywors: Vertical Integration, Bilateral Oligopoly, Firms Matching, Consmer Welfare. JEL Classification: D43, L13, L49, C78. I acknowlege with thanks the financial spport of my joint Ph.D. from Université Libre e Brxelles (Mini-ARC grant) an from Università La Sapienza. I am gratefl to my spervisors Estelle Cantillon an Patrick Legros from ECARES, Ligi Ventra from Università La Sapienza, to Bram De Rock, Stefano Trento, Matthew Olczak an the participants of Jornaas e Economía Instrial 011 (Universia e Valencia), Italian Doctoral Workshop 011 (Collegio Carlo Alberto of Trin), ENTER Jamboree 011 (University of Tilbrg), Doctoral Workshop 10 (Université Catholiqe e Lovain) for their sggestions. All the remaining errors an omissions are my own responsibility. The paper previosly circlate ner the title Si parva licet componere magnis. Matchings between vertically integrating firms. The latin qotation is taken from the Georgics by Virgil, book IV, 176, an it means If small with great things may compare. ECARES, SBS-EM, Université Libre e Brxelles an Università La Sapienza i Roma. loclemen@lb.ac.be. 1

2 1 Introction The Eropean Commission (EC) consiers many aspects in assessing vertical integrations. One of the major concerns is represente by the exercise of market power by the new entity in the p- an ownstream markets, throgh foreclosre strategies. A vertical integration is blocke as anticompetitive if the EC reckons that the merging firms or as well their competitors may be able to raise the price charge to consmers after the operation, throgh inpt or cstomer foreclosre strategies 1. This concern is greater when the integrating firms have a prominent size in the relevant markets. The literatre, so far, mainly focse on the effects of vertical integration an on the incentives to foreclosre, mostly in Bertran competition environments, while a really low attention has been given to what type of firms is willing to play a foreclosre or a raising rivals cost strategy, an ner what conitions. 3 In this paper I arge that the post vertical merger foreclosre strategy an its implications epen both on the size of the integrating firms an on the size of the inepenent firms left in the market. The prpose of this paper is to sty ner what conitions firms vertically integrate in centralize markets with presence of oble marginalization of price, what is the best partner choice, what is the reslting foreclosre strategy an what are the implications for consmer welfare, case by the firms matching. In the linear final goo eman example, I sty whether (an when) preventing a vertical integration between big firms an favoring a ifferent matching, is effectively beneficial to consmers, or etrimental. In fact, the choice of the partner with whom to vertically integrate is qite crcial for the ex-post merger competition an the consmer welfare, becase vertical integration creates some (positive or negative) externalities for all the agents active on the markets an for consmers. Some papers pose the qestion of the partnership choice (Behler an Schmtzler (005), art an Tirole (1990) for example) bt they i not link the vertical integration matching to the foreclosre strategy that can be playe by the new entity, mainly becase of a symmetric setting in one of the two markets. 4 The main contribtion of this paper is to provie a simple argment that shows that a policy that consiers vertical integrations between big firms anti-competitive per se, an that forbis them in favor of other types of vertical mergers, may have etrimental implications for consmers. In fact, sch a policy may lea to vertical integration patterns that can be worse than the ones that wol have happene withot the restriction. The rationale of my analysis relies on the fact that a vertically integrate firm chooses the foreclosre strategy to play - inpt foreclosre or raising rivals costs - epening on its own costs an on the profitability of serving the market after the merger, hence on the size of the competitors left into the pstream an ownstream markets. I consier a market strctre in which firms compete in qantities at both proction stages an cannot inivially agree on the inpt qantity to exchange, bt they eal only at aggregate level. I moel a bilateral oligopoly moel with two firms in each market, facing asymmetric an exogenos costs, an competing à la Cornot. The pstream opoly proces an intermeiate homogeneos goo that is prchase by the ownstream firms an se in orer to manfactre the final goo, in a one-to-one 1 Gielines on the assessment of non-horizontal mergers, Section I.18. The [Eropean] Commission is nlikely to fin concern in a non-horizontal merger,[...], where the market share post-merger of the new entity in each of the markets concerne is below 30% an the post-merger I is below 000,Gielines on the assessment of non-horizontal mergers, Press Release IP/07/1780, Section III.5. 3 Jst few papers consier Cornot competitions settings, for example: Salinger (1988), Behler an Schmtzler (005), Gaet an Long (1996). 4 The next sbsection proposes a relate literatre review on the topic.

3 relationship. The Cornot competition implies that both pstream an ownstream firms can exert market power in ifferent egrees, epening on the level of market integration an on the instry s cost strctre. The reslting oble marginalization of price harms consmers, bt it can be (partly, or completely) eliminate throgh vertical integration(s). Sch a type of centralize market offers several avantages for this analysis, becase it emboies both the pro-efficiency motives an the foreclosre concerns that mst be taken into accont by the Competition Athorities in assessing a vertical integration. 5 The moel I propose can fit both backwar an forwar integration, becase the integration ecision is moele with two istinctive featres: first, the integration ecision is enogenos an epens on the realization of the inpt market price: if it is higher than the partner s cost, then the ownstream integrate firm oes no longer sorce inpt in the market an it flly exploits its affiliate proction capacity; otherwise, it prchases the inpt on the market; secon, the pstream integrate firm can eventally spply to the ownstream inepenent competitor, bt, at the same time, it mst maximize the ownstream integration profits. Vertical integration on the one han may have positive effects on consmer welfare, becase it reces (or eliminates) oble marginalization an increases the final goo qantity available on the market; on the other han, given the limite nmber of firms involve in the market an the asymmetric pstream an ownstream Cornot competition, may have strong effects on competition. This paper provies three general reslts. The first concerns the participation of an integrate pstream firm to the inpt market, following the merger: a vertically integrate pstream firm withraws from the market if an only if the post-merger ownstream mark-p is greater than the postmerger pstream one (Proposition ). Therefore, an integrate pstream firm may renonce to serve its affiliate s competitor if the market withrawal helps obtaining higher ownstream profits. It implies that, espite the evental post-integration pstream profitability, an inpt foreclosre strategy may be chosen in orer to maximize the joint integration profits. Seconly, the final goo qantity, an conseqently the consmer welfare, increases after the merger if the efficiency gain (i.e. the cost rection) of the integrate ownstream firm is greater than the efficiency loss (i.e. the inpt price increase) face by its rival. Eqivalently, the vertical integration benefits consmers if the average ex-post merger instry cost of inpt provision is lower than the vertical separation one. Proposition 6 states the general conitions ner which consmer welfare increases after the integration, both when the inpt foreclosre strategy is playe, or not. These conitions involve the (observable) vertical separation qantities an the final goo an inpt eman fnction, pre- an post-integration. Therefore ner some assmption on the final goo eman form the conitions can provie a preview of the implications of the vertical integration. In thir place, I show that each firm wants to vertically integrate with a low (i.e. efficient) cost partner (Proposition 7). owever, if the vertical integration involves the low cost (i.e. big) ownstream firm, the pstream partner will always play an inpt foreclosre strategy. Sch strategy may force the remaining firms to exit the markets, whenever their cost are too high (Proposition 8). Using these three reslts an assming an explicit final goo inverse eman fnction, in Section 5 I sty when a policy too restrictive towars sch type of mergers may lea to scenarios that o not enhance consmer welfare, an when allowing a too efficient vertical integration may make the inepenent competitors exit the market, 5 Doble marginalization exists when there is market power p- an ownstream pre-merger. Chrch (008), Vertical Mergers, Isses in Competition Policy. 3

4 leaing to a market monopolization. The paper is organize as follows. Section 1.1 briefly smmarizes the relate literatre. Section introces the moel an shows the characterization of the eqilibria, both in the vertical separation an in the vertical integration settings. Section 3 erives an analyzes the pstream-ownstream eqilibrim following a vertical integration. Section 4 is the core of the paper an provies the main reslts abot consmer welfare, competition, an partnership choice. Section 5 provies some examples of the effects of vertical integration ner the assmption of linear final goo eman. Section 6 concles. 1.1 Relate Literatre A very large literatre focses on the effects of a vertical integration an a nmber of papers are relate to my analysis. The theories evelope are mainly three. The first one, raise between the 1950 s an the 1970 s an known as (cstomer or inpt) foreclosre strategy theory, asserts that a vertical merger reces the qantity of intermeiate goo exchange in the market, an, conseqently the final goo qantity, by precling the inepenent firms the access either to a spplier or a byer. Contrariwise, the Chicago School approach points ot the beneficial effects an the pro-competitive implications of a vertical integration. A thir approach, theorize first by Salop an Scheffman (1987), focse on market strctres, reckons that a vertical integration neither anti- nor pro-competitive per se, bt it can have anti-competitive effects ner some conitions. Salop an Scheffman (1983) also pose a new theory of harm, ifferent from the foreclosre theory, showing that a vertically integrate firm can have incentive to provie inpt to the competitors, bt at a higher price (raising rivals cost theory). Salop an Scheffman (1987) has inflence the ensing research. Salinger (1988) moels the interaction between a ownstream Cornot market an a perfect competition pstream market, where nintegrate an integrate firms coexists. In this setting, a vertical integration can both increase the final goo proction an strengthen the pstream inepenent firms market power. art an Tirole (1990) sty the conitions ner which market foreclosre is a conseqence or a prpose or both of the integration. They consier three possible variants of the moel (ex-post monopolization, scarce nees an scarce spplies), in a Bertran context, assessing possible social losses an gains. Orover et al. (1990) show that the vertical integration changes the integrate entity s incentives to spply to competitors, concling that vertical foreclosre arises in eqilibrim. Chen (001), biling on Orover et al. (1990), shows that the vertical integration changes not only moifies the integrate firms incentives to serve the rivals, bt also the competitors incentives in choosing the inpt sppliers. Moreover, he proves that ner some conitions, the vertical integration may facilitate collsion between firms. Behler an Schmtzler (005) show that efficient ownstream firms are more likely to vertically integrate. owever, they assme that the post-merger pstream withrawal an symmetric pstream costs. In my paper, the absence of any assmption regaring the integrate pstream post-merger withrawal is a key featre of the moel in nerstaning the pro- an anti-competitive implications of the operation. Gaet an Long (1996) consier a Cornot bilateral oligopoly framework, assming linear eman. In their moel, all firms proce at no cost bt a positive intermeiate goo price is paye from ownstream to pstream firms in orer to prchase the reqire inpt. In their paper, as in Salinger (1988), the only sorce of asymmetry is the nmber of firms active in each market. Their prpose is to show that vertical 4

5 integration affects the markets epening on the nmber of pstream an ownstream firms active in it. They also allow the integrate firms to trae with ownstream an pstream nintegrate firms. In this sitation, srprisingly, an integrate firm may ecie to prchase some inpt from inepenent pstream firms at a price that excee their marginal pstream cost. Choi an Yi (000) bil p a moel of vertical foreclosre where the pstream opoly competes in price, while the ownstream opoly can compete both in qantity or price, an the choice of inpt specificity is enogenosly erive. Inpt costs are ranomly realize, ownstream proction costs are nll. If the pstream ivision of the integrate firm provies a specialize inpt, the vertical integration can case foreclosre. Gans an De Fontenay (004) analyze the effects of a vertical integration in markets with pstream competition or monopoly. They show that vertical integration can affect the joint payoff of integrating parties in ex post bargaining. When there is pstream competition, the bargaining effect is stronger, while ex-post monopolization is more likely to occr ner pstream monopoly. Finally, Gans (007) an enricks an McAfee (010), bil p a measre of vertical concentration in orer to assess market power changes, moving from the Salinger s moel. Both papers provie an empirical analysis (horizontal an vertical) of a merger between two same size oil firms, Exxon an Mobil, in orer to preict the concentration change in refining (pstream) an retailing (ownstream) gasoline market. The latter work give the inspiration for some featres of the strctre of my moel. General framework of the moel an characterization of eqilibrim Consier a market in which the pstream firms proce a homogeneos intermeiate goo an spply it to ownstream firms. The ownstream procers transform the inpt into a final goo, sing a fixe coefficient technology. Both markets are Cornot opolies an all firms face an inivial non-negative exogenos cost. Let D an U respectively be the generic ownstream an pstream firms. The inexes, can take vale L or accoring whether the cost is low or high. The inivial proction cost characterizes the firm s size. Therefore, there is a big firm (D L, U L ) an a small firm (D, U ) in each proction stage. Each D proces a non-negative qantity q of final goo at constant marginal cost c, with = L,. The final goo eman is a fnction of the aggregate qantity Q (with Q = =L q ) an the price p. Assme that the inverse eman fnction p(q) has the following properties: i) p(q) is a ecreasing fnction in the argment: p (Q) < 0; ii) the ownstream firm s marginal revene fnction is a eclining fnction of the competitors proction: p (Q) + p (Q)q 0; iii) there exists a positive aggregate qantity Q sch that the final goo price p(q) is positive. 6 Each U proces an intermeiate goo qantity x an faces a constant marginal cost of proction f, with = L,. The aggregate spply schele is fnction of the total inpt proction X (where X = =L x ) an of the intermeiate goo price r, i.e. X(r). The pstream procers are able to anticipate that the inpt price clearing fnction r(x) lies in the set of ownstream Cornot eqilibria. The inpt market is moele as follows. On the one han, ownstream procers face a final goo eman p(q) an nee the intermeiate goo X in orer to proce the inivial qantity of final goo, q, for = L,. On the other han, pstream 6 By Novshek (1985), ner these conitions, the existence of Cornot eqilibrim in a market for a single homogeneos commoity is ensre. 5

6 U L, U jointly sbmit the inpt spply schele X(r) U L, U anticipate that D L, D jointly the inpt price clearing sbmit the inpt fnction r(x) is in the set of eman schele Q(r) Downstream Cornot Eqilibria Inpt Market clears for r s.t. Q(r ) = X(r ) Upstream Downstream Proction Proction Figre 1: Timeline of the moel. procers anticipate that the inpt market clearing price fnction is well efine fnction in the set of ownstream Cornot eqilibria. Then ownstream procers sbmit an aggregate inpt eman schele Q(r) to an (invisible) walrasian actioneer, while, simltaneosly, the inpt proviers sbmit her their aggregate spply schele X(r). The actioneer matches the inpt eman an spply an fins the market clearing price, r. Figre 1 escribes the timing of the moel. 7. The eqilibrim concept we consier will be the same throgh the whole paper. We efine the Nash-Cornot eqilibrim for the moel as follows. Definition 1. The Nash-Cornot eqilibrim of the moel is a vector of optimal intermeiate goo qantities (x L, x ), optimal final goo qantities (q L, q ) an an intermeiate goo eqilibrim price r, sch that, for r, every firm maximizes profits, an the intermeiate goo market clears. The soltion of the moel is achieve by backwar inction, both in the vertical separation an in the vertical integration scenarios..1 Vertical separation Consier the vertical separation scenario, in which each firm inivially maximizes its profits. The generic D firm s profit maximization problem is: max q 0 πvs = [p(q) c r]q, = L, (1) Uner vertical separation, the first orer conition of a generic ownstream firm D is the following: π vs q vs = p (Q)q vs + p(q) c r = 0, = L, () By the stanar Nash-Cornot analysis, the solving qantities are fnction of the ownstream marginal costs an of the intermeiate goo price r: q vs = qvs (c, c, r), = L,. Both firms eman inpt, therefore the inpt aggregate eman schele is Q vs (r) = i=l qvs (r). The market mechanism eqates the inpt eman an spply scheles, therefore, by imposing the market clearing conition q vs (c, c, r) = X, the intermeiate goo eman is a ecreasing fnction of ownstream costs an qantity: r(x vs ) = Q 1 (X; c, c ). The inpt eman inherits the final goo eman properties (r (X) < 0; r (X) + r (X)X 0, X > 0 s.t. r(x) > 0). 7 Given that r(x) = q 1 (X) an, by the market clearing conition, Q(X) is an increasing relation, then r (X) < 0, r (X)+r (X)X 0 an X > 0 s.t. r(x) > 0. Therefore, also the pstream Cornot market eqilibrim exists. 6

7 The generic pstream firm s profit maximization problem is then is: max x πvs = [r(x vs ) f ]x, = L, 0 The first orer conition with respect to the qantity x : π vs x vs = r (X vs )x vs + r(x vs ) f = 0, = L, (3) The Nash Cornot eqilibrim qantities are all fnctions of the exogenos parameters (c L, c, f L, f ). Denote by {x vs L, x vs } the pstream eqilibrim qantities, by {ql vs, q vs } the ownstream eqilibrim qantities an by r vs the optimal inpt price sch that every firm maximizes its profits an the intermeiate goo market clears, i.e. X vs = Q vs. The oble marginalization e to the p- an ownstream market power limits the qantity proce an it raises the final goo price.. Vertical Integration between D an U Consier now a single vertical integration between two firms. Withot loss of generality, let s enote as D an U the merging firms, an call D an U the remaining inepenent firms. The vertical integration allows the firms involve to transfer inpt internally at no cost an to share the joint profits. In orer to respect the timing of the moel an to leave the market mechanism nchange, let s assme that the proction ivisions of the new entity cannot commit ex-ante on the inpt qantity to exchange in-hose. The ownstream integrating procer D maximizes its profits by choosing the minimal cost between the complete in-hose inpt proction an the prchase on the market. Therefore, he will flly exploits the pstream integrate capacity, when the intermeiate goo price realize by the market is greater than its affiliate marginal cost; he will by the inpt on the market, otherwise. Uner vertical integration scenario, the generic ownstream integrate firm s profit fnction becomes the following: π vi = [p(q) c min(r, f )]q (4) Differently from most of the previos literatre I o not state an assmption on market withrawal by U. This featre provies a triple avantage to the moel. First, it is possible to fin what type of vertical integrations leaves the pstream integrate procer active on the market; secon, in this way, the moel can mimic both backwar an forwar vertical integrations. Finally, the moel allows to sty how the pstream an ownstream integrate firms proction incentives change. Let s then assme that U maximizes the joint profits of the vertically integrate entity accoring the following formla: max π vi }{{} x 0 U s V.I. profits = [r(x vi ) f ] x }{{} Inpt market profits + π vi (r(x vi )) }{{} Downstream affiliate s profits (5) Coherently to the vertical separation setting, ner vertical integration, the invisible walrasian actioneer matches inpt spply an eman an fins the market clearing price, accoring to what type of vertical merger takes place 8. Depening on the internal an external eman, the pstream integrate firm ecies whether procing for the market or not. Therefore, let s isentangle two ifferent 8 The extensive soltion of each scenario can be fon in the Appenix. 7

8 sitations: in the first, when r(x)) > f, the ownstream affiliate flly exploits the partner s proction capacity; in the secon, when r(x) < f, the integrate ownstream firm still aresses itself to the market an the pstream integrate firm s proction for the market represents a profit loss Vertical integration between D an U if r(x) > f Sppose that r(x) > f. The ownstream market first orer conitions are the following: π vi = 0 p (Q)q q vi vi + p(q) c f = 0 (6) = 0 p (Q)q vi + p(q) c r = 0 π vi q vi By the stanar Nash-Cornot analysis, the optimal qantities are fnctions of own costs an the competitor s costs: q vi = qvi (c, c, r, f ) an q vi = qvi (c, c, r, f ). By the assmption on integrate firm cstomer foreclosre, only the inepenent firm D emans inpt on the market. Therefore, by imposing the market clearing conition, q vi (r) = X, the intermeiate goo eman is r(xvi ) = q 1 (X; c, c, f ). 10 Before solving the pstream firms maximization problem, it mst be notice that each nit of inpt proce by the integrate pstream firm, ecreases its partner s proction. In fact, sing the Chain Rle an the eman properties: q vi x vi = π vi q vi qvi q vi r(x) r(x) x vi < 0 The nintegrate firm U still maximizes its profits as in vertical separation, therefore the first orer conition is eqivalent to eqation (3): π vi x vi = 0 r (X vi )x vi + r(x vi ) f = 0 Conversely, by eqation (5), the integrate pstream firm U solves the following maximization problem: max x 0 πvi = [r(x vi ) f ]x + [p(q) c f ]q (r(x vi )) Since D oes no longer sorce inpt on the market, q vi (r(x)) is increasing in the inpt price 11. The first orer conition for the integrate pstream firm elivers the following relation between marginal revenes an costs: 1 ɛ q vi r (X)x vi + r(x vi ) = f p (Q vi ) q vi (r(x)) + /xvi q vi x vi [p (Q vi ) q vi (r(x)) + p(q vi ) c f ] (7) 9 This case is clearly nrealistic, becase the pstream integrate firm s proction increases not only the the affiliate s proction, bt also the competitor s one. Proposition 1 rles ot this scenario showing that there is no incentive to vertically integrate ner this marginal cost-price conition. 10 Notice that this is a ifferent fnction from the vertical separation one. 11 An increase of the inpt price reces the qantity proce by D s competitor, i.e. D. 1 Recall that, in this case, the market clearing conition is q vi (r(x)) = Xvi, becase D oes not eman inpt on the market, an therefore q vi r(x vi ) r(x vi ) x vi = Xvi r(x) r(x vi ) x vi = 1. 8

9 where ɛ q vi is the elasticity of D s proction to the affiliate U /xvi s proction, ner this type of vertical integration, an it is negative. 13 The pstream integrate firm faces two aitional costs, represente by the last two terms of the left han sie of eqation (7). The first term escribes the effect on the ownstream profits of a ecrease in the intermeiate goo price: an increase in x vi ecreases the intermeiate price an reces the ownstream profits (recall that ner this scenario qvi x < 0). The secon term represents the marginal effect of a ecrease of vi the final goo price on the ownstream qantity proce. Therefore, U s participation to the market epens the level of proction that eqates the marginal revenes coming from the inpt market, to the marginal costs of proction an of being integrate with D. In eqilibrim, the D s first orer conition hols with the eqality sign, an by sbstitting it twice, the optimal x vi mst satisfy the following relation: { x vi > 0 if r (X vi )x vi + r(x vi ) = p(q vi ) c x vi = 0 otherwise ence, in eqilibrim, U participates to the market only if the market marginal revenes are high enogh to eqal the ownstream price-marginal cost ifference. Otherwise, U has no benefit from serving the market. This is an important relationship in the paper. In fact, accoring to (8), U, as part of the integrate firm, may prefer to not spply its partner s competitor espite the positive profits eriving from the inpt market. The evental foreclosre of the inpt may affect both markets. Section 3 will explore the implications of it. (8).. Vertical integration between D an U if r(x) < f To the contrary, consier a vertical integration ner the assmption that r(x) < f. In this case, the ownstream integrating firm D oes not benefit from the integration with U, an then it minimizes its costs by prchasing the inpt goo on the market (recall eqation (4)). ence, at ownstream level, nothing changes from the vertical separation scenario: both procers eman inpt to the pstream market. 14 At pstream level, if the integrating firm U participates to the market, it selles inpt at a price lower than the marginal cost by assmption. Accoring to eqation (5), U s maximization problem ner this scenario is: max x 0 πvi = [r(x) f ]x + [p(q) c r(x)]q (r(x)) (9) From the first orer conition πvi x vi = 0, the following relation can be obtaine: r (X)x vi + r(x) f }{{} = F.o.c. of U ner V.S. r(x) x vi q vi (r(x)) }{{} Positive irect effect on q + q (r(x)) r(x) r(x) x vi [p (Q)q vi (r(x)) + p(q) r(x) c ] }{{} = F.o.c. of D + + q (r(x)) r(x) r(x) x vi p (Q)q (r(x)) = 0 (10) }{{} Negative strategic effect on q 13 For frther etails on the elasticities check the Appenix. 14 Differently from the previos sbsection, where the market clearing conition was the matching between D s eman an the spply, i.e. q vi (r) = X, ner this scenario the market clearing conition is the vertical separation one, therefore both ownstream firm proctions epen negatively from the price, Q(r) = X, where Q(r) = q vi (r(x)) + qvi (r(x)) 9

10 From this relation it can be arge that, by serving the market, the integrate firm benefits not only its partner D, bt also the competitor D In fact eqation 10 is compose by for elements, respectively: the inpt market profit maximization conition, the final goo market profit maximization conition, the irect effect of a rection in the inpt price on the D s proction, an the strategic effect on D s proction of an increase of its competitor s qantity e to the ecrease in inpt price, weighte by the ecrease in final goo price. With respect to the U s first orer conition ner the r(x) > f (eqation (7)), an increase in the optimal x vi, ner the r(x) < f, benefits also the inepenent ownstream firm. Then, by the properties of r(x), p(q), an q (r(x)) r(x), the irect effect on q increases the profits, while the strategic effect on q ecreases it. Eqivalently, introcing the elasticities of ownstream proction to x vi, the following expression can be obtaine: r (X)x vi + r(x) r(x) ɛ q vi x vi /xvi q vi q vi (r(x)) = ɛ q vi /xvi q vi x vi x vi In eqilibrim, D s first orer conition eqals zero, then: r (X vi )[x vi or eqivalently: q vi ] + r(x vi ) = f + ɛ q vi p (X)q (r(x))f + [p (Q)q vi (r(x)) + p(q) r(x) c ] + f (11) /xvi r (X vi )x vi + r(x vi ) f = r(x) }{{} x vi Negative becase r(x) < f by assmption }{{} Negative q vi x vi q vi [p(q vi ) r(x vi ) c ] }{{} D s Mark-p [1 p (X vi ) q (r(x)) ] r(x) Therefore x vi > 0 only if the impact of the rection of the inpt price benefits the ownstream partner D more than inepenent competitor D, meaning that the strategic effect oes not affect too mch the D s mark-p, that happens if 1 > p (X vi ) q (r(x)) r(x). In other wors, the benefits of being integrate with D mst be high enogh to cover the negative profits of being active on the market an of serving the inepenent competitor.from a strategic point of view, if sch a merger involves the low cost firm U L, the integration wol case the exit of the inefficient pstream firm U. Paraoxically, a vertical integration of this type wol floo the intermeiate goo market of low price inpt, allowing the ownstream firms to increase the final goo proction. The final effect wol therefore benefit consmers. owever, a vertical integration ner this assmptions is not profitable for the firms involve, as showe in next Section. 3 Eqilibrim analysis A first reslt important to state is that vertical integration cannot take place in eqilibrim whenever f > r(x). In fact, this type vertical integration wol not be profitable for the merging firms: the integrate pstream firm wol have negative profits from the inpt market, an the sbseqent ecrease in the inpt price wol benefit not only the ownstream affiliate D, bt also the inepenent rival D. owever, as a paraox, a vertical integration this scenario wol make the market eliver the highest possible final goo qantity, since U wol serve the market gaining negative profits, 10

11 nless U achieves positive profits. This reslt can be easily achieve by comparing the integrate firms profits ner merger when f > r(x), an the vertical separation ones. Firms have no incentives to vertically integrate in this case. 15 Proposition 1. Uner the assmptions on firms inivial costs, a single vertical integration oes not take place if f > r(x). The secon reslt neee in orer to provie the eqilibrim analysis is the one concerning the participation of U. Next proposition states that, whenever the ownstream post integration mark-p is greater than the pstream post-integration mark-p, U withraws from the inpt market in orer to allocate its fll capacity internally. Uner vertical integration, the pstream integrate firm U has the task to maximize the joint profits eriving from the integration. Uner the assmption that r(x) > f, the ownstream integrate firm D flly emans the inpt neee internally. The self-spply is not irectly observable, bt it eqates D s proction, q vi. By the sal Cornot properties, q vi is a fnction of the firm s mark-p: q vi = p(q vi ) c f p (Q vi ). If D bears a high proction cost c, the internal eman of inpt will be low, an U may have some capacity left to spply the market. Then U mst balance the rection in ownstream profits from spplying to its partner s competitor with the increase in pstream profit. An important avantage from this approach is that the moel can cover both backwar an forwar vertical integration. Proposition. The integrate pstream firm U withraws from the intermeiate goo market if ownstream partner s market mark-p is greater than its own market mark-p. Proof. From the first orer conition of U ner the r(x) > f vertical integration, an recalling eqations (4), in eqilibrim, x vi mst solve the following expressions: { x vi >0 if r (X vi )x vi + r(x vi ) = p(q vi ) c x vi =0 otherwise By smming an sbtracting the pstream inivial marginal cost f, an rearranging the eqation, the following necessary an sfficient conition can be state: x vi > 0 if an only if x vi = [r(x vi ) f ] [p(q vi ) c f ] r (X vi > 0 (1) ) The nmerator of expression in (1) emboies the ifference between the post-merger pstream mark-p an the ownstream partner s mark-p after the vertical integration. Given that r(x) is a ecreasing fnction, the nmerator of the fraction mst be smaller than zero: [p(q vi ) c f ] [r(x vi ) f ] < 0, in orer to have a positive proction. Therefore, the reslt is achieve: { x vi >0 if p(q vi ) c f < r(x vi ) f x vi =0 if p(q vi ) c f (X vi ) f By the Khn-Tcker conitions the relation is necessary an sfficient. Proposition also efines the inverse relationship between the final goo qantity proce by D an the intermeiate goo qantity sol by U in the intermeiate goo market. 15 The formal proof is state in the Appenix. 11

12 Lemma 1. Uner the assmption r(x) > f, consier a vertical integration between a generic ownstream D an a generic pstream U. The qantity proce by the integrate firms at pstream an ownstream level are inversely proportional. x vi = r(x vi ) f r (X vi ) p (Q vi ) vi r (X vi q ) With these reslts it is then possible to analyze the effects of the vertical integration on the pstream an ownstream market. 3.1 The ownstream market after a D U vertical integration At ownstream level, the effect of the vertical integration can be analyze with the traitional Cornot comparative statics analysis. In fact, the rection in the integrate firm s cost of laying in spplies of the inpt makes D s an D s best response fnctions move in parallel an opposite sense: while the integrate firm s best response fnction shifts pwar, the nintegrate firm s one shift ownwar. Therefore, on the one han the rection in the cost of inpt allows D to increase its final goo proction, on the other han reces D s proction. Moreover, the nintegrate firm s proction D s proction can ecrease even more if the postmerger inpt price is higher than the vertical separation one. Proposition 3. Uner the assmption r(x) > f, consier a vertical integration between a generic ownstream D an a generic pstream U. The vertical integration increases the final goo qantity proce by the ownstream integrate firm D. Given that only the nintegrate firm D is left to eman inpt, after the vertical integration, the inpt eman schele shift ownwars, becase it eqates the only inepenent ownstream procer s (D ) best response fnction. By the Nash- Cornot properties, the inpt eman schele preserves the same slope of the vertical separation scenario s eman. Lemma. Uner the assmption r(x) > f, consier a vertical integration between a generic ownstream D an a generic pstream U. The vertical integration makes the inpt eman shift in a ownwar irection. Then it is also possible to fin the relation between the optimal final goo qantities proce by D an D. Lemma 3. Uner the assmption r(x) > f, consier a vertical integration between a generic ownstream D an a generic pstream U. In eqilibrim, the optimal qantity proce by D can be expresse as fnction of : 1. The ownstream verticals separation proction ifference an of the inepenent pstream firm U s post-integration proction, if the integrate pstream firm U keeps participating to the market: q vi = p (Q vs )[q vs vs q ] r vi (X vi )x vi p (Q vi ) r vi (X vi ) if x vi > 0 (13). The inepenent pstream firm U s post-integration proction, if the integrate pstream firm U j withraws from the market: q vi = x vi if x vi = 0 (14) 1

13 If the final goo eman is linear, the slopes of the eman fnctions - p ( ), r vi ( ), r vs ( ) - are constant an inepenent from the optimal qantities, an r vi ( ) = r vs ( ), then the expressions in eqations (14) an (13) become jst fnction of the inivial vertical separation qantities. Lemma 4. Uner the assmption r(x) > f, consier a vertical integration between a generic ownstream D an a generic pstream U. In eqilibrim, the optimal qantity proce by D can be expresse as fnction of the ownstream an pstream vertical separation proction ifferences an of the nintegrate pstream firm U s postintegration proction. 1. If the integrate pstream firm U keeps participating to the market: q vi = + p (Q vs )r vi (X vi ) vs p (Q vi )[p (Q vi )+r vi (X vi )][q q vs] rvi (X vi ) vs p (Q vi ) [x r vi (X vi )[p (Q vi )+r vi (X vi )] p (Q vi )[p (Q vi )+r vi (X vi )] x vi x vs ] + if x vi > 0 (15). if U withraws from the market: q vi = p (Q vs ) vs p (Q vi ) [q + q vs] rvi (X vi ) vs [x p (Q vi ) r vi (X vi )[p (Q vi )+r vi (X vi )] p (Q vi ) x vi x vs ] + if x vi = 0 (16) An important isse is whether the consmer welfare increase after the vertical integration. The vertical merger benefits the consmer if it increases the final goo qantity proce by the instry. Sch qantity epens on the new inpt cost face by the ownstream firms. From the previos analysis, after the vertical integration, while D can increase its final goo proction, D is harme by the increase market power of the pstream firms. Therefore, the final goo qantity increases when the average post-merger cost face by the final goo procers to lay in spplies of the intermeiate goo is lower than the vertical separation price, or, eqivalently, when the post-merger efficiency gain of D is greater than the post-merger efficiency loss of D. The conitions are: Q vs <Q vi, p(q vs ) > p(q vi ) if r vs (X vs ) > r vi (X vi ) + f Q vs >Q vi, p(q vs ) < p(q vi ) if r vs (X vs ) < r vi (X vi ) + f (17) In other wors, the final goo qantity proce by the instry increases after the vertical integration whenever the efficiency gain of being integrate obtaine by D is greater than the post-merger inpt price increase: i.e. r vs f > r vi r vs. { Q vs <Q vi, p(q vs ) > p(q vi ) if r vs (X vs ) f > r vi (X vi ) r vs (X vs ) Q vs >Q vi, p(q vs ) < p(q vi ) if r vs (X vs ) f < r vi (X vi ) r vs (X vs ) (18) From this relations, I can fin the post-integration optimal inepenent firm U s proction that allows the vertical integration between U an D to be consmer welfare enhancing. 13

14 Lemma 5. Uner the r(x) > f conition, consier the vertical integration between a generic pstream firm U an a generic ownstream firm D. The vertical integration is consmer welfare enhancing if the inepenent pstream firm U proces a inpt qantity that is greater than the sm of the pre-integration qantities weighte times the final goo an inpt eman slopes. Accoring to U participation to the inpt market: 1. If the integrate pstream firm U keeps participating to the market, i.e. x vi > 0, then Q vi > Q vs if an only if x vi is sch that: x vi = p (Q vi )[p (Q vi )+r vi (X vi )] p (Q vs )[p(q vi ) r vi (X vi )] r vi (X vi )[3p (Q vi )+r vi (X vi )] q vs + p (Q vs )[r vi (X vi ) p(q vi )]+p (Q vi )[r vi (X vi )+p (Q vi )] r vi (X vi )[3p (Q vi )+r vi (X vi )] q vs + r (X vs )[p (Q vi )+r vi (X vi )] vs r vi (X vi )[3p (Q vi )+r vi (X vi )][x x vs ] > 0 (19). if U withraws from the market, i.e. x vi is sch that: x vi = 0, then Q vi > Q vs if an only if x vi = q vs [p (Q vi ) p (Q vs )]+q vs [p (Q vi )+p (Q vs )] r (X vs )[x vs vs x ] p (Q vi )+r vi (X vi ) > 0 Notice that, if the final goo eman p( ) is linear, the slopes of the eman fnctions are inepenent from the qantities proce. Let s now consier the effects of the vertical merger on the pstream market. 3. The pstream market after a D U vertical integration After the vertical integration, the integrate firm oes not eman anymore inpt to the market. Therefore the new inpt eman face by the pstream firms is lower: as a final reslt, the optimal inpt qantity exchange on the market ecreases. owever the effect on the optimal price is not nambigos. In fact, if the integrate pstream firm U participates to the market, then the pstream market cost strctre increases, e to the new relation within the vertically integrate firms (Proposition 4).The increase or ecrease in the optimal inpt price epens on the eman variation, represente by the ownwars eman shift, an on U s cost increase, represente by the impact of the integrate firm self-spply. Conversely, if U withraws from the market, then the inepenent pstream procer U can charge monopoly price on the nintegrate ownstream procer D. This competition effect may raise the optimal inpt price above the vertical separation level whenever it is greater than the eman variation. Proposition 4. Uner the assmption of r(x) > f, consier a generic vertical integration between U an D. The optimal intermeiate goo qantity X exchange in the inpt market ecreases after the vertical integration. The effect on the price r(x) is not nambigos. Finally, before concling this Section on the p- an ownstream vertical integration eqilibrim, Proposition 5 states that a vertical integration can take place in eqilibrim an it happens if an only if when the post-merger price is greater than the marginal cost of U, i.e. r(x) > f. The reslt is consistent with Chen (001). Proposition 5. A vertical integration takes place in eqilibrim if an only if the post integration inpt price is greater than the pstream integrating firm marginal cost, i.e. r(x) > f. 14

15 4 Vertical Integration Matchings an Competition Isses This section presents the main reslts of the paper abot consmer welfare an competition. The vertical integration is likely to be cleare if the merger increases the final goo qantity proce (or eqivalently it ecreases the final goo price). Proposition 6 presents the conitions ner which a vertical integration is beneficial to consmers, both whether the integrate firm plays an inpt foreclosre strategy or it oes not. These conitions epens on the vertical separation optimal qantities an on the final goo an inpt emans, pre- an post-integration. Therefore, Proposition 6 sets a threshol ner which the vertical integration can be cleare. Proposition 6. Consier a vertical integration between a generic pstream U an D. The vertical integration increases the consmer welfare, i.e. Q vi > Q vs if an only if: 1. If U keeps participating to the market: p (Q vi ) r (X vs ) p (Q vs )[p(q vi ) r vi (X vi )] r (X vs )[p (Q vi ) + r vi (X vi )] > x vs q vs x vs q vs (0). If U withraws from the market: q vs [p (Q vi ) p (Q vs )] + q vs[p (Q vi ) + p (Q vs )] r (X vs > x vs x vs (1) ) Provie that the vertical separation eman slopes an optimal qantities are observable, in eqilibrim the only nknown variable of eqation (0) is the inpt eman slope r vi (X vi ) of the nintegrate ownstream firm D. This is an implicit measre of the level of proction that the instry mst reach ner the vertical integration. Given that the integrate ownstream increases its proction after the merger (Proposition 3), the formla gives a proxy of the level of proction that D mst maintain when the integrate firm D U oes not play inpt foreclosre strategy. The formla is slightly simpler if U plays inpt foreclosre strategy. In fact in eqation (1) both the vertical separation optimal qantities an the final goo an inpt eman slopes are known. It mst be notice that, if the final goo eman p( ) is linear, then the slopes of all the eman fnctions are inepenent from the qantities proce, an constant. Regaring the partnership choice, each firm can vertically integrate with a similar size partner (positive assortative matching), or join a ifferent size one (negative assortative matching). ence, on aggregate, for matchings are possible: two are positive assortative - big ownstream with big pstream (enote this case as D L U L ), small ownstream with small pstream (D U ) - an two are negative assortative - big ownstream with small pstream (D L U ), small ownstream with big pstream (D U ). Next Proposition states that firms prefer to vertically integrate with efficient partners. Therefore, in eqilibrim, the vertical integration matching that wol take place is the low cost (i.e. big firms) one, D L U L. Proposition 7. Uner the assmption on firms inivial cost, the eqilibrim vertical integration involves the low cost firms, i.e. D L an U L. 15

16 Proof. Consier the joint profit ner of a vertically integrate cople of firms, U an D : π vi= (c, f, r vi (X vi )) = [p(q vi ) c f ]q vi + [r vi (X vi ) f ]x vi () Accoring to the assmption on the inivial costs, big firms face low costs, while small firms bear high cost: at ownstream level c L < c, while at pstream level f L < f. Sppose that in eqilibrim the optimal proction of the integrate pstream firm U is zero, i.e. x vi = 0. Therefore the integrate pstream firm U oes not sell inpt to its partner s competitor. Since the pstream an ownstream competitors cost c, f affect only the final goo qantity (negatively), an the integration profits are inepenent of the intermeiate price r vi (X vi= ) - that is fnction of the for cost parameters - then they can be expresse in the sal Cornot way: π vi= (c, f ) = [p(qvi= ) c f ]. p (Q vi= ) Then, by reveale preferences, the integration profits will be greater as the cost bear by the integrate firms are lower. Therefore: πl vi=ll π vi=ll L (c L, f L ) > π vi=l (c L, f ) > π vi= (c, f ) (c L, f L ) > π vi=l (c, f L ) > π vi= (c, f ) L Moreover, if x vi = 0, the remaining inepenent pstream firm U is allowe to charge the monopoly price to the only byer left, D. Therefore, whenever these two inepenent firms are the inefficient ones, U, D, the inpt price is the highest, becase of the low qantity emane by D an the increase market power of U. Sppose now that in eqilibrim x vi > 0, meaning that the integrate pstream firm U oes sell inpt to its partner s competitor. Then, by Lemma 1 an Proposition, an recalling the market clearing conition X vi = q vi, eqation () can be conveniently rewritten as follows: π vi= (c, f, r vi (X vi )) = = [p(qvi ) c f ] p (Q vi ) + [p(qvi ) c f ] r vi (q vi ) p(q vi ) c f r vi (q vi ) Now, if the cost c, f bear by the ownstream ivision of the integrate firm D increase, then D can increase its proction, an the inpt price gets lower. 16 Therefore, on the one han D has incentives to integrate with a low cost partner rather than an inefficient partner in orer to increase the final goo proction; on the other han U has incentive to integrate with a low cost partner in orer to enjoy a higher market power an price. 17 Then, by reveale preferences, an given the properties of the inpt price fnction: πl vi=ll (c L, f L, r vi=ll ) > π vi= (c, f, r vi= ) = L, an = L, sch that LL 16 By the traitional Cornot analysis q vi is increasing in D s costs. Then r(q vi ) is ecreasing in the qantity, therefore as q vi increases, the enominator r (q vi ) ecreases. 17 Notice that if c, f increase than [p(qvi ) c f ] ecreases more qickly than p(q vi ) c f increases in the same cost variables. r (q vi ) r (q vi ) 16

17 This last reslt expans the conclsion state by Behler an Schmtzler (005) abot the willingness of ownstream firms in vertically integrating. Provie that D L U L is the vertical integration that firms wol prefer, Proposition 8 sties the effect of sch a merger on competition. Given that a vertically integrate ownstream firm flly exploits its partner s capacity, the pstream an ownstream competition may be affecte. In fact, any vertical integration that involves the low cost ownstream firm inces the pstream integrate firm to withraw from the market. This may imply the exit of the remaining competitors an a rection in consmer welfare. Proposition 8. Uner the assmption on firms inivial cost, consier a vertical integration that involves the low cost ownstram firms, D L, an a (generic) pstream partner U. Sch a vertical integration matching makes the pstream firm U withraw from the market, for any = L,. Moreover, if the remaining inepenent firms D an U are highly inefficient, they both may exit the market. Proof. Recall the eqilibrim efinition of x vi, in eqation 4. If D L vertically integrates with any U, then: { x vi=lj > 0 if r(x vi=lj ) > p(q vi=lj ) c L x vi=lj = 0, otherwise The reasoning is the following. If U wol have the capacity left to proce for the market, x vi=lj > 0, then the inpt price shol be sch that r(x vi=lj ) > p(q vi=lj ) c L, an since c L < c the inepenent an inefficient ownstream firm, D, wol pay a price so high to exit the market. In fact, it is easy to show: q vi=lj p(q vi=lj ) c r vi=lj p (Q vi=lj ) < p(q vi=lj ) c L r vi=lj p (Q vi=lj ) Then the integrate pstream offers its whole capacity to the ownstream affiliate. In this case the nintegrate pstream firm U charges the monopoly price on the inpt eman of D. owever if the pstream an ownstream costs relation hols exactly r(x vi=lj ) = p(q vi=lj ) c L then the inpt price impose by the nintegrate pstream firm U is too high for the nintegrate ownstream firm D, an they both proce zero, an exit. Therefore there exist a level of inpt price r vi (X vi=lj ) sch that the inepenent an inefficient ownstream firm, D oes not eman anymore the intermeiate goo an exits the market. ence, also the inepenent pstream firm U exits, an the integrate firm D L U L becomes a monopolistic final goo procer. Propositions 7 an 8 together sggest that a vertical integration between the two low cost firms D L, U L can ince an ex-post monopolization of the final goo market (as in art an Tirole (1990)). This eventality relies on the instry costs. In fact, if the proction cost ifference between the firms at pstream an ownstream level is small, the new entity composte by D L, U L will harly monopolize the final goo market. On the contrary, if the proction cost ifference is pretty high, the inepenent ownstream firm D will be force to exit both by the competitor s proction sol an by the market power gain of the inepenent pstream U. Competition Athorities shol then be worrie by top firms vertical integrations when the operation takes place in markets alreay heavily concentrate. Whenever the merger D L U L is not allowe, what type of vertical integration is preferable epens on the level of the inivial costs. As seen above, each firm has < 0 17

18 incentive to vertically integrate with an efficient, i.e. low cost, partner. Therefore, the matching D U is atomatically the least preferre vertical integration becase both high cost firms wol rather prefer to be integrate with an efficient partner. What vertical integration takes place between D U L an D L U epens on the inivial costs of the firms an on the final goo inverse eman fnction assme. In Section 5 I present some reslts assming a linear final goo eman. 5 An example with linear eman Along this section, let s assme a linear an ecreasing final goo eman, i.e. p(q) = 1 Q. The slopes of the final goo an inpt inverse eman fnctions are then inepenent of the qantity proce an constant: p (Q vs ) = p (Q vi ) = 1 an r (X vs ) = r (X vi ) = 3. I can then analyze the effect of the vertical integration on the consmer welfare by means of the reslts of Proposition 6: Q vi > Q vs Q vi > Q vs if an only if if an only if 3 > x vs q vs x vs q vs 4 vs q > x vs x vs 3 when x vi > 0 (3) when x vi = 0 (4) Figre proposes three examples of vertically separate markets. In the first two cases - (a) an (b) - a market is highly ominate by a firm, while the other has a more intense competition; case (c) preserves the same ownstream strctre of case (b) bt it has slightly less pstream competition. In all cases the vertical separation inpt market clearing conition is respecte: Q vs = X vs = 100. Consier the vertical integration between the two big firms, U L an D L. By Proposition 8, the new entity forme by the low cost ownstream firm an any pstream firm (i.e. low or high) will play a vertical foreclosre strategy. By means of eqation (4), it easy to check that ner case (a) the vertical integration wol ecrease consmer welfare, while ner case (b) the D L U L matching wol have a positive effect on it, by increasing the final goo qantity elivere to consmers. In case (c), again, the vertical integration between top firms wol harm consmers. Sppose that a Competition Athority prohibits the vertical integration between top firms, bt it is willing to allow a ifferent vertical integration. 1. The integration between D L U (with inpt foreclosre) is etrimental for consmers ner case (a), while beneficial ner the other two cases.. The integration between D U L is generally beneficial ner the three cases, except if the merge entity plays inpt foreclosre strategy ner case (a). 3. The integration between D U is generally beneficial ner the cases (b) an (c), bt is etrimental ner case (a). The matchings an the implications can be analyze also in analytical way. Uner the assmption of linear eman, let s normalize the low costs to zero, c L = 0 an f L = 0, an setting the high cost to a positive constant c = c an f = f, with c c, f > 0. Let s assme also the following conitions on cost parameters: 4 > f > 0 an 4 f 11 > c > 0, in orer to ensre an interior soltion ner vertical separation, by making the less efficient firms always proce a positive qantity in the vertical separation eqilibrim, i.e. x vs vs > 0, q > 0. These assmptions etermine a space of parameters that can be represente in two-imensions (c, f), as in Figre 3. 18

19 Low igh Low igh Low igh U x vs L = 90 x vs = 10 U x vs L = 55 x vs = 45 U x vs L = 60 x vs = 40 D ql vs = 55 q vs = 45 D q vs L = 90 q vs = 10 D q vs L = 90 q vs = 10 Case (a) Case (b) Case (c) Figre : Three examples of vertically separate markets. Uner these assmptions: The top firms vertical integration D L U L is a stable an it is the only stable matching;. Inpt foreclosre is always a ominant strategy for any matching; 3. The top firms vertical integration D L U L is the consmer welfare maximizing matching if the sm of the inefficient firms costs is smaller than one half, c+f 1. Over that costs, the high cost firms D an U exit the market an the integrate entity compose by the low cost firms monopolizes the final goo market, harming consmers. Therefore top firms vertical integration might be challenge by the Competition Athorities, e the evental loss in competition etermine by the nintegrate competitors exit. By Proposition 7, the vertical integration between low cost firms D U is the least preferre by firms, therefore let s consier the two negative assortative matchings (big ownstream with small pstream D L U, an, vice-versa, D U L ) as the ones likely to happen. Given that both matchings involve an inefficient firm, the comparison between them relies on the level of inefficiency of the small (pstream or ownstream) firm involve in the merger. Proposition 9. Whenever the top firms vertical integration D L U L is blocke then the matching realize will be negative assortative an the integrate firm will withraw from the market. In particlar: 1. if f > c, then the matching will be D U L ;. if f < c, then the matching will be D L U ; owever, for high levels of instry costs, these matchings can make the markets eliver a sboptimal final goo qantity becase of the inpt foreclosre strategy. Figre 3 shows the parameter conitions ner which a negative assortative vertical integration can take place, an the reslting effect on consmer welfare. The prple ashe line (f = 1 c) represents the threshol pon which the vertical integration between the low cost firms D L U L makes the inepenent firms exit. Therefore, below f = 1 c, the vertical integration matching D L U L effectively benefit consmers, also becase the inepenent competitor keep procing, while above this 18 This Section has a separate part in the Appenix. 19

20 Figre 3: Secon-best vertical integration matchings. line, D an U face too high costs an exit, an the new entity monopolizes the final goo qantity. The re line, that represents f = c, istingishes the area of parameters where D integrates with U L (above the line), or D L with U (below the line). The ble line, f = 1 5c, istingishes the effect of the inpt foreclosre strategy playe by a new entity compose by D U L on consmer welfare: if costs are below the line, the foreclosre strategy oes not harm consmers. Same resoning apply for a merger between D U L : below the green line, f = 1+7c 14, inpt foreclosre strategy oes not harm consmers. Therefore, whenever the vertical integration between the low cost (zero cost accoring the assmption) firms D L U L, is blocke, the negative assortative matchings that can take place, form for ifferent areas in the parameter space, epening also on the effect on consmer welfare an on the matching type that takes place: In the area enote by A ( 1 5c > f > c), the matching D U L takes place an it elivers the highest final goo qantity, becase c = c is not so high compare to c L = 0. In B, (c < f an 1+7c 14 > f), the matching D L U takes place an elivers the highest final goo qantity. In C, (c > f > 1 5c ), the matching D L U takes place bt it elivers a final goo qantity that the one proce by ner a D U L withot inpt foreclosre. Therefore, in this case, the vertical integration that is stable is not the consmer welfare maximizing one. In D, (f > c an f > 1 5c ), a similar reasoning applies: the stable matching is D U L bt the inpt foreclosre strategy playe reces the final goo qantity available on the market, harming consmers. Therefore, if in an instry with cost similar to the area below the prple ashe line, the vertical integration between the low cost firm D L U L is prevente in orer to favor a negative assortative vertical integration (for example D U L in A, or D L U in B), both consmers an merging firms wol benefit less of the operation, or jst 0