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09 – 10 March 2012 CESifo Conference Centre, Munich
Vertical Integration with Complementary Inputs
Markus Reisinger and Emanuele Tarantino
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Vertical Integration with Complementary Inputs∗
Markus Reisinger†
Emanuele Tarantino‡
February 2012
Abstract
We analyze the profitability and welfare consequences of vertical integration when downstream firms deal with suppliers of complementary input goods. In contrast to the received
literature, we find that vertical integration is not necessarily profitable because a complementary input provider extracts part of the larger profits earned by the integrated chain. Interestingly, this effect is particularly strong if the integrated firm is highly efficient. We also study
the incentives for a counter-merger and obtain two main results: First, the counter-merger is
generally profitable. Second, although the first merger is anticompetitive, a counter-merger is
often procompetitive and involves below cost pricing. We also show that a horizontal merger
between complementary input suppliers harms welfare. These results have profound consequences for antitrust policy.
JEL codes: K21, L15, L24, L42.
Keywords: Vertical Integration, Foreclosure, Complementary Inputs, Secret Offers, countermerger.
∗
This paper benefited from comments by Cédric Argenton, Felix Bierbrauer, Massimo Motta, Volker Nocke,
Marco Pagnozzi, Emmanuel Petrakis, Salvatore Piccolo, Patrick Rey, Nicolas Schutz, Elu von-Thadden and Gijsbert
Zwart. We thank the participants at the University of Bologna, CSEF (Naples), Max Planck Institute for Collective
Goods (Bonn) and TILEC - Tilburg University seminars, and at the 2010 Workshop for new Researchers (Centre
for Competition Policy - University of East Anglia), Jornadas de Economı̀a Industrial 2010 (Madrid), International
Industrial Organization Conference 2011 (Boston) and the European Economic Association Annual Meeting 2011
(Oslo).
†
WHU - Otto Besiheim School of Management, Department of Economics, Burgplatz 2, 56179 Vallendar, Germany. E-Mail: [email protected]. Also affiliated with CESifo.
‡
University of Bologna, Department of Economics, Piazza Scaravilli 1, I-40126, Bologna, Italy.
[email protected]. Also affiliated with TILEC.
1
E-mail:
1
Introduction
The combination of complementary inputs is a pervasive characteristic of the production process
in many industries. Downstream firms usually purchase several intermediate goods from different
wholesale markets and employ them to produce their final products. For example, in the information and communication technology sector many products are based on technological standards
and require the use of multiple specialized inputs that are produced by different firms. In addition,
high technology products can often only be produced when having access to multiple patents that
are owned by different IP holders. All these inputs—and patents—are perfect complements. Another example is the supermarket industry. Here the presence of shopping costs on the consumers’
side induces them to bundle their purchases. This creates a complementarity in the demand of
several goods which requires supermarkets to supply a large number of them.
In these industries vertical integration is also a prevalent feature. In the communication industry
several handset makers like Nokia or Sony Ericsson develop and produce many parts of their
handheld devices on their own while stand-alone firms hold essential patents for other technologies,
e.g., Qualcomm for transmission of data packages. This can also be observed in the computer
manufacturing industry, where manufacturers produce several inputs on their own but buy their
microprocessors from Intel and AMD, firms that do not produce computers themselves. Finally,
supermarket chains often offer private label products produced by integrated manufacturers and
at the same time buy other products from specialized firms that are not active in the distribution
industry.
Thus, the question arises what the consequences of vertical integration for consumers and welfare
are, and under which conditions firms find it profitable to integrate given that complementary
inputs are required. Surprisingly, although the need of two or more essential inputs is widespread,
the received literature so far has almost exclusively focussed on the case where manufacturers need
only one input. In particular, the theory of harm behind vertical integration and the resulting
conclusions on antitrust policy are based on settings where only one input is needed. A prominent
idea behind this theory is that with downstream competition it can be difficult for a dominant
upstream firm to extract the monopoly profit since it cannot commit to restrict its output to
the monopoly level. However, via vertical integration, the firm can restore its monopoly power by
foreclosing downstream rivals, thereby reducing output. This renders vertical integration profitable
but anticompetitive. The theory behind the raising rivals’ costs effect was brought forward by Salop
and Scheffman (1983, 1987), Hart and Tirole (1990), McAfee and Schwartz (1994) and has been
extensively discussed in the recent survey by Rey and Tirole (2007).1
1
Notice that the use of the term foreclosure to identify such a conduct may be misleading. The legal definition of
foreclosure is relatively broad and includes all the strategic practices undertaken by a firm to limit the competitive
pressure it faces on the market. Instead, here the term foreclosure is used for the specific practice of excessive
2
The aim of this paper is to fill the aforementioned gap in the literature. We provide a model
with complementary input producers that exert market power vis-à-vis downstream firms but also
face competition from producers of substitute goods. In this framework, we assess the profitability
and the welfare consequences of vertical integration.
The effects arising from vertical integration in an industry with complementary inputs are largely
distinct from those characterizing a model with only substitute intermediate goods. To begin with,
our analysis of vertical mergers highlights that an integrated chain is vulnerable to an expropriation
threat exerted by complementary input producers. In a set-up without complementary inputs,
integration raises the product market profits of the merging downstream unit and is therefore
profitable. The presence of a complementary input implies that part of this larger profit can
be extracted by the suppliers of this complementary input. This expropriation effect can render
vertical integration unprofitable.
The upstream pricing decisions of a vertical chain in a framework with complementary inputs
raise several interesting insights. In analogy to the case with substitute inputs, an integrated firm
aims at weakening the position of its downstream rivals via increased input prices (foreclosure
motive). This implies that integration has anticompetitive effects also in our set-up. However,
differently from the case with only substitute inputs, this foreclosure motive dominates only when
one downstream firm is integrated. Instead, if both downstream firms are integrated with a
complementary input supplier, at least one of the integrated firms sets its wholesale price below
the foreclosure level. Indeed, we show that for a wide array of circumstances it can be optimal
for a vertical chain to price its input below marginal costs, thereby rendering two vertical mergers
procompetitive compared to a single merger. Moreover, we find that if demand is sufficiently
concave two mergers can even be welfare enhancing with respect to an unintegrated industry. The
intuition behind these results is that an integrated firm that is foreclosed by its integrated rival
has an incentive to let the downstream unit of the rival produce a large amount of the final good
so to extract part of the rival’s profits via the fixed fee (production-shifting effect). To reach this
goal, the integrated firm might set the intermediate good’s price below marginal costs.
Given that the employment of complementary inputs is pervasive in many industries, our results have profound implications for antitrust policy. In particular, they show that the welfare
consequences of vertical integration crucially depend on the industry structure: if the industry is
already partially integrated, a vertical merger is likely to have a procompetitive impact. If instead
the industry is unintegrated at the outset, a vertical merger raises anticompetitive concerns.
More specifically, our framework embeds two upstream firms that provide perfectly complementary products. Each input supplier faces a competitive and less efficient fringe of firms and
formulates secret offers to downstream firms by means of two-part tariff contracts. On the downwholesale pricing at the expenses of competitors.
3
stream market, two firms compete and need both intermediate goods to produce the final good.
Finally, suppliers serve downstream firms on order and the latter produce the output good.
We obtain the following results. First, firms may abstain from integration since vertical integration can be less profitable than staying independent. Interestingly, this result occurs if an upstream
firm is “particularly efficient”, i.e., its cost advantage over the fringe firms is large. The intuition
for this result is as follows. In a framework without integration, the share of a downstream firm’s
profit that can be extracted by an efficient input provider depends on the efficiency advantage of
the provider with respect to the fringe firms. This implies that, when highly efficient, a stand-alone
upstream firm can extract a large part of the downstream rents. After an upstream firm integrates,
it internally trades its input at marginal costs, whereby losing the power to extract profits from
the affiliated downstream unit via the fixed fee. In turn, the provider of the second essential input
can fully exploit its power and obtain more profits from the integrated chain. In other words,
by staying separated, an efficient upstream firm shields some of its profits from complementary
input providers. This prediction is opposite to the one delivered by the received literature, which
concludes that vertical mergers are particularly profitable for very efficient firms, see e.g., Rey and
Tirole (2007).
We then analyze the profitability and the welfare consequences of a counter-merger. First,
we find that a counter-merger is always profitable. The newly merged firm is not hit by the
expropriation effect insofar as the first integrated chooses to (at least partially) foreclose the
downstream unit of the second integrated firm. By undertaking a foreclosure strategy, the first
integrated chain increases its downstream profit at the cost of charging a lower fixed fee to the
downstream rival. Consequently, the more the first firm forecloses the rival, the lower is the rent
that it can extract from the rival.
Moreover, we show that a counter-merger can be procompetitive. This is because, if the first
integrated firm chooses to foreclose, the second integrated does not necessarily find it optimal to
foreclose itself. The intuition relies on the interplay between the production-shifting effect and
the expropriation effect. By lowering the wholesale price charged to its downstream rival, the
second integrated can induce it to produce a larger quantity and obtain a greater product market
profit. This production-shifting strategy is optimal for the newly integrated firms because it allows
the firms to increase the fixed fee set on the rival (expropriation effect). We show that under a
wide array of circumstances this mechanism is so strong that the newly integrated firm sets its
price below marginal costs, thereby rendering the counter-merger welfare enhancing with respect
to a framework with single integration, and, provided demand is sufficiently concave, even with
respect to a framework featuring no merger. This yields important insights for antitrust policy
and shows that the results of previous models without complementary inputs cannot be extended
to these industries. It also shows that evaluating a vertical merger on myopic grounds can be
4
misleading.
We then analyze how the combined impact of the effects outlined above play out in a simple
two-stage model in which each upstream firm has the possibility to integrate at one stage.2 We
show that due to the expropriation effect a firm may want to forego the possibility to merge and
let the complementary input provider integrate at a later point in time. This can explain why an
asymmetric market structure with only one vertical merger arises in an industry that is comprised
of ex-ante identical firms.
Finally, an important implication of our analysis is that a horizontal merger between complementary input providers is anticompetitive. Such a merger would transform the industry into one
with a sole input provider, since the merged upstream firm controls the prices of both complementary inputs. Therefore, a horizontal merger renders vertical integration always profitable and
anticompetitive because the integrated firm will foreclose its downstream rival. This shows that the
competitive effects of a merger between firms that produce complements are very different if final
good producers merge than if intermediate good suppliers merge. This result stands in contrast
with the widespread opinion that horizontal mergers between firms producing complements are
procompetitive, an opinion reflected e.g., by the lenient stance kept by major Antitrust authorities
with respect to this type of transactions.
The expropriation conduct identified by our analysis helps explain the conflict between standalone upstream firms and integrated companies that lies at the heart of two recent high-profile
antitrust cases in the information and communication technology industry: The FTC v. Rambus
case and the EC v. Qualcomm case.
In 2006, the FTC found Rambus guilty of having manipulated the works in JEDEC, the Standard Setting Organization that was deciding on the development of the DRAM and DDR-SDRAM
standards.3 Rambus was a stand-alone upstream firm active in the development of Intellectual
Property Rights (IPRs). Micron, IBM and other vertically integrate firms claimed that, had they
known that Rambus owned relevant IPRs, they would have strongly opposed the inclusion of its
technology in the standard. This supports our intuition that at the roots of the conflict between integrated companies and pure innovators is the fear that stand-alone firms may extract a significant
chunk of integrated firms’ profits.
In the second case, Qualcomm is a pure developer that contributed to the development of the
3G technology: Qualcomm’s patents portfolio includes essential patents for the development of
the CDMA2000 and WCDMA, the former 3G standards in the United States and Europe, respec2
This can be seen as short-cut for a richer model in which merging opportunities do not arise continuously but
only at rare points in time.
3
The FTC alleged that the deceptive conduct kept by Rambus allowed the firm to include some of its patented
technologies in the final version of the standard. See In the Matter of Rambus Inc., Docket No. 9302.
5
tively. In the EC v. Qualcomm case,4 Qualcomm has been accused by Nokia and other vertically
integrated firms, which produce handsets and develop IPRs, to have infringed its obligation to
negotiate prices on fair, reasonable and nondiscriminatory terms.
Summarizing, in both cases vertically integrated firms were put in difficulty by stand-alone
upstream suppliers that hold essential inputs for downstream production technology, a result that
is consistent with the predictions of our model and that affects the conclusions on vertical mergers
that are important for antitrust policy.
The rest of the paper is organized as follows. The next Section provides an overview over the
related literature. Section 3 sets out the model and Section 4 analyzes the case without integration.
Section 5 provides the analysis and the results of the case with a vertical merger. In Section 6
we analyze the incentives for and the consequences of a counter-merger. Section 7 considers the
equilibrium market structure while Section 8 analyzes the effects of a horizontal merger. Section
9 provides an extension with public offers and Section 10 concludes.
2
Literature Review
The problem of a dominant upstream firm to be unable to commit to the monopoly quantity when
selling via multiple competing downstream firms was first pointed out by Hart and Tirole (1990)
and is summarized in the survey by Rey and Tirole (2007). In their framework, upstream firms’
offers are made by means of secret contracts and downstream firms adopt passive beliefs to infer
the offers received by their competitors when they face out-of-equilibrium offers. We take the
same approach as in Rey and Tirole (2007) when modeling the structure of the contracting game
between upstream and downstream firms. Consequently, the same commitment problem arises in
our framework. Instead, the crucial twist in our framework compared to Rey and Tirole (2007)
consists in the presence of producers of complementary inputs, which are rivals in extracting the
surplus from downstream manufacturers.
The role of retailers’ beliefs has been highly debated by the literature on vertical restraints. More
specifically, the paradox inherent to the commitment problem has been investigated by O’Brien and
Shaffer (1992), McAfee and Schwartz (1994) and Marx and Shaffer (2004). The general conclusion
from these papers is that via vertical integration a dominant firm is able to restrict its quantity,
thereby moving industry output closer to the monopoly level, which renders vertical integration
profitable but anticompetitive.
White (2007) relaxes the assumption that that downstream firms have perfect information on
the marginal cost of the intermediate goods’ suppliers. However, she finds that even in this case it is
still necessary to specify the downstream firms’ beliefs. She also shows that with uncertainty about
4
EC MEMO/07/389, 01/10/2007.
6
upstream marginal costs, vertical integration can result in high-cost types selling to downstream
firms at lower prices than they would set if vertically separated.
Baake, Kamecke, and Normann (2004) analyze the case in which a wholesaler can publicly
commit to a capacity level. In this way, the wholesaler can partly solve the Coasian conjecture
problem, commit to underinvest in capacity and produce at a level that can even be below the
monopoly output. Thus, with vertical integration output will increase to the monopoly level.
The driving forces in our model are markedly different from the ones in the articles above. More
specifically, we show that because of the presence of a complementary input provider endowed with
market power vertical integration may no longer be profitable, and this is particularly true for very
efficient firms. In addition, we consider the case of a counter-merger and find that two mergers
can be procompetitive while a single one always leads to foreclosure.
Pagnozzi and Piccolo (2011) analyze the effects of different assumptions on retailers’ beliefs in a
model with two competing supply chains, i.e., each manufacturer supplies to just one retailer. They
show that with symmetric beliefs supply chains prefer to stay separated while they are indifferent
between integration and separation with passive beliefs. By contrast, in our model manufacturers sell their respective inputs to both downstream firms, implying that the above mentioned
commitment problem is of major importance and can be solved by vertical integration.
There are several other papers that analyze the effects of vertical integration in different setups. For example, Ordover, Saloner and Salop (1990) or Chen (2001) consider the case of Bertrand
competition between upstream producers with public offers in linear prices. They determine under
which conditions vertical integration is profitable and analyze if counter-mergers can occur. Choi
and Yi (2003) provide a model in which upstream firms can choose to specialize their inputs to the
needs of downstream firms and analyze the consequences of vertical integration in this environment. Inderst and Valletti (2011) analyze the case in which upstream firms produce imperfectly
substitutable inputs and demonstrate how incentives to foreclose vary with changes in the competitiveness of input and output markets. Riordan (1998) considers a model with a dominant firm
that has market power in the final and the intermediate good market. He shows that vertical
integration of the dominant firm is anticompetitive due to foreclosure although production costs
of the dominant fall.5 In contrast to our set-up, these papers just consider a single input and are
not concerned with complementary inputs. In addition, they all show that integrated firms have
an incentive to foreclose their downstream rivals.
Finally, papers that consider the case of complementary inputs usually look at markets where
5
Nocke and White (2007) analyze the effects of vertical integration on the sustainability of collusion between
upstream firms in a repeated game framework. They show that vertical integration facilitates collusion because,
after integration, non-integrated upstream firms have a smaller incentive to deviate since they can no longer sell
via the downstream unit of the integrated firm. For a similar result with a different wholesale pricing regime, see
Normann (2009).
7
upstream firms hold essential patents that are required for the production of a final good, see e.g.,
Shapiro (2001). However, this literature is not concerned with the consequences of vertical integration. The only exception is Schmidt (2007). He considers a model in which each patent holder
is monopolist for its patent while there are several downstream firms competing on the product
market. Patent holders offer symmetric public contracts. Schmidt (2007) shows that vertical integration leads to foreclosure of rival downstream firms and to a reduction of output although the
integrated firm produces more due to the avoidance of double-marginalization. In contrast to his
model, we consider the case of private contracts and allow for a richer market structure in the
upstream market where a (less efficient) fringe firm exists for each input. As mentioned, in this
set-up we obtain starkly different results with respect to the previous literature.
3
The Model
There are two downstream firms, denoted by D1 and D2 , that produce a homogeneous good. To
produce one unit of the output good each downstream firm needs one unit of two input goods (or
intermediate goods). In other words, the two input goods are perfect complements and used in
fixed proportions for the production of the final good. In the following, we denote the output of
firm Di by qi , i = 1, 2.
Each input good j is produced by upstream firm Uj , j = A, B, and by a less efficient competitive
fringe. Specifically, we assume that firm Uj produces input j at a marginal cost of cj , while the
fringe firms incur a marginal cost of production given by ĉj ≥ cj . Since the fringe firms behave
competitively, they are willing to offer the input at a per-unit price of ĉj to each downstream firm.6
Therefore, the difference ĉj − cj can be interpreted as a measure for the market power of Uj . The
framework is given in Figure 1, in which we denote the fringe firms by Ûj .
[FIGURE 1 ABOUT HERE]
Downstream firms face an inverse demand function p = P (q1 + q2 ). We assume that, for any
Q = q1 + q2 > 0 such that P (Q) > 0, the demand function is downward-sloping (P (Q) < 0), and
each firm’s maximization problem is strictly concave (P (Q) + qi P (Q) < 0). Both assumptions
are standard in the literature (e.g., Vives, 1999).
The game proceeds as follows:
1. In the first stage, each upstream firm Uj simultaneously makes a take-it-or-leave-it offer to
Di
Di
i
each Di consisting of two-part tariffs, which are denoted by TUDji = wUDji xD
Uj + FUj , where xUj
6
The same result obtains if the fringe would consist only of one firm that competes with Uj to serve the
downstream firms. Since there is Bertrand competition between this firm and Uj , the firm will also set a per-unit
price of ĉj .
8
denotes the quantity of input j that Di buys from Uj . After having observed these offers,
Di decides whether to buy an intermediate good from Uj , j = A, B, or from the respective
competitive fringe, orders input quantities of goods j and −j, and pays the respective tariffs.
2. In the second stage, each downstream firm transforms the intermediate goods into output,
observes the output of its rival and sets its price on the product market.
The structure of the game implies that downstream firms play a Bertrand-Edgeworth game of
price competition with capacity constraints à la Kreps and Scheinkman (1983).
The equilibrium concept we employ is perfect Bayesian equilibrium. Given the quantity purchased in the first stage, in the second stage downstream firms transform their purchased input
units into output. We assume that if firms purchase an amount of the two inputs in the viable
range, it is optimal for them to transform all units into output. The downstream price is then
given by P (q1 + q2 ).
As for the first stage, the game is solved under the assumption that upstream firms supply
on order and that wholesale contracts are secret. The latter assumption implies that each Di
does not observe the contracts that are offered to D−i . In particular, by using the common agency
taxonomy, we restrict our attention to a bidding game with passive (out-of-equilibrium) beliefs. The
assumption of passive beliefs implies that if a downstream firm faces an out-of-equilibrium offer
by a supplier, it does not revise its beliefs concerning the offers made to its rival. Consequently,
a downstream firm Di expects its downstream rival D−i to produce the candidate equilibrium
quantity, regardless of the offer received by a supplier. As explained by Rey and Tirole (2007),
the assumption of passive beliefs is particularly plausible if upstream firms supply on order. This
is because, from the point of view of an upstream firm, the downstream firms D1 and D2 form
separate markets: downstream firms have already chosen their quantities and paid the respective
tariffs before competing on the product market.
As will become evident later, the presence of complementary inputs implies that a multiplicity
of equilibria arises in the first stage of the game. This is because for the two upstream firms a
continuum of combinations of FUDji and FUD−ji exists, such that Di accepts both offers. In this respect
our game with Uj and U−j making offers to Di is similar to a two-player bargaining game, with the
exception that in our framework there is a third player, Di , who has no bargaining power but can
reject the offers received by the upstream firms. Due to this analogy, a natural way to resolve the
equilibrium multiplicity is to employ an adaptation of the Nash bargaining solution to our specific
setting.
In the Nash bargaining solution, each player j obtains 1/2(F S − OOj ) + OOj = 1/2(F S + OOj ),
where F S denotes the full surplus from the bargaining relationship and OOj is the outside option
of player j. In our framework, the full surplus of firm j in the offer game with each Di , i = 1, 2,
9
is equivalent to the contribution of Uj to the profit of Di , that is, the profit that Di can obtain
by accepting Uj ’s offer minus the profit that Di obtains when buying from the competitive fringe.
This implies that the full surpluses of Uj and U−j can be different, because the cost structure in
the two input markets can differ. Therefore, in our framework the full surplus is contingent on
the input market it refers to (F Sj ). We determine the outside option of Uj (OOj ) in an analogous
manner. OOj is given by the offer that Uj needs to formulate to make sure that Di will accept.
This offer must be such that Di has no incentive to reject neither the offers of both Uj and U−j
(given that the latter demands respective full surplus, F S−j ) nor just Uj ’s offer. We will later show
that, in equilibrium, OOj is equal to the rent that Uj can extract from Di after U−j has extracted
F S−j , where Di ’s relevant threat is to turn down both the offers of Uj and U−j , and buy from the
fringe firms.
This equilibrium selection criterion is the natural adaptation of the two-player Nash bargaining
solution to our three-player game, where one agent has no bargaining power but only the possibility
of rejecting offers. Note that the same outcome emerges as the unqiue equilibrium in a game in
which both upstream firms make offers in a sequential order, each being the first to approach Di
with probability 1/2, and in which the second firm to offer knows the offer formulated by the first
due to communication with Di . Such a sequential offer game provides a micro-foundation for our
selected equilibrium.
Before solving the model, it is useful to introduce some additional notation. We denote by
qi (CDi , CD−i ) firm Di ’s downstream quantity when it faces costs of CDi while firm D−i faces costs
of CD−i . More formally,
q1 (CD1 , CD2 ) ≡ arg max {(P (q + q2 (CD2 , CD1 )) − CD1 )q}
q
and
q2 (CD2 , CD1 ) ≡ arg max {(P (q + q1 (CD1 , CD2 )) − CD2 )q}.
q
Similarly, the aggregate quantity Q will be written as Q(CD1 , CD2 ) = q1 (CD1 , CD2 ) + q2 (CD2 , CD1 ).
For example, if both firms face marginal costs of production equal to cA + cB , then the associated
downstream quantities are q1 (cA + cB , cA + cB ) and q2 (cA + cB , cA + cB ).
Finally, we assume that fringe firms are effective in constraining the market power of Uj , j = 1, 2,
by imposing that ĉj is low enough. This implies that qi (ĉj + c−j , cj + c−j ) > 0 i.e., the quantity
of Di when resorting to the fringe firms for input j, with j = A, B, is positive, even when D−i
obtains both inputs at marginal costs.
10
4
Set-up without Integration
In this section, we present the case in which no firm is vertically integrated. We obtain the following
result:
Proposition 1.
In equilibrium both upstream firms charge a per-unit price equal to marginal costs, that is, wUDji =
cj , with i = 1, 2 and j = A, B, leading to equilibrium quantities equal to q1 = q2 = q c ≡
qi (cA + cB , cA + cB ).
Proof See the appendix.
The perfect Bayesian equilibrium of the game without integration features the same commitment
problem that arises in Rey and Tirole (2007). Both downstream firms buy the inputs from the
efficient upstream firms at marginal cost and produce respective Cournot quantities while the
fringe firms stay inactive. Since a downstream firm does not observe the contract offers to its rival
and holds passive beliefs, upstream firms cannot commit to sell the monopoly quantity.
Turning to the determination of the fixed fees, we obtain that the fixed fee that Uj , j = A, B,
proposes to Di , i = 1, 2, is given by
1
Di
P (q c + q c ) − cA − cB )q c − max {(P (q + q c ) − ĉj − c−j ) q}
FUj =
q
2
c
c
+ max {(P (q + q ) − cj − ĉ−j ) q} − max max {(P (q + q ) − ĉj − ĉ−j ) q} , 0 .
q
(1)
q
The presence of the competitive fringe constrains the ability of upstream firms to extract profits
from the downstream firms. As can be seen from (1), the fixed fee that Uj can extract is the larger,
the more efficient Uj is relative to the fringe firms. Thus, the fixed fee is increasing in Δj = ĉj − cj .
Moreover, the fixed fee of Uj is the larger the smaller is Δ−j = ĉ−j − c−j . This is because Uj and
U−j extract as much as possible from a downstream firm, taking into account that the downstream
firm can also reject one or both offers, and split the extracted rent according to the adapted Nash
bargaining solution. Hence, if U−j ’s efficiency advantage becomes smaller, it receives a smaller
share of the extracted surplus and Uj ’s share becomes larger.
Before analyzing the profitability of a vertical merger between firms Uj and Di , we have to
determine the profits that Uj and Di receive when staying independent. As shown in the Appendix,
the profit of Di in case of non-integration is given by max [maxq {(P (q + q c ) − ĉj − ĉ−j )q, 0]. The
expected profit of Uj under non-integration is given by 2FUDji . Thus, the sum of the profits of Uj
and Di is given below:
(P (q c + q c ) − cA − cB )q c + max{(P (q + q c ) − cj − ĉ−j )q} − max{(P (q + q c ) − ĉj − c−j )q}.
q
q
11
(2)
We will use (2) to assess the profitability of a vertical merger.
5
Vertical Merger between UA and D1
Suppose that UA and D1 are now integrated and the other firms stay independent. The new
framework is given in Figure 2.
[FIGURE 2 ABOUT HERE]
The integrated firm trades the input good internally at marginal cost. This assumption is
standard in the literature (see e.g., Ordover, Saloner and Salop, 1990, Chen, 2001, or Choi and Yi,
2001) and is justified by the fact that pricing at marginal cost is (ex-post) the optimal strategy
for the integrated firm.7
As in the case without integration, the firms delivering the inputs still have all the bargaining
power. This implies that UB makes a take-it-or-leave-it-offer to the newly integrated firm. In
other words, vertical integration does not imply a change of the bargaining power positions, it
only changes the strategic position of the newly integrated firm, which now maximizes the joint
profit of UA and D1 .
We first determine the optimal wholesale price that the integrated firm charges to D2 . As is
well-known from Rey and Tirole (2007), in the absence of suppliers of the complementary good the
integrated firm finds it optimal to soften downstream product market competition via foreclosure.
This means that the integrated firm sets its downstream rival’s marginal costs of production to the
UA
= ĉA . Proposition 2 shows that this is also the case in a setting
highest possible value, i.e., wD
2
with complementary inputs.
Proposition 2.
In equilibrium, the integrated firm UA − D1 forecloses the downstream market, that is, wUDA2 = ĉA ,
while UB offers to both downstream firms a per-unit price equal to its marginal cost, that is,
wUDBi = cB , i = 1, 2.
The equilibrium quantities are asymmetric and given by q1 = q1c ≡ q1 (cA + cB , ĉA + cB ) and
q2 = q2c ≡ q2 (ĉA + cB , cA + cB ).
Proof See the appendix.
7
Even if the integrated firm would like to credibly commit to outsiders that the internal wholesale price is above
marginal costs, it cannot do so since the firm has an incentive to secretly change the price afterwards. This is
possible e.g., via exchanging payments between the upstream and the downstream unit, which are unobservable to
outsiders.
12
As in Rey and Tirole (2007), in equilibrium the integrated firm forecloses the downstream
market by setting the wholesale price wUDA2 = ĉA . Accordingly, its fixed fee is nil. The stand-alone
upstream firm UB sets wholesale prices equal to marginal cost and fully extracts its contribution
to downstream firms’ profits via the fixed fees. As a consequence, D1 expands its market share at
the expenses of D2 (q2c < q1c ).
Rey and Tirole (2007) also show that integration is more profitable than staying independent,
and it is the more so, the less competitive the fringe firms are, i.e., the higher ĉA is. In what follows,
we analyze the profitability of the vertical merger to verify whether analogous results emerge in a
setting with complementary inputs.
The profit of the integrated firm UA − D1 is equal to
ΠUA −D1 = max{(P (q + q2c ) − cA − ĉB )q} + (ĉA − cA )q2c .
q
(3)
In a setting without a provider of a complementary input with market power, the integrated firm
would receive its full downstream profit maxq {(P (q+q2c )−cA −cB )q}. Instead, here UA −D1 obtains
a profit from the downstream market that corresponds to the profit it can raise when buying from
the fringe firms for input B, maxq {(P (q + q2c ) − cA − ĉB )q} < maxq {(P (q + q2c ) − cA − cB )q}. This
reduces the profitability of the merger.
By comparing (3) with the profits of UA and D1 under non-integration—given in (2)—, we
obtain that integration is more profitable than non-integration if and only if
ΠUA −D1 − (ΠUA + ΠD1 ) = max{(P (q + q2c ) − cA − ĉB )q} + (ĉA − cA )q2c
q
c
c
c
−(P (q + q ) − cA − cB )q − max{(P (q + q c ) − cA − ĉB )q} + max{(P (q + q c ) − ĉA − cB )q} >(4)
0.
q
q
Let us start with the case in which the market for input B is perfectly competitive, i.e. ĉB = cB ,
which implies that UB has no bargaining power. In that case, a simple revealed-preference argument
shows that vertical integration is profitable. To begin with, it is easy to show that by setting
wUDA2 = cA the integrated firm could achieve the same outcome as in the case of non-integration.
However, we know that UA − D1 optimally sets wUDA2 = ĉA > cA . Given that it prefers to depart
from pricing at marginal cost, its profit from integration must be higher than the profit from
staying non-integrated. Therefore, vertical integration is profitable at ĉB = cB . Hence, if the
complementary input provider has no market power, the results in our framework are the same as
the ones for the case with only one input, i.e., our framework embeds the one of Rey and Tirole
(2007).
The picture differs if UB has market power, i.e., if ĉB > cB . To see this let us assess the impact
of an increase in ĉB on the profitability of integration. Taking the derivative of (4) with respect to
ĉB , we obtain that
∂ΠUA −D1 − (ΠUA + ΠD1 )
= −qI + qN
I,
∂ĉB
13
c
with qI ≡ arg maxq {(P (q + q2c ) − cA − ĉB )q} and qN
I ≡ arg maxq {(P (q + q ) − cA − ĉB )q}. Since
q2c < q c and downstream quantities are strategic substitutes, we have that qI > qN
I . This leads us
to the result in the proposition below.
Proposition 3.
The profitability of the vertical merger between UA and D1 falls in the market power of UB .
The result in Proposition 3 shows that the expropriation threat exerted by UB vis-à-vis D1
becomes more detrimental if D1 is integrated. The intuition is as in what follows: In the case
without integration, UA and UB are in conflict with each other when extracting the downstream
firms’ surplus. Ceteris paribus, if ĉB increases, i.e., the market power of UB increases, UB obtains
a larger surplus from D1 and D2 at the expense of UA . This increase in UB ’s surplus is overall
equal to the reduced profit that each downstream firm receives when buying from the competitive
fringe for input B.8 In the case with integration, UA − D1 forecloses D2 , implying that there is no
conflict between UA and UB when extracting rent from D2 . Moreover, the fact that UB has full
grip on the profit of D1 and that D2 is foreclosed by the integrated firm imply that the amount
that UB can extract from D1 is larger than what UB can extract in the benchmark case without
integration (where each downstream firm produces the symmetric Cournot quantity).9 hence, the
profitability of vertical integration falls as ĉB gets larger.
As will be shown by means of an example with linear demand, the larger is ΔB , the more
likely it is that vertical integration is unprofitable. In this case, the expropriation conduct of the
complementary input supplier is so large that it dominates the effect that industry profits rise
due to foreclosure. This result is peculiar to a setting with complementary inputs and it provides
a rationale for the examples presented in the Introduction, in which integrated firms face the
problem of expropriation conduct exerted by stand-alone firms whose inputs are included in a
standard.
We now analyze how a change in the market power of integrating firm UA affects the profitability
of integration. Here we find that the comparative statics are not clear-cut over the whole range of
ĉA . However, the result is unambiguous if ĉA is relatively large.
Proposition 4.
If the market power of UA is sufficiently large, then the profitability of the vertical merger between
UA and D1 falls in ĉA .
8
Analytically, the increase in profits is given by qN
I /2 per downstream unit.
9
More specifically, the larger rent extracted by UB in the case with integration is given by qI > qN
I.
14
Proof See the appendix.
Proposition 4 implies that—perhaps surprisingly—if UA is much more profitable than the competitive fringe for input A, vertical integration may not be profitable. The reason for this result
is the following. In the case without integration, UA can extract a large part of the profit of D1 ,
because the difference between ĉA and cA is large. After the vertical merger, we know that UA
loses its grip on the profit of the downstream unit and UB is the only one exerting bargaining
power vis-à-vis D1 . Due to the vulnerable position of D1 in the negotiation with UB , the profit
of the integrated firm might fall by more than in the case in which D1 and UA are stand-alone
firms. This implies that vertical integration does not pay off if the fringe is very inefficient.10 If ĉA
is only slightly above cA , the result is ambiguous because the possibility to foreclose D2 and raise
industry profits can dominate the expropriation conduct of UB .
This last result is markedly different from the conclusion in Rey and Tirole (2007) that vertical integration is particularly profitable for efficient firms. An interesting—and perhaps counterintuitive—implication of our analysis is that, in an industry with highly complementary inputs, very
efficient firms are less likely to vertically integrate than firms that are only slightly more efficient
than their competitors.
Example with Linear Demand We illustrate our analysis by employing a standard linear
demand function, P (q1 + q2 ) = max{0, 1 − q1 − q2 }. In this case the assumption that fringe firms
are effective in constraining market power is fulfilled if Δj < (1 − cA − cB )/2, where Δj = ĉj − cj .
We obtain that for firms UA and D1 it is more profitable to merge and foreclose the downstream
market than to stand alone if
ΔA ≤
8(1 − cA − cB ) − 12ΔB
.
11
This result shows that with linear demand there exists a unique threshold for ΔA above which
vertical integration is no longer profitable. Hence, very efficient firms prefer to stay unintegrated.
As is evident, this threshold is decreasing ΔB , i.e., the more efficient the complementary input
provider, the larger the range for which vertical integration is unprofitable. This illustrates the
expropriation effect exerted by UB at the expense of UA − D1 and its impact on the profitability
of integration.
10
Note that if an integrated firm could credibly commit to set its internal wholesale price above marginal costs,
this result would not occur. However, since this is impeded by secret internal renegotiation’s incentives, vertical
integration can be unprofitable.
15
6
Counter-merger Between UB and D2
In this section, we analyze the market outcome for the case in which both upstream firms are
merged. More specifically, we study the incentives for a counter-merger to form and its competitive
effects. Suppose therefore that not only UA and D1 are merged but also UB and D2 are. The
framework is displayed in Figure 3. Like in the case with single integration, both integrated firms
trade the input internally at marginal costs.
In the last section, we have shown that the merged firm has an incentive to fully foreclose its
downstream rival by setting the input price as high as possible. The question is if this result
emerges also in a set-up with two vertical mergers. As will turn out, this is not necessarily the
case because at least one vertically integrated firm, e.g., UB − D2 , has an incentive to depart from
foreclosure if ĉB is not too large.
Let us define a condition, denoted condition S, that will be helpful to state the equilibrium in
the counter-merger analysis:
S:
2(P (Q))2 > P (Q) (3P (Q)(qi − q−i ) + q−i P (Q)(2qi − q−i )) ,
D
i = 1, 2,
D
D
i
i
i
, w̃Uj−i + c−j ), q−i = q−i (w̃Uj−i + c−j , cj + w̃UD−j
) and Q = Q(cj + w̃UD−j
, w̃Uj−i +
where qi = qi (cj + w̃UD−j
c−j ), with
i
w̃UD−j
= c−j −
P (Q) (P (Q)(2q−i − qi ) + P (Q)q−i (q−i − qi ))
,
2P (Q) + q−i P (Q)
j = A, B, i = 1, 2.
(5)
We can now state the equilibrium in the counter-merger case:
Proposition 5.
(i) Suppose condition S is satisfied for at least one firm. Then, in any stable equilibrium one
D
integrated firm, e.g., Uj − Di , forecloses by setting wUj−i = ĉj , whereas U−j − D−i sets a wholesale
i
i
i
price given by wUD−j
, where w̌UD−j
= min ĉ−j , w̌UD−j
is defined as in (5) but with quantities qi =
i
i
, ĉj + c−j ) and q−i = q−i (ĉj + c−j , cj + w̌UD−j
).
qi (cj + w̌UD−j
i
i
Firms’ quantities are given by qi = qi (cj + w̌UD−j
, ĉj + c−j ) and q−i = q−i (ĉj + c−j , cj + w̌UD−j
).
(ii) Suppose condition S is not satisfied for both firms. Then, there
exists a stableequilibrium
D
D
i
i
= min ĉ−j , w̃UD−j
,
in which the wholesale prices are given by wUj−i = min ĉj , w̃Uj−i and wUD−j
D
i
are defined by (5).
where w̃Uj−i and w̃UD−j
D
D
i
i
Firms’ quantities are given by qi = qi (cj +wUD−j
, wUj−i +c−j ) and q−i = q−i (wUj−i +c−j , cj +wUD−j
).
Proof See the appendix.
In what follows, we first discuss the general properties of strategic interaction in the game
between the two integrated firms. Then, we turn to the analysis of the existence and the characteristics of each class of equilibria.
16
As is evident from Proposition 5, in the case of a counter-merger, it is not necessarily the optimal
strategy for both firms to foreclose its rival. The intuition behind this result can be grasped by
looking at the maximization problem of an integrated firm U−j − D−i :
D−i
Di
Di
max max{(P (q + qi ) − wUj − c−j )q} + qi (wU−j − c−j ) + max{(P (q + q−i ) − cj − wU−j )q} ,
D
wU i
−j
q
q
(6)
with qi = qi (cj +
When setting
D
i
w̃UD−j
, w̃Uj−i
i
, firm
wUD−j
+ c−j ) and q−i =
D
q−i (w̃Uj−i
+ c−j , cj +
i
w̃UD−j
).
U−j − D−i takes into account the impact of the wholesale price on
the profit of its downstream unit (first term), the profit that the upstream firm obtains through
the per-unit sales to the downstream rival (second term) and the downstream profit of the rival,
i
(third term). The structure of this
which is the part of the fixed fee (FUD−ji ) that depends on wUD−j
problem is the same as the one of firm UA − D1 in the case with single integration. There, firm
UA − D1 obtains the complementary input at marginal cost, i.e., wUDB1 = cB , and forecloses its rival
to the extent possible (that is, given the presence of the competitive fringe). In this way, it gets
as close to market monopolization as possible.
In the case of a counter-merger, an integrated firm does not necessarily receive the complemenD
tary input at marginal cost. Let an integrated firm Uj − Di set wUj−i > cj , i.e., its wholesale price
is above marginal costs. If this is the case, it is not necessarily optimal for U−j − D−i to restrict the
quantity set by its downstream rival as much as possible. This is because the rival Uj − Di pays
only the marginal cost on input j and, therefore, has a competitive advantage on the downstream
market. For this reason, it can be profitable for U−j − D−i to deflect from pricing at ĉ−j , shift
part of its downstream profits to the downstream rival and then extract them through the fixed
fee. The result is that if a firm pays for an input a price at which it is (partially) foreclosed, it
has less of an incentive to foreclose itself. We obtain that if Uj − Di increases its wholesale price
D
i
(wUj−i ), firm U−j − D−i has an incentive to reduce its wholesale price (wUD−j
), i.e., wholesale prices
are strategic substitutes.
At a first glance, this result looks similar to the standard result that the prices set by firms
that produce complementary goods are strategic substitutes. However, the intuition here is very
different. In the standard analysis only the downstream market is considered and strategic substitutability is due to a problem of negative externalities on the demand side: a firm receives
a smaller demand if the complementary good producer raises its price. As a consequence, each
firm finds it optimal to react to a price increase of its rival by lowering its own price. We study
the case in which the two complementary inputs are provided by upstream firms that operate in
two vertical chains. In such a framework, an integrated firm’s downstream profit becomes smaller
when the upstream rival charges a higher wholesale price. Consequently, the optimal response of
an integrated firm to an increase of the rival’s wholesale price is to reduce its own wholesale price,
17
shift part of the downstream rent to the downstream competitor, and then extract part of this rent
through the fixed fee. Hence, our analysis presents a new explanation for why wholesale prices of
complementary-goods producers are strategic substitutes.
We now analyze in greater detail the properties of the two different classes of equilibria in
the counter-merger game. In the first class, class (i) in Proposition 5, one integrated firm fully
forecloses whereas the other one plays its optimal reaction to the foreclosure strategy of the rival.
In the second class, class (ii) in Proposition 5, both integrated firms do not necessarily foreclose
and play their best responses to the strategy of the rival. In this respect, the equilibria in the first
class are extremal equilibria and the equilibria in the second class are interior equilibria. However,
D
as is evident from Proposition 5, if at an interior equilibrium one integrated firm sets wUj−i = ĉj ,
then the interior and the extremal equilibria coincide.
If condition S is satisfied for at least one firm, the extremal equilibria in class (i) are the unique
stable equilibria of the game. Condition S implies that the reaction function of an integrated firm’s
wholesale
price with
respect to its rival wholesale price overshoots at an interior equilibrium, that
D−i
Di is, dwUj /dwU−j > 1 at a point in which the two reaction functions intersect. This means, any
interior equilibrium is unstable and, since upstream prices are bounded by ĉj , there can only exist
extremal equilibria. It is easy to show that condition S is fulfilled if the demand function is not
too concave.11 Hence, with linear or close to linear demand functions extremal equilibria are the
only stable equilibria.12 Conversely, an interior equilibrium is stable only if the demand function
is highly concave. This is the case because for highly concave demand functions the downstream
i
. As a result, firm need to adapt their
quantity of firm Di reacts strongly to a change in wUD−j
wholesale
prices
only slightly to induce a large impact on the downstream quantity, implying
D−i
Di dwUj /dwU−j ≤ 1.
In the following, we discuss the features of interior and extremal equilibria and show that
they share important properties with respect to the profitability of the counter-merger and its
competitive effects. Thus, the implications of our analysis for competition policy do not rely
crucially on the class of stable equilibria of the game.
We start with extremal equilibria. The characteristics of an extremal equilibrium in Proposition
5 resemble the ones of the equilibrium obtained under single integration, in which the integrated
firm chooses to foreclose the downstream market. The crucial difference with respect to the single
integration case is that the second integrated firm may want to depart from setting a wholesale price
equal to marginal costs, due to the above-mentioned trade-off between own foreclosure incentives
and to extract rent from the rival.
11
The proof is contained in the proof of Proposition 6.
One can also show that for the class of demand functions of the form P (Q) = 1 − Qb , with b > 0, condition S
√
is satisfied if b ≤ 1 + 2.
12
18
We first look at the profitability of the counter-merger and the associated competitive outcome
under an extremal equilibrium. In this class of equilibria, the integrated firms face a coordination
problem with respect to which firm sets its wholesale price at the foreclosure level and which firm
deflects from it. However, a clear selection criterion stands out. Since one firm, denote it UA − D1 ,
is already integrated and forecloses the market if it were the only integrated firm, it is natural to
conjecture that in the counter-merger case this firm sets wUDA2 = ĉA and the other firm, the newly
integrated one, plays according to its best reaction.
In the last section, we have shown that the first merger (between UA and D1 ) is not necessarily
profitable. As we show in the next proposition, this is not the case for the counter-merger.
Proposition 6.
Given that firms UA and D1 are integrated, a counter-merger between UB and D2 is always profitable.
Proof See the appendix.
The intuition behind this result is in what follows. There are two sources of gains associated
with the counter-merger for firms UB and D2 . The first is that after the counter-merger the
downstream unit D2 is informed by UB on whether its rival D1 buys input B from UB or from
the competitive fringe (ÛB ). Accordingly, D2 can optimally adjust its downstream quantity to the
price of input B faced by its rival. In other words, if D1 is buying from the fringe firm, firm D2
can expand its downstream quantity at the expense of D1 and the integrated firm UB − D2 can
extract a larger fixed fee from the downstream unit of UA − D1 than in the case in which UB and
D2 are not integrated. We denote this effect information effect.13
In addition to the information effect, the other source of gain from the counter-merger is that
UB − D2 can now choose the optimal reaction to the foreclosure strategy undertaken by the rival
and maximize industry profits. This can be easily grasped from expression (6), which can be
written as
max
D
wU 1
B
max{(P (q + q1 ) − ĉA − cB )q} + max{(P (q + q2 ) − cA − cB )q} .
q
q
(7)
By revealed preference, if wUDB1 = cB , the integrated firm UB − D2 sets a different wholesale price
than the one set by UB when UB and D2 stand alone, implying that wUDB1 = cB does not optimize
the industry profit. The combined impact of the information effect and the optimization effect
implies that the counter-merger is always profitable. In contrast to the first merger, there is no
13
Notice that the information effect emerges in a clear manner only if an integrated firm finds it optimal not to
D1
< ĉB . If instead full foreclosure were optimal for UB − D2 , its fixed
foreclose its downstream rival entirely, i.e., wU
B
fee FUDB1 would be zero.
19
expropriation problem for UB − D1 because UA optimally sets its wholesale price at foreclosure
level. Hence, its fixed fee equals zero.
We can now study whether the counter-merger increases consumer surplus by expanding downstream quantity. We obtain that, in contrast to the result for single integration, a counter-merger
can indeed be procompetitive:
Proposition 7.
A counter-merger between UB and D2 is procompetitive relative to a framework featuring a single
merged firm (UA − D1 ) if and only if
−2(P (Q))2 + P (Q)(P − cA − cB ) + (P (Q))2 (P (Q) − cA − cB )2 + 4(P (Q))2
ΔA >
,
2P (Q)
(8)
with Q = Q(cA + w̌UDB1 , ĉA +cB ). For P (Q) = 0, condition (8) simplifies to ΔA > (P (Q)−cA −cB )/2.
Proof See the appendix.
The reason why the counter-merger can be procompetitive is that the newly integrated firm
sets a wholesale price wUDB1 that might be below marginal costs. As shown in Proposition 7, if ΔA
is large enough, the counter-merger can lead to a rise in overall quantity as compared to single
integration. The reason for this lies in the fact that wholesale prices are strategic substitutes: As
is evident from (7), firm UB − D2 optimally sets wUDB1 to maximize industry profits, given that
it faces marginal costs of ĉA + cB while its rival faces marginal costs of cA + cB . Firm UB − D2
takes this decision facing the following trade-off. On the one hand, it has the incentive to set wUDB1
relatively low to shift profits to the firm with the lower marginal costs (D1 ). By that UB − D2
is able to extract a higher rent from D1 . On the other hand, by setting wUDB1 relatively high, the
aggregate quantity falls, implying that the quantity moves closer to the monopoly quantity. If ΔA
is large, firm D2 ’s quantity is low, implying that the aggregate quantity is already close to the
monopoly one. In this case, the first effect, which dictates to shift profits to D1 , dominates, i.e., if
wUDA2 is large, firm UB − D2 optimally sets its wholesale price wUDB1 at a relatively low level due to
strategic substitutability. Since UB − D2 can control the downstream quantity only indirectly via
the wholesale price wUDB1 , it can be optimal for UB − D2 to set wUDB1 below marginal costs to induce
its downstream unit D2 to produce less.
Given that the counter-merger is always profitable, that is, a single merger always triggers a
counter-merger, it is of particular interest to analyze whether two vertical mergers can lead to
a procompetitive outcome with respect to a framework in which none of the firms is integrated.
Proposition 8.
20
Two mergers are procompetitive relative to no merger if an only if
−P (Q)ΔA > (P (Q))2 ,
(9)
with Q = Q(cA + w̌UDB1 , ĉA + cB ).
Proof See the appendix.
Proposition 8 shows that if ΔA is large and the demand function is sufficiently concave, two
mergers can be welfare enhancing although a single merger is welfare reducing.
The economic intuition of the result that two mergers can be procompetitive with respect to no
merger rests upon the arguments used to explain Proposition 7: If ΔA is large enough, it is optimal
for UB to set its wholesale price below marginal costs to induce D1 , the downstream firm with the
lower marginal costs, to produce more. More specifically, Proposition 8 shows that the condition on
the value of ΔA that renders the counter-merger procompetitive relative to a benchmark without
integration is more likely to be satisfied the more concave the demand function is. The rationale
for this result can be grasped by looking at the reaction functions in the downstream market,
which are given by dqi /dq−i = −(P (Q) + P (Q)qi )/(2P (Q) + P (Q)qi ). The larger is P (Q),
the larger is the value of the reactions in absolute terms. Consequently, if P large, by lowering
wUDB1 firm UB − D2 does not induce a large increase in aggregate quantity, because D2 cuts back
on its quantity on a similar scale as D1 increases its quantity. Therefore, the dominating effect of
a reduction in wUDB1 is that the firm with the lower marginal costs, UA − D1 , produces more and
this increases industry profits. As a consequence, the more concave the demand, the larger is the
incentive for UB − D2 to set its wholesale price below marginal costs. This implies that the range
for ΔA such that two mergers are procompetitive relative to no merger.
The result in Proposition 8 has implications for competition policy because it shows that the
competitive effects of a vertical merger depend on the market structure in the status-quo. We
show that if in the status-quo firms stand alone, a single vertical merger is anticompetitive. By
contrast, a new vertical merger in an industry that already features vertical integration can have
welfare enhancing consequences. In addition, the procompetitive effects can even overturn the
anticompetitive effects of a single merger, implying that two mergers are procompetitive relative
to no merger. This result contrasts with the finding in Nocke and Whinston (2010) on horizontal
mergers, that the competitive effects of a first merger are not overturned by following mergers.
This is not necessarily true in the case of vertical integration in a framework with complementary
inputs.
We now briefly turn to the case in which an interior equilibrium is stable. With regard to
profitability, the effects that underpin the result on the profitability of the counter-merger are
analogous to the ones that characterize the case with extremal equilibria. In particular, the
21
information effect is now even stronger, because firm UA − D1 sets its wholesale price wUDA2 below
ĉA . This implies that firm UB − D2 ’s downstream unit can expand its quantity even more when
UA − D1 buys input B from the competitive fringe (ÛB ). Hence, the rent that the newly integrated
firm has to leave to UA − D1 is lower than in the case in which UA − D1 undertakes a foreclosure
strategy. Moreover, as in the analysis of extremal equilibria, the newly integrated firm optimizes
over its wholesale price and does not necessarily need to set it equal to cB (as UB does when
unintegrated).
What complicates the comparison between the profits of firms UB and D2 under single integration and the profits of UB − D2 in the counter-merger at an interior equilibrium is the fact that
both UA − D1 and UB − D2 set a different wholesale price with respect to the single merger case.14
So the downstream quantities change in two respects. However, as long as wUDA2 is not far below ĉA
and due to the greater incidence of the information effect at an internal equilibrium, the incentives
to counter merge are even more pronounced than at an extremal equilibrium.15
With regard to the competitive effects of the counter-merger in an interior equilibrium, we
obtain the following result.
Proposition 9.
Suppose that condition S is not satisfied for both firms. A counter-merger is procompetitive
relative to a single merger if ΔA is large, while a counter-merger is always anticompetitive relative
to no merger.
Proof See the appendix.
The intuition behind the result that a counter-merger can be welfare enhancing with respect to
a single merger at an interior equilibrium is similar to the intuition for the result in Proposition 7.
If ΔA is large enough, the foreclosure price set by UA − D1 in the single merger case is particularly
large, leading to a small industry quantity. Instead, in the counter-merger case, although both
wholesale prices may lie above marginal costs, they might do so to a smaller extent leading to a
larger industry quantity.
The reason for the result that two mergers can never be procompetitive with respect to a
framework with no merger, given that the equilibrium is an interior one, is that in this
class of
D
i equilibria strategic substitutability between wholesale prices is low, i.e., dwUj−i /dwUD−j
< 1. Let
one of the two integrated firms set its wholesale price above marginal costs. The other integrated
14
D2
D1
At an interior equilibrium in the counter-merger case we have wU
≤ ĉA and wU
≤ ĉB , while in the single
A
B
D2
D1
= ĉA and wU
= cA .
integration case wU
A
B
D2
15
Note that even if wUA were way below the foreclosure level, the fact that the incidence of the information effect
becomes particularly important under an interior equilibrium implies that the counter-merger is very likely to be
profitable for UB and D2 .
22
firm, even if it finds it optimal to price below marginal costs, will never reduce its wholesale price
by the same amount as the first firm raises its wholesale price above marginal costs. This implies
that the sum of the marginal costs in the case with two mergers is larger than the marginal costs’
sum in the case with no merger. This leads to a reduced aggregate quantity.
The discussion shows that the same major effects characterize the analysis of the counter-merger
at an interior equilibrium and at an extremal equilibrium. Moreover, the result that a countermerger can be procompetitive with respect to the single merger case holds in both situations.
Instead, two mergers can be welfare enhancing as compered to a no merger scenario only at an
extremal equilibrium.
We conclude the section by looking at the linear demand example.
Example with linear demand (cont.): Let firm UA −D1 be integrated. With linear demand,
condition S implies that any interior equilibrium in the counter-merger case is unstable. So the
only stable equilibria are extremal equilibria. As shown above, the counter-merger between UB
and D2 is always profitable. At the equilibrium, UA − D1 sets wUDA2 = ĉA and UB − D2 sets
wUDB1 = (1 + 4cA + cB − 5ĉA )/2.
The counter-merger is procompetitive with respect to a single merger if wUDB1 < cB . We find
that this is the case if and only if
1 − cA − cB
.
5
Since ΔA is restricted to be below (1 − cA − cB )/2, for (1 − cA − cB )/5 < ΔA ≤ (1 − cA − cB )/2,
ΔA >
the counter-merger is procompetitive.
Finally, as shown in Proposition 8, if the demand function is linear, i.e., P (Q) = 0, two mergers
can never be procompetitive with respect to a scenario featuring no vertical merger.
7
Equilibrium Market Structure
So far we have analyzed the incentives to integrate and its competitive effects for a given market
structure, that is, when the industry is either unintegrated or when the rival firm is already
integrated. Such an analysis is of major importance for competition policy, because it provides
guidance to antitrust authorities that need to decide in favor or against a proposed merger. It is also
of interest, though, to determine how the interplay between the effects highlighted by our model
shape the equilibrium market structure in an industry in which firms produce complementary
inputs. To do so, we consider a simple merger game in which merging opportunities come up
sequentially for upstream firms UA and UB .
Suppose that the timing of the game is as in what follows: In the first stage, upstream firm UA
can merge with a downstream firm. Since downstream firms are ex-ante symmetric, we assume
that the target is D1 . In the second stage, after the decision on the first merger has been taken,
23
upstream firm UB can merge with D2 .
To simplify the exposition we restrict our attention to the linear demand case. As in the previous
section, we suppose that if both firms are integrated, the first integrated firm UA − D1 is the one
that charges wUDA2 = ĉA .16 We obtain the following subgame perfect equilibrium market structure:
Proposition 10.
• If ΔB ≤ (14(1 − cA − cB ) − 41ΔA ) /28, in equilibrium UA merges with D1 and UB merges
with D2 .
• If (14(1 − cA − cB ) − 41ΔA ) /28 < ΔB ≤ (8(1 − cA − cB ) − 12ΔA ) /11, UB merges with D2
but UA and D1 stay non-integrated.
• If ΔB > (8(1 − cA − cB ) − 12ΔA ) /11, no merger takes place in equilibrium.
Proof See the appendix.
If UB is particularly efficient compared to the fringe firms, no merger takes place, while if UB
is only slightly more efficient than the fringe firms, two mergers obtain result in equilibrium. The
intuition behind these results follows from the discussion in the previous sections. If ΔB is large,
firms UA and D1 have no incentive to integrate in the first stage because of the expropriation effect.
Moreover, consistently with the result in Proposition 4 that the expropriation effect is particular
detrimental for very efficient upstream firms, UB and D2 can also not profitably integrate due to
the rent extraction by UA . By contrast, if ΔB is small, the rents that UB can expropriate from an
integrated firm UA − D1 is limited, so it is profitable for UA and D1 to merge in the first stage and
firms UB and D2 will follow suit in the second stage.
By contrast, if ΔB is in an intermediate range, an asymmetric outcome obtains in which only
UB and D2 integrate. For intermediate values of ΔB , firm UA prefers not to integrate with D1 and
prefers UB to integrate with D2 in the second stage. This is optimal for UA to be able to extract
more rents from D2 . Since UB is not particularly efficient, the expropriation conduct undertaken by
firm UA does not prevent the merger between UB and D2 , because this raises industry profits.
The analysis in this section shows that the expropriation effect plays a crucial role in shaping
the equilibrium structure of an industry with complementary inputs. In particular, we show that
an upstream firm might forego a merging opportunity to profit from an integrated rival. In turn,
the rival will have an incentive to merge in order to raise industry profit. This can lead to an
asymmetric market structure although upstream firms are symmetric.
This insight is also consistent with anecdotal evidence regarding Qualcomm strategic decision
to run its business as a pure stand-alone company in the 1990’s. Qualcomm divested its hardware
16
Remember that under linear demand any stable equilibrium under double integration is an extremal equilibrium.
24
business of producing mobile phones in 1999 and concentrated on the upstream market of developing technology.17 As mentioned in the introduction, integrated mobile phones producers like
Ericsson and Nokia complained a few years later about the “excessive and disproportionate royalties” charged by Qualcomm for licensing of its patents (source: Ericsson.com, October 28 2005
press release). So Qualcomm chose to not interfere with the downstream business of its former
rivals, which allowed the firm to extract a significant part of its rival’s downstream profits.
A restrictive assumption in the merger game above is that firms UA and D1 only have the
possibility to merge in the first stage, but cannot merge after firms UB and D2 have merged.
From the analysis in the counter-merger section, we know that this incentive is present. However,
typically merging possibilities do not arise at any point in time: If two firms have foregone their
opportunity to merge, it is often difficult to revert this decision due to, e.g., organizational or
credibility reasons. Hence, our game structure reflects an industry in which merging opportunities
arise very rarely.
Note that a very similar result to the one of Proposition 10 holds if we considered a more
elaborate merger game, e.g., one in which upstream firms engage in a bidding game to merge with
downstream firms along the lines of Ordover, Saloner and Salop (1990) or Chen (2001). In this
respect, our merger game, although simple, captures the relevant trade-offs that also arise in a
more sophisticated game.
8
Horizontal Merger
In this section, we briefly discuss the consequences of horizontal integration between the efficient
upstream firms (UA and UB ).
It is well-known that consumer surplus rises in industries in which firms that sell complementary
goods to final consumers integrate. Interestingly, the opposite results holds after an horizontal
merger between upstream firms UA and UB in our set-up. To see this, consider the case with one
merged firm UA − D1 . If this firm merges with UB , the new firm would foreclose the downstream
market by imposing to the downstream firm that stands alone a wholesale price as high as ĉA + ĉB .
Clearly, this horizontal upstream merger would be anticompetitive, because the downstream market
would move closer to monopoly. An analogous result arises in the counter-merger case. If UA − D1
and UB − D2 would merge then the new entity would fully monopolize the downstream market,
with detrimental consequences on consumers’ welfare.18
17
18
See e.g., Krippendorf (2008).
In the framework without integration a merger between UA and UB has no effect on consumer surplus, since
the newly merged firm would still incur in the commitment problem in Rey and Tirole (2007). This drives the
per-unit prices paid by downstream firms down to marginal costs of production.
25
Our analysis shows that the competitive impact of a horizontal merger greatly differs depending
on whether the two merging units that produce complementary inputs are upstream or downstream
firms. If two producers of intermediate complementary goods merge, this has no direct effect on
the price borne by final consumers but it has an impact on the input prices paid by downstream
producers. If downstream firms produce substitute goods, as in our model, the incentive for
the integrated firm to foreclose the downstream market leads to an anticompetitive outcome.
This contrasts with the procompetitive effects imparted by a merger between the producers of
complementary final goods, which allows for a better coordination on, and therefore commands a
reduction of, the downstream market price.
Antitrust authorities typically deem a merger between producers of complements as procompetitive. In the United States, policy makers’ stance toward horizontal mergers between complementary product providers was laid out in the DOJ-FTC Horizontal Merger Guidelines (2010),
which acknowledge that “merger-generated efficiencies may enhance competition by [...] combining
complementary assets.”19 A similar argument can be found in the Horizontal Merger Guidelines
in Europe20
Also in the UK the CC-OFT Merger Assessment Guidelines (2010) recognize the potential for
welfare enhancing consequences due to mergers among complementary goods producers. However,
it is also stated that the Authority assesses whether a conglomerate merger “of two suppliers of
goods which do not lie within the same market, [...] for example because they are complements”
may generate a substantial lessening of competition.21 We provide a clear theoretical rationale
behind the idea that these kind of mergers may turn out to be anticompetitive.
9
Public offers
In this section, we briefly discuss the case in which offers are public, that is, each downstream firm
observes not only the offers to itself but also the ones to its rival. As discussed in e.g., Rey and
Tirole (2007), in many circumstances public offers are less realistic than secrete offers, because
negotiations often take place privately and hard information about these contracts is relatively
difficult to communicate.
The goal of this section is to show in a simple way that vertical integration is never profitable in
the case of public offers. The reason is that upstream firms can extract as much as possible from
19
See the U.S. Department of Justice and Federal Trade Commission Horizontal Merger Guidelines, August 2010,
page 29.
20
See the Guidelines on the Assessment of Horizontal Mergers under the Council Regulation on the Control of
Concentrations Between Undertakings, February 2004.
21
See the Competition Commission and Office of Fair Trading Merger Assessment Guidelines, September 2010,
page 50.
26
the downstream firms already under non-integration, given the constraint that downstream firms
can buy from the competitive fringe. Thus, vertical integration and foreclosure are not necessary
to increase industry profits. Moreover, under integration the integrated firm would face in the
expropriation conduct exerted by the complementary input provider. As a consequence, vertical
integration yields weakly lower profits to the integrating firms as opposed to non-integration and
vertical integration does not occur in equilibrium. All this implies that the main insights of our
analysis under secret offers remain valid in the case of public offers.
To see this formally, consider the same framework as above with the difference that offers are
public, that is, each downstream firm, before deciding which offer to accept, can not only observe
the offer that it receives but also the one received by its rival. Assume first that no firm is
integrated. The goal of the upstream firms is to maximize industry profits and extract them from
the downstream firms via the fixed fees, given the presence of alternative sources. The easiest way
to do so is to offer per-unit prices of wUDAi = cA and wUDBi = cB to firm Di and very high wholesale
prices to firm D−i . If there were no fringe firms, Di would buy the monopoly quantity and each
upstream firm would receive expected profits of half of the monopoly profit via the fixed fees.
However, firm D−i would buy from the fringe firms in this case. Therefore, it is optimal for the
D
D
upstream firms to serve D−i themselves at wholesale prices of wUA−i = ĉA and wUB−i = ĉB . This
implies that downstream firms play an asymmetric Cournot game in the downstream market in
which they produce quantities of
qi = arg max {(P (q + q−i ) − cA − cB )q}
q
and
q−i
= max arg max {(P (q + qi ) − ĉA − ĉB )q} , 0 .
q
In this way, the downstream market is as close as possible to the monopoly outcome. As a
consequence, the fixed fee set to firm D−i is nil whereas the fixed fee set by a firm Uj , j = A, B,
to Di is equal to
1
max {(P (q + q−i ) − cj − c−j )q} − max {(P (q + q−i ) − ĉj − c−j )q} +
q
q
2
+ max {(P (q + q−i ) − cj − ĉ−j )q} − max max {(P (q + q−i ) − ĉj − ĉ−j )q} , 0 .
FUDji =
q
q
Here the first two terms represent the full surplus of firm Uj in the bargaining with Di while the
last two terms represent firm Uj ’s outside option.
Now, let us look the case of a vertical merger between Uj and Di . Since D−i is operating with
the highest possible marginal costs in case of no integration, such a vertical merger cannot raise
industry profits because they are already as close as possible to the monopoly profits. In addition,
there information effect plays no role, because D−i buys input i at a price of ĉj in case of no
27
integration already, and Di knows this. The only effect that is still present is the expropriation
effect: U−j can extract more profits from Di than in the case of no integration, because Uj cannot
exert its bargaining power on Di . Overall, the result is that the merged firm is put in difficulty as
much as in a framework with secret offers, due to the expropriation conduct of the complementary
input provider, but the degree of downstream competition is not affected by the vertical merger.
As a consequence, a vertical merger cannot be profitable.
10
Conclusion
This paper analyzes the profitability and welfare consequences of vertical mergers when downstream firms need complementary inputs, and these inputs are supplied by producers with market
power. We showed that the presence of the complementary input suppliers gives rise to an expropriation conduct that is not present in a framework where only one input is necessary for final
good production. A consequence of this is that vertically related firms may not find it profitable
to integrate since integration makes the profit of the downstream unit fully vulnerable to this
expropriation conduct. The expropriation effect is particularly strong if the upstream firm is very
efficient, implying that it is less profitable for efficient firms to integrate. We also showed that
a single merger triggers a second merger which has interesting welfare consequences. While a
single merger has anticompetitive effects, two mergers may involve one firm setting its wholesale
price below marginal costs, implying that a fully integrated industry can lead to welfare superior
outcomes than a non-integrated one. In this respect our results provide markedly different policy
implications compared to those of models without complementary inputs.
A natural direction for future research is to consider the case of Bertrand competition in the
downstream market. In our analysis we focussed on the case of Cournot competition—in line
with Rey and Tirole (2007)—which implies that firms’ strategy variables are strategic substitutes.
The case of strategic complements could be analyzed, for example, by considering a model with
differentiated Bertrand competition as in O’Brien and Shaffer (1992). It is of interest to understand
whether the expropriation problem is attenuated once the mode of competition in the downstream
market changes and whether new competitive effects of vertical integration arise.
Another research question regards the choice of the competition authority to approve or disapprove a vertical merger in a model with several merging opportunities. In our model, while
disapproving a single vertical merger is optimal for the competition authority, this merger triggers
a second merger which may renders an integrated industry procompetitive as compared to an unintegrated one. This implies that a myopic decision of the competition authority may not be optimal.
This result contrasts with the finding of Nocke and Whinston (2010) on horizontal mergers. It is
therefore of interest to explore if our result also holds in a dynamic merger game.
28
A
Appendix
Proof of Proposition 1
We first show that upstream firm Uj sets the per-unit price equal to marginal costs when making
an offer to a downstream firm Di .
We solve the game by backward induction. Thus, we start with the second stage, the downstream
stage. Since contract offers to a downstream firm Di , i = 1, 2, are not observable to the rival firm
D−i and downstream firms hold passive conjectures, Di expects D−i to produce the candidate
equilibrium quantity q−i independent of the contract offers it receives. Therefore, if Di accepts
offers such that its input costs are wj for input j and w−j for input −j, due to the one-to-one
technology it will produce a quantity qi that is given by
qi = arg max {(P (q + q−i ) − wj − w−j ) q} .
q
(10)
In the following of this proof, to simplify the exposition, we denote the downstream profit of firm
i, (P (qi + q−i ) − wj − w−j ) qi , by Πi (qi (wj , w−j ), q−i ).
Now we turn to the first stage, the offer game. Due to our assumption of passive beliefs, the
maximization problem of Uj in the game with Di , is given by
max
D
D
wU i ,FU i
j
j
Di
Di
Di
Di
i
xD
Uj (wUj , wU−j )(wUj − cj ) + FUj ,
(11)
subject to the constraint that FUDji is chosen in such a way that Di accepts the offer. As mentioned,
since there exists a continuum of combinations of FUDji and FUD−ji that fulfill this property, our
equilibrium selection technique is such that FUDji = 1/2(F SUDji + OOUDji ). We will now determine
F SUDji and OOUDji .
We start with F SUDji . The full surplus that Uj contributes to the profit of Di is the profit that
Di can obtain when accepting Uj ’s offer minus the profit it can obtain when rejecting the offer and
buying from the competitive fringe. When Di accepts Uj ’s offer, it obtains a profit net of the fixed
fee of
i
), q−i ),
Πi (qi (wUDji , wUD−j
Di
is the optimal wholesale price that Di pays for input −j. If Di rejects Uj ’s offer and
where w−j
buys from the competitive fringe, it obtains a profit of
i
Πi (qi (ĉj , wUD−j
), q−i ),
because the fringe just offers a wholesale price of ĉj and fixed fee of zero. Taken together, this
implies that the full surplus that Uj can receive from the relationship with Di is given by
i
i
F SUDji = Πi (qi (wUDji , wUD−j
), q−i ) − Πi (qi (ĉj , wUD−j
), q−i ).
29
Now we turn to the determination of OOUDji . As explained in the main text, OOUDji is given by
the offer that Uj needs to make to be sure that Di accepts. For this to hold true, two constraints
must be satisfied. First, Di accepts the offer from Uj if its profit from accepting is larger than
i
for input −j and U−j demands the
buying from the competitive fringe, given that it pays wUD−j
i
. This leads to the following constraint
full surplus F SUD−j
i
i
i
i
Πi (qi (wUDji , wUD−j
), q−i ) − F SUD−j
− FUDji ≥ Πi (qi (ĉj , wUD−j
), q−i ) − F SUD−j
.
To be sure that Di accepts, Uj ’s offer must also satisfy a second constraint. This is, that Di ’s
i
profit from accepting the offers from Uj and U−j , given that U−j demands the full surplus F SUD−j
,
is larger than the maximum of the profits when either accepting the offers from both fringe firms,
which is Πi (qi (ĉj , ĉ−j ), q−i ), or when not producing at all, which gives a profit of zero. Therefore,
the second constraint is given by
i
i
), q−i ) − F SUD−j
− FUDji ≥ max [Πi (qi (ĉj , ĉ−j ), q−i ), 0] .
Πi (qi (wUDji , wUD−j
As a consequence, OOUDji is given by the minimum fixed fee arising from these two constraints,
i.e.,
OOUDji
i
i
= min Πi (qi (wUDji , wUD−j
), q−i ) − Πi (qi (ĉj , wUD−j
), q−i ),
i
i
Πi (qi (wUDji , wUD−j
), q−i ) − F SUD−j
− max [Πi (qi (ĉj , ĉ−j ), q−i ), 0] .
We can now look at the maximization problem for wUDji . We know that FUDji = 1/2(F SUDji +
OOUDji ). From the expressions determining F SUDji and OOUDji it is evident that, independent of
Di
which expression is relevant for OOUDji , the only term that depends on wUDji is Πi (qi (wUDji , w−j
), q−i ).
Therefore, the maximization problem with respect to wUDji reduces to
Di
Di
Di
Di
Di
i
max xD
Uj (wUj , w−j )(wUj − cj ) + Πi (qi (wUj , w−j ), q−i ),
D
(12)
wU i
j
Because of the envelope theorem, the effect of a change of qi in response to a change in wUDji on
the profit of Di is zero. Thus, differentiating (12) with respect to wUDji gives
(wUDji
− cj )
i
∂xD
Uj
∂wUDji
i
+ xD
Uj − qi = 0.
(13)
Since the downstream transformation technology is one-to-one and downstream firms transform
Di
Di
i
all input into output we have that qi = xD
Uj . Moreover, given that (∂xUj /∂wUj ) < 0, we obtain
wUDji = cj , i.e. Uj optimally sets the per-unit price equal to marginal cost.
Since this holds for both offers of Uj , j = A, B, to Di , i = 1, 2, we obtain that both upstream
firms set their wholesale price equal to marginal costs.
30
As a consequence, both downstream firms face marginal costs of cA + cB . Therefore, the maximization problem of downstream firm i is given by
max {(P (q + q−i ) − cA − cB ) q} , i = 1, 2.
q
It follows that each downstream firm produces the Cournot quantity for marginal costs of cA + cB ,
that is,
q1 = q2 = q c ≡ qi (cA + cB , cA + cB ).
D
Finally, we turn to the determination of the fixed fees. Inserting wUDji = wUj−i = cj and q1 = q2 = q c
into F SUDji we obtain
F SUDji = Πi (q c , q c ) − Πi (qi (ĉj , c−j ), q c ).
(14)
For OOUDji , the value depends on the first or the second expression in the minimum term being the
tighter one. The first term is the same as F SUDji and is given by
Πi (q c , q c ) − Πi (qi (ĉj , c−j ), q c ).
(15)
Turning to the second expression we know that the full surplus of U−j is given by
i
F SUD−j
= Πi (q c , q c ) − Πi (qi (cj , ĉ−j ), q c ).
Inserting this into the second term and rearranging, we obtain
Πi (qi (cj , ĉ−j ), q c ) − max [Πi (qi (ĉj , ĉ−j ), q c ), 0] .
(16)
To determine which of the two expressions is smaller, we need to compare (15) with (16).
Subtracting (16) from (15) and rearranging, we obtain that (15) is larger than (16) if
Πi (q c , q c ) + max [Πi (qi (ĉj , ĉ−j ), q c ), 0] > Πi (qi (ĉj , c−j ), q c ) + Πi (qi (cj , ĉ−j ), q c ).
(17)
Now suppose that Πi (qi (ĉj , ĉ−j ), q c ) > 0. Then (17) writes as
Πi (q c , q c ) + Πi (qi (ĉj , ĉ−j ), q c ) > Πi (qi (ĉj , c−j ), q c ) + Πi (qi (cj , ĉ−j ), q c ).
(18)
The first term on the left-hand side of (18) is the profit of firm Di given that its marginal cost is
cA + cB , while the second term on the left-hand side is the profit of firm Di given that its marginal
cost is ĉA + ĉB . Instead, the two terms on the right-hand side of (18) represent firm Di ’s profit
given that its marginal cost is ĉA + cB and cA + ĉB , respectively.22 By Jensen’s inequality, (18)
is fulfilled if the profit function of Di is convex in marginal costs. Now, differentiating Πi with
respect to marginal cost C := cA + cB and using the envelope theorem, we obtain
∂Πi
= −qi < 0
∂C
22
Note that in each of the four terms in (18) firm D−i produces a quantity of q c .
31
and
∂ 2 Πi
∂qi
> 0.
=−
2
∂C
∂C
Thus, Πi is convex in marginal costs and (18) holds. The only difference between (17) and (18) is
that in (17) the second term is given by the maximum between Πi (qi (ĉj , ĉ−j ), q c ) and 0 while in
(18) it is just Πi (qi (ĉj , ĉ−j ), q c ). Therefore, the left-hand side of (17) is weakly larger than the one
of (18) implying that (17) is fulfilled as well. This implies that (15) is larger than (16) and the
second term in the minimum expression is the smaller one. Thus, OOUDji is given by (16).
To conclude, we have that firm Uj , j = A, B, proposes a contract in which the wholesale price
is given by
wUDji = cj
and the fixed fee is given by
1
Π(q c , q c ) − max {(P (q + q c ) − ĉj − c−j ) q}
=
q
2
c
c
+ max {(P (q + q ) − cj − ĉ−j ) q} − max max {(P (q + q ) − ĉj − ĉ−j ) q} , 0 .
FUDji
q
q
Proof of Proposition 2
We first consider the maximization problem of the non-integrated firm UB . Due to the assumption of passive beliefs, the maximization problem of UB in the game with Di , i = 1, 2, is given
by
max
D
D
wU i ,FU i
B
B
Di
Di
Di
Di
i
xD
UB (wA , wUB )(wUB − cB ) + FUB ,
(19)
subject to the constraint that FUDBi is chosen in such a way that Di accepts the offer.
Let us first look at the offer to the integrated firm UA − D1 . Since UB has full bargaining power
vis-à-vis D1 , FUDB1 is only subject to the constraint that determines the full surplus. That is, the
profit that D1 can obtain when accepting UB ’s offer minus the profit it can obtain when rejecting
the offer and buying from the competitive fringe. Since we know that UA − D1 trades the input
good internally at marginal cost, the full surplus, and therefore the fixed fee, is equal to
FUDB1 = F SUDB1 = max{(P (q + q2 ) − cA − wUDB1 )q} − max{(P (q + q2 ) − cA − ĉB )q}.
q
q
Now we turn to the fixed fee of UB on D2 , FUDB2 . This fixed fee is subject to both the constraint
for the determination of the full surplus (F SUDB2 ) and for the determination of the outside option
32
(OOUDB2 ). By the same token as in the proof of Proposition 1, we have that FUDB2 = 1/2(F SUDB2 +
OOUDB2 ), with
F SUDB2 = max{(P (q + q1 ) − wUDA2 − wUDB2 )q} − max{(P (q + q1 ) − wUDA2 − ĉB )q}
q
q
and
OOUDB2
= max{(P (q + q1 ) −
q
wUDA2
−
wUDB2 )q}
−
− max max{(P (q + q1 ) − ĉA − ĉB )q}, 0 .
FUDA2
q
By inspecting the expressions for FUDB1 and FUDB2 and using the terms that directly depend on
wUDB1 and wUDB2 , the problem of UB when setting wUDB1 and wUDB2 are given by
D1
D1
D1
1
xD
UB (cA , wUB )(wUB − cB ) + max{(P (q + q2 ) − cA − wUB )q}
max
q
D
wU 1
B
and
max
D
wU 2
B
D2
D2
D2
D2
D2
2
xD
UB (wUA , wUB )(wUB − cB ) + max{(P (q + q1 ) − wUA − wUB )q},
q
respectively. As above, solving these problems leads to wUDB1 = wUDB2 = cB .
We now characterize the optimization problem of the integrated firm UA − D1 with respect to
wUDA2 and use wUDBi = cB . UA − D1 solves
max max{(P (q + q2 ) − cA − cB )q} + q2 (wUDA2 − cA ) + FUDA2 − FUDB1 ,
q
D
D
wU 2 ,FU 2
A
A
(20)
where q2 solves maxq {(P (q + q1 ) − wUDA2 − cB )q}, with q1 ≡ arg maxq {(P (q + q2 ) − cA − cB )q}.
As for UB , the fixed fee that UA − D1 sets on D2 is given by FUDA2 = 1/2(F SUDA2 + OOUDA2 ),
with
F SUDA2 = max{(P (q + q1 ) − wUDA2 − cB )q} − max{(P (q + q1 ) − ĉA − cB )q}
q
q
and
OOUDA2
= max{(P (q + q1 ) −
q
wUDA2
− cB )q} −
FUDB2
− max max{(P (q + q1 ) − ĉA − ĉB )q}, 0 .
q
Therefore, we can rewrite the optimization program of the integrated firm by using only the
terms that are directly affected by wUDA2 as
max
D
wU 2
A
max{(P (q + q2 ) − cB − cA )q} + q2 (wUDA2 − cA ) + max{(P (q + q1 ) − wUDA2 − cB )q}.
q
q
(21)
As shown by Hart and Tirole (1990), (21) is maximized by raising the wholesale price as much
as possible, i.e., wUDA2 = ĉA . This is because the highest profit that can be reaped in the industry is
maxq {(P (q) − cB − cA )q}, and setting wUDA2 as high as possible comes closest to this profit.
33
Summarizing, at equilibrium UA − D1 sets wUDA2 = ĉA and FUDB2 = 0 while UB sets wUDBi = cB and
the fixed fees are equal to
FUDB1 = max{(P (q + q2c ) − cA − cB )q} − max{(P (q + q2c ) − cA − ĉB )q}
q
q
and
FUDB2 = max{(P (q + q1c ) − ĉA − cB )q} − max{(P (q + q1c ) − ĉA − ĉB )q},
q
q
where q1c ≡ q1 (cA + cB , ĉA + cB ) and q2c ≡ q2 (ĉA + cB , cA + cB ). Proof of Proposition 4
We take the derivative of the difference between the profit in case of integration ΠUA −D1 and the
sum of the profits in case of non-integration, ΠUA and ΠD1 , with respect to ĉA . Using the envelope
theorem, we obtain
∂q2c
∂ [ΠUA −D1 − (ΠUA + ΠD1 )]
c
c c
= (P (q1 + q2 )q1 + ĉA − cA )
+ q2c − q ,
∂ĉB
∂ĉA
(22)
with q ≡ arg maxq {(P (q + q c ) − ĉA − cB ) q}.
Quantities are strategic substitutes, so q > q2c and the sum of the last two terms is negative.
We now turn to the first term. Suppose that ĉA is just large enough so that q2c = 0. This is the case
if P (q1c ) = ĉA + cB . The first-order condition for the integrated firm determining q1c is then given
by q1c P (q1c ) + P (q1c ) − cA − cB = 0. Inserting P (q1c ) = ĉA + cB into this first-order condition yields
P (q1c )q1c + ĉA − cA = 0. But this implies that the first term of (22) equals zero at q2c = 0. Therefore,
at the value of ĉA at which q2c is equal to zero, we have (∂ [ΠUA −D1 − (ΠUA + ΠD1 )]) /∂ĉB < 0. By
continuity, if ĉA is sufficiently large but q2c is now slightly positive, the last two terms still dominate
the first, implying that the profitability of integration is still decreasing in ĉA . Proof of Proposition 5
i
The optimization problem of integrated firm U−j − D−i with respect to wUD−j
is
max
D
D
wU i ,FU i
−j
−j
D
D
i
max{(P (q + qi ) − wUj−i − c−j )q} + qi (wUD−j
− c−j ) + FUD−ji − FUj−i ,
q
D
i
)q} and q−i arg maxq {(P (q + qi ) − wUj−i − c−j )q}.
where qi ≡ arg maxq {(P (q + q−i ) − cj − wUD−j
Since Uj and Di trade input j at cost cj , the largest fixed fee that U−j can set is
i
FUD−ji = max{(P (q + q−i ) − cj − wUD−j
)q} − max{(P (q + q−i ) − cj − ĉ−j )q}.
q
q
34
(23)
We can now rewrite the optimization program of U−j − D−i by using only the terms that are
i
directly affected by wUD−j
max
D
wU i
−j
D
i
i
max{(P (q + qi ) − wUj−i − c−j )q} + qi (wUD−j
− c−j ) + max{(P (q + q−i ) − cj − wUD−j
)q}. (24)
q
q
i
is
The first-order condition with respect to wUD−j
i
P (Q)q−i + wUD−j
− c−j
∂q
∂q−i
i
+ P (Q)qi Di = 0.
Di
∂wU−j
∂wU−j
(25)
From the first-order conditions for the downstream quantities, which are given by P (Q) − cj −
D
i
i
wUD−j
+ P (Q)qi = 0 and P (Q) − c−j − wUj−i + P (Q)q−i = 0 we can calculate ∂qi /∂wUD−j
and
i
to get
∂q−i /∂wUD−j
and
∂qi
2P (Q) + q−i P (Q)
= i
P (Q) (3P (Q) + QP (Q))
∂wUD−j
(26)
∂q−i
P (Q) + q−i P (Q)
=
−
.
i
P (Q) (3P (Q) + QP (Q))
∂wUD−j
(27)
Inserting this into (25) we obtain
i
wUD−j
= c−j −
P (Q) (P (Q)(2q−i − qi ) + P (Q)q−i (q−i − qi ))
,
2P (Q) + q−i P (Q)
(28)
with j = A, B, i = 1, 2.
i
< ĉj ) is stable in this
We can now analyze for which conditions an interior equilibrium (wUD−j
Di
D−i framework. We know that an equilibrium is unstable if dwU−j /dwUj > 1 for at least one firm.
D
i
/dwUj−i , we apply the Implicit Function Theorem to the first-order condition
To determine dwUD−j
i
i
in (25). Calculating the second derivatives of ∂qi /∂wUD−j
and ∂q−i /∂wUD−j
from (26) and (27),
i
respectively, und ssing that wUD−j
is given by (28) at an interior equilibrium, we obtain
D dw i U−j D−i > 1 ⇐⇒ 2(P (Q))2 > P (Q) (3P (Q)(qi − q−i ) + q−i P (Q)(2qi − q−i )) ,
dwU j
D
(29)
D
i
i
, wUj−i + c−j ) and q−i = q−i (wUj−i + c−j , cj + wUD−j
)
with qi = qi (cj + wUD−j
D
i
implicitly
Consequently, if the inequality in (29) holds for the wholesale prices wUj−i and wUD−j
defined by (28) and the respective quantities qi and q−i , an interior equilibrium is unstable. In
this case, the only stable equilibria of the game feature one integrated firm, e.g., Uj − Di , setting
D
i
equal to the
its wholesale price wUj−i at ĉj and firm U−j − D−i optimally replying by setting wUD−j
minimum between ĉ−j and
i
= c−j −
w̌UD−j
P (Q) (P (Q)(2q−i − qi ) + P (Q)q−i (q−i − qi ))
,
2P (Q) + q−i P (Q)
35
i
i
with qi = qi (cj + w̌UD−j
, ĉj + c−j ) and q−i = q−i (ĉj + c−j , cj + w̌UD−j
).
The optimal fixed fees are equal to
D
FUj−i = 0
and
i
FUD−ji = max{(P (q + q−i ) − wUD−j
− cj )q} − max{(P (q + q−i ) − ĉ−j − cj )q},
q
q
i
where q−i = q−i (ĉj + c−j , cj + wUD−j
).
Conversely, if the inequality in (29) does not hold true, then an interior equilibrium is stable.
At this equilibrium, linear prices are given by the minimum between (28) and ĉj , with j = A, B.
The optimal fixed fees are equal to
D
D
FUj−i = max{(P (q + qi ) − wUj−i − c−j )q} − max{(P (q + qi ) − ĉj − c−j )q}
q
q
and
i
FUD−ji = max{(P (q + q−i ) − wUD−j
− cj )q} − max{(P (q + q−i ) − ĉ−j − cj )q},
q
where qi = qi (cj +
q
D
i
wUD−j
, wUj−i
+ c−j ) and q−i =
D
q−i (wUj−i
i
+ c−j , cj + wUD−j
). Proof of Proposition 6
The sum of the profits of firms UB and D2 in the case with a single merger is given by
ΠUB +ΠD2 = q2c (P (q1c +q2c )−ĉA −cB )+q1c (P (q1c +q2c )−cA −cB )−max {(P (q + q2c ) − cA − ĉB )q} , (30)
q
where q1c = q1 (cA + cB , ĉA + cB ) and q2c = q2 (ĉA + cB , cA + cB ).
The profit of an integrated firm UB − D2 in the counter-merger case is equal to
ΠUB −D2 = q2 (P (q1 +q2 )−ĉA −cB )+(wUDB1 −cB )q1 +q1 (P (q1 +q2 )−cA −wUDB1 )−(P (q1 +q2 )−cA −ĉB )q1 .
(31)
In this equation, q1 =
q1 (cA + wUDB1 , ĉA
+ cB ), q2 =
q2 (ĉA + cB , cA + wUDB1 ),
q1 = q1 (cA + ĉB , ĉA + cB ),
and q2 = q2 (ĉA + cB , cA + ĉB ), where wUDB1 is given by (28). Rewriting (31) yields
ΠUB −D2 = q2 (P (q1 + q2 ) − ĉA − cB ) + q1 (P (q1 + q2 ) − cA − cB ) − (P (q1 + q2 ) − cA − ĉB )q1 . (32)
We can now compare (30) with (32). It is evident that the last term in (32) is smaller than
the last term in (30): q2 is larger than q2c because firm UB − D2 observes that firm D1 ’s wholesale
price for input 2 is ĉB and not cB (information effect). Turning to the first two terms, they differ
insofar as both firms produce a different quantity in each equation, that is, qic is not equal to qi ,
i = 1, 2. From (24) we know that in the counter-merger case wUDB1 is chosen to maximize
q2 (P (q1 + q2 ) − wUDA2 − cB ) + q1 (wUDB1 − cB ) + q1 (P (q1 + q2 ) − cA − wUDB1 ) =
36
q2 (P (q1 + q2 ) − wUDA2 − cB ) + q1 (P (q1 + q2 ) − cA − cB ).
Since we are at an extremal equilibrium, firm UA sets wUDA2 = ĉA . Therefore, wUDB1 maximizes
q2 (P (q1 + q2 ) − ĉA − cB ) + q1 (P (q1 + q2 ) − cA − cB ),
which is exactly equal to the first two terms in (32). Hence, the first two terms in (32) must be
weakly larger than the corresponding two terms in (30), because in the latter expression wUDB1 = cA ,
which does not necessarily maximize the expression. This allows us to conclude that (32) is weakly
larger than (30), implying that the counter-merger is always profitable at an extremal equilibrium.
Proof of Proposition 7
To evaluate the competitive effect of a counter-merger, it is sufficient to determine whether wUDB1
lies above or below cB . This is because all other input prices are the same in the two scenarios
featuring single integration and two merged firms. From above, we know that
wUDB1 = cB −
P (Q) (P (Q)(2q2 − q1 ) + P (Q)q2 (q2 − q1 ))
,
2P (Q) + q2 P (Q)
with q1 = q1 (cA + wUDB1 , ĉA + cB ) and q2 = q2 (ĉA + cB , cA + wUDB1 ). From the first-order conditions for
q1 and q2 we have q1 = −(P (Q) − cA − wUDB1 )/P (Q) and q2 = −(P (Q) − ĉA − cB )/P (Q). Plugging
these expressions into wUDB1 we find that
wUDB1 = P (Q) − 2ĉA + cA +
P (Q)(ĉA − cA )(P − ĉA − cB )
.
(P (Q))2
Subtracting cB from the last expression, using ĉA = ΔA + cA , and solving for ΔA yields that
wUDB1 < cB if and only if
1
2
2
2
2
ΔA >
−2(P (Q)) + P (Q)(P (Q) − cA − cB ) + (P (Q)) (P (Q) − cA − cB ) + 4(P (Q)) .
2P (Q)
Applying L’Hospital’s rule we obtain that for P (Q) → 0, the last expression boils down to
ΔA >
P (Q) − cA − cB
.
2
Proof of Proposition 8
Since the two downstream firms produce homogeneous products and face the same technology,
they are symmetric up to potential differences in the marginal costs. In this case, if both firms
37
are active, the aggregate quantity is determined by the sum of the marginal costs, but not by how
these costs are distributed across firms (see, e.g., Bergstrom and Varian, 1985).
If no firm is integrated, the sum of marginal costs is 2(cA + cB ), instead if both firms are
integrated, the sum is cA + wUDB1 + ĉA + cB . Subtracting the latter expression from the former and
simplifying, we obtain
−
(P (Q) − ĉA − cB )(P (Q)(ĉA − cA ) + (P (Q))2 )
.
(P (Q))2
The first term of the numerator is positive by our assumption that both firms are active even
if one of them purchases one of its input from the competitive fringe. The second term of the
numerator is positive for P (Q) ≥ 0. Hence, if the demand function is linear or convex, the whole
expression is negative implying that two mergers are anticompetitive as compared to no merger.
By contrast, if P (Q) < 0, two mergers are procompetitive as compared to no merger if and only
if −P (Q)(ĉA − cA ) > (P (Q))2 or −P (Q)ΔA > (P (Q))2 . Proof of Proposition 9
From the proof of Proposition 5 we know that if condition S does not hold, a stable interior
equilibrium exists. At this equilibrium wholesale prices are given by (5). From the downstream
market’s first-order conditions, we know that the equilibrium quantities are implicitly defined by
i
qi = −(P (Q) − cj − wUD−j
)/P (Q), i = 1, 2, j = A, B. Inserting these quantities into the wholesale
D
prices given by (5) and solving for wUj−i , i = 1, 2, j = A, B, yields
D
wUj−i
1
=
2
P (Q) + cj − c−j
3(P (Q))2 +
−
P (Q)
√ ξ
,
with
2
ξ ≡ 9(P (Q))4 + (P (Q)) (P (Q) − cA − cB )2 − 2P (Q)(P (Q))2 (P (Q) − cA − cB ).
Therefore, at an interior equilibrium the sum of the the downstream firms’ marginal costs is given
by cA + wUDB1 + wUDA2 + cB , which equals
1 2
3(P (Q)) − ξ .
cA + cB + P (Q) − P (Q)
We can now turn to the assessment of the competitive effects of the counter-merger, given that
an interior equilibrium exists. First, we start with the comparison between the single integration
case and the counter-merger case. In the single-integration case, the sum of marginal costs is equal
to cA + 2cB + ĉA , which is increasing in ĉA . Instead, the sum of marginal costs at an interior
38
equilibrium of the counter-merger game is independent of ĉA . Hence, if ΔA is large enough, the
counter-merger is clearly procompetitive relative to a single merger.
We can now compare the sum of marginal costs in the counter-merger case with the one under
no merger. In the latter case we know that marginal costs are equal to 2(cA + cB ). Subtracting the
sum of marginal costs in the no merger case from the sum of marginal costs in the counter-merger
case we obtain that two mergers are anticompetitive as compared to no merger if and only if
√
ξ
3(P (Q))2
+ > 0.
P (Q) − cA − cB −
P (Q)
P (Q)
Suppose first that P (Q) < 0.23 Multiplying the inequality above by P (Q) yields P (Q)(P (Q) −
√
cA − cB ) − 3(P (Q))2 + ξ < 0. Bringing the square root the right-hand side and multiplying
√
by −1 then yields 3(P (Q))2 − P (Q)(P (Q) − cA − cB ) > ξ. Since P (Q) < 0 both sides are
strictly positive and we can square the inequality without changing the inequality sign. Doing so
and simplifying the resulting expression gives
0 > 4(P (Q))2 P (Q)(P (Q) − cA − cB ),
which is fulfilled since P (Q) < 0. Hence, a counter-merger resulting in a stable interior equilibrium
is anticompetitive as compared to no merger.
We now show that an interior equilibrium cannot be stable for P (Q) > 0. For an interior to
be stable we must have that condition S is satisfied for both firms, that is
2(P (Q))2 − (P (Q))2 q−i (2qi − q−i ) − 3P (Q)P (Q)(qi − q−i ) < 0
and
2(P (Q))2 − (P (Q))2 qi (2q−i − qi ) − 3P (Q)P (Q)(q−i − qi ) < 0.
Summing up these two conditions and simplifying yields
4(P (Q))2 − (P (Q))2 (qi − q−i )2 − 2(P (Q))2 qi q−i < 0.
Now suppose that downstream firm Di produces a share α of the downstream quantity while D−i
produces a share 1 − α, with 0 ≤ α ≤ 1. Inserting this into the above inequality, we obtain
4(P (Q))2 − (P (Q))2 (1 − 2α)2 Q2 − 2(P (Q))2 α(1 − α)Q2 < 0.
(33)
By our assumption that the profit functions are concave we have P (Q) + qi P (Q) < 0, i = 1, 2,
which implies for P (Q) > 0, that qi < −P /P . Hence, Q < −2P /P . Inserting Q = −2P /P into the left-hand side of (33), and simplifying yields
8(P (Q))2 α(1 − α),
23
Remember that the interior equilibrium is unstable for P (Q) = 0.
39
which is weakly positive for 0 ≤ α ≤ 1. But since Q is in fact smaller than −2P /P , the
left-hand side of (33) must be strictly positive for P (Q) > 0. This contradicts that condition S
is satisfied for both firms, implying that an interior equilibrium must be unstable for P (Q) > 0. Proof of Proposition 10
We solve the game by backward induction. In the second stage, firm UB can decide to integrate
with firm D2 after UA and D1 decide on whether to integrate or not. From the section that
analyzes the counter-merger incentives, we know that if UA and D1 integrate in the first stage, it
is always profitable for UB and D2 to integrate in the second stage. Suppose now that UA and
D1 did not integrate in the first stage. Then, the integration decision of UB and D2 in the second
stage depends on the same condition that determines UA and D1 decision in Section 4, mutatis
mutandis. It follows that, if UA and D1 do not merge in the first stage, UB and D2 integrate if and
only if
8(1 − cA − cB ) − 12ΔA
.
11
Having pinned down the condition that prescribes the integration decision of UB and D2 in the
ΔB ≤
second stage, let us turn to the first stage. Suppose first that ΔB ≤ (8(1 − cA − cB ) − 12ΔA ) /11,
i.e., firms UB and D2 will always integrate in the second stage. Two cases must be distinguished:
1. If firms UA and D1 stay unintegrated in the first stage, their joint profit is equal to
2 2 2
1
1
1
(1 − cA − 2ĉB + cB ) +
(1 − cA − 2cB + ĉB ) −
(2 − 3ĉA − 4cB + cA + 2ĉB ) .
3
3
6
Here, the first term is the profit of D1 gross of the fixed fee paid to UB , whereas the other
two terms represent the fixed fee that UA charges to D2 .
2. If firms UA and D1 integrate, they obtain a profit of
2
1
1
(1 − 2cA − 2ĉB + ĉA + cB ) + (ĉA − cA ) (1 + 2cA − cB − 3ĉA ).
3
2
Here, the first term is again the profit raised by D1 and the second term is UA ’s margin on
the quantity that it sells to D2 , times this quantity.
Comparing these two profits, we obtain that the profit under integration is larger if and only
if
14(1 − cA − cB ) − 41ΔA
.
28
It is readily verified, that (14(1 − cA − cB ) − 41ΔA ) /28 < (8(1 − cA − cB ) − 12ΔA ) /11. Hence,
ΔB ≤
for ΔB ≤ (14(1 − cA − cB ) − 41ΔA ) /28 two mergers obtain in equilibrium. Instead, for (14(1 −
40
cA − cB ) − 41ΔA )/28 < ΔB ≤ (8(1 − cA − cB ) − 12ΔA ) /11 only UB and D2 integrate, while UA
and D1 stay unintegrated.
Finally, we turn to the case in which ΔB > (8(1 − cA − cB ) − 12ΔA ) /11. Again, two cases can
occur:
1. If UA and D1 do not integrate, UB and D2 will also not integrate. The joint profit of UA and
D1 is then equal to
2 2 2
1
1
1
(1 − cA − cB ) +
(2 − 2cA + cB − 3ĉB ) −
(2 − 2cB + cA − 3ĉA ) .
3
6
6
2. The profit of an integrated UA − D1 is as above,
2
1
1
(1 − 2cA − 2ĉB + ĉA + cB ) + (ĉA − cA ) (1 + 2cA − cB − 3ĉA ),
3
2
because UB and D2 will also decide to integrate if UA and D1 merge.
Comparing these two expressions, we obtain that integration is profitable for UA and D1 if and
only if
1
ΔB ≤
7
8ΔA + 2(1 − cA − cB ) −
351Δ2A
+ 4(1 − 2cA − 2cB ) − 66ΔA (1 − cA − cB ) + 4(cA + cB )2 .
It is easy to check that the right-hand side of the last inequality is smaller than (8(1 − cA −
cB ) − 12ΔA )/11. It follows that for ΔB > (8(1 − cA − cB ) − 12ΔA ) /11, no merger takes place in
equilibrium. References
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Down- stream Competition with Capital Precommitment”, International Journal of Industrial
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[2] Bergstrom, Theodore C. and Hal R. Varian (1985): “When are Nash Equilibria Independent
of Agents’ Characteristics?”, Review of Economic Studies, 52, 715-718.
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of Economics, 32, 667-685.
[4] Choi, Jay Pil and Sang-Seung Yi (2000): “Vertical Foreclosure with the Choice of Input
Specifications”, Rand Journal of Economics, 30, 717-743.
[5] Hart, Oliver and Jean Tirole (1990): “Vertical Integration and Market Foreclosure”, Brookings
Papers on Economic Activity (Microeconomics), 205-285.
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Economic Review, 55, 820-831.
41
[7] Kreps, David and Jose Scheinkman (1983): “Quantity Pre-Commitment and Bertrand Competition yield Cournot Outcomes”, Bell Journal of Economics, 14, 326-337.
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H. Porter (eds.), Handbook of Industrial Organization, Vol. 3, North-Holland, 2145-2220.
[18] Riordan, Michael (1996): “Anticompetitive Vertical Integration by a Dominant Firm”, American Economic Review, 88, 1232-1248.
[19] Salop, Steven C. and Scheffman, David T. (1983): “Raising Rivals’ Costs”, American Economic Review, Papers and Proceedings, 73, 267-271.
[20] Salop, Steven C. and Scheffman, David T. (1987): “Cost-Raising Strategies”, The Journal
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[21] Schmidt, Klaus M. (2008): “Complementary Patents and Market Structure”, CEPR Discussion Paper No. DP7005.
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Management Strategy, 16, 507-535.
42
[24] Vives, Xavier (1999): Oligopoly Pricing: Old Ideas and New Tools, MIT Press, Cambridge:
Massachusetts.
ÛA
UA
UB
ÛB
Q
Q
Q
A
@
A
A
A
QQ @
Q @
A A @
Q
AU AU R
Q
+ s D1
D2
@
R
@
Consumers
Figure 1: Framework.
'$
ÛA
UA
UB
ÛB
Q
Q
Q @
A
A
Q @
Q
Q @
A Q
@
AU R
+ Q
s
D2
D1
&%
@
R
@
Consumers
Figure 2: Framework with Integration.
43
'$'$
ÛA
UA
UB
D1
D2
ÛB
Q
Q
Q @
Q @
Q
Q @
@
Q
R
+ Q
s
&%&%
@
R
@
Consumers
Figure 3: Framework with Counter-merger.
44