Vertical Integration, Foreclosure, and Productive
Efficiency∗
Markus Reisinger†
Emanuele Tarantino‡
WHU - Otto Beisheim School of Management
University of Bologna
July 2014
Abstract
We analyze the competitive consequences of vertical integration in a model featuring
a monopoly producer dealing with asymmetric retailers via secret two-part tariffs.
When integrated with the inefficient retailer, the monopoly producer keeps the rival
retailer active on the product market due to an output-shifting effect. This effect
can induce the integrated firm to engage in below-cost pricing at the wholesale level,
thereby rendering integration procompetitive. We study how information transmission
within a vertically integrated organization affects these results, and extend the model
to show that integration with an inefficient retailer emerges in a model with uncertainty
over retailers’ costs.
JEL classification: K21, L12, L13, L41, L42
Keywords: vertical relations, vertical integration, foreclosure, output shifting, antitrust policy.
∗
We are indebted to the editor (Benjamin Hermalin) and three anonymous referees for insightful comments and suggestions.
The article also benefited from comments by Cédric Argenton, Heski Bar-Isaac, Felix Bierbrauer, Giacomo Calzolari, Simon
Cowan, Vincenzo Denicolò, Chiara Fumagalli, Massimo Motta, Volker Nocke, Marco Pagnozzi, Emmanuel Petrakis, Salvatore
Piccolo, Patrick Rey, Armin Schmutzler, Nicolas Schutz, Marius Schwartz, Greg Shaffer, Kathryn Spier, Elu von-Thadden, Ali
Yurukoglu, and Gijsbert Zwart. We also thank participants at the University of Bologna, Max Planck Institute for Collective
Goods (Bonn), Pontificia Universidad Católica de Chile, University of Luxemburg, University of Mannheim, CSEF (Naples),
University of Rochester (Simon), TILEC - Tilburg University, ETH Zurich seminars, and at the 2012 Annual Searle Center
Conference on Antitrust Economics and Competition Policy (Northwestern University), the 2011 International Industrial Organization Conference (Boston), 2011 European Economic Association Annual Meeting (Oslo), the 2010 Workshop for new
Researchers (Centre for Competition Policy - University of East Anglia), and XXV Jornadas de Economı̀a Industrial (Madrid).
†
WHU - Otto Beisheim School of Management, Department of Economics, Burgplatz 2, 56179 Vallendar, Germany. E-Mail:
Markus.reisinger@whu.edu. Also affiliated with CESifo.
‡
University of Bologna, Department of Economics, piazza Scaravilli 1, I-40126, Bologna, Italy; Phone: +39-051-20-98885;
emanuele.tarantino@unibo.it. Also affiliated with TILEC.
1
Introduction
How does vertical integration affect economic outcomes such as prices, quantities, and consumer surplus? Is vertical integration motivated by the desire to increase market power, or
is it a device that enhances productive efficiency and welfare? On the one hand, a large
theoretical literature shows that when a manufacturer deals with equally efficient retailers,
vertical integration allows it to increase its market power by foreclosing rival retailers’ access
to the input it produces (see, e.g., Rey and Tirole, 2007). On the other hand, the empirical
literature presents evidence suggesting that efficiency-based mechanisms are behind vertical
integration (e.g., Hortaçsu and Syverson, 2007; Lafontaine and Slade, 2007).1
This paper theoretically analyzes the welfare consequences of vertical integration using
a model in which a dominant producer sells through retailers that have different marginal
costs of production. We find that vertical integration with the less efficient retailer can
raise consumer surplus and total welfare by improving productive efficiency. The mechanism
we put forward features the integrated firm’s upstream unit selling its input at favorable
conditions to the nonintegrated efficient retailer, to induce this retailer to expand its output.
This output-shifting effect gives rise to a procompetitive outcome whenever the integrated
firm engages in below-cost pricing at the wholesale level.
The evidence on the competitive effects of vertical integration is not fully conclusive.
There is some evidence in support of foreclosure (Chipty, 2001; Hastings and Gilbert, 2005;
among others). However, several empirical studies show that the efficiency gains produced
by vertical integration can outweigh the welfare losses caused by foreclosure. For example,
in their survey of empirical research, Lafontaine and Slade (2007) conclude that in most circumstances profit-maximizing vertical integration also raises consumer welfare. Our results
suggest that vertical integration enhances productive efficiency thanks to the output-shifting
strategy undertaken by an integrated company that acts as a merchant supplier (McAfee,
2002) by serving competing buyers on favorable terms.2 This novel channel for procompetitive vertical integration arises in a framework that builds on the literature studying the
anticompetitive effects of vertical restraints.
In our model, a monopoly producer offers an intermediate good by means of secret twopart tariffs to two competing retailers that transform the input into a homogeneous final
product. In contrast to the standard setting employed in the literature, we allow retailers to
carry different marginal costs of production.
Without integration, the monopoly producer’s limited commitment to observable contracts prevents it from monopolizing the final product’s market (e.g., Hart and Tirole, 1990;
O’Brien and Shaffer, 1992; McAfee and Schwartz, 1994; Rey and Tirole, 2007; White, 2007)
(Lemma 1). As is well-established in the literature, integration with the more efficient retailer allows the monopoly producer to restore its market power by formulating an unfeasible
1
Specifically, the property-rights theories (e.g., Grossman and Hart, 1986), the theories highlighting the
role of transaction costs (Williamson, 1971, among others) and those looking at the elimination of double
marginalization (e.g., Salinger, 1988) show that vertical integration can raise efficiency.
2
Following McAfee (2002), a merchant supplier is an integrated firm that treats nonintegrated buyers
on equal or favorable terms. McAfee (2002) provides a discussion and examples of merchant buyers and
suppliers. Block, Bock, and Henkel (2010) document that selling to a competitor is a widespread practice in
business, and report evidence from construction, tea packaging, and mining industries.
1
offer to the nonintegrated firm (the foreclosure effect of vertical integration) (Lemma 2). If
instead the monopolist were integrated with the less efficient retailer, would the integrated
firm still engage in a foreclosure strategy? We show that the newly integrated firm will
reduce the unit-price offer to the nonintegrated but more efficient retailer (Proposition 1).
The trade-off faced by the integrated upstream firm is as follows. With respect to a foreclosure strategy, a reduction of the unit price to the nonintegrated retailer raises the industry
quantity and thus implies a reduction of industry revenue. However, a countervailing effect
arises in our framework. Differently from the case with vertical separation, the reduction of
the unit price to the nonintegrated retailer is observed by the integrated firm’s downstream
unit, which responds by reducing its quantity. A lower unit price then triggers an increase of
the nonintegrated and more efficient retailer’s quantity and profit. The upstream integrated
firm can extract this higher profit via the fixed component of the two-part tariff. We show
that the reduction of industry revenue is outweighed by the fact that the industry quantity
is produced more efficiently. Indeed, we find that the nonintegrated retailer will be active on
the final good’s market as long as it is strictly more efficient than the integrated downstream
unit (Proposition 2), thereby establishing the output-shifting effect of vertical integration.
A question of great importance for antitrust policy is the magnitude of the output-shifting
effect. Specifically, can the incentive to reduce the unit price to the nonintegrated firm be
so strong as to render vertical integration procompetitive with respect to a nonintegrated
industry? We show in Proposition 3 that the output-shifting effect can induce the upstream
unit of the integrated firm to engage in below-cost pricing at the wholesale level. Below-cost
pricing produces an expansion of industry output relative to vertical separation, and renders
vertical integration procompetitive. It is optimal because it allows the integrated firm to
restrict the quantity of the inefficient integrated retailer and obtain larger profits from the
more efficient nonintegrated one. Indeed, we find that below-cost pricing is more likely to
occur when cost differences between retailers are particularly large.
In the industry structure of our main model, the upstream producer is an unconstrained
monopolist. We then extend our main model to consider an industry in which a dominant
producer competes with a fringe of less efficient firms. These fringe firms act as a bottleneck
alternative that constrains the market power of the dominant producer. We show that the
outcome mirrors that of our main model: when the dominant producer is merged with
the inefficient retailer, the integrated firm will engage in output shifting that can result in
below-cost pricing at the wholesale level (Proposition 4).
The analysis of the model with a bottleneck alternative allows us to study the impact
of information transmission within a vertically integrated firm on market prices and allocations.3 We show that the intensity of the output-shifting effect depends on whether the
upstream unit of the integrated firm can inform its downstream unit about the acceptance
decision of the rival retailer. Interestingly, we find that if information transmission is possible, vertical integration results in a more competitive outcome (Proposition 5). When
information transmission is possible, the downstream affiliate of the integrated firm can adjust its quantity based on the rival retailer’s decision to reject the upstream affiliate’s offer.
3
With the exception of Nocke and Rey (2012), studies of information sharing in models with vertical
hierarchies typically analyze the implications of information exchange between organizations rather than
within an organization (Bonanno and Vickers, 1988; Pagnozzi and Piccolo, 2012; Arya and Mittendorf,
2011).
2
In this case, the outside option of the rival retailer, i.e., the profits it obtains when buying
from the bottleneck alternative, does not depend on the terms of the upstream unit firm’s
offer. Instead, when information transmission is not possible, the downstream unit believes
that the rival retailer buys at the equilibrium per-unit price. The upstream unit can then
reduce the outside option of the nonintegrated retailer by setting a higher unit price.
The imposition of behavioral remedies such as information firewalls is one of the most
common forms of conduct relief in merger decisions.4 Our analysis of information transmission within a vertical organization allows us to assess the consequences of such firewalls on
economic outcomes and welfare. We show that, within our model, this type of remedy can
be detrimental to consumer surplus.
To check the robustness of our results, we solve the model under a number of alternative
assumptions. Motivated by the empirical evidence on manufacturer-retailer relationships
in vertically related industries (e.g., Villas-Boas, 2007; Bonnet and Dubois, 2010), we employ two-part tariff contracts in our main set-up. We first extend the model by letting
the upstream monopoly use quantity-forcing contracts, and find that the equilibrium allocations with quantity-forcing arrangements coincide with those obtained using two-part tariffs.
Second, although in our main model we assume that retailers hold passive beliefs, our conclusions are robust to the adoption of wary beliefs. Third, in the main model, we assume
that retailers offer a homogeneous good. We extend the model to consider retailers offering
differentiated products and again find that when the monopoly producer is integrated with
the inefficient retailer, it engages in output shifting, which can result in a procompetitive
outcome. Finally, we argue that the asymmetry in retailers’ marginal costs that we exogenously impose can be endogenized in a setup with providers of complementary intermediate
goods (consistently with the results in Reisinger and Tarantino, 2013).5
When the monopoly producer is integrated with the inefficient retailer, it would like to
shutter its downstream unit and let the efficient retailer produce the monopoly quantity.
However, it lacks the commitment to internally transfer the input good at any price above
marginal cost. This commitment problem prevents the integrated firm from monopolizing
the final good market. To show that this result survives even when the upstream unit of
the integrated organization can commit to shutting its downstream subsidiary down, we
introduce a variant of the main model in which we assume that the integrated firm deals
with the nonintegrated retailer via Nash bargaining (Horn and Wolinsky, 1988; O’Brien and
Shaffer, 2005; Milliou and Petrakis, 2007). We show that the integrated firm wants to keep
4
For example, firewall provisions have been recently adopted by the FTC in vertical merger cases like
Lockheed Martin (Lockheed Martin Corp., FTC File No. 9610026 ), Raytheon (Raytheon Co., FTC File
No. 9610057 ) and PepsiCo (PepsiCo Inc., FTC Docket No. 0910133 ). See the 2008 UK Competition
Commission guidelines or the 2011 US Department of Justice guidelines to merger remedies for further
references.
5
Other papers focusing on settings that feature complementarity in the provision of inputs or services
are Laussel (2008), Laussel and Van Long (2012), Matsushima and Mizuno (2012), and Hermalin and Katz
(2013). The analysis in Reisinger and Tarantino (2013) departs from these papers by analyzing vertical
integration rather than exclusivity arrangements of service providers, which is the main focus of Hermalin
and Katz (2013). In addition, the results in Reisinger and Tarantino (2013) do not rely on the efficiency
incentives for vertical integration such as the elimination of double marginalization, a common feature of
Laussel (2008), Laussel and Long (2012) and Matsushima and Mizuno (2012).
3
its downstream affiliate alive to improve its bargaining position vis-à-vis the independent
retailer.
The discussion thus far begs the question why would the upstream monopolist merge
with the inefficient retailer. This might happen because the antitrust authority forbids a
merger with the efficient retailer because it leads to market monopolization (as shown in
Lemma 2). In this respect, the model provides a motivation why the upstream monopolist
can only merge with the less efficient retailer. There are also several alternative explanations.
For example, the efficient retailer might be part of a large conglomerate, which prevents the
upstream monopolist from acquiring it. Also, due to historical reasons the upstream firm
might be integrated with a retailer when the industry is liberalized, and a more efficient
retailer enters the market. In addition, we demonstrate that a market structure in which
the monopoly producer is integrated with the inefficient retailer can arise in a model where
the upstream monopolist’s integration decision is taken under uncertainty over retailers’
marginal cost of production. Specifically, we provide the conditions such that the monopoly
producer merges with a retailer that is less efficient in expectation and this merger leads to
a procompetitive outcome (Proposition 6).
The “Chicago School” has challenged the view that an upstream monopolist needs to
integrate in order to monopolize a competitive downstream market (e.g., Bork, 1978; Posner, 1976; among others). It has also disputed that an integrated monopoly producer has
an incentive to exclude competing firms that can be the source of extra rents thanks to, say,
cost efficiency. The post-Chicago School literature has noted that, when wholesale contracts
are secret, the upstream monopolist’s market power is eroded by a commitment problem
that prevents it from monopolizing the final good market.6 In this literature, vertical integration allows a dominant supplier to restore its market power by foreclosing the competing
retailer’s access to the intermediate good. We build on the post-Chicago School literature by
embracing its approach. At the same time, we borrow from the Chicago School the idea that
the dominant producer might deal with retailers with different marginal costs of production.
We show that in line with the Chicago School argument, these differences in marginal costs
give rise to an output-shifting effect that can render vertical integration procompetitive with
respect to nonintegration.
This result suggests that, for example, policies of divestiture imposed by regulatory agencies to prevent foreclosure can have unintended consequences and may well be misguided.
Rey and Tirole (2007) list some of the major decisions of divestitures taken by antitrust
authorities, from the 1984 breakup of AT&T to the separation of electricity generation systems from high voltage electricity transmission systems in most countries. Consistent with
our conclusions, Lafontaine and Slade (2007) document that studies assessing the implications of these forced vertical separations generally find that such legal decisions lead to price
increases.
Other papers have analyzed vertical integration in different, but related, contexts. For
example, Ordover, Saloner, and Salop (1990) and Chen (2001) consider the case of public
offers in linear prices. Choi and Yi (2000) develop a model in which upstream firms can
6
This commitment problem was first noticed by Hart and Tirole (1990), and then further analyzed by
O’Brien and Shaffer (1992), McAfee and Schwartz (1994), Rey and Vergé (2004), and Marx and Shaffer
(2004), among others. Recently, Nocke and Rey (2012) have proposed a model featuring upstream manufacturers that produce differentiated goods.
4
choose to customize their inputs to fit the needs of downstream firms, and Riordan (1998)
considers a model in which a dominant firm has market power in the final and intermediate
goods market. Finally, Nocke and White (2007) analyze the effects of vertical integration
on the sustainability of upstream collusion. These papers find that integrated firms have
an incentive to foreclose their downstream rivals. Instead, we show that an integrated firm
wants to keep a more efficient rival retailer alive, and serve it at favorable conditions when
particularly efficient.
Our paper is also related to the literature on vertical relationships that emphasizes the
role of differences among retailers. Inderst and Shaffer (2009) and Inderst and Valletti
(2009) study the implications of price discrimination in input markets when buyers are
asymmetric.7 Relatedly, Chen and Schwartz (2013) analyze the welfare effects of monopoly
price discrimination when costs of service differ across consumer groups. Finally, Spiegel and
Yehezkel (2003) analyze the case in which retailers are vertically differentiated. However,
this literature does not study vertical integration and therefore does not examine the forces
at work in our model.
The article is structured as follows: Section 2 presents the main model, and Section
3 provides the equilibrium analysis. In Section 4, we extend the model to include a less
efficient fringe of alternative producers of the intermediate good, and in Section 5, we test
the robustness of our main results to alternative assumptions. Section 6 presents a model
with equilibrium vertical integration when retail costs are uncertain, and Section 7 concludes.
The formal proofs of the results are relegated to the Appendix.
2
The Model
We study a vertically related industry along the lines of McAfee and Schwartz (1994) and Rey
and Tirole (2007). An upstream firm, U , is a monopoly producer of an intermediate good
with marginal production cost c. It supplies two retailers, D1 and D2 , that are Cournot rivals
in a downstream market (see Figure 1). The retailers transform the intermediate good into a
homogeneous final product on a one-to-one basis. In contrast to previous literature, we allow
retailers to carry different marginal costs of production. Specifically, retailer D1 ’s constant
marginal cost of production is µ1 , and retailer D2 ’s marginal cost is µ2 , with µ2 ≥ µ1 . We
assume that the difference between µ2 and µ1 is small enough that both retailers are active
when they obtain the intermediate good at marginal cost c.
Each retailer produces a quantity of qi , i = 1, 2, resulting in an aggregate retail output
of Q = q1 + q2 . The (inverse) demand function for the final good is p = P (Q). It is
strictly decreasing and thrice continuously differentiable whenever P (Q) > 0. Moreover,
we employ the standard assumption that P 0 (Q) + QP 00 (Q) < 0, which guarantees that the
profit functions are (strictly) quasi-concave and that the Cournot game exhibits strategic
substitutability (e.g., Vives, 1999). We also assume that P 000 (Q) is not too negative. As we
show below, this ensures concavity of the monopoly producer’s profit function.
7
Hansen and Motta (2013) consider a model in which retailers differ in their production costs due to
cost shocks, but neither the manufacturer nor rival retailers observe the cost realization. They show that if
retailers are sufficiently risk averse, the manufacturer optimally sells through a single retailer.
5
U
A
A
A
AU
D1
D2
@
R
@
Consumers
Figure 1: Framework with Vertical Separation.
When contracting with retailer Di , i = 1, 2, the upstream monopolist offers a take-itor-leave-it two-part tariff contract consisting of a fixed component, Fi , and a unit price, wi .
Upon accepting the monopolist’s offer, retailer Di ’s total marginal costs are µi + wi .
We consider two scenarios. In the first, firms are not integrated (vertical separation),
whereas in the second, the monopoly producer U is integrated with either retailer D1 or
retailer D2 (vertical integration).
In the scenario with vertical separation, the game proceeds as follows:
1. U secretly offers to each retailer Di a two-part tariff contract denoted by Ti = {wi , Fi }.
2. Retailers simultaneously and secretly accept or reject the contract offer.
3. Retailers order a quantity of the intermediate good, qi , and pay the tariff. Then,
they transform the intermediate good into the final good and simultaneously choose
quantities.
Afterwards, retail purchases are made, and profits are realized.
We assume that the offers formulated by the monopoly producer U are secret: retailer
Di observes the contract it is offered by U but not the contract that U offers to retailer D−i ,
and vice versa. Moreover, the upstream monopolist U produces on order because retailers
accept their offers and pay the respective tariff before competing in the final good market.
We solve for the perfect Bayesian Nash equilibrium that satisfies the standard passive
beliefs’ refinement (e.g., Hart and Tirole, 1990; O’Brien and Shaffer, 1992; McAfee and
Schwartz, 1994; Rey and Tirole, 2007; Arya and Mittendorf, 2011). With passive beliefs,
a retailer’s conjecture about the contract offered to the rival is not influenced by an offequilibrium contract offer to itself. This is a natural restriction on the potential equilibria
of a game with secret offers and production on order because, from the perspective of the
upstream monopolist, under these two assumptions retailers D1 and D2 form two separate
6
markets (Rey and Tirole, 2007). In Section 5, we discuss the impact of alternative belief
assumptions on our results.
In stage 2, we assume that retailers secretly decide on the contract offer made by U . Note,
though, that the analysis would not change if these decisions were public. This is because
each retailer Di correctly anticipates the equilibrium action of the rival retailer (which is to
accept U ’s offer) and because the out-of-equilibrium action to reject U ’s offer leads to zero
profits and is therefore independent of the unit price w−i .
In the scenario with vertical integration, the monopoly producer U and its downstream
affiliate maximize joint profits. The game proceeds as laid out above, with the exception
that, as is natural and in line with Hart and Tirole (1990) and Rey and Tirole (2007), the
downstream affiliate of the integrated firm is informed about the terms of U ’s offer to the
rival retailer. We also assume that the downstream affiliate is informed about the acceptance
decision of its rival. However, for the same reasons as above, the outcome would be identical
if the downstream affiliate was not informed about this decision.8
Before solving the model, it is useful to introduce some notation. First, we denote by
m
qi the monopoly quantity produced by retailer Di when it obtains the intermediate good at
marginal cost (wi = c),
qim ≡ arg max {(P (q) − c − µi )q},
q
whereas πim denotes retailer Di ’s monopoly profit when producing qim :
πim ≡ max {(P (q) − c − µi )q}.
q
Analogously, we use qic and πic to denote the Cournot quantity and profit of Di when both
retailers obtain the intermediate good at marginal cost
c
qic ≡ arg max {(P (q + q−i
) − c − µi )q},
q
πic
c
≡ max {(P (q + q−i
) − c − µi )q}.
q
Finally, we denote by qi (wi , w−i ) the Cournot quantity produced by retailer Di when it pays
a unit price of wi for the intermediate good, and the unit price paid by retailer D−i is w−i ,
qi (wi , w−i ) ≡ arg max {(P (q + q−i ) − µi − wi ) q} .
q
3
(1)
Equilibrium Analysis
We now examine the equilibrium allocations with vertical separation and integration.
8
As we will show in Section 4, considering the case in which the transmission of this information is not
possible becomes relevant in the analysis of the model with a bottleneck alternative.
7
3.1
Vertical Separation
We start with the case in which no firm is vertically integrated. Because retailer D1 is
(weakly) more efficient than retailer D2 , U will seek to monopolize the product market by
inducing D1 to sell the monopoly output (q1m ). It can attain this outcome by making an
unfeasible offer to D2 and an offer such as T1m = {c, π1m } to D1 . However, D1 anticipates
that when offers are secret, U ’s unfeasible offer to D2 is not robust to secret renegotiation.
The reason is that the monopoly producer has the incentive to sell an additional amount to
D2 , thus causing D1 to incur a loss.9 Retailer D1 will then turn T1m down, implying that
the equilibrium of the game without integration features the same commitment problem as
in Hart and Tirole (1990).
In Lemma 1 we show that with passive beliefs, the monopoly producer offers contracts
such that the unit price of the intermediate good is equal to its marginal cost of production
(c) and these contracts are accepted by both retailers in equilibrium.
LEMMA 1. The monopoly producer (U ) offers a contract Ti = {c, πic } to retailer Di with
i = 1, 2. Thus, the equilibrium quantities with vertical separation are given by q1c and q2c .
The result in Lemma 1 is well-known in the literature, and its intuition is as follows. A
retailer’s decisions (contract acceptance and intermediate good purchases) are unaffected by
an unobserved change in the input price to the rival retailer. Therefore, when the monopoly
producer contracts with each retailer Di , i = 1, 2, it acts as if the two are integrated, given the
contract to retailer D−i . This pairwise maximization problem requires that the contractual
arrangements between U and Di maximize bilateral profits. Bilateral profit maximization
yields a unit price equal to the monopoly producer’s marginal cost (c). Consequently, each retailer produces its Cournot quantity, and the upstream monopolist reaps the sum of Cournot
profits π1c + π2c via the fixed components of the two-part tariff.
3.2
Vertical Integration
With vertical separation, the equilibrium production profile features retailer D1 producing
q1c and retailer D2 producing q2c . From the monopoly producer’s perspective, it would be
better if the profile were instead given by q1c + 1 and q2c − 1. This alternative production
profile would allow the upstream monopolist U to reap greater profits from the final good
market. The aggregate retail output and, therefore, the retail price are the same, so industry
revenues are the same, but costs have fallen, because D1 is (weakly) more efficient than D2 .
Integration is an obvious way for U to implement a more profitable production profile. Thus,
we proceed by analyzing a framework in which the monopoly producer U is integrated with
one of the retailers. We consider two distinct cases: first, U is integrated with the efficient
retailer (D1 ), and second, U is integrated with the inefficient retailer (D2 ).
9
This result follows from the observation that given q1 = q1m ,
q2 = arg max {(P (q + q1m ) − c − µ2 )q} = RC (q1m ) > 0,
q
where RC denotes the standard Cournot reaction function. Given T1m , when secretly renegotiating with
D2 , the upstream monopolist maximizes the value of the contractual relationship with this retailer, and the
profits that U can extract from D2 are positive.
8
Vertical Integration between U and D1 The monopoly producer U can acquire the
efficient retailer (D1 ) and foreclose the inefficient retailer’s (D2 ) access to the intermediate
good. Hart and Tirole (1990) and Rey and Tirole (2007) establish this result in a framework
with symmetric retailers; the intuition is that with vertical integration, U internalizes the
effect of selling to the rival retailer via the reduced downstream profits made by its own
affiliate. Therefore, the temptation of opportunism vanishes and the monopolist can credibly
commit itself to reducing supplies to the rival retailer. This is the foreclosure effect of vertical
integration.
In our framework with asymmetric retailers, the same result occurs. The argument is
standard. Suppose that the monopoly producer supplies the monopoly quantity q1m to its
downstream affiliate (D1 ) and denies access to the intermediate good to the nonintegrated
retailer D2 . The integrated firm U − D1 then receives the monopoly profit π1m , so that any
deviation to supply the nonintegrated retailer D2 , which is less efficient than D1 , will result
in a lower profit for the integrated firm.10
LEMMA 2. Suppose the upstream monopolist U is integrated with retailer D1 . In equilibrium, the integrated firm U − D1 forecloses retailer D2 ’s access to the intermediate good.
Hence, retailer D1 produces q1 = q1m , retailer D2 remains inactive (q2 = 0), and U − D1
obtains the monopoly profit π m .
Clearly, for the upstream producer (U ), integration with retailer D1 is more profitable
than remaining separated, because it allows the integrated firm to monopolize the final good’s
market.
Vertical Integration between U and D2 Let the monopoly producer be integrated
with the inefficient retailer (D2 ). As explained in the introduction, a reason for this could
be that regulation prohibits U from merging with D1 on the grounds that such a merger
is anticompetitive. In Section 6, we show that a market structure in which U is integrated
with the inefficient retailer can arise when retailers’ costs are uncertain. Here we analyze
U ’s pricing decisions when integrated with retailer D2 .
We solve for U ’s optimal offer to its downstream unit D2 and the competing retailer D1 ,
leading us to Proposition 1.
PROPOSITION 1. Suppose the upstream monopolist U is integrated with retailer D2 . The
unique equilibrium features firm U − D2 trading the intermediate good internally at marginal
cost (w2? = c) and setting
w1? = P (Q) − 2µ2 + µ1 +
P 00 (Q) (µ2 − µ1 ) (P (Q) − c − µ2 )
(P 0 (Q))2
(2)
and
F1? = q1 (P (Q) − µ1 − w1? ) ,
with q1 = q1 (w1? , c), q2 = q2 (c, w1? ) and Q = q1 + q2 .
10
The proof of Lemma 2 follows the same lines as in Hart and Tirole (1990) and is therefore omitted.
9
(3)
Proposition 1 shows that, if U and D2 are vertically integrated, they internally trade the
intermediate good at a price equal to marginal cost. The two firms maximize joint profits;
thus, any alternative internal pricing policy is not robust to secret renegotiation (Ordover,
Saloner, and Salop, 1990, Chen, 2001, Choi and Yi, 2001). It follows that the downstream
unit of the integrated firm (D2 ), although inefficient, will be active in the final good’s market.
As the following discussion suggests, the upstream unit of the integrated firm is better off
seeing its inefficient downstream unit shuttered. However, in our setting, U cannot credibly
commit to such a policy. Note, though, that U would not shut D2 down in more elaborate
models that dispense of the commitment assumption. As we show in Section 5.4, if D1 has
some bargaining power vis-á-vis U − D2 , then U − D2 keeps the downstream unit alive to
reduce D1 ’s outside option. Another reason why U might not want to end D2 ’s business
could be because D2 allows U to capture valuable information about demand conditions.
Alternatively, D2 might retail multiple products; thus, shuttering D2 would reduce the profits
of the integrated firm across its full range of products.
More importantly, Proposition 1 shows that the integrated firm U −D2 does not necessarily foreclose the competing retailer’s access to the intermediate good. To grasp the incentives
of U −D2 when setting w1 , note that the integrated firm fixes w1 to maximize bilateral profits
with D1 , subject to the constraint that the internal trading price is at marginal cost. The
trade-off it faces is as follows. By raising the unit-price offer to the competing retailer D1 , U
forecloses D1 ’s access to the intermediate good and monopolizes the market for the final good
via D2 (the foreclosure effect). However, a countervailing effect emerges in our framework.
U benefits if the quantity produced by the efficient retailer D1 increases, at the expenses of
D2 . How can the upstream firm achieve this, given that it cannot commit to restricting its
downstream affiliate’s quantity? Since with vertical integration D2 observes U ’s offer to D1 ,
U needs to lower the unit price w1 : in this way, D1 ’s quantity increases and D2 responds by
reducing its output. U then extracts the profit of D1 via the fixed component of the two-part
tariff. This establishes the output-shifting effect of vertical integration.
Note that a reduction in w1 implies a reduction in industry revenue: a lower value of
w1 triggers an increase of q1 that is larger than the consequent decrease in q2 , because
|∂q1 /∂w1 | > ∂q2 /∂w1 > 0. This hints that lowering w1 benefits the integrated firm only to
the extent that the fall in industry revenue is compensated by the cost-efficiency of D1 . In
light of this discussion, we next ask under which conditions w1? will be low enough that D1
produces a positive quantity.
PROPOSITION 2. The integrated firm U − D2 sets a unit price w1? such that the efficient
nonintegrated retailer D1 is active on the market for the final good (q1 (w1? , c) > 0) if, and
only if, µ1 < µ2 . If µ1 = µ2 , U − D2 forecloses retailer D1 ’s access to the intermediate good.
The proposition shows that if the two retailers D1 and D2 were equally efficient (µ1 = µ2 ),
then U − D2 would foreclose D1 ’s access to the intermediate good (Hart and Tirole, 1990;
Rey and Tirole, 2007). Indeed, turning back to (2), for µ1 = µ2 = µ, the equilibrium value
of w1? is equal to P (Q) − µ. At this unit price, the marginal costs of D1 (w1? + µ1 ) are equal
to the equilibrium price, implying that q1 = 0. This shows that the expression for w1? in
Proposition 1 provides a generalization of the equilibrium unit price obtained in models with
equally efficient retailers to the case of asymmetric retailers.
10
Proposition 2 also shows that retailer D1 is active as long as it is strictly more efficient
than retailer D2 (µ1 < µ2 ). To understand this, look at the first-order condition for w1 ,
which, as we show in the Appendix, is given by
(P 0 (Q)q2 + w1 − c)
∂q2
∂q1
+ P 0 (Q)q1
= 0.
∂w1
∂w1
(4)
U can foreclose D1 ’s access to the intermediate good by setting w1 = P (Q) − µ1 , so that q1
equals zero. In this case, Q = q2m , with q2m implicitly determined by the downstream firstorder condition q2m = −(P (q2m ) − µ2 − c)/P 0 (q2m ). Inserting these values into the left-hand
side of (4) and simplifying yields
(µ2 − µ1 )
∂q1
≤ 0,
∂w1
(5)
because ∂q1 /∂w1 < 0. Therefore, if µ2 − µ1 > 0 and w1 = P (q2m ) − µ1 , U finds it profitable
to decrease w1 from the foreclosure level and induce D1 to produce a positive quantity.
The intuition is as follows. The monopoly quantity q2m maximizes industry profits when
the integrated firm is bound to produce the final good at a marginal cost of µ2 . However,
if µ1 < µ2 , letting D1 participate in the final good market implies that the final good
is produced more efficiently. The latter effect outweighs the decrease in industry revenue
because it reflects the direct impact of lower production costs on industry profitability.11
Then, even if the difference between µ1 and µ2 is tiny, it is not optimal for D1 to foreclose
D1 ’s access to the intermediate good.
A question of great importance for competition policy regards the magnitude of the
output-shifting effect. Specifically, can the incentive to reduce w1 be so strong as to induce
U to formulate a unit-price offer below its marginal cost of production (i.e., w1? < c)? This
would render vertical integration procompetitive because the sum of the per-unit prices at
the wholesale level would be lower than with vertical separation, leading to larger industry
output and consumer surplus. Proposition 3 shows that this can indeed occur.
PROPOSITION 3. The integrated firm U − D2 sets a unit price w1? below its marginal
cost of production c if, and only if,
µ2 − µ1 >
(P (Q) − µ2 − c)(P 0 (Q))2
.
(P 0 (Q))2 − (P (Q) − µ2 − c)P 00 (Q)
(6)
If condition (6) is satisfied, the aggregate quantity with vertical integration is larger than with
vertical separation.
Proposition 3 shows that the output-shifting effect can induce the upstream unit of
the integrated firm to engage in below-cost pricing at the wholesale level. Specifically, the
proposition indicates that below-cost pricing is more likely to occur, the more efficient D1 is
11
Technically, the increase in industry quantity (and the consequent fall in revenue) is of second-order
importance, because q2 is at the equilibrium level when D2 ’s marginal cost is µ2 , due to the envelope
theorem.
11
relative to D2 (i.e., if µ2 − µ1 is large). Indeed, the more efficient D1 , the higher the profit
increase that the integrated firm obtains when shifting output to D1 .
For the intuition behind the result in Proposition 3, recall that the unit price w1 is
set to maximize the bilateral profits of the integrated firm and D1 . The question is why
a value of w1 lower than c can maximize these profits. Below-cost pricing produces an
expansion of industry output relative to vertical separation, and thus a fall in industry
revenue. However, setting w1 below c induces retailer D2 to reduce its quantity compared to
vertical separation. If D1 is particularly efficient, the fall in industry revenue is outweighed
by the fact that output is produced more efficiently, thereby rendering a strategy of belowcost pricing optimal. Consistent with the intuition for the results above, lowering w1 acts
as a commitment to tame the incentive of the integrated firm’s downstream unit to expand
its output. In the scenario with vertical separation, lowering w1 below c was not optimal
because retailer D2 did not observe the offer to D1 , and therefore could not respond to a
change of the value of w1 . This last observation explains why the strategy of pricing below
cost, although available, does not arise at equilibrium with vertical separation.
Regarding the profitability of vertical integration, a revealed preference argument shows
that for U , a vertical merger with retailer D2 is more profitable than remaining separated,
even when the value of the unit price w1 lies below marginal cost c. With integration
the intermediate good is traded internally at marginal cost. Therefore, setting also w1 at
marginal cost would allow the monopoly producer U to replicate the outcome with vertical
separation (Lemma 1). Yet, U prefers to depart from pricing at cost, which implies that this
departure must be profitable.
The output-shifting effect can also be linked to the rate of cost pass-through. A cost
increase is shifted to consumers at a rate that depends on the curvature of consumer demand
(Bulow and Pfleiderer, 1983; Weyl and Fabinger, 2013). Specifically, the pass-through rate
is larger if the demand function is relatively convex. These considerations suggest that
the below-cost pricing result is more likely to occur when the demand function is concave,
because the nonintegrated retailer D1 then adjusts its quantity only slightly in reaction to a
change in w1 . Thus, U must reduce its unit price by a large amount to induce D1 to expand
its quantity. Indeed, in the formula for w1? in (2), w1? is particularly low if the demand
function is relatively concave (P 00 (Q) is small).
To illustrate these results, we solve an example using the linear demand function P (Q) =
α − βQ, with β > 0 and α > c + µ2 . We find that if the monopoly producer U is integrated
with the less efficient retailer D2 , it sets a unit price w1? equal to (α + c + 4µ1 − 5µ2 ) /2. Let
us denote by ∆ the difference between retailers’ marginal costs of production (µ2 −µ1 ). Both
¯ ≡ (α − c − µ1 ) /3. In line
D1 and D2 are active provided ∆ is smaller than the threshold ∆
with Proposition 3, we find that vertical integration between U and D2 is procompetitive
(w1? < c) if, and only if, ∆ ≥ ∆c ≡ (α − c − µ1 ) /5, whereas it is anticompetitive for values
¯ 12 Figure 2 plots these conditions using α = β = 1 and c = .25.
of ∆ below ∆c , with ∆c < ∆.
The shaded area shows when vertical integration between U and D2 is procompetitive, as
the equilibrium value of w1 lies below the upstream monopolist’s marginal cost c.
12
Note that, for all ∆ > 0 the profits of U − D2 when the unit price is equal to w1? are strictly larger than
the profits of the integrated firm when foreclosing D1 ’s access to the intermediate good.
12
D
0.25
¯
∆
0.20
Procompetitive Vertical Integration
0.15
∆c
0.10
Anticompetitive Vertical Integration
0.05
0.05
0.10
0.15
0.20
Ð1
Figure 2: Linear Demand Example
4
Model with a Competitive Bottleneck Alternative
In the industry structure of Section 2, the upstream producer U is an unconstrained monopolist. In this section, we extend that model to consider an industry in which U competes
with a fringe of less efficient firms, denoted by Û . These fringe firms bear unit costs ĉ to
produce the intermediate good, with ĉ ≥ c, and behave competitively, as they are willing
to offer the intermediate good at a unit price of ĉ.13 Thus, fringe firms Û act as a bottleneck alternative that constrains the market power of the dominant producer (U ). The game
proceeds as described in Section 2.
We introduce an additional piece of notation:
π̂ic ≡ max {(P (q + q−i ) − ĉ − µi ) q} ,
q
with q−i = q(c, ĉ) as defined in (1). Thus, π̂ic is the profit of a retailer Di that buys the
intermediate good from the fringe firms at cost ĉ, whereas its rival obtains the same good at
marginal cost c. We assume that fringe firms are effective in constraining the market power
of U , i.e., ĉ is low enough. This implies that the quantity of retailer Di when buying the
intermediate good at ĉ is positive, even if retailer D−i obtains the intermediate good at c
(qi (ĉ, c) > 0).
In what follows, we first show that the competitive consequences of vertical integration
are the same as those obtained in the main model of Section 2. Then, we analyze how
information transmission within an integrated company affects the final good’s quantity and
price.
13
We obtain the same results if the fringe consists of only a single firm competing à la Bertrand with U
to serve retailers. Bertrand competition implies that Û would set a per-unit price of ĉ and a fixed payment
of zero.
13
4.1
Vertical Separation
Consider first an industry featuring vertical separation.
LEMMA 3. The dominant producer (U ) offers a contract Ti = {c, πic − π̂ic } to retailer Di
with i = 1, 2. Thus, the equilibrium quantities with vertical separation are given by q1c and
q2c .
The result in Lemma 3 follows Hart and Tirole (1990) and Rey and Tirole (2007). We
omit the proof because it mirrors the one of Lemma 1. First, note that the equilibrium
quantities are the same as in the model without a bottleneck alternative. Because the
dominant producer is more efficient than the bottleneck alternative (Û ), it supplies both
retailers. Moreover, pairwise profit maximization between U and each retailer Di requires
fixing the per-unit price at c, leading to equilibrium Cournot quantities of q1c and q2c . However,
differently from the unrestricted monopoly model, when a bottleneck alternative is available
to retailers, U cannot ask for the payment of the full Cournot profit in the fixed component of
the two-part tariff, because D1 and D2 can buy from Û and secure a profit of π̂ic . Therefore,
U will set a fixed payment of Fi equal to πic − π̂ic .
4.2
Vertical Integration
We now consider the vertically integrated industry. We focus on the case in which U is
integrated with the inefficient retailer (D2 ). The reason is that when U is integrated with the
more efficient retailer (D1 ), it will foreclose the competing retailer’s access to the intermediate
good to the extent possible. Differently from the analysis in Section 3.2, in a model with
a bottleneck alternative, U needs to take into account that retailer D2 can turn to Û and
buy the intermediate good at a unit price of ĉ. Therefore, in equilibrium U − D1 will set
w2 = ĉ, both retailers D1 and D2 are active and produce q1 = q1 (c, ĉ) and q2 = q2 (ĉ, c),
respectively.14
We now let the dominant producer U be integrated with the less efficient retailer D2 ,
leading to the results in the following proposition.
PROPOSITION 4. The integrated firm U − D2 sets the same unit price as in the set-up
without the bottleneck alternative, that is, w1BA = w1? , if w1? < ĉ, and w1BA = ĉ otherwise
(wiBA = min{w1? , ĉ}). The fixed component of the two-part tariff is equal to
F1BA = q1 (w1BA , c)(P (Q) − µ1 − w1BA ) − q1 (ĉ, c)(P (q1 (ĉ, c) + q2 (c, ĉ)) − µ1 − ĉ),
with q1 = q1 (w1BA , c), q2 = q2 (c, w1BA ), and Q = q1 + q2 .
The proposition shows that, as long as w1? < ĉ, the optimal unit price is the same with and
without a bottleneck alternative. The intuition is as follows. The presence of the bottleneck
alternative affects the outside option of retailer D1 (i.e., the profits D1 raises when rejecting
U ’s offer). In the scenario without the bottleneck alternative this outside option was zero,
whereas with the bottleneck alternative it is given by the profit of D1 when buying from Û .
However, since the downstream affiliate of the integrated firm (D2 ) knows when D1 buys
14
The formal proof of this result follows Rey and Tirole (2007).
14
the intermediate good at ĉ, the outside option of D1 does not depend on w1BA . Therefore,
the optimal value of w1 coincides with the one derived in Proposition 1 as long as w1∗ < ĉ.
When w1? ≥ ĉ, the unit price with the bottleneck alternative (w1BA ) is equal to ĉ. The reason
is that raising the unit price above ĉ cannot be profitable for U , because D1 can then turn
to the bottleneck alternative Û .
This discussion implies that the competitive implications of vertical integration are as
in Section 3. Specifically, the results in Proposition 3 carry over to the model with the
bottleneck alternative.
4.3
Information Transmission
The results in Proposition 4 rely on the assumption that, as in the main model, the downstream affiliate of the integrated firm is informed about the rival retailer’s acceptance decision. However, this information transmission might not be possible, for example, due to
the regulatory imposition of an information firewall between the two units of the integrated
firm. In fact, the imposition of behavioral remedies of this type is one of the most common
forms of conduct relief in merger decisions. For this reason, in this section we propose a
variant of the model with a bottleneck alternative to analyze the effects of the imposition of
an information firewall on economic outcomes.15
Following Nocke and Rey (2102), we model the situation in which information transmission is not feasible by assuming that stages 2 and 3 of the game in Section 2 take place
simultaneously. When stages 2 and 3 occur simultaneously, the downstream unit of the integrated firm cannot know the rival retailer’s decision when it chooses its quantity. Instead,
when stages 2 and 3 take place sequentially (as in the main model), the upstream unit of the
integrated company can inform the downstream unit about the acceptance decision of the
rival retailer before the integrated retailer chooses its quantity. The analysis of the model
without information transmission leads to the results in Proposition 5.
PROPOSITION 5. If the dominant producer U is integrated with the less efficient retailer
D2 , the unit price in the regime with information transmission is lower than the unit price
in the regime without information transmission.
We find that when information transmission is not possible, the integrated firm has
an incentive to raise the unit price w1 above the one with information transmission. The
intuition behind this result is as follows. If information transmission is not possible, D2
does not know whether D1 accepted U ’s offer when setting its quantity. Due to passive
conjectures, D2 believes that D1 buys from U at w1 , as it does on the equilibrium path.
This implies that D2 produces the equilibrium output q2 (c, w1 ) even when D1 buys from Û .
Consequently, the outside option of D1 decreases in w1 , because an increase of w1 leads to an
increase of D2 ’s quantity. Realizing this, the upstream unit of the integrated firm increases
w1 to reduce the outside option of D1 and demand a higher fixed payment.16
15
The imposition of information firewalls is typically justified by the possibility that the downstream unit
of an integrated firm obtains private information of a rival’s production process via its upstream unit. We
show that even in a model that abstracts from these considerations, information firewalls affect consumer
surplus.
16
This effect is not present when information transmission is possible because the outside option of D1
does not depend on the value of w1 .
15
Proposition 5 shows that behavioral remedies like information firewalls can be detrimental to consumer surplus. Typically, the literature on information transmission has focused
on the impact of information exchange between vertical hierarchies (Rey and Stiglitz, 1995;
Arya and Mittendorf, 2011; among others). The general wisdom is that this sort of information exchange results in an anticompetitive outcome. Conversely, Proposition 5 shows that
information transmission within vertically integrated firms can be procompetitive.
5
Additional Results
In this section, we test the robustness of the results of our main model to alternative assumptions. Specifically, we show that the output-shifting effect also arises in settings with
quantity-forcing contracts, wary beliefs, or differentiated products. We also solve for a setting in which U −D2 deals with the nonintegrated retailer via Nash bargaining and show that
the integrated firm wants to keep its downstream affiliate alive so to improve its bargaining position. Finally, we argue that the difference in retailers’ marginal costs endogenously
emerges in a model with complementary intermediate goods.
5.1
Quantity-forcing Contracts
First, we consider the scenarios with vertical separation and vertical integration, assuming
that the upstream monopolist U offers a contract Ti (·) to each retailer Di , i = 1, 2, that
specifies the retailer’s payment for any quantity purchased. Following e.g., Rey and Tirole
(2007), within this class of contracts the optimal contract is a quantity-forcing contract
featuring Ti (0) = 0, Ti (qi ) = F̄i if qi = qi0 , and Ti (qi ) = ∞ otherwise. Essentially, the
upstream manufacturer determines the output that retailer Di is allowed to produce. Here,
F̄i denotes the fixed payment that retailer Di needs to pay to obtain the intermediate good.
We next show that the equilibrium allocations with quantity-forcing contracts and two-part
tariffs coincide.
Vertical Separation Assume first that no firm is vertically integrated. As argued in
Section 3.2, U contracts with each retailer Di , acting as if the two were integrated. This
pairwise maximization problem requires a contractual arrangement that maximizes U ’s and
Di ’s bilateral gains, given the candidate equilibrium quantity produced by D−i . That is, U
fixes qi to maximize the product-market profits of Di : maxq {(P (q + q−i ) − µi − c)q}. As a
consequence, the equilibrium contract that U offers to Di requires this retailer to produce
the Cournot quantity qic and pay a tariff F̄i = qic (P (q1c + q2c ) − µi ), leading to a profit of πic
for U . Thus, the equilibrium allocations and profits with vertical separation are the same as
in Lemma 1.
Vertical Integration Under the assumption of two-part tariffs, if the monopoly producer
is integrated with the efficient retailer (D1 ), then the equilibrium outcome features the
integrated firm U − D1 monopolizing the final good market (Lemma 2) and raising industry
profits to π1m . To replicate this outcome through quantity-forcing arrangements, U − D1 can
deny the nonintegrated retailer D2 access to the intermediate good and supply the monopoly
16
quantity q1m to its downstream affiliate. Clearly, any deviation to supply the nonintegrated
retailer can only result in a lower profit for the integrated firm.
We next consider the case of U being integrated with the inefficient retailer (D2 ). Lemma
4 presents the equilibrium quantity-forcing contracts. We then show that these contracts give
rise to the same equilibrium allocations as those obtained in Proposition 1 using two-part
tariffs.
LEMMA 4. Suppose U is integrated with D2 . The unique equilibrium features firm U − D2
trading the intermediate good internally at marginal cost (c) and setting
2
0
00
(µ
−
µ
)
2
(P
(Q))
−
(P
(Q)
−
c
−
µ
)P
(Q)
2
1
2
q1? =
− (P 0 (Q))3
and
F̄1? (q1? ) = (P (q2? + q1? ) − µ1 ) q1? ,
with q2? = arg maxq {(P (q + q1? ) − c − µ2 )q}.
Similarly to Proposition 1, Lemma 4 shows that U − D2 does not necessarily foreclose
the competing retailer’s access to the intermediate good. To establish that the equilibrium
allocations with quantity-forcing contracts coincide with those obtained with two-part tariffs,
we note that the value of q1 obtained using w1? in Proposition 1 is equal to17
(µ2 − µ1 ) 2 (P 0 (Q))2 − (P (Q) − c − µ2 )P 00 (Q)
?
= q1? .
q1 (w1 , c) =
3
0
− (P (Q))
It follows that the output-shifting effect of vertical integration arises when using quantityforcing contracts with the same intensity as with two-part tariffs.
5.2
Wary Beliefs
In our main model, we assume that retailers hold passive out-of-equilibrium beliefs. This is
a particularly plausible assumption when the monopoly producer supplies on order (Rey and
Tirole, 2007), as it does in our setting. It implies that when a retailer receives an unexpected
offer, it does not revise its beliefs about the offer made to its rival.
An alternative assumption is that retailers hold wary beliefs. Under wary beliefs (McAfee
and Schwartz, 1994; Rey and Vergé, 2004), if a retailer receives an offer different from the
expected one in the candidate equilibrium, it believes that the incumbent will act optimally
with the other retailer. The adoption of wary beliefs in our model does not change our
conclusions because, given that the monopoly producer supplies on order, a contract with
retailer Di has no impact on U ’s gains from trade with the competing retailer D−i . Wary
beliefs then prescribe that Di must expect the monopoly producer U to behave as if it were
integrated with D−i and thus that D−i will receive the intermediate good at a unit price
equal to U ’s marginal cost c, regardless of the offer received by Di .
17
We show this result in the proof of Proposition 2. See the Appendix for the details.
17
5.3
Differentiated Products
The main model considers competition between retailers offering a homogeneous final good.
In this section, we analyze the case in which D1 and D2 offer differentiated products. We
assume that retailer Di ’s inverse demand function is equal to Pi (qi , q2 ) = α − βqi − γq−i , with
i, j = 1, 2 and i 6= j, where the parameter γ ∈ [0, β) reflects the degree of substitutability
between D1 ’s and D2 ’s products. This is a widely employed demand function (e.g., Vives,
1999). Because β > γ ≥ 0, inverting the system of inverse demand functions yields the direct
demand functions we will use in the analysis with price competition. We further impose that
α > c + µi .
We will analyze whether, due to the output-shifting effect, vertical integration between
the upstream monopolist (U ) and the inefficient retailer (D2 ) results in an outcome that is
procompetitive relative to the one with vertical separation. Before proceeding, note that if
retailers offer differentiated products, vertical integration may not lead to full monopolization
of the final good’s market even in a setting with equally efficient retailers. Indeed, the integrated company will generally maintain the rival retailer active, although in a discriminatory
way. In what follows, we denote by ∆ the difference µ2 − µ1 .
Retail quantity competition Let us first consider the case of quantity competition between D1 and D2 . In line with Lemma 1, when no firm is integrated, the intermediate good
is supplied by the upstream monopolist at a unit price of wi = c, so that retailers produce
respective Cournot outputs. However, when U is merged with retailer D2 , we proceed in the
same way as in Proposition 1 and find that the integrated firm sets a unit price w1 equal to
w1C =
(4β 2 + γ 2 )γ(α − c − µ1 − ∆) + 8β 3 c − 2βγ 2 (2α + c − 2µ1 )
.
2β(4β 2 − 3γ 2 )
The equilibrium value of w1C decreases in ∆ for all β > γ ≥ 0. This shows that at equilibrium,
U − D2 engages in output shifting. That is, for all positive values of ∆, the integrated firm
sets a strictly lower unit price for D1 than in a setting with equally efficient retailers.
Can output shifting result in a unit price below U ’s marginal cost of production (c)?
¯d
In Figure 3 we illustrate that w1C lies below c for all values of ∆ ≥ ∆cd . In the figure, ∆
represents the threshold below which both firms are active. Thus, the shaded area is the
one in which vertical integration between U and D2 leads to a procompetitive outcome with
respect to vertical separation.18
Retail price competition We now analyze whether procompetitive vertical integration
can arise when retailers are Bertrand competitors that offer differentiated products. Differently from the case of quantity competition, where retailers order quantities and pay the
tariff before competing on the final good market, with price competition each retailer Di
first sets its final good price and then orders the quantity qi of the intermediate good so
as to satisfy demand (Rey and Vergé, 2004). This implies that with Bertrand competition
the monopoly producer U produces on demand (and not on order, as in case of quantity
competition). We therefore follow the literature and modify the third stage of the timing in
18
For the figure, we use the following parameter values: α = 1, β = .6, γ = .3 and c = .25.
18
D
0.5
¯d
∆
0.4
Procompetitive Vertical Integration
∆cd
0.3
0.2
Anticompetitive Vertical Integration
0.1
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Ð1
Figure 3: Linear Demand Example — Differentiated Products
Section 2 by letting retailers first simultaneously choose final good prices and then order the
quantity to satisfy their demand, transform the intermediate good into the final good, and
pay the tariffs.
This change in the timing of the game implies that with downstream Bertrand competition the assumption of passive beliefs is not as reasonable as with Cournot competition (Rey
and Vergé, 2004; Rey and Tirole, 2007). The reason is that contracts are more interdependent: retailers pay their tariffs to U only after their demand is realized; thus, a change in
the unit price to retailer Di affects the payment that U receives from retailer D−i , thereby
invalidating the approach with passive beliefs. In addition, if products are close substitutes,
an equilibrium fails to exist with passive beliefs. We therefore focus on wary beliefs, which
circumvent these problems. Rey and Vergé (2004) solve for the equilibrium of a game with
Bertrand competition and wary beliefs. They find that, in the vertically separated industry,
the unit price offered by the upstream monopolist lies above marginal cost (c).19 Although
the equilibrium cannot be solved for analytically, this can be done numerically. For example, for α = β = 1, γ = 0.7, c = 0.2, and µ1 = µ2 = 0.1, we obtain unit prices given by
w1 = w2 = 0.311.
Let then U be integrated with D2 . Since the problem of conjectures does not arise with
vertical integration, we obtain an explicit solution. Proceeding as in Proposition 1,20 We
19
Instead, with passive beliefs, if an equilibrium exists, it leads to a unit price offer equal to marginal cost.
Indeed, Rey and Tirole (2007) show that the analysis with integration follows the same lines as with
Cournot competition downstream, with the exception that retailer D2 takes into account that a change in
its downstream price affects the quantity of retailer D1 and thus the payment that its upstream affiliate
receives. This is a consequence of tariffs being paid after the product market competition stage.
20
19
find that U − D2 sets a unit price w1 equal to
w1B =
(4β 2 + γ 2 )γ(α − c − µ1 − ∆) + 8β 3 c + 2βγ 2 (2α + 3c − 2µ1 )
.
2β(4β 2 + 5γ 2 )
Clearly, w1B decreases in ∆ for all β > γ ≥ 0. This shows that U − D2 sets a lower unit
price to D1 than in a setting in which µ2 and µ1 coincide, thereby inducing retailer D1 to
expand its output at the expenses of the integrated downstream unit. Moreover, it can be
shown that w1B > c for β > γ > 0. Therefore, U − D2 never engages in below-cost pricing.
For instance, using values of the parameters as above, we obtain w1B = 0.477 (and w2 = 0.2
due to internal transfer pricing at marginal cost).
Although the wholesale price to D1 is never below marginal cost, vertical integration with
D2 can still be procompetitive. This is because, with wary beliefs and vertical separation, w1
and w2 are larger than c. To shed light on the question whether vertical integration increases
consumer surplus with respect to vertical separation, we use numerical computations. We
find that there are several parameter constellations for which vertical integration is anticompetitive when retailers are equally efficient but procompetitive when D1 is more efficient
than D2 . For example, using the same parameter values as above, in which retailers are
equally efficient, vertical integration is detrimental to consumer surplus, whereas it increases
consumer surplus if µ1 = 0.05 < 0.1 = µ2 . The reason is again that, as retailer D1 becomes
more efficient, U ’s unit price to D1 with vertical integration falls relative to the one with
vertical separation. This result is in line with what we obtain in the main model: productive
efficiency increases and this makes vertical integration procompetitive.
5.4
Nash Bargaining between U − D2 and D1
In Proposition 1 we show that the vertically integrated firm U − D2 internally trades the
input good at marginal cost, because any alternative internal pricing policy is not robust to
secret renegotiation. The upstream unit of the integrated firm would like to see its inefficient
downstream unit shuttered; however, it lacks the power to credibly commit to such a policy.
Here, we propose a variant of the main model in which we assume that the integrated
firm U − D2 deals with D1 via Nash bargaining (Horn and Wolinsky, 1988; O’Brien and
Shaffer, 2005; Milliou and Petrakis, 2007). The upstream unit of the integrated firm has the
commitment power to shut down its downstream subsidiary. We show that U − D2 will want
to keep D2 alive to improve its bargaining position vis-à-vis D1 .
Let us assume, as in Horn and Wolinsky (1988), that the outcomes of bargaining are
determined by the set of simultaneous and asymmetric Nash bargaining solutions between
U − D2 and D1 .21 If a negotiation breaks down, each firm earns its disagreement payoff. For
the retailer who can only acquire the input good from the monopoly producer, this payoff
equals zero. In the case of the integrated firm, we assume that its disagreement payoff is the
profit it could earn by operating via its downstream unit only. Then, the Nash bargaining
21
Although the Nash bargaining solution is a concept borrowed from cooperative game theory, the same
solution can be obtained as a subgame perfect equilibrium of a non-cooperative game in which U and D1
formulate alternating offers (Binmore, Rubinstein, and Wolinsky, 1986). Nash bargaining can therefore be
seen as a shortcut for such an alternating offers game.
20
solution between U − D2 and D1 solves
t
)1−δ ,
max (πUt −D2 − dtU −D2 )δ (πD
1
T1
(7)
t
under the constraints that πUt −D2 ≥ dtU −D2 and πD
≥ 0. In (7), δ ∈ [0, 1] is the bargaining
1
power of the integrated firm, and T1 = {w1 , F } is the two-part tariff contract offered by
t
U − D2 to retailer D1 . Instead, πD
is the profit of the nonintegrated retailer, and πUt −D2 and
1
dtU −D2 are, respectively, the profit and the disagreement payoff of the integrated firm. The
subscript t = s, n stands for the two scenarios we consider. In the first, U − D2 commits to
shuttering D2 before negotiating with D1 (t = s), whereas in the second U − D2 keeps D2
alive (t = n).
We start with the first scenario. When U − D2 commits to shuttering D2 ,
πUs −D2 = F1 + q1 (w1 )(w1 − c)
dsU −D2 = 0
s
πD
= π1 (w1 ) − F1 ,
1
where πi (wi ) = maxq {(P (q) − wi − µi )q} is the monopoly profit of retailer Di as a function of
wi . Clearly, if D2 is inactive, the disagreement payoff of U − D2 is zero and D1 is monopolist
on the final good market.
In the second scenario, U − D2 keeps D2 alive (t = n). Hence,
πUn −D2 = F1 + q1 (w1 , c)(w1 − c) + π2 (c, w1 )
dnU −D2 = π2m
n
πD
= π1 (w1 , c) − F1 .
1
We use the result in Proposition 1 that when D2 has not been shuttered, U − D2 will serve
its downstream affiliate at w2 = c. Moreover, we denote by πi (wi , w−i ) = maxq {(P (q +q−i )−
wi − µi )q} the Cournot profit of retailer Di when it obtains the input good at wi , whereas
its competitor obtains it at w−i . Finally, in this second scenario, the disagreement payoff of
U − D2 is the monopoly profit that U − D2 raises via its downstream unit (π2m ).
Solving (7) for these two cases yields Lemma 5.
LEMMA 5. If U − D2 shutters its downstream unit, then the unique solution of Nash
bargaining features w1B = c and F1B = δπ1m . The profits of U − D2 are πUs −D2 = δπ1m .
Instead, if U − D2 keeps its downstream unit alive, the unique solution of Nash bargaining
features w1B = w1? as in (2) and
F1B = δπ1 (w1? , c) − (1 − δ)[q1 (w1? , c)(w1? − c) + π1 (w1? , c) + π2 (c, w1? ) − π2m ].
In this case, the profit of U − D2 is πUn −D2 = δ[π1 (w1? , c) + q1 (w1? , c)(w1? − c) + π2 (c, w1? )] +
(1 − δ)π2m .
The bargaining equilibria have the same unit prices and therefore equilibrium outputs as
in the model outlined in Section 2, in which the monopoly producer U has all the bargaining
power. This result is well-known (e.g., O’Brien and Shaffer, 2005), and the intuition is
21
simple: negotiating parties choose the unit price w1 to maximize bilateral profits and then
divide the surplus via a non-distortionary transfer (F1 ).
Looking at the profits of the integrated firm U −D2 in the two scenarios, we obtain that an
intuitive trade-off determines U − D2 ’s decision to keep its downstream unit alive. If U − D2
had full bargaining power (δ = 1), then it would prefer to commit to keeping D2 inactive
and letting D1 operate as monopolist because π1m > π1 (w1? , c) + q1 (w1? , c)(w1? − c) + π2 (c, w1? ).
On the other hand, if D1 had full bargaining power (δ = 0), then U − D2 would want to
keep D2 alive and obtain π2m . Since the difference in profits is monotone and continuous in
δ, there will be a unique threshold value for δ below which it is optimal for the integrated
firm not to shutter its downstream unit, as we show in Corollary 1.
COROLLARY 1. U − D2 keeps its downstream unit alive if, and only if,
δ ≤ δB ≡
π2m
∈ (0, 1).
π2m + π1m − [q1 (w1? , c)(w1? − c) + π1 (w1? , c) + π2 (c, w1? )]
Note that, using the linear demand P (Q) = α − βQ, we find that
(α − c − µ2 )2
δ =
.
(α − c − 2µ2 + µ1 )(α − c + 2µ2 − 3µ1 )
B
Employing the same parameterization as used in Figure 2 (α = β = 1, c = .25) together
with µ1 = .15 and µ2 = .3, we obtain that δ B = .75.
5.5
The Difference in Retailers’ Marginal Costs
In our model, retailers hold different marginal costs of production. This assumption drives
our main result that vertical integration between the upstream manufacturer and the inefficient retailer can be procompetitive due to an output-shifting effect.
The asymmetry in retailers’ marginal costs can be endogenized in a model with providers
of complementary intermediate goods (Reisinger and Tarantino, 2013). Assume there are
two vertically integrated firms, U 0 − D0 and U 00 − D00 . Producers U 0 and U 00 operate in
separate markets, as they supply intermediate goods that are in a relationship of perfect
complementarity. Assume further that for each intermediate good, there exists a less efficient
bottleneck alternative. Finally, let downstream units (D0 and D00 ) be competitors that sell
to final consumers.
Reisinger and Tarantino (2013) show that the equilibrium in the intermediate good market involves one integrated firm, say U 0 −D0 , setting its unit price at the foreclosure level (i.e.,
at the marginal cost of the bottleneck alternative). Then, the two retailers D0 and D00 hold
asymmetric marginal costs of production: D00 pays a high price for the intermediate good
supplied by U 0 , whereas D0 , because being integrated, obtains that same input at marginal
cost. This asymmetry induces U 00 − D00 to engage in output shifting: U 00 replies to U 0 − D0 ’s
foreclosure strategy by lowering the unit price offered to D0 , thereby inducing D0 to expand
its quantity.
22
Prob. ρ
Prob. 1 − ρ
D̃ marginal cost
µ+λ
µ−λ
D̄ marginal cost
µ
µ
Table 1: Realizations of Retailers’ Marginal Cost of Production
6
Vertical Merger under Uncertainty
In this section, we study the monopoly producer’s integration decision in a setup where
retailers’ marginal costs of production are uncertain, reflecting the idea that vertical integration is a long-term decision (e.g., Harrigan, 1984; Williamson, 1985). We show that a
market structure in which the monopoly producer is integrated with an inefficient retailer
might arise in equilibrium, and vertical integration is in fact procompetitive in expectations.
6.1
Set-up with Uncertainty
As in our base model in Section 2, the monopoly producer U deals with two retailers, D̄ and
D̃. The marginal cost of retailer D̄ is certain and equal to µ. Conversely, the marginal cost
of retailer D̃ is stochastic. Specifically, it can take on two values: µ + λ with probability
ρ, and µ − λ with probability 1 − ρ. Thus, the expected value of D̃’s marginal costs is
ρ(µ + λ) + (1 − ρ)(µ − λ). For reasons that will become clear later on, we restrict attention
to ρ ∈ [1/2, 1). If ρ = 1/2, then D̄ and D̃ are equally efficient in expectation. Instead, D̄ is
more efficient in expectation than D̃ for all values of ρ larger than 1/2. Table 1 shows the
possible combinations of retailers’ marginal costs of production.
The game develops in two stages:
I. The monopoly producer U decides whether to merge with retailer D̄ or D̃.
I’. Uncertainty over retailer D̃’s marginal cost realizes.
II. The game of Section 2 takes place.
We solve the game by backward induction. In stage II, absent integration, the results in
Section 3.1 and Lemma 1 regarding U ’s pricing decisions apply. Instead, if U is integrated,
the results in Section 3.2 apply. Specifically, if U is integrated with the more efficient
retailer, then its pricing decisions follow Lemma 2. Alternatively, if U is integrated with the
less efficient retailer, then its pricing decisions are as in Proposition 1. Finally, the merger
decision takes place in stage I, before uncertainty over D̃’s marginal cost realizes in the
intermediate stage I’.
In what follows, we assume that the consumer demand function is linear and equal to
P (Q) = α − βQ, with β > 0 and α > c + µ + λ.
6.2
Merger Decision
First note that regardless of whether U is merged with the more efficient retailer, vertical
integration is profitable. This follows directly from the results in Sections 3.1 and 3.2 and
23
simplifies the rest of the analysis, because it implies that we can focus on the monopoly
producer’s merger decision. Will U merge with the retailer whose marginal cost of production
is certain (D̄) or with the one whose marginal cost is stochastic (D̃)? To answer this question,
we first consider the case in which the two retailers are equally efficient in expectation
(ρ = 1/2).
LEMMA 6. If ρ = 1/2, the monopoly producer integrates with the retailer whose marginal
cost of production is stochastic (D̃).
To build the intuition for the result in Lemma 6, we first note that a retailer’s profit
function is convex in its marginal costs of production, holding fixed the quantity of the
rival retailer. Let us denote by π(C) the profit of a retailer when facing marginal cost of
C, with π(C) = maxq {(P (Q) − C)q}. Differentiating π(C) twice with respect to C yields
∂π/∂C = −q < 0 and ∂ 2 π/∂C 2 = −∂q/∂C > 0, thereby showing that a retailer’s monopoly
profit is convex with respect to C. This property implies that for ρ = 1/2, the following
holds:
1/2π̃ m (µ + λ) + 1/2π̃ m (µ − λ) ≥ π̄ m (µ),
(8)
where the superscript m denotes the monopoly profit of a retailer. Expression (8) implies
that the expected value of the monopoly profits of retailer D̃ evaluated at costs µ + λ and
µ − λ is higher than the value of the monopoly profit of the retailer with certain marginal
cost µ (D̄).
The condition in (8) exploits the convexity of profit functions in costs. If the integrated
firm foreclosed the rival retailer’s access to the intermediate good independent of the rival’s
cost efficiency, then (8) would be sufficient to establish the result in Lemma 6. By following
this strategy, if U is integrated with D̃, then the integrated company obtains the expected
value of this retailer’s monopoly profits (left-hand side of (8)). If instead U is integrated
with D̄, then the integrated firm raises π̄ m in all states (right-hand side of (8)).
However, we show in Proposition 1 that if the downstream unit of the integrated firm
is less efficient than the rival retailer, then the upstream unit engages in output shifting.
Consequently, if U is integrated with D̃ (D), the profit it raises when its downstream unit is
less efficient than D (D̃) is larger than π m (µ + λ) (π m (µ)). Output shifting thus raises the
value of both sides of (8). Yet, Lemma 6 shows that with linear demand and ρ = 1/2, the
profit increase arising when U merges with D̃ dominates.
What happens when ρ rises above 1/2? Because the merger with D̃ is profitable when
ρ = 1/2, by a continuity argument, the same result holds true when ρ is (slightly) larger
than 1/2. However, as ρ increases, it is also more likely that the monopoly producer will be
integrated with the inefficient retailer and engage in output shifting. This renders the final
good market more competitive in expectation than if the two retailers were equally efficient
(ρ = 1/2). Can output shifting be so effective that it makes the vertical merger with D̃
procompetitive with respect to vertical separation? Proposition 6 addresses this question.
In this proposition, ρV I = 1/2 + λ/4(α − c − µ − 2λ) > 1/2 and ρc = (α − c − µ − λ)/4λ.
PROPOSITION 6. If ρ ≥ ρV I , the monopoly producer (U ) integrates with retailer D̄ and
the merger is anticompetitive in expectation. Instead, if ρ < ρV I , then U integrates with
24
Ρ
1.0
0.9
0.8
ρV I
0.7
0.6
ρc
Λ
0.00
0.05
0.10
0.15
λ̃
Figure 4: Linear Demand Example
retailer D̃. The merger with D̃ is procompetitive in expectation if ρ is also larger than ρc ,
and anticompetitive otherwise.
Proposition 6 shows that vertical integration can result in a procompetitive outcome
in expectation in a model with uncertainty over retailers’ marginal costs of production.
Specifically, this result holds true if ρ lies in an intermediate range (i.e., ρc ≤ ρ ≤ ρV I ).
First, for values of ρ below ρV I , U finds it more profitable to merge with retailer D̃, which
is less efficient in expectation, due to convexity of the profit function in costs. Second, for
values of ρ above ρc , the probability that the integrated firm engages in output shifting is
sufficiently high that the aggregate quantity in the vertically integrated industry is larger
than the quantity in the vertically separated industry.
We illustrate the results of Proposition 6 using a parametric example with α = β = 1
and c = µ = .25. The shaded area in Figure 4 illustrates when vertical integration between
U and D̃ is profitable and procompetitive in expectation. The width of the interval [ρc , ρV I )
increases with λ: as the difference in retailers’ efficiencies rises, the impact of output shifting
increases, making vertical integration procompetitive for ρ between 0.5 and 0.75. This is
the case at λ̃, the upper-bound of λ derived in the proof of Lemma 6, below which both
retailers are active with vertical integration. Note that these results are not unique to a
setting with linear demand. For example, vertical integration with D̃ is procompetitive and
more profitable than a merger with D̄ for values of ρ between 0.59 and 0.7 when using the
demand function P (Q) = α − Qβ together with the following parameter values: α = 1,
β = 2, c = .2, µ = .3, and λ = .2.
25
7
Conclusions
This paper examines a standard model in which a monopoly producer deals with competing retailers by means of secret two-part tariffs. We build on the setting employed by the
literature on anticompetitive vertical restraints and introduce a crucial element: we allow
retailers to carry different marginal costs of production. Our central finding is that when
the upstream monopolist is integrated with the less efficient retailer, it will depart from the
foreclosure strategy by reducing the unit-price offer to the nonintegrated but more efficient
retailer. This output-shifting effect makes vertical integration procompetitive when compared to vertical separation if the nonintegrated retailer is sufficiently more efficient. This
shows that, for example, policies of divestiture imposed by regulatory agencies to prevent
foreclosure can have unintended consequences and may be misguided. Consistent with our
conclusions, Lafontaine and Slade (2007) document that studies assessing the implications
of forced vertical separations generally find that these legal decisions lead to price increases.
We extend the model to consider an intermediate good market that features a competitive bottleneck alternative to the dominant producer consisting of a fringe of less efficient
firms. First, we show that the competitive consequences of vertical integration are the same
as those obtained in our main model. Then, we analyze how information transmission within
an integrated company affects the final good’s market quantity and price. Specifically, we
find that when the upstream unit of an integrated firm is able to transmit to its downstream
affiliate information regarding the competing retailer’s decision on the contract offer, the
downstream market is more competitive. In regard to competition policy, this result implies that the employment of behavioral remedies like information firewalls within vertically
integrated companies can be detrimental to consumer surplus.
Our results are robust to several alternative assumptions. Moreover, we provide a model
in which the upstream monopolist’s integration decision is taken under uncertainty over
retailers’ marginal costs of production, reflecting the consideration that vertical integration
is a long-term decision. There, we determine the conditions such that the monopoly producer
integrates with a retailer that is less efficient in expectation and this merger gives rise to a
procompetitive outcome.
Our results have implications for public policy formulation. Specifically, the model shows
that vertical integration might not necessarily result in a foreclosure strategy when the
integrated company deals with more efficient retailers, a claim that is reminiscent of the
Chicago School argument against the anticompetitive theories of vertical integration and
foreclosure. An avenue for future research could be to extend the analysis in this paper to a
framework where opportunities for vertical mergers arise dynamically (e.g., along the lines of
Nocke and Whinston, 2013). This would allow to further study the conditions under which
vertical integration is procompetitive.
26
Appendix
Proof of Lemma 1. Let Ti = {wi , Fi } and T−i = {w−i , F−i } denote the monopoly producer’s
candidate equilibrium offers in the game with vertical separation, with i = 1, 2. Moreover,
let qi and q−i denote the candidate equilibrium quantities produced in the third stage by
retailers Di and D−i at such candidate equilibrium offers. Finally, recall that, because offers
are secret and retailers hold passive beliefs, when a retailer receives an offer different from
what it expects in the candidate equilibrium, it does not revise its belief about the offer
made to the competing retailer.
We solve the game by backward induction. In the third stage, retailer Di produces
q(wi , w−i ) as defined in (1). Accordingly, one-to-one production technology implies that Di
orders qi (wi , w−i ) from the monopoly producer U .
We now determine the monopoly producer U ’s tariffs. With passive beliefs, the equilibrium contract offered by U to each retailer Di must maximize their joint profits, i.e., the
sum of the intermediate good’s sales to Di plus the profits of Di , holding fixed the contract
offered to retailer D−i (McAfee and Schwartz, 1994). Therefore, the first-stage maximization
problem of U can be written as
max qi (wi , w−i )(wi − c) + (P (qi (wi , w−i ) + q−i ) − µi − wi ) qi (wi , w−i ).
wi
Taking the first-order condition with respect to wi and invoking the Envelope Theorem, we
obtain (functional notation is dropped, for simplicity)
∂qi
+ q i − qi
∂wi
∂qi
= (wi − c)
= 0.
∂wi
(wi − c)
Because ∂qi /∂wi < 0, at the equilibrium wi = c, i.e., U optimally sets the unit price equal
to its marginal cost (c) to maximize the revenues from the bilateral relationship with Di
(i = 1, 2). At this unit price, both retailers are active and produce the respective Cournot
quantity, q1c and q2c , to obtain Cournot profits of π1c and π2c . In turn, the monopoly producer
fully extracts retailers’ Cournot profits by setting the fixed component of the two-part tariff
equal to Fi = πic , i = 1, 2.
Thus, the unique equilibrium of the game with vertical separation features the monopoly
producer making the following take-it-or-leave-it offers to retailers D1 and D2 :
T1 = {c, π1c },
T2 = {c, π2c }.
Q.E.D.
Proof of Proposition 1. We first show that the integrated firm U −D2 trades the intermediate
good internally at marginal cost (w2? = c). After D1 ordered its quantity of the intermediate
good and paid the tariff, firm U − D2 maximizes
max
w2
(P (q2 (w2 , w1 ) + q1 ) − w2 − µ2 ) q2 (w2 , w1 ) + q2 (w2 , w1 )(w2 − c),
27
which can be rewritten as
max
w2
q2 (w2 , w1 ) (P (q2 (w2 , w1 ) + q1 ) − c − µ2 ) .
(9)
That is, it is always in the best interest of U and D2 to maximize their joint profits in (9),
given the contract acceptance decision of retailer D1 . The maximum of the expression in (9)
is reached at an internal unit price equal to marginal cost, so that the downstream unit of
the integrated firm sets its quantity to solve maxq {(P (q + q1 ) − c − µ2 )q}. This shows that U
cannot commit itself to an internal trading price other than marginal cost, since this pricing
strategy is not robust to secret renegotiation.
Then, the optimization problem of U − D2 when dealing with retailer D1 can be written
as follows:
max
w1 ,F1
max{(P (q + q1 (w1 , c)) − µ2 − c)q} + q1 (w1 , c)(w1 − c) + F1 ,
q
with
F1 ≤ max{(P (q + q2 (c, w1 )) − µ1 − w1 )q}.
q
(10)
At equilibrium, U − D2 formulates a take-it-or-leave-it offer to fully extract retailer D1 ’s
profit. Thus, the constraint in (10) is binding, and we can rewrite the optimization program
of U − D2 as
maxw1 max{(P (q + q1 (w1 , c)) − µ2 − c)q} + q1 (w1 , c)(w1 − c) +
q
max{(P (q + q2 (c, w1 )) − µ1 − w1 )q}.
q
The ensuing first-order condition with respect to w1 is equal to
(P 0 (Q)q2 + w1 − c)
∂q1
∂q2
+ P 0 (Q)q1
= 0.
∂w1
∂w1
(11)
To derive the expressions for ∂q1 /∂w1 and ∂q2 /∂w1 , we take the total derivative of the
first-order conditions for downstream quantities q1 and q2 , which are given by, respectively,
P (Q) − µ1 − w1 + P 0 (Q)q1 = 0 and P (Q) − µ2 − c + P 0 (Q)q2 = 0. We obtain
∂q1
2P 0 (Q) + q2 P 00 (Q)
= 0
∂w1
P (Q) (3P 0 (Q) + QP 00 (Q))
(12)
∂q2
P 0 (Q) + q2 P 00 (Q)
=− 0
.
∂w1
P (Q) (3P 0 (Q) + QP 00 (Q))
(13)
and
28
Plugging expressions (12) and (13) into (11), and rearranging, we obtain
P 0 (Q)q1
0
.
w1 = c + P (Q) q1 − q2 −
2P 0 (Q) + q2 P 00 (Q)
(14)
To show that the value of w1 resulting from (14) is unique, we compute the derivative of
the right-hand side of the equation with respect to w1 and find that under our assumption
P 0 (Q) + QP 00 (Q) < 0, i = 1, 2, this derivative is strictly smaller than one. Hence, since the
slope of the left-hand side of the expression is equal to one, there is a unique value of w1
that solves (14). We denote this value w1? .
The first-order conditions for q1 and q2 imply that q1 = −(P (Q) − µ1 − w1 )/P 0 (Q) and
q2 = −(P (Q) − µ2 − c)/P 0 (Q). Plugging these expressions into (14) yields
w1? = P (Q) − 2µ2 + µ1 +
P 00 (Q) (µ2 − µ1 ) (P (Q) − c − µ2 )
.
(P 0 (Q))2
(15)
We now turn to the second-order condition. Taking the derivative of the first-order
condition (11) with respect to w1 and inserting q1 = −(P (Q) − µ1 − w1 )/P 0 (Q), q2 =
−(P (Q) − µ2 − c)/P 0 (Q) and the value of w1? into (15), we obtain
(3P 0 (Q)
Φ(Ψ + Γ)
.
+ QP 00 (Q))3 (P 0 (Q))6
(16)
The denominator of (16) and
Φ ≡ −3(P 0 (Q))4 + (P 0 (Q))2 P 00 (Q)(P (Q) − c − µ1 + (µ2 − µ1 )) −
(µ2 − µ1 )(P (Q) − c − µ2 )(P 00 (Q))2
are negative due to the assumption that P 0 (Q) + QP 00 (Q) < 0.22 This assumption also
implies that
Ψ ≡ −2(P 0 (Q))4 + (P 0 (Q))2 P 00 (Q)(P (Q) − c − µ1 + 2(µ2 − µ1 )) −
3(µ2 − µ1 )(P (Q) − c − µ2 )(P 00 (Q))2
is negative. Finally, the sign of
Γ ≡ P 000 (Q)P 0 (Q)(P (Q) − c − µ2 )(µ2 − µ1 )
depends on the sign of P 000 (Q), which we assume to be positive or not too negative. It follows
that Ψ + Γ is negative, which implies that the numerator Φ(Ψ + Γ) is positive. Because the
denominator is negative, the whole expression is negative.
This discussion shows that the profit function is locally concave at w1 = w1? . However,
To show this, we note that our assumption implies that P 00 (Q) < P 0 (Q)/(q1 + q2 ). We then insert
q2 = −(P (Q) − µ2 − c)/P 0 (Q) and q1 = −(P (Q) − µ1 − w1 )/P 0 (Q) evaluated at w1? as in (15), and solve
the resulting expression for P 00 (Q). This gives us an upper bound for P 00 (Q) that, once plugged into the
expression for Φ, shows that Φ is negative. We proceed analogously with Ψ.
22
29
because the first-order condition (11) has a unique solution, we can conclude that the profit
function is strictly quasi-concave. That is, w1? is a global maximum of the integrated firm’s
maximization problem.
In sum, the unique equilibrium value of the per-unit price is equal to w1? and the ensuing
equilibrium value of the fixed component of the two-part tariff is F1? = q1 (P (Q) − µ1 − w1? ),
with q1 = q1 (w1? , c), q2 = q2 (c, w1? ), Q = q1 + q2 and w1? as in (15).
Q.E.D.
Proof of Proposition 2. First, we take the value of q1 as determined by retailer D1 ’s firstorder condition, q1 = −(P (Q) − µ1 − w1 )/P 0 (Q). Plugging w1 = w1? as in (2) into this
expression, we get
(µ2 − µ1 ) 2 (P 0 (Q))2 − (P (Q) − c − µ2 )P 00 (Q)
?
.
(17)
q1 (w1 ) =
− (P 0 (Q))3
Our working assumption (P 0 (Q)+QP 00 (Q) < 0) implies that P 0 (Q)+P 00 (Q)q2 < 0. Plugging
q2 = −(P (Q) − µ2 − c)/P 0 (Q) into P 0 (Q) + P 00 (Q)q2 < 0 yields (P 0 (Q))2 − (P (Q) − c −
µ2 )P 00 (Q) > 0. It follows that the numerator is positive for all µ2 ≥ µ1 . Finally, recall
that the demand function is strictly decreasing (P (Q)0 < 0), implying that the denominator
of (17) is strictly positive. Consequently, the value of q1 in (17) is strictly positive for all
µ2 > µ1 , and zero otherwise.
Q.E.D.
Proof of Proposition 3. Subtracting w1? from c and rearranging, we obtain that c > w1? if,
and only if,
µ2 − µ1 >
(P (Q) − µ2 − c)(P 0 (Q))2
.
(P 0 (Q))2 − (P (Q) − µ2 − c)P 00 (Q)
For the second part of the claim, note that retailer D2 obtains the intermediate good at a
per-unit price of c both with vertical separation and vertical integration. Thus, changes in
the aggregate output Q are only due to changes in the per-unit price of D1 . Summing (12)
and (13), we get
∂Q
∂q1
∂q2
1
=
+
=
< 0.
0
∂w1
∂w1 ∂w1
3P (Q) + QP 00 (Q)
Because ∂Q/∂w1 < 0, aggregate output is larger under vertical separation than under vertical
integration if, and only if, w1? < c.
Q.E.D.
Proof of Proposition 4. The maximization problem of the integrated firm is equal to23
max
w1 ,F1
max{(P (q + q1 (w1 , c)) − µ2 − c)q} + q1 (w1 , c)(w1 − c) + F1 .
q
(18)
23
Note that we are using the result that the downstream unit of the integrated firm (D2 ) obtains the input
good at marginal cost (see the proof of Proposition 1 for details).
30
Retailer D1 accepts U −D2 ’s offer if the fixed component of the two-part tariff F1 satisfies
the following constraint:
F1 ≤ max{(P (q + q2 (c, w1 )) − µ1 − w1 )q} − max{(P (q + q2 (c, ĉ)) − µ1 − ĉ)q},
q
q
(19)
which will bind at equilibrium. In the first term on the right-hand side of (19), q2 is the
quantity produced by D2 given that D1 obtains the intermediate good at a unit price equal
to w1 , that is, q2 (c, w1 ). In the second term, the quantity produced by D2 is q2 (c, ĉ). This
reflects the fact that D2 is informed by U that D1 has rejected the integrated firm’s offer
and thus pays ĉ for the input good. D2 responds by adjusting its quantity of the final good
accordingly.
Using the value of F1 resulting from (19) and realizing that the last term in (19) is
independent of w1 , the problem in (18) becomes
max
w1
max{(P (q + q1 (w1 , c)) − µ2 − c)q} + q1 (w1 , c)(w1 − c) +
q
max{(P (q + q2 (c, w1 )) − µ1 − w1 )q}.
q
This optimization problem is the same as in the case without a bottleneck alternative, so
the solution coincides and is given by the value of w1? in (15). However, because D1 can
turn to Û when U sets a unit price higher than ĉ, the equilibrium unit price is equal to
w1BA = min{w1? , ĉ}. The equilibrium value of the fixed component of the two-part tariff can
be determined from a binding (19):
F1BA = q1 (w1BA , c)(P (Q) − µ1 − w1BA ) − q1 (ĉ, c)(P (q1 (ĉ, c) + q2 (c, ĉ)) − µ1 − ĉ),
with Q = q1 (w1BA , c) + q2 (c, w1BA ).
Q.E.D.
Proof of Proposition 5. Suppose that U cannot transmit to D2 the information regarding
D1 ’s decision on U ’s offer. The offer of U − D2 is accepted by D1 if F1 is such that:
F1 ≤ max{(P (q + q2 (c, w1 )) − µ1 − w1 )q} − max{(P (q + q2 (c, w1 )) − µ1 − ĉ)q},
q
q
(20)
with equality at equilibrium. Because D2 is not informed about D1 ’s decision, D2 believes
that D1 will accept U ’s offer and produce a quantity q2 (c, w1 ), as it does on the equilibrium
path. That is, in contrast to the case with information transmission, D2 cannot condition
the value of its production on D1 ’s rejection of U ’s offer.
The integrated firm’s optimization program is given by (18) also in the case without
information transmission. Using (20), the program can be rewritten as
maxw1 max{(P (q + q1 (w1 , c)) − µ2 − c)q} + q1 (w1 , c)(w1 − c) +
q
max{(P (q + q2 (c, w1 )) − µ1 − w1 )q} − max{(P (q + q2 (c, w1 )) − µ1 − ĉ)q}.
q
q
31
The ensuing first-order condition with respect to w1 is
(P 0 (Q)q2 + w1 − c)
∂q2
∂q1
+ (P 0 (Q)q1 − P 0 (Q̂)q̂1 )
= 0,
∂w1
∂w1
with q̂1 = arg maxq {(P (q + q2 (c, w1 )) − µ1 − ĉ)q}, q2 = q2 (c, w1 ), and Q̂ = q̂1 + q2 (c, w1 ).
Proceeding as in the proof of Proposition 1, we obtain
w1 = c −
P 0 (Q) (P 0 (Q)(2q2 − q1 ) + P 00 (Q)q2 (q2 − q1 )) + P 0 (Q̂)q̂1 (P 0 (Q) + P 00 (Q)q2 )
. (21)
2P 0 (Q) + q2 P 00 (Q)
Inserting q1 = −(P (Q) − µ1 − w1 )/P 0 (Q), q2 = −(P (Q) − µ2 − c)/P 0 (Q), and q̂1 = −(P (Q̂) −
µ1 − ĉ)/P 0 (Q̂) into the expression for w1 given by (21) yields
P 00 (Q) (P (Q) − c − µ2 ) (P (Q̂) − ĉ − µ2 )
ŵ1 = P (Q) + P (Q̂) − ĉ − 2µ2 −
,
(P 0 (Q))2
implying that ŵiBA = min{ŵ1 , ĉ} at equilibrium. Note that, as in Proposition 1, standard
computations show that the second-order condition is satisfied at w1 = ŵ1 . Finally, the fixed
component of the two-part tariff set by U − D2 to
D1 when information transmission is not
BA
BA
possible is equal to F̂1 = q1 P (Q) − µ1 − ŵ1 − q̂1 (P (Q̂) − µ1 − ĉ).
Comparing the formula for ŵ1 with w1? , we obtain
ŵ1 − w1? =
(P (Q̂) − µ1 − ĉ)((P 0 (Q))2 − P 00 (Q)(P (Q) − c − µ2 ))
,
(P 0 (Q))2
(22)
where Q̂ is defined above. Our working assumption (P 0 (Q) + QP 00 (Q) < 0) implies that
P 0 (Q) + P 00 (Q)q2 < 0. Thus, P 00 (Q) < −P 0 (Q)/q2 . Inserting q2 = −(P (Q) − c − µ2 )/(P 0 (Q))
into this inequality yields
P 00 (Q) <
(P 0 (Q))2
,
P (Q) − c − µ2
which implies that the second term in the numerator of the right-hand side of (22) is positive.
Since the first term of the numerator and the denominator are also positive, we obtain
ŵ1 − w1? > 0. Therefore, the difference between ŵ1BA and w1BA is strictly positive for all
values of w1BA < ĉ, whereas ŵiBA = w1BA if w1BA = ĉ.
Q.E.D.
Proof of Lemma 4. First note that, following the same arguments as in the proof of Proposition 1, the integrated firm trades the intermediate good internally at marginal cost and the
downstream unit is active. Then, the optimization problem of U − D2 when dealing with
retailer D1 can be written as follows:
max
q1 ,F̄1
max{(P (q + q1 ) − c − µ2 )q} + F̄1 − q1 c,
q
32
(23)
subject to the constraint that F̄1 ≤ (P (q1 + q2 ) − µ1 )q1 , with q2 = arg maxq {(P (q + q1 ) − c −
µ2 )q}. The constraint is binding at equilibrium. Thus, we can rewrite (23) as
max
q1
max{(P (q + q1 ) − c − µ2 )q} + (P (q1 + q2 ) − c − µ1 )q1 ,
q
(24)
The first term in (24) corresponds to the profit of U − D2 ’s downstream affiliate. The second
term is the profit of the nonintegrated retailer D1 , given the output produced by D2 . Taking
the first-order condition of (24) with respect to q1 yields the following expression:
∂q2
0
0
= 0.
(25)
P (Q)q2 + P (Q) − c − µ1 + P (Q)q1 1 +
∂q1
To derive ∂q2 /∂q1 , we take the total derivative of the first-order condition for q2 , which is
given by P (Q) − µ2 − c + P 0 (Q)q2 = 0, and obtain
∂q2
P 0 (Q) + q2 P 00 (Q)
=− 0
.
∂q1
2P (Q) + q2 P 00 (Q)
Plugging this expression into (25) yields
P 0 (Q)(q1 + q2 ) + P (Q) − c − µ1 − q1 P 0 (Q)
P 0 (Q) + q2 P 00 (Q)
= 0.
2P 0 (Q) + q2 P 00 (Q)
Finally, inserting q2 = −(P (Q) − µ2 − c)/P 0 (Q) and solving for q1 , we find that
2
0
00
(µ
−
µ
)
2
(P
(Q))
−
(P
(Q)
−
c
−
µ
)P
(Q)
2
1
2
.
q1? =
− (P 0 (Q))3
Accordingly, the equilibrium value of F̄1 is equal to
F̄1? = (P (q2? + q1? ) − µ1 ) q1? ,
with q2? = arg maxq {(P (q + q1? ) − c − µ2 )q}.
Q.E.D.
Proof of Lemma 5. The first-order conditions of the program in (7) with respect to w1 and
F1 , respectively, are given by
t
∂πUt −D2
t
−δ ∂πD1
)
+ (1 − δ)(πUt −D2 − dtU −D2 )δ (πD
= 0,
(26)
1
∂w1
∂w1
t
t
t
1−δ ∂πU −D2
t
t
δ
t
−δ ∂πD1
− dtU −D2 )δ−1 (πD
)
+
(1
−
δ)(π
−
d
)
(π
)
= 0.
(27)
U −D2
U −D2
D1
1
∂F1
∂F1
t
)1−δ
δ(πUt −D2 − dtU −D2 )δ−1 (πD
1
δ(πUt −D2
We first consider the case in which U − D2 shuts its downstream unit down (t = s).
s
Inserting the expressions for πUs −D2 , πD
and dsU −D2 given in the main text into (26) and
1
33
(27), and simplifying, we obtain
∂q1
δ(π1 (w1 ) − F1 ) q1 (w1 ) +
(w1 − c) − (1 − δ) (F1 + q1 (w1 )(w1 − c)) q1 (w1 ) = 0
∂w1
δ(π1 (w1 ) − F1 ) − (1 − δ) (F1 + q1 (w1 )(w1 − c)) = 0.
Solving the second expression for F1 and inserting it into the first yields (∂q1 /∂w1 )(w1 − c) =
0. Because ∂q1 /∂w1 < 0, the last equation implies that the unique value of the unit price
that solves the problem in (7) features w1B = c. Plugging w1B = c into the rearranged firstorder condition for F1 yields F1B = δπ1m . Hence, the profits of U − D2 when it shuts D2 down
(t = s) are πUs −D2 = δπ1m .
We next analyze the case in which U − D2 keeps D2 alive (t = n). Inserting πUn −D2 ,
n
πD
, and dnU −D2 into (26) and (27) and proceeding in the same way as above, we obtain the
1
following expressions:
∂q2
∂q1
+ P 0 (Q)q1
=0
∂w1
∂w1
δ(π1 (w1 , c) − F1 ) − (1 − δ)(F1 + q1 (w1 , c)(w1 − c) + π2 (c, w1 ) − π2m ) = 0.
(P 0 (Q)q2 + w1 − c)
The rearranged first-order condition for w1 coincides with (11), which is the expression that
determines the value of the unit price in the proof of Proposition 1. Therefore, the unique
optimal Nash bargaining solution features a unit price w1B = w1? and a fixed component of
the two-part tariff equal to
F1B = δπ1 (w1? , c) − (1 − δ)[q1 (w1? , c)(w1? − c) + π2 (c, w1? ) − π2m ].
It follows that the profits of U − D2 when it keeps D2 alive (t = n) are πUn −D2 = δ[π1 (w1? , c) +
Q.E.D.
q1 (w1? , c)(w1? − c) + π2 (c, w1? )] + (1 − δ)π2m .
Proof of Lemma 6. First, we determine the profits of the vertically integrated firm when U
merges with D̄, the retailer whose marginal cost of production is certain and equal to µ.
U merges with D̄ With probability ρ, retailer D̄ is the more efficient retailer (Table
1). Thus, the results in Lemma 2 apply: the upstream unit of the integrated firm forecloses retailer D̃’s access to the intermediate good, and the downstream unit produces the
monopoly quantity. Using the assumption of linear demand, the quantity and the profit of
the integrated firm U − D̄ in this state are equal to (α − c − µ)/2β and (α − c − µ)2 /4β,
respectively.
However, with probability 1 − ρ, retailer D̃ has the lower marginal cost of production
µ − λ < µ. Thus, the results in Proposition 1 apply. Specifically, U sets the internal
trading price equal to its marginal cost of production (c) and a unit price w̃ to D̃ following
Proposition 1, mutatis mutandis. With linear demand, the equilibrium value of w̃ is given
by
w̃ =
α + c − µ − 4λ
.
2
34
Denote by q̄ and q̃ the quantity produced by retailer D̄ and D̃, respectively. Using the results
above, the equilibrium quantities are equal to
q̄ =
α − c − µ − 2λ
2β
and q̃ =
2λ
.
β
Note that for the quantity of the integrated firm’s downstream unit (q̄) to be positive, it
must be that λ < λ̄ ≡ (α − c − µ)/2, a condition that defines the upper bound on λ when
U is integrated with retailer D̄. In this case, the profit of the integrated firm U − D̄ is equal
to β q̄ 2 + β q̃ 2 + q̃(w̃ − c) = [(α − c − µ)2 + 4λ2 ]/4β.
Thus, the expected profit of U − D̄ is given by
(α − c − µ)2
(α − c − µ)2 + 4λ2
+ (1 − ρ)
4β
4β
2
2
(α − c − µ) + 4(1 − ρ)λ
.
=
4β
I
ΠVU −
D̄ = ρ
I
For ρ = 1/2, ΠVU −
simplifies to [(α − c − µ)2 + 2λ2 ] /4β.
D̄
U merges with D̃ Let U be integrated with retailer D̃. With probability ρ, retailer D̃
is less efficient than D̄. Therefore, the results in Proposition 1 apply: U sets the internal
trading price equal to its marginal cost of production (c) and a unit-price offer w̄ to D̄ as
in the claim of the proposition, mutatis mutandis. Specifically, the upstream unit of the
integrated firm will charge a unit price w̄ = (α + c − µ − 5λ)/2, and the resulting equilibrium
quantities are equal to
q̄ =
2λ
β
and q̃ =
α − c − µ − 3λ
.
2β
The profit of U − D̃ in this case is
(α − c − µ)2 − 2λ(α − c − µ) + 5λ2
.
4β
Before proceeding to the case in which D̃’s marginal cost is µ − λ, note that q̃ is positive
if, and only if, λ < λ̃ ≡ (α − c − µ)/3, which represents the upper bound on λ when U is
integrated with D̃.
With probability 1 − ρ, retailer D̃ is more efficient. Thus, the upstream unit of the
integrated firm forecloses D̄’s access to the intermediate good, and the downstream unit
produces the monopoly quantity. Accordingly, the equilibrium value of D̃’s quantity and the
profit of the integrated firm U − D̃ are equal to (α − c − µ + λ)/2β and (α − c − µ + λ)2 /4β,
respectively.
In sum, the expected profits of U − D̃ are
I
ΠVU −
=
D̃
(α − c − µ)2 − 2(2ρ − 1)λ(α − c − µ) + (1 + 4ρ)λ2
.
4β
35
I
If ρ = 1/2, the value of ΠVU −
simplifies to [(α − c − µ)2 + 3λ2 ] /4β. Therefore, the difference
D̃
I
I
between ΠVU −
and ΠVU −
at ρ = 1/2 is equal to
D̄
D̃
I
I
ΠVU −
− ΠVU −
D̄ =
D̃
λ2
> 0 ∀λ > 0,
4β
which establishes the claim in the proposition.
Q.E.D.
Proof of Proposition 6. First, recall that in the proof of Lemma 6 we established that when
U is integrated with D̄, retailers are all active if λ < λ̄ = (α − c − µ)/2. However, if U
is integrated with D̃, the condition is λ < λ̃ = (α − c − µ)/3, with λ̃ < λ̄. We start by
determining the threshold above which a merger with D̄ is more profitable than a merger
I
I
with D̃. Using the values of ΠVU −
and ΠVU −
obtained in the proof of Lemma 6, we find
D̄
D̃
that
I
I
ΠVU −
− ΠVU −
D̄ =
D̃
λ
[2(α − c − µ − 2λ) + λ − 4ρ(α − c − µ − 2λ)] .
4β
I
I
It follows that ΠVU −
≥ ΠVU −
if, and only if,
D̄
D̃
ρ ≥ ρV I ≡
1
λ
+
,
2 4(α − c − µ − 2λ)
with ρV I > 1/2 for λ < λ̃.
We next show that vertical integration between U and D̄ is, in fact, anticompetitive.
Determining the aggregate expected quantity under vertical separation and under vertical
integration between U and D̄, we obtain
QV S =
2α − 2c − 2µ + λ(1 − 2ρ)
3β
I
and QVU −
D̄ =
α − c − µ + 2λ(1 − ρ)
.
2β
I
The difference between QVU −
and QV S is equal to
D̄
4λ − (α − c − µ) − 2λρ
,
6β
I
implying that QVU −
− QV S ≥ 0 if, and only if,
D̄
ρ ≤ ρc1 ≡
4λ − (α − c − µ)
1 α−c−µ
= −
.
2λ
2
2λ
Were ρc1 to lie above ρV I , then there would be values of ρ such that integration between U
and D̄ would be profitable and procompetitive. However, it is easy to see that
ρc1 < ρV I
∀λ < λ̃,
implying that the merger between U and D̄ is anticompetitive for all values of ρ ∈ [1/2, 1).
36
Finally, we establish the condition such that the merger between U and D̃ is procompetitive. The aggregate expected quantity when U is integrated with D̃ is equal to
I
I
QVU −
= (α − c − µ + λ)/2β. Then the difference between QVU −
and QV S is
D̃
D̃
I
QVU −
− QV S =
D̃
α−c−µ−λ
4λρ − (α − c − µ − λ)
≥ 0 ⇐⇒ ρ ≥ ρc ≡
.
6β
4λ
It is easy to show that ρc ≤ ρV I for all values of λ such that
√ 1
λ∈
5 − 5 (α − c − µ), λ̃ .
10
Therefore, the merger between U and D̃ is procompetitive if ρ lies in the interval [ρc , ρV I ).
Q.E.D.
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