RAND Journal of Economics
Vol. 46, No. 3, Fall 2015
pp. 461–479
Vertical integration, foreclosure,
and productive efficiency
Markus Reisinger∗
and
Emanuele Tarantino∗∗
We analyze the consequences of vertical integration by a monopoly producer dealing with two
retailers (downstream firms) of varying efficiency via secret two-part tariffs. When integrated
with the inefficient retailer, the monopoly producer does not foreclose the rival retailer 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. Output shifting
arises with homogeneous and differentiated products. Moreover, we show that integration with
an inefficient retailer emerges in a model with uncertainty over retailers’ costs, and this merger
can be procompetitive in expectation.
1. Introduction
How does vertical integration affect economic outcomes such as prices, quantities, and
consumer surplus? Is vertical integration solely about increasing market power, or can it enhance
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).1 On the other hand, the empirical literature presents evidence suggesting that
∗ WHU - Otto Beisheim School of Management; [email protected].
∗∗ University of Mannheim; [email protected].
We are indebted to the Editor (Benjamin Hermalin) for very insightful comments and suggestions. The article also
benefited from comments by three anonymous referees, Cédric Argenton, Heski Bar-Isaac, Felix Bierbrauer, Giacomo
Calzolari, Simon Cowan, Vincenzo Denicolò, Chiara Fumagalli, Massimo Motta, Volker Nocke, Marco Pagnozzi, Martin
Peitz, 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 Bayreuth, University of Bologna, Pontificia Universidad Católica de Chile, University of St. Gallen, UCL (Louvain),
University of Oxford, CREST (Paris), University of Rochester (Simon), TILEC - Tilburg University, ETH Zurich
seminars, and at the 2014 MaCCI Competition and Regulation Day (Mannheim), 2014 Industrial Organization Workshop
(Alberobello), and 2012 Annual Searle Center Conference on Antitrust Economics and Competition Policy (Northwestern
University).
1
There is also evidence that integration leads to foreclosure of competitors (Chipty, 2001; Hastings and Gilbert,
2005; among others).
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efficiency-based mechanisms are behind vertical integration (e.g., Hortaçsu and Syverson, 2007).2
Several studies show that the efficiency gains produced by vertical integration can outweigh the
welfare losses caused by foreclosure: in their survey of empirical research, Lafontaine and
Slade (2007) conclude that, in most circumstances, profit-maximizing vertical integration raises
consumer welfare.
This article 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 on favorable terms to the unintegrated 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.
In our model, a monopoly producer sells an intermediate good via secret two-part tariffs
to competing retailers. In contrast to the standard setting employed in the literature, we allow
retailers to have different marginal costs of production.
Without integration, a monopoly producer’s limited commitment with unobservable contracts prevents it from monopolizing the final-product market (e.g., Hart and Tirole, 1990; O’Brien
and Shaffer, 1992; McAfee and Schwartz, 1994; Rey and Tirole, 2007; White, 2007): see our
Lemma 1. Consistent with earlier findings, integration with the more efficient retailer yields the
monopolist full market power because it can foreclose the inefficient (and competing) retailer:
see Proposition 1. If, instead, the monopolist were integrated with the less efficient retailer,
would the integrated firm still pursue a foreclosure strategy? We show that the answer can be
“more than no”: the newly integrated firm may reduce the unit price to the unintegrated, but
more efficient, retailer even below the marginal cost of production.
The upstream firm faces the following trade-off. A reduction in the unit price to the unintegrated retailer raises industry quantity and thus reduces industry revenue. However, a countervailing effect arises in our framework: the reduction in the unit price to the unintegrated retailer
is known by the integrated firm’s downstream unit, which responds by reducing its quantity. A
lower unit price then triggers an increase in the unintegrated and more efficient retailer’s quantity and profit. The integrated firm can extract this higher profit via the fixed component of the
two-part tariff. We show that the reduction in industry revenue is outweighed by the fact that the
industry produces more efficiently. Indeed, we find that the unintegrated retailer will be active in
the final-good market as long as it is strictly more efficient than the integrated downstream unit:
see Proposition 2. This establishes the output-shifting effect of vertical integration.
A question of importance for competition policy is the magnitude of the output-shifting
effect. Specifically, can the incentive to reduce the unit price to the unintegrated firm be so strong
as to render vertical integration procompetitive with respect to an unintegrated 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. This leads to an expansion of industry
output relative to vertical separation and renders vertical integration procompetitive. We find that
below-cost pricing is more likely to occur when cost differences between retailers are particularly
large.
The integrated upstream producer’s problem can be seen as a particular Stackelberg game,
in which the upstream firm moves first and the integrated retailer follows. In the first stage,
the upstream monopolist sets the input price to the unintegrated retailer, thereby determining
this retailer’s output. Because the upstream firm can extract the competing retailer’s profit via
its two-part tariff, it effectively chooses the rival’s output to maximize industry profit, under the
constraint that its inefficient downstream subsidiary seeks to maximize its own profits. If retailers’
2
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.
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cost differences are large enough, the profitability of output shifting is particularly large, inducing
the upstream firm to set a unit price below marginal cost. By this strategy, the integrated firm
commits to a reduction of its downstream firm’s output.
In the main model, we assume that retailers offer a homogeneous good. We show that
our results are robust to retailers offering differentiated products. We 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.
The discussion thus far begs the question of 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 Proposition 1).
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 4).
The “Chicago School” has challenged the view that an upstream monopolist needs to
integrate 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.3 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 separation.4
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 highvoltage 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 articles 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 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-good market.
Finally, Nocke and White (2007) analyze the effects of vertical integration on the sustainability
3
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), Marx and Shaffer (2004), and Nocke and Rey
(2014).
4
Asymmetry in retailers’ marginal costs can arise endogenously in a setup with providers of complementary inputs
(as in, among others, Laussel, 2008; Laussel and Van Long, 2012; Matsushima and Mizuno, 2012; Hermalin and Katz,
2013; Reisinger and Tarantino, 2013).
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of upstream collusion. These articles 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 on favorable conditions when particularly efficient.
Our article 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.5 Relatedly,
Chen and Schwartz (2015) 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.6
The article is structured as follows. Section 2 presents the model, and Section 3 provides
the equilibrium analysis. Section 4 shows the robustness of our main results when retailers offer
differentiated products. Section 5 presents a model with equilibrium vertical integration when
retail costs are uncertain. Section 6 concludes. Proofs can be found in the Appendix.
2. The model
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. The retailers transform the intermediate good into a homogeneous final product on a oneto-one basis. In contrast to the previous literature, we allow retailers to have 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 (Q) + Q P (Q) < 0, which guarantees that the profit functions
are (strictly) quasi-concave and that the Cournot game exhibits strategic substitutability (Vives,
1999). We also assume that P (Q) is not too negative. This ensures concavity of the monopoly
producer’s profit function.
When contracting with retailer Di , i = 1, 2, the upstream monopolist makes a take-it-orleave-it offer of a two-part tariff contract consisting of a fixed component, Fi , and a unit price,
wi .7 If it accepts, retailer Di ’s total marginal cost is μi + wi .
We consider two scenarios: firms are not integrated (vertical separation) or the monopoly
producer U is integrated with one of the two retailers (vertical integration). Given vertical
separation, the game proceeds as follows:
(i) U secretly offers to each retailer Di a two-part tariff {wi , Fi } ≡ Ti .
(ii) Retailers simultaneously accept or reject the contract offer.
(iii) Retailers order a quantity of the intermediate good, qi , and pay the tariff. Then, they transform
the intermediate good into the final good and bring output to the market.
Afterward, retail purchases are made, and profits are realized.
5
Hansen and Motta (2012) 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.
6
An exception is Linnemer (2003), who studies the implications of vertical integration on welfare in a model with
asymmetric retailers. However, he restricts his attention to the analysis of the impact of foreclosure on market structure.
7
In the web Appendix, we show that letting the upstream monopoly use quantity-forcing contracts instead of
two-part tariffs yields the same equilibrium allocation.
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We solve for the perfect Bayesian Nash equilibrium that satisfies the standard “passive beliefs” refinement (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 out-of-equilibrium contract offer
it receives. This is a natural restriction on the potential equilibria of a game with secret offers
and supply to order because, from the perspective of the upstream monopolist, under these two
assumptions retailers D1 and D2 form two separate markets (Rey and Tirole, 2007).8
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 knows the terms of U ’s offer to the rival retailer. We also assume that
the downstream affiliate knows the acceptance decision of its rival. However, the outcome would
be identical if the downstream affiliate was not informed about this decision. This is because the
integrated retailer correctly anticipates the equilibrium action of the rival (which is to accept U ’s
offer).
We denote by qim the monopoly quantity produced by retailer Di when it alone 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
The Cournot equilibrium is the solution to the system
q1 = arg max (P(q + q2 ) − w1 − μ1 )q
q
and q2 = arg max (P(q1 + q) − w2 − μ2 )q.
q
Let q1 (w1 , w2 ) and q2 (w2 , w1 ) denote the solution, which is unique given the assumed properties
of inverse demand. For convenience, we denote qi (c, c) by qic . Let
c
− c − μi qic
(1)
πic ≡ P qic + q−i
denote Di ’s equilibrium profit when U sets the unit price uniformly and equal to its marginal cost.
3. Equilibrium analysis
Vertical separation. Suppose neither retailer is vertically integrated. Because retailer D1
is (weakly) more efficient than D2 , U would ideally like to monopolize the product market by
inducing D1 to sell the monopoly output (q1m ). It might seem it can achieve this outcome by
making an unacceptable offer to D2 and offering T1m = {c, π1m } to D1 . However, D1 understands
that, because offers are secret, U ’s offer to D2 is not credible. The reason is that U has an incentive
to sell an additional amount to D2 .9 So D1 would incur a loss if it accepts. Retailer D1 will thus
turn T1m down.
Lemma 1. With passive beliefs, the upstream monopoly (U ) offers, in equilibrium, each retailer
a two-part tariff with the unit price equal to U ’s marginal cost and the fixed component equal to
the retailer’s resulting profit in the ensuing Cournot competition; that is, the equilibrium tariff
offered to retailer Di is Ti = {c, πic }.
8
All our results hold true under the alternative assumption that retailers hold wary beliefs (McAfee and Schwartz,
1994; Rey and Vergé, 2004).
9
This result follows from the observation that given q1 = q1m , q2 = arg maxq (P(q + q1m ) − c − μ2 )q > 0. 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.
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This result is well known. Intuitively, a retailer’s decisions cannot change if U deviates in its
offer to the rival retailer. Therefore, when the monopoly producer contracts with each retailer, it
acts as if the two are integrated. This pairwise maximization problem requires that the contractual
arrangements between U and Di maximize bilateral profits. This entails 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.
Vertical integration. With vertical separation, the equilibrium has retailer D1 producing
q1c and D2 producing q2c . From U ’s perspective, a better profile is q1c + 1 and q2c − 1. Industry
revenues are the same, but costs have fallen, because D1 is more efficient than D2 and, via the
fixed component, U can capture this increase in surplus. Integration is a way for U to implement
a more profitable production profile. We consider two cases: U is integrated with the efficient
retailer (D1 ), or it is integrated with the inefficient retailer (D2 ).
Vertical integration between U and D1 . As shown by Hart and Tirole (1990) and Rey and
Tirole (2007), U internalizes the effect selling to the rival retailer has on the profits made by its
own affiliate. Therefore, the temptation of opportunism vanishes and U can credibly commit to
reducing supplies to the rival retailer. This is the foreclosure effect of vertical integration.
Proposition 1. 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 the monopoly output (q1 = q1m ), retailer D2 is inactive (q2 = 0), and U -D1 obtains
the monopoly profit π1m .10
For U , integration with retailer D1 is more profitable than remaining separate, because it
allows the integrated firm to monopolize the market for the final good.
Vertical integration between U and D2 (the inefficient retailer). We solve for U ’s optimal offer
to its downstream unit D2 and the competing retailer D1 , leading us to Proposition 2.
Proposition 2. Suppose the upstream monopolist, U , is integrated with the inefficient retailer D2 .
The unique equilibrium has firm U -D2 trading the intermediate good internally at marginal cost
(w2 = c) and setting a unit price w1 such that the efficient unintegrated retailer D1 is active on
the market for the final good (q1 (w1 , c) > 0) if and only if D1 is strictly more efficient than D2 ;
that is, iff μ1 < μ2 .
As will be seen, the integrated firm is better off shuttering its inefficient downstream unit.
However, in our setting, U cannot do so credibly.11 Critically, Proposition 2 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 , consider the trade-offs it faces. By raising
the unit price to the competing retailer D1 , U forecloses D1 ’s access to the intermediate good
and monopolizes the market for the final good (the foreclosure effect). However, a countervailing
effect exists in our framework. U benefits if the quantity produced by the efficient retailer D1
increases, at the expense of D2 . How can the upstream firm achieve this, given that it cannot commit to restricting its downstream affiliate’s quantity? As, 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.
10
The proof of Proposition 1 follows the same lines as in Hart and Tirole (1990), and is therefore omitted.
We discuss in the conclusions the motives that lead U not to shut down D2 in models that dispense with the
commitment assumption.
11
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The integrated upstream producer’s problem can be seen as a choice between a market
structure featuring a monopoly and one featuring a particular Stackelberg game. When choosing
monopoly, U ’s choice is constrained insofar as it can make only its own affiliate the downstream
monopolist. When choosing Stackelberg, U acts as a leader by setting w1 , thereby determining
D1 ’s output. The downstream affiliate acts as the follower with marginal cost μ2 + c. Because
U can capture all of the downstream profits via its two-part tariff, unlike a standard Stackelberg
game, the Stackelberg leader chooses D1 ’s output to maximize industry profit, under the constraint
that its inefficient downstream affiliate (the follower) seeks to maximize its own profits.
In equilibrium, D1 is foreclosed if and only if the two retailers are equally efficient (μ1 = μ2 ).
Then, D2 produces the monopoly quantity q2m . At q2m , the marginal revenue of output is equal
to the marginal cost of D2 ; thus, a small reduction in D2 ’s output has no effect on the profit of
U -D2 . If μ1 < μ2 , it is profitable for U -D2 to expand D1 ’s output even though this increases
industry output and therefore reduces industry revenue.12 The reason is that at q2m the marginal
revenue of output exceeds D1 ’s marginal cost of production. The increase in industry output from
the monopoly level yields a second-order loss, but there is a first-order gain from having the
additional output unit produced more efficiently.
Finally, a revealed preference argument shows that for U , a vertical merger with retailer D2
is more profitable than no integration at all. As the Stackelberg structure encompasses Cournot
competition with a common unit price of c, which is the outcome with vertical separation, and
monopoly by D2 , it must be weakly more profitable than either. That it is strictly more so than
Cournot follows because industry profit rises in a Cournot equilibrium if one firm reduces its
output or if output is produced more efficiently. That it is strictly superior to monopoly by D2
follows from the first-order gain due to the production expansion by the more efficient retailer.
A question of importance for competition policy regards the magnitude of the output-shifting
effect. Specifically, can the incentive to reduce w1 be so strong that U offers a unit price below its
marginal cost of production (i.e., w1 < c)? This would render vertical integration procompetitive
because wholesale prices would be lower for one retailer and no greater for the other compared to
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, thus rendering vertical integration procompetitive, as long as the difference between μ2 and μ1
is sufficiently large.
Proposition 3 indicates that below-cost pricing is more likely to occur, the more efficient D1
is relative to D2 (i.e., if μ2 − μ1 is large). The more efficient D1 , the higher the profit increase
that the integrated firm obtains when shifting output to D1 . Therefore, reducing the per-unit price
is particularly valuable to the upstream monopolist.13
The output-shifting effect can 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. As a result, below-cost pricing is more likely to occur when
the demand function is concave, because the unintegrated 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.
To illustrate these results, suppose P(Q) = α − β Q, with β > 0 and α > c + μ2 . Denote
by the difference between retailers’ marginal costs of production, μ2 − μ1 . With vertical
12
Indeed, a reduction in w1 triggers an increase in q1 that is larger than the consequent decrease in q2 , implying
that aggregate output rises.
13
The conditions leading to procompetitive vertical integration are the same in a model in which U competes with
a fringe of less efficient upstream firms. The proof is in the web Appendix.
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FIGURE 1
LINEAR DEMAND EXAMPLE
w1
Δ
0.35
0.5
Δ
0.30
0.4
w1
0.25
Procompetitive vertical integration
0.3
0.20
c
0.15
Δc
0.2
0.10
Anticompetitive vertical integration
0.1
0.05
0.05
0.10
0.15
0.20
μ1
0.05
0.10
Δc
0.15
0.20
Δ
0.25
0.30
Δ
Δ
separation, if D1 and D2 receive the input at marginal cost c, both are active on the final-good
market provided is smaller than ≡ (α − c − μ1 )/2. If U is integrated with the less efficient
≡ (α − c − μ1 )/3 and
retailer D2 , it sets a unit price w1 equal to (α + c + 4μ1 − 5μ2 )/2 if ≤ 2μ2 − μ1 + 2c − α if > , where is the threshold such that D2 is inactive when receiving
the input at w2 = c, given that D1 receives the input at w1 = w1 . In line 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 of below c , with
c < .14 The left panel of Figure 1 plots these conditions using α = β = 1 and c = .25.
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.
The right panel of Figure 1 plots the value of w1 as function of using α = β = 1, c = .25,
and μ1 = .15. If retailers are equally efficient, U sets a unit price such that the unintegrated
retailer’s access to the downstream market is foreclosed. As increases, U reduces the unit price
to induce its downstream affiliate (D2 ) to reduce its output at the advantage of the more efficient
and unintegrated retailer (D1 ). Once becomes so large that, given w1 , D2 remains inactive (i.e.,
), U can raise w1 to limit the distortion of industry output. This shows that w1 changes
if ≥ nonmonotonically in the difference between the costs of D2 and D1 .
4. 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 , q−i ) = α − βqi − γ q−i ,
with i, j = 1, 2 and i = j, where the parameter γ ∈ [0, β) reflects the degree of substitutability
between D1 ’s and D2 ’s products. 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 + μ2 .
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
14
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.
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FIGURE 2
LINEAR DEMAND EXAMPLE
Δ
w1
0.35
0.5
Δd
0.4
Procompetitive vertical integration
c
0.25
Δcd
0.3
w1C
0.30
0.20
0.15
0.2
Anticompetitive vertical integration
0.10
0.1
0.05
0.02
0.04
0.06
0.08
0.10
0.12
0.14
μ1
0.1
0.2
0.3
Δcd
0.4
Δ
Δd
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 finalgood market even in a setting with equally efficient retailers. Indeed, the integrated firm will
generally maintain the rival retailer active, although in a discriminatory way.
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. When U is merged with retailer D2 , we proceed in the same way as in Proposition 2 and
find that the integrated firm sets a unit price equal to
w1C =
(4β 2 + γ 2 )γ (α − c − μ1 − ) + 8β 3 c − 2βγ 2 (2α + c − 2μ1 )
,
2β(4β 2 − 3γ 2 )
which decreases in for all β > γ ≥ 0.15 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 to
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)? In
Figure 2, we illustrate that w1C lies below c for all values of ≥ cd . In the figure, d represents
the threshold below which both firms are active when receiving the intermediate good at marginal
cost. 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.16
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). We therefore follow the literature and modify the third stage of the timing in Section 2 by
15
The value of w1C when the integrated unit is inactive is equal to [γ (α − μ1 ) + 2β(c − α + μ1 + )]/γ , which,
consistent with the results for the case of homogeneous goods (Figure 1) and the intuition developed there, is increasing
in .
16
For the figure, we use the following parameter values: α = 1, β = .6, γ = .35, and c = .25. For the right panel,
we also assume that μ1 = .15.
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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 interdependent from the perspective
of the upstream monopoly: 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 a vertically separated industry, the unit
price offered by the upstream monopolist lies above marginal cost (c).17 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 . As the problem of conjectures does not arise with vertical
integration, we obtain an explicit solution. Proceeding as in Proposition 2,18 we 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 ≥ 0.19
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 expense 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 show that vertical integration can increase 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. 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. 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 expectation.
17
Instead, with passive beliefs, if an equilibrium exists, it leads to a unit price offer equal to marginal cost.
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.
19
The value of w1B when the integrated unit is inactive is [β(2β 2 − γ 2 )(α − c − μ1 − ) − γβ 2 (α − 2c − μ1 ) −
cγ 3 ]/[γ (β + γ )(β − γ )].
18
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Setup with uncertainty. As in our base model in Section 2, the monopoly producer U
deals with two retailers, Dk and Ds . The marginal cost of retailer Dk is known and equal to μ.
Conversely, the marginal cost of retailer Ds is stochastic. Specifically, it can take on two values:
μ + λ with probability ρ, and μ − λ with probability 1 − ρ, with λ > 0. Thus, the expected
value of Ds ’s marginal cost 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 Dk and Ds are equally efficient in
expectation. Instead, Dk is more efficient in expectation than Ds for all values of ρ larger than
1/2.
The game develops in two stages:
I. The monopoly producer U decides whether to merge with retailer Dk or Ds .
I’. Uncertainty over retailer Ds ’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
Lemma 1 regarding U ’s pricing decisions apply. Instead, if U is integrated, the results in Propositions 1, 2, and 3 apply. Finally, the merger decision takes place in stage I, before uncertainty over
Ds ’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 − β Q, with β > 0 and α > c + μ + λ. Moreover, we assume that λ < λ ≡ (α − c − μ)/2, the
threshold below which both retailers are active when receiving the input at marginal cost.
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 Section 3 and
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 (Dk )
or with the one whose marginal cost is stochastic (Ds )? To answer this question, we first consider
the case in which the two retailers are equally efficient in expectation (ρ = 1/2).
Lemma 2. If ρ = 1/2, the monopoly producer integrates with the retailer whose marginal cost
of production is stochastic (Ds ).
One might expect that, given that U captures the industry profit with two-part tariffs, it is
irrelevant which retailer it owns. However, there are two reasons why expected profits differ. The
first is due to the fact that profits are convex in costs. Let us denote by π m (C) the monopoly profit
of a retailer when facing marginal cost of C.20 If the unit prices set by U under integration were
at the foreclosure level, the difference in expected profits between merging with Ds and Dk is
1 m
[π (μ − λ + c) + π m (μ + λ + c)] − π m (μ + c),
2
which is positive by the convexity of profits in costs. This observation alone is not enough to
explain the result in the lemma, because setting the unit price at the foreclosure level is not optimal
if U happens to be integrated with the less efficient firm.
The second reason is related to the market structure that the integrated firm can implement
downstream when it engages in production shifting.21 Assume that λ = λ and that U is merged
with the inefficient firm. Then, if the integrated firm’s subsidiary is Ds , U can serve Dk at a unit
price equal to marginal cost. At this unit price, Ds remains inactive and the integrated firm can
extract the highest possible profits from the downstream market. Instead, if U is integrated with
20
Because π m (C) is equal to maxq {(P(Q) − C)q}, differentiating π m (C) twice with respect to C yields ∂π m /∂C =
−q < 0 and ∂ 2 π m /∂C 2 = −∂q/∂C > 0.
21
If it is merged with the efficient retailer, U can implement the monopoly outcome via foreclosure of the rival
retailer.
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FIGURE 3
LINEAR DEMAND EXAMPLE
1.0
ρ
0.9
0.8
0.7
ρV I
0.6
ρc
0.00
0.05
0.10
0.15
0.20
0.25
λ
λ
Dk , then Dk stays active even if the rival obtains the good at marginal cost. Thus, the integrated
firm cannot extract the monopoly profits in this case.
This asymmetry arises because it is relatively more costly to keep an inefficient downstream
firm inactive when this firm has lower cost in absolute value. Indeed, if U is integrated with an
inefficient Ds , then Ds ’s marginal cost of production is μ + λ + c. If the firm is integrated with
an inefficient Dk , then Dk ’s marginal cost of production is μ + c. In the latter case, U must reduce
the unit price below c to keep its downstream unit inactive when λ = λ. As a consequence, the
distortion of the unit price is higher if U is integrated with Dk rather than Ds .
As these examples illustrate, both forces point in the same direction; that is, to make vertical
integration with Ds more profitable, although in two extreme cases. Lemma 2 shows that the two
forces lead to the same conclusion for all values of λ < λ when ρ = 1/2 and the demand is linear.
What happens when ρ rises above 1/2? Because the merger with Ds is profitable when
ρ = 1/2, by a continuity argument, the same result holds true when ρ is (slightly) larger than
1/2. As ρ increases, it is also more likely that the monopoly producer will be integrated with the
inefficient retailer and engage in output shifting. Can output shifting be so effective that it makes
the vertical merger with Ds profitable and procompetitive with respect to vertical separation?
Proposition 4 addresses this question. The expressions for ρ V I and ρ c are given in the Appendix.
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Proposition 4. If ρ ≥ ρ V I , the monopoly producer (U ) integrates with retailer Dk and the merger
is anticompetitive in expectation. Instead, U integrates with retailer Ds if ρ < ρ V I . The merger
with Ds is procompetitive in expectation if ρ is also larger than ρ c , and anticompetitive otherwise.
Proposition 4 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., ρ c ≤ ρ ≤ ρ V I ). First, for values
of ρ below ρ V I , U finds it more profitable to merge with retailer Ds , which is less efficient in
expectation. Second, for values of ρ above ρ c , 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 4 using a parametric example with α = β = 1 and
c = μ = .25. The shaded area in Figure 3 shows when vertical integration with Ds is profitable
and procompetitive in expectation. This can happen even for high values of ρ, and the intuition
relies on the same insights developed below Lemma 2. If, for instance, ρ is close to 1 and λ is
close to λ, then U -Ds can shift output to Dk at a unit price close to marginal cost. This allows the
integrated firm to implement an outcome close to the monopoly one on the downstream market.
This is more profitable than merging with Dk , because in that case shifting output to Ds requires
reducing the unit price below marginal cost (if Ds happens to be more efficient). Note that these
results are not unique to a setting with linear demand. For example, vertical integration with Ds
is procompetitive and more profitable than a merger with Dk for values of ρ between 0.59 and 0.7
when using the demand function P(Q) = α − Q β together with the following parameter values:
α = 1, β = 2, c = .2, μ = .3, and λ = .2.
6. Conclusions
This article examines a standard model in which a monopoly producer deals with competing
retailers by means of secret two-part tariffs. We show that a crucial element in any such analysis
is whether the retailers (downstream firms) have 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 it offers to the unintegrated
but more efficient retailer. This output-shifting effect makes vertical integration procompetitive
when compared to vertical separation if the unintegrated 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.
Our results are robust to a setting with differentiated products. 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.
A crucial assumption in our setting is that the monopoly producer cannot commit to shutter its
integrated less efficient retailer. However, there are various reasons why an integrate entity might
not want to do so. For example, if the efficient unintegrated retailer has some bargaining power,
the integrated firm wants to keep its downstream affiliate active to improve its bargaining position
vis-à-vis the independent retailer.22 Another reason could be that the downstream subsidiary
allows the monopoly producer to capture valuable information about demand conditions, or that
it allows the upstream producer to sell other product lines.
22
In the web Appendix, we provide a formal analysis of this situation, using a bargaining game along the lines of
O’Brien and Shaffer (2005).
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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 article to a framework where opportunities
for vertical mergers arise dynamically (e.g., along the lines of Nocke and Whinston, 2010). This
would allow to further study the conditions under which vertical integration is procompetitive.
Appendix: The Appendix contains the proofs of Lemma 1–2 and Propositions
2–4.
Proof of Proposition 1. We solve the game by backward induction. In the last stage, retailer Di produces qi (wi , w−i )
as defined by (1). Accordingly, one-to-one production technology implies that Di orders qi (wi , w−i ) from the monopoly
producer U .
We now determine U ’s tariffs. With passive beliefs, the equilibrium contract offered by U to each retailer Di
must maximize their joint profits (McAfee and Schwartz, 1994). Therefore, U ’s first-stage maximization problem 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)
(wi − c)
∂qi
∂qi
+ qi − qi = (wi − c)
= 0.
∂wi
∂wi
Because ∂qi /∂wi < 0, at the equilibrium wi = c. 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.
Proof of Proposition 2. Note first, that when D2 is integrated with U , its output q2 (w2 , w1 ) is a strictly monotone function
in w2 for any w1 . It follows that U -D2 ’s choosing w2 is isomorphic to its choosing q2 . The integrated firm chooses q2 to
maximize its profit given q1 :
max (P(q1 + q) − μ2 − c) q.
q
The optimal solution results from the first-order condition P(Q) − μ2 − c + P (Q)q2 = 0.23 Denote this solution by
q2 (q1 ). Applying the Implicit Function Theorem yields
dq2
P (Q) + P (Q)q2
1
,
−1
,
(A1)
=− ∈
−
dq1
2P (Q) + P (Q)q2
2
implying that q2 (·) is a strictly decreasing function. Technically, w2 is indeterminant. However, if D2 decides about q2 to
maximize its own profit, it requires the instruction by U that w2 = c.
We now turn to the optimal output choice of D1 . This choice is implicitly defined by P(Q) − μ1 − w1 + P (Q)q1 =
0. It follows that q1 (w1 , c) is strictly decreasing in w1 , and we can define the inverse function ŵ1 (q1 ) as the w1 that solves
q1 = q1 (w1 , c). Then, the optimization problem of U -D2 when dealing with retailer D1 can be written as
max F1 + (ŵ1 (q1 ) − c) q1 + P q1 + q2 (q1 ) − μ2 − c q2 (q1 ),
q1
subject to
F1 ≤ P q1 + q2 (q1 ) − μ1 − ŵ1 (q1 ) q1 .
(A2)
At equilibrium, U -D2 formulates a take-it-or-leave-it offer to fully extract retailer D1 ’s profit. Thus, the constraint in (A2)
is binding, and we can rewrite the optimization program of U -D2 as
P q1 + q2 (q1 ) − μ1 − ŵ1 (q1 ) q1 + (ŵ1 (q1 ) − c) q1 + P q1 + q2 (q1 ) − μ2 − c q2 (q1 ).
max
q1
23
Because of our assumption P (Q) + P Q < 0, the second-order condition is satisfied.
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This can be rewritten as q1 chosen to maximize
P q1 + q2 (q1 ) − c q1 + q2 (q1 ) − μ1 q1 − μ2 q2 (q1 ),
which is industry profit. Using the Envelope Theorem, the first-order condition is
dq P q1 + q2 (q1 ) q1 1 + 2 + μ2 − μ1 = 0.
dq1
(A3)
We show below that the second-order condition is satisfied. As dq2 /dq1 ∈ [−1/2, −1), it follows that
1 + dq2 /dq1 > 0. Hence, (A3) yields q1 > 0 if and only if μ2 > μ1 .
We now turn to the second-order condition. Taking the derivative of the first-order condition (A3) with respect to
q1 yields
d 2 q2
dq dq P (q1 + q2 (q1 ))q1 1 + 2 + P (q1 + q2 (q1 )) + P (q1 + q2 (q1 ))q1
1+ 2
,
(A4)
dq1
dq1
dq12
where d 2 q2 /dq12 , as determined from (A1), is equal to
P (Q ) 2q2 (q1 )(P (Q ))2 + P (Q )(P (Q ) − q2 (q1 )P (Q ))
d 2 q2
=
,
dq12
(2P (Q ) + P (Q )q2 (q1 ))3
with Q ≡ q1 + q2 (q1 ). Inserting the last equation into (A4), and ignoring functional arguments, yields
(P )2 4(P )2 + (P )2 q2 (3q1 + q2 ) + P (3P q1 + 4P q2 − P q1 q2 )
.
(2P + P q2 )3
(A5)
Because P + P Q < 0, the denominator is negative. The numerator is positive as long as P is positive or not too
negative, which is our working assumption. As a consequence, the whole expression in (A5) is negative, implying that
the second-order condition is satisfied.
Proof of Proposition 3. To determine w1 , we use the first-order conditions for q1 and q2 . They imply q1 = −(P(Q ) −
μ1 − w1 )/P (Q ) and q2 = −(P(Q ) − μ2 − c)/P (Q ). Plugging these expressions together with dq2 /dq1 into (A3)
and rearranging yields
w1 = P(Q ) − 2μ2 + μ1 +
P (Q ) (μ2 − μ1 ) (P(Q ) − c − μ2 )
.
(P (Q ))2
(A6)
Then, w1 < c if and only if
μ2 − μ1 >
(P(Q ) − μ2 − c)(P (Q ))2
.
(P (Q ))2 − (P(Q ) − μ2 − c)P (Q )
If w1 is sufficiently smaller than c, the value of q2 is equal to zero. In this case, U sets w1 such that its downstream
affiliate is inactive. Therefore, the first-order conditions in the downstream market are P(q1 ) − μ2 − c = 0 and P(q1 ) −
μ1 − w1 + P (q1 )q1 = 0, yielding w1 = μ2 − μ1 + c + P (q1 )q1 < c.
Finally, note that changes in the aggregate output Q are due only to changes in the per-unit price of D1 , because
D2 obtains the intermediate good at a per-unit price of c both with vertical separation and vertical integration. Using the
first-order conditions for the downstream quantities q1 and q2 to determine ∂q1 /∂w1 and ∂q2 /∂w1 yields
∂ Q
∂q1
∂q2
1
< 0.
=
+
=
∂w1
∂w1
∂w1
3P (Q ) + Q P (Q )
Because ∂ Q /∂w1 < 0, aggregate output is larger under vertical integration than under vertical separation if and only if
w1 < c.
Proof of Lemma 2. Note that our assumption that firms are both active when receiving the input at marginal cost implies
that λ < λ ≡ (α − c − μ)/2. So we assume that this condition holds in this proof. To begin with, we determine the profits
of the vertically integrated firm when U merges with Dk , the retailer whose marginal cost of production is certain and
equal to μ.
U merges with Dk . With probability ρ, retailer Dk is the more efficient retailer. Thus, the results in Proposition 1
apply: the upstream unit of the integrated firm forecloses retailer Ds ’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 -Dk in this state are equal to (α − c − μ)/2β and (α − c − μ)2 /4β, respectively.
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With probability 1 − ρ, retailer Ds has the lower marginal cost of production μ − λ < μ. U sets the internal trading
price equal to its marginal cost of production (c) and a unit price ws as in (A6) as long as Dk is active. Invoking linear
demand yields
ws =
α + c − μ − 4λ
.
2
The quantity produced by retailer Dk (qk ) and Ds (qs ) at equilibrium are
qk =
α − c − μ − 2λ
2β
and qs =
2λ
.
β
The quantity of the integrated firm’s downstream unit (qk ) is positive for all λ < λ. Then, the expected profits of U -Dk
I
(UV −D
) are given by
k
ρ
(α − c − μ)2 + 4λ2
(α − c − μ)2 + 4(1 − ρ)λ2
(α − c − μ)2
+ (1 − ρ)
=
.
4β
4β
4β
For ρ = 1/2, this expression reduces to [(α − c − μ)2 + 2λ2 ]/4β.
U merges with Ds . Let U be integrated with retailer Ds . With probability ρ, retailer Ds is less efficient than Dk . In
this case, production shifting occurs. If Ds is active, wk = (α + c − μ − 5λ)/2 and the resulting equilibrium quantities
are
qk =
2λ
β
and qs =
α − c − μ − 3λ
.
2β
Then, the profit of U -Ds is
(α − c − μ)2 − 2λ(α − c − μ) + 5λ2
.
4β
λ ≡ (α − c − μ)/3. For λ < λ < λ, the value of wk is 2c − α + 2λ + μ.
The value of qs is positive if and only if λ ≤ This gives equilibrium quantities of
qk =
α−c−μ−λ
β
and qs = 0,
and U -Ds ’s profit equal to
λ(α − c − μ − λ)
.
β
With probability 1 − ρ, retailer Ds is more efficient. Thus, the upstream unit of the integrated firm forecloses
Dk ’s access to the intermediate good, and the downstream unit produces the monopoly quantity. Accordingly, the
equilibrium value of Ds ’s quantity and the profit of the integrated firm U -Ds are equal to (α − c − μ + λ)/2β and
(α − c − μ + λ)2 /4β, respectively.
I
In sum, the expected profits of U -Ds (UV −D
) are
s
(α − c − μ + λ)2 − 4λρ(α − c − μ − λ)
4β
if λ ≤ λ, and
(α − c − μ + λ)2 − [(α − c − μ)(α − c − 2λ − μ) + 5λ2 ]ρ
4β
for λ ∈ (
λ, λ).
If ρ = 1/2, these expressions reduce to
(α − c − μ)2 + 3λ2 /4β
(α − c − μ)(α − c − μ + 6λ) − 3λ2 /8β
if
λ ≤
λ,
if
λ ∈ (
λ, λ).
I
I
At ρ = 1/2, the difference between UV −D
and UV −D
is strictly positive for all values of λ ∈ (0, λ), which establishes
s
k
the claim in the proposition.
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Proof of Proposition 4. We start by determining the threshold above which a merger with Dk is more profitable than a
I
I
merger with Ds . Using the values of UV −D
and UV −D
obtained in the proof of Lemma 2, we find that
s
k
⎧
λ [2(α − c − μ − 2λ) + λ − 4ρ(α − c − μ − 2λ)]
⎪
⎪
if λ ≤ λ,
⎪
⎨
4β
VI
VI
U −Ds − U −Dk =
⎪
(1 − ρ)[(α − c − μ + λ)2 − 4λ2 ] − (α − c − μ)2 + 4λρ(α − c − μ − λ)
⎪
⎪
⎩
if λ ∈ (
λ, λ).
4β
I
I
≥ UV −D
if and only if
It follows that UV −D
s
k
⎧
λ
1
⎪
⎪
⎪ +
⎨
2
4(α
−
c
−
μ − 2λ)
ρ ≥ ρV I ≡
⎪
λ[2(α
−
c
−
μ)
− 3λ]
⎪
⎪
⎩
(α − c − μ − λ)2
if
λ ≤
λ,
if
λ ∈ (
λ, λ).
We next analyze whether vertical integration between U and Dk is procompetitive. Determining the aggregate expected
quantity under vertical separation and under vertical integration between U and Dk , we obtain
QV S =
2α − 2c − 2μ + λ(1 − 2ρ)
3β
and
I
Q UV −D
=
k
α − c − μ + 2λ(1 − ρ)
.
2β
I
and Q V S is equal to
The difference between Q UV −D
k
4λ − (α − c − μ) − 2λρ
,
6β
I
− Q V S ≥ 0 if and only if
implying that Q UV −D
k
ρ ≤ ρc ≡
α−c−μ
4λ − (α − c − μ)
=2−
.
2λ
2λ
If ρ c were to lie above ρ V I , then there would be values of ρ such that integration between U and Dk is profitable and
procompetitive. We find that
ρc < ρV I
∀λ < λ,
implying that the merger between U and Dk is anticompetitive for all values of ρ ∈ [1/2, 1).
Finally, we establish the condition such that the merger between U and Ds is procompetitive. The aggregate expected
quantity when U is integrated with Ds is equal to
⎧ α−c−μ+λ
⎪
if λ ≤ λ,
⎪
⎨
2β
I
Q UV −D
=
s
⎪
⎪
⎩ α + λ − (1 + ρ)(μ + c) + ρ(α − 3λ)
if λ ∈ (
λ, λ).
2β
I
and Q V S is positive if and only if
Then, the difference between Q UV −D
s
⎧
α−c−μ−λ
⎪
⎪
⎨
4λ
c
ρ≥ρ ≡
⎪
α−c−μ−λ
⎪
⎩
3(α − c − μ − λ) − 2λ
if
λ ≤
λ,
if
λ ∈ (
λ, λ).
It is easy to show that ρ c ≤ ρ V I for all values of λ such that
√
1
(5 − 5)(α − c − μ), λ .
λ∈
10
Therefore, the merger between U and Ds is procompetitive if ρ lies in the interval [ρ c , ρ V I ).
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