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International Journal of Industrial Organization
26 (2008) 1 – 16
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Outsourcing, vertical integration, and price vs.
quantity competition ☆
Anil Arya a , Brian Mittendorf b , David E.M. Sappington c,⁎
a
b
The Ohio State University, United States
Yale School of Management, United States
c
University of Florida, United States
Received 12 December 2005; received in revised form 2 October 2006; accepted 14 October 2006
Available online 28 November 2006
Abstract
We show that standard conclusions about duopoly competition can be reversed when the production of
key inputs is outsourced to a vertically integrated retail competitor with upstream market power. Under such
outsourcing, Bertrand competition can produce higher prices, higher industry profit, lower consumer
surplus, and lower total surplus than Cournot competition. In addition to limiting the intensity of retail
competition, Bertrand competition can limit the extent of wholesale competition by reducing the incentive
of retail providers to produce key inputs themselves.
© 2006 Published by Elsevier B.V.
JEL classification: D43; L13; L40
Keywords: Competition; Outsourcing; Vertical integration
1. Introduction
Outsourcing the production of key inputs to external suppliers is ubiquitous in today's
economy, and outsourcing to retail competitors is common in many important industries. For
example, in the telecommunications industry, vertically integrated incumbent operators routinely
supply key inputs (e.g., telephone loops)1 to retail competitors. In addition, soft-drink producers,
☆
We thank the co-editor, Roman Inderst, and two anonymous referees for very helpful comments.
⁎ Corresponding author.
E-mail address: [email protected] (D.E.M. Sappington).
1
Telephone loops are the wires that connect a telephone customer's residence to the central office of a
telecommunications supplier.
0167-7187/$ - see front matter © 2006 Published by Elsevier B.V.
doi:10.1016/j.ijindorg.2006.10.006
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A. Arya et al. / Int. J. Ind. Organ. 26 (2008) 1–16
cereal manufacturers, and gasoline refiners have long supplied key inputs both to their
downstream affiliates and to retail competitors. More recently, the explosion in online commerce
has brought manufacturers into direct competition with their own retailers.
We demonstrate that standard conclusions about price and quantity competition (e.g., Singh and
Vives, 1984) can be altered when the production of inputs is outsourced to retail rivals. We show that
when the supplier of an input is also a retail rival, the vertically integrated producer (VIP) may set a
higher input price under Bertrand competition than under Cournot competition. The higher input
price is not designed to limit the competitive strength of the VIP’s retail rival or to drive the rival from
the market. Rather, the higher input price increases the VIP's opportunity cost of aggressive retail
price competition by increasing the wholesale profit the VIP foregoes when the retail output of its
competitor declines. The higher opportunity cost serves as a credible commitment on the part of the
VIP to refrain from aggressive retail competition. The higher retail prices that ensue can generate a
lower level of consumer surplus under Bertrand competition than under Cournot competition, which
reverses the standard conclusion regarding the welfare implications of price and quantity competition.
Our demonstration of this conclusion and related observations proceeds as follows. Section 2
describes the key elements of our baseline model, in which a VIP is the monopoly supplier of an
essential input to a non-integrated retail rival. Section 3 demonstrates that retail prices and
industry profit are higher while consumer surplus and total surplus are lower under Bertrand
competition than under Cournot competition in this setting.
Section 4 demonstrates that the industry structure presumed in the baseline model can emerge
naturally as the equilibrium of a simple game. The analysis in Section 4 reveals that a more
efficient retail competitor will outbid its less efficient counterpart for the right to merge with a
monopoly input supplier. The resulting vertical integration increases both consumer surplus and
total surplus. The increase is more pronounced under Cournot retail competition than under
Bertrand competition because of the more intense competition that prevails in the former regime.
Section 5 extends the model to admit upstream competition. Entrants are presumed able to
achieve the same upstream cost structure as the VIP by incurring an investment I N 0. Intense
upstream pricing competition deters non-integrated firms from entering the input market.
However, if I is sufficiently small, the retail rival will undertake the investment in order to ensure
low-cost access to the input. The retail rival is not willing to pay as much under Bertrand as under
Cournot competition to secure such access because of the less intense retail competition that
prevails under Bertrand competition when production is outsourced to the VIP. As a result, the
decision to seek alternate input sources may actually be less prevalent when external input prices
are high (as under Bertrand competition) than when they are low (as under Cournot competition).
Thus, in addition to reducing the intensity of retail competition, Bertrand competition can limit
the extent of wholesale competition.
As noted above, our analysis extends the extensive literature that compares outcomes under
Bertrand and Cournot competition in standard duopoly settings (e.g., Singh and Vives, 1984;
Okuguchi, 1987; Vives, 2005) by examining the role of outsourcing to a rival. As such, our
analysis complements the standard literature on outsourcing (e.g., Shy and Stenbacka, 2003; Van
Long, 2005) by analyzing outsourcing to a vertically integrated retail rival. Kamien et al. (1989),
Spiegel (1993), and Baake et al. (1999), among others, consider outsourcing (or subcontracting)
to a rival. In contrast to our model, these analyses focus on the effects of nonlinear cost structures
and do not compare outcomes under alternative forms of retail competition.
We also contribute to the literature on vertical integration (e.g., Salinger, 1988; Hart and Tirole, 1990;
Ordover et al., 1990) by analyzing the incentives for and the effects of vertical integration in the presence
of monopoly supply of an essential input. Biglaiser and DeGraba (2001) also consider monopoly supply
A. Arya et al. / Int. J. Ind. Organ. 26 (2008) 1–16
3
of an input, but assume the input price is regulated. Consequently, the equilibrium changes in input
prices that are a central feature of vertical integration in our model do not arise in their model.
Chen's (2001) analysis of vertical integration may be closest to our own in that it identifies the
role of high input prices in relaxing downstream price competition. Other authors have noted that
wholesale profit margins can affect the intensity of retail price competition. For example, Armstrong
(1998) and Laffont et al. (1998a,b) do so in a setting where network operators set a reciprocal
interconnection fee; Armstrong (1999) and Harbord and Ottaviani (2002) do so in a setting where
cable television operators can sell programming to competitors; and Sappington (2005) does so in a
setting where entrants decide whether to make an essential input themselves or purchase the input
from a vertically-integrated incumbent supplier. In contrast to the present analysis, these studies do
not contrast outcomes and welfare implications of Cournot and Bertrand competition.
2. The baseline model
We modify the classic model of duopoly competition by allowing one of the retail competitors
(firm 1) to be a VIP. In the baseline model of primary interest, firm 1 is the sole producer of an
input that is essential for retail production. Each unit of retail output requires exactly one unit of
the input. Firm 1 charges its retail rival, firm 2, unit price w for the input. We normalize firm 1's
upstream cost to zero. After securing the required input supply, firm i's incremental cost of
producing and selling its retail product is ci (i = 1,2).
Consumer demand for the retail product of firm i is given by the (inverse) demand function
pi = α − qi − γqj, where pi is the price of firm i's retail product, α is a strictly positive constant, and
qi and qj are the outputs of firms i and j, respectively (i, j ∈ {1,2}, i ≠ j). The parameter γ ∈ (0,1)
represents the degree of product homogeneity. As γ approaches 0, the products of the two retail
providers become independent. As γ approaches 1, the products of the firms become completely
homogeneous.
The timing in the model is as follows. First, firm 1 sets the input price it will charge to firm 2.
Then, under Cournot competition, firms 1 and 2 choose their retail output levels simultaneously
and independently. Under Bertrand competition, the two firms set retail prices simultaneously and
independently. Finally, under both forms of retail competition, consumers make their purchase
decisions and all realized consumer demand is satisfied.
The profits of firms 1 and 2 when firm i produces retail output qi, firm i's retail price is pi, and
the input price is w are, respectively:
P1 ¼ wq2 þ ½ p1 −c1 q1 ; and
ð1Þ
P2 ¼ ½ p2 −w−c2 q2 :
ð2Þ
The first term to the right of the equality in Eq. (1) captures firm 1's profit from selling the
input to firm 2. The second term reflects firm 1's profit from its retail sales, just as Eq. (2) reflects
firm 2's profit from its retail sales. Retail profit is the product of sales volume and the relevant
retail profit margin. Firm 1's retail profit margin is the difference between the price ( p1) it charges
for its retail product and its downstream production cost (c1). Firm 2's profit margin is the
difference between the price of its product ( p2) and the sum of its input cost (w) and incremental
downstream production cost (c2).
Consumer surplus given retail outputs q1 and q2 is:
CS ¼ ð½q1 2 þ 2cq1 q2 þ ½q2 2 Þ=2:
ð3Þ
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The quadratic expression in Eq. (3) arises from the linear retail demand. Total surplus is the
sum of CS, Π1, and Π2.
Ensuing calculations are simplified by introducing the parameters α1 ≡ α − c1 and α2 ≡ α − c2.
In words, αi is the difference between the intercept of firm i's inverse demand curve and its
downstream marginal cost of production. The larger is αi, the more efficient is firm i in its retail
operations. We assume α1 ≥ α2, so the VIP (firm 1) is at least as efficient a retail provider as its
rival (firm 2). The analysis in Section 4 demonstrates that this industry structure arises
endogenously as the equilibrium of a simple game.
3. Outcomes in the baseline model
Because firm 1 is the only supplier of the essential input in the baseline model, firm 1 has the
ability to foreclose firm 2 (i.e., to set the input price so high that firm 2 finds it unprofitable to
participate in the retail market). Firm 1 can enhance its success in the retail market by foreclosing
its only retail competitor. However, foreclosure also terminates firm 1's sale of the input to firm 2
and thereby eliminates firm 1's profit from wholesale operations. When the potential for
wholesale profit is sufficiently large, firm 1 will not foreclose firm 2. As Lemma 1 indicates, firm
1 will prefer to maintain firm 2 as an active buyer of the input rather than foreclose firm 2 under
both Bertrand and Cournot retail competition when firm 2 is a sufficiently efficient competitor
(i.e., when α2 / α1 N γ).
Lemma 1. Under both Bertrand and Cournot retail competition, firm 1 forecloses firm 2 (i.e.,
q2 = 0) if and only if α2 / α1 ≤ γ.
Lemma 1 reveals that foreclosure of firm 2 is less likely when the firms' retail products are
more heterogeneous (i.e., when γ is smaller). Greater product heterogeneity implies that firm 1's
retail profit declines less rapidly as firm 2 enjoys greater retail success. Consequently, because
firm 2's retail success secures greater wholesale profit for firm 1, firm 1 is more willing to support
firm 2's operation.
If firm 2 is foreclosed, the setting reduces to one where a vertically integrated monopolist supplies
a retail market. Because our primary purpose is to study the competitive interactions between a
vertically integrated producer and its retail rival, we will abstract from the foreclosure setting in the
ensuing discussion by assuming that the following non-foreclosure (NF) condition holds:
a2 =a1 Nc:
ðNFÞ
In the absence of foreclosure, firm 1 finds a high input price particularly appealing under
Bertrand competition. A high input price increases firm 1's wholesale profit margin. The greater
wholesale profit margin increases the (opportunity) cost that firm 1 incurs if it reduces firm 2's
demand for the input by limiting firm 2's retail success. The opportunity cost of competing
vigorously in the retail market endows firm 1 with a credible commitment to set a higher retail
price. Because prices are strategic complements under Bertrand competition, firm 1's credible
commitment to set a higher retail price induces firm 2 to set a higher retail price. Firm 1 benefits
from the increased demand for its retail product that results from firm 2's higher price. Firm 1 also
gains from the higher retail price that it sets because the higher price increases firm 2's retail
output and thereby increases firm 1 wholesale profit by increasing firm 2's demand for the input.
Firm 1 does not experience a corresponding commitment benefit under Cournot competition.
When firms set retail quantities rather than prices, firm 1 takes the demand for its input (q2) as
A. Arya et al. / Int. J. Ind. Organ. 26 (2008) 1–16
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given when it chooses its retail output. Therefore, firm 1 does not perceive that variations in its
retail output will affect its wholesale profit under Cournot retail competition. Consequently, firm 1
will set a higher input price under Bertrand competition than under Cournot competition, as
Lemma 2 states.
Lemma
2. Given
(NF), firm 1 sets a higher
price
input
under Bertrand competition
2
½a2 −ca1 a2 c2 ½a2 −ca1 than under Cournot competition a22 − c2½8−3c
.
2
2 − 2½8þc2 The higher input price and reduced intensity of retail competition under Bertrand competition
reverses the standard welfare comparisons of Bertrand and Cournot competition. As long as firm
2 is sufficiently efficient that it is not foreclosed by firm 1, both firms secure greater profit under
Bertrand competition than under Cournot competition. In contrast, consumer surplus is higher
under Cournot competition because it does not provide firm 1 with an incentive to raise the input
price and thereby reduce the intensity of retail competition. The loss consumers suffer under
Bertrand competition (relative to Cournot competition) outweighs the corresponding increase in
industry profit. Consequently, total surplus is lower under Bertrand than under Cournot
competition, as Proposition 1 reports.
Proposition 1. Given (NF): (i) both firms secure greater profit under Bertrand competition than
under Cournot competition; and (ii) consumer surplus and total surplus are both lower under
Bertrand competition than under Cournot competition.
In summary, standard conclusions regarding the effects of price and quantity competition can
be reversed when a retail provider purchases an essential input from a vertically integrated retail
rival. In particular, price competition can be less intense than quantity competition in the presence
of such outsourcing.
Before proceeding to consider extensions of this baseline model, we briefly discuss three natural
modeling variations. First, the key qualitative conclusions drawn above persist when firm 1 does
not have unrestricted ability to set its preferred input price. In particular, the equilibrium input price
continues to be higher under Bertrand competition than under Cournot competition when the input
price that firm 1 charges to firm 2 is determined by Nash bargaining, where the disagreement
outcome is that in which firm 1 is the monopoly supplier of the retail product. Also, it can be shown
that firm 1 and firm 2 both secure higher profit while consumer surplus is lower under Bertrand
competition than under Cournot competition in this setting with Nash bargaining.2
Second, the main qualitative conclusions drawn above also can arise in the presence of
nonlinear pricing. When nonlinear pricing of the input is feasible and the upstream VIP has all of
the relevant bargaining power, it can foreclose the least efficient retail provider and extract the full
monopoly retail surplus. Such a policy can be optimal when the retail products are sufficiently
homogeneous (Chemla, 2003), much as foreclosure arises in the baseline model when γ is
sufficiently large (recall Lemma 1). However, as long as the firms' retail products are sufficiently
heterogeneous, the integrated supplier will sell the input to its rival and set the marginal price in a
two-part input tariff above marginal cost. Furthermore, this marginal price will be higher under
Bertrand competition than under Cournot competition. In this case, the higher marginal price
under Bertrand competition translates into higher profit for the firms, lower consumer surplus,
and lower total surplus under Bertrand competition than under Cournot competition.
2
See Chemla (2003), Inderst and Wey (2003), and de Fontenay and Gans (2004, 2005), for example, for related models
of bargaining between upstream and downstream firms.
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Finally, we note that Proposition 1 is derived in a setting in which the duopolists' products are
substitutes. When the products are independent, the two retailers effectively do not compete.
Consequently, the input price plays no meaningful role in reducing the intensity of retail
competition. When the duopolists' products are complements and the firms engage in Bertrand
competition, the VIP realizes that by reducing the price of its retail product it can enhance the
demand for its input by increasing the output of the non-integrated retail rival. The lower
equilibrium prices that emerge under Bertrand competition secure a higher level of consumer
surplus than arises under Cournot competition.
4. Endogenous market structure
The analysis to this point has taken the industry structure as given. We now demonstrate that the
industry structure in the baseline model can arise naturally as the equilibrium of a simple game.
The game we consider proceeds as follows. Initially, an independent upstream supplier (U)
announces that it will merge with the retail operator that bids the most for this opportunity, provided
the winning bid exceeds the maximum profit (Π̄U) U can secure under vertical separation. The two
independent retail operators, D1 and D2, each announce simultaneously the amount they are
willing to pay to merge with U. If at least one of the announced bids exceeds Π̄U, U accepts the
highest bid and the merger of U and the highest retail bidder is consummated (costlessly).3 If
neither bid exceeds Π̄U, no merger takes place, and vertical separation prevails. Under vertical
separation, U sets the profit-maximizing prices at which it will sell the input to each of the nonintegrated firms, D1 and D2. The two retail providers then compete to serve customers.
Lemma 3 reports that under vertical separation, U charges the same input prices regardless of
the form of the prevailing downstream competition.
Lemma 3. Given (NF) and vertical separation, the independent upstream supplier sets its input
price for Di equal to αi / 2 for i = 1,2 under both Bertrand and Cournot retail competition.
It is well known that, holding constant the cost structures of the retail competitors, Bertrand
competition produces lower retail prices than Cournot competition and thereby secures greater
consumer surplus and total surplus while reducing industry profit (e.g., Singh and Vives, 1984).
Lemma 3 states that the form of retail competition (Bertrand vs. Cournot) does not affect the cost
structures of the retail competitors under vertical separation in the present setting. Hence, the
standard welfare comparisons emerge if the retail rivals procure inputs from an independent
wholesale producer.
However, in part because it eliminates the double marginalization problem,4 vertical
integration increases the joint profit of the merging parties. Therefore, vertical integration, not
vertical separation, will arise in equilibrium. Because D1 has a lower downstream cost than D2
(i.e., α1 ≥ α2), D1 will produce more retail output than D2 under both Bertrand and Cournot
competition if the two firms incur the same input costs. Because of its relatively large retail
3
Following the merger of U and Di, the combined entity acts to maximize the profit from its wholesale and retail
operations, just as in the baseline model. The entire profit earned by the vertically integrated firm accrues to Di. Residual
rights of control are not an issue in the simple game analyzed here because all relevant industry information is common
knowledge and there are no contracting frictions. Bolton and Whinston (1993) and Heavner (2004), among others,
provide interesting analyses of vertical integration in which non-trivial residual control rights arise.
4
A double marginalization problem arises when a retailer pays more for an input than the supplier's marginal cost of
production and then raises its retail price above its own (already inflated) marginal cost of production to a level that is
higher than that which maximizes the joint profits of the retailer and the supplier.
A. Arya et al. / Int. J. Ind. Organ. 26 (2008) 1–16
7
output, D1 derives a greater increase in profit than D2 from the ability to secure the input at
marginal cost. Consequently, as Proposition 2 reports, D1 will bid more than D2 for the right
to merge with U.
Proposition 2. Given (NF), D1 successfully outbids D2 for the right to merge with U under both
Bertrand and Cournot competition.
Other authors have reported similar conclusions. For example, Armstrong (1999) provides a
corresponding finding in a setting where an independent supplier markets a premium product to
retailers of a basic product. Armstrong demonstrates that the conclusion persists under a variety of
allocation procedures that admit different distributions of bargaining power.5 Chen (2001)
considers mergers between a retail provider and multiple upstream producers. He finds that if a
merger occurs, it will involve the most efficient upstream producer. Such a merger will occur if
and only if the production cost of the most efficient upstream producer is strictly less than the
production costs of the other upstream producers.
The equilibrium merger between the upstream producer and the most efficient retail operator in
the present setting helps to justify our focus on the baseline model. After U and D1 merge, the
combined entity (the VIP) will sell the essential input to D2 as long as D2 is sufficiently efficient
(i.e., as long as (NF) holds). Therefore, Proposition 2 implies that the outsourcing to a more
efficient vertically integrated rival that was assumed to occur in the baseline model arises
naturally as the equilibrium of a simple game that admits vertical mergers.
Proposition 3 reports welfare implications of the merger between U and D1.
Proposition 3. Given (NF), the merger of U and D1 increases consumer surplus and total
surplus under both Bertrand competition and Cournot competition. The increases are most
pronounced under Cournot competition.
The merger of U and D1 increases surplus under both Bertrand and Cournot competition for
two primary reasons. First, the merged entity avoids the double marginalization problem because
the retail division of the VIP receives the essential input at cost. Second and perhaps more
surprisingly, the VIP sets the input price for D2 below the corresponding price that D2 faces
under vertical separation. One might suspect the VIP would charge its retail rival a particularly
high price for the input in order to limit the rival’s competitive strength. However, as Lemmas 2
and 3 indicate, this is not the case in the absence of foreclosure. The VIP charges D2 a relatively
low input price in order to partially offset the advantage the retail division of the VIP secures from
obtaining the input at cost. The increase in D2’s competitive strength increases its retail sales and
thereby increases D2's retail sales and so increases D2's purchase of the input from the VIP. The
resulting increase in the VIP's wholesale profit outweighs the decline in its retail profit caused by
the input price reduction.6
The lower input prices that both retailers face under vertical integration lead to an increase in
both consumer surplus and total surplus, as Proposition 3 reports. It can also be shown that the
merger of U and D1 produces greater total surplus than a merger of U and D2 would generate.
5
In Armstrong's analysis, the retailer that acquires the premium content does not sub-license the content to its retail
rival. Harbord and Ottaviani (2002) show that sub-licensing will occur if per-subscriber license fees are admitted because
such fees reduce the intensity of retail price competition and thereby enhance industry profit.
6
Although D2 enjoys a lower input price under vertical integration, it also faces a more formidable competitor that
secures the input at cost. The latter drawback dominates the former benefit, leaving D2 with lower profit under vertical
integration than under vertical separation.
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Therefore, in pursuing their own private interests, the industry participants secure a welfaremaximizing outcome.
The larger increases in consumer surplus and total surplus that arise under Cournot
competition reflect two factors. First, the integrated supplier sets a higher input price under
Bertrand competition than under Cournot competition (recall Lemma 2). As explained above, the
higher input price helps the VIP establish a credible commitment to compete less vigorously
against D2. This high input price introduces a relatively severe double marginalization problem,
which reduces welfare. Second, because the VIP is able to establish a credible commitment to
compete less vigorously and because D2 faces a relatively high input price, retail prices are
relatively high under Bertrand competition. The resulting decline in consumer surplus below the
level achieved under Cournot competition outweighs the corresponding increase in industry
profit. Consequently, the increases in surplus from vertical integration are less pronounced under
Bertrand competition than under Cournot competition.
Together, Lemma 3 and Proposition 3 reveal that the identified reversal of the standard welfare
conclusions about Bertrand and Cournot competition is not due simply to the presence of
outsourcing. Outsourcing is present under both vertical separation and vertical integration, and
the standard conclusions prevail under vertical separation. Instead, the reversal results from
outsourcing to a vertically integrated retail rival.
5. Upstream competition
The analysis to this point has abstracted from competition among suppliers of the essential
input. To assess the effects of upstream competition most simply, consider the following
modification of the baseline model. Suppose that, as in the baseline model, firm 1 (the VIP) and
firm 2 are the retail providers. Now, though, suppose that new upstream competitors can replicate
the upstream cost structure of firm 1 by incurring a fixed investment cost, I N 0. After upstream
participation is determined, firm 1 and any active upstream competitors simultaneously and
independently choose the unit price at which they will sell the essential input to firm 2. (Firm 1
continues to supply all of its own input requirements.) Firm 2 then makes its procurement
decision and competes with firm 1 for retail customers.
A non-integrated producer will not enter the upstream industry by incurring I in this setting.
The ensuing price competition with an equally efficient upstream rival (firm 1) would dissipate
all variable profit, making it impossible for the non-integrated entrant to recover its investment
cost, I. In contrast, firm 2 may find it profitable to incur I and then “in-source” production of the
essential input at the same marginal cost as firm 1. If the net benefit associated with the
reduction in input costs exceeds I, firm 2 will find it profitable to incur the up-front investment
cost.
Under Bertrand competition, firm 2's net benefit from reduced input costs is diminished by an
opportunity cost of in-sourcing. Recall that under Bertrand competition, firm 1 sets a relatively
high input price in the baseline model. The high input price reduces the intensity of retail
competition and thereby increases firm 2's profit. Firm 2 foregoes this profit enhancement from
softened competition if it chooses to in-source production of the input. Consequently, firm 2
views this foregone profit as an additional (opportunity) cost of in-sourcing.
Proposition 4 characterizes firm 2's sourcing decision. The proposition refers to I B, which
denotes the minimum investment cost such that firm 2 optimally relies on outsourcing under
Bertrand competition. I C denotes the corresponding investment cost under Cournot competition.
(The Appendix provides closed-form expressions for IB and I C.)
A. Arya et al. / Int. J. Ind. Organ. 26 (2008) 1–16
9
Proposition 4. Given (NF) in the setting with potential upstream entry, 0 b IB b IC. Consequently,
firm 2 will always in-source production of the input if I is sufficiently small (I b IB) and always
outsource production of the input to firm 1 if I is sufficiently large (I ≥ I C). For intermediate
values of I (I ∈ [IB,I C)), firm 2 will in-source production of the input under Cournot retail
competition and outsource production of the input to firm 1 under Bertrand competition.
Proposition 4 reports that firm 2 is less likely to in-source production of the input under
Bertrand competition than under Cournot competition. This is the case even though firm 2 faces a
higher input price under Bertrand competition than under Cournot competition when it outsources
production to firm 1. The reluctance to in-source production despite the presence of higher input
prices reflects the aforementioned opportunity cost of in-sourcing (i.e., the foregone benefit of
less intense retail competition) under Bertrand competition. Proposition 4 implies that in addition
to promoting less intense retail competition, Bertrand competition can limit the extent of upstream
competition. It can do so by reducing the incentive of retail providers to in-source production of
essential inputs.
These findings imply that standard welfare conclusions continue to be reversed (as they are in
the baseline model) in the presence of potential upstream competition as long as upstream entry
costs (I ) are sufficiently pronounced. Clearly, if the requisite entry cost (I ) is at least I C, firm 2
will outsource production of the input to firm 1, and so the baseline model prevails and all of the
conclusions derived in the baseline model hold. Furthermore, as Proposition 5 reports, there exists
w
an up-front investment level I aðI B ; I C Þ such that the key conclusions of the baseline model also
w C
hold for all IaðI ; I Þ.
Proposition 5. Given (NF) in the setting with potential upstream entry, there exists an
w
w
I aðI B ; I C Þ such that for IN I : (i) firms 1 and 2 both secure greater profit under Bertrand
competition than under Cournot competition; and (ii) consumer surplus is lower under Bertrand
competition than under Cournot competition.
To understand the conclusions drawn in Proposition 5, recall from Proposition 4 that if I ∈ [IB,
I ), firm 2 will in-source production of the input under Cournot competition and outsource
production of the input under Bertrand competition. The lower input costs that arise under
Cournot competition and the less intense retail competition and lower fixed costs that arise under
Bertrand competition (with outsourcing of production to a vertically-integrated retail rival) are the
driving forces underlying Proposition 5.7
C
6. Conclusions
It is well known that Bertrand competition typically produces lower retail prices, lower
industry profit, and higher levels of consumer surplus and total surplus than Cournot competition.
We have shown that these standard conclusions can be reversed when a retail competitor secures
an essential input from a vertically integrated provider of substitute goods. In the presence of such
7
Of course, if entry costs are sufficiently low, firm 2 will always in-source production of the input. This is the case
even when (NF) does not hold, as firm 2 can avoid foreclosure by producing the input itself. In the absence of
w
outsourcing to a vertically-integrated rival, the standard welfare conclusions prevail. Even when IaðI ; I C Þ , the
standard conclusion that total surplus is higher under Bertrand competition than under Cournot competition persists.
This is the case because the loss from the duplication of fixed costs outweighs the gains realized by consumers under
Cournot competition.
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outsourcing, Bertrand competition can produce higher retail prices, higher industry profit, and
lower levels of consumer surplus and total surplus than Cournot competition. These outcomes
arise because the vertically integrated producer (VIP) sets a relatively high input price under
Bertrand competition in order to establish a high opportunity cost of aggressive retail competition.
The high opportunity cost provides the VIP with a credible commitment to engage in less
aggressive retail competition, which serves to increase the retail rival's output and thus its demand
for the VIP's (lucrative) input. The resulting diminished intensity of retail competition and the
relatively high input price under Bertrand competition produce the higher retail prices that cause
the reduction in consumer surplus and total surplus (and the increase in industry profit) relative to
Cournot competition.
These effects are particularly pronounced when the VIP is a monopoly supplier of the input.
However, the effects continue to operate in the presence of potential upstream competition.
Furthermore, when upstream competition is possible, Bertrand retail competition can reduce both
the intensity of retail competition and the extent of wholesale competition.
Future research might consider more general demand and cost structures, economies/
diseconomies of integration, relevant information asymmetries, and alternative forms of
wholesale and retail competition. Although these extensions may provide new insights of
interest, they seem unlikely to reverse the finding that outsourcing to vertically integrated rivals
can alter standard conclusions about outcomes under price and quantity competition.
Appendix A
Proof of Lemma 1. We first characterize the equilibrium outcomes under both Cournot and
Bertrand competition. The equilibria are derived using backward induction. The firms' strategic
decision variables in the retail market are derived first, taking input prices as given. Then profitmaximizing input prices are determined.
A.1. Cournot Competition
When α2 / α1 N γ, equilibrium outcomes are as follows:
pC1 ¼
½8−c2 a1 −2ca2
þ c1 ;
2½8−3c2 qC1 ¼
½8−c2 a1 −2ca2
;
2½8−3c2 PC1 ¼
pC2 ¼
qC2 ¼
2½a2 −ca1 ;
8−3c2
½8 þ c2 a21 þ 4a22 −8ca1 a2
;
4½8−3c2 CS C ¼
½4−c2 ca1 −4½3−c2 a2
þ c2 ;
2½8−3c2 PC2 ¼
wC ¼
a2 c2 ½a2 −ca1 −
;
2½8−3c2 2
4½a2 −ca1 2
½8−3c2 2
; and
½64−64c2 þ 9c4 a21 þ 4½4−3c2 a22 þ 12c3 a1 a2 :
8½8−3c2 2
ðA1Þ
When α2 / α1 = γ, q2C is 0. When α2 / α1 b γ, firm 2 again produces no retail output in
equilibrium, and it is readily verified that firm 1 produces α1 / 2 units of output.
A. Arya et al. / Int. J. Ind. Organ. 26 (2008) 1–16
11
A.2. Bertrand Competition
Again, when α2 / α1 N γ, there is no foreclosure (so q2 N 0) and equilibrium outcomes are
readily shown to be the following:
P1B ¼
½8−c2 a1 þ 2ca2
þ c1 ;
2½8 þ c2 qB1 ¼
½8−c2 −c4 a1 −6ca2
½2 þ c2 ½a2 −ca1 a2 c2 ½a2 −ca1 ; qB2 ¼
;
; wB ¼ −
2
4
2
4
8−7c −c
2½8−7c −c 2 2½8 þ c2 PB1 ¼
2½6 þ c2 a2 −c½4 þ c2 a1
þ c2 ;
2½8 þ c2 pB2 ¼
½8−3c2 −c4 a21 þ 4a22 −8ca1 a2
½2 þ c2 2 ½a2 −ca1 2
B
;
P
¼
; and
2
4½8−7c2 −c4 ½1−c2 ½8 þ c2 2
CS B ¼
½64−23c4 −5c6 a21 þ 4½4 þ 5c2 a22 −4½16 þ 3c2 −c4 ca1 a2 :
8½1−c2 ½8 þ c2 2
ðA2Þ
When α2 / α1 = γ, q2B is 0. When α2 / α1 b γ, firm 2 again produces no retail output in
equilibrium, and it is readily verified that firm 1 produces α1 / 2 units in this case.
Proof of Lemma 2. The lemma follows directly from the expressions for input prices in Eq. (A1)
and Eq. (A2).
Proof of Proposition 1. From Eq. (A1) and Eq. (A2):
PB1 −PC1 ¼
PB2 −PC2 ¼
c2 ½4 þ c2 ½a2 −ca1 2
N0;
64−80c2 þ 13c4 þ 3c6
c2 ½256−32c2 −8c4 þ 9c6 ½a2 −ca1 2
CS B −CS C ¼
½1−c2 ½64−16c2 −3c4 2
c½a2 −ca1 ½ca2 A−a1 B
2½1−c2 ½64−16c2 −3c4 2
N0; and
b0;
ðA3Þ
where A = 512 − 288γ2 + 4γ4 − 3γ6 and B = 1024 − 896γ2 + 80γ4 + 26γ6 − 9γ8. The inequalities in
Eq. (A3) hold because γ b 1, γα1 b α2 ≤ α1, and γA b B.
From the expressions for profit and consumer surplus in Eq. (A1) and Eq. (A2):
TS B −TS C ¼
c½a2 −ca1 ½ca2 C−a1 D
2½1−c2 ½64−16c2 −3c4 2
b0; where
ðA4Þ
C = 1536 − 352γ2 − 68γ4 + 9γ6 and D = 1024 + 128γ2 + 16γ4 − 46γ6 + 3γ8 . The inequality in
Eq. (A4) holds because γ b 1, γα1 b α2 ≤ α1, and γC b D.
12
A. Arya et al. / Int. J. Ind. Organ. 26 (2008) 1–16
Proof of Lemma 3. The proof follows from examining equilibrium outcomes under vertical
separation (VS) for both Cournot and Bertrand retail competition.
A.3. VS and Cournot Competition
Denote the equilibrium retail prices under VS and Cournot competition by p̄iC, quantities
PC
by q̄iC, input prices by w̄iC, firm profits by Π̄1C, Π̄2C, and Π̄UC, and consumer surplus by CS .
Equilibrium outcomes are readily shown to be (for i, j ∈ {1,2}, i ≠ j):
C
p̄i ¼
C
P̄ i ¼
½6−c2 ai −caj
þ ci ;
2½4−c2 ½2ai −caj 2
4½4−c2 2
;
C
C
q̄i ¼
P̄ U ¼
2ai −caj
;
2½4−c2 C
w̄i ¼
ai
;
2
a21 þ a22 −ca1 a2
; and
2½4−c2 P C ½4−3c2 ½a21 þ a22 þ 2c3 a1 a2 :
CS ¼
8½4−c2 2
ðA5Þ
A.4. VS and Bertrand Competition
Now employ the same notation as above, but replace the superscript C with the superscript B
to denote Bertrand competition. Equilibrium outcomes are:
B
p̄i ¼
B
P̄ i ¼
2½3−c2 ai −caj
þ ci ;
2½4−c2 ½ð2−c2 Þai −caj 2
4½1−c2 ½4−c2 2
;
B
q̄i ¼
B
P̄ U ¼
½2−c2 ai −caj
;
2½4−5c2 þ c4 B
w̄i ¼
ai
;
2
½2−c2 ½a21 þ a22 −2ca1 a2
; and
4½4−5c2 þ c4 P B ½4−3c2 ½a21 þ a22 −2c3 a1 a2 :
CS ¼
8½1−c2 ½4−c2 2
ðA6Þ
Proof of Proposition 2. Initially, consider the setting with Cournot retail competition. We show
first that D1 will merge with U if, in the absence of their merger, U would not merge with D2.
The profit of the merged firm (U and D1) is as specified in Eq. (A1). If U merges with neither D1
nor D2, the profits of U and D1 are as specified in Eq. (A5). Therefore, the incremental joint
profit U and D1 secure from a merger in this case is:
C
C
PC1 −½P̄ U þ P̄ 1 ¼
½32 þ 4c2 −6c4 þ c6 a21 þ c4 a22 −2½16−6c2 þ c4 ca1 a2 :
4½8−3c2 ½4−c2 2
This expression is a convex function of α2. Furthermore, both its roots exceed α1 Therefore,
the expression is positive for all relevant values of α2, since α2 ≤ α1.
A. Arya et al. / Int. J. Ind. Organ. 26 (2008) 1–16
13
We now show that D1 also will merge with U if, in the absence of their merger, U would
merge with D2. If D1 merges with U, their combined profit is as specified in Eq. (A1). If D2
merges with U, D1's profit is as given by Π2C in Eq. (A1) with α1 and α2 interchanged. Denote
∼
this profit by Π 1C. These expressions reveal that D1's maximum bid in this case is:
∼C ½48−16c2 −3c4 a21 þ 4½8−7c2 a22 −8½4−3c2 ca1 a2 :
PC1 −P1 ¼
4½8−3c2 2
Analogous logic reveals that D2's maximum bid in this case is this same expression, but with
α1 and α2 interchanged. This implies that the difference between D1's maximum bid and D2's
½16þ12c2 −3c4 ½a21 −a22 z0:
maximum bid is
4½8−3c2 2
This inequality implies D1 is willing to bid at least as much as D2 to merge with U.
The analysis for the setting with Bertrand competition is analogous. In this case, firm 1's
maximum willingness to bid when firm 2 would not merge with U is:
½32−12c2 −2c4 þ c6 a21 þ c4 a22 −4½8−3c2 ca1 a2
4½8 þ c2 ½4−c2 2
N0:
When firm 2 would merge with U, firm 1's maximum bid is readily shown to exceed the
½16−4c2 −3c4 ½a21 −a22 maximum bid of firm 2 by
z0:
4½8þc2 2
Proof of Proposition 3. Finding 1. The merger of U and D1 increases total surplus under Cournot
competition. Under vertical integration (VI), the surpluses of the relevant parties are as specified in
Eq. (A1). Under VS, the surpluses are as specified in Eq. (A5). Using these expressions:
P C Ea2 þ Fa22 −2Gca1 a2
TS C − TS ¼ 1
N0;
8½32−20c2 þ 3c4 2
ðA7Þ
where E = 1280 − 896γ2 + 260γ4 − 43γ6 + 3γ8, F = [192 − 92γ2 + 9γ4]γ2, and G = 512 − 288γ2+
48γ4 − 3γ6. The expression in Eq. (A7) is convex in α2 and both roots exceed α1. Therefore, the
inequality in Eq. (A7) holds since α2 ≤ α1.
Finding 2. The merger of U and D1 increases total surplus under Bertrand competition.
Eq. (A2) and Eq. (A6) imply that:
P B Ha21 þ J a22 −2Kca1 a2
TS B ¼ TS ¼
N0;
8½1−c2 ½32−4c2 −c4 2
ðA8Þ
where H = 1280 − 640γ 2 + 36γ 4 + 77γ 6 − 25γ 8 + γ 10 , J = [960 − 188γ 2 − 47γ 4 + 4γ 6 ]γ 2 , and
K = 1024 − 224γ2 − 80γ4 + 3γ6 + 6γ8. The inequality in Eq. (A8) holds because the expression
in Eq. (A8) is convex in α2 and has no real roots.
Finding 3. The merger of U and D1 increases total surplus more under Cournot competition
than under Bertrand competition.
From Eq. (A7) and Eq. (A8):
PC
PB
TS C − TS −½TS B − TS ¼
c½−Lca21 −M ca22 þ 2N a1 a2 8½1−c2 ½256−128c2 þ 4c4 þ 3c6 2
N0;
ðA9Þ
14
A. Arya et al. / Int. J. Ind. Organ. 26 (2008) 1–16
where L = 16,384 + 4096γ2 + 512γ4 − 4224γ6 + 1716γ8 − 271γ10 + 12γ12, M = 49,152− 43,008γ2 +
12,544γ4 + 64γ6 − 572γ8 + 45γ10, and N = 32,768− 20,480γ2 + 5632γ4+ 704γ6 − 192γ8 −258γ10 +
51γ12.
The expression in Eq. (A9) is a concave in α2 and is positive at the two boundary values for α2.
Therefore, the expression is positive for all relevant values of α2.
Finding 4. The merger of U and D1 increases consumer surplus under Cournot. Under VI,
consumer surplus is as specified in Eq. (A1); under VS, it is as specified in Eq. (A5). These
expressions imply:
P C Pa2 þ Qa22 þ 2Rca1 a2
CS C − CS ¼ 1
N0;
8½32−20c2 þ 3c4 2
ðA10Þ
where P = 768 − 1152γ2 + 540γ4 − 109γ6 + 9γ8, Q = [64−68γ2 + 15γ4]γ2, and R = [32 − 3γ4]γ2.
The inequality in Eq. (A10) holds because the expression in Eq. (A10) is convex in α2 and has no
real roots.
Finding 5. The merger of U and D1 increases consumer surplus under Bertrand. From Eqs.
(A2) and (A6):
P B Sa2 þ T a22 −2U ca1 a2
CS B − CS ¼ 1
N0;
8½1−c2 ½32−4c2 −c4 2
ðA11Þ
where S = 768 − 384γ2 − 260γ4 + 107γ6 + 17γ8 − 5γ10, T = [320 − 100γ2 + 23γ4]γ2, and U = [512−
224γ2 − 64γ4 + 21γ6 − 2γ8 ]. The expression in Eq. (A11) is convex in α2, and both roots of the
expression exceed α1. Therefore, the inequality holds for all relevant values of α2, since
α2 ≤ α 1 .
Finding 6. The merger of U and D1 increases consumer surplus more under Cournot
competition than under Bertrand competition.
From Eqs. (A10) and (A11):
h
i
PC
PB
CS C − CS − CS B − CS ¼
c½−V ca21 −W ca22 þ 2X a1 a2 8½1−c2 ½256−128c2 þ 4c4 þ 3c6 2
N0;
ðA12Þ
where V = 49,152 − 69,632γ 2 + 32,256γ 4 − 5248γ 6 − 836γ 8 + 419γ 10 − 36γ 12 , W = 16,384−
14,336γ2 + 5888γ4 − 2240γ6 + 364γ8 + 15γ10 , and X = 32,768 − 36,864γ2 + 9728γ4 + 1728γ6 −
1600γ8 + 330γ10 − 15γ12.
The expression in Eq. (A12) is a concave function in α2 and is positive at the two boundary
values. Therefore, the expression is positive for all relevant values of α2.
Proof of Proposition 4. First, we show that under Cournot competition, firm 2 produces the
input itself if and only if I b IC, where
ICu
−c4 ½16−5c2 a21 þ 32½6−5c2 þ c4 a22 −4c½32−32c2 þ 7c4 a1 a2 :
½32−20c2 þ 3c4 2
A. Arya et al. / Int. J. Ind. Organ. 26 (2008) 1–16
15
If firm 2 produces the input itself (at zero marginal cost), the equilibrium outcome is the
outcome under VS with w = 0 . Corresponding profits and consumer surplus are:
2
C ½2a1 −ca2 ;
P1 ¼
½4−c2 2
2
C ½2a2 −ca1 P2 ¼
−I; and
½4−c2 2
j C ½4−3c2 a21 þ ½4−3c2 a22 þ 2c3 a1 a2 :
CS ¼
2½4−c2 2
C
I C is derived by equating P 2 and Π2C.
Similarly, under Bertrand competition, firm 2 will produce the input itself if and only if
IbI B u
−c4 ½16 þ 3c2 −c4 a21 þ 16½12−2c2 −c4 a22 −2c½64−2c4 þ c6 a1 a2
½32−4c2 −c4 2
:
As above, if firm 2 produces the input itself, the equilibrium outcome is the outcome under VS
with w = 0, yielding profits for firms 1 and 2 and consumer surplus:
2
B ½ð2−c2 Þa1 −ca2 P1 ¼
;
½4−c2 2 ½1−c2 2
B ½ð2−c2 Þa2 −ca1 P2 ¼
−I; and
½4−c2 2 ½1−c2 j B ½4−3c2 a21 þ ½4−3c2 a22 −2c3 a1 a2 :
CS ¼
2½4−c2 2 ½1−c2 B
Equating P 2 and Π2B yields IB . Finally, observe that:
I C −I B ¼
−c6 Y a21 þ 16c2 Za22 þ 2c5 Y a1 a2
½256−128c2 þ 4c4 þ 3c6 2
N0;
ðA13Þ
where Y = 512 − 80γ4 + 9γ6 and Z = 256 − 160γ2 − 8γ4 + 11γ6. It is readily verified that the
inequality in Eq. (A13) holds, given (NF).
C
C B
Proof of Proposition 5. Let Î be the value of I that equates Π2B and P 2 . Î N IB because P 2 NP 2 .
C
B
C
Also, Î b I because Π2 N Π2 .
Recall that firm 2 does not produce the input itself under either Cournot or Bertrand competition if
I ≥ IC. In this case, conclusions (i) and (ii) follow from Proposition 1. If Î b I b I C, firm 2 produces the
input itself only under Cournot competition, in which case conclusions (i) and (ii) are confirmed as
C by construction of Î. Second, a comparison of firm 1's profits
̂ PB NP
follows. First, when IbI;
2
2
yields:
C c4 A Va21 þ 4B Va22 −24c3 ½2 þ c2 a1 a2
PB1 −P 1 ¼
N0;
4½4−c2 2 ½8−7c2 −c4 ðA14Þ
where A′ = 32 + 5γ2 − γ4 and B′ = 16 − 16γ2 + 8γ4 + γ6. The inequality in Eq. (A14) holds because the
expression in Eq. (A14) is convex in α2 and has no real roots. Finally:
−c2 C Va21 þ 12D Va22 þ 4cE Va1 a2
jC
N0;
CS −CS B ¼
8½1−c2 ½32−4c2 −c4 2
ðA15Þ
16
A. Arya et al. / Int. J. Ind. Organ. 26 (2008) 1–16
where C′ = 1024 − 640γ2 − 60γ4 + 5γ6 − 5γ8, D′ = 64 − 144γ2 + 40γ4 + 12γ6 + γ8, and E′ = 256+
48γ2 − 120γ4 − 19γ6 − 3γ8. The term in Eq. (A15) is concave in α1 and positive at the boundary
values. Therefore, it is positive for all relevant values of α1.
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