International Journal
of Industrial
28 (2010) 244–253
Int. J. Ind.
Organ. 28Organization
(2010) 244–253
Contents lists available at ScienceDirect
International Journal of Industrial Organization
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i j i o
Complements and potential competition
Mikko Packalen
Department of Economics, University of Waterloo, Ontario Canada N2L 3G1
a r t i c l e
i n f o
Article history:
Received 5 January 2008
Received in revised form 27 August 2009
Accepted 31 August 2009
Available online 6 September 2009
JEL classification:
K11
K21
L24
L41
Keywords:
Complements
Monopoly
Entry inducement
Cooperation
Double marginalization
Double monopoly
a b s t r a c t
In this paper we examine the effect of cooperation between complementary incumbent monopolists on
consumer welfare. While divided technical leadership makes it difficult for firms to integrate into
complementary markets, firms induce entry in complementary markets by reducing the cost of entry in
those markets. This is accomplished through, for example, the development and dissemination of royaltyfree intellectual property. We present and analyze a model in which incumbents can influence the ease of
entry in complementary markets. Cooperation between complementary monopolists decreases consumer
welfare by reducing or even eliminating the entry inducement incentive but increases consumer welfare by
eliminating double marginalization. We show that cooperation may decrease consumer welfare, contrary to
Cournot's celebrated double monopoly result, and that the welfare comparison can be determined in terms
of straightforward economic concepts. We also present and analyze a model in which each incumbent can
induce entry in the complementary market through long-term price commitments which are common in
patent licensing.
© 2009 Elsevier B.V. All rights reserved.
1. Introduction
In this paper we examine the effect of cooperation between
complementary monopolists on consumer welfare. The importance of
the complementary monopoly problem has increased due to both the
fragmentation of patent ownership and the divided technical leadership
in the computer industry. Recent analyses of cooperation between
complementary patent owners have followed the double monopoly
analysis of Cournot (1838) by focusing on the effect that double
marginalization has on consumer welfare (see e.g. Shapiro (2001),
Gilbert (2004) and Lerner and Tirole (2004)). In contrast, during United
States v. Microsoft several analyses suggested that the divestiture of
Microsoft's operating system and applications monopolies might
benefit consumers because the two complementary monopolists
would then have an incentive to facilitate entry against each other.1
The total effect of cooperation between complementors on
consumer welfare depends on the impacts that cooperation has on
consumer welfare through the elimination of double marginalization
and through the reduction or elimination of the entry inducement
incentive. However, the combined effect of cooperation through
E-mail address: [email protected].
See the declaration by Rebecca M. Henderson (http://www.usdoj.gov/atr/cases/
f219100/219129.htm), the declaration by Paul M. Romer (http://www.usdoj.gov/atr/cases/
f219100/219128.htm) and, the declaration by Carl Shapiro (http://www.usdoj.gov/atr/cases/
f219100/219127.htm).
1
0167-7187/$ – see front matter © 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.ijindorg.2009.08.005
these two channels has not been formally explored in the case that
both monopolists face potential competition. To fill this gap, we
formally examine the impact of cooperation between two complementary monopolists in two models that incorporate both the static
double marginalization phenomenon and the dynamic entry inducement mechanism.
The cooperative case in our analysis corresponds to the integrated
case with the restriction that an integrated firm does not commit to
bundling of its products. If an integrated firm can commit to bundling of
its products through technical bundling, integration can create the twolevel entry problem (see Choi and Stefanadis, 2001; Dennis and
Waldman, 2002). This potential effect of integration is separate from
the effect that cooperation (or integration) has on the equilibrium
outcome through the internalization of the entry inducement externality, which can be achieved without even contractual bundling.2
Among firms, the means to induce entry are numerous, and the
incentive to induce entry into complementary markets appears to be
well understood at least in the computer industry. For example, as
Gawer and Henderson (2007) discussed, Intel has a long history of
inducing entry into complementary markets by lowering the cost of
entry in these markets. Intel has achieved this largely through the
2
Because an integrated firm would in some cases choose technical bundling over
entry inducement, and technical bundling lowers entry even relative to the case
without entry inducement, our cooperative case represents an upper bound for
consumer welfare in the integrated case.
M. Packalen / Int. J. Ind. Organ. 28 (2010) 244–253
development and royalty-free dissemination of intellectual property.
Another example of entry inducement is IBM's support of Linux. As
Yoffie and Kwak (2006) describe, IBM has donated vast amounts of
money, people and intellectual property to advance the development of
Linux and thereby reduce IBM's dependence on Microsoft's products. A
third example of entry inducement in the computer industry is
Microsoft's continued support of AMD in an effort to maintain and
strengthen AMD as a competitor of Intel. Microsoft has successfully
demanded that Intel share technologies such as the MMX multimedia
technology with AMD (see Yoffie and Kwak, 2006) and Microsoft and
AMD have collaborated on the development of new products.3
In our formal analysis we examine two models, an ease of entry
model and a price commitment model. In both models two complementary monopolists face potential entry, and entry is uncertain.
The entry inducement mechanism is the only difference between the
models. Together the analyses demonstrate that the impact that
cooperation (or integration) has on consumer welfare is very modelspecific when also entry inducement is considered. The optimal policy
must therefore be solved on a case-by-case basis. Our analysis offers
constructive guidance on how such policy analyses can be conducted
in practice as we show that the equilibrium comparison can be
characterized in terms of straightforward and empirically malleable
economic concepts.
In the ease of entry model each incumbent monopolist can induce
entry into the complementary market by lowering the marginal cost of
entry in the complementary market. As the aforementioned examples
suggest, this reduction in the marginal cost of entry is achieved by, for
example, developing and disseminating royalty-free intellectual
property or other technological know-how. We compare the expected
consumer surplus when the two complementary incumbent monopolists set the entry cost parameters and prices non-cooperatively and
when the incumbent monopolists set the entry cost parameters and
prices cooperatively. We show that consumer surplus can be higher in
the non-cooperative case than in the cooperative case, contrary to the
celebrated double monopoly result of Cournot (1838). Importantly,
while the comparisons of equilibrium outcomes in the cooperative and
non-cooperative cases are ambiguous, throughout the paper we
demonstrate that the comparison can be characterized in terms of
straightforward economic concepts, namely the demand curve, the
probability of entry without entry inducement, and the rewardelasticity of the probability of entry.
The non-cooperative and cooperative cases yield different equilibrium outcomes because of the presence of two externalities: the entry
inducement externality and the double marginalization (pricing)
externality. These externalities are internalized only in the cooperative
case. Internalization of these externalities has generally opposing
effects on consumer welfare. If neither incumbent is displaced by an
entrant, cooperation avoids double marginalization in pricing, and
thereby decreases prices and increases consumer surplus relative to
the non-cooperative case. In contrast, internalization of the entry
inducement externality generally reduces and may even eliminate the
entry inducement incentive, and thereby decreases the probability of
entry and the expected consumer surplus.
In the price commitment model each incumbent monopolist can
induce entry into the complementary market through a long-term price
commitment in their own market. Long-term price commitments are
common in patent licensing. For example, the license to patents that
cover the ATCS standard does not expire until the end of the year 2016.
In addition to the royalty rate commitment until the end of the year
2016, the license includes a commitment not to increase royalty rates to
current licensees by more than 10% during each 5-year renewal period
3
See e.g. the press release “AMD Collaborates With Microsoft on Windows Server
2008 to Deliver Industry Leading Solution to Customers” (available at http://www.
microsoft.com/presspass/events/HHH launch/docs/amd.doc; last accessed 27 March
2009).
245
after the year 2016.4 We emphasize that while our analysis shows that
entry inducement is one rationale for long-term price commitments, we
neither develop an empirical test for whether entry inducement is an
important factor in patent licensing nor offer direct evidence on the
motives of licensors in setting the licensing terms.
In the analysis of the price commitment model, we compare the
expected consumer surplus when the incumbents set prices noncooperatively and when the incumbents set prices cooperatively.
These two pricing arrangements yield different outcomes because of
the presence of the entry inducement externality and double marginalization (pricing) externality. We show that non-cooperative pricing
can yield higher expected consumer surplus than cooperative pricing.
However, this happens only for a limited set of parameterizations
of the model. Hence, while the analysis shows that entry inducement
can again overturn Cournot's celebrated double monopoly result, the
results do not warrant prohibiting complementary patent owners
from forming patent pools that set prices cooperatively.5
Our analysis departs from the earlier analyses of complementary
monopolists in the presence of entry by Farrell and Katz (2000), Choi et al.
(2003), Heeb (2003), and Cheng and Nahm (2007) in two ways. First, we
assume that both incumbent monopolists face potential entry. When an
incumbent monopolist itself also faces potential entry its incentive to
induce entry into its complementary market is limited because inducing
entry against the complementary incumbent monopolist indirectly also
increases entry against the monopolist itself. We show that inducing entry
against the complementary incumbent can nevertheless be profitable
if entry is uncertain. Second, we assume that the complementary
incumbent monopolists themselves do not enter their complementary
markets but instead induce entry by other firms. Due to divided technical
leadership entry into complementary markets by incumbent monopolists
is often difficult (see Bresnahan, 2004), and therefore entry inducement is
often a more realistic scenario. Casadedus-Masanell et al. (2007) provide a
related analysis but focus on the effect of actual competition in one market
on the complementary monopolist's profit in another market.
Our analysis is obviously closely related to the Chicago School's
single-monopoly-rent theorem and the more recent analyses on
4
The license is offered by the patent pool organization MPEG LA. The License
Agreement is available at http://www.mpegla.com/atsc/atsc-agreement.cfm; last
accessed 27 March 2009.
5
In the working paper version of our paper we examine two extensions of the price
commitment model that demonstrate that forms of cooperation that fall short of full
integration can have significant pro-competitive benefits compared to full integration.
In the first extension we compare the cooperative outcome with the outcome when the
incumbents cooperatively set the price of the bundle of both incumbent goods but noncooperatively set the price of each individual good. Such independent pricing
provisions are common in patent licensing (see Lerner et al., 2007). In the second
extension we examine the impact of independent pricing provisions on ex-ante
consumer welfare which takes into account the effect that the pricing regime has on the
incumbents' expected profit and, consequently, on the probability that the incumbent
innovations are invented in the first place. The results indicate that if cooperation does
not have pro-competitive effects other than the elimination of double marginalization,
antitrust authorities should require that patent pooling be accomplished through a
patent pool that allows for independent licensing rather than a cooperative
arrangement that does not allow for independent licensing such as a merger, the
acquisition of patents, or a patent pool that does not allow for independent licensing.
This policy conclusion illustrates the importance of taking the entry inducement
incentive into account as the current policy is to allow both types of cooperative arrangements between the owners of complementary patents and independent licensing provisions are seen only as protection against pooling of patents that are substitutes (see U.S.
F.T.C./D.O.J. Antitrust Guidelines for Licensing of Intellectual Property, 1995, and U.S. D.O.J.
Business Review Letters, 1997, 1998, 1999). The policy is largely based on the analysis of
Cournot (1838) which recent articles on cooperative pricing of complements (Economides
and Salop, 1992, Economides, 1999, Shapiro 2001, Gilbert, 2004, Lerner and Tirole, 2004)
have reiterated and reinforced. Pooling of intellectual property has also been addressed in
many law review articles (see e.g. Gollier, 1968, Tom and Newberg, 1997, Merges, 1996,
2001, and Carlson, 1999). These contributions echo the existing economics literature by
distinguishing between competing and complementary patents. A novel conceptual
contribution of our analysis is that the same two patents can be ex-ante substitutes but expost complements. According to Nalebuff (2003), the case when two goods are ex-ante
substitutes but ex-post complements has not come up in the antitrust literature.
246
M. Packalen / Int. J. Ind. Organ. 28 (2010) 244–253
monopoly leveraging.6 Whinston (1990) notes that a monopolist
benefits from increased competition in a complementary market but
does not examine the incentives for cooperation between firms with
market power in complementary markets or the case with entry
threats in both markets.7 Dennis and Waldman (2002) note that entry
in a complementary market can affect entry in the essential good
market but their analysis relies on inter-temporal cost efficiencies
whereas our analysis centers on the effect that entry in one market has
on the rewards for entry in a complementary market.8
Choi and Stefanadis (2001) and Gilbert and Katz (2006) examine
tying of two complementary goods by a monopolist that faces
potential entry. Neither analysis examines cooperation, integration,
or double marginalization. Bresnahan and Greenstein (1999) argue
that in the computer industry divided technical leadership across
vertical components of platforms has been inevitable and made entry
easier. We examine and provide formal analyses of specific channels
that allow firms to influence entry in complementary markets.
In the analyses of bundling by a monopolist by Nalebuff (2004)
and Gilbert and Katz (2006), price commitments provide the means
for entry deterrence, and in Gilbert and Katz (2006) price commitments also facilitate complementary investments by buyers. In our
analysis price commitments facilitate the inducement of entry. Our
analysis of the price commitment model differs also from the analysis
of forward contracting and quantity competition by Allaz and Vila
(1993) as we consider option contracting and complements.9
This paper is organized as follows. In the next section we present
and analyze the ease of entry model with homogenous potential
buyers. With homogenous potential buyers only the entry inducement externality is present. In the third section we then extend the
ease of entry model to the case with heterogenous potential buyers so
that both the double marginalization externality and the entry
inducement externality are present. In the fourth section we present
and analyze the price commitment model. The fifth section concludes.
2. The ease of entry model with homogenous buyers
In this section we compare the non-cooperative and cooperative
outcomes in a model with homogenous potential buyers. Only the
entry inducement externality is present in the model.
2.1. The model with homogenous potential buyers
We consider a setting with two incumbent monopolists, two
potential entrants, and a unit mass of homogenous potential buyers.
Incumbent 1 sells good 1, and incumbent 2 sells good 2. Absent entry,
the incumbents' goods are perfect complements, and each potential
buyer's valuation for the two incumbent goods is υ. The number of
potential buyers is large enough to justify the assumption that the
potential buyers are price-takers.
6
The single-monopoly-rent theorem states that a monopolist cannot increase its
profits by monopolizing a competitive complementary market. The single-monopolyrent theorem was presented in Director and Levi (1956, p. 290) and Bowman (1957,
p. 21), and later in Posner (1976, p. 173) and Bork (1978, p. 373).
7
See also Judge Richard Posner's opinion in Olympia Equipment Leasing Co. v. Western
Union Telegraph Co., 797 F.2d 370, 374 (7th Cir. 1986) and Farrell and Weiser (2003).
8
Bresnahan (2004) notes that with network externalities the sudden development
of a complementary market can increase the demand for the essential good so
dramatically that the incumbent monopolist's essential good installed base advantage
is threatened. This effect of increased provision of complementary goods was first
understood by the Microsoft Corporation (see Bresnahan, 2004).
9
In contrast with the contestable markets literature (see e.g. Baumol et al., 1982,
and Stefanadis, 2003), we assume that the potential entrants cannot contract with
potential buyers before they enter. Asymmetric information about the potential
entrants' capabilities and the quality of their final product makes such contracting
between buyers and potential entrants unlikely in the type of innovative environments that motivate our analysis.
Potential entrants 1 and 2 attempt entry against the incumbents 1
and 2, respectively. Each buyer's marginal valuation for a successful
entrant's good instead of the corresponding incumbent good is Δ.10
Therefore, each buyer's valuation for one incumbent's good and the
complementary entrant's good is υ + Δ, and each buyer's valuation for
both entrants' goods is υ + 2Δ. For notational convenience we assume
that production costs are zero.
R&D investments by the potential entrant i increase its probability
of successful entry against the corresponding incumbent, which is
denoted by μ i. The R&D investment cost C(μ i) of the potential entrant
i is an increasing function of its probability of success:
Cðμ i Þ = F + ai μ i +
b 2
μ ;
2 i
where ai > 0 and b > 0:
ð1Þ
The assumptions ai > 0 and b > 0 together imply that each incumbent's
marginal
cost of increasing the probability of entry is positive
dCðμ i Þ
> 0 . This assumption captures the feature that a firm must
dμ i
expand the scale of its R&D investments to increase its probability of
entry. The assumption
(b > 0) ensures that the cost function is convex
d2 Cðμ i Þ
dμ 2i
> 0 . This assumption captures the feature that the expansion
of R&D investments is increasingly more expensive for a firm because
the limited supply of suitable R&D capabilities makes adding the
required additional resources increasingly more costly.
As was indicated by the examples discussed in the Introduction,
firms influence potential entrants' entry costs in complementary
markets by, for example, developing and disseminating royalty-free
intellectual property and other technological know-how. We model
this particular entry inducement mechanism by assuming that the
incumbent j can influence the entry cost in the complementary
P
market i ≠ j by setting ai ∈ (0, a].11,12
Timing in the model is as follows. In stage 1 the incumbents set the
entry cost parameters a1 and a2. In stage 2 potential entrants noncooperatively choose their R&D investment levels as measured by the
associated probabilities of success μ1 and μ2, and nature subsequently reveals the outcome of these R&D investments. In stage 3 the
10
An analytically equivalent alternative is to assume that each incumbent's marginal
cost of production is c̄¯, and each successful entrant's marginal cost of production cost
is c̄¯ − Δ.
11
Decreasing the entry cost parameter aj of the potential entrant j ≠i is a more profitable
means of inducing entry for the incumbent i than decreasing the entry cost parameter ai of
the potential entrant i. This can be seen by comparing the expressions Eqs. (5) and (6),
which imply that an increase in αj increases μ j more than μ i whereas an increase in αi
increases μ i more than μ j. Each incumbent of course prefers increasing entry against the
complementary incumbent more than increasing entry against itself.
12
The assumption that the incumbents do not influence b is justified if increasing the
probability of success requires that a potential entrant implements another research
project, and entry inducement (say, in the form of provision of technical know-how) has
the same impact on the cost of implementing each additional research project (as reflected
by a decrease in αj) whereas the entry inducement has little or no impact on the parameter
b, which reflects the feature that adding the resources required to complete each
additional research project is increasingly more expensive due to the limited supply of
suitable resources.
When an incumbent j decreases aj, it shifts the best-response curve Eq. (2) of the
potential entrant j ≠i upward. If each incumbent i could instead decrease the parameter b
for the potential entrant j ≠i, doing so would simultaneously shift the best-response curve
Eq. (2) of the potential entrant j ≠i upward and increase the slope of that best-response
curve. The increase in the slope of the best-response curve for the potential entrant j ≠i
makes inducing entry less attractive for the complementary incumbent j ≠i because any
entry inducement by the incumbent j ≠i now has a higher indirect impact (see the end of
Section 2.2) on the probability that the incumbent j ≠i itself is displaced by an entrant.
Consequently, the incumbents would induce less entry against one another in equilibrium
compared to the case when each incumbent i can influence the cost parameter aj.
The assumption that the incumbents do not influence F is inconsequential as we have
assumed that F is such that the equilibrium probability of entry without entry inducement
is strictly positive. Without the latter assumption, at least one incumbent i would always
prefer to decrease F of the potential entrant j ≠i so that the equilibrium probability of
entry against the incumbent j ≠i is positive.
M. Packalen / Int. J. Ind. Organ. 28 (2010) 244–253
incumbents and successful entrants set prices. If entry against the
incumbent i is successful, the incumbent i and the successful entrant i
engage in Bertrand competition.
potential entrant i to increase its probability of success against the
incumbent i. The parameter α i therefore measures the ease of entry.
Combining the two potential entrants' best-response curves Eq. (3)
yields each potential entrant's equilibrium probability of entry18
2.2. Equilibrium analysis: the non-cooperative case
In this subsection we solve by backward induction the equilibrium
when the incumbents non-cooperatively set prices and the parameters a1 and a2 that influence the ease of entry. We focus on perfectly
coalition-proof subgame-perfect pure strategy Nash equilibria.13
2.2.1. Stage 3: incumbents and successful entrants set prices
When neither potential entrant is successful in entry, the price of
υ
each good is .14 When both potential entrants are successful in entry,
2
the prices of each entrant good and the corresponding incumbent
good are Δ and 0, respectively.15 When only the potential entrant i is
successful in entry, the entrant i and the corresponding incumbent i
Δ
set prices and 0, respectively, and the price of the complementary
2
incumbent good j is υ + Δ2 .16
2.2.2. Stage 2: potential entrants choose R&D investments
Each potential entrant chooses its probability of entry to maximize
its expected profit ПRi = μ iRi − C(μ i), where C(μ i) is given by Eq. (1)
and Ri is the expected reward for successful entry. The first-order
condition for the optimum yields
μ ⁎i =
1
a
R− ia:
b i b i
ð2Þ
Δ
Substituting Ri = ð1−μ j Þ + μ j Δ for Ri in the Eq. (2) yields the
2
best-response curve17
μ i⁎ ðμ j Þ
Δ
=
+ 2 μj
b
b
|{z} |{z}
Δ
2 −αi
≡αi
ð3Þ
≡β
for the potential entrant i. In the above expression (3) we introduce the
P
definitions of αi and β. When the incumbent j ≠i chooses ai ∈ (0, a], it
Δ
Δ
−
−
a
P
α≡2
and −
α ≡ 2 . The higher
effectively chooses αi ∈ [α
__ , α), where −
b
b
the incumbent j sets the parameter α i, the less costly it is for the
μi⁎ =
2
not all of the surplus created by entry.
Δ
17
When only the potential entrant i is successful in entry, it receives the reward for
2
successful entry. When both potential entrants are successful in entry, each entrant
receives the reward Δ for entry.
αi + βαj
1−β2
:
ð4Þ
The result (4) implies that
dμj⁎
=
dαj
1
1−β2
ð5Þ
and
dμi⁎
β
=
:
dαj
1−β2
dμ ⁎j
dαj
ð6Þ
Because β ∈ (0.1), the results (5) and (6), respectively, imply that
> 0 and
dμ ⁎i
dαj
> 0. The result
dμ ⁎j
dαj
> 0 represents the direct effect of
increasing the ease of entry against the complementary incumbent j.
Lower marginal entry cost of the potential entrant j increases the
probability of entry against the complementary incumbent j. The
result
dμ ⁎i
dαj
> 0 represents the indirect effect of increasing the ease of
entry against the complementary incumbent j. The indirect effect
Δ
arises because an entrant's reward for entry increases from to Δ
2
when also the complementary entrant is successful in entry as neither
entrant then has to share the surplus Δ generated by its entry with the
complementary incumbent. This introduces a strategic complementary in the potential entrants' entry decisions. Consequently, when an
incumbent i increases the ease of entry against the complementary
incumbent j, the incumbent indirectly increases also the probability of
entry against the incumbent i itself.
2.2.3. Stage 1: incumbents set the ease of entry
Each incumbent i chooses αj to maximize its expected profit19
υ
Δ
I
Πi = ð1−μ i⁎ Þ ð1−μ j⁎ Þ + μ j⁎ υ +
:
2
2
ð7Þ
The first-order condition for the optimum yields20
−
13
In the non-cooperative case we further focus only on symmetric equilibria. This
rules out equilibria in which one incumbent engages in full entry inducement against
the complementary incumbent (the complementary incumbent is displaced by an
entrant with certainty). Coalition-proofness eliminates the equilibrium in which the
incumbents engage in full entry inducement against each other (it is an equilibrium as
each incumbent's action then has no impact on its expected profit).
14
All price combinations that satisfy pI1 + pI2 = υ, where pI1 is the price set by the
incumbent i, form an equilibrium. We select the symmetric equilibrium with
υ
pI1 = pI2 = .
2
15
Each incumbent and the corresponding entrant now compete on price. This drives
the incumbents' prices to zero, and each entrant sets its price equal to the buyers'
marginal valuation Δ.
16
Price competition between the successful entrant i and the corresponding
incumbent i drives the price of the incumbent good i to zero. Let pIj and pEi denote
the prices set by the incumbent j and the successful entrant i, respectively. All price
combinations that satisfy the conditions pEi ≤ Δ and pIj + pEi − υ + Δ now form an
equilibrium. We select the equilibrium in which the entrant i and the incumbent j
Δ
divide the surplus generated by the successful entry equally, so that pIj = υ + and
2
Δ
pEi = . See Gawer and Henderson (2007) on how Intel seeks to appropriate some but
247
dμ j⁎ υ
dμ i⁎ υ
υ
Δ
Δ
+ μ j⁎
+
+ ð1−μ i⁎ Þ
+
= 0:
dαj 2
dαj 2
2
2
2
ð8Þ
An increase in the ease of entry αj against the complementary
incumbent j increases the probability of entry against the incumbent j.
This benefits the incumbent i because successful entry against the
complementary incumbent j increases the revenue of the incumbent i
υ
Δ
from to υ + when the incumbent i itself is not displaced by an
2
2
entrant. The second term on the left-hand side of the expression (8)
represents this direct effect of a change in αj. Because the potential
entrants' entry decisions are strategic complements, increased entry
against the complementary incumbent j indirectly also increases
the probability of entry against the incumbent i, which lowers the
18
To limit the scope of the analysis we assume that the fixed cost F, ai, bi and Δ are
such that the equilibrium probability of entry without any entry inducement is strictly
positive and less than one, which requires that α
__ > 0 and α
__ + β < 1. Together these two
conditions imply that β < 1.
19
When neither potential entrant is successful in entry, the profit of the incumbent i
υ
is . When only entry against the incumbent j is successful, the profit of the incumbent
2
Δ
i is υ + .
⁎ 2
d 2 ΠIi
d μ i⁎ dμ j
Δ
2β
υ
Δ
20
Because
= −2
+
υ+
=−
< 0, also the second2
2 2
dαj
order condition holds.
dαj dαj
2
ð1−β Þ
2
2
248
M. Packalen / Int. J. Ind. Organ. 28 (2010) 244–253
expected profit of the incumbent i. The first term on the left-hand side of
the expression (8) represents this indirect effect of a change in αj.
Solving the first-order condition (8) yields the (candidate) equi-
incumbents never induce entry against both incumbent goods.25
Hence, α1⁎ = α . The first-order condition for the optimal α2 is26
librium probability of entry μ ⁎ =
−
ðυ + ΔÞ−βv 21
.
ð1 + βÞðυ + ΔÞ
In order to charac-
"
terize the equilibrium in terms of straightforward and empirically
malleable economic concepts, we replace the parameters Δ and υ
γð1 + μ Þ− μ ε
μ
−
− − 23
ð1 + μ + μ ε μ Þγ.
−
γ>
− −
The incumbents induce entry in equilibrium (μ⁎ > μ_) if
1
1 + εμ
μ < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi. The intuition for the former
and
−
−
condition is that each incumbent induces entry if the relative size
of the additional surplus that each incumbent captures from entry
against the complementary incumbent is high (which is reflected in the
model by a high value of γ) because then each incumbent's benefit from
successful entry against the complementary incumbent is high.24
We now collect the results obtained in this subsection in a
proposition.
Proposition 1. When the incumbents choose the entry parameters and
prices non-cooperatively, there exists a unique symmetric equilibrium. The
equilibrium level of entry is
μ = max
8
<
:
μ;
−
9
γð1 + μ− Þ− μ ε μ =
−
−
ð1 + μ + μ ε μ Þγ;
−
:
ð9Þ
− −
The incumbents induce entry (μ⁎ > μ_) if and only if γ >
1
1
and μ < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi .
−
ð10Þ
−
μ εμ
−−
1− μ 2 ð1 + εμ Þ
−
−
⁎
"
#
#
dμ ⁎2
dμ ⁎1
Δ
Δ
+
= 0:
ð1−μ ⁎1 Þ +
ð1−μ ⁎2 Þ
υ+
2
2
dα2
dα2
μ
υ+Δ
,
υ
εμ , where μ
and the parameter β with
_ denotes
1 +μ −
the equilibrium probability of entry without entry inducement and
εμ_ denotes the reward-elasticity of a potential entrant's probability
of entry when the probability of entry equals μ_.22 This yields μ ⁎ =
with γ ≡
dμ ⁎1 ⁎
dμ ⁎2 ⁎
μ2 +
μ
dα2
dα2 1
1 + εμ
μ εμ
−−
μ 2 ð1 + ε μ Þ
−
−
−
2.3. Equilibrium analysis: the cooperative case
We now consider the case when the incumbents set prices and the
parameters α1 and α2 that influence the ease of entry cooperatively.
Because there is no double marginalization, cooperation alters neither the
equilibrium pricing in stage 3 nor the entry equilibrium in stage 2 as a
function of the parameters α1 and α2, compared to the non-cooperative
case.
In stage 1 the incumbents set α1 and α2 to maximize their combined
Δ
expected profit ΠI = ð1−μ1 μ2 Þυ + ½ð1−μ1 Þμ2 + ð1−μ2 Þμ1 . The
An increase in the ease of entry against the incumbent good 2
increases the probability that both incumbents are displaced by
entrants, and that consequently the incumbents do not receive the
Δ
revenue υ + . This effect of an increase in α2 is represented by the
2
first term on the left-hand side of the condition (10). An increase in α2
also increases the probability that only entry against one of the
incumbents is successful, and that consequently the incumbents
Δ
receive as their share of the surplus generated by entry. This effect is
2
represented by the second term on the left-hand side of the condition
(10).
Δ
½1 + β
2 − μ 1−β :27
Solving the first-order condition (10) yields μ2⁎ = 2βðυ + ΔÞ
− 2β
This can be rewritten as
μ2⁎ =
1
4
1
γ
1−
1+ μ
μ εμ
−
− −
incumbents thus induce entry (μ2⁎ > μ_) if γ
!
+ 1 − 12 μ
−
1
1−2 μ
1+ μ
μ εμ
!
−
−1
. The
− −
1
2
and μ < . The in−
−
tuition is that the incumbents induce entry if the surplus that they
capture from entry is sufficiently high (as is reflected by a high value
1
of γ) except when μ ≥ because then any increase in the probability
− 2
of entry μ 2⁎ increases the probability that both incumbents are
displaced more than it increases the probability that only one
incumbent is displaced. The result that the incumbents induce entry
in the cooperative case may seem counterintuitive, or even unrealistic. However, the observation that the cooperative case can be
interpreted as the integrated case (one firm is the incumbent in both
markets) and the observation that Intel has actively induced entry
into complementary markets even when it has been present in those
markets (see Gawer and Henderson, 2007) together demonstrate that
this prediction of the model is a plausible scenario.
We now collect the results obtained in this subsection in a
proposition.
Proposition 2. When the incumbents choose the entry parameters and
prices cooperatively, in equilibrium the incumbents induce entry against at
most one of the two incumbent goods. The equilibrium entry probabilities
are
μ ⁎2
8
8
0
1
!99
<
<
==
μ
1 +μ
1
1 @1 + −
1
−
1−
= min 1; max
μ;
+ 1A− μ
−1
:
: − 4
;;
γ
2 − μ εμ
μ εμ
− −
−
−
ð11Þ
2
21
To derive this result, substitute the expressions (5) and (6) for
dμ ⁎j
dαj
and
dμ ⁎i
,
dαj
respectively, into the Eq. (8), set μ ⁎i = μ ⁎j = μ ⁎, and solve the resulting expression for μ⁎.
22
Hence γ denotes the ratio of the buyers' valuations for an entrant good and for the
corresponding incumbent good. The reward-elasticity of entry describes what happens
to a potential entrant's probability of entry when its reward for entry increases, and is
dμ ⁎i Ri
j
: Using the expression (2) for the probability of entry
dRi μ i μ i = μ ;μ j = μ
−
−
Δ
expression Ri = ð1−μ j Þ + μ j Δ, and the definition of the parameter β given in
2
defined as ε μ ≡
−
μ ⁎i , the
and μ ⁎1 = −μ +
if γ >
1
1−2 μ
1+ μ
23
−
, the condition μ⁎ ≤ 1 is never
Because β ∈ (0, 1) implies that ε μ ∈ 0;
μ
−
−
binding.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
24
The intuition for the condition μ < 1 = 1 + ε μ is that when μ_ is high, the
−
−
υ
Δ
marginal gain ð1− μ Þ 2 + 2 of the incumbent i from an increase in entry against the
−
complementary incumbent j is low (as the incumbent itself is often displaced by an
entrant and thereby receives no revenue) and the marginal loss υ2 + μ υ2 + Δ2 of the
−
incumbent i from an increase in entry against itself (through the indirect effect) is
high (as the incumbent j is often displaced by an entrant and the incumbent i thus
often receives the additional surplus 2v + Δ2 ).
1
2
and μ < .
−
−
Fig. 1 shows the percentage difference between the expected
consumer surplus μ1⁎μ2⁎v in the non-cooperative and cooperative
−
reward-elasticity of entry in innovative environments.
!
⁎
μ Þ. The incumbents induce entry if and only
1 + μ ðμ 2 − −
−
2.4. Comparison of non-cooperative and cooperative outcomes
the expression (3), the reward-elasticity of entry can be written as εμ = β (1 + μ_) / μ_ .
See Acemoglu and Linn (2004) and Popp (2002) for recent empirical analyses of the
μ εμ
− −
25
Because
d2 ∏I
dα2j
2
−
d2 ∏I
dαj dαi
2
= −ðv + ΔÞ < 0, there are no interior maxima for
(α1, α2).
26
Because
d2 ΠI
dα22
dμ1⁎ dμ2⁎
Þðv
dα2 dα2
= −2ð
+ ΔÞ = −2
β
ð1−β2 Þ2
ðv + ΔÞ < 0, also the second-
order condition holds.
27
To derive this result, first substitute the expressions for
correspond to the expressions (5) and (6) for
dμ ⁎2
dα2
and
dμ ⁎1
,
dα2
dμ ⁎2
dα2
and
dμ 1⁎
dα2
that
and substitute μ⁎1 = α
_+
⁎
βμ ⁎2 for μ⁎
1 , into the Eq. (10), and solve the resulting expression for μ 2 . Next substitute
α
μ =
_ for αi and αj in Eq. (4) to get −
previously derived expression for μ ⁎2 .
ð1−βÞ
.
α
−
Finally, substitute ð1−βÞ μ for α
_ in the
−
M. Packalen / Int. J. Ind. Organ. 28 (2010) 244–253
249
With these assumptions, the demand for one entrant good and the
complementary incumbent good is D(p − Δ), where p is the total
price of the two goods, provided that the quality-adjusted total price
p − Δ is lower than the quality-adjusted total price of any other
available combination of two complementary goods. The demand for
two entrant goods is D(p − Δ − Δ), where p is the total price of the
two goods, provided that the quality-adjusted price of each good is
lower than the price of the corresponding incumbent good.
3.2. Equilibrium analysis: the non-cooperative case
Fig. 1. Percentage difference in expected consumer surplus in the non-cooperative case
vs. the cooperative case when μ̄ = 0.1 and γ = 2.
cases for different values of the reward-elasticity of entry when γ = 2
and μ = 0.1. In this example consumer welfare is higher in the noncooperative case than in the cooperative case (we have not found
examples of the reverse result).28 Establishing analytical conditions is
especially cumbersome due to the non-linearities in the equilibrium
entry probabilities. Instead we emphasize that the results show that
cooperation/integration between complementary incumbents can have
a large (negative) impact on consumer welfare, which implies that it is
important that the impact of cooperation/integration on entry inducement is considered in the relevant policy analyses, and that our analysis
offers constructive guidance on how the policy analyses can be
conducted in practice as it shows that the equilibrium comparison can
be determined in terms of the straightforward economic concepts that
are represented by the parameters γ, μ_ and εμ_.
We now restate as a proposition the results obtained in this
subsection.
Proposition 3. In the ease of entry model with homogenous potential
buyers, expected consumer surplus can be higher in the non-cooperative case
than in the cooperative case. Equilibrium comparisons can be determined in
terms of the probability of entry without entry inducement, the rewardelasticity of the probability of entry, and the ratio of the potential buyers'
valuation for an entrant good and for the corresponding incumbent good.
In stage 3 the incumbents and successful entrants set prices. When
neither potential entrant is successful in entry, each incumbent sets its
price equal to the double monopoly price pDM.30 When only the
potential entrant i is successful in entry, the entrant i sets its price
equal to Δ and the complementary incumbent j sets its price equal to
the monopoly price pM.31 When both potential entrants are successful
in entry, each incumbent sets its price equal to zero and each entrant
sets its price equal to Δ.
In stage 2 each potential entrant again chooses μi to maximize its
expected profit. Substituting Ri = (1 − μ j)ΔD(pM) + μ jΔD(0) for Ri in
the Eq. (2) yields the best-response curve32
μ i⁎ ðμ j Þ =
ΔDðpM Þ−ai
Δ½Dð0Þ−DðpM Þ
μj:
+
b
b
|{z}
|{z}
≡αi
≡β
ð12Þ
for the potential entrant i. With the definitions of the parameters αi
and β given in the above expression (12), the rest of the analysis of the
stage 2 is the same as the analysis of the model with homogenous
buyers that follows the expression (3).
In stage 1 each incumbent i sets αj to maximize its expected profit
h
i
π
I
Πi = ð1−μ ⁎i Þ ð1−μ ⁎j Þ DM + μ ⁎j πM ;
2
ð13Þ
3. The ease of entry model with heterogenous buyers
where πDM ≡ (pDM + pDM)D(pDM) and πM ≡ pMD(pM).33 Defining v and
π
υ
Δ
Δ such that DM = and πM = υ + , we can see that the equilibrium
In this section we extend the ease of entry model to the case with
heterogenous buyers, and compare the cooperative and non-cooperative
outcomes in the model. With heterogenous buyers also the double
marginalization externality is present.
analysis based on the above expression (13) will be identical to the
equilibrium analysis of the model with homogenous buyers that
follows the expression (7). Consequently, the equilibrium probability
of entry against each incumbent is given by the expression (9) with
2π −π
γ = M DM .
3.1. The model with heterogenous potential buyers
We keep all other aspects of the model the same as in the model
presented in Section 2.1, except that we now assume that absent entry
the potential buyers' valuation for the two incumbent goods is
heterogenous. The number of buyers with a valuation greater than p
for the two incumbent goods is denoted by D(p). The buyers' marginal
valuation for each potential entrant's good remains Δ for all buyers.29
28
For small values of the reward-elasticity consumer welfare is higher in the noncooperative case because entry inducement by the incumbent i then has only a
negligible indirect effect on the probability that the incumbent i itself is displaced and
it thus becomes optimal for each incumbent to engage in almost full entry inducement.
The kink in Fig. 1 signifies the point to the left of which the incumbents engage in full
entry inducement against one of the incumbent goods in the cooperative case (μ⁎
2 = 1).
For high values of the reward-elasticity entry inducement by the incumbent i has
almost the same impact on the probability that the incumbent j is displaced and on the
probability that the incumbent i itself is displaced, and hence also almost the same
impact impact on each incumbent's expected profit. Cooperation can thus have only a
small effect on the outcome when the reward-elasticity is high.
29
We assume that the improvement Δ is non-drastic, so that an entrant always sets
1
its price equal to Δ. With linear demand D(p) = 1 − p, the condition Δ < is a
2
sufficient condition for the potential entrant to always set its price equal to Δ.
2
2
2
πDM
3.3. Equilibrium analysis: the cooperative case
In stage 3 the incumbents and successful entrants set prices. When
neither potential entrant enters, the incumbents set the total price of the
two incumbent goods equal to the monopoly price pM. A successful
entrant always sets its price equal to Δ, and an incumbent that is not
displaced by an entrant sets its price equal to the monopoly price pM.
30
The double monopoly price pDM satisfies pDM = p̂(pDM), where p̂(pj) ≡ arg maxp{pD
(p + pj)} is the best-response curve of the incumbent i in stage 3. As was established
formally by Cournot (1838), due to “double marginalization” the total price pDM + pDM
exceeds the monopoly price pM ≡ arg maxp{pD(p)}.
31
The monopoly price is defined by pM ≡ arg maxp{pD(p)}. Price competition between
the successful entrant and the corresponding incumbent drives the incumbent's price to
zero.
32
When only one potential entrant is successful in entry, the successful entrant
receives the reward ΔD(pM) for entry. When both entrants are successful in entry, each
entrant receives the reward ΔD(0) for entry. Each entrant's expected reward for
successful entry is therefore (1 − μj)ΔD(pM) + μjΔD(0).
33
When neither potential entrant is successful in entry, the revenue of the
π
incumbent i is DM , where πDM ≡ (pDM + pDM)D(pDM). When only entry against the
2
complementary incumbent j is successful, the revenue of the incumbent i is πM, where
πM ≡ pMD(pM). Whenever entry against the incumbent i is successful, the incumbent i
receives no revenue.
250
M. Packalen / Int. J. Ind. Organ. 28 (2010) 244–253
Proposition 4. In the ease of entry model with heterogenous potential
buyers, expected consumer surplus can be higher in the non-cooperative
case than in the cooperative case, and vice versa. For a given demand curve,
equilibrium comparisons can be determined in terms of the probability of
entry without entry inducement and the reward-elasticity of the probability
of entry.
4. The price commitment model
Fig. 2. Percentage difference in expected consumer surplus in the non-cooperative
case vs. the cooperative case when μ̄ = 0.2 and demand is linear.
The analysis of the stage 2 is the same as in the non-cooperative
case because cooperation does not alter equilibrium pricing when one
or both entrants are successful in entry.
In stage 1 the incumbents set α1 and α2 to maximize their combined
expected profit ПI = (1−μ1μ2)πM, where πM ≡ pMD(pM). The incumbents receive the same combined profit πM in stage 3 when one
potential entrant is successful in entry and when neither potential
entrant is successful in entry. Consequently, the incumbents never
induce entry.
In this section we present and examine the price commitment
model. Both the entry inducement externality and the double
marginalization externality are present in the model.
4.1. The model with price Commitment
We retain all other aspects of the ease of entry model with
heterogenous buyers (see Section 3.1), except that now we assume
that the incumbents cannot influence the potential entrants' entry
costs, so that a1 = a2 = a . Instead, each incumbent can commit to a
price for its own good in stage 1. Such long-term price commitments
are common in patent licensing (see the Introduction). In stage 3 each
incumbent can decrease but not increase its price.
4.2. Equilibrium analysis: the non-cooperative case
3.4. Comparison of non-cooperative and cooperative outcomes
Fig. 2 shows the expected consumer welfare comparison for
different values of the reward-elasticity of entry when the probability
of entry without entry inducement is μ_ = 0.2 and the demand curve is
linear.34 For low values of the reward-elasticity of entry, the expected
consumer surplus is higher in the non-cooperative case than in the
cooperative case because entry inducement by the incumbent i has
only a negligible indirect effect on the probability that the incumbent i
itself is displaced by an entrant and, consequently, each incumbent
engages in almost full entry inducement in the non-cooperative
case.35 When the reward-elasticity of entry is high, the expected
consumer surplus is lower in the non-cooperative case than in the
cooperative case because neither incumbent induces entry against the
complementary incumbent in the non-cooperative case as any entry
inducement would have a large indirect impact on the probability that
the incumbent itself is displaced by an entrant.36
We now restate as a proposition these results, which show that
entry inducement can overturn Cournot's celebrated double monopoly result even in the presence double marginalization.
34
Let csM, csDM and cs0 denote the consumer surplus in stage 3 when the qualityadjusted equilibrium price in stage 3 is the monopoly price pM, the double monopoly
price pDM + pDM, and 0, respectively. The expected consumer surplus in stage 1 is
CSN = (μ⁎)2cs0 + 2 μ⁎(1 − μ⁎)csM + (1 − μ⁎)2csDM in the non-cooperative case, where
μ⁎ is given by Eq. (9) with γ =
2πM −πDM
,
πDM
2
and CSC = μ_2cs0 + (1 – μ
_ )csM in the
cooperative case, where _μ is the probability of entry without entry inducement. We
use these expressions for CSN and CSC to calculate the percentage difference (CSN
− CSC)/CSC × 100 in the expected consumer surplus between the non-cooperative and
cooperative cases.
35
There is no entry inducement in the cooperative case. The expression (9) implies
that in the non-cooperative case μ⁎ → 1 as εμ
_ → 0. With full entry inducement against
both incumbent goods, double marginalization is avoided also in the non-cooperative
case. These observations imply that the expected consumer surplus is higher in the
non-cooperative case than in the cooperative case when εμ
_ is small enough.
36
There is no entry
inducement in the cooperative case. The expression (9) implies that
μ ⁎ → max μ ;
−
γ−1
2γ
as ε μ →
−
μ
−
1+ μ
(the restriction β ∈ (0, 1) implies that ε μ ∈ð0;
−
With linear demand γ = 1.25 and, therefore, for all μ = 0:2 >
−
1:25 1
2 × 1:25
−
μ
−
1+ μ
ÞÞ.
−
elasticity of entry. Double marginalization is avoided only in the cooperative case. These
observations imply that the expected consumer surplus is higher in the cooperative case
μ
−
and the demand curve is linear.
than in the non-cooperative case when _μ >0.1, ε−
μ →
−
37
We again focus on perfectly coalition-proof subgame-perfect pure strategy Nash
equilibria. Coalition-proofness refers to the assumption that in the non-cooperative
case the incumbents coordinate on their most profitable equilibrium.
38
Price competition between the entrant and incumbent i drives the incumbent's
price to zero and the entrant sets its price equal to Δ. Absent any price commitments
by the incumbent j in stage 1, the incumbent j will set its price equal to the monopoly
price pM, where pM ≡ arg maxp{pD (p)}. Any price commitment pj by the incumbent j
in stage 1 limits the pricing of the incumbent j to p⁎j = min{pj,pM} in stage 3.
39
When both incumbents are displaced by an entrant the number of active
consumers is D(0), and each entrant's reward for successful entry is ΔD(0). When only
the incumbent i is displaced by a successful entrant, the number of active consumers is
D(pj), where pj denotes the price that a buyer must then pay for the complementary
incumbent good j in stage 3, and the entrant's reward for successful entry is ΔD(pj).
40
Combining the best-response curves for both potential entrants yields the
equilibrium entry threat
= 0:1 there is no
entry inducement in the non-cooperative case either for large enough values of the reward-
1+ μ
In stage 3 the incumbents and successful entrants set prices.37
Consider first the case when neither incumbent is displaced by an
entrant. If neither incumbent has committed to a price in stage 1, each
incumbent sets its price equal to the double monopoly price pDM, which
satisfies pDM =p̂ (pDM), where p̂ (pi) ≡ arg maxp{pD(p +pi)}. If only the
incumbent i has committed in stage 1 to a price pi that satisfies pi <pDM,
the incumbents i and j set prices p⁎i =pi and p⁎j = min {p̂ (pi), pj},
respectively. If both incumbents have committed in stage 1 to prices that
satisfy p1 <pDM and p2 <pDM, the incumbents set prices p1⁎ =p1 and
p2⁎ =p2. When only the incumbent i is displaced by an entrant, the
entrant sets its price equal to Δ and the complementary incumbent j sets
its price equal to p⁎j = min {pj, pM}.38 When both incumbents are
displaced by entrants, the price of each entrant good is Δ.
In stage 2 each potential entrant again chooses μ i to maximize its
expected profit. By substituting Ri = (1 − μj)ΔD(pj) + μ jΔD(0) for Ri in
the first-order condition (2) gives the best-response curve μ i⁎ðμ i Þ =
a
Δ
Δ
Dðpj Þ− − + ½Dð0Þ−Dðpj Þμ j for the potential entrant i.39 The
b
b
b
equilibrium is denoted by (μ1⁎, μ2⁎).40 As in the analysis of the ease
of entry model, we assume that the parameters of the model are such
that without entry inducement μ ⁎i ∈ (0.1).41 This implies that that the
μ ⁎i =
41
Δ
b
Dðpj Þ−
a
−
b
Δ
b
+ ½Dð0Þ−Dðpj Þ
Δ
b
Δ
b
Dðpi Þ−
a
−
b
This requires that
Δ
b
DðpM Þ−
a
−
b
> 0 and −
ð14Þ
:
Δ
b
1− ½Dð0Þ−Dðpj Þ ½Dð0Þ−Dðpi Þ
a
−
b
Δ
b
+ Dð0Þ < 1.
M. Packalen / Int. J. Ind. Organ. 28 (2010) 244–253
251
Δ
slope ½Dð0Þ−Dðpj Þ of each potential entrant's best-response curve is
b
positive and less than one, which in turn implies that each equilibrium
entry threat μ ⁎i is decreasing in the price commitment pj by the
dμ⁎
complementary incumbent, i < 0, and that each equilibrium entry
dpj
threat μ ⁎i is also decreasing in the price commitment pi by the
corresponding incumbent,
dμ i
dpi
< 0.42
In stage 1 each incumbent can commit to a lower than ex-post
optimal price. We assume that absent entry the goods are not strategic
complements, so that the ex-post optimal price in stage 3 never exceeds
the monopoly price pM. We can therefore restrict attention to price
commitments that are smaller than the monopoly price, p1 ≤pM and
(pj) denote the best-response correspondence of the
p2 ≤pM. Let pNC
i
⁎
incumbent i in stage 1.43 A symmetric equilibrium satisfies p⁎ =pNC
1 (p )
(p⁎). A price commitment increases an incumbent's
and p⁎ =pNC
2
expected profit by inducing entry against the complementary incumbent but also decreases the incumbent's expected profit because the
price commitment indirectly increases entry against the incumbent
itself and lowers the incumbent's price. The equilibrium impact of price
commitments on consumer welfare is examined in Subsection 4.4.
4.3. Equilibrium analysis: the cooperative case
In stage 3 the incumbents set their prices cooperatively to maximize
their combined expected profit. When neither incumbent is displaced by
an entrant, the ex-post optimal price for the bundle of both goods is the
monopoly price pM ≡arg maxp pD(p). If the incumbents have committed
in stage 1 to prices p1 and p2 for the good 1 and the good 2, respectively,
and to the price pB for the bundle of both goods, the incumbents set the
price of the bundle of both goods at pB⁎ ≡min {pM,pB,p1 +p2} in stage 3.
When at least one potential entrant is successful in entry, cooperation
in stage 3 does not impact equilibrium pricing in stage 3.44 This also
implies that the equilibrium analysis of the stage 2 is the same as in the
non-cooperative case.
In stage 1 the incumbents set p1,p2 and pB to maximize their combined
expected profit ПI = (1 − μ1⁎)(1 − μ2⁎)pBD(pB) + (1 − μ1⁎)μ2⁎p1D(p1) +
42
These two properties of the model correspond to the properties (5) and (6),
respectively, of the ease of entry model and also the intuitions are the same (see the
end of Section 2.2.2). These properties of the model imply that by committing to a
lower than ex-post optimal price pi for its good, each incumbent i can directly induce
entry against the complementary incumbent j, and that such entry inducement will
indirectly induce entry also against the incumbent i itself. The analytical proof of these
results is a straightforward calculation of
dμi⁎
dpi
and
However, the most accessible proof of the results
dμi⁎
dpj
dμi⁎
dpj
using the expression (14).
< 0 and
dμi⁎
dpi
< 0 is graphical. A
price commitment by the incumbent j decreases pj, which increases D(pj) and thereby
increases the value of the best-response curve of potential entrant i for any level of μj.
Hence, a decrease in pj shifts this best-response curve upward. Because the
corresponding best-response curve μ⁎j (μi) of the potential entrant j is increasing in μi
and remains unchanged when the price pj changes, this implies that when pj
decreases, the equilibrium probabilities of entry μ⁎i and μ⁎j must both increase.
43
The best-response correspondence pNC
i (pj) of the incumbent i in stage 1 is given by
I
I
⁎
⁎
⁎
pNC
i (pj) ≡ argmaxpi ≤ pMПi(pi, pj), where Пi(pi, pj) = (1 −μi )[(1−μj )π(pi, pj) +μi piD(pi)],
where
8
pDM DðpDM + pDM Þ if pi ≥pDM and pj ≥pDM
>
>
<
pi Dðpi + p̂ðpi ÞÞ
if pi < pDM and pj ≥ p̂ðpi Þ
πðpi ; pj Þ =
p̂ðpj ÞDð p̂ðpj Þ + pj Þ if pj < pDM and pi ≥ p̂ðpj Þ
>
>
:
pi Dðpi + pj Þ
otherwise:
The separate expressions for π(pi,pj) in the four cases arise because in the event that
neither incumbent is displaced by an entrant the equilibrium pricing in stage 3 depends on
whether either incumbent has committed to a price lower than pDM in stage 1.
44
When only the incumbent i is displaced by an entrant, in equilibrium the
incumbent sets its price equal to zero and the entrant sets its price equal to Δ. For the
complementary incumbent j, the ex-post optimal price in stage 3 is the monopoly
price pM. If the incumbents have committed to prices pj and pB in stage 1, the
equilibrium price of the good j is pj* = min {pM,pB,pj} in stage 3. When both incumbents
are displaced by entrants, each entrant sets its price equal to Δ.
Fig. 3. Percentage difference in expected consumer surplus in the non-cooperative case
vs. the cooperative case when μ̄ = 0.05 and demand is linear.
μ1⁎(1−μ2⁎)p2D(p2) subject to the constraints p1 ≤pB,p2 ≤pB, and pB ≤
p1 +p2. Without price commitments the incumbents receive the same
profit in stage 3 when one incumbent is displaced by an entrant
and when both incumbents are displaced by entrants. Therefore, the
incumbents have no incentive to induce entry and, consequently, the
equilibrium price commitments in the cooperative case are trivial, so
that p1 =p2 =pB =pM.
4.4. Comparison of non-cooperative and cooperative outcomes
Fig. 3 shows the expected consumer welfare comparison for the
case when the probability of entry against each incumbent is μ_ = 0.05
without entry inducement and the demand curve is linear.45 The
equilibrium comparisons in Fig. 3 show that entry inducement by
price commitment has the greatest effect on the equilibrium outcome
for intermediate values of the reward-elasticity of entry. The incentive
to induce entry is low for small values of the reward-elasticity because
a price commitment then has only a small effect on entry. The
incentive to induce entry is small also for large values of the rewardelasticity of entry because then the ratio of the effect that a price
commitment has on the probability of entry against the incumbent
itself and the effect that the price commitment has on the probability
of entry against the complementary incumbent is almost one.
The equilibrium comparisons in Fig. 3 also show that for some
values of the reward-elasticity of entry Cournot's celebrated double
monopoly result is overturned as consumer welfare is higher in the
non-cooperative case than in the cooperative case. The region of
parameterizations of the entry threat for which consumer welfare is
higher in the non-cooperative case than in the cooperative case is
indicated in Fig. 4. The results in Fig. 4 show that Cournot's double
monopoly result is overturned only for a very limited (almost a knifeedge) set of parameterizations of the entry threat. Moreover, consumer
45
The equilibrium in the non-cooperative case is determined by first finding all
prices p that satisfy p = pNC
i (p) and then choosing the highest price among these prices
for which the price is also a global optimum in stage 1 for each incumbent. The
expression (14) shows that the effect of cooperation on the equilibrium outcome
Δ
a
that govern the
depends on the demand function D(p) and on parameters ¼ and
b
b
Δ
a
entry threats. Because the parameters ¼ and themselves are not intuitive economic
b
b
concepts, we instead express the results in terms of the probability of entry without
price commitments, μ ≡ μi⁎ ðpM ; pM Þ =
−
probability of entry
j
a
ΔDðp Þ−¼
M
b
b
,
1−Δb ½Dð0Þ−DðpMÞ
dμi Ri
ε μ ≡ dR
i μi pi = pM ; pj = pM
−
=
1
b
and the reward-elasticity of the
ð1− μ ÞΔDðpM Þ + μ ΔDð0Þ
−
μ
−
−
, at the equilibrium
probability of entry without price commitment μ
_.
In a previous version of the paper we examined the comparison when the demand
function is log-linear (so that absent entry the goods are neither strategic substitutes
nor strategic complements, and hence pM = pDM). The welfare comparisons for the
admissible range of the parameters μ_ and εμ indicated that the expected consumer
¯¯
surplus is always higher in the cooperative case than in the non-cooperative case.
252
M. Packalen / Int. J. Ind. Organ. 28 (2010) 244–253
the probability of entry without entry inducement, and the rewardelasticity of the probability of entry, which are straightforward and
empirically malleable economic concepts.
Acknowledgments
I thank Roger Noll, Jay Bhattacharya and Jon Levin for advice, and
Susan Athey, Tim Bresnahan, Jeremy Bulow, Peter Coles, Liran Einav,
Rob McMillan, Bruce Owen, Paul Riskind, Greg Rosston, Steven Tadelis
and anonymous referees for comments and discussions.
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Fig. 4. The set of entry threats for which the expected consumer surplus is higher in the
non-cooperative case than in the cooperative case.
welfare is never more than 2% higher in the non-cooperative case than
in the cooperative case. Therefore, from a policy perspective our analysis
confirms that cooperative pricing in the form of patent pools is generally
welfare-improving; price commitment is generally not a strong enough
entry inducement mechanism to overturn this result.
We now summarize the results obtained in this subsection.
Proposition 5. With a linear demand curve, in the price commitment
model the expected consumer surplus is generally though not always
higher in the cooperative case than in the non-cooperative case.
While this analysis only shows the results for linear demand, the
analysis demonstrates how the effect of cooperation on consumer
welfare can be determined in the price commitment model for a given
demand function in terms of the probability of entry without entry
inducement and the reward-elasticity of the probability of entry.
5. Conclusion
Several commentators suggested during the Microsoft case that the
divestiture of a firm that is a monopolist in two complementary markets
might benefit consumers because two separate complementary monopolists have an incentive to induce entry against each other. An
incumbent can induce entry against a complementary incumbent by
developing and disseminating technological know-how that decreases
the cost of entry in the complementary market. This has been a common
strategy for incumbent (near) monopolists in the computer industry.
Entry inducement can also be achieved through long-term price
commitments, which are common in patent licensing. Cooperative
patent licensing through patent pools may therefore decrease consumer
welfare by eliminating the entry inducement incentive.
Our analysis provides the first formal analysis of the impact that
entry inducement has on the equilibrium comparison between the noncooperative/non-integrated outcome and the cooperative/integrated
outcome. We show that because cooperation/integration between
complementary incumbent monopolists reduces or even eliminates
the incumbents' entry inducement incentive, cooperation/integration
may decrease entry and, consequently, consumer welfare, contrary to
Cournot's celebrated double monopoly result.
We find that the equilibrium impact of entry inducement is different
when the incumbents influence the ease of entry and when they use
price commitments to induce entry. This demonstrates that when the
impacts of cooperation/integration on both entry inducement and
double marginalization are considered, the optimal policy must be
solved on a case-by-case basis and requires careful modelling of the
demand structure and the entry technology. While detailed analysis of a
particular case — such as the Microsoft case or a specific patent pool — is
beyond the scope of this paper, our analysis offers constructive guidance
for how such analysis can be conducted because we show that the
equilibrium comparison can be characterized in terms of the demand,
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