Cross-Licensing and Competition
Doh-Shin Jeon and Yassine Lefouiliy
June 7, 2013
Very preliminary and incomplete - Please do not circulate
Abstract
We study bilateral cross-licensing agreements among N (> 2) …rms that engage in competition after the licensing phase. It is shown that the most collusive cross-licensing royalty,
i.e. the one that allows the industry to achieve the monopoly pro…t, is sustainable as the
outcome of bilaterally e¢ cient agreements. When the terms of cross-licensing agreements
are not observable to third parties, the monopoly royalty is the unique symmetric bilaterally
e¢ cient royalty. However, when the terms of the agreements are public, the most competitive
royalty (i.e. zero) can also be bilaterally e¢ cient. Policy implications regading the antitrust
treatment of cross-licensing agreements are derived from these results.
Keywords: Cross-Licensing, Collusion, Antitrust and Intellectual Property.
y
Toulouse School of Economics. Email: doh-shin.jeon@tse-fr.eu
Toulouse School of Economics. Email: yassine.lefouili@tse-fr.eu
1
1
Introduction
In several industries, the intellectual property rights necessary to market a product are held by a
large number of parties. Such a situation, known as a patent thicket, arises for multiple reasons:
…rms’desire to strengthen and broaden their patent rights (Hall and Ziedonis, 2001, and Galasso
and Schankerman, 2010), the cumulative nature of technology (von Graevenitz et al., 2012), the
lack of resources and misaligned incentives at patent o¢ ces (Ja¤e and Lerner, 2004, and Bessen
and Meurer, 2008), etc. Patent thickets are particularly prevalent in areas related to information
technology such as semiconductors, audio-visual technology, telecommunications and optics (von
Graevenitz et al., 2011). For instance, according to an estimate of Intel, by 2002 over 90,000 US
patents related to central processing unit technologies had been awarded to more than 10,000
…rms, universities, government labs and independent inventors (Detkin, 2002). Patent thickets
raise many concerns and are considered as one of the most crucial intellectual property issues
of the day (Shapiro, 2007, and Régibeau and Rockett, 2011).
One well-known solution to the patent thicket problem is cross-licensing, an agreement between two …rms that grants each the right to practice the other’s patents (Shapiro, 2001, and
Régibeau and Rockett, 2011). For instance, cross-licensing is widespread in the semiconductor
industry (Grindley and Teece, 1997, Hall and Ziedonis, 2001 and Galasso, 2012). However, like
other solutions to the patent thicket problem such as patent pools and cooperative standard setting, cross-licensing may negatively a¤ect competition. More precisely, it can reduce competition
through high royalties. In particular, …rms in a duopoly can sign a cross-licensing agreement
with royalties high enough to replicate the monopoly pro…t (Fershtman and Kamien, 1992). For
this reason, competition authorities often prohibit the use of royalties that are disproportionate
with respect to the market value of the license.1 However, as long as bilateral cross-licensing
agreements are signed in a industry with more than two competing …rms, it is unclear whether
two …rms would agree on high royalties since this might weaken their competitive positions with
respect to their rivals.
In this paper, we investigate the formation of bilateral cross-licensing agreements among
N (> 2) competing …rms and assess their e¤ects on competition. Each …rm is initially endowed
with a patent portfolio and can get access to its rivals’patents through cross-licensing agreements
involving the payment of …xed fees and royalties. After the cross-licensing phase, …rms engage
in quantity competition. As a …rst step, this paper considers symmetric …rms with symmetric
patent portfolios and focuses on symmetric equilibria where all cross-licensing agreements specify
the same royalty.
We investigate the strategic e¤ects that drive the choice of royalties in a bilateral cross1
For instance, according to the Guidelines on the application of Article 81 of the EC Treaty to technology
transfer agreements (2004), “. . . Article 81(1) may be applicable where competitors cross license and impose
running royalties that are clearly disproportionate compared to the market value of the licence and where such
royalties have a signi…cant impact on market prices.”
2
licensing agreement. In the benchmark case of a multilateral licensing agreement signed by
all N …rms, the …rms would agree on the royalty that allows them to achieve the monopoly
outcome. When we consider a bilateral cross-licensing, the two …rms in an agreement will also
internalize the externalities exerted on each other. Is this the only e¤ect that matters when there
are N (> 2) competing …rms? If not, under which conditions is this e¤ect dominated by other
(potentially pro-competitive) strategic e¤ects? The answers to these questions can generate
interesting policy implications regarding the antitrust treatment of cross-licensing.
In our model, we assume that the larger the set of patents to which a …rm has access, the
lower its marginal cost. Alternatively, we can assume that the larger the set of patents to which
a …rm has access, the higher the value of its product. We actually show that our model of
cost-reducing patents can be equivalently interpreted as a model of value-increasing patents.
We distinguish publicly observable cross-licensing agreements from secret agreements and
focus on bilaterally e¢ cient agreements. A set of cross-licensing agreements is said to be bilaterally e¢ cient if each agreement maximizes the joint pro…t of the pair of …rms who signed
the agreement given all other cross-licensing agreements in the industry. In the case of public
agreements, we …nd that when two …rms decide bilateral royalties to maximize their joint pro…t,
they take into account two opposite e¤ects: the coordination e¤ ect and the Stackelberg e¤ ect.
The coordination e¤ect refers to the fact that the coalition chooses its royalties to internalize the
externalities to its members. The Stackelberg e¤ect captures the fact that …rms can in‡uence
the output levels chosen by their rivals through their choice of royalties. This e¤ect disappears
when agreements are secret.
Our main results are as follows. If agreements are secret, the unique bilaterally e¢ cient set
of symmetric cross-licensing agreements is the one involving the monopoly royalty, i.e. the one
that induces the …rms to achieve the monopoly outcome. If agreements are public, the outcome
depends on which of the Stackelberg e¤ect and the coordination e¤ect dominates the other one:
if the coordination e¤ect dominates, the unique symmetric bilaterally e¢ cient set of agreements
is the one involving the monopoly royalty; if the Stackelberg e¤ect dominates, there are two
symmetric bilaterally e¢ cient set of agreements: the one involving the monopoly royalty and
the one involving zero royalty. In both cases, the Pareto-dominant bilaterally e¢ cient royalty
is the monopoly royalty. This implies that the most collusive outcome can be sustained even
if cross-licensing agreements are decided bilaterally. We show that this result still holds in the
case of cross-licensing agreements decided by coalitions of any size comprising more than two
…rms.
Finally, we provide policy remedies to restore the most competitive outcome, i.e. the outcome
with zero royalty. The remedies consist of three elements. First, the antitrust authorities should
impose an upper bound on royalties, which is lower than the monopoly royalty. Second, the
authorities should make disclosure of royalty terms compulsory. Third, the authorities should
impose an appropriate upper bound on the size of coalitions, i.e. the number of …rms who
3
can sign a cross-licensing agreement together. These remedies allow to induce cross-licensing
agreements with zero royalty as the unique symmetric equilibrium outcome.
2
Related literature
Our paper contributes to the literature on the competitive e¤ects of cross-licensing agreements
and patent pools. In a pioneering paper, Priest (1977) shows how these two practices can be
used as a disguise for cartel arrangements. Fershtman and Kamien (1992) develop a model in
which two …rms engage in a patent race for two complementary patents and use it to shed light
on the social trade-o¤ underlying cross-licensing agreements: One the one hand, cross-licensing
improves the e¢ ciency of the R&D investments by eliminating the duplication of e¤orts. But
on the other hand, it favors price collusion between the …rms. Eswaran (1994) shows that crosslicensing can enhance the degree of collusion achieved in a repeated game by credibly introducing
the threat of increased rivalry in the market for each …rm’s product. Shapiro (2001) argues that
patent pools tend to raise (lower) welfare when patents are perfect complements (substitutes),
an idea which is generalized to intermediate levels of substitutability/complementarity by Lerner
and Tirole (2004) and further extended to the case of uncertain patents by Choi (2010). To the
best of our knowledge, the present paper is the …rst formalized study of the competitive e¤ects
of bilateral cross-licensing agreements in an industry comprised of more than two …rms.
Our paper is also related to the literature on strategic formation of networks surveyed in
Goyal (2007) and Jackson (2008). In particular, the concept of bilateral e¢ ciency we use is
similar to the widespread re…nement of pairwise stability in the network literature (Jackson
and Wolinsky, 1996).2 Goyal and Moraga-Gonzales (2001) and Goyal, Moraga-Gonzales, and
Konovalov (2008) also study two-stage games where a network formation stage is followed by a
competition stage. However, they examine ex ante R&D cooperation, while we study ex post
licensing agreements, and, contrary to the present paper, they do not allow for transfers among
…rms. Bloch and Jackson (2007) develop a general framework to examine the issue of network
formation with transfers among players. A crucial di¤erence between their framework and our
setting is that we allow the agreements among …rms to involve the payment of per-unit royalties
on top of …xed fees while they only consider the case of lump-sum transfers.
More generally, our paper is related to the literature on negotiations and cooperative arrangements in industrial organization (e.g. Gans and Inderst, 2001, 2003, and, for a detailed survey,
Gans and Inderst, 2007). Of particular interest to us is the paper by De Fontenay and Gans
(2012) who provide an analysis of a non-cooperative pairwise bargaining game between agents
in a network and show that the equilibria of their bargaining game are those that result in bilaterally e¢ cient agreements. This …nding supports our focus on bilaterally e¢ cient agreements.
2
Our concept of bilateral e¢ ciency is also similar to the bilateral contracting principle used in the vertical
relations literature (Hart and Tirole, 1990; Whinston, 2006).
4
3
The Model
3.1
The model of cost-reducing patents
Consider an industry consisting of N
Each …rm owns one
patent3
3 symmetric …rms producing a homogeneous good.
covering a cost-reducing technology and can get access to its rivals’
patents through cross-licensing agreements. We assume that the patents are symmetric in the
sense that the marginal cost of a …rm only depends on the number of patents it has access to.
Let c(n) be a …rm’s marginal cost when it has access to a number n 2 f1; :::; N g of patents with
c(N )(
c)
c(N
1)
:::
c(1)(
c) and let c
marginal costs of all …rms in the industry. Denote
(c1 ; :::; cN ) be the vector representing the
c
c
c.
Let us assume that, prior to engaging in Cournot competition, each pair of …rms can sign
a cross-licensing agreement whereby each party gets access to the patented technology of the
other one. More precisely, the …rms play the following two-stage game:
Stage 1: Cross-licensing.
Each pair of …rms (i; j) decides whether to sign a cross-licensing agreement and determine
the terms of the agreement if any. We assume that a bilateral cross-licensing agreement between
2 and a lump-sum transfer F
…rm i and …rm j speci…es a pair of royalties (ri!j ; rj!i ) 2 R+
i!j 2 R;
where ri!j (resp. rj!i ) is the per-unit royalty paid by …rm i (resp. …rm j) to …rm j (resp. …rm
i) and Fi!j is a transfer made by …rm i to …rm j (which is negative if …rm i receives a transfer
from …rm j). Assume that all bilateral negotiations occur simultaneously. For the purpose of
this paper we do not need to specify the underlying pairwise bargaining game: we will focus on
the negotiation outcomes (and whether these satisfy a given property) and set aside the way
such outcomes are achieved.
Stage 2: Cournot competition.
Firms compete à la Cournot with the cost structure inherited from Stage 1.
Depending on the observability of royalties signed by i and j to other …rms, we distinguish
between public cross-licensing and secret cross-licensing. In the case of public cross-licensing,
all …rms observe all the cross-licensing agreements signed at stage 1 before they engage in
Coutnot competition. In contrast, in the case of secret cross-licensing, the terms of crosslicensing agreement signed between i and j remain known only to i and j and therefore each
…rm k should form a conjecture about the licensing terms signed between i and j.
We assume that the …rms face an inverse demand function P ( ) satisfying the following
standard conditions (Novshek, 1985):
3
Given that we consider that …rms have symmetric patent portfolios, we can assume, without loss of generality,
that each …rm has one patent.
5
A1 P ( ) is twice continuously di¤erentiable and P 0 ( ) < 0 whenever P ( ) > 0:
A2 P (0) > c > c > P (Q) for Q su¢ ciently high.
A3 P 0 (Q) + QP 00 (Q) < 0 for all Q
0 with P (Q) > 0.
These mild assumptions ensure the existence and uniqueness of a Cournot equilibrium
(qi )i=1;:::;n satisfying the following (intuitive) comparative statics properties, where ci denote
X
@q
@q
…rm i’s marginal cost: @cii < 0 and @cij > 0 for any j 6= i; @Q
<
0
for
any
i,
where
Q
=
qi
@ci
i
is the total equilibrium output; and
@ i
@ci
< 0 and
@ i
@cj
equilibrium pro…t (see e.g. Amir et al., 2012).
3.2
> 0 for any j 6= i, where
i
is …rm i’s
Alternative equivalent formulation: value-increasing patents
Instead of assuming that access to more patents reduces a …rm’s marginal cost, we can assume
that access to more patents increases the value of the product produced by the …rm. We below
show that our model of cost-reducing patents can be equivalently interpreted as a model of
value-increasing patents.
We consider a constant symmetric marginal cost c for all …rms. Each …rm has one patent.
Let v(n) represent the value of the product produced by a …rm when the …rm has access to
n 2 f1; :::; N g number of distinct patents with v(N )
v
v(N
1)
:::
v(1)(
v). Let
(v1 ; :::; vN ) be the vector representing the value of each …rm’s product after the licensing
stage.
We de…ne Cournot competition for given v
ously chooses its quantity qi . Given v
(v1 ; :::; vN ) as follows. Each …rm i simultane-
(v1 ; :::; vN ), q
(q1 ; :::; qN ) and Q = q1 + ::: + qN the
quality-adjusted equilibrium prices are determined by the following two conditions:
indi¤erence condition:
vi
pi = vj
pj
for all (i; j) 2 f1; :::; N g2 ;
market-clearing condition:
Q = D(p) where pi = p + vi
v:
In other words, p is the price for the product of a …rm which has access to its own patent only.
The market clearing condition means that this price is adjusted to make the total supply equal to
the demand. The indi¤erence condition implies that the price each …rm charges is adjusted such
that all consumers who buy any product are indi¤erent among all products. A microfoundation
of this setup can be provided as follows. There is a mass one of consumers. Each consumer has
a unit demand and hence buys at most one unit among all products. A consumer’s gross utility
6
from having a unit of product of …rm i is given by u + vi : u is speci…c to the consumer while vi is
common to all consumers. Let F (u) represent the cumulative distribution function of u: Then,
by construction of quality-adjusted prices, any consumer is indi¤erent among all products and
the marginal consumer indi¤erent between buying any product and not buying is characterized
by u + v
p = 0, implying
D(p) = 1
In equilibrium, p is adjusted such that 1
F (p
F (p
v):
v) = Q.
Let P (Q) be the inverse demand curve. In the equilibrium, a …rm’s pro…t is given by
i
0
= @P (Q) + vi
v
X
c
j6=i
1
ri!j A qi +
X
rj!i qj :
j6=i
After making a change of variables as follows
c
(vi
v) = ci ;
the pro…t can be equivalently written as
i
0
= @P (Q)
ci
X
j6=i
1
ri!j A qi +
X
rj!i qj ;
j6=i
which is the pro…t expression in our original model of cost-reducing patents. In particular, the
change of variables implies
c
(v(n)
v) = c(n);
which in turn implies
v(n + 1)
v(n) = c(n)
c(n + 1):
Therefore, our model of cost-reducing patents can be equivalently interpreted as a model
of value-increasing patents. In what follows, for our formal analysis, we stick to the model of
cost-reducing patents.
4
Benchmark: multilateral licensing agreement
We here consider the case of a multilateral licensing agreement signed by all N …rms. We focus
on a symmetric outcome where all …rms pay the same royalty r to each other.
7
Let P m (c) be the monopoly price when each …rm’s marginal cost is c. It is characterized by
P m (c) c
1
=
:
m
m
P (c)
"(P (c))
(1)
where "(:) is the elasticity of demand.
Given a symmetric r, each …rm’s marginal cost is c + (N
1)r. The …rms will agree on a
royalty to achieve the monopoly price. Given a symmetric r, at stage 2, …rm i chooses qi to
maximize [P (Q
i
+ qi )
c
(N
1)r] qi + rQ
where Q
i
…rms. Note that this is the same as maximizing [P (Q
considered by …rm i as given. Let
P m (c).
rm
i
i
is the quantity chosen by all other
+ qi )
c
(N
1)r] qi when Q
i
is
be the royalty that allows to achieve the monopoly price
Then, from the …rst-order condition associated to …rm i’s maximization program, we
have
P m (c)
c (N
P m (c)
1)rm
=
1
"(P m (c))N
:
(2)
From (1) and (2), rm is determined by
N [P m (c)
which is equivalent to
c
1)rm ] = P m (c)
(N
P m (c)
N
c
c;
= rm :
(3)
Now we examine under which condition the multilateral licensing agreement will lead to a
higher downstream price compared to the situation before cross-licensing. Let Q (N; c) represent
the industry output when N …rms with the marginal cost c compete in quantity. Therefore, the
equilibrium price prior to licensing is P (Q (N; c)). At P (Q (N; c))., we have
P (Q (N; c)) c
1
=
P (Q (N; c))
"(P (Q (N; c)))N
Therefore, P m (c) = P (Q (N; c)) if
N [P (Q (N; c))
c] = P (Q (N; c))
c;
which is equivalent to
c = (N
1) [P (Q (N; c))
Note that the right hand side of (4) does not depend on
c] :
(4)
c. As the cost reduction from
licensing increases, c is smaller and the monopoly price is smaller. Therefore, we must have
P m (c) T P (Q (N; c)) if and only if
c S (N
1) [P (Q (N; c)):
c].
Proposition 1 (Multilateral licensing). Assume that A1-A3 hold and that all …rms in the
8
industry jointly agree on a symmetric royalty. Then they agree on rm = (P m (c)
c) =N to
achieve the monopoly price P m (c). Moreover:
(i) If
c < (N
1) (P (Q (N; c))
c), the post-licensing monopoly price is higher than the
pre-licensing equilibrium price.
(ii) If
c = (N
1) (P (Q (N; c))
c), the post-licensing monopoly price is the same as the
pre-licensing equilibrium price.
(iii) If
c > (N
1) (P (Q (N; c))
c), the post-licensing monopoly price is lower than the
pre-licensing equilibrium price.
Note that when
c = 0, i.e. when patents are perfect substitutes, the multilateral licensing
agreement described above serves as a purely collusive device.
5
Characterization of the bilaterally e¢ cient public agreements
We now characterize the bilaterally e¢ cient cross-licensing outcomes, i.e. the outcomes such
that that each pair of …rms (i; j) does not …nd it jointly optimal to modify the terms of the
agreement they signed. In other words, each cross-licensing agreement is required to maximize
the joint payo¤ of the two parties involved in it, given all other cross-licensing agreements.
In this section we focus on cross-licensing agreements that are observable to all …rms in the
industry.
5.1
Preliminaries
Assume that all …rms are always active (i.e. produce a positive output) whatever the crosslicensing agreements that are signed. Under this assumption, any given pair of …rms …nd it
(joitly) optimal to license each other. To see why, assume that …rm i does not license its patent
to j. These two …rms can (weakly) increase their joint pro…ts if i licenses its patent to j by
specifying the payment of a per-unit royalty rj!i equal to the reduction in marginal cost allowed
by j’s use of i’s patent. Such licensing agreement would not a¤ect the level of joint output but
will (weakly) decrease the cost of …rm j. It will therefore (weakly) increase their joint pro…ts.
Consider now a symmetric situation where all …rms cross-license each other and let r denote
the (common) per-unit royalty paid by any …rm i to have access to the patent of any …rm j 6= i
with i; j = 1; :::; N . Let S(r; N ) denote such a set of cross-licensing agreements. We below study
the joint incentives of a coalition of two …rms, say 1 and 2, to deviate from the symmetric royalty
r. The next lemma shows that it is su¢ cient to focus on deviations specifying identical royalties
between the two …rms in the coalition. Indeed the joint payo¤ of any asymmetric deviation can
be replicated by a symmetric one as the joint payo¤ depends on the royalties paid by each …rm
of the coalition to the other only through their sum.
9
Lemma 1 Consider a symmetric set of cross-licensing agreements S(r; N ). The joint payo¤ a
coalition fi; jg gets from a deviation to a cross-licensing agreement in which …rm i (resp. …rm
j) pays a royalty ri!j (resp. rj!i ) to …rm j (resp. …rm j) depends on (ri!j ; rj!i ) only through
the sum ri!j + rj!i .
Proof. See Appendix.
This lemma shows that to determine whether the set S(r; N ) of cross-licensing agreements
is bilaterally e¢ cient, it is su¢ cient to investigate the incentives of the coalition to deviate by
changing the royalty they charge each other from r to r^ 6= r.
For given (r; r^), let Q (r; r^) denote the total industry output and Q12 (r; r^) denote the sum
of the outputs of …rms 1 and 2 in the (second-stage) equilibrium of Cournot competition. Let
Q
^)
12 (r; r
Q (r; r^)
Q12 (r; r^) denote the total equilibrium output of their rivals. Then, the
considered set of symmetric agreements is bilaterally e¢ cient if and only if:
r 2 Arg max (
1
r^ 0
+
^)
2 ) (r; r
where:
(
1
+
^)
2 ) (r; r
= [P (Q (r; r^))
(c + (N
= [P (Q12 (r; r^) + BR
where BR
12 (:)
2) r)] Q12 (r; r^) + 2rQ
^)))
1;2 (Q12 (r; r
(c + (N
is de…ned as follows. If N = 3; then BR
of …rm 3. If N
4, then BR
12 (:)
12 (:)
^)
12 (r; r
2) r)] Q12 (r; r^) + 2rBR
is the best-response function
is the aggregate response of the coalition’s rivals, de…ned
as follows: For any joint output Q12 = q1 + q2 of …rms 1 and 2, BR
real number such that
BR
12 (Q12 )
N 2
producing qi and each …rm j
^))
12 (Q12 (r; r
is the best-response of any …rm j
3 producing
BR
12 (Q12 )
N 2
12 (Q12 )
is the (unique)
3 to each …rm i = 1; 2
:
After observing the coalition’s deviation to r^ 6= r, the rivals of the coalition expect the coali-
tion to produce Q12 (r; r^) and will best respond to this quantity by producing BR
^))
12 (Q12 (r; r
in aggregate. In other words, from a strategic point of view, the coalition’s deviation to r^ 6= r is
equivalent to its committment to produce Q12 (r; r^) as a Stackelberg leader. We therefore de…ne
the following two-stage Stackelberg game.
De…nition: For any r
0 and N
3, the Stackelberg game G(r; N ) is de…ned by the
following elements:
Players: Coalition {1,2} and …rms j = 3; :::; N ; each player has a marginal production cost
c (excluding royalties) and pays a per-unit royalty r to each of the other players.
Actions: The coalition {1,2} chooses its (total) output Q12 within the interval I(r; N )
[Q12 (r; 1); Q12 (r; 0)] and each …rm j
3 can choose any quantity qj
Timing: There are two stages.
10
0.
- Stage 1: The coalition {1,2} acts as a Stackelberg leader and chooses Q12 within the interval
[Q12 (r; 1); Q12 (r; 0)] :
- Stage 2: If N = 3, then …rm 3 chooses its output q3 (given Q12 ). If N
j
4, then …rms
3 compete in quantities (given Q12 ).
The following lemma shows that the set of cross-licensing agreements S(r; N ) is bilaterally
e¢ cient if and only if choosing Q12 (r; r) is optimal for the coalition in the Stackelberg game
G(r; N ):
Lemma 2 The symmetric set of cross-licensing agreements S(r; N ) is bilaterally e¢ cient if and
only if choosing to produce Q12 (r; r) is a subgame-perfect equilibrium strategy of the coalition
{1,2} in the Stackelberg game G(r; N ).
Proof. See Appendix.
5.2
Incentives to deviate: downstream and upstream pro…ts
In what follows we identify two opposite e¤ects on the incentives of the coalition f1; 2g to
marginally expand or contract its output with respect to Q12 (r; r). The coalition’s pro…t can be
rewritten as
12 (Q12 ; r)
The term
= [P (Q12 + BR
|
D (Q ; r)
12
12
12 (Q12 ))
{z
(c + (N
D (Q ;r)
12
12
1) r)] Q12 + r [Q12 + 2BR
} |
{z
12 (Q12 )] :
U (Q ;r)
12
12
}
(5)
represents the coalition’s downstream market pro…t from selling its prod-
ucts when its marginal cost is given by c + (N
1) r. The term
U (Q ; r)
12
12
represents its
upstream market pro…t from licensing its patents and includes the payment made by each …rm
within the coalition to the other one. We below study the e¤ect of a (local) variation of Q12 on
each of the two sources of pro…t.
E¤ect on the downstream market pro…t
Starting with the e¤ect on the coalition’s downstream market pro…t
@ D
12
(Q12 ; r) = P 0 (Q12 + BR
@Q12
[P (Q12 + BR
0
12 (Q12 )) Q12 BR 12 (Q12 )
12 (Q12 ))
(c + (N
D (Q ; r),
12
12
+ P 0 (Q12 + BR
we have
12 (Q12 )) Q12
+(6)
1) r)]
which, when evaluated at Q12 (r; r), is given by
@ D
12
(Q (r; r); r) = P 0 (Q (r; r)) Q12 (r; r)BR0 12 (Q12 (r; r)) + P 0 (Q (r; r)) Q12 (r; r) + (7)
@Q12 12
[P (Q (r; r)) (c + (N 1) r)] :
11
The term P 0 (Q (r; r)) Q12 (r; r)BR0
12 (Q12 (r; r))
> 0 in (7) captures the (usual) Stackelberg
e¤ ect: the leader has an incentive to increase its output Q12 above the Cournot level Q12 (r; r)
because such an increase will be met with a decrease in the aggregate output of the followers
(one can easily check that BR0
term as
2 [P (Q (r; r))
12 (Q12 (r; r))
(c + (N
< 0). It will turn out to be useful to rewrite this
1) r)] BR0
12 (Q12 ),
which is obtained from the F.O.C. of
…rm 1 (or 2) in the Cournot game:
P 0 (Q (r; r))
Q12 (r; r)
+ P (Q (r; r))
2
The term P 0 (Q (r; r)) Q12 (r; r) + [P (Q (r; r))
(c + (N
(c + (N
1) r) = 0:
(8)
1) r)] in (7) represents the marginal
downstream pro…t of the coalition in a setting where it would play a simultaneous quantitysetting game. This term captures a coordination e¤ ect: the coalition has an incentive to reduce
output below the Cournot level Q12 (r; r) since the joint output of the coalition when each member
chooses its quantity in a non-cooperative way is too high with respect to what maximizes its
joint downstream pro…t (in a simultaneous quantity-setting game). Indeed, using again (8), we
have:
P 0 (Q (r; r)) Q12 (r; r) + [P (Q (r; r))
(c + (N
1) r)] =
[P (Q (r; r))
(c + (N
1) r)] < 0:
Therefore, the overall marginal e¤ect of a local increase of Q12 (above the Cournot level
Q12 (r; r)) on the coalition’s downstream pro…t is given by:
@ D
12
(Q (r; r); r) =
@Q12 12
[P (Q (r; r))
(c + (N
1) r)] [1 + 2BR0
12 (Q12 (r; r))]:
(9)
The …rst term between brackets in (9) is positive while the sign of the second term between
brackets in (9) is ambiguous. If the aggregate response of the coalition’s rival is relatively
strong, such that 1 + 2BR0
12 (Q12 (r; r))
< 0, then the Stackelberg e¤ect dominates the coor-
dination e¤ect and the coalition has an incentive to increase its quantity above Q12 (r; r), i.e.
@ D
12
@Q12 (Q12 (r; r); r)
> 0. In contrast, if the aggregate response of the coalition’s rival is rela-
tively weak, such that 1 + 2BR0
12 (Q12 (r; r))
< 0, then the coordination e¤ect dominates the
Stackelberg e¤ect and the coalition has an incentive to reduce its quantity below Q12 (r; r), i.e.
@ D
12
@Q12 (Q12 (r; r); r)
< 0.
E¤ect on the upstream market pro…t
Let us now turn to the e¤ect of a local variation in Q12 on the coalition’s upstream market
pro…t
U (Q ):
12
12
We have:
@ U
12
(Q (r; r); r) = r 1 + 2BR0
@Q12 12
12
12 (Q12 (r; r))
:
(10)
One the one hand, a marginal increase in Q12 results in an increase of r=2 in the licensing
revenues each …rm of the coalition gets from the other member. On the other hand, a marginal
increase in Q12 induces the rivals of the coalition to reduce their output by BR0
hence results in a reduction of r
BR0
12 (Q12 )
12 (Q12 )
and
in the licensing revenues that each …rm of the
coalition gets from the rivals. If the Stackelberg e¤ect dominates the coordination e¤ect such
that 1 + 2BR0
12 (Q12 )
< 0 holds, then the e¤ect on the "extra-coalition" licensing revenues
dominates the e¤ect on the "intra-coalition" licensing revenues such that an increase in Q12
reduces the upstream licensing revenue. The reverse holds if 1 + 2BR0
12 (Q12 )
> 0.
The e¤ect of an increase in Q12 (above the Cournot level Q12 (r; r)) on each of the coalition’s downstream and upstream market pro…ts is ambiguous. What is very important to notice
in (9) and (10) is that the e¤ ect on the downstream pro…t is always opposite to the e¤ ect on
the upstream pro…t (whenever 1 + 2BR0
12 (Q12 (r; r))
6= 0). For instance, when the Stackel-
berg e¤ect dominates the coordination e¤ect (i.e., 1 + 2BR0
1;2 (Q12 (r; r))
< 0), the marginal
e¤ect of Q12 on the downstream pro…t gives output-expanding incentives to the coalition, i.e.
@ D
12
@Q12 (Q12 (r; r); r) > 0, while the marginal e¤ect on the upstream pro…t yields output-contracting
@ U
incentives, i.e. @Q12
(Q12 (r; r); r) < 0. This is because the contraction in the output of the rivals
12
increases the coalition’s downstream pro…t while reducing its upstream pro…t. The reverse holds
when when the coordination e¤ect dominates the Stackelberg e¤ect.
5.3
Incentives to deviate: four cases
From the analysis of the previous subsection, by summing up (9) and (10), the total e¤ect of a
marginal increase in Q12 on the coalition’s pro…t can be described in a simple way as:
@ 12
(Q (r; r); r) = [c + N r
@Q12 12
P (Q (r; r))] [1 + 2BR0
12 (Q12 (r; r))]:
(11)
We can distinguish four cases depending on whether or not the Stackelberg e¤ect dominates the
coordination e¤ect and which is more important between the e¤ect of a local deviation on the
downstream pro…t and the one on the upstream pro…t.
Stackelberg e¤ect vs. coordination e¤ect
Let us …rst examine the term 1 + 2BR0
12 (Q12 (r; r))
which determines whether or not the
Stackelberg e¤ect dominates the coordination e¤ect. The FOC for the maximization program
of any …rm j
3, when the coalition produces a given quantity Q12 , can be written as:
P 0 (Q12 + BR
12 (Q12 )
BR 12 (Q12 )
+ P (Q12 + BR
N 2
13
12 (Q12 )
(c + (N
1) r) = 0:
Di¤erentiating the latter with respect to Q12 yields (dropping the argument (r; r))
BR0
12 (Q12 )
=
P 00 (Q)BR
P 00 (Q)BR
which, combined with P 0 (Q) < 0, proves that
2)P 0 (Q)
1)P 0 (Q)
12 (Q12 )
+ (N
12 (Q12 ) + (N
1 < BR0
12 (Q12 )
< 0 (a result that will be useful
later), and, when evaluated at Q12 = Q12 , also yields that
1 + 2BR0
P 00 (Q )BR
P 00 (Q )BR
12 (Q12 ) =
N 2 00
N P (Q
N 2 00
N P (Q
=
because BR
N 2
N Q as
scenarios P 00 (Q )
12 (Q12 )
between the two
=
12 (Q12 )
+ (N
12 (Q12 ) + (N
3)P 0 (Q )
1)P 0 (Q )
)Q + (N
3)P 0 (Q )
)Q + (N
1)P 0 (Q )
the corresponding equilibrium is symmetric. Distinguishing
0 and P 00 (Q ) > 0 and using the fact that BR
12 (Q12 )
Q , one can easily show that A3 implies that the denominator is always negative. Therefore,
from P 0 (Q ) < 0, it follows that
1 + 2BR0
12 (Q12 ) Q 0 ()
Q P 00 (Q )
R
P 0 (Q )
N (N 3)
N 2
This shows that the coordination e¤ect dominates the Stackelberg e¤ect (1 + 2BR0
if and only if the slope of the inverse demand is positive
00
( QPP0 (Q(Q) )
<
N (N 3)
N 2 ).
(P 00 (Q
(12)
12 (Q12 )
> 0)
) > 0) and su¢ ciently elastic
The main intuition behind this result follows from the fact that a local
increase in Q12 a¤ects the marginal revenue P 0 (Q) qj +P (Q) of a …rm j
3 outside the coalition
through two channels: it has an e¤ect on both the inverse demand P (Q) and its slope P 0 (Q).
If the magnitude of the e¤ect on P 0 (Q), which is proportional to P 00 (Q), is high relative to the
magnitude of the e¤ect on P (Q), which is proportional to P 0 (Q), then the magnitude of the
adjustment that each rival of the coalition has to make to equalize its marginal revenue to its
marginal cost will be relatively low. In particular, for N
4, under A3, the Stackelberg e¤ect
always dominates the coordination e¤ect.
Downstream pro…t vs. upstream pro…t
Let us now examine the term f (r; N )
c+N r P (Q (r; r)). The e¤ect of an increase in Q12
on the downstream pro…t is more important than the e¤ect on the upstream pro…t if f (r; N ) < 0.
For instance, when r = 0, there is no upstream pro…t and we have f (0; N ) = c P (Q (0; 0)) < 0.
Intuitively, we expect that the upstream pro…t becomes more important as r increases, which
turns out to be true as we below show that
@f
@r (r; N )
=N
@Q
@r
P 0 (Q ) > 0: Adding the FOCs
for all …rms’ maximization program, when all of them produce at the e¤ective marginal cost
14
c + (N
1)r, yields:
P 0 (Q ) Q + N P (Q )
N (c + (N
1)r) = 0:
Di¤erentiating the latter with respect to r, we get
@Q
@r
P 0 (Q ) + P 00 (Q ) Q
+ N P 0 (Q )
From P 0 (Q ) + P 00 (Q ) Q < 0 (by A3) and
which implies that
@f
@r (r; N )
> 0 for any N
@Q
@r
@Q
@r
(N
1) = 0:
< 0, it follows that P 0 (Q ) @Q
@r
(N
1) < 0,
3. Since f (r; N ) strictly increases with r, we
expect that there exists a unique r > 0 at which c + N r
P (Q (r; r)) = 0.
Surprisingly, it turns out that the unique r at which f (r; N ) = 0 is rm in (3). At r = rm ,
we have that P (Q (rm ; rm )) = P m and, therefore, c + N rm
P (Q (rm ; rm )) = 0. Thus, for
r < rm , the downstream pro…t is more important than the upstream one and the reverse holds
for r > rm .
5.4
Bilaterally e¢ cient royalties
From the previous analysis of local deviations, we know that there are four possible cases. First,
when the Stackelberg e¤ect dominates the coordination e¤ect and the downstream pro…t e¤ect
is more important than the upstream pro…t e¤ect (i.e. r belongs to [0; rm ), the coalition has an
incentive to decrease its royalty in order to reduce the rivals’ outputs, which generates r = 0
as the unique potential bilateral e¢ cient royalty among royalties r in [0; rm ). Second, when
the Stackelberg e¤ect dominates the coordination e¤ect but the upstream pro…t e¤ect is more
important than the downstream pro…t e¤ect (i.e. r > rm ), the coalition has an incentive to
increase its royalty to boost the rivals’production and thereby the licensing revenue from the
rivals. At r = rm , the downstream pro…t e¤ect is equal to the upstream pro…t e¤ect and the
coalition has no incentive to deviate at least locally. In summary, when the Stackelberg e¤ect
dominates the coordination e¤ect, there are two potential bilaterally e¢ cient royalties: r = 0
and r = rm .
Third, when the coordination e¤ect dominates the Stackelberg e¤ect and the downstream
pro…t e¤ect is more important than the upstream pro…t e¤ect (i.e. r belongs to (0; rm )), the
coalition has an incentive to increase its royalty to make the retail price as close as possible to the
monopoly price. In a symmetric way, when the coordination e¤ect dominates the Stackelberg
e¤ect but the upstream pro…t e¤ect is more important than the downstream pro…t e¤ect (i.e.
r > rm ), the coalition has an incentive to decrease its royalty to make the retail price as close
as possible to the monopoly price. In summary, when the coordination e¤ect dominates the
Stackelberg e¤ect, we have a unique potential bilaterally e¢ cient royalty: r = rm . Summarizing,
we have:
15
Lemma 3 Under A1-A3, from the local analysis of the Stackelberg game G(r; N ), we obtain the
following candidates for the bilaterally e¢ cient symmetric royalties:
(i) If the Stackelberg e¤ ect dominates the coordination e¤ ect
there are two candidate royalties, r = 0 and r =
rm ,
r=
>
N (N 3)
N 2
, then
for which S(r; N ) can be bilaterally e¢ cient.
(ii) If the coordination e¤ ect dominates the Stackelberg e¤ ect
rm
QP 00 (Q)
P 0 (Q)
QP 00 (Q)
P 0 (Q)
<
N (N 3)
N 2
, then
is the only possible value of r for which S(r; N ) can be bilaterally e¢ cient.
In order to determine the values of r for which S(r; N ) is indeed bilaterally e¢ cient, we need
to switch from a local analysis (which provides necessary conditions for S(r; N ) to be bilaterally e¢ cient) to a global analysis (which con…rms or denies that the potential candidates are
bilaterally e¢ cient). The following proposition characterizes the bilaterally e¢ cient symmetric
agreements depending on whether the Stackelberg e¤ect dominates the coordination e¤ect.
Proposition 2 (public cross-licensing) Under A1-A3, the following holds:
(i) If the Stackelberg e¤ ect dominates the coordination e¤ ect
S(r; N ) is bilaterally e¢ cient if and only if r 2
f0; rm g.
S(r; N ) is bilaterally e¢ cient if and only if r =
rm .
(ii) If the coordination e¤ ect dominates the Stackelberg e¤ ect
QP 00 (Q)
P 0 (Q)
>
N (N 3)
N 2
, then
QP 00 (Q)
P 0 (Q)
<
N (N 3)
N 2
, then
Proof. See Appendix.
This proposition shows that the monopoly royalty is the Pareto-dominant bilaterally e¢ cient royalty regardless of whether the Stackelberg e¤ect dominates or is dominated by the
coordination e¤ect.
6
Secret cross-licensing
In the previous section, we considered public cross-licensing agreements. In this section, we
consider the alternative scenario of secret cross-licensing: each bilateral cross-licensing agreement
is only observable to the two …rms involved in it. Hence, at the beginning of stage 2, …rm i is
aware of only the royalties of the licensing contracts that it itself signed. Regarding the royalties
agreed on between …rm j(6= i) and k (6= i), i should form an expectation. In a symmetric
equilibrium in which every pair of …rms agree on the same royalty r, every …rm should believe
that every pair of …rms agreed on r on the equilibrium path. We here make an assumption of
passive beliefs: regardless of the terms of cross-licensing signed between i and j in the licensing
stage, i maintains the same belief about the terms signed by any pair of rivals and believes that
every pair of rivals agreed on r.
Then, we can show that Lemma 1 continues to hold in the case of secret cross-licensing and
hence, without loss of generaltiy, we can restrict a coalition to deviate with a symmetric royalty.
Then, a result similar to Lemma 2 holds in that the symmetric set of cross-licensing agreements
16
S(r; N ) is bilaterally e¢ cient if and only if choosing to produce Q12 (r; r) is optimal to the
coalition {1,2}. The major di¤erence between public cross-licensing and secret cross-licensing
is that in the latter case, there is no Stackelberg e¤ect: formally speaking, the analysis under
secret cross-licensing can be derived from that of public cross-licensing by setting BR0
zero. This implies that
S(rm ; N )
12 (Q12 )
to
is the unique bilaterally e¢ cient set of symmetric agreements.
Proposition 3 (secret cross-licensing) Under A1-A3, in the two-stage game of secret crosslicensing followed by Cournot competition, S(rm ; N ) is the unique bilaterally e¢ cient set of
symmetric cross-licensing agreements.
Therefore, secret cross-licensing strengthens the …nding under public cross-licensing in Proposition 2 since the monopoly outcome is the unique symmetric outcome under secret crosslicensing.
7
Robustness to coalitions of any size
In this section, we show that the main results previously obtained by considering a coalition of
size 2 extend to a coalition of any given size k in f3; :::; N g. We will say that a set of cross-
licensing agreements is k-e¢ cient if no coalition of size k …nds it optimal to change the terms of
the cross-licensing agreements between the members of the coalition. Since the case of k = N
was studied in the benchmark of multilateral licensing by all …rms in the industry, we study a
coalition of size k in f3; :::; N
1g, …rst in the case of public cross-licensing and then consider
the case of secret cross-licensing.
Consider the deviation of a coalition composed of f1; :::; kg in the licensing stage. Lemma
1 continues to hold in the case of coalition of size k and hence, without loss of generality,
we can restrict attention to deviations involving a symmetric royalty r^. For given (r; r^), let
Q (r; r^) denote the total industry output and Qk (r; r^) denote the sum of the outputs of the
…rms in the coalition in the (second-stage) equilibrium of Cournot competition. Let Q
Q (r; r^)
^)
k (r; r
Qk (r; r^) denote the total equilibrium output of the coalition’s rivals.
Denoting Qk the total quantity produced by the considered coalition and r the common
royalty paid to the …rms outside the coalition, the coalition’s pro…t can be rewritten as
k
(Qk ; r) = [P (Qk + BR
|
k (Qk ))
{z
(c + (N
1) r)] Qk + r [(k
} |
D (Q ;r)
k
k
1)Qk + kBR
{z
U (Q ;r)
k
k
k (Qk )] :
}
(13)
Expression (13) generalizes (5). Suppose that the coalition marginally expands or contracts its
output Qk with respect to Qk (r; r). Then, we have
@ D
k
(Q (r; r); r) =
@Qk k
[P (Q (r; r))
(c + (N
17
1) r)] [k
1 + kBR0 k (Qk (r; r))]:
@ U
k
(Q (r; r); r) = r k
@Qk k
1 + kBR0 k (Qk (r; r)) :
Summing up the two terms leads to
@ k
(Q (r; r); r) = [c + N r
@Qk k
P (Q (r; r))] [k
1 + kBR0 k (Qk (r; r))]:
(14)
Expression (14) generalizes (11). In particular, the …rst bracket term is the same in both
equations does not depend on k while the second bracket term in (14) depends on k. The
Stackelberg e¤ect dominates the coordination e¤ect if and only if k
We have
k
1 + kBR0 k (Qk (r; r)) S 0 i¤
QP 00 (Q)
T
P 0 (Q)
1 + kBR0 k (Qk (r; r)) < 0.
N (N 2k + 1)
:
N k
The important point is that at r = rm , the …rst bracket term is zero regardless of the coalition
size: c + N rm
P (Q (rm ; rm )) = 0. Therefore, we have the following result
Theorem 1 Assume that A1-A3 hold. In the two-stage game of public cross-licensing followed
by Cournot competition;
(i) If the Stackelberg e¤ ect dominates the coordination e¤ ect
S(rm ; N )
is k-e¢ cient if and only if r 2
f0; rm g.
(ii) If the coordination e¤ ect dominates the Stackelberg e¤ ect
S(rm ; N )
QP 00 (Q)
P 0 (Q)
>
N (N 2k+1)
N k
, then
QP 00 (Q)
P 0 (Q)
<
N (N 2k+1)
N k
, then
is k-e¢ cient if and only if r = 0.
Theorem 1 generalizes Proposition 2 to any given size of coalition. In addition, this theorem
implies that even if we allow for coalitions of di¤erent sizes (for instance, …rm 1 signs a licensing
agreement with …rm 2 and at the same time signs an agreement with …rms f3; 4g), the monopoly
outcome survives.
The theorem also implies that Proposition 3 of secret cross-licensing generalizes to coalitions
of any size since secret cross-licensing is a particular case of public cross-licensing in which the
Stackelberg e¤ect is absent:
Theorem 2 Assume that A1-A3 hold. In the two-stage game of secret cross-licensing followed
by Cournot competition, S(rm ; N ) is the unique k-e¢ cient set of symmetric agreements.
8
Policy implications
The previous analysis reveals that market forces induce …rms to sign cross-licensing agreements
allowing them to achieve the monopoly outcome regardless of the size of coalitions and the
information structure (i.e. whether the licensing terms are public or secret). In this section, we
derive two di¤erent policy implications. First, we interpret our results in terms of desirability
18
(or undesirability) of cross-licensing of complementary patents (substituable patents). Second,
we provide simple policy remedies to restore the most competitive outcome (i.e. cross-licensing
with zero royalties).
In our model,
c
c c has some close connection with whether patents are close substitutes
or complements. In the case of close substitutes,
strong complements,
c is expected to be very small: in the case of
c is expected to be large. When we combine this interpretation with the
main result that under laissez-faire cross-licensing leads to the monopoly outcome, we obtain
the following result from Proposition 1.
Proposition 4 (i) When patents are relatively substitutes (i.e.
c < (N 1) (P (Q (N; c))
c)),
cross-licensing leads to a decrease in consumer surplus.
(ii) When patents are relatively complements (i.e.
c > (N
1) (P (Q (N; c))
c)), cross-
licensing leads to an increase in consumer surplus.
Therefore, absent any other policy remedies, a consumer-surplus-maximizing competition
authority should allow cross-licensing only if patents are complementary enough.
We now turn to simple policy remedies which take advantage of competitive forces existing
in the market. Our policy remedies have three components. First, the antitrust authorities
should prohibit any royalty above the monopoly royalty rm . More precisely, the authority
should impose an upper bound on royalty, denoted by r satisfying r < rm . However, using our
previous analysis of unconstrained agreements, we can show that this restrcition is not enough
to restore the competitive outcome: under secret cross-licensing, the …rms will end up choosing
r = r. Therefore, antitrust authorities should make the disclosure of royalty terms compulsory.
Then, as long as the Stackelberg e¤ect dominates the coordination e¤ect, competition among
the …rms induces all …rms to end up sigining zero royalty. Third, from Theorem 1, it follows that
the Stackelberg e¤ect becomes less likely to dominate the coordination e¤ect as the coalition
size increases. Note also that, if only coalitions made of two …rms are allowed, the Stackelberg
e¤ect dominates the coordination e¤ect for any N
4 (due to assumption A3). Therefore, it is
important to allow only coalitions made of small number of …rms.
4: Imposing an upper bound on royalty r satisfying r < rm ,
Proposition 5 Assume that N
making disclosure of royalty terms compulsory and allowing agreements only among a small
number of …rms (i.e. k satisfying
QP 00 (Q)
P 0 (Q)
>
N (N 2k+1)
)
N k
induces the most competitive outcome:
S(0; N ) is then the unique k-e¢ cient set of symmetric agreements.
9
Concluding remarks
This paper is a …rst step toward understanding bilateral cross-licensing agreements among N
competing …rms. We plan to address several extensions in future research.
19
3
One extension is to introduce some asymmetries in our setting to study issues which cannot
be investigated with a symmetric model. In particular, we can include in the set of players nonpracticing entities (NPEs) which do not compete in the downstream market. This would allow
us to study the conditions under which NPEs weaken competition and (when these conditions
are met) to isolate the anticompetitive e¤ects generated by NPEs from the e¤ects resulting
from cross-licensing in the absence of NPEs. The issue of how NPEs a¤ect competition and
innovation is of substantial current interest to policymakers (Morton, 2012).
Another interesting asymmetry we can introduce is to consider, in addition to incumbent
…rms, entrants with no (or weak) patent portfolios. This would allow us to study whether crosslicensing can be used to raise barriers to entry (Hall et al. , 2012). Note that NPEs and entrants
involve completely opposite asymmetries. The former are present in the upstream market of
patent licensing and are absent in the downstream (product) market while the second are absent
(or have very weak presence) in the upstream market and are present in the downstream market.
10
Appendix
Proof of Lemma 1
Assume, without loss of generality, that (i; j) = (1; 2):The joint payo¤ …rms 1 and 2 derive from
a deviation to a licensing agreement involving the payment of (r1!2 ; r2!1 ) is:
1
+
2
= [P (Q )
+ [P (Q )
= [P (Q )
r1!2
c
c
c
(N
r2!1
2) r] q1 + r2!1 q2 + rQ
(N
12
2) r] q2 + r1!2 q1 + rQ
r] (q1 + q2 ) + 2rQ
12
12
which can be rewritten as
1
+
2
= [P (Q )
c
(N
2) r] Q
Q
12
+ 2rQ
12
(15)
Denoting ci the marginal cost of …rm i (including the royalties paid to the other …rms), the FOC
for …rm i0 s maximization program is:
P (Q )
ci + qi P 0 (Q ) = 0
Summing the FOCs for i = 1; 2; ::; N yields:
N P (Q )
X
ci + Q P 0 (Q ) = 0
i 1
which shows that Q depends on (c1 ; c2 ; ::; cN ) only through
X
i 1
20
ci : Moreover, summing the FOCs
for i = 3; ::; N yields
(N
2) P (Q )
X
ci + Q
12 P
0
(Q ) = 0
i 3
which implies that Q
12
depends on (c1 ; c2 ; :::; cN ) only through
X
ci and
i 1
(c + r1!2 + (N
2) r; c + r2!1 + (N
both Q and Q
12
(15), implies that
2) r; c + (N
X
ci . From (c1 ; c2 ; c3 ; :::; cN ) =
i 3
1) r; :::; c + (N
1) r) it then follows that
depend on (r1!2 ; r2!1 ) only through r1!2 + r2!1 , which, combined with
1
+
2
depends on (r1!2 ; r2!1 ) only through r1!2 + r2!1 :
Proof of Lemma 2
Since Q12 (r; r^) is strictly decreasing and continuous in r^ then r 2 Arg max (
only if:
Q12 (r; r) 2
1
r^2[0; ]
Arg max
12 (Q12 )
[P (Q12 + BR
12 (Q12 ))
+
(c + (N
^)
2 ) (r; r
if and
2) r)] Q12 +2rBR
Q12 2[Q12 (r;r);Q12 (r;0)]
which means that Q12 (r; r) is a subgame-perfect equilibrium strategy of the coalition f1; 2g in
the game G(r; N ):
Proof of Proposition 2
Let us …rst show some general preliminary results which will be useful for the subsequent speci…c
analysis of the four considered scenarios. We have:
@ 12
(Q12 ; r) = P 0 (Q12 + BR
@Q12
[P (Q12 + BR
= [c + N r
2 P 0 (Q12 + BR
|
|
1;2 (Q12 ))
1;2 (Q12 )) Q12 (1
1;2 (Q12 ))
J(Q12 ;r)
1;2 (Q12 ))]
1;2 (Q12 ))
12 (Q12 ))
+
1) r)] + r 1 + 2BR0
(c + (N
P (Q12 + BR
Q12
+ P (Q12 + BR
2
{z
+ BR0
1 + 2BR
(c + (N
0
12 (Q12 )
+
1 + BR0
1) r)
}
{z
D((Q12 ;r)
12 (Q12 )
1;2 (Q12 )
}
Let us show that J(Q12 ; r) is decreasing in Q12 . We have
Q12
@J(Q12 ; r)
P 0 (Q)
= P 00 (Q)
+ P 0 (Q) 1 + BR0 1;2 (Q12 ) +
@Q12
2
2 }
|
{z
} | {z
>0
<0
Since P 00 (Q) Q212 + P 0 (Q) < max [P 00 (Q) Q + P 0 (Q) ; P 0 (Q)] < 0 then J(Q12 ; r) is decreasing in
Q12 : This, combined with the fact that the FOC for each …rm i = 1; 2, satis…ed at the symmetric
21
12 (Q12 )
Cournot equilibrium equilibrium, is given by J(Q12 (r; r) ; r) = 0, yields that
J(Q12 ; r) Q 0 () Q12 R Q12 (r; r)
Since 1 + BR0
1;2 (Q12 )
(16)
> 0, it follows that
D(Q12 ; r) Q 0 () Q12 R Q12 (r; r)
(17)
Moreover, from
BR0
P 00 (Q)BR
P 00 (Q)BR
12 (Q12 ) =
2)P 0 (Q)
1)P 0 (Q)
12 (Q12 )
+ (N
12 (Q12 ) + (N
it follows that
1 + 2BR0
12 (Q12 )
=
P 00 (Q)BR
P 00 (Q)BR
+ (N
12 (Q12 ) + (N
3)P 0 (Q)
1)P 0 (Q)
QP 00 (Q)
1)P 0 (Q) P 0 (Q)
N
12 (Q12 )
(18)
which can be rewritten as
1+2BR0
12 (Q12 )
=
Since P 00 (Q)BR
N 2 0
N P (Q)
P 00 (Q)BR
12 (Q12 )
12 (Q12 )+(N
+ (N
1)P 0 (Q)
max ((N
N
2
:
BR
12 (Q12 )
Q
1)P 0 (Q); P 00 (Q)Q + (N
+
N (N 3)
N 2
1)P 0 (Q)) <
0, it follows that
1 + 2BR0
12 (Q12 ) Q 0 ()
QP 00 (Q)
R
P 0 (Q)
N (N 3) N 2
Q
:
N 2
N BR 12 (Q12 )
(for any Q12 such that BR
12 (Q12 )
N 2
N
is decreasing in Q12 and BR
(Q12 + BR
12 (Q12 ))
6= 0). From the fact that BR
12 (Q12 )
12 (Q12 (r; r))
=
(19)
N 2
N Q = BR 12 (Q12 )
N 2
N Q (r; r) (by sym-
metry of the considered Cournot equilibrium), it follows that
N
2
N
:
Q
R 1 () Q12 R Q12 (r; r)
BR 12 (Q12 )
In particular we obtain the following result which will be useful for the next steps of the proof:
If
QP 00 (Q)
P 0 (Q)
Q12
>
N (N 3)
N 2
Q12 (r; r) :
(for any Q such that P 0 (Q) 6= 0) then 1 + 2BR0
12 (Q12 )
< 0 for any
- Let us now show that r = rm is bilaterally e¢ cient regardless of whether the Stackelberg
e¤ect dominates or is dominated by the coordination e¤ect.
22
@ 12
(Q12 ; rm ) = P 0 (Q12 + BR
@Q12
[P (Q12 + BR
12 (Q12 )) Q12 (1
+ BR0
12 (Q12 ))
+
(c + (N 1) rm )] + rm 1 + 2BR0
Q12
+ rm (1 + BR0 12 (Q12 )) +
= 2 P 0 (Q12 + BR 12 (Q12 ))
2
[P (Q12 + BR 12 (Q12 )) (c + N rm )]
Since c + N rm
P (Q12 + BR
12 (Q12 ))
12 (Q12 )
P (Q (rm ; rm )) = 0 then
12 (Q12 ))
(c + N rm ) = P (Q12 + BR
Moreover, conbining c + N rm
1;2 (Q12 ))
P (Q12 (rm ; rm ) + BR
12 (Q12 (r
m
; rm )))
P (Q (rm ; rm )) = 0 with the FOC
Q12 (rm ;r m )
m m
+
12 (Q12 (r ; r )))
2
m
m
c (N 1) rm
12 (Q12 (r ; r )))
P 0 (Q12 (rm ; rm ) + BR
P (Q12 (rm ; rm ) + BR
=0
yields
P 0 (Q12 + BR
12 (Q12 ))
Q12
+ rm = P 0 (Q12 + BR
2
Q12
2
12 (Q12 ))
P 0 (Q12 (rm ; rm ) + BR
12 (Q12 (r
m
; rm )))
Q12 (rm ; rm )
2
Therefore,
@ 12
Q12
Q (rm ; rm )
(Q12 ; rm ) = 2 P 0 (Q12 + BR 12 (Q12 ))
P 0 (Q12 (rm ; rm + BR 12 (Q12 (rm ; rm ))) 12
+
@Q12
2
2
[P (Q12 + BR 12 (Q12 )) P (Q12 (rm ; rm ) + BR 12 (Q12 (rm ; rm )))]
Using the fact that P 0 (Q) < 0 and 1 + BR0
functions
P 0 (Q12
+ BR
12 (Q12 ))
Q12
2
12 (Q12 )
> 0, it is straightforward to show that both
and P (Q12 + BR
12 (Q12 ))
are decreasing in Q12 , which
implies that
@ 12
(Q12 ; rm ) R 0 () Q12 Q Q12 (rm ; rm )
@Q12
Therefore, Q12 (rm ; rm ) maximizes
12 (Q12 ; r
m)
over [Q12 (rm ; r); Q12 (rm ; 0)], which is equivalent
to the fact that S(rm ; N ) is bilaterally e¢ cient.
- Consider now the case
that is Q12 (0; 0) maximizes
@ 12
(Q12 ; 0) = (c
@Q12
QP 00 (Q)
P 0 (Q)
>
12 (Q12 ; 0)
N (N 3)
N 2
and let us show that r = 0 is bilaterally e¢ cient,
over [Q12 (0; 1); Q12 (0; 0)]. We have
P (Q12 + BR
1;2 (Q12 ))) (1
23
+ 2BR0
12 (Q12 ))
+ D(Q12 ; 0)
Let us show that P 00 (Q12 + BR
12 (Q12 ))BR 12 (Q12 )
+ (N
3)P 0 (Q12 + BR
any Q12 2 [Q12 (0; 1); Q12 (0; 0)], which, by (18)), is su¢ cient to state that
12 (Q12 )) < 0 for
1 + 2BR0 12 (Q12 ) < 0
for any Q12 2 [Q12 (0; 1); Q12 (0; 0)]. On the one hand, if P 00 (Q12 + BR
it follows from BR
hand, if
P 00 (Q12
12 (Q12 )
+ BR
0 and
12 (Q12 ))
P 0 (Q)
< 0 that 1 +
0 then from BR
2BR0
12 (Q12 )
12 (Q12 )
1;2 (Q12 ))
< 0 then
< 0. On the other
Q, it follows that P 00 (Q12 +
3)P 0 (Q12 + BR 12 (Q12 )) P 00 (Q)Q + (N 3)P 0 (Q).
12 (Q12 ))BR 12 (Q12 ) + (N
Note …rst that if P 00 (Q12 + BR 12 (Q12 )) 0, it must hold that N 4; otherwise the condi00 (Q)
N (N 3)
tion QP
N 2 (which is one of the two conditions de…ning the present scenario) would
P 0 (Q) >
be violated. This implies that P 00 (Q)Q + (N 3)P 0 (Q)
P 00 (Q)Q + P 0 (Q), which combined
BR
with A3 yields P 00 (Q)Q + (N
3)P 0 (Q) < 0 and, therefore, that 1 + 2BR0
12 (Q12 )
< 0. We
are now in position to state that, for any Q12 2 [Q12 (0; 1); Q12 (0; 0)], the latter inequality
holds independently of whether P 00 (Q12 + BR
12 (Q12 ))
< 0 or P 00 (Q12 + BR
Combining that with the fact that the two inequalities c
P (Q12 + BR
12 (Q12 ))
1;2 (Q12 ))
0.
c
P (Q12 (0; 0) + BR
< 0 and D(Q12 ; 0)
we get that
1;2 (Q12 (0; 0)))
@ 12
0 for
@Q12 (Q12 ; 0)
maximizes
12 (Q12 ; 0)
any Q12 2 [Q12 (0; r); Q12 (0; 0)]. This implies that Q12 (0; 0)
11
0 hold for any Q12 2 [Q12 (0; r); Q12 (0; 0)],
over [Q12 (0; 1); Q12 (0; 0)]. Therefore, S(0; N ) is bilaterally e¢ cient.
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