Vertical Licensing, Input Pricing, and Entry
Elpiniki Bakaouka and Chrysovalantou Milliou
May 2016
Preliminary Draft
Abstract
In this paper, we explore the incentives of an incumbent …rm to license the production technology of its core input to an external …rm. We …nd that licensing always
arises in equilibrium independently of whether or not it induces the entry of the licensee
into the …nal goods market. Interestingly, we also …nd that although the entry of the
licensee increases the intensity of market competition faced by the licensor, it reinforces the latter’s licensing incentives. Both with and without entry, licensing enhances
welfare.
Keywords: licensing; vertical relations, entry; two-part tari¤s; outsourcing
JEL classi…cation: L22; L24; L13; L42; D45
Bakaouka: Department of International and European Economic Studies, Athens University of Economics and Business, Athens 10434, Greece, e-mail: [email protected]. Chrysovalantou Milliou: Department of International and European Economic Studies, Athens University of Economics and Business,
Athens 10434, Greece, e-mail: [email protected]. We would like to thank Maria Alipranti, Christos Constantatos, Lampros Pechlivanos, Emmanuel Petrakis, Joel Sandonis, and Spyros Vassilakis for their useful
comments. This research has been co-…nanced by the European Union (European Social Fund - ESF) and
Greek national funds through the Operational Program "Education and Lifelong Learning" of the National
Strategic Reference Framework (NSRF) - Research Funding Program: Thalis - Athens University of Economics and Business - "New Methods in the Analysis of Market Competition: Oligopoly, Networks and
Regulation". Full responsibility for all shortcomings is ours.
1
Introduction
Original brand manufacturers often license the production technology of their core inputs to
external …rms.1 Such licensing activities may trigger the entry of the latter - the licensees into direct competition with their licensors. An illustration of this business practice comes
from the commercial aircraft market, where, in 2003, Boeing, the US aircraft manufacturer,
licensed its wing technology to Mitsubishi Heavy Industries, a Japanese engineering and
electrical equipment …rm, after agreeing to source the wings of its planes from the latter.
Boeing, as well as its main competitor in the market, Airbus, had been already subcontracting the production of various aircraft components to external suppliers. However, never
before a commercial aircraft manufacturer had subcontracted its wing production. Boeing’s
decision to do so, was accepted with wide spread criticism. First, because the wing technology of Boeing has been regarded as its crown jewel, and second, because the transfer
of wing manufacturing and assembly expertise to Mitsubishi e¤ectively gave to the latter
‘total production competence’with regard to commercial aircrafts. Mitsubishi is now ready
to launch its own passenger jet, which is due to enter service in 2018.2
A similar illustration can be found in the electronic appliances market, where, Haier,
a Chinese …rm which was initially manufacturing electrical equipment for original brand
manufacturers, acquired in 1984 the technology for producing high-quality refrigerators
from Liebherr, the German manufacturer of refrigerators. A year after the licensing of
Liebherr’s technology, Haier introduced its own …rst four-star refrigerator into the market.
That is, Haier transformed from an original equipment manufacturer to an original brand
manufacturer. It soon became the leader refrigerator producer in China and, eventually, it
also became a major competitor in the global market.
The above illustrations give rise to a number of important questions regarding vertical
licensing: Does a …rm have incentives to license its input technology to an external supplier
when the latter can turn into its competitor in the …nal goods market? How does the
pricing of the input a¤ect the licensing decision? Does the entry of the licensee into the
1
According to empirical evidence, technology licensing is a common practice among …rms (see e.g., Caves
et al., 1983, Nadiri, 1993, Anand and Khanna, 2000). Many well-known …rms, such as IBM, Hitachi, Kodak,
and Procter & Gamble, have licensed their technology to others, earning millions of dollars in licensing
revenue (see e.g., A market for ideas, The Economist (October 20, 2005)).
2
Note that without the ability to make its own wings, Mitsubishi could never become an independent
player in the passenger jet industry. For more information regarding this case, see e.g., Secret Wounds Of
Globalism: Boeing Sells Its Technology –Cheap –To Japan, Forbes (January 2014), and Boeing Outsourcing
Gives Wing to Concerns, Chicago Tribune (December 21, 2003).
1
licensor’s market weaken the licensing incentives? What are the welfare implications of
vertical licensing? Does the type of vertical contract and the commitment of the licensor
in‡uence the licensing incentives and implications? In this paper, we attempt to address
these questions.
To this end, we consider a framework in which two incumbent …rms produce two competing …nal goods using an input that initially they produce in-house. The …rm which is
more e¢ cient in input production considers licensing its input technology to a potential
input supplier for a …xed licensing fee. Once the licensing agreement is signed, the licensor
can source the input from the licensee after bargaining with the latter over the terms of a
two-part tari¤ contract. Moreover, once the licensing agreement is signed, the licensee can
enter into the …nal goods market and compete with the two incumbents. We assume that
competition in the …nal goods market takes place in quantities and examine what happens
when the licensee does not enter into the …nal goods market (the ‘no entry case’) as well as
when it enters (the ‘entry case’).
We …nd that the incumbent always opts for licensing. This holds independently of
whether the licensee enters into the …nal goods market or not. The driving force behind
licensing, however, di¤ers substantially among the entry and the no entry case. In the no
entry case, licensing is mainly driven by e¢ ciency reasons: when the incumbent licenses
its input technology, it becomes more e¢ cient, and thus, it enjoys a larger competitive
advantage in the …nal goods market. The higher e¢ ciency of the licensor is not due to
a cost advantage of the licensee. It is due, instead, to input pricing. In particular, when
the licensee does not enter into the …nal goods market, it subsidizes the production of its
customer - it charges a wholesale price which is below the input’s marginal cost. It has
incentives to do so because it can extract part of the resulting higher pro…ts of the licensor
through the …xed fee included in the two-part tari¤ contract. Interestingly, due to the
higher e¢ ciency that the incumbent enjoys under licensing, it is willing to license its input
technology even for free. This holds as long as the share of the licensor’s pro…ts that the
licensee extracts is su¢ ciently enough, i.e., when the bargaining power of the licensee is low
enough.
When the licensee enters into the …nal goods market, it no longer has incentives to
subsidize the licensor since the latter is now both its customer and its rival. On the contrary,
in order to extract higher pro…ts from its own sales in the …nal goods market, the licensee
now raises its rival’s cost by charging a wholesale price that exceeds the input’s marginal
2
cost. Therefore, in the entry case, in contrast to the no entry case, licensing deteriorates the
e¢ ciency of the licensor, leading to a decrease in the licensor’s pro…ts from its own sales in
the …nal goods market. Moreover, in the entry case, licensing results in an increase in the
number of downstream competitors, and thus, in an increase in the intensity of competition
in the …nal goods market. This contributes to the further decrease in the licensor’s pro…ts
from its own sales in the …nal goods market. Why then does the licensor opt for licensing
in the entry case? There are two e¤ects that both are pro…t increasing for the licensor.
First, the entry of the licensee into the …nal goods market, gives rise to a market expansion
e¤ ect. That is, it expands the demand for the …nal goods. Second, the wholesale price
that exceeds the input’s marginal cost gives rise to a strategic e¤ ect. Although, the greater
wholesale price decreases the licensor’s pro…ts, it increases the licensee’s pro…ts. Thus, this
tradeo¤ results to a greater pro…t margin for the licensor under licensing with entry than
under no licensing. In equilibrium, the licensor manages to extract, through the licensing
fee, not only all of the pro…ts that are generated by its own sales, but also all the pro…ts
that the licensee generates in the …nal goods market. In other words, in equilibrium, the
licensor takes full advantage of both the market expansion e¤ ect and the strategic e¤ ect. As
long as a licensing fee can be used, the latter e¤ects outweigh the decreased e¢ ciency and
the increased competition intensity and leads to the emergence of licensing in equilibrium.
Importantly, the entry of the licensee into the …nal goods market, although it intensi…es
the market competition and reduces the e¢ ciency of the licensor, it strengthens instead of
weakens the licensing incentives. Even the fact that the licensor’s wholesale price is greater
than its rival’s cost, it turns to be an advantage for the licensor instead of a drawback.
Stated in di¤erent words, the incumbent has stronger licensing incentives when the licensee
becomes its competitor and faces a greater marginal cost. The intuition for this somewhat
surprising result lies again on the market expansion e¤ ect and the strategic e¤ ect of licensing.
Vertical licensing turns out to be desirable not only for the licensor, but also for the
consumers and the society as a whole. This holds both with entry and without entry of
the licensee in the …nal goods market. In fact, in the former case, licensing is even more
desirable. Intuitively, licensing results in lower …nal prices, and thus, in higher consumers’
surplus, either because it enhances the e¢ ciency of the licensor (in the no entry case) or
because it intensi…es the market competition (in the entry case). The positive impact of
licensing on consumers’surplus and on the licensor’s pro…ts dominate its negative impact
on the pro…ts of the other incumbent, rendering vertical licensing to a welfare-enhancing
3
practice.
We demonstrate that our main …ndings are robust when we relax the assumption that the
e¢ cient incumbent commits to not producing the input in-house after licensing. Moreover,
examining what happens when the licensor and the licensee trade through a wholesale price
contract, we …nd that licensing incentives exist if and only if the licensee enters into the
…nal goods market. This occurs because double marginalization is present both in the
entry and in the no entry case; hence, the licensor does not enjoy higher e¢ ciency under
licensing in either case. In the entry case though, the licensor bene…ts again from the market
expansion e¤ ect which is in place. Finally, considering what happens when licensing takes
place through a per-unit of output royalty, we …nd that, in contrast to the …xed licensing
fee case, the …rm has incentives to license its input technology only for free and only in the
no entry case.3
Our work is related to the vast theoretical literature on technology licensing. This
literature has analyzed various aspects of licensing such as, the choice among royalties and
licensing fees (e.g., Wang, 1998, Kamien and Tauman, 1986, Muto, 1993), the impact of
licensing on innovation (e.g., Gallini and Winter, 1985), the role of information asymmetries
(e.g., Gallini and Wright, 1990, Beggs, 1992), the choice among merger and licensing (e.g.,
Fauli-Oller and Sandonis, 2003). The majority of this literature has done so assuming that
the licensor and the licensee(s) operate in the same one-tier market or that the licensor
is not active in the market (an outsider). Some recent exceptions include the papers of
Mukherjee (2003), Arya and Mittendorf (2006), Mukherjee and Ray (2007), Rey and Salant
(2012), which, similarly to ours, have examined licensing within a vertically related market.
The latter papers, however, di¤er from ours in three important aspects. First, most of
them have considered licensing either among downstream …rms or among upstream …rms,
and not among vertically related …rms.4 Second, they have analyzed settings in which, after
the signing of the licensing agreement, the licensor is not a customer of the licensee. Third,
they have examined the possibility that the licensing triggers the entry of a new …rm into
the market, but not the entry of the licensee into the licensor’s market. It follows from
3
It should be noted that free licensing is observed quite often. It is observed not only in the open source
software market, but also in other markets. For instance, in January 2015, Toyota announced that it would
make more than 5,600 patents on fuel-cell technologies available for use, free of royalty payments, to a
wide array of companies in the transportation sector. For more on this, see e.g. Toyota O¤ers To License
Hydrogen Fuel-Cell Patents, For Free, Green Car Reports (January 5, 2015), and Toyota To Share Hydrogen
Fuel Cell Patents, Forbes (January 5, 2015).
4
An exception in this respect is the paper of Rey and Salant (2012) which does consider vertical licensing.
4
this, that our paper complements the existing literature on technology licensing. It also
follows that our paper, in contrast to the existing ones, is more appropriate for the analysis
of situations such as the ones that we described in the beginning of this section (see e.g.,
Boeing case).
Our paper is also related to the literature on outsourcing. A number of papers within
this literature (e.g., Pack and Saggi, 2001, Shy and Stenbacka, 2003, Sappington 2005,
Arya et al., 2008a and 2008b) have analyzed a …nal product manufacturer’s ‘make-or-buy’
decision. That is, its choice among the production of an input in-house and the sourcing
of the input from an external …rm - outsourcing.5 Some of these papers have assumed that
the input production is outsourced to a vertically integrated rival (i.e., to a …rm which is
already in the downstream market). Others, instead, have assumed that it is outsourced to
an independent upstream …rm. One of the latter, the paper of Pack and Saggi (2001), similar
to ours, has taken into account the possibility that outsourcing can result in downstream
entry and examined that implications of entry.
However, Pack and Saggi (2001) have
allowed for the downstream entry of an external independent …rm, and not of the upstream
…rm to which the input production is outsourced. We, instead, have considered the entry
of the input supplier into the downstream market - that is, in our setting, the entrant is
both a downstream rival and an upstream supplier of the incumbent …rm and not just a
downstream rival.
The remainder of the paper is organized as follows. In Section 2, we describe our main
model. In Sections 3, we examine the licensing incentives in both the no entry case and the
entry case, and we characterize the impact of entry on these incentives. In the following
Section, we evaluate the welfare implications of vertical licensing. In Section 5, we examine
the robustness of our main …ndings when vertical trading takes place through a wholesale
price contract. In Section 6, we discuss a number of other extensions of our main model.
Finally, in Section 7, we conclude. All the proofs are included in the Appendix.
5
The practice of outsourcing is obviously quite similar to the practice of licensing. In fact, the only
distinction among them is the fact that outsourcing takes place for free (i.e., there are no fees or royalties
involved).
5
2
The Model
We consider a market consisting initially of two …rms, …rm 1 and …rm 2. Each …rm i,
with i = 1; 2, produces a di¤erentiated …nal good using, in a one-to-one proportion, a core
input that it produces in-house.6 We assume that both …rms have patented their input
productions technologies and that the marginal cost that they face is equal to c.
Firm 1 considers licensing its input technology to a potential input supplier, …rm S.
We assume that licensing takes place through a …xed licensing fee, F
0, that …rm S has
to pay to …rm 1.7 After the licensing agreement is signed, the licensee (…rm S) is in the
position to produce the licensor’s (…rm 1’s) patented input, facing the same cost with the
licensor, i.e., at marginal cost c.8 The signing of the licensing agreement gives rise to the
possibility that the licensee also produces the …nal good. That is, it allows …rm S to enter
into the …nal goods market and become a competitor of …rms 1 and 2. In what follows, we
consider the case in which …rm S enters into the …nal goods market as well as the case in
which it does not enter. In both cases, we assume that …rm 1 stops producing the input
in-house and commits to sourcing it from …rm S.9 ’10 We also assume that …rm S and …rm
1 negotiate over the terms of a two-part tari¤ contract, that is, over a …xed fee, T , and a
wholesale price per unit of input, w, in a Nash bargaining fashion. The bargaining power
of …rm S and …rm 1 is given, respectively, by
and 1
, with 0 <
< 1:
The (inverse) demand function for …rm i’s …nal good is:
pi (qi ; Q i ) = a
qi
Q i ; a > c > 0; 0
< 1;
where pi and qi are, respectively, the price and the quantity of …rm i’s …nal good, and Q
is the quantity of its rival(s)’…nal goods. In particular, Q
in the no entry case, while Q
case. The parameter
i
i
i
= qj , with i; j = 1; 2 and i 6= j,
= qj + qk , with i; j:k = 1; 2; S and i 6= j 6= k, in the entry
measures the degree of product di¤erentiation; namely, the higher
is , the closer substitutes the …nal goods are.
6
The production of the …nal good is impossible without the use of this input.
In Section 6, we discuss what happens when licensing takes place through a variable royalty instead.
8
We abstract from assuming that the input supplier will be more e¢ cient in input production than …rm
1 since then the incentives for vertical licensing would be straightforward.
9
It can commit, for instance, by shutting-down its input producing facilities.
10
In Section 6, we examine the robustness of our main …ndings in the alternative scenario in which …rm
1 does not commit to souring the input from its licensee and it keeps instead open the option of producing
the input in-house.
7
6
The timing of moves is as follows. First, …rm 1 decides whether or not to license its input
technology to S. In case of licensing, it sets the licensing fee F and, in turn, …rm S decides
to sign or not the licensing agreement. If the agreement is signed, then in the following
stage, …rm 1 and …rm S negotiate over (w; T ). Finally, in the last stage, …rm 1 and …rm 2
(as well as …rm S in case of licensing and entry) choose their quantities simultaneously and
separately. We solve for the subgame perfect Nash equilibrium of this game.
We start our analysis by considering the benchmark case of no licensing, in which, …rm
1 and …rm 2, compete among them in the standard Cournot way. In particular, each …rm
i chooses its output in order to maximize its pro…ts:
i (qi ; qj )
= (a
qi
qj )qi
cqi , with
i; j = 1; 2 and i 6= j. Solving the resulting system of …rst order conditions, we obtain the
Cournot-Nash equilibrium quantities, q1B and q2B and the respective equilibrium pro…ts,
and
3
B,
2
B
1
included in Table 1 of Appendix.
Entry and Licensing Incentives
In this Section, …rst, we analyze the licensing incentives both in the no entry case and in
the entry case, and then we examine how entry a¤ects the licensing incentives.
3.1
Licensing and No Entry
We examine here the case in which the licensing agreement has been signed and …rm S has
stayed out of the …nal goods market.11
In the last stage, …rm 2 faces the same maximization problem as in the benchmark case,
while …rm 1 chooses q1 in order to maximize the following (gross from the …xed fee and the
licensing fee) pro…ts:
1 (q1 ; q2 ; w)
= (a
q1
q2 )q1
wq1 . Solving the system of the two
…rst order conditions, we derive the equilibrium quantities in terms of w:
q1 (w) =
a(2
)+ c
2
4
2w
and q2 (w) =
a(2
)
4
2c + w
2
:
(1)
Obviously, a decrease in the wholesale price results in a higher output for …rm 1 and a lower
output for its rival.
In the following stage, …rm S and …rm 1 negotiate over (w, T ). In particular, they solve
11
This could be so because, for instance, the entry costs are prohibitive high, and thus, entry is blocked.
7
the following generalized Nash bargaining problem:
max [
w;T
where
S (w)
= (w
S (w)
+ T] [
1 (w)
T ]1
c)q1 (w) are the pro…ts of …rm S, and
;
(2)
1 (w)
=
1 (q1 (w); q2 (w); w).
Note that the disagreement payo¤s of both …rms are equal to zero since neither …rm has an
outside option. Maximizing (2) with respect to T we get:
T =
1 (w)
(1
)
S (w):
(3)
Using (3), the gross (from the licensing fee) pro…ts of …rm S and …rm 1 can be rewritten
as:
S (w)
+T = (
S (w)
+
1 (w))
and
1 (w)
T = (1
)(
S (w)
+
1 (w)):
(4)
Substituting the above expressions into (2), we obtain an expression which is proportional
to the joint pro…ts of …rm S and …rm 1. It follows that w is chosen to maximize these
pro…ts:
max
w
S (w)
+
1 (w)
= (a
q1 (w)
q2 (w))q1 (w)
cq1 (w).
(5)
From the …rst order condition of (5), we …nd the equilibrium wholesale price and, after
substituting the latter into (3), we also …nd the equilibrium …xed fee:
wLN =
8c
2(a + c)
4(2
2
+ (a
2)
c)
3
and T LN =
(a
c)2 (2
)2 (2 + (1
2 )2
8(2
)
2)
.
(6)
One can easily check that wLN < c. That is, …rm S subsidizes, through the wholesale price,
the production of its customer, …rm 1.12 As we saw above, by charging a lower wholesale
price, …rm S increases the aggressiveness of …rm 1 in the …nal goods market and enhances
its output at the expense of …rm 2’s output. Firm S has incentives to do so because it can
use, in turn, the …xed fee T in order to capture part of the resulting higher …rm 1’s pro…ts.13
Clearly, the higher is …rm S’s bargaining power, the larger is the share of …rm 1’s pro…ts
12
The subsidization of the downstream production arises also in settings in which an upstream monopolist
faces a ‘commitment problem’, i.e., in settings in which, in contrast to here, it trades with more than one
downstream …rms and cannot commit to any of them that it will not o¤er better trading terms to the
other(s). For more on this, see e.g., Milliou and Petrakis (2007), Alipranti et al. (2014).
13
A similar rationale exists in the delegation literature (e.g., Vickers, 1985, Sklivas, 1987, Fershtmam and
Judd, 1987).
8
that it captures through T .14
Furthermore, one can also easily check that the higher is product di¤erentiation, the
higher is the equilibrium wholesale price (i.e.,
#wLN
#
< 0), and thus, the smaller is the
subsidy that …rm S o¤ers to …rm 1. In fact, in the extreme case in which …rm 1 and …rm
= 0), we have that wLN = c. In other words, the
2 do not compete in the market (i.e.,
weaker is the competition in the …nal goods market, the weaker are the incentives of …rm S
to enhance the competitive position of its customer by decreasing the latter’s variable cost
(i.e., the wholesale price).
In the …rst stage of the game, …rm 1 chooses between licensing and no licensing. Substituting (6) into
S (w)
+ T and
1 (w)
T from above, we obtain the gross (from the
licensing fee) equilibrium pro…ts of the licensor and the licensee. Firm 1 knows that …rm
S will reject the licensing agreement if and only if its pro…ts without the agreement exceed
its pro…ts with the agreement. Since the former pro…ts are equal to 0, it follows that …rm
1 will optimally set F LN =
S (w
under licensing and no entry are:
LN )
+ T LN . As a result, …rm 1’s net equilibrium pro…ts
LN
1
=
1 (w
LN )
T LN + F LN =
1 (w
LN )
+
S (w
LN ).
It is obvious that …rm 1 ends up obtaining, through the licensing fee, not only the pro…ts
from its own sales in the …nal goods market but also the pro…ts of …rm S from the input
sales, i.e., it obtains all of its joint pro…ts with …rm S. Comparing the equilibrium pro…ts
of …rm 1 in the case of licensing and no entry (included in Table 1 of the Appendix) with
its respective pro…ts in the benchmark case, we reach the following conclusion.
Proposition 1 When the input supplier (the licensee) does not enter into the …nal goods
market, …rm 1 always has incentives to license its input technology.
As Proposition 1 informs us, …rm 1 always licenses its input technology to …rm S when the
latter does not enter into the …nal goods market. It is important to note that this holds not
only when …rm 1 charges a positive …xed fee for the licensing agreement, but also when it
o¤ers the licensing agreement for free (F = 0) as long as its bargaining power is su¢ ciently
high (i.e., if
is su¢ ciently low). The latter is stated formally in the following Corollary.
Corollary 1 When the input supplier (the licensee) does not enter into the …nal goods
market and there is no licensing fee (F = 0), …rm 1 has incentives to license its input
technology if and only if
is su¢ ciently low.
14
The wholesale price, as we saw above, is chosen in order to maximize the joint pro…ts of …rm S and …rm
1; hence, the equilibrium wholesale price is independent of the bargaining power distribution.
9
Why does …rm 1 have incentives to transfer its input technology to …rm S even for free?
Recall that when …rm 1 produces the input in-house, the marginal cost that it faces is c.
When, instead, it licenses its input technology to …rm S, its marginal cost is lower since
wLN < c. We refer to this as the input pricing e¤ ect of licensing. It follows then that
licensing results in an increase in …rm 1’s e¢ ciency, and thus, it reinforces …rm 1’s cost
advantage in the …nal goods market. A straightforward implication of this is that the gross
from T LN pro…ts of …rm 1 are larger under licensing. When the bargaining power of the
input supplier is not too large (i.e., when
is low enough), the latter obtains only a small
share of these pro…ts through the T LN . As a consequence, when
willing to license its input technology even for
free.15
is low enough, …rm 1 is
As we know from our analysis above,
when licensing takes place through a licensing fee, …rm 1 enjoys all of its pro…ts in the …nal
goods market; hence, its licensing incentives then are independent of the bargaining power
distribution.
3.2
Licensing and Entry
We turn now to the examination of the case in which after the signing of the licensing
agreement, …rm S enters into the …nal goods market.
In the last stage of the game, three …rms, instead of two, are active in the …nal goods
market. In particular, …rms 1, 2, and S choose their outputs in order to maximize their
respective pro…ts:
1 (q1 ; q2 ; qS ; w)
= (a
q1
q2
qS )q1
wq1 ;
(7)
2 (q1 ; q2 ; qS ; w)
= (a
q2
q1
qS )q2
cq2 ;
(8)
qS
q1
q2 )qS
c(q1 + qS ) + wq1 :
(9)
S (q1 ; q2 ; qS ; w)
= (a
Solving the system of the …rst order conditions, we …nd the equilibrium quantities as a
function of w:
q1 (w) =
a(2
q2 (w) =
15
) + 2c
(2 + )w
;
2(2
)(1 + )
2(a
c)
4+2
(a
2
w)
2
;
When licensing is for free, then there is no distinction between licensing and outsouring.
10
(10)
(11)
qS (w) =
2(a
c)
4+2
(a
2
w)
2
:
(12)
In the second stage, …rm S and …rm 1 solve the following maximization problem:
max [
w;T
where
1 (w)
and
S (w)
S (w)
dS + T ] [
1 (w)
T ]1
,
(13)
are found after substituting (10), (11) and (12), respectively, into
(7) and (9). It is important to note that while the disagreement payo¤ of …rm 1 continues to
be null, the same does not longer hold for …rm S’s disagreement payo¤. This is so because
…rm S now has an outside option in its bargaining with …rm 1: in case of disagreement,
S can still have pro…ts from its own sales in the …nal goods market. In particular, its
disagreement payo¤ is given by dS = (a
qSD
q2B )qSD
cqS , where qSD = q1B .
Maximizing (13) with respect to T , we obtain:
T =
1 (w)
(1
)[
S (w)
dS ]:
(14)
Using the above expression, we get the following:
S (w)
dS + T = [
S (w)
+
1 (w)
dS ];
(15)
1 (w)
T = (1
S (w)
+
1 (w)
dS ]:
(16)
)[
Substituting in turn (15) and (16) into (13), we note that the latter reduces to an expression
proportional to the joint pro…ts of …rm S and …rm 1 minus …rm S’s disagreement payo¤.
The wholesale price that maximizes this expression is:
wLE =
a(2
)(1
) + c(4 + (2
3)
2(2 + 2
2 2
3
2 ))
.
(17)
One can check that the equilibrium wholesale price exceeds the marginal cost of producing
the input, wLE > c. In other words, in contrast to the no entry case, when …rm S enters
into the …nal goods market, it no longer subsidizes …rm 1. The reason for the change in …rm
S’s behavior is quite clear. In case of entry, the pro…ts of …rm S accrue not only from its
sales to …rm 1 but also from its own sales in the …nal goods market. The less e¢ cient are the
rivals that …rm S faces in the …nal goods market, the larger is its own market share, and in
turn, the pro…ts that it makes from its own sales in the …nal goods market. Because of this,
11
…rm S now has incentives to raise its rivals’cost. In other words, it has incentives to charge
a higher wholesale price to its customer-rival …rm 1. However, an opposite force is also in
action, since …rm S has incentives to decrease the wholesale price in order to keep …rm 1
in the market and obtain pro…ts from the input sales. This e¤ect softens the incentives of
…rm S to set a very high wholesale price. Clearly, in this case too the equilibrium wholesale
price is independent of the bargaining power distribution, while the equilibrium …xed fee,
T LE , which can be found after substituting (17) into (14), increases with the bargaining
power of the input supplier.
One can also check that the relationship between the equilibrium wholesale price and
product di¤erentiation is U-shaped in this case. In particular,
#wLE
#
< 0 if and only if
> 0:437924. Therefore, a decrease in product di¤erentiation leads to a decrease in the
wholesale price when the …nal goods are initially close enough substitutes, and to an increase
otherwise. Why a further intensi…cation of competition when competition is already quite
…erce results in a decrease in the rival’s cost? Intuitively, …rm S, by decreasing the wholesale
price, on the one hand, it reduces its own market share and on the other hand, it increases its
pro…ts from its input sales to …rm 1. The latter pro…ts are quite low when the competition in
the …nal goods market is already too …erce. Therefore, a further increase in the intensity of
competition (a decrease in ) can result in the market foreclosure of …rm 1 (since wLE > c,
…rm 1 faces a cost-disadvantage relative to …rm S) and result in the disappearance of …rm
S’s pro…ts from the input sales.16
It remains to determine the presence or absence of licensing incentives in the …rst stage
of the game. Substituting wLE and T LE into (15) and (16), we obtain the gross from the
licensing fee equilibrium pro…ts of …rm 1 and …rm S. For the reasons that we explained
in subsection 3.1, …rm 1 optimally sets F LE =
S (w
LE )
+ T LE . Therefore, …rm 1’s net
equilibrium pro…ts (included in Table 2) under licensing and entry are:
S (w
LE
1
=
1 (w
LE )
+
LE ).
Comparing
LE
1
with
B,
1
we …nd that …rm 1 has incentives to license its input technology
even when licensing reinforces the intensity of competition. This is formally stated in
Proposition 2.
Proposition 2 When the input supplier (licensee) enters into the …nal goods market, …rm
16
For a review of the market foreclosure issues that arise in the presence of vertical integration see Rey
and Tirole (2007).
12
1 always has incentives to license its input technology.
The intuition for this surprising result is as follows. Under no licensing, …rm 1’s equilibrium
pro…ts consist only of its own pro…ts in the …nal goods market. Under licensing and entry
instead, …rm 1’s equilibrium pro…ts consist of three parts: (i) its own pro…ts from the …nal
goods market, (ii) …rm S’s (gross) pro…ts from its input sales to …rm 1, and (iii) …rm S’s
pro…ts from its own sales in the …nal goods market. The …rst part of these pro…ts is clearly
lower with licensing and entry than without licensing. This is so because in the former case,
the input pricing e¤ ect is negative instead of positive (since wLE > c); hence, …rm 1 is less
e¢ cient when it licenses its input technology than when it produces the input in-house.
Moreover, this is so because when …rm 1 licenses its technology, there are more …rms in the
…nal goods market. Since the pro…ts of …rm 1 from its own sales in the …nal goods market
are lower under licensing, in this case, in contrast to the no entry case, …rm 1 does not have
incentives to license its technology when licensing is for free. When a licensing fee though
is used, then, as mentioned above, …rm 1’s pro…ts under licensing and entry include two
more parts ((ii) and (iii)), both of which are positive. The pro…ts of …rm S from its input
sales to …rm 1 alone are not in the position to compensate …rm 1 for the reduction in its
own pro…ts in the …nal goods market. This so due to the fact that …rm 1’s output does not
remain constant: …rm 1’s output is lower in the licensing with entry case than in the no
licensing case.
It follows from the above discussion that …rm S’s pro…ts from the …nal goods market
constitute the main driver of licensing in this case. In particular, the entry of …rm S in the
…nal goods market, gives rise to two e¤ects, the market expansion e¤ ect and the strategic
e¤ ect. The market expansion e¤ ect arises from the impact that the entry of …rm S has on
the market demand. In particular, although …rm S’s entry has a negative impact on the
quantities of the incumbent …rms, its overall impact on the total market quantity is positive
(i.e., QLE > QB ). The strategic e¤ ect arises from the impact that the entry of …rm S has
on the wholesale price. Although the licensor’s revenues from sales decrease due to its lower
…nal production, the greater licensee’s revenues due to the increased level of output lead the
pro…t margin to be greater compared to the no licensing case. So, the fact that the licensee
becomes more e¢ cient seems to be an advantage for the licensor instead of a drawback.
Thus, the entry of the licensee encourages instead of discouraging the licensing incentives.
Firm 1, by extracting all of the pro…ts that accrue from the sales of …rm S in the …nal goods
13
market, it takes the full advantage of both the market expansion and the strategic bene…t.
These e¤ects make licensing attractive even when licensing intensi…es market competition.
One might wonder why the wholesale price is chosen to exceed the input’s marginal
cost and not to be equal to it, so as the licensor generate the same e¢ ciency with its rival
and thus more market quantity and pro…ts. This is due to the strategic e¤ect, the impact
of which is more pronounced in this comparison because it increases the …nal price of the
licensee’s output. Since the licensor cannot a¤ect directly the …nal price of the licensee’s
…nal good, due to the nature of the …nal competition, it signs a contract where its wholesale
price increases in order to lead indirectly the licensee to increase its …nal price. Although the
licensor’s revenues from sales decrease due to its lower …nal production, the greater licensee’s
revenues, due to the increased level of output and the increased …nal price, lead the pro…t
margin to be greater compared to the case where the wholesale price was charged equal to
the input’s marginal cost. So, the fact that the licensee becomes more e¢ cient seems to be
an advantage for the licensor instead of a drawback. Since, the licensor extracts all licensee’s
greater pro…ts, it fully exploits the strategic bene…t. Thus, the entry of the licensee which
leads to the charge of a greater wholesale price, encourages instead of discouraging the
licensing incentives.
3.3
The Impact of Entry
Having explored the incentives for licensing both with and without entry, we are now in the
position to characterize the impact of entry on these incentives.
Comparing …rm 1’s pro…ts under licensing and no entry with its respective pro…ts under
licensing and entry, we …nd the following.
Proposition 3 The entry of the input supplier (licensee) into the …nal goods market reinforces …rm 1’s licensing incentives.
Interestingly, the entry of the licensee into the market of the licensor has a positive instead
of a negative impact on the latter’s incentives to license its technology. Stated in di¤erent
words, when licensing intensi…es the competition faced by the licensor, the licensing incentives of the latter are stronger. Why is that? The drawbacks as well as the advantages
that licensing with entry has relative to licensing without entry are similar to its respective
drawbacks and advantage relative to no licensing. More speci…cally, its …rst drawback is the
presence of more intense market competition due to the existence of three instead of two
14
…rms in the …nal goods market. Its second drawback is the negative input pricing e¤ ect,
which translates into lower e¢ ciency for …rm 1: However, this e¤ect drives the licensee’s
pro…ts to be greater than …rm 1’s pro…ts. The extraction of these pro…ts via the licensing
fee arises the strategic bene…t that …rm 1 has from the greater e¢ ciency of …rm S and thus
its greater pro…ts. In fact, the inverse input pricing e¤ect, turns to become an advantage
since it increases the joint pro…ts of …rms 1 and S. Along with the greater market expansion e¤ ect which occurs in the case of licensing with entry compared to the no entry case,
…rm 1 has stronger incentives for licensing. As Proposition 3 reveals, the market expansion
e¤ ect and the strategic e¤ ect dominate and the entry of the licensee reinforces the licensing
incentives.
4
The Welfare Implications of Vertical Licensing
We have seen that vertical licensing is desirable for …rm 1. What we have not seen yet,
and we are going to see in this section, is whether vertical licensing is also desirable from a
welfare point of view.
Proposition 4 Vertical licensing both with and without the entry of the licensee in the …nal
goods market, has a positive impact on the consumers’surplus and on the total welfare. Its
positive impact is larger with entry than without entry.
Proposition 4 informs us that vertical licensing is bene…cial both for the consumers and
for the society as a whole. Intuitively, in the no entry case, the positive impact of vertical
licensing on the consumers’surplus is due to the positive input pricing e¤ ect. In the entry
case instead, the positive impact of vertical licensing on the consumers’ surplus is due to
the increase in the number of competitors in the …nal goods market. As for the impact of
licensing on the producers’surplus, we know from Propositions 1 and 2 that the pro…ts of
…rm 1 are higher under licensing than under no licensing. The opposite clearly holds for
…rm 1’s rival: …rm 2 is worse o¤ under licensing since the latter either leads to the increase
in the e¢ ciency of its competitor or to an increase in the number of its competitors. Still,
the negative impact of licensing on …rm 2’s pro…ts is dominated by its positive impact on
the consumers’surplus and on the licensor’s pro…ts.
Proposition 4 also informs us that the positive impact of vertical licensing on consumer
welfare and on total welfare is greater when the licensee enters into the …nal goods market.
15
In light of our previous …ndings this is not so surprising. More speci…cally, even though
…rm 1 is more e¢ cient under no entry than under entry, market competition is still …ercer
in the latter case because there are more competitors in the market, Consequently, the
consumers’surplus is larger under entry than under no entry. In turn, the total welfare is
also larger under entry not only because of the higher consumers’surplus, but also because
of the higher pro…ts that …rm 1 enjoys under entry (Proposition 3).
Recently, the EU revised its rules for the assessment of technology licensing agreements
under the EU competition law. The new regulation that it adopted, Regulation N. 316/2014
on the application of Article 101(3) of the Treaty on the Functioning of the European
Union to categories of technology transfer agreements, facilitates licensing and continues
to re‡ect the view that licensing is in most cases pro-competitive. Based on this view,
a block exemption applies to all the licensing agreements between …rms that have limited
market shares. In particular, according to the regulation, in the case of licensing agreements
between non-competitors, the block exemption applies when the individual market share of
each party does not exceed 30%. Our analysis suggests that vertical licensing, especially in
the case in which it triggers entry into more than one stages of the vertical chain, facilitates
the di¤usion of technology and generates more intense product market competition even
when the market share of the licensor exceeds 50%.17
5
Wholesale Price Contract
In this Section, we consider what would happen if in the second stage of the game, the
licensor and the licensee trade through a wholesale price contract instead of through a
two-part tari¤ contract. In order to simplify our analysis, we assume here that there is no
bargaining - …rm S makes a take-it-or-leave-it o¤er regarding w.18
Under trading through a wholesale price contract, …rm S is not in the position to extract
part of …rm 1’s pro…ts through the …xed fee, since the latter is not available. Because of
this, both in the no entry case and in the entry case, double marginalization is present, i.e.,
w
bLN > c and w
bLE > c. Comparing w
bLN and w
bLE , we …nd that w
bLE > w
bLN if and only if
> 0:5. This occurs because when products are close substitutes, the entrant faces …erce
17
As we show in Section 6, licensing can be welfare-enhancing even when the licensor is initially a monopolist and/or when it has a cost advantage relative to the other incumbent.
18
As we discuss later on the inclusion of bargaining would not a¤ect our results.
16
competition, and thus, it has stronger incentives to raise its rival’s cost by charging a higher
wholesale price. Clearly, once again, in equilibrium, the licensor extracts fully, through the
licensing fee, both the pro…ts that arise from its own sales and the pro…ts of the licensee.
The comparison of its net equilibrium pro…ts in the case of licensing without entry as well
as of those in the case of licensing with entry (included in Table 3) with its respective pro…ts
in the case of no licensing, results in the following.
Proposition 5 When vertical trading takes place through a wholesale price contract, …rm
1 has incentives to license its input technology if and only if the input supplier (licensee)
enters into the …nal goods market.
When …rms trade through a wholesale price contract, in contrast to when they trade
through a two-part tari¤, licensing does not arise in equilibrium in the no entry case. This
is so because, due to double marginalization, the input pricing e¤ ect is negative instead of
positive. In other words, the licensor is less e¢ cient in the presence than in the absence of
licensing. Its lower e¢ ciency translates into a lower market share, and, in turn, into lower
pro…ts from its own sales. When the licensee does not enter into the market, and thus, the
pro…ts from …rm S’s sales in the …nal goods market are zero, the licensor has no reason to
license its technology. When instead the licensee enters into the market, and as result the
…nal market expands, the licensing incentives are restored just like in our main analysis, as
long as a licensing fee can be optimally applied.19
It follows from the above that when vertical trading takes place through a wholesale
price contract, the entry of the licensee not only does not discourage …rm 1 from licensing,
but in fact it constitutes the driving force behind licensing.
6
Extensions - Discussion
In this Section, we discuss brie‡y a number of further extensions of our model in order to
examine the robustness of our main …ndings as well as in order to extract some additional
insights.
19
If we included bargaining, the equilibrium wholesale prices would be lower than w
bLN and w
bLE . In fact
they would be decreasing with the bargaining power of …rm 1 (i.e., with 1
). Clearly, this means, that
the licensing incentives in the entry case would not only be present but that they would be even stronger
then. Moreover, …rm 1 would continue to not have licensing incentives in the no entry case. In fact, in the
latter case, even if …rm 1 had all the bargaining power ( = 0), it would be indi¤erent among no licensing
and licensing.
17
(i) Licensing without Commitment
Throughout our analysis, we have assumed that after the signing of the licensing agreement,
…rm 1 commits to sourcing the input from the licensee. We relax this assumption now. That
is, we allow for the possibility that …rm 1 produces the input in-house in case of disagreement
with …rm S during the negotiations.
In the no entry without commitment case, the last stage of the game is the same as the
one in the no entry with commitment case. The second stage of the game though is now
di¤erent. In particular, …rm 1 now has the outside option to produce the input on its own.
As a result, the maximization problem that …rm S and …rm 1 face is the following:
max [
w;T
where d1 = (a
q1B
q2B )q1B
S (w)
+ T] [
1 (w)
T ]1
d1
;
(18)
cq1B is …rm 1’s disagreement payo¤. Maximizing (18) with
respect to T , we get: T (w) = [
1 (w)
d1 ]
(1
)
S (w):
Using the latter, we rewrite
(18) and we note that w is chosen in order to maximize the joint pro…ts of S and …rm 1
minus the disagreement payo¤ of the latter:
max [
w
S (w)
+
1 (w)
d1 ]:
(19)
From the …rst order condition of (19), we obtain the equilibrium wholesale price, w
eLN =
wLN < c. Substituting w
eLN into T (w), we also obtain the equilibrium …xed fee, TeLN . We
note that TeLN < T LN . This is a straightforward implication of the positive disagreement
payo¤ of …rm 1 under no commitment.
In the entry without commitment case, the last stage of the game as well as the resulting
equilibrium quantities and pro…ts in terms of w are the same as in the case of entry with
commitment. In the second stage of the game now, both …rm 1 and S have outside options.
Firm 1 has the outside option to produce the input on its own, while S has the outside
option to operate as a …nal product competitor without making any input sales to …rm 1.
In particular, …rm S and …rm 1 face now the following maximization problem:
max [
w;T
where dS = (a
qS
(a
qS )q1
q1
q2
q1
S (w)
q2 )qS
dS + T ] [
1 (w)
d1
T ]1
;
(20)
cqS , is the disagreement payo¤ of …rm S and d1 =
cq1 is …rm 1’s disagreement payo¤, where q1 = q2 = qS =
18
(a c)
(2+2 ) .
We maximize (20) with respect to T and then substituting the resulting T (w) back into
(20), we observe that the maximization problem reduces to an expression proportional to
the joint surplus generated by …rm S and …rm 1, minus their disagreement payo¤s:
max [
w
S (w)
+
1 (w)
dS
d1 ]:
(21)
The resulting equilibrium wholesale price is: w
eLE = wLE . Therefore, the equilibrium
wholesale price without commitment coincides again with the equilibrium wholesale price
with commitment; hence, it exceeds …rm S’s marginal production cost. The equilibrium
…xed fees though di¤er among the two cases. In particular, substituting w
eLE into T (w), we
…nd that the resulting TeLE satis…es the following TeLE < 0 < T LE . Therefore, we have here
subsidization of …rm 1’s production via the …xed fee and not via the wholesale price.
Since the wholesale prices without commitment coincide with the respective ones with
commitment, it follows that the equilibrium quantities, as well as the net equilibrium pro…ts
also coincide. A straightforward implication of this is that Propositions 1, 2, 3, and 4 hold
in this case too. The only thing that di¤ers now is Corollary 1. In particular, recall that
according to Corollary 1, when licensing is for free (F = 0), then in the case of no entry
with commitment, …rm 1 opts for licensing if and only if
was su¢ ciently low. Here,
instead, when licensing is for free, …rm 1 always opts for licensing in the no entry case. This
di¤erence arises from the fact that now …rm 1 has an outside option during the negotiations,
and thus, …rm S ends up compensating for it.
(ii) Licensing through Royalties
Throughout our analysis, we have assumed that licensing takes place via a …xed licensing
fee. Now, we consider what happens when …rm 1 charges instead a per-unit of output
royalty, r
0, for the licensed input technology. Clearly, in the absence of the licensing
…xed fee, …rm 1 is not in the position to extract the potential pro…ts of the licensee.
When r is imposed on the output of …rm 1 (and not only on the output of …rm S as it
could be the case under entry), the marginal cost of …rm S increases from c to r + c. The
increase in the input supplier’s marginal cost can translate into worse input sourcing terms
for …rm 1, and thus, into less …rm 1’s pro…ts from its own sales in the …nal goods market.
Not surprisingly then, …rm 1 optimally sets rN = rE = 0 both with and without entry.
Given this, does it have incentives to license its technology? The answer to this question
is already provided in our main analysis. More speci…cally, recall that when F = 0, …rm
19
1 opts for licensing if and only if its bargaining power is su¢ ciently high and the licensee
does not enter into the …nal goods market. When either
is high or entry occurs, licensing
incentives are absent.
When instead the royalty is imposed only in the case of entry and only on the output
of …rm S, we …nd that as r increases, the wholesale price decreases. This happens, because
the increase of …rm S’s marginal cost due to r decrease …rm S’s competitiveness in the
…nal goods market. However, the negative e¤ect that r has on the wholesale price, can not
dominate the increase on the wholesale price due to …rm S’s incentives to raise its rival’s
cost. In this case, thus, we …nd that rE > 0 and in turn that licensing does not arise in
equilibrium.
It follows from the above that when licensing takes place through a royalty, then, in
contrast to what happens when it takes place through a licensing …xed fee, the entry of the
licensee into the …nal goods market discourages licensing.
(iii) Ex-ante Monopoly
In our main model, we have assumed that in the market there are initially two vertically
integrated incumbents, …rm 1 and …rm 2. One might wonder whether this assumption is
innocuous or not. In order to examine this, we consider here the alternative case in which
…rm 1 is the only …rm that it is initially in the market - it is an ex-ante monopolist. We
…nd that in the no entry case, i.e., in the case in which …rm 1 is also an ex-post monopolist,
the licensee, in contrast to our main analysis, does not subsidize the production of …rm 1.
This occurs because …rm 1 does not face any competition, then and thus, …rm S has no
reason to increase its aggressiveness in the …nal goods market. An implication of this is
that …rm 1 and …rm S operate then as a vertically integrated …rm; hence, as …rm 1 in the
benchmark case of no licensing. It follows then that …rm 1 is indi¤erent between licensing
without entry and no licensing. It follows from this that when licensing does not trigger
entry of the licensor into the …nal goods market, the market structure can be crucial for the
licensing incentives: when the licensor is a monopolist it does not opt for licensing, while
when instead it faces competition, it opts for licensing.
When the licensee enters into competition with the licensor, the former, just like in our
main analysis, sets a wholesale price that exceeds the input marginal cost. Still, the fact
that …rm 1 extracts, through the licensing fee, the pro…ts of …rm S from the …nal goods
market, along with the fact that the market is expanded and that …rm S’s pro…ts increase
20
due to …rm 1’s strategic choice, dominate the negative impact of the increased competition
and the licensing incentives are restored.
7
Concluding Remarks
We have examined the incentives of a vertically integrated incumbent to license its input
technology to an external input supplier in a setting in which after the signing of the licensing
agreement, the licensor sources the input from the licensee and the latter can enter into the
…nal goods market and compete with the licensor. We have assumed that initially there are
two vertically integrated incumbents in the market, that the licensing takes place through
a …xed licensing fee, and that vertical trading is conducted via a two-part tari¤ contract.
We have shown that licensing always emerges in equilibrium. This occurs not only when
the licensee does not enter into direct competition with the licensor, but also when it does
enter. In fact, we have shown that in the latter case, although market competition is more
intense, the licensing incentives are stronger. Intuitively, in the absence of entry, vertical
licensing is motivated by the low input price at which the licensor sources the input from the
licensee. The low input price translates into higher e¢ ciency, and thus, into larger market
share and pro…ts for the licensor. When, instead, the licensee enters into the …nal goods
market, …rst, the licensor is less e¢ cient - it pays a high input price - and second, it competes
with more …rms than in the no licensing case. Still, it opts for licensing because the entry
of the licensee results in the expansion of the …nal goods market and in the strategic bene…t
of increasing the licensee’s pro…ts at the expense of its own pro…ts, since the licensor takes
full advantage of these two e¤ects by extracting the resulting pro…ts of the licensee through
the licensing fee.
Our welfare analysis has revealed, …rst, that vertical licensing is always bene…cial both
for the consumers and for the society as a whole, and second, that it is even more bene…cial
in the entry case. Extending our main analysis in various directions, we have also shown that
in many instances the emergence of licensing in equilibrium would be impossible without
the entry of the licensee in the …nal goods market. This is, for instance, the case when the
licensor and the licensee trade via a wholesale price contract or when the licensor is initially
the only …rm in the market.
Summing up, we have provided an explanation for the commonly observed practice of
vertical licensing in markets where licensing can transform the licensee from an input sup21
plier to a direct competitor of the licensor. Our explanation lies on strategic considerations
and not on exogenously assumed e¢ ciencies of the licensee. Clearly, if we had assumed
that the licensee is either more e¢ cient in input production than the licensor or that the
input production is characterized by economies of scale, then vertical licensing and the positive impact of entry on the licensing incentives would be much more expected than in our
setting.
Still, we should mention that our analysis is just a …rst step in the direction of understanding licensing in vertically related markets. In future work, we plan to extend our work
by endogenizing the licensing and the input sourcing decisions of all the incumbents, by considering the interactions that might exist between the various inputs that are used jointly
in the production of …nal goods, by examining what happens under price competition, as
well as by incorporating bargaining over the licensing fee.
8
Appendix
Table 1: Equilibrium Values in the Benchmark Case and in the Licensing with No Entry
Case
2
B = (a c) ;
2
(2+ )2
(2+ ))
q2LN = (a c)(4
q1LN =
4(2 2 )
c)2 (2 )2
LN
T LN = (1 )(a
1 (q1 )
8(2 2 )
2
LN = (a c)(4 (2+ ))
2
16(2 2 )2
2
2
F LN = (a8(2c) (22 ) )
2
2
LN = (a c) (2 )
1
8(2 2 )
a c
2+ ;
(a c)(2 )
;
2(2 2 )
q1B = q2B =
B
1
=
Table 2: Equilibrium Values in the Licensing with Entry Case
2)
(a c)(4 2
4(2+2 2 2 3 )
(a c)(4 3 2 )
q2LE = qSLE = 4(2+2
2 2 3)
2
(1
)(a
c)
LE
T LE = 8(2+ )2 (2+2(4 22 2
1 (q1 )
2
2 2
LE = (a c) (4 3 )
2
16(2+2 2 2 3 )2
2
) 8 2 (2+ )+ (4 2
F LE = (a c) (16(1+
8(2+ )2 (2+2 2 2 3 )
2
2
LE = (a c) (8 8 + )
1
8(2+2 2 2 3 )
q1LE =
2 )2
3)
2 )2 )
Table 3: Equilibrium Values in the Licensing with Entry and Wholesale Price Contract
Case
22
2)
(a c)(4 2
2(8+8 3 2 2 3 )
c)(2 )(4+3 2 )
qb2LE = qbSLE = (a
2(8+8 3 2 2 3 )
2 )2
(a c)2 (4 2
q1LE ) = 4(8+8
1 (b
3 2 2 3 )2
(a c)2 (2 )2 (4+3 2 )2
bLE
2 = 4(8+8 3 2 2 3 )2
(a c)2 (6 )(2 )
FbLE = 4(8+8
3 2 2 3)
(a c)2 (56+8 +48 2 6 3 7 4 +
LE
b1 =
2(8+8 3 2 2 3 )2
qb1LE =
5)
Proof of Proposition 1: Calculating the di¤erence in the pro…ts of …rm 1 in the case of
licensing and no entry and in the benchmark case, we …nd:
LN
1
B
1
=
(a c)2
8(2 + )2 (2
4
2)
> 0:
Thus, …rm 1 always has incentives to license its input technology under no entry.
Proof of Proposition 2: Calculating the di¤erence in the pro…ts of …rm 1 in the case of
licensing and entry and in the benchmark case, we …nd:
LE
1
B
1
=
(a c)2 (4 2
8(2 + )2 (2 + 2
2
2 )2
2
3)
> 0:
Therefore, …rm 1 always has incentives to license its input technology under entry.
Proof of Proposition 3: Calculating the di¤erence in the pro…ts of …rm 1 in the case of
licensing with entry and in the case of licensing without entry,we …nd:
LE
1
LN
1
=
(a
c)2 (1
)(8 8 + 2
2
8(2
)(2 + 2
2 2
3
4)
3)
> 0:
It follows that …rm 1 has stronger incentives to license its input technology with entry than
without entry.
Proof of Proposition 4: In the benchmark case, the consumers’surplus is given by:
CS B = aq1B + aq2B
1 B 2
[(q ) + (q2B )2 + 2 q1B q2B ]
2 1
23
p1 q1B
p2 q2B =
(a
c)2 (1 + )
:
(2 + )2
In the case of licensing with no entry, the consumers’surplus is:
CS LN = aq1LN + aq2LN
1 LN 2
[(q ) + (q2LN )2 + 2 q1LN q2LN ]
2 1
2 ) + 3 (4 + 5 )
(a c)2 32(1
:
2 )2
32(2
p1 q1LN
p2 q2LN =
In the case of licensing with entry, the consumers’surplus is:
1 LE 2
[(q ) + (q2LE )2 + (qSLE )2 +
2 1
CS LE = aq1LE + aq2LE + aqSLE
2 q1LE q2LE + 2 q1LE qSLE + 2 q2LE qSLE ]
(a
c)2 48 + 80
84
32(2 + 2
Calculating, we …nd that CS LN
2
108
2
p1 q1LE
3
2
p2 q2LE
+ 43
3)
CS B > 0 and CS LE
4
pS qSLE =
5
+ 30
:
CS B > 0.
Total welfare is de…ned as the sum of consumers and producers’surplus, namely: W k =
CS k +
k
1
and W LE
+
k
2
+
k,
S
where k = B, LN and LE. Calculating, we …nd that W LN
WB > 0
W B > 0. Thus, both the consumers’surplus and total welfare are greater under
licensing than under no licensing.
Moreover, calculating, we …nd that CS LE CS LN > 0 and W LE W LN > 0. Therefore,
both the consumers’surplus and total welfare are greater with entry than without entry.
Proof of Proposition 5: Calculating the di¤erence in the pro…ts of …rm 1 in the case of
licensing with entry and in the case of licensing without entry, we …nd:
bLE
1
where A = 512 + 256
bLN
=
1
448
2
(a c)2 A
8(2 + )2 (8 + 8
3
32
3 + 170 4
32
5
2
35
+2
3 )2
> 0;
6 + 8 7 + 4 8.
Therefore, …rm 1
has always stronger incentives to license its input technology with entry than without entry.
9
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