Package licenses in patent pools with basic and optional patents ∗Kenji AZETSUOsaka University of Health and Sport Sciences †Seiji YAMADA
Graduate School of Economics, Kobe University ‡December. 2010

Abstract
Recently, patent pools have been often established by the patent holders in order topromote research and development (R&D) and set technological standards. Patent pools are
one of the administration methods for licensing, whereby many patent holders assemble theirown patents and the administrator of the patent pools oﬀers the licenses to users. This paper
investigates the patent holders’ incentive to form a patent pool, the patent pool’s licensing
behavior, and the anticompetitive eﬀect of a patent pool. Our model is characterized by thefollowing two features. First, we consider diﬀerent two types of patent: basic and optional.
Second, we consider not only a patent pool that oﬀers a single package license, but alsoa patent pool that oﬀers multiple package licenses. The single package license includes
both basic and optional patents in the patent pool. The multiple package licenses involve
two licenses: one includes only basic patents, whereas the other includes both basic andoptional patents. The results of our analysis yield some implications for a patent pool thatis characterized by the strength of complementarity between basic and optional patents.
Keywords: patent pools, multiple package licenses, antitrust laws, bundling goods
JEL Classification: K11, K21, L4∗
We would like to thank Kenichi Amaya, Reiko Aoki, Sadao Nagaoka, Taiji Hagiwara, Toshihiro Matsumura,
Takao Okawa, Kotaro Suzumura, and Takashi Yanagawa. The ﬁnancial support oﬀered by KAKENHI (19830108)
is also gratefully acknowledged.
†
Correspondence to: Kenji Azetsu, Osaka University of Health and Sport Sciences, 1-1 Asashirodai, Kumatoricho, Sennan-gun,Osaka 590-0496, JAPAN. E-mail:azetsu@ouhs.ac.jp. Phone:072-453-8900 (Int’l +81-72-4538900).
Fax:072-453-8900 (Int’l +81-72-4538900).
‡
E-mail: sei.yamada@lion.kobe-u.ac.jp1

1

Introduction

This paper investigates the anticompetitive eﬀect of a patent pool that oﬀers a package license.
Our analysis is characterized by the following two features. First, we consider diﬀerent two
types of patents: basic and optional. The basic technology can be used by itself, it becomes
more valuable when used in combination with the optional technology. The optional technology
cannot be used by itself. Second, we consider not only a patent pool that oﬀers a single package
license, but also a patent pool that oﬀers multiple package licenses. The single package license
includes both basic and optional patents in the patent pool. In the multiple package case there
are two package licenses: one includes only basic patents, whereas the other includes both basic
and optional patents.
A patent pool refers to organizations where patent holders concentrate their own patents
for commercializing new innovations or for setting standards, and oﬀer a package license thatis inclusive of many patents in the pool1 A patent pool plays an important role in solving the“tragedy of the anticommons”, which is discussed in Heller and Eisenberg (1998). The wellknown “tragedy of the commons” is the situation wherein a resource can be overused when it
is not protected by property rights. “Tragedy of the anticommons”, as Heller and Eisenberg
indicated, refers to a situation wherein “excessive” property rights render the resource underused
when there are multiple property rights holders. In the case of patents, excessive property rights
can have the perverse eﬀect of stiﬂing or discouraging innovation. A patent pool is expected to bea useful means to solve this “tragedy of the anticommons” that arises, particularly in advanced
technology ﬁelds. A patent pool enables ﬁrms to reduce the cost of seeking technologies andto negotiate by simplifying the license contract. Furthermore, a patent pool can avoid patent
litigations and can help establish standardization committees such as MPEG-LA, DVD 6C
Licensing Agency, and 3G2 .
Unfortunately, the competition authorities in many countries have a deep-rooted suspicion ofpatent pools, which involve cooperative activity between patent holders. There is the possibility
1
2

Examples of comprehensive surveys on patent pools are Shapiro (2001) and Gilbert (2004).
The standardization committees are the organizations that set international technological standards. Tech-

nological standards aim at the wide adoption of technologies in the marketplace. However, this wide adoption oftechnologies may bring about patent conﬂict between patent holders or ﬁrms. Standardization committees establish patent pools in order to avoid these conﬂicts. MPEG-LA, DVD 6C Licensing Agency, and 3G are examples
of such standardization committees.

2that a patent pool can exercise monopoly power as a major cartel3 . Historically, patent pools
have been abused since the early 1900s4 . Priest (1977) indicated that it is possible that a patentpool is a means to disguise a cartel, formed by using a cross-license between the members in thepool. Thus, many discussions have been held between economists, legal scholars, and antitrust
enforcement leagues as to whether patent pools beneﬁt both intellectual property owners andconsumers. Our concern is to determine whether patent pools are competitive or anticompetitive.
The US competition authority focuses primarily on the technical relationships between thepatents included in the pool5 . Its viewpoint is that a pool of technical substitute patents ismore suspicious than a pool of technical complementary patents. In addition to this view of the
US antitrust enforcement agency, Shapiro (2001) and Lerner and Tirole (2004) focused on thetechnical relationships between the patents included in a pool, and investigated whether patentpools have an anticompetitive eﬀect. They concluded that a patent pool is pro-competitive when
the patents are of the technical complementary type, whereas a patent pool always operates as acartel when the patents are technical substitutes. These results are consistent with the current
US and European policies (see Lerner and Tirole 2007).
Most of the literature on patent pools has sought to determine the social implications of
the pool under the situation where all ﬁrms join the pool (Lerner and Tirole 2004). Recently,
some works have focused on the ﬁrm’s incentive problem in relation to participation in the pool
(Aoki and Nagaoka 2004; Brenner 2009; Langinier 2006; Lerner and Tirole 2007). Our paper
considers both the patent pool that oﬀers only a single package license and the patent poolthat oﬀers the multiple package licenses. There are no theoretical discussions of the patentpool in the literature that oﬀered a choice of single package and multiple package licenses. The
multiple package licenses are packaged within the subsets of all the patents in the pool. In
practice, it is observed that about 12% of the pools surveyed by Lerner et al. (2003) oﬀer
multiple package licenses (that is, about 88% of the pools oﬀer the single package license). For
example, MPEG-LA, which is the patent pool administrator in regard to MPEG compression
technological standards, oﬀers multiple package licenses. One enormous advantage of multiple
package licenses is that they give users various choices of patents, because a single package
3

See Antitrust Guidelines for the Licensing of Intellectual Property. April 1995.
See Gilbert (2004) for historical perspectives on the antitrust treatment of patent pools.
5
The US Department of Justice focused on the technical relationship between the pools in three business review
4

letters regarding an MPEG patent pool and two DVD patent pools. See Shapiro (2001) for details.

3license inclusive of all the patents in the pool could be tie-in sale. The recent guidelines of theEuropean Commission encourage patent pools to oﬀer the multiple package licenses as a usefulway to provide users with a broader choice 6 .
In this paper, we investigate the patent holders’ incentive to form a pool, the patent pool’s
licensing behavior, and the anticompetitive eﬀect of a pool. In particular, we focus on the patentpool that oﬀers multiple package licenses. When does a patent pool oﬀer the multiple package
licenses and when does it not do so? When does the patent pool have an anticompetitive eﬀect?
Should the competition authorities, such as the European Commission, encourage patent pools
to oﬀer the multiple package licenses? We ﬁnd that the technical complementarity between thepatented technologies in a pool plays a critical role in answering the above questions.
The paper is organized as follows. Section 2 deﬁnes patented technologies and the users’
gross surplus for using patent in our models. Section 3 characterizes the licensing fees in thecase where the patent pool licenses monopolistically. Section 4 characterizes the licensing feesin the case where patent holders license individually. Section 5 analyzes the anticompetitiveeﬀect of a patent pool, using the outcomes of Sections 3 and 4, and then investigates patent
holder’s incentives to form a patent pool. Finally, Section 6 discusses the results derived fromthe analysis conducted in the paper.

2

Basic set-up

We suppose that there are two patents, A and B. Patent A relates to basic technology, whereas
patent B involves optional technology, which is valuable only when used in combination with thebasic technology. Although the basic technology can be used by itself, it becomes more valuable
when used in combination with the optional technology. The optional technology cannot be
used by itself. Each patent is owned by a patent holder, A and B. Following Shapiro (2001) and
Lerner and Tirole (2004), we distinguish between patent holders and users. Patent holders do
not have the ability to commercialize the patented technology on their own. The patent holder
obtains the licensing fee from patent users, and the users obtain a surplus using the patents.
Users make their products by using patents, which is represented by U (m, θ). m ∈ {0, 1, 2}
denotes the number of patents employed by the user to make a product, and θ represents
6

See the Guidelines on the application of Article 81 of the EC Treaty to technology transfer agreements (2004/C

101/02).

4

the heterogeneity between users. Note that the users buy license A ﬁrst because it is basic.
Therefore, m = 1 implies that the user uses only patent A and m = 2 implies that the user uses
both patents A and B. For simplicity, we assume that users are distributed uniformly on theinterval [0, 1].
Our model speciﬁes the user’s gross surplus function as the following quadratic form:
U (m, θ) = θm + sm2 , (m = 0, 1, 2)(1)where s > −1/3. The levels of gross surplus for each number of patents are U (0, θ) = 0,
U (1, θ) = θ + s, and U (2, θ) = 2θ + 4s. We ﬁnd that a user with a high θ obtains a high level ofgross surplus for any number of patents used. The heterogeneity of the gross surplus between
users is also assumed by Lerner and Tirole (2004). They explained heterogeneity in terms of (a)the ﬁxed cost for the user to adopt the patent, (b) the user’s opportunity costs of choosing thepatent, and (c) the beneﬁts that the user derives from the patent.
The initial diﬀerences in the gross surplus are △U (1, θ) = θ + s and △U (2, θ) = θ + 3s.
The diﬀerence, △U (1, θ), implies the user’s willingness to pay for patent A when the user does
not access any patent, and △U (2, θ) is the user’s willingness to pay for patent B when the user
already has access to patent A. Note that the willingness to pay for an additional patent depends
on θ, and the user with a high θ obtains a high additional surplus7 . Lerner and Tirole (2004)
assumed that the willingness to pay for an additional patent is the same between users. Under
this assumption, a patent pool solely oﬀers single package licenses because all users demand
the same number of patents. In practice, however, 12% of the patent pools in the Lerner etal. (2003) sample oﬀered multiple package licenses. To explain the observed oﬀering of multiple
package licenses by 12% of the patent pools, we must allow for diﬀerences in the willingness topay for an additional patent between users. In reality, because of the variations in the ability
or knowledge of each user, it is no wonder that there are diﬀerences in their beneﬁts from using
an additional patent.
The second diﬀerence in the gross surplus is △2 U (2, θ) = 2s. The value of s represents
the strength of complementarity between basic and optional patents. When an optional patent
7This speciﬁcation satisﬁes the discrete form of what we refer to as the Spence? Mirrlees single-crossing

condition in the Contract Theory literature; see, for example, Salanie (2002) and Bolton and Dewatripont (2005).
The condition is known for separating diﬀerent types of agents by oﬀering larger allocations to higher types andmaking them pay for the privilege.

5signiﬁcantly enhances the usefulness of a basic patent, the value of s cannot be negative (thestrength of complementarity is strong ). Then, the gross surplus function is convex for thenumber of patents. The user’s additional gross surplus for an optional patent is larger than thatfor a basic patent: △U (1, θ) ≤ △U (2, θ). An example is the combination of a PC and MS Oﬃce,
containing MS Word and Excel, etc. The PC is the basic patent and MS Oﬃce is the optional
patent. If we use MS Oﬃce installed on a PC, we would have a signiﬁcantly larger additionalgross surplus.
When an optional patent does not signiﬁcantly enhance the usefulness of the basic patent,
then the value of s could be negative (the strength of complementarity is weak ). Then, the grosssurplus function is strictly concave for the number of patents. In this case, △U (1, θ) > △U (2, θ)
is satisﬁed. We can consider a PC and typing software as an example. Typing software is anapplication software program for teaching touch typing. However, installing typing software,
which is optional, would not result in a signiﬁcantly large additional gross surplus.

3

Patent pool pricing

3.1

Users’ decisions and the demand for package licenses

In this section, we characterize the equilibrium for the case where patents A and B are monopolistically licensed only by the patent pool formed by patent holders A and B. The patent pool
could oﬀer users two package licenses, A and AB. In package A, the pool licenses patent A forusers for a licensing fee, pA . In package AB, the pool licenses patents A and B for users for the
licensing fee, pAB ; the package AB is a bundled good. If a user wants only the basic patent, heor she can buy package A, whereas if a user wants both the basic and the optional patent, heor she can buy the package AB. In addition, we assume that a pool must pay very low costs foroﬀering a package license. The pool does not oﬀer the package that no users buy.
Given the licensing fees, pA and pAB , the users choose the number of patents m to maximize
their net surplus. When packages A and AB are oﬀered by the patent pool, the type θ user
chooses which package to buy in the following manner:• The user does not buy any packages (the user chooses m = 0) if θ + s − pA < 0.
• The user buys package A (the user chooses m = 1) if θ + s − pA ≥ 0 and θ + 3s < pB ,where pB ≡ pAB − pA .

6• The user buys package AB (the user chooses m = 2) if θ + 3s ≥ pB .
As the surpluses diﬀer across diﬀerent users, the number of patents chosen by each user also
diﬀers across users. Now, we deﬁne θA ≡ pA − s, θB ≡ pB − 3s. The type θA user is indiﬀerent
between buying package A and not buying any package. The type θB user is indiﬀerent between
buying package A and package AB. Each user behaves as follows:• The users on interval [0, θA ] where θA ≡ pA − s do not buy any packages.
• The users on interval [θA , θB ] where θB ≡ pB − 3s buy package A.
• The users on interval [θB , 1] buy package AB.
Lemma 1. When given licensing fees satisfy the inequality pB − pA > 2s, there are users who
buy package A.
Proof. When θA < θB , there are users who buy package A. From the deﬁnition of θA and θB ,we can ﬁnd that θA < θB ⇔ pB − pA > 2s.

■

When licensing fees satisfy the inequality pB −pA > 2s, there are both users who buy package
A and users who buy package AB (θA < θB ). Then, the demand for the package A is:
DA = θB − θA .

(2)

DAB = 1 − θB .

(3)

The demand for the package AB is:

When licensing fees do not satisfy the inequality pB − pA > 2s, then no user buys package
A (DA = 0). The type θ user decides whether to buy the package AB in the following manner:• The user does not buy any packages (the user chooses m = 0) if 2θ + 4s < pAB .
• The user buys package AB (the user chooses m = 2) if 2θ + 4s ≥ pAB .
Then, deﬁning θC ≡ (pAB /2) − 2s, the demand for the package AB is:
DAB = 1 − θC .

7

(4)

3.2

Patent pool equilibrium

For simplicity, we ignore the costs paid by the pool members (the patent holders) for developing
their own patents. Given the demand function for each package license, the patent pool’s proﬁt
is Πpool = pA DA + pAB DAB = pA (DA + DAB ) + pB DAB . Note that DA + DAB is the demandfor patent A. The patent pool determines pA and pB to maximize its proﬁt.
For simplicity, we ignore the costs paid by the pool members (the patent holders) for developing their own patents. Under the demand function of each package license, the patent pool’s
proﬁt is Πpool = pA DA + pAB DAB = pA (DA + DAB ) + pB DAB . Note that DA + DAB is thedemand for patent A. The patent pool decides pA and pB to maximize the proﬁt. From lemma
1 and the deﬁnition of demand (2), (3) and (4), the proﬁt is rewritten as the follows.

Πpool =pA (1 − θA ) + pB (1 − θB ),if pB − pA > 2s (θA < θB ).

pAB (1 − θC ),if otherwise.

The equilibrium licensing fees are given by the solution of above proﬁt maximization problem.
Now we consider the case where patent pool oﬀers both package A and AB. The optimal licensing
fees are the solution of the following proﬁt maximization problem.
max pA (1 − θA ) + pB (1 − θB ),pA ,pB

where pB − pA > 2s. The interior solution should satisfy the following equations.

∂Πpool= (1 − pA + s) − pA = 0∂pA∂Πpool= (1 − pA + s) − pA = 0
∂pA

(5)
(6)

Since the second order condition is warranted, the licensing fees which satisfy (5) and (6)are p∗A = (1 + s)/2, p∗B = (1 + 3s)/2. Checking these licensing fees satisfy the inequalitypB − pA > 2s, we ﬁnd that the optimal solution is p∗A = (1 + s)/2 and p∗B = (1 + 3s)/2 (that isp∗AB = p∗A + p∗B = 1 + 2s) if s is negative value.
The equation (5) and (6) is rewritten as the following forms:

8

pA
=1
1 − pA + spB
ψA (pA ) ≡=1
1 − pB + 3s
ψA (pA ) ≡

(7)
(8)

where ψA (pA ) and ψB (pB ) are the price elasticity of the demand for patent A, and of thedemand for patent B, respectively. Both ψA (pA ) and ψB (pB ) are increasing for each licensing
fee. The licensing fees are determined in such that ψA (pA ) and ψB (pB ) are equal to one. WhenψA (pA ) is located above ψB (pB ) as Fig. 1, there are users who buy the package A (θA < θB ).
In the case where s is negative value, patent B is not really attractive for users compared withpatent A. Then the price elasticity of demand for patent B is smaller than that of demand forpatent A; ψB (pB ) > ψA (pA ). Similarly to the mechanism of price discrimination, pA is higher
than pB ; pB − pA < 2s.
[Insert Fig. 1]
If s is not negative value, then the patent pool does not oﬀer the package A. The patentpool’s problem ismax pAB (1 − θC ),pAB

The optimal licensing fee satisﬁes
ΠAB
1
1= (1 − pAB + 2s) − pAB = 0
pAB
2
2

(9)

Since the second order condition is warranted, the licensing fees which satisfy (9) is p∗AB =1 + 2s. Then we get the following proposition.
Proposition 1. Patent pool pricing(1) Multiple package licenses:

When s is negative value, the patent pool oﬀers two package

licenses A and AB, the licensing fees are p∗A = (1 + s)/2, p∗AB = 1 + 2s (p∗B = (1 + 3s)/2).
(2) Single package license: When s is not negative value, the patent pool oﬀers only single
package AB, and the licensing fee is p∗AB = 1 + 2s.
Proposition 1 shows that patent pool pricing is characterized by the value of s. When thestrength of complementarity between A and B is weak (s < 0), both the package license A,

9inclusive of only basic, and the package license AB, inclusive of basic and optional, are oﬀered
by the patent pool. Each package licensing fee is p∗A = (1 + s)/2 and p∗AB = 1 + 2s. In thiscase, there are both users who buy the package license A and who buy the package license AB.
The patent pool is willing to oﬀer the multiple package licenses by oﬀering not only the package
license AB but also the package license A in order to maximize his/her own proﬁt. On theother hand, when the strength of complementarity is strong (s ≥ 0), Only the package license
AB, inclusive of basic and optional, is oﬀered by the patent pool. The package licensing fees isp∗AB = 1 + 2s. In this case, there are users who buy only the package license AB, if they buy thelicense. Since the strength of complementarity is strong, there are not the users who use only
patent A. This case corresponds to the patent pool pricing in Shapiro (2001) and the demandmargins bind in Lerner and Tirole (2004).
The patent pool oﬀers only the package AB which is similar to the tie-in sale and is notwilling to oﬀer multiple package licenses in order to maximize his/her own proﬁt. In order toavoid the tie-in sale, EU committee encourages patent pools to oﬀer multiple package licenses.
But our result is that none of users buy package licenses when is negative. Therefore, therecommendation of the competition authority makes no senses in the case s ≥ 0 under ourmodel.

4

Individual pricing

4.1

Demand for each patent

In this section, we consider the case where a patent pool is not established. Patent holder A
and B individually licenses the users to his/her own patent. If a user wants to use patent A(patent B), the user must access to patent holder A (patent holder B). If a user wants to use
both patent, then the user must access with both the patent holders.
Given the licensing fee pi (i = A, B), the users decide whether or not to choose each patent
to maximize their net surplus. In the same manner of section 3, we describe the users behavior
as the follows. When given licensing fees satisfy the inequality pB − pA > 2s,• The users on interval [0, θA ] where θA ≡ pA − s do not buy any packages.
• The users on interval [θA , θB ] where θB ≡ pB − 3s buy the package A.

10• The users on interval [θB , 1] buy both license A and B.
When given licensing fees satisfy the inequality pB − pA ≤ 2s,• The users on interval [0, θC ] where θC ≡ (pAB /2) − 2s do not buy any packages.
• The users on interval [θC , 1] buy both license A and B.
Given the licensing fee pi (i = A, B), the users decide whether or not to choose each patent
to maximize their net surplus. The demands for patent A and B are

dA =

dB =


1 − θA

if pB − pA > 2s

 1 − θC

if pB − pA ≤ 2s
1 − θ

B

if pB − pA > 2s

 1 − θC

if pB − pA ≤ 2s

(10)

(11)

Under each demand of patent, the proﬁts of patent holder A and B are πA = pA dA andπB = pB dB , respectively.

4.2

Individual pricing equilibrium

The patent holders decide their licensing fees to maximizing their own proﬁt, as a given otherlicensing fee. Individual pricing equilibrium is characterized by the follows:
Proposition 2. Individual pricing∗∗(1) When s is negative value, Nash equilibrium licensing fees are p∗∗
A = (1+s)/2, pB = (1+3s)/2.
∗∗(2) When s is not negative value, Nash equilibrium licensing fees are p∗∗
A = pB = 2(1 + 2s)/3.
∗∗∗∗∗∗
Proof. If the equilibrium licensing fees p∗∗
A and pB satisfy the inequality pB − pA > 2s, thepatent holder A and B are faced on the demands of their patents 1 − θA and 1 − θB , respectively.
Then the licensing fee of patent A is p∗∗
A = (1 + s)/2, which maximizes the proﬁt of patent holder
A, for any licensing fee of patent B, as far as pB − pA > 2s is satisﬁed. The licensing fee ofpatent B is p∗∗
B = (1 + 3s)/2, which maximizes the proﬁt of patent holder B, for any licensing fee∗∗of patent A, as far as pB − pA > 2s is satisﬁed. Substituting p∗∗
A = (1 + s)/2 and pB = (1 + 3s)/2

is equilibrium licensing fees when s < 0.

11

∗∗
∗∗
∗∗
Next, if the equilibrium licensing fees p∗∗
A and pB satisfy the inequality pB − pA ≤ 2s, thepatent holder A and B are faced on the same demands 1 − θC . The licensing fees of patent A
and B must maximize each proﬁt of patent holders for each licensing fee. Then optimal response∗∗of patent holder A is p∗∗
A = −(1/2)pB + 1 + 2s, and the optimal response of patent holder B is
∗∗
∗∗
∗∗
p∗∗
B = −(1/2)pA +1+2s. Therefore Nash equilibrium is pA = pB = (2+4s)/3. Substituting the∗∗∗∗∗∗licensing fees into the inequality p∗∗
B − pA ≤ 2s, we can ﬁnd pA = pB = (2 + 4s)/3 is equilibrium

licensing fees when s ≥ 0.
■
Similar to proposition 1, proposition 2 shows that individual pricing is characterized by thevalue of s. When the strength of complementarity is weak (s < 0), patent holder A of the basic
oﬀers the equilibrium licensing fee p∗∗
A = (1 + s)/2 and patent holder B of the optional oﬀers theequilibrium licensing fee p∗∗
B = (1 + 3s)/2. In the case, there are the users who buy only basic
and who buy both basic and optional. Since the strength of complementarity is weak (s < 0),
the demand for patent A is independent of the demand for patent B. Then each patent holder
sets his/her licensing fee, regardless of the strategic interaction between patent holders. The
licensing fee of basic patent is set higher than that of optional, because price elasticity of demandfor patent holder A is lower than that of demand for patent holder B. As the result, when s is∗∗negative value, these licensing fee is asymmetric; p∗∗
A > pB .

When the strength of complementarity is strong (s ≥ 0), each patent holder A and B oﬀers∗∗
the equilibrium licensing fee p∗∗
A = pB = (2 + 4s)/3. In the case, the users decide whether they

buy two licenses from patent holder A and patent holder B, since the strength of complementarity
is strong . Since the users who buy any license buy both paten A and B, the demand for thepatents depends on the sum of licensing fees (pA + pB ). Therefore each patent holder must set
his/her licensing fee under the strategic interaction between patent holders. The concern of theusers who buy both patent A and B is only the sum of licensing fees. Then patent holder A
does not have technical advantage as the patent holder of basic patent, and these licensing fee∗∗
is symmetric ; p∗∗
A = pB . This case corresponds to the Cournot-Shapiro argument in Shapiro(2001) and Demand margin bind in Lerner and Tirole (2004).

12

5

Welfare analysis

In section 3 and 4, we characterize the licensing fees in the equilibrium under a patent pooland in the absence of a patent pool. Comparing the welfare of patent pool pricing with that ofindividual pricing, we analyze whether or not a patent pool enhances social welfare.
We deﬁne the social welfare as the sum of users’ net surplus and patent holders’ proﬁt. Since
users’ payment for the patent is equal to the proﬁt of patent holder A and B, the social welfare
is equal to the gross surplus of users. If there are the users who buy only patent A, then thesocial welfare (that is the sum of users’ net surplus) is
∫ θB

∫ 1
(θ + s)dθ +θA(2θ + 4s)dθ

(12)

θB

where θA ≡ pA − s and θB ≡ pB − 3s. Therefore the social welfare depends on the licensing
fee of patent A and B (or licensing fees of patent packages), pA and pB 8 .
If there are not the users who buy only patent A, then the social welfare is∫ 1(2θ + 4s)dθ

(13)

θC

where θC ≡ (pA + pB /2) − 2s. The social welfare depends on the sum of licensing fees (orlicensing fee of the package AB), pA + pB .
Proposition 3. A patent pool does not aﬀect the welfare when s is negative value, but enhances
welfare when s is not negative value.
Proof. Now, we consider the case where the value of s is negative. From proposition 1-(1) andproposition 2-(1), we know that the licensing fees in patent pool pricing are same as in theindividual pricing ( p∗A = p∗∗
A ). Then we can ﬁnd that a patent pool does not aﬀect the welfare(12) when s is negative value.
8

Our model does not deﬁne R&D cost of patent holders. As the result, the social welfare is same as the sum of

all users’ gross surplus, and the social welfare does not include the social cost for R&D. Once information goods
such as technical information are developed, the additional cost for licensing information goods is zero (R&D costis lump cost). The social welfare is all users’ gross surplus minus R&D lump cost. But this does not inﬂuence
our results.

13

Next we consider the case where the value of s is not negative. When s is not negative, lower
the sum of the licensing fees pA + pB , higher the social welfare (13). From proposition 1-(2),the licensing fee is p∗AB = p∗A + p∗B = 1 + 2s in the patent pool pricing. And the licensing fee is∗∗p∗∗
A + pB = 4(1 + 2s)/3 n the individual pricing, from proposition 2-(2). Then we can ﬁnd thata patent pool enhances social welfare when is not negative value.
■
Next we investigate whether the patent holder A and B have the incentive to form a patentpool. As the standard assumption of the corporate merger theory, we assume that the patent
holders A and B form a patent pool if the patent pool’s proﬁt is larger than the sum of the
proﬁt of the patent holders A and B (Πpool > πA + πB ) 9 , with the proﬁt of patent pool, Πpool ,we can get the following proposition.
Proposition 4. When s is negative value, Πpool = πA + πB is satisﬁed. Then the patent holder
A and B does not have the incentive to form a patent pool. When s is not negative value,Πpool > πA + πB is satisﬁed. Then the patent holder A and B have the incentive to form a patent
pool.
Proof. Firstly we consider the case where s is negative value. We already know the licensing∗∗∗
fees in patent pool pricing are same as in the individual pricing (p∗A = p∗∗
A and pB = pB . The

proﬁt of the patent pool is same as the sum of proﬁt of patent holders A and B.

Πpool = πA + πB =(1 + s)2 + (1 + 3s)2
4

Then the patent holder A and B does not have the incentive to form a patent pool.
Next we consider the case where s is not negative value. From the proposition 1-(2), we
get the patent pool proﬁt is Πpool = (1 + 2s)2 /2. On the other hand, the proﬁts of patent
holders A and B are πA = πB = 2(1 + 2s)2 /9, from the proposition 2-(2). Then we can ﬁndthat Πpool > πA + πB is satisﬁed. When s is not negative, the patent holders A and B have the
incentive to form a patent pool.
■9

We do not focus on the process of patent pool formation and the stability of pools, as Brenner (2005) andsome other studies have undertaken. In our paper, the patent holders have the incentive to form a patent pool if
the proﬁt of the pool is larger than the sum of the patent holders’ proﬁts, since the equilibrium in the pool caseis equal to the corporative solution of the patent holders (the pool maximizes their joined proﬁts).

14

Proposition 3 and 4 show that the eﬀect of a patent pool on the welfare and the patent
holders’ incentive to form a pool are characterized by the value of s. When the strength ofcomplementarity is weak (s < 0), a patent pool does not aﬀect the welfare and patent holders
do not have the incentive to form a patent pool. From proposition 2, we ﬁnd that each patent
holder sets his/her licensing fee without inﬂuencing each other in ex-post, when s is negative.
∗∗∗∗∗
The patent pool pricing is equal to the individual pricing: p∗A = p∗∗
A , pAB = pA + pB , Then thepatent pool does not aﬀect the welfare. Further, the patent holders do not have the incentive
to form patent pool, since the corporative pricing does not change their own proﬁt.
On the other hand, a patent pool enhances the welfare and patent holders have the incentive
to form a patent pool when the strength of complementarity is strong (s ≥ 0). From proposition
2, we ﬁnd that each patent holder sets his/her higher licensing fee in order to get his/her more
proﬁts. As the result, the licensing fee of the patent pool is lower than the sum of licensing
fees of individual patent holders: p∗AB < p∗A + p∗B . Then the patent pool enhances the welfare.
Further the patent holders have the incentive to form a pool, since the corporative pricing byforming a pool can reduces the licensing fees to the levels which maximize both proﬁt of patent
holders.

6

Conclusion

This paper investigates the anti-competitive eﬀect of the patent pool which oﬀers the package
license. The most important work of the paper is to consider the strength of complementarity
between basic and optional patents. We ﬁnd that the strength of complementarity characterizes
the licensing behavior of the patent pool, the social welfare, and the patent holders’ incentive
to form the patent pool. Our main results are as follows.
(1) If the strength of complementarity between basic and optional is weak, the patent holders
do not have the incentive to form the patent pool; otherwise, there is other patent holders’
incentive beyond our model. If the patent pool is formed for any reasons, both the package
license inclusive of only basic , and the package license inclusive of basic and optional are oﬀered
by the patent pool (multiple package licenses). Then the form of the patent pool does not change
the licensing fee which user pay to patent holders. As result, the patent pool does not aﬀect thewelfare.
(2) If the strength of complementarity between basic and optional is strong, the patent holders

15

have the incentive to form the patent pool. Only the package license inclusive of basic andoptional is oﬀered by the patent pool (single package license). Then the form of the patentpool reduces the licensing fee which users pay to patent holders. As the result, the patent poolenhances the welfare.
These results lead to four suggestions. First, we suggest that actually-observed many patentpools could include only strong complementary patents. It is observed that about 88% of thepatent pools, surveyed by Lerner et al. (2003), oﬀers the single package license. From our model,
it is when the strength of complementarity is strong that the single package license is oﬀered by
the patent pool. Therefore, we can guess that actually-observed many patent pools include only
strong complementary patents.
Second, we suggest the reason why actually-observed many patent pools include only strong
complementary patents. Our model explains that the patent holders have strong incentive toform a patent pool, if the strength of complementarity is strong, conversely the patent holders
do not have the incentive to form a patent pool, if the strength of complementarity is weak.
Furthermore, the antitrust guidelines require the patent pools only to include ”essential patents”
which implies strong complementary among the patents.
Third, it is less important on our model that the competition authorities, such as the European Commission etc, encourage patent pools to oﬀer the multiple package licenses. The
licensing behavior of patent pools is determined by the proﬁt maximization of the patent pools.
If multiple package licenses increase the proﬁt of a patent pool, the patent pool is willing tooﬀer the multiple package licenses. Otherwise, the patent pool does not oﬀer the multiple package licenses as long as the competition authorities enforce the patent pool to oﬀer the multiple
package licenses. Even if the competition authorities enforce the patent pool which oﬀers the
single package license to oﬀer the multiple package licenses, any users dose not buy the package
license which include only basic as long as the licensing fee of the single package license change.
Finally, we suggest the criterion for pro-competitive patent pools. The antitrust guidelines
indicate that the patent pools should not include patents except ”essential patents” which implies strong complementary among patents. On our model, the strength of complementarity
characterizes the anti-competitive eﬀect of a patent pool on the social welfare. We ﬁnd that thepatent pool is pro-competitive when the strength of complementarity is strong. This corresponds
to the case where the patent pool includes complementary patents in Shapiro (2001) and the16demand margins bind in Lerner and Tirole (2004). Further, the theoretical criterion is consistent
to the antitrust guidelines which require the patent pools to include only ”essential patents”.
On the other hand, the patent pool is neither pro-competitive nor anti-competitive whenthe
strength of complementarity is weak. Lerner and Tirole (2004) concludes that a patent pool isanti-competitive when the patent pool includes only substitutes which can replace other patents,that is to say similar patents. Our paper ﬁnds that a patent pool does not aﬀect the social
welfare if the patent pool includes only patents which are weak complementary between thepatents. Therefore weak complementary patents need not be included in patent pools, since
there is not social beneﬁt. Furthermore other factors outside our model might give negativeeﬀect to the social welfare. We conclude that the antitrust guidelines which require the patentpools only to include ”essential patents” are appropriate for competition policy.
In this paper, we ﬁnd that patent holders do not have the incentive to form the patentpool which oﬀers the multiple package licenses voluntarily. But some patent pools which oﬀer
multiple package licenses are observed actually. That is the problem which remains to be solvedin our paper.

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18

the

Figure 1: ψA (p) and ψB (p) (the case where the value of s is negative) The value of and are theprice elasticity of demand for patent A, and of the demand for patent B. The ﬁgure shows thatthe price elasticity of demand for patent A is larger than that of the demand for patent B for
any same, when is negative value.

19