Electronic copy available at: http://ssrn.com/abstract=2818008
Patent Licensing and Bargaining
with Innovative Complements and Substitutes
NORTHWESTERN UNIVERSITY SCHOOL OF LAW
LAW AND ECONOMICS SERIES • NO. 16-12
Daniel F. Spulber Northwestern University School of Law
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Patent Licensing and Bargaining with Innovative
Complements and Substitutes
Daniel F. Spulber�
Northwestern University
July 15, 2016
Abstract
Inventors and producers bargain over royalties to license multiple patented
inventions. In the �rst stage of the bargaining game, inventors o¤er licenses
to producers and producers demand licenses. In the second stage of the game,
inventors and producers engage in bilateral bargaining over licensing royalties.
The analysis shows that there is a unique weakly dominant strategy equi-
librium in license o¤ers. The main result is that this bargaining procedure
maximizes the joint pro�ts of inventors and producers. Licensing royalties are
less than bundled monopoly royalties. The e¢ ciency of the bargaining out-
come contrasts with the ine¢ ciency of patent royalties in the Cournot model.
The analysis explores the implications of the main results for antitrust policy
concerns including Standard Essential Patent holdup, royalty stacking, patent
thickets, the tragedy of the anticommons, and justi�cation for patent pools.
�Elinor Hobbs Distinguished Professor of International Business, Professor of Strategy, KelloggSchool of Management, Northwestern University, 2001 Sheridan Road, Evanston, IL, 60208. E-mail: [email protected]. I gratefully acknowledge research support from Qualcommand the Kellogg School of Management. I thank the editor Federico Etro for helpful commentsthat improved the paper. I also thank Alexei Alexandrov, Pere Arque-Castells, Justus Baron, JohnHowells, and Joaquin Poblete for their helpful comments. The opinions expressed in this paper aresolely those of the author.
1
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The discussion also considers how imperfect intellectual property rights a¤ect
bargaining over royalties.
Keywords: bargaining, complements, substitutes, competition, royalties, li-
censing, patents, invention, innovation
JEL Codes: C7, O3, D4
I Introduction
Complex innovations often combine multiple inventions.1 For example, smart phones
apply inventions in telecommunications networks, radio communications, micropro-
cessors, memory, product design, software, batteries, and screen displays. This im-
plies that inventors and producers engage in multilateral transactions that combine
inventions and allocate economic returns to innovation.
To examine the economic implications of complex innovations, I study patent
licensing when multiple inventors and producers bargain over patent licensing royal-
ties. I show that bargaining is economically e¢ cient and generates royalties, outputs,
and combinations of inventions that maximize the joint pro�ts of inventors and pro-
ducers. This has signi�cant implications for public policy including antitrust and
intellectual property (IP) rights.
Patent licensing usually involves bargaining rather than posted prices. There is
considerable evidence that licensors and licensees bargain over royalties; see Gold-
scheider (1995-1996), Stasik (2010), Sakakibara (2010), Epstein andMalherbe (2011),
and Radauer and Dudenbostel (2013). Bargaining is necessary because patent licens-
ing involves long-term contractual relationships between IP owners and producers
1Innovative products, production processes, transaction methods, or business models typicallyapply many inventions. Schumpeter�s (1934, p. 66) entrepreneur is an innovator who makes �newcombinations.�Alfred Chandler (1990, p. 597) observes �The �rst movers � those entrepreneursthat established the �rst modern industrial enterprises in the new industries of the Second IndustrialRevolution �had to innovate in all of these activities. They had to be aware of the potential ofnew technologies and then get the funds and make investments large enough to exploit fully theeconomies of scale and scope existing in the new technologies.�Kline and Rosenberg (1986, p. 279)point out that �There is no single, simple dimensionality to innovation. There are, rather, manysorts of dimensions covering a variety of activities.�
2
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rather than immediate exchange (Krattiger et al., 2007). Indeed, as Anand and
Khanna (2000) point out, patent licensing is "one of the most commonly observed
inter-�rm contractual agreements." Firms and IP owners generally bargain over the
division of incremental pro�t or cost savings from the invention (Schroeder, 1985-
1986). According to the World Intellectual Property Organization (WIPO, 2005, p.
57), "It is important that the rate results in a good business proposition for both
parties, and so negotiation of the royalty rate is fundamental to the success of the
agreement." The courts recognize that royalties generally are established through
bargaining between licensors and licensees. Bargaining provides a better description
of markets for IP than do standard posted prices models used in industrial organiz-
ation.
Understanding how markets for IP function is important because IP makes ma-
jor contributions to innovation and economic growth.2 An indication of the size
of markets for IP is that total worldwide royalty payments for IP (patents, trade-
marks, copyrights) exceed $289 billion.3 Markets for IP continue to expand because
inventors and producers increasingly use patents to facilitate technology transfers
(Yanagisawa and Guellec, 2009; Spulber, 2015). Producers rely extensively on ac-
quisition and licensing of IP to obtain their product designs and manufacturing
technology (Chesbrough, et al., 2006).
I introduce a two-stage model of patent licensing that provides some insights
into bargaining procedures and outcomes. In the �rst stage, inventors make non-
cooperative patent licensing o¤ers to producers and producers make technology ad-
option decisions. In the second stage, inventors and producers engage in bargaining
over licensing royalties and producers compete in the downstream product market.
The model addresses several important institutional features including coordination
between inventors and producers, adoption of multiple inventions, allocation of rents
across inventions, and negotiation of licensing contracts.
To study the allocation of rents across inventions, I examine two inventions and
2A report by the Economics and Statistics Administration and United States Patent and Trade-mark O¢ ce (2012) found that IP-intensive industries contribute over a third of U.S. GDP.
3World Bank, Table 5.13 World Development Indicators: Science and technology, for 2014 basedon 2013 data, http://wdi.worldbank.org/table/5.13, Accessed August 2, 2016.
3
allow producers to adopt both inventions, only one of the inventions, or neither of the
inventions. I show that in equilibrium, producers choose to adopt both inventions
whether inventions are innovative substitutes, innovative complements, or perfect
innovative complements. An important feature of the analysis is the interaction
between the number of licenses and licensing royalties. The analysis considers pro-
ducer participation constraints associated with the option of relying on the existing
technology and the option of adopting only one invention. The main results of the
analysis are as follows.
First, I examine bargaining between inventors and producers when the down-
stream market is perfectly competitive. I �nd that the bargaining outcome is eco-
nomically e¢ cient because bargaining eliminates multiple marginalization. The e¢ -
ciency result holds whether inventions are innovative substitutes, innovative comple-
ments, or perfect innovative complements. I show that when producer participation
constraints are not binding, the market output corresponds to that with a bundled
monopoly inventor. When producer participation constraints are binding, the market
output is greater than or equal to that with a bundled monopoly inventor.
Most signi�cantly, bargaining implies that total patent royalties are less than
what would be charged by a bundled monopoly inventor. This holds because of
several forces. Equilibrium output is greater than or equal to the bundled mono-
poly output even though inventors choose license o¤ers non-cooperatively. Also, the
competitive entry of producers generates a total demand for patent licenses that is
equal to minimum of the license o¤ers. In addition, negotiation of licensing royalties
means that inventors only obtain a share of economic rents, so that patent royalties
are less than the bundled monopoly level even if the �nal output equals the bundled
monopoly output.
Second, I examine bargaining over patent royalties when the producer has a
monopoly in the downstream market. Here also, I �nd that patent royalties with
bargaining are less than what would be charged by a bundled monopoly inventor.
Extending Arrow�s (1962) analysis to multiple inventions, I show that the incent-
ive to invent is greater with perfect competition downstream than with monopoly
downstream.
4
Third, the analysis shows that the main results are robust to imperfect IP enforce-
ment. I consider the possibility that patents need not be valid and the possibility
that the courts �nd that producers using a patent without licensing are found to
have infringed on a valid patent. Producers have the option of licensing one of
the inventions, both of the inventions, or neither of the inventions. With imperfect
enforcement of IP rights, I �nd that the equilibrium maximum licensing o¤ers are
greater than or equal the pro�t-maximizing bundled monopoly output. Total roy-
alties are less than the bundled monopoly level even if the �nal output equals the
bundled monopoly output. Imperfect patent enforcement only a¤ects �nal output
through the producer participation constraints. It is not necessary to weaken IP as a
means of reducing the �nal price below what would result with a bundled monopoly
inventor.
Fourth, I apply innovative complements and substitutes to extend Cournot�s
(1838) posted prices model beyond perfect complements. Posted prices generate
horizontal multiple marginalization known as the "Cournot e¤ect." When applied to
patents, total royalties are greater than the joint pro�t maximum. This is because
inventors do not take into account the e¤ect of their royalties on the pro�ts of other
inventors. I show that "Cournot e¤ect" continues to hold with innovative comple-
ments and substitutes and with producer participation constraints. This implies that
the "Cournot e¤ect" depends on the assumption of posted prices.
The main results obtained here show that bargaining over patent royalties elim-
inates the "Cournot e¤ect." Because patent licensing usually is based on bargaining,
this suggests that public policy in antitrust and IP should not rely on the "Cournot
e¤ect." I review the literature on issues in antitrust policy that are based on the
"Cournot e¤ect." These issues include: Standard Essential Patent (SEP) holdup,
royalty stacking, patent thickets, the Tragedy of the Anticommons, and justi�cation
for patent pools.4 I �nd that such concerns derive from the posted prices assumption
in the Cournot model. The discussion suggests that public policy makers should re-
cognize the prevalence of bargaining in patent licensing and reevaluate these policy
concerns.4For an overview of the literature, see Geradin, and Rato (2007).
5
I de�ne inventions as innovative complements (innovative substitutes) when there
are increasing (decreasing) di¤erences in the production of innovations. This frame-
work is helpful in understanding how bargaining addresses the allocation of economic
rents when there are combinations of inventions. Innovative complements and sub-
stitutes a¤ect whether producer participation constraints depend on the existing
technology or the incremental contributions of inventions. This analysis is related
to that of Lerner and Tirole (2004, 2015) in which the returns to the number of in-
ventions can be either concave or convex. Lerner and Tirole (2004, 2015) generalize
the Cournot model to allow patents that are complements and substitutes in terms
of the relative prices of licenses. They show that the Cournot e¤ect holds when the
incremental bene�ts from an inventions are increasing or decreasing in the number
of inventions.
The present analysis extends that of Spulber (2016) in which suppliers of perfectly
complementary inputs choose quantities to o¤er producers in �rst stage and then
bargain with producers over the division of economic returns in the second stage.
Spulber (2016) �nds that the dominant strategy equilibrium in quantities is the
joint pro�t maximum. Spulber (2016) considers only perfect complements in inputs,
whereas the present analysis examines inventions that are innovative substitutes
and complements in addition to inventions that are perfect innovative complements.
Because the present analysis allows for imperfect complements and substitutes, the
present analysis examines the e¤ects of constraints due to the initial technology and
also the possibility that producers only adopt one invention. These constraints are
not part of the previous analysis of complementary monopolies.
The two-stage model of bargaining presented here is di¤erent from standard
cooperative bargaining models. Rather than having unrestricted bargaining, the
present model considers non-cooperative licensing o¤ers in the �rst stage and bilat-
eral simultaneous bargaining over royalties in the second stage. The second stage
bargaining game is based on bilateral Nash bargaining between inventors and produ-
cers. The second-stage bargaining draws upon the discussion of multiplayer games
with bilateral Nash bargaining in Harsanyi (1959, 1963).
In the present setting, inventors bargain with producers, so that there is no bar-
6
gaining among inventors. Inventors do not explicitly coordinate with each other
because they choose license o¤ers non-cooperatively. Royalties are established by bi-
lateral bargaining between an inventor and a producer, so inventors do not cooperate
with each other in choosing royalties. The traditional critique of the Cournot model
has been that complementary monopolists would achieve an e¢ cient outcome if they
could bargain among themselves. In particular, Bowley (1928) argued that cooperat-
ive bargaining among complementary monopolists would generate e¢ cient outcomes,
see also Wicksell (1934), Tintner (1939), Henderson (1940), Leontief (1946), and Fell-
ner (1947).5 The present analysis shows that the market outcome is e¢ cient even if
inventors only bargaining with producers and even if inventors choose licensing o¤ers
non-cooperatively.
Even with these restrictions, the e¢ ciency of the outcome is consistent with
unrestricted cooperative bargaining models, which generally yield Pareto e¢ cient
outcomes.6 Axiomatic game theory further suggests that unrestricted cooperative
behavior should lead to maximization of joint bene�ts.7 The axiomatic approach
includes for example the Shapley value, the Core, and Nash (1950, 1953) bargaining.
Rubinstein�s (1982) non-cooperative bilateral bargaining framework and other dis-
cussions of non-cooperative bargaining also suggest that the outcome will be Pareto
e¢ cient.8
The market for patent licensing in the present setting involves decentralized bar-
gaining. Inventors and producers engage in simultaneous pairwise bargaining. There
many studies of markets with various types of pairwise bargaining.9 These studies
5Schumpeter (1928) suggests that Cournot duopolists (or complementary monopolists) wouldmaximize joint pro�ts through tacit collusion. Machlup and Taber (1960, p. 111) note: "negotiationsbetween separate monopolists would, in the case of intermediate products, necessarily be carried onin terms of both quantity and price, and that the quantity agreed upon between the parties wouldbe the same as that produced by an integrated monopolist."
6Edgeworth (1881) and Pareto (1903, 1927) emphasize e¢ ciency of bilateral exchange along thecontract curve.
7On bargaining in game theory, see Shapley (1952), Harsanyi (1959, 1963), Aumann (1987),Shubik (1982, 1984), and Kalai and Smorodinsky (1975).
8There is an extensive literature on non-cooperative multilateral bargaining, see Krishna andSerrano (1996), Britz et al. (2010), and the references therein.
9On markets with decentralized bargaining, see for example Diamond (1981, 1982), Diamond
7
generally di¤er from the present model because in those setting buyers and sellers
only carry out bilateral exchange. In the present setting, inventors deal with multiple
producers and producers deal with multiple inventors.
The present discussion of innovative complements and substitutes has precedents
in addition to Cournot. Edgeworth (1925) extends Cournot�s models of competi-
tion to allow imperfect complements and substitutes, and points out that perfect
complementarity is a limiting case of complementary goods. Discrete choice models
in characteristics space are related to innovative complements and substitutes if we
interpret inventions as characteristics of alternative technologies (Anderson et al.,
1989, 1992). Arrow (1962) considers inventions as vertically di¤erentiated substi-
tutes, see also Spulber (2013). Various studies consider complementary inventions,
see for example Layne-Farrar and Schmidt (2011), Schmidt (2014), and Llanes and
Poblete (2014). Laussel (2008) examines Nash bargaining over prices of complement-
ary components in automobiles and aircraft, see also Laussel and Van Long (2012)
for a dynamic equilibrium analysis.
The discussion is organized as follows. Section II introduces the non-cooperative
bargaining model. Section III considers the equilibrium when the downstreammarket
is perfectly competitive. Section IV extends the non-cooperative bargaining model to
the situation in which the producer is downstream monopolist. Section V examines
the implications of imperfect IP rights for technology licensing. Section VI considers
the public policy implications of the analysis. Section VII concludes the discussion.
II A model of patent licensing
This section introduces a model of patent licensing in which inventors and producers
bargain over patent royalties. The downstream product market is perfectly com-
petitive. In the �rst stage, inventors make patent license o¤ers to producers and
producers enter the downstream market. In the second stage, inventors bargain with
producers over patent license royalties. Then, producers innovate by applying the
and Maskin (1979), Rubinstein and Wolinsky (1985), Gale (1986a, 1986b), and Bester (1993).
8
inventions and the product market clears. A later section examines bargaining when
the downstream product market is a monopoly.
II.1 Inventors
There are two inventors each with a distinct invention. Inventions refer to discoveries
in the sciences, technology, engineering, and mathematics and in commercial studies
such as economics, �nance, operations, marketing, accounting, and business strategy.
Inventions are non-divisible and non-rivalrous. Producers apply the inventions to
innovate in the downstream product market.
The inventions generate technological change �(t1; t2), where ti = 1 if a producer
uses invention i and ti = 0 otherwise, i = 1; 2. We represent technological change by
�(t1; t2) =
8>>>><>>>>:a if t1 = t2 = 1;
b if t1 = 1 and t2 = 0;
b if t1 = 0 and t2 = 1;
0 if t1 = t2 = 0;
(1)
where a > b and b � 0. The assumption that there are bene�ts from using both
inventions, a > b, rules out inventions that are perfect substitutes, that is, a = b.
The symmetry assumption, �(1; 0) = �(0; 1), rules out inventions that are vertic-
ally di¤erentiated substitutes such as those considered in Arrow (1962) and Spulber
(2013). Lerner and Tirole (2004) consider the e¤ects of the number of patents on in-
cremental bene�ts, and consider both increasing and decreasing incremental bene�ts.
The present analysis can be extended to many inventions.
We characterize technological change as follows.
DEFINITION 1. Inventions are innovative complements (innovative substitutes) iftechnological change �(t1; t2) exhibits increasing (decreasing) di¤erences. Inventions
are perfect innovative complements if �(1; 1) > �(1; 0) = �(0; 1) = 0.
9
The cross e¤ects of inventions on technological change are
[�(1; 1)��(1; 0)]� [�(0; 1)��(0; 0)] = a� 2b: (2)
So, inventions are innovative substitutes if a � 2b, innovative complements if a � 2band b > 0, and perfect innovative complements if b = 0.
Every producer has unit capacity and requires a license to use an invention. In
the �rst stage of the game, each inventor i makes a binding commitment to supply
patent licenses to every producer that demands a license up to a maximum number
of licenses yi. Let qi denote the number of patent licenses for invention i demanded
by producers. So, each inventor o¤ers a license schedule Yi(qi) given by
Yi(qi) = minfqi; yig; i = 1; 2: (3)
Each inventor i earns royalties ri, i = 1; 2 for each license. Each inventor i earns
pro�t
V (qi; ri) = riYi(qi); i = 1; 2: (4)
Inventors are active if and only if V (qi; ri) � 0.When choosing their license schedule o¤ers, inventors do not know either the
license schedule of the other inventor Y�i(q�i), the number of producers qi, i = 1; 2
that demand licenses, or downstream market output q. We consider weakly domin-
ant strategy equilibria in license o¤ers. To simplify notation, let the maximum levels
y1 and y2 represent the o¤ers of license schedules Y1(q1) and Y2(q2). After invent-
ors o¤er license schedules, producers enter the market and choose license demands.
We assume that downstream producers enter the market sequentially so that each
producer is able to obtain both inventions up to minfy1; y2g.In the second stage of the game, each inventor i bargains bilaterally with each
downstream producer over the license royalty ri. Bilateral bargaining follows the
Nash cooperative bargaining solution.10 The cooperative approach simpli�es the
10See Nash (1950, 1953), Harsanyi and Selten (1972), Kalai (1977), Binmore et al. (1986), Roth(1979), and Binmore (1987).
10
discussion. We could extend the analysis to allow bilateral noncooperative bargaining
as in Rubinstein (1982) and Binmore et al. (1986).11
The basic Nash cooperative bargaining equilibrium is modi�ed because bilateral
bargaining between the inventor-producer pairs occurs simultaneously. Each bar-
gaining pair chooses a royalty ri in response to the equilibrium outcomes of other
negotiations r��i, i = 1; 2 as in a Nash noncooperative equilibrium. Harsanyi (1959,
1963) considers multilateral Nash bargaining in terms of bilateral bargaining between
every pair of players.12 In the present setting, we only have Nash bargaining between
inventors and producers.
Let � denote the bargaining power of an inventor relative to a downstream pro-
ducer and assume that 0 < � < 1. The division of economic rents among inventors
and producers depends on the bargaining power parameter and interdependence of
the bilateral bargains in equilibrium.
Producers are symmetric so that it is reasonable to suppose that every producer
pays the same royalty for a particular invention. In practice, many contracts for tech-
nology transfers include most favored nation or most favored licensee clauses that
require similar producers to pay the same royalties.13 Technology standard-setting
organizations (SSOs) have rules that prevent price discrimination when producers are
the same, such as Fair, Reasonable and Non-discriminatory (FRAND) restrictions
on royalties (Sidak, 2013, 2015). Regulation and antitrust may rule out price dis-
crimination across producers when producers are identical and other contract terms
are the same.
II.2 Producers
With perfect competition, the product market consists of a homogeneous good q, a
product price p, and market demand q = D(p). The market demand D(p) is strictly
11On multilateral non-cooperative bargaining, see also Chae and Yang (1984) and Sutton (1986).12See also Lensberg (1988) and Bennett (1997) on multilateral Nash bargaining.13There are also most favored licensee (MFL) clauses: "MFL clauses suggest that licensed compet-
itors should be treated equally, so long as they bear equivalent obligations to the licensor/patentee"(Sneddon, 2007, p. 805).
11
decreasing and twice continuously di¤erentiable. Let P (q) denote inverse demand.
Competitive producers produce a unit of output with production costs c. At the
market output q, the pro�t of a competitive producer equals P (q) � c excludingpatent royalties.
Consider the e¤ects of technological change on downstream costs. A process
innovation reduces production cost relative to initial cost c0 using the existing tech-
nology,
c = c0 ��(t1; t2): (5)
The same basic framework can be applied to other types of innovations such as
product innovations and transaction innovations.14
At the market output q, the pro�t of a competitive producer equals
�(q; t1; t2) = P (q)� (c0 ��(t1; t2))� r1t1 � r2t2: (6)
Producers have the option of licensing both inventions, only one of the inventions,
or neither invention. If producers license both inventions they have production cost
c0 � a and if they license only one invention they have production cost c0 � b. Ifproducers license neither invention, producers use the initial technology and have
cost c0. A later section considers imperfect patent enforcement so that producers
can choose to use the patent technologies without licensing and risk legal damages
for patent infringement.
For producers to have incentives to license both inventions, equilibrium royalties
must satisfy two participation constraints. First, producers must have incentives to
license both inventions instead of using the existing technology, r1+ r2+ c0�a � c0.I refer to this as the initial technology constraint. This constraint is equivalent to
the requirement that total royalties do not exceed the technological change from the
combination of inventions,
r1 + r2 � a:14A product innovation increases bene�ts from production relative to bene�ts v0 using the existing
technology, v = v0+�. A transaction innovation as improvements in the e¢ ciency of transactions,such as lower search costs, or more convenient transactions.
12
If royalties are equal, the constraint requires each royalty to be less than or equal to
the average technological change across inventions, r � a2.
Second, royalties must be such that producers have incentives to adopt both
inventions instead of just one invention. The incremental innovation constraint is
r1+r2+c0�a � c0�b+minfr1; r2g. This constraint is equivalent to the requirementthat the maximum of the two royalties is less than or equal to the incremental
innovation,
maxfr1; r2g � a� b:
If royalties are equal, the constraint requires each royalty to be less than or equal to
incremental technological change for the second invention, r � a� b.For inventors to deter competitive entry, industry output must satisfy two market
constraints. First, producers have the option of entering the market by using the
initial technology c0 at no cost. Then, industry output would equal q0,
P (q0) = c0: (7)
This possibility is ruled out if the �nal price is below cost at the initial technology,
P (q) � c0, or equivalently if industry output is such that q � q0 = D(c0). I refer tothis as the initial technology market constraint.
Second, producers can enter the market by adopting only one invention. To deter
entry of producers using only one invention, the market price must be less than the
cost of using one invention plus the lowest royalty, P (q) � c0 � b+minfr1; r2g. Letq(r1; r2) be such that the market price equals the cost of using one invention plus the
lowest royalty,
P (q(r1; r2)) = c0 � b+minfr1; r2g: (8)
The critical output level q(r1; r2) will be determined endogenously in equilibrium.
The possibility of entry with only one invention is ruled out if P (q) � P (q(r1; r2)),or equivalently if industry output is such that q � q(r1; r2) = D(c0�b+minfr1; r2g).I refer to this as the incremental innovation market constraint.
13
Royalties are said to be feasible for producers at some output q if
P (q)� (c0 � a)� r1 � r2 � 0:
The following result spells out the connections between the market constraints on
industry output and the participation constraints on royalties.
LEMMA 1. A. If industry output satis�es the initial technology market constraintand the incremental innovation market constraint, q � maxfq0; q(r1; r2)g then feas-ible royalties satisfy both the initial technology constraint and the incremental in-
novation constraint, r1 + r2 � a and maxfr1; r2g � a � b. B. When royalties aresymmetric, with innovative complements or perfect innovative complements, the ini-
tial technology constraint is the relevant constraint, and with innovative substitutes,
the incremental innovation constraint is the relevant constraint.
Notice that feasibility and the market constraints together imply that
c0 � a+ r1 + r2 � P (q) � minfc0; c0 � b+minfr1; r2gg:
This implies that the initial technology and the incremental innovation constraints
are satis�ed. This establishes the �rst part of Lemma 1.
When royalties are symmetric, each royalty is subject to the constraints r �fa2; a � bg. With innovative complements, a � 2b, a
2� a � b. This also holds with
perfect innovative complements, b = 0. With innovative substitutes, the converse
holds, a2� a� b. This establishes the second part of Lemma 1.
II.3 A benchmark: the bundled monopoly inventor
As a benchmark for the two-stage game, consider a monopolist inventor that sells a
bundle of the two inventions to producers. The monopolist posts a royalty � for the
bundle of inventions. In a perfectly competitive downstream market, producers enter
the market until average returns equal the royalty, P (q) � c = �, where c = c0 � a.
14
The bundled monopoly inventor earns pro�t
V M(q) = �q = (P (q)� c)q: (9)
The monopolist chooses downstream output to maximize pro�t subject to the
initial technology market constraint,
maxqV M(q) subject to q � q0:
Downstream output is then
qM(c) = maxfbqM(c); q0g; (10)
where bqM(c) is unconstrained pro�t-maximizing monopoly output. Assume that
there exists an interior solution bqM(c) > 0 to the unconstrained monopoly problem,P 0(bqM)bqM+P (bqM)�c = 0. The unconstrained monopoly pro�t is positive, (P (bqM)�c)bqM = �P 0(bqM)(bqM)2 > 0.The monopolist�s output choice need not be unique. If there are multiple solu-
tions, then for ease of notation let qM(c) denote the smallest pro�t-maximizing out-
put. We will show that the main result holds whether or not the unconstrained
monopoly output is unique.
With a bundled monopoly inventor, the downstream market price is P (qM(c)).
The monopolist�s price for the bundle of inventions equals the marginal return to
producers evaluated at the monopoly output,
�M(c) = P (qM(c))� c: (11)
By bundling inventions, the monopoly inventor is not subject to the incremental
technology market constraint. Suppose that the bundled monopoly inventor was
required to engage in "mixed bundling", that is, the monopolist was required to o¤er
both the bundle and the two inventions separately. Then, the monopolist would be
subject to the incremental innovation market constraint P (q) � c0� b+minfr1; r2g.
15
Suppose that the monopolist chooses symmetric royalties so that the incremental
innovation market constraint is P (q) � c0 � b + �2. Substituting for the bundled
royalty � = P (q)� (c0 � a) gives the critical output level qM ,
P (qM) = (c0 � b) + (a� b): (12)
The monopolist satis�es the incremental innovation market constraint if P (q) �P (qM) or equivalently q � qM = D((c0 � b) + (a � b)). The monopolist bene�tsfrom bundling the inventions only if the incremental innovation market constraint
is binding, that is when the incremental innovation constraint is the relevant con-
straint qM � q0 and the unconstrained monopoly output is too low, bqM(c0�a) < qM .Notice that qM � (�)q0 and (c0 � b) + (a� b) � (�)c0, if the inventions are innov-ative substitutes (innovative complements and perfect innovative complements). So,
the incremental innovation constraint can be binding on the monopolist only if the
inventions are innovative substitutes.
III Equilibrium of the two-stage game
In the �rst stage, inventors choose license o¤ers represented by y�1, y�2 and entry of
producers determines demands for licenses q�1, q�2: In the second stage, the equilibrium
bargaining outcome is represented by license royalties r�1; r�2. We solve the model by
backward induction.
III.1 Bargaining over royalties
In the second stage of the game, inventors and producers bargain over royalties. By
Lemma 1, when the two market constraints hold, the two producer participation
constraints are satis�ed. This means that when the two market constraints hold,
producers purchase both inventions. Producers cannot earn positive returns to entry
with either the initial technology or with only one invention. This implies that the
16
disagreement point is zero for producers. The disagreement point also is zero for
inventors.
Inventor i�s royalty ri solves the asymmetric Nash cooperative bargaining problem
taking as given the royalty chosen by bargaining between the other inventor and
producers r��i,
maxri(P (q)� (c0 � a)� r��i � ri)1��(ri)�; i = 1; 2:
The �rst-order conditions simplify to
�(P (q)� (c0 � a)� r��i � ri) = (1� �)ri; i = 1; 2: (13)
In equilibrium, royalties are symmetric across inventors and across producers. The
bargaining equilibrium exists and is unique for any q satisfying the market con-
straints,
r�(q) =�
1 + �[P (q)� (c0 � a)]: (14)
Equilibrium royalties are decreasing in output because inverse demand is strictly
decreasing in output, r�0(q) = �1+�P 0(q) < 0.
III.2 Patent licensing
In the �rst stage, inventors o¤er licenses to producers and producers enter the market.
When the market output is q, the return for a producer equals
�(q; 1; 1) = P (q)� (c0 � a)� 2r�(q) =1� �1 + �
[P (q)� (c0 � a)]: (15)
Inventors have net bene�ts
V (q; r�(q)) = r�(q)q =�
1 + �[P (q)� (c0 � a)]q: (16)
Inventors and producers divide the total rents, 1��1+�
+ 2 �1+�
= 1.
17
Bargaining in the second stage implies that all active producers earn pro�t for any
output q such that P (q) > c0� a. Because the downstream market is competitive, ifP (minfy1; y2g) � c0�a, entry of downstream producers continues until total demandfor licenses equals the minimum of the maximum o¤ers, q�1 = q
�2 = minfy�1; y�2g. So,
entry of downstream producers continues until output equals the minimum of the
maximum license o¤ers,
q = minfy1; y2g: (17)
This is an important e¤ect of competitive entry because it means that license o¤ers
are perfect complements in the determination of �nal output. This property holds
even if the inventions are innovative complements or substitutes, and thus not perfect
innovative complements.
Given equilibrium royalties, we can derive the critical output needed to deter
entry of producers using only one invention. The critical output level solves
P (q) = c0 � b+ r�(q): (18)
Using the form of the equilibrium royalty, we obtain
P (q) = c0 � b+�
1 + �[P (q)� (c0 � a)]: (19)
Simplifying this expression the critical output solves P (q) = c0 � b+ �(a� b), or
q = D(c0 � b+ �(a� b)): (20)
This implies that the critical output is greater than the critical output for the mixed-
bundling monopoly, q > qM , and as � approaches one, q approaches qM .
Notice that q0 � (�)q as � � (�) ba�b . With innovative substitutes, b
a�b �1 > � so the incremental innovation market constraint is the relevant constraint.
With perfect innovative complements, only the initial technology market constraint
is relevant.
Inventors choose license o¤ers y1 and y2 to solve the following strategic optimiz-
18
ation problems,
maxyiV (q; r�(q))
subject to the entry condition q = minfy1; y2g and the market constraints q �max fq0; qg, i = 1; 2.I consider the weakly dominant strategy equilibrium of the licensing game. The
following result characterizes the equilibrium in the �rst stage of the game.
PROPOSITION 1. In the �rst stage, the weakly dominant strategy equilibriumin license o¤ers (y�1; y
�2) exists and is unique and symmetric. A. With innovative
complements and � � ba�b or with perfect innovative complements, the equilibrium
maximum license o¤ers equal the pro�t-maximizing bundled monopoly output,
y�1 = y�2 = maxfbqM(c0 � a); q0g = qM(c0 � a):
B. With innovative complements and � � ba�b or with innovative substitutes, the
equilibrium maximum license o¤ers are greater than or equal to the pro�t-maximizing
bundled monopoly output,
y�1 = y�2 = maxfbqM(c0 � a); qg � qM(c0 � a):
PROOF. A. Consider the equilibrium with innovative complements and � � ba�b ,
or perfect innovative complements. The relevant constraint is the initial technology
constraint. Suppose �rst that bqM(c0�a) � q0. The argument with a nonbinding ini-tial technology market constraint follows the discussion in Spulber (2016). Inventor
i�s pro�t in the �rst stage of the game equals v(yi; y�i) = �1+�[P (q)� (c0�a)]q where
q = minfy1; y2g. Notice that
V (q; r�(q)) =�
1 + �V M(q);
so the pro�t maximizing output for a bundled monopoly also maximizes pro�t for an
inventor.15 Consider �rst the possibility that y�i � qM . Then, because the bundled15The pro�t-maximizing output may or may not be unique. As shown in Spulber (2016), pro�t
19
monopolist maximizes pro�t, it follows that v(qM ; y�i) � v(yi; y�i) for all yi. If yi = q,v(yi; y�i) =
�1+�[P (q) � (c0 � a)]q. So, if y�i � qM , inventor i maximizes pro�t by
choosing the monopoly output, y�i = qM . Conversely, if y�i < qM , then because
the bundled monopolist maximizes pro�t it follows that v(qM ; y�i) � v(yi; y�i) for
all yi and strictly for yi < y�i. Again, inventor i maximizes pro�t by choosing
the monopoly output, y�i = qM . This implies that the monopoly output is the
weakly dominant strategy for each inventor i, and thus the weakly dominant strategy
equilibrium is equal to the unconstrained monopoly output.
Suppose next that bqM(c0 � a) < q0, so that the initial technology constraint
is binding on the monopolist. Then, the pro�t-maximizing output for each of the
inventors is q0. By the same reasoning, the weakly dominant strategy equilibrium is
unique and equals y�1 = y�2 = q0. This implies that with strict complements or perfect
innovative complements, y�1 = y�2 = maxfbqM(c0 � a); q0g = qM(c0 � a).
B. Suppose that the inventions are innovative complements and � � ba�b or
innovative substitutes. The relevant constraint is q � q. Suppose �rst that the
constraint is not binding on the monopolist, bqM(c0 � a) � q. Then, by the same
reasoning, it follows that the unique weakly dominant strategy equilibrium is y�1 =
y�2 = bqM(c0 � a). If qM(c0 � a) < q, the weakly dominant strategy equilibrium is
unique and equals y�1 = y�2 = q. This implies with innovative substitutes, y
�1 = y
�2 =
maxfbqM(c0 � a); qg � maxfbqM(c0 � a); q0g = qM(c0 � a). �Proposition 1 generalizes the complementary monopolies model in Spulber (2016) in
two ways. Proposition 1 does not require perfect complements but also holds with
innovative substitutes and innovative complements. This is because competitive
entry of producers implies that output equals the minimum of the maximum o¤ers.
Also, Proposition 1 allows inventors to choose licensing o¤ers subject to market
constraints needed to deter entry of producers using the initial technology or the
adoption of only one invention.
Proposition 1 shows that the bene�ts of combining inventions provide coordina-
maximization also holds if the pro�t-maximizing monopoly output is not unique, with the outcomebeing the smallest bundled monopoly output.
20
tion incentives, with either innovative complements or innovative substitutes. This
is because with bargaining over patent royalties, inventors obtain shares of industry
pro�t. Also, because there are bene�ts to combining inventions, inventors recognize
the e¤ects of their licensing o¤ers on total industry output, thus giving them an
incentive to take into account the external e¤ects of their licensing o¤ers. Regardless
of the licensing o¤er of the other inventor, each inventor has an incentive to come
as close as possible to the monopoly outcome. If the other inventor were to make
a maximum o¤er above the monopoly outcome, an inventor would prefer to restrict
industry output. Conversely, if the other inventor were to make a maximum o¤er
below the monopoly outcome, an inventor would prefer to make a maximum o¤er
that was greater than or equal to the other inventor�s o¤er, which would include the
maximum o¤er corresponding to the monopoly outcome.
Proposition 1 establishes that the unique weakly dominant strategy equilibrium
generates the joint-pro�t maximum with innovative complements and � � ba�b or
with perfect innovative complements. The proposition also establishes that with in-
novative complements and � � ba�b or with innovative substitutes, the unique weakly
dominant strategy equilibrium output is greater than or equal to the joint pro�t max-
imum. This is because with non-cooperative license o¤ers and a binding incremental
innovation market constraint, inventors attain the mixed bundling equilibrium and
cannot achieve the bundled monopoly outcome.
The proposition shows that the equilibrium is unique even though the bundled
monopoly outcome need not be unique. This is because non-negative rents for pro-
ducers at the bargaining outcome and entry of competitive producers imply that
the downstream output equals the minimum of the maximum licensing o¤ers. The
equilibrium with licensing o¤ers is unique even though there are many Nash non-
cooperative equilibria with �xed-quantity license o¤ers.
The market equilibrium price with a bundled monopoly inventor is P (qM(c0 �a)) = �M(c0 � a) + c0 � a. Proposition 1 implies that the price in the downstreammarket with bargaining between inventors and producers is less than or equal to the
21
price with a bundled monopoly inventor,
P (q�) � P (qM):
This holds because with innovative complements or perfect innovative complements,
the �nal output with bargaining is equal to the bundled monopoly output. With
innovative substitutes, inventors are subject to the mixed bundling e¤ect so that the
output with bargaining is greater than or equal to that of the bundled monopoly
inventor.
Compare total royalties when there is bargaining with the bundled monopoly
royalty. With bargaining over licensing royalties, output equals q� = maxfbqM(c0 �a); q0; qg. This implies the following result.PROPOSITION 2. Equilibrium patent license royalties with bargaining are unique,
r� =�
1 + �[P (q�)� (c0 � a)]: (21)
Total patent license royalties with bargaining are less than the bundled monopoly
royalty,
2r� < �M :
PROOF. From the form of the equilibrium license royalties,
2r� < P (q�)� (c0 � a) � P (qM(c0 � a))� (c0 � a) = �M((c0 � a)):
So, 2r� < �M . �
Total royalties with bargaining are less that the bundled monopoly royalty whether
inventions are innovative complements, perfect innovative complements, or innovat-
ive substitutes. The e¤ects of bargaining are magni�ed when inventions are innov-
ative substitutes. This is because the �nal output with two inventors can be strictly
greater than the bundled monopoly output when the monopolist is not subject to the
mixed bundling limitation. Total royalties are constrained with bargaining because
downstream producers have the option of only adopting one invention.
22
It is interesting to compare the bargaining outcome with Arrow�s (1962) result.
He considers a monopoly inventor with a single invention that competes with a freely
available existing technology. Arrow de�nes the invention as non-drastic (drastic)
if the monopoly price at the new cost is greater than or equal to (less than) the
unit cost under the existing technology. When the downstream market is perfectly
competitive, there are two possible outcomes. When the invention is non-drastic,
the market equilibrium royalty equals the unit cost di¤erence between the existing
technology and the new technology. When the invention is drastic, the market equi-
librium royalty equals the di¤erence between the monopoly price at the new cost
minus the unit cost under the existing technology.
In the present bargaining model, when inventions are innovative complements
with � � ba�b or perfect innovative complements, the innovation that combines the
two inventions competes with the existing technology. We can say that the combina-
tion of the two inventions is non-drastic (drastic) compared to the initial technology
when the �nal output is less than or equal to (greater than) the output under the
existing technology. With innovative complements with � � ba�b or innovative sub-
stitutes, the innovation using both inventions competes with the option of using only
one invention. We can say that the incremental innovation is non-drastic (drastic)
when the �nal output with both inventions is less than or equal to (greater than)
output with one invention.
We can compare the present model with the market outcome when competing
inventors have vertically di¤erentiated inventions. In Spulber (2013), competing
inventions are vertically di¤erentiated substitutes and only one invention is used.
Royalties then depend on cost di¤erences that result from using the best invention in
comparison with the second-best invention. When cost di¤erences are non-drastic,
the second-best invention constrains royalties on the best invention. When cost
di¤erences are drastic, the second-best invention does not constrain the royalties on
the best invention. In the present model with innovative substitutes, the incremental
innovation from the combination of inventions in comparison to a single invention
corresponds to the cost di¤erences between the best and the second-best invention
with vertically di¤erentiated inventions.
23
IV Bargaining over royalties when the downstream
market is a monopoly
Consider bargaining between the two inventors and a downstream monopoly pro-
ducer. The solution is related to the discussion of bilateral Nash bargaining with
multiple players in Harsanyi (1959, 1963). The game with a downstream monopoly
producer di¤ers from the game with a competitive downstream market. Because
the downstream monopoly does not face competitive entry, there are no market con-
straints on the �nal price. However, the downstream monopoly has the option of
using the initial technology and also has the option of adopting only one invention.
This implies that the monopolist�s disagreement point in bargaining should re�ect
these two outside options.
The pro�t-maximizing output of the downstream monopoly with unit costs c isbqM(c) > 0. Monopoly pro�t equals �M(c) = (P (bqM(c))� c)bqM(c). With technologylicensing, the monopoly producer receives net pro�t
�M(t1; t2) = �M(c0 ��(t1; t2))� t1R1 � t2R2: (22)
Suppose �rst that the producer only licenses one inventor�s technology. The
monopoly producer�s disagreement point is to use the initial technology. The royalty
solves the asymmetric Nash cooperative bargaining problem
maxR[�M(c0 � b)�R� �M(c0)]1��R�:
The �rst-order condition simpli�es to
�[�M(c0 � b)�R� �M(c0)] = (1� �)R: (23)
So, the equilibrium royalty is
eR = �[�M(c0 � b)� �M(c0)]: (24)
24
The producer obtains pro�t e� = �M(c0 � b)� eR,e� = (1� �)�M(c0 � b) + ��M(c0): (25)
The disagreement points in bargaining when the producer bargains with both
inventors depend on what could be obtained if the producer only bargained with
one inventor. When the monopoly producer bargains bilaterally with each of the
inventors, the producer�s disagreement point is e�. 16Inventor i�s royalty Ri solves the asymmetric Nash cooperative bargaining prob-
lem given the equilibrium royalty chosen by bargaining between the other inventor
and the producer R��i,
maxRi[�M(c0 � a)�R��i �Ri � e�]1��(Ri)�; i = 1; 2:
The �rst-order conditions simplify to
�[�M(c0 � a)�R��i �Ri � e�] = (1� �)Ri; i = 1; 2: (26)
In equilibrium, royalties are symmetric and the bargaining equilibrium exists and is
unique. Substituting for e� we obtainR� =
�
1 + �[�M(c0 � a)� (1� �)�M(c0 � b)� ��M(c0)]: (27)
The producer receives the equilibrium pro�t �� = �M(c0 � a) � 2R�. Substitutingfor R� yields
�� =1
1 + �[(1� �)�M(c0 � a) + 2�(1� �)�M(c0 � b) + 2�2�M(c0))]: (28)
Royalties satisfy the initial technology constraint, 2R� < �M(c0�a)��M(c0). This isanalogous to Arrow�s (1962) inertia e¤ect from an existing technology. The equilib-
16The inventors�disagreement points are zero. It is not possible to guarantee eR to each inventorbecause e�+2 eR = (1+�)�M (c0�b)���M (c0) � �M (c0�a) if and only if �M (c0�a)��M (c0�b) ��[�M (c0 � b)� �M (c0)].
25
rium royalty is positive, R� > 0. The producer obtains greater returns when dealing
with both inventors rather than only one inventor, �� > e�.The next result gives conditions such that competition among producers in the
downstream market increases the expected returns for inventors in comparison to
returns for inventors when the downstream market is a monopoly. The proof applies
arguments made by Arrow (1962) when an inventor is a monopolist with a single
invention.
PROPOSITION 3. Incentives to invent are greater with downstream competition
than with downstream monopoly,
r�q� > R�;
if either of the following hold. A. Inventions are innovative complements or perfect
innovative complements. B. Inventions are innovative substitutes, where either the
incremental innovation market constraint is not binding, bqM(c0 � a) � q, or the
incremental innovation market constraint is binding, bqM(c0 � a) < q, and (1 +
�)(a� b)q >R c0�bc0�a bqM(c)dc+ � R c0c0�b bqM(c)dc.
PROOF. A. Let inventions be innovative complements and � � ba�b or perfect
innovative complements. The relevant constraint is the initial technology market
constraint. Suppose that the initial technology market constraint is not binding,bqM(c0 � a) � q0. Then, an inventor obtains revenues equal tor�q� =
�
1 + �[P (bqM(c0 � a))� (c0 � a)]bqM(c0 � a) = �
1 + ��M(c0 � a): (29)
So, r�q� > R�.
Suppose that the initial technology market constraint is binding, bqM(c0 � a) <q0. Then, an inventor obtains revenues equal to
r�q� =�
1 + �[P (q0)� (c0 � a)]q0 =
�
1 + �aq0: (30)
26
The royalty with downstream monopoly can be written as
R� =�
1 + �[�M(c0 � a)� �M(c0 � b) + ��M(c0 � b)� ��M(c0)]: (31)
By Arrow�s (1962) arguments, because �M 0(c) = �bqM(c), it follows thatR� =
�
1 + �
�Z c0�b
c0�abqM(c)dc+ � Z c0
c0�bbqM(c)dc� : (32)
For all c 2 [c0 � a; c0], bqM(c) � bqM(c0 � a) < q0 so thatR� <
�
1 + �[a� b+ �b]q0 <
�
1 + �aq0 = r
�q�:
Let inventions be innovative complements and � � ba�b . The relevant constraint
is the incremental innovation market constraint. Suppose that the incremental in-
novation market constraint is not binding, bqM(c0 � a) � q. Then, as in part A,
r�q� > R�. Suppose that the incremental innovation market constraint is binding,bqM(c0�a) < q. Then, because P (q) = c0� b+�(a� b), an inventor�s revenues equalr�q� =
�
1 + �[P (q)� (c0 � a)]q = �(a� b)q: (33)
If the incremental innovation market constraint is binding, the royalty R� is such
that
R� <�
1 + �[a� b+ �b]q:
Because inventions are innovative complements, a � 2b, it follows that
�
1 + �[a� b+ �b]q � �(a� b)q = r�q�:
So, r�q� > R�.
B. Let inventions be innovative substitutes. The relevant constraint is the incre-
mental innovation market constraint. If the constraint is not binding, bqM(c0�a) � q,then as in part A, r�q� = �
1+��M(c0 � a) > R�. If the constraint is binding,
27
bqM(c0 � a) < q, then r�q� > R� if(1 + �)(a� b)q >
Z c0�b
c0�abqM(c)dc+ � Z c0
c0�bbqM(c)dc:
�
When market constraints are non-binding, inventors obtain a greater share of
pro�ts when the downstream market is competitive than they do when the down-
stream market is a monopoly. This e¤ect is due to the greater rents provided to
inventors by a competitive downstream market, as in Arrow�s analysis. This e¤ect
holds for perfect innovative complements, innovative complements, or innovative
substitutes.
When inventions are innovative complements or perfect innovative complements,
and the initial technology market constraint is binding on the downstream market,
inventors have greater expected returns with downstream competition because �-
nal prices are lower than with monopoly. Because prices are lower in a competitive
product market, the returns to using both inventions increase relative to either ad-
opting one invention or using the initial technology. The price bene�ts of competition
reinforce the bargaining power e¤ects of competition. When inventions are innovat-
ive complements and � � ba�b and the incremental innovation market constraint is
binding, inventors have greater returns because output is greater with downstream
competition and the returns per unit of output with downstream competition are
su¢ ciently high.
When inventions are innovative substitutes, an issue arises when the incremental
innovation market constraint is binding. This constrains inventor royalties in the
competitive case and lowers pro�ts. Inventors obtain greater expected returns with
downstream competition than with downstream monopoly when the output e¤ect
outweighs the constraint on pro�ts. This is more likely the greater are the incremental
bene�ts of the technology a�b and the greater is constrained output q in comparisonwith the monopoly output.
28
V Intellectual property rights
The analysis thus far assumes that inventors have valid patents and producers always
choose to license technologies rather than to infringe on the patents. This assumes
that patents are valid and enforced. Consider instead the possibility that patents
need not be valid and are imperfectly enforced.
Let � represent the strength of the patent.17 The strength of the patent represents
the probability that the courts �nd a particular patent to be valid, that infringement
is detected, and that the courts �nd that infringing producers should pay damages.
Suppose that patent strength is the same for both inventions. Suppose that there is
independence of these events across the two patents.
If producers are found to have infringed, they must pay damages equal to "reas-
onable royalties." This is a general legal term that is subject to determination in
individual cases.18 For purposes of discussion, suppose that reasonable royalties for
infringing are greater than or equal to bene�ts of the technology, and let g � 1 de-note the damage multiple. If a producer infringes on one patent, the bene�ts of the
technology equal a � b, and if a producer infringes on both patents, the bene�ts ofthe technology equal a. So, damages are g(a � b) for a single invention and ga forboth inventions.
Producers that use both patented inventions but do not license either invention
may be found to infringe either or both of the patents. This means that producers
have costs c0�a and expected damages 2(1��)�g(a�b)+�2ga. Producers have thesame costs whether or not they license the two inventions. This implies that total
17Farrell and Shapiro (2008) consider the bene�ts of determining patent validity before negotiatingroyalties.18On the subject of reasonable royalties, the U.S. Code states "Upon �nding for the claimant
the court shall award the claimant damages adequate to compensate for the infringement but inno event less than a reasonable royalty for the use made of the invention by the infringer, togetherwith interest and costs as �xed by the court. When the damages are not found by a jury, the courtshall assess them. In either event the court may increase the damages up to three times the amountfound or assessed." 35 U.S.C. § 284 para. 1 (1994).
29
royalties must satisfy the following constraint,
r1 + r2 � 2(1� �)�g(a� b) + �2ga:
This constraint replaces the initial technology constraint, r1+ r2+ c0�a � c0, if andonly if expected damages are less than the total contribution of the two inventions,
2(1��)�g(a�b)+�2ga < a. To examine the e¤ects of imperfect patent enforcement,assume that this condition holds so that this constraint replaces the initial technology
constraint.
Producers have the same costs whether they license one invention or both of
the inventions. For producers to license both inventions instead of licensing just
one invention, royalties must satisfy the modi�ed incremental innovation constraint
r1 + r2 � minfr1; r2g + �g(a � b). This constraint is equivalent to the requirementthat the higher royalty is less than or equal to expected damages per patent,
maxfr1; r2g � �g(a� b):
So, imperfect patents alter the incremental innovation constraint by replacing the
incremental returns to an additional invention a� b with the expected damages forinfringing on one of the patents �g(a� b). This constraint replaces the incrementalinnovation constraint if and only if �g(a � b) < a � b. To examine the e¤ects
of imperfect patent enforcement, assume that the strength of the patent times the
expected damage multiple is not too large, that is, �g < 1, so that this constraint
replaces the incremental innovation constraint.
With symmetric royalties, the initial technology constraint is r � (1� �)�g(a�b) + �2g a
2: With symmetric royalties, the incremental innovation constraint is r �
�g(a � b). It can be shown that with innovative complements (substitutes), therelevant constraint is the initial technology (incremental innovation) constraint.
In a competitive market, there is the possibility of competitive entry by producers
using the two inventions at no cost. Then, industry output equals qD,
P (qD) = c0 � a+ 2(1� �)�g(a� b) + �2ga: (34)
30
This possibility is ruled out if q � qD = D(c0 � a+ 2(1� �)�g(a� b) + �2ga).Consider the e¤ects patent enforcement on the output constraints. The greater
is patent strength �, the weaker is the output constraint,
@qD@�
= D0(c0 � a+ 2(1� �)�g(a� b) + �2ga)2g[(a� b)� �(a� 2b)] < 0:
This is because (a � b) � �(a � 2b) = (1 � �)(a � b) + �b > 0. The greater is thedamage multiple g, the weaker is the output constraint,
@qD@g
= D0(c0 � a+ 2(1� �)�g(a� b) + �2ga)[2(1� �)�(a� b) + �2a] < 0:
In a competitive market, there is also the possibility that producers will enter the
market by using both inventions but only licensing only one invention. To rule this
out, de�ne the constrained output qD by equating the price to the production costs
when using one invention, the royalty for that invention, plus the expected damages
for infringing on one of the patents �g(a� b),
P (qD) = c0 � a+ r + �g(a� b): (35)
As before, the royalty will be determined in equilibrium.
Because of the two market constraints, the disagreement points are zero for pro-
ducers. This implies that the outcome of bargaining is the same as with perfect
patent enforcement, r�(q) = �1+�[P (q)� (c0 � a)]. We substitute for the equilibrium
royalty to solve for the constrained output qD,
P (qD) = c0 � a+ (1 + �)�g(a� b): (36)
The possibility of entry with only one invention is ruled out if q � qD = D(c0 � a+(1+�)�g(a�b)). The greater is patent strength, the weaker is the output constraint,
@qD@�
= D0(c0 � a+ (1 + �)�g(a� b))(1 + �)g(a� b) < 0:
31
The greater is the damage multiple, the weaker is the output constraint, @qD@g
< 0.
Finally, the greater is the inventor�s bargaining power, the weaker is the output
constraint, @qD@�< 0.
The relevant market constraint is the initial technology constraint if qD > qD, or
equivalently
c0 � a+ 2(1� �)�g(a� b) + �2ga < c0 � a+ (1 + �)�g(a� b):
This simpli�es to
�a < [2�� (1� �)](a� b):
If the inequality is reversed, the relevant market constraint is the incremental innov-
ation market constraint, qD < qD.
With innovative substitutes, 2(a� b) � a so that
[2�� (1� �)](a� b) < 2�(a� b) � �a:
With innovative substitutes, the relevant market constraint is the incremental innova-
tion market constraint, qD < qD. With perfect innovative complements or innovative
complements, the relevant market constraint is the initial technology (incremental
innovation) constraint if �a < (>)[2�� (1� �)](a� b):As before, equilibrium royalties r�(q) are symmetric across inventors and across
producers. The bargaining equilibrium exists and is unique for any q. We characterize
the equilibrium license o¤ers using Proposition 1. The bundled monopoly output
is constrained by the initial technology constraint with damages for infringement,
qM(c0 � a) = maxfbqM(c0 � a); qDg.COROLLARY 1. In the �rst stage, the weakly dominant strategy equilibrium in
license o¤ers (y�1; y�2) exists and is unique and symmetric. The equilibrium maximum
licensing o¤ers are greater than or equal to the pro�t-maximizing bundled monopoly
output,
y�1 = y�2 = maxfbqM(c0 � a); qD; qDg � qM(c0 � a):
32
With e¤ective patent enforcement, downstream output is greater than or equal
to the bundled monopoly output. With imperfect patent enforcement, downstream
output also is greater or equal to the bundled monopoly output. Imperfect patent
enforcement does not a¤ect the equilibrium output unless one of the market con-
straints is binding. So, imperfect patent enforcement raises downstream output only
through the market constraints.
This does not imply that public policy should weaken IP as a means of increasing
output. Output increases because inventors increase license o¤ers as a means of
deterring entry on producers that infringe on one or both patents. This serves to
reduce incentives to invent by lowering the returns obtained by inventors.
The main implication of the analysis is that it is not necessary to weaken patent
enforcement as a means of increasing �nal output. With e¤ect patent enforcement,
we have already shown that bargaining results in output that is greater than or equal
to the bundled monopoly output.
The bundled monopoly royalty is �M = P (qM(c0�a))�(c0�a), where qM(c0�a) =maxfbqM(c0 � a); qDg. By Proposition 2, we obtain the following.COROLLARY 2. Equilibrium license royalties with bargaining are unique, r� =�1+�[P (q�) � (c0 � a)], where q� = maxfbqM(c0 � a); qD; qDg. Total royalties with
bargaining are less than the bundled monopoly royalty, 2r� < �M .
The result does not imply that imperfect patent enforcement should be used as a
method of lowering prices. With e¤ective patent enforcement, bargaining is su¢ cient
to lower royalties below the bundled monopoly level. E¤ective patent enforcement
has additional bene�ts because patents lower transaction costs in markets for in-
vention, provide incentives for innovation, and facilitate �nancing of invention and
innovation (Spulber, 2015).
33
VI Public policy implications
It is informative to compare the present bargaining game to Cournot�s (1838) posted-
prices game.19 Cournot�s (1838) pricing model is used widely in studies of innovation
and public policy. Critics of the patent system present a long list of policy concerns
that typically rely on an application of Cournot�s complementary monopolies model.
These include: (1) Standard Essential Patent (SEP) holdup; (2) royalty stacking;
(3) patent thickets; (4) the Tragedy of the Anticommons, and (5) justi�cation for
patent pools. However, bargaining between IP owners and producers is an important
institutional feature of the market for inventions. Because bargaining blocks the
"Cournot e¤ect", these alleged policy concerns need not occur. The discussion in
this section suggests that antitrust and IP policy should not be based on the "Cournot
e¤ect."
VI.1 Comparison of bargaining and Cournot�s complement-
ary monopolies
This section compares the bargaining model presented here with Cournot�s comple-
mentary monopolies model. In Cournot�s model, inputs are perfect complements.
The discussion extends Cournot�s result to innovative complements and innovative
substitutes. The analysis shows that Cournot�s result depends on the assumption
that suppliers compete with posted prices.
The "Cournot e¤ect" is the result that complementary monopolists will choose
higher total prices than would a bundled monopoly.20 This e¤ect occurs in Cournot�s
19Cournot (1838) considers hypothetical complementary monopolists who sell copper and zinc toperfectly competitive producers of brass. Copper and zinc are perfect complements and are used in�xed proportions. The upstream monopolists chooses posted prices for copper and zinc and �naloutput is determined by downstream market demand evaluated at the total of input prices.20Cournot (1838, p. 103) observes that "the composite commodity will always be made more
expensive, by reason of separation of interests than by reason of the fusion of monopolies." Cournot(1838, p. 103) further observes "An association of monopolists, working for their own interest, inthis instance will also work for the interest of consumers, which is exactly the opposite of whathappens with competing producers."
34
model because input suppliers do not take into account the e¤ect of their price
increases on the pro�ts of other input suppliers. The "Cournot e¤ect" has been
applied to patent licensing. When inventors choose royalties non-cooperatively, they
will choose higher total royalties than would a bundled monopoly inventor.
The "Cournot e¤ect" is obtained as follows. Let c = c0 � a be the producers�unit cost when adopting both inventions. Inventors choose royalties ri, i = 1; 2.
Following Cournot, suppose for the moment that the inventions are perfect innovative
complements, b = 0. The downstream industry is perfectly competitive so that the
�nal output price in the downstream market equals production costs plus royalties,
p = c+ r1 + r2.
Input prices in Cournot�s non-cooperative equilibrium rCi , i = 1; 2 solve
rCi = argmax riD(c+ ri + r�i); i = 1; 2:
To characterize the Cournot posted price game, assume that market demand D(p) is
log concave, d2 lnD(p)dp2
� 0. Assume also that royalties are bounded, r � r. We havealready assumed that D(p) is decreasing and twice-continuously di¤erentiable. With
constant unit costs, this implies that there exists a unique and symmetric Cournot
equilibrium in prices. On the properties of the Cournot equilibrium, see Vives (1999),
Amir and Lambson (2000) and the references therein.
At a symmetric equilibrium, the �rst-order conditions in Cournot�s model simplify
to rCD0(c+ 2rC) +D(c+ 2rC) = 0, or
rC = �D(c+ 2rC)
D0(c+ 2rC): (37)
The bundled monopoly inventor�s �rst-order condition for total royalties is �MD0(c+
�M) +D(c+ �M) = 0. At the bundled monopoly price, we have
�M
2< �D(c+ �
M)
D0(c+ �M).
Compare the bundled monopoly royalty with total royalties chosen by posted prices
35
using the log concavity of market demand. This yields the "Cournot e¤ect,"
�M < 2rC :
The Cournot outcome yields lower output than the bundled monopoly, qM = D(c+
�M) > D(c + 2rC) = qC . The Cournot outcome also yields a higher �nal output
price, pM = c+ �M > c+ 2rC = pC .
We can extend Cournot�s model to include innovative complements and innovat-
ive substitutes. We restrict attention to symmetric equilibria. Then, when inventions
are innovative complements, royalties are subject to the initial technology constraint,
r � a2. When inventions are innovative substitutes, royalties are subject to the in-
cremental technology constraint, r � a� b. The posted price equilibrium results in
total output equal to qC = maxfD(c0 � a + 2brC); q0; qMg, where brC is the uncon-strained Cournot royalty. See Lerner and Tirole (2004, 2015) for a related extension
of Cournot.
Compare the Cournot output qC to the monopoly output qM . Suppose that inven-
tions are perfect innovative complements or innovative complements. Then, because
the relevant constraint is the initial technology constraint, the Cournot output is less
than or equal to the bundled monopoly output, qC � qM and 2rC � �M .Suppose that inventions are innovative substitutes. Recall that the bundling
monopolist is not subject to the incremental innovation constraint unless mixed
bundling were to be required. With innovative substitutes, the Cournot output is
less than or equal to the bundled monopoly output, qC � qM , if the incremental
innovation constraint is not binding on the mixed bundling monopolist. If the inven-
tions are innovative substitutes and the incremental innovation constraint is binding
on the mixed bundling monopolist, it would also be binding on the Cournot �rms, so
that qC = qM > qM . Under these conditions, the Cournot output would be greater
than the bundled monopoly output, and the Cournot e¤ect would not hold. This
discussion implies that the "Cournot e¤ect" holds, except for the situation in which
the inventions are innovative substitutes and a mixed bundling monopolist was con-
strained by the incremental innovation constraint. In this situation, the Cournot
36
output would exceed the bundled monopoly output.
We now compare the outcome of the two-stage bargaining model with the outcome
of the posted-prices Cournot model. De�ne social welfare as the sum of consumers�
and producers�surplusW (p) = CS(p)+PS(p), where consumers�surplus is CS(p) =R1pD(z)dz and total producers� surplus is PS(p) = [p� c]D(p). This gives the
following result.
PROPOSITION 4. A. With bargaining, the �nal output in the product market isgreater than or equal to output with a bundled monopoly inventor and thus greater
than or equal to output in Cournot�s posted prices game
q� � maxfqM ; qMg � qC :
B. With bargaining, the �nal market price is less than or equal to the price with
a bundled monopoly inventor and thus less than or equal to the price in Cournot�s
posted prices game,
P (q�) � P (maxfqM ; qMg) � P (qC):
C. With bargaining, total royalties are lower than in Cournot�s posted prices game,
2r� < 2rC :
D. With bargaining, consumers�surplus, total producers�surplus, and social welfare
are greater than or equal to those in Cournot�s posted-prices game,
CS(P (q�)) � CS(P (qC));
PS(P (q�)) � PS(P (qC));
W (P (q�)) � W (P (qC)):
PROOF.A.When inventions are perfect innovative complements or innovative com-plements and the Cournot equilibrium is not constrained, we know that �M < 2brC .Suppose now that inventions are perfect innovative complements or innovative com-
37
plements and the initial technology constraint is binding, brC > a2. At the symmetric
equilibrium, royalties will equal 2rC = a � �M . Suppose now that inventions are in-novative substitutes and the incremental innovation constraint is binding, brC > a�b.At the symmetric equilibrium, royalties will equal 2rC = 2(a � b) � �M . It followsfrom qC = D(c+2rC) and downward-sloping demand that q� � maxfqM ; qMg � qC .B. This further implies that P (q�) � P (maxfqM ; qMg) � P (qC). C. By Propos-
ition 2, 2r� < �M . Suppose that inventions are perfect innovative complements
or innovative complements. Then, �M � 2rC implies that 2r� < 2rC . Suppose
that inventions are innovative substitutes and the incremental innovation constraint
would not be binding on the mixed bundling monopolist, so �M � 2rC still holds.
Then, �M � 2rC again implies that 2r� < 2rC . Suppose that inventions are in-
novative substitutes and the incremental innovation constraint would be binding
on the mixed bundling monopolist. Then, qC = qM < q � q�, which implies that
2r� < 2rC . D. Consumers surplus is decreasing in price so consumer surplus with bar-
gaining is greater than or equal to that with posted prices, CS(P (q�)) � CS(P (qC)).Suppose that the incremental innovation market constraint, q � q, is not binding
on the bargaining equilibrium, so that q� = qM . Then, by pro�t maximization
PS(P (q�)) = PS(P (qM)) � PS(P (qC)). Suppose that the incremental innovation
market constraint is binding on the bargaining equilibrium so that q� = q. Because
q� � qC , it follows that the Cournot output must also be constrained, qC = q. So,PS(P (q�)) = PS(P (q)) = PS(P (qC)). It follows that producers�s surplus is always
greater with bargaining than with posted prices, PS(P (q�)) � PS(P (qC)). Using
CS(P (q�)) � CS(P (qC)), it follows that W (P (q�)) � W (P (qC)). �
The result depends on a comparison of the properties of the bargaining equilibrium
and those of the symmetric Cournot posted-price equilibrium. The result does not
depend directly on the assumptions of bounded prices and log concavity of demand,
which were used to characterize the Cournot posted-price equilibrium.
The result compares the e¢ ciency of the outcomes with bargaining and posted
prices. When inventors make non-cooperative license o¤ers and inventors and pro-
ducers bargain over royalties, inventors and producers achieve an e¢ cient outcome.
38
Inventors choosing licensing o¤ers non-cooperatively take into account the e¤ect of
their o¤ers on �nal output. Inventors choose the output that maximizes joint pro�t
when royalties are chosen by bargaining between inventors and producers. This
avoids the free rider problem that occurs when inventors post prices. So, bargaining
lowers market prices relative to the Cournot outcome and increases consumers�sur-
plus, producers�surplus, and social welfare. The discussion in this section considers
some public policy implications of this comparison.
VI.2 Standard Essential Patent holdup
Standard Setting Organizations (SSOs) are voluntary organizations that establish
and disseminate technology standards for industries. Patent owners may declare that
their patents are essential to manufacturing products that conform to the standard.
Many critics of SSOs suggest that inclusion of SEPs in technology standards allows
patent owners to charge much higher royalties than if the SEPs were not included
in the standard. SEPs are said to cause a form of "holdup" if producers using the
patented technology would incur high costs of switching to alternative technologies.21
Lemley and Shapiro�s (2007, p. 2013) analysis of patent holdup applies the
Cournot model: "The Cournot-complements e¤ect arises when multiple input owners
each charge more than marginal cost for their input, thereby raising the price of the
downstream product and reducing sales of that product." Lemley and Shapiro con-
sider bargaining between patent owners and producers, but express concerns that
patent owners enforcing their patents by injunctions will extract higher royalties.
Lemley and Shapiro (2007, p. 1993) state that "Injunction threats often involve a
strong element of holdup in the common circumstance in which the defendant has
already invested heavily to design, manufacture, market, and sell the product with
the allegedly infringing feature."22
Llanes and Poblete (2014) examine SEPs when IP owners choose posted prices
using the Cournot model. They suggest that making pricing agreements before a
21 See Geradin and Rato (2007) for an overview of the literature.22See Elhauge (2008) for additional discussion of Lemley and Shapiro (2007).
39
standard is chosen can lower prices and improve e¢ ciency in standard setting.
Lerner and Tirole (2015) develop a model of SEPs in which patent owners choose
posted prices based on Cournot�s model. The Cournot prices are subject to con-
straints from substitution both within and across technology functionalities. Lerner
and Tirole (2015, p. 558, n. 19) �nd the Cournot e¤ect: "The patents need not be
perfect complements as in Cournot�s model in order to generate prices that exceed
their monopoly level." Lerner and Tirole (2015, p. 564) observe that price setting
by patent owners can create SEP holdup: "Ex post price setting creates scope for
opportunism by IP holders."
These discussions of the e¤ects of SEPs depend on patent owners choosing roy-
alties using posted prices, generating total royalties above the bundled monopoly
level. When IP owners and producers engage in bargaining, the present analysis sug-
gests that total royalties will be less than the bundled monopoly level. E¢ ciencies
in choosing licensing royalties should mitigate concerns about the e¤ects of SEPs on
total royalties when patent licensing involves bargaining. The present analysis fur-
ther suggests bargaining should reduce or eliminate concerns about SEP "holdup".
E¢ ciencies in choosing patent licensing royalties also should help mitigate concerns
about whether or not SSOs choose e¢ cient technology standards.
VI.3 Royalty Stacking
"Royalty stacking" refers to the situation in which total royalties are excessive in
comparison to some benchmark, typically the bundled monopoly rate. Lemley and
Shapiro (2007) apply the Cournot posted prices model and predict that royalty
stacking occurs when there are complementary inventions, see also Shapiro (2001).23
As Geradin et al. (2008) point out "Royalty stacking is at its heart a reincarn-
ation of the �complements problem��rst studied by the French engineer Augustin
Cournot in 1838." For a critical discussion of royalty stacking see Geradin and Rato
23Lemley & Shapiro (2007, p. 2013) state that �The theory of Cournot complements teachesus that the royalty stacking problem is likely to be worse the greater the number of independentowners of patents that read on a product.�
40
(2007), Geradin et al. (2008), Sidak (2008), Elhauge (2008), and Schmalensee (2009).
Méniere and Parlane (2010) generalize the analysis of patent royalties with comple-
ments to include entry, di¤erentiated products, and two-part tari¤s.
The present analysis shows that the perceived royalty stacking problem is due to
the posted prices assumption in Cournot�s model. Royalty stacking is not due to the
complements problem. Cournot�s posted prices model yields royalty stacking with
innovation complements and substitutes. The problem as Cournot observed is free
riding with non-cooperative posted price setting.
The present analysis shows that royalty stacking need not occur with di¤erent
market institutions, notably bargaining between IP owners and producers. In partic-
ular, with non-cooperative licensing o¤ers and negotiation of royalty rates between
IP owners and producers, total royalties will be less than the royalties chosen by a
bundled monopoly IP owner. The result that total royalties are less than the bundled
monopoly benchmark holds even if there are many patented inventions. Total royal-
ties are less than the benchmark with innovative complements and substitutes.
VI.4 Patent Thickets
Shapiro (2001, p. 119) de�nes a patent thicket as �an overlapping set of patent rights
requiring that those seeking to commercialize new technology obtain licenses from
multiple patentees.� Shapiro (2001, p. 123) applies the Cournot e¤ect to identify
various policy implications of patent thickets: �This basic theory of complements
(used in �xed proportions) gives strong support for businesses to adopt, and for
competition authorities to welcome, either cross licensees, package licenses, or patent
pools to clear such blocking positions.�24
The patent thickets view considers patents as deterrents to innovation. This view
di¤ers substantially from the view that patents function as property rights that stim-
ulate innovation. For example, Ayres and Parchomovsky (2007) argue that "Patent
24Shapiro (2001, p. 123) state that �In order to produce the chip as designed, the company needsto obtain licenses from a number, call it N, of separate rights holders. This situation is preciselythe classic complements problem originally studied by Cournot in 1838.�
41
thickets are especially harmful in cumulative innovation settings." By analogy to
regulation of pollution, they suggest that the government should "weed out" patent
thickets by reducing the number of patents, either by raising renewal fees on patents
or setting limits on the total number of patents by tradeable patenting rights.
The bargaining analysis presented here suggests that multiple patents should not
be viewed as deterring innovation. Multiple inventors can coordinate with producers
through market transactions. This means that by making licensing o¤ers to produ-
cers and negotiating patent royalties, inventors and producers can achieve e¢ cient
outcomes. There is no need for government regulation to restrict the total number
of patents. Arbitrarily limiting the total number of patents by various regulatory
mechanisms would likely discourage invention and innovation.
VI.5 The Tragedy of the Anticommons
The "Tragedy of the Anticommons" describes the situation in which dispersed own-
ership of complementary inventions results in underuse of resources; see Heller (1998,
2008) and Heller and Eisenberg (1998). According to this view, patents and other
forms of IP create excess property rights that lead to economic ine¢ ciency. Heller
and Eisenberg (1998, p. 700) argue that "When owners have con�icting goals and
each can deploy its rights to block the strategies of the others, they may not be able
to reach an agreement that leaves enough private value for downstream developers
to bring products to the market."
Buchanan and Yoon (2000) provide a theoretical foundation of the "Tragedy of
the Anticommons" based on the Cournot posted prices model. In Cournot�s posted-
price model, an increase in the number of complementary inventions increases the
sum of input prices. This is because a greater number of inventors worsens the free-
rider e¤ects of posted price competition. This means that in Cournot�s model, a
greater number of inventors reduces both equilibrium output and social welfare.
In Buchanan and Yoon (2000), multiple rights to exclude such as patents result in
underutilization of resources. They assume that entry does not provide any bene�ts.
The only e¤ects of entry are to further divide ownership of scarce resources. In
42
addition, Buchanan and Yoon (2000) assume that there are no costs of entry, so
entry continues until �nal output is driven to zero. Buchanan and Yoon (2000, p. 10)
conclude that "The common facility or resource tends toward total abandonment, its
potential value being wasted in idleness." The total abandonment of the downstream
industry is the result of a combination of free riding with posted prices, no bene�ts
from entry, and an absence of entry costs.
The present analysis shows that patents need not create excess property rights
when there is bargaining between IP owners and producers. Bargaining results in
a total output that maximizes the joint returns of inventors and producers. Social
welfare and �nal output are greater with bargaining than in Cournot�s posted prices
model. This contradicts the "Tragedy of the Anticommons" result and shows that
there need not be underutilization of resources due to high royalties.
Another important di¤erence with Cournot�s model is the e¤ect of entry. With
bargaining between IP owners and producers, the entry of additional inventors does
not a¤ect the �nal output. This is because the bargaining outcome is not subject
to the free rider e¤ects that are observed in Cournot�s posted prices model. This
means that the entry of additional inventors need not diminish social welfare. More
inventors will divide the rents from innovation but will not diminish the e¢ ciency of
the outcome.
A more complete analysis of the entry of inventors requires consideration of the
bene�ts of the entry of inventors. In the present setting, one invention lowers produc-
tion costs, and the second invention generates an additional reduction in production
costs. More generally, a greater number of inventors should improve the quality of
inventions, resulting in greater social welfare. So, the entry of inventors is likely to
result in the creation of more IP rather than just subdividing a �xed amount of IP.
This contrasts with the anticommons view that patents are just a subdivision of a
given amount of intellectual property.
A more complete analysis of IP also should consider the costs of market entry.
In practice, inventors must incur the costs of R&D to develop inventions. Inventors
incur additional costs to obtain patent licenses. These costs are market entry costs for
inventors, which limits the entry of inventors. In addition, R&D is inherently risky
43
so that inventors need not succeed in creating inventions. Also, there is competition
among inventors so that not all inventors obtain patents. These risks further limit
the entry of inventors. This di¤ers from the anticommons view that unlimited entry
of inventors might occur, resulting in the abandonment of the downstream industry.
The present analysis shows that with bargaining between IP owners and produ-
cers, the downstream industry will combine inventions e¢ ciently. The overall e¤ects
of inventor entry on social welfare will depend on the trade-o¤s between the bene�ts
and costs of invention.
VI.6 Justi�cation for patent pools
Patent pools provide a mechanism for patent owners to choose patent royalties co-
operatively.25 The "Cournot e¤ect" often is used as a justi�cation for the formation
of patent pools. Without forming a patent pool, owners of complementary inven-
tions generate total royalties that exceed the bundled monopoly level. A patent pool
chooses the bundled monopoly royalty for the inventions, thereby reducing royalties
and increasing social welfare. For example, Shapiro (2001, p. 123) states "Cournot�s
theory of complements cast in terms of blocking patents ... gives strong support
for businesses to adopt, and for competition authorities to welcome, either cross
licensees, package licenses, or patent pools to clear such blocking positions."
A number of other studies apply the Cournot e¤ect to explain the need for cooper-
ation through patent pools. Lerner and Tirole (2004) generalize Cournot�s setting to
include increasing or decreasing incremental bene�ts from the number of inventions,
see also Kim (2004), Kato (2004), Brenner (2009), Gilbert (2010), Llanes and Trento
(2012), Dequiedt and Versaevel (2013), and Llanes and Poblete (2014). Schmidt
(2014) considers both patent pools and the e¤ects of vertical integration as a means
of reducing double marginalization with posted prices and �nds that vertical integ-
ration may reduce welfare.
25According to Carlson (1999, p. 367). "Patent pools are private contractual agreements wherebyrival patentees transfer their rights into a common holding company for the purpose of jointlylicensing their patent portfolios."
44
Our analysis suggests that patent pools need not be used as a mechanism for
reducing total royalties. The present analysis shows that inventors can coordinate
through the market using non-cooperative licensing o¤ers and bilateral bargaining
with producers over royalties. This eliminates the "Cournot e¤ect" whether in-
ventions are perfect innovative complements, innovative complements, or innovative
substitutes. This does not mean that antitrust authorities should reverse their fa-
vorable treatment of patent pools. The analysis suggests that it might be useful to
consider more fully the economic functions of patent pools.
VII Conclusion
Patent licensing in practice generally involves bargaining between IP owners and
licensees. The present analysis shows that bargaining generates industry output and
licensing royalties that maximize the joint pro�ts of inventors and producers. Even
when inventors make licensing o¤ers in a non-cooperative game, bargaining results in
an e¢ cient combination of inventions. Bargaining is e¢ cient whether inventions are
perfect innovative complements, innovative complements, or innovative substitutes.
With bargaining, total royalties are less than monopoly prices for the bundle
of inventions. Royalties are lowered when producer participation constraints are
binding because royalties must re�ect the incremental contribution of additional
inventions and the contribution of the inventions relative to the initial technology.
Also, royalties are lowered because inventors and producers have bargaining power.
The e¢ ciency of the bargaining outcome di¤ers from the outcome of the Cournot
posted prices model. Understanding the role of bargaining helps address a host of
public policy concerns, including SEP holdup, royalty stacking, patent thickets, the
tragedy of the anticommons, and justi�cation for patent pools. The e¢ ciency of the
bargaining outcome suggests the need for antitrust forbearance toward industries
that combine multiple inventions, including SEPs.
The bargaining framework helps explain why markets for technology tend to di¤er
from neoclassical auctioneer markets and industrial organization oligopoly markets.
45
Bargaining helps in the development of complex innovations such as smart phones
that require combinations of inventions and coordination among many inventors
and producers. With bargaining, royalties re�ect returns to combining inventions
and adjusts to inventions that are innovative complements or substitutes. With
bargaining, royalties depend on the extent of competition among producers in the
downstreammarket. Overall, bargaining between IP owners and producers is e¢ cient
because it tailors patent licensing contracts to the characteristics of inventions and
producers.
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