Licensing downstream technology when upstream firms are
capacity constrained
Eva-Maria Scholz∗
June 2014
Abstract
The market conditions in unlicensed, but economically connected industry sectors may
have an important influence on the design of a licensing contract. To analyse this issue we
study the licensing behaviour of an outside patentee who licenses a cost-reducing technology
to the downstream sector of a vertical Cournot oligopoly via either a per-unit royalty or
a fixed-fee contract. Downstream firms source their input requirements from an upstream
industry and it is assumed that some of the upstream firms are unable to expand their
production levels in reaction to the increased degree of downstream efficiency which follows
the technology transfer. In this framework we show that as a consequence of such upstream
capacity constraints a per-unit royalty contract may dominate a fixed-fee contract in terms of
licensing revenues. Here we further explore the welfare implications of the optimal licensing
contract and analyse how the latter depend on the number of capacity constrained upstream
firms. As a final point we study the optimal long run licensing contract in a two-period
setting and discuss its static and dynamic welfare properties.
Keywords patent licensing, vertical Cournot oligopoly, capacity constraints.
JEL classification D43, L13, O31, O34.
∗
Université catholique de Louvain, CORE, B-1348 Louvain-la-Neuve, Belgium.
1
1
Introduction
Knowledge intensive industries are characterised by two main features: complex supply chains
as well as a high importance of external knowledge sources. Regarding the development or the
transfer of technology, one increasingly important way of accessing those external sources of
knowledge are licensing agreements. This trend is reflected in the data which indicates that
technology licensing activities have increased over the last two decades (Arora and Gambardella
(2010), Robbins (2006), Sheehan et al. (2004), Zuniga and Guellec (2009)). For example, Sheehan et al. (2004) find that almost 60% of the surveyed firms in Europe, the United States,
and Asia-Pacific reported increased licensing activities during the 1990s. Similarly, Zuniga and
Guellec (2009) state that 45% of European firms in their sample reported increased licensing
revenues or activities between 2003 and 2006. On the aggregate, Robbins (2006) estimates that
the licensing of industrial processes amounted to $66 billion in 2002 in the United States (as
compared to $27.4 billion in 1995), indicating a global market for technology of around $100
billion (Arora and Gambardella (2010)).
An important player in this development are independent (outside) innovators. Meaning,
patent holding firms or institutions, which themselves are not active in the potentially licensed
industry and transfer technology in exchange for royalties or upfront fee payments. Depending
on the industry considered, those outside innovators are called specialised engineering firms
(Chemicals), technology or research and development specialist firms (Biotechnology), fabless or
chipless firms (Semiconductors) or software specialist firms (IT) (Arora and Gambardella (2010),
Arora et al. (2004), Cesaroni (2003), Linden and Somaya (2003)). According to Cesaroni (2003)
those independent technology suppliers cover almost 70 % of the total market for licensing in
the Chemical industry. Similarly, most of the technology transfer in functional design inventions
or design modules in the Semiconductor industry involves chipless firms (Linden and Somaya
(2003)).
There is by now a vast theoretical literature, which studies licensing agreements involving
such outside innovators. However, the majority of this literature assumes that the licensed
industry exists in autonomy and is thus not economically connected to other industry sectors.
Or, alternatively, that such supply-chain relationships to other industry sectors are not important
for the design of the optimal licensing contract. The few papers which address the technology
transfer or licensing to vertically separated markets assume that the other sectors of the vertical
structure are able to perfectly adapt their production to the higher degree of efficiency in the
licensed industry (see Section 2 for a review of the literature). It is in our eyes intuitive that
this assumption does not need to apply in reality. For instance, the accommodation of a higher
production efficiency in another market may necessitate complementary investments, innovation
or learning processes. Alternatively, those industries may be unable to adjust their production of,
for example, input goods that are required by the licensed industry due to the presence of fixed
contracts with the licensed industry or other parts of the supply chain. In the same vein, supply
shortages on various layers of an industry’s supply chain may hinder a sector’s instantaneous
adjustment to the expanding demand stemming from a licensed downstream industry.
2
The following discussion illustrates our point. Over the past years, technological innovations
made increasingly use of rare earth elements (REEs). This is particularly true for so-called green
technologies such as energy efficient lighting systems, solar panels, high performance batteries
or (hybrid) electronic cars (Schüler et al. (2011)). But also consumer products like smartphones,
computers or flat panel television heavily rely on REEs (U.S. Department of Energy (2011)).
REEs form a group of seventeen chemical elements and derive their rareness from the fact
that their occurrence in economically mineable deposits is limited. Recently, various studies
highlighted the critical, or near critical, supply status of some of the latter elements (Schüler et al.
(2011), U.S. Department of Energy (2011)). These actual or potential supply shortages may not
only originate from geological factors, but may equally be due to geopolitical or economic reasons.
A prominent example of the last point are the export restrictions introduced by China in 2010.
At present, China is the world’s leading exporter of REEs with its major importers, Europe,
Japan and the United States, sourcing more than 90 percent of their input requirements from
the country. The dependency on Chinese, or more generally foreign, supplies is strengthened
by the fact that a substitution of REEs by other materials is rarely feasible; also their large
scale recycling faces several constraints (Schüler et al. (2011)). As a result, the introduction of
export quotas and duties on these input factors by the Chinese government was followed by a
significant price increase for REEs in 2011.
In this paper we acknowledge those impediments to capacity expansion and show that it is
indeed crucial to not only analyse the industry which is the potential target of a technology
transfer when designing a licensing contract. Instead, the market conditions in other industry
sectors, which are economically connected to the target industry, exert an important influence
on the optimal design of a licensing agreement.
To be more precise, we analyse the licensing behaviour of an independent innovator who
transfers a cost-reducing technology to the downstream sector of a vertical Cournot oligopoly.
The innovation may be transferred either by means of a fixed-fee or a per-unit royalty contract.
We assume that some of the upstream firms are unable to expand their production levels in
reaction to the increased efficiency that follows a downstream technology transfer. In this
framework it is then shown that, in contrast to the traditional licensing literature (Kamien
et al. (1992), Kamien and Tauman (1986)), a per-unit royalty contract may provide larger
licensing revenues than a fixed-fee scheme. By this our paper provides another rationale for the
empirically observed popularity of royalty licensing agreements (Bousquet et al. (1998), Macho
Stadler et al. (1996), Rostoker (1983-1984), Sakakibara (2010)).1
Our results are based on the presence of a sufficiently strong capacity effect, which reduces
the patentee’s fixed-fee licensing income for a given size of the innovation. In the absence of
capacity constraints, the transfer of a cost-reducing innovation is followed by an expansion of
aggregate industry output. In the presence of capacity constraints, some of the upstream firms
cannot accommodate the higher level of downstream demand and instead react by increasing the
1
Empirical evidence on firms’ licensing strategies is scarce. Firms are reluctant to share such information as
it gives important insight in their overall strategy and competitiveness. More recent empirical studies try to
explain why firms favour one licensing policy over the other (Vishwasrao (2007)) or why firms engage in licensing
agreements (Anand and Khanna (2000)).
3
intermediary input price. More generally, what drives our results is thus the presence of some
inelastically supplied input factor which is essential for the licensed industry. The higher level of
the input price then reduces the downstream firms’ willingness to pay for the innovation, as well
as the number of licensed firms, and by this equally the patentee’s fixed-fee licensing income.
Under a per-unit royalty contract this effect is absent. It follows that capacity constraints reduce
the attractiveness of fixed-fee licensing agreements, whereas they do not affect the patentee’s
royalty licensing revenues.
The main part of our paper concentrates on the case of a full capacity constraint in which
the entire upstream industry is unable to expand its production levels in reaction to an increased
efficiency of the downstream market. Although this scenario serves well to illustrate our main
results and the mechanisms underlying the latter, a full capacity constraint may be seen as
an extreme assumption. This is why we also address the case of a partial capacity constraint
in which only some upstream firms face such constraints. Our analysis reveals that a full
capacity constraint is not required for our results to hold (qualitatively). In fact, starting from a
framework in which the patentee prefers a fixed-fee contract in absence of capacity constraints,
we show that the introduction of a single capacity constrained upstream firm may be a sufficient
condition for the patentee to realise higher licensing revenues under a per-unit royalty scheme.
We further explore the welfare implications of the optimal licensing contract. We derive a
result, which at first glance may appear puzzling. Namely, under a full capacity constraint,
a privately optimal royalty contract is all the time optimal in terms of aggregate welfare. In
contrast, with a single capacity constrained upstream firm, a privately optimal royalty contract
almost never maximises welfare. Hence, the apparent stability of the patentee’s private incentives
to a reduction in the the number of capacity constrained upstream firms cannot be replicated
in terms of aggregate welfare.
As a final point we study the design of the optimal long run licensing contract in a twoperiod setting. It is assumed that the upstream market is fully capacity constrained in the short
run, while capacity constraints are absent in the long run. In this framework, long run perunit royalty contracts may be optimal only for sufficiently concentrated downstream industries.
Despite their potential superiority in terms of licensing income, we further show that long run
per-unit royalty contracts are characterised by poor static and dynamic welfare properties which
make room for public policy interventions.
Our paper is structured as follows. Section 2 provides a brief review of the relevant literature. In Section 3 we introduce the general modelling framework. Subsequently, we solve
the per-unit royalty (Section 3.1) and the fixed-fee (Section 3.2) licensing game and state the
respective subgame perfect Nash equilibria. This is done for a general number of capacity constrained upstream firms. Section 4 then derives the optimal licensing contract by contrasting
the patentee’s licensing income for the two licensing policies. Here, Section 4.1 studies the case
of a full capacity constraint, Section 4.2 addresses the case of a partial capacity constraint. In
each section we first illustrate our main result via a duopoly example and then extend the results
to an oligopolistic framework. Section 4.3 discusses some welfare implications of the optimal
licensing contract. Section 5 studies the optimal long run licensing contract in a two-period
4
setting. Section 6 concludes.
2
Literature Review
The traditional licensing literature predicts that (for a non-drastic innovation2 ) a fixed-fee policy provides higher licensing revenues to an outside patentee than a per-unit royalty contract
(Kamien et al. (1992), Kamien and Tauman (1986)). This theoretical result stands somewhat
at odds to the findings of the empirical literature, which documents a widespread use of royalty
licensing schemes (Bousquet et al. (1998), Macho Stadler et al. (1996), Rostoker (1983-1984),
Sakakibara (2010)). This apparent contradiction of theoretical prediction by empirical evidence
led to a newly awakened interest in the study of licensing behaviour. Today, there is a large
theoretical literature which tries to rationalise the empirically observed popularity of per-unit
royalty licensing agreements. This is achieved by either relaxing assumptions of the standard
framework or by introducing new considerations into it. Approaches range from the consideration of different modes of competition (Filippini (2005), Kabiraj (2004)) to the introduction of
product differentiation (Erkal (2005), Kabiraj and Lee (2011), Wang (2002), Wang and Yang
(1999)) or asymmetries (Beggs (1992), Gallini and Wright (1990)). Other authors rationalise
the popularity of royalty licensing agreements by taking into account that the patentee may
compete in the product market (inside patentee) (Kamien and Tauman (2002), Wang (1998)
or by acknowledging the fact that the number of licensing contracts offered may only take on
integer values (Sen (2005), Sen and Tauman (2012)).
The theoretical literature on licensing considerations in vertically related market environments is scarce. Most related to this paper are Chang et al. (2013), Mukherjee (2002) or Scholz
(2014). Mukherjee (2002) studies the licensing strategy of a producing (inside) innovator within
a downstream Cournot duopoly under a fixed-fee policy. It is shown that licensing to the
downstream competitor is profitable provided it encourages entry in the previously monopolised
upstream market. The case of a non-producing (outside) innovator is addressed in Chang et al.
(2013) and Scholz (2014). In Chang et al. (2013) an outside patentee transfers a cost-reducing
technology to a downstream Cournot duopoly. As in Mukherjee (2002) the upstream firm holds
a monopoly position. The authors then focus on the superiority of two different licensing policies
(fixed-fee and per-unit royalty) in terms of the patentee’s licensing revenues as well as on welfare considerations. In Scholz (2014) we study the strategy of an outside innovator who licenses
a cost-reducing, generic technology to a vertical Cournot oligopoly. The vertical structure is
composed of an oligopolistic upstream and downstream market which may differ in terms of the
benefit they derive from the innovation. It is shown that the optimal licensing contract may
involve licensing to the sector which derives the lowest benefit from the technology, downstream
per unit-royalty contracts and one-sector licensing of a single sector or two-sector licensing of
both sectors.
2
The distinction between a drastic and a non-drastic innovation is due to Arrow (1962). A drastic innovation
is optimally licensed to a single firm, which monopolises the market.
5
3
The Model
We consider a vertical Cournot oligopoly with an upstream (m) and a downstream (n) sector.
On the upstream sector M firms are active and produce a homogeneous product. This product
serves as an input for the downstream industry on a one-to-one basis. For simplicity we set the
upstream marginal cost of production equal to zero. It is assumed that upstream firms move first
and compete à la Cournot. Their individual quantities are denoted by xi . Quantity competition
upstream yields the market clearing input price w which is taken as given by the downstream
firms. On the downstream market N firms produce a final homogeneous good. Next to the input
price w, downstream firms face their constant marginal cost of production cαi where α ∈ {l, nl}
denotes the licensing status of a firm (l for licensed and nl for not licensed). Intuitively,
α
cli < cnl
i . Downstream firms compete à la Cournot where qi refers to the individual downstream
quantities. The inverse demand function is of a standard form and given by P (Q) = a − Q with
P
α
Q= N
i=1 qi .
Downstream firms have potential access to a cost-reducing (process) innovation which lowers
their marginal cost of production by some strictly positive θ with θ ≤ c. It is assumed that the
technology is supplied by an outside patentee which is not active in either industry sector. This
corresponds to the case of an independent research laboratory or think tank. The patentee may
transfer the technology via either a per-unit royalty (r) or a fixed-fee contract (f ). It follows
that the marginal cost of production of a downstream agent is specified as either cli = c − θ + r
or as cnl
i = c, depending on the firm’s licensing status. Intuitively, r = 0 in the case of a fixed-fee
policy (we do not consider two-part tariffs).
The licensing game is modelled as the following well-known three stage game. On the first
stage the patentee announces the licensing contract, which, depending on the licensing policy
chosen, specifies the royalty rate r or the fixed-fee f . Subsequently, on the second stage of the
game, downstream firms independently and simultaneously decide about acceptance or refusal
of the proposed contract. On the last stage of the game Cournot competition takes place. This
is the two stage game described above. The licensing game is solved by employing the solution
concept of a subgame perfect Nash Equilibrium (SPNE).
As such, on the last stage of the game downstream firms compete in quantities and maximise
a profit function of the form πin,α,k = (P (Qk ) − w − cαi )qiα,k with respect to qiα,k . In the latter
expression, k ∈ [0, M ] denotes the number of capacity constrained upstream firms. Summing
the corresponding first order conditions for the group of licensed and unlicensed downstream
firms we obtain the derived demand for input goods as
w(Qk ) = a −
with C =
PN
α
i=1 ci .
C
N +1 k
−
Q
N
N
(1)
Note that as soon as one of the downstream firms is licensed C < N c.
Subsequently, w(Qk ) is observed by the upstream firms. It is assumed that k of the M
upstream firms are unable to expand their production of the input good. More precisely, we
impose xki ≤ xi,pre where xi,pre denotes the pre-innovation output of an upstream firm. Capacity
6
constraints of this form may be motivated by the presence of a capacity cost which is zero up to
xi,pre , but becomes infinitely large for any level of production above this threshold. As we argue
in the introduction to this paper, such capacity costs may stem from the presence of supply
shortages that are due to the finiteness of an input factor on this or earlier levels of the supply
chain (geological, geopolitical or economic factors). Alternatively, the scale-up of upstream
production processes may necessitate further complementary innovations or investments. As
a corollary, the setting may be seen as embedded in a dynamic, two-period framework with
different supply elasticities across periods. As such, in the short run, i.e., in the first period,
supply is (fully) inelastic, while it becomes fully or partially elastic in the long run. We address
this last point in more detail in Section 5.
As a result, capacity constrained upstream firms maximise πim,k = (w(X k ))xki with xki ≤
xi,pre . Similarly, unconstrained firms maximise πim = (w(X k ))xi with respect to xi .3
In absence of capacity constraints (k = 0) this yields
X0 =
C
M N (a − N
)
.
(M + 1)(N + 1)
(2)
As a consequence,
w0 =
C
a− N
M +1
P0 =
and
a(M + N + 1) + M C
.
(M + 1)(N + 1)
(3)
Lemma 1 As the sum of the marginal costs of the downstream industry decrease, aggregate
industry output and the intermediary input price increase, whereas the final good price decreases.
In other words,
∂X 0
∂C
< 0,
∂w0
∂C
< 0, while
∂P 0
∂C
> 0.4
From Lemma 1 it follows that the innovation features a capacity effect. Meaning, aggregate
industry output expands proportionally to a decrease in the marginal costs of production.
For k = M the aggregate solution to the constrained optimisation problem is given by
XM =
M N (a − c)
(M + 1)(N + 1)
(4)
together with
wM =
C
a − (M + 1) N
+ Mc
M +1
and
PM =
a(M + N + 1) + M N c
.
(M + 1)(N + 1)
(5)
Lemma 2 Assume that at least one downstream firm is licensed so that C < N c. Then, for
a fully constrained upstream market the intermediary input price increases compared to the
pre-licensing equilibrium, aggregate industry output and the final good price are unaltered by the
3
Notice that our assumption of Cournot competition allows us to abstract from rationing considerations. The
Cournotian auctioneer distributes the available supply efficiently, according to the relative demand of licensed
and unlicensed downstream firms.
4
For a general demand function the increase in the intermediary input price is identified by Banerjee and
Lin (2003) and termed raising rival’s cost effect. Following the acceptance of a licensing contract by a single
downstream firm the input price increases, not only for this firm, but equally for this firm’s rival.
7
transfer of the cost-reducing technology. That is, wM > wpre , while P M = Ppre and X M = Xpre .
In the remainder of this section we solve the second and the first stage of the licensing game
for either a per-unit royalty or a fixed-fee policy. We do this for a general number of capacity
constrained upstream firms (k ∈ [0, M ]). Then, in Section 4, we derive the optimal licensing
policy. In this context we first concentrate on deriving the optimal licensing contract under a
full capacity constraint (k = M ). With this benchmark in place, we then address the case of a
partial capacity constraint (k ∈ (0, M )).
3.1
The Royalty Licensing Game
Assume first that the patentee transfers the innovation via a per-unit royalty contract. That is,
by means of a fee per unit of output produced with the innovation, denoted by rk . Under such
a contract, a licensee’s profit function takes on the form of πin,l,k (c + rk − θ). It follows that the
maximum rk a firm is willing to pay for the innovation is θ. Otherwise, the benefit embodied in
the innovation (reduced marginal cost of production) is outweighed by the cost of obtaining it
(royalty rate). As a corollary, every downstream firm accepts a licensing contract for any rk ≤ θ
and the patentee’s per-unit royalty licensing income amounts to π P,r,k = rk X k (c − θ + rk ). It is
easily seen that the latter is strictly increasing in the royalty rate for any non-drastic innovation.
Hence, the patentee optimally sets rk = θ. Note that in such a SPNE the entire downstream
industry obtains a licensing contract, denoted by lk = N , due to the symmetry of the setting.
Lemma 3 For a non-drastic innovation the royalty licensing game has a SPNE in pure strategies in which lk = N and rk = θ. The patentee’s associated licensing income is given by
π P,r,k = θXpre .
At this point two things are worth pointing out. First, a per-unit royalty is a variable part
in the optimisation problem of a licensee. For any strictly positive royalty rate this leads to
a relative downward distortion of the optimal (downstream) production levels as the royalty
rate partially offsets the cost reduction embodied in the innovation. In fact, at the licensing
equilibrium the patentee equates royalty rate and cost reduction. As a result, downstream firms
are indifferent between accepting and rejecting the proposed licensing contract and continue
producing as they did prior to the transfer of the technology. To put it differently, under a
per-unit royalty scheme, the innovation does not result in an expansion of industry output.
The patentee’s licensing strategy and income are consequently not affected by the introduction
of capacity constraints under a per-unit royalty scheme. That is, rk = θ and π P,r,k = θXpre
∀k ∈ [0, M ].
Second, notice that under a fixed-fee policy this effect is absent; the technology transfer
features an expansion of production levels (in the absence of capacity constraints) and by this
leads to increased gross market revenues of upstream and licensed downstream firms. This is
the intuition behind why an outside patentee generally maximises licensing profits by offering a
fixed-fee instead of a per-unit royalty contract in the absence of capacity constraints.5
5
For completeness it has to be pointed out that when licensing to the downstream sector of a vertical structure
a new effect is present (raising rival’s cost effect), which may render a downstream royalty contract optimal. This
8
3.2
The Fixed-Fee Licensing Game
We now assume that the patentee offers a fixed-fee contract to the downstream industry.
Downstream firms accept any proposed contract as long as the demanded upfront fee does
not exceed their willingness to pay for the innovation or, in other words, as long as f k (lk ) ≤
πin,l,k (lk )−πin,nl,k (lk −1) (fixed-fee licensing contracts cannot be re-offered). With the previously
derived results in mind we derive the market revenues of licensed and unlicensed firms as

MA
lk
2 α = l,
(
n,α,k
(M +1)(N +1) + θ − N θλk )
πi
=
k
MA
(
− l θλ )2
α = nl
(M +1)(N +1)
with λk =
(M −k)N
N +1 ). For notational convenience we define A = a − c. The patentee
optimally at f k (lk ) = πin,l,k (lk )−πin,nl,k (lk −1) and extracts the entire willingness
1
M −k+1 (1
sets the fixed-fee
N
(6)
k
+
to pay of the downstream firms. Given (6) the fixed-fee therefore amounts to
f k (lk ) = θ2 (1 −
2M A
2lk
λk
λk
)(
−
λk + 1 + ).
N (M + 1)(N + 1)θ
N
N
(7)
Note that f k (lk ) > 0 ∀k ∈ [0, M ] requires N > 1. For instance, for N = 1 and k = M the
increase in the input price due to the presence of capacity constraints exactly offsets the benefit
from the innovation so that f M (lM ) = 0 at N = 1. This is why we assume N > 1 in the
following.6
In order to determine the optimal number of licensing contracts the patentee solves
max π P,f,k = f k (lk )lk .
(8)
lk
Next to an interior solution at l∗,k (a strict subset of the downstream market is licensed; licensed
and unlicensed firms realise strictly positive market revenues), the optimisation problem has two
other equilibria at the boundary. One at lk = N (the entire industry is licensed) and one at
lk = Lk (essential innovation; only licensed firms realise strictly positive market revenues).
Lemma 4 summarises the SPNE of the fixed-fee licensing game.
Lemma 4 For a non-drastic innovation the fixed-fee licensing game has a SPNE in pure strategies in which, depending on the size of the cost reduction, lk licenses are offered with
lk =


N


1
4λk



Lk
θ ≤ θk ,
2M N A
( (M +1)(N
+1)θ + N + λk )
θ ∈ (θk , θ̄k ),
θ ∈ [θ̄k , θ̄¯k )
(9)
is for instance shown in Chang et al. (2013) or Scholz (2014). However, the scenarios in which a per-unit royalty
contract dominates a fixed-fee contract in terms of downstream licensing revenues are few. In Section C of the
appendix we provide some evidence on how the introduction of capacity constraints considerable enlarges the set
of cases in which a per-unit royalty contract is optimal.
6
For N = 1 and k = M it is trivial to show that a per-unit royalty yields larger licensing revenues than a
fixed-fee policy. For k = M λk = 1. Thus, at k = M and N = 1, f k (lk ) = 0 and consequently π P,f,k = 0, whereas
π P,r,k > 0 at k = M and N = 1.
9
MNA
(N +1)(M +1)λk θ ,
MNA
(M +1)(N +1)λk .
where Lk =
and θ̄¯k =
θk =
2M A(M −k+1)
1 ,
(M +1)((M −k+1)(3N −2)+4− N
)
θ̄k =
2M N A(M −k+1)
(M +1)((M −k+1)N (N +2)+1) ,
Note that as λk increases in k the optimal lk decreases in the number of capacity constrained
upstream firms for a given size of the innovation. Capacity constraints thus reduce the degree
of diffusion of the innovation under a fixed-fee policy. This result stands in contrast to the one
obtained for a per-unit royalty policy (lk = N ∀k ∈ [0, M ]).
4
The Optimal Licensing Contract
Having derived the SPNE of the per-unit royalty and the fixed-fee licensing game we are now
able to compare the patentee’s licensing income under either policy. To do so, we start with
a scenario in which the entire upstream industry is unable to expand its production levels in
reaction to an increased downstream efficiency. With this benchmark in place, we then study
how our results are affected by a reduction in the number of capacity constrained firms. In
each section we begin with a simple duopoly example. Subsequently, we analyse an oligopolistic
market environment.
4.1
Full Capacity Constraint
Duopoly example
To illustrate the main result of our paper we initially restrict our attention to the case of a
vertical Cournot duopoly (i.e., M = N = 2). Under full capacity constraints, i.e., k = 2, it is
assumed that both upstream firms cannot produce more than in the pre-licensing equilibrium.
Let us first consider the case of a per-unit royalty contract. From Lemma 3 it follows directly
that the optimal royalty contract specifies l2 = 2 and r2 = θ. The associated royalty licensing
revenues are thus given by π P,r,2 =
4Aθ
9
(see (2)). As it has been argued previously, this result
applies in the presence as well as in the absence of capacity constraints due to the relative
downward distortion of industry output under a per-unit royalty scheme.
Under a fixed-fee policy, the patentee may transfer the innovation either to both or to a single
downstream firm. It can be shown that the patentee optimally licenses to both downstream firms
whenever the innovation is sufficiently small. For larger cost reductions, a single firm is licensed.
The optimal licensing strategy and the associated licensing revenues for respectively full (10) or
no (11) capacity constraints consequently amount to
πlP,f,2

 18θ2 ( 2A − 81 )
81
θ
36
=
 9θ2 ( 2A + 81 )
81
or
πlP,f,0
θ
36

 22θ2 ( 2A − 3 )
81
θ
4
=
 11θ2 ( 2A + 11 )
81
θ
4
10
θ≤
θ∈
θ≤
θ∈
8A
27 ,
4A
( 8A
27 , 9 ),
(10)
8A
17 ,
4A
( 8A
17 , 7 ).
(11)
Proof Here we show how to obtain (10), (11) may then be derived in a similar fashion. Plugging
in M = N = k = 2 and either l2 = 1 or l2 = 2 in f 2 (l2 ) (see (8)) one can obtain the relevant
P,f,2
P,f,2
yields
≥ πl=1
fixed-fee. πlP,f,2 is then easily derived as f 2 (l2 )l2 with l2 ∈ {1, 2}. Then, πl=2
θ≤
8A
27
(and vice versa). For the innovation to be non-drastic πin,nl,2 (l2 = 1) > 0 is required.
Given (6), this implies θ <
4A
9
for M = N = 2.
Comparing π P,r,0 to π P,f,0 one can first observe that in the absence of capacity constraints
the patentee maximises licensing revenues by offering a fixed-fee contract. This result is due
to the previously mentioned relative downward distortion of production levels under a per-unit
royalty contract (Section 3.1). However, comparing π P,r,2 to π P,f,2 shows that the introduction of
capacity constraints reverses this result. Under full capacity constraints, the patentee maximises
licensing revenues by offering a per-unit royalty contract.
Proposition 5 Assume that an outside patentee licenses to the downstream sector of a vertical
Cournot duopoly. Then, in the absence of upstream capacity constraints a fixed-fee contract
dominates a per-unit royalty contract in terms of licensing revenues. When both upstream firms
are capacity constrained, the result is reversed in favour of a per-unit royalty scheme. Hence,
π P,f,0 > π P,r,0 = π P,r,2 > π P,f,2 for M = N = 2 and a fully constrained upstream market.
This result may be rationalised by the presence of a capacity effect, which reduces the patentee’s fixed-fee licensing income for a given size of the innovation. It was argued previously that,
in absence of capacity constraints, the transfer of a cost-reducing innovation is followed by an
expansion of production levels. However, under capacity constraints upstream firms cannot accommodate the higher level of downstream demand and react by increasing the intermediary
input price. This reduces the downstream firms’ willingness to pay for the innovation and by
this lowers the patentee’s fixed-fee licensing income. Intuitively, this capacity effect increases in
the size of the innovation.
In this context we also want to point to another result which illustrates the working of the
capacity effect. Note that under a fixed-fee contract the patentee prefers to license to both firms
if and only if the innovation is small. In this context it can be observed that the upper bound on
θ is lower in the presence of upstream capacity constraints (θ ≤
8A
27
as compared to θ ≤
8A
17 ).
For
a given size of innovation, both firms producing with the superior technology implies a larger
level of downstream derived demand and by this a larger capacity effect. It follows that under
capacity constraints licensing to both firms is only optimal for sufficiently small innovations. As
a corollary, for a given size of innovation, a full diffusion of the innovation is less likely under
capacity constraints.
Oligopolistic Industries
We now show how the previously derived results may be generalised for an industry with an
oligopolistic upstream and downstream sector. While in a vertical duopoly it is all the time
optimal to offer a per-unit royalty contract (under capacity constraints), this result does not
necessarily apply in a vertical oligopoly. Instead, our findings show that the superiority of a
11
fixed-fee over a per-unit royalty contract in an oligopolistic environment depends on the presence
of a sufficiently strong capacity effect.
From Lemma 4 it is clear that under a fixed-fee policy we have three subcases (lk = Lk ,
lk = l∗,k , lk = N ) that depend on the size of the innovation. Consequently, we need to compare
the patentee’s licensing revenues under a per-unit royalty contract and a fixed-fee contract for
each of them. Graphical illustrations are provided in Section A of the appendix.
For sufficiently large values of the cost reduction the innovation is essential. Meaning, only
licensed firms realise strictly positive market revenues. Here we restrict our attention to nondrastic innovations. This implies that the innovation is not sufficiently important for the industry
to become monopolised after the technology transfer.
Proposition 6 Assume lM = LM . Then π P,r,M ≥ π P,f,M ∀N > 1.
Essential innovations feature the largest values of a non-drastic process innovation and by
this, intuitively, the most pronounced capacity effects (largest output expansion in the absence
of capacity constraints). Consequently, in the given framework, a royalty contract is all the time
licensing revenue maximising for an essential innovation.
For intermediate values of the cost reduction the industry is one of a mixed technology in
which firms with and without the innovation realise strictly positive market revenues.
Proposition 7 Assume lM = l∗,M . Then π P,r,M ≥ π P,f,M if and only if
• N ≤ 3 and θ ∈ (θM , θ̄M ),
• N > 3 and θ ∈ [θ0 , θ̄M ) where θ0 =
2M N A(N −1)
.
(M +1)(N +1)3
Proposition 7 implies that for a sufficiently concentrated downstream market it is all the
time optimal to transfer the innovation by means of a per-unit royalty contract. For more
competitive downstream markets the innovation has to be of a certain size in order to ensure
a minimal capacity effect. For lM = l∗,M , the innovation, and by this also the capacity effect,
are of an intermediate size. As N increases π P,r,M and π P,f,M both increase, however, π P,f,M
does so at a faster rate (under a fixed-fee contract, l∗,M and f M (l∗,M ) both increase in N , while
under a royalty contract only X increases). Therefore, in order to render a per-unit royalty
contract optimal, a lower bound on θ is required.
For sufficiently small innovations, the entire downstream industry is licensed.
Proposition 8 Assume lM = N . Then π P,r,M ≥ π P,f,M if and only if
• N ≤ 2 and θ ∈ (0, θM ],
• N ∈ (2, 3) and θ ∈ [θ00 , θM ] where θ00 =
M N A(N −2)
,
(M +1)(N +1)(N −1)2
• N = 3 and θ = θ00 .
12
Similar to Proposition 7, Proposition 8 states that a per-unit royalty contract is licensing income maximising for sufficiently concentrated downstream industries. As the size of the
downstream market increases, the innovation has to be of a certain size for a per-unit royalty
contract to be optimal. What distinguishes Proposition 8 from Proposition 7 is the fact that
for any N > 3 a royalty contract never maximises licensing revenues. How may this result be
rationalised? Notice that for lM = N the innovation is small. Consequently, the capacity effect,
which may render a per-unit royalty contract the licensing revenue maximising option, is insignificant. Also, as N increases, π P,f,M increases at a very fast rate (lM = N ). Taken together,
this implies that for lM = N and N > 3 the patentee optimally offers a fixed-fee contract to the
downstream industry.
Proposition 9 summarises the results so far.
Proposition 9 In a vertical oligopoly an outside patentee optimally transfers a process innovation to the downstream industry by means of a per-unit royalty contract. This result applies
provided the cost reduction embodied in the innovation is sufficiently large. For insignificant cost
reductions together with a sufficiently competitive downstream market offering a per-unit royalty
contract is never optimal. In other words, the superiority of a per-unit royalty over a fixed-fee
contract in terms of licensing revenues depends on the presence of a sufficiently strong capacity
effect.
Partial Capacity Constraints
The previous results were derived under the assumption of a full capacity constraint. This
assumption may bee seen as an extreme restriction. That is why the present section investigates
the case in which a strict subset of upstream firms faces capacity constraints. Put differently,
we assume that k ∈ (0, M ).
In the following we will argue that the superiority of a fixed-fee over a per-unit royalty has
the flavour of a knife-edge result. The smallest introduction of capacity constraints may reverse
the classic result and yield larger licensing revenues under a per-unit royalty contract. As before
we start with a duopoly example and then move to an oligopolistic industry.
Duopoly Example
It has been stated repeatedly throughout this paper that the patentee’s royalty licensing income is unaffected by the introduction of capacity constraints. Hence, Lemma 3 applies and
summarises the SPNE of the per-unit royalty licensing game.
The outcome of the fixed-fee licensing game for a general k is summarised in (12). Depending
on the size of the cost reduction, the patentee’s licensing income amounts to πlP,f,k where
πlP,f,k

2θ2 (1 − λk )( 4A + 1 − 3λk )
2
9θ
2
=
θ2 (1 − λk )( 4A + 1 − λk )
2
with λk =
2(3−k)+1
3(3−k)
9
2
at M = N = 2.
13
θ≤
θ∈
8(3−k)A
12(3−k)+15 ,
8(3−k)A
4(3−k)A
( 12(3−k)+15
, 3(2(3−k)+1)
)
(12)
Proof The proof of (12) is equivalent to the one of (10) or (11).
Consider now the integer case of k = 1, meaning the case in which one of the two upstream
firms is unable to expand its capacities. It is easy to show that a per-unit royalty contract still
dominates a fixed-fee policy in terms of licensing revenues. This indicates that no full capacity
constraint is required for the previously derived results to hold (qualitatively). In fact, at the
largest magnitude of the innovation under which licensing to both downstream firms is optimal,
i.e. at θ =
8(3−k)A
12(3−k)+15 ,
π P,r,k ≥ π2P,f,k for any k ≥ 0.079. The result illustrates nicely that in
the absence of capacity constraints a fixed-fee dominates a per-unit royalty contract, while the
slightest introduction of capacity constraints reverses the result.
Oligopolistic Industries
The previous example indicates that full capacity constraints may not be necessary for our results
to hold. To illustrate this point further, let us assume that π P,r,M > π P,f,M and π P,r,0 < π P,f,0 ,
meaning that a per-unit royalty contract is optimal for a fully constrained upstream market,
while a fixed-fee contract maximises licensing revenues in the absence of capacity constraints.
Then, the fact that the patentee’s fixed-fee licensing income is strictly decreasing in k implies
∗
∗
the existence of a unique k ∗ , with k ∗ < M , such that π P,r,k = π P,f,k . Or in other words, a fully
constraint upstream market may not be required for our previously derived results to apply.
Let us thus focus on the other extreme case of k = 1 in which only one of the upstream firms
is unable to expand its level of production. Here we state the result qualitatively, the full result
is derived in the appendix (Section B).
First off, the general intuition still applies. For a sufficiently strong capacity effect, which,
depending on the size of the downstream industry translates into a lower bound on the efficiency
parameter, the patentee optimally offers a per-unit royalty contract. However, there is a new
element. The superiority of a per-unit royalty over a fixed-fee policy necessitates an upper
bound on the size of the upstream industry. This result is intuitive when we take into account
that the capacity effect works via an upward shift in the intermediary input price. Under
a full capacity constraint the upstream market is unable to accommodate the expansion in
downstream production levels, but reacts to the increased downstream efficiency by increasing
the input price. When only a part of the upstream industry is capacity constrained, this implies
that a sufficiently large unconstrained segment may counteract this effect; as M increases for
a given k we approach the unconstrained solution in which a fixed-fee contract is likely to be
optimal.
Proposition 10 Assume that a strict subset of the upstream market is capacity constrained.
Then a per-unit royalty provides larger licensing revenues than a fixed-fee contract provided the
capacity effect embodied in the innovation is sufficiently large. For a general number of capacity
constrained upstream firms a sufficiently strong capacity effect not only translates into a lower
bound on the efficiency parameter, but also in an upper bound on the size of the upstream market.
That is, a full capacity constraint is not a necessary conditions for the patentee to prefer a perunit royalty over a fixed-fee contract.
14
To end this section we want to point to one other result which underlines the strength of the
capacity effect in rendering a per-unit royalty the preferred licensing contract. Assume that we
are in a situation in which none of the upstream firms faces restrictions regarding its production
levels. Assume further that the patentee strictly maximises licensing revenues by offering a
fixed-fee contract. That is to say π P,f,0 > π P,r,0 . Starting from such a situation it can then
be shown (Section C) that the introduction of a single capacity constrained upstream firm may
be sufficient to reverse this result in favour of a per-unit royalty based scheme. This gives the
superiority of fixed-fee contracts in terms of licensing revenues the flavour of a knife-edge result.
4.2
Welfare Implications
In this section we briefly discuss some welfare implications of the privately optimal licensing
contract. We begin with the case of a full capacity constraint (k = M ) and then address the
case of a single capacity constrained upstream firm (k = 1).
n
Proposition 11 Denote by CSpre (Πm
pre , Πpre ) the consumer surplus (producer surplus) in the
pre-licensing equilibrium. It can then be shown that for k = M
• consumers are indifferent to the type of licensing contract offered; CSroyalty = CSfMee =
CSpre ,7
m
• upstream firms strictly prefer a fixed-fee over a per-unit royalty contract; Πm,M
f ee > Πroyalty =
Πm
pre ,
• downstream firms strictly prefer a per-unit royalty contract over a fixed-fee contract; Πnpre =
> Πn,nl,M
.
Πnroyalty > Πn,l,M
f ee
f ee
Proposition 11 illustrates that upstream and downstream firms generally have conflicting
interests regarding the optimal, surplus maximising, licensing contract. While upstream firms
prefer a fixed-fee policy, downstream firms are better off under a per-unit royalty contract. What
is more, the patentee’s private incentives are aligned with those of the firms in the licensed
industry whenever a royalty contract is privately optimal.
To determine the implications of the chosen licensing policy in terms of aggregate welfare, we
define the latter as the sum of aggregate producer surplus and licensing revenues. Proposition
12 summarises the result.
Proposition 12 Assume that k = M . Then, whenever a per-unit royalty contract maximises
licensing revenues, such a contract is also optimal in terms of economic welfare. For k = 1,
a privately optimal licensing contract is never optimal in terms of aggregate welfare whenever
M > 1 and l1 = N or l1 = L1 . For l1 = l∗,1 the result is ambiguous.
Proposition 12 has two main implications. First, for k = 1, the welfare results are close to the
ones of the unconstrained scenario (k = 0). In the absence of capacity constraints offering a perunit royalty contract to the downstream industry is never optimal in terms of aggregate welfare.
7
This follows from our definition of capacity constraints. See also Lemma 2.
15
This result is mainly driven by consumer surplus. Consumers strictly prefer a fixed-fee over a
per-unit royalty contract for any value of k.8 Second, although the patentee’s private incentives
appear to be robust to a reduction in the number of capacity constrained upstream firms, a
similar conclusion cannot be drawn for aggregate welfare. Under a full capacity constraint
the patentee’s choice of a per-unit royalty contract maximises economic welfare. Conversely,
under a partial capacity constraint, one is likely to observe a conflict between private and social
incentives.
5
A Two-Period Model
At an earlier point, more precisely when introducing the vertical Cournot oligopoly in Section 3,
we mentioned that this model may equally be seen as part of a two-period framework with different supply elasticities across periods. Indeed, it appears intuitive that (some of the) upstream
firms may eventually overcome their output restrictions. Coming back to our introductory example of rare earth elements, this view appears to be in line with the assessment of the U.S.
Department of Energy, which in its 2011 report concludes that ”several clean technologies [. . . ]
use materials at risk of supply disruptions in the short term. Those risks will generally decrease
in the medium and long terms” (U.S. Department of Energy (2011), p.3).
We take this observation as motivation to discuss the implications of dynamic aspects for
the design of the optimal, long run licensing contract and economic welfare. For simplicity we
consider a two-period setting and assume that in the first period, t = 1, all upstream firms
face capacity constraints (k1 = M ), while in the second period, t = 2, none of the upstream
players does (k2 = 0). For intermediate values of kt our results are shifted in favour of a fixed-fee
contract (k1 < M ) or a per-unit royalty contract (k2 > 0).
Regarding the licensing policy, we impose that the type of licensing contract cannot be
renegotiated in the second period. That is, the patentee either offers a per-unit royalty or
a fixed-fee contract for both periods. This assumption appears reasonable given that a) the
licensees always prefer a per-unit royalty over a fixed-fee contract and b) the patentee is likely
to prefer a per-unit royalty contract in t = 1, but a fixed-fee contract in t = 2. Nevertheless,
in case of a fixed-fee contract, the second period fee as well as the second period number of
licensing contracts may be adjusted to the market conditions.9 The fixed-fee may consequently
be seen as e.g. a per-period rental fee. Here, we focus on symmetric long run equilibria, i.e. on
l1 = N and l2 = N , l1 = l∗,M and l2 = l∗,0 or l1 = LM and l2 = L0 . Finally, we do not make
any assumptions regarding the patentee’s preferred licensing strategy in the absence of capacity
constraints. Given that k2 = 0, this would bias our results in favour of either policy.
8
The associated expansion of industry output leads to a drop in the final good price and by this to an increase
in consumer surplus.
9
If the fixed-fee and the number of licensing contracts cannot be renegotiated, the two-period comparison of
the two-licensing schemes is equivalent to the analysis of a full capacity constraint in Section 4.1.
16
The (privately) optimal long run licensing contract
Knowing that in the future upstream firms may overcome their capacity constraints may provide the patentee with incentives to offer fixed-fee, instead of per-unit royalty, contracts. By
comparing the patentee’s aggregate fixed-fee licensing revenues, i.e., π p,f,M + π P,f,0 , with the
corresponding royalty licensing income, i.e., π P,r,M + π P,r,0 = 2π P,r,M = 2π P,r,0 , one can derive
conditions under which the patentee prefers either policy.
We derive a result, which, in terms of its basic intuition, is highly similar to our previous
findings, however, with one important qualification. As such, it should by now not come as a
surprise that a per-unit royalty contract yields larger licensing revenues than a fixed-fee contract
given a sufficiently strong capacity effect. Here, the latter translates into an upper bound on M ,
together with a lower bound on the efficiency parameter for intermediate and minor innovations.
Hence, the discussion we provided in the case of a partial capacity constraints carries over to
the current scenario. However, in contrast to the previous analysis, the result only applies for a
sufficiently concentrated downstream market. As soon as the downstream market is composed
of at least five firms (three for minor innovations) a fixed-fee contract becomes optimal. Put
differently, for sufficiently concentrated target industries, per-unit royalty licensing contracts may
be optimal, not only in the short run, but also in terms of their aggregate long run licensing
revenues.
Welfare Implications
As it is suggested in Proposition 12, the presence of private incentives for offering a particular
type of licensing contract does not guarantee the social desirability of the latter. Also in the
given context there is a potential conflict between the privately and the socially optimal licensing
contract. From Proposition 12 it follows that for k = M a privately optimal per-unit royalty
contract is always optimal in terms of economic welfare, while for k = 0 such a licensing policy
never maximises economic welfare (compared to a fixed-fee). Applied to the present scenario
this implies that although in the short run a privately optimal per-unit royalty contract always
maximises economic welfare, it is never the socially optimal long run policy. In fact, its poor
performance in the long run dominates, and a per-unit royalty scheme rarely maximises economic
welfare across periods.
Proposition 13 For sufficiently concentrated downstream markets a per-unit royalty contract
may be the optimal licensing scheme, not only in terms of its short run, but also in terms of
its aggregate long run licensing revenues. However, per-unit royalties rarely maximise aggregate
welfare across periods. As such, a privately optimal per-unit royalty contract is never optimal in
terms of aggregate welfare for (l1 = N , l2 = N ) and (l1 = LM , l2 = L0 ). For (l1 = l∗,M , l2 = l∗,0 )
the result is ambiguous.
So far we exclusively focused on a static welfare analysis. To end this section we now want
to draw attention to dynamic welfare aspects.
17
Notice that under a per-unit royalty contract, upstream firms are given no incentives to
invest in overcoming capacity constraints. This is due to the fact that per-unit royalties do
not feature any capacity expansion effects. Put differently, under a per-unit royalty contract,
overcoming capacity constraints in t = 1 does not yield any benefits in t = 2 as an upstream
firm’s profit is the same regardless of whether it is capacity constrained or not. In contrast, under
a fixed-fee contract, upstream firms may gain from overcoming restraints to capacity expansion
in the long run. In particular, under a fixed-fee contract, an upstream firm is willing to invest a
m,M −1
m,M
maximal amount of IN Vf ee = πi,uc
− πi,c
in t = 1 in order to face no capacity restrictions
in t = 2. Here we employ a slightly different notation to highlight that in t = 1 firm i is capacity
constrained (c), as are all of its downstream competitors, i.e., k1 = M . In t = 2, firm i faces
no capacity constraints (uc), whereas its downstream competitors do, i.e., k2 = M − 1. This
gives us the maximal amount an upstream firm is willing to invest in order to overcome any
constraints to capacity expansion under a fixed-fee policy. Under a per-unit royalty contract,
m,M −1
m,M
IN Vroyalty = 0 as πi,uc
= πi,c
for a per-unit royalty policy.
A similar discussion applies for the patentee and his incentives to participate in research and
development projects aimed at overcoming upstream capacity constraints. All this is to say that
per-unit royalty contracts may provide an outside patentee with larger licensing revenues, may
it be in the short run, the long run or both. However, at the same time, this type of contract is
likely to be characterised by poor static and dynamic welfare properties which make room for
public policy interventions.
6
Discussion
Despite its vastness, the theoretical licensing literature suffers from shortcomings. Although the
vertical industry structure inherent to most knowledge intensive industries has an important
impact on the design of licensing contracts, licensing considerations in or to such vertically
separated market environments are rarely studied. In this paper we address one complication
outside innovators may face when licensing downstream technology in such market environments.
Namely, the presence of capacity constraints in economically connected upstream industries.
That is, assuming that the licensed downstream market is connected to an upstream industry
via some input-output relationship, we focus on how the optimal downstream licensing contract
depends on the market conditions in the upstream industry. Our results show that it is indeed
crucial to see industries which are the target of a technology transfer not as isolated entities,
but as part of a supply chain (and by this as connected to other industry sectors).
To be more specific, we analyse a vertical Cournot oligopoly with an upstream and a downstream sector. Focus is on the licensing strategy of an outside patentee who licenses a costreducing technology to the downstream market via either a per-unit royalty or a fixed-fee contract. Regarding the upstream industry it is assumed that not all upstream firms are able
to expand their production levels in reaction to the increased degree of downstream efficiency,
which follows the technology transfer. We then show that as a consequence of such upstream
capacity constraints, a per-unit royalty contract may dominate a fixed-fee contract in terms of
18
licensing revenues. Our results are driven by the presence of a sufficiently strong capacity effect
(potential expansion of industry output in the absence of capacity constraints), which reduces
the patentee’s fixed-fee licensing income for a given size of the innovation. Here a sufficiently
strong capacity effect does not imply that all upstream firms operate under capacity constraints.
Instead, our findings reveal that even when only one of the upstream firms is unable to expand
its production levels, the patentee may still maximise licensing revenues by offering a per-unit
royalty contract. Nevertheless, the apparent stability of the patentee’s optimal licensing strategy
to a reduction in the number of capacity constrained upstream firms cannot be replicated in
terms of aggregate welfare. Whereas a privately optimal per-unit royalty contract is all the time
superior to a fixed-fee contract in terms of aggregate welfare for a fully constrained upstream
market, a similar conclusion can rarely be drawn when only one upstream firm is capacity constrained. As a final point we explore the design of the optimal, long run licensing contract in
a two-period setting. It is assumed that the upstream market is fully capacity constrained in
the short run, while capacity constraints are absent in the long run. In this framework, perunit royalty contracts may be optimal only for sufficiently concentrated downstream industries.
However, a privately optimal per-unit royalty contract is rarely optimal in terms of aggregate
welfare across periods.
On a more general note, the primary driver behind our results is the presence of some inelastically supplied input factor which is essential for the licensed industry. This opens the door
for a wide range of applications of this model and its results. This is one of the reasons for
why it is in our eyes important to address some more general implications of our findings. Most
importantly, constraints to output expansion in one part of a supply chain may have important
consequences for the innovative climate in downstream sectors. First, the presence of upstream
capacity constraints reduces the degree of diffusion of the innovation on the downstream market
under a fixed-fee policy. This makes a widespread adoption of innovative technology less likely.
Second, by reducing licensing revenues (or incentives to innovate), upstream capacity constraints
may function as a barrier to efficiency enhancing investments or efforts on downstream markets.
In the same vein, they potentially reduce the innovation incentives of outside innovators regarding downstream technology. Nevertheless, at the same time, capacity constraints may give
greater incentives to outside innovators (or upstream firms) to invest in efficiency enhancing
upstream technology. Related to this point is our finding that fixed-fee contracts may provide
outside innovators or upstream firms with long run incentives to participate in research and
development projects aimed at overcoming upstream capacity constraints.
As a final remark, in this paper we assumed that upstream capacity constraints affect all
downstream firms in a similar fashion. What if this is not the case? Meaning, what if only some
of the downstream firms face capacity constrained suppliers? Assume for simplicity that there
are two downstream firms and one of them (firm 1) has a capacity constrained supplier. Assume
further that the patentee cannot price discriminate between the two firms. Then, the patentee
has three options. First, he may offer a per-unit royalty contract to the entire industry. In that
case, both downstream firms are equally efficient. Second, the patentee may offer a fixed-fee
contract based on the unconstrained firms willingness to pay. In that case, firm 1 does not accept
19
such a licensing contract and its supplier’s capacity constraints do not bind. Firm 2 is thus the
single licensee. Third, the patentee may offer a fixed-fee contract based on the constrained firms
willingness to pay. In that case, both firms are potential licensees. Further, firm 1’s capacity
constraints bind, making it a weaker rival. In such a scenario, what is the patentee’s optimal
licensing contract?
20
A
Graphical Analysis
Graphical analysis for a fully constrained upstream market (k = M ) with M = 10 and A =
a − c = 1.
ΠP
0.7
0.6
0.5
0.4
royalty
fee
0.3
0.6
0.8
1.0
1.2
Θ
Figure 1: lM = LM , N = 2
ΠP
ΠP
0.24
0.18
0.22
0.16
0.20
0.14
0.18
0.16
0.12
royalty
fee
0.14
0.25
0.30
0.35
0.40
royalty
fee
0.10
Θ
0.12 0.14 0.16 0.18 0.20 0.22 0.24
Θ
Figure 3: lM = l∗,M , N = 5
Figure 2: lM = l∗,M , N = 2
ΠP
ΠP
0.14
0.12
0.12
0.10
0.10
0.08
0.08
0.06
0.06
royalty
fee
0.04
royalty
fee
0.04
0.02
0.02
0.05
0.10
0.15
0.20
Θ
0.05
Figure 4: lM = N, N = 2
0.10
0.15
Figure 5: lM = N, N = 2.5
21
0.20
Θ
ΠP
ΠP
0.10
0.08
0.08
0.06
0.06
0.04
0.04
royalty
fee
0.02
0.05
0.10
0.15
Θ
0.02
Figure 6: lM = N, N = 3
B
royalty
fee
0.02
0.04
0.06
0.08
0.10
Θ
Figure 7: lM = N, N = 5
The Optimal Licensing Contract (Partial Capacity Constraint)
Here we assume that a single upstream firms is capacity constrained (i.e., k = 1). Similar to
Section 4 we derive the optimal licensing contract by comparing the patentee’s per-unit royalty
and fixed-fee licensing income. Lemma 14 to Lemma 16 summarise the results of our analysis
for the three relevant subcases. In order to be able to present the results in a clean way, we
restrict our attention to the integer solutions.
Lemma 14 Assume l1 = L1 . Then, π P,r,1 ≥ π P,f,1 for
• N ≥ 2, M ≤ M̄ (N ) and θ ∈ [θ̄1 , θ̄¯1 ).
Here M̄ (N ) =
√
(N +1)(N + 4+N 2 )+2
.
2N 2
Lemma 15 Assume l1 = l∗,1 . Then, π P,r,1 ≥ π P,f,1 for
• N = 2, M = 1 or N = 3, M = 1 and θ ∈ (θ1 , θ̄1 ),
• N = 2, M = 2 or N ≥ 4, M < M̄ (N ) and θ ∈ [θa , θ̄1 ).
Here M̄ (N ) is given by Lemma 14 and
√
2AM 2 N [1+M 2 N 4 +2M N (N 2 +N +1)−2 M N 2 (N +1)(1−M 3 N 4 +M 2 N 2 (N 2 +2)+M N (N 2 +N +3))]
θa =
.
(M +1)(M N 2 −1)(M N (N +2)+1)2
Lemma 16 Assume l1 = N . Then, π P,r,1 ≥ π P,f,1 for
• N = 2, M = 1 and θ ≤ θ1 ,
• N = 3, M = 1 and θ = θ1 .
C
The Optimal Licensing Contract (k = 0 versus k = 1)
In this section we re-do the analysis of Section B, however, under an additional constraint. We
impose that in the absence of capacity constraints (k = 0) the patentee strictly prefers to offer a
fixed-fee contract. Lemma 17 to Lemma 19 thus present the cases in which the patentee prefers
22
a per-unit royalty to a fixed-fee contract for k = 1, but not for k = 0. Those cases correspond to
the scenarios in which a per-unit royalty contract is superior to a fixed-fee contract solely due
to the introduction of a single capacity constrained upstream firm. As in Section B we state the
integer solutions.
Lemma 17 Assume l1 = L1 . Then, π P,r,1 ≥ π P,f,1 for
• N = 2, M = 2 and θ ∈ [θ̄1 , θ̄¯1 ),
• N ≥ 3, M ≤ M̄ (N ) and θ ∈ [θ̄1 , θ̄¯1 ).
¯ ) is given by Lemma 14.
Here M (N
Lemma 18 Assume l1 = l∗,1 . Then, π P,r,1 ≥ π P,f,1 for
• N = 2, M = 1 and θ ∈ (θ1 , θb ),
• N = 3, M = 1 and θ ∈ (θ1 , θ̄1 ),
• N = 2, M = 2 or N ≥ 4, M < M̄ (N ) and θ ∈ [θa , θ̄1 ).
Here M̄ (N ) is given by Lemma 14, θa is given by Lemma 15, and θb = 0.196A.10
Lemma 19 Assume l1 = N . Then, π P,r,1 ≥ π P,f,1 for
• N = 2, M = 1 and θ ≤ θ1 ,
• N = 3, M = 1 and θ = θ1 .
A comparison of Section B and Section C shows that there are indeed cases in which the
patentee prefers a per-unit royalty to a fixed-fee contract in the absence of capacity constraints.11
However, those cases are few and the introduction of a single capacity constrained upstream
firm considerably widens the set of scenarios in which a per-unit royalty contract yields larger
licensing revenues than a fixed-fee contract.
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