Optimal Licensing Contracts
and the Value of a Patent
CAN ERUTKU
Economics Department
York University, Glendon Campus
Toronto (On), Canada
[email protected]
YVES RICHELLE
Département de sciences économiques
Université de Montréal
Montréal (Qc), Canada
[email protected]
We extend Kamien and Tauman’s (1986) analysis of the value of a patent. We
find that an inventor can always design a fixed fee plus royalty contract such
that his revenue is equal to the profit a monopoly endowed with the innovation
could make on the market. This implies that the social value of a patent can
be strictly negative whenever the patented innovation is of bad quality. We
also explain why a principal can have an interest in using performance-based
contracts although the principal and the agents are risk-neutral, information
is symmetric, and agents’ actions are verifiable.
1. Introduction
Because innovation activities are mainly undertaken by private agents,
two issues arise: What are the private incentives to innovate? Do social
incentives to conduct innovation activities exceed private incentives?
To analyze these two questions, we adopt the framework proposed by
Arrow (1962) and extended by Kamien and Tauman (1984, 1986) where
the interactions between an inventor who owns and seeks to license
a patent of a cost-reducing innovation and firms of an oligopolistic
industry are described by a noncooperative three-stage game.1 At the first
The authors thank the editor, a coeditor as well as two anonymous referees for their
valuable and insightful comments and suggestions. We are solely responsible for all
remaining errors.
1. The inventor is not a market participant. This is the case when the inventor is an
independent research lab or a domestic firm licensing its technology to foreign firms that
are active on markets where the domestic firm is absent.
C 2007, The Author(s)
C 2007 Blackwell Publishing
Journal Compilation Journal of Economics & Management Strategy, Volume 16, Number 2, Summer 2007, 407–436
408
Journal of Economics & Management Strategy
stage, the inventor proposes a licensing contract. The contract and the
innovation quality are publicly observable. At the second stage, firms
decide simultaneously to accept or to reject the contract. And, at the
third stage, firms compete à la Cournot knowing the set of licensees. In
this setting, private incentives to innovate are given by the private value
of the patent, which coincides with the inventor’s licensing revenue.
Kamien and Tauman (1986) focus on contracts that specify either a
fixed fee or a royalty. They show how a patent’s private value depends
on the number of firms and on the innovation quality. When royalty
contracts are used, the private value of a patent increases with the
number of firms: the inventor’s incentives to innovate are maximized
when he faces a perfectly competitive industry. If the inventor uses
fixed-fee contracts and the innovation is nondrastic, there may exist
a finite number of firms that maximize a patent’s private value. Kamien
and Tauman (1986) also find that the inventor’s licensing revenue is
larger with fixed-fee contracts and falls short of the profit a monopolist
endowed with the innovation could obtain. The only exception is the
case of perfect competition: by using a royalty or a fixed-fee contract,
the inventor obtains the monopoly profit if the innovation is drastic. In
addition, Kamien and Tauman (1986)’s analysis provides insights on the
comparison between private and social incentives to innovate. Although
the consumers’ surplus always increases with the adoption of the
innovation, the inventor is unable to capture this increase and the social
value of a patent exceeds the private value if all firms become licensees.
If fixed-fee contracts are used and the innovation is of relatively high
quality, the inventor maximizes his revenue by proposing a contract that
is not accepted by all firms. As the nonlicensees’ profit decreases with
the introduction of the innovation, the private value of a patent may
exceed the social one.2
Many variants of this model have been analyzed (see Kamien,
1992 for a survey). Wang (1998) studies the case where the inventor is a
member of the industry whereas Muto (1993) examines the case where
firms compete in price with differentiated goods. Gallini and Wright
(1990) introduce asymmetric information between the inventor and
firms as well as the possibility of imitation. Bousquet et al. (1998) allow
for cost or demand uncertainty. These studies do not alter significantly
Kamien and Tauman (1986)’s conclusions on the private and social value
of a patent. However, they identify situations where, in contrast to
Kamien and Tauman (1986), royalty contracts generate a larger revenue
for the inventor than fixed-fee contracts. Rostoker (1983) finds that 46%
2. A similar conclusion has been obtained by Katz and Shapiro (1986) when licenses
are auctioned.
The Value of a Patent
409
of licensing contracts specify a fixed fee and a royalty whereas 39% of
them rely only on royalties.
In this paper, we extend Kamien and Tauman (1986)’s analysis
by allowing the inventor to design contracts that specify the payment
of a fixed fee plus a royalty. A first attempt in this direction can be
found in Kamien and Tauman (1984) where the analysis is restricted to
linear royalties and no definite results are provided except for the case
of perfect competition. We also develop further and formally the idea
suggested by Shapiro (1985) that licensing has the potential to induce
oligopolistic firms to produce at the monopoly level.3
We obtain the following results. First, for any innovation quality
and for any number of firms in the oligopolistic industry,4 the inventor
can design a fixed fee plus royalty contract such that his revenue equals
the profit a monopolist endowed with the innovation could make on the
market. Thus, the private value of a patent is independent of the number
of firms on the market. Second, there exists a critical level of innovation
quality such that the consumers’ surplus decreases (increases) with
the adoption of the new technology when the innovation quality falls
short of (exceeds) this critical level. The social value of a patent may
thus be smaller than its private value even if it is licensed to all firms.
Moreover, this critical value is increasing with the number of firms. This
implies that when the number of firms tends to infinity any nondrastic
innovation has a private value exceeding its social value. Third, if the
social value of a patent is measured by the sum of consumers’ and
producers’ surpluses and of the inventor’s revenue, the social value
of a patent is strictly negative when the innovation does not lead to
a significant cost reduction. Finally, contracts that allow the inventor
to obtain the monopoly profit, called optimal (from the inventor’s
point of view) licensing contracts, specify the payment of a fixed fee
and royalties. More generally, the paper contributes to contract theory
because we identify situations where, and explain why, a principal
may use performance-based contracts although the principal and the
agents are risk-neutral, information is symmetric, and agents’ actions
are verifiable.
These results are obtained by decomposing the licensor’s revenue
into two components: his benefit and the sum of licensees’ reservation
profits. Any contract leading to a licensor’s benefit equal to the profit a
monopoly using the innovation would obtain and to a sum of licensees’
3. The underlying assumptions behind Shapiro (1985)’s idea are that the licensor participates in the output market, competes against only one rival and shares the monopoly
profit with the licensee.
4. Note that we assume throughout the analysis that there are at least two firms in the
industry.
410
Journal of Economics & Management Strategy
reservation profits equal to zero would be equivalent to our fixed fee
plus royalty contracts. We do not pretend that fixed fee plus royalty
contracts are the only kind of contracts that lead to a licensor’s revenue
equal to the monopoly profit. In particular, this level of revenue could be
obtained by an inventor using a forcing contract à la Holmstrom (1982).
This type of contract specifies that (1) if all firms accept the contract,
the inventor returns to the licensees a large up-front payment when the
monopoly quantity is produced and returns nothing otherwise; (2) if
some but not all firms accept the contract, the inventor induces the
firms to profitably flood the market through the use of a simple upfront payment scheme; and, (3) if some but not all firms accept the
contract, the inventor provides a reward above the Cournot profit to
the licensees.5 Point (2) ensures that the licensees’ reservation profit
equals zero without any incidence on the licensor’s revenue because
the stipulated payments are made on an off-equilibrium path while
point (3) prevents the rejection of the contract by all firms to be part
of an equilibrium. Finally, point (1) allows the licensor to obtain a
revenue equal to the monopoly profit. The major characteristic of this
forcing contract is that it disentangles the maximization of the licensor’s
benefit and the minimization of the licensees’ reservation profits through
conditioning payments between the parties involved in the contract on
the acceptance decision of the whole set of firms. This characteristic is
a necessary condition for a forcing contract to lead a licensor’s revenue
equal to the monopoly profit. Indeed, whenever the licensor wants
to increase his benefit, he must constrain a licensee’s production in
the forcing contract to be smaller than the Cournot quantity. But, this
increases the licensee’s reservation profit because firms that do not
accept the contract produce more than their Cournot quantity and obtain
a larger profit when licensees produce less than their Cournot quantity.6
Our paper shows that conditioning payments on acceptance decisions
taken by firms not directly involved in the contract are not necessary for
the licensor’s revenue to be equal to the monopoly profit whenever fixed
fee plus royalty contracts are considered. Thus, fixed fee plus royalty
contracts are less demanding than forcing contracts.
The paper is organized as follows. Section 2 presents the model.
In Section 3, we begin the analysis by considering fixed fee plus linear
royalty contracts. Such contracts generate licensing revenues equal to
the monopoly profit whenever the innovation is of sufficient quality.
The initial focus on relatively simple contracts helps in understanding
how to construct optimal licensing contracts. Section 4 is devoted to the
5. We thank the coeditor for the precise construction of this contract.
6. A formal proof of this argument is given in Erutku and Richelle (2000).
411
The Value of a Patent
analysis of the existence of optimal contracts for any kind of innovation
and to the derivation of our conclusions on the private and social value
of a patent. In Section 5, we relate our findings with some existing works
in contract theory. Section 6 concludes. Proofs of Section 3 are provided
in the Appendix.
2. The Model
An inventor owns a patent on an innovation that reduces the constant
marginal cost to produce some good from c > 0 to c − , with 0 < ≤ c.
The market where the good is being produced consists of a finite set
N = {1, . . . , n} of firms,
with n ≥ 2, that face an inverse demand function
P(Q), where Q = h∈N q h stands for total output and qh denotes firm h’s
output. We suppose that P(Q) = a − Q, with a > c, whenever a > Q and
P(Q) = 0 otherwise. Using Arrow (1962)’s definition, an innovation is
said to be drastic if the price a monopolist using the innovation would
set, PM = [a + (c − )]/2, is smaller than or equal to the competitive price
under the old technology c; an innovation is said to be nondrastic if
PM > c. With our functional forms, an innovation is drastic whenever
(a − c)/ ≤ 1.
Interactions between firms and the inventor or licensor are described by a three-stage game. At the first stage, the licensor offers a
licensing contract consisting of an up-front fixed fee α and a royalty
scheme τ (Q, q). Royalties paid by firm h may depend on firm h’s output
and total output Q. This would be the case if royalties were based
on firm h’s sales revenues P(Q)qh .7 Royalties are not restricted to be
positive. Contracts are costlessly enforceable by courts and include a
clause that prohibits a licensee from reselling the license.8 At the second
stage, firms observe the cost reduction allowed by the innovation and
the proposed contract (α, τ ). They then decide simultaneously to accept
or to reject the contract and the fixed fee is paid by licensees. The set
of firms is therefore partioned into two subsets: the set of licensees L
and the set of nonlicensees N\L. Let i and j denote an element of L and
N\L, respectively. At the third stage, firms observe the set of licensees
before they decide simultaneously how much to produce. After their
production has been sold on the market, licensees pay the appropriate
7. For instance, Mortimer (2004) states that revenue-sharing contracts consisting of an
upfront fee per unit of inventory and a revenue split paid on the basis of rental revenue,
P(Q)qh in our notation, started to be adopted in the late 1990s between video stores and
movie distributors.
8. This is innocuous since we shall prove that there always exists a subgame perfect
equilibrium where the licensor proposes a contract that is accepted by all firms along the
equilibrium path.
412
Journal of Economics & Management Strategy
royalties. Firm i’s profit is [P(Q) − c + ]qi −τ (Q, qi ) − α, firm j’s profit
is [P(Q) − c]qj and the licensor’s revenue is i∈L [α + τ (Q, q i )].
A contract proposed at a subgame-perfect equilibrium (SPE) of the
game is referred to as an equilibrium contract. An equilibrium contract
is said to be optimal (from the licensor’s point of view) if it leads to
a revenue of M for the licensor where M = maxQ [P(Q) − c + ]Q is
the profit a monopolist with a marginal cost of c − would achieve
on the market.9 An optimal contract is said to be strongly optimal if
the subgame that follows the proposal of this contract has a unique
equilibrium outcome. One of our goals is to show that there exists a
contract (α, τ ) that is strongly optimal for almost all values of the model
parameters.
We proceed in two steps. In the next section, we restrict the set
of contracts to those specifying a fixed fee and a linear royalty and
we characterize the equilibrium contract. This is done for two reasons.
First, while such contracts are commonly observed, their performance
has not been theoretically assessed except for the case of a perfectly
competitive market. Second, the analysis of such contracts facilitates the
understanding of how to construct optimal contracts as they are strongly
optimal when the innovation is sufficiently drastic. The existence of
optimal contracts for any kind of innovation is analyzed in Section 4.
3. Fixed Fee Plus Linear Royalty Contracts
We suppose in this section that the licensor offers a contract specifying a
fixed fee α and a per-unit of output royalty τ (Q, qh ) = ρqh for all (Q, qh ) ∈
R2+ . The royalty rate ρ is such that c − + ρ > 0 and ρ < because, as
we shall see, this latter constraint is never binding.10
To find an equilibrium contract, we must first analyze the last
two stages of the game. At the third stage, each firm, knowing the
characteristics of the contract proposed at the first stage and the set of
licensees, chooses a quantity qh ∈ [0, +∞) to maximize its profit. Letting
l = |L| and K = (a − c)/, quantities at the unique Cournot equilibrium
are
9. Because the monopoly profit is the largest industry profit that can be achieved on
the market, our optimality criterion is more demanding than the one adopted by Kamien,
Oren, and Tauman (1992).
10. If the licensor were to propose a contract with a fixed fee α and a royalty rate
ρ < − c ≤ 0, then it would make a loss: the marginal production cost of a licensee would
become strictly negative and all firms would accept the contract and produce a quantity
strictly greater than −α/ρ.
413
The Value of a Patent

K + (n + 1 − l)( − ρ)



n+1
q iC (l, ρ) =

K + ( − ρ)


l +1

 K − l( − ρ)
n+1
q Cj (l, ρ) =

0
if l( − ρ) ≤ K
∀i ∈ L
(1)
if l( − ρ) ≥ K
if l( − ρ) ≤ K
∀ j ∈ N\L .
(2)
if l( − ρ) ≥ K
Firm h’s Cournot equilibrium profit, πhC (l, ρ), is equal to [qCh (l, ρ)]2 ∀h ∈ N
with πiC (l, ρ) representing a licensee’s Cournot equilibrium profit gross
of the fixed fee.
At the second stage, firms simultaneously choose to accept or to
reject the contract. We suppose that when a firm obtains the same profit
by accepting or rejecting the contract, it accepts the contract. Assume that
l − 1 firms have accepted the contract and n − l firms have rejected it.
The remaining firm accepts the contract if its profit as a licensee is greater
than or equal to its profit as a nonlicensee, that is, if α ≤ πiC (l, ρ) − πjC (l −
1, ρ). Similarly, when l firms have accepted the contract while n − l − 1
firms have rejected it, the remaining firm rejects the contract if its profit
as a nonlicensee is strictly greater than its profit as a licensee, i.e., if
α > πiC (l + 1, ρ) − πjC (l, ρ). Defining w(l, ρ) as
w(l, ρ) = πiC (l, ρ) − π Cj (l − 1, ρ),
(3)
and using (1) and (2) we obtain the following claim.
Claim 1: For all ρ ∈ [ − c, ) and for all l ∈ {1, . . . , n − 1}, w(l + 1, ρ) <
w(l, ρ). If ρ = then w(l, ) = 0 for all l ∈ {1, . . . , n}.
At an equilibrium of the subgame starting after the proposal of
some contract (α, ρ), l̂ ∈ [1, n − 1] firms accept the contract if w(l̂ +
1, ρ) < α ≤ w(l̂, ρ). Claim 1 implies that there exists at most one l̂
satisfying w(l̂ + 1, ρ) < α ≤ w(l̂, ρ) for any contract (α, ρ). Furthermore,
it implies that all firms become licensees if α ≤ w(n, ρ). However, all
firms remain unlicensed if α > w(1, ρ). In addition, Claim 1 ensures
that the subgame starting after the proposal of any contract (α, ρ) has
a unique equilibrium outcome. Denoting by l̂(α, ρ) the equilibrium
number of firms that accept the contract (α, ρ), it then follows that


n if w(n, ρ) ≥ α
l̂(α, ρ) = l̂ if w(l̂ + 1, ρ) < α ≤ w(l̂, ρ)
(4)


0 if w(1, ρ) < α.
414
Journal of Economics & Management Strategy
We can now look at the choice of a contract by the licensor. An
equilibrium fixed fee and linear royalty contract (α ∗ , ρ ∗ ) maximizes the
licensor’s revenue when
(α ∗ , ρ ∗ ) ∈ arg max l̂(α, ρ) α + ρq iC (l̂(α, ρ), ρ) .
(5)
α,ρ
Proposition 1: First, suppose that K ≤ (n2 − 1)/(n2 + 1). Then, any
equilibrium contract (α ∗ , ρ ∗ ) is such that α ∗ = w(l0 , ρ ∗ ) and ρ ∗ =
that belongs to
[(K
+ 1)(l0 − 1)]/2l0 with l0 being any integer
2
[ (K + 1)/(1 − K ), n]. Second, suppose that K ≥ (n − 1)/(n2 + 1). Then,
the equilibrium contract is unique and given by α ∗ = w(n, ρ ∗ ) and

2

 (n − 1)(2n − 1 − K ) if K ≤ 2c(n − n + 1) − (n + 1)
2(n2 − n + 1)
(n − 1)
ρ∗ =
(6)

 − c
otherwise.
Kamien and Tauman (1986) suggest that to any royalty based
contract, there exists a fixed fee contract that generates a larger revenue.
Proposition 1 shows that the equilibrium contract stipulates a royalty
rate different from zero except when K = 2n − 1 [see (6)].
Corollary 1: Let K = 2n − 1. To any fixed fee only contract (α, 0), there
exists a fixed fee plus linear royalty contract (α, ρ) that leads to a strictly greater
licensor’s revenue.
To develop the intuition underlying our results, note that
max l̂(α, ρ) α + ρq iC (l̂(α, ρ), ρ) = max l w(l, ρ) + ρq iC (l, ρ)
α,ρ
l,ρ
(7)
with w(l, ρ) given by (3). The right-hand side of (7) can be written as
l w(l, ρ) + ρq iC (l, ρ) = B(l, ρ) − lπ Cj (l − 1, ρ),
(8)
where B(l, ρ) stands for the licensor’s benefit and consists of the sum
of licensees’ profits and royalties paid by licensees while lπjC (l − 1, ρ) is
the sum of firms’ reservation profits. The equilibrium contract must be
such that the number of firms that accept the contract and the royalty
rate maximize the difference between B(l, ρ) and lπjC (l − 1, ρ).
Let us start by looking at the licensor’s benefit which can be written
as
B(l, ρ) = P QCN (l, ρ) − c + QCL (l, ρ),
(9)
where
QCN (l, ρ) = QCL (l, ρ) + QCN\L (l, ρ), QCL (l, ρ) = i∈L qCi (l, ρ)
and
C
QN\L (l, ρ) = j∈N\L qCj (l, ρ) with qCi and qCj given by (1) and (2). From
The Value of a Patent
415
(9), the highest benefit the licensor can obtain is the monopoly profit
M = [P(QM ) − c + ]QM , where Q M = arg max Q [P(Q) − c + ]Q is
the monopoly quantity. The monopoly profit can be obtained by
choosing (l, ρ) such that (i) the sum of nonlicensees’ Cournot quantities
QCN\L (l, ρ) = 0, and (ii) the sum of licensees’ Cournot quantities
QCL (l, ρ) = QM . Requirement (i) can be satisfied in two ways. First, and
using (2), we can choose (l, ρ) such that l( − ρ) ≥ K. This would imply
that a nonlicensee’s Cournot equilibrium quantity is equal to zero.
Second, we can choose l = n so that QCN\L = 0 for any ρ. Accordingly, the
condition QCN\L = 0 does not impose any restriction on the choice of the
royalty rate but requires that the chosen number of licensees be greater
than or equal to the minimum between n and K/( − ρ). Hence, for
l ≥ min{n, K/( − ρ)}, we can examine the condition on ρ for the sum
of licensees’ Cournot quantities to be equal to QM . Whenever l ≥ 2, the
royalty rate such that QCL (l, ρ) = QM must be strictly positive. Indeed,
the sum of Cournot quantities is strictly greater than the monopoly
quantity for a zero royalty rate and is strictly decreasing in ρ. Denoting
by ρ M (l), the royalty rate such that QCL (l, ρ) = QM with QCN\L = 0, we
have
ρ M (l) =
(K + 1)(l − 1)
.
2l
(10)
To sum up, the licensor’s benefit equals the monopoly profit if and only
if (l, ρ) is such that l ≥ min{n, K/( − ρ)} and ρ = ρ M (l). The use of a
strictly positive royalty rate comes from the licensor’s desire to reduce
the Cournot quantity to the monopoly quantity.
We can turn to the reservation profit πjC (l − 1, ρ), that is, a
nonlicensee’s Cournot profit when l − 1 firms are licensees. Because
[qCj (l, ρ)]2 = πjC (l, ρ), we have from (2) that the reservation profit is
decreasing in l and increasing in ρ. Also, πjC (l − 1, ρ) = 0 if and only
if (l, ρ) is such that (l − 1)( − ρ) ≥ K. Thus, if there exists an integer
l0 ≤ n such that
(l0 − 1) − ρ M (l0 ) ≥ K
(11)
then B(l0 , ρ M (l0 )) is equal to the monopoly profit. Indeed, any l0 ≤ n
satisfying (11) is greater than or equal to min{n, K/( − ρ M (l0 ))} and
πjC (l0 − 1, ρ M (l0 )) = 0. Consequently, whenever there exists l0 ≤ n satisfying (11), the contract (w(l0 , ρ M (l0 )), ρ M (l0 )) is strongly optimal, while if
(l − 1)( − ρ M (l)) < K for all l ≤ n, then no fixed fee plus linear royalty
contract is optimal. From (10) and (11), it is possible to verify that such
l0 exists if and only if K ≤ (n2 − 1)/(n2 + 1), that is if and only if the
innovation is sufficiently drastic
416
Journal of Economics & Management Strategy
Proposition 2: (i) For all K ≤ (n2 − 1)/(n2 + 1), any equilibrium contract (α ∗ , ρ ∗ ) given in Proposition 1 is strongly optimal. (ii) If K > (n2 − 1)/
(n2 + 1), then no fixed fee plus linear royalty contract is optimal.
Whenever K > (n2 − 1)/(n2 + 1), the equilibrium royalty rate results from a compromise between the licensor’s desire to keep his benefit
as close as possible to the monopoly profit by choosing a large royalty
rate and his desire to keep firms’ reservation profits as close as possible
to zero by choosing a small (or even negative) royalty rate.
Corollary 2: Let K > (n2 − 1)/(n2 + 1) and let (α ∗ , ρ ∗ ) be the equilibrium contract given in Proposition 1. Then, B(n, ρ ∗ ) < M and πjC (n −
1, ρ ∗ ) > 0.11
Corollary 3: Letting l∗ be the equilibrium number of licensees corresponding to an equilibrium contract (α ∗ , ρ ∗ ) given in Proposition 1, where
l ∗ = l̂(α ∗ , ρ ∗ ) with l̂(α, ρ) given in (4), we have l∗ ≥ 2.
In comparison to what is obtained with fixed fee only contracts
(see Kamien and Tauman, 1986), licensing a drastic innovation does not
result in market monopolization. Our decomposition of the licensor’s
revenue in terms of benefit and reservation profit helps in providing
some intuition. From the definition of a drastic innovation, the licensor’s
benefit is equal to the monopoly profit if the licensor proposes a contract
(ᾱ, 0) with ᾱ such that only one firm accepts the contract. In this case,
a nonlicensee’s best-response to the monopoly quantity in the Cournot
game is to produce nothing, so that the licensee’s Cournot equilibrium
quantity coincides with QM . But the reservation profit of the sole licensee
is equal to the Cournot equilibrium profit in a market where no firm has
bought the license and is strictly positive when the number of firms
is finite. Because any contract that is accepted by only one firm gives
the licensor a revenue strictly less than the monopoly profit, then an
optimal contract cannot lead to market monopolization. By proposing
a contract that is accepted by at least two firms, the licensor is able to
decrease the reservation profit of any firm as well as the sum of the
licensees’ reservation profits.12 The licensor is also able to control the
licensees’ Cournot quantity by choosing an appropriate royalty rate.
We saw in Proposition 2 that the innovator can find a strongly optimal
contract if the innovation is sufficiently drastic.
When the market is perfectly competitive, the licensor’s benefit
cannot be maximized for a royalty rate ρ > while, for any (l, ρ) such
that ρ ≤ and l ≥ 1, the sum of the reservation profits tends to zero as
11. Note that, for K > (n2 − 1)/(n2 + 1), the contract (α ∗ , ρ ∗ ) is accepted by all firms.
C
2
12. From (2), l[qC
j (l, ρ)] = lπj (l, ρ) is decreasing in l whenever ρ is not too large.
The Value of a Patent
417
n → ∞. Because the sum of reservation profits becomes independent
of the specifics of the proposed contract, any contract maximizing the
licensor’s benefit is an equilibrium contract and we obtain Kamien and
Tauman (1984)’s result.13
Corollary 4: Suppose that n → +∞. There exists an equilibrium fixed
fee only contract that gives to the licensor a revenue equal to the one he obtains
by proposing any equilibrium contract (α ∗ , ρ ∗ ) specified in Proposition 1.14
Before we examine the existence of optimal licensing contracts
for any kind of innovation, two remarks are in order. The first one
refers to the first part of Proposition 1 where the royalty rate in the
optimal contract is increasing with respect to the innovation quality.
This positive correlation is also a conclusion of the analysis of licensing
in a setting with asymmetric information and imitation. In such a case,
the licensor wants to signal the quality of his innovation but is limited by
the possibility of imitation by the licensee who can decide not to use the
innovation. Gallini and Wright (1990) show that the high-quality licensor
can signal his type by offering a separating equilibrium contract that
specifies output-based royalties and a fixed fee, while the low-quality
licensor offers a fixed fee only contract.
The second remark refers to the licensor’s revenue in (8) which is
similar to the principal’s objective function in a standard relationship
between a principal and multiple agents with moral hazard and contractible individual effort levels. There is, however, a crucial difference
between the licensor’s problem and that of the principal. In the standard
principal-agent framework, agents’ reservation utility level is constant,
while, here, firms’ reservation profits depend on the number of licensees
and on the specifics of the contract. Thus, it is no longer true, as is the
case in a standard principal-agent setting, that the principal can always
extract the whole surplus by using a contract specifying an appropriate
linear output-based payment.
4. Optimal Licensing Contracts and the Value
of a Patent
In Section 3, we identified conditions for a licensing contract to be
optimal. We showed that a fixed fee plus linear royalty contract can
13. This explains why no optimal contract exists and the equilibrium contract specifies
a fixed fee only when the licensor faces a monopolist. Because the monopolist’s reservation
profit is a strictly positive constant, the best the licensor can do is to maximize his benefit
B by proposing an appropriate fixed fee only contract.
14. As the careful reader will note, an equilibrium contract (α ∗ , ρ ∗ ) specified in
Proposition 1 does not lead to a fixed fee only contract whenever n → +∞. This comes
from the fact that the proof of Proposition 1 relies explicitly on the assumption that n is
finite.
418
Journal of Economics & Management Strategy
be strongly optimal when the innovation is sufficiently drastic. Here,
we show that whatever the number of firms, the demand level, and the
cost reduction allowed by the innovation, the licensor can always find
an optimal contract. Thus, the private value of a patent is the profit a
monopolist makes by using the innovation. We shall then be able to
compare the private and social value of a patent.
4.1 Optimal Licensing Contracts
Let us consider situations where K > (n2 − 1)/(n2 + 1). From our decomposition of the licensor’s revenue in equation (8), the following
conditions are necessary and sufficient for a contract (α o , τ o ) to be
optimal at the corresponding Cournot equilibrium: (i) when lo firms have
of licensees’ quantities must be
accepted the contract (α o , τ o ), the sum equal to the monopoly quantity, that is, i∈L o q iC (l o ; τ o ) = Q M , and each
nonlicensee must produce nothing, that is, qCj (lo ; τ o ) = 0 for all j ∈ N\Lo ;
(ii) when lo − 1 firms have accepted the contract (α o , τ o ), no nonlicensee
must find it profitable to produce a strictly positive quantity, that is,
qCj (lo − 1; τ o ) = 0; and, (iii) when lo − 1 firms have accepted the contract,
it must be profitable for another firm to accept it, while, if the contract
has been accepted by lo firms, it must not be profitable for another
firm to accept it, that is, α o = w(lo ; τ o ) > w(lo + 1; τ o ), where w(l; τ o ) =
πiC (l; τ o ) − πjC (l − 1; τ o ) and πhC (l; τ o ) stands for firm h’s Cournot equilibrium profit when l firms have accepted the contract (α o , τ o ). Condition (i)
ensures that the licensor’s benefit equals the monopoly profit; condition
(ii) ensures that each firm’s reservation profit equals zero; and, condition
(iii) ensures that the equilibrium number of licensees equals lo .
We begin with contracts specifying sliding scale per-unit of output
royalties such as τ (Q, q) = [ρ − µq]q with ρ > 0 and µ > 0. With such
contracts, the first-order condition (FOC) by a licensee and nonlicensee
are respectively
(K + 1) −
q h − ρ − (1 − 2µ)q i ≤ 0 with equality if q i > 0
(12)
h∈N
K −
q h − q j ≤ 0 with equality if q j > 0.
(13)
h∈N
From (12), if any licensee produces a strictly positive quantity at the
Cournot equilibrium, then all licensees produce the same quantity
qCi (l; τ ). Thus, the first part of condition (i) which requires that qCi (l; τ ) =
QM /l = (K + 1)/2l ∀i ∈ L is verified when ρ and µ satisfy
(K + 1)
(14)
[l − 1 + 2µ] − ρ = 0.
2l
419
The Value of a Patent
Condition (ii) requires that qCj (l − 1; τ ) = 0. From (13), this is the
case if (l − 1)qCi (l − 1; τ ) ≥ K. Hence, qCj (l − 1; τ ) = 0 if and only if ρ and
µ are such that
2µ < l
(15)
− ρ − (1 − 2µ)
K
≥ 0.
l −1
(16)
Using (14) and (16), the first part of condition (i) and condition (ii)
are met if
(K + 1)(l − 1 + 2µ)
2l
l
µ ∈ µ̂(l),
,
2
ρ=
(17)
(18)
with
µ̂(l) =
K (l 2 + 1) − (l 2 − 1)
.
2[K (l + 1) − (l − 1)]
(19)
Remark that 0 < µ̂(l) < l/2 for all K > (n2 − 1)/(n2 + 1) and all l ≥ 2.
For simplicity, let us then consider a contract (α o , τ o ) such that
o
α = w(lo ; τ o ) and τ o = [ρ o − µo q]q with µo = µ̂(l) and ρ o = ρ̂(l o ) where
ρ̂(l) =
(K + 1)[l − 1 + 2µ̂(l)]
(K + 1)[l K − (l − 1)]
=
.
2l
K (l + 1) − (l − 1)
(20)
Such contract is optimal if the second part of condition (i) and condition
(iii) are satisfied. This is true for drastic innovations because QM ≥ K
and, from the definition of τ o , lo qCj (lo ; τ o ) = QM ≥ K so that qCj (lo ; τ o ) = 0
as required by the second part of condition (i). Furthermore, condition
(iii) holds for all lo ∈ {2, . . . , n}.
Proposition 3: Let lo ∈ {2, . . . , n}. (A) If K ∈ ((n2 − 1)/(n2 + 1), 1), then
the contract (α o , τ o ), where α o = w(lo ; τ o ) and τ o (Q, q) = [ρ o − µo q]q, with
ρ o = ρ̂(l o ) and µo = µ̂(l o ), is strongly optimal. (B) If K = 1 then the contract
(α o , τ o ), where α o = w(lo ; τ o ) and τ o (Q, q) = [ρ o − µo q]q, with ρ o = and
µo = 1/2, is optimal.
Note that when K = 1, the contract (α o , τ o ) is an optimal one. This
is due to the fact that when µo = 1/2 and ρ o = , the FOC by a licensee,
equation (12), is satisfied for any (q i )i∈L o such that i∈L o q i = K . Hence,
in this case, there exists an infinity of Cournot equilibria where any
licensee produces a strictly positive quantity.
420
Journal of Economics & Management Strategy
Consider now nondrastic innovations. A first difference with the
case of drastic innovations is that the equilibrium number of licensee lo
must be equal to the number of firms n for the second part of condition
(i) to be satisfied.15 A second difference is that we cannot find an
optimal contract (α, τ ) with τ = [ρ − µq]q for (K, n) such that K > 1 and
n > 1 + 2K /(K − 1) since µ̂(n) > 1. From (18), conditions (i) and (ii)
are satisfied only if µ > 1. However, for such values of µ the profit of
a licensee is no longer concave with respect to qi and the FOC by a
licensee is no longer sufficient.16 A third difference is that µ̂(n) > 1/2
for all K > 1. This entails that there exists a Cournot equilibrium at
which some licensee does not produce for any contract (α o , τ o ) such
that l o = n, τ o = [ρ o − µo q ]q , ρ o = ρ̂(n) and µo = µ̂(n).17 To eliminate
this kind of equilibrium, the licensor can slightly modify the contract
by imposing a penalty f > 0 as small as he wants if the licensee does
not produce. A last difference with the case of a drastic innovation is the
possibility that, when n − 1 firms accept the contract, the corresponding
game played at the third stage has more than one Cournot equilibrium.
However, this is the case only when K = (n + 1)/(n − 1).18 With these
remarks, we can state the following proposition.
Proposition 4: Let ρ o = ρ̂(n) and µo = µ̂(n) with ρ̂ and µ̂ given by
(20) and (19), respectively. Let (α o , τ o ) be given by α o = w(n; τ o ), τ o (Q, q) =
o
[ρ o − µo q]q if q >
0 and τ (Q, q) = f > 0 if q = 0. (A) For allo (K,o n) such that
K > 1, n < 1 + 2K /(K − 1) and K = (n + 1)/(n − 1), (α , τ ) is strongly
optimal. (B) For all (K, n) such that K > 1, and either n = 1 + 2K /(K − 1)
or K = (n + 1)/(n − 1), (α o , τ o ) is optimal.
√
Note that 1 + 2K /(K − 1) → 1 + 2 > 2 when K → +∞ and is strictly
decreasing in K.
Corollary 5: Let K > 1 and n = 2. If K = 3, then the contract given in
Proposition 4 is strongly optimal, while this contract is optimal if K = 3.
C
M
M
15. Indeed, if lqC
i (l; τ ) = Q then qj (l; τ ) > 0 because K > Q = (K + 1)/2 when K >
1.
16. From (12), the second-order condition for profit maximization is −2(1 − µ) < 0
which requires µ < 1.
17. If we let q0i = [(K + 1) − ρ o ]/(n − 2µo ) for i = 1, . . . , n − 1 and q0n = 0, then the
FOC is satisfied for all i = 1, . . . , n − 1. With qi = q0i for i = 1, . . . , n − 1, the FOC of the
last firm is [1 − 2µo ][(K + 1) − ρ o ]/(n − 2µo ) − 2(1 − µo )qn < 0 for all qn when µo > 1/2.
Thus, ((q0i )n−1
i=1 , 0) is a Cournot equilibrium.
18. Let ρ o = ρ̂(n) and µo = µ̂(n). Then, (12), (13), and (16) all hold with equality. This
implies that any licensee produces the same quantity at a Cournot equilibrium with qC
i
K
such that [n + 1 − 4µ̂(n)][ n−1
− q iC ] = 0. If n − 1 firms accept the contract, the Cournot
equilibrium is unique if and only if 4µ̂(n) = n + 1 which holds, using (19), when K =
(n + 1)/(n − 1). If K = (n + 1)/(n − 1), the Cournot game played by the (n − 1) licensees
and the nonlicensee has a continuum of equilibrium outcomes.
The Value of a Patent
421
The next question to examine is the form taken by optimal contracts
whenever n ≥ 3 and K is relatively large. A natural way to extend our
previous analysis is to consider more sophisticated per-unit of output
royalty scheme. Indeed, we have the following.19
Proposition 5: Let T be the set of functions τ : R2+ → R that (a) are
continuously differentiable on R2++ , and (b) satisfy limq→0 [∂τ/∂Q + ∂τ/∂q] =
γ , with γ ∈ R, for all Q ∈ R+ . For any K > 1 there exists n̄(K ) < +∞ such
that if n ≥ n̄ and (α o , τ o ) is an optimal contract then τ o ∈ T .
This result implies that if (α o , τ o ) is an optimal contract for n → +∞
then τ o cannot be a polynomial that depends only on a single firm’s
output. In other words, optimal contracts must specify, in some cases,
royalties that are based on the industry output Q, as is the case when
royalties are based on a firm’s sales P(Q)q. As computations become
rapidly complex, we shall not try to find a contract form approximating
some observed contracts but report only the following existence result.20
Proposition
6: Let (K, n) be such that K > 1, n ≥ 3, and n ≥ 1 +
2K /(K − 1), that is, let n ≥ 3 and K ≥ (n − 1)2 /[(n − 1)2 − 2]. (A) Let
(α o , τ o ) with α o = w(n; τ o ) and τ o = [ρ o + µo q − ν o Q]q if q > 0 and τ o =
f > 0 if q = 0. For any K > (n + 1)/(n − 1), there exist ρ o > 0, µo > 0, and
ν o > 0 such that the contract (α o , τ o ) is strongly optimal. If K = (n + 1)/(n −
1), there exist ρ o > 0, µo > 0, and ν o > 0 such that the contract (α o , τ o ) is
optimal. (B) Let (α o , τ o ) be given by α o = w(n; τ o ) and
 o
o
o

[ρ + µ q − ν Q]q if q > 0 and Q ≥ Q̄,
if q > 0 and Q < Q̄,
τ o (Q, q ) = [σ o + µo q − Q]q


f >0
if q = 0.
For all K < (n + 1)/(n − 1), there exist ρ o > 0, µo > 0, ν o > 0, σ o > 0, and
Q̄ < Q M such that (α o , τ o ) is strongly optimal.
4.2 The Value of a Patent
A consequence of the previous results is as follows.
Proposition 7:
[(a − c + )/2]2 .
A patent’s private value equals the monopoly profit M =
19. See Erutku and Richelle (2000) for a proof.
20. The interested reader can find a proof of this Proposition in Erutku and Richelle
(2000). Note that the observation that a combination of output-based royalties and salesbased royalties can lead to a larger revenue for the licensor than either output-based
or sales-based royalties has already been made by Bousquet et al. (1998) but in a quite
different context.
422
Journal of Economics & Management Strategy
The licensor’s revenue is a strictly increasing and strictly convex
function in the innovation quality. Also, as long as the market is not
monopolized, that is, n ≥ 2, the private value of a patent does not depend
on the number of firms. This contrasts with Kamien and Tauman (1986)
where there is a most profitable size of the industry for the licensor when
fixed-fee only contracts are used. In our analysis, the number of firms
has an impact on the optimal contract’s characteristics. For a nondrastic
innovation, optimal contracts must specify a royalty based on sales if
the number of firms is large although, if the number of firms is small,
optimal contracts can use a royalty based only on a firm’s individual
output.
Another implication of Proposition 7 is that the licensor has no
interest in entering the industry either by becoming an additional
producer or by merging with an existing firm because he is able to
obtain the largest industry profit by designing optimal contracts.21
Furthermore, even if the innovation allows for an infinitesimal cost
reduction, the licensor’s revenue exceeds the profit a monopoly would
make without the innovation.
Corollary 6: For any > 0, the private value of a patent is strictly greater
than [(a − c)/2]2 and is therefore strictly positive.
This differs from Kamien and Tauman (1986)’s findings under the
assumption that the licensor can use either fixed fee only or linear
output-based royalty contracts. In these cases, the licensor’s revenue
tends to zero as the cost reduction allowed by the innovation tends
to zero. This can shed some doubts on our analysis. However, this
phenomenon is quite intuitive. Indeed, when the licensor can use fixed
fee plus royalty contracts, he is able to manipulate a licensee’s marginal
cost of whatever the quality of his innovation. The licensor can therefore
design a contract such that, for each firm, “to accept the contract”
provides a larger profit than “to reject the contract.” But if all firms accept
the contract, they obtain a profit strictly less (actually a zero profit) than
the one they would make if no firm accepts it. Hence, the contract places
firms in a Prisoner’s Dilemma and the licensor can extract the whole
industry profit whatever the cost reduction allowed by his innovation.22
We can now turn to the social value of a patent measured by
the sum of consumers’ and producers’ surpluses and of the licensor’s
revenue. Our analysis shows that the social value of a patent may be
strictly negative. Indeed, an innovation that reduces firms’ marginal
cost by an infinitesimal amount leads firms to produce the monopoly
21. This stands in contrast to Shapiro (1985).
22. We shall pursue the discussion on this particular feature in the next section.
The Value of a Patent
423
quantity with almost the same marginal cost as before the innovation.
It is as if the market shifts from oligopoly to monopoly which entails
a fall in total surplus. For innovations of better quality, the consumers’
surplus can still decrease with the adoption of the innovation so that the
social value of the innovation falls short of the private one.
Proposition 8: The social value of a patent, as measured by the sum of
consumers’ and producers’ surpluses and of the licensor’s revenue, is
(i) strictly negative for all < ¯ ,
(ii) strictly smaller than the patent’s private value if < (a − c)(n − 1)/(n +
1),
(iii) greater than or equal to the patent’s private value if ≥ (a − c)(n −
1)/(n + 1),
strictly increasing with respect the market size a and the number of firms n,
and strictly decreasing in the level of the pre-innovation marginal cost c, and
where ¯ > 0.
According to conventional wisdom, the social value of a patent
is larger than the private one. Our analysis points otherwise. Thus,
innovation activities should not be subsidized per-se when the sum
of surpluses is an appropriate measure of the social value of a patent.
In addition, the kind of innovation for which the social value exceeds
the private one depends on the market demand and on the number of
firms. One particular case needs to be noted.
Corollary 7: Let n → +∞ and let the social value of a patent be measured
by the sum of consumers’ and producers’ surpluses and of the licensor’s revenue.
Then the social value of a patent is strictly smaller than its private value if and
only if the innovation is nondrastic.
Finally, the possibility for an innovation to have a negative social
value justifies the common practice of patent systems granting patents
only for innovations that lead to significant improvements of the current
technology.23 The analysis also suggests that the criteria used by patent
systems should ideally depend on the characteristics of the market to
which the innovation is addressed. For instance, the larger and more
competitive a market, the better the innovation quality should be for
this innovation to be patented.
23. For instance, in the US Code, one condition for patentability is stated as follows: “A
patent may not be obtained though the invention is not identically disclosed or described
as set forth in section 102 of this title, if the differences between the subject matter sought
to be patented and the prior art are such that the subject matter as a whole would have
been obvious at the time the invention was made to a person having ordinary skill in the
art to which said subject matter pertains” (US Code, Title 35, Section 103).
424
Journal of Economics & Management Strategy
5. Related Literature
We showed how the licensor’s revenue can be decomposed in terms of
benefit and firms’ reservation profits. This implies that our analysis can
be recast to belong to the class of contracting games with externalities
developed in Segal (1999). In these games, the principal (the licensor)
offers contracts to n agents (the firms); the principal’s profit can be
written as the difference between total surplus (the licensor’s benefit)
and the sum of agents’ reservation utilities (firms’ reservation profits)
which depend on all agents’ trades with the principal. A contract in Segal
(1999) consists of a trade xi from the principal to agent i and a monetary
transfer ti from agent i to the principal. He shows that total surplus is
not maximized when contracts are public. Contracting distortions come
from the dependence of agents’ reservation utilities on other agents’
trade with the principal: contracts offered by the principal result from
an arbitrage between total surplus and the sum of agents’ reservation
utilities.
In our framework, the trade between the principal and an agent
corresponds to the technological transfer from the licensor to a firm with
xi = 1 if such transfer takes place and xi = 0 otherwise. In Segal (1999)’s
set-up, a license contract corresponds to a contract (xi , ti ) with
xi = 1
and the number of licensees l coincides with the sum of trades i∈N xi .
Thus, contracts in Segal (1999) are analogous to fixed fee only licensing
contracts.
As total surplus in Segal (1999) stands for the licensor’s benefit in
our model, contracting distortions arise whenever the licensor cannot
design an optimal contract such that his revenue equals the monopoly
profit. Our analysis reveals that fixed fee only contracts cannot be
optimal and that optimal contracts exist for all value of the model
parameters. Thus, if contractual inefficiencies arise, they do not come
from the presence of externalities, as Segal’s analysis implies, but from
the restrictions on the kind of contracts that can be offered by the
principal. When more general contracts are allowed, agents’ reservation
utilities depend on the principal’s trade with other agents and also
on the specifics of the contracts proposed by the principal to other
agents. The principal can exploit this dependence to his advantage by
designing optimal contracts so that contractual efficiency is restored.
This means that contractual inefficiencies obtained in many applications
cited in Segal (1999) should disappear if performance-based contracts
are allowed.
Nevertheless, situations exist where there are no contractual inefficiencies in Segal’s (1999) framework. Rasmusen et al. (1991) and its
“correction” by Segal and Whinston (2000a) are one example. They show
The Value of a Patent
425
that an upstream supplier can achieve the monopoly profit by deterring the entry of an equally efficient rival through exclusive contracts
when the manufacturing process is subject to economies of scale and
when there is a lack of coordination between buyers. Buyers, who can
be seen as end-customers, firms of a perfectly competitive industry or
local monopolies with no monopsony power, must decide to accept or
to reject the exclusionary agreement proposed by the incumbent. The
agreement specifies a bonus, that takes the form of a fixed fee, to any
buyer who signs it. After observing the number of buyers that have
been locked-in by the incumbent, the entrant decides to enter or not.
If enough buyers sign the agreement, entry is deterred as the entrant
cannot achieve the necessary economies of scale to become competitive
and buyers who have not signed the agreement face a monopolist. By
signing the exclusive contract, a buyer imposes a negative externality
on other buyers.24 There exists an equilibrium where all buyers sign the
exclusionary agreement even if they must pay the monopoly price and
if the bonus offered by the incumbent tends to zero. As such, buyers
are put in a Prisoner’s Dilemma: each accepts the incumbent’s contract
while they would have all been better off by refusing it.
In our analysis, firms also face a Prisoner’s Dilemma. Indeed,
the licensor’s revenue tends to [(a − c)/2]2 whenever the reduction in
marginal cost allowed by his innovation tends to zero. Thus, even if the
licensor does not own an innovation, it is able to collect the monopoly
profit and to drive firms’ profit to zero. To do so, the licensor proposes
contracts with a royalty scheme that artificially reduces the marginal
cost of a licensee. Although each firm has an interest in accepting the
contract, each firm makes a zero profit if all firms accept it.
Because this result may look strange, let us illustrate it from another
point of view. Suppose that firms engage in a research joint venture (JV)
which disseminates the results of its activities through license contracts
to its owners, the firms. Even if the research project is a complete failure,
firms can use the JV as a device to ensure that market competition leads
to the cooperative outcome.25 This points to a close relationship between
our framework and the “common agency” one studied in Bernheim and
Whinston (1985, 1986). Bernheim and Whinston (1985) examine why
24. Note that although each buyer’s reservation utility depends on other buyers’ trade
with the incumbent, it does not depend on the characteristics of the other buyers.
25. Naturally, such a JV could raise antitrust concerns. If the licensing scheme used by
the JV, that is, the ancillary restraint imposed upon licensees, is not deemed reasonably
necessary to the accomplishment of the JV’s efficiency enhancing purpose and, in addition,
leads to prices set at the cooperative level, then it could be challenged by antitrust
authorities (see Werden, 1998). This would be particularly true if the JV results in the
creation of a sham innovation. Under antitrust laws, JV’s participants have the burden of
showing the type and size of efficiencies that can be achieved through the JV.
426
Journal of Economics & Management Strategy
firms voluntarily delegate control over certain aspects of marketing
to common agents and why such agents are typically compensated
on the basis of sales rather than fee-for-service. They show that there
exists a noncooperative equilibrium where, by choosing to deal with
a common agent, firms can ensure that all strategic variables are set
at their cooperative level. At this equilibrium, agent’s compensation
scheme consists of a fee and a commission based on sales, and the agent
(expected) revenue equals zero, although firms obtain the (expected)
monopoly profit. It is not surprising that this can also be the case in our
set-up because it could be viewed as a “common principal” one.
6. Concluding Remarks
We examine the revenue a licensor can obtain from licensing his innovation to an oligopolistic industry through fixed fee plus royalty contracts.
Whatever the number of firms and the innovation quality, the licensor
is able to design contracts such that his revenue is equal to the profit a
monopoly endowed with the innovation could achieve on the market.
This result can lead, in turn, to situations where the private value of a
patent exceeds its social value, that is, total surplus can decrease after
the adoption of the cost reducing innovation.
We obtained our findings by decomposing the licensor’s revenue
into his benefit and the sum of licensees’ reservation profit. When the
licensing contract specifies royalties and a fixed fee, the reservation
profit of a licensee depends on the number of licensees and on the
royalty scheme’s characteristics. The licensor can therefore manipulate
licensees’ reservation profit to its own interest and contractual inefficiencies disappear. Thus, our work points out that Segal (1999)’s contractual
inefficiency result can disappear when more general contracts (than
fixed fee only) are allowed and when multiple strategically competing
agents are present.
Appendix
A.1 Proof of Proposition 1
The proof of the first part of Proposition 1 has been given in the text and
we shall suppose in the sequel that K ≥ (n2 − 1)/(n2 + 1). Let us define
R(l, ρ) as
R(l, ρ) = l w(l, ρ) + ρq iC (l, ρ)
(A1)
with w(l, ρ) given in (3). (α ∗ , ρ ∗ ) is an equilibrium contract if and only if
α ∗ = w(l∗ , ρ ∗ ) and
427
The Value of a Patent
(l ∗ , ρ ∗ ) = arg max R(l, ρ) subject to l ∈ N and − c ≤ ρ ≤ .
l,ρ
To prove part 2, we must show that l∗ = n and ρ ∗ = arg maxρ
{R(n, ρ) subject to − c ≤ ρ ≤ } with ρ ∗ given by (6). Using (1) and
(2), R(n, ρ) writes as
R(n, ρ)

n


[(K + 1) − ρ][(K + 1) + nρ]
if ρ ≤ −



(n + 1)2




n( − ρ)
=
[ K (n − 1) − ( − ρ)(n2 − n + 1)]
2

(n
+
1)





n


[ K + ( − ρ)]
if ρ ≥ −
 +
(n + 1)
K
n−1
K
.
n−1
(A2)
It is then possible to verify the following claim.
Claim 2: Let K ≥ (n2 − 1)/(n2 + 1). ρ ∗ = arg maxρ {R(n, ρ) subject to
− c ≤ ρ ≤ } if and only if ρ ∗ is given by (6).
To begin with, we shall prove the following:
Lemma 1: Let K ≥ (n2 − 1)/(n2 + 1). Then, maxρ {R(1, ρ) subject to −
c ≤ ρ ≤ } is strictly smaller than R(n, ρ ∗ ).
Proof. Using (1) and (2), R(1, ρ) writes as
R(1, ρ)

1
K 2



[(K
+
1)
−
ρ][(K
+
1)
+
ρ]
−
 4
n+1
= 
−ρ



[(n − 1) K − n( − ρ)] +
(n + 1)2
n+1
if ρ ≤ (1 − K )
if ρ ≥ (1 − K ).
(A3)
Remark that R(1, ρ) is continuously differentiable with respect to ρ.
Letting ρ 1 stand for arg maxρ {R(1, ρ) subject to − c ≤ ρ ≤ }, we can
obtain the following claim
Claim 3: Let K ≥ (n2 − 1)/(n2 + 1). We have

0
if K ≤ 1




max{(1 − K ), − c}
if 1 ≤ K ≤ n
ρ1 =

(n
−
1)(K
+
n)


max −
,−c
if K ≥ n.
2n
(A4)
428
Journal of Economics & Management Strategy
Now let (1, n) = R(n, ρ ∗ ) − R(1, ρ 1 ). We shall show that (1, n) <
0 for all K ≥ (n2 − 1)/(n2 + 1). Consider first that K ≤ 1. We have that
2
∂(1, n)
n(n − 1)(K − 2n + 1)
(n − 1)
−K
(3
+
n)
+
(3n
+
1)
+
=
∂K
2(n + 1)2
n2 − n + 1
∂ 2 (1, n)
n(n − 1)2
2
(1
−
n)(3
+
n)
+
=
.
∂K2
2(n + 1)2
n2 − n + 1
It can be verified that ∂ 2 (1, n)/∂K2 is strictly negative so that
∂(1, n)
2
∂(1, n) = 2
≥
> 0.
∂K
∂K
n −n+1
K =1
It follows that (1, n) < 0 for all K ∈ [(n2 − 1)/(n2 + 1), 1] because, for
K = (n2 − 1)/(n2 + 1), R(n, ρ ∗ ) is equal to the monopoly profit 2 (K +
1)2 /4 > R(1, 0) = 2 (K + 1)2 /4 − 2 K2 /(n + 1)2 . Suppose then that K ∈
[1, n]. (1, n) ≥ R(n, ρ ∗ ) − R(1, (1 − K)) with ρ ∗ > − c because K ≤
˜
n ≤ 2n − 1 (see (6)). Let us denote R(n, ρ ∗ ) − R(1, (1 − K)) by (1,
n).
We have
˜
∂ (1,
n)
2
=
∂K
2(n + 1)2 (n2 − n + 1)
× {K [4(n2 − n + 1) + n(n − 1)2 ] − (n + 1)[n2 − n + 2]}
˜
∂ 2 (1,
n)
2
n(n − 1]2
=
2
+
.
∂K2
2(n + 1)2 (n2 − n + 1)
2(n2 − n + 1)
˜
∂ 2 (1,
n)/∂ K 2 is thus strictly positive and we obtain that
2
˜
˜
∂ (1,
n)
n2
(n − 1)
∂ (1,
n) =
≥
− 1 > 0.
∂K
∂ K K =1
(n + 1)2
n2 − n + 1
˜
Consequently, (1,
n) > 0, because it is strictly increasing in K and,
˜
for K = 1, (1, n) = (1, n) > 0. Therefore, (1, n) > 0 for all K ∈ [1, n]
˜
since (1, n) ≥ (1,
n).
Suppose now that K ≥ n. From (6) and (24) we have that ρ ∗ > ρ 1
when ρ ∗ > − c so that if ρ ∗ = − c then ρ 1 = − c. To prove that
(1, n) > 0 for all K ≥ n, we distinguish between two cases: the case
where ρ ∗ > − c and the case where ρ ∗ = ρ 1 = − c. Consider first
def.
¯
the case where ρ ∗ > − c. We have that (1, n) ≥ (1,
n) = R(n, ρ ∗ ) −
R(1, −(n − 1)(K + n)/2n). We obtain that
429
The Value of a Patent
¯
n[(n + 1) + (n − 1)K ]
2 (n − 1)
∂ (1,
n)
(n + 1) +
=
∂K
(n + 1)2
2(n2 − n + 1)
(n − 1)K + n(n + 1)
−
2n
¯
∂ 2 (1,
n)
2 (n − 1)3
=
> 0.
∂K2
2n(n + 1)2 (n2 − n + 1)
Accordingly, we find
¯
¯
∂ (1,
n)
∂ (1,
n) ≥
∂K
∂ K K =n
2
2 (n − 1)n
=
n(n + 1) + 2(n2 − n + 1) > 0.
2
2
2(n + 1) (n − n + 1)
¯
˜
Because we have (1,
n) = (1,
n) > 0 for K = n, then (1, n) > 0 for
all K ≥ n such that ρ ∗ > − c. Consider now the case where ρ ∗ = −
c, i.e, the case where K ≥ [2c(n2 − n + 1) − (n + 1)]/(n − 1). We have
(1, n) = R(n, − c) − R(1, − c). Accordingly,
(n − 1) (1, n) =
c[ K (n − 1) − cn2 ] + 2 K (n + 1)
2
(n + 1)
(n − 1)
c 2 (n2 − 2n + 2)
>
(n + 1)2
(n + 1)
2
+
[c(2n − 3n + 3) − (n + 1)] .
n−1
However, ≤ c so that
(n − 1)
(1, n) >
{c 2 (n2 − 2n + 1) + 2(n2 − 1)} > 0.
(n + 1)2
The proof of the lemma is therefore complete.
Using (1) and (2), the licensor’s revenue for 2 ≤ l ≤ n − 1 is

K


R
(l,
ρ)
if
ρ
≤
1
−

ll


l −1




K
K
R(l, ρ) = Rnl (l, ρ) if 1 −
≤ρ ≤ 1−
(A5)

l −1
l





K


,
 Rnn (l, ρ) if ρ ≥ 1 −
l
430
where
Journal of Economics & Management Strategy
l
Rll (l, ρ) =
[(K + 1) − ρ] [(K + 1) + lρ]
(l + 1)2
Rnl (l, ρ) = Rll (l, ρ) −
l
[ K − (l − 1)( − ρ)]2
(n + 1)2
(A6)
(A7)
l
{( − ρ) [ K (n − 1) − ( − ρ) (l(n − 1) + 1)]
Rnn (l, ρ) =
(n + 1)2
+ (n + 1) [ K + (n + 1 − l)( − ρ)]} .
(A8)
Claim 4: If ρ ≥ [1 − K/(n − 1)], Rnl (n, ρ) = Rnn (n, ρ) = R(n, ρ) with
R(n, ρ) given by (A2).
Let denote {(l, ρ) | 2 ≤ l ≤ n, − c ≤ ρ ≤ } and let us define
∗∗
∗∗
R∗∗
,
R
ll
nl , and Rnn as
Rll∗∗ = max {Rll (l, ρ) s. t. l ≤ n − 1, ρ ≤ [1 − K /(l − 1)]}
(l,ρ)∈
(A9)
∗∗
Rnl
= max {Rnl (l, ρ) s. t. l ≤ n − 1, [1 − K /(l − 1)]
(l,ρ)∈
≤ ρ ≤ [1 − K /l]}
∗∗
Rnn
= max {Rnn (l, ρ) s. t. l ≤ n − 1, ρ ≥ [1 − K /l]} .
(l,ρ)∈
(A10)
(A11)
We shall then immediately state the following lemma
∗∗
Lemma 2: Let K ≥ (n2 − 1)/(n2 + 1). Then R∗∗
ll < Rnl .
Proof.
From (A6), we have
∂ Rll (l, ρ)
l
=
[(K + 1)(l − 1) − 2lρ]
∂ρ
(l + 1)2
2
∂ 2 Rll (l, ρ)
l
=
−2
.
∂ρ 2
l +1
(A12)
It follows that, for all (l, ρ) ∈ such that ρ ≤ [1 − K/(l − 1)] and K ≥
(n2 − 1)/(n2 + 1),
∂ Rll (l, ρ)
∂ Rll (l, ρ) =
≥
K (l 2 + 1) − (l 2 − 1) > 0.
∂ρ
∂ρ
l
−
1
ρ=[1−K /(l−1)]
431
The Value of a Patent
Using (A7), we can then verify that, for all l ∈ {2, . . . , n − 1},
∂ Rnl (l, ρ) ∂ Rll (l, ρ) =
> 0.
∂ρ
∂ρ
ρ=[1−K /(l−1)]
ρ=[1−K /(l−1)]
(A13)
We therefore obtain
∗∗
Rll∗∗ = max Rll (l, [1 − K /(l − 1)]) = max Rnl (l, [1 − K /(l − 1)]) < Rnl
.
l
l
We shall then complete the proof of Proposition 1 by stating the
following.
∗∗
Lemma 3: Let K ≥ (n2 − 1)/(n2 + 1). Then R∗∗
nl and Rnn are both strictly
smaller than maxρ {R(n, ρ) subject to − c ≤ ρ ≤ }.
Proof. To simplify the presentation, we shall treat hereafter the number
¯ and
of licensees, l, as a continuous variable. We begin by defining R̃nl (l)
¯
¯
R̃nn (l), for l ∈ [2, n], as
¯ = max {Rnl (l, ρ) s.t. l ≤ l,
¯ [1 − K /(l − 1)] ≤ ρ ≤ [1 − K /l]}
R̃nl (l)
(l,ρ)∈
(A14)
¯ = max {Rnn (l, ρ) s.t. l ≤ l,
¯ ρ ≥ [1 − K /l]}.
R̃nn (l)
(l,ρ)∈
(A15)
∗∗
∗∗
We have Rnl
≤ R̃ll (n − 1) and Rnl
≤ R̃ll (n − 1). To prove the lemma we
shall prove that the constraint l ≤ l̄ in the maximization problem in
(A14) and (A15) is binding. This has two implications. First, it leads to
R̃nl (n − 1) < R̃nl (n) and R̃nn (n − 1) < R̃nn (n). Second, using Claim 4
R̃nl (n) = max {Rnl (n, ρ) s.t. [1 − K /(n − 1)] ≤ ρ ≤ [1 − K /n]}
ρ∈[−c,]
= max {R(n, ρ) s.t. [1 − K /(n − 1)] ≤ ρ ≤ [1 − K /n]}
ρ∈[−c,]
≤ max R(n, ρ)
ρ∈[−c,]
R̃nn (n) = max {Rnn (n, ρ) s.t. ρ ≥ [1 − K /n]}
ρ∈[−c,]
= max {R(n, ρ) s.t. ρ ≥ [1 − K /n]}
ρ∈[−c,]
≤ max R(n, ρ).
ρ∈[−c,]
1. Let us begin by considering the FOC of the maximization problem
given in (A14). From (A13) and the requirement that ρ ≤ [1 − K/n],
432
Journal of Economics & Management Strategy
we know that the constraints ρ ≥ [1 − K /(n − 1)] and ρ ≤ are nonbinding so that these FOC can be written as
∂ Rnl
+ ν − µl = 0
∂ρ
(A16)
∂ Rnl
+ µ( − ρ) + γ − λ = 0
∂l
(A17)
ν[ − c − ρ] = 0,
µ[ρ − (1 − K /n)] = 0,
¯ = 0.
γ [2 − l] = 0 λ[l − l]
From (A7), we obtain
∂ Rnl
∂ Rll
2l(l − 1)
=
−
[ K − (l − 1)( − ρ)]
∂ρ
∂ρ
(n + 1)2
∂ Rnl
∂ Rll
1
=
−
∂l
∂l
(n + 1)2
(A18)
(A19)
(A20)
× [ K − (l − 1)( − ρ)] [ K − (3l − 1)( − ρ)] .
Then, using (A6) and (A12), it is possible to verify that
∂ Rll
1
∂ Rll
=−
[ K + ( − ρ)]
.
∂l
l(l + 1)
∂ρ
(A21)
We can next introduce (A21) and (A19) into (A20) to obtain
∂ Rnl
1
∂ Rnl
l
=−
[ K + ( − ρ)]
−
∂l
l(l + 1)
∂ρ
(n + 1)2 (l + 1)
× [ K − (l − 1)( − ρ)][2 K − (3l + 1)( − ρ)]
∂ Rnl
∂ Rnl
l
+ ( − ρ)
∂l
∂ρ
K − l( − ρ) ∂ Rnl
l2
=−
−
l +1
∂ρ
(n + 1)2 (l + 1)
(A22)
× [ K − (l − 1)( − ρ)][2 K − (3l + 1)( − ρ)].
(A23)
Remark that ρ ≤ [1 − K/l] implies 2K − (3l + 1)( − ρ) < 0. Accordingly, (A22) leads to
∂ Rnl (l, ρ)
∂ Rnl (l, ρ)
> 0 for all (l, ρ) such that
≤ 0.
∂l
∂ρ
(A24)
433
The Value of a Patent
We also have that
k
∂ Rnl
∂ Rnl
+ ( − ρ)
> for all (l, ρ) such that
∂l
∂ρ
ρ = [1 − K /l].
(A25)
Suppose now that the solution of the maximization problem in (A14)
is µ = 0. From (A16)
∂ Rnl
= −ν ≤ 0.
∂ρ
This implies, using (A17) and (A24), that λ > 0. If the solution of the
maximization problem given in (A14) is such that µ > 0 then ρ =
[1 − K/l]. Multiplying (A16) and (A17) by ( − ρ) and l respectively
and summing the resulting equalities, we obtain
∂ Rnl
∂ Rnl
lλ = l
+ ( − ρ)
+ lγ + ( − ρ)ν.
∂l
∂ρ
From (A24), we know that the term in brackets of the right-hand
side (RHS) is strictly positive while the other terms of the RHS are
nonnegative. It follows that λ > 0.
2. Consider the maximization problem in (A15). Before writing the FOC
of this problem, let us compute and make some observations about
the partial derivatives of Rnn . From (A8)
∂ Rnn
l
{ K (n − 1) − 2( − ρ)[l(n − 1) + 1]
=−
∂ρ
(n + 1)2
+ (n + 1)(n + 1 − l)}
(A26)
∂ Rnn
1
{( − ρ)[ K (n − 1) − ( − ρ)[2l(n − 1) + 1]]
=
∂l
(n + 1)2
+ (n + 1)[ K + ( − ρ)(n + 1 − l)]}.
(A27)
Accordingly, we have
∂ 2 Rnn ∂ρ 2 < 0
∂ Rnn /∂ρ < 0 for all ρ ≥ and all l ∈ [2, n].
(A28)
434
Journal of Economics & Management Strategy
Moreover, (A26) and (A27) lead to
l
∂ Rnn
∂ Rnn
+ ( − ρ)
∂l
∂ρ
l
=
[( − ρ)2 + (n + 1)[ K − l( − ρ)]]
(n + 1)2
> 0 for all (l, ρ) such that ρ ≥ [1 − K /l].
(A29)
Let us now consider the FOC of the maximization problem in (A15).
We have
∂ Rnn
(A30)
+ lµ − ν1 + ν0 = 0
∂ρ
∂ Rnn
− ( − ρ)µ − λ + γ = 0
∂l
ν0 [ − c − ρ] = 0, ν1 [ρ − ] = 0, µ[(1 − K /l) − ρ] = 0
γ [2 − l] = 0, λ[l − l̄] = 0.
(A31)
(A32)
(A33)
Remark that if, at the solution, we have ρ = then ν0 = 0, µ = 0 and
∂Rnn /∂ρ = ν1 . But this contradicts (A28) and hence, at the solution,
we have ρ < and ν1 = 0. Multiplying (A30) and (A31) by ( − ρ)
and l, respectively, and summing the resulting equations, we have
lλ = l
∂ Rnn
∂ Rnn
+ ( − ρ)
+ ( − ρ)ν0 + lγ .
∂l
∂ρ
(A34)
Hence, using (A29), λ must be strictly positive at a solution of
the maximization problem. The proof of the lemma is therefore
complete.
A.2 Proof of Proposition 3
(A) To prove Proposition 3-(A), it remains to show that, for all K ∈
((n2 − 1)/(n2 + 1), 1), the subgame that starts after the proposition of
the contract (α o , τ o ) has a unique equilibrium. We proceed in two steps.
We first show that, whatever the number of firms that accept the contract
(α o , τ o ) at the second stage of the game, the Cournot game played at the
third stage has a unique equilibrium. We then prove that, at the second
stage, the equilibrium number of firms that accept the contract (α o , τ o ), lo ,
is unique. To prove the uniqueness of the Cournot equilibrium for
any number of licensee, we use Gaudet and Salant (1991)’s uniqueness
result. Given the specifities of our model, only Assumption 4 in Gaudet
and Salant (1991) must be checked and a proof that this assumption is
satisfied is available upon request. To prove that the number of firms
The Value of a Patent
435
that accept the proposed contract is equal to lo , we use the following
result.
Lemma 4: Let K ∈ ((n2 − 1)/(n2 + 1), 1), let (α o , ρ o ) and lo be given
in Proposition 3 and, for all l ≥ 1, let w(l; τ o ) = πiC (l; τ o ) − πjC (l −
1, τ o ). For all l ∈ {1, . . . , lo − 1}, w(l; τ o ) > w(lo ; τ o ) and, for all l ∈ {lo +
1, . . . , n}, w(l; τ o ) < w(lo ; τ o ).
Proof. Available upon request.
(B) Suppose now that K = 1. In this case, 2µo = 1 and ρ o = . We
can find that,
for any l ∈ {1, . . . , n}, the Cournot equilibrium quantities
satisfy (i) i∈L q i (l; τ o ) = K and qCj (l; τ o ) = 0 for all j ∈ N\L. Suppose
that for any l, the equilibrium with symmetric licensee’s quantities
is played. We have that w(l; τ o ) = (/l)2 /2 for all l ∈ {2, . . . , n} and
w(1; τ o ) = 2 /2 − 2 /(n + 1)2 leading to w(l; τ o ) > w(l + 1; τ o ) for all l ∈
{1, . . . , n}.
A.3 Proof of Proposition 8
(i) Pre-innovation, total surplus is
Q0
(Q0 )2
n−2
[a − Q] d Q +
(A35)
= a−
Q0 Q0 ,
n
2n
0
with Q0 = n(a − c)/(n + 1). Post-innovation, total surplus equals
QM
QM
[a − Q] d Q + [Q M ]2 = a +
(A36)
Q M,
2
0
with QM = (a − c + )/2. Using (A35) and (A36), the social value of an
innovation is strictly negative whenever
<2
(a + Q0 )2 − (n − 1)Q20 n − (a + Q0 ) + (n − 1)Q0 /2n . (A37)
We then denote by ¯ the RHS of (A37). It can be verified that ¯ > 0 and
that
−1/2
∂ ¯
n+1
+ 2 (a + Q0 )2 − (n − 1)Q20 n
=−
∂ Q0
n
× [a + Q0 − (n − 1)Q0 /n] > 0,
−1/2 Q20
∂ ¯
Q0 2
2
> 0,
= 2 + (a + Q0 ) − (n − 1)Q0 n
∂n
n
n2
−1/2
∂ ¯
(a + Q0 ) > 0.
= −2 + 2 (a + Q0 )2 − (n − 1)Q20 n
∂a
Thus ¯ is strictly decreasing with respect to c and is strictly increasing in
n and a. (ii) and (iii). The private value of a patent exceeds its social value
436
Journal of Economics & Management Strategy
when consumers’ surplus decreases with the adoption of the innovation.
This is the case when Q0 > QM , that is, when < (n − 1)(a − c)/(n + 1).
References
Arrow, K., 1962, “Economic Welfare and the Allocation of Ressources for Inventions,” in
R.R. Nelson, ed., The Rate and Direction of Inventive Activity, Princeton, NJ: Princeton
University Press.
Bernheim, B. D. and M. Whinston, 1985, “Common Marketing Agency as a Device for
Falicitating Collusion,” RAND Journal of Economics, 16, 269–281.
—— and ——, 1986, “Common Agency,” Econometrica, LIV, 923–942.
Bousquet, A., H. Cremer, M. Ivaldi, and, M. Wolkovicz, 1998, “Risk Sharing in Licensing,”
International Journal of Industrial Economics, 16, 535–554.
Erutku, C. and Y. Richelle, 2000, “Optimal Licensing Contracts and the Value of a Patent,”
Working Paper 2000-07, Université de Montréal.
Gallini, N. and B. Wright, 1990, “Technology Transfer under Asymmetric Information,”
RAND Journal of Economics, 21, 147–160.
Gaudet, G. and S. Salant, 1991, “Uniqueness of Cournot Equilibrium: New Results from
Old Methods,” Review of Economic Studies, 58, 399–404.
Holmstrom, B., 1982, “Moral Hazard in Teams,” Bell Journal of Economics, 13, 324–340.
Katz, M. and C. Shapiro, 1986, “How to License Intangible Property,” Quarterly Journal of
Economics, 101, 567–589.
Kamien, M., 1992, “Patent Licensing, ” in R. Aumann and S. Hart, eds., Handbook of Game
Theory, vol. I, Amsterdam: North-Holland.
—— and Y. Tauman, 1984, “The Private Value of a Patent: A Game Theoretic Analysis,”
Journal of Economics (Supplement), 4, 93–118.
—— and ——, 1986, “Fees versus Royalties and the Private Value of a Patent,” Quaterly
Journal of Economics, 101, 471–491.
——, S. Oren, and Y. Tauman, 1992, “Optimal Licensing of Cost Reducing Innovation,”
Journal of Mathematical Economics, 21, 483–508.
——,Y. Tauman, and I. Zang, 1988, “Optimal Licensing Fees for a New Product,” Mathematical Social Sciences, 16, 77–106.
Mortimer, J. H., 2004, “Vertical Contracts in the Video Rental Industry, ” Mimeo, Harvard
University.
Muto, S., 1993,“ On Licensing Policies in Bertrand Competition,” Games and Economic
Behavior, 5, 257–267.
Rasmusen, E., M. Ramseyer, and J. Wiley Jr., 1991, “Naked Exclusion,” American Economic
Review, 81, 1137–1145.
Rostocker, M., 1983, “PTC Research Report: A Survey of Corporate Licensing,” IDEA—The
Journal of Law and Technology, 24, 59–92.
Segal, I., 1999, “Contracting with Externalities,” Quarterly Journal of Economics, 114, 337–
388.
—— and M. Whinston, 2000, “Naked Exclusion: Comment,” American Economic Review,
90, 296–309.
Shapiro, C., 1985, “Patent Licensing and R&D Rivalry,” American Economic Review, 75,
25–30.
Tirole, J., 1988, The Theory of Industrial Organization, Cambridge, MA: The MIT Press.
Wang, X., 1998, “Fee versus Royalty in a Cournot Duopoly Model,” Economics Letters, 60,
55–62.
Werden, G., 1998, “Antitrust Analysis of Joint Ventures: An Overview,” Antitrust Law
Journal, 66, 701–735.