Content uploaded by Cheng-Hau Peng
Author content
All content in this area was uploaded by Cheng-Hau Peng on Nov 20, 2015
Content may be subject to copyright.
Signaling in Technology Licensing
Cheng-Tai Wuand Cheng-Hau Pengy
November 4, 2015
Abstract
We analyze licensing contracts in an oligopolistic industry in the pres-
ence of asymmetric information. In the case where the innovator has to
license to all downstream …rms, we …nd that the innovator with superior
technology will use a higher but suboptimal royalty to signal his type.
This result generalizes the …nding of Macho-Stadler and Perez-Castrillo
(1991) in a monopolistic industry. Furthermore, if the assumption that the
innovator has to license to all …rms is relaxed and the number of licenses
sold is endogenously determined, we …nd that the e¢ cient innovator in
general uses the number of licenses sold, instead of a higher royalty, to
signal his type. That is, in the separating equilibrium that maximizes
the payo¤ of the e¢ cient innovator, the number of licenses o¤ered by the
e¢ cient innovator is less than that o¤ered by the ine¢ cient one.
JEL classi…cation: D45; D82
Keywords: Signaling, Technology Licensing
1 Introduction
Technology licensing has drawn a lot of attention and has been debated exten-
sively in the literature. In their seminal paper, Kamien and Tauman (1986)
show that it is more pro…table for an outsider innovator to license a non-drastic
innovation by means of a …xed fee than a royalty under symmetric informa-
tion.1Instead of comparing the contract with a pure …xed fee with the contract
involving a pure royalty, Erutku and Richelle (2007) show that the optimal
contract of an outsider is a two-part contract, i.e., an up-front …xed fee plus
Departm ent of Economics, Fu Jen Catholic University, New Taipei City 24205, Taiwan.
Tel: +886-2-2905-2956; Fax: +886-2-2905-2811; E mail: ctaiwu@gmail.com .
yCorresp onding author: Department of Economics, Fu Jen Catholic University, New
Taipei City 24205, Taiwan. Tel: +886-2-2905-2876; Fax: +886-2-2905-2811; E-mail: ch-
pon@mail.fju.edu.tw.
1A similar result can also b e found in Katz and Shap iro (1985, 1986) and Kamien et al.
(1992).
1
a royalty, and it optimally sells the license to all downstream …rms.2Sen and
Tauman (2007) obtain a similar result in an environment where the …xed fee
is determined by auction. However, empirical studies …nd that arrangements
with a pure …xed fee, pure royalty, or two-part contract coexist in industries.
For example, Rostoker (1984) documents that in the US, 46% of licensing con-
tracts are in the form of a …xed fee plus a royalty, 39% are royalty only, and
13% are fee only. More recently, Yanagawa and Wada (2000) document that
48% of licensing contracts from US …rms to Japanese …rms are in the form of a
…xed fee plus a royalty. The results under symmetric information have di¢ culty
explaining the coexistence of di¤erent licensing schemes.
In reality, it is commonly observed that the licensor may have a better idea
of its value than its licensees. This is especially plausible if one thinks of the
innovator as a more experienced company with R&D ability, which will have a
very clear idea of the reduction in costs from a new process, and of the buy-
ers as small …rms, that may have a very hazy idea of such things. With the
rapid development of incentive theory in the past two decades, many theoretical
papers point out that asymmetric information between the innovator and the
licensees plays an important role in terms of the form of the licensing contracts;
see, for example, Gallini and Wright (1990), Macho-Stadler and Perez-Castrillo
(1991), Beggs (1992), Sen (2005), and Antelo (2012), among others. In their
paper, Gallini and Wright (1990) investigate the e¤ects of the seller’s private
information on the optimal form of the contracts. The authors consider an envi-
ronment where an informed innovator owns a product innovation and can design
an exclusive contract, i.e., the contract dictates the number of downstream …rms
he wishes to sell the license to, and only the …rms with a license can produce.
The authors show that the high quality innovations are signalled through two-
part contracts, i.e., involving a …xed fee and a variable fee. Macho-Stadler and
Perez-Castrillo (1991), on the other hand, consider an environment where an
informed innovator owns a process innovation (a cost-reduction technology) and
wants to sell the license to a downstream monopoly producer. The authors show
that the innovator with superior technology can use a linear royalty to signal his
type. Furthermore, they characterize the best separating equilibrium, i.e., the
separating equilibrium that maximizes the payo¤ of the innovator with superior
technology, in which the innovator with superior technology uses a two-part
contract involving a …xed fee plus a linear royalty, to separate himself from the
inferior one (who uses a …xed fee only). Beggs (1992) assumes that the down-
stream licensee …rm, which has private information, makes a contract o¤er to
the innovator. He …nds that by using royalty contracts, which relate payment
to observed output, a separating equilibrium is made possible and may allow
a more e¢ cient outcome than a …xed fee. The author also shows that there
exists a unique equilibrium satisfying Cho and Kreps’intuitive criterion. More
recently, Sen (2005), in the context of adverse selection, investigates the optimal
2More precisely, the authors show that when the innovation is not very drastic, the op timal
contract is a two-part contract and the license is sold to all downstream …rm. On the other
hand, when the innovation is very drastic, the innovator can design a contract to secure the
monopoly pro…t of the industry.
2
two-part contract o¤ered by an outside innovator under the assumption that the
downstream monopolist has private information. Antelo (2012) analyses in a
signaling game the e¤ects of asymmetric information (the contract duration and
the choice of the royalties) on the design of royalties-only contracts.
To the best of our knowledge, all of the discussions in the aforementioned
literature on process innovation with imperfect information assume a down-
stream monopolistic industry. The problem of the optimal separating licensing
contracts in an oligopolistic downstream market has not received adequate at-
tention. The consideration of an oligopolistic downstream market is important
because in a monopolistic downstream market, the instruments that can be used
by the innovator as a signaling or a screening device are restricted to a …xed-fee
or a royalty, while in an oligopolistic market, the innovator has an additional
instrument, the number of licenses sold, to use. Indeed, in an optimal licensing
contract, the number of licenses sold is endogenously determined, and it can
be a substitute for the royalty or the …xed fee as a signaling or a screening
device. The assumption of a downstream monopolistic industry rules out this
possibility.
The purpose of this paper is to develop an oligopolistic model with imperfect
information to highlight how an outside innovator with a superior process (cost
reduction) innovation can separate himself from the inferior one via a two-
part licensing contract. Speci…cally, we assume that the innovation can be
either e¢ cient or ine¢ cient. Futhermore, it is the private information of the
innovator. After observing its type, the innovator o¤ers a take-it-or-leave-it two-
part contract to the downstream …rms. The contract o¤ered by the innovator
may convey information regarding the technology. The main …ndings of this
paper are as follows. First, we consider the case where the innovator has to
license to all downstream …rms. In this case, we …nd that the royalty rate
proposed by the innovator of the e¢ cient type is de…nitely positive and higher
than that proposed by the ine¢ cient one. This result implies that the separating
contract o¤ered by the e¢ cient type can never be a pure …xed fee contract.
Furthermore, when the di¤erence in innovation is small, the contract o¤ered by
the e¢ cient type is a two-part contract, while when the di¤erence in innovation
is large, the contract o¤ered by the e¢ cient type is a pure royalty contract in the
best separating equilibrium.3These results generalize the …ndings of Macho-
Stadler and Perez-Castrillo (1991) in a monopolistic industry: the e¢ cient type
uses a higher but suboptimal royalty to signal his type even in an oligopolistic
industry. Second, we drop the assumption that the innovator has to license
to all downstream …rms, and consider the case where the number of licenses
sold is endogenously determined. We characterize the separating equilibrium
that maximizes the payo¤ of the e¢ cient innovator, and …nd that the e¢ cient
innovator in general uses the number of licenses sold instead of a higher royalty,
to signal his type. In the equilibrium, the number of licenses o¤ered by the
e¢ cient innovator is less than that o¤ered by the ine¢ cient one.
3The b est separating equilibrium is the separating equilibrium tha t maximizes the e¢ cient
type’s payo¤. Note that here we consider the case where the license is sold to all downstream
…rms.
3
The remainder of this paper is organized as follows. Section 2 sets out the
basic model. Section 3 investigates the separating equilibria of the e¢ cient out-
sider innovator and the ine¢ cient outsider innovator. Section 4 studies the best
separating equilibrium for the high-tech outsider innovator. Section 5 concludes
the paper.
2 Model
An innovator owns a patent for a new technology that reduces the constant
marginal cost of producing some good from cto c". The amount of the
cost reduction can be either high, denoted by "H;or low, denoted by "L;with
0< "L< "H< c: The type 2 fH; Lgof the technology is the private
information of the innovator.
We assume that the innovator has a comparative disadvantage in the produc-
tion of the product and can only license the patent to some (or all) downstream
…rms in an industry. There are npotential …rms producing the same product,
where N=f1; :::; ngdenotes the set of …rms. For m2N, let qmbe the quan-
tity produced by …rm mand let Q=Pn
m=1 qmbe the total quantity produced.
Let pdenote the industry price. The inverse demand function of the market
is p=maxf0; a Qg. The …rms with and without a patent have constant
marginal costs of c"and c; respectively. Furthermore, the downstream …rms
compete in Cournot fashion.
Interactions between the innovator and the downstream …rms are described
by a three-stage signaling game. In the …rst stage, after type is realized,
the informed innovator o¤ers a licensing contract (r; F)consisting of a lin-
ear royalty rand an up-front …xed fee F:The contract may reveal the type
of the innovator. In the second stage, all downstream …rms observe the pro-
posed contract (r; F)and form (the same) beliefs about whether the contract
is o¤ered by an e¢ cient =Hor ine¢ cient =Linnovator. They then de-
cide simultaneously to accept or to reject the contract. As a result, the set of
downstream …rms is partitioned into two subsets: the set of licensees Kand
the set of non-licensees NnK. Let iand jdenote an element of Kand NnK,
respectively. The …rms (licensees, hereafter) that accept the contract pay the
up-front …xed fee. In the third stage, …rms infer the set of licensees and non-
licensees before they decide simultaneously how much to produce. After their
production has been sold on the market, licensees pay the corresponding roy-
alties. Therefore, licensee i0s pro…t is (p(Q)(c"+r))qiFfor i2K,4
non-licensee j0s pro…t is (p(Q)c)qjfor j2NnK, and the innovator’s payo¤
is (r; F) = rPi2Kqi+kF, where k=jKjdenotes the number of licenses
sold by the innovator, or equivalently the number of licensees.
4We assum e that as long as the licensing contract is signed, the licensees have to pay the
per-unit royalty charged by the licensor even if r> ". In this context, although the e¤ective
production cost becomes higher after licensing, the licensees shall never use their original
technology because they will produce at the un it cost of c+rand this will reduce their
pro…t.
4
We begin our analysis by considering the subgame in the third stage. Given
that the innovator o¤ers a contract (r; F )and there are kdownstream …rms
accepting the contract (i.e., there are klicensees) whose marginal cost is c("r)
and nk…rms rejecting (i.e., there are nknon-licensees) whose marginal
cost is c, the quantity that the …rms produce under Cournot competition is as
follows
qi=qI(k; " r)((ac)+(n+1k)("r)
n+1 ;for ("r)ac
k
(ac)+("r)
k+1 ;for ("r)ac
k
for i2K; and
qj=qJ(k; " r)(ac)k("r)
n+1 ;for ("r)ac
k
0;for ("r)ac
k
for j2NnK:
The corresponding Cournot pro…t for the licensees and the non-licensees,
when the number of licenses sold is k; is
m(k; " r) = I(k; " r) = [qI(k; " r)]2;for m2K;
J(k; " r) = [qJ(k; " r)]2;for m2NnK:
Note that, given r; k; when (ac)=(k1) ("r), i.e., ("r)is large,
the non-licensees are driven out of the market. When (ac)=k ("r)<
(ac)=(k1), i.e., ("r)is in the middle range, the non-licensees are driven
out of the market (qJ(k; " r) = 0) if they compete with klicensees, but not
(qJ(k1; " r)>0);if they compete with k1licensees. When ("r)<
(ac)=k, i.e., ("r)is small, the non-licensees are not driven out of the market
(qJ(k; " r)>0) and (qJ(k1; " r)>0) if they compete with klicensees or
k1licensees.
Given that there have already been k1…rms accepting the contract, let
w(k; " r)I(k ; " r)J(k1; " r)denote the di¤erence in the Cournot
pro…t of the …rm between accepting and rejecting the contract: for 1< k n;
w(k; "r)8
>
<
>
:
((ac)+(n+1k)("r)
n+1 )2((ac)(k1)("r)
n+1 )2;for ("r)ac
k
((ac)+("r)
k+1 )2((ac)(k1)("r)
n+1 )2;for ac
k("r)ac
k1
((ac)+("r)
k+1 )2;for ac
k1("r)
:
Furthermore, for k= 1; qL(0; " r) = (ac)
n+1 ;
w(1; " r) = (((ac)+n("r)
n+1 )2((ac)
n+1 )2;for ("r)ac
((ac)+("r)
2)2((ac)
n+1 )2;for ac("r).
Note that w(k; "r)is the upper bound of the …xed fee that the innovator can
charge and it can be easily shown that w(k; " r)is decreasing in k; increasing
in "; and w(k; " r) = 0 when "=r:5
5See Claim 1 in E rutku and R ichelle (2007).
5
In the second stage, given that the innovator proposes a contract (r; F );the
contract endogenously determines the number of licensees and non-licensees:
There are exactly k…rms that accept the contract and nk…rms that reject
it, where the number of licensees is
k=8
<
:
n; if 0Fw(n; " r)
k; if w(k+ 1; " r)Fw(k; " r)
0;if F > w(1; " r)
:
In the …rst stage, the innovator will choose contract (r; F),2H; L; to
maximize the payo¤
(r; F) = k(rqI(k; "r) + F):
Throughout the whole paper, we should focus on the separating equilibria of
the game. Furthermore, we wish to characterize the separating equilibria that
maximize the payo¤ of the e¢ cient type (we refer to this as the best separating
equilibrium).
We now de…ne the equilibrium condition. Given a pair of contracts (r; F)
o¤ered by the innovator of type ; 2H; L; the participation constraints (IRs)
are satis…ed for selling the license to kdownstream …rms, if the following con-
ditions hold. De…ne w(n+ 1; " r) = 0:
IR:w(k+ 1; "r)Fw(k; "r);for 1kn; and 2H; L:
When the IRare satis…ed, there are exactly kdownstream …rms that are
willing to accept the contract o¤ered by the innovator. Note that the di¤erent
types might induce di¤erent numbers of k.
Recall that the (r; F )denote the payo¤of the in type by o¤ering contract
(r; F ):The incentive compatibility (IC) constraints are satis…ed for inducing k
downstream …rms to accept the contract, if the following conditions hold:
IC: (r; F)(r0; F0);for ; 02H; L:
A strategy pro…le ((rH; FH);(rL; FL)) and a belief system constitute a sep-
arating equilibrium if both IRand IChold, and the belief of the downstream
…rms on the H-type (L-type) innovator, upon observing contract (rH; FH)
((rH; FH)) is 1(0):6
6Following the literature on technology licensing under imp erfect information, in this pap er
we focus only on the characterization of the separating equilib ria of the gam e and analyze
their prop erties. We do not discuss the p ooling equilibria of the game.
6
3 Separating Equilibrium
We …rst consider the case of symmetric information in which the type of the
innovator is common knowledge. In this case, the problem faced by the innovator
can be regarded as a two-stage maximization problem. In the second stage, for
each k; the number of licenses sold, the innovator chooses (r; F )to solve the
following problem
max
r;F k[rqI(k; " r) + F]
s.t. r2[0; "]and w(k+ 1; " r)Fw(k; " r):
Since for a given "r; w(k+ 1; " r)< w(k; " r);the innovator optimally
chooses F=w(k; " r);given that it wants to sell the license to kdownstream
…rms. Hence, the problem can be rewritten as
("; k) = max
r2[0;"]k[rqI(k; " r) + w(k ; " r)]:
Note that for any given k; the value ("; k)is uniquely determined. In the
…rst stage, the innovator decides the optimal number of licenses to be sold by
solving the following problem
(") = max
1kn("; k):
Erutku and Richelle (2007) solve this symmetric information problem for
the case where r2["c; "]:In this paper, we restrict r2[0; "], i.e., we do not
allow the innovator to "subsidize" the downstream …rms by o¤ering a negative
royalty. With this modi…cation, we rewrite Proposition 1 in Erutku and Richelle
(2007) below. De…ne "=(n2+1)(ac)
n21and "=(ac)
2n1:
Proposition 1 (Erutku and Richelle (2007)) When the innovation is large,
i.e., ""; there exist two-part contracts allowing the innovator to secure the
monopoly pro…t.7When the innovation is moderate i.e., " < " < "; the opti-
mal contract is a two-part contract: a positive royalty plus a …xed fee, and all
downstream …rms will accept the contract, i.e., the number of licenses sold is n:
When the innovation is small, i.e., ""; the optimal contract is fee-only, and
all downstream …rms will accept the contract.
7Note that in this case by o¤erin g the contract (r; F ), where r="(+1)(k01)
2k0,F=
w(k0; "r);and k=k0;with k0is any integer in [q(ac
"+ 1)=(1 ac
"); n]:The innovator
can control the total amount of the quantity produced to be the monopoly quantity with cost
c"and secure the monop oly pro…t.
7
Several notes are in order. First, both the optimal royalty rand the optimal
…xed fee Fare increasing in ":8A more e¢ cient type will set a higher royalty
and charge a higher …xed fee. Second, the optimal contract will be sold to
all ndownstream …rms in the case where the cost-reduction technology is not
very drastic, i.e., when " < ": Kamien and Tauman (1986) show that when
comparing fee-only with royalty-only contracts, the former always dominate the
latter. In the optimal (…xed fee) contract, when "is large, the innovator will
only sell the license to k < n downstream …rms, while when "is small, the
innovator will sell the license to all ndownstream …rms. Since the authors do
not consider general two-part contracts, the results in Proposition 1 complement
their …ndings. Third, Sen and Tauman (2007) study the problem arising when
the outside innovator o¤ers a contract with a linear royalty plus an up-front fee
determined by auction. The authors obtain similar results: the optimal contract
is also two-part contract and the license is practically sold to all downstream
…rms. However, the optimal royalty and …xed fee in their model are more
complicated depending on the parameters.
Throughout the paper we will consider the case where the innovation is
not very large, i.e., " < ": Note that since " > (ac);the case we consider
includes the range in which the innovation is drastic ( ac
"1) and non-drastic
(ac
">1).
We now investigate the case of asymmetric information in which downstream
…rms do not know the type of the innovator. The following result is standard.
Proposition 2 For any separating equilibria, the contract o¤ered by the ine¢ -
cient type is the symmetric information contract.
Proof. See the appendix.
Let kHdenote the number of licenses sold for the e¢ cient type. In what
follows, we discuss the case of kH=n; i.e., the separating equilibria in which
the e¢ cient type sells the license to all ndownstream …rms. Since kL=n; we
compare the contract (royalty and fee) of the e¢ cient type with the one of the
ine¢ cient type when the number of licenses sold is the same (n)in equilibrium.
The following result characterizes the separating equilibria.
Proposition 3 In any separating equilibria with kH=n, (1) rHrL(2)
rH>0.
Proof. See the appendix.
8Note that in the case where " < "; the optimal royalty r(") =
((n1)((2n1)"(ac))
2(n2n+1) ;for " < " < "
0;for ""and F(") = w(n; " r):Both are increas-
ing in ":
8
In words;the e¢ cient type shall use a higher royalty to signal his type.9
Moreover, the e¢ cient type can not use a fee-only contract to separate himself
from the ine¢ cient type in equilibrium. Note that, in the proof, we use the
condition that r(qL(k; "Hr)qL(k; "Lr)) is increasing in rto show that
rHrL:This is the single cross property and it is automatically satis…ed under
our assumptions (linear demand and constant marginal cost) when both types
sell the license to all downstream …rms, i.e., kH=kL=n.10 In the next section
we will show that it is possible that there exist separating equilibria such that
kH< n: In this situation, there is no general relationship between rHand rL;
i.e., it is possible that rH< rLin equilibrium.
From Proposition 1, we observe that the e¢ cient type uses a higher royalty
in the optimal contract under symmetric information. This raises the question
as to whether the …rst best is achievable when information is asymmetric, i.e.,
no type will want to mimic the other type when both types o¤er the contracts
under symmetric information. The next proposition shows that this is not the
case: the …rst best is not achievable under asymmetric information.
Proposition 4 The e¢ cient type cannot use the contract under symmetric in-
formation to separate himelf from the ine¢ cient type.
Proof. See the appendix.
Since there is a continuum of separating equilibria, in the next proposition,
we shall characterize the best separating equilibrium under the assumption that
kH=n, i.e., the separating equilibrium that maximizes the payo¤ of the e¢ cient
type when kH=n: Let b
L(r)L(r; w(n; "Hr)) denote the payo¤ of
the ine¢ cient type when he mimics the e¢ cient type who o¤ers the contract
(rH; FH) = (r; w(n; "Hr)) and recall that ("L)is the payo¤of the ine¢ cient
type under symmetric information. Note that when the e¢ cient type sets r=
"H; w(n; "Hr) = w(n; 0) = 0.11
Proposition 5 Suppose kH=n. If the condition b
L("H)("L)holds;
the contract for the e¢ cient type (rH; FH)is a royalty plus an up-front …xed
fee with rHand FHjointly determined by L(rH; FH) = ("L)and FH=
w(n; "HrH);i.e., both IRHand ICLbind. On the other hand, if the condition
fails, i.e., b
L("H)>("L), the contract for the e¢ cient type is a pure royalty
with rHbeing determined by L(rH;0) = nrHqI(n; "LrH) = ("L):
9Note that w hen kH< n; i.e., the numb er of licenses granted is smaller for the e¢ cient
type, rHmight not b e greater than rLin general. On the oth er hand, the fee for the e¢ cient
type FHis not necessarily smaller than the fee for the ine¢ cient type FL:For instance, when
q0
L=qI(n; "LrH) = 0;it is possible that FH> FLin equilibrium (if ICLis binding and
qL=qI(n; "LrL)>0).
10 N ote that qL(k; "HrH)qL(k ; "LrH) = ((n+1k)("H"L)
n+1 ;when "rac
("H"L)
k+1 ;when "rac
which is independent of r:
11 W hen setting r=rH;the e¢ cient innovato r can not charge any p ositive …xed fee, i.e.,
w(n; "Hr) = w(n; 0) = 0:Therefore, (rH; FH) = ("H;0) is a royalty-only contract.
9
Proof. See the appendix.
The condition b
L("H)("L)12 will be satis…ed when the di¤erence in
innovation "H"Lis small. In this case, the e¢ cient type can optimally use a
two-part contract to signal his type. However, when the di¤erence in innovation
"H"Lis large, (for example "Lis close to 0), b
L(r)>("L)for all r2[0; "H]:
It is impossible for the e¢ cient type to use a two-part contract to signal his type.
He can only use a royalty-only contract. Intuitively, given "H;the e¢ cient type
can use a two-part contract to signal his type when the di¤erence between "H
and "Lis small. When "Lbecomes smaller, i.e., the di¤erence becomes larger,
the e¢ cient type has to use a higher (more suboptimal) royalty to prevent the
mimicking from the ine¢ cient type. When "Lbecomes so small (i.e., close to
0), the ine¢ cient type will want to mimic the e¢ cient type (since ("L)is
very small) if the contract o¤ered by the e¢ cient type can guarantee him a
strictly positive payo¤ i.e., if the …xed fee is strictly positive. In this situation, a
two-part contract is not incentive feasible. The only incentive feasible contract
is a royalty-only contract. (Indeed, the e¢ cient type can set the royalty rvery
small to prevent the ine¢ cient type from mimicking it.) The results in Propo-
sition 5 generalize the …ndings of Macho-Stadler and Perez-Castrillo (1991) in
a monopolistic industry: the e¢ cient type uses a higher but suboptimal royalty
to signal its type even in an oligopolistic industry.
12 T he condition will be satis…ed when either "H"Lis small or ac > "H. For the case
where "H"Lis small, b
L("H)("H)("L):On the other hand, for the case where
ac>"H;since the Courn ot quantity qI(n; "L"H) = (ac+("L"H))
n+1 = 0 for "H> a c
and "L!0, if the L type mim ics the H type, the downstream …rms that accept the contract
will produce 0quantity. Therefore, the L typ e will receive only …xed fees w(n; "Hr). Sin ce
w(n; "Hr)=0when r="H;by cho osing royalty rclose to "H;b
L(r)can b e arbitrarily
close to 0by continuity:That is, by setting the royalty rlarge enough, the H typ e can p revent
the L type from mimicking him.
The condition will fail w hen ac"Hand "Lis small . Note that th e Cournot quantity
qI(n; "L"H) = (ac+("L"H))
n+1 >0if "H< a c: In this situation, if the L typ e mim ics
the H type, the downstream …rms that accept the contract will pro duce a strictly p ositive
quantity and the L typ e will receive a strictly p ositive payo¤. O n the other hand, ("L)!0
as "L!0:Hence, the H typ e can not prevent the L type from m imicking h im.
10
In the following, we use two graphs to portray the best separating equilibrium
for the case where ("H"L)is small, and for the case where ("H"L)is large.
Consider the parameters: n= 2; a = 3; c = 2; "H= 0:8:
1. The case where ("H"L)is small: "L= 0:5:
0.80.60.40.20
0.8
0.6
0.4
0.2
0
r
Π(r)
r
Π(r)
Figure 1: The di¤erence in innovation ("H"L)is small
In Figure 1, the red line is the graph of b
L(r) = L(r; w(2; "Hr));the
payo¤for which the L type mimics the H type when the H type o¤ers a two-part
contract. The dash red line is the graph of H(r; w(2; "Hr)), the corresponding
payo¤ of the H type. The green line is the graph of 2rqI(2; "Lr);the payo¤
for which the L type mimics the H type when the H type o¤ers a pure royalty
contract. The dash green line is the graph of 2rqI(2; "Hr);the corresponding
payo¤ of the H type. Furthermore, the black line is the graph of ("L), the
payo¤ of the L type under symmetric information.
Note that in this case, the best separating equilibrium is characterized by
the intersection of the Red and Black lines. Moreover, if the H type uses a pure
royalty contract, the L type will not mimic him since the green line lies under
the black one. Therefore, any pure royalty rhigher than er(the royalty where
the dash green line and the black line intersect) can be supported as a separating
equilibrium.
11
2. The case where ("H"L)is large: "L= 0:1:
0.80.60.40.20
0.8
0.6
0.4
0.2
0
r
Π(r)
r
Π(r)
Figure 2: The di¤erence in innovation ("H"L)is big
Figure 2 demonstrates the best separating equilibrium, which is character-
ized by the intersection of the green and the black lines, when ("H"L)is large.
Using a two-part contract to induce 2 licenses by the H type is not incentive
feasible (note that the red line lies above the black line for all r2[0; "H]).
Therefore, any pure royalty rhigher than er(the royalty where the dash green
line and the black line intersect) can be supported as a separating equilibrium.
12
Let b"Ldenote the smallest solution of n"H((ac+("L"H))
n+1 ) = ("L).13 Note
that when "Lb"L;the e¢ cient type optimally uses a royalty-only contract to
signal his type.14 Note also that from Proposition 1 we know that when "L";
r("L) = 0, i.e., the contract of the ine¢ cient type under symmetric information
is fee-only. We have the following result which provides an economic explanation
for the coexistence of di¤erent licensing schemes in reality.
Corollary 6 When "Hacand "Lminfb"L; "g;the ine¢ cient type uses
a fee-only contract and the e¢ cient type uses a royalty-only contract in the best
separating equilibrium with kH=n.
In addition, by di¤erentiating the equilibrium royalty rate with respect to
"L;we can observe that when the e¢ cient type uses a royalty-only contract in
the best equilibrium, the equilibrium royalty rHincreases in "L:On the other
hand, when the e¢ cient type uses a two-part contract in the best equilibrium,
the equilibrium royalty rHdecreases in "L:
Proposition 7 Suppose kH=n. When b
L("H)>("); rH("L)is increasing
in "L:On the other hand, when b
L("H)("); rH("L)is decreasing in "L:
Proof. See the appendix.
4 Best separating equilibrium
In this section, we relax the assumption of kH=nand consider the possibility
that the e¢ cient type may optimally o¤er a contact that only kdownstream
…rms will accept in equilibrium. Indeed, the number of licenses sold can be
another instrument (traditionally, the literature on technology licensing focuses
on instruments like royalties or …xed fees for signaling) for the e¢ cient type to
signal his type. Speci…cally, we want to characterize the equilibrium that maxi-
mizes the e¢ cient type’s payo¤in the case where kis endogenously determined.
Hence, the problem now faced by the innovator can also be considered as a two-
stage maximization problem. In the second stage, given that the e¢ cient type
sells the license to kdownstream …rms, the e¢ cient type solves the following
problem:
max
r;F H(r; F ) = k[rqI(k; "Hr) + F]
13 N ote that n"H((ac+("L"H))
n+1 )!n"H(ac"H
n+1 )>0and ("L)!0;as "L!0:
Furthermore, n"H((ac+("L"H))
n+1 )<("L)as "L!"Hby the de…nition of :Therefore,
the intersection of the graph of n"H((ac+("L"H))
n+1 )and that of ("L)(as a function of "L)
must b e non-em pty in the range of "L2(0; "H);based on the continuity of both functions.
14 N ote that b
L("H)is a linear function of "Land ("L)is a qu adratic function of "L:
Therefore, the equ ation has (at most) two solutions. b"Ldenotes the smaller one.
13
s.t. IRH, IRL;ICLand ICHare satis…ed.
We can rewrite the problem as:
max
r;F H(r; F ) = k[rqI(k; "Hr) + F]
s.t. L(r; F )("L)and w(k+ 1; "Hr)Fw(k; "Hr):
The …rst constraint L(r; F ) = k(rq0
L+F)("L)is ICL;where q0
L=
qI(k; "Lr);and the second constraint w(k+ 1; "Hr)Fw(k; "Hr)is
IRH:Note that IRLis satis…ed in the contract of symmetric information for the
L type. Furthermore, as we have already argued in Proposition 5, if ICLbinds,
then ICHmust be satis…ed.1 5 Hence, we drop the constraint in the problem.
Let V(k)denote the value function of the above problem. In the …rst stage,
the e¢ cient type chooses an optimal number of licenses sold to maximize V(k);
i.e., he solves
max
k2[1;n]V(k):
Let b
L(r; k)L(r; w(k; "Hr)) denote the payo¤ of the ine¢ cient type
when mimicking the e¢ cient type, given that the e¢ cient type sells the li-
cense to kdownstream …rms and sets the …xed fee FH=w(k; "Hr):16 Let
brL(k)denote the unique maximizer of b
L(; k):Similarly, de…ne b
H(r; k)
H(r; w(k; "Hr)) as the payo¤ of the e¢ cient type, and let brH(k)denote the
unique maximizer of b
H(; k);which is also the optimal royalty for the e¢ cient
type under symmetric information. We impose the following two conditions:
Condition 1: b
L(brH(k); k)>("L)for all 1kn: When condition 1
does not hold, the e¢ cient type can ignore the ICLconstraint since the optimal
royalty brH(k)(when the number of licenses sold is k) under symmetric infor-
mation can be used to separate the ine¢ cient type. Note that we show in the
previous section that condition 1 will hold when k=n.
Condition 2: b
L("H; k)("L)for all 1kn. When condition
2 is satis…ed, the graph of b
L(r; k)intersects the graph of ("L)(a constant
line) at least once in the range of r2[0; "H]. This condition implies that there
exists r2[0; rH]such that b
L(r; k)("L). Therefore, a two-part contract is
15 N ote that we sh ould show later that ICLis binding. Since H(r("L); F ("L)) =
kr("L)(q0
Hq
L)+("L)<H(r; F ) = kr(qHq0
L)ICHmust be slack. Note that
q0
Hq
L=(ac)+(nk+1)("Hr)
n+1 (ac)+(nk+1)("Lr)
n+1 =(nk+1)("H"L)
n+1 =qHq0
L:
16 N ote that
b
L(r; k) = k(rq0
L+w(k; "Hr))
=k(r((ac)+(n+1k)("Lr)
n+1 ) + ( (ac)+(n+1k)("Hr)
n+1 )2((ac)(k1)("Hr)
n+1 )2)
=k(r((ac)+(n+1k)("Lr)
n+1 ) + n(2(ac)+(n+2(1k))("Hr))("Hr)
(n+1)2)is a quadratic function
of r, and @2(b
L)=@r2<0:
14
incentive feasible, i.e., the e¢ cient type can use a two-part contract to separate
himself from the ine¢ cient one.
We should claim that under conditions 1 and 2, the …rst constraint (ICL)
will bind, i.e., L(r; F )=("L)and the second constraint (IRH) will bind
on the right side, i.e., F=w(k; "Hr):Both equalities jointly determine the
optimal rand F: The following proposition characterizes the best separating
equilibrium when the number of licenses sold is k.
Proposition 8 Suppose condition 1 and condition 2 hold. In the best separating
equilibrium in which the number of licenses sold is k, both IRHand ICLare
binding:Furthermore, the optimal contract (rH; FH)is determined by FH=
w(k; "HrH)and L(rH; FH) = ("L):
Proof. See the appendix.
As discussed in the previous section, when k=n; condition 2 will hold
when "Hac; or the di¤erence in innovation between the e¢ cient and the
ine¢ cient type "H"Lis small. Note that since b
L("H; k)is increasing in k,17
if the condition holds for k=n, it will hold for all 1kn.
From the above proposition, in the best separating equilibrium, the equilib-
rium royalty of the e¢ cient type rHis determined by L(rH; w(k; "HrH)) =
("L):By di¤erentiating kon both sides of the equation, we obtain the fol-
lowing result.
Proposition 9 Suppose condition 1 and condition 2 hold. The equilibrium roy-
alty of the e¢ cient type rHis increasing in kif the condition @L
@r jr=rH<0
holds.18
By substituting the solution rH(k)into the payo¤ function, the e¢ cient type
now solves for the optimal number of licenses sold:
max
kH(rH; FH) = b
H(rH(k); k);
where rH(k)is determined by b
L(r; k) = ("L):
Note that since ICLbinds, b
H(r; k) = kr((n+1k)("H"L)
n+1 )+("L).19 Sub-
stituting it into the objective function and di¤erentiating it by k; we obtain
17 N ote that @2b
L("H:k)=@k2>0and b
L("H:0) = 0:Hence, b
L("H:k) = k(rq0
L+w(k; "H
"H)) = k"H((ac)+(n+1k)("L"H)
n+1 )is increasing in k:
18 N ote that Lis a q uadratic function of r, and rHis deterimined by the intersection of
the curve of Lan d the parallel line of ("L). T his condition m eans that they intersect at
the p oint where Lis decreasing. As dem onstrated in …gure 1 and …gure 2, the condition
holds when "H"Lis small.
19 b
H(r; k)
=k(rqI+w(k; "Hr))
=k(r(qIq0
I) + rq0
I+w(k; "Hr))
=kr(qIq0
I) + k(rq0
I+w(k; "Hr))
=kr((n+1k)("H"L)
n+1 )+("L), where the third equality is from the constraint IC L
binding.
15
d
dk b
H(rH(k); k)
= ( @
@r (kr((n+1k)("H"L)
n+1 ) + ("L)) @r
@k +@
@k (kr((n+1k)("H"L)
n+1 )+("L))
= (k((n+1k)("H"L)
n+1 )@rH
@k +r(n+12k)("H"L)
n+1
= (k(n+ 1 k)@r
@k +r(n+ 1 2k)) ("H"L)
n+1 ;with r=rH:
From the previous proposition, we know that @rH=@k > 0:Therefore, to
satisfy the …rst-order condition k(n+ 1 k)(@rH=@k) + rH(n+ 1 2k) = 0, we
must have (n+ 1 2k)0:That is, the optimal number of licenses sold has at
least to be (n+ 1)=2:We summarize the above discussion below.
Proposition 10 Suppose condition 1 and condition 2 hold. The optimal num-
ber of licenses sold, k;is determined by k(n+ 1 k)@rH
@k +rH(n+ 1 2k) = 0:
Furthermore, if @L
@r jr=rH(k)<0,klies in the range of n+1
2kn:
The result shows that the number of licenses sold kcan be an alternative
signaling device of the e¢ cient type. The assumption of a downstream monop-
olistic industry rules out this possibility. Indeed, in an oligopolistic industry,
kis endogenously determined and it can be a substitute for the royalty or the
…xed fee as a signaling device. The graph below shows that it is optimal for the
e¢ cient type to sell a two-part contract to some, but not all, of the downstream
…rms. For the case of n= 9; a = 3; c = 2; "L= 0:5;and "H= 0:8;Figure
3 below shows that the equilibrium payo¤ is not monotonic in k: In the best
separating equilibrium, the e¢ cient type o¤ers a two-part contract that only 6
downstream …rms will accept.
8642
1
0.75
0.5
0.25
0
k
Π_H
k
Π_H
Figure 3: The equilibrium payo¤ of the e¢ cient type
16
5 Conclusion
The studies on the optimal licensing contract for process innovation under asym-
metric information mainly focus on a special case where there is only one down-
stream producer. We extend the analysis to an oligopolistic industry. In the
case where the innovator has to license to all downstream …rms, we …nd that
the royalty rate proposed by the innovator with the e¢ cient type is de…nitely
positive and higher than that proposed by the ine¢ cient one. This result im-
plies that the separating contract o¤ered by the e¢ cient type can never be a
pure …xed fee contract. Moreover, when the di¤erence in innovation is small,
the contract o¤ered by the e¢ cient type is a two-part contract. However, when
the di¤erence in innovation is large, the contract o¤ered by the e¢ cient type is
a pure royalty contract in the best separating equilibrium. These results gener-
alize the …ndings of Macho-Stadler and Perez-Castrillo (1991) in a monopolistic
industry. On the other hand, if the assumption that the innovator has to license
to all …rms is relaxed and the number of licenses sold is endogenously deter-
mined, we …nd that the e¢ cient innovator in general uses the number of licenses
sold instead of a higher royalty, to signal its type.
In this paper, we have assumed that all the licensee …rms have the same
production e¢ ciency and produce a homogeneous product. For more exten-
sions, one can relax these assumptions and investigate how cost asymmetry and
product di¤erentiation a¤ect the optimal licensing contract under information
asymmetry. We hope that this paper can not only complement the existing li-
censing literature but also go some way in stimulating future studies along this
strand.
17
Appendix
Proof of Proposition 2
In any separating equilibrium, the payo¤ of the ine¢ cient type can not be
lower than ("L);otherwise it will have a pro…table deviation. On the other
hand, the payo¤ can not be higher than ("L)since this would imply that the
payo¤s of the licensees are negative. Since the optimal contract under symmetric
information is unique, the ine¢ cient type will o¤er the symmetric information
contract in equilibrium.
Proof of Proposition 3
We …rst show that rHrL:In equilibrium, the incentive constraints must
be satis…ed. ICHimplies that H(rH; FH) = n(rHqH+FH)H(rL; FL) =
n(rLq0
H+FL);where qH=qI(n; "HrH);and q0
H=qI(n; "HrL):Furthermore,
ICLimplies that L(rL; FL) = n(rLqL+FL)L(rH; FH) = n(rHq0
L+FH);
where qL=qI(n; "LrL);and q0
L=qI(n; "LrH):Therefore, rLqL+rHqH
rLq0
H+rHq0
L;which implies that
rL(ac+"LrL
n+1 ) + rH(ac+"HrH
n+1 )rL(ac+"HrL
n+1 ) + rH(ac+"LrH
n+1 )
)rH("H"L)rL("H"L)
)rHrL:
Second, we show that rH>0:Suppose that rH= 0:The above claim that
rHrLimplies rL= 0:Then from the IC constraints we obtain
H(rH; FH) = nFHH(rL; FL) = nFL= L(rL; FL)L(rH; FH) =
nFH:Hence, FL=FH:This is a contradiction to the presumption that we are
considering the equilibrium contracts of a separating equilibrium here.
Proof of Proposition 4
From Proposition 1, we have r(") = "(n1)(2n1ac
")
2(n2n+1) =(n1)((2n1)"(ac))
2(n2n+1) ;
and "r(") = (2(n2n+1)(n1)(2n1))"+(ac)
2(n2n+1) =(n+1))"+(ac)
2(n2n+1) :If the …rst best
is achievable, then L(r("H); F ("H)) ("L);i.e., the ine¢ cient type will
not mimic the e¢ cient one under the contract (r("H); F ("H)). Consider the
equality
L(r("H); F ("H)) = ("L)
,n(r("H)((ac)+("Lr("H))
n+1 ) + n(2(ac)+(2n)("Hr("H)))("Hr("H))
(n+1)2)
=n(r("L)((ac)+("Lr("L))
n+1 ) + n(2(ac)+(2n)("Lr("L)))("Lr("L))
(n+1)2):
By substituting rand "r(")into the equality, we obtain a quadratic
equation of "L:The roots of the quadratic equation are "L="H(this root
is ruled out) and "L=1
n+1 ((4n27n+ 1)"H4n2(ac)):The second root
"L=1
n+1 ((4n27n+ 1)"H4n2(ac)) 1
n+1 ((4n27n+ 1)( n2+1
n21(ac))
4n2(ac)) = (7n9n2+7n31)
n21(ac)<0when n1:
18
(Note that if we let f(n) = (7n9n2+ 7n31);then f0(n) = 7 + 18n
21n2<0when n1;and f(1) = 4<0:
Proof of Proposition 5
The e¢ cient type solves the following problem in the best equilibrium:
max
r;F H(r; F ) = n[rqI(n; "Hr) + F]
s.t. L(r; F )("L),H(r("L); F ("L)) L(r; F )and 0F
w(n; "Hr):
Note that the …rst, second, and third inequalities are ICL;ICH, and IRH;
respectively.
We should claim that if the condition b
L("H) = L("H; w(n; 0)) ("L)
holds, ICLhas to bind and IRHwill bind on the right side, i.e., F=w(n; "Hr).
On the other hand, if the condition b
L("H)=L("H; w (n; 0)) ("L)fails,
ICLstill binds and IRHwill instead bind on the left side, i.e., F= 0. This means
that the e¢ cient type uses a royalty-only contract in the best equilibrium.
We will temporally ignore the second constraint (ICH) and it is easy to
see that in the best equilibrium it must be slack.20 Recall that L(r; F ) =
n(rqI(n; "Lr) + F). For given any r; F = minf("L)nrqI(n;"Lr)
n; w(n; "H
r)g;i.e., either ICLor IRHhas to bind in the best equilibrium. There are two
possible cases:
1. F=("L)nrqI(n;"Lr)
nw(n; "Hr)
We should claim that F=w(n; "Hr). Otherwise, F= 0:
Note that when ICLbinds, the payo¤ function His increasing in r:
H(r; F )
=n(rqI(n; "Hr) + F)
=nr(qI(n; "Hr)qI(n; "Lr)) + ("L)
=nr("H"L)
n+1 )+("L):
Suppose 0< F < w(n; "Hr);i.e., IRHis slack. The e¢ cient type can
o¤er another contract (r0; F 0), where r0> r and F0=("L)nr0qI(n;"Lr0)
nso
that IRHstill holds, (since nw(n; "Hr) + nrqI(n; "Lr)("L)>0;by
the continuity of nw(n; "Hr) + nrqI(n; "Lr)("L),nw(n; "Hr0) +
nr0qI(n; "Lr0)("L)>0), a contradiction. Therefore, F=w(n; "Hr):
For the case of F= 0;IRHbinds on the left side.
2. F=w(n; "Hr)("L)nrqI(n; "Lr):
Since b
L(r)is a quadratic function of r, there are at most two solutions
fr1; r2gfor b
L(r) = ("L):Note that if the condition b
L("H)("L)holds,
20 N ote that from Proposition 4 we know that in any separating equilibriu m, the e¢ cient
type uses a higher royalty. We shall show that ICLbinds in the b est separating equilibrium.
This implies that (see below) the payo¤ of the e¢ cient typ e is strictly increasing in r: Th ere-
fore, ICHmust be slack. Speci…cally, H(r("L); F ("L)) = nr("L)( "H"L)
n+1 )+("L)
nr("H"L)
n+1 ) + ("L);since rr("L)in any separating equilibrium.
19
there must exist at least one r2[0; "H]such that b
L(r) = ("L). Let r1; r2
with r1< r2denote the solution of b
L(r) = ("L):Note that d(b
L)=dr > 0
at r1and d(b
L)=dr < 0at r2:We …rst claim that F= ("L)nrqI(n; "Lr);
i.e., ICLfor rihas to bind.
At ri;suppose that F < ("L)nrqI(n; "Lri)i.e., ICLis slack. The
e¢ cient type can o¤er another contract (r0
i; F 0
i), where r0
1> r1; r0
2< r2and
F0=w(n; "Hr0)so that ICLstill holds, (again by the continuity of ("L)
nrqI(n; "Lr)nw(n; "Hr)), a contradiction. Therefore, F= ("L)
nrqI(n; "Lr):
Since ICLbinds, which implies H(r; F )is increasing in r; the best equilib-
rium will occur at r2:Furthermore, @b
L(r)
@r <0at r2:
Note that ICHwill be slack: when the e¢ cient type mimics the ine¢ cient
type, it obtains
H(r("L); F ("L))
=n(rqI(n; "Hr) + F)
=n(rqI(n; "Hr)qI(n; "Lr) + qI(n; "Lr) + F)
=n(rqI(n; "Hr)qI(n; "Lr)) + ("L)
=nr("H"L)
n+1 )+("L);
which is smaller than H(r2; F ) = nr2("H"L)
n+1 )+("L):
Proof of Proposition 7
When b
L("H)>("); rHis determined by L(rH;0) = nrHqI(n; "L
rH) = nrH(ac+("LrH))
n+1 = ("L):The LHS is a quadratic function of rHand
its second derivative is negative. Hence, in general, there are two roots of the
equation. The condition b
L("H)>(")implies that there only one root of
the equation and the slope of the graph of LHS is positive. Note that when "L
increases, both the graph of nrHqI(n; "LrH)and ("L)shift up. Therefore,
the solution rHis increasing in "L:When b
L("H)("); rHis determined by
b
L(rH) = L(rH; w(n; "HrH)) = ("L):Di¤erentiating both sides by "L;
we obtain
@
@rHb
L(@rH
@"L) + @
@"Lb
L=@
@"L("L))@ rH
@"L=
@
@"L("L)@
@"Lb
L
@
@rHb
L:
Note that
b
L(r) = n(rq0
L+w(n; "Hr))
=n(r((ac)+("Lr)
n+1 ) + (((ac)+("Hr)
n+1 )2((ac)(n1)("Hr)
n+1 )2)
=n(r((ac)+("Lr)
n+1 ) + n(2(ac)(n2)("Hr))("Hr)
(n+1)2);and
("L) = n(r("L)qL+w(k; "Lr("L)))
=n(r((ac)+("Lr)
n+1 ) + n(2(ac)(n2)("Lr))("Lr)
(n+1)2):
Note that @
@rHb
L<0at r=rH("L);when the condition b
L("H)(")
holds. (The condition implies that the slope at the point of the intersection of
the graph of b
Land the graph of (")is negative.)
Note that
20
@
@"L("L)
=n(r(1
n+1 ) + 2n((ac)+(n+2(1n)))("Lr))
(n+1)2)
=n(r(1
n+1 ) + 2n((ac)(n2))("Lr))
(n+1)2)>0;
as r="(n1)(2n1)
2(n2n+1) ;and =ac
"n21
n2+1 :
Note that @
@"Lb
L=nr(1
n+1 )>0:
Furthermore,
@
@"L("L)@
@"Lb
L
=n(r(1
n+1 ) + 2n((ac)(n2))("Lr))
(n+1)2)nr(n+1n
n+1 )
=2n2((ac)(n2))("Lr))
(n+1)2
=2n2
(n+1)2[(ac)(n2)((n1)(ac)+(n+1)"L
2(n2n+1) )]
=2n2
(n+1)2[n(n+1)
2(n2n+1) (ac)(n2)(n+1)
2(n2n+1) "L]
>0;
where the third equality using
"r=""(n1)(2n1)
2(n2n+1) =""(n1)(2n1ac
")
2(n2n+1) =(n1)(ac)+(n+1)"
2(n2n+1) ;and
the last inequality using the condition
ac
"L>ac
"Hn21
n2+1 >(n2)(n+1)
n(n+1) =n2
n:(Note that n21
n2+1 n2
n=n21
n2+1
1 + 2
n=2
n2
n2+1 =1
2(1
n1
n2+1 )>0:
Proof of Proposition 8
The argument is similar to that in Proposition 5 with the following obser-
vation.
First, Condition 1 rules out the case that the graph of b
L(r; k)lies under
the constant line of ("L):Second, Condition 2 ensures that the graph of
b
L(r; k)and that of ("L)intersect at least once. Note that if the graph
of b
L(r; k)and that of ("L)intersect twice (called the solutions: r1< r2),
since b
H(r; k) = kr((n+1k)("H"L)
n+1 )+("L)is increasing in r; in the best
separating equilibrium, the optimal royalty must be rH=r2. Note that this
implies that @
@r b
L(r; k)jr=rH<0:
Proof of Proposition 9
Note that rHis determined by L(rH; w(k; ")) = ("L), where "="HrH:
Di¤erentiating by kon the both sides of the equation, we obtain
@rH
@k =(@L
@F
@w
@k )=(@L
@r jr=rH@w
@" ):
Note that @L
@F <0;@ w
@k <0;and @ w
@" >0:Therefore, the condition that
@L
@r jr=rH<0implies @ rH
@k >0:
21
References
Antelo, M. (2012), “Screening and Signalling for Technology Licensing: a Com-
parison,”R&D Management, 42, Issue 4, 289–302.
Beggs, A. W. (1992), “The Licensing Patent under Asymmetric Information,”
International Journal of Industrial Organization, 10, 171-191.
Erutku, C. and Y. Richelle (2007), “Optimal Licensing Contracts and the Value
of a Patent,”Journal of Economics and Management Strategy, 16, 407-436.
Gallini, N. Y. and B. D. Wright (1990), “Technology Transfer under Asymmetric
Information,”Rand Journal of Economics, 21, 147-160.
Kamien, M., S. Oren, and Y. Tauman (1992), “Optimal Licensing of a Cost
Reducing Innovation,”Journal of Mathematical Economics, 2, 483-508.
Kamien, M. and Y. Tauman (1984), “The Private Value of a Patent: A Game
Theoretic Analysis,”Journal of Economics (Supplement), 4, 93-118.
Kamien, M. and Y. Tauman (1986), “Fees versus Royalties and the Private
Value of a Patent,”Quarterly Journal of Economics, 101, 471-491.
Kamien, M. and Y. Tauman (2002), “Patent Licensing: the Inside Story,”The
Manchester School, 70, 7-15.
Katz, M. and C. Shapiro (1985), “On the Licensing of Innovations,”The RAND
Journal of Economics, 16, 504-520.
Katz, M. and C. Shapiro (1986), “How to License Intangible Property,”Quar-
terly Journal of Economics, 101, 567-590.
Macho-Stadler, I and J. D. Perez-Castrillo (1991), “Contrats de Licences et
Asymetrie d’Information,”Annales d’Economie et de Statistique 24, 189–208.
Rostoker, M. (1984), "A survey of corporate licensing," The Journal of Law and
Technology, 24, 59–92.
Sen, D. (2005), “On the Coexistence of Di¤erent Licensing Schemes,” Interna-
tional Review of Economics and Finance, 14.
Sen, D. and Y. Tauman (2007), "General Licensing Schemes for a Cost-reducing
Innovation," Games and Economic Behavior 59, 163-186.
Yanagawa N. and Wada T., (2000), "Post-Contracting Innovations and Con-
tingent Payment Scheme in Patent Licensing Contracts," Mimeo, University of
Tokyo & Gakushuin University.
22
