We analyze the incentives for technology transfer between two firms in a market characterized by a logit demand framework. The available licensing policies of the incumbent innovator are the up front fee, royalty and two-part tariff policies. We show that when the market is covered there is no equilibrium where technology transfer occurs.

Giorgos Stamatopoulos

English

Research Papers in Economics

On the possibility of licensing in a market with logit demand
functions 
Giorgos Stamatopoulos
Department of Economics, University of Crete
Abstract
We analyze the incentives for technology transfer between two firms in a market
characterized by a logit demand framework. The available licensing policies of the
incumbent innovator are the up front fee, royalty and two-part tariff policies. We show that
when the market is covered there is no equilibrium where technology transfer occurs.
Citation: Stamatopoulos, Giorgos, (2008) "On the possibility of licensing in a market with logit demand functions." Economics
Bulletin, Vol. 4, No. 17 pp. 1-11
Submitted: December 15, 2007.  Accepted: June 13, 2008.
URL: http://economicsbulletin.vanderbilt.edu/2008/volume4/EB-07D40028A.pdf1. Introduction
It is well known that the owner of a patented cost-reducing innovation may
increase the return of his innovation by allowing his competitors in the market
to acquire the new technology. In most cases, the transfer of technology is based
on a licensing policy that includes royalties as then the innovator maintains a
competitive advantage over his rivals. The purpose of this note is to identify a
simple context where such a possibility for technology transfer does not exist
–irrespective of the licensing policy used.
The patent licensing literature was initiated by Arrow (1962) who analyzed
licensing of a cost-reducing innovation in a perfectly competitive industry and
in a monopolistic industry. Subsequently, Katz and Shapiro (1985, 1986) and
Kamien and Tauman (1984, 1986) analyzed optimal licensing for the oligopoly
case. Later studies considered models with diﬀerentiated goods (Muto 1993,
Faul´ ı-Oller and Sandon´ ıs 2002), asymmetric information (Gallini and Wright
1990, Macho-Stadler and P´ erez-Castrillo 1991, Beggs 1992), strategic delega-
tion, (Mukherjee 2001, Saracho 2002), Stackelberg leader-follower (Filippini
2005).
Licensing of an innovation by an incumbent innovator was ﬁrst discussed
by Shapiro (1985). Wang (1998) analyzed the optimal policy of an incumbent
innovator in a Cournot duopoly with homogeneous goods and showed the su-
periority of the royalty policy over the fee policy. Kamien and Tauman (2002)
extended Wang’s model for the case of an arbitrary number of ﬁrms. Fauli-
Oller and Sandonis (2002) examined two-part tariﬀ policies in duopoly models
with diﬀerentiated goods and showed that the incumbent innovator licenses to
his rival even a drastic innovation. Filippini (2005) examined a model where
the incumbent innovator acts as the Stackelberg leader and showed that the
optimal policy includes only royalty. Sen and Tauman (2007) analyzed two-
part tariﬀ policies for an incumbent innovator in a Cournot oligopoly with n
ﬁrms and showed that the optimal policy depends on both the number of ﬁrms
and the magnitude of the innovation.
1In this note we examine the licensing of a quality-improving innovation by
an incumbent innovator too. We focus on a duopoly market where consumers
preferences depend on the quality level of the products and on an idiosyncratic
random term distributed according to a double exponential distribution. This
framework gives rise to the logit demand framework. The innovator either
uses the new technology on his own or can license it to his rival as well. The
licensing policies available for the latter purpose are the up front fee, royalty
and two-part tariﬀ policies.
We restrict attention to the covered market case, i.e., the case where all
consumers purchase one of the two products in the market. We show that there
is no equilibrium where transfer of technology occurs between the two ﬁrms. In
particular, we show that whenever the licensing policy involves royalties, the
resulting price equilibrium is such that the problem of computing the optimal
royalty has no solution. On the other hand, whenever restricted to a mere up
front fee policy, the innovator ﬁnds it optimal not to license the new technology
to his competitor.
2. The Model
We consider a market with N consumers and two ﬁrms. Each consumer in
the market purchases one unit of just one of the two products that ﬁrms oﬀer.
Consumer m’s evaluation for the product of ﬁrm i is given by
Vmi = y + θsi − pi + mi
where y is the income of the consumer, si is the quality of the product of ﬁrm
i, i = 1,2, θ is the marginal valuation of quality (common for all consumers),
pi is the price charged by ﬁrm i and mi is a random term distributed according
to the double exponential distribution
F(x) = Pr(mi ≤ x) = exp{−exp−[(x/µ) + γ]}
where γ and µ are positive constants. The mi random variables are indepen-
dent across both products and consumers, namely for every consumer m, m1
2and m2 are mutually independent and for every product i consumers obtain
independent signals 1i,2i,...,Ni. The mean and variance of the random term
are given respectively by E(mi) = 0 and V ar(mi) = µ2π2/6. Hence, param-
eter µ is a measure of the dispersion of consumers’ preferences and expresses
the degree of horizontal diﬀerentiation in the market.
Consumer m observes the realization of mi, i = 1,2 and selects the product
which maximizes his net utility. The probability that m will select the product
of ﬁrm i is Pr(Vmi ≥ Vmj). In the absence of an outside (no-purchase) option,
the expected demand of ﬁrm i is given by the following logit formula (see
Anderson et.al, 1992)
di(p) =
exp[(θsi − pi)/µ]
2 P
j=1
exp[(θsj − pj)/µ]
N, i,j = 1,2, i 6= j (1)
where p = (p1,p2).
The ﬁrms in the market compete in prices. The marginal cost of production
of ﬁrm i is independent of its quantity but depends on the quality level of its
product, namely ci = c(si), i = 1,2, where ci denotes the marginal cost of
ﬁrm i. Fixed costs are zero. Prior to any innovative activity ﬁrms produce
commodities of the same quality, s1 = s2 = s.
Let now ﬁrm 1 innovate (in a costless way) a product of higher quality
s∗ > s. To guarantee that the new quality level is proﬁt-enhancing we need
to assume that the quality-marginal cost diﬀerential (of any ﬁrm that uses the
new quality) increases in quality, i.e.,
θs
0−c(s
0) > θs−c(s), ∀ s
0 > s (A1)
Firm 1 can either use the new technology exclusively or it could license it to
ﬁrm 2 as well. The relevant decision of ﬁrm 1 is embedded in the following
three-stage game. In the ﬁrst stage, ﬁrm 1 decides whether to license or not
to ﬁrm 2 and in the former case it decides on the form of the licensing policy.
We consider the licensing policies of royalty, up front fee and two-part tariﬀs.
3If ﬁrm 1 does oﬀer a licensing contract (based on one of the three policies)
then in the second stage ﬁrm 2 responses to the oﬀer. Finally, in the last
stage, and given the outcomes of the previous stages, the two ﬁrms engage in
simultaneous price competition. Let G denote this interaction. Let also Gfr,
Gr and Gf denote the sub-games of G induced by the two-part tariﬀ, royalty
and up front fee policies respectively. Finally let G− denote the sub-game of G
corresponding to the case where licensing does not occur. In the next section
we analyze the price competition stage of each of these games.
2.1 The price stage
Let ﬁrm i enter the price stage with quality level si. Denote by ci = c(si) its
marginal cost, i = 1,2. Let
xi = exp[(θsi − pi)/µ], i = 1,2 (2)
By (1) and (2) the demand functions of the two ﬁrms are
d1(p) =
x1
P2
k=1 xk
N, d2(p) =
x2
P2
k=1 xk
N (3)
The payoﬀ functions are
π1(p) = (p1 − c1)
x1
P2
k=1 xk
N, π2(p) = (p2 − c2)
x2
P2
k=1 xk
N (4)
It can be veriﬁed that equilibrium prices are given by the unique solution of
the system1
p1 = c1 + µ + µ
x1
x2
, p2 = c2 + µ + µ
x2
x1
(5)
where x1 and x2 are given by (2). Let p∗
1(s1,s2), p∗
2(s1,s2) be the unique
solution2 of the system in (5). By (2) − (5) equilibrium proﬁts are
π
∗
1(s1,s2) = µN
x∗
1
x∗
2
, π
∗
2(s1,s2) = µN
x∗
2
x∗
1
(6)
1See Anderson et.al (1992).
2A closed-form solution for the equilibrium prices of the logit model cannot be computed
unless ﬁrms are symmetric.
4where x∗
1 and x∗
2 are evaluated at p∗
1(s1,s2),p∗
2(s1,s2) (using (2)).
Notice that in G−, s1 = s∗ and s2 = s. On the other hand, in Gf we have
s1 = s2 = s∗. In this case it is easy to show that
p
∗
1(s
∗,s
∗) = p
∗
2(s
∗,s
∗) = c
∗ + 2µ, π
∗
1(s
∗,s
∗) = π
∗
2(s
∗,s
∗) = µN (7)
where c∗ = c(s∗).
Consider next the price stage under the assumption that ﬁrm 1 has licensed
the new technology to ﬁrm 2 using a royalty rate r (which could be accompanied
or not by an up front fee). The payoﬀ function of ﬁrm 1 in the price stage of
either Gfr or Gr is
π1(p) = (p1 − c
∗)
x1
P2
k=1 xk
N + r
x2
P2
k=1 xk
N,
while that of ﬁrm 2 is
π2(p) = (p2 − c
∗ − r)
x2
P2
k=1 xk
N
Equilibrium prices are given implicitly by the (unique) solution of the system
p1 = c
∗ + µ + µ
x1
x2
+ r, p2 = c
∗ + µ + µ
x2
x1
+ r (8)
Let p∗
1(r) and p∗
2(r) denote the solution of (8). Equilibrium payoﬀs are3
π
∗
1(r) = µ
x∗
1(r)
x∗
2(r)
N + rN, π
∗
2(r) = µ
x∗
2(r)
x∗
1(r)
N (9)
where x∗
1(r) and x∗
2(r) are computed at the unique solution of (8) (using again
(2)).
2.2 The licensing stage
Let us now analyze the licensing stage of the interaction. Consider ﬁrst the
two-part tariﬀ licensing policy where ﬁrm 2, if acquires the new technology has
3We use diﬀerent notations for the equilibrium prices and payoﬀs in Gfr-Gr and in Gf
hoping that this won’t cause a confusion.
5to pay a royalty r per unit of its production plus an up front fee α. Clearly,
the fee cannot exceed the diﬀerence between the proﬁt of ﬁrm 2 when it uses
the new technology s∗ and pays r and its proﬁt when it unilaterally uses the
old technology s. Hence α = π∗
2(r) − π∗
2(s∗,s) where π∗
2(r) is given by (9) and
π∗
2(s∗,s) by (6). The problem facing ﬁrm 1 in the licensing stage of Gfr is to
ﬁnd the royalty rate that solves
maxr{π
∗
1(r) + π
∗
2(r) − π
∗
2(s
∗,s)} s.t. π
∗
2(r) ≥ π
∗
2(s
∗,s) (10)
Lemma 1. The optimization problem in (10) has no solution.
Proof. First we note that for all r ≥ 0, p∗
1(r) = p∗
2(r). This is proved in the
Appendix (Lemma A1). Given this equality of prices and since s1 = s2 = s∗,
(2) implies that x∗
1(r) = x∗
2(r). Hence by (6) and (9) the total payoﬀ of ﬁrm 1
in Gfr is
Π1(r) ≡ 2µN − µN
x∗
2
x∗
1
+ rN (11)
Note that x∗
1 and x∗
2 do not depend on r (as x∗
1,x∗
2 emerge in G−).
Claim 1. The constraint in (10) is always satisﬁed, i.e., irrespective of r,
π∗
2(r) ≥ π∗
2(s∗,s).
Proof. Appears in the Appendix.
Note in (11) that Π1(r) is strictly increasing in r. This fact combined with
Claim 1 imply that the problem in (10) has no solution in r.
It is straightforward to show that the optimal royalty cannot be computed in
Gr as well. On the other hand, under an up front fee policy the conclusion is
clear: under this policy ﬁrm 1 prefers not to sell its technology.
Lemma 2. Irrespective of s∗, ﬁrm 1 prefers G− over Gf, i.e., ﬁrm 1 does not
sell its technology under the up front fee policy.
Proof. The fee ﬁrm 1 can charge in Gf is given by π2(s∗,s∗)−π2(s∗,s). Hence
6by (6) and (7), ﬁrm 1 obtains in Gf a total payoﬀ equal to
π1(s
∗,s
∗) + π
∗
2(s
∗,s
∗) − π
∗
2(s
∗,s) = 2µN − µN
x∗
2
x∗
1
On the other hand, in G− ﬁrm 1 obtains µN
x∗
1
x∗
2
. Hence, licensing occurs if and
only if 2µN − µN
x∗
2
x∗
1
≥ µN
x∗
1
x∗
2
or if and only if 2x∗
1x∗
2 ≥ (x∗
1)2 + (x∗
2)2 which
cannot hold.
Lemmas 1 and 2 imply the following.
Proposition 1. The game G has no equilibrium where transfer of technology
between the two ﬁrms takes place.
7Appendix
Lemma A1. For all r ≥ 0, p∗
1(r) = p∗
2(r).
Proof. First note that if r = 0 then ﬁrms 1 and 2 produce commodities of the
same quality s∗ and with the same marginal cost; hence by (7), when r = 0
the two ﬁrms charge the same price. Let now r increase from 0. The eﬀects of
a marginal change of r in the prices of the two ﬁrms are given by the solution
of the system
p1r = 1 +
−x∗
1(r)p1r + x∗
1(r)p2r
x∗
2(r)
, p2r = 1 +
−x∗
2(r)p2r + x∗
2(r)p1r
x∗
1(r)
(12)
in p1r,p2r where pir =
∂p∗
i(r)
∂r
, i = 1,2. Note that in the unique solution of (12)
we have p1r = p2r = 1. By this last fact and by the fact that when r = 0 prices
are equal, we conclude that for all r ≥ 0, p∗
1(r) = p∗
2(r).
Proof of Claim 1. Note that π∗
2(r) = µN (by (9) and since x∗
1(r) = x∗
2(r)).
On the other hand π
∗
2(s
∗,s) = µN
x∗
1
x∗
2
. Hence to show the validity of the claim
it suﬃces to show that x∗
2 < x∗
1. This holds by Proposition 1 of Anderson and
de Palma (2001) according to which in the logit model the proﬁt of ﬁrm i is
higher than that of ﬁrm j if and only if θsi − ci > θsj − cj or if and only
if θsi − p∗
i > θsj − p∗
j. Notice that in G− we have θs1 − c1 = θs∗ − c∗ and
θs2 − c2 = θs − c, where c = c(s). By assumption (A1), θs∗ − c∗ > θs − c and
hence using Proposition 1 of Anderson and de Palma (2001), we conclude that
θs∗ − p∗
1 > θs − p∗
2. This last inequality implies that x∗
1 > x∗
2 (by (2)). This
proves Claim 1.
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10

Contrats de licence et asym´ etrie d’information.

Discrete Choice Theory of Product Diﬀerentiation,

Economic welfare and the allocation of resources for invention. In: R.R. Nelson (Ed.), The Rate and Direction of Inventive Activity: Economic and Social Factors,

Fee vs. royalty licensing in a Cournot duopoly model,

Fees versus royalties and the private value of a patent,

General licensing schemes for a cost-reducing innovation.

How to license intangible property,

Licensing contract in a Stackelberg model.

On licensing policies in Bertrand competition,

On the licensing of innovations,

Patent licensing and R&D rivalry,

Patent licensing under strategic delegation,

Patent licensing: The inside story,

Product diversity in asymmetric oligopoly: is 8the quality of consumer goods too low?,

On the possibility of licensing in a market with logit demand functions

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