Discussion Papers In Economics
And Business

Dynamic Analysis of Innovation and International Transfer ofTechnology through Licensing
Hitoshi Tanaka

Tatsuro Iwaisako

Koichi Futagami

Discussion Paper 06-06-Rev.

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANDynamic Analysis of Innovation and International Transfer ofTechnology through Licensing
Hitoshi Tanaka

Tatsuro Iwaisako

Koichi Futagami

Discussion Paper 06-06-Rev.
October 2006
この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。
Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANDynamic Analysis of Innovation and International Transfer ofTechnology through Licensing∗
Hitoshi Tanaka†

Tatsuro Iwaisako‡

Koichi Futagami§
Abstract
This paper develops a quality-ladder type dynamic general equilibrium model withendogenous innovation and technology licensing as a major source of international
technology transfer in developing countries. Examining the dynamic characteristicsof the model fully, we explore the short- and long-run effects of both an improvementin the probability of reaching a licensing agreement with a given effort and an increasein the license fee rate. The model shows that the former promotes innovation and technology transfers in both the long and short run, while the latter discourages them.
Keywords: Innovation; Licensing; Technology transfer
JEL classification: F43; O33∗
A previous version of this paper was entitled “Dynamic Analysis of Innovation and Licensing: The Effects
of Intellectual Property Rights Protection”. The authors are grateful to the co-editor, Robert Staiger, and two
anonymous referees for their valuable comments. The authors would also like to thank Yukiko Abe, Ichiroh
Daitoh, Tetsugen Haruyama, Ryo Horii, Jun-ichi Itaya, Noritaka Kudoh, Kazuo Mino, and seminar participantsat the European Trade Study Group (ETSG) 2005 Dublin Seventh Annual Conference, Hokkaido University
and the Kansai Macroeconomics Workshop for their helpful suggestions on an earlier version of this paper.
This study was partly supported by a Grant-in-Aid for Young Scientists (No.16730104) from the Ministry of
Education, Culture, Sports and Technology. All remaining errors are the authors’ responsibility.
†
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. E-mail: cg039th@srv.econ.osaka-u.ac.jp‡
Faculty of Economics, Ritsumeikan University, 1-1-1 Noji-higashi, Kusatsu, Shiga 525-8577, Japan.
Phone: +81-77-561-5044, E-mail: tiwai@ec.ritsumei.ac.jp§
Faculty of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043, Japan. Phone:+81-6-6850-5266, E-mail: futagami@econ.osaka-u.ac.jp

1 Introduction
Advanced technologies possessed by firms in developed countries are necessary for the industrialization of developing countries. Many countries that are now developed benefited
from imported advanced technologies. For example, countries in Europe and the US imported many technologies from Britain, while Japan and Korea acquired technologies from
Western economies. As these examples show, technology acquisition is indispensable fordevelopment.
Advanced foreign technologies are transferred with licensing, foreign direct investment(FDI), illegal imitation and trade as typical examples. Japan and Korea preferred licensing to
FDI, especially in the early stages of their development following World War II.1 In this paper, we construct a quality-ladder type dynamic general equilibrium model with endogenous
innovation and technology licensing to investigate the international transfer of technology
through licensing activities.
Based on the product-cycle model developed by Grossman and Helpman (1991), Yang
and Maskus (2001) have already constructed a dynamic general equilibrium model in which
the channel of technology transfer is licensing. Yang and Maskus (2001) explored how
strengthening intellectual property rights (IPR) protection affects innovation and technology
transfer through licensing. They showed that stronger IPR raise innovation and technology
transfer through the reduction of licensing costs and improvements in the licensor’s share ofrents. Their conclusions have important implications for developing countries that are eager
to learn from the experiences of Japan and Korea, as both tended to adopt the purchase offoreign advanced technologies; that is, licensing.
Although Yang and Maskus’ analysis has some interesting findings and makes a contribution to the theory of technology transfer, we further enhance their analysis in two respects.
First, differently to Yang and Maskus, we consider the situation where the effort of firms that1

See Peck (1976), Ozawa (1980), and Enos and Park (1988). See also Pack and Saggi (1997).

1

aim to be granted a license is important in reaching agreement on the license. In Yang and
Maskus’ model, it is assumed that the North, which possesses advanced technologies, uses
its resources for licensing activities. However, Peck (1976), Ozawa (1980), and Enos andPark (1988) argue that most licensing efforts are undertaken by recipient countries, as occurred in the case of Japan and Korea.2 Therefore, to ensure the model matches better withthe experiences of these countries, we modify Yang and Maskus’ model in the following
way: the parties who must make the effort to reach license agreement are firms in the recipient country (the South), and they must use the resources of the recipient country.
Second, and theoretically more importantly, we analyze not only the steady state but also
the transitional dynamics of the model. Yang and Maskus analyzed only the steady stateand consequently did not explore the dynamic nature of their model. With the exceptionof Helpman (1993) and Arnold (2002), other studies on international technology transfer
also often focus only on the steady state. To focus on the steady state can be allowed asa first approach if the dynamic system of the model has a stable equilibrium path to thesteady state. However, the dynamic system actually becomes totally unstable in Yang and
Maskus’ setting (where the North, which has advanced technologies, uses its resources forlicensing activities).3 As a result, their model has no equilibrium path converging to thesteady state. By contrast, we show that there exists a stable equilibrium path; that is, a stablesaddle path in our setting (where firms in the recipient country must make the effort to gain
licenses and use the resources of the recipient country). Thus, our setting where firms in therecipient country make the effort not only corresponds to actual experience but also appears
reasonable from the viewpoint of macroeconomic theory.
Because the instability of the steady state in Yang and Maskus’ model implies that theeconomy never leads to the steady state analyzed in their work, we reexamine their policy2
A famous episode is the ‘pilgrimage to Montecatini’. Many Japanese firms visited Montecatini—an Italiancompany that succeeded in converting propylene into a fiber-forming polypropylene—in order to obtain alicensing agreement. See Ozawa (1980).
3
The proof is available from the authors on request.

2

analysis using a model with a stable equilibrium path, as developed in this paper. By reinterpreting their model somewhat, we separately explore the effects of the following changes:
namely, an improvement in the “smoothness” of the licensing negotiation, which is represented by an increase in the probability of reaching license agreement with a given negotiation effort, and a rise in the license fee rate. Both are regarded by Yang and Maskus as aconsequence of strengthening IPR protection. Based on the dynamic analysis of the model,
we explore both the long- and short-run effects. Examining the short-run effect is one advantage of our approach because many studies, including Yang and Maskus, analyzed only
the steady state representing the long-run state of the economy.4
The short-run effects are well deserving of consideration for at least two reasons. First,
there is the possibility that the short-run effects will run in the opposite direction to thelong-run effects. If a policy has opposite effects in the long run and in the short run, thepolicy must be assessed more carefully.5 Second, the speed of convergence suggested byendogenous growth models may be slow.6 The low speed of convergence is also supported
empirically. For example, in the context of convergence across regions in some countries,
Barro and Sala-i-Martin (2004) conclude that “it takes 25–35 years to eliminate one-half of
an initial gap in per capita incomes” (p.496). The low speed of adjustment after a policychange implies that the short-run effects of the change can be more important than the longrun effects.7 Thus, the analyses of the short-run effects are quite meaningful to economists
and policy makers.
4

Dinopoulos and Segerstrom (2006) examined the short-run effects of strengthening IPR protection on
innovation, although they also only focused on the steady state.
5This is also true of Helpman’s (1993) model. In that model, strengthening intellectual property rights
protection enhances innovation in the short run but reduces innovation in the long run. Therefore Helpman(1993) fully examined the short-run effect.
6
See, for example, Steger (2003). He examined the speed of convergence in the quality-ladder type endogenous growth model in Segerstrom (1998) and showed that the speed of convergence is low through calibrationof the model. Our North–South growth model is also based on a quality-ladder model.
7
Many theoretical studies of macroeconomics regarded the speed of convergence as important and examined it in the context of the various types of growth models. In particular, Ortigueira and Santos (1997)
examined the speed of convergence in endogenous growth models with adjustment costs and showed that the
speed of convergence may be slow. This result implies that it is necessary to examine the transitional dynamics
as well as the steady states.

3

As a result of the analysis, we obtain two main conclusions. First, an improvement in
the smoothness of licensing negotiation encourages innovation and technology transfer inboth the long and short run. Second, an increase in the license fee rate is detrimental toinnovation and technology transfer in both the long and short run. In addition, we show thatan improvement in the smoothness of licensing negotiation increases the wage rate in theSouth, while the increase in the license fee rate reduces the wage rate. Because it is difficult
to determine the effects on the relative wage rate, we present some numerical examples
concerning the relationships between these changes and the relative wage rate.
Although we investigate technology transfer to developing countries through licensing,
the present paper also relates to earlier studies that deal with the issue of innovation andtechnology transfer through channels other than licensing. For instance, Helpman (1993)
constructed a dynamic general equilibrium model in which Northern firms innovate andSouthern firms imitate. He showed that although stronger IPR increase Northern innovationin the short run, they reduce it in the longer run.8 More recently, Dinopoulos and Segerstrom(2006) developed a dynamic North–South model with scale-invariant growth and endogenous imitation.9 On the other hand, some studies incorporated FDI into their models. Lai(1998) showed that strengthening Southern IPR increases innovation and technology transfers when FDI is the channel of transfer but reduces such transfers when imitation is thechannel of transfer. By contrast, Glass and Saggi (2002b) showed that stronger IPR decrease
the level of innovation and technology transfer in a dynamic model where both imitation and
FDI are the channels of production transfer. Because these models did not introduce licensing activities in their frameworks, our model will complement these studies and enable us tounderstand innovation and technology transfer better.10
8Arnold (2002) suggests that Helpman’s (1993) results concerning the long-run effect no longer hold whenlabor mobility among industries in the North is imperfect.
9
In order to remove scale effects, Dinopoulos and Segerstrom (2006) assumed like Segerstrom (1998) andLi (2001) that the difficulty of R&D increases with innovation (and imitation). Moreover, Sener (2006) extended the scale-invariant growth model of Dinopoulos and Syropoulos (2006), which includes rent protection
activities, into a North–South product-cycle model.
10
Glass and Saggi (2002a) examined a dynamic general equilibrium model in which firms can choose the4The remainder of the paper is structured as follows. Section 2 describes the model. In
section 3, we derive the equilibrium path of the model and show that there exists a uniqueequilibrium path converging to the steady state. In section 4, we consider the short- andlong-run effects of a change in the smoothness of licensing negotiations and the license fee
rate on the equilibrium path. In this section, we also compare the results with those in Yang
and Maskus. In section 5, we make a comparison between our technology licensing modeland the endogenous imitation models. Section 6 provides some concluding remarks.

2 The Model
We develop a dynamic general equilibrium model based on Yang and Maskus (2001), where
licensing is introduced into a quality-ladder model as a means of international technical diffusion. Our model has the same basic structure as Grossman and Helpman (1991, ch.12).11
Consider an economy consisting of two regions, North and South, denoted by N andS, respectively. There is a continuum of goods, indexed by ω ∈ [0, 1], that are producedin the North or the South. Each product is classified by a countable infinite number ofqualities j = 0, 1, · · · , and its quality improves if innovation occurs in the industry. Productω of quality j can be produced after the jth innovation in the industry ω, and the quality
is provided by qj (ω) = λj , where the increment of quality, λ > 1, is identical for all
products. As described below, the process of climbing the quality ladder requires researchand development by firms. We choose our units appropriately so that the quality at time
t = 0 equals unity in all industries.

mode of technology transfer (FDI or licensing). However, in contrast to our study and those discussed earlier,they assumed that the two countries were identical.
11
Although a ‘scale effect’ remains in our model, as in Grossman and Helpman (1991), we follow Yang and
Maskus (2001) in not attempting to remove it. Regarding the scale-effect problem, Temple (2003) concluded
that it is unlikely the debate will be resolved empirically.

5

Consumers living in both regions have identical preferences as follows:∫ ∞U=e−ρt log u(t)dt,(1)0where ρ is a common subjective discount rate and log u(t) represents the instantaneous utility at time t. We specify the instantaneous utility function as:[∫ 1
log u(t) =

log

∑

0

]
qj (ω)dj,t (ω) dω,jwhere dj,t (ω) denotes the consumption of good ω of quality j at time t. The representative consumer maximizes his or her intertemporal utility (1) under the following budget
constraint:

∫ ∞

Rt

e− 0 r(s)ds E(t)dt = A(0),0where r(t) is the interest rate that consumers in both countries face at time t and A(0) is thesum of initial asset holdings and discounted total labor income. The term E(t) represents
the flow of spending at time t, namely:

E(t) =∫ 1 [∑
0]pj,t (ω)dj,t (ω) dω,jwhere pj,t (ω) is the price of product ω of quality j at time t.
As is well established, the consumer’s utility maximization problem can be solved in
two stages. In the first stage, the consumer allocates his or her spending E(t) to maximize
log u(t), given prices at time t. To solve this static problem, the consumer allots identicalexpenditure shares to all products. Then, for each product, the consumer chooses the single
quality j = Jt (ω) that carries the lowest quality-adjusted price pj,t (ω)/qj,t (ω). This implies

6the following static demand function: E(t)/pj,t (ω) for j = Jt (ω),
dj,t (ω) =0otherwise.
In the second stage, the consumer chooses the time pattern of spending to maximize hisor her utility (1). This intertemporal utility maximization requires that Ė(t)/E(t) = r(t)−ρ.
By taking the aggregate spending as the numeraire, we normalize E(t) = 1 for all t so thatthe interest rate r(t) always corresponds to the subjective discount rate ρ.12
Turning to the production side, we assume that each economy has a single primary production factor in the form of labor. The amount of total labor supply is constant in eachcountry but varies between countries. We assume that one unit of output requires one unit oflabor input. In addition, research activities and licensing negotiations to win a license froma patent holder require labor inputs as discussed below.
Firms are separated into two types, ‘leaders’ and ‘followers’. Leaders are firms withthe ability to produce goods at the highest level of quality currently available, whereas allother firms are followers. A general feature of this kind of model is that industrial leaders
do not intend to invest in further research and development of their products as long as the
products are not imitated. In this model, we assume no imitation occurs in equilibrium,
so that industrial leaders have no incentive to invest in further R&D. Therefore, wheneverinnovation takes place in the industry, the incumbent leader must have been overtaken by afollower in terms of product quality.
Firms are distinguished in terms of their location; that is, whether they are in the Northor in the South. We assume that Northern firms only have the ability to conduct R&D andbring state-of-the-art products to the market. Hence, only Northern followers drive quality
improvements. The Northern firms that succeed in innovating and becoming quality leaders
12This normalization is a convenient method for examining the dynamic behavior of the economy. See
Grossman and Helpman (1991, ch.12).

7

acquire patents in the North. In addition, they can export their goods to the South without
facing any transportation costs or tariffs.
Southern firms can offer a Northern leader a license contract such that they acquire therights to produce and sell an invention of a Northern firm in exchange for royalty payments.
When granted the license, the Southern firm receives the blueprint of the highest-quality
product and acquires sufficient knowledge to manufacture it. Moreover, the firm can legallysell the product to the entire world. However, Southern licensees must pay a part of the rents
from the sale of the product to their licensors as a license fee, until the product is replaced
by a new product of higher quality. We assume that the Southern licensee receives an exogenously determined share of the rent from sales, which reflects the bargaining power between
licensees and licensors. That is, Southern firms can retain a greater fraction of the rent if
they possess greater bargaining power. The contracts between Northern and Southern firms
forbid every Southern licensee from breaking the agreement and imitating the blueprints toavoid paying the license fee, and close monitoring ensures that this does not occur.
In order to focus on the progress of licensing, we make the following two assumptions.
First, no imitation by Southern followers occurs in equilibrium. We posit that Northern firms
maintain confidentiality when manufacturing their state-of-the-art products in the North.
Therefore, even if IPR protection is not perfectly enforced in the South, it is economicallyand technologically impractical for Southern firms to copy the Northern firms’ products
manufactured in the North. On the other hand, weaker IPR protection in the South may
not prevent Southern followers from imitating a state-of-the-art product that is licensed to
a Southern firm.13 However, assuming that unauthorized imitators are obliged to competewith the rightful licensee in a Bertrand fashion, they earn no positive profits as both types offirm face the same marginal costs. This implies that imitators can never pay imitation costs
as long as they are strictly positive. Therefore, no imitator intends to enter the market. The

13
Grossman and Lai (2004) explored the reason why IPR tend to be more weakly protected in the Souththan in the North, and also examined methods of efficient patent protection in the global economy.

8second assumption is that inward FDI is banned by the Southern governmental authorities
or is unfeasible.14 Hence, licensing is the unique means of international technology transferfrom the North to the South.
For tractability, we assume that the second-highest-quality product is always in the public domain and that its specifications are freely available. This means that, at any time,
the nearest rivals of the leaders are Southern firms with the ability to produce goods of thesecond-highest quality. These products can potentially be competitive against the state-ofthe-art products, despite the lower quality, because the equilibrium wage rate in the South islower than in the North. Then, to exclude rivals, every leader charges the same limit price,
as follows:p = λwS ,(2)

where wS is the wage rate in the South.
In equilibrium, two possible types of market activity exist: either the Southern licensee
produces the highest-quality good under license, or the Northern leader produces the stateof-the-art variety of goods. Following Yang and Maskus (2001), we refer to the former as
the licensed South technology (S) market and the latter as the original North technology(N) market. Assuming that a licensor is obliged to compete with its licensee in a Bertrand
fashion if it enters the product market, no licensor has an incentive to continue producing thegood for himself or herself in equilibrium. That is, once a license contract has been made, theSouthern licensee supplies the product monopolistically in both the Northern and Southernmarkets. Whenever a Northern follower succeeds in innovation and produces a new higherquality product, the market would become N, which is independent of whether the targetedmarket is N or S. Therefore, the research efforts of entrepreneurs indiscriminately range overall ω because the expected gains from innovation are equal between industries, provided

14
In practice, Japanese authorities adopted a restrictive policy towards FDI in the early stages of the development process following World War II in order to encourage foreign firms to license advanced technology toJapanese firms. See Hoekman, Maskus, and Saggi (2005) and Peck (1976).

9

that the leaders in each market are symmetrical; that is, provided that all leaders are equally
exposed to the danger of replacement by the next higher-quality product and that all Northernleaders succeed equally in reaching agreement on licensing.
Under the pricing strategy (2), Northern leaders and Southern licensees make different
profits because costs differ. The price setting of each leader yields a demand per product ofE/λwS . Therefore, each Northern leader earns a flow of profits as follows:
πI = (λwS − wN )

wN
1
=
1
−
,
λwS
λwS

whereas each Southern licensee earns the following:
πL = (λwS − wS )

1
1
=1− ,
S
λw
λ

where wN is the wage rate in the North, which must be restricted to be below λwS so that
the Northern leaders can earn a strictly positive profit.
We assume that the R&D process is modeled as a Poisson process following Grossman
and Helpman (1991). If a Northern firm i uses aI I˜i units of the labor input in research for atime interval of length dt, it succeeds in innovation with a probability of I˜i dt, where aI is aparameter. The variable I˜i , which is the Poisson arrival rate at which new technology will beinnovated in the next instant, is the intensity of R&D chosen endogenously by entrepreneurs.
As usual, the success of R&D depends not on the cumulated resources that have been spentin the former period but only on the current spending resources. We let VI,t denote themarket value of representative leaders operating in the North at time t, i.e., the leaders thatbelong to the N market. As successful innovators attain this market value, each entrepreneur
maximizes the expected net benefit, VI,t I˜i dt − wtN aI I˜i dt. In equilibrium with a finite size of
R&D investment, we must have:
VI,t ≤ wtN aI

with equality whenever I˜i > 0.
10

(3)

Similarly, the formation of a license contract follows a Poisson process. The negotiations
leading to agreement on a new license contract between the licensor and the licensee andadaptations to a new technique after the agreement are most likely time consuming. However, the lengths of time involved may be uncertain. Therefore, we take licensing negotiation
to be costly and assume that a Southern firm i that wishes to be licensed must input aL ι̃i /κ
units of labor per unit of time in order to attain success with an instantaneous probability ofι̃i . Let aL be a parameter, while ι̃i denotes the intensity of a licensing agreement that is optimally chosen by the Southern firm.15 We let κ be a parameter representing circumstancesthat influence the speed of negotiations and subsequent adaptations to a licensed technique.
For example, the speed of progress in a license negotiation may depend on the degree of
establishment of laws and regulations on the contract in the developing country. Moreover,strict enforcement of punishment on infringement of a contract may ease the patent holder’sfear involved with licensing and thus enable swift agreement between the parties concerned.
One possibility is that the educational level in the developing country may affect the ability
to adapt an unknown technique. Hence, we regard κ as an index reflecting all of these factors
and representing the “smoothness” of negotiations.
If a Northern firm and a Southern firm agree on a license, they split the expected present
value of joint profits earned by the Southern firm, VL,t . The Southern firm i, which undertakes licensing negotiations at the intensity ι̃i during a time interval dt, receives an expected
gain of (1 − δ)VL,t ι̃i dt, where 0 < δ < 1 denotes the exogenously determined license fee
rate. Hence, a Southern firm under licensing negotiations decides on an intensity ι̃i to maximize its expected payoff, (1 − δ)VL,t ι̃i dt − wtS (aL /κ)ι̃i dt. In equilibrium, the Southernfirm’s decision requires a zero-profit condition as follows:
VL,t ≤ wtS

aL /κ
1−δ

with equality whenever ι̃i > 0.

(4)

15
In Yang and Maskus (2001), Northern leaders spend resources in order to transfer the technology to theSouth. However, as stated in the introduction, our setting appears more realistic, at least in the historical caseof Japan and Korea.

11

On the other hand, a Northern patent holder that has not yet granted a license may refuse
a Southern firm’s offer. If a Northern firm obtains a smaller expected market value from
granting a license than it obtains by continuing to operate in the North on its own account,then the Northern firm will prefer not to grant a license. Therefore, for a license contract totake place at time t, the stock value of a Northern licensor must exceed that of the Northerncurrent leader that has not yet granted a license: that is:
VI,t ≤ δVL,t .

(5)

The measures of products that belong to the S market at time t, nSt , change over time. A
measure of inflow into the S market is equal to the measure of newly licensed industries in
the N market at time t, whereas the outflow out of the S market is equal to the measure ofindustries in the S market where innovation occurs at time t. As in Grossman and Helpman(1991), we focus only on the symmetric equilibrium. In the equilibrium, every leader in∑
the N market reaches a licensing agreement at the same aggregate intensity ιt = i ι̃i , andevery incumbent leader in the economy is exposed to the danger of being replaced by the∑invention of a higher quality product at the same aggregate intensity It = i I˜i . For a timeinterval dt, a new agreement is made about licensing in ιt nN
t dt industries of the N market,
S
S
where nNt ≡ 1 − nt is a measure of the N market. In addition, innovation occurs in It nt dt

industries of the S market and It nNt dt industries of the N market in the same time interval.
Therefore, nSt must follow the following equation of motion:
SṅSt = ιt nN
t − It nt .

(6)

We now consider how the market value of each firm varies over time. Shareholders
of a firm in the S market earn dividends πL dt and capital gains V̇L dt over a time intervalof length dt if no follower succeeds in innovating a new state-of-the-art product in the in-12dustry. However, the stock of each firm becomes worthless if the product is replaced by ahigher-quality product during the interval dt. The probability that this occurs is equal to theprobability that innovation succeeds in the industry in the time interval, It dt. Provided thatthese idiosyncratic risks are properly diversified away by all investors, a stock should yieldexactly the same expected rate of return as the risk-free interest rate, r(t). The no-arbitrage
condition between the stock of a firm in the S market and a riskless asset is then:r(t)VL,t = πL,t + V̇L,t − It VL,t .

(7)

The no-arbitrage condition for the stock of Northern leaders in the N market is, however,more complex. The shareholders of a leader in the N market earn dividends πI dt and capitalgains V̇I dt if no innovation occurs in the industry, while suffering a total capital loss ofamount VI with a probability of It dt. In addition, the stock value transforms into δVL if
the firm succeeds in reaching an agreement with a Southern firm about licensing during dt,
the probability of which corresponds to ιt dt. Northern leaders in the N market take theinstantaneous probability ιt as given, notwithstanding its endogeneity, because it is selectedby Southern followers. The sum of these risky returns must be identical to the risk-freeinterest. Therefore, we obtain the no-arbitrage condition between the stock of a leader in the
N market and a riskless asset, as follows:r(t)VI,t = πI,t + V̇I,t − It VI,t + ιt (δVL,t − VI,t )if ιt > 0.

(8)

Finally, we close the model by describing the labor market-clearing conditions. Let thelabor supply be LN and LS in the North and South, respectively, where both are exogenously
given. The total manufacturing employment in the South equals nSt E(t)/(λwtS ), whereas inN
SS
the North it equals nN
t E(t)/(λwt ). The R&D sector in the North employs aI It (nt + nt )13

units of labor. Labor-market clearing in the Northern market requires that:
1 N
N
n + aI It (nSt + nNt ) = L .
λwtS t(9)

On the other hand, the labor input for licensing negotiations by Southern follower firms isequal to (aL /κ)ιt nN
t . Hence, the Southern labor-market-clearing condition becomes:
1 S aL N
n + ιt nt = LS .
κ
λwtS t(10)

3 The Equilibrium Path
We now derive the equilibrium path of the economy. First, we compute innovation andlicensing intensity in the equilibrium. Substituting the zero-profit condition in licensing (4)into the Northern labor-market-clearing condition (9), we have:

It =

LN
aL /κ 1 − nSt−aIaI (1 − δ)λ VL,twhenever It > 0 and ιt > 0.

(11)

Similarly, from the zero-profit condition in licensing (4) and the Southern labor-marketclearing condition (10), we obtain:
[ S
]
L
nSt
1
1
ιt =
−
1 − nSt aL /κ (1 − δ)λ VL,twhenever ιt > 0.

(12)Note that both innovation and licensing intensity depend only on the two endogenous variables, nSt and VL,t . No innovation (It = 0) and no licensing (ιt = 0) take place when theright-hand sides of (11) and (12), respectively, become negative. However, we focus ourattention on the region where both It > 0 and ιt > 0.
Next, we compute the evolution of variables nSt and VL,t . Substituting equations (11)14and (12) into (6), we can rewrite the equation of motion for nSt as follows:ṅSt =

LS
−
aL /κ

{

[
]}
aL /κ
LN
1
S
1−
+
(1 − nt ) nSt .
aI(1 − δ)λVL,t
aI

(13)

In addition, using r(t) = ρ for all t and combining (7) with (11), we derive the equation of
motion for VL,t as follows:(V̇L,t =

LN
ρ+
aI

)

[

)]
(
aL /κ
1
S
VL,t −(1 − nt ) + 1 −.
aI (1 − δ)λλ(14)Equations (13) and (14) form an autonomous system of two differential equations in nSt and
VL,t . Therefore, we can examine the dynamic behavior of these two variables separatelyfrom the remaining variables. In this system, nSt is a state variable, whereas VL,t is a jump
variable.
Figure 1 depicts the phase diagram for this system on the (nS , VL ) plane. The intersection of the two curves ṅSt = 0 and V̇L,t = 0 at point A is the fixed point of this system. The
shaded area represents a region in which both research and licensing do not occur. Recalling
equations (11) and (12), we focus on the region where the following two inequalities are
satisfied:
VL,t >aL /κ
(1 − nSt ),(1 − δ)λLN

(15)

aL /κ
nSt .
S
(1 − δ)λL

(16)

and
VL,t >The equation for the ṅSt = 0-locus is represented by:{}(aL /κ) (aL /κ)nS + [aI − (aL /κ)] nS
VL =,(1 − δ)λaI LS − (aL /κ)LN nS

15

(17)

whereas the equation for the V̇L,t = 0-locus is given by:
[
]
aL /κ
1
aL /κ
VL =
+
a
(λ
−
1)
−
nS .
I
λ(LN + aI ρ) 1 − δ(1 − δ)λ(LN + aI ρ)(18)

The ṅSt = 0-locus is upward sloping and remains in a finite region provided that innovation
requires more labor inputs than licensing in order to attain a certain probability of successand that the South is endowed with relatively abundant labor. In greater detail, the conditions
are:aL
aL /κ NL ≤ LS .
< aI and
κ
aI

(19)

Furthermore, to ensure that the two loci cross once, we assume that:(aL /κ)(LN + aI ρ) − (1 − δ)(λ − 1)[aI LS − (aL /κ)LN ] > 0.

(20)

The inequality is the condition such that the VL coordinate of the ṅSt = 0-locus exceeds thatof the V̇L,t = 0-locus at nS = 1. As the V̇L,t = 0-locus lies above the ṅSt = 0-locus at
nS = 0, the two loci cross at least once if the restriction is fulfilled. This economy may
then have a steady state that is a saddle point under appropriate additional assumptions.
Moreover, in the steady state, a strictly positive fraction of products is under license andmanufactured in the South.
To characterize the economy completely and seek out the steady state, we must investigate the evolution of the third variable, VI,t . Imposing It > 0 and ιt > 0, from equations(3), (4), (8), (11), and (12), we obtain the equation of motion for VI,t , as follows:
)
]
)
(
[(
aL /κ
LS
nSt
1
LN
1
+
VI,t
+
−
V̇I,t =
ρ+aI
(1 − δ)λVL,taIaL /κ 1 − nSt
1 − nStδLS VL,t
nSt
δ
−1−
+
.
aL /κ 1 − nSt(1 − δ)λ 1 − nSt(21)

Using the three variables, nSt , VL,t , and VI,t , we state some conditions under which a
16feasible steady state of the economy exists. Let n̄S , V̄L , and V̄I denote the values of the fixed
points in the differential equation system composed of (13), (14), and (21). The condition(5) must then be imposed on V̄L and V̄I so that the steady state is attainable. Moreover,in equilibrium, the Southern wage rate must be less than the Northern wage rate, while
the Northern wage rate cannot exceed the Southern wage rate multiplied by λ, i.e., wtS <wtN < λwtS . Under the assumptions that It > 0 and ιt > 0, from equations (3) and (4), thecondition is described as follows:aI (1 − δ)aI (1 − δ)λVL,t < VI,t <VL,t .
aL /κ
aL /κ

(22)

In addition, in order that both innovation and licensing take place in the steady state, n̄S andV̄L must take values that satisfy (15) and (16). This restriction corresponds to intersection
A in figure 1 falling outside the shaded area because the point represents the coordinate of(n̄S , V̄L ). If nSt , VL,t , and VI,t satisfy all of those conditions and the steady state is attainable,
it is a saddle point (see Appendix). These results are stated as the following proposition.
Proposition 1: Suppose that parameters are under conditions (19) and (20). Then, theeconomy has a steady state with positive innovation and licensing if the steady-state valuesn̄S , V̄L , and V̄I satisfy all of the conditions (5), (15), (16), and (22). Moreover, the steadystate is a saddle point.
A numerical example of parameters where the steady state exists is provided in the next
section. As nSt is a state variable, the saddle path converging to the steady state is theequilibrium trajectory. Along this saddle path, the fraction of licensed products increases
over time when the economy is below its steady-state value.

17

4 Effects of Changes in the Smoothness of Negotiation and
Rent Distribution
In this section, we investigate the effects of an improvement in the smoothness of license
negotiation and an increase in the license fee rate. In the first part of this section, we analyze
the long- and short-run effects of the improvement in the smoothness of license negotiation.
Later, we examine the long- and short-run effects of higher license fee rates.

4.1 An improvement in the smoothness of license negotiation
The improvement in the smoothness of license negotiation is expressed by an increase in κ.
First, we examine the long-run effect on innovation and licensing by conducting comparative statics. Combining equations (17) and (18), we derive n̄S as a positive solution of thefollowing equation:
a2L ρ(n̄S )2 + Bn̄S − C = 0,(23)

where:{}
B ≡ aL κ LS + aI ρ + LN [λ − (λ − 1)δ] − a2L ρ > 0,C ≡ κLS [aL + aI (1 − δ)κ(λ − 1)] > 0.

Taking a total differential of the equation (23), we obtain:
∂ n̄S
1
= 2 S
∂κ
2aL ρn̄ + B()∂C ∂B S
−
n̄ .
∂κ
∂κ

(24)

This equation shows that an increase in κ increases the fraction of licensed products aslong as (∂B/∂κ)n̄S < ∂C/∂κ. Noting that ∂B/∂κ = (B + a2L ρ)/κ and ∂C/∂κ = (C/κ) +aI (1−δ)κ(λ−1)LS , from the condition on parameters and equation (23), we can verify that∂C/∂κ − (∂B/∂κ)n̄S is greater than (aL /κ)[(1 − δ)κ(λ − 1)LN − aL ρ(1 − n̄S )]. Exploiting18

condition (15), which is necessary for the steady state with positive innovation, and equation(18) representing the V̇L = 0-locus, we obtain (1−δ)κ(λ−1)LN −aL ρ(1− n̄S ) > 0. Hence,∂C/∂κ is always larger than (∂B/∂κ)n̄S ; that is, ∂ n̄S /∂κ > 0 in any case.
The effect on the value of V̄L is computed by using the effect on n̄S . As the fixed point of
the system is located on the V̇L,t = 0-locus, V̄L is related to n̄S by equation (18). Therefore,
the long-run response of VL,t to a change in κ is given by:
∂ V̄L
aL /κ
=−
∂κ
(1 − δ)λ(LN + aI ρ)(

1 − n̄S ∂ n̄S
+
κ
∂κ

)
< 0.

(25)

Exploiting the above, we can examine how an increase in κ affects the remaining variables. First, we calculate the long-run effect on innovation. As the innovation intensity attime t satisfies (11), taking the derivative of I¯ with respect to κ, we obtain the following:
∂ I¯
aL /κ
=
∂κ
aI (1 − δ)λV̄L(∂ n̄S 1 − n̄S 1 − n̄S ∂ V̄L++∂κκ
V̄L ∂κ

).

(26)

The above equation shows that the change in κ affects I¯ through three channels: through thechange of n̄S , the direct effect, and the change of V̄L . These effects are competing because
whereas the first two effects encourage innovation, the third effect weakens the incentive
for innovation. However, using equations (18) and (25), we can immediately confirm that¯
∂ I/∂κ
> 0. That is, the first two positive effects dominate the third negative effect, and theincrease in κ induces greater innovation in the long run.
Intuitively, an increase in κ has two effects that lead to a decrease in Northern labor
employed in the production sector. First, more products are manufactured in the South underlicense. This is because Southern firms are more eager to engage in license negotiations
because they require less labor to attain a unit probability of successfully achieving a license
agreement. This first effect is expressed by the first term in parentheses in equation (26).
Second, there is less demand for each product and, therefore, less demand for labor from19

each incumbent leader. The reduction in demand occurs because the stronger incentives toundertake licensing negotiations caused by an increase in κ lead to a rise in the Southern
wage rate, as verified later. An increase in the Southern wage rate involves higher prices
for products as each leader adopts a limit-pricing strategy to compete with the Southernclosest rivals. This second effect is expressed by the second and third terms in parentheses
in equation (26). These two effects decrease labor demand from Northern incumbent leaders
and, consequently, increase the labor input for R&D activities in the steady state.
In addition, an increase in κ positively affects licensing intensity in the long run. As
equation (6) implies that ῑ = I¯n̄S /(1 − n̄S ) in the steady state, the effect of the rise in κ on
ῑ is given by:
∂ n̄S
∂ῑ
n̄S ∂ I¯
I¯
=
+
> 0.
∂κ
1 − n̄S ∂κ (1 − n̄S )2 ∂κ
Moreover, the aggregate amount of licenses, n̄N ῑ, is positively related to κ. We can summarize the above analysis as the following proposition.
Proposition 2: An improvement in the smoothness of license negotiation promotes both innovation and licensing in the long run.
Furthermore, our model can answer another related and important question: does the
improvement encourage innovation and licensing in the short run as well as the long run?
Many related studies are unable to respond to this kind of question because they focus only
on the steady state. In contrast, our analysis enables us to examine the short-run effectbecause it fully describes the progress of the economy.
To investigate the short-run effect, we exploit the same approach as Helpman (1993).16
For tractability, we restrict the analysis to an economy that initially stays in the steady state:
namely, nS0 = n̄S . Then suppose that an unanticipated marginal increase in κ occurs at time
0. We can calculate the first-order response of (nSt , VL,t ) to the marginal increase in κ froma linearized system of the differential equations (13) and (14) around the steady-state value.
16

Kwan and Lai (2003) have adopted the same method in their closed economy model.

20

In the Appendix, we show that:¯
S∂nSt ¯¯−xt ∂ n̄
=
(1
−
e
)
,
∂κ ¯nS =n̄S
∂κ

(27)

¯
S
∂ V̄L
∂VL,t ¯¯
−xt ∂ n̄=+Λe,∂κ ¯nS =n̄S
∂κ
∂κ

(28)

0

and

0

where x is the absolute value of the negative eigenvalue of the linearized coefficient matrix,and Λ, which represents the second element of the eigenvector associated with the negative
eigenvalue, is also positive. Because It follows equation (11), taking into consideration
the initial condition nS0 = n̄S and the condition ∂nS0 /∂κ = 0, we can derive the effect oninnovation intensity at time 0, as follows:)(¯¯(aL /κ)(1 − n̄S )κ ∂VL,0 ¯¯∂I0 ¯¯=.
1+∂κ ¯nS =n̄S ,VL,0 =V̄L
aI (1 − δ)κλV̄LV̄L ∂κ ¯nS =n̄S
00This equation suggests that the extent to which the innovation intensity responds to theincrease in κ depends on the elasticity of VL,0 with respect to κ. If the elasticity exceeds −1,then ∂I0 /∂κ|nS0 =n̄S ,VL,0 =V̄L is positive: that is, innovation is stimulated with an increase in κ
in the short and long term. To compute the elasticity, we must know the value of ∂VL,0 /∂κ.
However, equation (28) implies that ∂VL,0 /∂κ|nS0 =n̄S is greater than ∂ V̄L /∂κ. As we canverify that (κ/V̄L )(∂ V̄L /∂κ) > −1 (see the Appendix), we conclude that the elasticity of
VL,0 with respect to κ evaluated at the steady-state value also exceeds −1. As a result, weshow that ∂I0 /∂κ|nS0 =n̄S ,VL,0 =V̄L > 0.
Similarly, using equation (12), we have the short-run effect on licensing intensity as
follows:][¯¯∂ι0 ¯¯
LS
1n̄S∂VL,0 ¯¯=.
+∂κ ¯nS =n̄S ,VL,0 =V̄L
1 − n̄S aL(1 − δ)λ(V̄L )2 ∂κ ¯nS =n̄S
0

0

21

(29)

We can confirm that this ∂ι0 /∂κ|nS0 =n̄S ,VL,0 =V̄L is also greater than zero (see the Appendix).
Thus, these results prove the following proposition.
Proposition 3: The improvement in the smoothness of license negotiation promotes bothinnovation and licensing in the short run as in the long run.
How are the wages in both countries affected by an increase in κ in the steady state? As
discussed, the Southern wage rate in the steady state rises unambiguously. Using equation(4), we obtain:
∂ w̄S
1−δ
(1 − δ)κ ∂ V̄L
=
V̄L +
> 0.
∂κ
aL
aL
∂κ

(30)

Turning to the relative wage rate between the North and the South, we may have difficulty incomputing the effect because the impact on the Northern wage rate is unclear. Therefore, byusing some numerical examples of parameters, we have examined the effect on the relative
wage rate (see figure 2).17 As a result, we found that the relative wage rate of the South ismonotonically increasing with an increase in κ for all parameter values chosen.

4.2 A higher license fee rate
An increase in the license fee rate is represented by an increase in δ. The higher license fee
rate means that the Northern licensors can enjoy a larger monopolistic rent after the licenses,
whereas the Southern licensees receive a smaller rent. In the first part of this subsection, we
examine the comparative statics with respect to δ. Next, we show the short-run effect of theincrease in δ on innovation and licensing agreements by exploiting the same approach as
employed in the previous subsection.
In order to derive the long-run effects, we first compute the derivative of n̄S and V̄L with

17
Figure 2 is an output of the numerical calculation. In the figure, we specify the parameters as aI = 7, aL =3.5, λ = 1.5, LN = 1, LS = 2, and ρ = 0.05. Other examples are available from the authors upon request.

22

respect to δ. Totally differentiating equation (23) implies that:∂ n̄S
κ(λ − 1)(aI κLS − aL LN n̄S )
=−
< 0.
∂δ
2a2L ρn̄S + B(31)

Furthermore, using the V̇L = 0-locus, we can derive the following:
∂ V̄L
aL /κ
=
∂δ
(1 − δ)λ(LN + aI ρ)(

1 − n̄S
∂ n̄S
−
1−δ
∂δ

)
> 0.

(32)

The first term of this expression represents the direct effect of a change in δ, whereas thesecond represents the indirect effect that occurs through the change of n̄S . These effects
complement each other and shift V̄L in the same direction. Consequently, V̄L responds positively to an increase in share of Northern rents.
¯
Using the above derivatives, we compute ∂ I/∂δusing the same method as in the previous subsection. The effects of a change in δ on innovation intensity in the steady state
are:
∂ I¯
aL /κ
=
∂δ
aI (1 − δ)λV̄L

(

∂ n̄S
1 − n̄S 1 − n̄S ∂ V̄L
−
+
∂δ
1−δ
V̄L ∂δ

)
< 0.

(33)

The change in δ affects I¯ through three channels: a change in n̄S , a direct effect, and achange in V̄L . An intuitive interpretation is as follows. First, as shown, a higher δ results in alower n̄S ; that is, more leaders begin to operate in the North. The expansion of industries belonging to the N market creates additional Northern labor demand from incumbent leaders,which leads to lower innovation intensity. The first term in parentheses in (33) represents
this effect. Second, the higher δ discourages Southern followers from pursuing licensing
efforts and reduces the Southern wage rate because of its lower return, all other things remaining unchanged. The lower Southern wage rate obliges the incumbent leaders to charge
a lower price and generates additional product demand. As a result, Northern labor demand
from each incumbent leader is also increasing with δ. The second and the third terms inparentheses in (33) represent the second effect. These two effects increase the labor demand23¯from Northern incumbent leaders and, consequently, decrease I.
The effects on licensing intensity ῑ are computed in the same way as in the previous
subsection. ∂ῑ/∂δ is derived by:
∂ῑ
∂ n̄S
n̄S ∂ I¯
I¯
=
+
< 0.
∂δ
(1 − n̄S )2 ∂δ
1 − n̄S ∂δ¯As both ∂ n̄S /∂δ and ∂ I/∂δare negative, ∂ῑ/∂δ is also negative. Hence, a higher license fee
rate reduces the efforts of Southern followers to negotiate a license contract in the long run.
Next, we investigate the short-run effects of a change in rent sharing on innovationand licensing. Using analogues of equations (27) and (28), we obtain: ∂VL,0 /∂δ|nS0 =n̄S =(∂ V̄L /∂δ) + Λ(∂ n̄S /∂δ) and ∂nS0 /∂δ = 0 for the economy that initially stays in the steadystate. Hence, equations (11) and (12) imply that:¯)][((aL /κ)(1 − n̄S )1∂I0 ¯¯1 ∂ V̄L
∂ n̄S
=
,
−
+
+Λ
∂δ ¯nS =n̄S ,VL,0 =V̄L
1 − δ V̄L ∂δ∂δaI (1 − δ)λV̄L
0

and
¯
)]
(
[
∂ι0 ¯¯n̄S
1
1 ∂ V̄L
dn̄S
=
.
−
+
+Λ
∂δ ¯nS =n̄S ,VL,0 =V̄L
1 − δ V̄L ∂δdδ(1 − δ)λ(1 − n̄S )V̄L
0

Using equations (18), (31), and (32), and the definition of B, we can verify that −[1/(1 −δ)] + (1/V̄L )(∂ V̄L /∂δ) is less than zero. Thus, both innovation and licensing intensity attime zero respond negatively to an increase in the license fee rate.
Finally, we examine the effect of an increase in δ on the Southern wage rate and therelative wage rate in the steady state. Equation (4) implies that:∂ w̄S(1 − δ)V̄L
=
∂δ
aL /κ

(

1
1 ∂ V̄L−+
1 − δ V̄L ∂δ

)
< 0.

(34)

Therefore, a higher δ pushes the Southern wage rate down in the steady state. Moreover,24

we ascertain the tendency of the effects on the Southern relative wage rate by using somenumerical examples (see figure 2). From the results, we found under reasonable ranges ofparameters that the Southern relative wage rate is monotonically decreasing with δ.
The above results are summarized as the following proposition.
Proposition 4: A higher license fee rate reduces both innovation and licensing in both thelong and short run.
This proposition suggests that an excessively high license fee rate results in low licensing
efforts and interferes with the smooth transfer of production to the South. In addition, as
the Southern wage rate falls with the license fee rate, manufacturing per firm in the N market increases. These two effects lead to greater production in the North, which discourages
innovation through a decrease in labor inputs for research. As a result, lower quality improvements take place, and the expected duration of existing products increases.

4.3 Discussion
How can these results be compared with those of Yang and Maskus? Yang and Maskus haveargued that strengthening IPR protection in the South promotes innovation and technology
licensing in the long run. They do so by assuming that stronger IPR protection in the South
has two influences: an increase in the probability of success in licensing for a given resource
spending and an increase in the licensor’s share of rents. Yang and Maskus named theformer “the size effect” of strengthening IPR and the latter “the distribution effect”. In
addition, they concluded that “the size effect and the distribution effect would enhance eachother in encouraging more licensing and more innovation in response to stronger Southernprotection of intellectual property” (Yang and Maskus, 2001, p.182).
In comparison with Yang and Maskus, our analysis in section 4.1 shows that an improvement in the smoothness of licensing negotiation, corresponding to the size effect in Yang and
Maskus’ analysis, encourages innovation and licensing. Therefore, our analysis of proposi25

tion 2 can be interpreted as proof that Yang and Maskus’ conclusion that the size effect ofstrengthening IPR encourages innovation and licensing in the long run holds, even if theirmodel is modified to have a stable path converging to the steady state. Moreover, we obtaina richer conclusion on the short-run effects than Yang and Maskus. Thus, one can regard
our analysis in section 4.1 as reinforcing and extending the Yang and Maskus argument on
the size effect induced by stronger IPR protection.
In contrast, our analysis in section 4.2 shows that an increase in the license fee rate
discourages innovation and licensing in the long run, as opposed to the analysis of the distribution effect in Yang and Maskus. We can interpret causes of this significant difference
between our analysis and Yang and Maskus as follows: in a Yang and Maskus setting,Northern patent holders take an active part in the technology transfer to developing countries. Therefore, when tighter IPR induce a higher rent share for licensors, Northern leaders
have more incentive to engage in license activities. This accelerates both licensing and innovation. On the other hand, in our setting, Southern firms that hope to be granted a license
play an important role in the technology transfer. Because a higher license fee rate leads to alower rent share for licensees, Southern followers have fewer incentives to engage in license
negotiation activities under the higher license fee rate. This slows down both licensing andinnovation. Consequently, our results concerning an increase in the license fee rate lie opposite to that in Yang and Maskus. Thus, from our analysis in section 4.2, one can draw theconclusion that Yang and Maskus’ argument concerning the distribution effect is not robust
and is crucially dependent on their assumption that patent holders themselves make an effort to license the Southern firms to sell the products. From this comparison between ourmodel and that in Yang and Maskus, we can surmise that whether an increase in δ enhances
technology transfer and innovation depends largely on who plays an important role in technology transfer. In order to obtain a more general result, we need to examine the effect of
the licensing fee in a more general model where Northern firms, as well as Southern firms,
devote their resources to licensing activities. Examination of such a general model remains
26

an important task for future study. We provide a related discussion about this general modelin the conclusion.

5 A Comparison with Models with Imitation
In this subsection, in order to show the differences between licensing and imitation as alternative channels that transfer technology from the North to the South, we compare thecharacteristics and results of the present technology licensing model with such an imitation
model as Grossman and Helpman (1991, ch.12).
The basic structure of our model is similar to the North–South quality-ladder modelwith endogenous innovation and imitation.18 We can obtain a model where the channel oftechnology transfer is imitation by making a few changes in the present licensing model.
Envisage that Southern follower firms in our model spend labor not to negotiate a licensing
agreement with patent holders but to imitate products. We then interpret ι as the probabilityof success in imitation and interpret wS aL /κ as the unit cost of imitation. In the case oftechnology transfer through imitation, a Southern firm can earn all of the monopoly profits
without paying a license fee at each point of time if the firm succeeds in the technology
transfer, while a Northern innovator cannot obtain profits after the success of imitation in
the industry. If we set δ = 0 in equations (4) and (8), we could consider the equations
to be the zero-profit condition for the imitation activities and the no-arbitrage condition
concerning the stocks of the Northern firms, respectively. Moreover, we note that our model
can be regarded as a model with endogenous innovation and imitation by ignoring condition(5).
In fact, it is the existence of the license fee and the condition (5) that are a key feature
in separating our technology licensing model from other endogenous imitation models. In
the case that a Southern follower firm offers a Northern patent holder a license contract, we
18

The similarity between licensing and imitation in our model was pointed out by an anonymous referee.

27

consider the possibility that the Northern patent holder refuses the offer as discussed above.
For example, if the license fee rate δ were zero, a patent holder would never enter into alicense contract. As a result, no equilibrium with positive licensing would exist in the casewhere the license fee rate is too low. To exclude this possibility and ensure a sufficiently
high license fee rate, we imposed the condition (5). As long as this condition is satisfied,reaching license agreement is beneficial not only to the Southern firm granted the license
but also to the Northern patent holder. Meanwhile, if a Southern firm imitates the designof a state-of-the-art product without the permission of the patent holder, then we do nothave to take into account the possibility that the Southern firm fails in the imitation activity
through the objections of the patent holder. Therefore we do not need to impose a condition
corresponding to condition (5) in the technology licensing model. However, in the case oftechnology transfer through imitation, a Southern imitator benefits from succeeding in theimitation at the cost of the monopolistic rent of the Northern patent holder.
By utilizing the similarities between our technology licensing model and endogenous
imitation models, and the results of sections 4.1 and 4.2, we can compare the effects of anincrease in κ and δ in the case of licensing with those in the case of imitation that corresponds
to the model where δ is set at 0 and condition (5) is not imposed.19 Applying proposition 3 tothe case of δ = 0, we can show that a decrease in imitation cost (an increase in κ) promotes
innovation and imitation in both the short and long run. Hence, it is concluded that whether
the channel of technology transfer is licensing or imitation, a decrease in cost for technology
transfer enhances not only technology transfer but also innovation. In addition, according toproposition 4, the intensities of innovation and licensing, I¯ and ῑ, are decreasing functions of
δ. Therefore the intensities are maximized at δ = 0. This implies that innovation, as well as
technology transfer, is more active when the channel of technology transfer is imitation thanwhen the channel of technology transfer is licensing. It may be somewhat counterintuitive
19
Though we can obtain the results concerning innovation and imitation by using this similarity, we cannot
easily obtain results concerning the relative wage rate between the North and the South and welfare, because
we need to examine the dynamics of VI,t to derive the relative wage rate and welfare on the equilibrium paths.

28

that more innovation occurs in the case of imitation than in the case of licensing because
the Northern innovator can receive the rent after technology transfer in the case of licensing
but not in the case of imitation. The interpretation is similar to that of proposition 4: since
the existence of a license fee that partly discourages Southern follower firms from acquiring
advanced technologies leads to the lower frequency of technology transfer to the South, as
it causes more production and fewer research activities in the North.

6 Concluding Remarks
The international transfer of technology is brought about through various channels, including licensing. This paper has presented a quality-ladder type of product-cycle model where
licensing is introduced as the channel of technology diffusion. In this model, we supposedthat firms in developing countries must incur costs and input resources in their efforts to win
a license contract. In practice, such licensee firms often play an important role in reducing technological backwardness in recipient countries. Our model captures the activity offirms in the recipient countries and shows the existence of the steady state in which positive
innovation and licensing continue to take place.
An important advantage of our analysis is that we fully explore the dynamic nature of theeconomy. As a result, we have succeeded in verifying that the steady state in the economy
is a saddle point. Moreover, owing to the analysis of the dynamic system, our study has
yielded some results with respect to the short-run effects of a change in the parameters.
Many existing studies compromise such analysis by only drawing conclusions about thelong-run effects, whereas our dynamic analysis enables us to determine both the short-run
and the long-run effects.
Although the model developed in this paper will assist us to comprehend better innovation and international transfer of technology through licensing, some topics are left open
for future research. For instance, one could consider a “hybrid” between our model and29

that presented in Yang and Maskus. In that model, Northern firms could use their resources
for licensing activities, for example, to monitor the transfer of technology, and the Southernfirms could use their resources for licensing negotiations. To maintain simplicity, we haveassumed that only Southern firms use their resources to reach a license agreement, while
Yang and Maskus assumed that only Northern firms use their resources to succeed in licensing. Because the difference in assumptions generates the distinct stability property of thesteady state between our model and that in Yang and Maskus, examination of the stabilityof the hybrid model remains an important task for future studies. Perhaps one may be able
to discover a range of parameters that stand for the labor inputs of both parties necessary to
the licensing activities in which the stability of the hybrid model is ensured.20
One could also consider an extended version of our model in the sense that licensing
and another mode of technology transfer coexist. So as to focus on licensing, we haveexcluded channels of technology transfer other than licensing. Consequently, our analysis
is restricted in the following ways. First, our licensing model does not take into account
imitation activities, although these are widely observed and constitute a major source oftechnology acquisition in the early development stage. On that point, our results are more
likely to be applied to middle-income countries than less-developed countries. Second, we
have assumed that foreign direct investment is impractical. In reality, however, there are
two types of middle-income developing countries: one encourages domestic firms to learn
advanced technologies through licensing from firms in developed countries, whereas theother prefers FDI by multinational firms to licensing. Clearly, we take only the former as
the object of our analysis. Hence, to provide a North–South technology licensing model
where a mode of technology transfer exists in addition to licensing is also an important
topic for future study.21 Our model would probably then be a good starting point for these20

The authors thank an anonymous referee for these points.
For example, Antràs (2005) has constructed a simple model such that under the environment of incomplete
contracts, a Northern research firm can choose the mode of manufacturing: to transact with an independent
Southern firm (licensing) or to integrate the Southern manufacturing plant vertically (multinationalizing), although the model assumed the innovation process to be exogenous for the purpose of simplicity.
21

30

explorations.

A Appendix
In this appendix, we derive the effect of an increase in κ on innovation and licensing inthe short run. To do so, we first compute the negative eigenvalue and the corresponding
eigenvector of the system. The linearized system of (13), (14), and (21) is:










S
S
  nt − n̄  −b11 b12 0


 

  V − V̄  , V̇  =  b
b
0
L
22
  L,t
 L,t   21


 

VI,t − V̄Ib31 b32 b33
V̇I,t
S
 ṅt

where:

(35)

[
]
LN
aL /κ
1
S
b11 =
1−
+
(1 − 2n̄ ) > 0,aI
aI
(1 − δ)λV̄L
[
]
aL /κ
n̄S
S
1−
b12 =
(1 − n̄ ) > 0,aI
(1 − δ)λV̄L2
b21 =

aL /κ
> 0,
aI (1 − δ)λb22 = ρ +

LN
> 0,
aI

and
b33 = ρ +LN
> 0,
aI (1 − n̄S )while b31 and b32 are irrelevant to the analysis. The eigenequation associated with the coefficient matrix on the right-hand side is:(y − b33 )[y 2 + (b11 − b22 )y − b11 b22 − b12 b21 ] = 0.

31

(36)

As the equation y 2 + (b11 − b22 )y − b11 b22 − b12 b21 = 0 has one positive and one negative
solution, the eigenequation (36) has two positive and one negative solution. Therefore,
the steady state is a saddle point and there exists a stable saddle path converging to it. In
addition, recalling that nSt is a state variable, while VL,t and VI,t are jump variables, thenumber of negative eigenvalues corresponds to that of the state variable.
Next, we compute the approximate saddle path around the steady state, using the linearized system of differential equations (35). In integrating the linearized differential equations, we have to base our choice of free integral constants on the conditions that a stablesaddle path converges to the steady state and that nSt is a state variable whose initial value ishistorically given. This procedure yields the following expressions:nSt = n̄S + (nS0 − n̄S )e−xt ,(37)

VL,t = V̄L − Λ(nS0 − n̄S )e−xt ,(38)

where x is the absolute value of the negative eigenvalue, and Λ, which represents the absolute value of the second element of an eigenvector associated with the negative eigenvalue,
is also positive. The definition of an eigenvalue and an eigenvector implies the following:1
x = (D + b11 − b22 ),
2

Λ=

1
1
(x − b11 ) =(D − b11 − b22 ),
b12
2b12

where D ≡ [(b22 − b11 )2 + 4(b11 b22 + b12 b21 )]1/2 > b11 + b22 . Because equations (37)and (38), of which the linearized stable saddle path consists, are consolidated into VL,t =−ΛnSt + (V̄L + Λn̄S ), the stable saddle path projecting on the (nS , VL ) plane has a negative
slope.
Third, we compute the change of nSt and VL,t to an unexpected marginal rise of κ. Dif-32

ferentiating (37) and (38) with respect to κ, we obtain:
∂nSt
∂ n̄S
∂x
= (1 − e−xt )− (nS0 − n̄S )te−xt ,
∂κ
∂κ
∂κ

(39)

∂ V̄L
∂VL,t
∂ n̄S
∂Λ
∂x
=
+ Λe−xt− (nS0 − n̄S )e−xt+ Λ(nS0 − n̄S )te−xt .
∂κ
∂κ
∂κ
∂κ
∂κ

(40)

As we consider that the economy initially stays in the steady state, namely, nS0 = n̄S , thesecond term of (39) and the last two terms of (40) are equal to zero. Hence, the derivativesof nSt and VL,t with respect to κ on the steady state are given by (27) and (28) in the text. In
particular, the size of the initial jump responding the policy change is derived as:¯∂ V̄L∂VL,0 ¯¯
∂ n̄S
=
+
Λ
.
∂κ ¯nS =n̄S∂κ∂κ0Figure 3 depicts the situation where κ increases. The figure indicates that ∂VL,0 /∂κ|nS0 =n̄S
is greater than ∂ V̄L /∂κ because the stable saddle path inclines negatively. Therefore, theelasticity of VL,0 with respect to κ evaluating at the steady-state value is larger than that ofV̄L :¯κ ∂ V̄Lκ ∂VL,0 ¯¯.
<V̄L ∂κ
V̄L ∂κ ¯nS0 =n̄S
Fourth, we show that (κ/V̄L )(∂ V̄L /∂κ) is greater than −1 in order to prove ∂I0 /∂κ|nS0 =n̄S ,VL,0 =V̄L >0. Substituting (24) into (25), we obtain:
∂ V̄L
aL /κ
=− 2 S∂κ(2aL ρn̄ + B)(1 − δ)λ(LN + aI ρ)[]
1∂C ∂B SSS
2×
(2a ρn̄ + B)(1 − n̄ ) +
−
n̄ .
κ L
∂κ
∂κ
Then, let us notice the content of the square bracket of this equation. Using the equation (23),
we can readily verify that (2a2L ρn̄S + B)(1 − n̄S ) = (2a2L ρ + B)n̄S + (B − 2C). In addition,from the definitions of B and C, ∂B/∂κ = (a2L ρ + B)/κ and ∂C/∂κ = −aL LS + 2C/κ.

33

Hence, we can rewrite ∂ V̄L /∂κ as:
∂ V̄L
aL /κ
=− 2 S∂κ(2aL ρn̄ + B)(1 − δ)λ(LN + aI ρ)( 2)
aL ρ S B
Sn̄ + − aL L .
κ
κ

(41)

Therefore, exploiting (18) and (41), we obtain the following expression about the elasticityof V̄L with respect to κ:κ ∂ V̄LaL (a2L ρn̄S + B − aL κLS )=− 2 S.
(2aL ρn̄ + B) [aL (1 − n̄S ) + aI (1 − δ)κ(λ − 1)]V̄L ∂κ

(42)

By using (23) we can show that the denominator on the right-hand side of equation (42)is greater than the numerator and, thus, we conclude that (κ/V̄L )(∂ V̄L /∂κ) > −1 is true.
Hence, innovation intensity at time zero responds positively to a marginal increase in κ.
Finally, we confirm that ∂ι0 /∂κ|nS0 =n̄S ,VL,0 =V̄L is greater than zero. For the sake of theconfirmation, we show that (LS /aL )+[n̄S /(1−δ)λ(V̄L )2 ](∂ V̄L /∂κ) > 0. As ∂VL,0 /∂κ|nS0 =n̄S
is larger than ∂ V̄L /∂κ, the inequality and equation (29) imply ∂ι0 /∂κ|nS0 =n̄S ,VL,0 =V̄L > 0.
Using (18) and (41), together with the definition of B, we obtain:
[
]
n̄S
LS
∂ V̄L
+
2
aL
∂κ
(1 − δ)λ(V̄L )(C − aL κLS n̄S )=(2a2L ρn̄S + B) [aL (1 − n̄S ) + aI (1 − δ)κ(λ − 1)]2 κ2 LS{
× aL n̄S ρ(C − aL κLS n̄S ) + κ(LN + aI ρ)(C − aL κLS )}+C[(1 − δ)κ(λ − 1)LN − aL ρ(1 − n̄S )] .
Note that, from equations (15) and (18), the term (1 − δ)κ(λ − 1)LN − aL ρ(1 − n̄S ) is
positive under the situation with positive innovation in the steady state. Therefore, LS /aL +[n̄S /(1 − δ)λV̄L ](∂ V̄L /∂κ) is greater than zero. This means that licensing intensity at time
zero reacts positively to a marginal increase in κ; that is, ∂ι0 /∂κ|nS0 =n̄S ,VL,0 =V̄L > 0.
Thus, the proof of proposition 3 has been completed.
34

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35

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36

VL[(aL /κ)+aI (1−δ)(λ−1)](1−δ)λ(LN +aI ρ)
aL /κ(1−δ)λLN

V̇L,t = 0 locusṅSt = 0 locusaI (aL /κ)(1−δ)λ[aI LS −(aL /κ)LN ]It = 0A

V̄L
aI (λ−1)
λ(LN +aI ρ)
aL /κ(1−δ)λLSaL /κ(1−δ)λ(LS +LN )ιt = 0nS
LS
LS +LNO

Figure 1: Phase diagram

37

n̄S 1

1.3

1.2

1.1
0.5
0.65

0.6

0.6
0.7

0.55
0.5

0.8

Κ

∆

0.45

Figure 2: Relative wage rate corresponding to each value of δ and κ38w Nw SVL

the old steady state

initial response =

∂VL,0
V̄L
n̄S
= ∂∂κ
+ Λ ∂∂κ
∂κ

∂ V̄L
∂κ

stable saddle path: convergence speed = xSn̄−Λ ∂∂κ

−Λthe new steady state

∂ n̄S
∂κ

nS

O

Figure 3: Response to a rise of κ at the initial time and in the long run

39