Discussion Papers In Economics
And BusinessEconomic Growth with Imperfect Protection ofIntellectual Property Rights
Ryo Horii and Tatsuro Iwaisako

Discussion Paper 05-23-Rev.

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANEconomic Growth with Imperfect Protection ofIntellectual Property Rights
Ryo Horii and Tatsuro Iwaisako

Discussion Paper 05-23-Rev.

April 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Economic Growth with Imperfect Protection ofIntellectual Property Rights∗
Ryo Horii† and Tatsuro Iwaisako‡
April 18, 2006

Abstract
This paper examines the growth eﬀects of intellectual property right (IPR)
protection in a quality-ladder model of endogenous growth. Stronger IPR
protection, which reduces the imitation probability, increases the reward forinnovation. However, stronger protection also gradually reduces the number ofcompetitive sectors, in which innovation is easier than in monopolistic sectors.
With free entry to R&D, the number of researchers in each remaining competitive sector increases, but the concentration of R&D activity raises the possibility of unnecessary duplication of innovation, thereby hindering growth. Consequently, imperfect rather than perfect protection maximizes growth. Welfare
and scale eﬀects are also examined.
Keywords: intellectual property rights, endogenous growth, quality ladder,
imitation, leapfrogging, duplication.
JEL Classification: O31, O34, O41
Running Head: Growth with Imperfect IPR Protection.

∗

The authors would like to thank Taro Akiyama, Munetomo Ando, Koichi Futagami, Kazuo
Mino, Takao Ohkawa, Tetsuya Shinkai, Ryu-ichi Tanaka, participants at the Fall 2004 JEA Annual
Meeting, attendees at the Kansai Macroeconomic Workshop, Osaka University, the Policy ModelingWorkshop at GRIPS, and Ritsumeikan University, and three anonymous referees for their helpfulcomments and suggestions on an earlier version of this paper. This study was partly supported by
JSPS Grant-in-Aid for Scientific Research (Nos. 16730097 and 16730104). All remaining errors are,of course, our own.
†

Corresponding author: Graduate School of Economics, Osaka University, 1-7 Machikaneyama,
Toyonaka, Osaka, 560-0043, Japan. Tel: +81-6-6850-6111. Fax: +81-6-6850-5274. Email:horii@econ.osaka-u.ac.jp‡

Faculty of Economics, Ritsumeikan University, 1-1-1 Noji-higashi, Kusatsu, Shiga 525-8577,
Japan. Email: tiwai@ec.ritsumei.ac.jp1

1

Introduction

In the last two decades, a number of countries have reformed their patent systems inorder to strengthen the protection of intellectual property rights (hereafter, IPRs).
Such reforms are often justified by the view that stronger IPR protection should enhance economic growth by increasing the returns to innovation and, hence, incentives
to innovate. However, the relationship between IPR protection and growth is not asclear as widely believed.1 Figure 1 illustrates a scatter plot of the average growthrate and the level of IPR protection, using Rapp and Rozek’s (1990) cross-country
data on the level of patent protection. As shown, although there is substantial variability across countries in both the level of IPR protection and the rate of economic
growth, it is diﬃcult to determine a clear relationship between these two variables.2

Given that stronger IPRs unambiguously provide greater incentives to innovate,
the weak relationship observed between IPR protection and the rate of economic
growth suggests that IPRs may have a negative eﬀect on economic growth. A wellknown drawback of stronger IPRs is that they provide innovators, on average, withlonger monopoly periods and therefore tend to increase the number of monopolistic
sectors within the economy as a whole. In light of this tendency, earlier studies
have demonstrated that consumer welfare is not necessarily improved when IPRs
are strengthened, because monopolists limit their output below the socially desirable
level. In this paper, we present a mechanism through which stronger IPRs negatively
1

Applications for patents in the United States have increased drastically since 1985; however,there is little evidence that this increase was caused by the strengthening of IPR protection. Changes
in the management of R&D and the shift to more applied activities appear to have spurred patenting(Kortum and Lerner, 1998).
2

Gould and Gruben (1996) ran regressions using Rapp and Rozek ’s (1990) index and theaverage growth rate between 1960 and 1988. They found a positive but weak relationship between
IPR protection and economic growth (see Table 2 of their paper). In addition, they pointed out
that, on average, countries with level “three” IPR protection grew more slowly than those withweaker (level “two”) IPR protection.

2

Figure 1: Average per capita growth rate 1960—2000 and the level of patent protection. The horizontal axis ranks the level of patent protection from one to six, where one
corresponds to nations with no patent protection law and higher values indicate countries withpatent laws that are more consistent with the minimum standards proposed by the United States
Chamber of Commerce (1987). The average growth rate is from Heston et al. (2002).

aﬀect the long-term rate of economic growth, by focusing on the following two R&D
properties.
The first property is the diﬀerence in the environment for R&D provided by monopolistic and competitive sectors. In a monopolistic sector, where an incumbent
holds the exclusive right to produce a state-of-the-art good, the incumbent has little
incentive to improve the product further, because it can secure monopoly profits
without such eﬀorts (the Arrow eﬀect, 1962). Thus, in monopolistic sectors, innovative eﬀorts are made only by outside firms, which can succeed only when they create
a good of higher quality than the incumbent’s. Such leapfrogging innovations are
more diﬃcult to achieve than innovations in competitive sectors because the outsidefirms have no experience in producing state-of-the-art-quality goods. In addition,
production of a new superior good often involves a similar process to the productionof current state-of-the-art goods, and therefore, the incumbent’s IPRs may result in3the production of the new superior goods being banned. Thus, ceteris paribus, it
is more diﬃcult for outside firms to invent new high-quality goods in monopolistic
sectors than in competitive sectors.3
Another important property of R&D projects is that they take time, and theiroutcomes (successes or failures) are revealed only after the projects are completed.
This means that individual innovators must initiate R&D projects without knowing
whether other innovators’ projects will eventually succeed. Thus, there is a nonnegligible probability that more than two innovators may independently succeed indeveloping a similar intermediate good of the same quality. Given the nonrivalryof the knowledge obtained by innovation, this duplication of innovations not onlyis futile from the viewpoint of economic growth but also reduces the profits andincentives of innovators.4
Incorporating these two properties of R&D explicitly into the quality-laddermodel of endogenous growth, this paper shows that IPR protection inhibits growthwhen it is too strict. At the same time, a moderate degree of IPR protection isrequired to provide the incentives to innovate. As a result, it is shown that the longterm growth rate is maximized by the imperfect, rather than the perfect, protectionof IPRs.
The remainder of the paper is organized as follows. After intuitively explaining
in the next section the reason why stronger IPR may inhibit growth, Section 3 sets
up a formal model. Section 4 explains how tightening patent protection aﬀects thegrowth rate, assuming that leapfrogging is prohibitively diﬃcult. Section 5 derives

3

Empirically, this statement is confirmed by Blundell et al. (1995) and Nickell(1996), who foundthat innovation is less active in more concentrated sectors. See discussion in Section 5.
4

Earlier studies stressed the importance of duplication in R&D. For example, Kortum (1993)

found evidence of significant diminishing returns in R&D attributable to duplication. Lambson
and Phillips (2005) found that the probability of duplication is not low for most industries. The
possibility of duplication also plays an important role in R&D-based growth models by Jones (1995)
and Jones and Williams (2000), although they use a reduced form setting, whereas we explicitly
consider the time required to innovate.

4the balanced growth path (BGP) in a general setting and presents the main resultsof the paper. Section 6 compares our results with those obtained by earlier studies.
Section 7 discusses the scale eﬀects and the impact on welfare. Section 8 concludes
the paper. The proofs of the lemmas and propositions are provided in the appendix.

2

Overview of the Mechanism

Before presenting a formal model, this section provides a brief intuition for the reasoning why the two properties discussed in the introduction jointly imply that stronger
IPRs may inhibit growth.
Consider a quality-ladder model of endogenous growth and suppose that innovations are imitated over time. If IPR protection is strengthened, the probabilityof imitation declines. Thus, in the long run, stronger IPR protection increases thenumber of monopolistic sectors in which the state-of-the-art-quality goods have notyet been imitated. Because the environment for R&D is assumed to be better incompetitive sectors than in monopolistic sectors, the number of R&D workers per
sector (research intensity) is higher in competitive sectors. Therefore, as stronger IPR
protection reduces the number of competitive sectors, the aggregate R&D activity
declines if the research intensity in each type of sector is fixed (see the horizontal
arrow in Figure 2).
While this provides a central mechanism in this analysis, this finding is not conclusive since research intensity actually changes. Specifically, since increased monopolization lowers aggregate labor demand, the equilibrium market wage falls.5 Thismeans a lower cost of R&D, which induces more research firms to operate in every
sector employing new R&D workers, as illustrated by the vertical arrows in Figure
2. In a standard setting with an inelastic labor supply and constant returns to R&D
activities, entry would continue until it totally oﬀset the initial decline in aggre-

5This is an example of the general-equilibrium eﬀects of IPR policy discussed in O’Donoghue

and Zweimüller (2004, Section 5).

5

Figure 2: Reallocation of R&D workers when IPR protection is strengthened.

gate R&D activities caused by the increased monopolization. However, when thetime required for innovation is considered explicitly, the possibility of duplicationof innovations becomes more prominent as more researchers enter each field. Since
this reduces the expected return on R&D, entry stops before the negative eﬀect ofincreased monopolization on growth is canceled totally.
Accordingly, the net eﬀect of monopoly on growth tends to be negative. Of
course, stronger IPR protection not only intensifies monopolization but also spurs
innovation directly by guaranteeing more rewards for successful innovators. The aimof this paper is to examine the relative significance of the growth-enhancing and-reducing eﬀects of IPRs by explicitly incorporating the two key components into anotherwise standard quality-ladder model of endogenous growth.

3

Model

This section sets up a discrete-time version of a quality-ladder model as based on
Grossman and Helpman (1991). After describing, in the first subsection, households
and production sectors, in the second subsection, we explain how the environments
for R&D diﬀer between competitive and monopolistic sectors. The final subsection
considers the evolution of the value of innovation and the number of monopolistic
6sectors in the economy, allowing for the possibility of imitation.

3.1

Households and Production Technologies

We consider a closed economy consisting of homogeneous and infinitely lived households of size L. Each household is endowed with a unit of labor in each period.
The nominal market value of labor (i.e., the wage) is denoted by wt , which is to be
determined in equilibrium. The utility function of the representative household isgiven by:U=∞X

β t ln ct ,

(1)

t=0

where β ∈ (0, 1) is a constant subjective discount factor and ct is consumption of
the final good in period t. This is maximized subject to the intertemporal budget
constraint at+1 − at = rt at + wt − Pt ct , where at denotes the per capita nominal
financial asset,6 rt is the interest rate, and Pt is the price of the final good, which isnormalized such that the aggregate consumption expenditure becomes unity in eachperiod: Pt ct L = 1 for all t. The Euler equation under this normalization shows thatthe interest rate must be constant at rt = β −1 − 1 ≡ r.
The final good is produced competitively using a continuum of diﬀerent types ofintermediate goods, indexed by i ∈ [0, 1]. As a result of past product innovations,each type of intermediate good potentially has several quality grades, indexed byintegers j ≥ 0, which are perfectly substitutable as inputs to the production of the
final good and have diﬀerent marginal productivities. For each i and j, let x̃it (j)denote the units of the type i intermediate good of quality j, which are used in the
final-good production. Then, the output Yt is determined by the following production
function:
Yt = exp

(Z

0

6

1

ln

"q
it
X
j=0

#

)

λj x̃it (j) di ,(2)

The initial value of a0 is determined by the initial value of the IPRs: a0 = µ0 V0 /L, where thedefinitions of µ0 and V0 are given later. In fact, we do not need to keep track of the consumer’s
budget constraint owing to Walras’ law.

7

where λ > 1 represents the size of the quality improvement obtained from one innovation and qit ≥ 0 is the highest (state-of-the-art) quality of the type i intermediate
good (i.e., it represents the cumulative innovations that have occurred in sector iby date t). As the intermediate goods of the same type are perfect substitutes, final good producers use the single quality that has the lowest quality-adjusted price,p̃it (j)/λj , for every type of intermediate good.
Each type of intermediate good is produced in a separate intermediate good
sector. In every sector, production of one unit of the intermediate good of any
quality requires one unit of labor: this means that the marginal cost of production
equals the nominal wage, wt . As the costs are the same for all qualities, only thestate-of-the-art technology is used in equilibrium. Let xit ≡ x̃it (qit ) and pit ≡ p̃it (qit )denote the amount and price of the state-of-the-art good in sector i. Then the profit
maximization of the final goods sector, under the normalization of Pt as described,
implies that the demand function for each type of intermediate good is xit = 1/pit .
The equilibrium prices and quantities in the intermediate good sectors depend
on whether they are monopolistic or competitive. We call an intermediate goodcompetitive when the technology to produce the highest quality intermediate goodis publicly available. In a competitive sector, firms compete with each other, pushing
the price down to the marginal cost, pit = wt . Given the demand function xit = 1/pit ,the output of a competitive sector is xit = 1/wt . An intermediate good sector ismonopolistic when there is a firm that holds an IPR giving it the exclusive right toproduce the highest quality intermediate goods. For simplicity, we assume that only
the highest quality technology is protected by IPR, and the technology to produce
the next-best-quality good is always publicly available. In this case, the monopoly
firm maximizes profits by employing the limit-pricing strategy, pit = λwt , whichcannot be undercut by the next-best-quality good in terms of the quality-adjusted
price. Since the demand function is xit = 1/pit , the monopoly firm sells amount
xit = 1/(λwt ), obtaining the monopoly profit π = (λ − 1)/λ.

8

3.2

R&D and the Labor Market Equilibrium

In this model, economic growth is driven by innovations, which raise the quality ofintermediate goods. Innovation may occur in competitive and monopolistic sectors as
a result of R&D activities by workers. However, the environments for such activities
diﬀer, depending upon whether R&D is undertaken in a competitive or a monopolistic
sector.
Let us start by describing the environment for R&D in the competitive sectors.
When one worker conducts R&D activity for one period in a particular competitive
sector i, he or she has a small probability, b
a > 0, of successfully creating an innovatedgood. Any R&D activity is assumed to take one period to complete. Hence, theinnovated intermediate good can be produced only from the next period onwards.
We assume away coordinating behaviors among researchers. Then, given that the
total number of researchers in the competitive sector is nit , the probability that someof them successfully innovate is:a)nit = 1 − exp(−anit ),G(nit ) = 1 − (1 − b(3)where a ≡ − ln(1 − ba). As a ' b
a, given that b
a is suﬃciently small, we use parametera interchangeably with b
a in the following. Let Vt+1 denote the value of monopolizing
this innovation in period t+1. Note that this value must be discounted by β = 1/(1+r), because the fruits from innovation may be reaped only in the next period, whereas

the costs of the innovation must be incurred immediately. Then, the expected payoﬀ
per worker from R&D is written as βVt+1 g(nit ), where:g(nit ) ≡G(nit )
1 − exp(−anit )
=
≤ a,
nit
nit

(4)

with g 0 (·) < 0, g(0) = a, and limn→∞ g(n) = 0.
Equation (4) implies that the payoﬀ per worker decreases as the total numberof researchers competing with each other in the sector increases. This is due tothe possibility of duplication; even when two workers simultaneously innovate in thesame sector, their total payoﬀ is at most βVt+1 , rather than 2βVt+1 , because of thenonrival nature of knowledge. We may assume that successful innovators share the9profits from the innovation, βVt+1 , equally because they fear entering into Bertrand
competition from the next period onwards and ending with zero profits if they failto agree upon a division of profits. Alternatively, we can consider the case where
one fortunate innovator obtains the sole right to use that innovation, while others
are prohibited. In either case, the expected return from engaging in R&D activities
in sector i is βVt+1 g(nit ).
Innovation may also occur in monopolistic sectors. Note, however, that the incumbent monopolist has no incentive to innovate in its own sector, since its profit,
π = (λ − 1)/λ, does not change, even when innovation is successful.7 Therefore, any
R&D that occurs is carried out by outsiders who try to leapfrog the current monopolist without the experience of producing goods with state-of-the-art technology.
The lack of experience limits these firms’ ability to attain the information required
to improve upon the current state-of-the-art technology. In addition, outside researchers must restrict their methods of innovation to avoid infringing the patentsof the incumbent firm. For these reasons, the probability that an outside researcher
in a monopolistic sector succeeds in innovation, denoted by bb ∈ (0, ba), is well below
the probability of success of innovation in a competitive sector.

Let mit denote the total number of researchers in monopolistic sector i. Then,the probability that at least one of the researchers succeeds is:H(mit ) = 1 − (1 − bb)mit = 1 − exp(−bmit ),(5)

where b ≡ − ln(1 − bb) ' b. The expected payoﬀ that each researcher receives is

βVt+1 h(mit ). Here, h(mit ) ≡ H(mit )/mit , with h0 (·) < 0, h(0) = b, and limm→∞ h(m) =7This property, commonly called the Arrow eﬀect, depends on the simplifying assumption that

only state-of-the-art technology is protected. If the incumbent firm can retain the exclusive right touse the current technology, in addition to the innovated technology, it may be willing to continue
research activity in order to widen the technological advantage between itself and other firms (see,for example, Aghion et al. 2001). When the incumbent firm’s technology is j-steps ahead of otherfirms, the profit flow is (λj − 1)/λj , and an additional innovation can increase it by (λ − 1)/λj+1 .
However, since the profit increment gradually shrinks as j gets large, the incumbent eventually loses
the incentive to innovate. Thus, even in this generalized setting, the Arrow eﬀect holds.

10

0.
The number of researchers in each sector, competitive or monopolistic, is determined by the free-entry condition. We assume a complete insurance market suchthat the risk of engaging in R&D activities can be fully diversified. Free entry toresearch activities in each type of sector then implies that the following conditions
must be satisfied in equilibrium: βVt+1 g(nit ) ≤ wt with equality when nit > 0 for
all competitive intermediate good sectors, and βVt+1 h(mit ) ≤ wt with equality whenmit > 0 for all monopolistic sectors. These requirements determine nit and mit asfollows:8

nit = nt = N (wt /βVt+1 ) ≡mit = mt = M (wt /βVt+1 ) ≡g −1 (wt /βVt+1 ) if wt /βVt+1 < a,0
if wt /βVt+1 ≥ a,h−1 (wt /βVt+1 ) if wt /βVt+1 < b,

0

(6)

(7)

if wt /βVt+1 ≥ b.

Equations (6) and (7) show that the number of researchers in each sector is determined by wt /βVt+1 , which represents the cost of R&D relative to the discounted
value of innovation that a researcher can obtain if he or she succeeds. Researchers
participate in R&D activities in competitive sectors when this relative cost is lower
than the probability of success, a, and they carry out R&D also in monopolistic
sectors if the relative cost is lower than the probability of success there, b. Observethat given the discounted value of innovation, βVt+1 , the number of researchers inevery competitive or monopolistic sector, is a decreasing function of the market wage.
That is, if the cost of R&D rises, the number of R&D competitors must decrease sothat the reduced possibility of duplication compensates for the increase in the costof R&D.
The equilibrium market wage in each period is determined so that the aggregate
labor demand for both production and R&D is equalized to the aggregate labor8

As expression wt /βVt+1 appears frequently: to minimize notation, we write it this way rather

than wt /(βVt+1 ) .

11supply. Recall that the demand for production workers in each intermediate good
sector is determined by the amount of sales, which is 1/wt in the competitive sectors
and 1/(λwt ) in the monopolistic sectors. Let µt denote the number of monopolistic
sectors in the economy. Labor market clearing requires that:9(1 − πµt )/wt + (1 − µt )N (wt /βVt+1 ) + µt M (wt /βVt+1 ) = L.

(8)

The LHS of (8) gives the aggregate labor demand, with the first term representing thenumber of production workers and the second and third terms representing the total
number of researchers in the competitive and the monopolistic sectors, respectively.
It is easily confirmed that the aggregate labor demand is downward sloping withrespect to wt and that it increases unboundedly when wt → 0, and contracts toward
zero as wt → ∞. Those properties guarantee that, given µt and Vt+1 , there exists a
unique level of wt at which the aggregate labor demand coincides with the aggregate
labor supply, L. Once the equilibrium value of wt is obtained, nt and mt are foundfrom (6) and (7).

3.3

Evolution of the Economy

The state of the economy is characterized by the number of monopolistic sectors,µt , and the value of innovation, Vt . This subsection describes the evolution of theeconomy over time.
There are two reasons why a firm’s monopoly in a particular intermediate good
sector ends. The first is leapfrogging. If a successful innovation occurs in a monopolistic sector, the incumbent monopolist is replaced by the successful innovatorin the following period. As shown by (5), leapfrogging occurs with a probability ofH(mt ) during each period. The second cause is imitation. Even though the IPRs of
a monopoly firm are, to a certain extent, protected, other firms may find diﬀerent
methods of producing similar goods without infringing upon the monopolist’s patent.
9

Recall that π = (λ − 1)/λ. The total number of production workers is (1 − µt )/wt + µt /(λwt ) =(1 − µt + µt /λ)/wt = (1 − πµt )/wt .

12

We assume that such imitation occurs with a constant probability of δ ∈ [0, 1] ineach period and that it allows any firm to produce the state-of-the-art-quality intermediate good from the next period onwards. Parameter δ measures the weakness of
IPR protection; in an economy with strong IPR protection, δ is close to zero andimitation rarely occurs, whereas, with weak IPR protection, δ is large and imitation
is frequent.
If imitation occurs, it converts a monopolistic sector into a competitive sector.
A subtle point is that both leapfrogging and imitation may simultaneously occur
within one period. In this case, a new monopoly firm emerges in the following
period and the imitation of the current state-of-the-art product has no impact on
the new monopolist. Without systematic correlation between these two events, thecoincidence occurs with a probability of δH(mt ). Thus, the number of monopolistic
sectors that change to competitive ones in the following period is δ(1 − H(mt ))µt .
On the other hand, out of 1 − µt competitive intermediate good sectors, G(nt )(1 −µt ) sectors develop into monopolistic ones through successful innovations. The netchange in µt for one period is determined by the diﬀerence between these two flows:µt+1 − µt = (1 − µt )G(nt ) − δµt (1 − H(mt )).

(9)

Let us now examine how the value of innovation evolves over time. Recall that Vt
represents the value of holding a valid (not imitated) IPR at period t. If a monopoly
firm holds this right, it earns a profit of π = (λ − 1)/λ. In addition, this IPR isstill valid at period t + 1 if neither imitation nor leapfrogging has occurred during
period t. Its present value is βVt+1 , and this value is realized with a probability of(1 − δ)(1 − H(mt )). In sum, Vt is determined by a backward dynamics, as follows:
Vt = π + β(1 − δ)(1 − H(mt ))Vt+1 ,(10)

together with a transversality condition, limT →∞ β T VT = 0, which is required forruling out bubble prices.
The equilibrium dynamics is characterized by equations (9) and (10). Once nt
and mt are eliminated by (6), (7), and (8), we see that they constitute an autonomous
13

system of diﬀerence equations in terms of µt and Vt .10 The initial number of monopolistic sectors, µ0 , is historically given, whereas the initial value of an IPR, V0 , isdetermined such that the transversality condition is not violated. Along the equilibrium path, the growth rate of output is obtained by taking the diﬀerence in the logof Yt . From (2), it is:
γYt+1 ≡ ln Yt+1 − ln Yt = (ln λ)Z 10(qi t+1 − qit )di +Z 10(ln xi t+1 − ln xit )di.

(11)

Equation (11) shows that the growth rate can be decomposed into changes in qualities
and changes in quantities. On the right-hand side (RHS) of (11), the integral in thefirst term corresponds to the total number of sectors in which innovation occurs at
period t. It is the sum of the number of leapfroggings, µt H(mt ), and the numberof innovations in the competitive sectors, (1 − µt )G(nt ). The second term is theaggregate change of (the log of) output in the intermediate good sectors. Recall
that the intermediate good output is 1/(λwt ) in a monopolistic sector, and 1/wt in
a competitive sector. By substituting these quantities into (11), the growth rate canbe expressed as:
γYt+1 = (ln λ)[(1 − µt )G(nt ) + µt H(mt )] − (ln λ)(µt+1 − µt ) − (ln wt+1 − ln wt ). (12)The growth eﬀects of IPR policies can be studied by examining how changes inδ aﬀect the endogenous variables in expression (12). The following two sections are
devoted to this task; after analyzing a special but still informative case in the next
section, Section 5 examines the eﬀects of IPR protection in a general setting.

4

Growth without Leapfrogging

This section examines the growth eﬀects of IPR policies by focusing on an extreme
case in which leapfrogging is prohibitively diﬃcult. Note that, from (8) and (10),10

Let us define function W (µt , Vt+1 ) as the equilibrium level of wt given µt and Vt+1 , as deter£¡¢¤
mined by (8). Then, (9) and (10) can be written as µt+1 = 1−δ+δH M (W (µt , Vt+1 )/βVt+1 ) µt +
¡
¢
£
¡
¢¤
G N (W (µt , Vt+1 )/βVt+1 ) (1 − µt ), and Vt = π + β(1 − δ) 1 − H M (W (µt , Vt+1 )/βVt+1 ) Vt+1 .

14there is a lower bound for the cost of R&D relative to the discounted value of innovation:11
wt /βVt+1 ≥ r/((λ − 1)L) ≡ z min .

(13)

As a benchmark, this section considers a case in which the probability of successful
leapfrogging, b, is even lower than this lower bound.
Assumption 1. b < z min .
Under Assumption 1, (7) and (13) imply that mt = 0 for all t; that is, noresearchers operate in the monopolistic sectors, and therefore leapfrogging neveroccurs. Without the risk of being leapfrogged, the value of an innovation is influenced
only by the probability of being imitated, which is directly controlled by IPR policies.
Substituting mt = 0 for all t into (10) and then applying the transversality condition,
we find that the value of innovation is constant over time:Vt =π≡ V (δ) for all t.
1 − β(1 − δ)(14)

The number of researchers in each competitive sector, nt , is determined by thefree-entry condition (6) and the labor-market-clearing condition (8). Given that
mt = 0 and Vt = V (δ) for all t, these conditions can be restated as follows:
1 − πµt≥ βV (δ)g(nt ),L − (1 − µt )nt

with equality if nt > 0.

(15)

Condition (15) has an intuitive interpretation. The LHS of (15) represents the production workers’ wage. Note that the number of production workers, L − (1 − µt )nt
in the denominator, decreases as nt rises. Then, the nominal wage that each production worker receives increases because the total aggregate consumption expenditure
is normalized to unity. Hence, the production workers’ wage is upward sloping, asshown by Figure 3. The RHS of (15) is the expected payoﬀ of R&D in the competitive
11

Note that, from (8), the equilibrium wage cannot be lower than (1−π)/L, as otherwise the labordemand for production, (1−πµt )/wt , exceeds the labor supply, L. Note also that Vt+1 ≤ π/(1−β) =(λ − 1)/(λ(1 − β)) from (10) and the transversality condition. Therefore, using r = β −1 − 1 andπ ≡ (λ − 1)/λ, we have wt /βVt+1 ≥ r/((λ − 1)L) ≡ z min .

15

Figure 3: Determination of nt and wt when R&D occurs only in competitive sectors.
The dashed curve at the bottom of the figure shows that there is no interior solution when IPR
protection is too weak (δ ≥ δ(0)). The expected payoﬀ of R&D shifts up as δ decreases toward 0,
and the wage for production shifts down as µ increases toward 1. The result of both changes is anincrease of nt toward nmax ≡ g −1 (z min ).

sectors. It is downward sloping with respect to nt because of the increasing possibility of duplication. If IPR protection is very weak, or if the possibility of imitation
δ is very high, the expected payoﬀ of R&D is lower than the wage for any positive
value of nt . Therefore, no worker engages in R&D (i.e., nt = 0). A straightforward
calculation shows that this is the case when δ ≥ δ(µt ), where:12
δ(µt ) ≡aπL−r
1 − πµt(16)

gives the level of IPR protection required to activate R&D activities. Observe from(16) that if a ≤ r/(πL), no R&D occurs regardless of the degree of IPR protection,
which means that growth is not possible. Conversely, if a ≥ (1 + r)/(πL), innovation
is so easy that R&D take place without any IPR protection. Therefore, the realistic

12

Note that δ ≥ δ(µt ) is equivalent to (1 − πµt )/L ≥ aβV (δ) = βV (δ)g(0). As function g(nt ) isstrictly decreasing with respect to nt , the above inequality implies that (1 − πµt )/(L − (1 − µt )nt ) ≥(1 − πµt )/L > βV (δ)g(nt ) for all nt > 0.

16

case is:r/(πL) < a < (1 + r)/(πL),(17)

so that δ(0) ≡ aπL − r ∈ (0, 1). We assume (17) throughout the paper.
Under (17), R&D take place provided that IPR protection is reasonably tight.13
When δ < δ(µt ), a unique interior intersection exists between the two curves depicted
in Figure 3, because the expected return from R&D is higher than the production
workers’ wage when nt = 0, and it converges to zero as nt → ∞. At the intersecting
point, denoted by nt = ψ(µt , δ), workers are indiﬀerent between these two types ofactivity: the free-entry condition for R&D is satisfied. The figure also shows thatthe point of intersection, i.e., the equilibrium number of researchers, depends on
the number of monopolistic sectors in the economy, µt . Note that as the numberof monopolistic sectors increases, the aggregate labor demand declines because no
researcher is working in monopolistic sectors and because the monopoly firm hires
fewer production workers than those employed in a competitive sector. As a result,as µt increases, the curve representing the market wage in Figure 3 shifts down (seethe LHS of equation 15), and the point of intersection moves toward the right. Thismeans that the equilibrium number of researchers, nt = ψ(µt , δ), is increasing withrespect to µt .
With this result in hand, let us investigate the dynamics of the economy. The left
panel of Figure 4 depicts the phase diagram of the economy in (µt , nt ) space. Oneof the upward sloping curves represents the equilibrium relationship between µt and
nt , given by nt = ψ(µt , δ). Another upward sloping curve in the panel represents
the number of researchers where the number of monopolistic sectors is stationary, as
explained below. From (9) and mt = 0, µt is constant if and only if the aggregate
flow of innovation, (1 − µt )G(nt ), coincides with the aggregate flow of imitation, δµt .
Note that, since G(nt ) < 1, this condition can be satisfied only if µt < 1/(1 + δ).
Using the definition of G(nt ) = 1 − e−ant , the stationary condition can be solved for

130Note that, because δ (µt ) > 0, (17) implies δ(µt ) ≥ δ(0) > 0 for all µt ∈ [0, 1].

17

Figure 4: Phase diagram. The left panel shows the dynamics of µt and nt for a given δ < δ(0).
The right panel shows the transitional dynamics after IPR protection is strengthened.

n:
nt =

1
1
1 − µt≡ φ(µt , δ) for µt <log.
a
1 − (1 + δ)µt
1+δ

(18)

Equation (18) shows that the µt+1 = µt locus, given by nt = φ(µt , δ), is increasing inµt as depicted in Figure 4. When the equilibrium number of researchers nt = ψ(µt , δ)is above this locus, the aggregate flow of innovation dominates the flow of imitationand therefore µt increases over time. The opposite holds if nt = ψ(µt , δ) is below
the locus. Observe that, if δ < δ(0), the two curves have at least one intersecting
point in µt ∈ (0, 1/(1 + δ)) from ψ(0, δ) > 0 = φ(0, δ) and ψ(1/(1 + δ), δ) < ∞ =limµ→1/(1+δ) φ(µ, δ). In the following, we assume that the intersecting point is unique,and we denote it by µ∗ (δ).14 Then, as illustrated by the left panel of Figure 4, µt
converges to the steady-state value µ∗ (δ) from any initial µ0 .
We are now ready to examine the eﬀect of IPR policies on economic growth.
Suppose that the economy is initially at the steady state µt = µ∗ (δ) and then IPR
protection is strengthened so that the probability of imitation δ falls to δ 0 < δ.
Strengthened protection increases the value of innovation V (δ), given by (14), andhence the expected payoﬀ of R&D, given by the RHS of (15). Then the downward
sloping curve in Figure 3 shifts up, increasing the equilibrium number of researchers
in a competitive sector, nt . Since this raises the probability that innovation occurs
14

We experimented with various combinations of parameters and found that the intersecting

point is, in fact, unique for every combination of parameters chosen.

18

in each competitive sector, G(nt ), the rate of economic growth is enhanced, at least
temporarily.
However, to obtain the long-term eﬀect of stronger IPR protection, we must
consider the transition to the new steady state. Note that the above discussion
implies that function nt = ψ(µt , δ) shifts up when δ is decreased. In addition, theµt+1 = µt locus, given by (18), shifts down, since the stationarity of µt can be
maintained with fewer researchers when the probability of imitation is low. As aresult, the number of researchers in a competitive sector is now above the µt+1 = µt
locus, as depicted by the right panel of Figure 4. The number of monopolistic sectors,µt , increases gradually toward the new steady state level µ∗ (δ 0 ) > µ∗ (δ). This implies
shrinkage in the number of competitive sectors where R&D activities are taken place,which gradually inhibits growth.
At the same time, the number of researchers in each competitive sector gradually
increases during the transition, more or less oﬀsetting the negative eﬀect of increased
monopolization on growth. As explained, an increase in µt reduces the competitive
sectors where researchers are operating, lowering the aggregate labor demand andtherefore the equilibrium wage level. Given a lower wage level, the number of researchers in each competitive sector, nt , increases although there are fewer competitive sectors, 1 − µt . In eﬀect, the reduction in the number of competitive sectors
reallocates researchers into fewer sectors or research fields.
We can confirm those eﬀects in mathematical terms by substituting µt = µt+1 =µ∗ (δ), wt+1 = wt , nt = ψ(µ∗ (δ), δ), and mt = 0 into (12):γY∗ (δ)/(ln λ) = (1 − µ∗ (δ))G(ψ(µ∗ (δ), δ)).

(19)

Consistent with the above, equation (19) clarifies the three ways in which growth isaﬀected by stronger IPR protection: first, ψ(·, δ) is directly increased by smaller δ;
second, (1 − µ∗ (δ)) is reduced by larger µ∗ (δ); and, third, ψ(µ∗ (δ), ·) is increased bylarger µ∗ (δ). As a result, the overall eﬀect of stronger IPRs on growth can be either
positive or negative depending on the relative magnitude of those three eﬀects. In
fact, the following proposition states that the relationship between IPR protection
19and long-term growth is nonmonotonic:
Proposition 1. Under assumption 1, protection of IPRs should be neither too strict(δ = 0) nor too loose (δ ≥ δ(0)), because in either case, the resulting long-term
rate of growth would be zero. There exists a noncorner level of IPR protection,
δ max ∈ (0, δ(0)), which maximizes the total flow of innovation in competitive sectors
and therefore the long-term rate of growth.
The reason that IPR protection should not be too loose is obvious: if δ ≥ δ(0),the expected reward from innovation is so low that no one will find it profitable toparticipate in R&D activities.15 The reason that IPR protection should not be toostrict requires some clarification. When IPR protection is extremely strengthened sothat it shuts out the possibility of imitation (i.e., δ = 0), the µt+1 = µt locus falls tothe horizontal axis (φ(µt , 0) = 0 < ψ(µt , 0) for all µt < 1), and therefore the numberof monopolistic sectors gradually approaches 1. Although this means that the numberof sectors where researchers operate converges to zero, it does not imply per se thatgrowth is impossible if the number of researchers in each competitive sector can growwithout bound such that the aggregate number of researchers is unaﬀected. However,recall that as more researchers operate in the same intermediate good sector, the riskof duplication rises. In particular, because of the increased risk of duplication, we
can show that the positive eﬀects cannot overturn the negative eﬀect of increased
monopolization on growth when IPR protection becomes very strict. To see this,observe that the number of researchers per sector is bounded by a finite number,
nmax ≡ g −1 (z min ), from (6) and (13), because if more than nmax researchers operate
in a sector, the probability that duplication will occur is so high that the expected
return is below the lowest possible wage. This means that, if δ = 0, the number ofcompetitive sectors converges to zero, whereas the flow of innovation per sector isbounded, jointly implying that the aggregate flow of innovation is eliminated. The
15

Suppose that δ ≥ δ(0). Then, ψ(0, δ) = 0 = φ(0, δ) from the definition of δ(0) and (18). Thismeans that the unique steady state is µ∗ (δ) = 0, and therefore γY∗ (δ)/(ln λ) = G(ψ(0, δ)) = G(0) = 0
from (19).

20latter half of the proposition is confirmed by noting that the rate of economic growth
given by (19) is positive if and only if δ ∈ (0, δ(0)) and that it is continuous in δ.
In all, Proposition 1 states that the growth-maximizing IPRs policy should be anintermediate one under the assumption that leapfrogging is impossible. However, howdoes the conclusion depend on this assumption? The following section investigates
the general case in which R&D can be conducted in both the competitive and themonopolistic sectors and shows that the above result continues to hold in reasonablesettings.

5

Growth with Imitation and Leapfrogging

In this section, we dispense with Assumption 1 and examine the eﬀect of IPR protection on the long-term rate of growth in an economy where innovation can occur
in both the competitive and the monopolistic sectors. Since this general case is considerably more complex, in that we need to keep track of the diﬀerent intensities of
R&D between the monopolistic sectors and the competitive sectors, we restrict theanalysis to the long-run properties of the economy. On the BGP, the analysis canbe kept more tractable by focusing on a single variable, zt ≡ wt /βVt+1 > 0, which
represents the cost of R&D relative to the discounted value of innovation. The pointis that both intensities are functions only of zt ; i.e., nt = N (zt ) and mt = M (zt ) from
(6) and (7). This means that the flows of innovation in competitive or monopolistic
are given by G(N (zt )) and H(M (zt )), both of which are determined only by zt .
Given these flows, the number of monopolistic sectors in the long run is determined accordingly. Let z denote the long-run level of zt . Then, by substituting
G(N (z)) and H(M (z)) for (9) and equating µt+1 to µt , the value of µt in the longrun is obtained as a function of z and δ:16µ∗ (z, δ) =16

G(N (z)).
δ(1 − H(M (z))) + G(N (z))(20)

Here, we recycle the notation µ∗ (·). Its definition diﬀers from µ∗ (δ), as defined in the previous

section.

21

Similarly, from (6), (7), (8), and (10), the values of Vt and wt on the BGP arecalculated as follows:π,
1 − β(1 − δ)(1 − H(M (z)))
1 − πµ∗ (z, δ)w∗ (z, δ) =L − (1 − µ∗ (z, δ))N (z) − µ∗ (z, δ)M (z)V ∗ (z, δ) =(21)
for z > z(δ),(22)

where z(δ) ∈ (0, a) denotes the level of z at which the number of researchers, (1 −µ∗ (z, δ))N (z) − µ∗ (z, δ)M (z), coincides with L.17
Equations (20)-(22) express the BGP of the economy in terms of z and δ. As
z is an endogenous variable, its dependence on δ needs to be clarified. From thedefinition of z, it must satisfy an identity:z = w∗ (z, δ)/βV ∗ (z, δ) ≡ Z(z, δ).

(23)That is, the value of z on the BGP is determined as a fixed point of function Z(z, δ).
The following lemma establishes the existence of the fixed point and how it is aﬀectedby IPR policies.
Lemma 1. For every δ ∈ [0, 1], there exists a fixed point z > z(δ) that satisfies (23).

Suppose that the fixed point is unique and let it be denoted by z ∗ (δ). Then, z ∗ 0 (δ) > 0
for all δ ∈ [0, 1]. In addition, there exists a continuous function δ LF (b) such that:



≥ a
if δ ≥ δ(0);z ∗ (δ) ∈ [b, a) if δ ∈ [δ LF (b), δ(0));





< b
if δ < δ LF (b).

Function δ LF (b) is zero for all b ≤ z min , strictly increasing in b for all b ∈ (z min , a),and limb→a δ LF (b) = δ(0).
Proof: see the Appendix.
17

From (6) and (7), note that limz→0 N (z) = limz→0 M (z) = ∞ and N (a) = M (a) = 0. Thereforez(δ) ∈ (0, a) is well defined. In equilibrium, z cannot be smaller than z(δ), since otherwise thenumber of researchers exceeds the population.

22

Figure 5: The degree of IPR protection and R&D activities. Calculated numerically by
setting λ = 1.5, β = 0.95, L = 108 , and a = 1/L. Under these parameter values, δ(0) ≈ 0.28 andz min ≈ 0.11a.

Throughout this section, we assume that mapping (23) has a unique fixed point(we confirmed this under various parameter values). Then, Lemma 1 proves thatstronger IPR protection reduces the cost of innovation relative to the value of innovation (i.e., zt ≡ wt /βVt+1 falls when δ is reduced). Recall that, from (6) and (7),researchers in competitive and monopolistic sectors are willing to incur the cost of
R&D only when zt ≡ wt /βVt+1 is lower than a and b, respectively. Thus, as discussed
in the previous section, R&D occurs in competitive sectors only when IPR protection
is stronger than δ(0). Since b is smaller than a, the degree of IPR protection must be
even tighter (i.e., δ < δ LF (b) < δ(0)) to activate R&D activities in the monopolistic
sectors, as illustrated in Figure 5,
To see the long-run eﬀect of IPR policies on growth, we substitute µt+1 = µt =µ∗ (z ∗ (δ), δ), nt = N (z ∗ (δ)), mt = M (z ∗ (δ)), and wt+1 = wt into (12):γY∗ (δ)/ ln λ = (1 − µ∗ (z ∗ (δ), δ))G(N (z ∗ (δ))) + µ∗ (z ∗ (δ), δ)H(M (z ∗ (δ))),(24)

which is simply the sum of the innovation flow in the competitive sectors and theleapfrogging flow in the monopolistic sectors. Expression (24) depends on the degreeof IPR protection functions µ∗ (z ∗ (δ), δ)), N (z ∗ (δ)), and M (z ∗ (δ)). Observe that,since µ∗δ < 0, µ∗z < 0 from (20) and z ∗ 0 > 0 from Lemma 1, stronger IPR protec23

tion (or smaller δ) increases the number of monopolistic sectors, µ∗ (z ∗ (δ), δ). Since
the flow of innovation in a sector is higher when the sector is competitive rather
than monopolistic (i.e., G(N (z ∗ (δ))) > H(M (z ∗ (δ))) for any given δ < δ(0)), thischange in the composition of the economy reduces the aggregate research intensity
as illustrated by the horizontal arrow in Figure 2, which is detrimental for growth.
Stronger protection also has positive eﬀects on growth, since it reduces the ratioof the cost of innovation relative to the reward, z ∗ (δ) = w/βV , which stimulates theflows of innovation in both types of sector, N (z ∗ (δ)) and M (z ∗ (δ)), from z ∗ 0 > 0,N 0 < 0 and M 0 < 0. Intuitively, there are two reasons why stronger IPR protection
reduces this ratio. For one thing, since the demands for both production and R&D
workers are smaller in a monopolistic sector than in a competitive sector, increased
monopolization of the economy reduces the aggregate labor demand, and therefore
the equilibrium wage level falls (observe that wδ∗ > 0 from equation 22). Since thismakes working as a researcher more attractive than working as a production worker,
the number of R&D workers in every sector increases, as illustrated by the verticalarrows in Figure 2.
Therefore, the increased monopolization of the economy causes a reallocation ofresearch workers across sectors, through which the distribution of research workers
becomes more “skewed” (see, again, Figure 2). This reallocation would be growthneutral when both the aggregate number of researchers and the productivity of eachresearcher remain constant. However, if the possibility of duplication of innovations
is explicitly considered, such a possibility will be higher when the distribution ofresearchers across sectors is more skewed. Therefore, the aggregate productivityof R&D after subtracting the losses from duplication decreases as the number ofmonopolistic sectors increases. Furthermore, the fall in productivity diverts workers
from R&D activities.
As discussed, the net eﬀect of increased monopolization, which is caused bystronger IPR protection, on growth tends to be negative. However, recall that there
is yet another reason why stronger IPR protection stimulates R&D activities: it di-24

Panel 2: b = 0.8a.

Panel 1: b = 0.4a.

Figure 6: Growth rate on the BGP as a function of δ. When the parameters are given by
λ = 1.5, β = 0.95, L = 108 , and a = 1/L (the same as in Figure 5), the long-run rate of growth ismaximized by setting δ = δ max = 0.207 when b ≤ 0.578a, whereas it is maximized at δ = 0 whenb ≥ 0.578a.

rectly raises the reward for successful innovators. This positive eﬀect can be seen
by noting that the denominator of z ∗ (δ) = w/βV is increased by stronger IPR protection, since Vδ∗ < 0 from (21). Therefore, similarly to the analysis in the previous
section, the total eﬀect of stronger IPR on growth can be of either sign, dependingon the relative magnitude of positive and negative eﬀects. Note that the negativeeﬀect of monopoly on growth stems from the assumption that innovation is morediﬃcult in monopolistic sectors than in competitive sectors. Therefore, the relative
magnitude of opposing eﬀects hinges critically upon the diﬃculty of leapfrogging
compared with that of innovation in competitive sectors.
To evaluate the total eﬀect, we numerically calculate the long-term rate of growth
as a function of δ, for two diﬀerent values of the probability of success in R&D ina monopolistic sector b ∈ (z min , a).18 Panel 1 of Figure 6 shows that, when b isconsiderably small, the flow of leapfrogging is small even when IPR protection isstrongest (δ = 0). In this case, the economy can grow faster with an intermediatelevel of IPR protection, δ max ∈ (0, δ(0)), at which the total flow of innovation incompetitive sector is maximized (see Proposition 1), even though no leapfrogging
18

Recall that the case of b < z min is already analyzed in Section 4.

25

occurs at this level of IPR protection. Panel 2 of Figure 6 illustrates the relationshipbetween IPR protection and growth when b is high and there is only a small diﬀerencein the probability of success between the monopolistic and the competitive sectors.
In this case, the growth rate is maximized at δ = 0, which means that the economy
need not rely on imitation to promote growth because the monopoly is not very
harmful to R&D activities by outside researchers.
To interpret this result in more detail, recall from Lemma 1 and Figure 5 thatwhen b is small, δ LF (b) is small, and therefore 0 < δ LF (b) < δ max < δ(0) holds (notethat δ max does not depend on b). For δ > δ LF (b), leapfrogging does not occur, and as
discussed in the previous section, there is a level of IPR protection at which the total
flow of innovation in competitive sectors is maximized, which is δ max . The positiveeﬀect of stronger IPR dominates when δ > δ max , whereas the opposite holds whenδ ∈ (δ LF (b), δ max ). When IPR protection is further strengthened beyond δ LF (b), thesimulation result shows that the positive eﬀect again slightly dominates, but thisdoes not aﬀect the fact that δ = δ max is growth maximizing, since δ LF (b) is alreadynear zero. However, when b is large, δ LF (b) is higher than δ max , and therefore thepositive eﬀect of stronger IPR protection globally dominates the negative eﬀect.
Then, why is stronger IPR protection more favorable for growth when δ < δ LF (b)?
There are two reasons. First, stronger IPR protection now encourages R&D, notonly in competitive sectors but also in monopolistic sectors. Second, a worker who
loses a job in a formerly competitive sector can now find another as a researcher,
not only in a competitive sector but also in a monopolistic sector. This implies thatthe problem of the increased possibility of duplication is less serious than the case inwhich the job opportunities for R&D exist only in competitive sectors. From theseobservations, we obtain the following proposition.
Proposition 2. If the probability of successful leapfrogging is low (with b suﬃciently
close to z min ), the long-term rate of growth is maximized by allowing a certain positive
probability of imitation, δ max ∈ (0, δ(0)), so that the flow of imitation is maximized.
If the probability of successful leapfrogging is high (with b suﬃciently close to a), the26

long-term rate of growth is maximized by shutting out any imitation, δ = 0, so thatthe flow of leapfrogging is maximized.
Proof: see the Appendix.
Proposition 2 states that the relationship between IPR protection and growthis quite diﬀerent depending on the diﬃculty of leapfrogging in monopolistic sectors.
Specifically, our simulation results suggest that there is typically a threshold level of b,
below which growth is maximized by δ = δ max and above which growth is maximizedby δ = 0. Therefore, to recommend an intermediate degree of IPR protection as a
growth-maximizing policy, we must evaluate the extent to which monopoly is harmful
for R&D activities. Although the validity of this assumption can vary considerablyacross economies in general, evidence from previous studies appears to support rather
than reject it.
Using panel data from some 670 U.K. companies, Nickell (1996) examined how
total factor productivity (TFP) growth is related to a survey-based measure of competition, specified as a dummy variable taking a value of one if the manager answered
yes to the following question: “Have you more than five competitors in the market foryour products?” After controlling for various factors, Nickell (1996) found a robust
and significantly positive eﬀect of competition on growth. In addition, he attempted
to measure the size of this eﬀect by ranking firms on the basis of rents, which arepresumed to relate positively to the degree of monopolization. The result is that thegrowth of the TFP of a firm at the eightieth percentile in the ranking of rents is,ceteris paribus, 3.8—4.6 percentage points lower than that of a firm at the twentiethpercentile, suggesting that monopolization significantly inhibits TFP growth.
Our assumption is more directly supported by Blundell, Griﬃth and Van Reenen(1993, 1995), who estimated dynamic models of innovation using industry-level variables and the count of technologically significant and commercially important innovations commercialized by British firms between 1972 and 1982. Measuring thedegree of concentration of an industry by the proportion of sales that is represented
by the five largest domestic firms, they found a negative impact of concentration on

27

the probability of innovation. On this basis, they concluded that firms in competitive
industries tend to have a greater probability of innovation, and therefore firms withlarger market shares that increase the level of industry concentration would depress
the aggregate level of innovative activity. Geroski (1990) estimated a similar regression, which revealed that monopoly inhibits research activities, even after correcting
for interindustry variations in technological opportunity and postinnovation returns.
Finally, note that the diﬃculty of leapfrogging stems not only from technicalaspects but also from the incumbent’s entry-deterring activities. For example, Bresnahan (1985, p.16) documented that when IBM and Litton entered the PPC (plain
paper copier) market in 1972, Xerox sued to block entry with literally hundreds ofpatents. As a result, IBM spent millions to “invent around” Xerox’s major patents–
with 25 percent of the budget spent not on R&D, but on patent counsel. By 1974,
free access to PPC technology was ensured through the FTC’s antitrust action. After
this, many firms entered the PPC market, and the transition period saw a great dealof innovative activity from both new entrants and Xerox.

6

Relationship with the LiteratureThus far, it has been shown that the long-term rate of economic growth can be maximized under intermediate IPR protection strength on the assumption that leapfrogging is considerably more diﬃcult than innovation in competitive sectors. This study,of course, is not the first attempt to investigate the growth eﬀects of IPR policies.
In this section, we discuss the similarities and diﬀerences between our results andthose obtained from earlier studies.
Using variety expansion models of endogenous growth, Kwan and Lai (2003) andIwaisako and Futagami (2003) demonstrated that the long-term rate of growth ismaximized when IPR policy is as stringent as possible. From the above analysis, it
can be reckoned that their results follow from the fact that they did not consider thenegative eﬀects of monopoly on innovation, since the same result is obtained in ourmodel in the extreme case of b = a. Intuitively, in an environment where outsiders
28

can learn the state-of-the-art technology nearly as easily as insiders, stronger IPR
does not inhibit innovation and therefore growth. In addition, the second half of
Proposition 2 implies that their result is robust to small changes in that it continues
to hold as long as the negative eﬀects of monopoly on innovation are small (i.e., evenwhen b is slightly below a).
While the majority of theoretical studies of IPR have concluded that stronger IPR
protection enhances economic growth, there are notable exceptions. One important
study was made by Michel and Nyssen (1998), who examined the impact of IPRs
on growth in terms of patent length and found that the rate of economic growth ismaximized when the patent length is finite rather than infinite. This result comesfrom their assumption that the knowledge spillover from past R&D, which makes
creation of new goods easier, is limited during the term of patent protection. At
the level of the aggregate economy, their assumption works similarly to ours in that
IPR protection prevents outside firms from obtaining experience in the productionof state-of-the-art-quality goods. However, since Michel and Nyssen used a varietyexpansion setting in which R&D does not occur in existing sectors, they did not
obtain diﬀerent research intensities between monopolistic and competitive sectors,
as reported by Blundell, Griﬃth and Van Reenen (1993, 1995) and Geroski (1990).
A stronger assumption is made in a class of models called step-by-step innovation,in which product innovations by outsiders can occur only after the state-of-the-art
product has been imitated (i.e., leapfrogging is prohibited). Aghion et al. (1997,
2001) and Mukoyama (2003) used this setting and showed that the growth rate canincrease with the ease of imitation and the rate of subsidy on imitation. In thispaper, their setting corresponds to the analysis in Section 4, where leapfrogging is so
diﬃcult that trying to do so is not profitable, even when the degree of IPR protection
is strongest (i.e., b < z min ), and in Proposition 1 we obtained a result comparable
to theirs. Moreover, the first half of Proposition 2 shows that Proposition 1 can beextended for the case in which b is slightly above z min . Therefore, the results from thestep-by-step models, where facilitating imitation can be beneficial for growth when29

leapfrogging is impossible, is extended to the case in which leapfrogging is possible
but considerably more diﬃcult.
While in our model the probability of imitation is assumed to be directly controlled by the authority, studies of step-by-step innovation have an advantage in thatthey can explain the flow of imitation endogenously.19 However, this was achievedat the cost of abstracting from reallocation of R&D workers across sectors throughlabor market. Specifically, Aghion et al. (1997, 2001) assumed an infinitely elastic
supply of labor such that the wage level is constant, and therefore the number ofcompetitive sectors does not aﬀect research intensity in each sector.20 Mukoyama(2003) ingeniously set up an innovation race game with two firms, in which the research intensity in a symmetric Nash equilibrium is determined independently of thelabor market. In contrast, assuming free entry for innovative activities and inelastic aggregate labor supply, as is commonly assumed in standard endogenous growthmodels, we explicitly consider the process in which a reduction in the number ofcompetitive sectors lowers the market wage and therefore increases the research intensity in both types of sector. Such reallocation appears to be realistic, but it cancompletely eliminate the negative eﬀect of increased monopolization on growth, especially when innovation is subject to constant returns as is usually assumed. Thisis the reason why another realistic feature–the time required to innovate–must
be introduced. As explained in Section 3, even when individual R&D is subject
to constant returns, the time to innovate creates decreasing returns at the industry
level.21 In this setting, we have shown that widespread competition promotes growth19

It is possible to incorporate endogenous imitation into our model by assuming that IPR pro-tection, as in Dinopoulos and Segerstrom (2005), aﬀects the ease of intentional imitation. However,it seriously complicates the analysis and provides vaguer implications.
20

Aghion et al. (1997) briefly considered an extension with a fixed labor supply (see Section 5in their paper). However, they analyzed only extreme cases in which r + δ = 0 and r + δ → ∞.
21

Note that decreasing returns at the level of individual firms, as assumed in Aghion et al. (1997,

2001), does not inhibit innovation at the industry level if free entry is allowed. At the same time,reallocation of research workers across sectors will be neutral for growth if decreasing returns to30

by diversifying researchers across broader fields, thereby reducing the possibility of
unnecessary duplication at each industry.
Finally, while the studies discussed restrict attention to a closed economy, anothernegative eﬀect of IPR protection on growth can stem from the international divisionof labor. For example, in the framework of a North—South economy, Helpman (1993)
showed that strengthening IPR protection (i.e., reducing the probability of imitation)in the South reduces the rate of innovation in the North in the long run, since IPR
protection leads to more workers in the North in production activities rather thanin R&D. This depends on the assumption of no movement of labor between the twocountries, or equivalently, between the monopolistic (North) and competitive (South)
sectors, while this paper considers a closed economy model without any restriction
on the movement of workers across sectors. Obviously, the relevance of either model
depends on the actual economy in which one is trying to apply a theory.

7

Discussion

This section briefly discusses two important topics in the literature of endogenous
growth: scale eﬀects and welfare issues.

7.1

Scale Eﬀects

It is well known that the prototypes of R&D-based growth models exhibit strong
scale eﬀects in that the long-term rate of growth increases very sensitively with population size. For example, using our notations, the equilibrium growth rate derived
in Grossman and Helpman (1991) can be written as (ln λ) (aπL − r/λ). This meansthat if the size of population, L, is doubled, the long-term rate of economic growth

innovation operate with respect to aggregate innovative activities as measured by the total numberof R&D workers in the economy. Thus, on the assumption of free entry, the increased monopolizationof the economy inhibits growth only when there are some forms of decreasing returns in innovation
working at the industry level.

31

Figure 7: Relationship between IPR protection and growth with diﬀerent sizes of
population. Calculated numerically by changing L = 2i × 108 , where i = 0, 1, 2 . . . . Other

parameters are fixed at λ = 1.5, β = 0.95, L = 108 , a = 1/108 , and b = a/10. The growthmaximizing level of δ for each size of L is designated by a black point.

is more than doubled.
Also in our model, greater population enables more labor to be allocated to R&D
and therefore, given the fixed number of sectors, increases the possibility that innovation occurs in each sector.22 However, since we explicitly consider the possibilityof duplication, the rate of economic growth does not increase linearly with the aggregate number of R&D workers. Figure 7 illustrates how the relationship between thelong-term rate of economic growth and the degree of IPR protection changes as thesize of population is doubled, doubled again, and so on. Observe that although theeconomy with twice the population grows faster, it does not grow twice as fast. In
particular, as the size of population increases, the possibility of duplication becomes
more serious, and therefore the rate of economic growth cannot exceed a certain

22

It is possible to eliminate the scale eﬀects completely by incorporating the mechanism throughwhich the number of sectors increases with the size of population (e.g., Young 1998). Alternatively,
the scale eﬀect can be removed by assuming that R&D diﬃculty increases with cumulative R&D
eﬀorts (e.g., Segerstrom 1998).

32

upper bound.
Figure 7 also suggests that the growth-maximizing IPR policy changes nonmonotonically with the size of population. When the size of population, L, is small, themarket to which an innovated good can be sold is also small, and therefore strict
IPR protection is required to give enough incentives for innovators to conduct R&D(observe that δ(0) = aπL − r is increasing in L). Therefore, when the size of economy is small, the growth-maximizing level of δ is low, although it must be positivesince R&D occurs only in competitive sectors given the small market size. As L
increases, innovators become willing to conduct R&D at weaker levels of IPR protection, and therefore the rate of growth can be increased with weaker protection,
since it will increase the number of competitive sectors where innovation occurs. In
the example depicted in Figure 7, the growth-maximizing δ(= δ max ) increases with
L for L ≤ 16 × 108 . However, when the size of population becomes suﬃciently large(L ≥ 32 × 108 ), the market size is so large that innovators are also willing to conduct
R&D in monopolistic sectors, regardless of the level of δ (i.e., δ LF (b) ≥ 1). In thatcase, as discussed in Section 5, the positive eﬀects of stronger protection on growthtend to dominate the negative eﬀects, and therefore growth is maximized by thestrongest possible level of IPR protection, δ = 0.
This suggests that even when the technological circumstances are the same, thedesirable level of IPR protection can diﬀer significantly depending on the market
size.

7.2

WelfareAlthough we have so far discussed the desirability of IPR policies only in terms of
economic growth, previous studies showed that the strength of IPR protection thatis desirable in terms of welfare can diﬀer from those that maximize growth. Specifically, Kwan and Lai (2003) and Iwaisako and Futagami (2003) showed that, while
the long-term rate of growth is maximized by the strongest possible IPR protection,33

social welfare is higher when IPR protection is weaker.23 Our model provides a very
diﬀerent implication in that growth is not necessarily maximized by the strongest
possible IPR protection. What then is the relationship between the growth- andwelfare-maximizing IPR policies? We shed some light on this issue by considering
how the degree of IPR protection, represented by δ, aﬀects the utility of the representative consumer on the BGP.24
On the BGP, the future path of output Yt is given by ln Yτ = ln Yt + γY∗ (δ)(τ − t)for τ ≥ t, where γY∗ (δ) is given by (24). Substituting this expression and ct = Yt /L
into (1) gives:
Ut =

1
β
1
γY∗ (δ) −ln Yt +ln L.
21−β
(1 − β)1−β(25)Since the last term is constant, (25) shows that welfare is determined as the sum of alevel eﬀect (the first term) and a growth eﬀect (the second term). Since the growtheﬀect has been analyzed extensively in Section 5, here we examine the level eﬀect
of IPR protection. From (2), (22), and the fact that the outputs from competitiveand monopolistic sectors are respectively 1/wt and 1/(λwt ), the level of ln Yt on the
BGP is given by:
ln Yt = −(ln λ)µ − ln (1 − πµ) + ln (L − (1 − µ)n − µm) + (ln λ)Z 1

qit di,(26)0where we omit the subscripts on µ, n and m, since they are constant on the BGP.
The last term of (26) represents the current technological level of the economy, or
the aggregate TFP. Since this term is historically determined, we examine the eﬀects
of δ on the other terms.
Recall that, as shown in Section 5, if IPR protection is relaxed (if δ is increased),
the research intensities in both types of sector decline; i.e., dn/dδ, dm/dδ ≤ 0. Since
23

Grossman and Lai (2005) also showed that stronger IPR protection is not necessarily welfare

enhancing in a nonendogenous growth model.
24

Because the transitional dynamics of the present model are too complex for analysis, we focus

our analysis on the BGP following Iwaisako and Futagami (2003). The transitional dynamics ofKwan and Lai (2003) is more tractable and can be analyzed by linearizing the system around thesteady state.

34

(26) is decreasing in n and m, these changes have positive eﬀects on the current levelof output, Yt . Intuitively, since weaker IPR protection discourages R&D, a reallocation of labor from R&D to production occurs. Recall also that if IPR protection isrelaxed, the number of monopolistic sectors declines; i.e., dµ/dδ ≤ 0. Since goods
are sold at higher markups in monopolistic sectors than in competitive sectors, thischange aﬀects the extent of output distortions, which can be measured by:¯πn−m∂ ln Yt ¯¯= − ln λ ++.
¯∂µ BGP
1 − πµ L − (1 − µ)n − µm

(27)

To understand the diﬀerence between the growth- and welfare-maximizing IPR

policy, let us examine the eﬀect on welfare of a marginal increase in δ from the growthmaximizing level. As discussed in the previous sections, the growth-maximizing levelcan be either δ = 0 or δ = δ max , depending on the diﬃculty of leapfrogging. First,
consider the case in which the long-term rate of growth is maximized at δ = 0, as inPanel 2 of Figure 6. If IPR protection is marginally relaxed from δ = 0, the growtheﬀect is negative since γY∗ (δ) is already maximized at the corner level of δ = 0.
In addition, if relaxing IPR protection reduces µ from 1, industries that becomecompetitive start to sell goods at lower markups than other monopolistic sectors,
creating output distortions.25 In fact (27) is positive at µ = 1 (note that the second
term becomes λ − 1 > ln λ), which implies that output level declines if µ is reduced.
Since both the growth eﬀect and output distortions undermine welfare, relaxing IPR
protection can only be justified when the induced reallocation of labor from R&D
to production substantially improves welfare. In other words, in a circumstance
where the second half of Proposition 2 applies, the strongest possible IPR protection
is welfare maximizing unless economic growth is too fast in terms of the trade-oﬀ
between current consumption and future growth.26
Now consider the case in which δ = δ max is the growth-maximizing policy. Then,
25

Note that if all sectors are monopolized, markups are the same across all sectors, and there-

fore there are no output distortions. This is another general-equilibrium eﬀect pointed out by
O’Donoghue and Zweimüller (2004) and Judd (1985).
26

This trade-oﬀ is extensively analyzed by Kwan and Lai (2003). Note that, since both Kwan

35

a slight increase in δ from δ max yields no growth eﬀect in the first order, since γY∗ (δ)
is maximized at the interior level of δ = δ max . In addition, reducing µ does notnecessarily worsen output distortions, especially when µ is already substantially small(observe that the second term of (27) becomes smaller than ln λ when µ = 0).
Moreover, reallocation of labor from R&D to production increases current output.
As a result, it can be shown that the marginal relaxation of IPR protection at δ max
improves welfare on the BGP provided that:27∗µ < f (δmax¶µr + δ max
1
r
)≡
−2 .
+
πr
ln λ r + δ max(28)

To summarize, similarly to the growth-maximizing IPR policy, the desirable degree of IPR protection depends critically on the diﬃculty of leapfrogging relative tothat of innovation in competitive sectors. If leapfrogging is so diﬃcult that growthis maximized by an intermediate level of IPR protection (δ = δ max ), the welfaremaximizing IPR protection tends to be even weaker. Conversely, if leapfrogging is
not so diﬃcult and therefore δ = 0 is the growth-maximizing policy, then it is likelythat the strongest possible protection also maximizes social welfare.

8

Conclusion

In a quality-ladder model of endogenous growth, we examined the extent to which
growth is facilitated by stronger IPR protection by incorporating into our model
two notable features of R&D: (i) for both technical and legal reasons, R&D is easierin competitive sectors where any firm can produce state-of-the-art goods than it isin monopolistic sectors where outsiders cannot produce state-of-the-art goods, and(ii) R&D projects take time to complete, which creates the risk of duplication ofand Lai (2003) and Iwaisako and Futagami (2003) assume that intermediate goods are produced
from final goods rather than from labor, reductions in µ always improve welfare. Therefore, in theirmodels, weaker IPR protection is more likely to be beneficial in terms of welfare than in our model.
27

It can be confirmed that the RHS of (28) is positive from π = (λ − 1)/λ < ln λ. Derivation of(28) is available from the corresponding author upon request.

36

innovation. In this setting, it is shown that stronger IPR protection that lowers
the possibility of imitation has three eﬀects on growth. First, it raises the expected
profit from R&D, a positive eﬀect. Second, it gradually reduces the number ofcompetitive sectors where R&D is more active, a negative eﬀect. Third, it lowers
the market wage and therefore increases the number of R&D researchers employedin each competitive (or monopolistic) sector, a positive eﬀect partially oﬀsetting thesecond negative eﬀect. The overall eﬀect of stronger IPR on growth is determinedby the relative magnitudes of these three eﬀects.
The main finding is that the degree of IPR protection by which the long-term
rate of economic growth is maximized diﬀers significantly depending on the extent
to which the existence of an incumbent monopolist raises the diﬃculty of R&D. If
leapfrogging in monopolistic sectors is substantially more diﬃcult than innovation incompetitive sectors, it is shown that growth is maximized with an intermediate levelof IPR protection. Conversely, if the diﬀerence in diﬃculty is small, the strongest
possible protection maximizes growth. Both results are comparable to those fromearlier studies employing diﬀerent assumptions. In addition, we show that the dependence of a desirable level of IPR protection on the diﬃculty of leapfrogging iseven more critical when the desirability is measured in terms of welfare. These results illustrate the importance of quantifying the adverse eﬀect of monopoly, notonly on output distortions but also on innovative activities. So far, empirical studies have found significant negative eﬀects of monopoly on industry-level innovations,suggesting a case for the imperfect protection of IPR.

Appendix
Proof of Lemma 1
From (21)-(23), function Z(z, δ) is continuous, and it satisfies the conditions thatlimz→z(δ) Z(z, δ) − z = ∞ − z(δ) > 0 and limz→∞ Z(z, δ) − z = 1/(βV (δ)L) − ∞ < 0,where V (δ) is defined by (14). Thus, the intermediate value theorem guarantees
37that for every δ ∈ [0, 1], there is at least one level of z ∗ (δ) ∈ (z(δ), ∞) such thatZ(z ∗ (δ), δ) − z ∗ (δ) = 0 holds, which is a fixed point of (23). In addition, the assumption that the fixed point is unique means that Z(z, δ) − z cuts the horizontal z
axis from above and does so only once in z ∈ (z(δ), ∞). Note that Zδ (z, δ) > 0 from(20)-(23). Thus, the curve of Z(z, δ) − z shifts upward as δ increases. This implies

that the point of intersection moves rightward as δ increases: i.e., z ∗ 0 (δ) > 0.

Let us examine the condition under which z ∗ (δ) is smaller than a. Note that,
because Z(z, δ) − z cuts the horizontal z axis from above and does so only once, thepoint of intersection (i.e., z ∗ (δ)) is smaller than a if and only if Z(a, δ) − a < 0. As
Z(a, δ) = 1/(βV (δ)L) from (21)-(23), this condition is equivalent to δ < 1 − β −1 +aπL ≡ δ(0). Thus, z ∗ (δ) < a if and only if δ < δ(0).
Next, let us examine the condition under which z ∗ (δ) is smaller than b. Specifically, we want to find a threshold level δ LF (b) such that z ∗ (δ) < b if and only if
δ < δ LF (b), for a given level of b. Similarly to the above argument, z ∗ (δ) < b holds
if and only if Z(b, δ) − b < 0. From (22)-(23):Z(b, δ) =1
δ + (1 − π)G(N (b)).
βV (δ) (L − N (b))δ + G(N (b))L(29)

When δ = 0, (29) implies that Z(b, 0) = (1 − β)(1 − π)/(βπL) ≡ z min . Thus, whenb ≤ z min , Z(b, δ) − b ≥ Z(b, 0) − b ≥ 0 holds for all δ ∈ [0, 1] because Z(b, δ) is
increasing in δ. This means that z ∗ (δ) ≥ b for all δ ∈ [0, 1] and therefore that
δ LF (b) = 0 for b ≤ z min .
Now, consider the case of b > z min . Note that in this case, Z(b, 0) − b = z min − b <0. In addition, note that we have shown Z(a, δ(0))−a = 0, which implies Z(b, δ(0))−
b > 0, because b < a (recall that Z(z, δ(0)) − z > 0 cuts the horizontal z axis fromabove and does so only once). Thus, from the continuity of Z(z, δ) − z with respectto δ, the intermediate value theorem guarantees the existence of δ LF (b) ∈ (0, δ(0))
such that:
Z(b, δ LF (b)) − b = 0.

(30)

As Z(b, δ) is strictly increasing in δ, δ LF (b) is uniquely determined, and Z(b, δ)−b < 0
if and only if δ < δ LF (b). Thus, z ∗ (δ) < b if and only if δ < δ LF (b). The following
38

proves that δ LF (b) is increasing in b for all b > z min . Let us choose arbitrary valuesof b1 and b2 so that z min < b1 < b2 . From the definition of δ LF (b) in (30):Z(b1 , δ LF (b1 ); b1 ) − b1 = 0 = Z(b2 , δ LF (b2 ); b2 ) − b2 ,(31)

where we use expression Z(z, δ; b) to show explicitly the dependence of function
Z(z, δ) on b. As Z(z, δ LF (b1 ); b1 ) − z intersects the horizontal z axis from above anddoes so only once, the first equation of (31) implies that Z(z, δ LF (b1 ); b1 ) − z < 0 for
all z > b1 . Thus, from b1 < b2 and (31):Z(b2 , δ LF (b1 ); b1 ) − b2 < 0 = Z(b2 , δ LF (b2 ); b2 ) − b2 .

(32)

Note from (7) that M (z) = 0 whenever b ≤ z, which means that the value of function
Z(z, δ; b) does not depend on b whenever b ≤ z. Thus, when z = b2 (which meansthat b1 < b2 = z):Z(b2 , δ LF (b1 ); b1 ) = Z(b2 , δ LF (b1 ); b2 ).

(33)

From (32) and (33), we obtain Z(b2 , δ LF (b1 ); b2 ) < Z(b2 , δ LF (b2 ); b2 ). As Zδ (z, δ; b) >0, this implies that δ LF (b1 ) < δ LF (b2 ); i.e., δ LF (b) is increasing in b.
Finally, we show the continuity of δ LF (b) and its boundary property. As Z(b, δ; b)−b is continuous in δ and b, δ LF (b), as defined by (30), is continuous for all b ∈ (z min , a).
In addition, it is continuous at b = z min because δ LF (b) → 0 as b → z min fromZ(z min , 0) − z min = 0 (recall that δ LF (b) = 0 for all b ≤ z min ). When b → a,
δ LF (b) → δ(0) because Z(a, δ(0)) − a = 0. This completes the proof.

Proof of Proposition 2
From (4)-(7), function M (z) can be written as:
M (z) = (a/b)N ((a/b)z).

(34)

Note that, since N (·) is a continuous function, equation (34) shows that function
M (z) changes continuously with respect to b. In addition, it implies that both
functions µ∗ (z, δ) and z ∗ (δ) are continuous with respect to b, from (20)-(23) and the39

assumption that the fixed point of (23) is unique. Therefore, the long-term rate of
growth γY∗ (δ), defined by (24), is also continuous with respect to b.
Utilizing the continuity of γY∗ (δ) with respect to b, we first prove that γY∗ (δ) is
maximized at δ = δ max ∈ (0, δ(0)) whenever b is suﬃciently close to z min . From
Lemma 1 and (24):γY∗ (δ)/(ln λ) = (1 − µ∗ (z ∗ (δ), δ))G(N (z ∗ (δ))) whenever δ ≥ δ LF (b),(35)

since M (z ∗ (δ)) = 0 in this case. Note that equation (35) is exactly the same as(19) since expressions µ∗ (z ∗ (δ), δ)) and N (z ∗ (δ)) introduced in Section 5 correspondrespectively to µ∗ (δ) and ψ(µ∗ (δ), δ) in Section 4. Note also that δ LF (z min ) = 0 from
Lemma 1. Therefore, when b = z min , Proposition 1 shows that γY∗ (δ) is maximizedat δ max ∈ (0, δ(0)), and the maximized rate of growth g noleap ≡ γY∗ (δ max ) is strictly
positive. Now suppose that b is slightly larger than z min . Then δ LF (b) is slightly larger
than 0, since δ LF (b) is continuous and strictly increasing from Lemma 1. Given thatb is suﬃciently close to z min , we have δ LF (b) ≤ δ max , and therefore, from (35), thegrowth rate of g noleap > 0 is attainable by setting δ = δ max . This is the growthmaximizing IPR policy within the range of δ ≥ δ LF (b), relying only on the flow ofleapfrogging. On the other hand, given b, define η(b) as the maximum value of γY∗ (δ)
within the range of δ ∈ [0, δ LF (b)]. Then η(z min ) = γY∗ (0) = 0 from δ LF (z min ) = 0.
In addition, since γY∗ (δ) and δ LF (b) are continuous with respect to δ and b, η(b) is
continuous with respect to b. From these we obtain η(b) < g noleap whenever b is
suﬃciently close to z min . This establishes the first half of Proposition 2.
Let us turn to the case in which b is large and close to a. If b = a, M (z) = N (z)from (34), and therefore equation (24) reduces to γY∗ (δ)/(ln λ) = G(N (z ∗ (δ))). Recall
from Lemma 1 that z ∗ (δ) < a and z ∗ 0 (δ) > 0 for all δ ∈ (0, δ(0)). In addition,
N 0 (z) < 0 for all z < a and G0 (·) > 0. Therefore, γY∗ (δ) is strictly decreasing inδ for all δ ∈ (0, δ(0)), implying that δ = 0 is the unique growth-maximizing IPR
policy when b = a. Now suppose that b is slightly smaller than a. Since γY∗ (δ)
changes continuously with respect to b, the growth-maximizing choice of δ must bein the neighborhood of δ = 0 given that b is suﬃciently close to a. Therefore, to40

prove that δ = 0 is actually the growth-maximizing policy, it is suﬃcient to showb
≡ G(N (z ∗ (δ))) and similarlythat γY∗ (δ) is downward sloping at δ = 0. Define G(δ)b
H(δ)≡ H(M (z ∗ (δ))). Using (20), the long-term rate of growth (24) can be writtenas:Ãb − H(δ)
b
G(δ)
b
1−
γY∗ (δ)/(ln λ) = G(δ)
bb
δ(1 − H(δ))+ G(δ)!

.

Diﬀerentiating the above expression with respect to δ and substituting δ = 0 into
the result yields:
d
dδ

µ ∗
¶¯
b
b
γY (δ) ¯¯
b 0 (0) + G(0) − H(0) (1 − H(0)),
b=Hbln λ ¯δ=0
G(0)

(36)

b 0 (0) = H 0 (M (z ∗ (0))) · M 0 (z ∗ (0)) · z ∗ 0 (0). Note that, since we are considering
where Hthe case of b > z min , Lemma 1 implies δ LF (b) > 0 and thus z ∗ (0) < b. Thereforeb 0 (0), the first term in (36), is strictly
M 0 (z ∗ (0)) < 0 holds, which means that Hbnegative (this holds even at b = a). The second term is positive because 0 < H(0)
<

b
b
b
G(0)
< 1, but it approaches zero as b → a, since in that case H(0)→ G(0)
from (34).

Therefore, when b is suﬃciently close to a, the slope of γY∗ (δ) at δ = 0 is negative,
which completes the proof of the latter half of Proposition 2.

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