Discussion Papers In Economics
And BusinessIntellectual Property Rights and Foreign Direct Investment:
A Welfare Analysis

Hitoshi Tanaka and Tatsuro Iwaisako

Discussion Paper 13-15-Rev.

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANIntellectual Property Rights and Foreign Direct Investment:
A Welfare Analysis

Hitoshi Tanaka and Tatsuro Iwaisako

Discussion Paper 13-15-Rev.

November 2013

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANIntellectual Property Rights and Foreign Direct Investment:
A Welfare Analysis∗
Hitoshi Tanaka†

Tatsuro Iwaisako‡
Abstract

This paper examines how intellectual property rights (IPR) protection affects innovationand foreign direct investment (FDI) using a North–South quality-ladder model incorporating the exogenous and costless imitation of technology and subsidy policies for both R&D
and FDI. We show that for the interior steady state to be stable, either R&D or FDI subsidy rates must be positive. Our findings also indicate that strengthening IPR protection
promotes both innovation and FDI. Moreover, a strengthening of IPR protection can also
improve welfare if the initial IPR protection in the South is weak and the R&D subsidy rate
is not too high.
JEL classification: F43, O33, O34
Keywords: foreign direct investment, innovation, intellectual property rights protection∗

The authors would like to thank Koichi Futagami, Takeo Hori, Masaru Inaba, Hwan Lin, Makoto Nirei, Paul

Segerstrom, and seminar participants at Hitotsubashi University and Konan University for helpful suggestions on
this paper. The authors are especially grateful to Ichiroh Daitoh, Theo Eicher, Masao Oda, and two anonymousreferees for their useful comments. Any errors remaining are the authors’ responsibility. This research is partly
supported by a Grant-in-Aid for Young Scientists (No. 20730138) from the Japanese Ministry of Education,
Culture, Sports, and Technology.
†
Faculty of Economics, Hokkai-Gakuen University, 4-1-40, Asahi-machi, Toyohira, Sapporo, Hokkaido 0628605, Japan. E-mail: tanaka h@hgu.jp‡
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043, Japan.
E-mail: iwaisako@econ.osaka-u.ac.jp

1 Introduction
Since the Agreement on Trade-Related Aspects of Intellectual Property Rights (TRIPs) was
signed in the Uruguay Round, developing countries that are members of the World Trade Organization (WTO) have been under pressure to adopt a set of minimum standards on intellectual
property rights (IPR). To comply with these international agreements, some developing countries have recently strengthened their IPR protection. For example, according to the indexes inPark (2008), patent protection in Brazil, China, and India generally strengthened between 1990
and 2005 compared with before 1990.
This change toward strengthening IPR protection in developing countries is likely to havea great impact on innovation and foreign direct investment (FDI) in these countries for several
reasons. For example, strengthening IPR protection in a developing country makes it difficult
for local firms to copy products developed by other firms and decreases the risk of technology
imitation in that country. Thus, strengthening IPR protection is likely to influence the decisionof a firm with advanced technology on whether to transfer production to a developing country.
In addition, a decrease in imitation changes the monopolistic rent that the inventor of a goodcan earn, which is likely to influence R&D activities by firms in developed countries.
The present paper theoretically investigates the impact of strengthening IPR protection indeveloping countries using a dynamic general equilibrium model with two countries: the North,where new technology is invented, and the South, where new technology cannot be invented but
can be transferred from the North through FDI. From this analysis, we derive three results. First,
strengthening IPR protection in the South increases the wage in the South relative to that in the
North. Second, strengthening IPR protection increases innovation in the North and the flow of
FDI from the North to the South in both the long and the short run. Third, strengthening IPR
protection can improve the welfare of both Southern and Northern households if the initial IPR
protection in the South is sufficiently weak and the R&D subsidy rate is not too high.
A number of theoretical studies on technology transfer have examined the influence ofstrengthening IPR protection using North–South dynamic general equilibrium models where
the chosen channel for technology transfer is FDI. However, these studies are divided on the
results. For example, two of the most important studies in this field, Lai (1998) and Glass and–1–

Saggi (2002), obtained contrasting results. Lai (1998), using a model of variety-expandingtype innovation, concluded that strengthening IPR protection promotes both innovation and
FDI, whereas Glass and Saggi (2002), using a model of quality-improvement-type innovation,suggested the opposite. In related work, Glass and Wu (2007) (hereafter G-W) introduced costless imitation, as did Lai (1998), into a quality-improvement-type R&D model similar to thatof Glass and Saggi (2002), and examined how increasing the probability of imitation affects
innovation and FDI. Their results showed that strengthening IPR protection impedes both innovation and FDI. This finding lies contrary to Lai (1998) but is similar to that in Glass andSaggi (2002). By comparing the settings and results in these papers, G-W (2007) surmised that
we could attribute the different results in Lai (1998) and Glass and Saggi (2002) to whether
innovation is variety expanding or quality improving.
In this paper, we show that this presumption is not correct using a quality-ladder-type model.
More specifically, the present paper extends G-W’s (2007) model by introducing selected industrial policies into the model, i.e., subsidies for R&D and FDI, in order to reexamine the effectof strengthening IPR protection. Our model also includes the case of “inefficient followers”
from their paper as a particular case where both of these subsidies are zero. In terms of results,
our model shows that the unique interior steady state is necessarily unstable if both subsidies
are zero. Hence, there is no equilibrium path converging to the interior steady state in the caseof zero subsidies. In that case, following a policy change, the economy must move toward a“corner-solution equilibrium” in which some endogenous variables are zero. This result implies
that the conclusion on IPR protection in G-W’s (2007) inefficient followers’ case needs to bereexamined because we cannot apply comparative statics of the steady state to the evaluation ofa policy change. To address this issue, we prove that the unique steady state can be stable andcomparative statics are applicable if the subsidy rates are higher than some critical level. Ourmodel shows that if the interior steady state is stable, strengthening IPR protection necessarily
promotes both innovation and FDI. This central conclusion is the opposite of the result in GW (2007) and the same as that of the variety-expanding-type model in Lai (1998). Thus, ourresult proposes a counterexample to G-W’s (2007) conjecture that whether strengthening IPR
protection promotes innovation and FDI depends on the type of innovation.
As a more important topic, we also explore the welfare effects of strengthening IPR pro-–2–

tection. Many earlier studies in this area, including Lai (1998), Glass and Saggi (2002), and
G-W (2007), did not analyze the welfare effects because they focused on the effects on innovation and FDI. However, we cannot draw conclusions about the desirability of IPR policies
based only on their effects on innovation and FDI. Our welfare analysis shows that strengthening IPR protection in the South entails simultaneous dynamic and static effects on welfare.
The former dynamic effect arises from promoting innovation. An increase in innovation thenenables households to consume higher-quality goods over time and increases their welfare. But
strengthening IPR protection also accounts for static effects through changing the number ofimitated goods and the income of households. As the imitator firms produce the imitated goods
competitively, they sell at a lower price than the other goods produced monopolistically under
the patents. Therefore, strengthening IPR protection may reduce welfare through decreasing
imitation and the number of cheaper goods. In addition, it may affect welfare through changing
the wages and the values of shares owned by households. To evaluate the desirability of IPR
policies, we then need to compare the sizes of the dynamic effects with those of the static effects. In this paper, we show that the dynamic effects outweigh the static effects if the initial
IPR protection is sufficiently weak and the rate of R&D subsidy is not too high. This implies
that stronger protection of IPR can improve the welfare of the South and the North.
The remainder of the paper is structured as follows. Section 2 describes the model. In Section 3, we derive the equilibrium path of this model. In Sections 4 and 5, we show that strengthening IPR protection promotes both innovation and FDI. Section 6 shows that strengthening IPR
protection can improve welfare. In Section 7, we discuss the welfare effects of subsidy policies
and the welfare effects in the model where imitation and FDI are costly processes. Section 8
provides some concluding remarks.

2 The Model
Our model has the same basic structure as that of G-W (2007), which is a version of the North–South quality-ladder model developed by Grossman and Helpman (1991, Ch. 12).1 The main
1

Our model is based on the “fully endogenous” rather than the “semi-endogenous” theory of growth. The fully

endogenous growth model has often been criticized because of the “problem” of scale effects. However, Ha and–3–difference between our model and G-W (2007) is the existence of two subsidies, one for R&D
and the other for FDI. Consider an economy consisting of two countries, the North and theSouth, denoted by N and S, respectively. The population size of country i ∈ {N, S} is constantand given by Li . Each agent supplies one unit of labor inelastically at each point in time t ∈[0, ∞) at a wage of wi (t). We choose Southern labor as the numeraire and normalize wS (t)to one at any point in time t. We let w(t) denote the relative wage of the North to the South:
w(t) ≡ wN (t)/wS (t) = wN (t).
In this economy, there is a continuum of goods, indexed by j ∈ [0, 1], that are producedin the North or the South. One unit of good output requires one unit of labor input. Eachgood is classified by a number of “generations” m = 0, 1, 2, · · · . We normalize the generation
number of every good to be zero at time t = 0. A one-step newer generation of good j becomes
available if innovation takes place in industry j as a result of successful R&D efforts by a firm.
We assume that different generations of a good have different “qualities”. The quality of goodj of generation m is provided by qm (j) = λm .

2.1 Consumers
Consumers living in country i ∈ {N, S} have the following lifetime utility:Z ∞Ui =e−ρt log ui (t)dt,(1)0where ρ is a common subjective discount rate and log ui (t) represents instantaneous utility attime t. We specify the instantaneous utility function as:
"
#
Z 1
X
log ui (t) =log
qm (j)xi,m (j, t) dj,
0

(2)

m

where xi,m (j, t) denotes consumption by consumers living in country i of generation m of goodj at time t. The representative consumer in country i ∈ {N, S} maximizes his or her lifetime
Howitt (2007), for example, have argued that fully endogenous theory is more consistent than semi-endogenous
theory with the long-run data. Further, Aghion and Howitt (2006, p. 98) stated that “... there is no evidence
pointing to the absence of a scale effect at the world level or in small closed economies”. Because debate remains
as to whether endogenous or semi-endogenous theory is more appropriate, we adopt an endogenous growth model.
This generally has the benefit of a simpler dynamic structure than the semi-endogenous growth model, so we canobtain clearer results, particularly for welfare analysis.

–4–utility (1) under the following budget constraint:
Z ∞ R
Z ∞ R
Z ∞ R
t
− 0t r(s)ds− 0t r(s)dse
Ei (t)dt = Ai (0) +ewi (t)dt −e− 0 r(s)ds Ti (t)dt,000where r(t) is the interest rate that consumers in both countries face at time t, Ai (0) is the initialasset holdings of a consumer in country i, and Ti (t) is a lump-sum tax levied by the governmentof country i. The term Ei (t) represents the flow of expenditure by a consumer in countryi ∈ {N, S} at time t, namely:
Ei (t) =Z 1 "X0#pm (j, t)xi,m (j, t) dj,mwhere pm (j, t) is the price of generation m of good j at time t.
We can solve this consumer’s utility maximization problem in two stages. First, for eachproduct, the consumer chooses the single generation m̃(j, t) that carries the lowest qualityadjusted price pm (j, t)/qm (j). This implies the following static demand function: E (t)/p (j, t) for m = m̃(j, t),imxi,m (j, t) =0otherwise.

(3)

Second, intertemporal utility maximization requires that Ėi (t)/Ei (t) = r(t) − ρ. Therefore,
aggregate world expenditure, E(t) ≡ LN EN (t) + LS ES (t), also changes over time according
to the following condition:Ė(t)= r(t) − ρ.
E(t)

(4)

2.2 Production
In this model, we can classify firms into two types: “leaders”, which are firms that developed
the current latest generation of each good, and “followers”, which are firms other than theleaders. These firms are located in the North or the South. Regardless of location, a firm canfreely sell its good in both countries without incurring any transportation costs or tariffs. We
assume that only Northern firms have the ability to undertake R&D and bring about innovation.
If a Northern firm succeeds in developing a newer generation of good j, the firm can patentthat generation of good j in the North and monopolistically sell the patented good until it is–5–imitated. Hereafter, we refer to the leader producing a latest-generation good in the North as a
“Northern leader”.
A Northern leader can become a “multinational” firm by shifting production to the Souththrough FDI. By becoming a multinational firm, the firm can employ cheaper Southern laborfor production. However, a multinational firm faces the risk of imitation by Southern followers
because IPR protection is not perfect in the South. Once imitation takes place in industry j, theprocess for making the current latest generation of good j is completely revealed and perfectcompetition prevails in the industry until the next generation is developed. The imitated good
is sold at wS (t) = 1 in both countries and the multinational firm loses its monopoly rents. We
assume that follower firms cannot copy a good produced in the North by the Northern leader.
As in Grossman and Helpman (1991), a monopolistic leader (which refers to a leader whose
good has not yet been imitated) can maximize profits by pricing its good at the upper limit of
the prices such that rival firms are unable to operate. As described in Section 2.3, we focus on
the case where innovation takes place only in industries in which the current latest generation
has already been copied. In this case, the strongest rivals of each monopolistic leader are theSouthern followers that have the ability to produce the second-highest quality of each good,which could cut the price to the marginal cost, wS (t) = 1. Because a monopolistic leader needs
to set the lowest quality-adjusted price to eliminate rivals from its market, the optimal price foreach Northern leader and multinational firm is the marginal cost of the rivals multiplied by the
degree of advantage in quality: p = λ · wS (t) = λ. This price setting and the demand function
imply that the sales of a monopolistic leader are E(t)/λ. Therefore, the flow of profits for a
Northern leader is:E(t)=πN (t) = (λ − wN (t))
λ

µ

w(t)
1−
λ

¶
E(t).

(5)

Likewise, the flow of profits for a multinational firm is:
µ
¶
E(t)
1
πF (t) = (1 + sF )(λ − wS (t))= (1 + sF ) 1 −
E(t),
λ
λ

(6)

where sF ≥ 0 is the rate of FDI subsidy. That is, the Southern government pays each multinational firm 100 × sF percent of profits as “FDI subsidies”.2 Note that sF is assumed to be zeroin G-W’s (2007) model.
2

In reality, we can consider that the governments of some developing countries provide a “subsidy” in the form

–6–

2.3 R&D and FDI
We make the same two assumptions as G-W (2007) to focus on the case where no R&D isundertaken in industries where a Northern leader or a multinational firm produces a good. First,
the labor input required for one unit of R&D by followers is sufficiently larger than that for
the leader of the industry. This first assumption ensures that only leaders undertake R&D.
G-W (2007) referred to this situation as the case of “inefficient followers” because the R&D
productivity of followers is relatively low. Second, no leader has an incentive to further research
its own good until the good has been imitated.3 Under these two assumptions, only the leader
firm that made the previous innovation for the good undertakes R&D after the good has been
imitated.
As in Grossman and Helpman (1991), we assume that the success or failure of R&D follows a Poisson process: specifically, if the leader firm of industry j devotes aN ιN (t)dt unitsof Northern labor for a time interval of length dt to research on good j, it succeeds in developing the next generation of good j with probability ιN (t)dt. We refer to ιN (t) as “innovation
intensity”. A leader firm whose good has been imitated chooses ιN (t) to maximize expected
net gains from R&D. If a firm succeeds in innovation by R&D, it obtains vN (t), which denotes
the market value of a Northern leader. Meanwhile, R&D costs wN (t)aN ιN (t)dt for the wage
payments. Thus, the expected net gains from R&D are [vN (t) − (1 − sR )wN (t)aN ]ιN (t)dt,
where sR ∈ [0, 1) denotes the rate of R&D subsidy. That is, the Northern government bears
100 × sR percent of the R&D cost as subsidies. Note that G-W’s (2007) model corresponds tothe case of sR = 0 in our model. Because R&D activities have to be of positive but finite size,
the following zero-profit condition on R&D must be satisfied:vN (t) = (1 − sR )w(t)aN .

(7)

Following G-W (2007), we assume that a Northern leader can become a multinational firm
instantaneously without cost. Under this assumption, a Northern leader firm must be indifferent
as to whether it becomes a multinational firm or continues production only in the North. Thus,of tax incentives, including partial exemption from corporate taxes, to the FDI of multinational firms. See, forexample, UNCTAD (2001) for a broad survey of the tax incentives governments in developing countries use topromote FDI.
3
In equilibrium, this assumption is satisfied if the quality increment by innovation, λ, is sufficiently large.

–7–

the following equality must hold at each point of time:vN (t) = vF (t),(8)

where vF (t) is the market value of a multinational firm whose good has not yet been imitated.
Arbitrage between assets requires that a stock of a leader firm yields the same expected rateof return as the risk-free interest rate, r(t). Thus, in equilibrium, the following no-arbitrage
condition between the risk-free asset and the stock of a Northern leader is satisfied:r(t)vN (t) = πN (t) + v̇N (t),(9)

where the right-hand side is the return from holding the stock of a Northern leader and is equal
to the sum of dividends and capital gains. Meanwhile, the shareholders of a multinational firm
face the risk of imitation. If imitation takes place in industry j, the multinational firm in theindustry loses its monopoly rents. Following Lai (1998) and G-W (2007), we assume that every
multinational firm is equally exposed to the risk of imitation at an exogenous rate M (> 0) that
depends on IPR protection in the South. Thus, the shareholders of a multinational firm suffer acapital loss of amount vF (t) at rate M . This implies that the following no-arbitrage condition
between the risk-free asset and the stock of a multinational firm is satisfied in equilibrium:
r(t)vF (t) = πF (t) + v̇F (t) − M vF (t),(10)

where the right-hand side is the return from holding multinational firm stock.

2.4 Type of Industry
In this model, we classify every industry into the following three categories: (i) “type-N” industries where the Northern leader firm monopolistically produces the good; (ii) “type-F” industries
where the multinational firm monopolistically produces the good; and (iii) “type-S” industries
where Southern imitator firms produce the good under perfect competition. We represent themeasure (number) of industries belonging to each category by nN (t), nF (t), and nS (t), respectively. The sum of the measure of all industries is equal to one, so that nN (t)+nF (t)+nS (t) = 1.
Hereafter, we focus only on the equilibrium such that all industries in the same category
are symmetric. In this equilibrium, innovation intensity ιN (t) takes a common value in every–8–

type-S industry. Because all innovation takes place in type-S industries, the measure of industries in which innovation takes place in a time interval of length dt is given by ι(t)dt, whereι(t) ≡ ιN (t)nS (t) is the “aggregate rate of innovation” in the economy as a whole. These industries change the state from type-S to type-N in this time interval by successful innovation.
Meanwhile, the measure of type-F industries in which imitation takes place is M nF (t)dt. These
industries change the state from type-F to type-S in this time interval by successful imitation.
Therefore, the value of nS (t) changes over time with the following equation of motion:ṅS (t) = M nF (t) − ι(t).

(11)

The values of nN (t) and nF (t) are determined at each point in time between 0 and 1 − nS (t)depending on the Northern leader firms’ location choices.

2.5 Labor Market
Multinational and imitator firms employ Southern labor in the production of goods. Therefore,
the labor market-clearing condition in the South is:nF (t)

E(t)+ nS (t)E(t) = LS .
λ(12)Northern labor is devoted to R&D and production. Each leader firm whose good has been
imitated undertakes R&D with intensity ιN (t), so that the labor demand for R&D is given byaN ιN (t)nS (t). Therefore, the labor market-clearing condition in the North is:aN ι(t) + nN (t)E(t)= LN .
λ(13)

2.6 Government Budget Constraints
In this economy, the Northern and Southern governments subsidize R&D activities and multinational production, respectively. For simplicity, suppose that at each point in time each government runs a balanced budget where it finances its total subsidy payments with lump-sum taxes
levied on each country’s consumers. To achieve a balanced budget, the following constraints
must be satisfied in both countries: sF (1 − 1/λ) E(t)nF (t) = LS TS (t) and sR w(t)aN ι(t) =LN TN (t), where the left-hand sides are the total payments of subsidies by both governments–9–and the right-hand sides are the tax revenues. The governments determine taxation Ti (t) suchthat these budget constraints are satisfied.

3 Market Equilibrium PathIn this section, we characterize the equilibrium path of the economy. The detailed derivation of
the results in this section is given in Appendix A. In the following parts of the paper, we focus
only on the interior equilibrium in which the aggregate rate of innovation is strictly positive andthere are all types (type-N, type-F, and type-S) of industry at any time. Hereafter, let variables
with an upper bar, e.g., Ē, denote the steady-state values of the corresponding variables.
On the equilibrium path of this model, some endogenous variables become constant by
jumping to their steady-state values immediately at the initial time. In particular, E(t) and w(t)respectively take the following constant values for all t in the equilibrium:Ē =aN λ(1 − sR )[ρ + M λ − ρsF (λ − 1)],(1 + sF )(λ − 1)

(14)

ρ + M λ − ρsF (λ − 1)
.
ρ+M

(15)

w̄ =

We assume that w̄ is greater than 1 because sF is sufficiently small such that sF < M/ρ. As thevalues of E(t) and w(t) are constant, r(t), vN (t), and vF (t) also become constant for all t on
the equilibrium path. Equation (4) implies that r(t) = ρ for all t.
In the steady state of the model, nS (t), nN (t), nF (t), and ι(t) also take constant values. The
steady-state values of nS (t), nN (t), and nF (t) are given by:
aN M λ2 LS + Ē 2 − λ(LN + LS )Ēn̄S =,[aN M λ2 − (λ − 1)Ē]Ēλ(LS + λLN − Ē),aN M λ2 − (λ − 1)Ē(17)

λ[aN M λ(Ē − LS ) − LN Ē(λ − 1)].
[aN M λ2 − (λ − 1)Ē]Ē

(18)

n̄F =

n̄N =

(16)

– 10 –

These values must be positive if there is an interior steady state in the model. The aggregate
rate of innovation ι(t) is determined by the value of nS (t) as follows:¾½1Ēι(t) =
LN + LS − [1 + (λ − 1)nS (t)] .
aN
λ

(19)

However, nS (t), nN (t), nF (t), and ι(t) cannot jump to their steady-state values immediately
at the initial time because nS (t) is a state variable whose value is historically given, and nN (t),nF (t), ι(t) are determined depending on the value of nS (t). The equation of motion of nS (t) is:ṅS (t) =aN M λ2 LS + Ē 2 − λ(LN + LS )Ē− µnS (t),aN λĒ

(20)

where Ē is given by (14) and µ ≡ M λ − (λ − 1)Ē/(aN λ) = M λ − (1 − sR )[ρ + M λ − ρsF (λ −1)]/(1 + sF ) is the speed of convergence to (divergence from) the steady state if µ > 0 (µ < 0).
Note that this equation of motion is a linear function of nS (t) and depends only on the valuesof nS (t) itself and the exogenous variables. We solve the linear differential equation (20) as:n̄S + (nS (0) − n̄S )e−µt
for µ 6= 0,(21)
nS (t) = n (0) + aN M λ2 LS +Ē 2 −λ(LN +LS )Ē t for µ = 0,S

aN λĒ

where nS (0) denotes the initial value of nS (t). Because nS (t) is not jumpable, (21) implies
that µ must be strictly positive so that the steady state can be attained; otherwise, nS (t) cannever take the value of the interior steady state given by (16), except in the special case wherenS (0) = n̄S . Likewise, nF (t) and nN (t) approach the steady-state values of (17) and (18) overtime if and only if µ > 0 because the values of both variables depend on nS (t). In this sense, theinterior steady state of the model becomes unstable if µ is negative. We summarize this result
as follows.
Proposition 1. Suppose that there exists a unique steady state in which n̄S , n̄F , n̄N , and ῑ areall positive. Then the economy approaches the steady state over time if and only if µ is positive.
Proposition 1 shows that the key to the stability of the steady state is the sign of µ, being
the coefficient of the equation of motion of nS (t). Intuitively, we can interpret what determines
the value of µ and the stability of the steady state by equation (11). Other things being equal,
an increase in nS reduces the inflow into nS (the first term in (11)) because it reduces thetargets of imitation, nF , by the Southern labor constraint. This first effect is represented by– 11 –

the first term in the definition of µ. Meanwhile, other things being equal, an increase in nS
also reduces the outflow from nS (the second term in (11)) because it reduces ι.4 This second
effect is represented by the second term in the definition of µ. If the former effect outweighs thelatter, µ becomes positive. In this case, a change in nS decelerates over time and the economy
asymptotically approaches the steady state. If the latter effect outweighs the former, µ becomes
negative. In this case, a change in nS accelerates over time and the economy moves away fromthe steady state.
We hereafter assume that µ > 0 so that the interior steady state is attainable.5 Meanwhile,by the definition of µ, we can immediately prove the following corollary.
Corollary. Suppose that the rates of the subsidies on R&D and FDI are equal to zero: sR =sF = 0. Then, the interior steady state of the model is unstable.
This corollary implies that the economy in the case of the inefficient followers in G-W(2007), which corresponds to the specific case of sR = sF = 0 in our model, does not tend
to the interior steady state, except when it begins in the steady state by chance. In particular,when the government changes a policy, such as IPR protection, the economy without subsidies
no longer moves toward the new interior steady state, even if it were originally in the interior
steady state. Rather, it moves toward the “corner-solution equilibrium” such that the values ofsome endogenous variables are zero over time.6 For this reason, if sR = sF = 0, a comparisonof the interior steady states before and after a given policy change does not enable an accurate
judgment on the effect of that policy. In this case, we should not evaluate the influence ofa policy using analysis of the comparative statics. Thus, the conclusion on IPR protection
4

This is shown in equation (19). Because a type-S industry employs more labor than a type-F industry, thelabor constraint in the South requires that an increase in nS must reduce nS + nF by sharply decreasing nF . Thus,
an increase in nS increases the measure of industries producing in the North, nN = 1 − (nS + nF ), and thereby
decreases the labor devoted to R&D in terms of the labor constraint in the North. This is why ι decreases with nS .
5
If λ is sufficiently large, this condition is not very restrictive. For instance, following Glass and Saggi (2002),
we use ρ = 0.05, λ = 4, LN = 3, and LS = 6 as a numerical example. We further set M = 0.037, aN = 123.5,
and sF = 0.2. In this case, the interior steady state exists and is stable if 0.220 < sR < 0.487, which does notappear a very narrow range.
6More specifically, if nS (0) is greater than n̄S , then nS (t) gradually increases and either innovation or FDI
becomes zero at finite time. If nS (0) is smaller than n̄S , then nS (t) gradually decreases and either nS (t) or nN (t)becomes zero at finite time.

– 12 –

policy in G-W’s (2007) inefficient followers’ case should be reexamined because it draws on
comparative statics.

4 Long-Run Effects of Strengthening IPR Protection
In this section, we examine the influence of strengthening IPR protection on the steady-statevalues of the endogenous variables. Because we can interpret the strengthening of IPR protection in the South as a decrease in the imitation rate M , we carry out the comparative statics withrespect to M .

4.1 The Effect on the Relative Wage
Using the comparative statics, we can find that strengthening IPR protection (a decrease in M )increases the wage in the South relative to the North, 1/w(t). Differentiating w̄ given by (15)with respect to M , we have:(1 + sF )(λ − 1)ρ
∂ w̄
=
> 0.
∂M
(ρ + M )2
The reason why stronger IPR protection increases the Southern relative wage is as follows.
To start with, the no-arbitrage condition (10) implies that the market value of a multinationalfirm must equal the sum of the present value of its instantaneous profit flow, that is, v̄F =π̄F /(ρ + M ). This relation shows that, other things being equal, strengthening IPR protection(a decrease in M ) increases vF directly. This is because the decrease in the risk of imitation
enables a multinational firm to earn instantaneous profits πF for a longer time on average byextending the expected period of monopoly. The larger vF then provides the Northern leader
firms with greater incentive to transfer production to the South. Therefore, other things being
equal, the demand for Southern labor to produce a good would increase because more Northern leader firms would choose to convert to multinational firms. In equilibrium, the Northern(relative) wage w̄ unambiguously falls to increase the incentive for production in the North. We
summarize this result as follows.
Proposition 2. Strengthening IPR protection in the South increases the wage in the South relative to that in the North.
– 13 –

4.2 The Effect on Innovation
How then does strengthening IPR protection affect the “long-run” aggregate rate of innovation ῑ? G-W (2007) concluded that relaxing IPR protection (an increase in M ) increases the aggregate
rate of innovation in the case of inefficient followers. However, our model shows that thisconclusion is not true in the parameter ranges where the interior steady state becomes stable. In
fact, differentiating (19) with respect to M and evaluating the steady-state value, we can prove
the following proposition.
Proposition 3. Strengthening IPR protection in the South increases the long-run aggregate rateof innovation.
Proof. See Appendix B.
How does stronger IPR protection promote innovation in our model? Intuitively, it is usefulto recall the Northern labor market-clearing condition (13). This condition requires that thesum of the labor inputs into R&D and production by the Northern leaders must be equal to theconstant Northern labor supply. Accordingly, the lower the labor input into production by theNorthern leaders, the more abundant the labor input into R&D. In our model, strengthening IPR
protection decreases the labor input for production by Northern leaders through the following
channels. The first channel is through a decrease in aggregate spending Ē. By differentiating Ē
given by (14) with respect to M , the effect of strengthening IPR protection (a decrease in M )on Ē proves to be negative as follows:∂ Ē
aN λ2 (1 − sR )
=
> 0.
∂M
(1 + sF )(λ − 1)
Because the decrease in aggregate spending Ē reduces the demand for goods produced by
Northern leaders, each Northern leader decreases the labor input for production. The second
channel is through a decrease in the measure of type-N industries. In our model, the measureof type-N industries n̄N increases with the rate of imitation M ; that is, strengthening IPR protection decreases the number of Northern patent holders that choose to operate in the North. By
differentiating (18) with respect to M , we can verify this as follows:
½
·
¸
¾
∂ n̄N
1
M λLS (λ − 1)n̄N ∂ Ē=+(λ − 1)n̄F +
> 0.
∂M
µ
aN λ
∂M
Ē 2
– 14 –(22)

The fewer leaders operating in the North mean less use of Northern labor for production. Because both these effects decrease the labor input for production, strengthening IPR protectionin the South increases the labor input for R&D and thereby promotes innovation.
Note that the conclusion of Proposition 3 is completely opposite to that of the inefficient
followers’ case in G-W (2007). Why does the result of our model differ from that of G-W(2007)? The main reason is the stability of the steady state in the model. Recall that we restrict
the parameter ranges so that the steady state becomes stable. As discussed in Section 3, atleast µ must be positive if the steady state is stable. Because µ is positive, we find that n̄Nincreases with the rate of imitation M from (22). As a result, we can draw the conclusionthat strengthening IPR protection unambiguously decreases the labor input for production in
the North. However, in G-W’s (2007) model, the steady state is unstable and the value of µ
is negative. Therefore, from (22), n̄N must be decreasing with the rate of imitation M . Thismeans that the strengthening of IPR protection in their model necessarily increases the numberof leader firms operating in the North (see also equation 29 in G-W, 2007). Because the signof the abovementioned second channel lies opposite to that of our model, strengthening IPR
protection can increase the labor input for production in the North. In fact, the negative effectof the latter channel outweighs the positive effect of the former, so that strengthening IPR
protection decreases the labor input for R&D and the aggregate rate of innovation in G-W’s(2007) model.
The result of Proposition 3 is similar to that obtained with variety-expanding-type innovation models including an exogenous process of imitation in the South, such as in Lai (1998).
However, our result is contrary to Mondal and Gupta (2008) who employed a variety-expandingtype innovation model including an endogenous process of imitation in the South, unlike Lai(1998) and the present paper. Mondal and Gupta (2008) showed that strengthening IPR protection impedes innovation.

– 15 –

4.3 The Effect on FDI
Strengthening IPR protection in the South increases the measure of type-F industries as well as
the aggregate rate of innovation. By differentiating (17) with respect to M , we have:
·
¸
∂ n̄F
1
(1 − n̄F )λ + n̄F ∂ Ē= − λn̄F +
< 0.
∂M
µ
aN λ
∂M

(23)

We summarize this result as the following proposition.
Proposition 4. Strengthening IPR protection in the South increases FDI in the long run.
By using the equation of motion of nS (t), we obtain an interpretation of Proposition 4. From(11), the value of n̄F is given by the ratio of the aggregate rate of innovation to the imitation
rate, ῑ/M , in the steady state. Because strengthening IPR protection decreases the imitation rate
M while increasing the aggregate rate of innovation ῑ, it necessarily increases the measure of
type-F industries n̄F .
The result of Proposition 4 is also contrary to the conclusion of G-W’s (2007) inefficient
followers’ case. G-W (2007) concluded that a decrease in M necessarily decreases n̄F . However, our model shows that a decrease in M necessarily increases n̄F from (23) because µ is
positive. The difference in the result of our model and that of G-W (2007) is also attributable
to the difference in the stability of the steady states in the two models. Recall that under G-W’s(2007) setting, strengthening IPR protection decreases the aggregate rate of innovation because
the value of µ is negative, unlike in our model. As n̄F is equal to ῑ/M in both models, strengthening IPR protection influences n̄F through two channels: the first is a direct effect through thedecrease in the imitation rate, and the second is an indirect effect through the decrease in theaggregate rate of innovation. The former has a positive effect, whereas the latter has a negative
effect on the value of n̄F in their model. Because the latter indirect effect is sufficient to outweigh the former direct effect in their model, strengthening IPR protection decreases n̄F . Notethat this conclusion in G-W (2007) crucially depends on the result that ῑ is increasing with M
under their setting. The reason why ῑ is increasing with M is the instability of the steady stateof their model. Thus, their result on the effect of a change in M on n̄F also arises from the lackof stability of the steady state.
The result of Proposition 4 accords with that in some of the literature. Lai (1998) and
Gustafsson and Segerstrom (2011), who assumed variety-expanding type of innovation, also
– 16 –

concluded that strengthening IPR protection in the South increases technology transfer to theSouth within multinational firms. Dinopoulos and Segerstrom (2010) obtained the same result
by using a quality-improvement-type innovation model that exhibits scale-invariant growth suchas Segerstrom (1998), Li (2003), and Dinopoulos and Segerstrom (2007). However, there area few exceptions, including Glass and Saggi (2002) and Mondal and Gupta (2008), both ofwhich assumed an endogenous process of imitation. Contrary to our findings, they showed thatstrengthening IPR protection in the South impedes FDI.

5 Short-Run Effects of Strengthening IPR Protection
In Section 4, we concluded that strengthening IPR protection positively affects the aggregate
rate of innovation and FDI in the long run by comparing the steady-state values before and after
the policy change. However, it is also important to explore how strengthening IPR protection
influences the aggregate rate of innovation and FDI on the transitional path to the new steadystate. Unlike many existing studies in this literature, we can investigate this, so to speak, “shortrun” effect of strengthening IPR protection because the transitional dynamics are explicitly
analyzed in our model.7
Suppose that the economy is initially in the steady state and the Southern government
strengthens IPR protection at the initial time. We define the short-run effects of strengthening IPR protection as the magnitudes of ∂ι(t)/∂M and ∂nF (t)/∂M for any t ∈ [0, ∞). Notethat ∂ι(t)/∂M and ∂nF (t)/∂M are not necessarily equal to ∂ῑ/∂M and ∂ n̄F /∂M because ourmodel includes the transitional process to the steady state. If ∂ι(t)/∂M and ∂nF (t)/∂M are
negative for all t ∈ [0, ∞), we can conclude that innovation and FDI are promoted at any point
in time after strengthening IPR protection. On this short-run effect, we can prove the following
proposition.
Proposition 5. Suppose that the economy is initially in the steady state. Then, strengthening
7

Most North–South innovation models tend to have a complicated dynamic structure and so are unable toanalytically examine the transition of the equilibrium paths. The exceptions are Helpman (1993), Arnold (2002)(which is based on Helpman (1993)), and Tanaka, Iwaisako, and Futagami (2007). These studies analyzed thetransition of the equilibrium path by employing a relatively simple dynamic structure in their models.

– 17 –

IPR protection in the South increases innovation and FDI for all t ∈ [0, ∞).
Proof. See Appendix C.
Proposition 5 shows that ∂ι(t)/∂M < 0 and ∂nF (t)/∂M < 0 for all t. It therefore implies
that both innovation and FDI necessarily increase, even on the transitional path, after strengthening IPR protection. That is, strengthening IPR protection promotes innovation and FDI not
only in the long run, but also in the short run. Proposition 5 shows that the result of the short-run
analysis is the same as that of the long-run analysis. Thus, it reinforces our conclusion regarding
the effect of IPR protection.

6 Welfare Analysis
In Sections 4 and 5, we showed that strengthening IPR protection enhances innovation and
FDI in both the long and short run. However, we should judge a policy’s desirability by how
it affects welfare. In this section, we examine the welfare effects of IPR protection with theresults obtained in Section 5. By considering the short-run welfare effect, we can examine how
a marginal increase in M , that is, relaxing IPR protection in the South, affects welfare.
To conduct the welfare analysis, we first consider the instantaneous utility function. Substituting (3) into (2), we obtain the utility level of a consumer in country i at time t as folR1
R1
lows: log ui (t) = (log λ) 0 m̃(j, t)dj + 0 log xi (j, t)dj, where xi (j, t) denotes the quantity
demanded for the current latest generation of good j at time t. The first term in this equation represents the welfare brought by the qualities of all goods consumed. Because m̃(j, t)
R1
equals the current state-of-the-art generation number of good j at time t, 0 m̃(j, t)dj in thefirst term is equal to the aggregate number of innovations brought in the interval from time 0
Rt
to time t. Thus, the first term can be replaced with (log λ) 0 ι(τ )dτ . This means that welfare
positively depends on the aggregate rate of innovation, ι(t). The second term in the instantaneous utility function represents the welfare brought by the quantities of all goods consumed.
R1
Substituting (3) into the second term, we can rewrite the term as log Ei (t) − 0 log p(j, t)dj,
which means that welfare depends positively on total spending, Ei (t), and negatively on theprices of the state-of-the-art goods, p(j, t). The price of type-S goods is equal to one because of– 18 –

competition, whereas the price of the goods other than type-S is λ (> 1) because the monopolistic leaders supply them. Therefore, the part of welfare that depends on the prices reduces to
R1
log p(j, t)dj = nS (t) log 1 + (1 − nS (t)) log λ = (1 − nS (t)) log λ, which depends on the0measure of type-S industries. An increase in the measure of type-S industries lowers the pricesof goods and improves the welfare of consumers. Considering these results, we can rewrite
instantaneous utility as:

Z tι(τ )dτ + log Ei (t) − (1 − nS (t)) log λ.

log ui (t) = (log λ)0

By differentiating the lifetime utility function with respect to M , we obtain the change of
welfare from the marginal increase in M as follows:
µZ t
¶
Z ∞
∂Ui
∂
1 1 ∂Ei−ρte (log λ)ι(τ )dτ dt +=∂M
∂M
ρ Ei ∂M
0
| {z
}
|0
{z
}
nominal-spending effectinnovation-impeding effect(ambiguous)(–)
Z ∞∂nS (t)+e−ρt (log λ)
dt .
∂M
0
|
{z
}
competition effect(ambiguous)(24)

Equation (24) shows that the total welfare effect of relaxing IPR protection (an increase in M )can be decomposed into the following three parts.
First, relaxing IPR protection impedes innovation and thus reduces welfare. As shown in
Proposition 5, a marginal increase in the imitation rate necessarily reduces the aggregate rateof innovation for all t. We refer to the welfare effect through this channel as the innovationimpeding effect. In the remainder of the analysis, we assume that the economy is initially in thesteady state, namely nS (0) = n̄S . Under this assumption, we derive the innovation-impeding
effect from the results in Appendix C as follows:
µZ t
¶
Z ∞
∂
−ρt
e (log λ)ι(τ )dτ dt
∂M
0
0
½
¾
1
∂ Ē
µ
∂ῑ
−
[1 + (λ − 1)n̄S ]= (log λ)< 0,
ρ2 (ρ + µ) ∂M
aN λρ(ρ + µ)
∂M

(25)

which is necessarily negative because an increase in M reduces ῑ and increases Ē.
Second, relaxing IPR protection may affect nominal spending and welfare. We refer to thiseffect as the nominal-spending effect. Only these nominal-spending effects differ between theSouth and the North.
– 19 –

Finally, relaxing IPR protection affects the measure of type-S industries, nS (t). An increasein the measure of type-S industries where the price is lower improves welfare. We refer to thiswelfare effect as the competition effect. Using the results of the analysis of the short-run effectin Appendix C, we can compute the competition effect as follows:
Z ∞
µ
∂ n̄S
∂nS (t)dt = (log λ).
e−ρt (log λ)∂Mρ(ρ + µ) ∂M
0(26)

The sign of the competition effect is indeterminate because the sign of ∂nS /∂M is ambiguous.

6.1 The Effect on Welfare in the SouthWe now examine whether strengthening IPR protection in the South can increase welfare in
the South. In Sections 6.1 and 6.2, we focus on the case where Southern households initiallypossess no assets. Further, we assume no subsidy for FDI, sF = 0, in order to make the results of
the welfare analysis clearer.8 Based on these assumptions, the intertemporal budget constraint
for a Southern consumer is reduced to ES = 1, because the Southern wage is normalized to
one and no tax is levied on Southern consumers. As the nominal spending in the South, ES , isindependent of M , the nominal-spending effect is absent in the South in this case. Therefore,
strengthening IPR protection in the South improves the welfare of the South if the condition
stated in the following proposition is satisfied.
Proposition 6. Suppose that the economy is initially in the steady state. Strengthening IPR
protection in the South then increases the welfare of Southern consumers if M > ρsR /[λ(1 −sR )].
Proof. See Appendix D.
8In Appendix G, we briefly explore what happens when these assumptions are relaxed. If the Southern house-holds have some assets and the FDI subsidy rate is positive, the nominal spending of a Southern household is notequal to one. This implies that the effect of relaxing IPR protection on the welfare of the South becomes more
complex than is analyzed in Section 6.1 because it depends also on the nominal spending effect. Further, throughthe change in the value of the assets, a tax levied in the trading partner affects income and spending in both countries. Thus, the welfare effect of the North also becomes more complex than that analyzed in Section 6.2. However,
our numerical analysis in Appendix G indicates that the results for welfare do not change if both the subsidy rate
for FDI and the share of assets held by Southern households are sufficiently small.

– 20 –

The interpretation of the condition in Proposition 6 is somewhat complex. If relaxing IPR
protection increases the measure of type-S industries, it increases welfare through reducing
prices but decreases welfare by impeding innovation.9 One of the reasons why innovation
decreases is that an increase in nS reduces the labor input for R&D (see also footnote 4). The
effect of reducing R&D through the increase in nS intensifies as M is larger, sR is smaller, andλ is larger. In addition, the innovation-impeding effect worsens if the subjective discount rate
ρ is small. This is because the slowdown in quality improvement in the future is evaluated as aheavier loss if ρ is small. Thus, if the parameters satisfy the condition given in Proposition 6, theinnovation-impeding effect becomes large enough to dominate the competition effect. Because
the nominal spending effect is equal to zero given the assumptions, strengthening (relaxing)
IPR protection improves (harms) welfare in the South.
Proposition 6 implies a desirable IPR policy in the South. Until now, earlier studies examined only whether strengthening IPR protection enhances innovation and FDI and not whether
it could improve welfare. In contrast, we can evaluate the welfare effect of strengthening IPR
protection analytically because of the tractable dynamic structure of the present model. The
result of the analysis shows that strengthening IPR protection improves welfare in the Southif M is sufficiently high and sR is so low as to satisfy the inequality given in Proposition 6.10
In other words, we can conclude that strengthening IPR protection is a desirable policy change
for the South as long as the initial protection of IPR is so weak in the South as to satisfy thecondition.11

9

If relaxing IPR protection decreases the measure of type-S industries, it unambiguously decreases welfare in

the South through both decreasing those goods with a lower price and impeding innovation.
10
If IPR protection is initially strong and M is sufficiently low so as not to satisfy the inequality given in
Proposition 6, the effect of strengthening IPR protection is indeterminate. In Appendix H, we provide a numerical analysis of the case where the condition of Proposition 6 is not satisfied. According to the analysis, further
strengthening IPR under strong protection tends to worsen the welfare of the South if sR is considerably large, andvice versa. Thus, the policy recommendation for the South may depend on the R&D subsidy policy of its trading
partner, namely, the North. The authors appreciate the comment of an anonymous referee regarding this point.
11
The condition given in Proposition 6 is not so restrictive. For example, if we set ρ = 0.05, λ = 4, andsR = 0.3 as in footnote 5, the condition is satisfied when M > 0.0054.

– 21 –

6.2 The Effect on Welfare in the North
Next, we analyze the welfare effect in the North. Unlike the South, nominal spending in the
North, EN , depends on M . ¿From ES = 1 and Ē = LN EN + LS ES , the nominal spendingof a Northern household is derived as EN = (Ē − LS )/LN . An increase in M then increases
aggregate expenditure Ē but does not change ES , so it must increase the expenditure of a
Northern household, EN . Thus, the total welfare effect of a change in IPR protection in the
North is more complex than in the South. Nevertheless, by imposing a certain condition, we
can show that strengthening IPR protection in the South improves welfare in the North as shownby the following proposition.
Proposition 7. Suppose that the economy is initially in the steady state. Strengthening IPR proh
i
λ)LS
λ−1
tection in the South increases the welfare of Northern consumers if sR < 1− 1 + aN(log.
λ(ρ+M λ) λ−1+log λProof. See Appendix E.
If the condition provided in Proposition 7 is satisfied, the innovation-impeding effect becomes large enough to dominate the nominal spending effect and the competition effect.12 The
condition tends to be satisfied if LS is relatively small. This is because the nominal spending
effect in the North is weak in that case.13 Meanwhile, the effect of λ on the condition is rather
complex. First, an increase in λ intensifies the innovation-impeding effect because a quality
improvement is highly valued by households in the case of a large λ. Second, it also changes
the innovation-impeding effect through influencing Ē. A change in Ē affects the demand forgoods and thereby the labor input for production. It therefore changes the allocation of labor
resources between production and R&D, which affects the aggregate rate of innovation. Third,
an increase in λ affects income and spending in the North through changing the wage and the12

According to our numerical analysis, the condition given in Proposition 7 turns out to be comparatively

restrictive. However, it does not mean that stronger IPR protection tends to decrease the welfare of Northern
consumers because the condition is sufficient but not necessary. Even if the condition given in Proposition 7 is notsatisfied, the welfare of the North may improve.
13
The nominal spending effect in the North is proportionate to 1/EN . Therefore, it decreases with the nominal
spending of a Northern household, EN . As the nominal spending of a Southern household is fixed to one, the small
population size in the South leads to large spending in the North for a given value of total spending in the world,
Ē. In consequence, a small LS implies that the nominal spending effect in the North is weak.

– 22 –

value of share holdings. Finally, it also intensifies the competition effect because λ is equal
to the ratio of the price of monopolistic goods to that of competitive goods. Because these effects simultaneously arise from the change in λ, how the welfare effect of IPR depends on λ is
ambiguous.
As shown in the proof, the condition in Proposition 7 is stricter than that in Proposition
6.14 Thus, when the inequality given in Proposition 7 holds, strengthening IPR protection in theSouth improves welfare in both the South and the North. This means that strengthening IPR
protection in the South is a Pareto-improving policy as long as the initial protection of IPR issufficiently weak in the South and the rate of R&D subsidy is so low as to satisfy the inequality
in Proposition 7.

6.3

Comparison with the Literature

Although most of the studies in this field did not conduct a welfare analysis, there are some
exceptions. For example, Helpman (1993), Grinols and Lin (2006), and Iwaisako et al. (2011)
examined the welfare effect of IPR using a growth model. In the seminal study on IPR protection in the dynamic North–South model, Helpman (1993) conducted a welfare analysis using avariety-expanding innovation model. He showed that stronger IPR protection necessarily harms
welfare in the South. Helpman (1993) constructed a model including the choice of multinationalization in addition to a model where the only mode of technology transfer is imitation. However, in his model, imitation does not affect multinationalization directly because he assumed a
multinational firm equals a Northern firm in the risk of imitation. To correct this shortcoming,some additional research was subsequently undertaken.
In contrast, Grinols and Lin (2006) showed that stronger IPR protection may improve welfare in the South by introducing a group of goods consumed only in the South into Helpman’s
14

From (24) and the definition of E(t), ∂UN /∂M = (∂US /∂M )+(1/ρ)(1/EN )(∂EN /∂M ) and ∂EN /∂M =(1/LN )(∂ Ē/∂M ) > 0 if ES = 1. Thus, there may be a case where stronger IPR protection is favorable forSouthern consumers and unfavorable for Northern consumers. In this case, if the nominal spending effect towelfare in the North is sufficiently large (small) relative to the sum of the innovation-impeding effect and thecompetition effect, the magnitude of the decrease in the North’s welfare is larger (smaller) than the magnitude ofincrease in the South’s welfare. This is because ∂US /∂M is equal to the sum of the innovation-impeding effectand the competition effect if ES = 1.

– 23 –

(1993) imitation model. Although our results on the welfare of the South appear similar to
those of Grinols and Lin (2006), the assumption concerning the channel of technology transfer
differs. In brief, our model includes FDI, whereas Grinols and Lin (2006) focused on technology transfer through the direct imitation of Northern goods. In addition, we obtain the welfare
results analytically, unlike Grinols and Lin (2006) who employed a numerical method.
Similarly to our conclusion, Iwaisako et al. (2011) also showed the possibility that strengthening patent protection in the South can improve welfare in the South by using a qualityimprovement-type innovation model. However, the instrument of IPR protection examined inthat analysis differs from the present paper. In general, IPR authorities can control the protection of patents using two instruments: patent length and patent breadth. Patent length refers to
the duration for which a patentee can sell the patented product monopolistically, whereas patent
breadth refers to the scope of the products that patentees can prevent firms without patentsfrom producing and selling. Iwaisako et al. (2011) focused on the effects of broadening patent
breadth. In the present paper, we assume that IPR authorities can control the probability of imitation, which is associated with the expected duration of IPR. Therefore, this paper focuses on
the effects of extending patent length rather than patent breadth. Hence, our study and Iwaisako
et al. (2011) complement each other in that the welfare effects of strengthening IPR protectionin the South are examined more completely.15
By using a model where R&D activities are undertaken in both the North and the South,
Grossman and Lai (2004) also conducted a welfare analysis. They concluded that the levelof patent protection that maximizes welfare in the South tends to be weaker than in the North.
However, to obtain this result, they specified the utility function in quasilinear form and assumed
no knowledge spillovers in the R&D process. In addition, they assumed the “obsolescence” ofgoods; that is, a newly developed good provides utility to consumers for only a finite lengthof time. These assumptions imply no growth of utility in their model. Therefore, we cannot
simply compare their results with that from a growth model including the present paper.
15

Using a closed-economy framework, Iwaisako and Futagami (2003) and Palokangas (2011) both examined

the welfare effects of patent length and breadth. The welfare effects of other patent instruments, for example,
leading and lagging patent breadths, blocking patents, and the division rule governing profits between basic andapplied researchers, have also been examined using a closed-economy framework by Li (2001), O’Donoghue andZweimüller (2004), Chu (2009), and Chu and Furukawa (2012).

– 24 –

Furthermore, Lin (2010) conducted a welfare analysis in an extended North–South modelcomposed of three countries. In his model, the middle country’s firms can imitate the goods
invented in the North and thereafter shift production to the South through FDI. Lin (2010)
also assumed that Southern local firms might imitate the goods where production is shiftedto the South. However, the purpose of his paper was to evaluate not the welfare change ofstrengthening IPR protection in the South but rather that of tightening South-bound FDI in the
middle country.

7 Discussion
7.1 The Welfare Effects of Subsidy Policies
Using the same method as in Section 6, we can also investigate the welfare effects of a change inthe subsidy policies. Details of the analysis are given in Appendix I. We summarize the results
as follows. First, a marginal increase in the R&D subsidy necessarily improves the welfare of
the South. This is because the effect through promoting innovation necessarily outweighs theeffect through changing the proportion of imitated goods. Second, a marginal increase in the
R&D subsidy and the introduction of the FDI subsidy improve the welfare of the North if thecondition given in Proposition 7 is satisfied. Finally, introduction of the FDI subsidy improvesλLS (M +ρ)(log λ)− 1 > 0. The reason why the second and thethe welfare of the South if aNM(λ−1)ρ(ρ+µ)(ρ+M
λ)

third results are satisfied is also that the effect through promoting innovation is sufficiently large
if the conditions are satisfied.

7.2 Costly Imitation and FDI
In previous sections, we assumed that imitation is costless. However, imitation of high-technology
goods involves large costs in the real world. In addition, FDI also involves nonnegligible
cost. For instance, affiliates in developing countries often conduct R&D for the absorptionof parent-firm technology, as mentioned in Dinopoulos and Segerstrom (2010) and Gustafsson
and Segerstrom (2011). In this section, we introduce a few results on the effects of strengthening IPR protection in the South in an extended model where imitation and FDI are a costly– 25 –

process, as in Glass and Saggi (2002). The model is described in Appendix F and its analysis isin Appendix J.
The costly-imitation model shows that strengthening IPR protection (an increase in the imitation cost) tends to reduce innovation in the North. There are some changes from the basic model and thus we cannot strictly compare the results of the costly-imitation model withthose of the costless-imitation model. Nonetheless, the result is opposite to that obtained in the
costless-imitation model.16 Moreover, it is similar to the results in Mondal and Gupta (2008)
using an expanding-variety model with a costly-imitation process. Thus, also in the costlyimitation models, the claim in G-W (2007) that whether strengthening IPR protection promotes innovation or not depends on whether innovation is of the variety-expansion or qualityimproving type is not correct. This difference in results suggests that the effects of strengthening
IPR protection may then depend on whether imitation is costless or costly.
The numerical results of the costly-imitation model show that the relation between thestrength of IPR protection and the welfare of Southern consumers is ambiguous. Stronger IPR
protection reduces welfare under a lower innovation cost, whereas it increases the welfare under higher innovation cost. When the innovation cost is lower, innovation is larger, and thus thenegative welfare effect through impeding innovation also tends to be large enough to dominate
the other positive welfare effects and vice versa.17

8 Concluding Remarks
This paper examined the effects of strengthening IPR protection in developing countries using
a simple North–South quality-ladder-type R&D model where FDI is determined endogenously.
We find that strengthening IPR protection promotes innovation and FDI in both the long and16

We surmise the reason for this result in the costly-imitation model is as follows. In the costly-imitation model,

strengthening IPR protection increases the cost of imitation, and thus wastes Southern labor and crowds out FDI.
The decrease in FDI then increases the production sectors in the North and consequently reduces the innovation.
17
In the costly-imitation model, we consider only the effects on the steady state and obtain the results only
numerically because of the intractability of the model. To investigate the welfare effects under costly imitation
more precisely, we would need to examine the short-run welfare effects as in the analysis of the costless-imitation
model.

– 26 –

short run. This finding contrasts with the result in G-W (2007) but is identical to that in Lai(1998), who employed a variety-expansion-type North–South model. This result is thus important not only in the sense that it reverses the results of the comparative statics in G-W’s (2007)
model, but also because it shows that the type of innovation, whether as a quality improvement
or a variety expansion, does not play a key role in determining the effects of strengthening IPR
protection on innovation and FDI. Accordingly, hereafter we must examine carefully whichaspects determine the effects of IPR protection on innovation and FDI.
As a central issue, we analytically examined the welfare effect of strengthening IPR protection. We showed that the positive welfare effect through promoting innovation dominates
the other effects if the initial protection of IPR in the South is sufficiently weak and the rateof R&D subsidy is not too high. Thus, strengthening IPR protection can increase welfare inboth the South and the North. This is important because most related studies did not evaluate
the policy of strengthening IPR protection from the viewpoint of welfare. Hence, the resultof the present paper provides valuable information about desirable IPR policies in developing
countries.
In the basic model, we assumed costless imitation for tractability. In practice, imitatingproducts involving high-level technology can entail large costs. Thus, we need also to examine
the welfare effects of strengthening IPR protection in the case where imitation is costly. By
limiting the analysis to the steady state, Section 7.2 and Appendix J examined the effects numerically. This suggests it is necessary hereafter to examine not only the welfare effect in thesteady state but also that in the short run, as in the analysis of the costless-imitation model. Thisis far beyond the scope of the present paper and thus remains as future work.
The present paper also assumes no cost of trade, such as in the form of tariffs. However,the presence of a trade cost would certainly affect the incentives for FDI and thereby change
the effects of strengthening IPR protection on FDI and innovation. By extending the present
model, we would also be able to analyze the effects of a change in trade cost as in Grieben(2005), Dinopoulos and Segerstrom (2007), Grieben and Şener (2009), and Dinopoulos andUnel (2011). It is also well worth examining how the presence of a trade cost would change theeffects of strengthening IPR protection on FDI, innovation, and welfare.
Finally, as we examined the welfare effects of strengthening IPR in a model that exhibits– 27 –

scale effects, one direction for future research would be to remove the scale effects from themodel. However, if we extended the model in this direction, the dynamic structure of themodel would be too complex to obtain clear analytical results, obliging us to rely on numericalanalysis. Although this extension is beyond the scope of the present paper, it would be well
worth conducting in the future.

Appendix A Derivation of the Equilibrium PathIn this appendix, we derive the equilibrium path of the economy. From (7) and (8), the following
equality is satisfied in the equilibrium:vN (t) = vF (t) = (1 − sR )w(t)aN .

(27)

Because this equality implies that v̇N (t) = v̇F (t) for all t, substituting (9) and (10) into therelation leads to the following equality:
πF (t) − πN (t) = M vF (t),(28)

where the left-hand side represents the “benefit” of becoming a multinational firm for a Northernleader firm, which can be measured by the increment in profits at each point in time, and theright-hand side represents the “cost” of becoming a multinational firm for a Northern leader
firm, which corresponds to the risk of losing its monopolistic rents through imitation at eachpoint in time. Substituting (5), (6), and (27) into (28), we have the following relation between
E(t) and w(t):E(t) =

aN M λ(1 − sR )w(t),sF (λ − 1) + w(t) − 1(29)which implies that:sF (λ − 1) − 1
Ė(t)
ẇ(t)
=
.
E(t)
sF (λ − 1) + w(t) − 1 w(t)(30)

Noting that ẇ(t)/w(t) = v̇N (t)/vN (t) from (7), we obtain the following equality by substituting(5), (7), and (29) into (9):ẇ(t)
v̇N (t)M (λ − w(t))== r(t) −,w(t)vN (t)sF (λ − 1) + w(t) − 1– 28 –(31)

where r(t) can be computed from (4), (30), and (31) as:r(t) =[sF (λ − 1) − 1]M (λ − w(t))
sF (λ − 1) + w(t) − 1
ρ−
.
w(t)
w(t)[sF (λ − 1) + w(t) − 1](32)

Substituting (32) into (31), we can derive the equation of motion of w(t) as follows:ẇ(t) = (ρ + M )w(t) + ρsF (λ − 1) − (ρ + M λ).

(33)

The only endogenous variable in this equation is w(t), which is a jumpable variable. Thus,(33) implies that w(t) must jump to its steady-state value immediately; otherwise, w(t) would
never reach the steady-state value because the coefficient of w(t) is strictly positive. From thedifferential equation, the steady-state value of w(t) is given by:w̄ =ρ + M λ − ρsF (λ − 1).
ρ+M

As w(t) takes a constant value at all times, we can show that E(t), r(t), vN (t), and vF (t)must also be constant over time. From (29), E(t) must immediately jump to the following
steady-state value:Ē =aN λ(1 − sR )[ρ + M λ − ρsF (λ − 1)].
(1 + sF )(λ − 1)

Therefore, r(t) = ρ is satisfied for all t because Ė(t)/E(t) = r(t) − ρ = 0 at any time. In
addition, (27) shows that vN (t) and vF (t) must also take the following constant value of thesteady state for all t:v̄N = v̄F =(1 − sR )aN [ρ + M λ − ρsF (λ − 1)].
ρ+M

Meanwhile, we can show that the values of nN (t), nF (t), and ι(t) are determined dependingon the value of the only state variable in the model, nS (t). By rewriting (12), we have:¶µLS− nS (t) .
nF (t) = λ
Ē

(34)

Furthermore, substituting (34) into nN (t) + nF (t) + nS (t) = 1 yields:nN (t) = 1 −λLS+ (λ − 1)nS (t).
ĒUsing (13) and (35), we can compute ι(t) as follows:½¾
1Ēι(t) =
LN + LS − [1 + (λ − 1)nS (t)] .
aN
λ– 29 –(35)(36)

By using the above results, we find that the motion of nS (t) in this model is determined byonly the values of nS (t) itself and some exogenous variables. Substituting (34) and (36) into
(11), we have the following equation of motion of nS (t):·¸aN M λ2 LS + Ē 2 − λ(LN + LS )Ē(λ − 1)Ē− Mλ −nS (t).
ṅS (t) =aN λ
aN λĒ

(37)

Because nS (t) as well as the other variables must be constant in the steady state, the steady-statevalue of nS (t) can be computed from (37) as follows:n̄S =aN M λ2 LS + Ē 2 − λ(LN + LS )Ē.
[aN M λ2 − (λ − 1)Ē]Ē

(38)Then, substituting (38) into (34) and (35) yields the steady-state values of nF (t) and nN (t):
n̄F =

n̄N =

λ[LS + λLN − Ē],aN M λ2 − (λ − 1)Ēλ[aN M λ(Ē − LS ) − LN Ē(λ − 1)].
[aN M λ2 − (λ − 1)Ē]ĒAppendix B Proof of Proposition 3∂ῑ< 0.
In this appendix, we prove Proposition 3. It is sufficient for the proof to show that ∂M

Differentiating (19) with respect to M , we have:
½
¾
∂ῑ
1
∂ n̄S
∂ Ē
=−
(λ − 1)Ē+ [1 + (λ − 1)n̄S ]
,
∂M
aN λ
∂M
∂M
n̄S
where ∂∂M
can be computed from (16) as follows:
·
¸
∂ n̄S
n̄F
1
aN M λ2 LS ∂ Ē=+
1 + (λ − 1)n̄S −
.
∂M
µ
aN λµ
∂M
Ē 2

Substituting (40) into (39) yields:
¸
½
·
∂ῑ
1
(λ − 1)ĒaN M λ2 LS ∂ Ē= −(λ − 1)Ē n̄F +1 + (λ − 1)n̄S −
∂M
aN λµ
aN λ
∂M
Ē 2
¾
2
∂ Ē
aN M λ − (λ − 1)Ē
[1 + (λ − 1)n̄S ]
+
aN λ
∂M
¶
·
µ
¸
∂ Ē
1
λ−1
n̄F
= −
(λ − 1)Ē n̄F + M λ 1 −
aN λµ
λ
∂M
< 0,
– 30 –(39)(40)

where the second equality uses the Southern labor market-clearing condition, n̄S = LĒS − n̄λF ,and the last inequality uses 0 < n̄F < 1. Thus, we can confirm that stronger IPR protection
increases the long-run aggregate rate of innovation ῑ.

Appendix C Proof of Proposition 5
In this appendix, we prove Proposition 5. We first show that ∂ι(t)< 0 for all t ∈ [0, ∞) if∂MnS (0) = n̄S . Differentiating (19) with respect to M , we have:
½
¾
∂ι(t)
∂nS (t)1∂ Ē
=−(λ − 1)Ē+ [1 + (λ − 1)nS (t)]
.
∂M
aN λ
∂M
∂M

(41)

n̄S
S (t)
Note that ∂n∂M
is not necessarily equal to ∂∂M. By differentiating (21) with respect to M , weS (t)can compute ∂n∂M
as follows:∂nS (t)∂ n̄S∂µ= (1 − e−µt )− te−µt (nS (0) − n̄S )
.
∂M
∂M
∂M

(42)

If the economy is initially in the steady state and nS (0) = n̄S , the second term in (42) is equal
to zero. Thus, substituting (21), (39), and (42) into (41) and applying nS (0) = n̄S , we obtainthe following relation:¯½¾∂ι(t) ¯¯
1∂ Ē−µt ∂ n̄S= −(λ − 1)Ē(1 − e )+ [1 + (λ − 1)n̄S ]∂M ¯nS (0)=n̄S
aN λ
∂M
∂M
= (1 − e−µt )

∂ Ē
∂ῑ
1
−
[1 + (λ − 1)n̄S ]e−µt
∂M
aN λ
∂M

< 0,
∂ῑ
∂ Ē
where the inequality holds because ∂M< 0 and ∂M
> 0.
F (t)
Next, we prove that ∂n∂M
< 0 for all t ∈ [0, ∞) if nS (0) = n̄S . Differentiating (34) withrespect to M and substituting (42) and nS (0) = n̄S into the derivative, we have:¯µ¶LS ∂ Ē
∂nF (t) ¯¯∂nS (t)= λ − 2−∂M ¯nS (0)=n̄S
∂M
Ē ∂M
·
¸
LS ∂ Ē−µt ∂ n̄S
− (1 − e )= λ − 2∂MĒ ∂M
λLS ∂ Ē∂ n̄F− e−µt 2= (1 − e−µt )
∂M
Ē ∂M
< 0,
– 31 –

³
´
n̄F
n̄S
∂ Ē
where the third equality uses the relation that ∂∂M= λ − LĒS2 ∂M− ∂∂M
and the inequalityn̄F∂ Ēuses ∂∂M< 0 and ∂M> 0.

Thus, we have been able to show that strengthening IPR protection increases innovation and
FDI for all t ∈ [0, ∞) if nS (0) = n̄S .

Appendix D Proof of Proposition 6
ρsR
SS< 0 holds when M > λ(1−s. From (24) and ES = 1, ∂U
In this appendix, we prove that ∂U
∂M
∂M
R)

is equal to the sum of the innovation-impeding effect and the competition effect. The sign ofn̄S
the competition effect depends on that of ∂∂M.
n̄S
If ∂∂M≤ 0, the competition effect is nonpositive. Because the innovation-impeding effect isn̄SS< 0 necessarily holds if ∂∂M≤ 0.
negative, ∂U∂Mn̄S
If ∂∂M> 0, the competition effect is positive. Then, substituting (25), (26), and (39) into

(24), we obtain a simpler expression of the total welfare effect as follows:½·¸¾(λ − 1)Ē
∂ n̄S
∂US
µ
1
∂ Ē
= (log λ) −−1−
[1 + (λ − 1)n̄S ].
∂M
ρ(ρ + µ)aN λρ
∂MaN λρ2∂M
Ē
n̄S
From this relation, even if ∂∂M> 0, imposing (λ−1)> 1 guarantees that the innovationaN λρ
S
impeding effect outweighs the competition effect, namely ∂U< 0. Using (14) and sF = 0,∂M(λ−1)ĒρsR> 1 can be reduced to M > λ(1−s
.
aN λρ
R)

Appendix E Proof of Proposition 7
In this appendix, we prove Proposition 7. If the inequality in Proposition 7 holds, we obtain(Ē−LS )(log λ)N> 1 from (14). Therefore, we prove that ∂U
aN λ(ρ+µ)∂M

S )(log λ)< 0 if (Ē−L> 1.
aN λ(ρ+µ)1∂ Ē
S
Because EN = Ē−L
, the nominal spending effect in the North is given by ρ(Ē−L. SubLNS ) ∂M

stituting this equation, (25), and (26) into (24), the total welfare effect in the North is reduced
to:
∂UN
∂M

½
= (log λ)

¸
·
µ
1
∂ Ē
∂ῑ
(Ē − LS )(log λ)−−12
ρ (ρ + µ) ∂MaN λ(ρ + µ)∂Mρ(Ē − LS )(log λ)¾µ(λ − 1)n̄S ∂ Ē
∂ n̄S
+
−
.
aN λρ(ρ + µ) ∂Mρ(ρ + µ) ∂M– 32 –n̄S
The sign of the competition effect depends on that of ∂∂M.
n̄S
S )(log λ)If ∂∂M≤ 0, the competition effect is nonpositive. From (Ē−L> 1, the second termaN λ(ρ+µ)in the curly brackets on the right-hand side is negative. Thus, the total welfare effect is negativen̄S≤ 0.
if ∂∂Mn̄S
On the other hand, if ∂∂M> 0, the competition effect is positive. Then, from (39), we canrewrite the total welfare effect as follows:
·
¸
n
∂UN
µ
(λ − 1)Ē
µ
∂ Ē
∂ n̄S
= (log λ) −−1−
[1 + (λ − 1)n̄S ]2∂M
ρ(ρ + µ)aN λρ
∂MaN λρ (ρ + µ)∂M·¸1(Ē − LS )(log λ)∂ Ē
(λ − 1)n̄S ∂ Ē o−
−1−.
aN λ(ρ + µ)∂MaN λρ(ρ + µ) ∂Mρ(Ē − LS )(log λ)ĒBecause λ is greater than one, λ − 1 is greater than log λ necessarily. This implies that (λ−1)>
aN λρ(Ē−LS )(log λ)> 1. Therefore, the first and third terms in the curly brackets on the right-hand sideaN λ(ρ+µ)n̄SN
are negative, and we can confirm ∂U< 0 even if ∂∂M> 0.
∂M

Appendix F The Costly-Imitation ModelIn this appendix, we present an extended model where imitation and FDI are costly. An analysisof the extended model is given in Appendix J. We describe only the parts that are changed in
the extended model. For simplicity, we assume that the R&D and FDI subsidies are equal tozero, that is, sR = sF = 0.
Following Glass and Saggi (2002), we assume that multinational firms need ζ(> 1) unitsof Southern labor to produce one unit of goods, whereas Southern imitator firms need only one
unit of Southern labor. In this case, the marginal cost for the multinational firms is ζ, although
their optimal price is the same as in the original model, pF (t) = λ. Thus, the profits of a¡¢multinational firm change to πF (t) = 1 − λζ E(t). On the other hand, to maximize profits,
Southern imitator firms charge a price so that the multinational firms cannot earn a positive³´
profit; pS (t) = ζ. Therefore, the profits of a Southern imitator become πS (t) = 1 − ζ1 E(t).
In this model, we assume that imitation activities require labor inputs in contrast with theoriginal model. If a Southern follower firm devotes am M (t)dt units of Southern labor to copy
the current latest generation of good j that the multinational firm produces, the firm succeeds
in imitating the good with probability M (t)dt. Under this setting, we can interpret stronger– 33 –

IPR protection in the South as an increase in am . Given the value of am , a Southern follower
firm attempting to copy a good chooses M (t) optimally. Therefore, “imitation intensity” M (t)is endogenously determined in this extended model, whereas the imitation rate is constant andexogenous in the original model. Letting vS (t) denote the value of a Southern imitator firm, theequilibrium condition of imitation is given by:vS (t) = am .

(43)

The shareholders of a Southern imitator firm earn dividends πS (t)dt and capital gains v̇S (t)dt,
and face a capital loss of amount vS (t) with probability ιN (t)dt over a time interval dt. Therefore, the no-arbitrage condition between the shares of a Southern imitator firm and the risk-free
asset is:r(t)vS (t) = πS (t) + v̇S (t) − ιN (t)vS (t).

(44)

As another extension, we assume that FDI is also costly and requires labor inputs in thisextended model, whereas we abstract the FDI cost in the original model. More concretely,
we assume that the success or failure of FDI also follows a Poisson process: if a Northernleader devotes aF ιF (t)dt units of Southern labor to adaptation of technology, it succeeds inshifting production of its good to the South with probability ιF (t)dt. Under this assumption,
the equilibrium condition of FDI changes from (8) tovN (t) + aF = vF (t).

(45)

Next, we consider the labor market-equilibrium conditions. In contrast with the original model, a multinational firm needs ζ units of labor for the production of one unit of thegood. This implies that the aggregate labor demand of the multinational firms changes tonF (t)(E(t)/λ)ζ. The aggregate labor demand of the imitator firms also changes to nS (t)E(t)/ζ
because the price set by the imitator firms is ζ in this model. In addition, Southern labor is devoted to both imitation activities and FDI. Therefore, the labor market-clearing condition in theSouth changes from (12) toaF ιF (t)nN (t) + am M (t)nF (t) + nF (t)

E(t)
E(t)
ζ + nS (t)= LS .
λζThe labor market-clearing condition in the North is the same as (13).
– 34 –(46)

Finally, we describe the law of motion governing the measures of the industries belonging toeach category. In an infinitesimal time interval of length dt, leader firms succeed in developing
a newer generation of ιN (t)nS (t)dt goods and transferring production of ιF (t)nN (t)dt type-N
goods to the South. Moreover, the imitator firms succeed in copying M (t)nF (t)dt type-F goodsin the same time interval. Therefore, the measure of the industries where the Southern imitator
monopolistically produces the good changes over time with the following law of motion:ṅS (t) = M (t)nF (t) − ι(t),(47)

where ι(t) ≡ ιN (t)nS (t). Meanwhile, the law of motion of nN (t) is given by:ṅN (t) = ι(t) − ιF (t)nN (t).

(48)

Appendix G Generalization of Initial Distribution of Assets
and FDI Subsidy Rate
In Sections 6.1 and 6.2, we assumed no asset holding by the South and no FDI subsidy. In thisappendix, we briefly discuss how the welfare results change if these assumptions are relaxed.
To assess the consequences of the generalization, we draw on a numerical example because it
is difficult to compute the welfare effect analytically.
We first compute the nominal spending of households in the generalized case such that theSouthern households have some assets and the FDI subsidy rate is positive. As discussed in
Section 3 and Appendix A, r(t) = ρ for all t because E(t) is constant on the equilibrium path.
Thus, Ei (t) also takes a constant value Ei on the equilibrium path because of the intertemporalutility maximization. Considering these, the intertemporal budget constraint can be expressedas:ES = ρAS (0) + 1 − ρZ ∞e−ρt TS (t)dt,(49)

e−ρt TN (t)dt.

(50)

0

EN = ρAN (0) + w̄ − ρZ ∞0

By using the government budget constraints, (19), (21), and (34), the third terms of (49) and– 35 –

(50) are given by:Z ∞Z ∞sF (1 − 1/λ)Ē−ρte TS (t)dt = ρe−ρtnF (t)dt
ρ
LS
0
0
µ
¶
Z
ρsF (1 − 1/λ)Ē ∞ −ρtLS=e λ− nS (t) dt
LS
Ē
0
·
¸
ρ
sF (λ − 1)Ē LS=− n̄S − (nS (0) − n̄S )
,
LS
ρ+µ
Ē
Z ∞
Z ∞
sR w̄aN−ρt
ι(t)dtρe TN (t)dt = ρ
e−ρt
LN
0
0
½
¾
Z
ρsR w̄aN ∞ −ρt 1
Ē
=e
LN + LS − [1 + (λ − 1)nS (t)] dt
LN
aN
λ
0
¸
·
sR w̄
Ē (λ − 1)Ē(λ − 1)ρĒ
.
=
LN + LS − −n̄S − (nS (0) − n̄S )LNλλλ(ρ + µ)The values of Ai (0) are computed as follows. Letting A(0) ≡ AN (0)LN + AS (0)LS denote thetotal initial asset holdings of the world, we obtain the following equation by summing up thespending of every household:Ē = LS ES + LN EN= ρA(0) + LS + w̄LN − ρLS

Z ∞
e

−ρt

TS (t)dt − ρLN0or equivalently,11
A(0) = Ē − (LS + w̄LN ) + LS
ρ
ρ

Z ∞

Z ∞

e−ρt TN (t)dt,

0

−ρt

e
0

Z ∞
TS (t)dt + LN

e−ρt TN (t)dt.

0

Because country i’s share of asset holdings must be given as an initial condition, we let σ ∈ [0, 1]
denote the share of assets held by Southern households, that is, σ ≡ AS (0)LS /A(0). This meansthat AS (0) = σA(0)/LS and AN (0) = (1 − σ)A(0)/LN . Substituting these into (49) and (50),
we have:
ES

Z ∞
σA(0)
+1−ρ
e−ρt TS (t)dt
= ρ
LS
0
¸
·
Z ∞
Z ∞
σ
−ρt
−ρt
e TN (t)dt=e TS (t)dt + ρLN
Ē − (LS + w̄LN ) + ρLS
LS
0
0
Z ∞
+1 − ρ
e−ρt TS (t)dt0·¸Ē − LSσ w̄
sR Ē
= 1 − sF (1 − σ)(λ − 1) + σ−(1 − sR )LN − sR LS +
LS
LS
λ
·
¸
h
i
ρ
(λ − 1)Ē σsR w̄
− sF (1 − σ) n̄S + (nS (0) − n̄S )
,
(51)
−
LS
λ
ρ+µ
– 36 –

ENZ ∞(1 − σ)A(0)= ρ+ w̄ − ρe−ρt TN (t)dt
LN
0
·
¸
Z ∞
Z ∞
1−σ
−ρt
−ρt
=
Ē − (LS + w̄LN ) + ρLS
e TS (t)dt + ρLN
e TN (t)dt
LN
0
0
Z ∞
+w̄ − ρe−ρt TN (t)dt0·¸LSĒ − LS σ w̄sR Ē= sF (1 − σ)(λ − 1)
+ (1 − σ)+(1 − sR )LN − sR LS +
LN
LN
LN
λ
·
¸
h
i
(λ − 1)Ē σsR w̄
ρ+− sF (1 − σ) n̄S + (nS (0) − n̄S )
.
(52)
LN
λ
ρ+µ

Note that ES = 1 and EN = (Ē − LS )/LN are satisfied if σ = 0 (households in the South haveno asset) and sF = 0 (the FDI subsidy is zero), which is the same as analyzed in Sections 6.1
and 6.2. The derivatives of equations (51) and (52) with respect to M are:
¯
½
·
¸
¾
∂ES ¯¯σsR Ē(λ − 1)σsR Ē n̄S ∂ w̄=−(1 − sR )LN − sR LS +−∂M ¯nS (0)=n̄S
LS
λ
λLS
∂M
½
i ¾ ∂ Ēσ
σsR w̄ λ − 1 h σsR w̄
+−−− sF (1 − σ) n̄S
LS
λLS
LS
λ
∂M
i µ ∂ n̄(λ − 1)Ē h σsR w̄
S−− sF (1 − σ),LSλ
ρ + µ ∂M¯·¸¾½
∂EN ¯¯σsR Ē(λ − 1)σsR Ē n̄S ∂ w̄(1 − sR )LN − sR LS ++=∂M ¯nS (0)=n̄S
LN
λ
λLN
∂M
½
i ¾ ∂ Ē1 − σ σsR w̄ λ − 1 h σsR w̄
+++− sF (1 − σ) n̄S
LN
λLN
LN
λ
∂M
i µ ∂ n̄(λ − 1)Ē h σsR w̄
S+− sF (1 − σ).
LNλ
ρ + µ ∂M
Using these, we compute the nominal spending effect in the generalized case. The innovationimpeding effect and the competition effect are as in the text.
Figure 1 illustrates the welfare effects of relaxing IPR for each value of sF and σ. Following
Glass and Saggi (2002), we set ρ = 0.05, λ = 4, LN = 3, and LS = 6 in Figure 1. We further
set M = 0.037, aN = 123.5, and sR = 0.45. Under these parameter values, it is required that0 ≤ sF ≤ 0.24 to ensure the existence and the stability of the steady state. The sign of thewelfare effect is unchanged from the case of sF = σ = 0 if sF and σ are sufficiently small.
In this example, the effect on the welfare of the South ∂US /∂M is negative for any sF and σ.
– 37 –

0

500

߲ܷௌ
߲‫ܯ‬

0

-1000

-500

0.75

0.75

0.5
0.25

߲ܷே
߲‫ܯ‬

0.5

ߪ

0.25

0.2

ߪ

0.2
0.15

0.15
0.1

‫ݏ‬ி

0.1

0

0.05

‫ݏ‬ி

0

0

0.05
0

Figure 1: The welfare effect under positive FDI subsidy and positive asset holdings of the SouthThus, strengthening IPR protection in this case increases the welfare of Southern consumers
even if the FDI subsidy and the asset holdings of the South are positive. On the other hand,the sign of the effect on the welfare of the North ∂UN /∂M depends on sF and σ. If the FDI
subsidy rate and the asset holdings of the South are sufficiently large, ∂UN /∂M can be positive:
strengthening IPR protection in the South can harm the North. However, if they are sufficiently
small, ∂UN /∂M is negative, such that strengthening IPR protection in the South still benefits
the North.

Appendix H Welfare Effects in the Case of Initially Strong
IPR
In this appendix, we explore whether further strengthening IPR under strong protection is benSin the case where the condition of
eficial to the South. To do this, we examine the sign of ∂U∂M

Proposition 6 is not satisfied, by using two numerical examples. In the examples, we set λ = 4,
LS = 6, and LN = 3 as in footnote 5, and set aN = 400. Following the setting in Section 6.1,
we assume that the FDI subsidy is zero and the Southern households have no asset. Figures 2
S
and 3 show the value of ∂U
for each M in the two cases. We set ρ = 0.12 and sR = 0.83 in∂Mcase 1 (Figure 2), while ρ = 0.05 and sR = 0.65 in case 2 (Figure 3). To ensure the existence
and the stability of the steady state, it is required that 0.0097 < M < 0.0196 in case 1 and
0.009 < M < 0.0116 in case 2. Note that both cases do not satisfy the condition provided in– 38 –

Proposition 6; that is, IPR protection in the South is relatively strong.
According to the examples, the welfare effect of strengthening IPR is indeterminate if IPR
S
is initially strong. Figure 2 shows that the sign of ∂Uturns positive in case 1 if M is sufficiently∂Msmall. This implies that there is the optimal level of IPR protection inside the interval of M .
Sremains negative for any value of M in case
However, Figure 3 shows that the sign of ∂U∂M

2. Thus, unlike case 1, maximum IPR protection is beneficial to the South. Suppose, forexample, that the initial imitation rate is M = 0.01. Then, the welfare effect of strengthening
IPR protection differs between case 1 and case 2. In case 1, a further strengthening of IPR isunfavorable to the South, whereas in case 2, it is favorable to the South. Roughly speaking, itappears that further strengthening IPR under strong protection tends to worsen the welfare of
the South if sR is considerably large.

– 39 –

డ௎

Welfare effect ( డெೄ )

20
0

-20
-40
-60
0.0196 0.0184 0.0172

0.016

0.0148 0.0136 0.0124 0.0112

0.01

‫ܯ‬

Figure 2: Imitation rate and the effects of strengthening IPR on the welfare of the South: Thecase of ρ = 0.12 and sR = 0.83. It is required that 0.0097 < M < 0.0196 to ensure theexistence and the stability of the steady state.

డ௎

Welfare effect ( డெೄ )
0

-100
-200
-300
-400
-500
0.01160.01120.01080.01040.01

0.00960.0092‫ܯ‬Figure 3: Imitation rate and the effects of strengthening IPR on the welfare of the South: Thecase of ρ = 0.05 and sR = 0.65. It is required that 0.009 < M < 0.0116 to ensure the existence
and the stability of the steady state.
– 40 –

Appendix I Welfare Effects of the Subsidy Policies
In this appendix, we show the welfare effects of the subsidy policies for R&D and FDI. Differentiating the lifetime utility function with respect to sR and sF , we have:
¶
µZ t
Z ∞
Z ∞
∂Ui
∂
1 1 ∂Ei∂nS (t)−ρt=e (log λ)ι(τ )dτ dt ++e−ρt (log λ)dt, (53)∂sR∂sR
ρ Ei ∂sR
∂sR
0
0
0
µZ t
¶
Z ∞
Z ∞
∂
∂nS (t)∂Ui
1 1 ∂Ei−ρt=e (log λ)+e−ρt (log λ)dt. (54)
ι(τ )dτ dt +∂sF
∂sFρ Ei ∂sF∂sF
0
0
0
An interpretation of the three terms of (53) and (54) is the same as that of (24). The first terms
represent the positive welfare effect through promoting innovation. An increase in the subsidies stimulates R&D, so that it can improve welfare by speeding up the quality improvement.
The second terms represent the welfare effect through the effect on the nominal spending. An
increase in the subsidies changes the profitability of R&D and FDI, which influences the firms’
decision on R&D and the location choices. Thus, it affects the spending of both countries
through the change in the wages and the value of the shares of the leader firms. The thirdterms represent the welfare effect through changing the proportion of competitive goods. An
increase in the subsidies changes the measure of type-S industries because it affects the aggregate rate of innovation, which is equal to the outflow from type-S industries, and FDI, which
determines the inflow into type-S industries. It therefore influences welfare through the change
in the proportion of cheaper goods.
To compute the value of each term, we first carry out comparative statics. Differentiating(14) with respect to sR and sF , we have:
Ē
∂ Ē
=−
< 0,
∂sR
1 − sR(55)

λ(M + ρ)Ē
∂ Ē
=−
< 0.
∂sF
(1 + sF )[ρ + M λ − sF (λ − 1)ρ](56)

Using equation (16), we can compute ∂∂sn̄RS and ∂∂sn̄FS as follows:
¸
·
∂ n̄S
1
aN M λ2 LS ∂ Ē=,
1 + (λ − 1)n̄S −
∂sR
aN λµ
∂sR
Ē 2
¸
·
1
∂ n̄S
aN M λ2 LS ∂ Ē=,
1 + (λ − 1)n̄S −
∂sF
aN λµ
∂sF
Ē 2
– 41 –(57)(58)

which are analogous to (40). Therefore, we can compute the long-run effect of the subsidy
policies on the aggregate rate of innovation:
½
¾
∂ῑ
1
∂ n̄S
∂ Ē
(λ − 1)Ē= −+ [1 + (λ − 1)n̄S ]
∂sR
aN λ
∂sR
∂sR
½
·
¸
1
(λ − 1)ĒaN M λ2 LS ∂ Ē
1 + (λ − 1)n̄S −
= −
aN λµ
aN λ
∂sR
Ē 2
¾
2
aN M λ − (λ − 1)Ē∂ Ē+[1 + (λ − 1)n̄S ]
aN λ
∂sR
³
´
M
n̄F ∂ Ē= −1 − n̄F +
aN µ
λ ∂sR
> 0,
½
¾
∂ῑ
1
∂ n̄S
∂ Ē
= −
(λ − 1)Ē+ [1 + (λ − 1)n̄S ]
∂sF
aN λ
∂sF
∂sF
¸
½
·
1
(λ − 1)ĒaN M λ2 LS ∂ Ē= −1 + (λ − 1)n̄S −
aN λµ
aN λ
∂sF
Ē 2
¾
2
aN M λ − (λ − 1)Ē∂ Ē+[1 + (λ − 1)n̄S ]
aN λ
∂sF
³
´
M
n̄F ∂ Ē= −1 − n̄F +
aN µ
λ ∂sF
> 0,

(59)

(60)

where the second equalities in both equations use µ ≡ M λ − (λ − 1)Ē/(aN λ) and the inequalities hold from 0 < n̄F < 1, (55), and (56). By differentiating (17) with respect to sR and sF ,we have:∂ n̄F(1 − n̄F )λ + n̄F ∂ Ē
=−
> 0,
∂sR
aN λµ
∂sR

and

∂ n̄F
(1 − n̄F )λ + n̄F ∂ Ē
=−
> 0,
∂sF
aN λµ
∂sF

both of which are similar to equation (23).
Next, we analyze the dynamic effects of the change in the subsidies. Differentiating (19)with respect to sR and sF yields:
¾
½
1
∂ Ē
∂ι(t)
∂nS (t)=−+ [1 + (λ − 1)nS (t)],(λ − 1)Ē∂sR
aN λ ½
∂sR
∂sR ¾
1
∂ Ē
∂ι(t)
∂nS (t)=−+ [1 + (λ − 1)nS (t)],(λ − 1)Ē
∂sF
aN λ
∂sF
∂sF

(61)
(62)

which correspond to (41). Differentiating (21) with respect to sR and sF , and applying nS (0) =n̄S to them, we have:∂nS (t)∂ n̄S= (1 − e−µt )
,
∂sR
∂sR
– 42 –(63)∂nS (t)∂ n̄S= (1 − e−µt )
.
∂sF
∂sF

(64)

Thus, substituting these, (21), (59), and (60) into (61) and (62), we obtain the following relation:¯¾½∂ι(t) ¯¯
1∂ Ē−µt ∂ n̄S
+ [1 + (λ − 1)n̄S ]= −(λ − 1)Ē(1 − e )∂sR ¯nS (0)=n̄S
aN λ
∂sR
∂sR
= (1 − e−µt )

∂ῑ
1
∂ Ē
−
[1 + (λ − 1)n̄S ]e−µt∂sR aN λ
∂sR

> 0,
¯
½
¾
∂ι(t) ¯¯
1∂ Ē−µt ∂ n̄S= −+ [1 + (λ − 1)n̄S ](λ − 1)Ē(1 − e )∂sF ¯nS (0)=n̄S
aN λ
∂sF
∂sF
= (1 − e−µt )

∂ῑ
1
∂ Ē
−
[1 + (λ − 1)n̄S ]e−µt
∂sF
aN λ
∂sF

> 0,
where the inequalities hold from (55), (56), (59), and (60).
Using the above results, we can compute the welfare effects of the subsidy policies. First,
the welfare effects through promoting innovation are:
µZ t
¶
Z ∞
∂
−ρt
ι(τ )dτ dte (log λ)
∂sR
0
0
¾
Z ∞
Z t½
1
−ρt
−µτ ∂ Ē−µτ ∂ῑ= (log λ)e−[1 + (λ − 1)n̄S ]edτ dt(1 − e )∂sR aN λ
∂sR
0
0
½
¾
µ
∂ῑ
1
∂ Ē
= (log λ)−[1 + (λ − 1)n̄S ]
> 0,
(65)
ρ2 (ρ + µ) ∂sR aN λρ(ρ + µ)
∂sR
Z ∞
0

µZ t
¶
∂
ι(τ )dτ dte (log λ)
∂sF
0
¾
Z ∞
Z t½
1
−ρt
−µτ ∂ῑ−µτ ∂ Ē= (log λ)e(1 − e )−[1 + (λ − 1)n̄S ]e
dτ dt
∂sF
aN λ
∂sF
0
0
¾
½
1
∂ Ē
µ
∂ῑ
−
[1 + (λ − 1)n̄S ]
> 0.
(66)
= (log λ)ρ2 (ρ + µ) ∂sF
aN λρ(ρ + µ)∂sF
−ρtSecond, differentiating (51) and (52) with respect to sR and sF , and applying nS (0) = n̄S ,σ = 0, and sF = 0, we get:
∂ES
= 0,
∂sR

– 43 –

∂EN
1 ∂ Ē=,∂sR
LN ∂sR¯∂ES ¯¯(λ − 1)Ē(λ − 1)Ē n̄F= −(λ − 1) +n̄S = −,¯∂sF sF =0LS
λLS¯∂EN ¯¯(λ − 1)Ē
1 ∂ Ē
(λ − 1)Ē n̄F
1 ∂ ĒLS= (λ − 1)
−
n̄S +
=
+
.
¯
∂sF sF =0
LN
LN
LN ∂sFλLN
LN ∂sF
Thus, in the case of σ = 0 and sF = 0, the welfare effects through nominal spending are givenby:
1 1 ∂ES= 0,
ρ ES ∂sR(67)

1 1 ∂EN
1∂ Ē
=,ρ EN ∂sR
ρ(Ē − LS ) ∂sR(68)

1 1 ∂ES(λ − 1)Ē n̄F=−,
ρ ES ∂sF
ρλLS¸·
1 1 ∂EN1(λ − 1)Ē n̄F
∂ Ē
=
+
.
ρ EN ∂sF
λ∂sF
ρ(Ē − LS )

(69)
(70)

Third, from (63) and (64), the welfare effects through changing the proportion of competitive
goods are:Z ∞e−ρt (log λ)µ∂ n̄S∂nS (t)dt = (log λ),∂sRρ(ρ + µ) ∂sR(71)e−ρt (log λ)∂ n̄S∂nS (t)µdt = (log λ).
∂sF
ρ(ρ + µ) ∂sF

(72)

0

Z ∞
0

Using equation (57), we obtain the sum of the welfare effects through promoting innovationand through changing the proportion of competitive goods:
µZ t
¶
Z ∞
Z ∞
∂
∂nS (t)−ρte (log λ)ι(τ )dτ dt +e−ρt (log λ)
dt
∂sR
∂sR
0
0
0
½
¾
µ
1
∂ Ē
µ
∂ῑ
∂ n̄S
= (log λ)−[1 + (λ − 1)n̄S ]+ρ2 (ρ + µ) ∂sR aN λρ(ρ + µ)∂sR ρ(ρ + µ) ∂sR
½
¾
µ
1
∂ῑ
M λLS ∂ Ē= (log λ)−ρ2 (ρ + µ) ∂sR ρ(ρ + µ) Ē 2 ∂sR
> 0,
(73)
where the inequality holds from (55) and (59). Thus, from (53), (67), and (73), an increase in
the R&D subsidy necessarily improves the welfare of the South.
– 44 –Meanwhile, an increase in the R&D subsidy can improve the welfare of the North if thecondition in Proposition 7 holds, proved as follows. Substituting (65), (68), and (71) into (53),
we have:∂UN∂sR½= (log λ)·¸µ1(Ē − LS )(log λ)
∂ Ē
∂ῑ
−
−1
2
ρ (ρ + µ) ∂sR ρ(Ē − LS )(log λ)aN λ(ρ + µ)∂sR¾(λ − 1)n̄S ∂ Ē
µ
∂ n̄S
−
+
.
(74)
aN λρ(ρ + µ) ∂sR ρ(ρ + µ) ∂sR

S )(log λ)> 1 is satisfied from (14) if the condition of ProposiAs discussed in Appendix E, (Ē−LaN λ(ρ+µ)tion 7 holds. Thus, (55), (59), and (74) imply that the total welfare effect of increasing the R&D
subsidy is positive if ∂∂sn̄RS ≥ 0. Next, substituting (59) into (74), we have:
∂UN
∂sR

·
¸
µ
(λ − 1)Ē
∂ n̄S
µ
∂ Ē
= (log λ) −−1−
[1 + (λ − 1)n̄S ]2
ρ(ρ + µ)aN λρ∂sR aN λρ (ρ + µ)
∂sR
·
¸
1
(Ē − LS )(log λ)∂ Ē
(λ − 1)n̄S ∂ Ē o−
−1−.
aN λ(ρ + µ)∂sR aN λρ(ρ + µ) ∂sR
ρ(Ē − LS )(log λ)nĒS )(log λ)Because (λ−1)> (Ē−L> 1 as discussed in Appendix E, this equation shows that theaN λρ
aN λ(ρ+µ)

total welfare effect of increasing the R&D subsidy is positive even if ∂∂sn̄RS < 0.
Next, we analyze the welfare effect of a marginal increase in the FDI subsidy from zero. To
explore the effect on the welfare of the North, we use similar logic to the proof of the effect of
the R&D subsidy. More specifically, substituting (66), (70), and (72) into (54) yields:
½
·
¸
µ
∂ῑ
(Ē − LS )(log λ)
∂ Ē
1
∂UN
= (log λ)
−
−1
2
∂sF
ρ (ρ + µ) ∂sF
aN λ(ρ + µ)∂sF
ρ(Ē − LS )(log λ)¾∂ n̄S(λ − 1)Ē n̄F(λ − 1)n̄S ∂ Ē
µ
1
−
+
+
. (75)
aN λρ(ρ + µ) ∂sF
ρ(ρ + µ) ∂sF
λρ(Ē − LS )Thus, (56), (60), and (75) imply that the welfare effect of a marginal increase in the FDI subsidy
from zero is positive if ∂∂sn̄FS ≥ 0 and the condition in Proposition 7 holds. Further, substituting(60) into (75), we have:
∂UN
∂sF

¸
·
µ
∂ n̄S
µ
∂ Ē
(λ − 1)Ē
= (log λ) −−1−
[1 + (λ − 1)n̄S ]2
ρ(ρ + µ)aN λρ
∂sF
aN λρ (ρ + µ)
∂sF
¸
·
∂ Ē
(λ − 1)n̄S ∂ Ē o
1(Ē − LS )(log λ)−1−−aN λ(ρ + µ)∂sF
aN λρ(ρ + µ) ∂sF
ρ(Ē − LS )(log λ)1(λ − 1)Ē n̄F+.
λρ(Ē − LS )n

– 45 –Given the condition in Proposition 7, this equation shows that the total welfare effect of amarginal increase in the FDI subsidy from zero is positive, even if ∂∂sn̄FS < 0. Thus, the introduction of the FDI subsidy improves the welfare of the North if the condition in Proposition 7
holds.
Finally, we analyze the effect of the FDI subsidy on the welfare of the South. Substituting(58), (66), (69), and (72) into (54), we obtain:
·
¸
∂US
∂ῑ
1
M λLS ∂ Ē(λ − 1)Ē n̄Fµ= (log λ) 2
−
−
.
∂sF
ρ (ρ + µ) ∂sF
ρ(ρ + µ) Ē 2 ∂sF
ρλLS

(76)

Further, substituting (60) into (76) yields:
∂US
∂sF

= −
= −

M (log λ)(1 − n̄F ) ∂ ĒM (log λ)n̄F ∂ ĒM λLS (log λ) ∂ Ē(λ − 1)Ē n̄F
−
−
−
2
2
2
aN ρ (ρ + µ) ∂sF
aN λρ (ρ + µ) ∂sF
ρλLS
ρ(ρ + µ)Ē ∂sF
M (log λ)(1 − n̄F ) ∂ ĒM λLS (log λ) ∂ Ē−
2
aN ρ (ρ + µ) ∂sF
ρ(ρ + µ)Ē 2 ∂sF·¸(λ − 1)Ē n̄F
M λLS (M + ρ)(log λ)
+
−1 ,
ρλLS
aN (λ − 1)ρ(ρ + µ)(ρ + M λ)

(77)

∂ Ē
where the second equality uses (56) and sF = 0. Because ∂s< 0 from (56), equation (77)F

implies that the welfare effect of a marginal increase in the FDI subsidy from zero is positive if
M λLS (M +ρ)(log λ)− 1 > 0.
aN (λ−1)ρ(ρ+µ)(ρ+M λ)Appendix J Analysis of the Costly-Imitation ModelIn this appendix, we show the derivations of the equations that characterize the steady-state
equilibrium of the costly-imitation model in detail, and examine the effects of strengthening
IPR protection in the South. Because the model with costly imitation and FDI is more complex
than the costless-imitation model, we focus only on the steady state. Hereafter, we let variables
without time “t” denote the steady-state values of the variables.

Steady State
Using the Euler equation yields: r = ρ. Combining the equilibrium condition of R&D and theno-arbitrage condition (9) yields:³
w´E.
ρwaN = 1 −λ– 46 –(78)

Combining the equilibrium condition of imitation (43) and the no-arbitrage condition (44), we
get:

µ
ιN =

1
1−
ζ

¶

E
− ρ.
am

(79)

From (7) and (45), the following equality is satisfied in the equilibrium: vN (t) + aF = vF (t) =w(t)aN . Because this equality implies that v̇N (t) = v̇F (t) for all t, substituting (9) and (10) into
the relation leads to the following equality: πF (t) − πN (t) = M (t)vF (t) + r(t)aF . Thus, in thesteady state, we get:
Ã
M =ρ

!
1 − λζ
waN−1 .
1 − wλ waN + aF

(80)

Using the laws of motion (47) and (48) yields:
M nF = ιF nN = ιN nS = ι.

(81)

Finally, from (13), (46), and (81), the labor market-equilibrium conditions in the North and theSouth in the steady state are given by:
1= LN ,λ(82)ζ1+ nS E = LS .
λ
ζ

(83)

aN ι + n N E(aF + am )ι + nF E

Hereafter, we describe the steady-state equilibrium using only two variables, ι and w. First,
we derive nS E and nF E as functions of ι and w. ¿From (78), we can express E as a functionof w as follows:
ρwaN.
1 − wλ

(84)

1− w
1
λ
−
am
waN

≡ φ(w, am ). Because nS E = ( ιEN )ι fromE=
From (79) and (84), we get: ιEN³=

1 − ζ1´(81), we obtain:ι.
(85)
φ(w, am )
¡
¢
1− w
E
Next, from (80) and (84), we get: M= 1 − λζ waN1+aF − waNλ ≡ m(w). Because nF E = ( M)ιEnS E =

from (81), we have:nF E =

ι.
m(w)– 47 –(86)ι

LN curveam ↑LS curve

λ0w

Figure 4: The steady state in the costly-imitation and FDI model

Substituting (85) and (86) into (82) and (83), we obtain the following system of two equations
in two unknowns (ι, w):
ι=

ι=

1
λ

h

ρwaN
− LN
λ−w
1
1
+ m(w)
φ(w,am )

i

− aN

,

LS
.
ζ 1
1
aF + am + λ m(w) + ζ1 φ(w,a
m)

(87)

(88)

We refer to the curves that satisfy (87) and (88) as the LN curve and the LS curve, respectively.
These are both upward sloping, as shown in Figure 4. We assume that these curves intersect
once.

Effects of Stronger IPR Protection on Innovation
Next, we prove that an increase in am , which is interpreted as strengthening IPR protectionin the South, reduces the aggregate rate of innovation in the costly-imitation model. We let
ΨN (w, am ) and ΨS (w, am ) denote the right-hand sides of (87) and (88). Totally differentiating(87) and (88), we get:

A



∂ι
 ∂am 
∂w
∂am




=

ΨN
a

– 48 –

ΨSa

,

(89)

where


A=
i

1

−ΨN
w

1

−ΨSw


,

i

(w,am )m)
Ψiw ≡ ∂Ψ (w,afor i = {N, S}. By differentiating ΨN (w, am ) and, and Ψia ≡ ∂Ψ ∂a∂w
mΨS (w, am ) with respect to w respectively, we obtain:∂ΨN (w, am )
≡
ΨN
w
∂w

=

³

1
λ

1 ∂E
λ ∂w´

1
+ m1
φ

= h ³
1
λ

∂Ψ
ΨSw ≡

S

(w, am )
= −³
∂w
= ³

− aN

−h ³
1
λ

1
+ m1
φ

− aN

i2

1
λ

1
λ

1
i2
λ

´

µ

1
1
− 2 φw − 2 mw
φ
m

− aN
´
1
1
+ m − aN ∂E
φ

1
+ m1
φ

 ³

E
− LN
λ

´

E
− LN
λ

E
− LN
λ

∂w

³

´

¶



+ φ̂ + m̂  ,

µ
µ µ
¶
¶¶
ζ
1
1
1
− 2 mw +− 2 φw
´2
λ
m
ζ
φ
11

LS

am + aF + λζ m1 + ζ φLSam + aF + λζ m1 + ζ1 φ1

µ

´2

¶
ζ
1
m̂ + φ̂ ,λζ∂φwwhere φw ≡ ∂w
, mw ≡ ∂m, φ̂ ≡ φφw2 and m̂ ≡ m. By differentiating ΨN (w, am ) and∂w
m2

ΨS (w, am ) with respect to am respectively, we obtain:
∂Ψ
ΨN
a ≡

E
− LN
1
(w, am )= −h ³ λ ´
i2
∂am
λ
1
1
+ m1 − aN
λ φN= −h ³1
λ∂ΨS (w, am )
ΨSa ≡
∂am

= −³
= −³

E
− LN
λ
1
+ m1
φ

´

− aN

LS
am + aF + λζ m1 + ζ1 φ1LS
am + aF + λζ m1 + ζ1 φ1

1− 2 φaφ¶

1i2 φ̂a ,
λ
µ

´2
µ
´2

µ

1
1+
ζ

µ

1
− 2 φaφ¶¶¶
1
1 + φ̂a ,ζ∂φawhere φa ≡ ∂aand φ̂a ≡ −φ.
φ2m

Suppose the LN and LS curves intersect only once. Then the LN curve intersects the LS
NS
S
curve from below, as in Figure 4. Thus ΨN
w > Ψw , and detA = −Ψw + Ψw > 0. Applying– 49 –

Cremer’s rule to (89), we get:
¢
∂ι
1 ¡
S
S N
=
−ΨN
a Ψw + Ψa Ψw .
∂am
detA
¡
¢
S
S N
Substituting Ψiw and Ψia into −ΨNa Ψw + Ψa Ψw yields:
S
S N
−ΨN
a Ψw + Ψa Ψw = h ³
1
λ

E
− LN
λ
1
+ m1
φ

´

− aN

i2 ³

LS

1
´2 F,
λ
11

am + aF + λζ m1 + ζ φ

where
¶ µ
¶ ·³
¸
´
ζ
1
1
1 ∂E
m̂ + φ̂ + N
F ≡ φ̂am̂ + φ̂ − 1 + φ̂a
λ
ζ
ζ
Ψ ∂w
" µ
!# µ
¶ Ã
¶
ζ
1
φ̂
1
1 ∂E
− 1+
− 1 + φ̂a= m̂ φ̂a
−
.
λ ζ
m̂
ζ
ΨN ∂w
µ

a
From the definitions of φ(w, am ) and m(w) above equations (85) and (86), φ̂a ≡ −φ
> 0,
φ2
w
φ̂ ≡ φφw2 > 0 and m̂ ≡ m> 0. In addition, from (84), ∂E> 0. Therefore, if λζ − ζ1 < 0, F < 0m2∂w

and thus ∂a∂ιm < 0. This case is depicted in Figure 4.

Welfare Analysis
Finally, we conduct the welfare analysis in the steady state by using the results of the positive
analysis. In the same manner as in Section 6, we can derive the instantaneous utility. Only
R1
the part of welfare that depends on the prices changes to 0 log p(j, t)dj = nS (t) log ζ + (1 −nS (t)) log λ = −nS (t)(log λ − log ζ) + log λ. We can rewrite the instantaneous utility as:
Z t
log ui (t) = (log λ)ι(τ )dτ + log Ei (t) + nS (t)(log λ − log ζ) − log λ.
0Therefore, we obtain the lifetime utility in the steady state as follows:¸·
1 log λι + log Ei + nS (log λ − log ζ) − log λ .
Ui =ρρWe assume that Southern households possess only the stocks of the imitation firms, which theNorthern households cannot. This assumption implies that the budget constraint of the Southernhouseholds is ESρLS = wSρLS + vS nS . Using (43) and wS = 1, we can rewrite the budget– 50 –

constraint as ES = 1 + ρ amLSnS . By differentiating the lifetime utility function with respect toam , we obtain the change of welfare by the marginal increase in am as follows:µ¶ρ
log λ ∂ι
∂US
1h
∂nS
LS
=
+
nS + am
m
∂am
ρ
ρ ∂am
1 + ρa∂amnS
LS| {z }{z}|innovation-impeding effectnominal-spending effect(–)(ambiguous)i∂nS+ (log λ − log ζ)
.
∂am
|
{z
}
competition effect(ambiguous)This equation shows that the total welfare effect of strengthening IPR protection (an increasein am ) can be decomposed into the following three parts. First, strengthening IPR protection
reduces aggregate innovation and thus reduces welfare. We refer to this welfare effect as theinnovation-impeding effect. In contrast with the costless-imitation case, strengthening IPR protection through increasing the imitation cost reduces aggregate innovation, at least in the steadystate. Second, strengthening IPR protection affects nominal spending and welfare. We referto this as the nominal-spending effect. An increase in the imitation cost increases the stock
value of an imitation firm; however, the sign of ∂nS /∂am is ambiguous and thus the sign ofthe nominal-spending effect is indeterminate. Finally, strengthening IPR protection affects themeasure of type-S industries, nS (t). An increase in the measure of type-S industries, where
the price is lower because type-S goods are supplied by Southern firms with the lower marginal
cost, increases welfare. We refer to this welfare effect as the competition effect. The sign of∂nS /∂am is ambiguous and thus the sign of the competition effect is indeterminate.
Our numerical results indicate that strengthening IPR protection increases the measure of
type-S industries, nS ; therefore, both the nominal-spending effect and the competition effectare positive. The numerical results show that with higher innovation cost, strengthening IPR
protection increases the welfare, whereas it reduces the welfare with lower innovation cost.
For instance, if we set the parameter value of the innovation cost aN at 123.5, which is thesame as in the numerical examples of the text, the negative innovation-impeding effect outweighs those two positive effects, so that the welfare of the South increases with am , as shownin the upper panels of Figure 5. However, if we set aN at the lower value, 50, the negative
innovation-impeding effect overwhelms those two positive effects, and thus the welfare of the– 51 –

Ι

0.00305
0.003
0.00295
0.00290.00285
0.00280.00275

0.135

nS

0.125

US -21.6

0.12

-21.8
-22

0.11
112 114 116 118 120 122

112 114 116 118 120 122

112 114 116 118 120 122

am

am

am

0.016

nS

0.01560.01540.0152
45

-21.4

0.115

0.0158

Ι

-21.2

0.13

46

47

48

49

50

0.148
0.146
0.144
0.142
0.14
0.138
0.136
45

-15.33-15.34US -15.35-15.36-15.37
46

47

48

49

50

am

am

45

46

47

48

49

50

am

Figure 5: The effects of an increase in imitation cost
The horizontal axes represent am . The vertical axes represent aggregate innovation ι, measure of imitated sectors
nS , and the utility of the Southern consumer US , respectively. The parameter values are λ = 4, ζ = 1.6, ρ = 0.05,
LN = 3, LS = 6, and aF = 5. The upper three panels are the numerical results as aN = 123.5, and the lower
three panels are those as aN = 50.

South decreases with am , as shown in the lower panels of Figure 5. We interpret the result for
the welfare effect somewhat differently from that obtained in the costless-imitation model. In
the basic model where imitation is costless, strengthening IPR protection in the South increases
welfare if the positive welfare effect through promoting innovation outweighs the negative welfare effects. On the other hand, in the model where imitation is costly, strengthening IPR
protection impedes innovation. However, if the other positive welfare effects overwhelm thenegative effect on innovation, strengthening IPR protection in the South increases the welfareof Southern consumers.

– 52 –

Appendix K Notes on some results in the text
In this appendix, caluculations of some results in the text are explained in more detail.

Derivation of the result of (22)In this note, we derive the result of (22). By differentiating (18) with respect to M , we have
∂ n̄N
∂M

=

=

=

aN λ2 (Ē − LS )[aN M λ2 − (λ − 1)Ē]Ē − aN λ2 Ē[aN M λ2 (Ē − LS ) − λLN Ē(λ − 1)][aN M λ2 − (λ − 1)Ē]2 Ē 2½¾
2
[aN M λ2 − 2(λ − 1)Ē][aN M λ2 (Ē − LS ) − λLN Ē(λ − 1)] ∂ ĒaN M λ − λLN (λ − 1)−+∂M
[aN M λ2 − (λ − 1)Ē]Ē
[aN M λ2 − (λ − 1)Ē]2 Ē 2
−aN λ2 (Ē − LS )(λ − 1)Ē 2 + aN λ3 (λ − 1)LN Ē 2
[aN M λ2 − (λ − 1)Ē]2 Ē 2½
aN M λ2 − λLN (λ − 1)
[aN M λ2 − (λ − 1)Ē][aN M λ2 (Ē − LS ) − λLN Ē(λ − 1)]+−[aN M λ2 − (λ − 1)Ē]Ē
[aN M λ2 − (λ − 1)Ē]2 Ē 2¾
(λ − 1)Ē[aN M λ2 (Ē − LS ) − λLN Ē(λ − 1)] ∂ Ē+∂M
[aN M λ2 − (λ − 1)Ē]2 Ē 2
aN λ2 (λ − 1)[LS + λLN − Ē][aN M λ2 − (λ − 1)Ē]2½¾
aN M λ2 LS(λ − 1)Ē[aN M λ2 (Ē − LS ) − λLN Ē(λ − 1)] ∂ Ē++∂M
[aN M λ2 − (λ − 1)Ē]Ē 2[aN M λ2 − (λ − 1)Ē]2 Ē 2=·¸(λ − 1)n̄F
M λLS(λ − 1)n̄N ∂ Ē
+
+
µ
aN λµ
∂M
µĒ 2

=

1
µ

½
·
¸
¾
(λ − 1)n̄N ∂ ĒM λLS+(λ − 1)n̄F +
aN λ
∂M
Ē 2

> 0,

where the fourth equality uses (17), (18), and the definition of µ, and the inequality uses (22).

– 53 –

Derivation of the result of (23)In this note, we derive the result of (23). By differentiating (17) with respect to M , we have
∂ n̄F
∂M

=
=
=
=

½
¾
aN λ3 (LS + λLN − Ē)λλ(λ − 1)(LS + λLN − Ē) ∂ Ē
−
+ −
+
∂M
[aN M λ2 − (λ − 1)Ē]2
aN M λ2 − (λ − 1)Ē[aN M λ2 − (λ − 1)Ē]2
·
¸
λn̄F
1
(λ − 1)n̄F ∂ Ē
−
+ −
+
µ
aN µ
aN λµ
∂M
·
¸
1
λ − (λ − 1)n̄F ∂ Ē−λn̄F +
µ
aN λ
∂M
·
¸
1
(1 − n̄F )λ + n̄F ∂ Ē−λn̄F +
µ
aN λ
∂M

< 0,

where the second equality uses (17) and the definition of µ, and the inequality uses (22).

Derivation of the result of (25)¯¯In this note, we derive the result of (25). Using ∂ι(t)derived in Appendix C, we obtain∂M n (0)=n̄
S

S

µZ t
¶
∂
ι(τ )dτ dte (log λ)
∂M
0
0
¾
Z ∞
Z t½
1 + (λ − 1)n̄S −µτ ∂ Ē−ρt−µτ ∂ῑ= (log λ)
e
−
e
dτ dt
(1 − e )
∂M
aN λ
∂M
0
0
½
·
¸µ
¶¾
Z ∞
∂ῑ
∂ῑ
1 + (λ − 1)n̄S ∂ Ē1 −µt 1−ρt
= (log λ)
e
t×
−
+
− e +
dt
∂M
∂M
aN λ
∂M
µ
µ
0
·
¸Z ∞
Z ∞
£ −(ρ+µ)t¤log λ ∂ῑ
1 + (λ − 1)n̄S ∂ Ē−ρt ∂ῑ= (log λ)
te
dt +
+
e
− e−ρt dt
∂M
µ
∂M
aN λ
∂M 0
0
·
¸µ
¶¾
½
1 ∂ῑ
1 + (λ − 1)n̄S ∂ Ē
1
1 ∂ῑ
1
+
+
−
= (log λ)
ρ2 ∂M
µ ∂M
aN λ
∂M
ρ+µ ρ
½
¾
µ
1
∂ Ē
∂ῑ
= (log λ)−[1 + (λ − 1)n̄S ]ρ2 (ρ + µ) ∂M
aN λρ(ρ + µ)
∂M

Z ∞

−ρt

< 0,
where the inequality uses (22) and the result of Proposition 3.

– 54 –

Derivation of the result of (40)In this note, we derive the result of (40). By differentiating (16) with respect to M , we have
∂ n̄S
∂M

=

=

=

aN λ2 LS [aN M λ2 − (λ − 1)Ē]Ē − aN λ2 Ē[aN M λ2 LS + Ē 2 − λ(LN + LS )Ē][aN M λ2 − (λ − 1)Ē]2 Ē 2½¾
2Ē − λ(LN + LS )[aN M λ2 − 2(λ − 1)Ē][aN M λ2 LS + Ē 2 − λ(LN + LS )Ē] ∂ Ē+−∂M
[aN M λ2 − (λ − 1)Ē]Ē
[aN M λ2 − (λ − 1)Ē]2 Ē 2
aN λ2 LS Ē 2 − aN λ2 Ē(Ē 2 − λLN Ē)[aN M λ2 − (λ − 1)Ē]2 Ē 2½
[aN M λ2 − (λ − 1)Ē][aN M λ2 LS + Ē 2 − λ(LN + LS )Ē]
2Ē − λ(LN + LS )−+2
[aN M λ − (λ − 1)Ē]Ē
[aN M λ2 − (λ − 1)Ē]2 Ē 2¾
(λ − 1)Ē[aN M λ2 LS + Ē 2 − λ(LN + LS )Ē] ∂ Ē+∂M
[aN M λ2 − (λ − 1)Ē]2 Ē 2
aN λ2 (LS + λLN − Ē)[aN M λ2 − (λ − 1)Ē]2½¾
Ē 2 − aN M λ2 LS(λ − 1)Ē[aN M λ2 LS + Ē 2 − λ(LN + LS )Ē] ∂ Ē++∂M
[aN M λ2 − (λ − 1)Ē]Ē 2[aN M λ2 − (λ − 1)Ē]2 Ē 2½

Ē 2 − aN M λ2 LS(λ − 1)n̄S+
aN λµ
aN λµĒ 2

¾

=

n̄F
+
µ

∂ Ē
∂M

=

·
¸
n̄F
1
aN M λ2 LS ∂ Ē+
1 + (λ − 1)n̄S −
,
µ
aN λµ
∂M
Ē 2

where the fourth equality uses (16), (17), and the definition of µ.

– 55 –

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– 58 –