Discussion Papers In Economics
And Business
Tariffs and Foreign Direct Investment in a North–South
Product Cycle Model
Tatsuro Iwaisako
Hitoshi Tanaka

Discussion Paper 20-08

May

2020

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANTariffs and Foreign Direct Investment in a North–South Product Cycle Model∗
Tatsuro Iwaisako†

Hitoshi Tanaka‡

April 30, 2020
Abstract
This paper theoretically examines how import tariffs by a developed country (the North) and adeveloping country (the South) affect innovation, foreign direct investment (FDI), wages, and welfare
using a North–South quality ladder model. We show that a Northern import tariff raises the relative
wage of Northern labor to Southern labor, but impedes innovation and FDI. Because of the decrease
in innovation and increased prices, this may worsen Northern welfare. By contrast, a Southern import tariff raises the relative wage of Southern labor to Northern labor and promotes innovation and
FDI. As a result, it can improve Southern welfare. These results imply that the North has a weaker
incentive than the South to impose an import tariff, and this is consistent with actual experience.
Keywords: foreign direct investment, innovation, intellectual property rights protection
JEL classification: F43, O33, O34∗

The authors would like to thank Yuichi Furukawa and seminar participants at Chukyo University, Hitotsubashi University,and Kobe University for their useful comments and discussions on an earlier version of this paper. The authors also gratefully
acknowledge financial support from the Japan Society for the Promotion of Science (JSPS) (No. 17K03623, 20K01561) and the
Joint Usage/Research Center at the Institute of Social and Economic Research (ISER), Osaka University. The usual disclaimer
applies.
†
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043, Japan. E-mail:iwaisako@econ.osaka-u.ac.jp‡
Faculty of Economics, Hokkai-Gakuen University, 4-1-40, Asahi-machi, Toyohira-ku, Sapporo, Hokkaido 062-8605,
Japan. E-mail: tanaka h@hgu.jp

1

Introduction

Since the start of the Trump administration, the US government has largely adopted a protectionist policy
concerning some aspects of international trade. As evidence, Fajgelbaum et al. (2020, p. 9) reported thatin 2018, the US had increased average import tariffs on about $247 billion worth of Chinese products
from 3.0% to 15.5%. They also showed that, in the same year, China had increased the average tariff on
about $93 billion worth of US exports from 8.4% to 18.9% in retaliation. Regarding an explanation for
the initial US action, according to a presidential memorandum signed on March 22, 2018, the increased
tariffs by the USA targeting China were at least partly in response to allegedly insufficient protection by
China of the intellectual property of US companies. The question then naturally arises as to what effectincreased tariffs by a developed country such as the USA (hereafter, the North) and a developing country
such as China (hereafter, the South) have on innovation and international technology transfer, given that
intellectual property rights (IPR) are not well protected in the South.
In this paper, we theoretically investigate the effects of import tariffs on innovation, foreign direct
investment (FDI) from the North to the South, wages, and welfare using a North–South growth model
based on the quality ladder-type product-cycle model first developed by Grossman and Helpman (1991).
In our model, a higher-quality product invented in the North replaces the current product if innovation
occurs through successful R&D. The Northern inventor can then choose to produce in the North or
shift production to the South through FDI to employ its lower-wage labor. However, some of thesegoods produced in the South are subject to imitation because of its imperfect protection of IPR. Unlike
Grossman and Helpman (1991), we also assume that each government imposes an ad valorem tariff onimported goods.
We provide the following three main results. First, a unilateral increase in the Northern tariff impedes
innovation and FDI, although it also raises the relative wage of Northern to Southern labor. In the
Northern market, the higher Northern tariff raises the duty-inclusive price of the goods produced in
the South. For Northern firms, this reduces the competition pressure from Southern firms, so that theycan enjoy higher profits through raising the price in the North. On the other hand, the Northern tariff
also reduces the Northern demand for the goods produced by FDI firms in the South through raising
the duty-inclusive price. As a result, the higher Northern tariff decreases the profits of the FDI firms.
Through both these effects, an increased Northern tariff depresses FDI from the North to the South. In
addition, it raises the Northern relative wage because more firms choose to produce in the North thanin the South. The higher Northern wage also leads to a higher cost of R&D in the North, and thus, an1increased Northern tariff impedes innovation.
Second, contrary to the effects of a Northern tariff, a unilateral increase in the Southern tariff raises
the Southern relative wage and promotes innovation and FDI. When the Southern tariff increases, Northern firms need to lower the pre-tariff price of the goods sold in the South to gain the Southern market.
This reduces the profits of Northern firms, such that a higher Southern tariff promotes FDI in the South.
The increased production transfer from the North to the South reduces the Northern relative wage andthereby the cost of R&D in the North. Accordingly, the higher Southern tariff promotes innovation.
These two results imply that simultaneous tariff increases by the North and South decrease innovation
and FDI if the impact of the Northern tariff increase surpasses that of the Southern tariff. We show that
this is the case when the population is relatively large in the North and small in the South. Conversely,
if the relative population size of the North compared with the South is sufficiently small, the symmetrictariff rate between the two countries that maximizes innovation and FDI is not zero, but strictly positive.
Third, we also show the effects of a unilateral tariff increase on the country’s own welfare as follows.
A unilateral tariff increase by the South improves the welfare of the South if the initial tariff rate is lower
than a certain positive level. This is because an increase in the Southern tariff promotes innovation andincreases cheaper imitated goods through increasing production transfer to the South. This result implies
that the welfare-maximizing tariff rate for the South is strictly positive. By contrast, a unilateral tariffincrease by the North tends to worsen the welfare of the North if the Southern tariff is lower and theimitation rate in the South is higher. In fact, a tariff increase in the North exerts a positive effect on
the North’s welfare because it shifts production from the South to the North and raises the Northern
wage. However, if the Southern tariff is low and imitation is active in the South, production in the Southtends to be small because fewer Northern firms choose to undertake FDI. As a result, a lower Southerntariff and active imitation weakens the positive effect of a Northern tariff increase on the Northern wage
through the production shift to the North. Thus, the total welfare effects of the Northern tariff tend tobe negative in this case because the negative effects from the decreased innovation and increased prices
tend to dominate the positive effect because of the increased income of the North.
The effects of trade cost have already been examined in a number of studies using quality ladder-type
growth models.1 Seminal studies include Dinopoulos and Segerstrom (1999a, 1999b) and Dinopoulos
and Syropoulos (1997). However, both of these studies assumed two structurally identical countries,

1

Using expanding variety as opposed to quality ladder-type growth models, Baldwin and Robert-Nicoud (2008), Dinopoulos

and Unel (2011), Gustafsson and Segerstrom (2010), Naito (2017), and Sampson (2016) also investigate the effects of trade
costs.

2that is, a North–North setting. By contrast, we focus on the trade policy in two asymmetric countries: a
North that is innovative and a South that is the recipient of production transfer. Therefore, the analyses
are complementary. In addition, as summarized above, our results show that a tariff increase by a country
has differing effects depending on whether the country is the North or the South, which is not possible
in a North–North setting. Dinopoulos and Segerstrom (2007) constructed a North–South model where aSouthern firm can produce some state-of-the-art good through imitation, and considered how decreasing
trade cost affects both innovation and imitation. However, they assumed the trade cost to be symmetricbetween both countries and thus, did not analyze either the case of asymmetric trade costs or the effectof a unilateral tariff increase by just one country. By contrast, our model allows unequal tariff rates
between the North and the South so that we can readily investigate the effect of a unilateral tariff increase.
Grieben and Şener (2009) also examined the effects of Northern and Southern unilateral tariff reduction
in a North–South product-cycle model. However, they assumed imitation by Southern firms to be theonly channel for technology diffusion from the North to the South and did not consider FDI, much like
Dinopoulos and Segerstrom (2007).
Unlike these studies, we incorporate production transfer by Northern firms into the model. As pointed
out by Keller (2004), FDI is one of the major channels for technology diffusion across countries. Moreover, with so-called “tariff-jumping FDI”, tariffs are likely to affect the incentive for FDI and thus labordemand.2 Our model captures this tariff effect not considered in Dinopoulos and Segerstrom (2007) and
Grieben and Şener (2009). In addition, unlike either of these previous studies, we examine the welfare
effects of a unilateral tariff increase.3 To date, many theoretical studies, including Kennan and Riezman(1988) and Syropoulos (2002), have concluded that the optimal tariff for a large country tends to be high.
However, as discussed by Naito (2019), in practice, we frequently observe the opposite: economicallylarger countries tend to set lower tariffs. As discussed, our model shows that the Northern tariff worsens
the welfare of the North under a certain condition, while the Southern tariff improves the welfare of theSouth if the initial tariff rate is set sufficiently low. Therefore, the results of the welfare analysis in thispaper are consistent with the actual tendencies shown between country size and the optimal tariff.
The remainder of the paper is structured as follows. Section 2 develops the North–South quality
2

For example, Belderbos and Sleuwaegen (1998), Chen and Moore (2010), and Ghodsi (2020) empirically showed thattariffs imposed by a host country exert a significantly positive effect on FDI to that country, which is consistent with thetariff-jumping motive.
3
A number of studies have examined the welfare effects of changes in unilateral tariffs in a two-country general equilibrium
model, e.g., Gros (1987), Opp (2010), Felbermayr, Jung, and Larch (2013). However, these employ static as opposed to growthmodels.

3

ladder model with tariff and Section 3 derives the market equilibrium path. Section 4 presents the comparative statics and Section 5 provides the welfare analysis. Section 6 concludes.

2

The model

We introduce FDI into the two-country quality ladder model developed by Grossman and Helpman(1991). Consider an economy consisting of two countries, the North and the South, denoted N andS, respectively. We assume perfect capital mobility between the countries so that their interest rates
equalize. With a fixed number (measure) of identical households, let Li (0) denote the number of households in country i ∈ {N, S}. A member of each household supplies one unit of labor inelastically ateach time point. Assume that the member size of each household is unity at the initial time and grows
at a constant rate gL (≥ 0), such that the quantity of labor supplied in country i at time t is given by
Li (0)egL t ≡ Li (t). We select Southern labor as the numeraire and normalize the Southern wage to beunity at every time point.
There is a continuum of goods, indexed by ω ∈ [0, 1], produced in the North or the South. One unitof good output requires one unit of labor input. Each good is classified by a countable infinite number of“generations” j = 0, 1, 2, · · · . We normalize the generation number at the initial time to be zero for all
goods. If innovation occurs in industry ω, a one-step new generation of good ω is developed. Therefore,
generation j of good ω can be produced after the jth innovation in industry ω. As described in Section
2.3, innovation occurs because of successful R&D efforts by a Northern firm. Different generations of agood possess different “qualities”. The quality of generation j of good ω is q(j, ω) = λj , where the rateof quality increase between any two consecutive generations, λ (> 1), is identical for all goods.
We assume that the government of each country imposes an ad valorem tariff on imported goods. The
tariff rate of country i is τi (≥ 0), which is common to all imported goods. The government transfers allof the tariff revenues to the households of its country as a lump sum and runs a balanced budget at eachtime point.

2.1 Households
Each household in country i maximizes the following lifetime utility:
∫ ∞
Ui =

e−(ρ−gL )t log ui (t)dt,

04

where ρ (> gL ) is a common subjective discount rate and log ui (t) represents instantaneous utility attime t. We specify the instantaneous utility function as:


∫ 1
∑
log ui (t) =log q(j, ω)di (j, ω, t) dω,
0

(1)

j

where di (j, ω, t) denotes the per capita consumption of good ω of generation j at time t. The intertemporal budget constraint of each household in country i is given by
∫ ∞ ∫
∫ ∞ ∫
∫ ∞ ∫
t
− 0t r(s)ds+gL t− 0t r(s)ds+gL te
Ei (t)dt = Ai (0) +ewi (t)dt +e− 0 r(s)ds+gL t Ti (t)dt, (2)000where r(t) is the interest rate, Ei (t) and Ai (0) denote consumption expenditure per capita and initialasset holdings per capita, respectively, wi (t) and Ti (t) denote wages and a lump-sum transfer per capita
by the government, respectively.
We solve this utility maximization problem in two stages. First, for each product, a householdchooses a single generation J(ω, t) that carries the lowest quality-adjusted price p(j, ω, t)/q(j, ω). Thisimplies the following static demand function:
 Ei (t)/p(j, ω, t) for j = J(ω, t),di (j, ω, t) =0otherwise.

(3)

Second, the household chooses a time pattern of expenditure to maximize its lifetime utility. Suchintertemporal utility maximization requires thatĖi (t)= r(t) − ρ.
Ei (t)

2.2(4)

Production

The firm that developed the current latest generation of good ω (hereafter, the “leader” firm in industryω) can produce it monopolistically under IPR protection if the firm chooses to operate in the North.
A leader firm can become a “multinational” firm by shifting production to the South. Following Lai(1998), Glass and Wu (2007), and Tanaka and Iwaisako (2014), we assume that a Northern firm cantransfer production to the South instantaneously without cost. The multinational firm enjoys a lower
labor cost for production, but faces the risk of imitation because the South does not sufficiently enforce
IPR protection. If good ω is imitated at time t, the leader firm in industry ω cannot earn profits for timet because perfect competition with copied goods prevails in the industry at that time. For simplicity, we
assume that whether a good is under imitation is determined independently at each time point. Morespecifically, if a leader firm chooses to produce a good in the South, Southern firms imitate that good5

at some constant probability m ∈ [0, 1) at any time point.4 Therefore, under the law of large numbers,
m × 100 percent of all multinational firms are imitated at any time point. We interpret this imitation
probability m as the degree of IPR protection; higher m implies weaker IPR enforcement in the South.
Next, we consider how the price and quantity supplied of each good are determined. Each good is
produced by either (i) the leader firm in the North monopolistically; (ii) the multinational firm in theSouth monopolistically; or (iii) imitators in the South under perfect competition.5 From the demand
function (3), a leader firm can maximize profits by selling at the upper limit of the price such that rival
firms that could produce an old generation in the same product line cannot operate. For a leader firm, theoptimal price in the Northern and Southern markets can be different because of tariffs.
2.2.1 Northern firms
First, we consider what level of price a leader firm producing in the North charges in the Northern market.
We assume that any firm can freely produce generations older than the currently latest in each productline because of expired patents.6 In this case, leader firms do not undertake R&D because they cannot
take more than a one-step quality lead over the nearest follower firms in the same product line. Thus, thepotential strongest rivals for a Northern leader firm are the follower firms that can produce the current
second to newest generation of the same good. A follower firm could cut the (pre-tariff) price down to its
marginal cost, which would be wN (t) if produced in the North and wS (t) = 1 if produced in the South.
This implies that the lowest possible duty-inclusive price of a follower’s good imported from the Southto the North is 1 + τN . Therefore, the optimal price for a Northern leader firm in the Northern market is
λ min{wN (t), 1 + τN } because it needs to set the lowest quality-adjusted price to sell the good.
In this paper, we focus on the case where the tariff rate in the North is low enough to satisfy 1 + τN ≤wN (t) (see footnote 5). Under this assumption, a Northern leader firm prices its good at pN N (t) =λ(1 + τN ) in the North. The demand for a Northern leader’s good by Northern consumers is
xN N (t) =4
5

EN (t)LN (t).
λ(1 + τN )(5)

Grossman and Lai (2004) adopt a similar assumption.
We focus on the case where a leader firm does not divide the location of production between the North and the South.

In equilibrium, this is satisfied if and only if τN < [wN (t) − 1]/λ. The proof is provided in Appendix C available from theauthors upon request.
6
Even without this assumption, leader firms do not undertake R&D, and our results do not change at all if wN (t) < (1−m)λand there is no advantage over follower firms in the R&D process. The proof is provided in Appendix C, which is available
from the authors upon request.

6

The Northern leader’s profits from the sale in the North are given by
πN N (t) = [λ(1 + τN ) − wN (t)]EN (t)LN (t).
λ(1 + τN )(6)

In a similar way, we derive the price a leader firm producing in the North charges in the Southernmarket. As we focus on the case where the Northern wage is not lower than the Southern wage, afollower firm could set a lower (duty-inclusive) price in the Southern market when it produced in theSouth than when it produced in the North. To set the lowest quality-adjusted price in the Southern market,
a Northern leader firm needs to choose a duty-inclusive price not higher than λ because the marginal costof the follower firm that produced in the South would be equal to wS (t) = 1. Therefore, the optimal
pre-tariff price that a Northern leader firm charges Southern consumers is pN S (t) = λ/(1 + τS ). As theduty-inclusive price is λ, the demand for a Northern leader’s good by Southern consumers is
xN S (t) =ES (t)LS (t).
λThe Northern leader’s profits from the sale in the South are given by7[]
λES (t)LS (t)πN S (t) =− wN (t).
1 + τSλ
Adding (6) and (8), we obtain the total profits of a Northern leader firm as follows:[][]
wN (t)
1wN (t)πN (t) = 1 −EN (t)LN (t) +−ES (t)LS (t).
λ(1 + τN )1 + τS
λ

(7)

(8)

(9)

2.2.2 Multinationals
Next, we consider what level of price a multinational firm charges Northern consumers. For the same
reason as in the decision by a Northern leader firm, a multinational firm needs to choose a duty-inclusive
price not higher than λ min{wN (t), 1 + τN } in the North to set the lowest quality-adjusted price. Under
the assumption that 1 + τN ≤ wN (t), the optimal pre-tariff price in the North is pF N (t) = λ fora multinational firm. Because the duty-inclusive price is λ(1 + τN ), the demand for an unimitatedmultinational’s good by Northern consumers is
xF N (t) =EN (t)LN (t).
λ(1 + τN )(10)

The unimitated multinational’s profits from sale in the North are given by
πF N (t) = (λ − 1)
7

EN (t)LN (t).
λ(1 + τN )(11)

If 1 + τS > λ/wN (t), a Northern leader firm could not earn positive profits by selling the good in the South. In this case,no Northern leader firm would supply the good in the South, that is, xN S (t) = 0. However, we rule out this case.

7

We derive the price a multinational firm charges Southern consumers in a similar way. Just as we
considered the pricing of a Northern leader firm, a follower firm could set a lower (duty-inclusive) pricein the South when it produces in the South. To set the lowest quality-adjusted price, a multinational firmneeds to choose a price not higher than λ in the South because the marginal cost would be wS (t) = 1 if afollower firm produced in the South. Therefore, the optimal price in the Southern market is pF S (t) = λfor a multinational firm. The demand for an unimitated multinational’s good by Southern consumers is
xF S (t) =ES (t)LS (t)
.
λ

(12)

The unimitated multinational’s profits from sale in the South are given by
πF S (t) = (λ − 1)

ES (t)LS (t).
λConsequently, from (11) and (13), the total profits of a multinational firm are)[](
EN (t)LN (t)1+ ES (t)LS (t) .
πF (t) = 1 −λ
1 + τN(13)(14)

2.2.3 Imitated goods
If a good is imitated at time t, any firm in the South can produce and export the latest-generation good atthat time. In the Southern market, the price of the good falls to wS (t) = 1, which is equal to the marginal
cost of imitators. The demand for an imitated good by Southern consumers then becomesxM S (t) = ES (t)LS (t).

(15)

In the Northern market, an imitated good is imported from the South and sold at 1 + τN after tariff.8
Therefore, the demand for an imitated good by Northern consumers is
xM N (t) =2.3

EN (t)LN (t).
1 + τN(16)

R&D and FDI

Following Grossman and Helpman (1991), we assume an R&D process as follows: if a firm devotesaN X(t)I˜ units of Northern labor for a time interval of length dt to research good ω, it succeeds in
˜ where aN is a parameter and X(t)developing the next generation of good ω with probability Idt,
represents the difficulty of R&D. As in Dinopoulos and Segerstrom (1999a), Dinopoulos and Thompson
8

Under the assumption that 1 + τN ≤ wN (t), a leader firm whose good is imitated cannot earn profits, irrespective ofwhether it is produced in the North or the South.

8(2000), Şener and Zhao (2009), and others, we assume that the growth rate of X(t) is equal to the growthrate of the total labor supply, gL , so that the model is free from the scale effect.9 For a finite size of R&D
activities in equilibrium, the expected gain from R&D must not exceed the cost of R&D. Thus, lettingvN (t) denote the stock market value of a Northern leader firm, we have:vN (t) ≤ wN (t)aN X(t)with equality if I(t) > 0,(17)

where I(t) denotes the innovation rate at time t, which is assumed to be the same in every industry in thesymmetric equilibrium.
Once a Northern firm succeeds in inventing a new-generation good, it can become a multinationalfirm by shifting production to the South without cost. Therefore, as long as both Northern leaders andmultinational firms exist in equilibrium, the market values of a Northern leader and a multinational firmmust be equal; that is, the following equality must hold at each time point:vN (t) = vF (t),(18)

where vF (t) denotes the stock market value of a multinational firm.
Next, we consider the no-arbitrage conditions between the stocks of a leader firm and the risk-free
asset. Assuming that the shareholders of a firm hold a well-diversified portfolio, the expected returnfrom the stocks of a leader firm must be equal to the return from the risk-free asset. The shareholders
of a Northern leader firm then earn dividends πN (t)dt and capital gains v̇N (t)dt over a time interval oflength dt. At the same time, the Northern leader firm loses its monopolistic rent through the developmentof a new generation of the same good by another firm at the innovation rate I(t) over the time interval.
Thus, the shareholders are faced with a capital loss of vN (t) with probability I(t)dt. These imply thatthe no-arbitrage condition with respect to the stocks of a Northern leader firm is10
r(t)vN (t) = πN (t) + v̇N (t) − I(t)vN (t).

(19)

A multinational firm earns profits πF (t) if its good is not produced by imitators at time t. This event
occurs with probability 1 − m at each time point. Meanwhile, because of imitation, the multinationalfirm cannot earn any profits at time t with probability m. Thus, the expected dividends that shareholders
of a multinational firm obtain at time t is (1 − m)πF (t). In addition, over a time interval of length dt,
9

Our model can thus be interpreted as an extension of “first-generation” fully endogenous growth models such as Grossman

and Helpman (1991) because it becomes one if we assume X(t) = 1 and gL = 0 for all t.
10
If a Northern leader firm transfers production to the South and becomes a multinational firm, it can obtain the valuevF (t) − vN (t). However, from (18), this is zero in equilibrium.

9the shareholders obtain capital gains v̇F (t)dt, and are faced with a capital loss of vF (t) with probability
I(t)dt given the loss of monopolistic rent through the development of a new generation of the same goodby another firm. Thus, the no-arbitrage condition between the stocks of a multinational firm and therisk-free asset isr(t)vF (t) = (1 − m)πF (t) + v̇F (t) − I(t)vF (t).

(20)

2.4 Labor markets
In the North, labor is devoted to production and R&D activities. Letting nN (t) ∈ (0, 1) represent
the number (measure) of industries in which the Northern leader firms produce state-of-the-art goods,
the aggregate labor demand for production in the North is given by nN (t)(xN N (t) + xN S (t)). The
aggregate labor demand for R&D activities is given by aN X(t)I(t) because firms undertaking R&D
target all industries. From (5) and (7), the labor market-clearing condition in the North is[]
EN (t)LN (t) ES (t)LS (t)nN (t)++ aN X(t)I(t) = LN (t).
λ(1 + τN )λ(21)In the South, multinational firms and imitators demand labor for production. We define nS (t) ≡1 − nN (t), which is the number of industries where the goods are produced in the South. The goods
produced in the South are imitated at probability m at each time point. From the law of large numbers,
the goods are produced by the multinational firms monopolistically in (1 − m)nS (t) industries andproduced by Southern imitators in mnS (t) industries. The aggregate labor demand of multinationals is(1 − m)nS (t)(xF N (t) + xF S (t)) and that of Southern imitators is mnS (t)(xM N (t) + xM S (t)). From(10), (12), (15), and (16), the labor market-clearing condition in the South becomes[][]
EN (t)LN (t) ES (t)LS (t)EN (t)LN (t)(1 − m)nS (t)++ mnS (t)+ ES (t)LS (t) = LS (t). (22)λ(1 + τN )λ1 + τN

2.5

Government budget constraints

The Northern government imposes the tariff on imports from the South by the multinational firms andthe Southern imitators. In the Northern market, the sales of the good supplied by a multinationalfirm are pF N (t)xF N (t) = EN (t)LN (t)/(1 + τN ) and the sales of the good supplied by the Southern imitators are xM N (t) = EN (t)LN (t)/(1 + τN ). Thus, the Northern tariff revenue is given bynS (t)τN EN (t)LN (t)/(1 + τN ). As the Northern government transfers all tariff revenue to Northernhouseholds, it determines the lump-sum transfer per capita TN (t) to satisfy the following budget constraint at each time point:
TN (t)LN (t) = nS (t)τN
10

EN (t)LN (t).
1 + τNThe Southern government imposes the tariff on imports from the North by the Northern leader firms.
The sales of a Northern leader firm in the Southern market are pN S (t)xN S (t) = ES (t)LS (t)/(1 + τS ),which implies that the Southern tariff revenue is given by nN (t)τS ES (t)LS (t)/(1 + τS ). Therefore, theSouthern government determines TS (t) to satisfy the following budget constraint:
TS (t)LS (t) = nN (t)τS

3

ES (t)LS (t).
1 + τS(23)

The equilibrium

In this section, we discuss the market equilibrium. To simplify notation, we define world aggregate
expenditure as E(t) ≡ EN (t)LN (t) + ES (t)LS (t) and the share of Northern aggregate expenditure toworld aggregate expenditure ϕ ≡ EN (t)LN (t)/E(t). On the equilibrium path, ϕ becomes constant overtime because Northern expenditure EN (t) and Southern expenditure ES (t) always grow at the same rate
given the Euler equation (4). By using E(t) and ϕ, the labor market equilibrium conditions (21) and (22)are rewritten as(nN (t)

ϕ+1−ϕ1 + τN((1 − m + mλ)nS (t))E(t)+ aN X(t)I(t) = LN (t),
λ

ϕ
+1−ϕ
1 + τN)E(t)= LS (t).
λ

(24)

(25)

In this model, there is no state variable, except population size and R&D difficulty, whose growth
rates are exogenous and constant. Consequently, as shown in Appendix A, this model does not have atransitional process and the economy jumps to the steady state immediately at the initial time. In thesteady state, Ei (t), I(t), nN (t), nS (t), and wN (t) are constant over time. We therefore omit the timeindex of these variables hereafter. As Ei is constant, the interest rate r(t) is also constant and equal to ρ
all the time from (4).
For analytical tractability, we focus on the case where the Southern households initially have noassets. Then, from (2) and (23), the budget constraint of a Southern household is
ES = 1 + nN ES

τS,
1 + τS(26)

where the left-hand side (LHS) is expenditure per capita, the first term on the right-hand side (RHS) is
wage income, and the second term on the RHS is the per capita lump-sum transfer from the Southerngovernment’s tariff revenue. Because ES = (1 − ϕ)E(t)/LS (t) by the definition of E(t) and ϕ, the11budget constraint (26) can be rewritten as 1 − ϕ = (1 + τS )LS (t)/[(1 + τS nS )E(t)]. Substituting (25)
into this equation to eliminate E(t), we obtain ϕ as a decreasing function of nS as follows:[]ϕλ
1 + τS n S= (1 + τN )−1 .
1−ϕ
1 − m + mλ (1 + τS )nS(27)

This equation implies that an increase in nS has two effects on the share of Southern aggregate expenditure to world aggregate expenditure, 1 − ϕ, through the budget constraint of a Southern household.
First, as shown in (25), an increase in nS raises the Southern wage compared with the world aggregate
expenditure, 1/E(t), because it expands the demand for Southern labor. The increased wage income has
a positive effect on 1−ϕ. Second, an increase in nS reduces the transfer payment to Southern households
because it decreases the tariff revenue of the Southern government. The decreased transfer payment has
a negative effect on 1 − ϕ. Nevertheless, the first effect necessarily dominates the second, so that ϕ isdecreasing with nS , as shown in (27).
Next, to analyze the equilibrium, we describe two key equations with respect to I and nS . We derive
the first equation from the labor market-clearing conditions in the two countries. Combining (24) and(25), we have the following equation:
LN (t)I=−
aN X(t)()
1
1
LS (t)−1.
nS
1 − m + mλ aN X(t)(28)

As X(t) grows at the same rate as LN (t) and LS (t), the relation between nS and I satisfying (28)
is depicted as an upward sloping curve in Figure 1. We refer to this as the LC curve given the labor
constraint. It shows the combinations of nS and I that are consistent with equilibrium in the labormarkets of the two countries.
The LC curve is upward sloping because the innovation rate I satisfying the labor constraints increases with nS for two reasons. First, as nS increases, the number of industries producing in the North,nN , contracts, and thus, labor demand for production decreases in the North. Second, an increase in nS
reduces the quantity of labor demanded by the Northern leader firm, [ϕ/(1 + τN ) + 1 − ϕ](E(t)/λ).
This second effect results from the increase in the Southern wage compared with the world aggregate
expenditure, 1/E(t), in the Southern labor market, as shown in (25). A higher Southern wage increases
the marginal cost of a follower firm when produced in the South, so that the Northern leader firms raise
the relative price of their goods to the world aggregate expenditure and thereby decrease production.
Because the abovementioned two effects reduce the labor inputs for production in the North, the Northern labor resources available for R&D increase. This is why an innovation rate I consistent with theequilibrium in the labor markets increases with nS .
12I
LN (t)aN X(t)

LC

E

R&D
1

0

nS

Figure 1: The equilibrium

The second equation with respect to nS and I is derived from the free-entry condition in R&D.
Because vF (t) = wN aN X(t) and v̇F (t)/vF (t) = Ẋ(t)/X(t) from (17) and (18), substituting (14) and(20) into the former yields
)
)( ϕ
(
+
1
−
ϕ
E(t)
(1 − m) 1 − λ1 1+τN
ρ + I − Ẋ(t)X(t)= wN aN X(t).

(29)

The LHS of (29) represents the expected gain from R&D that is equal to the present value of expected
profits discounted by the interest rate, the hazard rate of monopolistic rent due to another firm’s innovation, and the capital gain. The RHS of (29) then represents the cost of R&D.
The Northern wage (compared with the Southern wage), which affects the cost of R&D, is determined by the condition on FDI. When production is carried out in both countries continuously,v̇N (t)/vN (t) = v̇F (t)/vF (t) must be satisfied because (18) holds at each time point. Therefore, from(19) and (20), the profits of a Northern leader firm and a multinational firm must satisfy πN (t) =(1 − m)πF (t) in equilibrium. Substituting (9), (14), and (27) into this equation, we obtain the Northern
wage as a decreasing function of nS :wN = τN (1 − m)(λ − 1) +1 + τN (1 − nS )(1 − m + mλ).
1 + τS nS(30)

The reason why the Northern wage is decreasing with nS can be explained as follows. For botha Northern leader firm and a multinational firm, profits per unit consumption expenditure in the local
market (the North for a Northern leader firm and the South for a multinational firm) are higher than those13

in the other market because the firms can differentiate their prices in the two markets.11 As shown in(27), an increase in nS raises the share of Southern aggregate expenditure to world aggregate expenditure,
1 − ϕ, and therefore increases the profits of a multinational firm compared with a Northern leader firm.
Because this increases the incentive for a production transfer to the South, to restore equilibrium, theNorthern wage must decline so that the incentive for production in the North can increase.
Combining (25) and (29), we have the following equation:(1−m)(λ−1) LS (t)1−m+mλ X(t)Ẋ(t)ρ + I − X(t)= n S wN a N .

(31)

From (30), the RHS of (31) is increasing with nS and tends to zero as nS → 0.12 Therefore, the relation
between nS and I satisfying (30) and (31) is negative and asymptotes to the vertical axis, as depicted
in Figure 1. We refer to this as the R&D curve, which shows the combinations of nS and I that areconsistent with an incentive to carry out R&D.
The R&D curve is downward sloping because the innovation rate I consistent with an incentive for
R&D decreases with nS . In the Southern labor market, an increase in nS pushes up the Southern wage
when compared with world aggregate expenditure, 1/E(t). This decreases the demand for multinationalfirms’ goods through increasing their prices compared with aggregate expenditure. As a result, it decreases the profits of multinational firms and thereby the expected gain by R&D. This effect appears as
nS on the RHS of (31). Meanwhile, an increase in nS also reduces the cost of R&D through lowering theNorthern wage wN , as shown by (30). However, the decrease in the expected gain necessarily dominates
the decrease in the cost. Therefore, as nS increases, the innovation rate I must be lower in terms of the
incentive for R&D.
The intersection of the LC and R&D curves provides the equilibrium values of nS and I. As depicted
in Figure 1, the LC and R&D curves intersect only once if the R&D curve lies below the LC curve aroundthe upper limit of nS . Then, there exists a unique interior equilibrium such that nS and I are positive.
Depending on the equilibrium value of nS , (25), (27), and (30) determine the equilibrium values of E(t),ϕ, and wN , respectively.

11

Equations (6), (8), (11), and (13) show that the profits per unit consumption expenditure of a Northern leader firm are

1 − wN (t)/[λ(1 + τN )] in the North and 1/(1 + τS ) − wN (t)/λ in the South, while those of a multinational firm are(1 − 1/λ)/(1 + τN ) in the North and 1 − 1/λ in the South.
12
For the proof that the RHS is increasing with nS , see (53) in Appendix B. A detailed derivation is also provided in Appendix
C, which is available from the authors upon request.

14

4Comparative statics

In this section, we conduct comparative statics using Figure 1.

4.1

A unilateral tariff increase

We first consider the effects of a unilateral tariff increase by the North. Given nS , an increase in τNraises wN , as shown in (30). This is because the Northern tariff affects the incentive for FDI. For
Northern leader firms, an increase in τN reduces competition pressure in the Northern market because it
raises the lowest price that the follower firms in the South could charge after the tariff. This enables theNorthern leader firms to raise the price in the Northern market, which increases their profits. Meanwhile,the higher Northern tariff pushes up the duty-inclusive price of goods produced by multinational firms.
Accordingly, it reduces the demand for the goods and consequently, the profits of multinational firms.
Because the increased profits of a Northern leader firm and the decreased profits of a multinational firm
decrease the incentive for FDI, the relative wage of Northern to Southern labor must become higher torestore equilibrium.
Given that the higher Northern wage leads to a higher cost of R&D, an increase in τN negatively
affects the incentive for R&D and the innovation rate I for a given nS from the R&D equilibrium condition (31). This means that an increase in τN shifts the R&D curve downward. However, the LC curve(28) does not change because the Northern leader firm, the multinational firm, and an imitator change
the labor input proportionately in response to the tariff increase, as shown in (24) and (25). As a result,an increased Northern tariff lowers both nS and I, as in Figure 2. In addition, from (30), it raises therelative wage of Northern labor to Southern labor through both the direct effect discussed above and an
indirect effect through the decrease in nS . These results are summarized as follows.
Proposition 1. A unilateral tariff increase by the North reduces innovation and FDI from the North to
the South, although it also raises the relative wage of Northern to Southern labor.
Next, we analyze the effects of a unilateral tariff increase by the South. As the Southern tariffincreases, the Northern leader firms need to lower their pre-tariff export prices to the South, pN S . GivennS , this reduces the Northern leaders’ profits, and consequently, increases the incentive for FDI. As
shown in (30), the Northern wage wN must decrease to restore equilibrium as τS increases. Because thedecreased Northern wage reduces the cost of R&D, a higher τS positively affects the incentive for R&D
and innovation rate I for a given nS from the R&D equilibrium condition (31). Therefore, the R&D
curve shifts upward. However, the LC curve does not change because an increase in τS does not affect
15

I
LC
E
E’

R&D
1

0

nS

Figure 2: The effect of a unilateral tariff increase by the Norththe labor input for production. As a result, a tariff increase by the South increases both nS and I, asin Figure 3. In addition, equation (30) shows that it reduces wN through both the direct effect and theindirect effect through the increase in nS .
Proposition 2. A unilateral tariff increase by the South promotes innovation and FDI from the North to
the South. Moreover, it raises the relative wage of Southern to Northern labor.
With a North–South innovation-imitation model not including FDI, Grieben and Şener (2009) concluded that a unilateral reduction of the Southern (Northern) import tariff has no effect on innovationin their basic model, but decreases (increases) innovation rate in their extended model with a perfectlycompetitive low-tech sector in the South. Our results for Propositions 1 and 2 on innovation are similar
to the results of their extended model, but different to those of their basic model.

4.2 Tariff increases by both countries
In the former section, we concluded that a tariff increase by the South promotes innovation and FDI.
Nevertheless, this does not necessarily imply that such a policy change is favorable. This is because atariff increase by the South may result in a retaliatory tariff by the North, which has a negative effect oninnovation and FDI. Next, we discuss the effects of simultaneous tariff increases by the North and South.
In Section 4.2, we assume that the initial tariff rate in the North is not higher than that in the South; that

16

I
LC
E’
E
R&D

1

0

nS

Figure 3: The effect of a unilateral tariff increase by the Southis, τN = τ and τS = τ + τ̄ where τ ≥ 0 and τ̄ ≥ 0.13 Under this setting, we analyze the effects of anincrease in τ .
We interpret the effects of tariff increases by both countries as the combined effects of unilateral
tariff increases by the North and South. Similarly to a unilateral tariff increase, neither the Northern nor
Southern tariffs affect the LC curve from (28). Meanwhile, (31) shows that the tariff increases by bothcountries affect the R&D curve only through the change in wN . If an increase in τ raises wN for a given
value of nS , it moves the R&D curve downward and vice versa. Partially differentiating (30) with respectto τ , we have∂wN=
∂τ nS =given

λ − 2nS [(1 − m + mλ) − (1 − m)(λ − 1)τS ](1 + τS nS )2+(1 − m)(λ − 1)τS2 n2S + (τS − τN )(1 − m + mλ)nS (1 − nS ).
(1 + τS nS )2(32)

This equation shows that given nS , an increase in τ raises wN if (i) τS ≥ (1−m+mλ)/[(1−m)(λ−1)];or (ii) τS < (1 − m + mλ)/[(1 − m)(λ − 1)] and nS < λ/{2[(1 − m + mλ) − (1 − m)(λ − 1)τS ]} ≡ n̂S .
Consequently, if the R&D and LC curves intersect at a value of nS lower than n̂S , an increase in τ moves
the R&D curve downward around the intersection of the two curves, and the equilibrium to the lower
left along the LC curve. This implies, in that case, that simultaneous tariff increases by both countries
reduce the equilibrium values of I and nS . This is the case if the following condition is satisfied.
13

In fact, according to the World Bank Open Data (https://data.worldbank.org/), the weighted mean tariff rate applied in 2017

was 4.28% in “low & middle income” countries, which was higher than the 2.02% in “high income” countries.

17

Proposition 3. Suppose that the initial tariff rate in the North is not higher than that in the South: τN ≤
τS . Then, simultaneous tariff increases by the North and South to the same degree reduce innovation and
FDI if LN (t)/[aN X(t)] + ρ − Ẋ(t)/X(t) > LS (t)/[(1 − m + mλ)aN X(t)].
Proof. See Appendix B.
Proposition 3 implies that the tariff increases by both countries tend to be detrimental to innovation
and FDI if the share of population size is relatively large in the North and small in the South. In such acase, the negative effects of an increase in the Northern tariff tend to dominate the positive effects of anincrease in the Southern tariff. However, the negative effects on innovation and FDI are quantitatively
smaller than for a unilateral tariff increase by the North. This is because the increase in the Southerntariff works towards alleviating the negative effects.
Using the proof of Proposition 3, we also have the following result.
Corollary. Suppose that the initial tariff rate is zero in the North and South: τN = τS = 0. Then,simultaneous tariff increases by the North and South to the same degree reduce innovation and FDI if
and only if LN (t)/[aN X(t)] + ρ − Ẋ(t)/X(t) > LS (t)/[(1 − m + mλ)aN X(t)].
Proof. See Appendix B.
Unlike Proposition 3, the condition in this corollary is necessary and sufficient for the result, as it
implies a symmetric tariff rate between the two countries that maximizes innovation and FDI is not zero,but strictly positive if the condition is not satisfied. Furthermore, the condition is more restrictive whenm is small through strong IPR protection in the South. This is because a smaller m results in a larger
equilibrium value of nS , as discussed in Section 4.3. As nS is large, the simultaneous tariff increases byboth countries tend to reduce wN from (32) and move the R&D curve upward around the equilibrium,
which increases innovation and FDI.

4.3 A change in IPR protection in the South
Next, we analyze how the imitation rate m influences innovation and FDI to consider the effects ofstrengthening IPR protection in the South.
Equation (28) shows that a decrease in m rotates the LC curve counterclockwise around the point(1, LN (t)/aN X(t)) on the nS -I plane. Given nS , a lower imitation rate decreases the number of imitatedgoods with larger production volumes, and thus decreases labor demand in the South. The smaller labordemand lowers the wage of the South compared with the world aggregate expenditure, 1/E(t), in the
18

I
LC

E’
E
R&D

1

0

nS

Figure 4: The effect of a decrease in the imitation rate

Southern labor market. Because this pushes down the marginal cost of follower firms if they produce
in the South, the Northern leader firms set a lower price and expand production, which increases labordemand in the North. As a result, for a given value of nS , a lower imitation rate decreases the quantityof labor available for R&D in the North and negatively affects the innovation rate I.
Equations (30) and (31) imply that a decrease in m shifts the R&D curve upward for two reasons.
First, a lower imitation rate increases the return from successful R&D. This is because it increases the
expected profits of a multinational firm through (i) raising the probability of earning profits and (ii)reducing the marginal cost of production (the wage of the South) compared with the world aggregate
expenditure 1/E(t). Second, a lower imitation rate has a negative effect on the Northern wage wN and
thereby the cost of R&D. This is because it stimulates production transfer to the South through increasing
the expected profits of a multinational firm. Partially differentiating (30) with respect to m verifies thiseffect:[]∂wN
1 + τN (1 − nS )= (λ − 1) −τN +≥ 0,∂m nS =given
1 + τS n S(33)

where the inequality holds because τN ≤ 1/(1 + τS nS ) must be satisfied to ensure nonnegative profits
of Northern leader firms in the Southern market, as shown in Appendix B. Through the abovementioned
effects, a lower m positively affects the incentive for R&D and the innovation rate I for a given valueof nS . In equation (31), the first effect is represented by a decrease in m on the LHS, while the second
effect is represented by a decrease in wN on the RHS, both of which show that a lower m must increase
I, given nS .
19

Figure 4 depicts the effects of a decrease in m on the equilibrium. The intersection of the LC andR&D curves moves from point E to point E’. Figure 4 shows that a lower m unambiguously increases
the equilibrium value of nS . Because (30) shows that wN is decreasing with nS , a decreased m lowers wN through both the direct effect represented by (33) and the indirect effect by the increase in nS .
Meanwhile, Figure 4 does not indicate whether a lower m increases the equilibrium value of I because
it includes both the negative effect through tightening the labor constraint in the North and the positive effect through improving the incentive for R&D. On this point, we conclude as follows using total
differentiation.
Proposition 4. A decrease in the imitation rate through strengthening IPR protection in the South (i)promotes innovation, (ii) increases FDI, and (iii) decreases the relative wage of Northern to Southernlabor.
Proof. See Appendix B.
Note that Proposition 4 holds regardless of whether the tariff in both countries is zero or positive, andimplies that tariffs do not qualitatively change the effects of strengthening IPR protection in the Southon innovation and FDI. In addition, the result for Proposition 4 is consistent with those for Gustafsson
and Segerstrom (2011), Lai (1998), and Tanaka and Iwaisako (2014) using North–South innovation–FDI
models with exogenous imitation not including tariffs.

5

Welfare analysis

In this section, we examine how unilateral tariff increases by the North and South impact their own
welfare. To this end, we first derive the instantaneous utility of a household in each country. From (1),the instantaneous utility can be decomposed into two parts, utility from quality and utility from quantity,
as follows:∫ 1
log ui (t) =

∫ 1
log λ

J(ω,t)dω +0log di (ω, t)dω,

(34)

0

where J(ω, t) is the generation number of the state-of-the-art quality of good ω at time t, and di (ω, t)denotes the demand for that good. Hereafter, we let log Q(t) and log Di denote the first and second termsof (34), respectively. The growth rate of Q(t) is given byd log Q(t)= (log λ)I.
dt

20

(35)

Meanwhile, substituting (3) and the prices into log Di yields the utility from quantity as follows:∫ 1
log DN =log dN (ω, t)dω
0

EN
EN
EN
+ (1 − m)nS log+ mnS log
λ(1 + τN )λ(1 + τN )1 + τN= log EN − log(1 + τN ) − (1 − mnS ) log λ,= nN log(36)∫ 1
log DS =log dS (ω, t)dω0

ESES
+ (1 − m)nS log+ mnS log ESλλ= log ES − (1 − mnS ) log λ.

= nN log(37)

By rewriting (22), the Northern expenditure in (36) is expressed as()
LS (t)
λ1
EN = (1 + τN )− ES .
LN (t) 1 − m + mλ nS(38)

The Southern expenditure in (37) and (38) is derived from (26) as follows:ES =1
1 + τS.
τS =1 − nN 1+τS
1 + τS n S(39)

Substituting (35) into (34), we have the lifetime utility of a household in country i as
∫ ∞Ui =e−(ρ−gL )t log ui (t)dt∫0 ∞=e−(ρ−gL )t [log Q(0) + (log λ)It + log Di ] dt
0[]1(log λ)I=log Q(0) ++ log Di ,
ρ − gLρ − gL
where log Di is given by (36) - (39). From this lifetime utility, we next derive the welfare change from atariff increase in each country.

5.1

Welfare effect of a Northern tariff increase

By differentiating the Northern household’s lifetime utility with respect to τN , we obtain the Northern
welfare change from a marginal increase in its tariff rate as follows:
∂UN
1 ∂EN
1 [
log λ ∂I+=∂τN
ρ − gLρ − gL ∂τNE ∂τ| N {z N}{z}|income effectinnovation-impeding effect
(+)
(–)
]
∂nS
+
(log λ)m
.
∂τN
|
{z
}
competition-weakening effect
(–)
21

1
−
1+τ
| {z N}
price-raising effect
(–)
(40)

Equation (40) shows that the total welfare effect of a Northern tariff increase (an increase in τN ) canbe decomposed into the following four parts. First, the increased Northern tariff impedes innovation,
as shown in Proposition 1, and thus reduces welfare. We refer to this welfare effect as the innovationimpeding effect. Second, the increased Northern tariff increases Northern income as we later demonstrate. We refer to this welfare effect as the income effect. Third, the Northern tariff increase raises prices
in the North and thus reduces welfare. We refer to this welfare effect as the price-raising effect. Finally,
the Northern tariff increase impedes FDI and consequently decreases the number of goods produced by
the Southern imitators mnS . The price of the imitated goods is lower than that of the goods produced by
Northern and multinational firms, so that a decrease in the number of the imitated goods reduces welfare.
We refer to this welfare effect as the competition-weakening effect.
E

1 1+τNThe sum of the income effect and price-raising effect is equal to ρ−g
L EN∂ 1+τNN∂τN. From (38) and(39), differentiating EN /(1 + τN ) with respect to τN yields
EN
∂ 1+τ
N

∂τN

[
]
LS (t)
λ
τS
1 ∂nS
2
=
−
+ (ES nS )> 0,
LN (t)
1 − m + mλ
1 + τS nS 2 ∂τN(41)

where the inequality holds because λ/(1 − m + mλ) > 1, ES nS = (1 + τS )nS /(1 + τS nS ) < 1, and∂nS /∂τN < 0 from Proposition 1. As the price-raising effect is necessarily negative, this means that theincome effect is necessarily positive. This is because the increased Northern tariff raises Northern tariff
revenue, the Northern wage, as shown in Proposition 1, and the value of the holding stocks proportionate
to the Northern wage from (17) and (18).
Although both positive and negative welfare effects exist, the total welfare effect is negative under a certain parameter condition because the sum of the innovation-impeding and price-raising effects
surpasses the positive income effect. This result is summarized as follows.
log λProposition 5. If ρ−g
L

[

Ẋ(t)
LN (t)aN X(t) + ρ − X(t)]λ> 1−m+mλ(1 + τS ), a unilateral tariff increase by theNorth worsens the North’s welfare.
Proof. See Appendix B.
Note that the condition in Proposition 5 does not depend on the value of τN as it implies that thewelfare-maximizing tariff rate for the North is zero if the condition is satisfied.
From the condition in Proposition 5, the welfare effect tends to be negative when Northern labor islarger, the imitation rate in the South is higher, and the Southern tariff rate is lower. A tariff increase by
the North has a positive effect on the North’s welfare because it shifts production from the South to theNorth and raises the Northern wage. However, this positive welfare effect is weak under large Northern22

labor, a high imitation rate in the South, and a low Southern tariff rate because the number of industries
producing in the South is relatively smaller in these cases. That is why the total welfare effects of theNorthern tariff tend to be negative.

5.2

Welfare effect of a Southern tariff increase

Next, by differentiating the Southern household’s lifetime utility with respect to τS , we obtain the Southern welfare change from a marginal increase in its tariff rate as follows:
∂US
1 [
=
∂τS
ρ − gLlog λ ∂I
ρ − gL ∂τS|{z}innovation-enhancing effect
(+)

+

1 ∂ES
+
ES ∂τS| {z }
income effect(+) or (-)]∂nS(log λ)m. (42)∂τS|{z}
competition-strengthening effect(+)

Equation (42) shows that the total welfare effect of a Southern tariff increase (an increase in τS ) canbe decomposed into the following three parts. First, the Southern tariff increase enhances innovation,
as shown in Proposition 2, and thus raises welfare. We refer to this welfare effect as the innovationenhancing effect. Second, the Southern tariff increase may increase or decrease Southern expenditure
because it affects the transfer payment from the tariff revenue of the Southern government. We referto this effect as the income effect. From equation (39), if the elasticity of nN with respect to τS islower than 1/(1 + τS ), a higher Southern tariff increases tariff revenue, and thus has a positive effecton Southern income and welfare.14 Finally, the Southern tariff increase promotes FDI and increases thenumber of the goods produced by Southern imitators mnS . The increase in imitated goods improves
welfare because they are cheaper than the goods produced by Northern leaders and multinationals. We
refer to this welfare effect as the competition-strengthening effect.
As shown in Appendix B, under a certain parameter condition, the positive welfare effects surpassthe negative part of the income effect, which is the negative effect on the tariff revenue from the decreasein Northern firms. Accordingly, the total welfare effect of an increase in the Southern tariff is positive inthat case, as in the following proposition.
[]
LS (t)LS (t)log λ
log λProposition 6. If ρ−g
+
(log
λ)m
+
−
1
τS > 0, a unilateral
ρ−gL (1−m+mλ)aN X(t)L (1−m+mλ)aN X(t)tariff increase by the South improves the South’s welfare.
Proof. See Appendix B.
14

Differentiating (39) with respect to τS , we have(ES )2 nN∂ES=∂τS
1 + τS[()]
1τS ∂nN− −.
1 + τSnN ∂τS

Therefore, ∂ES /∂τS > 0 if −(τS /nN )(∂nN /∂τS ) < 1/(1 + τS ).

23

LS (t)log λProposition 6 shows that a tariff increase by the South improves the South’s welfare if ρ−g≥
L (1−m+mλ)aN X(t)LS (t)log λ
1. Even if ρ−g< 1, it improves the South’s welfare as long as the Southern tariff is
L (1−m+mλ)aN X(t)[][]−1
LS (t)LS (t)log λ
log λso small as to satisfy τS < ρ−g
+
(log
λ)m
1
−
(> 0).
ρ−gL (1−m+mλ)aN X(t)L (1−m+mλ)aN X(t)That is, the condition in Proposition 6 is necessarily satisfied if τS is zero and this implies that thewelfare-maximizing tariff rate for the South is strictly positive. Thus, the result shows that, in contrast to
the North, the South has a stronger incentive to raise tariffs.
The results of the welfare analysis in this section have implications for the optimal tariff literature.
Many theoretical studies have concluded that larger countries tend to set higher tariffs, but the opposite
is actually observed.15 As discussed, our model shows that the optimal tariff for the North may be zero,
whereas that for the South is necessarily positive. The difference is due to the dissimilar effects of tariffs
in the two countries on innovation, FDI, and prices. In contrast to extant theoretical studies, our welfare
analysis can then explain the observation that larger countries tend to set lower tariffs.

6 Concluding remarks
Using a North–South quality ladder model, this paper investigated how import tariffs affect innovation,
FDI, and welfare under imperfect IPR protection in a developing country. The conclusion is that a
unilateral tariff increase by a developed country reduces innovation and FDI, while that by a developing
country promotes innovation and FDI. In addition, because of the decrease in innovation, a unilateral
tariff increase by the developed country tends to worsen its welfare if the country is large, the tariff
rate of the developing country is low, and the protection of IPR in the developing country is weak. By
contrast, a unilateral tariff increase by the developing country improves its welfare if the initial tariff rateis sufficiently low, which implies that the optimal tariff rate for the developing country is strictly positive.
The effects of simultaneous tariff increases by both countries then depend on the relative country size andthe degree of IPR protection in the developing country: they tend to reduce innovation and FDI when therelative size of the developed country to the developing country is large and the protection of IPR in thedeveloping country is weak. In addition, we showed that strengthening IPR protection in a developing
country promotes innovation and FDI even when tariffs are imposed, which is the same as in the case of
free trade.
Possible directions of further research include extensions to address the following issues. First, we
15

The exception is Naito (2019), who obtained a result consistent with the actual tendency using a two-country growthmodel. However, the engine of growth in that model is not R&D, but rather capital accumulation by both countries, nor is it aNorth–South model.

24

assumed that the process of FDI is costless for analytical tractability. This could be justified if production
startup costs are small or at least similar between the two countries. However, it would be useful to examine whether our results continue to hold, even with a cost of FDI. Second, we assumed harmonizationin the patent system, such as breadth of coverage, between the developed and developing countries. To
investigate the effects of tariffs when the breadth of patent coverage differs between the two countries
would also be interesting.16 Third, we did not consider the differences between skilled and unskilledlabor. In reality, the R&D sector is likely to require more skilled labor than the production sector. To
examine how a tariff increase affects innovation, FDI, and wage gap in such a setup would also be important. Moreover, we could explore how increases in skilled and unskilled labor affect innovation andskill premium, as in Chu, Cozzi, and Furukawa (2015), or endogenize skill acquisition, as in Dinopoulos
and Segerstrom (1999a) and Cozzi and Impullitti (2010). Fourth, we ruled out R&D activities by thedeveloping country in inventing a new product. In fact, R&D spending has recently increased in a few
emerging countries such as China. International trade between a developed country and a “developing”
country that undertakes both imitation and R&D may then be an intermediate case between North–South
trade and North–North trade. It may be interesting to investigate how tariffs affect innovation and technology transfer in this case. As all of these extensions are worth examining, but beyond the scope of thispaper, we defer them to future research.

Appendix A

Dynamics of the model

In this appendix, we show that this model does not have transitional dynamics.
Rewriting the Southern labor market-clearing condition (25) and applying nS (t) ≡ 1 − nN (t), we
obtain([1 − nN (t)]ϕ+1−ϕ1 + τN)E(t)LS (t)=.
λ1 − m + mλ(43)

Note that ϕ must be constant on the equilibrium path because EN (t) and ES (t) grow at the same ratefrom (4). Adding both sides of this equation to those of (24) and rewriting, we have the following
equilibrium innovation rate:
I(t) =

16

LN (t)LS (t)
1+−aN X(t) (1 − m + mλ)aN X(t) aN λ

(

ϕ
+1−ϕ
1 + τN

)

E(t)
.
X(t)

(44)

Iwaisako, Tanaka, and Futagami (2010) considered this situation and examined how extending patent breadth in the Southaffects innovation and FDI with no tariff.

25

For both a Northern leader firm and a multinational firm to exist at each time point, (18) must be
satisfied over time. Differentiating both sides of (18) with respect to t yields v̇N (t) = v̇F (t). Substituting
this and (18) into (19) and (20), we have πN (t) = (1 − m)πF (t). Therefore, from (9), (14), and thedefinitions of ϕ and E(t), we obtain
[
]
[
]
(
)[
]
wN (t)
1wN (t)
1ϕE(t)1−ϕE(t) +−
(1 − ϕ)E(t) = (1 − m) 1 −+ (1 − ϕ)E(t) .
λ(1 + τN )1 + τSλλ
1 + τN
Rewriting this equation, we have the Northern wage (compared with the Southern wage) as follows.
)(1−ϕ
λ ϕ + 1+τS
wN (t) = ϕ− (1 − m)(λ − 1)
1+τN + (1 − ϕ)Hence, the Northern wage must be constant over time on the equilibrium path because ϕ is constant. As
the Northern wage wN is constant, (17) and (18) imply that the market values of a Northern leader firm
and a multinational firm, vN (t) and vF (t), grow at the same rate as X(t). Accordingly, from (9), (17),and (19), the equilibrium interest rate is expressed as follows.
{[
]
(
)
}
wN
1
wN
E(t) Ẋ(t)
1
1−
ϕ+
−
(1 − ϕ)+− I(t)
r(t) =
wN a Nλ(1 + τN )1 + τSλX(t) X(t)Substituting (44) into this equation, we have()
1
1 − ϕ E(t) Ẋ(t)LN (t)LS (t)
r(t) =
ϕ++−−.
wN a N
1 + τS X(t) X(t) aN X(t) (1 − m + mλ)aN X(t)(45)

Next, we compute the equation of motion with respect to E(t)/X(t). Taking the logarithm of
E(t)/X(t) and differentiating it with respect to t yields
˙[E(t)/X(t)]E(t)/X(t)=Ė(t) Ẋ(t)−E(t) X(t)=ĖN (t) EN (t)LN (t) L̇N (t) EN (t)LN (t)+EN (t)E(t)LN (t)E(t)ĖS (t) ES (t)LS (t) L̇S (t) ES (t)LS (t) Ẋ(t)+−ES (t)E( t)
LS (t)
E(t)
X(t)
(
)
EN (t)LN (t) ES (t)LS (t)Ẋ(t)= [r(t) − ρ + gL ]
+
−
E(t)
E(t)
X(t)
+

= r(t) − ρ + gL −

Ẋ(t)
,
X(t)

(46)

where the second and fourth equalities use the definition of E(t) and the third equality uses the Euler
equation (4). Substituting (45) into (46), we obtain()
˙[E(t)/X(t)]1
LN (t)LS (t)
1 − ϕ E(t)=−−− (ρ − gL ). (47)ϕ+
E(t)/X(t)wN aN
1 + τS X(t) aN X(t) (1 − m + mλ)aN X(t)26

As Ẋ(t)/X(t) = gL , LN (t)/X(t) and LS (t)/X(t) are constant. Thus, (47) has a unique interior steadystate that is unstable. In the equilibrium, E(t)/X(t) must jump to this steady-state value at the initialtime point and then become constant because E(t)/X(t) is jumpable. Otherwise, either (43) or (44)would be violated at a certain finite time point. This result implies that r(t) = ρ for all t from (46).
Then, Ei (t) must be constant over time from the Euler equation (4). Also, (25) and (44) show that nS (t),nN (t), and I(t) must be constant over time because E(t), X(t), LN (t), and LS (t) grow at the same rateon the equilibrium path.
Therefore, we conclude that the equilibrium path of this model does not have a transitional process
and immediately jumps to the steady state.

Appendix B

Proofs

Proof of Proposition 3
For the proof, it is sufficient to show that the LC curve is above the R&D curve at n̂S because the former
is upward sloping and the latter is downward sloping. From (28), the LC curve is given by()
LN (t)
1
1
LS (t)I=−−1≡ f (nS ).
aN X(t)nS
1 − m + mλ aN X(t)Substituting (30) into (31), we have the R&D curve as follows:[](1 − m)(λ − 1) LS (t)
1
Ẋ(t)
I =
− ρ−
1 − m + mλ aN X(t) nS wN
X(t)=(1 − m)(λ − 1) LS (t)1 + τS n S
1 − m + mλ aN X(t) nS {τN λ − τN [(1 − m + mλ) − (1 − m)(λ − 1)τS ] nS + (1 − m + mλ)}
[
]
Ẋ(t)
− ρ−
X(t)

≡ h(nS ).

27

Therefore, the following inequality holds from the definition of n̂S :f (n̂S ) − h(n̂S )()
LN (t)
1
1
LS (t)=−−1aN X(t)n̂S
1 − m + mλ aN X(t)
−
=

=

≥

1 + τS n̂SẊ(t)(1 − m)(λ − 1) LS (t)+ρ−
1 − m + mλ aN X(t) n̂S [(τN λ/2) + (1 − m + mλ)]X(t)LN (t)Ẋ(t)+ρ−aN X(t)X(t)LS (t) (λ/n̂S )[(τN /2) + 1] − [(τN λ/2) + (1 − m + mλ)] + (1 − m)(λ − 1)τS
1−
1 − m + mλ aN X(t)(τN λ/2) + (1 − m + mλ)
LN (t)Ẋ(t)+ρ−aN X(t)X(t)1
LS (t) (τN λ/2) + (1 − m + mλ) − (1 − m)(λ − 1)(τN + τN τS + τS )−1 − m + mλ aN X(t)(τN λ/2) + (1 − m + mλ)Ẋ(t)1LS (t)
LN (t)+ρ−−.
aN X(t)X(t) 1 − m + mλ aN X(t)(48)

This inequality shows that, if LN (t)/[aN X(t)] + ρ − Ẋ(t)/X(t) > LS (t)/[(1 − m + mλ)aN X(t)],then f (n̂S ) > h(n̂S ) and thus the LC curve is above the R&D curve at n̂S .

Proof of Corollary
As discussed in the main text, the condition under which the simultaneous tariff increases reduce innovation and FDI is equivalent to the condition to satisfy ∂wN /∂τ n =given > 0. Equation (32) implies thatS

under τS = τN = 0, ∂wN /∂τ n =given > 0 if and only if nS < λ/[2(1 − m + mλ)] = n̂S . Because (48)Sholds with equality if τS = τN = 0, the LC curve is above the R&D curve at n̂S and the equilibrium valueof nS is lower than n̂S if and only if LN (t)/[aN X(t)]+ρ− Ẋ(t)/X(t) > LS (t)/[(1−m+mλ)aN X(t)].

Proof of Proposition 4
The proofs on (ii) and (iii) are provided in the text and Figure 4. In this appendix, we show (i).
For the proof, we first show that parameters need to satisfy 1/(1 + τS nS ) ≥ τN . In the equilibrium,
the profit a Northern leader firm can obtain from a sale in the South must be nonnegative. Therefore,

28

from (8) and (30), the following condition must be satisfied:λ− wN
1 + τSλ1 + τN (1 − nS )− τN (1 − m)(λ − 1) −(1 − m + mλ)
1 + τS
1 + τS nS(1 − m)(λ − 1)(1 + τS ) − τS λ(1 − nS )
τN (1 − nS )=− τN (1 − m)(λ − 1) −(1 − m + mλ)(1 + τS )(1 + τS nS )1 + τS n S)([]
1 − nS1τS λ= (1 − m)(λ − 1)
− τN −+ τN (1 − m + mλ)
1 + τS nS
1 + τS n S 1 + τS
≥ 0.

=

For this condition to be satisfied, we need to assume1≥ τN .
1 + τS nS(49)

Next, combining (28) and (31) yields(1 − m)(λ − 1) LS (t)1 − m + mλ aN X(t)
{[
]
}
Ẋ(t)
LN (t)1
LS (t)LS (t)
1
= wN
ρ−
+
+
nS −
.
X(t) aN X(t) 1 − m + mλ aN X(t)1 − m + mλ aN X(t)Totally differentiating this equation, we have{[]}
(1 − m)(λ − 1) LS (t) 1 ∂wN(1 − m)(λ − 1) LS (t)1
LS (t)
1++wN
dnS
1 − m + mλ aN X(t) wN ∂nS
1 − m + mλ aN X(t) 1 − m + mλ aN X(t)nS{}
(1 − m)(λ − 1) LS (t) 1 ∂wN
λ−1
LS (t)++[wN (1 − nS ) + λ] dm = 0. (50)
1 − m + mλ aN X(t) wN ∂m(1 − m + mλ)2 aN X(t)From (30), we obtain[]∂wN
1 + τN (1 − nS )= (λ − 1) −τN +≥ 0,∂m
1 + τS n S

(51)

∂wN
τN + τS (1 + τN )=−(1 − m + mλ) ≤ 0,∂nS
(1 + τS nS )2

(52)

∂wN
1 + τN (1 − nS ) − τN nS (1 + τS nS )nS + wN =(1 − m + mλ) + τN (1 − m)(λ − 1) > 0, (53)∂nS
(1 + τS nS )2
where the first and the third inequalities use (49). Therefore, (50), (51), and (53) imply thatλ−1N
(1 − m)(λ − 1) w1N ∂w
dnS∂m + 1−m+mλ [wN (1 − nS ) + λ]
(
)
=−
< 0.
∂wN
dm
(1 − m)(λ − 1) 1
n +w
+ wN
nS w N

∂nS

29

S

N

nS

Differentiating (28) with respect to m, we obtain[]dI
1
LS (t)1dnS
λ−1=+ nS (1 − nS )dm
1 − m + mλ aN X(t) (nS )2 dm
1 − m + mλ= −LS (t)λ−111−m+mλ aN X(t) (nS )2()
N(1 − m)(λ − 1) nS1wN ∂w
n+w+ wnNSS
N∂nS[]
1 − m ∂wN
(1 − m)(λ − 1)
(1 − m)(λ − 1) 1 ∂wN×+1+nS − nS (1 − nS )
< 0,
wN ∂m
1 − m + mλ
1 − m + mλ wN ∂nS

where the last inequality uses (51) - (53). Thus, a decrease in m through strengthening IPR protection in
the South increases the innovation rate.

Proof of Proposition 5
From (28), (38), (39), and (41), we have
LS (t)
1 ∂nS
∂I
=
,
∂τN
(1 − m + mλ)aN X(t) nS 2 ∂τNE

1 + τN ∂ 1+τNN
EN
∂τN

=

(54)

τS
λ
2
1 ∂nS
1−m+mλ − 1+τS (ES nS )−nS 2
λnS ∂τN
1−m+mλ − ES nS= −λ1−m+mλ −()
1 − 1+τ1S nS (ES nS )λ1−m+mλ − ES nS= −A(nS )

nS

1 ∂nS
nS 2 ∂τN

1 ∂nS,
nS 2 ∂τN

(55)

where
1
1+τS nS (ES nS )A(nS ) ≡ nS + nS.
λ
1−m+mλ − ES nS(56)

We can show that the second term of (56) is an increasing function of nS as follows:][1
1∂(ES nS )∂τS
1 ∂(ES nS )1
1+τS nS (ES nS )=log nS
−
+
+
λ
λ
∂nS
nS
1 + τS n S
ES nS ∂nS
∂nS
1−m+mλ − ES nS
1−m+mλ − ES nS
λ=1∂(ES nS ))( 1−m+mλ+λnS (1 + τS nS ) E n
∂nS
−E n
S S

1−m+mλS S> 0,where the inequality holds because λ/(1 − m + mλ) > 1 > ES nS and ∂(ES nS )/∂nS > 0 from (39).
Therefore, A(·) is an increasing function of nS . Because A(nS ) depends on τN only through nS and30

ES , and ∂nS /∂τN < 0 from Proposition 1, A(·) is a decreasing function of τN . From (28), (30) and(31), nS equalsnS ∗ ≡
−1
LN (t)
Ẋ(t)
+
ρ
−
λ
(1
−
m)(λ
−
1)
X(t)
 aN X(t)+1−τS LS (t)
1 − m + mλ
1 − m + mλ(57)

(1−m+mλ)aN X(t)at τN = 0, so that nS ≤ nS ∗ and A(nS ) ≤ A(nS ∗ ) for any τN ≥ 0.
Substituting (54) and (55) into (40) yields the Northern welfare change due to the marginal increase
in its tariff as follows:
∂UN
∂τN

=
=

[
]
log λ ∂I
1
1 ∂EN
1
∂nS
+
−
+ (log λ)m
ρ − gL ρ − gL ∂τNEN ∂τN
1 + τN∂τN
1
1 ∂nS
F (nS ),
ρ − gL nS 2 ∂τN

where
F (nS ) ≡log λLS (t)− A(nS ) + m(log λ)n2S .
ρ − gL (1 − m + mλ)aN X(t)(58)

Next, we derive a sufficient condition that the first term of F (nS ) surpasses the second term A(nS ),which ensures ∂UN /∂τN < 0. To this end, we rewrite A(nS ) as follows:[]1
1+τS nS (ES nS )A(nS ) = 1 +nS
λ1−m+mλ − ES nS
[
]
τS nS
λ
(E
n
)
−
S
S
1−m+mλ
1+τS nS=nS
λ1−m+mλ − ES nS
]
[
τS nS
1+τS
λ
1−m+mλ − 1+τS nS 1+τS nS nS
nS
=
1+τS
λ
1−m+mλ − 1+τS nS nS
[
]
λ
2
2
1−m+mλ (1 + τS nS ) − τS (1 + τS )nS=nS ,λ2 − (1 + τ )(1 + τ n )n
(1
+
τ
n
)
S
S
S
S
S
S
1−m+mλwhere the third equality uses (39). For any τN ≥ 0,
A(nS ) ≤ A(nS ∗ )
=
≤
=

λ
∗ 2
∗
∗3
1−m+mλ (1 + τS nS ) nS − τS (1 + τS )nS
λ
∗ 2
∗
∗
1−m+mλ (1 + τS nS ) − (1 + τS )(1 + τS nS )nS
λ∗ 2∗
1−m+mλ (1 + τS nS ) nS
λ
∗ 2
∗
∗
1−m+mλ (1 + τS nS ) − (1 + τS )(1 + τS nS )nS
λ∗∗1−m+mλ (1 + τS nS )nS,
λ∗∗1−m+mλ (1 + τS nS ) − (1 + τS )nS

31

(59)

where nS ∗ is given by (57). From (57), the denominator of the RHS of (59) is rewritten as follows:=λ(1 + τS nS ∗ ) − (1 + τS )nS ∗1 − m + mλLS (t){}λλ(1−m+mλ)aN X(t)[]1 + τSLS (t)
1 − m + mλ
1 − m + mλ LN (t) + ρ − Ẋ(t) +1 − (1−m)(λ−1) τaN X(t)X(t)1−m+mλ(1−m+mλ)aN X(t)S

LS (t)

λ(1−m+mλ)aN X(t)[]−(1 + τS )
L
(t)
Ẋ(t)
LS (t)
1 − m + mλ N+ρ−+
1 − (1−m)(λ−1) τaN X(t)X(t)1−m+mλ(1−m+mλ)aN X(t)S

LN (t)Ẋ(t)λaN X(t) + ρ − X(t)[]=(1−m)(λ−1)
LS (t)
1 − m + mλ LN (t) + ρ − Ẋ(t) +1−τS
1−m+mλaN X(t)X(t)(1−m+mλ)aN X(t)[][]−1
LN (t)Ẋ(t)LS (t)= nS ∗.
+ρ−aN X(t)X(t) (1 − m + mλ)aN X(t)Substituting this into the RHS of (59), we obtain
A(nS ) ≤

nS ∗

[

λ
∗
∗
1−m+mλ (1 + τS nS )nS
Ẋ(t)LN (t)aN X(t) + ρ − X(t)][LS (t)(1−m+mλ)aN X(t)]−1

LS (t)

λ(1−m+mλ)aN X(t)(1 + τS nS ∗ )LN (t)
1 − m + mλ+ ρ − Ẋ(t)=aN X(t)X(t)LS (t)(1−m+mλ)aN X(t)λ
(1 + τS ).
LN (t)
1 − m + mλ+ ρ − Ẋ(t)≤aN X(t)X(t)By applying this to (58), we have
LS (t)F (nS ) ≥=LS (t)λlog λ(1−m+mλ)aN X(t)−(1 + τS )+ m(log λ)n2SẊ(t)LN (t)
ρ − gL (1 − m + mλ)aN X(t) 1 − m + mλ+ρ−LS (t)(1−m+mλ)aN X(t)LN (t)Ẋ(t)aN X(t) + ρ − X(t)log λTherefore, if ρ−g
L

[

{

log λ
ρ − gL

LN (t)Ẋ(t)aN X(t) + ρ − X(t)

]

[

]

aN X(t)X(t)}LN (t)Ẋ(t)λ+ρ−(1 + τS ) + m(log λ)n2S .
−aN X(t)X(t)1 − m + mλλ
> 1−m+mλ(1 + τS ), the sum of the innovation-impeding andprice-raising effects dominates the income effect, and F (nS ) > 0. Because ∂nS /∂τN < 0 from[]
LN (t)
Ẋ(t)
log λ
+
ρ
−
>
Proposition 1, we conclude that ∂UN /∂τN < 0 for any τN ≥ 0 if ρ−gaN X(t)X(t)Lλ1−m+mλ (1 + τS ).

32

Proof of Proposition 6
Differentiating (28) and (39) with respect to τS , we derive the effect of the marginal increase in Southerntariff as follows:∂I
LS (t)
1 ∂nS
=
,
∂τS
(1 − m + mλ)aN X(t) nS 2 ∂τS∂ES(ES )2 nN=∂τS
1 + τS(1τS ∂nS−
1 + τSnN ∂τS).

Substituting these into (42) and rewriting, we obtain
[
(
)
]
∂US
B ∂nS
1
∂nS
1
ES nN
τS ∂nS+ (log λ)m
=
+
−
∂τS
ρ − gL nS 2 ∂τS
1 + τS 1 + τS
nN ∂τS
∂τS
{[
]
}
1
B
τS
∂nS
ES nN
=
−
+ (log λ)m+2
ρ − gLnS
1 + τS n S∂τS(1 + τS )2{}
B + BτS nS + [(log λ)m − τS ]nS 2 + (log λ)mτS nS 3 ∂nS
E S nN1+, (60)=
ρ − gLnS 2 (1 + τS nS )∂τS
(1 + τS )2
LS (t)
log λwhere B ≡ ρ−g> 0 and the second equality uses (39). If the coefficient of ∂nS /∂τS
L (1−m+mλ)aN X(t)in (60) is positive, ∂US /∂τS > 0 because ∂nS /∂τS > 0 from Proposition 2. A sufficient condition forthis is that the quadratic part G(nS ) ≡ B+BτS nS +[(log λ)m−τS ]nS 2 is positive. (i) If (log λ)m−τS ≥0, then G(nS ) > 0 for all nS ∈ (0, 1). (ii) If (log λ)m − τS < 0, the quadratic function G(·) is concave.
Therefore, if G(1) > 0, then G(nS ) is positive for all nS ∈ (0, 1) because G(0) > 0. The condition forG(1) > 0 is rewritten as B + (log λ)m + (B − 1)τS > 0.
In conclusion, ∂US /∂τS > 0 if B + (log λ)m + (B − 1)τS > 0.

33

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