Discussion Papers In Economics
And BusinessKnowledge spillovers, location of industry, andendogenous growth
Kyoko Hirose

Kazuhiro Yamamoto

Discussion Paper 05-15

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANKnowledge spillovers, location of industry, andendogenous growth
Kyoko Hirose

Kazuhiro Yamamoto

Discussion Paper 05-15

June 2005
この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。
Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANKnowledge spillovers, location of industry, andendogenous growth∗
Kyoko Hirose†

Kazuhiro Yamamoto‡
June 2005

Abstract
A Grossman–Helpman–Romer-type endogenous-growth model isdeveloped in this study. This model has two countries in which there
are knowledge spillovers that are partially local. Owing to thesespillovers, innovation cost in a particular country decreases as thenumber of firms locating in both that country and the other country
increases. If international knowledge spillovers are symmetric, innovation cost is lower in the country that has the larger market. However, if a small-market country can absorb the international knowledge
spillovers better than a large-market country, the innovation cost may
be lower in the small-market country. When the innovation cost islower in the country that has a large market, the growth rate increases with agglomeration, which is generated by a reduction in thetransportation costs. However, when the innovation cost is lower inthe country that has a small market, the growth rate decreases withthe reduction in the transportation costs.
Key words: knowledge spillovers, growth rate, transportation costs,
market scale
JEL classification: F43: O30: R12∗

We are grateful to Marcus Berliant, Masahisa Fujita, Koichi Futagami, Tatsuo Kinugasa, Takuo Imagawa, Tomoya Mori, Se-il Mun, Hisahiro Naito, Ryoji Ohdoi, Tatsuyoshi
Saijyo, and Akihisa Shibata for their helpful comments and suggestions. Of course, we areresponsible for any remaining errors.
†
Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka,
Osaka 560-0043, Japan. E-mail address: bg008hk@srv.econ.osaka-u.ac.jp‡
Corresponding author. Faculty of Economics, Osaka University, Address: Graduate
school of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. E-mail address: yamamoto@econ.osaka-u.ac.jp1

1

Introduction

In this paper, a Grossman–Helpman–Romer-type endogenous-growth model
is developed that has two countries. The activity of the innovation sector
results in an increased variety of differentiated goods. In our model, there
are asymmetric knowledge spillovers that are partially local between the twocountries. Given these knowledge spillovers, the innovation cost in a particular country decreases as number of firms locating in both that country
and the other country increases. We assume that intra-national spillovers
are complete but international spillovers are incomplete. In addition, we assume that international knowledge spillovers between the two countries are
not symmetric. 1
Coe and Helpman (1995) and Eaton and Kortum (1996) produced empirical studies about international knowledge spillovers in OECD countries andfound that these spillovers were not symmetric. Coe and Helpman (1995)
showed that, in developed nations, intra-national knowledge spillovers played
an important role in economic growth, although international knowledge
spillovers had only small effects on economic growth. They also found that,in developing countries, international knowledge spillovers from developed
countries played an important role in economic growth, as did intra-national
knowledge spillovers. These results suggest that international knowledge
spillovers between developed and developing countries are asymmetric. Thus,
we think it important to construct a model in which there are asymmetricinternational knowledge spillovers. In this context, we interpret asymmetry in international knowledge spillovers as the difference between developed
and developing countries in their ability to absorb international knowledge
spillovers. 2
Our model is closely related to those of Martin (1999) and Martin andOttaviano (1999), who constructed models in which international knowledge
spillovers are symmetric. When the shipping of differentiated goods incurs
transportation costs, more firms locate in the large-market country than in
the small-market country to economize on transportation costs. In addition,
1

Baldwin et al. (2001) and Fujita and Thisse (2003) assumed that knowledge spillovers
are partially local. However, in their models, knowledge spillovers between two countries
are symmetric.
2
Asymmetries in knowledge spillovers may be due to language. For example, Coe andHelpman (1995) found that, while Japan absorbed international spillovers from the U.S.,
there were no international knowledge spillovers from Japan to the U.S. The explanation
may be that most Japanese recognized English as the international language and tried
to learn it, but Americans did not learn Japanese. Consequently, Japan can absorb international knowledge spillovers from the U.S., but there are no international knowledge
spillovers from Japan to the U.S.

2

if international knowledge spillovers are symmetric, the innovation cost isalways lower in the country that has the larger income, since the country
that has the larger income has a larger market than the other country. Thisis because the larger the number of firms locating in a particular country, thelower the innovation cost in that country due to partially local knowledge
spillovers. Thus, when international knowledge spillovers are symmetric, thegrowth rate increases monotonically with agglomeration in the large-income
country.
However, this result may not be valid if international knowledge spillovers
are asymmetric. If the small-market country can absorb international knowledge spillovers better than the large-market country, the innovation costs
may be lower in the small-market country. In this case, the growth rate
decreases as the share of differentiated firms in the large-market country increases, since the innovation cost in the small-market country increases as
the share of differentiated firms in the large-market country increases. In
Martin (1999) and Martin and Ottaviano (1999), owing to symmetric international spillovers, there is always a trade-off between international income
equality and a high growth rate. This trade-off arises in our model for someof the parameter values even if international spillovers are not symmetric.
However, in our model, we show that there is a situation in which greater
income equality raises the growth rate. Hence, although our model is closely
related to those of Martin (1999) and Martin and Ottaviano (1999), it has
richer implications. 3
Given this framework, our model yields two patterns relating to the innovation sector’s location. 4 In one pattern, the innovation sector locatesin the large-income country. In the other pattern, the innovation sector locates in the small-income country. Which location pattern emerges depends
on the ability to absorb international knowledge spillovers, international income inequality, and transportation costs. In the former case, the growthrate increases as the transportation costs decrease and the international income inequality increases. In the latter case, the growth rate decreases as
the transportation costs decrease and the international income inequality decreases. It must be remembered that, as the transportation costs decrease,3

Martin (1999) focused on the relationship between intra-national transportation costs
and international transportation costs. However, in his model, there is always a trade-off
between international income equality and the growth rate.
4
Barro and Sala-i-Martin (1997) explained the movement of innovation activity between
countries by another mechanism. Their model showed that this movement occurs because
agents in a particular country who imitate others (by copying their innovations) find that
innovation is preferable to imitation when some parameters change owing to, for example,
changes in government policy.

3

agglomeration in the large-income country increases. Therefore, there is a
critical value of the transportation costs. Below this critical value, the former pattern applies. Above the critical value, the economy follows the latterpattern. In the economy, there are two cases, one in which only the former
pattern emerges, and another one, in which both location patterns occur
simultaneously. When both location patterns occur, the effect of reduced
transportation costs on the growth rate changes at the critical value.
Many papers have shown that trade liberalization increases growth rates.
However, there are cases in which, in the process of economic development,
the regulation of trade has increased the growth and industrialization undercertain conditions. For example, Komiya et al. (1984) have argued that theJapanese government regulated trade for protection and to expand domestic
industry in the high-growth period between 1950 and 1970. By doing so,
Japan became industrialized and achieved a high growth rate. In addition,
Bernstein and Mohnen (1994), Coe and Helpman (1995), and Eaton andKortum (1996) reported that, from the 1960s to the 1980s, there were substantial international knowledge spillovers from the U.S. to Japan. Between
the 1950s and the 1980s, the Japanese government is perceived to have substantially regulated international trade. That is, transportation costs werehigh, and Japan could easily absorb international spillovers. We suggest that
our model can help to explain why high transportation costs and the ability
to absorb international knowledge spillovers led to high growth between the
1950s and the 1970s.
However, since the 1980s, international trade has been liberalized inJapan. Moreover, Japan has not intensively innovated. Coe and Helpman(1995) found that the elasticity of total factor productivity with respect toforeign R&D in Japan fell from 0.037 to 0.027 between 1980 and 1990. Bernstein and Mohnen (1994) recently showed that intensive R&D was not taking
place in Japan. This suggests that, after the 1980s, the transportation costs
fell and the ability to absorb international knowledge spillovers declined. Ourmodel suggests that, when the transportation costs are low, it is desirable
for the innovation sector to be located in a technologically advanced country. In fact, since the 1980s, by comparison with the U.S., Japan’s R&D
performance has been relatively poor.
This paper is organized as follows. In the next section, we present thebasic model and distinguish between two patterns relating to the innovation
sector’s location. In section 3, we examine the steady state and the growthrate in the context of these two patterns. In section 4, we present the resultsfrom the model and their interpretation. Section 5 is the conclusion.

4

2

The model

2.1

The economy

A basic setting in our model follows Martin and Ottaviano (1999). Consider
two countries, called Home and Foreign. Each country is endowed with a fixedamount of labor, L, which is immobile between countries. We normalize sothat L = 1. 5 The countries are symmetric except for the amount of initialpatent holding and the ability to absorb international knowledge spillovers.
Suppose that Home and Foreign initially own H0 and H0∗ with H0 > H0∗ .
Given symmetry, we consider only Home (with Foreign’s variables labeled*).
Consumer preferences are as follows:
Z ∞U=
log[D(t)α Y (t)1−α ]e−ρt dt,(1)0

where Y is the consumption of a homogeneous good, ρ is the rate of time preference, and α ∈ (0, 1) is the share of expenditure devoted to the composite
good, D(t). D(t) is given by· Z N (t)
¸ 11
1− σ
1− σ1
Di (t)
D(t) =di, σ > 1,(2)i=0where Di is the consumption of differentiated goods indexed by i, N is thetotal number of varieties available in the economy, and σ is the elasticity of
substitution between varieties.
The value of expenditure E is
Z n
Z N
E=
pi Di di +
τ p∗j Dj dj + pY Y,
(3)
0

j=n

where pY is the price of the homogeneous good, pi is the price of the ithvariety, and n is the number of varieties produced in Home and N = n + n∗ .
The international shipping of differentiated goods incurs transportation costs
taking the form of iceberg costs, as in Samuelson (1954). If a quantity ofdifferentiated goods is shipped, then a 1/τ (τ > 1) proportion of this quantity
reaches the customer. 6 There is no transportation cost for the homogeneous
good.
5With this normalization, our model lacks a scale effect. However, our focus is not on
the scale effect on growth, but on geography and growth. The geography of firms and theinnovation sector is independent of the scale effect.
6
These costs are interpreted as transportation costs that are affected by trade policy,
such as tariffs and quotas.

5

On the supply side, the homogeneous good is produced using only laborwith constant returns to scale technology. For simplicity, one unit of labor
is required to produce one unit of the homogenous good. We assume that,since the demand for this good is sufficiently large, it is produced in bothcountries in equilibrium. Because there are no transportation costs for the
homogenous good, the nominal wage is equal between the two countries.
Solving the optimization problem shows that pY = w.
For all varieties, differentiated goods are produced in a monopolisticallycompetitive sector with increasing returns to scale technology. If there is nocost of discrimination, each firm produces its own variety. We assume thata patent is required for starting production and β units of labor are used
to produce one unit of these goods. Once the patent is made, differentiated goods firms have monopoly power over the variety produced and canfreely relocate production facilities across countries. Under this assumption,
optimal pricing implies pi = p∗i = wβσ/(σ − 1). The operating profit isπ = px − wβx =

wβx
,
σ−1

(4)

where x is the size of the firm.
In contrast to firms, workers are immobile. Thus, their incomes are geographically fixed. Solving the first-order condition, we find the demand for
differentiated goods:
Di =

σ−1
αE
σ − 1 τ −σ αE
,
D
=
.
j
wβσ n + τ 1−σ n∗wβσ n + τ 1−σ n∗Suppose that there is perfect global financial market, which yields theinterest rate, r. Suppose the patent lives infinitely. Therefore, firms thathave a patent have a perpetual monopoly power. The value of differentiatedfirms is the present discounted value of all future operating profits. Thus,
the value of the firms, v, isZ ∞wβx(s)
v(t) =
e−[R(s)−R(t)]
,
(5)
σ−1
t
Rt
where R(t) = 0 r(τ )dτ . By differentiating with respect to t, we find theno-arbitrage condition:v̇(t) = r(t)v(t) − π(t).

(6)

Solving the consumer’s intertemporal optimization problem yields Ė/E =Ė ∗ /E ∗ = r − ρ. As in Grossman and Helpman (1991), our economy does nothave a monetary instrument, so there is nothing to pin down the price level.
Therefore, we can normalize aggregate income at any time; i.e., E + E ∗ = 1.
On this basis, the solution to the optimization problem derives that r = ρ.
6

2.2

Equilibrium

Owing to the free movement of patents, operating profits are equal between
countries. Thus, from (4), x = x∗ .
There is an equilibrium condition for the market clearing of goods, which
incorporates transportation costs. Using the consumer’s demand for differentiated goods, the market-clearing conditions are as follows:µ¶α(σ − 1)
E
E ∗δ
x =
+
,
βσ
n + δn∗ nδ + n∗µ¶α(σ − 1)
Eδ
E∗
∗
x =
+
.
βσ
n + δn∗ δn + n∗
For simplicity, we define the parameter δ as δ ≡ τ 1−σ . It is clear that
0 < δ < 1 and a higher δ implies lower transportation costs since τ > 1 andσ > 1. In addition, we denote the share of differentiated goods firms in Homeas γ.
Given the equilibrium condition, x = x∗ , the equilibrium share of differentiated goods firms in Home isγ=

θE(1 − θE )δ
−
,
1−δ
1−δ

(7)

where θE = E/(E + E ∗ ), which is the share of total expenditure of Home.
This equation implies that a higher share of expenditure leads to greater
agglomeration in Home, owing to increasing returns to scale in the differentiated goods sector. When the transportation costs are low, i.e. δ is high,the firms tend to locate in Home since the firms desire shipping to Foreign
while locating in Home.
From x = x∗ , the size of the firms, wherever they and the innovation
sector locate, isσ − 1 E + E∗x = x∗ = α
.
(8)
wβσ
N
It must be recalled that one patent is needed to produce a new differentiated good. Thus, the number of varieties and the number of firms in
the economy are equal to the number of patents in the economy; hence,n + n∗ = H + H ∗ at each point of the time.

7

3

The steady state and the equilibrium growthrateConsider the innovation sector, which produces a patent by using only labor. This sector operates as described in Grossman and Helpman (1991).
Suppose that the number of labor inputs required to produce the patent isη/(²n + ²∗ n∗ ). The parameter η represents the constant labor productivity
implied per patent produced. The parameters 0≤²∗ ≤1 and 0≤²≤1 represent
the degree of knowledge spillovers from Foreign to Home and from Home toForeign, respectively. We interpret ² (²∗ ) as the ability of Foreign (Home) toabsorb international knowledge spillovers. Note that the innovation cost isa decreasing function of the number of firms located in own and the othercountries. 7 We refer to knowledge spillovers within a country as local
knowledge spillovers, as do Martin and Ottaviano (1999), and assume that
local knowledge spillovers are perfect. However, it is assumed that the effectof the other country’s diversity on the innovation cost may be not perfect:(0≤²≤1 and 0≤²∗ ≤1). 8 Owing to free entry and zero profits in the innovation sector, vH = wη/(n + ²∗ n∗ ) when innovation takes place in Home.
Similarly, vF = wη/(n∗ +²n) when the innovation takes place in Foreign. The
terms vH and vF denote the value of a patent when the innovation sector is
located in Home and Foreign, respectively.
Note that γ is constant in equilibrium. Therefore, in the steady state, n,n∗ , and N grow at the same constant rate, g = Ṅ /N . The other equilibrium
conditions, which ensure that vH and vF decrease at the same rate as thegrowth rate of N , are as follows:
g=

3.1

Ṅ
v̇H
v̇F
=−
=− .
N
vH
vF

Location of the innovation sector

The innovation sector operates in the country with the lowest innovation
cost. From the setting of the innovation cost, the following patterns of thelocation of the innovation sector are identified.
• (Pattern 1) The innovation sector operates in Home (n+²∗ n∗ > ²n+n∗ ).
7Martin (1999) refers to this type of knowledge spillover as the Jacobs type of knowledge
spillover. The author suggested that the source of these benefits is, for example, direct
observation of the production process: researchers observe the production process and find
new goods easier to invent.
8
The production costs for the patent in Home and Foreign are wη/(n + ²∗ n∗ ) andwη/(n∗ + ²n), respectively.

8• (Pattern 2) The innovation activity operates in Foreign (n + ²∗ n∗ <²n + n∗ ).
When the location pattern is in pattern 1, the share of differentiated firms
in Home is as follows:
1 − ²∗
γ > γ̂ ≡.
(9)
2 − ² − ²∗This discussion implies the property of the economy described by the following lemma.
Lemma 1 When γ > γ̂ (γ < γ̂), pattern 1 (pattern 2) applies.
Next, we consider income. Given v = π/(ρ + g), (4) and (8), the equilibrium aggregate income in each country is
E =w+

2αρwh
2αρw(1 − h), E∗ = w +,(ρ + g)σ − αρ(ρ + g)σ − αρ(10)where h = H/N = H/(H + H ∗ ) and H is the number of patents in Home.
That is, h is the share of patents owned by the households in Home. In ourmodel, this share is constant because patents are free to move between thetwo countries. 9 Thus, H and N grow at the same rate. Next, we determine
the consumer’s share of income in Home, θE .
From (10)σ(ρ + g) + αρ(2h − 1)
θE =
.
(11)
2σ(ρ + g)
Equation (11) shows that international income inequality, θE , is a decreasing
function of the growth rate. This is because a higher growth rate implies ahigher rate of entry of firms, which lowers both the monopoly profits and thevalue of patents. The decrease in income from patent holdings is greater inHome than in Foreign because Home owns more patents. Thus, international
income inequality decreases as the growth rate increases.
It is found that θE > 1/2 provided that h > 1/2. In other words, whenHome has more patents than Foreign, the aggregate income is higher in Home
than in Foreign. (7) should be kept in mind. From assumption h > 1/2, it
is found that γ > 1/2 since, if h > 1/2, θE > 1/2. From (9), it is clear thatγ̂≤1/2 when ²∗ ≥². This generates the property of the economy described in
the next lemma.
9Baldwin (1999) and Baldwin et al. (2001) constructed models in which h is an endogenous variable. In these models, assets are not free to move internationally, and this assumption generates the cumulative causation mechanism for agglomeration. In our model,
the focus is not on the cumulative causation mechanism but on the effect of partially local
knowledge spillovers on the relationship between geography and growth.

9Lemma 2 With the assumption h > 1/2, pattern 2 cannot arise if ²∗ ≥².
This lemma states that, if Foreign is less able to absorb international knowledge spillovers than Home, pattern 2 cannot arise. 10
Wherever the innovation sector is located, Home’s nominal GDP, definedas the sum of value added in the three sectors, is GDPH = w +γβxN/(σ −1).
Home’s nominal GDP exceeds Foreign’s nominal GDP, which is GDPF =w + (1 − γ)βxN/(σ − 1), because γ > 1/2. Each country’s GDP is anincreasing function of its share of firms producing differentiated goods.

3.2

The steady state and the equilibrium growth rate
under pattern 1.

When γ > γ̂, this pattern emerges. Since labor is employed in three sectors,
when the innovation sector is in Home, the labor-market clearing condition
is
η
2σw
σw
E + E∗ =
−
gH ,
(12)
∗
σ − α (σ − α) γ + ² (1 − γ)
where gH is the growth rate when the innovation sector operates in Home. It
must be remembered that the total expenditure is constant due to normalization.
Given free entry in the innovation sector, r = ρ, (12) and (6), the growthrate is2α[γ + ²∗ (1 − γ)] σ − α
gH =
−
ρ.
(13)
ση
σ
Note that the increased agglomeration in Home raises the equilibrium growthrate when the innovation activity takes place in Home. Since agglomerationin Home reduces the innovation cost in Home, agglomeration in Home has apositive effect on the growth rate. From (7) and (13), the equilibrium growthrate as a function of the share of Home’s total income is¸·
σ−α2α 1 − ²∗∗[(1 + δ)θE − δ] + ² −
ρ.
(14)
gH =
ση 1 − δσSubstituting (13) into (11) yields the following relationship between international income inequality and the location of firms:
θEH =(1 − ²∗ )γ + ²∗ + ηρh,2(1 − ²∗ )γ + 2²∗ + ηρ

10

(15)

In Martin and Ottaviano (1999), ² = ²∗ = 1 or ² = ²∗ = 0. Hence, in their model,
the innovation sector always locates in Home. However, in our model, the case in which
innovation activity is conducted in Foreign may emerge when ² < ²∗ .

10

where θEH is the consumer’s share of income in Home when the innovation
sector is in Home. It is clear that the consumer’s share of income in Home
decreases as the number of differentiated firms in Home increases. This isbecause agglomeration in Home reduces the innovation cost in Home andraises the growth rate, which reduces the monopoly power of existing firms.
Since Home owns more patents, the fall in patent income is greater in Home.
Therefore, international income inequality is reduced when the growth rateincreases. This is known as the competition effect: see Martin (1999).
The value of γ that satisfies both (7) and (15) is the steady-state solution of the model. Combining (7) and (15) and imposing γ > 0 yields theequilibrium value of γ. 11 We denote γH as the equilibrium value of γ when
the innovation sector is located in Home.
√(1 − 3²∗ − ηρ) + ∆
γH =
,
(16)
4(1 − ²∗ )
where

¸
·
ηρ
∗
(h − (1 − h)δ) .
∆ = (1 − 3² − ηρ) + 8(1 − ² ) ² +
1−δ
∗

2

∗

The equilibrium growth rate can be computed from (13).
It is clear that ∂γH /∂h > 0 from (16), and, thus, ∂gH /∂h > 0 from (14).
In addition, ∂γH /∂δ > 0, and, thus, ∂gH /∂δ > 0.
Let us define δ̄ such that γH = 1 if δ≥δ̄. From (16), we find thatδ̄ =1 + ηρ(1 − h).
1 + ηρh(17)

It is obvious that δ̄ < 1 and ∂ δ̄/∂h < 0 since h > 1/2. From (14), if δ≥δ̄, thengH is constant at the maximum growth rate, gH = (2α/ση) − (σ − α)ρ/σ,
among all potential growth rates. These results are described in the next
lemma.
Lemma 3• When γH ≥γ̂ and δ < δ̄, the equilibrium growth rate, gH , isan increasing function of the share of patent holdings in Home, h, anda decreasing function of the transportation costs τ .
• When δ≥δ̄, gH is constant at the maximum growth rate, gH = (2α/ση)−(σ − α)ρ/σ, among all potential growth rates.
11

The equation that we find by combining (7) and (15) is2(1 − δ)(1 − ²∗ )γ 2 + (1 − δ)(3²∗ − 1 + ηρ)γ − {(1 − δ)²∗ + ηρ(h − (1 − h)δ)} = 0.

11

In this location pattern, the effect of the share of patent holdings in
Home on the equilibrium growth is positive. However, the effect of the highinequality of patent holding on international income inequality is ambiguous.
This is because there are two effects. First, the high h simply indicates thehigh international income inequality. Second, there is the competition effect.
Decreasing transportation costs raises the equilibrium growth rate. The
decline in the transportation costs induces agglomeration in Home, which
results in a reduction in the innovation costs in Home because the strengthof the degree of spillovers from local knowledge is greater than that from international knowledge. The increase in the growth rate reduces international
income inequality because of the competition effect. Hence, the decreasein the transportation costs raises the equilibrium growth rate and reduces
international income inequality.
The effect of an increase in ²∗ is ambiguous. We show that ∂γH /∂²∗ < 0from 1/2 < γH ≤ 1. Thus, it is found that the increase in ²∗ reduces thegrowth rate. However, it is found that an increase in ²∗ increases the growthrate for a given γH from (14). The increase in the growth rate reduces
international income inequality, which reduces the share of firms because of
the competition effect. These effects reduce the equilibrium growth rate. The
increase in ²∗ reduces international income inequality, but the effect on thegrowth rate is ambiguous.
The stability properties of the equilibrium are analyzed following Grossman and Helpman (1991). First, we define the variable W ≡ 1/w. That
is, W is the inverse of the wage rate. In addition, g = Ṅ /N represents theinstantaneous rate of innovation in the economy.
(6) is rewritten as a function of the wage rate as follows:gH =α[γH + ²∗ (1 − γH )]
− ρ.
σηw

(18)

Given the definition of W , (12) is rewritten as
W =

2σ
σ
η
−
gH .
∗
σ − α (σ − α) γH + ² (1 − γH )(19)Note that (19) must be satisfied at all times because this equation expresses
the resource constraint. The higher the innovation rate, the greater theemployment in the innovation sector, which lowers employment in the differentiated goods sector and, therefore, the supply of differentiated goods. This
increases the price of these goods as well as the wage. This is why the curve
representing (19) slopes downward.
The steady-state properties of the model are illustrated in Figure 1. (20)and (19) intersect at point M. At this point, dynamic forces in the economy
12

do not affect W and g. In other words, if the economy reaches point M,
innovation continues at a constant rate, and the division of resources among
the innovation sector, the homogenous goods sector, and the differentiatedgoods sector remains constant. The wage rate does not change. The upwardsloping curve represents the combinations of g and W .
W =ση(gH + ρ).
α[γH + ²∗ (1 − γH )](20)At all points above the line representing (20), an increase in the number ofvarieties reduces the wage rate, so W rises. Below this line, the opposite
applies. These movements follow (19).
As in Grossman and Helpman (1991), expectations can be fulfilled only
if the economy jumps immediately to this steady state. Expectations thatdiffer from those implied by point M cannot be fulfilled. In the long run,initial beliefs that contradict those implied by point M lead to either g = 0
or W = 0. In the first case, N eventually stops growing at a finite value,
while W grows infinitely. The finite value of N could only arise were w to
approach zero. However, this value of N requires a strictly positive wage
rate, not one of zero. Since the wage rate remains positive, this path requires
unfulfilled expectations. In the second case, g reaches its maximum value,
while the value of W falls arbitrarily close to zero. Owing to an unbounded
and continual increase in the number of varieties, the wage rate eventually
falls to zero. Hence, beliefs are clearly contradicted. The only remaining
possibility is for the economy to start at point M and stay there forever.
Suppose that the economy starts with the innovation cost being lower
in Home. If the transportation costs decrease or the share of patents in
Home increases, the share of differentiated firms rises because the demandfor differentiated goods increases. This lowers the innovation cost in Home
further. The lower innovation cost raises the growth rate. This movement
continues until profits in the innovation sector are zero and firm profits inthe differentiated goods sector are equal in both countries. Stability analysis
indicates that this process is instantaneous.

3.3

The steady state and the equilibrium growth rate
under pattern 2.

As in subsection 3.2, we examine the steady state and the growth rate.
The labor-market clearing condition is
E + E∗ =

2σw
σw
η
−
gF ,
σ − α (σ − α) (1 − γ) + ²γ
13

(21)

where gF is the growth rate when the innovation sector operates in Foreign.
The equilibrium growth rate is described as a function of γ as follows:gF =2α[²γ + (1 − γ)] σ − α
−
ρ.
ση
σ

(22)

Unlike (13), (22) shows that the equilibrium growth rate is a decreasing
function of γ. This is because the progress of agglomeration in Home increases the innovation cost in Foreign. Thus, the growth rate decreases as
the progress of agglomeration in Home.
Substituting (7) into (22) yields the equilibrium growth rate described asa function of θE as follows:·¸
2α ² − 1
σ−αgF =[(1 + δ)θE − δ] + 1 −ρ.
(23)ση 1 − δσCombining (22) and (11) shows the following relationship between international income inequality and the location of firms:
θEF =(² − 1)γ + 1 + ηρh,2(² − 1)γ + 2 + ηρ(24)

where θEF is the household’s share of income in Home when the innovation
sector operates in Foreign. Compared with (15), θEF is an increasing functionof γ. This is because agglomeration in Home raises the innovation cost in
Foreign, thereby reducing the equilibrium growth rate, and, thus, reinforces
the monopoly power of existing firms. Since Home owns more patents, theincome increases more in Home than in Foreign. Hence, the income gap
widens.
The value of γ that satisfies (7) and (15) is the steady-state solution of
the model. The equilibrium share of firms in Home, γF , is as follows: 12q(3 − ² + ηρ) ±γF =ηρ(3 − ² + ηρ)2 − 8(1 − ²){1 + 1−δ
(h − (1 − h)δ)}4(1 − ²)For the existence of a solution for γ, we impose the following condition:¸·ηρ2(h − (1 − h)δ) > 0.
(3 − ² + ηρ) − 8(1 − ²) 1 +1−δ
12The equation that we find by combining (7) and (15) is2(1 − δ)(1 − ²)γ 2 + (1 − δ)(3 − ² + ηρ)γ − {(1 − δ)² + ηρ(h − (1 − h)δ)} = 0.

14.

(7) shows that, if h = 1/2, γ = 1/2. Therefore, we find that the value ofγF is√(3 − ² + ηρ) − Φ
γF =
,
(25)
4(1 − ²)
where

·
¸
ηρ
Φ = (3 − ² + ηρ) − 8(1 − ²) 1 +(h − (1 − h)δ) .
1−δ
2

We calculate the equilibrium growth rate by substituting (25) into (22).
It is shown that ∂γF /∂² < 0. Thus, ∂gF /∂² > 0 from (22). In addition,from (25), ∂γF /∂δ > 0, and ∂γF /∂h > 0. Therefore, ∂gF /∂δ < 0, and∂gF /∂h < 0. We summarize these results in the next lemma.
Lemma 4 If γF < γ̂, the equilibrium growth rate is an increasing function of the transportation costs, the ability of Foreign to absorb international
knowledge spillovers, and a decreasing function of Home’s share of patents.
In this location pattern, the equilibrium growth rate rises if the share ofpatent holdings in Home decreases. In addition, the increase in the equilibrium growth rate reduces the international income inequality. This is because
there are two effects. First, the small h simply implies low international income inequality. A small h raises the share of firms locating in Foreign andthus reduces the innovation cost in Foreign. Second, there is the competition
effect. Hence, there is no trade-off between international income equality andthe growth rate. 13
Lower transportation costs reduce the equilibrium growth rate. The decline in the transportation costs increases the agglomeration in Home. Given
increasing returns to scale, firms in Foreign relocate to Home due to a reduction in the transportation costs. This relocation of firms raises the innovation
cost in Foreign, and the growth rate declines. The fall in the growth rate
raises international income inequality because the lower growth rate weakens
the competition effect. Hence, lower transportation costs reduce the equilibrium growth rate and raise international income inequality.
An increase in ² raises the equilibrium growth rate and reduces international income inequality. The increase in ² directly increases the growth rate
for a given γF from (22). The increase in the growth rate reduces Home’sshare of income and raises Foreign’s share of income because of the competition effect. Thus, the increase in ² raises Foreign’s share of income. This13

In Martin (1999), there is a trade-off between international income equality and thegrowth rate. This is because, in his model, innovation activities locate in the country withthe larger market.

15effect reinforces the direct effect of ² on the growth rate and raises the equilibrium growth rate. The increase in ² reduces international income inequality
and raises the equilibrium growth rate.
The stability properties are investigated as in subsection 3.2. 14
Here, we investigate the factors that determine the location of the innovation sector. As we explained earlier, if γ < γ̂, the innovation sector is in
Home and, if γ < γ̂, the innovation sector operates in Foreign. From (7), it isclear that the lower the transportation costs are, the higher the share of firmsthat locate in Home is. Thus, there is a critical value of δ that represents
the level of the transportation costs. We define δ̂ as the value that leads toγ = γ̂. We find the value of δ̂ using the equation to derive the steady-stateγ and the critical value γ̂: 15δ̂ =(² − ²∗ )(1 − ²²∗ ) − (2 − ² − ²∗ ){(1 − ²∗ ) − (2 − ² − ²∗ )h}ηρ. (26)(² − ²∗ )(1 − ²²∗ ) − (2 − ² − ²∗ ){(1 − ²∗ ) − (2 − ² − ²∗ )(1 − h)}ηρ

In this context, ∂γH /∂δ > 0 and 1 > γ̂ = (1 − ²∗ )/(2 − ² − ²∗ ). Hence, δ̄ > δ̂.
It must be recalled that δ̄ is the value of δ that implies γ = 1. In addition,
we find that, when γ = γ̂, gH = gF and θEH = θEF . 16
From the definition, the economy has three cases. If δ̂ is not positive, theinnovation sector is always in Home regardless of the level of transportation
costs. If δ̂ ≥ 1, the innovation sector is always in Foreign regardless of thelevel of the transportation costs. If 0 < δ̂ < 1, both location patterns of theinnovation sector exist in the economy, and which location pattern emerges
depends on the level of the transportation costs.
For simplicity, we focus on the following two cases: the case in which∗² = 1, 0≤²≤1 and that in which ²∗ = 0, 0≤²≤1. 17
14

We derive a labor-market clearing condition and a no-arbitrage condition. These are
W =

2σ
σ
η
−
gF ,
σ − α (σ − α) ²γF + (1 − γF )W =ση(gF + ρ).
α[²γF + (1 − γF )]15

Substituting (25) into γF = γ̂ yields the same value of δ̂.
We calculate the growth rate and the share of expenditure of Home. These are,respectively,
2α(1 − ²²∗ )σ−αgH = gF =−ρ,ση(2 − ² − ²∗ )σ
16

θEH = θEF =17(1 − ²²∗ ) + ηρh(2 − ² − ²∗ ).
2(1 − ²²∗ ) + ηρ(2 − ² − ²∗ )If we analyze the cases of 0 < ²∗ < 1 and 0 < ² < 1, the results are qualitatively the16When ²∗ =1 and 0≤²≤1, it is clear that ²∗ ≥². Thus, in this case, thelocation of the innovation sector is described by pattern 1 from Lemma 2.
The location of pattern 1 was analyzed in subsection 3.1. 18
Hence, we focus on the cases of ²∗ = 0 and 0≤²≤1. Substituting ²∗ = 0
into (26) yields the following value of δ̂:δ̂ =² + ηρ(2 − ²){1 − (2 − ²)h}.
² + ηρ(2 − ²){1 − (2 − ²)(1 − h)}(27)

Using δ̂, we restate the relationship between the location of the innovation
sector and the transportation costs from (7) and Lemma 1. When δ > δ̂, theinnovation sector is in Home (pattern 1). When δ < δ̂, the innovation sector
is in Foreign (pattern 2).
From (27), we find that h > 1/2 implies that δ̂ < 1. Thus, we find that,in the economy, cases such as δ̂ ≤ 0 and 0 < δ̂ < 1 emerge. We denote theformer and the latter as case A and case B, respectively. 19 Now, we haveto investigate the sign of δ̂. Since the denominator of δ̂ is positive provided
that 0≤²≤1 and h > 1/2, the sign of δ̂ is determined by the sign of thenumerator of δ̂. Here, we have to examine how the sign of the numerator ofδ̂ is determined.
When the numerator of δ̂ is not positive, the following relationship holds:h ≥ ĥ ≡1²+.
2 − ² ηρ(2 − ²)2(28)

From this equation, we find that, when ² = 0, ĥ = 1/2, and, when ² = 1,ĥ > 1 and ĥ is an increasing function of ². Next, let us define ²̄ as the value
associated with ĥ = 1. We derive the value of ²̄ as follows:p3ηρ + 1 + (3ηρ + 1)2 − 8(ηρ)2
.
(29)
²̄ ≡
2ηρ
same as those in our model. If ²∗ ≥², the innovation sector’s location is always pattern 1.
If ² > ²∗ , the relationship between the growth rate and the transportation costs may not
be monotonic, as shown later.
18
When ² = ²∗ = 0, δ̂ = −1. Therefore, only pattern 1 arises in the economy. This casecorresponds to the case of local spillovers in Martin and Ottaviano (1999). In addition,
when ² = ²∗ = 1, the growth rate is independent of the transportation costs and reaches
its maximum value,
2α σ − α−ρ.
gH = gF =ση
σ
This is consistent with the case of global spillovers in Martin and Ottaviano (1999).
In case A, only pattern 1 emerges in the economy. In case B, the economy has both
location pattern.
1917

It is clear that, if ² > ²̄, then ĥ > 1 because h increases with ². Then, we
define ²̂ as the value associated with δ̂ = 0. Deriving ²̂, it is shown that, from0 ≤ ² ≤ 1,p
(1 − ηρ + 4ηρh) − (1 − ηρ + 4ηρh)2 − 8(ηρ)2 h(2h − 1)
²̂ =
.
(30)
2ηρh
We find that 0 < ²̂ < 1. Here, we show from (27) that, when ² = 0, δ̂ = −1
and, when ² = 1, 0 < δ̂ < 1. In addition, ∂ δ̂/∂² > 0. Thus, it is found that,if ² > ²̂, then 0 < δ̂ < 1.
In short, (28) implies that, if Home’s share of patent holdings is high, theeconomy is in case A. This is because there is high-income inequality due tohigh inequality. However, a higher ² directly reduces the innovation cost in
Foreign. Therefore, the tendency that the innovation sector is in Foreign isstrong. (29) implies that, for any value of 1/2 < h≤1, case A is not observed,
provided ² > ²̄. From (30), it is shown that, if ² > ²̂, the economy is in case
B due to that high ² leads to a reduction in the innovation cost in Foreign.
We show that, if ² is sufficiently low and if the share of patent holding inHome is high, the economy is in case A.
Consider the relationship between δ̂ and h. It is clear that ∂ δ̂/∂h < 0 and,
when h = 1/2, δ̂ = 1. This implies that, if h is high, higher transportation
costs are necessary for a decline in the innovation cost in Foreign. This isbecause, when the transportation costs are high, differentiated firms relocate
in Foreign, which leads to a reduction in the innovation cost in Foreign. Thus,
it is possible for the innovation sector to locate in Foreign due to an increasein the transportation costs. In addition, it can be shown that an increase in² raises δ̂ from (27). In particular, if ² > ²̄, the economy is in case B for any
values of 1/2 < h≤1.
Given Lemmas 1 through 4 and the discussion in this section, we present
the following proposition, which summarizes the main results of our model.
Proposition 1 With the assumption h > 1/2, the following conditions are
satisfied:• The economy is in case A when h ≥ ĥ, when ²∗ ≥ ², or when ²∗ = 0 and² ≤ ²̂. In case A economy, the equilibrium growth rate is gH , and gH isan increasing function of Home’s share of the patent and a decreasing
function of the transportation costs.
• The economy is in case B when ² > ²̄ or when ² > ²̂. In case B
economy, the growth rate is not monotonic.

18

– When the innovation sector is in Home, the equilibrium growthrate is gH , and gH is an increasing function of Home’s share ofpatents and a decreasing function of the transportation costs.
– When the innovation sector is in Foreign, the equilibrium growthrate is gF , and gF is an increasing function of the transportation costs, the ability of Foreign to absorb international knowledge
spillovers, and a decreasing function of Home’s share of patent
holdings.
• When δ≥δ̄, gH is constant at the maximum growth rate, gH = (2α/ση)−(σ − α)ρ/σ, among all potential growth rates.
Here, we investigate the relationship between the equilibrium growth rate
and the transportation costs. If case A is observed, the equilibrium growthrate monotonically increases with the transportation costs. This case is illustrated in Figure 2.
However, if case B is observed, the relationship between the equilibrium
growth rate and the transportation costs is not monotonic. It must be recalled that, when δ̂ < δ, the innovation sector is in Home, and, when δ̂ > δ,
the innovation sector is in Foreign. The lower the transportation costs are,the higher the equilibrium growth rate is, provided that δ̂ < δ. However,lower transportation costs reduce the equilibrium growth rate if δ̂ > δ. Thisis because lower transportation costs increase the agglomeration of firms in
Home. The agglomeration has different effects on the innovation cost ineach country. Thus, the effect of the transportation costs on the equilibrium
growth rate changes at δ = δ̂. This relationship is illustrated in Figure 3.
Next, we show the effects of a change in ² on the equilibrium growth rate.
It must be recalled that Lemma 4 and ∂ δ̂/∂² > 0. This implies that, if theability of Foreign to absorb international knowledge spillovers improves, it
is easier for the innovation sector to locate in Foreign. As Lemma 4 shows,an increase in ² raises the equilibrium growth rate provided that δ < δ̂.
However, it is clear from (13) and (14) that, although ² varies, the equilibrium
growth rate is constant if the innovation sector is in Home. Moreover, when
δ = δ̂, gH = gF . 20 This discussion reveals the effect of an increase in ² on
the relationship between the equilibrium growth rate and the transportation
20

We find that, when γ = γ̂ (δ = δ̂),gH = gF =σ−α
2α(1 − ²²∗ )−ρση(2 − ² − ²∗ )
σ

.

19

costs when the economy is observed in case B. This is illustrated in Figure
4.
Next, consider the effects of a change in h on δ̂, δ̄, and the equilibrium
growth rate. From (39) and (21), we show that, if Home’s share of patent
holdings increases, it is easier for the innovation sector to locate in Home,and for the economy to achieves its maximum growth rate. As Lemma 4
shows, in pattern 2, the equilibrium growth rate, gF , is a decreasing functionof h. Due to these effects, the equilibrium growth rate falls when the locationof the innovation sector is in pattern 2, gF . On the contrary, the equilibrium
growth rate rises when the location of the innovation sector is gH , as shownin Lemma 3. The effect of an increase in h is illustrated in Figure 5.
Next, we discuss some implications for welfare in the two countries. Priceindices for differentiated goods in Home and Foreign are as follows:1

GH = N σ−1

1
1
1
σβw
σβw
(γ +(1−γ)δ) σ−1 , GF = N σ−1(γδ +(1−γ)) σ−1 , (31)σ−1σ−1where GH and GF represent the price indices for Home and Foreign, respectively. It is clear that GH is a decreasing function of γ and that GF is anincreasing function of γ. Thus, with respect to the price index, agglomeration in Home improves welfare in Home and reduces welfare in Foreign. Thisis because shipping differentiated goods incurs transportation costs in ourmodel. It must be noted that, if there are no transportation costs, both GH
and GF are independent of γ. Given this discussion, we refer to these effectsof the transportation costs as transportation cost effects.
Clearly, welfare in Home improves as agglomeration increases in Home
provided that δ≥δ̂ because of the high growth rate and the low price index in Home. In particular, when δ≥δ̄, the economy achieves its maximum
growth rate and γ = 1, and welfare in Home is maximized. However, when
δ < δ̂, the effect of agglomeration in Home on welhare in Home is ambiguous. The reason is as follows. The equilibrium growth rate decreases as
agglomeration in Home increases, which lowers Home’s welfare. However, increased transportation costs raise Home’s welfare as agglomeration increases
in Home.
Welfare in Foreign decreases as agglomeration in Home increases provided that δ < δ̂. In this range, the equilibrium growth rate decreases withincreased agglomeration in Home. When δ≥δ̂, the equilibrium growth rateincreases with increased agglomeration in Home. This raises welfare in Foreign. However, higher transportation costs reduce Foreign’s welfare as agglomeration in Home increases.
Our results have important implications, for example, for trade policy.
When the ability of a small-market country to absorb international knowl20

edge spillovers is high and there is little international income inequality, theeconomy has two location patterns of the innovation sector, and the growthrate does not monotonically increase as the transportation costs decline. In
this case, if this economy initially has high transportation costs, the innovation sector locates in that country. As the transportation costs decrease, thegrowth rate decreases. When the transportation costs continue to decrease
and are below some critical level, the innovation sector relocates from thesmall-market country to a large-market country. Once the innovation sector
relocates to a large-market country as the agglomeration of firms producing
differentiated goods increases in this country, the growth rate increases as
the transportation costs decrease. 21
Considering a real economy, Komiya et al. (1984) argued that the Japanese
government regulated trade to protect and expand the domestic industry. By
doing so, Japan accomplished industrialization and a high growth rate between the 1950s and the 1970s. In addition, Bernstein and Mohnen (1994),
Coe and Helpman (1995), and Eaton and Kortum (1996) reported that, between the 1960s and the 1980s, there were substantial international knowledge spillovers from the U.S. to Japan. Our results suggest that, at that time,
the Japanese economy was described by pattern 2 and that highly regulatedtrade and the Japan’s ability to absorb international knowledge spillovers led
to technological development in Japan and contributed to its high economic
growth between the 1950s and the 1970s.
However, since the 1980s, international trade has been liberalized in Japanand has not intensively innovated. 22 Our results suggest that, after the
1980s, the economy was in pattern 1 and that low transportation costs andJapan’s reduced ability to absorb international knowledge spillovers led theinnovation sector to locate in the U.S. From our result, in that pattern, as
the transportation costs fall, it is more favorable for the innovation sector tooperate in a country that has more advanced technology (the U.S.). In fact,
after the 1980s, relative to the U.S., Japan’s R&D sector has not performed
well.

21

These results explain only one of the many mechanisms affecting the relationshipbetween the transportation costs and the growth rate. For example, Yamamoto (2003)
has considered other mechanisms.
22
Coe and Helpman (1995) found that the elasticity of total factor productivity withrespect to foreign R&D in Japan decreased from 0.037 to 0.027 between 1980 and 1990.
Bernstein and Mohnen (1994) found that, recently, intensive R&D has not been taking
place in Japan.

21

4Conclusions

In this paper, we have constructed a model in which knowledge spillovers
are partially local and international knowledge spillovers are asymmetric.
Partially local and asymmetric international spillovers generate particularpatterns of growth and agglomeration in the economy.
The effect on the growth of agglomeration has been analyzed using partially local knowledge spillovers with agglomeration: agglomeration in acountry reduces the innovation cost in that country and stimulates growth.
With agglomeration, the interaction between the transportation costs andthe extent of international knowledge spillovers leads the innovation sector
to relocate. The effect of a reduction in the transportation costs on thegrowth rate is not monotonous due to this relocation. In our model, in thecontext of a growing economy, two key factors are the ability of the country
to absorb international knowledge spillovers and the level of the transportation costs. By focusing on these two points, our model analysis of growth
due to agglomeration has provided numerous implications.
In the modern world, local knowledge spillovers play an important role,and the development of telecommunications technology has facilitated international knowledge spillovers. Thus, an investigation of the influence ofinternational knowledge spillovers on the economy is worthwhile. Our study
has contributed to this objective.

References[1] Baldwin, R. E., 1999, Agglomeration and endogenous capital. European
Economic Review 43, 253-280.
[2] Baldwin, R. E., P. Martin, and G. I. P. Ottaviano, 2001, Global income
divergence, trade, and industrialization: The geography of the growth
take-offs. Journal of Economic Growth 6, 5–37.
[3] Barro, R., and X. Sala-i-Martin, 1997, Technological diffusion, convergence, and growth. Journal of Economic Growth 2, 1–26.
[4] Bernstein, J. I., and P. Mohnen, 1994, International R&D spillovers between U.S. and Japanese R&D intensive sectors. NBER Working Paper#4682.
[5] Coe, D. T., and E. Helpman, 1995, International R&D spillovers. European Economics Review 39, 859–887.
22

[6] Eaton, J., and S. Kortum, 1996, Trade in ideas: Patenting and productivity in the OECD. Journal of International Economics 40, 251–278.
[7] Fujita, M., and J.-F. Thisse, 2003, Does geographical agglomeration
foster economic growth, and who gains and loses from it? Japanese
Economic Review 54, 121–145.
[8] Grossman, G., and E. Helpman, 1991, Innovation and growth in theworld economy (MIT Press, Cambridge, MA).
[9] Komiya, R., M. Okuno, and K. Suzumura, 1984, Nihonnosangyouseisaku, (Industrial policies in Japan) (University of Tokyo Press,
Tokyo). in Japanese[10] Martin, P., 1999, Public policies, regional inequalities, and growth. Journal of Public Economics 73, 85–105.
[11] Martin, P., and G. I. P. Ottaviano, 1999, Growing locations: Industry
location in a model of endogenous growth. European Economic Review
43, 281–302.
[12] Romer, P., 1990, Endogenous technical change. Journal of Political
Economy 98, 71–102.
[13] Samuelson, P., 1954, The transfer problem and transport costs,:
Analysis of effects of trade impediments. Economic Journal 64, 264–289.
[14] Yamamoto, K., 2003, Agglomeration and growth with innovation in theintermediate goods sector. Regional Science and Urban Economics 33,
335–360.

23

W

M

gH
Figure 1:

g

0

δ̄
Figure 2:

1

δ

g

0

δ̄

δ̂
Figure 3:

1

δ

g

δ=0

δ̂
Figure 4:

δ̄

1

δ

g

δ=0

δ̂

δ̄
Figure 5:1δ