Discussion Papers In Economics
And Business

Migration and agglomeration with knowledge
spillovers
Kyoko Hirose
Discussion Paper 05-16

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANMigration and agglomeration with knowledge
spillovers
Kyoko Hirose
Discussion Paper 05-16

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Migration and agglomeration with knowledge
spillovers ∗Kyoko Hirose†
June 2005

Abstract
In this paper, a Grossman-Helpman-Romer-type endogenous growthmodel is developed with two regions in which there are mobile workers
and linkage between consumption goods and differentiated intermediate goods. The economy has the potential to reach the following
spatial configuration: full agglomeration, partial agglomeration, andsegmented agglomeration. In perfect agglomeration, the innovation
sector and intermediate goods sector agglomerate in one region. In
partial agglomeration, intermediate goods firms partially agglomerate
in the region where the innovation sector agglomerates perfectly. In
segmented agglomeration, the innovation sector agglomerates in theregion where both intermediate goods sector and final good sector do
not agglomerate perfectly.
In addition, we show the comparison of the welfare of skilled workers in each steady state. Not surprisingly, the welfare of the skilled
in full agglomeration is always the highest. However, even though
there are transportation costs of final good, the welfare in segmented
agglomeration is not necessarily the lowest.
Key words: knowledge spillovers, transportation costs, inter-regional
trade,
JEL classification: F43: O18: R11∗

We are grateful to Kristian Behrens, Masahisa Fujita, Mototsugu Fukushige, Ryo
Horii, Masako Ikefuji, Shingo Ishiguro, Kazuo Mino, Yuki Miyamoto, Tomoya Mori, Se-il
Mun, Yasusada Murata, Ryoji Ohdoi, Tetsuo Ono, Katsuya Takii, Kazuhiro Yamamoto for
their helpful comments and suggestions. Of course, we are responsible 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.jp1

1

Introduction

In this paper, a Grossman-Helpman-Romer-type endogenous growth model
is developed with two regions, in which there are two types of workers (i.e.,skilled and unskilled) and a linkage between the production of consumption
good and that of differentiated intermediate inputs. The activity of the innovation sector results in the expansion of the differentiated goods. In thispaper, the term growth means the expansion of differentiated goods. The
innovation sector uses only skilled workers. We assume that the productivityof each skilled worker residing in a certain region is affected by the knowledge
capital, which is determined by an interaction among the skilled. Knowledge
capital in one region increases as the number of skilled workers residing inthis region increases. The productivity of the skilled increases as knowledge
capital increases. Thus, the productivity of skilled workers increases as their
number increases. Due to this assumption, skilled workers tend to agglomerate in one region. In this setting, we investigate spatial configurations andthe welfare of skilled workers.
Agglomeration of innovation activity is notable in economic activity. Furthermore, agglomeration of the innovation sector does not always occur inthe region where manufacturing sectors agglomerate. The statistics presented by the Japan Statistics Bureau show that the share of the numberof workers who work at academic research institutions in three metropolitan
areas (i.e., Tokyo and Kanagawa Prefectures, Aichi Prefecture, and Osakaand Hyogo Prefectures) is about 37%. In addition, the data presented at
the Japan Patent Office shows that the share of patent registrations in thethree metropolitan areas is more than 80%. Moreover, an examination of
the research and development sector of the Japanese motor industry showsthat the innovation sectors of the automobile industry are located in Kantoor Tokai area.
Turning to the United States, Audretsh and Feldman (1996) report thatthe distribution of innovation is not uniform and that the cause of the agglomeration of innovation is not merely the agglomeration of production.
The industry where knowledge spillovers are more widespread has a stronger
propensity for agglomeration in one region. However, the region in which agglomeration of the innovation activity occurs might not be necessary where
manufacturing activity is occurring. This is shown by comparing Audretsh
and Feldman (1996) with Hanson (1998). Both papers discuss the distribution of innovation activity in the United States. Audretsh and Feldman(1996) indicate that the agglomeration of the innovation sector in New England is notable. On the other hand, Hanson (1998) shows that the shareof the manufacturing sector of New England is low. In short, this implies2

that the places where innovation occurs may be different from those where
production takes place.
From a theoretical viewpoint, in terms of the migration of skilled workers and the production of patents, our model is close to that of Fujita andThisse (2003). Moreover, in terms of the supply side, our model is close
to that of Yamamoto (2003). In Fujita and Thisse (2003), the linkage between manufacturing firms and workers and knowledge spillovers lead to theagglomeration of firms and workers. On the other hand, we show that thelinkage between final good firms and intermediate goods firms leads to agglomeration of firms in both sectors. In Yamamoto (2003), the transaction
between intermediate goods sector and manufactured goods sector creates alinkage. In addition, there is a linkage between the innovation sector andmanufactured goods sector due to the assumption that the innovation sector
uses manufactured goods as inputs. However, there is no linkage between
the innovation sector and manufactured goods sector in our model. Furthermore, there is a difference between Yamamoto (2003) and our model interms of mobility. Although, in Yamamoto (2003), workers are identical andimmobile, we assume that there are two types of workers and mobile skilledworkers. By integrating Fujita and Thisse (2003) and Yamamoto (2003), aricher implication is obtained.
In our model, the economy has two trade patterns, one in which interregional trade of final good occurs, and one in which it does not. The economy has both trade patterns simultaneously or only the latter trade pattern.
Which economy will emerge depends on the relationship between transportation costs of final good and those of intermediate goods. When transportation
costs of intermediate goods are relatively high, the former economy occurs.
When transportation costs of intermediate goods are relatively low, the latter
economy emerges. It is possible for the economy to reach the following spatial configurations: full agglomeration, partial agglomeration, and segmented
agglomeration. In full agglomeration, the innovation sector and intermediate
goods sector agglomerate in one region. In partial agglomeration, intermediate goods firms partially agglomerate in the region where the innovation
sector agglomerates perfectly. In segmented agglomeration, the innovation
sector agglomerates in one region, and all intermediate goods firms agglomerate perfectly in the other. The innovation sector always agglomerates in one
region. This is because the more skilled workers agglomerate in one region,
the higher their wage rate due to knowledge spillovers among skilled workers.
Partial agglomeration and full agglomeration occur in both trade patterns. However, segmented agglomeration does not always emerge. This
agglomeration is accomplished when transportation costs of final good are
sufficiently low and those of intermediate goods are relatively high. The
3

reason that this agglomeration might occur is as follows. Since there aretransportation costs for intermediate goods, final good firms and intermediate goods firms tend to locate in one region. When transportation costs offinal good are sufficiently low, under some situations, it might be profitable
for skilled workers to agglomerate in the region where final good firms do not
locate and to consume final good while paying transportation costs.
We investigate the welfare of skilled workers in each agglomeration. The
welfare in full agglomeration is always the highest. However, the welfare
in segmented agglomeration may not necessarily be the lowest in spite ofrequiring transportation costs for final good. When transportation costs offinal good are not too low and if the initial distribution of skilled workers
in the region where the final good sector does not operate is high, skilledworkers will tend to agglomerate in this region, even if they can increase
their welfare by means of agglomeration in the region where final good firms
agglomerate.
This paper is organized as follows. In the next section, the basic model ispresented, and the two patterns relating to final good trade are described. In
section 3, the migration behavior of skilled workers is described. In section 4,
a spatial configuration under fixed distribution of skilled workers is presented.
In section 5, a spatial configuration when migration is allowed is shown. In
section 6, the welfare of the skilled workers is investigated. Section 7 is theconclusion.

2

The model with migration

2.1

The model

The economy consists of regions A and B. There are four production sectors:
agricultural, final good, intermediate goods, and innovation. Moreover, there
are two types of labor, skilled and unskilled. The innovation sector uses only
skilled labor to produce patents, which are needed for production of intermediate goods. While agricultural sector uses only unskilled worker, final goodis produced by using unskilled labor and a set composed of intermediate
goods. Intermediate goods are also produced by using unskilled workers andpatent. We chose the price of agricultural goods as the numeraire. Shippingof final good and intermediate goods incurs transportation costs that are of
the ice-berg cost form.
There exists L units of unskilled labor in the economy, which is immobile
between two regions, and L is constant over time. Each region is an identical amount of unskilled labor. Each skilled worker can move between two
4

regions by incurring positive cost. The total number of skilled workers in theeconomy is constant over time. Without loss of generality, the number ofskilled workers is normalized to 1. Although the number of skilled workers
is constant, growth occurs by knowledge capital.
Since our model is symmetric, we focus on the specification of region A.
The inter-temporal utility function of the consumer who consumes agricultural goods and final good is given by the following form:
Z ∞U=
e−ρt ln u(t)dt,(1)0where u(t) is the instantaneous utility at time t and ρ is the subject discountrate. The instantaneous utility function is as follows: u(t) = M (t)α A(t)1−α ,where M (t) and A(t) represent the consumption of homogeneous final good
and that of agricultural good, respectively, α(0 < α < 1) is the share of
expenditures devoted to final good. Solving consumer’s problem derives theindirect utility function of a type j consumer residing in A, vjA :vjA = αα (1 − α)1−α p−αM A ²jA ,(2)

where ²jA represents the expenditure for a type j worker residing in A.
Moving from one region to the other at time t incurs cost C(t), which isexpressed in terms of an individual lifetime utility. We define the lifetime
utility of consumer j at time 0 as follows:Xe−ρt C(th ),(3)
Uj (0) ≡ Vj (0) −hwhere th represents the sequence of time in which a consumer moves andZ ∞e−ρt ln[vj (t)]dt.
(4)
Vj (0) ≡0This expresses lifetime utility obtained by consumption of goods.
We assume that there exists a global and perfectly competitive capital
market in which bonds bear an interest rate equal to r(t) at time t. This
capital market makes interest rates in both regions equate: rA (t) = rB (t) =r(t). We have to specify consumer j’s intertemporal budget constraint. First,
the present value of wage income of consumer j is given as follows:Z ∞
Wj (0) =e−r̄(t) wjA(t) (t)dt,(5)0where wjA(t) (t) is the wage of consumer j residing in region A at time t andRtr̄ is the average interest rate between 0 and t: r̄ = (1/t) 0 r(τ )dτ . Using
5

this presents an inter-temporal budget constraint from the flow of the budget
constraint:Z ∞²j (t)e−rt dt = aj (0) + Wj (0),(6)
0where aj (0) is the value of initial assets. Given any location path sj (t),s = A, B, if ²j (t) is the expenditure path, such that the lifetime utility
is maximized subject to an inter-temporal budget constraint, ²̇j (t)/²j (t) =r(t) − ρ, at time t. Since this relationship applies to any consumer, we showĖ(t)= r(t) − ρ,E(t)t ≥ 0,(7)

where E(t) is an expenditure in the economy at time t.
Turn to the supply side. First, agricultural good is produced by constantreturns to scale technology. For simplicity, suppose that the input coefficient in the production of this good is 1. Because there is no transportation
cost, the nominal wages equate between two regions, and w = 1, where w
represents the wage rate of unskilled labor. We assume that the demand
for agricultural good is sufficiently large so that the demand for agricultural
good is not satisfied when this good is produced exclusively in one region.
From this assumption, this good is produced in both regions at equilibrium.
Then, final good is produced by constant returns to scale technology under perfect competition: MA (t) = IAβ (t)L1−β
M A (t), where MA (t) is the quantityof final good produced in region A, LM A (t) is the demand for unskilled laborin final good sector in region A, and β(0 < β < 1) is the share of the expenditure for the set of intermediate goods to the total expenditure for input
requirement. IA (t) is the composite of intermediate goods, which is given as
follows:h Z N (t)i1/θθIA (t) =zi (t) di,(8)0

where zi (t) is an intermediate input indexed by i, N (t) is the total numberof varieties available in the economy in time t, and θ represents the intensityof the input requirement for a variety of differentiated intermediate goods. If
we set σ = 1/(1 − θ), σ is the elasticity of substitution between each variety
of intermediate goods. 1
Finally, differentiated intermediate goods are produced in a monopolistically competitive sector with an increasing return-to-scale technology in eachvariety. Suppose that there is no cost of discrimination. This makes each1

The sufficient condition such that agricultural good is produced in both regions inequilibrium is α < σ/(2σ − β).

6firm produce only one variety. This implies that the number of the firms isequal to the number of varieties available. We assume that a unit of a patentis required when production is started and that a µ unit of unskilled labor isalso needed marginally in the production. Hence, the optimal pricing is
piA = piB =

σµ
,
σ−1

(9)

where piA (piB ) represents the prices of intermediate goods produced in region
A (B). Because of equality in the price between the two regions, we denotethat piA = piB = p. The operating profit is as follows:
π = px − µx =

µx
,
σ−1

(10)

where x is the size of the firms in intermediate goods sector.
Solving problems of final good firm yields the demand for intermediate
goods locating in region A:
ziA = β[pM A MA p−σ Gσ−1
+ τI1−σ pM B MB−σ Gσ−1
A
B ],

(11)

where pM A (pM B ) is the price of final good produced in region A (B), τI istransportation costs of intermediate goods, and GA (GB ) represents the priceindex of intermediate goods in region A (B), which is given as follows:
GA = [nA p1−σ + τI1−σ nB p1−σ ]1/(1−σ) ,(12)

where nA (nB ) is the number of intermediate goods firms locating in region
A (B) and nA + nB = N . (12) says that, given the number of intermediate
goods firms, the region with more intermediate goods firms has the lower
price index of intermediate goods. Due to the zero profit of final good firm,
the price of final good produced in region A is given as follows:
pM A = β −β (1 − β)−(1−β) GβA .

(13)

From our assumption on trade cost and the demand for agricultural goods,
the wage rates of unskilled workers in two regions are 1. Thus, final good is
produced in the region that has more intermediate goods firms since the moreintermediate goods firms locate, the lower the price index of intermediate
goods is.
An innovation is needed to produce a new variety. We assume that, once
the innovation is performed, an entrepreneur can have monopoly power on
the variety produced and the choice to relocate the production facilities freely
across regions at no cost. Since a unit of patent is needed to start production
intermediate goods, the total number of varieties is equal to that of patents.
7

2.2

The innovation sector

We will now turn to the innovation sector, which produces a patent by using only skilled labor. Following Fujita and Thisse (2003), the production
function of a patent in region A takes the following form:
ṅA =

KA
λA , ,b(14)

where KA represents the existing knowledge in region A, b is the requirementof skilled labor for the production of a unit of patent, and λA represents
the number (share) of skilled workers residing in region A. Furthermore,
the knowledge stock existing in each region is determined by the interaction
among skilled workers. When we denote hj as the knowledge which worker
j has, the available knowledge in region A is 2
Z λB
i ψ1
h Z λAψhψj dj ,hj dj + η
KA =
(15)
0

0

where ψ(0 < ψ < 1) represents a measurement of the complementarity among
the skilled with regard to the creation of knowledge and η(0 ≤ η ≤ 1) is theindex of knowledge spillovers between two regions. For simplicity, we assume
that the knowledge that worker j has is proportionate to the number ofexisting patents in the economy, i.e., hj = N . 3 From this assumption, we
can rewrite (15) as follows: 4
1

KA = N [λA + ηλB ] ψ .

(16)

To simplify the notation, we define kA = [λA + ηλB ]1/ψ . Substituting (16)
into (14), we find that ṅA = (N/b)[λA + η(λB )]1/ψ λA . Supposing that thelength of a patent is infinite, the firm producing differentiate intermediate
goods has monopoly power forever. This leads to the next motion:Ṅ = ṅA + ṅB =N[λkA (λ) + (1 − λ)kB (λ)],
b

(17)

2

Audretsh and Feldman (1996) show that industries in which knowledge plays a more
important role tend to cluster spatially. Therefore, we think that this formation is notunnatural.
3This is interpreted as the knowledge that an individual has is the entire stock accumulated in the past.
4
When η = 1, KA = N . This implies that knowledge is interpreted as a pure public
1

good. On the contrary, when η = 0, KA = N λAψ . This describes a situation in whichknowledge is a local public good. Accordingly, η is interpreted as a measure of the localnessof knowledge.

8where λ ≡ λA . For simplicity of notation, we set f (λ) = λkA (λ) + (1 −λ)kB (λ). Using f (λ), we rewrite (17) as Ṅ = [f (λ)/b]N . Thus, when the
distribution of the skilled is λ, the growth rate of the number of varieties is:
5

f (λ)
.
(18)
b
Recalling (14), since KA is exogenous for the innovation sector, the marginal
product of skilled workers (or the marginal product of migration) is KA /b.
Therefore, the unit cost for the production of a patent is bwA /N kA . Free
entry and zero profit in the innovation sector make the value of a patentdeveloped in region A ΠA = bwA /N kA . Denoting the wage rate of skilled∗workers in region A on equilibrium as wA, the wage is represented as follows:g(λ) =∗wA=

ΠA N kA.
b(19)

In the intermediate goods sector, the value of a patent is equal to that of thefirm due to free entry at every moment t.

2.3

Trade patternNow, we investigate the location of the final good sector. As shown in Yamamoto (2003), there are two patterns regarding the location or trade of the
final good sector in the economy: final good is produced in both regions, orfinal good is produced in one region.
Since the economy is symmetric, we focus on the case 1/2 ≤ γ ≤ 1, whereγ is the share of intermediate goods firms locating in region A. This implies
that, if the agglomeration of a final good firm (perfect or not) emerges, itwill occur in region A. This is because, when 1/2 ≤ γ ≤ 1, the price index ofintermediate goods in region A is lower. From (13), it is clear that pM A ≤pM B
if GA ≤ GB . Moreover, if final good is produced in both regions, the priceof final good produced in region B must be lower than the price includingtransportation costs of this good produced in region A. In short, the tradepattern of final good is stated as follows.
1. (pattern 1) If pM A τM ≥pM B , final good is produced in both regions andthere is no trade of final good. 6
5

It is verified that g(λ) is symmetric about 1/2 and such that g(0) = g(1) = 1/b. In
addition, it is clear that ∂g(λ)/∂η > 0 and ∂ 2 g(λ)/∂η 2 > 0. Therefore, the growth rate of
the number of varieties is the increasing function of η, and, when η = 1, the growth rate
reaches its maximum value. That is, the strength of inter-regional knowledge spillovers
affects the growth rate.
6
More precisely, we have to state that, if pM A > pM B , final good is produced in both9

2. (pattern 2) If pM A τM < pM B , the production of final good is operated
only in region A and there is trade of final good.
From the above discussion, rearranging pM A τM < pM B clarifies that, if
γ is larger than a critical value, γ̂, then final goods sector operates only inregion A, where γ̂ is:
1−σ

γ̂ ≡

1 − τI1−σ τMβ
1−σ

,

(20)

(1 + τMβ )(1 − τI1−σ )This demonstrates that γ̂ > 1/2. Moreover, it is obvious that, if γ̂ > 1, thenthe following relationship holds:
τM > τIβ .

(21)

When (21) holds, perfect agglomeration of final good firm in region A cannot
occur. This is due to the relatively high transportation costs of final goodcompared to those of intermediate goods.
Then, we investigate the prospect for the economy when the trade is inpattern 2. If intermediate goods firms work in both regions, the operating
profit (the output) must be equal between two regions. In short, 1/2 < γ < 1
if xA = xB , and γ = 1 otherwise. Substituting nA > 0 and nB = 0 into (11)indicates that πA > πB (xA > xB ). Therefore, when the trade is in pattern
2, γ = 1 holds.
Now, we examine what happens when the trade is in pattern 1. In pattern
1, intermediate goods firms locate in both regions, which implies that theoutput is equal between two regions. This shows the following relationship:
β+σ−1
Gβ+σ−1
EA = GB
EB ,
A

(22)

where EA (EB ) is the expenditure of region A (B). If γ = 1/2 and λ = 1/2,
GA = GB and EA = EB . That is, nA = nB . Therefore, it is obvious that,when γ = 1/2 and λ = 1/2, (22) holds. However, we would like to focus on
the equilibrium that 1/2 < γ ≤ 1 and 1/2 < λ ≤ 1. Here, note that theexpenditure in each region cannot be determined until λ is satisfied.
Rearranging (22) reveals the share of intermediate goods firms located ina region, γ, which is given as follows.
γ=

EA− τI1−σ
EB
.
EA
+
1)
(1 − τI1−σ )( EB

(23)

regions, and there is no trade of final good. However, suppose that if pM A = pM B , final
good is also produced in both regions and there is no trade final good.

10

This shows that, if the expenditure ratio and transportation costs of intermediate goods rise, the share of intermediate goods firms locating in region
A increases. This is due to transportation costs of intermediate goods andscale economy of producing intermediate goods.
From the above discussion, it is found that all equilibria are in pattern 1 orpattern 2. This implies that there is no equilibrium such that intermediate
goods firms agglomerate in a region in which final good sector does notagglomerate. This is because final good can be produced with lower cost if
final good firm locates in the region in which the number of intermediate
goods firms is higher.
We will now turn to the labor market. Since unskilled labor is used in all
sectors except the innovation sector, the clearing condition for the unskilledlabor market is
LA + LM A + LM B + LIA + LIB = L,
(24)
where LA is the demand for unskilled labor in the agricultural sector, LM A
(LM B ) is the demand for unskilled labor in the final good sector in segion A(B), LIA (LIB ) is the demand for unskilled labor in the intermediate goods
sector in region A (B), and L is the amount of unskilled labor supply in theeconomy. Deriving the clearing condition for unskilled labor markets in eachsector and substituting these outcomes into (24), we show the relationshipbetween unskilled labor and expenditure in the economy: E = σL/(σ − αβ).
This implies that, if L is constant, then E is also constant. Hence, E isconstant due to the assumption that L is constant. This leads to r(t) = ρ
from (7). We find that the equilibrium interest rate is equal to the subjective
discount rate.
Here, we need to specify the income of each worker. An intertemporalbudget constraint derives ²j = ρ[aj (0)+Wj (0)]. We assume that intermediate
goods firms are equally possessed among only skilled workers. 7 This implies
aL R= 0. From (5), the present value of the wage income of the unskilled∞
is 0 e−ρt w(t)dt = 1/ρ, since w = 1. The income of skilled workers is also
represented as follows:²H = ρ[aH (0) + WH (0)].

(25)

From the assumption, the initial endowment of the skilled is given the following form:aH (0) = nA (0)ΠA + nB (0)ΠB .
(26)
7

Picard et al. (2004) have shown that the spatial configuration is affected by the
structure of the ownership of firms working under monopolistic competition. In our model,
the simplest case as to ownership of the firms is considered.

11Note that ²H (0) is determined by the present value of income and the equilibrium wage rate of skilled workers.

3

Migration behavior

Recall the assumption that migration between two regions incurs the lossof utility. Following Fujita and Thisse (2003), the migration cost takes thefollowing form: 8|λ̇(t)|
C(t) =.
(27)
ΨThe left-hand side represents the cost in terms of the utility loss for migranttime at t. As for the right-hand side, λ̇ represents the flow of skilled workers
moving from one region to the other and Ψ is a positive constant interpreted
as an adjustment cost. When skilled workers move from region B to A, λ̇ ispositive. On the contrary, when skilled workers move from region A to B, λ̇
is negative.
Consider the following case, which will be relevant for the stability ofanalysis of the steady state at λ̃ ∈ [0, 1]. Let the initial distribution of theskilled be lower than λ̃. Suppose that T > 0 exists such that the flow ofskilled workers from B to A starts at time 0 and stops at time T . Briefly,
this is stated as follows:λ̇(t) > 0λ(t) = λ̃t ∈ (0, T )
t≤T

(28)

In this case, the equilibrium dynamics of skilled workers under the expectation (28) is derived as the following form:
·
¸
Z T
Ψeρt
aH (0) + WH (0; t)
pY A (s)ρtλ̇(t) =ln
− αΨee−ρs lnds(29)ρaH (0) + WH (0; T )pY B (s)tfor any t ∈ (0, T ). The process of deriving (29) is shown in Appendix 1.
In the next section, we investigate the equilibrium when λ is fixed.

4

The steady-state growth path when λ is
fixed.

In this section, we investigate the steady-state growth path under fixed λ.
8

The migration cost is interpreted as a congestion cost.

12

4.1

Location of the intermediate goods sector

The objective in this subsection is to find the location of intermediate sector,γ, given the distribution of the skilled, λ. To show this, some preparation isrequired.
Here, we find the profit (the output) of intermediate goods firms. Notethat intermediate goods firms are free to chose the region where they work
because patents are perfectly mobile. First, we derive the profits when intermediate goods firms locate in both regions. In this case, xA = xB holds.
Using (22), nA + nB = N and EA + EB = E yield the number of intermediate
goods firms in each region:
EA − τI1−σ EBnA =N,(1 − τI1−σ )EEB − τI1−σ EAnB =N.
(1 − τI1−σ )E(30)

It is obvious that nA > 0 and nB > 0 if and only if τI1−σ < (EA /EB ) < 1/τI1−σ .
(11) reveals the output of intermediate goods firms in both regions. This isxA = xB =αβ(σ − 1)E.
σµN

Then, it is shown that nA = N and nB = 0 if and only if 1/τI1−σ ≤ (EA /EB ).
In this case, the output of intermediate goods firms in each region isxA =αβ(σ − 1)αβ(σ − 1) 1−σE≥(τI EA + τIσ−1 EB ) = xB .
σµN
σµN

Finally, it is also found that nA = 0 and nB = N if and only if τI1−σ ≥(EA /EB ). In this case, the output of intermediate goods firms in each regionisαβ(σ − 1) σ−1αβ(σ − 1)xA =(τI EA + τI1−σ EB ) ≤E = xB .
σµN
σµN
Since intermediate goods firms choose the region in which their own profit islarger, the equilibrium profits of intermediate goods firms are
π = max{πA , πB } =

αβ
E.
σN

(31)

From (17) and (18), it is clear that the total number of intermediate
goods in the economy is N (t) = N (0)eg(λ)t , where N (0) is the initial number of varieties Rin the economy. Since the value of each intermediate goods∞
firm is Π(t) = t e−ρ(τ −t) π(τ )dτ and E is constant, the aggregate value of13

intermediate goods firms in the economy is given as follows and denoted asã(λ);
αβE
N (t)Π(t) =≡ ã(λ).
(32)σ(ρ + g(λ))
Substituting (32) into (19) derives the wage rate of skilled workers in eachregion. Note that the aggregate expenditure in each region at any timeis Es (λ) = (L/2) + λs ²s (λ). Using this yields the ratio of the aggregate
expenditure in region A to that in region B as follows:
£
1/ψ ¤
L
+ λã(λ) ρ + (λ+η(1−λ))EA (λ)
2
b
=
.
£
1/ψ ¤
L
EB (λ)+ (1 − λ)ã(λ) ρ + (ηλ+1−λ)
2

(33)

b

Substituting λ = 1, 1/2, and 0 into this shows that EA (1)/EB (1) > 1,
EA (1/2)/EB (1/2) = 1, EA (0)/EB (0) < 1. 9
Here, we show the relationship between the expenditure ratio and λ. The
process used to explore this relationship is shown in Appendix 2. As a result,the following relationship holds:
d[EA (λ)/EB (λ)]> 0,dλλ ∈ (0, 1).

(34)This suggests that γ is an increasing function of the expenditure ratio andthe expenditure ratio increases monotonously as λ rises. This is described in−(1−σ)
the figures. Figure 1 describes the case in which τI< (σ +αβ)/(σ −αβ).
In this case, intermediate goods firms locate in both regions for any λ. On−(1−σ)
the other hand, Figure 2 represents the case τI≥ (σ + αβ)/(σ − αβ).
In this case, there are two configurations. One is that in which intermediate
goods are produced in both regions. The other is that in which intermediate
goods sector works in one region. The former will emerge if λ is sufficiently
close to 1/2, (i.e., λ00 < λ < λ0 ). 10 The latter will occur if λ is sufficiently
close to 1 or 0 (i.e., λ00 < λ or λ < λ0 ). In other words, if the difference of thenumber of skilled workers is large, intermediate goods firms will agglomerate
in the region in which more skilled workers reside, and if the difference is notlarge, intermediate goods firms will locate in both regions and will partially9

Writing more precisely, it is clear that
EA (1)
σ + αβEA (1/2)
EA (0)
σ − αβ
=
> 1,
= 1,
=
< 1.
EB (1)
σ − αβEB (1/2)
EB (0)
σ + αβ10 0

λ and λ00 are the values associated with EA (λ0 )/EB (λ0 )EA (λ00 )/EB (λ00 ) = τI1−σ , respectively.

14=−(1−σ)

τI

andagglomerate in the region in which more skilled workers reside. Agglomeration of intermediate goods firms is derived from the home market effectthrough the demand for final good.
When the location is pattern 2, intermediate goods firms perfectly agglomerate in the region where final good firms agglomerate, as mentioned
before.

5

The steady-state growth path when migration is allowed.

In this section, we investigate the steady state when migration of skilledworkers is allowed. To do this, the lifetime utility of skilled workers is necessary in two regions associated with the growth path under any fixed λ. In
the course of this process, the values of λ in equilibrium are acquired. The
lifetime utility of skilled workers in region s is shown as follows:Z ∞Vs =e−ρt ln vs (t; λ, γ)dt,(35)0where Vs is the lifetime utility of skilled workers in region s and vs (t; λ, γ) is
the instantaneous utility at time t. From this, the difference of the lifetime
utility of skilled workers between the two regions is·¸
Z ∞vA (t; λ, γ)−ρt
VA (0; λ, γ) − VB (0; λ, γ) =
e ln
.
(36)
vB (t; λ, γ)
0The expenditure for skilled workers in each region is represented as follows:·¸ks (λ)²s (λ) = ã(λ) ρ +.
(37)bFirst, an effort is made to determine the locations in which pattern 1
appears.

5.1

The steady-state growth path when trade is in pattern 1

In this pattern, final good firm does not engage in trade. Deriving the ratio
between the indirect utility in region A to that in region B from (2) and (13)and setting Φ1 (λ, γ) = vA (t; λ, γ)/vB (t; λ, γ), the difference of the lifetime

15

utility is rewritten as follows: 11
Z ∞1
VA (0; λ, γ) − VB (0; λ, γ) =e−ρt ln Φ1 (λ, γ) = ln Φ1 (λ, γ).
ρ
0

(38)

This implies that VA will be larger (smaller) than VB if Φ1 is larger (smaller)
than 1. Recall that GA = GB and kA = kB when γ = 1/2 and λ = 1/2.
Thus, it is obvious that VA (0; 1/2, 1/2) = VB (0; 1/2, 1/2) holds. This implies
that full dispersion (λ = 1/2, γ = 1/2) is always the steady-state equilibrium.
Recall that ∂kA (λ)/∂λ > 0, ∂kB (λ)/∂λ < 0, and kA (λ) = kB (λ) = 1 forη = 1. Hence, for any η > 0,∂Φ1 (λ, γ)≥ 0,∂λ∂Φ1 (λ, γ)≥ 0,∂γ

λ ∈ (0, 1), γ ∈ (0, 1).

Hence, it is determined that VA will be larger (smaller) than VB if γ and λ arelarger (smaller) than 1/2. This states that the economy is on a steady-state
equilibrium under λ = 0, 1/2, 1. We denote λ in steady-state equilibrium as
λ̃. It is shown in Appendix 3 that λ̃ = 1/2 is unstable and λ̃ = 0, 1 is stable.
As shown in Fujita and Thisse (2001), the self-fulfilling nature of the
migration process makes stability more difficult to define. It is possible thatseveral perfect foresight solution under the same initial distribution of skilledworker, λ0 in our model. As a result, for a given steady-state growth pathunder λ̃(= 0, 1/2, 1), there might be a neighborhood Λ of λ̃ such that, for any
λ0 ∈ λ, the equilibrium path based on a particular expectation converges tothis steady-state growth path while another equilibrium path diverges fromthe same steady-state growth path. To avoid such a difficulty, following
Fujita and Thisse (2001), we impose some restrictions on the expectations
that have to be satisfied when an equilibrium path converges into a steadystate growth path. Due to perfect foresight, this is amounts to imposing a
restriction on the equilibrium path itself. Hence, we introduce the following
restrictions.
Suppose that λ̃ ∈ [0, 1] and λ0 ∈ [0, 1] are not equal. If λ(t) is an equilibrium path satisfying the initial condition, λ0 , this path satisfies the monotonic convergence hypothesis under λ̃ (we call this the mc-hypothesis) when
0 < T ≤ ∞ such that,when λ0 < λ̃,

λ̇(t) > 0,

11

f or t ∈ (0, T ),From (2) and (13), we find that the ratio of the indirect utility in region A to that inregion B is as follows:¸−αβ/(1−σ)·γ + τI1−σ (1 − γ)vA (t; λ, γ)bρ + kA (λ)=.
vB (t; λ, γ)bρ + kB (λ)τI1−σ γ + 1 − γ

16when λ0 > λ̃,

λ(t) = λ̃,λ̇(t) < 0,λ(t) = λ̃,f or t ≤ T,
f or t ∈ (0, T ),f or t ≤ T.

(39)
(40)

The steady-state growth path under λ̃ will be stable if there is a neighborhood
Λ of λ̃ such that, for each λ0 ∈ Λ with λ0 6= λ̃, there is an equilibrium paththat satisfies the mc-hypothesis under λ̃. The steady-state growth path will
not be stable otherwise. We show in Appendix 3 that λ = 0, 1 is stable underλ̃.
The location of intermediate goods firms in the case of λ = 1 remains to bestudied. As mentioned in 4.1, when λ = 1, the configuration of intermediate
goods firms is determined by the relationship between the expenditure ratio
and transportation costs of intermediate goods.
First, we investigate the location of the intermediate goods sector whenτI1−σ < (σ − αβ)/(σ + αβ). In this case, the definitions of λ0 and λ00 expose
the following relationship:
EA (λ0 )1
σ + αβEA (1) EA (λ00 )σ − αβEA (0)
1−σ
=
>
=
,
=
τ
<
=
.
I
1−σ
EB (λ0 )σ − αβEB (1) EB (λ00 )σ + αβEB (0)
τI
Due to d[EA (λ)/EB (λ)]/dλ > 0, it is clear that λ0 > 1 and λ00 < 0. In−(1−σ)
holds. This situation
addition, for λ ∈ (0, 1), τI1−σ < EA (λ)/EB (λ) < τI
is described in Figure 3. Note that λ moves gradually and γ is able to jump
due to the assumption that skilled workers can move while incurring costs
and intermediate goods firms can move freely. The points on the curve in
Figure 3 satisfy the location equilibrium. This is so because of the following
MB holds, which implies that
MA < Gβ+σ−1
reason. Below the curve, Gβ+σ−1
BAxA < xB holds. That is, the profits of intermediate goods firms in region B
are higher than those in region A. Hence, the vertical line through γ = 1 does
not satisfy the location equilibrium. Above the curve, the opposite is true.
12
In short, we find that, when τI1−σ < (σ − αβ)/(σ + αβ), a configurationin which intermediate goods firms always locate in both regions is chosen as
the steady state. We denote this steady state as partial agglomeration underno trade in region s, (s=A, B) (PAN in s), where s represents the region inwhich the innovation sector locates.
Next, we consider the case τI1−σ ≥ (σ − αβ)/(σ + αβ) in the same way. It
is obvious that λ0 ≤ 1 and λ00 ≥ 0. 13 This situation is described in Figure 4.
12

Of course, due to the symmetry of the model, the points satisfy γ = 0 but not thelocation equilibrium.
13
Written more precisely, it is clear that
EA (1) EA (λ00 )σ − αβEA (0)EA (λ0 )σ + αβ−(1−σ)= τI=,= τI1−σ ≥=.
≤0EB (λ )
σ − αβEB (1) EB (λ00 )σ + αβEB (0)

17

The points on the curve satisfy the location equilibrium. The reason is given
above. In this case, the steady state occurs when all intermediate goods firms
and the innovation sector agglomerate in one region. We denote this steadystate as full agglomeration under no trade in region s, (s=A, B) (FAN in s).
We summarize the above discussion as a proposition.
Proposition 1 When there is no trade of final good (τIβ ≤ τM ), the stableequilibrium is as follows:• if transportation costs of intermediate goods are sufficiently high (τI1−σ <[σ − αβ]/[σ + αβ]), PAN in s (s=A, B) occurs.
• if transportation costs of intermediate goods are sufficiently low (τI1−σ ≥[σ − αβ]/[σ + αβ]), FAN in s (s=A, B) emerges.
• As transportation costs of intermediate goods become lower, this change
occurs more smoothly.

5.2

The steady-state growth path when trade is in pattern 2

We investigate this in the same manner shown in the previous subsection.
Setting Φ2 (λ) = vA (t; λ, γ)/vB (t; λ, γ), the difference of the utility of skilledworkers in region A and region B is represented as follows: 14Z ∞1e−ρt ln Φ2 (λ)dt = ln Φ2 (λ).
VA (0; λ, γ) − VB (0; λ, γ) =(41)
ρ0It is clear that VA is larger (smaller) than VB if Φ2 (λ) is larger (smaller) than
1. We denote the value associated with Φ2 (λ) = 1 as λ̂. As mentioned before,
it is shown that ∂kA (λ)/∂λ > 0 and ∂kB (λ)/∂λ < 0 and that, when η = 1,
kA (λ) = kB (λ) = 1 for all λ. Thus, it is clear that, for any η > 0,dΦ2 (λ)≥ 0,dλλ ∈ (0, 1).

14

(42)

When the difference in the indirect utility of skilled workers in each region is derived
by the same manner presented in previous subsection, this is represented as follows:²jA
vA (t; λ, γ)bρ + kA (λ) 1
= −α
=
−α .
vB (t; λ, γ)bρ + kB (λ) τMτM ²jB
Note that this is dependent on only λ.

18This implies that VA (λ) is larger (smaller) than VB (λ) if λ is larger (smaller)
than λ̂.
Now, we need to investigate the property of λ̂. From the definition of−αλ̂, bρ + kA (λ̂) = [bρ + kB (λ̂)]τM
must hold. Rearranging this by using thedefinition of ks , s = A, B yields the following relationship:
1

1

−α
−α
(1 − τM)bρ = τM(η λ̂ + 1 − λ̂) ψ − (λ̂ + η(1 − λ̂)) ψ .

Differentiating this equation reveals the following relationship:
dλ̂
< 0.
dτM

(43)

In addition, it is clear that, when τM ≥ τ̂M , λ̂ ≤ 0, where
µ
τ̂M =

1

bρ + η ψbρ + 1

¶− α1

(44)

If transportation costs of final good are higher than τ̂M , λ̂ will be less than 0,that is, λ̂ will not exist. In this case, residing in region B is not profitable forthe skilled, since transportation costs of final good are not sufficiently low.
Hence, the skilled do not locate in region B. However, if transportation costsof final good decrease so that the value of τM exceeds τ̂M , λ̂ will be larger thanzero. When this happens, there is a range of λ such that locating in region
B is profitable for skilled workers due to sufficiently low transportation costsof final good. In this case, if the initial share of the skilled workers in region
A, λ0 , is larger than λ̂, the steady state will be a case in which all sectors
agglomerate in region A, and, if that is lower than λ̂, final good sector andall intermediate goods firms will agglomerate in one region, and the skilledworkers will agglomerate in the other. We denote the former steady state asfull agglomeration under trade in region A (FAT in A) and the latter one assegmented agglomeration under trade in region B (SAT in B).
From the above discussion, it is clear that the economy in this patternis on a steady state equilibrium under λ = 0, λ̂, and 1 when τM ≤ τ̂M andon a steady state under λ = 1 when τM > τ̂M . We denote the value of λ on
steady state equilibrium as λ̃. Using the same technique shown in Appendix
3, it is demonstrated that, if τM ≤ τ̂M , λ̃ = λ̂ is unstable and λ̃ = 0 and
1 are stable; if τM > τ̂M , λ = 1 is stable. That is, the stable steady state
is FTA or SAT. The steady state that the economy reaches is determinedby transportation costs of final good and the initial distribution of skilledworkers. Due to the symmetry of the model, there is the value of λ, which
resembled λ̂. We denote this value as λ̄.
19We summarize the above discussion as the following proposition. 15
Proposition 2 When there is trade of final good, the stable steady state is
FAT in A or SAT in B. Which steady state occurs depends on transportation
costs of final good and the initial distribution of skilled workers.
• If τM ≥ τ̂M , FAT in A occurs regardless of the initial distribution of
the skilled.
• If τM < τ̂M and λ̂ < λ0 , FAT in A will emerge.
• If τM < τ̂M and λ̂ > λ0 , SAT in B will occur.
It is interesting to investigate what happens in the entire economy. It
must be recalled that the trade is only in pattern 1 when τIβ ≤ τM and bothtrade patterns occur simultaneously in the economy when τIβ > τM . The
economy under pattern 1 of trade has already been shown in the previous
subsection. Therefore, studying an economy in which both trade patterns
occur simultaneously is a very interesting proposition.
First, the relationship among τI , τM , and γ̂ is shown. From (20), it isobvious that ∂γ̂/∂τI < 0, ∂γ̂/∂τM > 0. In addition, (23) and EA (1)/EB (1) =(σ − αβ)/(σ + αβ) yield that the values of γ = γ̂ and λ = 1 hold if τM = τ̄M ,where¶β/(1−σ)µσ − αβ.
(45)τ̄M ≡σ + αβAs mentioned before, if the value of γ reaches γ̂, then the trade pattern
will change instantaneously, which implies the number of the possible steadystates which the economy can reach is determined by transportation costsof final good. That is, the relationship among τM , τ̂M , and τ̄M affects thenumber of steady states that the economy reaches.
The relationship between τ̂M and τ̄M is then presented. Using (44) and(45), it is shown that τ̂M > τ̄M holds if the expenditure share for final good, α,
the subject discount rate, ρ, and the expenditure share for intermediate goods
in final good sector, β, are low and the elasticity of substitution between any
varieties of intermediate goods, σ, is high. In this paper, we focus on adeveloped economy. Therefore, we assume τ̂M ≤ τ̄M .
The above relationships present the economy under τ̂M ≤ τ̄M < τM in
Figure 5. Recall that λ moves gradually and γ can jump. In this case, the15Consider the pattern in which the final good sector locates in region B. When τM > τ̂M ,the steady state is FAT in B regardless of the initial distribution of skilled workers. WhenτM < τ̂M , if λ0 < λ̄, the steady state is FAT in B, and, if λ0 > λ̄, the steady state is SAT
in A.

20

economy has four multiple equilibria (PAN in A, PAN in B, FAT in A, and
FAT in B) regardless of the initial distribution of the skilled. The steadystate chosen by the economy is determined by expectation. 16
Proposition 3 When τIβ > τM , if τ̂M ≤ τ̄M < τM , the stable steady state is
PAN in s, or FAT in s (s=A, B), regardless of the initial distribution of theskilled. 17
Finally, the value λ associated with γ = γ̂ when τM < τ̄M is found.
Denote this value as λ∗ . It is clear from (23) that 18− 1−σ
EA (λ∗ )β.
=τMEB (λ∗ )This implies that EA (λ∗ )/EB (λ∗ ) < EA (1)/EB (1), which derives λ∗ < 1 dueto the fact that d[EA (λ)/EB (λ)]/dλ > 0. 19 In addition, it is found that
there is a value of λ that is analogous to λ∗ , which is called λ∗∗ , and it isobvious that λ∗∗ > 0 since the model is symmetric. 20
In Figure 7, we show the economy when τ̂M ≤ τM ≤ τ̄M . In this case,the economy has two multiple equlibria (FAT in A and FAT in B) regardlessof the initial distribution of the skilled. The steady state that the economy
chooses is determined by expectation.
In Figure 8, we show the economy when τM < τ̂M ≤ τ̄M . In this case, theeconomy has four steady states (FAT in A, FAT in B, SAT in A, and SAT in
B). When the initial distribution of the skilled is lower than λ̂, it is possible
16

We describe the economy in Figure 6 under the condition that τ̄M < τM < τ̂M holds.
In this case, the economy has six multiple equilibria (PAN in A, PAN in B, FAT in A,
FAT in B, SAT in A, and SAT in B). The steady state that the economy might reachis determined by the initial distribution of skilled workers and the expectation. Whenλ0 ≤ λ̂, there are three steady states that the economy might reach (PAN in B, FAN in B,
and SAT in A). When λ0 ≥ λ̄, the number of steady states that the economy might reachmight be three (PAN in A, FAN in A, and SAT in B). When λ̂ < λ0 < λ̄, the economy has
the potential to reach each steady state. Given the initial distribution of skilled workers,
the steady state is determined by expectation.
17
When τIβ ≤ τM , if τ̄M < τM < τ̂M and λ0 ≤ λ̂, the steady state is FAT in B, PAN in
B, or SAT in A; if τ̄M < τM < τ̂M and λ0 ≥ λ̄, the steady state is FAT in A, PAN in A,
or SAT in B; if τ̄M < τM < τ̂M and λ̂ < λ0 < λ̄, the economy has the potential to reacheach steady state.
18
Of course, from this equation, it is also shown that, when τM = τ̄M , EA (λ∗ )/EB (λ∗ ) =(σ + αβ)/(σ − αβ) holds. Hence, we find that λ∗ = 1, in this case.
19This is because of EA (λ∗ )/EB (λ∗ ) < (σ + αβ)/(σ − αβ) and EA (1)/EB (1) = (σ +αβ)/(σ − αβ).
20More precisely, λ∗∗ is the value at which intermediate goods firms begin to agglomerate
in region B and the trade pattern changes in the case that τM < τ̄M .

21

for the economy to reach one of two steady states (FAT in B and SAT in
A). When the initial distribution of the skilled is higher than λ̄, there are
two steady states that the economy might reach (FAT in A and SAT in B).
When λ̂ < λ0 < λ̄, the economy has the potential to reach four steady states.
Given the initial distribution of the skilled, the steady state is determinedby expectation.
We summarize the above discussion as follows.
Proposition 4 When τIβ > τM ,• if τ̂M ≤ τM ≤ τ̄M , the stable steady state is FAT in s (s=A, B), regardless of the initial distribution of the skilled;• if τM < τ̂M ≤ τ̄M and λ0 < λ̂, steady state is FAT in B or SAT in A;• if τM < τ̂M ≤ τ̄M and λ0 > λ̄, the steady state is FAT in A or SAT in
B;• if τM < τ̂M ≤ τ̄M and λ̂ < λ0 < λ̄, the economy will have the potential
to reach four steady states.

5.3

Discussion

In this section, we discuss the effect of changes in transportation costs of final
good, τM , and those of intermediate goods, τI , on the spatial configuration.
When transportation costs of intermediate goods are relatively higher thanthose of final good, τIβ > τM . In this case, both trade patterns emerge
in the economy. Then, the economy will reach partial agglomeration or
full agglomeration (Figure 5). Which spatial configuration it will reach isdetermined by expectation.
Suppose that transportation costs of final good are reduced. The reductions of those costs results in partial agglomeration ceasing to exist. Thus,
the economy will reach only full agglomeration (Figure 7). The relatively
high transportation costs of intermediate goods cause the linkage between
final good firm and intermediate goods firms to be strong, which leads to full
agglomeration. Moreover, skilled workers think that it is preferable to residein the region where final good sector locates since transportation costs of final
good are not sufficiently low. We think that this situation is representativeof the fact that urban population growth is linked with the development oftransportation technology, shown in Bairoch (1988).
Now, suppose that transportation costs of intermediate goods decline.
We think that this is possible because of the development of communication22

technology. When the reductions in transportation costs of intermediate
goods are sufficient to hold τIβ ≤ τM , the trade pattern changes, and tradeof final good, consequently, there is no trade of final good. This is because
transportation costs of final good are relatively high. Thus, final good sector
tends to locate in both regions to save on transportation costs. The reductionof transportation costs of intermediate goods in 1950 is shown in Baldwin etal. (2001). In addition, a regional distribution of manufacturing employment
from 1850-1990 is seen in Hanson (1993). It was then shown that, in 1950,
the progress of dispersion in manufacturing was faster.
How the economy is affected by the change of transportation costs in thissituation is stated. At first, the economy will accomplish partial agglomeration under no trade (Figure 3) due to the linkage between final good firm andintermediate goods firms. However, as transportation costs of intermediate
goods continue to decline, the steady state is only full agglomeration underno trade (Figure 4). This is because it is profitable for intermediate goods
firms to agglomerate in the region where skilled workers locate. This is derived from the indirect effect through final good, which are consumed morein the region where skilled workers agglomerate. Now, suppose that transportation costs of final good decline and τIβ > τM thus holds. This might bederived by, for instance, the improvements in the technology associated withthe delivery of goods. The steady state is full agglomeration under tradeor segmented agglomeration under trade (Figure 8). Which steady stateis accomplished is determined by expectation. If the expectation leads tosegmented agglomeration in the economy, this spatial configuration will be
fulfilled. This case is described in Audretsh and Feldman (1996), who showedthat the agglomeration of production does not attract the agglomeration of
innovation. Workers who have a special ability tend to agglomerate in one
region to enjoy spillovers that are the most important for innovation activity.
The region in which the innovation sector agglomerates is not necessarily the
one in which the activities of production of goods agglomerate.

6

WelfareIn this subsection, we investigate the welfare under FAT in A, PAN in A,
and SAT in B when both trade patterns emerge. 21 The welfare gap between
each equilibrium is shown by using (2) and (13). 22 For simplicity, we omit
21

Since the model is symmetric, we focus on the case in which final good sector agglomerates in region A when this sector agglomerates.
22
Note that workers in the region in which final good firms locate can always acquire
final good without transportation costs and the wage rate of skilled workers does not23the superscript, which represents skilled workers, H. First, the welfare under
FAT in A with that under PAN in A is compared. The ratio of the welfare
between two steady states is given by the following form;
¸−αβ
·
αβ
vA (t; 1, 1)
GA (1)== [γ + τI1−σ (1 − γ)] 1−σ > 1.
vA (t; 1, γ)
GA (γ)
Thus, it is clear that VA (0; 1, 1) − VA (0; 1, γ) > 0. This implies that, forskilled workers residing in region A, welfare in a case in which intermediate
goods firms agglomerate in region A is higher than that in a case in whichintermediate goods firms locate in both regions. This is because the priceindex is lower in the region that has more intermediate goods firms.
Next, the relationship between the welfare of PAN in A and that of SAT
in B is as follows. 23·¸−αvA (t; 1, 1)
pY A
α
=
= τM
> 1.
vB (t; 0, 1)
pY B
Therefore, it is obvious that VA (0; 1, 1) − VB (0; 0, 1) > 0. This is due tothe fact that workers residing in region A can acquire final good withouttransportation costs.
Finally, the welfare of PAN in A with that of SAT in B is shown. This is
written as follows:αβvA (t; 1, γ)
α= [γ + τI1−σ (1 − γ)] σ−1 τM≥ (<)1vB (t; 0, 1)

whereas γ ≥ (<)γ ∗ .

1−σ

τMβ − τI1−σγ =.
1 − τI1−σ
Whether the ratio of the welfare in this case is larger than 1 or not depends onthe value of γ at PAN in A. Using (23) and EA (1)/EB (1) = (σ+αβ)/(σ−αβ)derives the value of γ under λ = 1:∗γ=

σ + αβ − τI1−σ (σ − αβ).
2σ(1 − τI1−σ )Computing using the two equations above reveals the following relationship:as τI1−σ ≥ (<)

VA (0; 1, γ) − VB (0; 0, 1) ≤ (>)01−σ2στMβ − 1.
σ + αβchange if the value of λ does not vary. In addition, the price of final good is affected by
the location of intermediate goods firms.
23
We have already shown that the wage rate of the skilled when λ = 1 is equal to thatwhen λ = 0.

24

1−σ

This is interpreted as follows. τI1−σ > [2σ/(σ + αβ)]τMbeta − 1 means transportation costs of intermediate goods are relatively low. While skilled workers residing in region A can acquire final good without paying transportation
costs, those residing in region B can consume final good while incurringtransportation costs. In addition, the cost of final good producer when theeconomy is under PAN in A is higher than when the economy is under SAT
in B. From the discussion above, the inequality is determined by the relationship between transportation costs of final good and those of intermediate
goods.
(1−σ)/β− 1, the welfare under SAT in B
Note that if τI1−σ < [2σ/(σ + αβ]τM
is lowest. However, the skilled workers tend to agglomerate in region B even
if they can increase their welfare by means of agglomeration in region A.
The discussion above is summarized as follows:
Proposition 5 The welfare of the skilled on FAT in A is the highest. The
welfare on SAT in B is higher than that on PAN in A, if the following relationship holds:
1−σ
2σ
τI1−σ >
τMβ − 1.
σ + αβ7Conclusion

In this paper, a Grossman-Helpman-Romer-type endogenous growth model
is developed with two regions. We show that there are multiple equilibria in
our economy. In addition, we investigate the welfare of skilled workers at thesteady state.
In our economy, there always exist multiple equilibria. Note that theeconomy has the potential to achieve segmented agglomeration. This describes the situation in which agglomeration of an innovation sector andagglomeration of production in manufacturing sector are spatially separate.
The relationship between transportation costs of final good and those ofintermediate goods determines the number of steady states that there are in
the economy. The steady state that the economy achieves is determined by
expectation and the initial distribution of the skilled. This implies that, inthe case in which expectation plays an important role, the government of aregion that would like to have agglomeration of firms and workers needs toimplement some coordination.
We investigate the level of welfare of skilled workers. Not surprisingly,
the welfare in full agglomeration is the highest. However, the relative magnitude between the welfare in partial agglomeration and that in segmented
agglomeration depends on the relationship between the transportation costs
25

of final good and those of intermediate goods. It is possible for the economy
to reach segmented agglomeration even if their welfare at this agglomeration is lowest. This shows that coordination failure occurs. From the pointof welfare analysis, some policies are needed to acquire the highest level of
welfare.

Appendix
Appendix 1
First, using (2), (4), and (25), we present the lifetime wage of the skilled who
migrates from B to A at time t as follows:
Z t
Z ∞
−ρτ
W (0; t) =e wB (τ )dτ +e−ρτ wA (τ )dτ.
(A.1)
0

t

Then, using (3) and (27), the lifetime utility is given by
U (0; t) = V (0; t) − eρt

λ̇(t)
,
Ψ

(A.2)

where V (0; t) is the lifetime utility gross of migration costs. V (0; t) is shownas follows:
¸
·Z t
Z ∞
1
1
−ρτ
−ρτ
e
ln[pY B (τ )]dτ +eln[pY A (τ )]dτV (0; t) = ln ρ+ ln[aH (0)+WH (0)]−α
ρ
ρ
0
t
α
1−α
+ ln α +ln(1 − α). (A.3)ρρ
From Fukao and Benabou (1993), it is found thatlim C(t) = 0.

t→T

(A.4)

Therefore, by using (A.2), (A.3), and (A.4), we haveU (0; T ) = V (0; T )
·Z T
¸
Z ∞
1
1
−ρτ
−ρτ
e
ln[pY B (τ )]dτ +eln[pY A (τ )]dτU (0; T ) = ln ρ + ln[aH (0) + WH (0)] − α
ρ
ρ
0
T
Z ∞
α
1−α
+(1 − α)e−ρτ ln pA (τ )dτ + ln α +ln(1 − α).
(A.5)
ρ
ρ
0

26

Since, in equilibrium, all migrants are indifferent to their migration time, it
is shown that U (0; t) = V (0; t) for all t ∈ (0, T ). Thus, rearranging (A.2)
and substituting (A.5) and U (0; t) = V (0; t) into (A.2), we find thatΨeρt
[V (0; t) − V (0; T )]
ρ
¸
·
Z T
pY A (τ )ΨeρtaH (0) + WH (0; t)ρt− αΨee−ρτ ln=lndτ,ρaH (0) + WH (0; T )pY B (τ )t(A.6)

λ̇(t) =

for any t ∈ (0, T ).

Appendix 2
We investigate the relationship between the expenditure ratio, EA /EB , andthe distribution of skilled workers, λ. Recall that the aggregate expenditurein the economy is constant. Thus, it is sufficient to investigate the relationship between EA (λ) and λ. To investigate this relationship, it is necessary
to specify ã(λ).
First, we derive the aggregate expenditure in region A;·¸
LkA
EA = + ã(λ) ρ +
(A.7)
2
b
Now, using (32) for finding ã(λ), we can identify it as follows:ã(λ) =αβL.
(σ − αβ)(ρ + g(λ))(A.8)

Next, we show the derivative of g(λ) and ã(λ) in terms of λ:10g 0 (λ) = [kA (λ) + λkA0 (λ) − kB (λ) + (1 − λ)kB
(λ)],
b
and

αβLg 0 (λ)
dã(λ)=−.
dλ(σ − αβ)(ρ + g(λ))2From (A.7) and these equations, it is verified that dEA (λ)/dλ > 0, λ ∈ (0, 1).
Therefore, it is shown that 24d(EA /EB )
> 0,
dλ
24

λ ∈ (0, 1).

Since EA + EB = E is constant, it is sufficient to verify dEA /dλ > 0.

27

Appendix 3
As shown before, the economy follows a steady-state growth path under
different values of λ. When the economy is in pattern 1, the values of λ at
the steady state are 0, 1/2, and 1. On the contrary, when the economy isin pattern 2, the values of λ at the steady state are 0, λ̂, and 1. Here, theanalysis of stability is shown.
Now, we show that, when trade is in pattern 1, the steady-state growth
path under λ̃ = 1/2 is unstable, while the steady-state growth path underλ̃ = 1 is stable. Consider the case λ̃ = 1/2. Since two regions are insymmetry, we focus on a value of λ0 that is lower than 1/2. In this case, theequilibrium migration dynamics of the skilled worker is represented by (55).
Now, using the asset value of intermediate goods firms and the growthrate, we find thatRτN (t)= e− t g[λ(v)]dv .
N (τ )
Using this equation, at each t ≥ 0, the following relationship reveals
Z
αβE ∞ − R τ [ρ+g[λ(v)]]dv
a(t) = N (t)Π(t) =
dτ.
(A.9)
e t
σ
t
In addition, from dynamics,a(T ) =αβE1.
σ ρ + g(λ̃)(A.10)

Hence, it is found that a(t) is not dependent on N (0) = N0 . This implies
that N0 does not affect the equilibrium values of variables except N (t).
Then, the wage of skilled workers in region s at time t is as follows:
ks (t).
(A.11)b
By definition, we write as follows, for t ≥ T ,
Z t
Z T
−ρτ
W (0; t) =e wB (τ )dτ +e−ρτ wA (τ )dτ
0
t
Z T
Z T
−ρτ
= W (0; T ) −e wB (τ )dτ +e−ρτ wA (τ )dτtt
Z T
kA (τ ) − kB (τ )= W (0; T ) +e−ρτ a(τ )dτ,(A.12)bt
ws (t) = a(t)

where

Z T
W (0; T ) =0e−ρτ a(τ )kA [λ(τ )]a(T )kA (λ̂)e−ρT
dτ +
.
b
b
28

If the workers who locate in region s do not migrate, then, it is shown thatZ ∞ks [λ(τ )]Ws (0) =dτ.
(A.13)e−ρτ a(τ )b0Now, we write the expenditure of region A as follows:
Z t
L
EA (t) = + λ(0)ρ[a(0) + WA (0)] +λ̇(τ )ρ[a(0) + W (0; τ )]dτ.
20
The last term in this equation represents the expenditure of workers thatmove from region B to region A between 0 and t. Since E(t) = EA (t)+EB (t),EB (t) = E − EA (t). Therefore, the following equations hold:EB (T ) =L
+ (1 − λ̃)ρ[a(0) + WB (0)].
2andĖB (t) = −ĖA (t) = −λ̇(t)ρ[a(0) + W (0; t)].
In addition, we write the expenditure of region B for each t ≥ T ,Z T
EB (t) = EB (T ) −ĖB (τ )dτt
Z TL+ (1 − λ̃)ρ[a(0) + WB (0)] +=λ̇(τ )ρ[a(0) + W (0; τ )]dτ.
2tFrom the discussion above, we show that
Z t
EA (t) − EB (t) = λ(0)ρ[a(0) + WA (0)] +λ̇(τ )ρ[a(0) + W (0; τ )]dτ0
Z T
− (1 − λ̃)ρ[a(0) + WB (0)] −λ̇(τ )ρ[a(0) + W (0; τ )]dτ. (A.14)
t

Proposition A.1 The steady-state growth path under λ̃ is stable.
Proof. Under dynamics and λ̃ = 1/2,
1
λ(t) < ,2
1
λ(t) = ,2

f or t < T,f or t ≥ T.

(A.15)

Since 1/2 ≤ λ ≤ 1, it is clear thatkA [λ(t)] ≡ [λ(t) + η(1 − λ(t))]1/ψ ≤ [1 − λ(t) + ηλ(t)]1/ψ ≡ kB [λ(t)]. (A.16)
29

Since kA is an increasing function of λ, η ≤ 1, and a(t) > 0 for t ≥ 0, it isfound from (A.12) that
W (0; t) ≤ W (0; T ).
This implies thataH + W (0; t)a(0) + W (0; t)=≤ 1.
aH + W (0; T )a(0) + W (0; T )(A.17)

From (A.12) and (A.16), it is obvious that WA (0) ≤ WB (0). Hence, (A.14)
is rewritten as follows:
1
EA (t) − EB (t) < λ(0)ρ[a(0) + WA (0)] − (1 − )ρ[a(0) + WB (0)]2Z Tλ̇(τ )ρ[a(0) + W (0; τ )]dτ
+
0
Z T
1
< (λ0 − )ρ[a(0) + WB (0)] +λ̇(τ )ρ[a(0) + W (0; τ )]dτ.
20(A.18)In addition, from the definition, it is shown that, for t < T ,Z ∞WB (0) − W (0; τ ) =e−ρs [wB (v) − wA (v)]dvZτ ∞kB (λ(v)) − kA (λ(v))
e−ρv a(v)=dv ≥ 0.(A.19)bτGiven λ(T ) = 1/2, we rewrite (A.18) as follows:µZ T¶
1
EA (t) − EB (t) < (λ0 − )ρ[a(0) + WB (0)] +λ̇(τ )dτ ρ[a(0) + WB (0)]201
1
= (λ0 − )ρ[a(0) + WB (0)] + ( − λ0 )ρ[a(0) + WB (0)] = 0.
22Therefore, we state that
EA (t) < EB (t)f or t ≤ T(A.20)

From the discussion in 4.1, (13) and (23), we obtain thatpY A (t)= minpY B (t)½µEA (t)EB (t)

30

β
¶ 1−σ

, τIβ

¾
.

Thus, we rewrite (A.6) as follows:
¾¸
·
¸
·
½µ
¶
Z
Ψeρt
αβΨeρt T −ρv
a(0) + W (0; t)EA
1−σλ̇(t) =ln+
e ln min, τIdv.
ρ
a(0) + W (0; τ )σ−1 tEB(A.21)
Therefore, it is found that RHS in (A.21) is negative from (A.20). Hence,
this implies that, under λ̃ = 1/2, there is no equilibrium to satisfy the mc
hypothesis, that is, the steady-state growth path under λ̃ = 1/2 is unstable.
Q.E.D.
Then, consider the case λ̃ = 1. Suppose that the equilibrium path thatsatisfies the dynamics under λ̃ = 1 given λ0 ∈ [1/2, 1). Under the mc hypothesis and λ̃ = 1, for λ0 ≥ 1/2, we write1< λ(t) < 1,
f or t ∈ (0, T ),2λ(T ) = 1.

(A.22)

Note that, since 1/2 < λ ≤ 1, kA [λ(t)] ≥ kB [λ(t)]. This implies thatWA (0) ≡ W (0; 0) ≥ W (0; t) ≥ W (0; T ),t ≥ T.

From this relationship, we show thataH + W (0; t)a(0) + W (0; t)=≥ 1,
aH + W (0; T )a(0) + W (0; T )t ≥ T.

(A.23)

From (A.12) and λ̃ = 1, it is shown that, for t ∈ (0, T ),Z t
EA (t) − EB (t) = λ0 ρ[a(0) + WA (0)] +λ̇(τ ρ[a(0) + W (0; τ )]dτ0Z T−λ̇(τ )ρ[a(0) + W (0; τ )]dτtZ Tλ̇(τ )ρ[a(0) + WA (0)]dτ> λ0 ρ[a(0) + WA (0)] −0¶µZ Tλ̇(τ )dτ ρ[a(0) + WA (0)].
≥ λ0 ρ[a(0) + WA (0)] −
0

Since

RT
0

λ̇(τ )dτ = 1 − λ0 , this equation is rewritten as follows:
EA (t) − EB (t) = (2λ0 − 1)ρ[a(0) + WA (0)] ≥ 0.
31

This implies that, when λ0 ≥ 1/2,
EA (t) ≥ EB (t)(A.24)In the same approach, since EA (t)/EB (t) ≥ 1, the difference in the price of
final good produced in each region is as follows:½µ¶ β µ ¶β ¾
pY A
EA (t) 1−σ
1
= min
≤1
,
pY B
EB (t)
τI

f or t ∈ (0, T ](A.25)

Here, we define as follows: for t ∈ [0, T ]∆V (t) ≡ eρt [V (0; t) − V (0; T )]·¸·½µ ¶β ¾¸Zeρta(0) + W (0; t)αβΨeρt T −ρτ
EA (t)
1=ln+eln min,dτ.
ρa(0) + W (0; T )σ−1 t
EB (t) τI(A.26)
Using this definition, we find that, given each λ0 ∈ (1/2, 1),∆V (t) > 0
f or t ∈ [0, T )∆V (T ) = 0(A.27)

Hence, from (A.6) and (A.30), it is shown that
λ̇(t) = Ψ∆V (t) > 0(A.28)

By assumption λ0 ∈ (1/2, 1], and even though λ̇(t) is positive when t is closeto 0:
λ̇(0) ≡ lim λ̇(t) > 0.
t→0

(A.29)

Therefore, when λ0 = 1/2, λ̇ > 0. This implies that expectation plays animportant role in the economy.
Next, we show that, starting from λ0 ∈ (1/2, 1], the economy reaches thepoint λ = 1 for a finite time.
From (A.9) and (A.10), we have the following representation:
αβE
1
αβE 1
≤ a(t) ≤σ ρ+1
σ ρ + g( 12 )anda(T ) =

µE 1
.
σ ρ+1
32

(A.30)(A.31)

In addition, by using (A.13), we rewrite the following form:Z ∞Z ∞kA [λ(τ )]αβE 1kA [λ(τ )]−ρτWA (0) =e a(τ )
dτ ≥
e−ρτ
dτ
b
σ ρ+1 0b
0αβE 1 12 (1 + η)≥.
(A.32)σ ρ+1ρbAs shown before, EA (t) ≥ EB (t), which implies that EB (t) ≤ E/2 for t ≤ T .
Using (7), the ratio between the expenditure of region A and that of region
B is
1

(1+η)

2(λ0 − 1)ρ[a(0) + WA (0)]αβE ρ + 2 b
EA (t)>1+≥ 1 + 2(2λ0 − 1).
EB (t)EB (t)σρ+1(A.33)
By the definition of ∆V , (A.30) is represented by the following equation:·¸eρta(0) + W (0; t)
∆V =
ln
ρ
a(0) + W (0; T )
1
·
½
¾¸
Z
(1+η)
αβE ρ + 2 b1αβΨeρt T −ρτeln min 1 + 2(2λ0 − 1)
, 1−σ dτ
+
σ−1 t
σ
ρ+1
τI
1
·
½
¾¸
Z
(1+η)
αβE ρ + 2 bαβΨeρt T −ρτ1eln min 1 + 2(2λ0 − 1)>
, 1−σ dτ
σ−1 t
σ
ρ+1
τI
−ρ(T −t)1−e=J(λ(0)),(A.34)
ρ
where
1
·
½
¾¸
(1+η)
αβΨ
αβE ρ + 2 b1
J(λ0 ) =ln min 1 + 2(2λ0 − 1)
, 1−σ .
σ−1
σ
ρ+1
τI

(A.35)

It is clear that J(λ0 ) > 0 when λ(0) > 1/2. In addition, it is shown thatdJ(λ(0))/dλ0 ≥ 0. Rewriting (A.30) by using (A.38), it is shown that
λ̇(t) = Ψ∆V (t) >ΨJ(λ0 )
[1 − e−ρ(T −t) ].
ρ(A.36)

Integrating both sides in (A.36) from t to T expose that
ρ2 1 − λ0> ρT − (1 − e−ρT ).
Φ J(λ0 )33(A.37)

Here, we denote the solution of the following equation as Tsup (λ0 ):ρ2 1 − λ0= ρT − (1 − e−ρT ).
Φ J(λ0 )(A.38)

It is illustrated that, for each λ0 ∈ (1/2, 1), there is a single solution, Tsup (λ0 ),which is positive, continuous, and decreasing on (1/2, 1), while
lim Tsup (λ0 ) = 0.

λ0 →1

(A.39)

Therefore, it is said that the value of T associated with the equilibrium paththat starts from λ0 is less than Tsup (λ0 ). Thus, we conclude as follows.
Lemma A.1 Suppose λ̃ = 1, and assume that the mc hypothesis holds.
There is a function Tsup (λ0 ) defined on (1/2, 1). This is positive, continuous, and decreasing. In addition, this function is such that the equilibrium
path that starts from λ0 ∈ (1/2, 1) at time 0 reaches λ̃ = 1 before Tsup (λ0 ),where limλ0 →1 Tsup (λ0 ) = 0.
Due to the fact that λ(1/2), the function Tsup (λ0 ) has the following property.
lim Tsup (λ0 ) = ∞λ0 →1/2(A.40)

However, the time that λ reaches λ̃ = 1 is finite. Since, when λ0 = 1/2,
∆V (t) > 0 and ∆V (t) is continuous on (0, T ), it is shown that λ̇(t) > 0, which
implies that λ(t) > 1/2 along the equilibrium path starting from λ0 = 1/2.
Now, we show the existence of a neighborhood, Λ, in such a way that
there is an equilibrium path that leads to λ̃ = 1 for any λ0 ∈ Λ. We define
this as follows:²(t) ≡ ρ[a(0) + W (0; t)].
(A.41)If (λ(t), ∆V (t), a(t), ²(t), EA (t))Tt=0 is the equilibrium path that reaches λ̃ = 1
at time T and starts from the initial distribution, λ0 , at time 0, then the
dynamics is as follows: for t ∈ (0, T ),λ̇ = Φ∆V,
·
½µ
¶ µ
¶¾¸
a(t)
αβρΨ
E
1
A
˙ = ρ∆V −∆V[kA (λ) − kB (λ)] −ln min
,
.
²(t)
σ−1
EB
τI1−σ
ȧ = [ρ + g(λ)]a −
34

αβE
,
σ

²̇ = −ρe−ρt akA (t) − kB (t),bĖA = λ̇ρ[a(0) + W (0; t)] = Φ∆V ².
The terminal conditions are as follows:λ(0) = λ0 ,λ(T ) = 1,

(A.42)V (T ) = 0,
αβE 1
,
σ ρ+1
1
EA (T ) = E − ,2a(T ) =·¸
Z T
kB (τ )
αβE e−ρT−ρτ
²(T ) = ρ a(0) +e a(τ )dτ +.
bσρ ρ + 1
0(A.43)The set of terminal conditions is unusual in two points. First, λ is specified
at both end points. Second, ²(T ) is a complex condition involving integrals.
Hence, it is not straightforward to show the existence of an equilibrium paththat starts from each λ0 ∈ Λ, whose Λ is neighborhood of λ̃ = 1. Therefore,
we take a different way to reach the desired result. This is that given terminal
conditions are specified at time T , and we move backward from t = T to t = 0
by introducing new time variables:τ ≡T −t
In addition, instead of specifying λ0 , we specify T and obtain the associated
λ0 . Using the new variable, we rewrite the dynamics as follows: for τ ∈ (0, T )λ̇ = −Φ∆V,(A.44)
·
½µ
¶ µ
¶¾¸
a
αβρΨ
EA
1
˙
∆V = −ρ∆V + [kA (λ) − kB (λ)] +ln min
,
,
²
σ−1
EB
τI1−σ
αβE
,
σ
kA (λ) − kB (λ)²̇ = ρe−ρ(T −t) a,bȧ = −(ρ + g(λ))a +ĖA = −Φ∆V ²,
where
λ(0) = 1,
35

·

Z T

²(0) = ρ a(T ) +∆V (0) = 0,αβE 1a(0) =,σ ρ+1
L
EA (0) = E − ,
2
−ρτ

e
0

¸
αβE e−ρT.
a(τ )kB [λ(τ )]dτ +σρ ρ + 1(A.45)

Since ²(T ) is complex, we denote it as ²(0) = ²0 , where ² is a parameter thatcan be chosen appropriately. It can be shown that, for each sufficiently small
T , there is a closed interval, I² (T ), in the positive part of a real number, suchthat, for each ²0 ∈ I² (T ), the above system and ²0 have a unique solution,
denoted as
λ[(s; T, ²0 ), ∆(s; T, ²0 ), a(s; T, ²0 ), EA (s; T, ²0 )]Ts=0 .
Let us denote ²(0; T, ²0 ) as the value associated with RHS in ²(0). It is shown
that
·
¸
Z T
αβE e−ρT−ρτ
e a(τ ; T, ²0 )kB [λ(τ ; T, ²0 )]dτ +²(0; T, ²0 ) ≡ ρ a(T ; T, ²0 )+.
σρ ρ + 10
It is shown that the equation, ²(0; T, ²0 ) = ²0 , has a unique solution, denoted
as ²0 (T ). This derives the associated value of λ at τ = T , denoted as λ0 (T ) ≡λ(T ; T, ²0 ). Finally, by showing that λ0 (T ) is a continuous function on theinterval (0, T̂ ] and has the following property,
lim λ0 (T ) = 0,

T →0

we obtain the desired neighborhood of λ̃ = 1, [λ0 (T̂ ), 1). This is sufficient toestablish the stability of the steady-state growth path under λ̃ = 1. We may
conclude the following:
Proposition A.2 The steady-state growth path under λ̃ = 1 is stable.

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38

EA (λ)EB (λ)

1
τI1−σ
σ+αβ
σ−αβ

1

0

1
1
2

σ−αβ
σ+αβ
τI1−σ

Figure 1: The case of τI < σ−αβwhen λ is fixedσ+αβλEA (λ)EB (λ)

σ+αβ
σ−αβ
1
τI1−σ

1

0

λ0

1
1
2

00

λ

τI1−σ
σ−αβ
σ+αβ

when λ is fixed
Figure 2: The case of τI ≥ σ−αβ
σ+αβ

λ

λ
λ

0

1

1
2
1
2

1

0
λ00
Figure 3: The case of τI1−σ < σ−αβwhen the trade of final good does not
σ+αβ
occur

γ

(λ0 = 1)τI1−σ = σ−αβ
σ+αβ

λ
1
λ0

reduction of τI

1
2
1
2

1

λ00
0

Figure 4: The case of τI1−σ ≥ σ−αβwhen the trade of final good does not
σ+αβ
occur

γ

λ
1

1
2

0

γ̄

1
2

γ̂

0
Figure 5: The case of τ̄M < τ̂M < τM or τ̂M < τ̄M < τM when both tradepattern occur.

1

γ

λ
1

λ̄

1
2

0

γ̄

γ̂

λ̂

0
Figure 6: The case of τM < τ̄M < τ̂M when both trade pattern occur.

1

γ

λ
1

1
2

γ̄

1
2

γ̂

0
Figure 7: The case of τ̂M < τM < τ̄M when both trade patterns occur.

1

γ

λ
1

λ̄

1
2

γ̄

1
2

γ̂

λ̂

0
Figure 8: The case of τM < τ̂M < τ̄M or τM < τ̄M < τ̂M when both tradepatterns occur.

1γ