Title

Market size in globalization

Author(s)Kato, Hayato; Okubo, Toshihiro

Citation

Journal of International Economics. 111 P.34P.60

Issue Date 2018-03
Text Version author
URL

http://hdl.handle.net/11094/81782

DOI

10.1016/j.jinteco.2017.12.003

rights©2017. This manuscript version is made available
under the CC-BY-NC-ND 4.0 license

Note

Osaka University Knowledge Archive : OUKA
https://ir.library.osaka-u.ac.jp/
Osaka University

Market size in globalization∗

Hayato Kato†Toshihiro Okubo‡

December, 2017

Abstract
A salient feature of the current globalization is a loss of manufacturing in developed countries and rapid industrialization in middle-sized developing countries.
This paper aims to construct a simple three-country trade and geography modelwith different market sizes and non-constant wage rates. The large country fosters industrial agglomeration (geographical concentration) in the early stage ofglobalization, but loses manufacturing in the later stage of globalization. Whenlosing manufacturing, the large country might be worse off. Thus, the large country might have an incentive to implement welfare-maintaining policies to preventa loss of manufacturing. All of these results can be explained by market sizes.
JEL classification: F12; F15; F20
Keywords: Agglomeration; Market size; Middle-sized country; Non-constant wages;
Industry/welfare maintaining policy∗

We wish to thank Richard Baldwin, Kiminori Matsuyama, Hajime Takatsuka, Kazuhiro Yamamoto
and Dao-Zhi Zeng for helpful discussions. Our thanks also go to seminar and conference participantsat the Asia Pacific Trade Seminars 2016 (National Taiwan), the Asian Meeting of the EconometricSociety 2016 (Doshisya), the Applied Regional Science Conference (Kobe), Keio and Osaka for usefulcomments. This paper was presented under a different title, “The Industrial Development in the
Middle-sized Country” in several occasions. Financial support was gratefully received from the Japan
Society for the Promotion of Science (Grant Number: JP16J01228). All remaining errors are our sole
responsibility.
†
Corresponding author. Faculty of Economics, Keio Univerisity. E-mail address:
hayato.kato2@gmail.com‡
Faculty of Economics, Keio Univerisity. E-mail address: okubo@econ.keio.ac.jp1

1

Introduction

In the early nineteenth century, the development of manufacturing caused a sudden
shift in hegemony towards the today’s wealthy countries such as those in Western Europe, and away from the empires of China, India and the Middle East, which had tothat point dominated the world economy for thousands of years. Pomeranz (2000)
called this the “Great Divergence” and discussed the growth acceleration in Europe
and the U.S., where manufacturing had been developed. These countries created industrial clusters/cities that lead to high economic growth, and dominated economics,
politics, military power and culture all over the world. The economies of developed
countries such as the U.S., Japan and those in Europe have grown substantially during
the twentieth century, particularly in terms of per-capita income and GDP. However,according to Baldwin (2016), the Great Divergence ended in the 1990s, at which point
the global shares of income and manufacturing of the developed countries began to
decline. By contrast, some middle-income countries have developed industries rapidly,
resulting in strong economic growth. He called this the “Great Convergence” as his
book title suggests.
The current trends in globalization are characterized by large international tradeflows and high capital mobility, facilitated by a substantial decline in transport costs
and tariff barriers and the revolution in information and communication technology.
Firms are mobile between countries, and the cites where production takes place are geographically concentrated. Some middle-income countries such as Newly Industrialized
Economies (NIEs) and the Association of Southeast Asian Nations (ASEAN) attract
productive industries and create a high degree of industrial agglomeration. Amid theGreat Convergence in the current globalization, growth paths across middle-income
countries have diverged; some middle-income countries have experienced the convergence process and joined the group of developed countries, while other countries havebecome caught in the middle income trap with low economic growth (Jones, 1997;
Bairoch and Kozul-Wright, 1998; Baldwin and Martin, 1999).
Behind the drastic shifts as mentioned above, a serious concern of globalizationin developed countries is the loss of manufacturing to developing countries, known as
offshoring in North America, delocation in Europe and hollowing-out in Japan. Many2

firms have ceased operations in developed countries and moved their manufacturing
to developing countries in search of large workforces with lower wages. In the U.S.,
manufacturing industries facing severe competition by increased imports from low-wagecountries saw higher exit rates of plants from the late 1970s to the 1990s (Bernard
et al., 2006). Furthermore, the rise of China in the last few decades has negatively
impacted the U.S. manufacturing employment and wages (Autor et al., 2013; Autoret al., 2014; Ebenstein et al., 2014; Acemoglu et al., 2016).1 Autor et al. (2013) findthat, of the decline in manufacturing employment between 1990 and 2007, one-quarter
could be due to a surge in imports from China. Autor et al. (2014) report that workers
in manufacturing industries facing import competition from China earn lower income
over the period of 1992 to 2007 than those in other sectors. Political debate on antiglobalism addresses the issue of how to stop firm relocation and keep jobs in developed
countries.
To illustrate the rise and fall of manufacturing across countries, we construct a
simple three-country trade and geography model with different market sizes. We showthat in the early stages of globalization (i.e., high or intermediate levels of trade costs),
manufacturing firms are concentrated in the large country, but further progressionof globalization (i.e., low trade costs) causes offshoring from the large to the smaller
countries. Offshoring might worsen welfare in the large country, which might justify
policy intervention. On the other hand, the outcome for the middle country is mixed
and depends on its market size.
Relation to the literature.

The literature on trade and economic geography has

addressed the question of how trade liberalization affects firm locations across countries.
The common finding using a variety of standard trade and geography models (Fujita
et al., 1999; Baldwin et al., 2003; Fujita and Thisse, 2013) is that lowering trade costs
results in geographical concentration of all firms in one region, which is the so-called
core-periphery structure. Once all firms are concentrated at the core by agglomeration
1
Section 7.4 links our theoretical results with these empirical findings. The negative impact ofincreased Chinese import competition on firm performance and labor-market outcome is found in other
Western countries; Belgium (Mion and Zhu, 2013), Norway (Balsvik et al., 2015), and 12 European
countries (Bloom et al., 2016). See Haskel et al. (2012) and Autor et al. (2016) for comprehensive
surveys.

3

forces, which always dominates dispersion forces, all firms remain at the core even in thecase of extremely low trade costs. This standard outcome cannot perfectly explain theabove-mentioned consequences of recent globalization; globalization triggers collapseof industrial clusters in developed countries and facilitates industrial development in
middle-sized countries. One reason why the standard trade and geography model fails
to explain these phenomena comes from its basic theoretical structure: the two-country
setting and constant wage rates. To characterize the recent globalization, we relax theseassumptions and extend our analysis to a three-country model with wage rates varying
in market size and firm share.
The three-country setting in our model can highlight the role of intermediate-sizedcountries in the agglomeration process. The trade and economic geography literature
to date ignores asymmetric country size in a three-country framework, apart from a few
studies. A limited number of three country/region models (e.g., Krugman and Livas
Elizondo, 1996; Takahashi, 2003; Ago et al., 2006; Saito et al., 2011; Forslid, 2011;
Brülhart et al., 2012; Gaspar et al., 2017) have provided numerous interesting results
not found in two-country models. Krugman and Livas Elizondo (1996) develop a modelwith two domestic regions and one foreign country, and find that lower trade costs
against the foreign country leads industries to spread across the two domestic regions.2
The closest paper to ours is Forslid (2011).3 He extends the footloose capital (FC) modelof Martin and Rogers (1995) to a three-country setting in which the three countries havedifferent market sizes, and firms are mobile across countries. He studies the impact ofmarket size difference on the agglomeration process. As trade costs fall, firms in thesmall country first relocate to the large country. After all firms in the small country have2

Extensions to three-country models often provide richer insights than two-country models. Takahashi (2003) finds the possibility of inefficient locations driven by factor mobility in the three-country
model. Using a model with a linear demand function, Ago et al. (2006) find that the hub country withgood transport access from the other countries could lose manufacturing because of severe competition. The three-country model by Saito et al. (2011) incorporates firm heterogeneity into the modelof Krugman and Livas Elizondo (1996). They discuss how a fall in trade costs affects firm locations aswell as regional average productivities in the two domestic regions. Brülhart et al. (2012) empiricallyexamine how regional employment and wages in Austria were affected by the opening of Central andEastern European markets after the end of the cold war. Then they show that their empirical findings
can be supported by a three-region economic geography model. Gaspar et al. (2017) investigate thebifurcation of equilibrium in a symmetric multi-region footloose entrepreneur model.
3
See also Matsuyama (2017)’s multi-region model with constant wages and a more general spatial
structure.

4

relocated to the large country, firms in the middle country start relocating. Finally, allfirms end up relocating to the large country. In his model, wage rates are normalized
and thus agglomeration is simply caused by the interaction of market size difference
and trade costs. An implication of Forslid (2011) is how substantial reductions in trade
costs and development of infrastructure affect firm location patterns within Europe.
Another important aspect of our paper is non-constant wage rates, i.e., wage rates
varying in market size and firm share. The standard economic geography models use themodel of Helpman and Krugman (1985), i.e., two-country and two-sector model with the
Dixit-Stiglitz monopolistic competition. The model has one monopolistic competitive
sector with trade costs (manufacturing sector) and one perfectly competitive sector
without trade costs (agricultural or numéraire sector). A crucial mechanism is that thepresence of an agricultural good can normalize wage rates between the two countries.
The wage equalization can simplify the analysis, but it ignores wage disparities in the
globalization process. Thus, to characterize wage rates varying in market size andfirm share, we relax the standard assumption by assuming away the tradable numéraire
good with no trade costs. Instead, our model introduces a non-tradable numéraire good(infinite trade costs for the agricultural good). In other words, this is an extreme caseof Davis (1998), who imposes trade costs on the agricultural sector, thus allowing fornon-constant wage rates.4 The labor market clearing process determines wage rates.
As firms geographically concentrate in one country, a rise of labor demand boosts wage
rates, which moderates the agglomeration process. In short, non-constant wages operate
as a dispersion force.
Non-constant wages in the Dixit-Stiglitz monopolistic competition model have beenstudied mainly in the literature on the home market effect (Davis, 1998; Head andRies, 2001; Brülhart et al., 2004; Davis and Weinstein, 1999, 2002; Crozet and Trionfetti, 2008; Takatsuka and Zeng, 2012). The definition of the home market effect is
4

To our knowledge, there are three standard approaches to non-constant wage rates in the literature.
One is using a one-sector model: monopolistic competition sector à la Krugman (1980). Recent
applications include Takahashi et al. (2013), Zeng and Uchikawa (2014) and Mossay and Tabuchi(2015). Under this approach, the trade balance endogenously determines wage rates between two
countries. The second method is to allow for trade costs in the numéraire sector (agriculture) à laDavis (1998) and Takatsuka and Zeng (2012). This drops the assumption of costless trade in thenuméraire sector. The third method is to introduce differentiated products in a constant-returns-toscale perfect competition sector (Head and Ries, 2001). Our model adopts the second approach.

5

twofold: (1) firm shares in large countries are greater than their market-size (i.e., population or GDP) shares, which is in line with Helpman and Krugman (1985); and (2)wage rates are increasing in market size shares, which is line with Krugman (1980). A
main objective of these studies is to determine whether the home market effect exists,
whether it is dampened or strengthened by trade cost reduction and how the model
assumptions influence the home market effect. In this literature, a model is first constructed, the home market effect is tested using data and then the results are comparedacross model specifications. By contrast, we are interested in the agglomeration process in the three-country model, the impact of market size difference on firm location
patterns (geographical concentration and dispersion), welfare analysis and policies.
Our model is a three-country FC model à la Forslid (2011) with non-constant wage
rates and different market sizes. The model can help us understand the consequencesof recent globalization, particularly industrial development in middle-sized countries
and offshoring in developed countries. We obtain the following results. First, themiddle-sized country might develop manufacturing as part of globalization, but thisdepends on its market size. Second, the large country (and the middle-sized country insome cases) can attract manufacturing despite increased wage rates. This moderates
agglomeration process, resulting in loss of manufacturing when trade costs are small.
Third, a fall in trade costs causes a collapse of agglomeration and might worsen welfare
in the large country. Fourth, to prevent the collapse of agglomeration and worsening
welfare, the large country has an incentive to use bilateral trade agreements with thesmall country rather than the middle country. All of the results are characterized by themarket sizes of countries. It is worth emphasizing that these results are not obtainedin the multi-region geography model with constant wages by Forslid (2011), where thecollapse of full agglomeration is never observed. Likewise, two-region geography models
with non-constant wages cannot characterize middle-sized countries (e.g., Crozet andTrionfetti, 2008; Takatsuka and Zeng, 2012). Although there is a limited number ofstudies dealing with multiple regions and non-constant wages (e.g., Krugman and Livas
Elizondo, 1996; Fujita et al., 1999, Ch 15; Brülhart et al., 2012), all of them assume
away or do not investigate the impact of market size difference on location patterns andwelfare in analytical forms.

6

The rest of the paper proceeds as follows. The next section constructs the threecountry model with non-constant wages. Section 3 derives the long-run equilibrium.
Section 4 explores industrial concentration in one or two countries. Section 5 conducts
welfare analysis and then Section 6 investigates welfare-maintaining policies. Section 7
discusses our results in details. The final section concludes.

2

Basic model

We construct a simple three-country economic geography model based on the footloose
capital (FC) model of Martin and Rogers (1995). The FC model is marked by internationally mobile capital and immobile capital owners and workers. The economy
has three countries, indexed by 1, 2 and 3, with two sectors, a tradable manufacturingsector and a non-tradable agricultural sector. As in the standard FC model, there are
two factors of production, capital used in the manufacturing sector, and labor used inboth the manufacturing and agricultural sectors. The agricultural sector produces ahomogeneous good using constant-returns-to-scale technology so that it is subject toperfect competition. Importantly, in contrast to the standard FC model, the homogeneous good in our model is not internationally traded because of infinite trade costs.
The manufacturing sector is monopolistically competitive and produces differentiatedgoods, which are internationally traded with trade costs.
The total amount of the two factors in the world is expressed as L (labor) and K(capital) and Country i ∈ {1, 2, 3} is endowed with Li = si L and Ki = si K, where
the labor and capital shares of Country i, si ∈ (0, 1) are identical.5 Importantly, theshare of endowments, si , is exogenously given and different across the three countries.
We assume that Country 1 has the largest market size and that Country 3 has thesmallest market size, i.e., s1 > s2 > s3 . Each household holds labor and capital. The
household provides one unit of labor to either sector. Labor is freely mobile between
the two sectors so that sectoral wages in a country are equalized. The household in
5This implies that all countries face identical capital-labor ratios. To highlight the impact of country
size, different capital-labor ratios are not allowed in our model.

7

Country i owns Ki /Li units of capital and invests it to create firms and then receives
capital returns. Simply, one unit of capital makes one manufacturing firm and thus thetotal number of firms in the world, denoted as N , is equal to that of world capital, i.e.,
N = K. In the long-run equilibrium, capital (i.e., firm) moves to the country in which
the highest (operating) profits are made, although the household (i.e., capital owner)
cannot move between countries. Capital rewards are repatriated to the country oforigin. Consequently, the share of capital employed in Country i, denoted as ni ∈ [0, 1](the number is Ni = ni K), is generally different from the initial endowment share,
namely ni 6= si .

2.1

Demand side

Aggregate utility in Country i takes the following form:
1−µ,cQµi q0i

where Qi ≡

" 3 Z
X
j=1

σ
# σ−1

qji (θ)

σ−1
σ

dθ

,

c ≡ µ−µ (1 − µ)−(1−µ) ,θ∈Ωjwhere µ ∈ (0, 1) is the expenditure share on manufacturing goods, σ > 1 is the elasticity of substitution between varieties of manufacturing goods. θ indicates a brandof differentiated products and Ω represents a set of the varieties. q0 is consumptionof the non-tradable good, qji is the quantity of the variety produced in Country j
and consumed in Country i for i, j ∈ {1, 2, 3}, and Q is a real consumption index of
manufacturing goods.
From the first order conditions, aggregate demand in each variety can be given aspji (θ)−σ
µYi ,
Pi1−σ
" 3 Z
X
where Pi ≡qji (θ) =

j=1

1
# 1−σ

pji (θ)1−σ dθ

.

θ∈Ωj

Yi is national income, pji is the price of a variety produced in Country j and consumed
8

in Country i and P is a price index of manufacturing goods. Hereafter, the index ofeach brand, θ, is suppressed.
In the non-tradable agricultural sector, domestic demand must equal domestic supply:q0i =(1 − µ)Yi,
p0i

where p0 is the price of the agricultural good.
The national income consists of labor income and capital rewards. Letting wi be thewage rate in Country i and r be the capital reward, which is identical across countries,
national income in Country i becomes Yi ≡ wi Li + rKi = si (wi L + rK). Without lossof generality, the wage rate in Country 3 can be normalized to unity, i.e., w3 = 1.

2.2

Supply side

Non-tradable agricultural sector.

The non-tradable sector uses one unit of labor

to produce one unit of the good. The price is determined to eliminate excess profits,
implying p0i = wi .
Tradable manufacturing sector.

Manufacturing firms are subject to monopolisticcompetition. An individual firm requires one unit of capital as a fixed cost and uses
a = (σ − 1)/σ units of labor to produce one unit of a brand. Profit maximization by afirm yields a constant mark-up of price over marginal cost:pji = τji pj =

τji σwj a= τji wj .
σ−1

Although firms can supply their local market without incurring trade costs, i.e., τjj = 1,
firms in Country j have to export τji > 1 units of a brand to sell one unit in Countryi 6= j. For the moment, we assume τji to be symmetric in all country pairs, i.e., τji = τ
for j 6= i.

9

By substituting these equilibrium prices, operating profits are given by
πj =
=

3
X

(pji − τji wj a)qji

i=1
3
X


φji

i=1

=

pj
Pi

1−σ

µYi
σ

3
µwj1−σ X
φji Yi,σ∆ii=1where ∆i ≡3
X

φki wk1−σ Nk .

k=1

wk1−σ is the inverse measure of marginal costs and φji ≡ τji1−σ ∈ [0, 1] is called thefreeness of trade, where higher values mean low trade costs. φ = 1 indicates free tradeand φ = 0 indicates autarky. We note that there are no intra-national trade costs,
φji = 1 if j = i. Operating profits go to capital owners so that capital rewards are
given by r = maxj=1,2,3 {πj }.

3Long-run equilibrium

3.1

Location patterns and market size

As capital (i.e., firm) is mobile between countries, the long-run equilibrium is defined as
location patterns where international capital movements stop. In the interior equilibrium where firms are active in all countries, operating profits are equalized between thethree countries, i.e., π1 = π2 = π3 , which endogenously determines location patterns.
In the corner equilibrium, profit equalization partially holds if firms concentrate in two
countries, i.e., πi = πj > πk where ni , nj ∈ (0, 1) and nk = 0, or it never holds if firms
locate only in one country, i.e., πi > πj and πi > πk where ni = 1 and nj = nk = 0.
Capital rewards can be derived from the market clearing condition in the manufacturingsector:
10

σ

3
X

3
X
πj Nj = µ(wi Li + rKi ),

j=1

→K

i=1

σ

3
X

!
πj nj − µr

j=1

=µ

3
X

wi Li ,

i=1

P3

→r=

µ i=1 wi Li.
K(σ − µ)where we make use of

P3

i=1 pji qji = σπj and πj nj = rnj because of r = maxj∈{1,2,3} πj

for nj > 0.
Let us now consider the labor market. The non-tradable sector needs q0j workers.
While labor remaining available for the manufacturing sector is Lj − q0j = Lj − (1 −Pµ)Yj /wj , labor demand is given by Nj 3i=1 τji aqji . Using constant mark-up pricing,Plabor demand can be re-written as Nj 3i=1 τji aqji = Nj πj (σ − 1)/wj . The labor market
clearing condition in Country j is given by
Lj − (1 − µ)(wj Lj + rKj )/wj = (σ − 1)πj Nj /wj .
Given firm shares, the labor market clearing conditions determine wage rates, w1 andw2 . Plugging capital rewards into these clearing conditions yields the following wage
rates:s3 [(1 − µ)s1 + (σ − 1)n1 ],s1 [(1 − µ)s3 + (σ − 1)n3 ]s3 [(1 − µ)s2 + (σ − 1)n2 ]w2 =.
s2 [(1 − µ)s3 + (σ − 1)n3 ]

w1 =

(1)
(2)

The wage rates in Countries 1 and 2 are proportional to their firm shares. More firms
increase labor demand, raising wage rates. It is also worth noting that w3 is normalised
to one, and hence w1 and w2 can be interpreted as the relative wage rates of Countries
1 and 2 to Country 3. This explains the result that w1 and w2 are decreasing in n3 .
Now we consider the interior equilibrium where firms locate in all countries. Let
vij = πi − πj be the profit gap between Countries i and j. Firm shares in the long-run
equilibrium are determined by profit equalization:

11


µ
(w11−σ − φw21−σ )Y1(φw11−σ − w21−σ )Y2
v12 =
+
(3)
σK n1 w11−σ + φn2 w21−σ + φn3 φn1 w11−σ + n2 w21−σ + φn3φ(w11−σ − w21−σ )Y3+= 0,
φn1 w11−σ + φn2 w21−σ + n3µ(w11−σ − φ)Y1
φ(w11−σ − 1)Y2
v13 =
+
(4)
σK n1 w11−σ + φn2 w21−σ + φn3 φn1 w11−σ + n2 w21−σ + φn3
(φw11−σ − 1)Y3
= 0,+
φn1 w11−σ + φn2 w21−σ + n3µφ(w21−σ − 1)Y1
(w21−σ − φ)Y2
v23 =
+
(5)
σK n1 w11−σ + φn2 w21−σ + φn3 φn1 w11−σ + n2 w21−σ + φn3
(φw21−σ − 1)Y3+
= 0.
φn1 w11−σ + φn2 w21−σ + n3
To obtain the equilibrium firm shares, we substitute Eqs. (1) and (2) into the above
equilibrium conditions and solve any two equations among Eqs. (3), (4) and (5) forni .6 Although the firm shares in general cannot be derived in an explicit form, those atφ ∈ {0, 1} exceptionally take a simple explicit form as ni = si . That is, the firm sharesof a country in autarky and free trade are equal to its market-size share, i.e., no homemarket effect. We further investigate the marginal impact of trade cost reduction on
the firm shares. At φ = 0, we have dni /dφ|φ=0 = σ(3si − 1). As the order of country
size implies s1 > 1/3; s3 < 1/3; s2 R 1/3, Country 1 attracts firms from the othercountries (dn1 /dφ|φ=0 > 0), whereas firms in Country 3 relocate to the other countries(dn3 /dφ|φ=0 < 0). Country 2 may gain or lose firms (dn2 /dφ|φ=0 R 0), depending onwhether its market-size share is greater or smaller than one-third.
In the same manner, the marginal impact at φ = 1 is negative in Country 1,
dn1 /dφ|φ=1 < 0, positive in Country 3, dn3 /dφ|φ=1 > 0, and either positive or negative in Country 2, dn2 /dφ|φ=1 R 0.7 We note that unlike our model, the firm sharesare indeterminate at φ = 1 in the standard FC model with normalized wages.
67P3

i=1 ni = 1 always holds, and thus we only need to solve two out of the three equations.
The marginal effects at φ = 1 take a complex form. See Appendix B for the exact formulas.

12

From these signs of the marginal impact at φ ∈ {0, 1}, we expect that as φ becomes
higher, n1 first increases and then decreases, while n3 moves in an opposite way. Moreover, n2 is expected to show more complex patterns. As explicit form solutions cannot
be derived at φ ∈ (0, 1), we rely on numerical simulations. Fig. 1 shows the relationshipbetween φ and nj in the interior equilibrium for different combinations of market sizes,
confirming our expectations.8 A horizontal line for each ni represents the market-size
share of Country i. At any φ ∈ (0, 1), Country 1 (or Country 3) always has a greater (orsmaller) share manufacturing share than its market-size share, i.e., n1 > s1 (n3 < s3 ).
Country 2 may gain manufacturing share like Country 1 (case (I)), lose it like Country
3 (case (III)), or the pattern may be more complex than these two (case (II)). In any
case, n1 looks hump-shaped in terms of φ: trade liberalization first attracts firms toCountry 1 and then promotes relocation from Country 1 to the other countries. n3
behaves in an opposite way to n1 . Trade liberalization first accelerates relocation fromCountry 3 and then attracts some firms from the other countries. These results are in
sharp contrast with the standard FC model (e.g., Forslid, 2011): large countries always
attract firms and small countries generally lose firms as φ rises.
These location patterns are characterized by hump-shaped agglomeration rents andnon-constant wages. As thoroughly investigated by Baldwin et al. (2003) and Baldwin
and Krugman (2004), hump-shaped agglomeration rents are a key element for a better
understanding of agglomeration.9 Markets are substantially segmented at low φ. Firms
do not easily export and thus have an incentive to diversify their production to avoid
severe competition in domestic markets. With a high φ, firms can easily export anywhere and prefer to locate in the small country with cheaper costs (i.e., lower wages).
With an intermediate φ, firms prefer to locate in the large country to save trade costs.
8

The parameter values that produce these figures are given in Appendix H.
Agglomeration rents are formally defined as a firm’s loss associated with deviating from core to
periphery,when full agglomeration occurs. See Section 4 for more details on thins point.
9

13

Fig. 1. The impact of trade liberalization on firm shares.
Note: Market-size shares: (s1 , s2 , s3 ) =(I)(0.4, 0.37, 0.23); (II)(0.51, 0.4, 0.09);(III)(0.5, 0.3, 0.2).

The other key element in understanding location patterns is non-constant wage
rates. The inflow (or outflow) of firms in a country raises (or reduces) wage rates
by increasing (or decreasing) labor demand. Fig. 2 plots the relationship between
P3
φ and wage rates relative to the world average (w =i=1 si wi ), corresponding to
Fig. 1. The wage rate in a country is largely proportional to its firm share.10 w1 /w
is hump-shaped, while 1/w is U-shaped in terms of φ. This indicates that the wage
gap between countries first expands and then shrinks in terms of φ. The wage rates
10

If the world average wage rate is defined as the simple average (w =rate in each country is not proportional to its firm share.

14

P3

i=1 wi /3), the relative wage

are internationally equalized when trade costs are prohibitively high (φ = 0) or zero(φ = 1).

Fig. 2. The impact of trade liberalization on wages.
Note: Market-size shares: (s1 , s2 , s3 ) =(I)(0.4, 0.37, 0.23); (II)(0.51, 0.4, 0.09);(III)(0.5, 0.3, 0.2) as in Fig. 1.

These findings are summarized as
Proposition 1.

The firm share in Country 1 is always greater than its market-size

share, i.e., n1 > s1 for φ ∈ (0, 1). By contrast, the firm share in Country 3 is always
smaller than its market-size share, i.e., n3 < s3 for φ ∈ (0, 1). Country 2 has a greateror smaller firm share than its market-size share. Firm shares are equal to market-size
shares, i.e., ni = si at φ = 0 and φ = 1.
Proposition 2.

Country 1 always has the highest wage, while Country 3 always has

the lowest one, i.e., w1 > w2 > w3 = 1 for φ ∈ (0, 1). The wage rates between countries
are equalized at φ = 0 and φ = 1.
The proofs of both propositions are in Appendix A.
15

At any positive but finite trade costs, the large country always has a larger firm sharethan its market-size share, whereas the smallest one always has a smaller share than its
market-size share. This result is consistent with the standard two-country FC model.11
However, the location patterns in the middle-sized country are not straightforward. On
the demand side, the middle-sized country is more profitable than the small country
but less than the large country. On the supply side, the middle country can employ
workers at lower wages than the large country, but at higher wages than small country.
In sum, market-size and cost (dis-)advantages in the middle country are not decisive
enough to generate clear-cut location patterns.
Based on Propositions 1 and 2, our model finds two types of home market effectin terms of firm shares and wage rates.12 A larger country has (1) a greater shareof firms than its market-size share in a two-factor model with exogenous wages à laHelpman and Krugman (1985) and (2) higher wages in a one-factor model with nonconstant wages à la Krugman (1980). We can interpret our results using these two
types of home market effects as follows. When trade costs are high, the home market
effects both in terms of firm shares and wages are complement. That is, both homemarket effects accelerate concentration of firms in larger countries. The influx of firms
increases labor demand and pushes wages upward in larger countries, which increases
manufacturing expenditures. This promotes more relocation to larger countries. By
contrast, when trade costs are low, the two home market effects work in opposite ways.
The home market effect in terms of firm shares promotes agglomeration, whereas thehome market effect in terms of wages dampens agglomeration. This is why we observe
an inverted U-shaped (or U-shaped) relationship between firm share and trade freenessin the large (or small) country. Location patterns in the middle country are determinedby the counter-balance of the two home market effects. The mechanism of the two types
of home market effect is similar to the one argued by Crozet and Trionfetti (2008), who
11

In a two-country model with non-constant wages, Takatsuka and Zeng (2012) also confirm thispoint.
12
The home market effect in our three country setting can be formally stated as follows (Behrens
et al., 2009; Zeng and Uchikawa, 2014). (1) A country with a larger domestic market hosts a greater
share of firms than its market-size share, i.e., n1 /s1 > n2 /s2 > n3 /s3 ; (2) a country with a larger
domestic market has a higher wage rate: w1 > w2 > w3 = 1; and (3) a country with a larger domestic
market has a greater trade surplus of manufacturing goods (or the net outflow of capital rewards):n1 − s1 > n2 − s2 > n3 − s3 . Our model supports the first two definitions, (1) and (2).

16

allow for non-constant wages in a two-country model.13
A few comments are in order. Krugman and Livas Elizondo (1996) model three
regions (two domestic regions and one foreign country) and find that high trade costs
against the foreign country result in full agglomeration in one of the two domestic
regions, while low trade costs result in dispersion between the two regions. Although
their model and ours yield a similar result that dispersion force is dominant at low
trade costs, mechanisms generating dispersion force are totally different. Krugman andLivas Elizondo (1996) involve urban congestion costs arising from commuting to thecity center, while our focus is on wages determined by local labor market.14 Moreover,since their model hinges on a quite specific structure (e.g., symmetric regional size andno international factor mobility), it is impossible to highlight the role of asymmetricmarket size on location patterns and welfare. In a related vein, Brülhart et al. (2012)
develop a three-region model where wages depend on firm location patterns. They
introduce a non-tradable good, housing, as well as heterogeneous locational preferences
as dispersion forces. Although they do not give a full analytical characterization, we
expect that their model would generate more dispersed location patterns than ours.

3.2

Industrial development in the middle country

As shown in Fig. 1, there are three types of location patterns with different development
paths in Country 2. This section discusses the country of intermediate size more in moredetail. First, to display market size differences in the three-country model, we proposea “market size triangle” (Fig. 3). The vertical axis in Fig. 3 is Country 2’s market
size (share), and the horizontal axis is Country 1’s market size (share). Fig. 3 displays
all possible combinations of market sizes in the three countries. The shaded triangle
13

Crozet and Trionfetti (2008) theoretically predict and empirically confirm that when two countries
become more dissimilar in market size, the magnitude of the home market effect in terms of firm sharegets stronger because of the mixture of Helpman and Krugman (1985) and Krugman (1980).
14This difference gives rise to different welfare implications. If dispersion force comes from urban
congestion costs, it is evident that dispersion evenly mitigates congestion costs and thus raises welfare(Fujita et al., 1999, Ch 18). On the other hand, if wages work as dispersion force as in our model, ashift from the agglomerated to the dispersed configuration could reduce the welfare of some countries,
which will be shown in Section 5.

17

represents our setting of s1 > s2 > s3 .15 As a comparison, a set of market sizes in thestandard two-country FC model, i.e., s1 > s2 and s3 = 0 (Martin and Rogers, 1995;
Baldwin et al., 2003, Ch. 5), can be plotted as the dotted line in Fig. 3. Clearly, the line
implies a limited set of market sizes. The advantage of our model is that country-size
combinations across countries can be plotted in area rather than on-the-line. In thissense, our model allows for many more possible combinations of market sizes, and thus
market size differences can be discussed in more depth than in the standard two-country
FC model.
Using the market size triangle, Fig. 4 illustrates three types of location patterns
in Country 2. Now the shaded triangle area can be divided into (I), (II) and (III),each corresponding to the three location patterns in Figs. 1 and 2. Area (I) in Fig. 4
satisfies dn2 /dφ|φ=0 > 0 and dn2 /dφ|φ=1 < 0. When Countries 1 and 2 are similar inmarket size, their industrial evolutions take similar paths. Country 2 has a larger firmshare than its market-size share and n2 is hump-shaped in terms of φ, as shown in Fig.
1 (I). Area (II) satisfies dn2 /dφ|φ=0 > 0 and dn2 /dφ|φ=1 > 0. When two large countries
have similar market sizes but Country 3 is very small, n2 looks inverted S-shaped interms of φ: Country 2 first gains, then loses and finally regains firms, as shown in Fig.
1 (II). Area (III) satisfies dn2 /dφ|φ=0 < 0 and dn2 /dφ|φ=1 > 0. When Country 1 ismuch larger than the other countries, Country 2 behaves like Country 3. Country 2
has a smaller share of firms than its market-size share and n2 is U-shaped in terms of
φ, as shown in Fig. 1 (III).
Importantly, Fig. 4 plots all possible combinations of market sizes of the three
countries. The market size largely affects Country 2’s industrial development pathunder trade liberalization. One thought experiment is that when Country 2’s market
size is s2 = 0.4, we gradually increase Country 1’s size, s1 , from 0.4 (equivalently, agradual decrease in Country 3’s size, s3 ). For s1 ranging from 0.4 to 0.45, Country
2 has a greater share of firms than its market-size share under any trade costs (Area(I)). For s1 between 0.5 to 0.6, however, Country 2 has a greater share of firms thanits marker-size share under high trade costs, but loses many firms under low trade15

By using s1 + s2 + s3 = 1, the order of s1 > s2 > s3 > 0 implies s1 > 1/3; s2 < 1/2; s2 < 1 − s1 ;s2 < s1 ; and s2 > (1 − s1 )/2. All of these conditions are satisfied in the shaded triangle excluding its
borders.

18

costs (Area (II)). What this thought experiment tells us is that even if Country 2’smarket size is unchanged, a change in Country 1’s size may result in either a gainor loss of manufacturing in Country 2. The industrial development path in Country
2 is determined by its market size relative to those of other two countries. Detailed
conditions can be found in Appendix B.

Fig. 3. The market size triangle.

Fig. 4. Three patterns of industrial evolutions in Country 2.
194Core-periphery structure

Following on from the interior equilibrium, we now explore two types of corner solutions(core-periphery structure): (1) two-country agglomeration, where all firms are in two
countries, and (2) full agglomeration, where all firms are in one country. As Proposition
1 states that n1 ≥ s1 and n3 ≤ s3 hold for φ ∈ [0, 1], Country 3 never hosts all firms
while Country 1 may attract all firms (full agglomeration) or Countries 1 and 2 mayachieve an agglomeration of firms (two-country agglomeration).
Trade in our model only involves manufacturing goods. Once full agglomerationor two-country agglomeration occurs, how is trade balanced? We can use the analogy
of Takahashi et al. (2013), who use a two-country one-sector FC model. Their model
relies on the assumption of the standard FC model that one unit of capital creates one
manufacturing firm associated with international mobility. As discussed in Takahashi
et al. (2013, p.226), when full agglomeration occurs, the country hosting all manufacturing firms is the exporter of manufacturing goods and the importer of capital fromthe countries without manufacturing firms. Thus, the trade deficit in the countries
without firms is compensated by the surplus on the capital account, and vice versa forthe country with full agglomeration. More details on our three-country model can befound in Appendix G.

4.1

Two-country agglomeration and full agglomeration

First, two-country agglomeration is investigated. When firms concentrate in both Countries 1 and 2, operating profits are required to be equalize between the two countries,
π1 = π2 > π3 . The left panel of Fig. 5 plots ni . Trade freeness in which two-country
agglomeration is sustainable (or unsustainable) is denoted by φ3 (or φ3 ). When φ exceeds φ3 , the agglomeration process in Country 1 is accelerated, whereas Country 2
loses firms. No firms locate in Country 3 from φ3 to φ3 . Then above φ3 , the higher
20

wage rate in Country 1 leads to firm relocation from Country 1 to Countries 2 and 3.
Next, we explore full agglomeration, shown in the right panel of Fig. 5. When all
firms concentrate in Country 1 at φ2 < φ < φ2 , this requires the profit gaps to bev12 |n1 =1 > 0 and v13 |n1 =1 > 0. Different market sizes result in v12 |n1 =1 > v13 |n1 =1 at
φ ∈ [0, 1). This implies that the gains of firms by moving from Country 2 to Country
1 are always larger than those by moving from Country 3 to Country 1.16 Therefore,
using the standard method (Baldwin et al., 2003, Ch. 2), solving v12 |n1 =1 = 0 for φgives two critical points where firms are indifferent between two countries:
0 = v12 |n1 =1 = π1 − π2 |n1 =1µL=[(1 − s1 − σ)w1σ−1 φ2 + (σ − s3 w1σ−1 )φ − s2 w1σ−1 ],σ(1 − µ)K

(6)

where w1 = 1 + (σ − 1)/[s1 (1 − µ)]. v12 |n1 =1 indicates hump-shaped agglomeration
rents, i.e., a quadratic function in terms of φ. We call φ2 the sustain point and φ2 thebreak point.
Furthermore, one condition on σ is required for full agglomeration. Larger valuesof σ lead to diversification. This simply means that agglomeration is not possible if
increasing returns are small, which is an analogy of the so-called no-black-hole condition(Fujita et al., 1999, Ch. 4). Thus, the degree of differentiation should be high such thatσ ∈ (1, σ̃), where σ̃ is defined as the critical value of σ for full agglomeration.17 Onceσ ∈ (1, σ̃) is satisfied, full agglomeration occurs in Country 1 over a certain range of
trade costs, φ2 < φ < φ2 .
The right panel of Fig. 5 shows firm shares in terms of φ. Full agglomeration occurs
at an intermediate level of trade costs (φ2 < φ < φ2 ). Furthermore, the figure illustrates
the order of firm relocation. As trade liberalization proceeds, Country 3 loses all firms
first before Country 2 and it re-attracts firms after Country 2. This order reflects thefact that the agglomeration rents transferred from Country 1 to Country 3 are always
larger than those to Country 2, i.e., v13 |n1 =1 > v12 |n1 =1 .
16These relocation incentives evaluated at full agglomeration are related to agglomeration rents andrelocation costs. See e.g., Baldwin et al. (2003) and Baldwin and Okubo (2006).
17
Eq. (6) indicates that v12 |n1 =1 is a quadratic function in terms of φ. For v12 |n1 =1 = 0 to have twosolutions for φ ∈ [0, 1], it must hold that (1) the axis of symmetry is in [0, 1] and (2) the discriminantof the equation is positive. These two conditions reduce to σ ∈ (1, σ̃). See Appendix C for details.

21

Fig. 5. Two-country agglomeration (left) and full agglomeration (right).

Again, market size is a key element in our model. Fig. 6 shows the equilibrium
patterns in the market size triangle. The line dividing (a) and (b) is v13 |n1 =1 = 0 andthat dividing (b) and (c) is v12 |n1 =1 = 0.18 Full agglomeration is more likely to occur
when Country 1 dominates with a large market-size share. Two-country agglomeration
is more likely to arise when Country 2 is relatively large. When Countries 1 and 2 arenot large enough, equilibrium is an interior solution.

18

To draw Fig. 6, full agglomeration as well as two-country agglomeration should arise. From ourprevious discussion, we need to choose small σ in order for v12 |n1 =1 = 0 and v13 |n1 =1 = 0 to have
two solutions for φ ∈ [0, 1]. In area (a), it holds that v12 |n1 =1 < 0 and v13 |n1 =1 < 0; in area (b),v12 |n1 =1 < 0 and v13 |n1 =1 > 0; in area (c), v12 |n1 =1 > 0 and v13 |n1 =1 > 0. Note that v12 |n1 =1 < 0 and
v13 |n1 =1 > 0 are a sufficient condition for two-country agglomeration.

22

Fig. 6. Equilibrium features in the market size triangle.

Here we make a few comments. First, areas (b) and (c) in the market size triangle
indicate that there exists a range of trade costs such that two-country agglomerationand full agglomeration occur. The areas do not indicate that two-country or full agglomeration occurs for the whole range of trade costs. Second, our model supports
the result of Forslid (2011). In the case of wi = 1 for i ∈ {1, 2, 3}, the sustain andbreak points are given by φ2 = s2 /(σ + s1 − 1) < 1 and φ2 = 1, implying that full
agglomeration always occurs, even if trade costs are very low (close to zero). On theother hand, a critical point in our model involves the wage term w1σ−1 . As non-constantwages work as a dispersion force, the range of φ for full agglomeration is smaller than in
the standard FC model.19 Third, there always exists a range of φ for full agglomeration
if σ is in (1, σ̃). Lower values of σ mean that varieties of manufacturing goods are moredifferentiated and the agglomeration force is stronger. Even if Country 1 has a slightly
larger market-size share than Country 2, Country 1 can attract all firms at intermediatelevel of trade costs.
19

To be precise, it is easily checked√ that as long as v12 |n1 =1 = 0 has two solutions√ in φ ∈ [0, 1], wehave dφ2 /dw1 = σ(σ − 1)φ2 /(w1σ−1 D) > 0 and dφ2 /dw1 = −σ(σ − 1)φ2 /(w1σ−1 D) < 0 where D
stands for the discriminant of v12 |n1 =1 = 0.

23

Proposition 3. Our model involves two-country agglomeration where all firms concentrate in Countries 1 and 2 and full agglomeration where all firms locate in Country
1. The main drivers for these configurations are market sizes and the degree of differentiation.
Detailed conditions can be found in Appendix C.

5

Welfare analysis

Each household’s welfare in Country i, Ui , is measured by real income as follows:
Ui =wi + π(K/L),
Piµ p1−µ0iwhich consists of labor and capital incomes, the price of the non-tradable good and theprice index of the manufacturing goods. It is easy to analytically solve welfare at two
extreme cases, i.e., φ = 0 and φ = 1. As shown in Propositions 1 and 2, it holds thatni = si and w1 = w2 = w3 = 1 at these two points. When φ = 0, the only difference
between countries is the price index of the manufacturing goods, which is increasingin the domestic firm share. The country with the largest (or smallest) market size
enjoys the lowest (or highest) price index. On the other hand, when φ = 1, costlesstrade equalizes the price indices internationally, and thus all countries have the same
standard of living. In other words, the welfare gap among countries will eventuallydisappear in free trade.
Proposition 4.

If trade costs are prohibitively high, i.e., φ = 0, welfare is the highest

in Country 1 and the lowest in Country 3. If trade costs are zero, i.e., φ = 1, thewelfare levels in all countries converge.
The proof of the proposition can be found in Appendix D.

24

As welfare cannot be derived as explicit form solutions at φ ∈ (0, 1), we rely on
numerical simulations. The right panel of Fig. 7 plots welfare in the interior equilibrium.
Two results can be observed: (1) welfare levels in all countries monotonically increasein φ, and (2) Country 1 (or Country 3) always has the highest (or the lowest) welfare.

Fig. 7. Firm shares and welfare levels in the interior equilibrium.

Next, Fig. 8 plots firm shares and welfare levels in the case of full agglomeration.
At φ ∈ [φ2 , φ2 ] full agglomeration arises in Country 1. Its welfare is flat because all
manufacturing firms locate in Country 1 and thus its welfare is not affected by trade
costs. On the other hand, the welfare levels in Countries 2 and 3 are identical because
Countries 2 and 3 have no firms and import all their manufacturing goods from Country
1 under the same trade costs. Furthermore, the welfare levels in Countries 2 and 3
increase in φ by lowering trade costs on imports.

25

Fig. 8. Firm shares and welfare levels in the full-agglomeration case.

The most notable feature is that welfare in Country 1 could decline once full agglomeration collapses above the break point, i.e., φ > φ2 , and offshoring occurs: firms leave
Country 1. To investigate the deterioration in welfare, it is worthwhile to decompose
the impact of trade liberalization on welfare:d log U11
dw1 K dπ+=dφ φ=φ2
w1 + π(K/L) dφ
L dφ
|{z}
|{z}
0

(−)

!
−

µ dP1 1 − µ dp01
−
, (7)
P1 dφ
p01 dφ φ=φ2
|{z}
|{z}
(+)(−)(−)

where U1 is differentiated at the break point, φ2 . Firms start to leave Country 1
at φ2 (i.e., offshoring occurs), reducing labor demand and thus lowering the wage rate.
This reduces household income, which has a negative impact on welfare (the first term).
However, this lowers the price of the non-tradable good, which is beneficial in consumption (the fourth term). The capital rewards remain unchanged because the decrease inCountry 1’s expenditure on manufacturing goods owing to the decreased wage is offset
by the increased expenditure by Country 2 owing to the increased wage (the secondterm). Finally, the sign of the third term, the effect on the price index of manufacturing
goods, is vague. Thus, a further decomposition is given by

26

3
dP1
∂P1 dn1 ∂P1 X ∂w1 ∂ni ∂P1+=+.
dφ φ=φ2∂n dφ ∂w∂n ∂φ∂φ
| 1{z } | 1 i=1{z i } |{z} φ=φ2
(+)

(8)

0

(−)

The first term represents the impact of changes in firm shares: a decrease in thenumber of domestic varieties raises the price index.20 The second term indicates that
offshoring reduces domestic wage rates and thus the price index. The third term is the
direct impact of trade cost reductions. This is negligible because all firms are in Country
1 and thus no trade costs are involved.21 Offshoring raises the price index if the loss
from reducing domestic varieties outweighs the gain from lowering domestic production
costs and importing cheaper foreign varieties. In sum, the collapse of agglomerationand offshoring of firms might be beneficial in Country 1 by reducing the price of the
non-tradable good and decreasing the price index by lower production costs and cheaper
import varieties. On the other hand, offshoring could be harmful by reducing the wage
rate and increasing the price index by reducing the number of domestic varieties.
Eq. (7) can be re-written as"#
2
d log U1
µ
dn1
s
(σ
−
1)
1
=
1 − φ2 w1σ−1 −.
dφ φ=φ2σ−1(σ + s1 − 1) {σ + s1 (1 − µ) − 1} dφ φ=φ2|{z
} |{z}(+)(−)(−)(9)
A close inspection of the expression in the large square brackets reveals that Country 1
is worse off when σ is close to one and Country 1’s market size is not extremely large.
The market size condition for worse off is given by s1 < s2 + 1/2.
Using the market size triangle, Fig. 9 can illustrate patterns of welfare change byoffshoring in Country 1. The shaded area in Fig. 9 indicates that full agglomeration
occurs at φ ∈ [φ2 , φ2 ], corresponding to area (c) in Fig. 6. Now s1 = s2 + 1/2 splits
the full-agglomeration shaded area into two areas, (A) and (B). Area (A) satisfies
20

We note that once φ exceeds φ2 , firms in Country 1 start moving to Country 2: it holds that
dn1 /dφ|φ=φ2 < 0, dn2 /dφ|φ=φ2 > 0 and dn3 /dφ|φ=φ2 = 0.
P3 ∂P1 dni
21
Strictly speaking, the effect on the price index also involves
and
i=2 ∂ni dφ
P3 ∂P1 P3 ∂wi dnj
i=2 ∂wi
j=1 ∂nj dφ , but these terms disappear. Our differentiation is evaluated at the point
where there are no firms in the smaller countries. See Appendix E for details.

27

s1 > s2 + 1/2 and welfare in Country 1 decreases. Area (B) satisfies s1 < s2 + 1/2 andwelfare always (weakly) increases under trade liberalization.
The reason for the worsening welfare from offshoring in Country 1 can be explained
as follows. As the left panel of Fig. 10 shows, as Country 1 is smaller, full agglomeration is less likely to occur and φ2 falls. In other words, a collapse of agglomeration andoffshoring happens under smaller values of φ.22 Once full agglomeration in Country
1 collapses and firm relocation occurs, Country 1 starts importing goods from abroadwith trade costs. Lower values of φ2 indicate greater payments of trade costs associatedwith imports. In addition, as Country 1 is smaller and Country 2 is larger, the wage
(production costs) differential between the two countries is smaller. This reduces Country 1’s benefit from importing from offshoring firms; the products produced in Country
2 do not have low prices. The smaller size of Country 1 and the smaller wage gap
with the other countries increase Country 1’s import payments. Thus, greater import
payments result in a reduction in Country 1.
To summarize:
Proposition 5.

Above the break point, φ > φ2 , full agglomeration collapses andoffshoring happens in Country 1. If the size of Country 1 is not extremely large, i.e.,s1 < s2 + 1/2, and the degree of product differentiation is sufficiently high, i.e., σ isclose to one, Country 1 experiences a reduction in welfare.
See Appendix E for details and the proof.
22

This can be confirmed by numerical simulations.

28Fig. 9. Welfare change by offshoring in Country 1.

Fig. 10. Change in the market size of Country 1.

29

6

Welfare-maintaining policies and trade liberalization

When Country 1 is not substantially large, Country 1 might experience a reduction in
welfare following the collapse of full agglomeration at φ ' φ2 . One important questionis how Country 1 manages to hamper offshoring when trade costs decrease. One policy
solution might be to levy a prohibitive tax on firm relocation or prohibit capital mobility.
This could prevent welfare losses from offshoring in Country 1.
However, following a common assumption in the trade and geography literature,
we retain the assumption of free mobility of firms under trade liberalization. What(second-best) policies are feasible in Country 1? One feasible policy is a rise of the
break point, φ2 , without increasing trade costs or regulating capital mobility. This cansustain full agglomeration and postpone the collapse of agglomeration and offshoring.
More specifically, one possibility is to reduce trade costs with a specific country. Let us
imagine a situation where trade costs can be reduced more between a particular pairof countries by a bilateral trade agreement. If Country 1 ratifies a trade agreement, forexample, with Country 2 in further reducing trade costs, the bilateral trade freenessbetween Countries 1 and 2 is α(≥ 1) times higher than the freeness of trade between
Countries 1 and 3. Namely, we denote φ12 = φ21 = αφ; φ13 = φ31 = φ23 = φ32 = φ,where φij stands for freeness of trade from i to j.
Suppose that Country 1 ratifies bilateral trade agreements with Country j ∈ {2, 3}.
The bilateral agreement alters all sustain and break points, which we re-define as φ1j
21j

and φ2 . A marginal improvement in trade freeness between a pair of countries affects
the critical points as follows:
12

dφ12
2
dα

α=1

dφ2
dα

α=1

dφ2
dα

< 0,

13

dφ13
2
dα

< 0,
α=1

< 0,

> 0,
α=1

1j

where φ1j|
= φ2 and φ2 |α=1 = φ2 hold. Fig. 11 compares the industrial evolutions
2 α=1
associated with welfare levels in different trade agreements between different country
pairs. Fig. 11 illustrates that bilateral trade agreements with Countries 2 and 3 at the30

< φ12< φ2 .23 This implies that lowering
sustain point promote full agglomeration, φ1322bilateral trade costs the large and the smaller countries fosters agglomeration in thelarge country. Bilateral trade agreements between Countries 2 and 3 result in a similareffect, i.e., φ23< φ2 .
2
On the other hand, at the break point, φ2 , we see different outcomes across country
12

13

pairs, φ2 < φ2 < φ2 . Country 1’s trade agreement with Country 2 promotes thecollapse of full agglomeration by offshoring and then reduces welfare in Country 1.
As firms in Country 1 are concerned about higher wages, once trade costs between
Countries 1 and 2 fall further, they have greater incentive to relocate to Country 2
because of lower wages and the middle-country market size. By contrast, bilateral tradecosts reductions between Countries 1 and 3 postpone the collapse because Country 1 has
better access to Country 3 but has a much larger market. Considering the higher trade
costs with Country 2 and the small market size in Country 3, firms are more likely
to stay at Country 1 In sum, if Country 1 attempts to maintain full agglomeration,
Country 1 should agree on freer trade with the smallest country rather than the middle
country.
Finally we have one note. Country 1’s bilateral trade agreement always improves
welfare in Countries 2 and 3. Intuitively, both countries benefit from cheaper imports
from Country 1 because of lower trade costs. In addition, Country 2 attracts more firms
from Country 1, raising wage rates. This means that if Country 1 offers a bilateral trade
agreement, Countries 2 and 3 always accept it.
The discussion is summarized as follows:
Proposition 6.

At the sustain point, φ2 , Country 1’s bilateral trade agreements withCountry 2 or 3 both accelerate full agglomeration. By contrast, at the break point, φ2 ,Country 1’s bilateral trade agreement with Country 2 causes the collapse of agglomeration, while an agreement with Country 3 sustains full agglomeration. When trade costs
decline, Country 1’s trade agreement with Country 3 can prevent the deterioration of
its welfare.
The first inequality, φ13< φ12, does not generally hold. At α close to one, φ12
may be greater or222
13smaller than φ2 .
23

31

The proof of the proposition is in Appendix F.

Fig. 11. Bilateral trade agreements in Country 1.

7Discussions and Extensions

This section discuss the comparison of our model with a constant-wage model and thegenerality of our results. Our theoretical results are linked to recent experiences of the
U.S. economy.

7.1

Comparison with a three-country constant-wage model

To make our contribution perfectly clear, we compare our non-constant-wage modelwith Forslid (2011)’s constant-wage model. Our model can reduce to Forslid (2011)’smodel by setting w1 = w2 = 1. Figs. 12 and 13 show the evolution of equilibrium firm

32

share (left panel) and individual welfare (right panel) under non-constant wages (thickline) and constant wages (thin line).
A crucial difference from our result is that under constant wages Country 1 weakly
monotonically increases its firm share as trade costs decline. Since firms can always
hire workers at a constant wage rate, agglomeration in Country 1 does not discourage
further inflow of firms and thus Country 1 can maintain full agglomeration at low
trade costs. Consequently, different equilibrium paths between the two models give rise
to different welfare implications. Under non-constant wages, Country 1 could reduce
welfare when full agglomeration collapses (Proposition 5). By contrast, under constantwages, the collapse of full agglomeration never happens and thus Country 1 neverdeteriorates welfare (Forslid, 2011). Accordingly, Country 1 does not need to take any
welfare-maintaining policies as discussed in Section 6.
Another subtle but important difference is in equilibrium path at high trade costs.
Under non-constant wages, Country 1 attracts more firms than under constant wages
at low φ (roughly, φ ∈ [0, 0.15] in the left panel of Fig. 12 and φ ∈ [0, 0.55] in the left
panel of Fig. 13). As emphasized in Section 3.1, our model incorporates two types ofhome market effect (larger firm shares and higher wages in larger countries). When
trade costs are high, both of the two effects work in favor of agglomeration in the large
country. As trade costs decline, the home market effect of higher wages in the large
country turns to moderate agglomeration force and thus firms are spatially dispersed.

33

Fig. 12. Non-constant wage (thick lines) vs. constant wage (thin lines) in theinterior case.
Note: The left (or right) panel shows the equilibrium firm share (or welfare).
Horizontal lines in the left panel are the market-size shares.

Fig. 13. Non-constant wage (thick lines) vs. constant wage (thin lines) in the full
agglomeration case.
Note: The left (or right) panel shows the equilibrium firm share (or welfare).
Horizontal lines in the left panel are the market-size shares.

34

7.2

More-than three countries

Our main results highlight two points: (i) complex location patterns in the middle
country and (ii) the collapse of full agglomeration in the large country. Most of theconditions related to these points are analytically derived and characterized by market
size. These results can be obtained in a general model with more-than three countries.
Appendix I shows equilibrium location patterns in a four-country model. The middle countries are now defined as the second and third largest countries. In parallelto the three-country model, equilibrium location patterns are determined by market
size and summarized in the market-size triangle/rectangle. The largest (or smallest)
country always shows an inverted U-shaped (or U-shaped) equilibrium path, while themiddle-sized countries involve more patterns determined by their positions in the triangle/rectangle.
The results concerning full agglomeration also hold in a general J(> 3)-country
model (see Appendix J). We can confirm that the market sizes of the largest and thesecond largest countries determine the collapse of full agglomeration and its welfare
impact. When full agglomeration is possible, the second-largest country is the secondmost attractive location for firms following the largest country. As firms compare theprofits in these countries, the market sizes of the top-two countries are crucial for the
results.

7.3

Multiple industries with different labor intensities

Our model in the main analysis has one manufacturing industry subject to monopolisticcompetition. Another extension is multiple manufacturing industries with different labor intensities (different unit labor requirements). Our main results remain unchanged(see Appendix K). As long as consumers put an equal expenditure share on each manufacturing industries, equilibrium firm share in each industry exhibits qualitatively thesame pattern as in the single-manufacturing model (see Fig. A7). All firms in each35

industry have the same labor intensity and thus face the same marginal costs. Hence,
their equilibrium location patterns mimic those in the single-manufacturing model.
Furthermore, to highlight inter-industry linkages, one could introduce intermediate
goods and model an input-output structure (see Fujita et al., 1999, Ch 15). In thiscase, we expect that firms in more labor intensive industries would avoid agglomeratingin the largest country and move to the smallest country to seek lower wages. On theother hand, firms in less labor intensive industries would be less sensitive to a rise ofwages so that they would agglomerate in the largest country to seek greater demand.

7.4

The U.S. economy

Although our analysis is based on a highly stylized model, it helps better understand thecurrent U.S. economy facing globalization. We have shown that as trade liberalization
proceeds, the large country loses manufacturing, leading to lower wages. According to
the sectoral level analysis by Acemoglu et al. (2016), the import penetration from China
contributed around 10% of the decline in manufacturing employment in the 2000s. This
number could be around 20% when taking into account indirect impact from interindustry linkages. Using the individual worker data from 1984 to 2002, Ebenstein et al.
(2014) find that occupational switching from manufacturing led to relatively large wage
declines. These findings suggest that the proliferation of globalization since the mid1980s, which Baldwin (2016) calls the “second unbundling,” has put downward pressure
on employment and wages in the U.S. manufacturing industry.
Another issue is the U.S. attitude toward trade policies. Proposition 6 states thatthe large country facing the collapse of agglomeration prefers to reduce trade costswith the small country rather than the middle country. Evaluating the process of free
trade agreement negotiations in the early 2000s, Schott (2004) concludes that “theU.S. has put too much effort on low risk, low reward (in both economic and politicalterms) negotiations with small countries around the globe” (p.373). Our result could
rationalize the incentive.

36

8

Conclusion

We construct a three-country FC model with non-constant wages to illustrate the Great
Divergence and the Great Convergence in globalization. Focusing on market size differences across countries, our model can explain full agglomeration in large countries,
as in the Great Divergence, as well as offshoring in developed countries and industrial
development in middle-sized countries, as in the Great Convergence. We find three
types of location patterns in the middle-sized country as well as full and two-country
agglomerations. Our three-country model provides much richer location patterns than
the standard two-country model. Wage rates depending on market size and firm share
are a main driver for offshoring in developed countries, which raises a possibility ofdeclining welfare. All of these results are characterized by market size.
A possible extension is empirical analysis. One might test how relative market size
crucially affects the rise and fall of manufacturing in developing and developed countries
in interaction with trade costs and wage rates. We leave this topic to future research.

37

Appendix A. Proofs of Propositions 1 and 2
Following the previous literature (e.g., Fujita et al., 1999), the simple dynamics of firm
migration are given as follows:!
3Xṅi = ni (1 − ni ) πi −sj πj , for i ∈ {1, 2, 3},
j=1

where a dot represents the time derivative. Namely, ni increases if Country i offers
higher profits than an average profit weighted by market size.24 Using the expressionof vij , we can rearrange the above equation asṅ1 = n1 (1 − n1 )(s2 v12 + s3 v13 ),ṅ2 = n2 (1 − n2 )(s3 v23 − s1 v21 ),ṅ3 = −n3 (1 − n3 )(s1 v13 + s3 v23 ),where we make use of vij = −vji .
Firm shares.

Evaluating v12 , v13 and v23 at (n1 , n2 ) = (s1 , s2 ) givesφµL(1 − φ)(s2 − s1 )≥ 0,
K(σ − µ)[s1 + φ(s2 + s3 )][s2 + φ(s1 + s3 )]φµL(1 − φ)(s1 − s3 )≥ 0,
v13 |(n1 ,n2 )=(s1 ,s2 ) =K(σ − µ)[s1 + φ(s2 + s3 )][s3 + φ(s1 + s2 )]φµL(1 − φ)(s2 − s3 )≥ 0,
v23 |(n1 ,n2 )=(s1 ,s2 ) =K(σ − µ)[s2 + φ(s1 + s2 )][s2 + φ(s1 + s3 )]v12 |(n1 ,n2 )=(s1 ,s2 ) =where equality holds at φ = 0 and φ = 1. v12 ≥ 0 and v13 ≥ 0 imply that firms in
Countries 2 and 3 are ready to move to Country 1, i.e., ṅ1 ≥ 0, and thus it holds thatn1 ≥ s1 in the long-run (or steady-state) equilibrium. Similarly, v13 ≥ 0 and v23 ≥ 0
imply that firms in Country 3 always find it profitable to relocate to either Country 1
or 2, i.e., ṅ3 ≤ 0, and thus it holds that n3 ≤ s3 in the long-run equilibrium.
24

The following discussion goes through if one takes a simple average of profits, rather than aweighted average.

38

Firm shares at two extreme cases: ni |φ=0 and ni |φ=1 . Evaluating profits at φ = 0 givesµ(w1 L1 + rK1 )µs1 (w1 L + rK)µw11−σ (w1 L1 + rK1 )
=
=
,
1−σ
1−σ
σ
w1 N1
σw1 N1
σw11−σ n1 K
µw21−σ (w2 L2 + rK2 )µ(w2 L2 + rK2 )µs2 (w2 L + rK)
π2 =
=
=
,
1−σ
1−σ
σ
w2 N2
σw2 N2
σw21−σ n2 Kµ(L3 + rK3 )µs3 (L + rK)
π3 =
=
.
σN3
σn3 K
π1 =

Solving π1 − π2 = 0 and π1 − π3 = 0 for (n1 , n2 ) yields (n1 , n2 ) = (s1 , s2 ).
Similarly, evaluating profits at φ = 1 givesµw11−σ (w1 L1 + rK1 ) + (w2 L2 + rK2 ) + (L3 + rK3 )π1 =,σ
w11−σ N1 + w21−σ N2 + N3
µw21−σ (w1 L1 + rK1 ) + (w2 L2 + rK2 ) + (L3 + rK3 )π2 =,σ
w11−σ N1 + w21−σ N2 + N3
µ (w1 L1 + rK1 ) + (w2 L2 + rK2 ) + (L3 + rK3 )π3 =.
σ
w11−σ N1 + w21−σ N2 + N3
Equating π1 and π2 with π3 gives w1 = w2 = 1, which implies (n1 , n2 ) = (s1 , s2 ).
Wage rates.
The above discussion shows that w1 = w2 = 1 holds at φ = 0 and
φ = 1, In the following, we consider the case where φ ∈ (0, 1). Suppose w1 > w2 holds,
then we have
w1 > w2 ,→ [(1 − µ)s1 + (σ − 1)n1 ]/s1 > [(1 − µ)s2 + (σ − 1)n2 ]/s2 ,→ n1 s2 = n1 (1 − s1 − s3 ) > s1 (1 − n1 − n3 ) = s1 n2 .
where we make use of n1 > s1 and n3 < s3 . Noting that n1 (1 − s1 − s3 ) > s1 (1 − s1 − s3 )and s1 (1 − n1 − s3 ) > s1 (1 − n1 − n3 ), the above inequality is satisfied if
s1 (1 − s1 − s3 ) > s1 (1 − n1 − s3 ),→ n1 > s1 .
This inequality holds from the previous discussion, and thus w1 > w2 holds for φ ∈(0, 1).

39

Similarly, w2 is compared with w3 = 1:
w2 > 1,→ s3 [(1 − µ)s2 + (σ − 1)n2 ] > s2 [(1 − µ)s3 + (σ − 1)n3 ],→ s3 n2 = s3 (1 − n1 − n3 ) > n3 (1 − s1 − s3 ) = n3 s2 .
Noting that s3 (1 − n1 − n3 ) > n3 (1 − s1 − n3 ), the above inequality is satisfied if
n3 (1 − s1 − n3 ) > n3 (1 − s1 − s3 ),→ s3 > n3 .
This inequality holds from the previous discussion, and thus w2 > 1 holds for φ ∈ (0, 1).
We conclude that w1 ≥ w2 ≥ 1 holds for φ ∈ [0, 1].

Appendix B. The marginal impact of trade liberalization at φ ∈ {0, 1}
We derive the slope of equilibrium firm shares at the two endpoints, i.e., φ ∈ {0, 1}.
The path of industrial development in Country 2 follows from these signs.
Profit equalizations are given as follows:
v12 (n1 , n2 , φ) = π1 − π2 = 0,v13 (n1 , n2 , φ) = π1 − π3 = 0,v23 (n1 , n2 , φ) = π2 − π3 = 0.
As the firm share in Country 3 is given by n3 = 1 − n1 − n2 , we do not need to consider
v23 = π2 − π3 . Thus, the two equations are differentiated with respect to trade freenessand can be rearranged in matrix form:


 

∂v12 /∂n1 ∂v12 /∂n2 dn1 /dφ−∂v12 /∂φ=.
∂v13 /∂n1 ∂v13 /∂n2 dn2 /dφ−∂v13 /∂φThe system of equations is solved in terms of dn1 /dφ and dn2 /dφ. At the extreme,
when φ = 0, it holds that ni = si . This can simplify the expressions todn1= σ(3s1 − 1) > 0,dφ φ=0dn2= σ(3s2 − 1) R 0.
dφ φ=0By our assumption, the large country has more than one-third of the world endowment40

(s1 > 1/3). Intuitively, with very small φ(' 0), Country 2 gains firms if its market-size
share exceeds one-third.
P3
From
the
fact
that
the
sum
of
firm
shares
is
equal
to
one:
i=1 ni = 1, we have
P3
dn
/dφ
=
0
for
any
φ.
The
slope
of
the
firm
share
in
Country
3 is thus given by
i
i=1


dn1 dn2dn3+=−= σ(3s3 − 1) < 0.
dφ φ=0dφ
dφ
φ=0Next, the slopes at φ = 1 are derived in the same manner. By using the fact thats1 ∈ (1/3, 1), s2 ∈ (0, 1/2) and s3 ∈ (0, 1/3), the signs of the slopes are determined asfollows:
s1 (σ − µ)f (s1 , s2 )
dn1
=
< 0,
dφ φ=1(σ − 1)2

f (s1 , s2 ) = 2s21 + (2s2 − 3)s1 + 2s22 − 2s2 + 1 < 0,dn2
s2 (σ − µ)g(s1 , s2 )=R 0,
dφ φ=1(σ − 1)2g(s1 , s2 ) = 2s21 + (1 − s2 )[1 − 2(s1 + s2 )] R 0,

s3 (σ − µ)h(s1 , s2 )
dn3
=
> 0,
dφ φ=1(σ − 1)2h(s1 , s2 ) = 2s21 − (s1 + s2 )(1 − 2s2 ) > 0.

We note that the signs of the slopes at φ ∈ {0, 1} only depend on market sizes.

Appendix C. Proof of Proposition 3
Conditions for full agglomeration. Suppose that all firms locate in Country 1 andhave no incentive to relocate to the other countries. The operating profits evaluated at
N1 = K are given byµ(Y1 + Y2 + Y3 ),σK 1−σµ1π2 |n1 =1 =[φY1 + (Y2 /φ) + Y3 ],σK w1
 1−σµ1π3 |n1 =1 =[φY1 + Y2 + (Y3 /φ)],σK w1
π1 |n1 =1 =41

where w1 = 1 + (σ − 1)/[s1 (1 − µ)]. We note that w2 = 1 at n1 = 1. From theseequations, the necessary and sufficient conditions for full agglomeration are given by
v12 |n1 =1 = π1 − π2 |n1 =1 = Ω · F12 (φ) > 0,
v13 |n1 =1 = π1 − π3 |n1 =1 = Ω · F13 (φ) > 0,
where F12 (φ) ≡ −(σ + s1 − 1)w1σ−1 φ2 + (σ − w1σ−1 s3 )φ − w1σ−1 s2 ,F13 (φ) ≡ −(σ + s1 − 1)w1σ−1 φ2 + (σ − w1σ−1 s2 )φ − w1σ−1 s3 ,and where Ω ≡ µL/[φσ(1 − µ)K] is a positive constant.
As it holds that −(φs2 + s3 ) ≥ −(φs3 + s2 ) because of s2 > s3 , we have F13 (φ) ≥F12 (φ) for φ ∈ [0, 1]. This allows us to focus only on F12 (φ) > 0. Because it holds that
F12 (0) = −w1σ−1 s2 < 0 and F12 (1) = σ(1 − w1σ−1 ) < 0 and F12 is a quadratic function interms of φ, the condition for F12 (φ) > 0 is required to reach a maximum in φ ∈ (0, 1).
This is equivalent to two conditions: (i) the axis of symmetry of F12 is in φ ∈ (0, 1),
and (ii) the discriminant D for F12 (φ) = 0 is positive.
Condition (i) is given as follows:
σ − w1σ−1 s3∈ (0, 1),2w1σ−1 (σ + s1 − 1)
→ 0 < σ − w1σ−1 s3 < 2w1σ−1 (σ + s1 − 1),→ w1σ−1 s3 < σ < G(σ) ≡ w1σ−1 [s3 + 2(σ + s1 − 1)].
We can show that the second inequality, σ < G(σ), always holds for σ > 1. Using the
Taylor approximation in w1σ−1 ' 1 + (σ − 1)2 /[s1 (1 − µ)], we can confirm that G(σ) is
greater and steeper than σ at σ = 1, i.e., 1 < G(1) = 2s1 + s3 and 1 < G0 (1) = 2. It
suffices to check that the slope of G(σ) is always positive and increasing in σ > 1:2[3σ 2 − (5 + s2 − s1 )σ + s2 − µs1 + 2]
> 0,
s1 (1 − µ)2(6σ − 5 + s1 − s2 )G00 (σ) => 0.
s1 (1 − µ)G0 (σ) =Hence, condition (i) only requires the first inequality, w1σ−1 s3 < σ.
Condition (ii) is given by2(σ−1)D ≡ (σ − w1σ−1 s3 )2 − 4s2 (σ + s1 − 1)w1

These two conditions reduce tohipσ > H(σ) ≡ w1σ−1 s3 + 2 s2 (σ + s1 − 1) .

42

> 0.

Under the above condition, the smaller root of F12 (φ) = 0 corresponds to φ2 and thelarger one to φ2 . We can confirm that the following inequalities hold for σ > 1:p√
1 > H(1) = s3 + 2 s1 s2 , 1 > H 0 (1) = s2 /s1 ,p
s2 [5σ 2 − 2(5 − 2s1 )σ + 5 − s1 (3 + µ)] + 2s3 (σ − 1) s2 (σ + s1 − 1)0p
> 0,
H (σ) =s1 (1 − µ) s2 (σ + s1 − 1)ps2 [15σ 2 − 6(5 − 4s1 )σ + 8s21 − (25 − µ)s1 + 15] + 4s3 (σ + s1 − 1) s2 (σ + s1 − 1)00p
H (σ) => 0.
2s1 (1 − µ)(σ + s1 − 1) s2 (σ + s1 − 1)
Hence, H(σ) crosses σ from below at some σ > 1. We define such σ as σ̃:σ̃ ≡ min arg [σ − H(σ) = 0].
σ>1

In other words, if σ is in (1, σ̃), full agglomeration occurs at intermediate trade costs
such that φ ∈ [φ2 , φ2 ]. Note that σ̃ depends on the market sizes.
Conditions for two-country agglomeration.
A sufficient condition for two-country
agglomeration is (i) v12 |n1 =1 < 0 and (ii) v13 |n1 =1 > 0. From the previous discussion,
condition (i) requires σ < H(σ), i.e., σ > σ̃. By using the same analogy as before,
condition (ii) reduces to the following inequality:ihp
σ > H ∗ (σ) ≡ w1σ−1 s2 + 2 s3 (σ + s1 − 1) ,where w1 = 1 + (σ − 1)/[s1 (1 − µ)]. As it can be verified that H ∗ (σ) crosses σ frombelow at σ > 1 in the same manner as before, condition (ii) turns out to be σ ∈ (1, σ ∗ )where σ ∗ is defined asσ ∗ ≡ min arg [σ − H ∗ (σ) = 0].
σ>1

As it holds that H(σ) > H ∗ (σ) for σ > 1,25 σ ∗ is greater than σ̃. Combining the two
conditions yields σ ∈ [σ̃, σ ∗ ). If this condition holds, Countries 1 and 2 attract all firms
at intermediate trade costs: φ ∈ [φ3 , φ3 ].
Order of move.
The right panel of Fig. 5 shows that as trade gets freer, Country 3
loses firms before Country 2, i.e., φ3 < φ2 . Under further trade cost reduction, Country25pp
To establish this inequality, it suffices to check that s3 +2 s2 (σ + s1 − 1) > s2 +2 s3 (σ + s1 − 1):
h
i  √
p
p
√
√  √
√
s3 + 2 s2 (σ + s1 − 1) − s2 + 2 s3 (σ + s1 − 1) = 2 σ + s1 − 1 − ( s2 + s3 ) ( s2 − s3 ) > 0.

43

2 first regains manufacturing firms and then Country 3 follows. We can confirm theorder of the moves analytically by looking at the relocation tendencies of the firmsin Country 1. The previous discussion tells us that v13 |n1 =1 > v12 |n1 =1 holds for φ ∈(φ2 , φ2 ), which can be illustrated in Fig. A1. At φ close to but lower than φ2 , v12 |n1 =1 <0 and v13 |n1 =1 > 0 hold. This means that a firm in Country 1 is ready to move toCountry 2 but has no incentive to move to Country 3. The figure suggests that there
is a range of trade costs where Country 2 hosts some firms, and Country 3 does not.
Therefore we can conclude that all firms in Country 3 leave before those in Country 2
do so. The analogous argument applies to the case of φ close to but higher than φ2 ; wecan also conclude that firms move back to Country 2 before Country 3.

Fig. A1. Agglomeration rents.

Appendix D. Proof of Proposition 4
Welfare comparison in Country 1.
We prove that Country 1 has a higher welfare
level in free trade (φ = 1) than in full agglomeration (n1 = 1). With free trade, Country

44

1’s welfare is
U1 |φ=1 =w1 + π(K/L),
P1µ p1−µφ=101where w1 |φ=1 = 1,

π|φ=1 =
1

P1 |φ=1 = K 1−σ ,µL,(σ − µ)K

p01 |φ=1 = 1.

Under full agglomeration, Country 1’s welfare is given by
w1 + π(K/L),
P1µ p1−µ
n1 =1
01
σ−1
,
where w1 |n1 =1 = 1 +
s1 (1 − µ)U1 |n1 =1 =1

P1 |n1 =1 = K 1−σ w1 |n1 =1 ,π|n=1 =µL,
(1 − µ)K

p01 |n1 =1 = w1 |n1 =1 .

Compared with free trade, Country 1 in full agglomeration is better off in terms ofwages and capital rewards (w1 |n1 =1 > w1 |φ=1 ; π|n1 =1 > π|φ=1 ), but worse off in terms of
the prices of both manufacturing and non-tradable goods (P1 |n1 =1 > P1 |φ=1 ; p01 |n1 =1 >p01 |φ=1 ). Further inspection reveals that the former positive effects on wages and capital
rewards are always dominated by the latter negative effects on prices:
U1 |φ=1 − U1 |n1 =1 =µ(σ − 1)(1 − s1 )> 0.
(σ − µ)[σ + s1 (1 − µ) − 1]

Thus, welfare in free trade is always higher than under full agglomeration.
Welfare comparison between the three countries at φ = 0.
From the fact that wehave ni = si and wi = 1 for i ∈ {1, 2, 3} at φ = 0, country i’s welfare is calculated asµ
1 + π(K/L)
σ−1=N[1 + π(K/L)],i
Piµ · 11−µµL
where π =.
(σ − µ)K

Ui |φ=0 =As N1 > N2 > N3 holds at φ = 0, we have U1 > U2 > U3 at φ = 0.

45

Appendix E. Proof of Proposition 5
Derivation of Eq. (9).
φ = φ2 , we have

Noting that dn1 /dφ + dn2 /dφ = 0 and dn3 /dφ = 0 hold at

3X
dw1∂w1 dni=dφ φ=φ2∂ni dφ φ=φ2
i=1=

σ − 1 dn1
< 0,
s1 (1 − µ) dφ φ=φ2

3 X
3
X
µ
∂wi dnjdπ=Lidφ φ=φ2K(σ − µ) i=1 j=1 ∂nj dφ φ=φ2


µ
∂w1
∂w2 dn1
=
− L2
= 0,
L1
K(σ − µ)∂n1
∂n2 dφ φ=φ2

where the expressions for wages and capital rewards are given in Section 3.1. Substituting these into the first two terms and the fourth term in Eq. (7) yields

1
dw1 K dπσ − 1 dn1
+
=
< 0,
(E.1)
w1 + π(K/L) dφ
L dφ φ=φ2
σ + s1 − 1 dφ φ=φ2
1 − µ dp01
(1 − µ)(σ − 1) dn1=< 0.
p01 dφσ + s1 (1 − µ) − 1 dφ φ=φ2

(E.2)

As for the price index term P1 , we have
KP1σ dn1∂P1 dn1= (w11−σ − φ2 )> 0,
∂n1 dφ φ=φ2
1 − σ dφ φ=φ2
where we can verify that w11−σ −φ > 0 holds at φ = φ2 by showing that (i) F12 (w11−σ ) < 0
and (ii) w11−σ is greater than the axis of symmetry of F12 .
Further calculations reveal∂P1 dn2
KP1σ dn2= (φ2 w21−σ − φ2 )= 0,
∂n2 dφ φ=φ2
1 − σ dφ φ=φ246

3

∂P1 X ∂w1 dni∂P1 ∂w1 dn1=∂w1 i=1 ∂ni dφ φ=φ2
∂w1 ∂n1 dφ φ=φ2
1

K 1−σ (σ − 1) dn1=< 0,
s1 (1 − µ) dφ φ=φ2
3

∂P1 X ∂w2 dni∂P1 ∂w2 dn2=∂w2 i=1 ∂ni dφ φ=φ2
∂w1 ∂n2 dφ φ=φ2
= φ2 N2 w2−σ P1σ

dn2
= 0,
dφ φ=φ2∂P1
Pσ= (N2 w21−σ + N3 ) 1= 0,∂φ φ=φ2
1 − σ φ=φ2
where we make use of the fact that at φ = φ2 , it holds that N1 = K, N2 = N3 = 0 and
w2 = 1.
Substituting these into Eq. (8) gives"
1
KP1σ (w11−σ − φ2 ) K 1−σ (σ − 1) dn1
dP1
=
.
+
dφ φ=φ21−σs1 (1 − µ)dφ φ=φ2
Accordingly, the third term in Eq. (7) reduces toµ(σ − 1)2
dn1
µ dP1
σ−1
=
1 − φ2 w1 −.
P1 dφ φ=φ2
1−σσ + s1 (1 − µ) − 1 dφ φ=φ2

(E.3)

From Eqs. (E.1) to (E.3), Eq. (7) can be re-expressed as Eq. (9).
Conditions for worse off in Country 1.
given as follows:
I(σ) ≡ 1 − φ2 w1σ−1 −From Eq. (9), the condition for worse off is

s1 (σ − 1)2> 0.
(σ + s1 − 1)[σ + s1 (1 − µ) − 1]47

Thus, we get
I 0 (1) = 0,2[1 + 2(s2 − s1 )]R 0.
I 00 (1) =(1 − µ)s1 (s1 − s2 )I(1) = 0,

If s1 < s2 + 1/2 holds, then I > 0 immediately holds. That is, if Country 1 is not
extremely large and σ is close to unity, its welfare level starts decreasing once offshoring
starts.

Appendix F. Proof of Proposition 6
12

12
= 0:
and φ2 are derived by v12
φ12
2



12
v12
= Ω0 −α2 w1σ−1 (σ + s1 − 1)φ2 + α(σ − w1σ−1 s3 )φ − w1σ−1 s2 = 0,
where Ω0 ≡ µL/[αφ(1−µ)σK] is a positive constant. The marginal impact of a bilateral
trade agreement on the critical points are given by√dφ12
σ − w1σ−1 s3 − D
2= − σ−1= −φ2 < 0,dα α=12w1 (σ + s1 − 1)√12dφ2
σ − w1σ−1 s3 + D= − σ−1= −φ2 < 0,dα α=12w1 (σ + s1 − 1)2(σ−1)

where D ≡ (σ − w1σ−1 s3 )2 − 4w1

s2 (σ + s1 − 1).

13

13
Similarly, φ13= 0:
and φ2 are the solutions of v12
2



13
v12
= Ω0 −αw1σ−1 (σ + s1 − 1)φ2 + (ασ − w1σ−1 s3 )φ − αw1σ−1 s2 = 0.
We have√  σ−1 s3 σ − w1σ−1 s3 − D
w 1 s3√√=−=−
φ2 < 0,dα α=12 D(σ + s1 − 1)
D

√  
σ−1

13
D
s
σ
−
w
s
+
3
3
1
dφ2
w1σ−1 s3√√==
φ2 > 0.
dα α=12 D(σ + s1 − 1)
D
dφ13
2

48

Appendix G. Balance of payments
Following the analogy of Takahashi et al. (2013), we first consider the trade balanceof the manufacturing good in the world. We then show trade deficits in a country iscompensated by a surplus on the capital account. The income spent in Country 1 on
the manufacturing goods must equal its expenditures:µY1 = µ(w1 L1 + rK1 ) = N1 p11 q11 + N2 p21 q21 + N3 p31 q31 ,Isomorphic expressions are applied to Countries 2 and 3. Trade deficit terms are extracted as follows:µ(w1 L1 + rK1 ) = N1 p11 q11 + N2 p21 q21 + N3 p31 q31= N1 (p11 q11 + p12 q12 + p13 q13 ) + (N2 p21 q21 − N1 p12 q12 ) + (N3 p31 q31 − N1 p13 q13 )= N1 (σπ1 ) + (N2 p21 q21 − N1 p12 q12 ) + (N3 p31 q31 − N1 p13 q13 ),Note that the result of the constant mark-up pricing is used from the second to thethird line. Rearranging the above equation as well as the corresponding equations forCountries 2 and 3 gives(N2 p21 q21 − N1 p12 q12 ) + (N3 p31 q31 − N1 p13 q13 ) = µw1 L1 + r(µs1 − σn1 )K, (G.1)(N1 p12 q12 − N2 p21 q21 ) + (N3 p32 q32 − N2 p23 q23 ) = µw2 L2 + r(µs2 − σn2 )K, (G.2)(N1 p13 q13 − N3 p31 q31 ) + (N2 p23 q23 − N3 p32 q32 ) = µw3 L3 + r(µs3 − σn3 )K, (G.3)
where we make use of πi ni = rni because of r = maxi∈{1,2,3} {πi } for ni > 0. The left
hand side represents the trade deficit of Country i (the net imports of i).
The trade deficit in Country 1, for example, equals the sum of the trade surpluses ofCountries 2 and 3. The world trade surplus against Country 1 is the sum of Eq. (G.2)
and Eq. (G.3):(N1 p12 q12 − N2 p21 q21 ) + (N1 p13 q13 − N3 p31 q31 )= [µw2 L2 + r(µs2 − σn2 )K] + [µw3 L3 + r(µs3 − σn3 )K],(G.4)

By equating Eq. (G.1) with (−1)× Eq. (G.4), we have
µw1 L1 + r(µs1 − σn1 )K = −[µw2 L2 + r(µs2 − σn2 )K + µw3 L3 + r(µs3 − σn3 )K],P
which reduces to r = µ 3i=1 wi Li /K(σ − µ). This equation holds when the equilibrium
capital rewards clear the world manufacturing market (see Section 3.1). As the sum of49

all trade deficits equals the sum of all trade surpluses across countries, we can confirm
that world trade is balanced.
Next, we examine how capital account surplus is offset by trade deficit. The labormarket clearing condition in Country 1 is[(1 − µ)Y1 /p01 ] + N1 (q11 + τ q12 + τ q13 ) = L1 ,→ [(1 − µ)Y1 /w1 ] + [(σ − 1)π1 N1 /w1 ] = L1 ,
→ (1 − µ)(w1 L1 + rK1 ) + (σ − 1)π1 N1 = w1 L1 ,→ µw1 L1 = [(1 − µ)rs1 + (σ − 1)π1 n1 ]K
Substituting this into Eq. (G.1) yields(N2 p21 q21 − N1 p12 q12 ) + (N3 p31 q31 − N1 p13 q13 ) = µw1 L1 + r(µs1 − σn1 )K= r[(1 − µ)s1 + (σ − 1)n1 ]K + r(µs1 − σn1 )K= r(s1 − n1 )K,
where the right hand side in the last line represents the rewards to capital employed in
the other countries.

Appendix H. Parameter values
The figures in the main text are derived using the following parameter values:
Figs. 1 and 2: σ = 2; µ = 0.3; K = L = 20; (s1 , s2 , s3 ) =(I)(0.4, 0.37, 0.23);(II)(0.51, 0.4, 0.09); (III)(0.5, 0.3, 0.2).
Figs. 3 and 4 do not depend on specific parameter values, except for the ordering ofcountry size: s1 > s2 > s3 > 0.
Fig. 5: σ = 1.2; µ = 0.3; K = L = 20; (s1 , s2 , s3 ) = (0.45, 0.33, 0.22)(two-country
agglomeration, left panel); (0.56, 0.3, 0.14)(full agglomeration, right panel).
Fig. 6: σ = 1.2; µ = 0.3; K = L = 20.
Fig. 7: σ = 2; µ = 0.3; K = L = 20; (s1 , s2 , s3 ) = (0.5, 0.3, 0.2).
Fig. 8: σ = 1.2; µ = 0.3; K = L = 20; (s1 , s2 , s3 ) = (0.56, 0.3, 0.14).
Fig. 9 does not depend on specific parameter values.
Fig. 10: σ = 1.4; µ = 0.3; K = L = 20; (s1 , s2 , s3 ) = (0.63, 0.2, 0.17)(solid);(0.66, 0.2, 0.14)(dashed); (0.7, 0.2, 0.1)(dotted).
Fig. 11: σ = 1.2; µ = 0.3; K = L = 20; α = 1.2; (s1 , s2 , s3 ) = (0.53, 0.28, 0.19).
Fig. 12: σ = 2; µ = 0.3; K = L = 20; (s1 , s2 , s3 ) = (0.55, 0.3, 0.15).
Fig. 13: Parameter values are the same as those in Fig. 12 except for σ = 1.2.
50

Appendix I. Industrial evolution of middle countries
in a four-country modelThis appendix shows how equilibrium firm shares evolve as trade costs decline in thefour-country model. We can confirm that our main results are robust: middle-sizedcountries see complex location patterns depending on their relative market sizes.
To illustrate the feasible set of market size, we need to map the three-dimensional
plane, i.e., (s1 , s2 , s3 ) (note that s4 = 1 − s1 − s2 − s3 ), to a two-dimension plane. We
should pin down one of the three market sizes and here fix s3 as a constant denotedby s3 . Figs. A2, A3 and A4 show all possible combinations of market sizes in thefour countries, given s3 = 0.1, 0.2 and 0.24999..., respectively.26 We can observe amarket size triangle in Fig. A4 that was seen in Fig. 3. However, the feasible is not benecessarily triangular; we see a “market size rectangle” in Figs. A2 and A3.

Fig. A2. The market size rectangle at s3 = 0.1.

26

By using s1 +s2 +s3 +s4 = 1, the order of s1 > s2 > s3 > s4 implies s1 > 1/4; 2s3 < s2 +s3 < 2/3;
s2 + s3 < s1 + s3 ; s2 + s3 < 1 − s1 ; and s2 + s3 > 1 − s1 − s3 . All of these conditions are satisfied in
the shaded rectangle/ triangle excluding borders.

51Fig. A3. The market size rectangle at s3 = 0.2.

Fig. A4. The market size triangle at s3 = 0.24999....
Following the same manner as Appendix B, the slope of equilibrium firm shares at
two endpoints, φ = 0 and φ = 1 are derived. At φ = 0 (infinite trade costs), we havedn1
= σ(4s1 − 1) > 0,dφ φ=0dn2= σ(4s2 − 1) R 0,dφ φ=0dn3= σ(4s3 − 1) R 0,dφ φ=0dn4= σ(4s4 − 1) < 0,dφ φ=052

with s1 ∈ (1/4, 1); s2 ∈ (0, 1/2); s3 ∈ (0, 1/3); and s4 ∈ (0, 1/4). At the outset of
opening trade, the largest country gains while the smallest country loses firms. Whether
the middle countries gain or lose firms depends on their own market size. At φ = 1(zero trade costs), we havesi (σ − µ)fi (s1 , s2 , s3 )dni=
, i ∈ {1, 2, 3, 4},dφ φ=1(σ − 1)2
f1 (s1 , s2 , s3 ) = 2s21 − [3 − 2(s2 + s3 )]s1 + 2s22 − 2(1 − s3 )s2 + 2s32 − 2s3 + 1 < 0,
f2 (s1 , s2 , s3 ) = 2s21 − 2(s1 + s3 )(1 − s2 − s3 ) + (1 − s2 )(1 − 2s2 ) R 0,
f3 (s1 , s2 , s3 ) = 2s21 − 2(s1 + s3 )(1 − s2 − s3 ) + (1 − s3 )(1 − 2s3 ) R 0,
f4 (s1 , s2 , s3 ) = 2s21 − (s1 + s2 )[1 − 2(s2 + s3 )] − s2 (1 − 2s2 ) > 0.
When the economy is close to free trade, the largest country loses while the smallest
country gains firms. The middle countries may gain or lose firms depending on theirmarket sizes relative to those of other countries. We note that the signs of the slopesat φ ∈ {0, 1} only depend on market sizes.
From these conditions, we can characterize how firm shares evolve as trade costs
decline. Using the market size triangle/rectangle, the representative equilibrium paths
are reported in Figs. A5 and A6. In all combinations of market size in the two figures, weobserve an inverted-U shape (or U shape) in terms of φ in Country 1 (or Country 4). By
contrast, Countries 2 and 3 exhibit more complex patterns. In the four-country model,
the industrial evolution of middle countries is determined by their relative market size,
as we emphasized in the three-country model.

53Fig. A5. Patterns of equilibrium path in the market size rectangle at s3 = 0.1.

Fig. A6. Patterns of equilibrium path in the market seize triangle at s3 = 0.27.

54

Appendix J. The largest country in a J(> 3) country
modelThis appendix re-examines the equilibrium features of the largest country in a J(> 3)country model. The market sizes are sorted in descending order: s1 > s2 > · · · > sJ .
In the main text, we have shown that (i) there exist corner solutions where at least
one of the countries loses its industry (Proposition 3) and (ii) the largest country could
decrease its welfare (Proposition 5).
Given firm shares, wage rate clearing of labor market in country i ∈ {1, 2, . . . , J −1}
takes:wi =sJ [(1 − µ)si + (σ − 1)ni ],si [(1 − µ)sJ + (σ − 1)nJ ]while wage rate in country J is normalized to unity, wJ = 1.
Conditions for full agglomeration.
Country 1 are given as

The operating profits at full agglomeration inJµ X
π1 |n1 =1 =Yj ,σK j=1# 1−σ "JXµ
1
Yj , i ∈ {2, 3, . . . , J},
πi |n1 =1 =φY1 + (Yi /φ) +σK w1
j6=1,iwhere w1 = 1 + (σ − 1)/[s1 (1 − µ)]. Full agglomeration is the long-run equilibrium if
and only if the following conditions hold:v1i |n1 =1 = π1 − πi |n1 =1 = Ω · F1i (φ) > 0, i ∈ {2, 3, . . . , J},
where F1i (φ) ≡ −(σ + s1 − 1)w1σ−1 φ2 + [σ − w1σ−1 (1 − s1 − si )]φ − w1σ−1 si ,and where Ω ≡ µL/[φσ(1 − µ)K] is a positive constant. This implies that no firms in
Country 1 has an incentive to move. It can be verified that F1k+1 (φ) ≥ F1k (φ) holds
for φ ∈ [0, 1] and k ∈ {2, 3, . . . , J − 1}. Thus we just need to check F12 (φ) > 0 as inAppendix G. The small root of F1k (φ) = 0 is denoted as φk and the large one as φk . By
applying the same analogy, full agglomeration arises at intermediate trade costs [φ2 , φ2 ]55

if σ is in (1, σ1∗ ). σ1∗ is defined byσ1∗ ≡ min arg[σ − H2 (σ) = 0],
σ>1
h
i
p
where H2 (σ) ≡ w1σ−1 (1 − s1 − s2 ) + 2 s2 (σ + s1 − 1) .
We note that H2 (σ) and H(σ) defined in Appendix C are identical due to s3 = 1−s1 −s2in H(σ). A crucial determinant of σ1∗ is the market sizes of the largest and the second
largest countries. The same can be true for thresholds, φ2 and φ2 .
Conditions for j-country agglomeration.
We refer to the situation where countries
from 1 to j(< J) attract all firms as j-country agglomeration. j = 1-country agglomeration corresponds to full agglomeration in our main text. A sufficient condition for
j-country agglomeration is (i) v1k |n1 =1 < 0 for k ∈ {2, 3, . . . , j} and (ii) v1l |n1 =1 > 0
for l ∈ {j, j + 1, . . . , J}. Following the discussion in Appendix C, we obtain j-country∗, σj∗ ).
agglomeration at intermediate trade costs such that φ ∈ [φj+1 , φj+1 ] if σ is in [σj−1σj∗ is defined by∗σ̃j∗ ≡ min arg[σ − Hj+1(σ) = 0],
σ>1


q
∗
σ−1
where Hj+1 (σ) ≡ w1
(1 − s1 − sj+1 ) + 2 sj+1 (σ + s1 − 1) , j ∈ {2, 3, . . . , J − 1},and where w1 = 1 + (σ − 1)/[s1 (1 − µ)]. What determines j-country agglomeration is
the market sizes of Countries 1 and j + 1.
By applying the same analogy of the three-country model, the order of firm movement proceeds as follows; as trade costs decline, countries except for Country 1 lose
and then re-attract firms in a descending order of country index.
Conditions for worse off in Country 1. In the three-country model, whether Country
1 decreases its welfare in the collapse of agglomeration is determined by the sign of Eq.
(9). We note that dn1 /dφ + dn2 /dφ = 0 and dn3 /dφ = · · · = dnJ /dφ = 0 hold at
φ = φ2 . It can be verified that the counterpart of Eq. (9) in the J-country model takes
the same expression as Eq. (9):d log U1
µI(σ) dn1=,
dφ φ=φ2
σ − 1 dφ φ=φ2
|{z}
(−)

where I(σ) ≡ 1 − φ2 w1σ−1 −s1 (σ − 1)2R 0.
(σ + s1 − 1) {σ + s1 (1 − µ) − 1}56

φ2 is defined as the large root of
F12 (φ) ≡ −(σ + s1 − 1)w1σ−1 φ2 + [σ − w1σ−1 (1 − s1 − s2 )]φ − w1σ−1 s2 = 0,
P
where 1 − s1 − s2 = Jj=3 sj . We can check that I(1) = I 0 (1) = 0 and I 00 (1) > 0 hold,
implying that the conditions for Country 1 decreasing its welfare are identical as thosein the three-country model. That is, if s1 < s2 + 1/2 holds and σ is close to unity, I(σ)
is positive and thus the sign of the above derivative becomes negative. The condition
involves the market sizes of the largest and the second largest countries.

Appendix K. Multiple manufacturing industries withdifferent labor intensities
This appendix considers two manufacturing industries (subject to monopolistic competition), called A and B, with different labor intensities. Suppose that industry-A is
more labor intensive than industry-B. An industry-A (or B) firm have a unit labor
requirement of aA (σ − 1)/σ (or aB (σ − 1)/σ; aA > aB ). To highlight different labor
intensities, the expenditure share on each industrial goods is assumed to be identical,
i.e., µA = µB = µ/2. All other settings are the same as in the single-manufacturingmodel.
Given firm shares, the operating profit of an industry-k ∈ {A, B} firm is given by
3

πki =

µwi1−σ X φij Yj,
2σ j=1 ∆kj

where ∆kj ≡3
X

φlj wl1−σ Nkl ,l=1

where the new subscript k stands for the variables of industry-k. We note that labor
intensity, ak , does not show up in the above expression. Although ak appears in the priceP11−σ ),
of individual brand (pkij = τij ak wi ) and in the price index (Pkj = ( 3l=1 Nkl p1−σklj )these ak s are canceled out.
P
It can be verified that capital rewards are common to all countries, r = µ 3i=1 wi Li /(σ−µ)K. As in the main text, the equilibrium wage rate is derived from labor market clearing conditions:wi =s3 [(1 − µ)si + (σ − 1)(nAi + nBi )],si [(1 − µ)s3 + (σ − 1)(nA3 + nB3 )]for i ∈ {1, 2},while w3 is normalized to unity. ak does not enter the above expression because it isreflected on individual price.
57

Solving πk1 = πk2 = πk3 for (nk1 , nk2 ) gives firm shares in the long-run equilibrium.
nAi is expected to behave exactly in the same way as nBi because πAi and πBi havesymmetric expressions. In fact, this can be confirmed in the left panel of Fig. A7.
Moreover, the sum of firm shares over industry in each country (nAi + nBi ) coincides
with the firm share in the single-manufacturing model (ni ) as shown in the right panelof Fig. A7. As consumers equally split their expenditures to two industries, firm sharesare also divided equally regardless of different labor intensities.

Fig. A7. The impact of trade liberalization on firm shares in the two-manufacturingmodel.
Note: Parameter values are σ = 2; µ = 0.3; K = L = 20;(s1 , s2 , s3 ) = (0.4, 0.37, 0.27).

58

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