Discussion Papers In Economics
And Business

Elastic Labor Supply and Agglomeration
Takanori Ago, Tadashi Morita,
Takatoshi Tabuchi and Kazuhiro Yamamoto

Discussion Paper 17-06

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANElastic Labor Supply and Agglomeration
Takanori Ago, Tadashi Morita,
Takatoshi Tabuchi and Kazuhiro Yamamoto

Discussion Paper 17-06
March 2017

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANElastic Labor Supply and Agglomeration
Takanori Agoy Tadashi Moritaz
Takatoshi Tabuchix Kazuhiro Yamamoto{
26 March 2017

Abstract
This study analyzes the interplay between the agglomeration of economic activities
and interregional di¤erences in working hours, which are typically longer in large cities,
as they are normally more developed than small cities. For this purpose, we develop
a two-region model with endogenous labor supply. Although we assume a symmetric
distribution of immobile workers, the symmetric equilibrium breaks in the sense that…rms may agglomerate when trade costs are intermediate and labor supply is elastic.
We also show that the price index is always lower, while labor supply, per capita income,real wages, and welfare are always higher in the more agglomerated region.

JEL Classi…cation: R23, F16
Key words: elastic labor supply; agglomeration; symmetry break

We thank M. Morikawa, Y. Murata, J. Oshiro, P. Picard, and W. Xaio as well as the seminar audiences at
RIETI, Kagawa University, NARSC 2015 at Portland, and ERSA 2016 at Vienna for their helpful comments
and discussions.
ySchool of Commerce, Senshu University, E-mail address: ago@isc.senshu-u.ac.jpzFaculty of Economics, Kindai University, E-mail address: t-morita@kindai.ac.jpx
Faculty of Economics, University of Tokyo, E-mail address: ttabuchi@e.u-tokyo.ac.jp{
Graduate School of Economics, Osaka University. E-mail address: yamamoto@econ.osaka-u.ac.jp1

1

Introduction

Large cities often have more varieties of consumption goods and services, where people havefewer children and work longer hours (Morita and Yamamoto, 2014) earning more salaries inorder to enjoy such rich urban life.1 During the era of Industrial Revolution, we observe two
phenomena: the economic activities are agglomerated in cities and working hours in cities
are longer than that in rural area. Hohenberg and Lees (1985) showed that in the processof Industrial Revolution, geographic agglomeration of economic activities has progressed.
Urban agglomeration materialized in an increase of urbanization rate and the formation ofindustrial clusters in the core of Europe like London that have been by and large sustained(Martin and Ottaviano, 2001). On the other hand, Voth (2000) investigated the working
hours in England during the Industrial Revolution and reported that the total working hours
per year was 3,350 hours in London and 3,211 in the North parts of England in 1760. deVries (1994) called the increase in working hours in the 18th century in the United Kingdom
an “Industrious Revolution.” He argued that since the variety of consumption goods had
increased during this period, workers worked harder in order to earn more income to pay forthe growing number of consumption goods available.
Thus, in the process of Industrial Revolution, the working hours increased because of
the rise in the real wage, which enabled them to purchase consumption varieties. At thesame time, the industrial agglomeration was progressed, and the working hours was raised,especially in London. The purpose of this paper is to construct a model in which industrial
agglomeration brings about the increase in working hours, which becomes one reason for
agglomeration of economic activities in the process of Industrial Revolution.
Based on the foregoing, this study analyzes the interplay between the agglomeration of1

Rosenthal and Strange (2008) show that among professionals, working hours are longer in larger cities,
which are comparable with more developed countries. Gicheva (2013) also shows that young highly educatedworkers work longer hours to pursue career advancement and to earn higher wages based on the 1979 cohortof the National Longitudinal Survey of Youth. According to 2006 Survey on Time Use and Leisure Activities
in Japan, the working time (average time spent on work) is longer in denser prefectures which consist of(more developed) large cities.

2

economic activities and interregional di¤erences in working hours by using the frameworkof new economic geography pioneered by Krugman (1991). For this purpose, we develop
a model of new economic geography by introducing endogenous labor supply without theinterregional migration of labor. More speci…cally, we construct a two-region model with onedi¤erentiated good sector. Each agent is spatially immobile and chooses the optimal amountof labor supply as well as the consumption of the good. An increase in labor supply brings
about disutility due to the labor burden, whereas it raises wage income. Therefore, eachagent determines labor supply at which marginal disutility by labor equals the real wage,
which is de…ned by the nominal income over the price index in the region.
Our main …nding is that even if two regions are identical, the symmetric con…gurationof …rms breaks if the elasticity of labor supply with respect to real wage is su¢ ciently high.
That is, the emergence of an endogenous agglomeration is possible without assuming the
spatial mobility of labor. This …nding is in sharp contrast to studies of new trade theory
such as Krugman (1980), where this symmetry never breaks.
The mechanism that brings about endogenous agglomeration occurs is as follows. The
real wage is higher in a region that has more manufacturing …rms. If …rms agglomerate more,
the price index decreases further, and thus, the real wage rises further. That is, the relativevalue of nominal income to labor disutility goes up. Since our model assumes an elastic labor
supply unlike the familiar models that incorporate a …xed labor supply, the amount of labor
supply rises in the agglomerated region. This leads to higher per capita income and a larger
market size, which attracts manufacturing …rms to the region. In summary, labor supply
has a positive correlation with agglomeration of manufacturing …rms, whereas migration ofworkers leads to agglomeration of …rms in the new economic geography framework.
When the symmetry breaks, we have an asymmetric distribution of …rms as a stableequilibrium, where the amount of labor supply is shown to be larger, while the nominal wage
earning and per capita total income are higher in the agglomerated region. We also show that
individual welfare is higher in the agglomerated region, implying that the higher nominal

3

income and lower price index dominate the higher labor disutility in the agglomerated region.
From the theoretical perspective, we construct a model of endogenous agglomeration
without interregional migration. According to Combes et al. (2008, p. 166), “Moving
beyond the Krugman model in search of alternative explanations appears to be warranted inorder to understand the emergence of large industrial regions in economies characterized bya low spatial mobility of labor.”In this study, we consider that labor supply changes basedon workers’choice of working hours rather than because of the relocation of …rms. Workers
prefer to adjust working hours than changing …rms through interregional migration in the
short run when shocks occur in the labor market. According to Nakajima and Tabuchi (2011),
the annual gross migration between prefectures was 2.9% of the Japanese population for 19542005 and that between states was 1.1% of the U.S. population for 1989–2004. Braunerhjelm
et al. (2000) also report the existence of low spatial labor mobility in EU countries.
Some studies have examined the endogenous agglomeration of …rms without labor migration. Krugman and Venables (1995) introduce the input-output linkages that yield theagglomeration of …rms in the absence of migration. Amiti and Pissarides (2005) assume
training costs for skill formation, which serves as a proxy for labor migration, resulting in
the emergence of …rm agglomeration. Picard and Toulemonde (2006) consider labor unions
that introduce wage rigidities so that unionized and high-wage …rms agglomerate in a region
in the absence of labor mobility. Our study shows that an increase in working hours fosters
agglomeration of …rms under immobile labor, which is consistent with the above-mentioned
facts in the process of Industrial Revolution.
Corsetti et al. (2007) is closely related with our paper. They construct a two country,
monopolistic competition model with elastic labor supply. Their focus is on the e¤ects oftechnological progress on terms of trade and welfare, while our focus is on the endogenous
agglomeration induced by the elastic labor supply. Ago et al. (2016) also consider elastic
labor supply focusing on international di¤erentials in productivity and working hours in
the framework of international trade, whereas this paper considers interregional or intercity

4

trade with migration of …rms, so that agglomeration emerges.
The remainder of the paper is organized as follows. In section 2, we present the basicmodel. Sections 3 and 4 assume the same population size while section 5 assumes di¤erent
population sizes. In section 3, we characterize and examine the symmetric equilibrium of…rm distribution. We show that when the symmetric equilibrium is unstable, asymmetricequilibria exist. In section 4, we analyze such asymmetric equilibria. Section 5 considers
regional asymmetry in the sense that population size or productivity di¤ers between regions.
Section 6 concludes the paper.

2

The model

The economy consists of two regions, denoted r = 1; 2 and a manufacturing sector producing
a di¤erentiated good. Let Lr be the mass of immobile workers in region r, and n be themass of mobile capital in the economy. We assume that one unit of capital is needed as a…xed requirement to produce each variety meaning that the total number of varieties of adi¤erentiated good is n, which is exogenously given.
The preferences of an agent located in region r = 1; 2 are given by:

Ur =

Z n

xr (i)

1

1

+1

di

+1

0

lr

;

> 1;

> 0;

(1)

where xr (i) is the consumption of a variety indexed i in region r and lr is the amount oflabor supply, which reduces the utility since supplying labor reduces leisure time in regionr. Each agent supplies labor and earns hourly wage wr , which is used to purchase the good.
She chooses the amount of labor supply, lr , as well as the consumption of each variety, xr (i).
Therefore, labor supply is elastic. We follow Corsetti et al. (2007) and the elasticity of labor
supply with respect to real wage

is constant.

In addition to the wage, she receives rewards from capital holding, a. Her income con-5

straint is given by
a + wr lr =

Z n

(2)

pr (i)xr (i)di;0

where pr (i) is the price of variety i sold in region r.
From (1) and (2), we …nd the labor supply to be

where
Pr =

lr =

wr
Pr

Z n

1

(3)

1

pr (i)

1

di0

is the price index,
assumeis the elasticity of the substitution between di¤erentiated varieties. We> 1 and> 0 to satisfy the second-order conditions for utility maximization.

Equation (3) shows that labor supply is an increasing function of the real wage. On the
one hand, when the nominal wage wr increases, each agent raises labor supply in order topurchase the good. On the other hand, when price index Pr goes up, the value of real income
goes down, which reduces labor supply.
We also …nd the individual demand for variety i produced in region r and consumed inregion s as follows:

xrs (i) = (a + ws ls )prs (i)Ps1= a + ws1+ Ps

prs (i)
Ps1

;

(4)

where the second equality is derived from the substitution of (3). Because of the symmetryof each variety, we drop i hereafter.
The interregional trade of the good incurs an iceberg type trade cost. If> 1 unitsof the good are exported between two regions, only one unit reaches the destination. We
de…ne

rs =

1

< 1 if r 6= s andrs = 1 if r = s. The price index in region r can be

expressed as
1)1
Pr = (nr p1rr + ns psr

6

1

(5)

for r; s = 1; 2 (r 6= s).
To produce x units of a di¤erentiated good, mx units of labor are needed in addition to
one unit of capital. The rewards from capital holding are the pro…ts of …rms. We assume
that each agent has an equal share of capital, therefore, the total rewards from capital are
equally shared by all agents. The pro…t of a manufacturing …rm in region r is described asr = (prrmwr )xrr Lr + (prsm wr )xrs Ls ;(6)

where individual demand xrs is given by (4) and the reward from capital holding per agent
is given bya=n1 1 + n2 2:
L 1 + L2

(7)

Each manufacturing …rm sets prices, prr and prs , to maximize the pro…ts. The prices of
the good are computed as
m
wr ;
1

prr =

prs =

m
1

(8)

wr :

By substituting (8) into (6), we have

r

=
=

m
wr
1

mwr xrr Lr +

m
1

wr

m wr xrs Ls

mwr(xrr Lr + xrs Ls ) :1(9)

Total labor supply and the total labor demand in region r are lr Lr and nr m (xrr Lr + xrs Ls ),respectively. Thus, the labor market clearing condition in region r is expressed aslr Lr = nr m (xrr Lr + xrs Ls ) ;(10)where the LHS of this equation represents the labor supply and the RHS the labor demand.

7

By plugging (10) into (9), we obtainr =wr lr Lr
:
nr
1

(11)

Hence, we have shown that the pro…t of a …rm is proportional to the sales per …rm, wr lr =nr ,which comprises the wage bill wr lr and the number nr of …rms. The pro…t is in proportion
to the former, while inversely proportional to the latter. In the agglomerated region, thedenominator nr of (11) is larger, implying keen competition among …rms there. To attaina spatial equilibrium, the numerator wr lr of (11) should also be larger in the agglomerated
region. This fact means that …rms in the agglomerated region should o¤er a higher wage bill
wr lr to secure larger labor supply, which is due to a larger number of …rms.
In the spatial equilibrium, the pro…t of each …rm is the same between regions. That is,
the spatial equilibrium conditions are given by

2 = 0;

1(12)

and the labor market clearing condition (10). They lead to
w1 l1
n1 =L1=:n2 =L2w2 l2(13)

If the home market e¤ect n1 =L1 > n2 =L2 is exhibited, then per capita income a + w1 l1 ishigher in the larger region.
Lemma 1 Per capita income is higher in a region with more …rms relative to population.
If an agglomerated region is interpreted as a more developed region (i.e., a large city),
then this agrees with the stylized facts in the urban economy: income per capita is higher
in larger cities.
However, the wage is not necessarily higher in the larger region. As shown in Appendix

8

A, we have
dw
Q0,
d
where

n1 =n and w

Q

1
2

(14)

1

w1 =w2 , which are the endogenous variables to be determined by

the two spatial equilibrium conditions, (10) and (12), with (3), (4), (7), and (11). We choose
the labor in region 2 as numeràire so that w = w1 .2 Since (14) holds for an arbitrary given, we can say the following.
Proposition 2 If trade costs are highthe agglomerated region. Otherwise< 1= (21), the nominal wage, wr is lower in1), the nominal wage is higher in the> 1= (2agglomerated region.
It is somewhat surprising that the wage is lower in the agglomerated region, which usually
does not occur under new economic geography or new trade theory with immobile workers.
We explain in section 4.2 that excess labor supply because of the increase inwage near autarky. A similar intuition can be applied to high trade costs

reduces the< 1= (21).

We also explain in section 4.2 that excess labor demand because of the increase inraisesthe wage near free trade. A similar intuition is applied for low trade costs> 1= (21).3

Firms migrate to a region with a higher pro…t, meaning that ad hoc dynamics may be
given by=:Finally, plugging (3), (7), and (11) into utility (1) yields the indirect utility:a + wr lr
Vr =
Pr

+1

wr
Pr

+1

;

(15)

2
One may wonder if region 2 is empty. However, such a fully agglomerated equilibrium never arises as
we show in section 3.
3
In the real world, nominal wage is higher in the agglomerated regions. This is because domestic interregional trade costs are relatively low. Thus, > 1= (21) my be satis…ed in the real world.

9

where a and Pr are expressed by

3

and wr .

Symmetric equilibrium

To focus on the symmetric equilibrium, we set an equal population size of regions L1 = L2 ,which is normalized to 1 in this section and the next section. It is apparent that there always
exists a symmetric equilibrium for any values of the parameters. However, this equilibrium
can be stable or unstable depending on the parameter values. Substituting (3) and (11)into (12), totally di¤erentiating it with respect to , and evaluating it at the symmetricequilibrium, we can obtain the stability condition as follows:dd=( ;w)=(1=2;1)

@
@

+

@
dP1 @
dP2
+
@P1 d
@P2 d

+

@
dw
@w d ( ;w)=(1=2;1)(16)= Cg ( ) < 0;where dw=d

is computed by applying the implicit function theorem to (10), which is afunction of

and w. C is a positive constant and

g( )

(2

1) (2 + 1)

2

+ 2 (2

1) + 3

2 2

1:

From g ( ), we have the following.
Proposition 3 The symmetric equilibrium is stable if<

break and unstable if>

break ,

where
[(2
break

1) + 1] [(2
2 (2
1) (1

3) + 1])is the solution of g ( ) = 0.
In Figure 1, the blue curve represents=

break .

The symmetric equilibrium is stablebelow it and unstable above it. Furthermore, we can show that the symmetric equilibrium

10

never breaks for a su¢ ciently inelastic labor supply such that

<

p
2 (
1+
2

B

1)
:
1

This result corresponds to the familiar result under an inelastic labor supply
Krugman (1980), among others. When

= 0 inis small, labor supply is inelastic with respect tothe real wage. Suppose that some manufacturing …rms move to a region. The price index in
the region that attracts …rms decreases, which raises labor supply from (3). However, suchan expansion of labor supply is small because labor supply is inelastic. On the contrary,labor demand increases according to the number of …rms. Further, the tight labor market
forces wage to rise, and thus the pro…ts of …rms reduce, which ensures the stability of thesymmetric equilibrium.
Thus, the symmetry break requires an elastic labor supply (large ).4 Suppose

is large

enough and labor supply is elastic with respect to the real wage. Firms can expect large
labor supply and agents can expect a higher real wage, which expands the market size in thedestination region. More precisely, if>

is in the interval of ( B1 ;B1 and0<

B1 <B2 ), where

B , the symmetric equilibrium is unstable when
B2 are the solutions of g () = 0 and satisfy

B2 < 1. Otherwise, the symmetric con…guration is a stable equilibrium.

Next, we check the possibility of a fully agglomerated equilibrium,= 1. If this is thecase, the substitution of (4) into (9) yields the pro…t di¤erential

j =1 = ( 1

"
w1 l1
1
2 )j =1 =
(
1)n

w1
w2

1

#

:

However, because labor supply in region 2 is l2 = 0, the wage in region 2 is w2 = 0 from (3).
Hence,

j =1 =

1, which violates the equilibrium condition. Therefore, full agglomera-

4

According to the literature such as Tabuchi and Thisse (2011), is between 1:9 and 7:6 depending on
the de…nition of industry: the wider it is, the smaller is. If = 3, the threshold of the labor supply
elasticity is B = 2:98. The labor supply elasticity in the literature is between 1 (Domeij and Floden, 2006)
and 3:8 (Imai and Keane, 2004) implying that the symmetric equilibrium can be stable or unstable.

11tion is never an equilibrium.5 Stated di¤erently, manufacturing production is always carriedout in both regions by immobile workers, whose labor supply is positive. Otherwise, theyearn no income and consume no good.
We have seen that the symmetric equilibrium is unstable if

>

break

and that the

fully agglomerated equilibrium never exists. Nevertheless, an equilibrium for any continuous
utilities always exists, as shown by Ginsburgh et al. (1985), and a stable equilibrium always
exists, as shown by Tabuchi and Zeng (2004).This …nding suggests the existence of a partiallyagglomerated equilibrium that is stable if

4>

break .

Asymmetric equilibrium

We explore asymmetric equilibria by examining the conditions of labor market clearing andspatial equilibrium, Taking the logarithm of the ratio of (20) to (19) in Appendix A, we canreduce the equilibrium conditions to the following single equation

f (R)

log

"

1
R

R

1
+2

R

1

#
1) R
= 0:
1)

1 + (2
R + (2(17)

Because the numerator of f 0 (R) is quartic in R, there are at most …ve solutions. Becausef (R) is symmetric about R = 1, the number of solutions is one, three, or …ve, which isdetermined by the two thresholds

break andj . When=j(; ), there is a repeated rootof f (R) = 0 in the interval of (1; 1= ) as well as in the interval of ( ; 1). Then, we canestablish the following proposition.
Proposition 4 Let= 1=2 be the symmetric equilibrium and1 2 (0; 1=2) be the asym-

metric equilibrium.
(i) If<j , there is a unique stable equilibrium5= 1=2;

w2 = 0 implies zero marginal cost under the CES setting. That is, the pro…t-maximizing price is zero,
which leads to in…nite demand and pro…ts. Hence, each …rm has an incentive to migrate to the empty region.

12

(ii) If=<<

break , there are …ve equilibria, three of which are stable and given by1;

1 ; 1=2; 1(iii) If
1; 1

jbreak , there are three equilibria, two of which are stable and given by

>

=

1;

The proof is contained in Appendix B.
Next, consider a thought experiment that the trade freenesssteadily increases. As seenin Figure 1, as

increases, there are two transitions of stable equilibria:(a) whenB,>…rst symmetric equilibrium, then asymmetric equilibrium, and …nallysymmetric equilibrium again;(b) when<B , the equilibrium is always symmetric.

In case (a), there are two subcases:(a1) whenis su¢ ciently large, the …rst bifurcation is tomahawk while the second pitchfork;(a2) whenis intermediate, both bifurcations are pitchfork.

That is, as trade costs steadily fall, the spatial distribution of economic activities is
initially dispersed, then partially agglomerated, and then dispersed again given
above mentioned transition is drawn as the red arrow in Figure 1. For a given>>

B . TheB , fallingtrade costs move along the arrow, where the stable equilibrium distribution of …rms runs
from dispersion to partial agglomeration and then redispersion. It is worth noting that theagglomeration force is strong for intermediate trade costs compared with small and large
ones.

4.1

Decomposition into the four e¤ects

We can decompose the e¤ects of the relocation of manufacturing …rms to region 1 on the
pro…t di¤erential

in the neighborhood of the symmetric equilibrium ( ; w ) = (1=2; 1) inthe stability condition (16). The …rst term of (16) is negative. An increase in the number of…rms brings about the competition e¤ect: the higher number of …rms, the lower are pro…ts.
The second term of (16) is positive. An increase in the number of …rms also generates the13price index e¤ect: an increase in the number of …rms lowers the price index in region 1
and raises the price index in region 2. When the price index is lowered in region 1, agentsincrease labor supply, which expands the market and raises pro…ts. The third term of (16) isthrough the change in the wage. Since @=@w > 0 always holds, the change dw=d throughthe labor market clearing condition (10) matters. Figure 2 illustrates the labor market inregion 1, where the upward sloping curve is the labor supply function given by the LHS of(10) and where the downward sloping curve is the labor demand function derived from the
RHS of (10).6 Further, there is a unique intersection point of the two curves, which is theequilibrium (l1 ; w1 ). Figure 2(A) illustrates the shift in labor supply l1S due to the increasein , while Figure 2(B) presents the shift in labor demand l1D due to the increase in . The
supply curve l1S shifts right because @l1S =@0 and this decreases the wage rate. We namethis e¤ect the excess labor supply e¤ect. Whenincreases, the number of …rms in region

1 increases, which lowers the price index in region 1. When the price index in region 1 islowered, agents in region 1 increase labor supply, since at the given nominal wage, the real
wage in region 1 rises. Then, excess labor supply emerges with the increase in .
The demand curve l1D can shift right or left following the increase in the number of …rms
in region 1. We name this e¤ect the excess labor demand e¤ect. Whenincreases, thenumber of …rms is raised, which increases the labor demand. However, the increase inlowers the price index in region 1, which decreases labor demand there, since competition
among …rms in region 1 intensi…es. If the former e¤ect dominates the latter, the demand
curve l1D shifts right and excess labor demand emerges asincreases. On the contrary, if

the latter e¤ect outweighs the former, the demand curve shifts left. The increase inmay

increase or decrease the equilibrium wage depending on the shifts in the two curves.
It can be shown below that dw=d < 0 if the excess labor supply e¤ect is strong, whereas
dw=d > 0 if the excess labor demand e¤ect is positive and strong. These two e¤ects are new6

From (3), the labor supply function is l1S = k

1
m

0.

n

1

+ (1

)w

. Using (13), the labor demand function is given by l1D = 1k

14

1

, where @l1S =@w > 0 and k( w1 +1 ) 1, where @l1D =@w <w
1

and do not exist in standard models with exogenous labor supply. Analysis of the indirect
impact is somewhat complicated because we have to consider the labor market clearing
condition.
The strength of these four e¤ects depends on the trade freenessas shown below. We

examine the two extreme cases of near autarky and near free trade in the vicinity of thesymmetric equilibrium ( ; w ) = (1=2; 1).

4.2

0 and near free trade

Near autarky

1

When the trade cost is su¢ ciently high, (16) is given by
d
!0 d

=

lim

( ;w)=(1=2;1)

@
@

+

@
dP1 @
dP2
+
@P1 d
@P2 d

+

@
@w
@w @
+

+

< 0:
( ;w)=(1=2;1)

The bracketed term is positive, implying that the price index e¤ect dominates the competition e¤ect, which tends to break the symmetry. However, the product of the second terms
is negative, which implies that the excess labor supply e¤ect dominates the excess labordemand e¤ect. When trade costs are su¢ ciently large, in both countries the real wage islow. Then, an increase in the number of …rms raises the labor supply compared to labordemand and the excess labor supply e¤ect dominates the other e¤ects for a prohibitive trade
cost as a whole, meaning that the symmetry does not break near autarky

2 [0;

B1 ).

When the trade cost is su¢ ciently small, (16) is given by
d
!1 d

lim

=
( ;w)=(1=2;1)

@
@

+

@
dP1 @
dP2
+
@P1 d
@P2 d

+

@
@w
@w @
+

+

< 0:
( ;w)=(1=2;1)

The bracketed term is negative, meaning that the price index e¤ect is dominated by the competition e¤ect, which stabilizes the symmetric equilibrium. When trade costs are su¢ ciently
small, market competition in both countries becomes severe and then the competition e¤ect15

is large. The positive second term means that an increase inhardly a¤ects the price in-dex P1 but raises the wage w1 due to excess labor demand, which destabilizes the symmetricequilibrium. The inequality implies that the competition e¤ect outweighs the other e¤ects forcostless trade, and hence, the symmetry is a stable equilibrium near free trade

2 ( B2 ; 1].

Next, by examining the di¤erential indices, we are able to establish the following results.
Proposition 5 In the asymmetric equilibrium, price index Pr is always lower, while labor
supply lr , wage earning wr lr , per capita nominal income a + wr lr , real wage wr =Pr , andwelfare Vr are always higher in the agglomerated region.
The proof is presented in Appendix C. Price index Pr is lower in the agglomerated region
because more …rms supply varieties without trade costs. In this region, the relative value ofthe nominal wage to the price index is higher, which raises labor supply lr from (3).7
Consequently, wage earning wr lr , per capita income a + wr lr and real wage wr =Pr are
higher in the agglomerated region. These higher values outweigh the disutility from labor
supply, which leads to higher welfare Vr in this region.
Proposition 5 states that labor supply is larger in the agglomerated region with higher
nominal wage earning and per capita income. This is consistent with the facts presented in
the introduction. Large labor supply brings about higher per capita income in this region,
which expands its market size. This in turn attracts manufacturing …rms. As a result,
workers enjoy better access to a large market and are better o¤ with a higher real wage andwelfare in the agglomerated region.
Finally, we check if the home market e¤ect is exhibited in the presence of an elastic labor
supply. This e¤ect is normally de…ned as “a more-than-proportional relationship between
a country’s share of the world production of a good and its share of world demand for thesame good” (Crozet and Trionfetti, 2008). By using the equilibrium condition (13), we can7This result is consistent with Rosenthal and Strange (2008) as we mentioned in the introduction. Theyshow that working hours are longer in larger cities among professionals.

16

easily show that if> 1=2, thenn1
(a + w1 l1 ) L1>
n2
(a + w2 l2 ) L2

always holds. Furthermore, the home market e¤ect is also de…ned that “countries tend
to export those kinds of products for which they have relatively large domestic demand”(Krugman, 1980). This is true if the following ratio exceeds 1:
n1 p12 x12
n1 w11=n2 p21 x21
n2 w21
We know from Proposition 5 that P11
P11

P21

= (1

P1
P2

> P21

)

1

a + w1 l1
a + w2 l2(18), which implies n1 w11> n2 w21

because1

m1

n1 w11n2 w21:

We also know from Proposition 5 that w1 l1 > w2 l2 . Thus, the three terms on the RHS of (18)are greater than 1 for all> 1=2. Hence, the home market e¤ect is necessarily exhibitedeven under an elastic labor supply.

5

Asymmetric regions

5.1

Di¤erent sized regions

So far, the mass of immobile workers was the same between regions. In this section, we
consider the case of di¤erent population sizes between regions, L1 > L2 , to explore the sizee¤ect on the spatial distribution of economic activities.
By using the parameter values of (L1 ; L2 ; ; ; n; m) = (2; 1; 3; 2; 1; 1), the interregionaldi¤erential indices are plotted in Figures 3 and 4. In Figure 3, the blue curve is region 1’s…rm share (= n1 =n), the red curve is the nominal wage di¤erential w1 =w2 , and the yellow
curve is the real wage di¤erential (w1 =P1 ) = (w2 =P2 ). In Figure 4, the blue curve is the utility17di¤erential V1 =V2 , the red curve is the di¤erential in working hours l1 =l2 , and the yellow
curve is the price index di¤erential P2 =P1 . It is worth noting that all the curves are inverted
U-shaped. The property remains the same for di¤erent parameter values.
Several observations can be made from these …gures. First, as to the …rm share, we
observe n1 =n2 > L1 =L2 for all 0 << 1, implying that the home market e¤ect is always

exhibited, as con…rmed by most studies in new trade theory. Second, the nominal wage
in the larger region is smaller for small , but larger for large , which is in accord with
Proposition 2. Therefore, if trade costs are low, this result is consistent with the real world.
Third, the price index is always lower, while labor supply, wage earning, per capita nominal
income, real wage, and welfare are always higher in the larger region for all 0 << 1. This…nding is in accord with Proposition 5. Note that the second and third results are based ondi¤erent population sizes, while Propositions 2 and 5 are based on the same population size.
Finally, all the di¤erential indices converge when the two regions are fully integrated

5.2

= 1.

Di¤erent productivity regions

Next, we consider the case that the marginal productivity in region 1 is higher than region 2.
Setting the parameter values as (L; ; ; n; m1 ; m2 ) = (1; 3; 2; 1; 0:8; 1), several interregionaldi¤erential indices are drawn in Figures 5 and 6. In Figure 5, the blue curve is region 1’s…rm share (= n1 =n), the red curve is the nominal wage di¤erential w1 =w2 , and the yellow
curve is the real wage di¤erential (w1 =P1 ) = (w2 =P2 ). In Figure 6, the blue curve is the utilitydi¤erential V1 =V2 , the red curve is the di¤erential in working hours l1 =l2 , and the yellow
curve is the price index di¤erential P2 =P1 . As in the case of di¤erent productivity, all thecurves are inverted U-shaped. The property remains the same for di¤erent parameter values.
From Figures 5 and 6, we can obtain some observations. First, the nominal wage in thehigh productivity region is smaller for small , but larger for large . Second, the price index
is always lower, while labor supply, wage earning, per capita nominal income, real wage, andwelfare are always higher in the high productivity region for all 0 <18< 1. At last, from theblue curve in Figure 5, …rms agglomerate in the high productivity region.

6

Conclusion

In this study, we introduced an elastic labor supply into the framework of new economic
geography and examined the impacts of trade costs on the equilibrium outcomes of working
hours and the spatial distribution of economic activities. Despite the symmetric distributionof immobile workers between two regions, we found that when trade costs are intermediateand labor supply is su¢ ciently elastic, the symmetry breaks. This …nding is in sharp contrast
to the body of literature on new economic geography. We also showed that the price index isalways lower, whereas labor supply, wage earning, per capita income, real wage, and welfare
are always higher in the agglomerated region.
When the labor supply elasticity is small, symmetric equilibrium tends to be stable. Thisimplies that an elastic labor supply is not a strong agglomeration force. For the emergence of
agglomeration, labor migration may be needed as in the new economic geography framework.
However, when the labor supply elasticity is 3.8 as in Imai and Keane (2004), the symmetricequilibrium is unstable. This implies that the elastic labor supply is one of the important
agglomeration forces.
We can present some future directions of research. The introduction of labor migration
into our model would be an important future extension. It might also be important to incorporate commuting costs into our model because they may be regarded as a part of working
hours for workers. It is of interest how government policies on income tax or commuting
subsidy a¤ect the social welfare.

References[1] Ago, T., T. Morita, T. Tabuchi, and K. Yamamoto (2016) Endogenous labor supply
and international trade, forthcoming in International Journal of Economic Theory.

19[2] Amiti, M., and C. A. Pissarides (2005) Trade and industrial location with heterogeneous
labor, Journal of International Economics 67, 392-412.
[3] Braunerhjelm, P., R. Faini, V. Norman, F. Ruane, and P. Seabright (2000) Integrationand the Regions of Europe: How the Right Policy Can Prevent Polarization. London:
Center for Economic Policy Research.
[4] Combes, P.-P., T. Mayer, and J.-F. Thisse (2008) Economic Geography: The integrationof Regions and Nations, Princeton: Princeton University Press.
[5] Corsetti, G., P. Martin, and P. Pesenti (2007) Productivity, terms of trade and the’home market e¤ect’Journal of International Economics 73, 99-127[6] Crozet, M. and F. Trionfetti (2008) Trade costs and the Home Market E¤ect, Journalof International Economics 76, 309-321.
[7] de Vries, J. (1994) The industrial revolution and the industrious revolution, Journal of
Economic History 54, 249-70.
[8] Domeij, D. and M. Floden (2006) The labor-supply elasticity and borrowing constraints:
Why estimates are biased, Review of Economic Dynamics 9, 242-262.
[9] Ginsburgh, V., Y.Y. Papageorgiou, and J.-F. Thisse (1985) On existence and stability ofspatial equilibria and steady-states. Regional Science and Urban Economics 15, 149-158.
[10] Hohenberg, P. and Lees, L. (1985) The Making of Urban Europe 1000-1950, Cambridge,
MA: Harvard University Press.
[11] Imai, S. and Keane, M. P., (2004) Intertemporal labor supply and human capital accumulation International Economic Review, 45, 601-641.
[12] Krugman, P. (1980). Scale economies, product di¤erentiation and the pattern of trade.
American Economic Review 70, 950-59.
[13] Krugman, P. (1991) Increasing returns and economic geography. Journal of Political
Economy 99, 483-99.
[14] Krugman, P. and Venables, A. (1995) Globalization and the inequality of nations. Quarterly Journal of Economics 110, 857-880.
[15] Kuhn, P. and F. Lozano (2008) The expanding workweek? Understanding trends in longwork hours among U.S. men, 1979–2006. Journal of Labor Economics 26:2, 311-343.
[16] Martin, P. and G.I.P. Ottaviano (2001) Growth and agglomeration. International Economic Review, 42, 947–968.
[17] Morita, T. and K. Yamamoto (2017) Economic geography, endogenous fertility andagglomeration, forthcoming in Japanese Economic Review.
[18] Nakajima, K. and T. Tabuchi (2011) Estimating interregional utility di¤erentials. Journal of Regional Science 51, 31-46.
20[19] Picard, P. M. and E. Toulemonde (2006) Firms agglomeration and unions. European
Economic Review 50, 669-694.
[20] Rosenthal, S.S. and W.C. Strange (2008) Agglomeration and hours worked. Review of
Economics and Statistics 90, 105-118.
[21] Tabuchi, T. and J.-F. Thisse (2011) A new economic geography model of central places.
Journal of Urban Economics 69, 240-252.
[22] Tabuchi, T. and D.-Z. Zeng (2004) Stability of spatial equilibrium. Journal of Regional
Science 44, 641-660.
[23] Voth , H.-J. (2000) Time and Work in England 1750-1830, Oxford: Clarendon Press.

Appendix A: Derivation of (14)Solving (5) for n1 and n2 and substituting them into the spatial equilibrium condition(12) and the labor market clearing condition (10), we get the following two equations
R + (2
1 + (2

W =

1
R

W =
with two variables W

w

1

(19)
1
+2

R

1

(20)

)W
2 ( ; 1= ):
)W

(21)

and
1

P1
P2

R

R

1)
1) R

1

=

+ (1
+ (1

First, suppose = 1=(21). Then, W = R = 1 always holds from (19). Next, suppose
6= 1=(21). From (19) and (21), can be expressed as a function of W :
=

2 W
W2
2

2

2
2

2

2

+1 W
+1 W +(22)2

Di¤erentiating it, we have
d
=
dW
2

4

W
W2

2

2
2

2

2
2

+ 1 (W 2 + 1)

+1 W +

2

When < 1=(21), d =dW = 0 has no real solution of W . This implies the sign of d =dW
does not change. Since d =dW jW =1 < 0, d =dW should be negative for all W .
On the other hand, when > 1=(21), d =dW = 0 has two real solutions, which
are denoted by Wl and Wh , where Wl < Wh . Let W0 be the solution of = 0 in (22) andW1 be the solution of = 1 in (22). We readily have Wl < W0 < 1 < W1 < Wh . Since
d =dW jW =1 > 0, d =dW should be positive for all W .

Appendix B: Proof of Proposition 4
21

We getf (1) = 0f (1= ) =

1
+1

2>
2<

Computing the derivative of f (R) and factorizing it, we havef 0 (R) =dfn (R);
dfd (R)

where dfn (R) is quartic in R anddfd (R)

(

2

) (R

) (1

R) [R + (21) ] [1 + (21) R] R:

The sign of dfd (R) is determined by the sign of2.
Because break >2, three cases may arise.
(i)
2,
1 R
R 1 , should be equal
When =
2, the term in parentheses in (17), f1 (R)R
to 1. Otherwise, f (R) is either 1 or +1. Because f10 (R) < 0 for all R 2 ( ; 1= ) andf1 (1) = 0, there exists a unique equilibrium R = 1.
When <
2, we can show that @dfn (R)=@ < 0, we get dfn (R) > dfn (R)j = 2 > 0.
On the other hand, dfd (R) is positive, and thus f 0 (R) is positive for all R 2 ( ; 1= ). Hence,
there exists a unique equilibrium R = 1.
(ii) break > >2.
We know that f (1) = 0, f (1= ) = 1 for >2, and that dfd (R) < 0 for all>
2. We also have dfn (1) < 0 for all
2 < < break . Hence, the number of solutionsin the interval of (1; 1= ) is even. However, because dfn (R) is quartic in R, there are at
most …ve solutions in f (R) = 0. Since f (1) = 0 and since f (R) is symmetric about R = 1,
the number of solutions in f (R) = 0 in the interval of (1; 1= ) is either zero or two. Moreprecisely, the number of equilibria in the interval of ( ; 1= ) is one if < minf 1 ; break g and…ve if 1 < < break , where 1 is a function of and .
(iii) > break .
We know that f (1) = 0, f (1= ) = 1 for >2, and f 0 (1) > 0 for all > break .
Hence, the number of solutions in f (R) = 0 in the interval of (1; 1= ) is one, and thus, thenumber of equilibria is three.

Appendix C: Proof of Proposition 5
Assume> 1=2.
(i) Proof of P1 < P2 . From (21), we immediately have
sgn

1
2

= sgn (1R ) = sgn (P2(ii) Proof of l1 > l2 . From (3), we get
l1
=
l2

w P2
P1

=
22

W
R

1

;

P1 ) :

where W is a function of R given by (19). Because @ (W =R ) =@R < 0 and W =R jR =1 =1 hold, we have1sgn
= sgn (1 R ) = sgn (l1 l2 ) :2(iii) Proof of w1 l1 > w2 l2 . We showed in Lemma 1.
(iv) Proof of w1 =P1 > w2 =P2 . This is obvious from l1 > l2 together with (3).
(v) Proof of V1 > V2 . From (15), we get
V1

V2 =

P2
2(

1) ( + 1) (R )

1
1

n
h
( + 1) 1

(R )

2
1

(m )

+1

i

(2 +

1) (R )

where ml1 =l2 . Since @ (V1 V2 ) =@m > 0 holds for all R < 1, we have V1
V1 V2 jm =1 > 0 for all R < 1, i.e., for all> 1=2.

23

1
1

h

1

V2 >

(m )

+1

io

6
5

B
C

4
3
2
1

A
0.2

0.4

0.6

0.8

1.0

Figure 1: One symmetric stable equilibrium (A), three stableequilibria (B), and two asymmetric stable equilibria (C)

𝑤1
𝑆′

𝑆

𝐷

𝑙1
0
Figure 2(A): Supply shift due to excess labor supply effect

𝐷′

𝑤1

𝑆

𝐷

𝑙1
0
Figure 2(B): Demand shift due to excess labor demand effectratio
2.0

1.5

1.0

0.5

0.0

0.2

0.4

0.6

0.8

1.0

Figure 3: Interregional differentials when L/L= 2(λ blue, w/w red, (w/P)/(w/P) yellow)
ratio
5
4
3
2
1

0.0

0.2

0.4

0.6

0.8

1.0

Figure 4: Interregional differentials when L/L= 2(V/V blue, l/l red, P /P yellow)

ratio
2.0

1.5

1.0

0.5

0.0

0.2

0.4

0.6

0.8

1.0

Figure 5: Interregional differentials when m/m= 0.8(λ blue, w/w red, (w/P)/(w/P) yellow)
ratio
4

3

2

1

0.0

0.2

0.4

0.6

0.8

1.0

Figure 6: Interregional differentials when m/m= 0.8(V/V blue, l/l red, P /P yellow)