Discussion Papers In Economics
And BusinessSubsidy competition, imperfect labor markets,and the endogenous entry of firms
Tadashi Morita, Yukiko Sawada and Kazuhiro Yamamoto

Discussion Paper 17-07

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANSubsidy competition, imperfect labor markets,and the endogenous entry of firms
Tadashi Morita, Yukiko Sawada and Kazuhiro Yamamoto

Discussion Paper 17-07
March 2017

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANSubsidy competition, imperfect labor markets,and the endogenous entry of …rms
Tadashi MoritayYukiko SawadazKazuhiro Yamamoto x
March 27, 2017

Abstract
This paper constructs a model of subsidy competition for manufacturing …rms under labor market imperfections. Because subsidies a¤ect the
distribution of …rms, they in‡uence unemployment rates, the number of…rms, and welfare. In our model, governments always provide ine¢ ciently
high subsidy rates to manufacturing …rms. When labor market frictions
are high, subsidy competition is bene…cial, although subsidies under subsidy competition are ine¢ ciently high. We show that an increase in labormarket frictions always lowers welfare, whereas trade liberalization always
improves welfare. Finally, we …nd that a rise in labor market friction in acountry raises the equilibrium subsidy rate, a¤ects unemployment rates,
and lowers welfare.

JEL Classi…cation: F10, J64, R10.
Key words: Labor market friction, Unemployment, Subsidy competition.

We are grateful to Rainald Borck, Koichi Futagami, Naoto Jinji, Keisuke Kawata, Yoshifumi Kon, Noritaka Kudoh, Masayuki Morikawa, Hiroshi Mukunoki, Yasusada Murata, Hikaru
Ogawa, Toshihiro Okubo, Shin-Kun Peng, Yasuhiro Sato, Mototsugu Shintani, Hajime Takatsuka, Jacques-Franco Thisse, Koji Yamamoto, Makoto Yano, and Dao-Zhi Zeng. Morita acknowledges …nancial support from a JSPS Grant-in-Aid for Young Scientists B (JP25870925)
and RIETI. Yamamoto acknowledges …nancial support from a JSPS Grants-in-Aid for Scienti…c Research B (JP16H03615, JP15H03348) and RIETI.
y Faculty of Economics, Kindai University, E-mail address: t-morita@kindai.ac.jp
z Graduate
Schoolof
Economics,
Osaka
University,
E-mailaddress:nge012sy@student.econ.osaka-u.ac.jp
x Graduate
Schoolof
Economics,
Osaka
University.
E-mailaddress:
yamamoto@econ.osaka-u.ac.jp1

1

Introduction

In recent years, Active Labor Market Policies (ALMP), involving subsidies toprivate sector employers have been executed in many EU (Kluve (2010)) andOECD countries (Card, et al. (2010) and Martin (2015)). On the one hand,in many countries, governments provide subsidies to private …rms with the objective of lowering unemployment rates, and these subsidies have been considered to have only indirect e¤ects on foreign countries.1 On the other hand,
the Subsidies and Countervailing Measures (SCM) Agreement aims to impose
discipline on subsidies granted by WTO members, because subsidies may be
harmful for other countries.2 A subsidy in a country may result in anothercountry experiencing a negative externality.3 In other words, ALMPs, althoughaiming to internalize the distortions generated by labor market imperfections,
may result in negative externalities to other countries, whereas WTO prohibits
subsidy policies that may be harmful for other countries.4 This paper investigates whether the WTO’s prohibition of subsidy competition is bene…cial or
harmful for two countries that trade manufactured goods with each other andface labor market imperfections. The analysis shows that subsidy competition
is bene…cial (wasteful) when labor market frictions are large (small).5
We construct a two-country, two-sector (manufacturing and agriculture)model in which markets for manufactured goods are segmented between two
countries and the total number of manufacturing …rms is endogenous.6 One
speci…c feature of our model is that the labor market in the manufacturing sector is assumed to be imperfect. Firms entering the manufacturing sector search
for workers to employ, and these search activities are assumed to incur a positive search cost, which lowers the equilibrium number of …rms and raises the
pro…ts of …rms. Individual …rms’entries into the manufacturing sector raise the
1 OECD (2010) states that labor market interventions have an indirect bearing on international trade.
2 Mavroidis (2016) states that SCM Agreement aims to discipline subsidies granted by WTO
members. To this end, it requires that WTO members avoid using two types of prohibited
subsidies (local content and export subsidies) and other subsidies that may adversely a¤ectother WTO members. The current SCM Agreement does not make the treatment of subsidies
conditional on their rationale. That is, nowadays, subsidies can be counteracted regardless of
their rationale.
3 We will provide an explanation of how negative externalities are created by subsidy competition in the latter part of Introduction.
4 The SCM Agreements prohibit an export subsidy or a subsidy contingent on the use ofdomestic over imported goods. If the subsidy policy in our paper can be interpreted as anexport subsidy or a subsidy contingent on the use of domestic over imported goods, the SCM
Agreement would prohibit such a subsidy.
5 Boadway et al. (2002) show a case in which tax competition improves welfare in a modelwith labor market imperfections. Wilson (1999) and Wilson and Wildasin (2004) introduce
models in which tax competition improves e¢ ciency in their comprehensive surveys of tax
competition studies.
6 Several studies examine segmented product markets in which the total number of manufacturing …rms is exogenous. Baldwin and Krugman (2004), Borck and P‡üger (2006), Hau‡er
and Wooton (2010), Kind et al. (2000), and Ludema and Wooton (2004) construct modelsof tax competition under segmented markets. In these models, the total number of …rms isexogenous.

2

probability of unemployed workers …nding a job in a given period. Workers whoenter the manufacturing sector search for a job and pay opportunity costs equal
to wages in the agriculture sector. Then, the wage in the manufacturing sector
should be higher than that in the agriculture sector. If …rms’search costs arezero, an in…nite number of …rms enter in one period, which makes the expected
search duration for a worker equal to zero. In this case, the equilibrium wage
in the manufacturing sector equals the wage in the agriculture sector, whichmeans that the labor market is perfect and there is no ine¢ ciency. As searchcosts for …rms are positive, matched …rms obtain higher pro…ts and workers receive higher wages, both of which result in positive rents in the absence of policy
intervention by governments. Thus, under the existence of a positive search cost,governments have an incentive to provide subsidies to manufacturing …rms tointernalize the ine¢ ciency induced by labor market imperfections.
Each government is assumed to provide a subsidy to maximize the welfare
in its own country. In our model, there is an externality generated by subsidy competition, as in previous tax competition studies, including Borck et al.
(2012) and P‡üger and Suedekum (2013). The increase in subsidies speeds up
the entry of …rms. This entry of …rms into the country then intensi…es competition, which induces the exit of …rms from the other country. In this paper, theprovision of a subsidy in one country in‡uences the welfare level of the othercountry through three channels. First, the decrease in the number of …rms in theother country raises the equilibrium price in the other country and an increasein the volume of imports lowers the equilibrium price in the other country.7
We call this e¤ect, which can be observed in studies of segmented markets, theconsumer surplus e¤ ect. Second, the decrease in the number of …rms reduces
the number of matched …rms and workers in the manufacturing sector and lowers welfare in the country, which we term the labor market imperfections e¤ ect.
Third, the decrease in the number of …rms reduces the total expenditure on thesubsidy, which raises welfare. We call this the …scal e¤ ect. @The total sumof these three e¤ects are (dis)externality induced by subsidy competition. We
show that when we evaluate the costs or bene…ts of subsidy competition for
two countries, it is important to compare the size of (dis)externality induced bysubsidy competition with costs of labor market friction. When (dis)externality
induced by subsidy competition is smaller than costs of labor market friction,
subsidy competition is bene…cial for two countries.
In addition to above evaluation for costs and bene…ts of subsidy competition,
our simple model enables us to execute next three topics: e¤ects of the increasein labor market friction, e¤ects of trade liberalization, and the case of asymmetric countries. We study how an increase in labor market frictions a¤ects
unemployment rates. In our model, equilibrium unemployment rates increase
7 In our model, we assume that the total number of …rms is endogenous and increases witha rise in the subsidy rate in a country. We can show that the increase in the number of…rms in the country is larger than the decrease in the number of …rms in the other country.
This reduces the negative externality compared with models in which the number of …rms isexogenous. However, subsidy competition always results in a race to the bottom, even in aperfect labor market.

3with the number of manufacturing …rms, because a rise in the number of manufacturing …rms attracts workers away from the agricultural goods sector andincreases the number of workers searching for jobs in the manufacturing sector.
Our analysis shows that the increase in the labor market frictions reduces theequilibrium number of matched …rms, which lowers the unemployment rate. At
the same time, the increase in labor market frictions reduces the tightness of
the labor market, which raises the unemployment rate. When the former effect is stronger (weaker) than the latter e¤ect, the increase in the labor market
frictions lowers (raises) the unemployment rate. When the market size for manufactured goods is small (large), an increase in labor market frictions lowers(raises) unemployment rates.
The e¤ects of trade liberalization on welfare under tax competition are investigated in recent papers by Egger and Seidel (2011), Exbrayat et al. (2012), andHau‡er and Mittermaier (2011). Our simple model enables us to derive clear
results about the e¤ects of trade liberalization on unemployment rates and welfare under subsidy competition. When trade liberalization occurs, the market
competition among manufacturing …rms becomes intense, which decreases thenumber of …rms, whereas the volume of exports increases, which increases thenumber of …rms. When trade liberalization increases the number of …rms, italso raises unemployment rates. We further show that, when trade costs are
high, trade liberalization raises unemployment rates. When trade costs are low,
the in‡uence of trade liberalization on unemployment rates depends on market
size: when the market size is large (small), trade liberalization lowers (raises)
unemployment rates. As seen above, trade liberalization can therefore raise or
lower unemployment rates. However, our model shows that trade liberalization always improves welfare. Thus, policies that facilitate trade liberalization
improve welfare when two countries are under subsidy competition.
Finally, we study a case where two countries are asymmetric with respect
to labor market frictions. Our analysis shows that, in the country with higher
labor market frictions, the equilibrium subsidy rate is higher.8 Moreover, an
increase in labor market frictions in a country raises its subsidy rate and lowers
the subsidy rate in the other country. We analyze how an increase in labormarket frictions in a country a¤ects unemployment rates and welfare, and …ndthat it lowers the country’s welfare, whereas it reduces unemployment rates andimproves welfare in the other country.
Related works include Kind et al. (2000), Baldwin and Krugman (2004),
Ludema and Wooton (2004), Borck and P‡üger (2006), and Hau‡er and Wooton(2010), all of which present tax competition models with segmented markets.
However, in all these models, in contrast to our model, the number of manufacturing …rms is exogenous and the labor market is perfect. Similarly, although
Davies and Eckel (2010) and P‡üger and Suedekum (2013) construct models oftax (subsidy) competition with an endogenous number of …rms, they focus on
the e¤ects of heterogeneous …rms on tax competition rather than on the e¤ects
8 Egger and Seidel (2011), Hau‡er and Mittermaier (2011), and Exbrayat et al. (2012) also
show that the equilibrium tax rate is lower in the country with high labor market frictions.

4

of labor market frictions.
Some studies examine the e¤ects of an imperfect labor market on the resultsof tax competition. Fuest and Huber (1999), Ogawa et al. (2006, 2016), and Sato(2009), for instance, study how labor market frictions in‡uence the results of tax
competition in a model with perfect product markets. Fuest and Huber (1999)
introduce wage bargaining, Ogawa et al. (2006) introduce a minimum wage,
Ogawa et al. (2016) introduce labor unions, and Sato (2009) introduces search
frictions to study the e¤ects of labor market imperfection on tax competition. In
these papers, the number of …rms is exogenous and product markets are perfectly
integrated. Comparing these papers with the present study, the number of …rms
is endogenous and markets are segmented between two countries.
Egger and Seidel (2011), Hau‡er and Mittermaier (2011), and Exbrayat
et al. (2012) construct tax competition models of imperfect product markets
and labor markets. In the latter two studies, the presence of a labor unionbrings about labor market imperfections, whereas in the former, a fair-wage
preference produces labor market imperfections. In our model, search frictions
a la Pissarides (2000), bring about labor market frictions. In Egger and Seidel(2011), Hau‡er and Mittermaier (2011), and Exbrayat et al. (2012), the numberof …rms is exogenous; however, none of these studies investigates whether tax
competition is bene…cial or wasteful.
Some studies point out that tax competition may be bene…cial. Ottaviano
and van Ypersele (2005) present a tax competition model with monopolisticcompetition, showing that under certain conditions, tax competition enhancese¢ ciency. Borck et al. (2012) present a model in which the ine¢ cient lock-in of
agglomeration may be removed by subsidy competition. Boadway et al. (2002)
construct a tax and redistributive policy competition model with search frictions in which governments compete by implementing ine¢ cient redistributing
policies. They …nd that tax competition reduces such ine¢ cient redistributive
policy competition, which improves welfare. Although these papers show thattax competition may be bene…cial, they di¤er from our model in terms of how
tax competition improves e¢ ciency. In our model, the entry of manufacturing…rms becomes ine¢ ciently scarce in the case without subsidy competition because of the existence of positive search costs. Positive subsidies under subsidy
competition thus increase the number of …rms, which improves welfare. In Ottaviano and van Ypersele (2005) and Borck and P‡üger (2012), the labor market
is perfect and the number of …rms is exogenous. In Boadway et al. (2002),
ine¢ ciency is not induced by positive search costs, while the number of …rms isexogenous. As this summary of the literature indicates, our paper thus adds a
new channel that brings about bene…cial tax competition.
The seminal paper of Harris and Todaro (1970) presents a model of urban
unemployment in developing countries. Our model has a similar structure to
theirs, in which the labor market in the rural agriculture sector is assumed
to be perfect, whereas that in the urban manufacturing sector is imperfect.
Workers therefore migrate from rural to urban areas because expected real wages
in urban areas are higher than those in rural areas, although unemployment
also exists in the former. In the equilibrium, expected real wages in urban
5areas thus equal real wages in rural areas.9 Our model analyzes the e¤ect ofsubsidies in the urban manufacturing sector in developing countries. We showthat governments provide urban manufacturing subsidies to improve welfare.
Such a subsidy induces …rms’entry, which brings about the externality to theother country and the equilibrium subsidy rate becomes too high. This paper
shows that subsidy competition is bene…cial when the labor market frictions in
the urban manufacturing sector are large.
The remainder of this paper is organized as follows. Section 2 presents themodel and derives the equilibrium conditions. Section 3 studies the case of
perfect labor markets. Section 4 analyzes the case of imperfect labor markets.
Section 5 investigates the case of asymmetry between two countries with respect
to labor market frictions. Section 6 concludes the paper.

2

The model

2.1

Basic setup

There are two countries, 1 and 2. The variables that refer to country 1 (2)have the subscript 1 (2). Each country is endowed with a …xed amount of labor
L1 = L2 = 1.10 We assume that agents in both countries obtain utility fromthe consumption of agricultural goods and homogeneous manufactured goods.
In the agricultural goods sector, there is no labor market friction, whereas in
the manufactured goods sector, there is labor market friction. Although labor
can be mobile between sectors in the same country, it cannot be mobile betweendi¤erent countries. The utility function of the agent in country i is given by~i = zi + Aqi
Uqi2; i = 1; 2;
2where zi and qi represent the consumption levels of agricultural goods and homogeneous manufactured goods in country i, respectively. The budget constraintof the agent in country i is:zi + p i q i = y i ;where yi is total income. In this model, the agricultural goods are chosen to
be the numéraire. By maximizing the utility function, a demand function for
manufactured goods becomes:
qi = Api :2~i = yi + (A pi ) .
Then, the indirect utility level in country i is U
2
Technology in the agricultural goods sector requires one unit of labor to
produce one unit of output. With free trade of agricultural goods, the choice of
9 Harris and Todaro (1970) and subsequent studies that built on this seminal paper, including Krichel and Levine (1999), Yabuuchi (1993), and Zenou (2011) analyze the welfare e¤ects
of urban employment subsidies.
1 0 We assume that both countries have the same market size. Then, when the level of labormarket imperfection is the same in both countries, they are perfectly symmetric.

6

this good as the numéraire implies that the equilibrium wage is equal to one inboth regions, w1 = w2 = 1.
Our focus lies on the market for manufactured good qi , which is served byni …rms in country i. Following Hau‡er and Stähler (2013), we assume that amanufactured goods …rm can produce a …xed amount of goods. In addition, amanufactured goods …rm can produce one unit of goods for the domestic market
and one unit of goods for the foreign market. When …rms exports one unit ofmanufactured goods, 0 < t < 1 units of goods arrives in the foreign country.
We interpret 1 t as trade costs. When t is small (large), trade costs are high(low).11
The inverse demand functions in country i are given by:
pi = A(ni + tnj ) ; i; j = 1; 2; i 6= j:(1)

Therefore, the revenue of manufacturing …rms in country i is given by:
Ri = [A2.2(ni + tnj )] + t [A(tni + nj )] :(2)

Matching

The search and matching setting in this paper has a similar structure to that
presented by Pissarides (2000, Ch. 1). In the manufactured goods sector, there
are search and matching frictions. Let the matching function be Mi = g(ui ; vi ),where Mi denotes the number of job matches, ui denotes unemployed workers,
and vi denotes job vacancies engaged in the matching process. The probabilityof a manufactured goods …rm …nding a worker is q( i ) = Mi =vi , where i = vi =ui
and i represent the tightness of the labor market. An increase in i decreases
the probability of a …rm …nding a worker for its vacancy. The probability of aworker …nding a job is Mi =ui = q( i ) i . An increase in i raises the probabilityof a worker …nding a job.
Next, we focus on the value of workers and manufacturing …rms. Let Wi and
Ui be the present value of the expected incomes of an employed and unemployedworker, respectively. Although Pissarides (2000) assumes that the unemployedworker receives unemployment bene…ts, we assume, for analytical simplicity,that unemployment bene…ts are zero. Then, Ui is:
Ui = (z + ai

Ti +

pi )2

(A
2

) + q( i ) i (WiUi ) ;(3)

where is the discount rate, ai is the asset revenue, and Ti is the lump-sum headtax in country i. We assume that z units of agricultural goods are distributed
1 1 Under the assumption of manufacturing …rms having …xed outputs, we can derive theexplicit forms of the equilibrium subsidy rates and social welfare functions. In the variable
output case, however, we cannot derive the explicit forms of the equilibrium subsidy rates andsocial welfare functions. In the Appendix 6.1, therefore, numerical methods are used to show
that, in the variable output case, we can derive the same main result as in the …xed output
case.

7

to each agent in each period.12 The second term on the right-hand side of (3)represents capital gains from success in matching. The value of Wi is given by:
Wi = (z + wM i + ai

Ti +

pi )2

(A
2

) + (Ui

Wi );

(4)

where wM i denotes the wage rate in the manufactured goods sector in country
i and denotes the rate of job destruction, which is an exogenous variable. The
second term on the right-hand side of (4) represents the capital loss to workers
from losing their jobs.
Next, we describe …rms’activities. Let Ji and Vi be the present discounted
values of the expected pro…ts of an occupied job and a vacant job, respectively.
The value of a vacant job is given by:Vi =k + q( i )(Ji

Vi );

(5)

where k denotes the search cost, which is identical in both countries. The
second term represents the capital gain from success in matching. The value ofan occupied job is given by:
Ji = (RiwM i + si ) + (Vi

Ji ):

(6)

si represents the lump-sum subsidy rates in country i.13
We assume that workers and …rms engage in wage bargaining. Speci…cally,
the wage rate in the manufactured goods sector is determined by Nash bargaining. The worker’s share of the total surplus is and the …rm’s share of the total
surplus is 1
. Then, the following equation must hold:Wi

2.3

Ui = (Ji + Wi

Vi

Ui ):

(7)

Equilibrium

In the equilibrium, because the number of workers …nding a job is equal to thenumber of workers who lose a job, the following equation must hold:q( i ) i ui = ni :(8)Some workers succeed in matching and others become unemployed. Then, thelabor market equilibrium condition in the manufactured goods sector is givenby:
LM i = ui + ni ;(9)where LM i denotes the supply of workers in the manufactured goods sector incountry i. In this paper, we assume that the agents engaged in the agricultural
1 2 We assume that z is su¢ ciently large, which ensures that the post-tax income of all agents
becomes positive in the equilibrium.
1 3 In this paper, we assume that governments provide subsidies to manufacturing …rms. In
the case that governments provide subsidies to matched workers (wage subsidies), we canderive the same results as for the case of subsidies to manufacturing …rms. See Appendix 6.2.

8

goods sector cannot search for manufactured goods …rms. In addition, when
the agents move from the agricultural goods to the manufactured goods sector,
the agents in the manufactured goods sector become unemployed. Then, thevalue of an unemployed worker is equal to the value of a worker engaged in the
agricultural goods sector. Thus, the following equation can be obtained:
Ui = 1 + z + ai
From (3) and (4), Wi

Ti +

pi )2

(A
2

:

(10)

Ui is given by:
Wi

Ui =

wM i
:
+ + q( i ) i(11)Then, by substituting (10) and (11) into (3), we can obtain the wage rate in the
manufactured goods sector as follows:
wM i = 1 ++:q( i ) i(12)

The …rst term of the right-hand side, 1 represents the outside option of workers
engaged in the manufactured goods sector, and the second term is the risk
premium for workers entering the manufactured goods sector. By using (5) and(6), we can obtain Ji Vi as follows:
Ji

Vi =

(Ri

wM i + si ) + k:+ + q( i )(13)

By substituting (7), (11), and (12) into (5), the value of a vacant job is givenby:1
Vi = k +:(14)i

Therefore, the value of a vacant job is decreasing with the tightness of the labormarket. When the value of a vacant job is positive, …rms enter the market andthe tightness of the labor market becomes severe. When the value of a vacantjob is negative, …rms exit the market and this alleviates labor market tightness.
Therefore, the value of a vacant job becomes zero, Vi = 0, and the tightness of
the labor market in each country is given by:
1 =

2 =

=

1
k

:

(15)

Then, in this setting, the equilibrium labor market tightness i is independentof the subsidy rate si , which simpli…es the analysis. Because of the free-entryof manufacturing …rms and the arbitrage of workers between the manufacturing
and agricultural sectors, the subsidy to matched manufacturing …rms increases
both the number of vacant …rms and the labor supply in the manufacturingsector, which makes the tightness of labor market independent of the subsidy
9rate. When the search cost is large or the worker’s share of the total output islarge, the tightness of the labor market becomes small.14
Lemma 1 When labor market frictions (search costs) are large, the tightness of the labor market becomes small. The tightness of the labor market isindependent of the lump-sum subsidy rate.
By substituting (11), (12), (13), and Vi = 0 into (7), we can obtain the
pro…ts of the manufactured goods …rms in country i as follows:
Ri + si
+

q( )

1

= 1;

(16)

where the left-hand side of this equation is the expected bene…t for workers
once they can match with …rms, and the right-hand side represents the bene…twhen workers engage in the agricultural goods sector.15 Then, from the above
equation, the pro…t level in country i can be obtained as follows:
Ri + si = 1 +

+

r;

q( )

(17)

where r represents the after-subsidy pro…t rate and @r( )=@ < 0. Whensearch costs are large, the entry of …rms becomes small and the pro…t level incountry i becomes large.
Here, we focus on the interior equilibrium in which there is a positive numberof …rms in both countries (n1 > 0 and n2 > 0). Equations (17) determine theequilibrium number of …rms in both countries. By substituting (17) into (2),
we obtain:[A (ni + tnj )] + [A (tni + nj )] t + si = r:
(18)
Thus, the equilibrium number of …rms in country i is:ni =A (1 + t) (12t)(1 + t2 )(r(1 t2 )2si ) + 2t(r

sj )

:

(19)

We can see @ni =@si > 0, and @ni =@sj < 0. The subsidy rates in one countryin‡uence the number of …rms in the other, which is the externality that occurs@ni sjas a result of the subsidy. We de…ne "@sj ni , as the elasticity of thenumber of …rms in a country to the subsidy rate in the other country. When
" is large, a small increase in the subsidy rate in the other country brings
about a large decrease in the number of …rms, which means that the subsidy@"
generates a large externality. We can see that @r< 0. Thus, when r is large,
1 4 Policies such as subsidies to unemployed workers and subsidies for the search costs of …rms
a¤ect equilibrium labor market tightness i , which complicates the analysis. In the Appendix
6.3, we analyze a case in which governments subsidize the search costs of …rms.
1 5 We substitutei into (16) as follows:(1)q( i )

Ri

ti
+

1

= k:

This equation means that the expected bene…t of …rms equals the search costs.

10

the externality generated by subsidy competition becomes small. In addition,@"@A > 0 which means that, when the size of manufactured goods market, A islarge, the externality caused by subsidy competition is large.
From (19), the total number of …rms in this economy is given by:
n1 + n2 =2A(1 + t) + s1 + s2
2

(1 + t)

2r

:

(20)

Then, an increase in the subsidy rate raises the total number of …rms.
In this paper, we assume that agents in country i own …rms located in thatcountry. Then, the capital market equilibrium condition in country i is given
by: 16ai = ni Ji :The government budget constraint is Ti = si ni , where the left-hand side represents the tax revenue and the right-hand side represents the government expenditure on the subsidy.
In Appendix 6.4, we derive the social welfare in coutnry i as a functionof subsidy rates in two countries. The government chooses its subsidy rate tomaximize welfare in each country:17
2

SWi = ni Wi + (1ni )Ui = (1 + z) + ni (si ) +(ni + tnj )
;
2

(21)

where
(Ji Vi ) + (Wi Ui ) = q( ) represents the rents of matchedworkers and …rms in the manufacturing sector brought about by labor market
imperfections. Thus, is the extent of the labor market frictions. The term
ni represents aggregate rents in country i, si ni represents the total subsidy
expenditure, and the third term represents the consumer surplus.18@"
Note that r = 1 + + . From, @r< 0, we …nd that @@" < 0, which meansthat the externality caused by the subsidy decreases with the size of the labormarket frictions. When the labor market frictions are large, the entry of …rms
incurs high costs for …rms. Thus, the elasticity of the number of …rms to thesubsidy rate decreases with an increase in labor market frictions. This …ndingshows that, when is large, the externality caused by the subsidy becomes
small.
1 6 We assume that the total assets in a country are equally held by all agents in this country.
Under our assumption of symmetric countries, the results are the same if we assume that allagents in the world share equal amounts of the total assets in the world.
1 7 Note that unemployed workers and workers producing homogeneous goods have the same
instantaneous utility Ui in the equilibrium.
1 8 Notice thatinvolves si . From (17), we can observe:= Ri + s i

1

q(

)

;

where
and q( )are constant because= 1 k . In our model, when siq( )increases, ni increases, which lowers Ri because of tougher competition.

11

We can see that is an increasing function of the search cost, k. Whenthe search cost is zero, k = 0,
= 0. Thus, the rent, , which is sharedby a matched worker and a …rm, increases with the degree of labor market
friction, whereas rents do not exist in the perfect labor market. In our model,
we assume that …rms incur positive search costs to search for workers. Under
the positive search costs, k, the number of …rms’ entering the manufacturingsector becomes ine¢ ciently small. In this circumstance, a matched …rm cangenerate an ine¢ ciently high revenue involving rent , which is divided between
a matched worker and a matched …rm.
The government sets its subsidy rate to maximize (21). When is large, thegovernment has a strong concern about total rents ni relative to the consumer
surplus. On the contrary, when is small, the government has a strong concern
about the consumer surplus.

3

Subsidy competition

The reaction function of the government in country i is given by:si = si (sj ) =t(rsj ) + t(1t2 )A + (1 + t2 ) + t2 r:
1 + 2t2(22)Then, because 0 < @si =@sj < 1, subsidy rates are strategic complements andthe competitive equilibrium is stable.
From (19) and (22), the equilibrium number of …rms in each country is:n = 1 + t2

A(1 + t)1+

(1 + t)2 (1t + 2t2 ):(23)

An increase in labor market frictions decreases the number of …rms in bothcountries because large labor market frictions prevent manufacturing …rms fromentering the market. Then, from (23), the condition under which there exists a
positive number of manufacturing …rms in both countries (ni > 0) is:
A>

1+

A:

1+t

The equilibrium price in each country becomes:p =(1 + t2 )(1 +(1 + t)(1

)

t2 )A

t(1
t + 2t2 ):An increase in labor market frictions raises the price level because the supplyof manufactured goods is scarce. In the equilibrium, the condition that ensuresthat the price in both countries is positive (pi > 0) is given by:A<

1 + t2 1 +t(1 t) 1 + t
12A:We can observe that A < A. Hereafter, we assume that A < A < A.
The equilibrium lump-sum subsidy rates are:si =+t(1 t)
1 t + 2t2

s ;

(24)

where:= A(1 + t)

(1 +) > 0;

from A < A < A, which means that s > 0. Further, we can see that @s =@A >0.
Proposition 1 1) When A < A < A, governments subsidize manufacturing…rms. 2) The subsidy rate is an increasing function of the market size for
manufactured goods.
The …rst term on the right-hand side of (24) is the size of the labor market
frictions, which equals the rent of a matched …rm and a matched worker. The
second term represents the externality caused by the subsidy, which decreases
with an increase in . As we saw earlier, the externality caused by the subsidy
becomes small when increases, which induces @ =@ < 0. When labor market
frictions increase, the …rst term in (24) (labor market friction) increases, whereas
the second term (the externality caused by the subsidy) decreases.
Substituting (19) into (21) and di¤erentiating it with sj , we can derive thefollowing:@(ni si ) @
+
@sj

@SWi
@(ni )=@sj s =s =s
@sj
i

j

(ni +tnj )2
2

@sj

:

The rise in the subsidy increases the number of domestic …rms and decreases thenumber of foreign …rms, which in‡uences foreign welfare through three channels:
labor market imperfections e¤ ect, …scal e¤ ect, and consumer surplus e¤ ect. The…rst term represents the labor market imperfections e¤ ect, which lowers welfare
in the foreign country. The second term represents the …scal e¤ ect, which raises
the foreign welfare because the decrease in the number of …rms in the foreigncountry decreases the total subsidy expenditure. The last terms, the consumer
surplus e¤ ect, is ambiguous, because the number of …rms increases in the country, imposing the subsidy, whereas the number of …rms decreases in the othercountry.19 In the subsidy competition equilibrium, the negative e¤ects on the
1 9 The consumer surplus e¤ect can be represented by@(ni +tnj )2
2

@sj

=

t
(1

t2 )2

A(1

t2 )

1+

The labor imperfection e¤ect can be expressed as@(ni )
=
@sj

2t
(1

13

t2 )2

< 0;

+

(1

t) + sitsj :foreign country’s welfare outweighs the positive e¤ects, and the rise in subsidy
in the country implementing it results in the negative externality a¤ecting theother country, as follows:t@SWi=[(1 + t)(s@sj s =s =s(1 t2 )2i)

(1t) ] < 0:(25)

jThus, in the subsidy competition equilibrium, the rise of the subsidy in a country
results in the negative externality a¤ecting the other country.

3.1

Coordinated subsidy rate

In the coordinated equilibrium, a supranational authority maximizes the global
welfare, which is the sum of the welfare of the two countries, as follows:
SWW= SW1 + SW2
2=2(1 + z) + n1 (s1 ) + n2 (s2 ) +

2(tn1 + n2 )(n1 + tn2 )+;22where the number of …rms is given by (19). By substituting (19) into global
welfare and di¤erentiating it with s1 and s2 , we …nd the …rst-order conditions
for this problem are given by:
@SWW
2t(sj
=
@si

(1 + t2 )(si

)
(1

)

2
t2 )

= 0; i; j 2 f1; 2g; i 6= j:(26)

We also derive the subsidy rate that maximizes global welfare as follows:
sci = scj =

> 0;

(27)

where the superscript c stands for the coordinated equilibrium. From (27), thecoordinated equilibrium subsidy level equals , namely the rent of a matched…rm and a matched worker.20 We assume that governments provide a subsidy tomatched …rms. In our framework, the revenue of matched …rms when a subsidy
is involved is divided between a matched worker and a …rm. Thus, a subsidy
to a matched …rm can be thought of as a subsidy to a successful match. A
successful match makes a rent, , and an increase in the number of matches inand the …scal externality e¤ect can be described as@(ni si )
2si t
> 0:
=
@sj
(1 t2 )2
2 0 In our model, when …rms enter into the manufacturing market, they consider the value of(J V + W U ), whereas the value generated by a match is J V + W U . The workers
who choose the sector where they work consider the value of (J V + W U ). These e¤ects
may results in the number of manufacturing …rms and workers being too large or too small.
In addition, when …rms enter into the manufacturing market, they do not consider the e¤ectof their entry on the domestic and foreign consumer surplus. The workers who choose thesector where they work do not consider the e¤ects of their choice on consumer surplus. This
may also result in the too many or too few manufacturing …rms and workers.

14both countries raises the welfare level monotonically. Therefore, the coordinatedsubsidy rate is equal to the rents of a matched …rm and a worker.
Di¤erentiating the global welfare with respect to s1 and s2 , we can obtainthe marginal bene…t (MB) and the marginal cost (MC) of the subsidy. The(MB) can be described as follows:
MB =

2 +
(1 + t)2t) (1 + t) + si 1 + t2

A (1

(1

2tsj

2

r (1

t)

2

t2 )

:

(28)

The …rst term on the right-hand side of (28) is the increase in the rent generated
by the increase in matched workers and …rms. The second term represents theincrease in consumer surplus generated by the increase in …rms. The (MC) is:
21

MC =

A (1

2

t) (1 + t) + 2si 1 + t2
(1

4tsj

r (1

2

t)

2

t2 )

:

(29)

From (28) and (29), when = 0, si = sj = 0 is the subsidy rate that makes themarginal bene…t equal to the marginal costs. In this case, the marginal value of
an increase in consumer surplus equals the marginal costs.
By comparing the subsidy rates of the competitive equilibrium with the coordinated equilibrium subsidy rates, we …nd that s is always larger than sci . These
ine¢ ciently high subsidy rates are caused by the way in which governments subsidize the manufacturing …rms in their country. Each government subsidizingthese manufacturing …rms ignores the externality in the other country causedby the entry and exit of …rms. We noted above that the negative externality on
foreign welfare overwhelms the positive externality, and the equilibrium subsidy
rate is higher than the coordinated subsidy rate.
Comparing the number of …rms in the competitive equilibrium with thecoordinated equilibrium number of …rms yieldshi
2 t + 3t2 (1 + t)A (1 + )
n
nc =
> 0;
2
(1 + t) (1 t + 2t2 )because A < A < A and 0 < t < 1. Therefore, in the competitive equilibrium, the number of …rms is larger and market competition is …ercer than when
governments provide coordinated equilibrium subsidy rates. From (8), the unemployment rates in the competitive equilibrium are higher than those when thesubsidy rates are coordinated because the number of …rms and the probabilityof a worker …nding a job are higher. Then, the number of workers entering the
manufactured goods sector and the unemployment rates become larger.
Summarizing these results, we can obtain the following proposition.
Proposition 2 Subsidy competition results in an ine¢ ciently high subsidy rate
(race to the bottom), a larger number of …rms, and high unemployment rates.
2 1 Of course, the sum of (28) and (29) is equal to the right hand side of (26).

15

3.2

Unemployment rates and welfare with or without subsidy competition

Here, we study the case in which neither of the two countries provides a subsidy,
and as a result, they do not engage in subsidy competition (si = sj = 0). From(19), the equilibrium number of manufacturing …rms becomes:
nn1 = nn2 = nn =

A(1

t)

(1 +

+

)

(1 + t)2;

where the superscript n represents the economy when neither government subsidizes the manufacturing sector. We see that under our assumption of A < A <A, nn < n holds. From (8), the equilibrium unemployment rate is an increasing function of the number of manufacturing …rms. Thus, subsidy competition
raises unemployment rates.
Lemma 2 Subsidy competition raises the equilibrium number of …rms andunemployment rates.
In our model, the increase in the number of manufacturing …rms raises theunemployment rates, because the number of workers entering into the manufacturing sector and the number of workers searching for jobs in the manufacturingsector increase. Under subsidy competition, governments provide positive subsidies to manufacturing …rms, which increases the equilibrium number of …rms.
Thus, unemployment rates are higher with than without subsidy competition.
When two coutries set the same subsidy rates, s1 = s2 = s; it becomesn 1 = n2n(s), where n(s) is the number of …rms as a function of s, which
is given by (19). From (21), the welfare level in country i as a function of thesubsidy rate can be written as:
2

SWi (s) = 1 + z + n(s)(s) +(1 + t) n(s)2
:
2

(30)

Following some calculations, we can derive the following equation:s+@SWi (s)=2:@s(1 + t)SWi (s) is a quadratic function of s and has a maximum value at so = . From(30), we can recognize that:
SWi (s)js=0 = SWi (s)js=2
Thus, if 2 > (<)s , SWi (s)js=0 < (>)SW . See Figures 1a and 1b. From(24), we see that if 2 > s , the next inequality holds:>t(A(1 + t2 ) (1 t))(1 t + 2t2 ) + t(1 t2 )

:

(31)

If the labor market is perfect (k = = 0), the welfare level with subsidy competition is always lower than that without. If (31) is satis…ed, subsidy competition
improves welfare compared with the case without subsidy competition.
16

Proposition 3 When
ful).

>

( < ), subsidy competition is bene…cial (waste-This proposition states that when labor market frictions are su¢ ciently large,
subsidy competition is bene…cial. In our model, the ine¢ ciency induced by labormarket imperfections is internalized in the subsidy competition equilibrium,
because the government maximizes social welfare in a country given in (21),which involves this ine¢ ciency, as captured by the term ni (si ). However,subsidy competition brings about the externality, which lowers social welfare:
further, the externality caused by the subsidy becomes small with an increasein . In the equilibrium without subsidy competition, there is no subsidyrelated externality, and the ine¢ ciency induced by labor market imperfections
is not internalized. When is large, the degree of ine¢ ciency induced by labormarket imperfections is large, whereas the externality caused by the subsidy issmall. Thus, welfare under subsidy competition is higher than welfare without
subsidy competition. Conversely, when is small, the ine¢ ciency induced bylabor market imperfections is relatively small compared with the e¤ects of theexternality caused by the subsidy, and welfare under subsidy competition islower than that without subsidy competition.
Note that, when no labor market friction exists (k = 0 and = 0), thecoordinated subsidy rate becomes zero ( sc jk=0 = 0), and the equilibrium subsidy
A(1+t)]> 0 because A < A <rate is positive, that is sjk=0 = t(1 1t)[1t+2t
2
A. Thus, when labor markets are perfect, subsidy competition always lowers
welfare to below that in the case without subsidy competition. Our results show
that, because there is labor market imperfections, subsidy competition may bebene…cial.22

3.3

E¤ects of labor market frictions

In this subsection, we investigate the e¤ect of labor market frictions of k. Givensi and sj , from (19), the e¤ect of an increase in the labor market frictions in1@r @iboth countries on the number of …rms is given by @n@k =@k < 0(1+t)2 @because @@r @@k > 0. Then, given si and sj , an increase in the labor market
frictions decreases the number of …rms. The e¤ect of a decrease in the searchicosts on the unemployment rate given si and si is ambiguous because of @u@k =@(q( ) )ni@ni. The …rst term on the right-hand side of theq( )@k(q( ) )2@kequation is negative, and this represents the fact that a decrease in the numberof …rms reduces the number of employed workers because jobs are destroyed by
the closure of …rms. This has a negative e¤ect on the number of unemployedworker. The second term on the right-hand side of the equation represents thefact that a decrease in the labor market frictions reduces the probability that
2 2 In Appendix 6.5, we present numerical results of the comparison of the welfare in thecase of zero subsidy in two countries, in the case of a positive subsidy in country 1 while zerosubsidy in country 2, and in the case of subsidy competition.

17

unemployed workers can …nd a job, which increases the number of unemployedworkers. Next, we investigate the e¤ect of labor market frictions on subsidy
rates. From (24), we can derive thatt + 2t2 ) t (1(1 t + 2t2 )

(1
@s
=
@k

t) @
;
@k

where @ =@k > 0. The sign of @s =@k depends on the sign of the numerator on
the right-hand side of the above equation. Then, we can obtain the following
lemma.
Lemma 3 The subsidy rate increases (decreases) with labor market friction,
when (1 t + 2t2 ) t (1 t) > (<)0.
Lemma 3 shows that there is a case where a rise in labor market frictions
raises (lowers) the equilibrium subsidy rate. From (22), the rise in labor market
frictions in a country raises the equilibrium subsidy rate in that country. However, it also lowers the equilibrium subsidy rate in the other country. Thus, thee¤ect of a decrease in labor market frictions on the subsidy rate is ambiguous.
From (8), the unemployment rate in the country is given by:
u =

n
q( )

:

(32)

Note that unemployment rates are an increasing function of the number of …rms.
By substituting q( 1) =from the de…nition of into (32) and di¤erentiating
it with respect to k, the following equation can be obtained:
h
i
2
2
(1
+
t
)
1
+
A(1
+
t)
@
@u
=
:
2 (1 + t)2 (12@k@kt + 2t )1+ 2@
A1 , @u
When A < (>) 1+t@k < (>)0 holds because of @k > 0, and A < A1 <A holds.23 In our model, an increase in labor market frictions a¤ects unemployment rates in two opposite ways. On the one hand, it decreases the probabilityof a worker …nding a job, which transfers workers who migrated to the manufacturing sector back to the agriculture sector and reduces unemployment rates.
On the other hand, it reduces the entry of …rms, and thus the probability ofa worker …nding a …rm that will employ him/her becomes small, which raises
equilibrium unemployment rates. When market size is su¢ ciently small, theformer e¤ect is stronger than the latter e¤ect. Therefore, an increase in labormarket frictions decreases unemployment rates and we can obtain the following
lemma.
2 3 By subtracting from A to A, the following equation can be obtained:1

A1
because 0 < t < 1 and 1

A=(1

t + 2t2 ) +t (12t + 3t2 > 0.

18

(1
t2 )

2t + 3t2 )> 0;

Lemma 4 When A < (>)A1 , the increase in labor market frictions lowers(raises) equilibrium unemployment rates in each country.
By di¤erentiating the welfare level with respect to k, we can obtain thefollowing equation:
i
h
A(1
+
t)
1
+
@
@SWi
(1 + t2 ) 1 2t + 3t2
=
< 0;
2
2
3
@k
@k
(1 + t + 2t )because A < A < A and 1 2t+3t2 > 0 in 0 < t < 1. An increase in labor market
frictions decreases the welfare level monotonically in the equilibrium. Then, bysummarizing the above results, the following proposition can be obtained.
Proposition 4 An increase in labor market frictions decreases the welfare levelmonotonically.
An increase in labor market frictions decreases the number of matched …rms.
This reduces the number of matched workers, which lowers welfare through thelabor market imperfections e¤ ect. In addition, the decrease in the number ofmatched …rms raises the price level of manufactured goods, which also lowers
welfare through the consumer surplus e¤ ect.

3.4

E¤ects of trade costs

In this subsection, we investigate how a decrease in trade costs a¤ects unemployment rates and welfare. We interpret t as trade freeness, and an increase
in t means a decline in trade costs. We de…ne such a decline in trade costs as
trade liberalization. Given ni and nj , the e¤ect of trade liberalization on thenumber of …rms is given by:
@ni
=
@t

(1

3

t) [A(1 + t)

2r] + 2sj (1 + 3t2 )
(1

2 3

t )

2tsi (3 + t2 ):The sign of this equation is ambiguous. The e¤ects of trade liberalization on@nunemployment rates can be expressed as @u@t = q( )@t . Then, the sign@n@u
t = 0:144427 and A2of @t is the same as that of @t . Here, we de…ne b(1+ )(1 t+9t2 t3 +2t4 )and obtain the following lemma (see the Appendix 6.6
2t2 (1+t)(3+t2 )for the proof).
Lemma 5 When 0 < t < bt, trade liberalization always increases unemployment rates. When bt < t < 1, trade liberalization increases unemployment rates
in A < A < A2 and decreases unemployment rates in A2 < A < A.
Trade liberalization has opposing e¤ects on unemployment rates. The negative e¤ect is that it intensi…es competition among manufacturing …rms, reducing
the number of …rms and lowering unemployment rates. The positive e¤ect isthat a reduction in trade costs means that …rms grow their volume of exports,
which increases pro…ts. Then, the number of …rms increases and some workers
19move from the agricultural to the manufactured goods sector. Therefore, unemployment rates rise. When trade costs are su¢ ciently high (0 < t < bt), tradeliberalization increases the number of …rms and raises unemployment rates.
When trade costs are su¢ ciently low, the e¤ects of trade liberalization on unemployment rates depend on market size. When market size is small (large),
trade liberalization raises (lowers) unemployment rates.
When market size is small in both countries, the number of …rms is small andthe manufactured goods market becomes less competitive. Then, the positivee¤ect is stronger than the negative e¤ect and trade liberalization increases thenumber of …rms and unemployment rates. When the market size is large, thenumber of …rms is large and the market is competitive. Then, the negativee¤ect overcomes the positive one and trade liberalization decreases the numberof …rms and unemployment rates.
We also …nd that trade liberalization always improves welfare, as shown in
the following proposition (see the Appendix 6.7 for the proof).
Proposition 5 Trade liberalization always increases the welfare level.
In our model, trade liberalization may increase (decrease) the number of…rms and raise (lower) unemployment rates. The increase in the number of…rms improves welfare, whereas the decrease in the number of …rms reduces
welfare. From (30), trade liberalization raises the consumer surplus, because
consumers can obtain imported goods with lower trade costs. In our model, thee¤ect of the rise in consumer surplus because of the low imported goods’priceis strong enough that trade liberalization always improves the welfare.

4

Asymmetric labor market frictions

In this section, we study the e¤ects of asymmetric labor market frictions on
subsidy rates. Without loss of generality, we assume that the labor market incountry 2 is more e¢ cient than that in country 1, namely k1 = k2 and 1 = 2 .
The di¤erence between the subsidy rates is given by:24
s1as2a = 1 +s 1 + t2 + 2t3( 1
1 + 3t2 + 4t42 ) > 0;because 1 = 2 . Thus, the country with the more ine¢ cient labor market
provides a higher subsidy rate. In addition, we see that @ (s1a s2a ) =@t > 0.
Thus, a decline in trade costs increases the di¤erence in equilibrium subsidy
rates. From the analysis in the Appendix 6.8, we can derive the next lemma:
Lemma 6 In the case of asymmetric countries, the subsidy rate of the moreine¢ cient country is higher. Subsidy competition always results in a race to thebottom.
We next analyze the e¤ects of labor market frictions on unemployment rates
and welfare. As deriving clear results in the general case of asymmetric countries
2 4 See the Appendix for the analysis of asymmetric countries.

20

is di¢ cult, we focus our attention on the neighborhood of symmetric countries.
Then, we can obtain the following lemma (See the Appendix 6.9 for proof).
Lemma 7 Suppose that the two countries are symmetric and labor market
frictions in country 1 increase, whereas those in country 2 are constant.
1)The unemployment rate in country 2 always rises.
2)When A < A < Aa , the unemployment rate in country 1 falls.
3)When Aa < A < A, the unemployment rate in country 1 rises.
With the increase in labor market frictions in a country, the number ofmanufacturing …rms in this country decreases, whereas the number in the othercountry increases. Then, unemployment rates in the other country rise. When
A is small, the manufacturing sector is small. Hence, the equilibrium pro…ts of…rms and workers’wages are low. In this case, with an increase in labor market
frictions, a large number of workers searching for jobs in the manufacturingsector migrate to the agriculture sector. This migration lowers unemployment
rates. When A is large, equilibrium pro…ts and wages are high. Thus, onlya small number of workers switch from the manufacturing to the agriculture
sector when labor market frictions increase, which raises unemployment rates.
We now study the e¤ect of search costs on welfare. We di¤erentiate welfare
as follows:@SW1= C [A (1 + t)];
@k1 k1 =k2@SW2= D [A (1 + t)
@k1 k1 =k2];

where C > 0 and D > 0.25 As A < A < A, A (1 + t)
can observe that:> 0. Then, we

@SW2
@SW1
< 0 and> 0:@k1 k1 =k2@k1 k1 =k2
We can summarize these results in the next proposition.
Proposition 6 Suppose that the two countries are symmetric and labor market
frictions in country 1 increase, whereas they are constant in country 2. Welfare
in country 1 falls, whereas welfare in country 2 rises.
The increase in labor market frictions in a country reduces the number of…rms in that country, which lowers the welfare though the consumer surpluse¤ect and the labor market imperfection e¤ect. The rise in labor market frictions in the other country increases the number of …rms in that country, whichraises its welfare through the consumer surplus e¤ect and the labor market
imperfection e¤ect.
2 5 We de…ne C
t
2

(1

2

1 + t2t)3 1 + 3t2 + 2t4(1 t)2(1+2t2 +t4

4t6 )

2

1+t+2t(1+2t2 +t4

4t6 )

2

2

.

21

1 + t + 4t2 + 3t3 + 9t4 + 6t5 + 8t6 and D

5Conclusion

In this paper, we construct a two-country model with labor market frictions toinvestigate how subsidy competition a¤ects welfare. Each government provedea subsidy to the manufactured goods …rms in their own country to raise thenumber of …rms and employment. As the result of the subsidy, the number of…rms in the other country decreases. Then, in our model, there are three e¤ectsof the subsidy for …rms in the other country: the labor market imperfectionse¤ ect, the …scal e¤ ect, and the consumer surplus e¤ ect. First, the labor market
imperfections e¤ ect is a negative e¤ect on the welfare of the other country. A
decrease in the number of …rms in the other country reduces the number ofmatched …rms and workers. This lowers the welfare level in the other country.
Second, the …scal e¤ ect is a positive e¤ect, whereby a decrease in the number of…rms reduces the total expenditure on the subsidy, which raises the welfare level.
Third, the consumer surplus e¤ ect is an ambiguous e¤ect because there are two
opposing e¤ect. On the one hand, the positive e¤ect is that an increase in thenumber of …rms in a country increases the volume of exports, which decreases
the price level in the other country. On the other hand, because of the decreasein the number of …rms in the other country, its market competition becomesless intense, which raises its price level. Because the sum of these three e¤ects
is negative, the subsidy competition always results in a race to the bottom in
our model.
We also show that the subsidy competition is bene…cial when labor market frictions are large. In the case without subsidy competition, the number of
entries of manufacturing …rms becomes ine¢ ciently small, because the search
activity by unmatched …rms involves positive search costs. In our model, theine¢ ciency induced by labor market imperfections is internalized in the subsidy
competition equilibrium. However, subsidy competition brings about a negative externality induced by the subsidy, which reduces social welfare. In theequilibrium without subsidy competition, no externality caused subsidy exists,
whereas the ine¢ ciency induced by labor market imperfections is not internalized. When labor market frictions are large, the ine¢ ciency induced by labormarket imperfections is large and the welfare level under subsidy competitionis higher than that without subsidy competition. Conversely, when labor market frictions is small, the ine¢ ciency induced by labor market imperfections
is relatively small compared with the externality caused by the subsidy, andthe welfare level under subsidy competition is lower than that without subsidy
competition.
Further, we show that the increase in labor market frictions always reduces
welfare, whereas trade liberalization always improves welfare. The increase inlabor market frictions raises labor market ine¢ ciency and reduces the total
number of matched …rms and workers in the manufacturing sector as well as theconsumer surplus. Trade liberalization lowers the equilibrium price of imported
goods, which raises the consumer surplus. Hence, trade liberalization reduces
the number of workers employed in the manufacturing sector, which lowers
welfare. Even in this case, however, the rise in the consumer surplus exceeds
22the decrease in the number of matched …rms and workers.
Finally, in terms of asymmetric labor market frictions between countries,
we show that the equilibrium subsidy rate is lower in the country with larger
labor market frictions. In addition, the increase in labor market frictions in acountry lowers its subsidy rate and raises the subsidy rate in the other country.
Further, the increase in labor market frictions in a country in the neighborhoodof symmetric countries lowers its welfare and raises the welfare in the othercountry.
The model presented in this paper can be extended in a number of directions. One is that …rm productivity is heterogeneous. If manufacturing …rms areheterogeneous and governments provide subsidies to …rms, competition among…rms becomes intensive, which may lower or raise cuto¤ productivities. Then,
when …rms are heterogeneous, new externalities can be observed, which enriches
the model. Future research could aim to analyze labor market or redistribution
policies under subsidy competition. For example, unemployment fees could be…nanced by corporate tax. Hence, we could study the e¤ect of redistribution
policies on employed and unemployed workers.

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6

Appendix

6.1

Variable outputs of …rms

In the basic model of this paper, for analytical simplicity, we assume that theoutputs of the …rms are constant. In this subsection, we extend the model bymaking the outputs of the …rms variable. The setup of the model involving
the utility function, the agriculture sector, and the matching process in the
manufacturing sector is assumed to be the same as in our basic model. In
this subsection, the …rms, which are under Cournot competition, can choose
their optimal amounts of domestic and export outputs. We assume that …rms
employ one worker and share revenue with that worker if they are matched.
For simplicity, the marginal costs incurred in the production of manufactured
goods are assumed to be zero. We also assume that the export of manufactured
goods incurs trade costs. To export one unit of manufactured goods, …rms
incur t units of numéraire goods. Under these conditions, the equilibrium priceof manufactured goods in country i is
pi = Ani qiinj qji ;where qii represents the domestic supply of manufactured goods produced bya …rm in country i and qji is the exported manufactured goods produced by a…rm in country j. The revenue for a …rm in a country can be described as:
Ri = pi qii + (pj25t)qij ;

where t represents trade costs. We can derive the equilibrium amount of outputs
and substitute them into the above revenue functions to …nd the equilibrium
revenue of …rms. We assume that the matching process is the same as that inour basic model, so the condition of (17) should also hold:Ri + si = r

1+

+

:

We refer to Ri + si = r as the equilibrium condition of i. These equations
determine the equilibrium number of …rms in the two countries, ni and nj . We
can observe that
@Ri
=
@ni

@Ri
2Gi
2H
;
;
=
(1 + n1 + n2 )3 @nj(1 + n1 + n2 )3

where
Gi
H

2A(1

t) + [1 + 2nj (1 + nj )] t2 > 0;

2A2 + 2At + (ni + nj + 2ni nj )t2 ;because 0 < t < 1. In addition, we can see that H < 0, because @(pi qii )=@nj =qii qji < 0 and @((pj t)qij )=@nj = qij qjj < 0; which means that @Ri =@nj <0. We can write the gradient of the equilibrium condition of 1 as dn2 =dn1 =G1 =H, whereas the gradient of the equilibrium condition of 2 as dn2 =dn1 =H=G2 . Here, G1 =H H=G2 = (1 + n1 + n2 )2 t2 ( 2A + t)2 = [G2 H] < 0. See
Figure A1. The equilibrium condition 1 moves to the upper right with the riseof the subsidy s1 , because @R1 =@n1 < 0 and @R1 =@n2 < 0. Therefore, the riseof s1 reduces n2 : In the same way, the equilibrium condition 2 moves to theupper right with the rise of the subsidy s2 , which reduces n1 . Then, in the caseof variable products of …rms, the rise of the subsidy in one country, si reduces
the number of …rms in the other, nj .
We substitute these values into the next social welfare function:
SWi = ni Wi + (1ni )Ui = 1 + z + ni (si ) +

2

(A

pi )
2

:

The government in country i sets its subsidy rate to maximize the country’s
welfare.
The calculations in the variable output case are complex, so we cannot derive
the explicit form of the equilibrium subsidy rate or social welfare in a country.
Therefore, we apply numerical methods to compare the equilibrium social welfare with and without the case of subsidy competition (si = s = 0). Figure A2
describes the results of these numerical methods. This …gure shows that whenlabor market frictions are large ( is large), subsidy competition becomes bene…cial. Thus, we show that our main result, subsidy competition is bene…cial in
the case of large labor market frictions, can be derived in the general model ofvariable …rm outputs.

26

6.2

Subsidy for employed workers

When we assume that the government provides a subsidy for employed workers,
the value of Wi is given by
Wi = (z + wM i + sM i + ai

Ti +

pi )2

(A
2

) + (Ui

Wi );

(33)

where sM i denotes the wage subsidy to the employed worker. Under the wage
subsidy, from (3) and (33), Wi Ui is given by:
Wi

Ui =

wM i + sM i:
+ + q( i ) i(34)

Then, by substituting (10) and (34) into (3), we can obtain the wage rate in the
manufactured goods sector as follows:
wM i + sM i = 1 ++:q( i ) i(35)

The …rst term of 1 represents the outside option of the worker and the second
term is the risk premium. By substituting (7), (34), and (35) into (5), the valueof a vacant job becomes the same as (14). Then, the labor market tightness inthe case of the subsidy rate for the employed worker is
1 =

2 =

=

1
k

:

(36)

Thus, this result is the same as that in the case of a subsidy to manufacturing…rms, which is independent of the subsidy rates.
By substituting (34), (35), (13), and Vi = 0 into (7), we can obtain the
pro…ts of manufactured goods …rms in country i as follows:q( )Ri + sM i
+

1

= 1:

(37)

Then, from the above equation, the pro…t level in country i can be obtained asfollows:+
Ri + sM i = 1 +
rM :
(38)
q( )
Here, we focus on the interior equilibrium in which there are a positive
number of …rms in both countries (n1 > 0 and n2 > 0). Thus, the equilibrium
number of …rms in country i is:ni =A (1 + t) (12t)(1 + t2 )(r sM i ) + 2t(r(1 t2 )2

sM j ):(39)

The government chooses its subsidy rate to maximize welfare in each country:2

SWi = ni Wi + (1ni )Ui = 1 + z + ni (27

sM i ) +(ni + tnj )
:
2

(40)

Thus, we saw that the equilibrium number of subsidy rates in the case of a
subsidy for employed workers is the same as that in the case of a subsidy for
manufacturing …rms. In addition, the social welfare function is the same in bothcases. That is, in the case of a subsidy for employed workers, we can derive thesame results as in the case of a subsidy for manufacturing …rms.

6.3

Subsidy for …rms’search costs

In this subsection, we study a case where: governments provide a subsidy tocover …rms’search activities. In this case, the net search costs of a …rm become
k si in country i. The value of a vacant job is given by:Vi =k + si + q( i )(Ji

Vi ):

(41)

Ji ):

(42)

The value of an occupied job is given by:
Ji = (RiwM i ) + (ViBy using (41) and (42), we can obtain Ji
Ji

Vi =

(Ri

Vi as follows:

wM i ) + k si:
+ + q( i )(43)

By substituting (7), (11), and (12) into (41), the value of a vacant job is givenby:1
Vi = k + si +
:
(44)
i

In the equilibrium, the value of a vacant job becomes zero Vi = 0, and thetightness of the labor market in each country is given by:
i =

1
(k

si )

:

(45)

Thus, the increase in the subsidy rate raises equilibrium labor market tightness,
i . When si = k, i = 1, which means that labor market imperfections vanish.
By substituting (11), (12), (43), and Vi = 0 into (7), the pro…t level incountry i can be obtained as follows:
Ri = 1 ++q( i ) i

ri :

(46)

Here, we focus on the interior equilibrium in which there are a positive number of …rms in both countries (n1 > 0 and n2 > 0). Equations (16) determine
the equilibrium number of …rms in the two countries. By substituting (16) into(2), we obtain:[A(ni + tnj )] + [A28(tni + nj )] t = ri :(47)

Thus, the equilibrium number of …rms in country i isni =2A (1 + t) (1t)(1(1 + t2 )ri + 2trj
t2 )2

:

(48)

Equations (45), (46), and (48) show that @ni =@sj < 0. Thus, when governments
subsidize the search costs of …rms, the externality generated by the subsidy
exists.
The government chooses its subsidy rate to maximize welfare in each country:
SWi=

1 + z + ni J i(si vi )(49)
2+ + q( i ) i
q( i ) i+ni

1 +q( i) i2

(1)+
q( i) iq( i ) i=

1 + z + ni=

1 + z + ni ((ni + tnj )2siq( i )+(ni + tnj )
2

2

si

q( i) iq( i ))+(ni + tnj );2iwhere we use vi = i ui = q(n ) and the government’s budget constraint bei
comes Ti = si vi . We can see that when si = k, i = 1. In this case, ri = 1 andni has a …nite value. Thus, when si = k, ni

q( i ) i

si q(

i)

=

1. Thismeans that the equilibrium value of the subsidy rate is lower than k, si < k.
We specify q( i ) = i , where 0 < < 1. In this case,2

SWi = 1+z+ni (
1

1

1
k si

si

1

)+
1

1
k si

(ni + tnj ):2

It is impossible to derive an explicit solution of si . We use numerical methods
with A = 20, t = 1=2, = 1=2, = 1=2, = 1=2, and = 1=2. We show that
there is a case where subsidy competition is bene…cial (see Figure A3).

6.4

Derivation of the welfare level

The welfare level in country i is given bySWi=ni Wi + (1ni )Ui= ni z + wM i + ai
+(1

Ti +

ni ) 1 + z + ai

29

(Ai

Ti +

pi )2
2
(Ai

+ (Ui
pi )2

2

:

Wi )

By substituting (11), (12), (14), (15), (17), ai = ni Ji , and the government
budget constraint into the above equation, we can obtain the following equation:
SWi=

1 + z + ni J i(si ni )2+ + q( i ) i
q( i ) i+ni=

1 + z + ni=

1 + z + ni (q( i) i(1)+
q( i) iq( i ) i

1 +(ni + tnj )2si(ni + tnj )
2

2

+
2

si ) +

q( i) i(ni + tnj ):2Then, we can obtain the welfare level in country i.

6.5

Subsidy in one country and no subsidy in the othercountry

In this section, we consider a case where the government in country 1 provides
subsidies to …rms in country 1, whereas the government in country 2 does notprovide any subsidies to …rms in country 2. In this case, the equilibrium subsidy
rate in country 1 is given bys1 (s2 = 0) =t2 )A + (1 + t2 ) + t2 r:
1 + 2t2

tr + t(1(50)

Thus, the equilibrium number of …rms in country i is
n1 (s2 = 0) =A (1 + t) (12t)(1 + t2 )(r(1 t2 )2

s1 ) + 2tr

:

(51)

In this case, the equilibrium numbers of …rms and the equilibrium price of
manufactured goods in country 1 and 2 arehi(1 + t)A 1
1 t + 2t2 3t3 + t4 + t 1 + 3t2
n1 (s2 = 0) =;1 3t4 + 2t6hi( (1 + t)A 1(1 t) 1 t 2t3
1 + t2 + 2t4
n2 (s2 = 0) =;1 3t4 + 2t6
p1 (s2 = 0) =p2 (s2 = 0) =(1

(1

t) 1 + t + 2t2+ (1 + tA) 1
1 + t2t + t2t3 ) + t2 A 1

2t3 (1 + )

2t4

t2 +

1 + t2t2

2t4

1 + t2 (1 + (1

;

t))

:

When t = 0:1, = 0:6, = 0:5, and A = 2, we can see that n1 > 0, n2 > 0,
p1 > 0, and p2 > 0 holds in 0
0:55. Figure A4 depicts the total welfare30

di¤erence between the two countries in the case of a subsidy in country 1 and no
subsidy in country 2. It shows that the total welfare in this case is higher (lower)
than the total welfare in the case of a zero subsidy in the two countries. Figure
A5 depicts the total welfare di¤erence in the case of subsidy competition and in
the case of a subsidy in country 1 while zero subsidy in country 2. This …gurealso shows that the total welfare in this case in contrast to the case of a subsidy
competition and show that, again welfare is higher (lower) than the total welfare
in the case of subsidy competition. Then, subsidy "competition" may improve
the total welfare compared with the case without subsidy "competition".

6.6

Proof of Lemma 5Di¤erentiating the number of …rms with respect to t yields
t + 9t2

1

1+
@n
=
@t

t3 + 4t43

2t2 A(1 + t) 3 + t2:

2

t + 2t2 )(1 + t) (1(1+ )(1 t+9t2 t3 +2t4 )When A < (>)
A2 , @n
2t2 (1+t)(3+t2 )@t > (<)0 holds. By subtractingfrom A2 to A, the following equation can be obtained:
A2

A=

1

t + 5t2 t3
2t2

1+

2t5

(3 + t3 )(1 + t)

> 0;

because 1 t + 5t2 t3 2t5 > 0 in 0 < t < 1. By subtracting from A to A2 ,we obtain the following equation:
A

A2 =

2

3

+
2t2

3

where F (t) 1 8t+10t 18t +3t 4t and the denominator of 3 2t2 t4 > 0
in 0 < t < 1. When 0 < t < b
t, where F (bt) = 0 and b
t = 0:144427, A < A2 holds.
bWhen t < t < 1, A > A2 holds. Therefore, when 0 < t < bt, @n@t > 0 holds.
@n@nWhen bt < t < 1, @t > 0 holds in A < A < A2 and @t < 0 holds in
A2 < A < A.

6.7

4

F (t)
;
2t2 t4

5

Proof of Proposition 5

Di¤erentiating the welfare level with respect to t, we obtain the following equation:
h
i
A(1
+
t)
(1
+
)
@SWi
=
G(A);
3
@t
(1 + t2 + 2t3 )where
G(A) = (1 +)tA(1 + t2 )(131

3t + t2 + t3 );and
1 2t + 6t2 12t3 + 15t4 6t5 + 6t6 > 0 in 0 < t < 1. Becausei
A < A < A, A(1 + t) (1 + ) is positive, the sign of @SW
depends on the sign
@t
2
3
e
e
e
e
of G(A). When t < t < 1, where 1 3t + t + t = 0 and et = 0:414214, G(A)is positive. When 0 < t < et, G(A) is also positive in A < A < A. Therefore,i
G(A) > 0 and @SW@t > 0 in 0 < t < 1.

6.8

Proof of Lemma 6

In the asymmetric countries, k1 = k2 ,the subsidy rate is given by
sai =

i+

1 =

2 , and r1 5 r2 . In this equilibrium,i

1 + 3t2 + 4t4

;

where
i = tA

1 + t + t2

t3

2t4

2t4 i + 2t4 ri + t(1 + t2 )( j12

+ t + t + t2 + 2t2

rj ):We can derive that
sa1

sa2 =

1

2t2 + t + 1

> 0:

(52)

Substituting si into (19) yieldshi
2
4
1 + t2
1 + t + 2t2 (1 t)2 (A(1 + t) 1)+ t j 1 + 3t2
i 1 + t + 2t
ni a =:
2(1 t2 ) (1 + 3t2 + 4t4 )(53)
Subtracting n1a from n2a yields
n2 abecause1 >n1 a =

1 + t2 ( 12t) (1 + t + 2t2 )(1

2 . We can observe that n1

A>

1
+
1+t

1

a

> 0;

< n2a . For n1a > 0, we assume that

1 + t2 + 2t4(1

2)

t 2 1 + 3t2

2t) (1 + t) (1 + t + 2t2 )Aa :The equilibrium prices in country 1 and 2 arepi a =(1t) 1 + t + 2t2

1 + t2 tA 1 t2 + 1 + t2
(1 t2 ) (1 + 3t2 + 4t4 )We investigate the di¤erence in the price level as follows:
p1 a

p2 a =

1 + t2 ( 12)> 0;
1 + t2 2t3
32

2
i (1 + t )

2t3 j

:

Since 1 > 2 and 1 + t2 2t3 > 0 in 0 < t < 1, p1a > p2a . For p2a > 0, thefollowing inequality should hold:A<(1 + t2 )s (1 s2 )2
2 (1 + t )2t3 1 + (1 t) 1 + t + 2t2(1 t) (1 + t + 2t2 )

Aa :

(54)

By comparing A with A, we can obtain the following equation:hi(1 t + 2t2 ) ( 2
t)
+
(1
t)
1
Aa Aa =:
2t(1 t) (1 + t)
For the existence of the asymmetric equilibrium, we assume that the bracket oft) > 0.
the numerator is positive, that is, ( 21 t) + (11 (Aa ) =We can see that @ 1 (A)=@A > 0 and

t

1

2

1 t

1 + t2 + 2t3 >20. In addition, @ 2 (A)=@A > 0 and 2 (Aa ) = t 11 t 2 1 + t2 + 2t3 > 0.
Therefore, 1 (A) > 0 and 2 (A) > 0. These results show that
sac
1 =

a
ac
1 < s 1 ; s2 =a
2 < s2 :(55)

Equations (52) and (55) proove Lemma 6.

6.9

Proof of Lemma 7

By di¤erentiating the unemployment rate in countries 1 and 2 with respect tosearch costs in the symmetric countries, the following equations can be obtained:(1 + t2 )@u1=@k1 k1 =k2(1t2 )2 (1 + 3t2 + 4t4 )

(A)

@ 1
;
@k1

@u2
=
@k1 k1 =k2(1

1 + 4t2 + 3t4
t 2 2@ 1> 0;
2t2 )2 (1 + 3t2 + 4t4 )
@k1

where
(A)

A(1 t)2 (1+t)(1+t+2t2 ) (1 t)2 (1+t+2t2 ) (2 t+2t2 3t3 +4t4 )1
Because @@k11 > 0, the sign of @u@k1coe¢ cient of

k1 =k2is positive because 2

depends on the sign oft + 2t(2 t+2t2 3t3 +4t4 )1When A > (<) 1+t+ (1 t)2 (1+t)(1+t+2t2 )

2

3

Aa =

(1

3t + 4t > 0 holds in 0 < t < 1.
1
Aa , @u@k1

k1 =k2> (<)0 holds. By

1 + t2 + 2t4> 0:2t) 1 + 2t + 3t2 + 2t3
33(A). The

4

subtracting from Aa to Aa , the following equation can be obtained:
Aa:Then, Aa is larger than Aa . By subtracting from Aa to Aa , the following
equation can be obtained:
Aa
When
@B
@t =

Aa =

(1

t)(1 + 3t2 + 4t4 ) +t(112t + 3t2

4t3 + 4t4t)2 (1 + t)(1 + t + 2t2 )6t5:(1 t)(1+3t2 +4t4 )
1+2t 3t2 +4t3 4t4 +6t5 , Aa > (<)Aa holds. In addition,
1+4t3 (1 t4 )+t4 +8t5 +14t6 +8t8
< 0 and Bjt=0 = 1. Thus, B < 0 <is(1 2t+3t2 4t3 +4t4 6t5 )2> (<)B

always satis…ed. Therefore, Aa < Aa < Aa always holds.

34

SW 𝑜

SW(s)

𝑆𝑊 ∗

𝑆𝑊|𝑠=0s=η

s = 𝑠∗

s = 2η

Figure 1A: Subsidy competition is beneficial (η > 𝜂̅ ).

s

𝑆𝑊 𝑜

𝑆𝑊|𝑠=0𝑆𝑊 ∗s=ηs = 2η

s = 𝑠∗

Figure 1B: Subsidy competition is wasteful (η < 𝜂̅ ).

s

𝑛𝑛2n2𝑠𝑠1

𝑛𝑛2

Equilibrium condition 2

Equilibrium condition 1

𝑛𝑛1

𝑛𝑛1

Figure A1: the effects of the rise of subsidy in country 1 on the number offirms in country 1 and 2.

Welfare

1.000

0.995

0.05

0.10

0.15

0.20

0.25

1

A = 1, t = , 𝜌𝜌 = 0.3, 𝛿𝛿 = 0.3.
2(The condition η > 0.00425484 is necessary to get a real value solution of the subsidy rate).
Blue line: Welfare with subsidy competition, Red line: Welfare without subsidy competition
Figure A2: Welfare with or without subsidy competition

0.30



Welfare

80

60

40

1.5
1

1

2.0
1

A = 20, t = , β = , 𝛾𝛾 = , δ =
2

2

2

1
2

1

𝑎𝑎𝑎𝑎𝑎𝑎 𝜌𝜌 = ,Red line: welfare without subsidy competition
Blue line: welfare with subsidy competition

2

Figure A3: Subsidy to firms’ search costs

2.5

3.0

k

𝑆𝑆𝑆𝑆1 (𝑠𝑠1 = 𝑠𝑠2 = 0) + 𝑆𝑆𝑆𝑆2 (𝑠𝑠1 = 𝑠𝑠2 = 0) − 𝑆𝑆𝑆𝑆1 �𝑠𝑠1∗ (𝑠𝑠2 = 0)� − 𝑆𝑆𝑆𝑆2 �𝑠𝑠1∗ (𝑠𝑠2 = 0)�

0.08

0.06

0.04

0.02

0.1

0.2

0.3

0.4

0.5

t=0.1, ρ=0.6, δ=0.5, and A=2
Figure A4: The difference of total welfare in the case of zero subsidy in two
countries and in the case of a subsidy in country 1 while zero subsidy incountry 2.

η𝑆𝑆𝑆𝑆1 (𝑠𝑠1∗ = 𝑠𝑠2∗ ) + 𝑆𝑆𝑆𝑆2 (𝑠𝑠1∗ = 𝑠𝑠2∗ ) − 𝑆𝑆𝑆𝑆1 �𝑠𝑠1∗ (𝑠𝑠2 = 0)� − 𝑆𝑆𝑆𝑆2 �𝑠𝑠1∗ (𝑠𝑠2 = 0)�

0.1

0.2

0.3

η

0.4

0.05

0.10

0.15
t=0.1, ρ=0.6, δ=0.5, and A=2

Figure A5: The difference of total welfare in the case of subsidy competition andin the case of a subsidy in country 1 while zero subsidy in country 2.

0.5