Discussion Paper No. 1041

REGIONAL SUBSIDIESANDINTERREGIONAL LABOR MOVEMENT

Daisuke Matsuzaki
Yoshiyasu Ono

September 2018

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

September 2018

Regional subsidies and interregional labor movement*byDaisuke Matsuzaki† and Yoshiyasu Ono‡

Abstract
When a government considers a subsidy for an underdeveloped region, it has several
options: the subsidies can be for land, wages, employment, or production. While land
subsidy is a lump-sum transfer, the others are meant to promote local production or worker
immigration. Under full employment, replacing the lump-sum subsidy with the othersubsidies benefit (harm) the recipient region if it specializes in labor-intensive(land-intensive) activities. If unemployment prevails in both the recipient and non-recipient
regions, production and employment subsidies benefit, but a wage subsidy harms, therecipient region. We also analyze asymmetric cases where one region attains full
employment while the other region remains underemployed.
JEL classification: F22, H71, R51
Keywords: transfer, subsidies, unemployment, wage rigidity.

*

The authors wish to thank Ken-ichi Hashimoto, Shinsuke Ikeda, Tadashi Morita, Takayuki Ogawa,Yoshitomo Ogawa, and seminar participants at Yamaguchi and Osaka Universities for their valuablecomments. This work was financially supported by the JSPS KAKENHI Grants (JP15H05728, JP15K13016,
and JP16K03664), and the program of the Joint Usage/Research Center for “Behavioral Economics” at ISER,
Osaka University. The authors are solely responsible for any errors.
†

Faculty of Economics, Toyo University, 5-20-28 Hakusan, Bunkyo, Tokyo 112-8606, JAPAN. E-mail:matsuzaki@toyo.jp‡

Institute of Social and Economic Research, Osaka University, 6-1, Mihogaoka, Ibaraki, Osaka 567-0047,
JAPAN. E-mail: ono@iser.osaka-u.ac.jp

1. Introduction
When a government offers a subsidy to a region, it often makes a lump-sum transfer.
While this is purely a redistribution, some types of subsidies directly promote employment
and stimulate local economic activity. Such subsidies may be particularly important if therecipient region is underemployed. In this paper, we examine the welfare effects of
replacing lump-sum transfers by these stimulation subsidies on the recipient andnon-recipient regions. For doing so, we adopt a two-region two-factor two-commodity
model in which the non-recipient region produces both commodities, but the recipient
region produces only one commodity because it is less developed. Each region may achieve
full employment or face unemployment, and workers migrate between the two regions inpursuit of higher (expected) labor income. Thus, the welfare of each region is affected bysubsidies through changes in terms of trade, interregional migration, and local employment.
Various situations arise in which a government has to offer subsidies. Offering alocation for “Not in My Back Yard” (NIMBY) facilities, such as military bases and nuclear
power plants, is an example. This often provokes opposition from local communities andtheir vicinities, and the government has to offer subsidies as compensation (for e.g., see
Frey et al., 1996). A typical example in Japan is Okinawa, where most of the US militarybases in Japan are located. The Japanese government compensates the land owners in
Okinawa by providing them with a land subsidy. Another example is the areas where
nuclear power plants or waste repositories are located, with the government offeringvarious subsidies to them. These areas are typically less developed, specializing in the
primary industry such as agriculture and fishery, or tourism. The various subsidies include alump-sum subsidy and subsidies for promoting economic activities. The aim of the present
analysis is to compare the welfare effects of these subsidies.

1The regulations for NIMBY facilities have been examined in the literature; forexample, see Markusen et al. (1995), Levinson (1999), and Saveyn (2012). These studies
examine the effects of pollution taxes and environmental regulation on the location of a
NIMBY facility. Some studies examine the compensational schemes for locating a NIMBY
facility along with asymmetric information on the damages caused by the facility. For
example, Feinerman et al. (2004) consider a two-city model to investigate the lobbying
activities of land owners affected by government decisions on the location of facilities.
Besfamille and Lozachmeur (2010) adopt a mechanism-design approach to examine theoptimal interregional compensation scheme for the host and other regions with full andasymmetric information. While these studies focus on the location choice of NIMBY
facilities, this paper considers the case of predetermined recipient regions and total subsidy
payments, and compares the welfare effects of various subsidies between the recipient andnon-recipient regions.
The regional subsidies’ welfare effects in case of interregional migration are also
analyzed in the literature of equalization subsidies; for example, Flatters et al. (1974),
Boadway and Flatters (1982), and Wildasin (1986). These studies focus on the negative
externalities effect of congestion on local public goods, and explore the migration subsidy
resulting in optimal interregional allocation of households. Albouy (2012) extends
Boadway and Flatters’ model (1982) by introducing household income heterogeneity.
These studies ignore unemployment, whereas we examine the welfare effects of replacing
direct transfers with various subsidies promoting local employment and production on therecipient and non-recipient regions both with and without unemployment.
The local unemployment and interregional migration combination was first modelledby Harris and Todaro (1970). They assumed an economy with a rural area and an urban

2area, where both areas specialize in different commodities, the rural area achieving full
employment whereas the urban area facing unemployment due to wage rigidity. In thissetting, they found that urban job creation leads to higher urban unemployment (the Todaro
Paradox); they also analyzed the effects of urban employment subsidy and immigration
restriction on social welfare. Since then, several studies have considered this issue; forexample, Bhagwati and Srinivasan (1974), Corden and Findlay (1975), Batra and Naqvi(1987), Nakagome (1989), Raimandos (1993), Brueckner and Zenou (1999), Ortega (2000),
Brueckner and Kim (2001), Sato (2004), Choi and Yu (2010), and Zenou (2011). Thesestudies extended the Harris-Torado model by introducing such factors as perfect orimperfect capital mobility, labor market monopsony, land market, and somemicroeconomic foundations on wage rigidity, for example, efficiency wage and search
matching, and explored the validity of the Harris-Todaro model policy implications.
Our analysis differs from the Harris-Todaro model literature in the following three
aspects. First, that study focused on social welfare, but we consider local development andregional welfare distribution. Second, we consider both urban and rural unemployment,
because rural unemployment is often more serious than, if not as serious as, urban
unemployment in Japan. We consider all unemployment and full employment combinationsin the two regions. Third, rather than considering perfect specialization in both regions, we
assume that the rural region produces only one commodity whereas the non-recipient
region produces two commodities; the Heckscher-Ohlin mechanism works in this setting.
Furthermore, we find that replacing a lump-sum subsidy with economic activity-stimulating
subsidies creates the employment expansion effect and the terms of trade effect in one orboth regions, depending on the employment situations of the two regions, and yields
different welfare effects.

3

2. The model
Our model economy has two regions, W and E, and two commodities, 1 and 2. Whileregion W produces both commodities, region E produces only commodity 1 owing to lackof technology. The government offers a lump-sum subsidy amounting to 𝑆! to the landowners in region E. The land of region 𝑗 (𝑗 = E, W) is assumed to be uniformly owned by
all native households of the region. Now, the government considers replacing the lump-sum
subsidy with the following stimulative subsidies: production subsidy 𝜖 , employment
subsidy 𝑧, and wage subsidy 𝑠! . These subsidies affect production, employment, andmigration, and vary labor income. All households freely move between the two regions inpursuit of higher wages, but they retain land ownership in their native region.
!

The population of households born in region 𝑗, called household 𝑗 in this study, is 𝐿! ,!

whereas the present population of the households living in region 𝑗 is 𝐿! (𝑗 = E, W), These
populations satisfy!
!
𝐿 = 𝐿!! + 𝐿!
! = 𝐿! + 𝐿! ,(1)

where 𝐿 is the total population, which is constant. For simplicity, we assume that eachregion initially has no immigrants, and hence!

!

𝐿! = 𝐿! for 𝑗 = 𝐸, 𝑊.
Furthermore, all households have the same utility function!

!

𝑢 ! = 𝑢(𝑐! , 𝑐! ) for 𝑗 = 𝐸, 𝑊,

4

(2)

!

where 𝑐! (𝑖 = 1,2; 𝑗 = 𝐸, 𝑊) is household 𝑗’s consumption of commodity 𝑖. Assuming theutility function to be homothetic, we have!

!

𝑝! 𝜔 𝑐! = 𝜑 𝜔 𝑌! ,𝑝! 𝜔 𝑐! = 1 − 𝜑 𝜔 𝑌! ,(3)

where 𝑝! 𝜔 is the price of commodity 𝑖 measured in terms of the composite of the two
commodities, 𝜔 is the relative price of commodity 2 in terms of commodity 1, 𝑌! ishousehold 𝑗’s disposable income, and 𝜑 𝜔 is the ratio of consumption expenditure oncommodity 1, which satisfies
0 < 𝜑 𝜔 < 1, 𝜑 ! 𝜔 > 0.

(4)
!

!

Because the utility function is homothetic, the value of 𝑢(𝑐! , 𝑐! ), into which the!

!

optimal levels of 𝑐! and 𝑐! given in (3) are substituted, is independent of the relative price𝜔. Thus,
𝑢

! !
!! !

!!! !

! !

!

!! !

𝑌! , ! ! 𝑌! = 𝜙 𝑌! 𝑢

!!! !

, ! !
!

≡ 𝑢(𝑌! ),(5)

where 𝑢 . , . satisfies!!

! ! !!! !
,!! ! !! !

!"

= 𝑢!

Because 𝑢! /𝑢! = 𝜔 = 𝑝! 𝜔 /𝑝! 𝜔

!

! !
!! !

!"

+ 𝑢!

!

!!! !
!! !

!"

= 0.

under rational household behavior, the above

equation yields
𝑝!! 𝜔 = −(!!! ! )!!
!

< 0,

𝑝!! 𝜔 =

! ! !!
!

= 𝜑 𝜔 𝑝! > 0.

(6)

The production function of sector 𝑖 (𝑖 = 1, 2) displays constant returns to scale withrespect to two factors, land 𝐾! and labor 𝐿! , and hence is represented by5

𝑓! 𝑘! 𝐿! where 𝑓! ! ⋅ > 0, 𝑓! !! ⋅ < 0, 𝑘! = 𝐾! 𝐿! .

(7)

Because the government offers no subsidy to region W, the two sectors’ profit
maximization behavior yields
𝑟 ! = 𝑝! 𝜔 𝑓!! = 𝑝! 𝜔 𝑓!! ,𝑤!

= 𝑝! 𝜔 [𝑓! − 𝑓!! 𝑘! ] = 𝑝! 𝜔

(8)

𝑓! − 𝑓!! 𝑘! ,where 𝑟 ! and 𝑤 ! are respectively the land rental rate and wage level in region 𝑗. From (6)and (8), we derive!

𝑟! = 𝑟! 𝜔 , 𝑟! 𝜔 =!! ! !! ! !!! ! ! !! !!!
!(!! !!! ),! ! !! !! ! !!! ! ! (!! !!! )!

𝑤! = 𝑤! 𝜔 , 𝑤! 𝜔 = − !

!(!! !!! )

,
(9)

!!

𝑘! = 𝑘! 𝜔 , 𝑘!! 𝜔 = ! !! !!! ,
!

!

!

!

𝑘! = 𝑘! 𝜔 , 𝑘! ′(𝜔) = (! !! !)!! !!! .
!

!

!

Region E has only sector 1, and the government offers production subsidy 𝜖 andemployment subsidy 𝑧 to the firms in this region. Therefore, the optimal firm behavior
yields𝑟 ! = 1 + 𝜖 𝑝! 𝜔 𝑓!! 𝑘!! ,!

𝑤 − 𝑧 = 1 + 𝜖 𝑝! 𝜔

𝑓! 𝑘!!

− 𝑓!! 𝑘!! 𝑘!! ,(10)

where 𝑘!! is the land-labor input ratio in region E. The government also offers wage subsidy
𝑠! and land subsidy amounting to 𝑆! . These are financed by lump-sum taxes, whose total
value is 𝑇, imposed equally on all households in the two regions. The government’s budget
equation then becomes6𝑧 + 𝑠! + 𝜖𝑓! 𝑘!! 𝛾 ! 𝐿!! = 𝑇 − 𝑆! ,(11)

where 𝛾 ! is the employment rate in region 𝑗. Note that because 𝑆! is a lump-sum subsidy,
the present analysis is valid also where a general lump-sum subsidy, rather than a land
subsidy, is paid to household E.
With perfect labor mobility between the two regions, household 𝑗’s disposable income,𝑌! 𝑗 = 𝐸, 𝑊 , satisfies! !! ! ! !

𝑌 ! = (𝑠! + 𝑤 ! )𝛾 ! + ! !!
!

𝑌 ! = (𝑠! + 𝑤 ! )𝛾 ! +!

! !! ! ! !

!

!

− = 𝑤 ! 𝛾 ! + ! !!

! !!!
!!
!

!

! !!!

!

!!
!

− = 𝑤!𝛾! +

!

− ,
!

(12)
!

− ,
!

where the second and third expressions of each equation show respectively the expected
income when the household lives in region E, and that when the household lives in region
W. From (12),(𝑠! + 𝑤 ! )𝛾 ! = 𝑤 ! 𝛾 ! ,(13)where, from (8), 𝑤 ! is a function of 𝜔, and from (10), 𝑤 ! is a function of 𝜔, 𝑘!! , 𝑧, and 𝜖.
The factor markets in region W satisfy
𝑘! 𝐿! + 𝑘! 𝐿! = 𝐾 ! , 𝐿! + 𝐿! = 𝛾 ! 𝐿!
!,

(14)

where 𝐾! is the land endowment of region 𝑗. Thus, we obtain𝐿! =!
!! ! ! (!!!!
! )!!

!! !!!

, 𝐿! =! ! !!! ! ! (!!!!
!)!! !!!

.

(15)

Since region E includes only sector 1, we have𝛾 ! 𝐿!! 𝑘!! = 𝐾 ! .

7

(16)

From (3) and (16), we obtain the market equilibrium conditions for commodities 1 and 2 asfollows:!!

𝑝! 𝑓! 𝐿! + 𝑝! 𝑓! 𝑘!! ! ! = 𝜑 𝜔 𝑌 ! 𝐿!! + 𝑌 ! 𝐿!
! ,!

𝑝! 𝑓! 𝐿! = 1 − 𝜑 𝜔(17)

𝑌 ! 𝐿!! + 𝑌 ! 𝐿!
! .

These yield!! !! ! !! !!!
!! !!

!!
!!
!

− 𝛿 𝜔 = 0,
(18)

!" !

where 𝛿 𝜔 ≡ !!! ! , 𝛿 𝜔 > 0, 𝛿 ! 𝜔 > 0,
and we derive the properties of 𝛿 𝜔 from (4).

3. Policy analysis
Having established the model, we next analyze the welfare effects of subsidy
replacement. The government originally pays land subsidy 𝑆! to household E, and thenreplaces it with the wage, employment, or production subsidy (𝑠! , 𝑧 or 𝜖); this urges
workers to move into the region and firms to increase local production.
We assume that the subsidies initially satisfy
𝑠! = 0, 𝑧 = 0, 𝜀 = 0,(19)

and analyze the welfare effects of the replacement policies in the following four situations:(i) full employment in both regions, (ii) unemployment in both regions, (iii) unemployment
in region W and full employment in region E, and (iv) full employment in region W andunemployment in region E. For simplicity, we assume that unemployment is the result of

8

wage rigidity. Noting that 𝑤 ! = 𝑤 ! (𝜔), from (9), the endogenous and exogenous
variables in each case are
i 𝑤 ! and 𝜔, while 𝛾 ! = 1, 𝛾 ! = 1;(ii) 𝛾 ! (< 1) and 𝛾 ! (< 1), while 𝑤 ! = 𝑤! , 𝑤 ! (𝜔) = 𝑤 ! ;(20)

iii 𝑤 ! and 𝛾 ! (< 1), while 𝛾 ! = 1, 𝑤 ! (𝜔) = 𝑤 ! ;(iv) 𝛾 ! (< 1) and 𝜔, while 𝑤 ! = 𝑤! , 𝛾 ! = 1.

While the three subsidies raise the labor income, their effects on employment
significantly differ. Employment and production subsidies, 𝑧 and 𝜀, respectively, urge local
firms to produce more and thereby increase employment. Wage subsidy 𝑠! urges household
W to move into region E by raising the total labor income without stimulating local
production, and hence this may worsen the local employment.
By applying 𝐿! and 𝐿! in (15) to (18), substituting 𝐿!! in (16) to the result, and notingthat 𝑘! = 𝑘! 𝜔 and 𝑘! = 𝑘! 𝜔 from (9), we obtain 𝜔 as a function of 𝛾 ! , 𝛾 ! , and 𝑘!! ,where, from the second equation of (10), 𝑘!! is a function of 𝜔, 𝑤 ! , 𝜖, and 𝑧. Thus, by
totally differentiating these equations and rearranging the results, we obtain the relationshipbetween 𝜔, 𝑤 ! , 𝛾 ! , 𝛾 ! , 𝜖, and 𝑧. That is,(!! !! !!!! !! )!!
!

𝐴𝑑𝜔 +

!! !!!

!!! !! !! ! ! !!
!

!
!! !!! !!!! !!!

𝑑𝑤 ! = −!! !!!

(!!!)! ! !

𝑑𝛾 ! +! ! (!! !! !!!! !! )!!
!
! ! !! !!!

!!! !! !! ! ! !!
!
!! !!!

!!! ! ! !

! !! !
!! !!

! !
𝐴 = ! !! !! !!!! + ! !! ! !
− 𝛿 ! 𝑓! 𝐿! +! ! !!
!

!

!

!

!

!

!! !!!

𝑑𝛾 ! −
𝑤 ! 𝑑𝜖 + 𝑑𝑧 ,!
!!! !! !! ! ! !!
! ! !!!

!!! !!!

! !! !
!! !!

!! !!!

(21),

the derivation of which is given in Appendix A. By totally differentiating (13) and!

substituting 𝑤 ! 𝜔 into (9), we have9

!! !! !! !(!!!)! ! (!! !!! )! !! !!!

𝛾 ! 𝑑𝜔 + 𝑤 ! 𝑑𝛾 ! − 𝑤 ! 𝑑𝛾 ! + 𝛾 ! 𝑑𝑤 ! = −𝛾 ! 𝑑𝑠! .

(22)Equations (21) and (22) include four endogenous variables, 𝑑𝑤 ! , 𝑑𝛾 ! , 𝑑𝜔, and 𝑑𝛾 ! , andthree policy variables, 𝑑𝜀, 𝑑𝑧, and 𝑑𝑠! . Equation (20) shows that in each case, two of thefour variables are given. Thus, these equations completely determine the policy effects on
all the four endogenous variables.
We next derive the welfare effects from the above results. The government replaces
land subsidy 𝑆! by employment subsidy 𝑧, wage subsidy 𝑠! , or production subsidy 𝜖; thetotal amount of lump-sum tax T is fixed,
𝑑𝑇 = 0.
From (10), 𝑘!! is a function of 𝜔, 𝑤 ! , 𝜖, and 𝑧, and from (16), 𝐿!! is a function of 𝛾 ! and𝑘!! . By substituting these functions into 𝑌 ! and 𝑌 ! represented by (12), totallydifferentiating the results, and taking account of (2), (15), and the second equation of (17),
we have
𝑑𝑌 ! = −𝑑𝑌 ! =!!! ! ! !!! !!! !!! !!! ! !
!

!!! ! ! !!! !!! !!! !!! ! ! !!
!
!!!
!

𝑑𝜔 + 𝑤 ! 𝑑𝛾 ! ,(23)𝑑𝜔 + 𝑤 ! 𝑑𝛾 ! .

Equations (20), (21), and (22) lead to 𝑑𝑤 ! , 𝑑𝛾 ! , 𝑑𝜔, and 𝑑𝛾 ! . Therefore, by applying
them to 𝑑𝑌 ! and 𝑑𝑌 ! , we obtain the policy effects on both regions’ income. Theystraightforwardly imply the welfare effects of subsidy replacement, because the utility ofhousehold 𝑗 (for 𝑗 = E, W) is 𝑢(𝑌! ) in (5). We obtain them for the four cases summarizedin (20).

10Note that neither 𝑑𝑌 ! nor 𝑑𝑌 ! in (23) directly depends on the policy variables, 𝑆! , 𝑧,𝑠! , and 𝜖. This implies that the policy variables affect 𝑌 ! and 𝑌 ! only indirectly throughchanges in the terms of trade and each region’s rate of employment, which we call (i) theterms of trade effect and (ii) the employment expansion effect, respectively. In addition tothese two effects, replacement policies create another effect, (iii) the effect of redistribution
from land owners to firms or workers. When both (2), that is, 𝐿!! = 𝐿!! , and (19), that is,𝑠! = 0, 𝑧 = 0, and 𝜖 = 0, are valid, this redistribution does not yield any interregional
redistribution, as explained below.
By replacing a land subsidy by a wage subsidy, the subsidy payment shifts from landowners to workers in region E. From (19) which implies that the initial wage subsidy iszero, an increase in immigrants creates no transfer to them. From (2), the number of
immigrants is negligible, and therefore almost the entire subsidy goes to native workers.
Thus, the subsidy transfers due to replacement of the land subsidy by the wage subsidy arealmost fully taken by native workers. The land subsidy replacement by employment andproduction subsidies for local firms go to local land owners as well as native and immigrant
workers. However, assuming no immigrants initially, all the increases in employment andproduction subsidies at the cost of decreasing the land subsidy go to the local land owners
and native workers. Thus, the three subsidy replacements create no interregional subsidy
redistribution; that is, we find only the terms of trade and employment expansion effects.
The mathematical expressions 𝑑𝑌 ! and 𝑑𝑌 ! in (23) clarify this property, which issummarized as follows:

Lemma 1: When (2) and (19) hold, there is no interregional redistribution effect; there are
only the terms of trade and employment expansion effects.

11

Therefore, we focus on the terms of trade and employment expansion effects in the fourcases given in (20).
In case of full employment, no employment expansion takes place for native workers
in the region. Thus, if the total employment involving native and immigrant workers in theregion expands owing to the subsidy policies, the additional revenue goes to the immigrants
as wages, while the total income of native households remain invariant. This property isapparent from (23); that is, if the employment rate 𝛾 ! (𝑗 = E, W) equals 1, 𝑑𝛾 ! = 0, andtherefore we have only the terms of trade effect. This can be restated as follows:

Lemma 2: The employment expansion effect is zero for household 𝑗 (𝑗 = E, W) if region 𝑗 isfully employed.

Finally, one naturally assumes that region E specializes in the sector that intensively
uses the abundant factor. This property is formally presented as follows:
Case A: 𝑘! >!

Case B: 𝑘! <

!

!

!

!!

!!

!!

!

!

!!
!

> !! ;

!!

!!

!!

!!

!

!

!!
!

> 𝑘! ≥ !! , 𝑘! > !! > 𝑘! ,< 𝑘! ≤ !! , 𝑘! < !! < 𝑘! ,

!!
!

(24)

< !! .
!

3-1. Full employment in both regions
In (20) case (i), both regions achieve full employment (𝛾 ! = 1, 𝛾 ! = 1); therefore,
from (13), (21), and (22), we obtain12𝑑𝜔 = −!! ! !! !! !!
!
!
!! !!! !!!! !!! !! !!! !!

𝑤 ! 𝑑𝜖 + 𝑑𝑠! + 𝑑𝑧 ,𝑑𝑤 ! = 𝑑𝑤 ! + 𝑑𝑠!
=!!! !! !! !!
!

!! !! !! ! !!! ! ! !! !!!
!! !!!

! !! !
!! !!

!! !!! ! !!

(25)
𝑤 ! 𝑑𝜖 + 𝑑𝑧 + 𝑑𝑠! ,𝛾 ! = 1, 𝛾 ! = 1,
where(!!!)! ! !

!!! ! ! !

!! ! !!

!

!

!

!

!

!

!

!

!

!!

!
! !
𝛺! = ! !! !! !!!! + ! !! ! !
+ !!! ! ! ! ! ! !! !!
! ! !!
!

!

!

− 𝛿 ! 𝑓! 𝐿! < 0.

The first equation of (25) implies that𝑠! ↑ , 𝜖 ↑, 𝑧 ↑ ⇒ 𝜔 ↓ (or ↑)if 𝑘! > 𝑘! (or 𝑘! < 𝑘! ).

Therefore, from (23), where 𝛾 ! = 1 and 𝛾 ! = 1, we have
!! !

=−
!"

!! !

=
!"

!!! ! ! !!! !!! !!! !!! ! ! !"!

!"!!! ! ! !!! !!! !!! !!! ! ! !!
! !"
!!!
!

!"

≷ 0,
≶ 0,

if 𝑘! ≷ 𝑘! for 𝜎 = 𝑠! , 𝜖, 𝑧.
Because the utility of household 𝑗(= 𝐸, 𝑊) is 𝑢(𝑌! ), as shown in (5), we have thefollowing proposition.

Proposition 1: If full employment prevails in both regions, replacing a land subsidy by
wage, employment, and production subsidies makes household E better (worse) off andhousehold W worse (better) off when region E specializes in the labor-intensive(land-intensive) commodity.

13

When full employment prevails, from Lemmas 1 and 2, there is neither theinterregional redistribution effect nor the employment expansion effect. Hence, we have
only the terms of trade effect. Replacement policies first raise the labor income in region E,
and then increase the wage in region W, as shown in (25); this raises the price of thelabor-intensive commodity. Therefore, if region E’s product is labor-intensive, it benefits
region E but harms region W through the terms of trade effect. If region E’s product island-intensive, it harms region E but benefits region W.

3-2. Unemployment in both regions
In (20) case (ii), where the wages in both regions are fixed, we have from (21) and(22),
𝑑𝛾 ! = −! ! !!

!! !! ! ! !!
! !!!
!
!
!!
!
!! !! !! !! !! !! !!!! !! !

𝑤! 𝑑𝜖 + 𝑑𝑧 − ! !!! 𝑑𝑠! ,!
!!
! ! !! !! !!!
! !! !
!
!! !! !! !! !! !! !!!! !! !

𝑤! 𝑑𝜖 + 𝑑𝑧 + ! ! !! 𝑑𝑠! ,𝑑𝛾 ! = −! ! !!

(26)

𝑑𝑤 ! = 0, 𝑑𝜔 = 0.
By substituting 𝑑𝛾 ! and 𝑑𝛾 ! in (26) to (23), we obtain the income effect on household E
as well as that on household W:
𝑑𝑌 ! = −! !
!!
! ! ! !! !! !!!

!
!! !!! !!!! !!!

𝑑𝑌 ! = −𝑤! 𝑑𝜖 + 𝑑𝑧 −!! !! !!!! !! !

! !
!!
! ! ! !! !! !!!

!
!! !!! !!!! !!!

!! !! !!!! !! !

! ! !!
!

𝑤! 𝑑𝜖 + 𝑑𝑧 +

!

𝑑𝑠! ,

! ! ! ! !!
!
!!!

Because 𝑓!!! < 0, the above equations imply that𝑑𝑌 ! 𝑑𝜖 > 0, 𝑑𝑌 ! 𝑑𝑧 > 0, 𝑑𝑌 ! 𝑑𝑠! < 0;
𝑑𝑌 ! 𝑑𝜖 > 0, 𝑑𝑌 ! 𝑑𝑧 > 0, 𝑑𝑌 ! 𝑑𝑠! > 0.
These properties are formally restated in the following proposition:
14

𝑑𝑠! .

Proposition 2: In case of unemployment in regions W and E, replacing a land subsidy by
production and employment subsidies makes both households better off, whereas replacing
it by a wage subsidy makes household E worse off and household W better off.

In region W, where both commodities are produced, the relative price does not change
because of wage rigidity; this implies that the terms of trade effect does not exist. From
Lemma 1, there is no interregional redistribution effect. Thus, only the employment
expansion effect exists. Replacement with production and employment subsidies expands
region E’s production as well as the native and immigrant employment, making both theregions better off. The wage subsidy does not change the wages of both regions’ firms, andso the production of neither region is affected. However, some workers in region W may
move to region E and take up some employment there. Thus, the employment expansion
effect is positive for household W but negative for household E, making the former better
off and the latter worse off.

3-3. Full employment in region E and unemployment in region W
In (20) case (iii), where the wage in region W is fixed, from (21) and (22), we obtain𝑑𝛾 ! =
𝑑𝑤 ! =!!! !! !! ! ! !!
!

!
!! !!! !!!! !!!

!! !! !!!! !! !!
! !!

!!! !! !! ! ! ! ! !!
!
!
!! !!! !!!! !!! !! !! !!!! !! !!
! !!

𝛾 ! = 1, 𝑑𝜔 = 0,
where

15

𝑤 ! 𝑑𝜖 + 𝑑𝑠! + 𝑑𝑧 ,!!

𝑤 ! 𝑑𝜖 + 𝑑𝑧 + !! ,
!

(27)

𝛺! =

!
!!! !! !! ! ! !!
!!

!! !!!

! !! !
!! !!

!! !! !!!! !! !!
!

− 1 < 0.

Therefore, the first equation in (27) implies that𝑠! ↑, 𝜖 ↑, 𝑧 ↑ → 𝛾 ! ↑.
By substituting (27) into (23), we obtain the effects of subsidy replacements on theincome of household E (or W). They satisfy
𝑑𝑌 ! = 0, 𝑑𝑌 ! > 0.
Because the wage rate in region W is fixed and both commodities are produced, the relative
commodity price is fixed. Thus, the terms of trade effect does not exist. Furthermore,
Lemma 1 implies that the interregional redistribution effect does not exist. As regards theemployment expansion effect, replacing land subsidy with the other subsidies urges
workers to come to region E, and this improves the employment situation of household W,
implying a positive employment expansion effect for household W. However, noemployment expansion occurs for household E because it is fully employed. Thus, there isno effect on household E’s utility.

Proposition 3: In case of full employment in region E and unemployment in region W, all
the subsidy replacements increase the utility of household W while the utility of household E
remains unaffected.

3-4. Unemployment in region E and full employment in region W
In (20) case (iv), where 𝑤 ! = 𝑤! , from (21) and (22), we obtain16𝑑𝜔 = −!!! !!
! !! !!
!
!! !!! !!!! !!! !! !!! !!

! ! !!! ! !!

𝑤! 𝑑𝜖 + 𝑑𝑧 + !! !!! !!! !! ! 𝑑𝑠! ,!

!!! !! ! !!! !! !!!! !! !!! !! !!! !! !! !!
!

!

𝑑𝛾 =

!

!

𝑤! 𝑑𝜖 + 𝑑𝑧 −!
!!! !!!! !!! ! !! !!! ! !!

!! !
! ! !!

𝑑𝑠! ,

(28)

𝑑𝑤 ! = 0, 𝛾 ! = 1,
where!!! ! ! !

!!! ! ! !

!

!

! !
𝛺! = ! !! !! !!!! + ! !! ! !
− 𝛿 ! 𝑓! 𝐿!
! ! !!
!

!

!

! ! ! ! !!! ! !!
!

!!

!

− ! ! !! !!! !! !
!

!

!!! ! ! !

!!! ! ! !

!

!

+!! ! !

!! !!! !!
!

! !
𝐵 = ! !! !! !!!! + ! !! ! !
− 𝛿 ! 𝑓! 𝐿! −! ! !!
!

!

!

!

−

!
!! ! !! !!
! !!! ! !!

!!!

! !! !
!! !!

!
!! ! !! !!
! !!! ! !!

!!!

! !! !
!! !!

!! !!! !

!! !!! !

< 0,(< 0 if 𝑘! < 𝑘! ),and 𝛺! is the derivative of the left-hand side of (18) with respect to 𝜔. Because theleft-hand side of (18) is the supply-value ratio of commodity 1 to commodity 2 minus thedemand-value ratio, it must decline as 𝜔 rises; that is, 𝛺! is negative. Therefore,𝑠! ↑, 𝜖 ↑, 𝑧 ↑ ⇒ 𝜔 ↓ (or ↑) if 𝑘! > 𝑘! (or 𝑘! < 𝑘! );𝜖 ↑, 𝑧 ↑ ⇒ 𝛾 ! ↑;(29)

𝑠! ↑ ⇒ 𝛾 ! ↓ if 𝑘! < 𝑘! .
Because an increase in production, employment, and wage subsidies raises the
expected labor income in region E, the wage in region W also rises. Therefore, the relative
price of the labor-intensive commodity increases, as the first line in (29) shows. As for theeffect on the employment rate in region E, an increase in production and employment
subsidies expands the employment and production in region E, and hence raises 𝛾 ! . An
increase in wage subsidy 𝑠! urges the workers in region W to move to region E without
directly stimulating the production in region E, and so the employment rate 𝛾 ! declines.
Moreover, if 𝑘! < 𝑘! , 𝜔 increases, as shown in (29), which lowers the price of the local

17

commodity in region E and urges the local firms to reduce 𝛾 ! further. If 𝑘! > 𝑘! , however,𝜔 decreases, which increases 𝑝! (𝜔), urges the local firms to increase employment, andhence makes the sign of 𝑑𝛾 ! /𝑑𝑠! ambiguous.
Using (29), one finds that 𝑑𝑌 ! and 𝑑𝑌 ! given in (23) satisfy
!! !

=
!"
!! !

!!! ! ! !!! !!! !!! !!! ! ! !!
! !"
!!!
!

=−
!"

!"

≶ 0 if 𝑘! ≷ 𝑘! ;!!! ! ! !!! !!! !!! !!! ! ! !"
!

!"

!" !

+ 𝑤 ! !" ;(30)

for 𝜎 = 𝑠! , 𝜖, 𝑧.
From Lemma 1, there is no interregional redistribution. Because of full employment inregion W, the employment expansion effect for household W is zero, as Lemma 2 shows.
Thus, only the terms of trade effect exists for household W, which is explicitly exhibited by
the first equation in (30).
Because region E is underemployed, both the terms of trade effect and employment
expansion effect exist, as the second equation in (30) shows. To find the total welfare effecton household E, we compare the two effects. Applying the properties in (29) to the second
equation in (30) yields
𝑑𝑌 ! 𝑑𝑧 > 0, 𝑑𝑌 ! 𝑑𝜖 > 0 if 𝑘! > 𝑘! ;
𝑑𝑌

!

(31)

𝑑𝑠! < 0 if 𝑘! < 𝑘! .

From (29), the terms of trade effect on household E is positive (or negative) when the local
commodity is labor-intensive (land-intensive). Employment and production subsidies, 𝑧and 𝜖, expand the employment in region E and make household E better off. Thus, if thelocal commodity is labor-intensive (i.e., 𝑘! > 𝑘! ), both the terms of trade effect andemployment expansion effect of the two subsidies are positive, making household E better

18

off, as the first line in (31) shows. If the local commodity is land-intensive, however, thetwo effects have opposite directions and the welfare effect becomes ambiguous.
A wage subsidy urges household W to move to region E, decreasing the employment
rate in region E. This raises the expected labor income in region E and the wage in region
W, and hence increases the relative price of the labor-intensive commodity. Thus, if thelocal commodity in region E is land-intensive, the wage subsidy not only lowers the local
employment rate, but also worsens the terms of trade, making household E worse off. If thelocal commodity in region E is labor-intensive, however, the terms of trade improves
whereas the employment rate worsens, and the welfare effect on household E becomesambiguous.

Specific utility and production functions: In order to explicitly show that the effects ofthe employment and production subsidies on household E’s utility can indeed be positive
and negative when the local commodity is land-intensive, and that the effect of the wage
subsidy on household E’s utility can be positive and negative when the local commodity islabor-intensive, we assume the following specific utility and production functions:! !!!

𝑢 = 𝑐! 𝑐!

!

, 𝑓! (𝑘! ) = 𝑘!! , 𝑓! (𝑘! ) = 𝑘! where 𝛼 , 𝛽 , 𝜑 ∈ (0,1),(32)

and consider the case where the employment rate in region E is almost unity,𝛾 ! ≈ 1.
We show below that in this specific case, the effects on household E’s utility can bepositive and negative depending on some parameter values. Thus, generally, the effects canindeed be positive and negative.

19

The welfare effect of the wage subsidy on household E when 𝒌𝟐 > 𝒌𝟏 : In this case, thesign of 𝑑𝑌 ! 𝑑𝑠! is ambiguous. As proved in Appendix B, the 𝐿! = 0 curve is!" !

𝜑 = 𝜑 ! ≡ !!! !!"! ,ԭೄิॿۚʢ౔஍ิॿ Sr Λ‫ݮ‬গͤͯ͞ɼϵ, z, sw ΛҾ্͖͛Δʣ

(33)

which is shown in Figure 1, and the boundary that makes 𝑑𝑌 ! 𝑑𝑠! positive or negative isԣ͕࣠ β
as depicted in Figure 1.
ϕcase B1case A
L2 = 0dY E
dsw < 0L1 = 0dY E
dsw > 0

0

1

α

β

Figure 1. The sign of 𝑑𝑌 ! ⁄𝑑𝑠!

Replacing a land subsidy with a wage subsidy urges the workers to move from region
W to region E, and this worsens the employment rate in region E, harming household E. It
however raises the wage in region W and the relative price of the labor-intensive
commodity, that is, commodity 1 in the present case. Thus, the terms of trade of household
E improves, benefiting household E. If the beneficial effect of the improvement in terms of
ϕ dominates the negative effecttrade
L2 =of0 the decrease in employment, household E is bettercase Bcase A1
off. The magnitude of the two effects depends on the values of 𝜑 and 𝛽. If 𝜑 is low, the1ratio
2−α of consumption expenditure on commodity 2 is large, and hence the benefit due to the
dY E
dY E
dϵ > 0, dz > 0
decline in 𝜔 is large, as is clear from the expression of 𝑑𝑌 ! 𝑑𝑠! in (30). Moreover, if 𝛽 isEclose todY1Ebut<𝛼0,isdYnot,<the0 production of commodity 2 does not require much labor, but that

dϵ

0

dz

L1 = 0

α
2−α

α

20

1

β

1
L2 = 0dY E
dsw < 0dY E
dsw > 0L1 = 0of commodity 1 does. Thus, an increase in wage raises the production cost of commodity 2
a 0little, while the cost of commodity 1 significantly rises, making 𝜔 broadly βdecline.
1αBecause the benefit and magnitude of the decline in 𝜔 are both large, the positive terms oftrade effect is significant, and dominates the negative employment expansion effect.

The welfare effect of the production and employment subsidies on household E when𝒌𝟐 < 𝒌𝟏 : As shown in Appendix C, the boundary that makes 𝑑𝑌 ! 𝑑𝜎 (for 𝜎 = 𝜖, 𝑧)positive or negative is as illustrated in Figure 2.

ϕ1

case B

12−αL2 = 0case AdY E
dY E
dϵ > 0, dz > 0

dY E
dY E
dϵ < 0, dz < 00

L1 = 0

α
2−α

α

1

β

Figure 2. The sign of 𝑑𝑌 ! ⁄𝑑𝜎 (for 𝜎 = 𝜖, 𝑧)

Replacing a land subsidy with production and employment subsidies expands theemployment in region E, benefiting household E. It however raises the wage in region W as
well as the relative price of the labor-intensive commodity, that is, commodity 2 in thepresent case. Thus, the terms of trade of household E deteriorates. The magnitude of theemployment expansion and terms of trade effects depends on the values of 𝜑 and 𝛽. If 𝜑 islow, the ratio of consumption expenditure on commodity 2 is large, and hence the harm due

211

to the rise in 𝜔 is large, as mentioned in the previous subsection. Moreover, if 𝛽 is small
and 𝛼 is large, the production of commodity 2 requires much more labor than that ofcommodity 1, and the wage increase raises the production cost of commodity 2 much morethan the production cost of commodity 1, significantly raising 𝜔. Thus, both the harm andmagnitude of the rise in 𝜔 are large, and the negative terms of trade effect is so large that it
dominates the positive employment expansion effect, making household E worse off.

Summary: The results of (30), (31), and the two figures are summarized as follows:
Household E
𝑑𝑢!

𝑑𝜖 > 0, 𝑑𝑢!

𝑑𝑧 > 0case A:𝑘! > 𝑘!

𝑑𝑢! 𝑑𝑠! < 0𝜑≈1!

case B:𝑘! < 𝑘!

!

𝑑𝑠! > 0Household W
𝑑𝑢! 𝑑𝜖 < 0
𝑑𝑢! 𝑑𝑧 < 0
𝑑𝑢! 𝑑𝑠! < 0𝜑 ≈𝜑 ,𝛽 ≈1

𝑑𝑢

𝜑≈1

𝑑𝑢! 𝑑𝜖 ≥ 0, 𝑑𝑢! 𝑑𝑧 ≥ 0

𝑑𝑢! 𝑑𝜖 > 0𝜑 ≈ 𝜑! , 𝛽 ≈ 0

𝑑𝑢! 𝑑𝜖 < 0, 𝑑𝑢! 𝑑𝑧 < 0

𝑑𝑢! 𝑑𝑧 > 0

𝑑𝑢! 𝑑𝑠! < 0

𝑑𝑢! 𝑑𝑠! > 0

In the above expressions, 𝜑 ! is given by (33). Therefore, we have the following
proposition:

Proposition 4: In case of unemployment in region E and full employment in region W,
subsidy replacements possess the following properties:

22

Case A (𝑘! > 𝑘! ): All the replacement policies decrease 𝜔 and household W's utility.
Production and employment subsidies increase household E's utility. A wage subsidy canmake household E better off and worse off. Under the utility and production functions givenin (32), the welfare effect on household E is negative when the ratio of consumption
expenditure on commodity 1 is large. However, it is positive when 𝛽 is close to 1 and theratio of consumption expenditure on commodity 1 is so small that region W almost
specializes in commodity 2.
Case B (𝑘! < 𝑘! ): All the replacement policies increase 𝜔 and household W's utility. A
wage subsidy makes household E worse off. The welfare effects of production andemployment subsidies on household E are ambiguous. Under the utility and production
functions given in (32), the effects are positive when the ratio of consumption expenditure oncommodity 1 is large. However, they are negative when 𝛽 is small and the ratio ofconsumption expenditure on commodity 1 is so small that region W almost specializes incommodity 2.

4. Concluding Remarks
A government offering a subsidy to a local region can choose between a lump-sumtransfer and various other subsidies that affect local economic activities. Using a two-region
two-commodity two-factor model in which the subsidy recipient region produces only oneof the two commodities because of lack of technology, we analyzed the welfare effects of
replacing a lump-sum transfer by production, employment, and wage subsidies on therecipient and non-recipient regions under various full employment and underemployment
combinations.

23

If the recipient region’s wage is flexible and full employment prevails, the replacement
policies urge workers to migrate to the recipient region because the labor income in that
region increases. The wage of the non-recipient region also increases if it is flexible, andthe price of the labor-intensive commodity rises. If the recipient region specializes in thelabor-intensive (land-intensive) commodity, the terms of trade in this region improves(deteriorates), and the native households of the region is better (worse) off. If
unemployment prevails in the non-recipient region, the relative price does not vary andsubsidy replacements have no effect on the recipient region.
If unemployment prevails in the recipient region, replacing a lump-sum subsidy withproduction and employment subsidies expands employment and benefit the region.
However, a wage subsidy does not stimulate local production, but urges workers to moveinto the region, and this worsens local employment and harms the recipient region. If the
non-recipient region also suffers from underemployment, the relative commodity priceshows no change. Therefore, only the employment effect matters; that is, production andemployment subsidies benefit, and a wage subsidy harms, the recipient region.
If the non-recipient region achieves full employment and the recipient region remains
underemployed, the three subsidies raise the wage in the non-recipient region and increase
the labor-intensive commodity’s price. Thus, if the recipient region’s local product islabor-intensive, the region’s terms of trade improves. Moreover, production andemployment subsidies increase local employment. Therefore, they definitely make therecipient region better off. A wage subsidy, however, worsens the region’s local
employment. Thus, it reduces, or even dominates, the benefit due to the improvement interms of trade. If the local product is land-intensive, however, the three subsidies worsen
the recipient region’s terms of trade. Therefore, a wage subsidy always harms the recipient24

region through two channels: deteriorating the terms of trade and decreasing theemployment rate. Production and employment subsidies, however, expand local
employment and alleviate, or even dominate, the harm due to the deterioration in the termsof trade.
Nowadays, several developed countries, including the EU and Japan, show declining
birth rates, especially in urban areas, and migrations from rural to urban areas. For a future
study, the present model can be extended to these problems and applied to analyze thewelfare effects on rural and urban areas of population changes and subsidy policies
affecting interregional immigrations.

Appendices
Appendix A
By totally differentiating (18), where 𝐿! and 𝐿! are given by (15) and 𝐿!! is given by(16), we obtain𝐴 ∙ 𝑑𝜔 +(!! !! !!!! !! )!!
!
!! !!!

𝑑𝛾 ! +! ! (!! !! !!!! !! )!!
!
! ! !! !!!

𝑑𝛾 ! +(!!!)!! !! ! ! !!
!
!!! !! !!!

where, because 𝑝! 𝑓! = 𝑟 ! 𝑘! + 𝑤 ! from (8), we have(!!!)! ! !

!!! ! ! !

! !
𝐴 = ! !! !! !!!! + ! !! ! !
− 𝛿 ! 𝑓! 𝐿! < 0.
! ! !!
!

!

!

!

!

!

By totally differentiating the second equation of (10), we have𝑑𝑘!! =! ! !"! ! ! (!!!)/! ∙!"!!! ! !!"!! !!! !!!! !!!

.

Substituting this 𝑑𝑘!! into (A1) and rearranging the result yields (21).

Appendix B

25

𝑑𝑘!! = 0,(A1)

Applying the utility and production functions in (32) to the two equations in (8) yields!! !! !
!!

=

!!

!
!

,

Because 𝐿! = (𝐾 ! − 𝑘! 𝐿! )/𝑘! , from the first equation in (14), substituting 𝐿! and theabove equation into (18) when 𝛾 ! ≈ 1 gives!

𝐿! = !!!
!

!
!

!!! !!

!
!
!
! !!! !

.

Since both commodities are produced in region W, 𝐿! , given above, must be positive, and
thus
!" !

𝜑 > !!! !!"! ≡ 𝜑 ! .
Note that region W produces only commodity 2 (i.e., 𝐿! = 0) when 𝜑 = 𝜑 ! . Thus, the
𝐿! = 0 curve in Figures 1 and 2 shows the curve of 𝜑 = 𝜑 ! given in (33).
Substituting 𝑑𝜔 and 𝑑𝛾 ! in (28) into 𝑑𝑌 ! in (30) gives
!! !
!!!

! ! !!! ! !!

! ! !
= − !!! !! ! !!
𝜂! ,
!!
!

!

!

where
𝜂! = 1 − 𝜑 𝑘! − 𝑘!
!! ! ! ! (! !! )!

+ ! ! !!!!! !!!
! !

! !

!

!!! ! ! ! (! ! )!

!
𝑤 ! + 𝑟 ! 𝛾 ! 𝑘!! − ! ! ! !!!
! ! ! !!
!

(!!!)!! ! !!
!! !!! ! !!!!

! !

! !

! !

!!! ! ! !

! !
+ ! !! ! !
− 𝛿 ! 𝑓! 𝐿! .
! ! !!
!

!

!

Applying the production functions in (32) to 𝜂! yields
𝜂! =!!!!! !! !!𝜑 !!!

!!! !!𝜑 !! !!
! ! ! !!! !! !! !!! !!
! !!! ! !!𝜑 !!𝜑 !!
!

.

The boundary curve in Figure 1 is the relationship between 𝛽 and 𝜑, which makes 𝜂! equal
to zero:26

1−𝜑 𝛽−𝛼

𝛽 − 𝛼 1 − 𝜑 + 1 𝐿!! = 𝛼 1 − 𝛼 𝐿! + 𝛽 1 − 𝛽 𝐿! .

In case 𝑘! < 𝑘! , it holds that 𝛼 < 𝛽 when the production functions are given by (32). Thus,
we find that 𝑑𝑌 ! 𝑑𝑠! < 0 above the boundary curve and 𝑑𝑌 ! 𝑑𝑠! > 0 below the curve.

Appendix C
In case 𝑘! > 𝑘! , we obtain 𝛼 > 𝛽 when the production functions are given by (32).
The effects of 𝜖 and 𝑧 on 𝑌 ! given in (30) become
𝑑𝑌 ! =!!!
!!
(!!!)(! ! !"!!")! !! !!

! !!! ! !!!

! !! !
!! !!

!! !!! ! !!

𝜂! ,where 𝜂! = 𝛼 1 − 𝛼 + 1 − 𝜑 𝛽 − 𝛼 2 − 𝛼 , 𝛼 > 𝛽,
implying that the sign of 𝑑𝑌 ! 𝑑𝑧 and 𝑑𝑌 ! 𝑑𝜖 is the same as the sign of 𝜂! . The condition
under which 𝜂! ≷ 0 is! !!!

𝜑 ≷ 1 + !!! !!! .
Figure 2 illustrates this property.

References
Albouy, D. (2012) “Evaluating the Efficiency and Equity of Federal Fiscal Equalization”,
Journal of Public Economics, 96(9-10), pp. 824–839.
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