Discussion Papers In Economics
And Business

Regional redistribution policy and welfare in a
two-region endogenous growth model
Yutaro Murakami
Discussion Paper 05-07

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANRegional redistribution policy and welfare in a
two-region endogenous growth model
Yutaro Murakami
Discussion Paper 05-07

March 2005

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANRegional redistribution policy and welfare in a
two-region endogenous growth model∗
Yutaro Murakami †Abstract
This paper constructs a two-region endogenous growth model withproductive government expenditure to analyze the relationship between regional redistribution of public input and the welfare of residents in each region. This paper shows that the redistribution policy
may be Pareto improving if the distribution rate of a more populous
region is increased because it raises the equilibrium growth rate. Furthermore, the higher the inequalities between the labor populations
are, the greater the possibility of a Pareto improving policy.
Key words: Endogenous growth; Government expenditure; Regional
distribution; Welfare; Pareto improving policy
JEL classification: H53; O41; R58∗

I am most grateful to Shin Saito. I also thank Nobuo Akai, Koichi Futagami, Tatsuro
Iwaisako, Ryoji Ohdoi and seminar participants at Osaka University and the 2002 Japanese
Economic Association Autumn Meeting at Hiroshima University. Of course all errors are
my own.
†Address: Graduate School of Economics, Osaka University 1-7, Machikaneyama, Toyonaka, Osaka, 560-0043, JAPAN. E-mail address: murakamiyutaro@srv.econ.osaka-u.ac.jp1

1

Introduction

Why does an increase in regional disparities cause some serious problems?
Some insist that regional inequalities hurt immobile residents in the poorer
regions, others state that an agglomeration of eﬃcient regions has positive
externalities. Thus, although issues about regional inequalities have been
controversial for a long time, there have only been a few theoretical studies.
Furthermore, despite limited theoretical understanding, in many countries
inter-regional redistribution policies that include tied and untied subsidies
are believed to be justiﬁed, as well as income redistribution policies between
individuals. Therefore, it is important to construct a theoretical model interms of regional inequalities and resident welfare.
The purpose of this paper is to investigate the relationship between theinter-regional redistribution of government expenditure and the welfare ofresidents in each region. Barro (1990) ﬁrst formulated an endogenous growthmodel that incorporated public services as a productive input into private
production. This paper extends Barro to two regions and examines a redistribution policy between the regions. Since Barro, although there are many
endogenous growth models with productive public expenditure,1 there has
been hardly any extension to two-region or multi-region models.
This paper is also related to the literature on international economics
that has examined eﬀects of transfers on welfare. They have examined threeeﬀects when the transfer is done. The ﬁrst is the normal case, in which adonor becomes worse oﬀ and a recipient becomes better oﬀ. The second isthat a donor becomes better oﬀ and a recipient becomes worse oﬀ, called thetransfer paradox. The third is that both of them are better (worse) oﬀ, called
Pareto improving (worsening). Although these studies are closely related toour model, they are almost all static2 and has supposed that the transfers
have no eﬀect on productivity.3
The structure of our model is similar to the models that analyze therelationship between the composition of public expenditure and economic
growth in a one-region endogenous growth model, such as Devarajan et al.
1

See Barro and Sala-i-Martin (1992), Futagami et al. (1993), Glomm and Ravikumar(1994), Turnovsky (1996), and others.
2
In international economics, there are many studies that analyze the relationship between forms of aid and welfare, for example Bhagwati et al. (1982). Ihori (1996) introduces
public goods that are voluntarily provided to these frameworks.
3
By way of exception, Galor and Polemarchakis (1987) analyze the existence of a transfer paradox in an overlapping generations model, and Yanagihara (1998) extends Galor
and Polemarchakis to introduce public goods that have externalities only on domestic
private production. However, those studies are not endogenous growth models in which
there is a possibility that the transfer aﬀects an equilibrium growth rate.

2

(1996) and Davoodi et al. (1998, 1999). They have shown that a change inthe composition of public expenditure leads to an increase in the equilibrium
growth rate. Thus, it is possible to say that our model is a two-region andmulti-region expansion of their model.
However, our model is distinct from others in the respect that it incorporates an inter-regional redistribution policy and a simple endogenous growthmodel. Moreover, this is the ﬁrst paper to investigate Pareto improving
public policy in a two-region Barro’s framework.4
The main results in this paper are as follows. A redistribution policy may
be Pareto improving if the distribution rate of the more populated region israised. On the contrary, it may be Pareto worsening if the distribution rateof the less populated region is raised. Furthermore, the higher the inequalities in labor population between the regions are, the possibility of a Pareto
improving policy is more enhanced. This is because a change in the distribution rate has eﬀects not only on initial consumption, but also on theeconomy-wide growth rate, which has a positive correlation with a change inthe distribution rate as long as that of the more populous region is raised.
The reasons why the economy-wide growth rate increases are because of scaleeﬀects in endogenous growth models5 and spillover eﬀects in government expenditure. Moreover, we extend this model to a multi-regional case. Similar
to the two-region case, we show the possibility for Pareto improvement if the
distribution rate of regions with higher than average population is raised.
The remainder of this paper is organized as follows. The basic model ispresented in Section 2. Section 3 examines comparative statics with respect
to policy variables of governments. Section 4 analyses the welfare eﬀectof changes in the distribution rate and shows the possibility of a Pareto
improving policy. Section 5 brieﬂy considers the case of a lump-sum transfer.
Section 6 extends the model to a multi-region model. Section 7 discusses theimplications and concludes this paper.

2

The model

We ﬁrst present the basic structure of the model. There are two regions,regions 1 and 2, in a country. The country has a central government thatcollects taxes from individuals in each region and distributes them to local
governments, which supply local public goods that enhance productivity as
4

Martin (1999) and Martin and Ottaviano (1999) have a similar idea to ours, but thestructure of their models are more complicated since they consider industrial geography
and they have analyzed public policies that lower intra- or inter-regional transaction costs.
5
See Barro and Sala-i-Martin (1995) for more detail.

3

an externality in each region. The regions produce a homogeneous good andthey are identical except for their initial assets and labor populations. We
assume that capital is perfectly mobile between regions, but labor migration
is prohibited.

2.1

Firms

Following Barro(1990), the production sector in each region has a linearlyhomogeneous production function, that isYi = F (Ki , Gi Li ),i = 1, 2,(1)where Yi is output in region i, Ki is physical capital devoted to the production
sector, Gi is local government expenditure in each region, and Li denotes
labor, which is identiﬁed as the size of the population, and is constant overtime.
By taking the price of the good as numeraire, let ri , wi represent therent on capital and the wage in region i, respectively. Since the production
function in each region is linearly homogeneous, we can express the proﬁt
maximization problem of competitive ﬁrms as follows:
ri = f  (ki),f (ki )Ki
Ki
− f (ki )= ω(ki) .
wi =
ki
Li
Li

(2)
(3)

In the above equations, ki ≡ Ki /Li Gi is the quantity of capital per unitof eﬀective labor, ω(ki) = f (ki )/ki − f  (ki ) and f (ki ) = F (Ki /Li Gi , 1). We
assume that f (ki ) satisﬁes the standard neoclassical properties and the Inada
conditions:f  (ki ) > 0, f  (ki ) < 0, lim f  (ki ) = ∞,
ki →0



lim f  (ki ) = 0.

ki →∞



where f and f denote the ﬁrst derivative and the second derivative withrespect to the argument of the function, respectively.

2.2

Households

We assume that many identical households live in two regions and that eachhousehold provides its assets in both regions and inelastically one unit of labor in the region where they live. The objective function of the representative
household in region i is
4

 ∞
0

u(ci)e−ρt dt,(4)where ci is the per capita consumption of the household in region i and ρ isthe constant rate of time preference. The felicity function is assumed to takethe constant relative risk aversion form:
c1−σi, for σ > 0, σ = 1,
1−σ= ln ci ,for σ = 1,u(ci ) =(5)(6)

where σ is the degree of relative risk aversion (the reciprocal of the intertemporal elasticity of substitution), which is assumed to be a positive constant.
Due to the perfect capital mobility between the regions, a no-arbitrage
condition r1 = r2 = r must always hold without the case of the corner
solution. Thus, the ﬂow budget constraint of the household in region i isȧi = (1 − τ )(rai + wi ) − ci ,(7)where ai , τ denotes the total asset per capita in region i, the rate of ﬂat-rate
income tax, respectively.6 Given the amount of initial holdings ai (0) > 0, thehousehold maximizes (4) subject to (7). It is assumed that the household
is endowed with perfect foresight, and it accurately anticipates the whole∞
sequences of real rents {r(t)}∞t=0 and the wage rate {wi (t)}t=0 . As a resultof the utility maximization problem of the household endowed with perfectforesight, the optimal consumption path and the transversality condition are
given by1c˙i= [(1 − τ )r∗ − ρ] ,ci
σlim λi (t)ai (t)e−ρt = 0,t→∞where λ represents the co-state variable of ai .

6

A dot above a variable represents taking the time-derivative.

5

(8)
(9)

2.3

Governments

There are three governments in the economy. The central government levies
ﬂat-rate income taxes in each region.7 The budget constraints of each government readG = τ (Y1 + Y2 ),G1 = βG, G2 = (1 − β)G,β ∈ (0, 1),

(10)
(11)

where G is the central government’s revenue (total tax revenue in the economy8 ), Gi is the local government expenditure in each region, and β is the
distribution rate for region 1. For simplicity, we assume that τ and β aretime-invariant and that distribution from the central government is tied togovernment expenditure.9 Moreover, note that equations (10) and (11) meanthat each government runs a balanced budget.

2.4

Market equilibrium

There are two integrated markets, asset and goods,10 and two disintegrated
markets, labor in each region, in the economy. Each market equilibrium is
described by
A1 + A2 = K1 + K2 ,Y1 + Y2 = C1 + C2 + K̇1 + K̇2 + G1 + G2 ,

(12)
(13)

where A1 and A2 respectively denote the aggregate assets in regions 1 and 2.
Let us next deﬁne the new variable x which represents the GDP ratio(Y2 /Y1 ), then from (1), (11) the quantity of capital per unit of eﬀective laborin each region is as follows:


k1
k2


1
,
= f
βL1 τ (1 + x)


1
−1
= f
.
(1 − β)L2 τ (1 + x−1 )
−1

7

(14)
(15)

The details of other taxation methods are given in Atkinson and Stiglitz (1980)
For the sake of an exhaustion of product, Y i = rKi + wi Li holds in each region. Using
the ﬂow budget constraint of the household and asset market clearing condition, we derive(10).
9
If the distribution includes tied and untied forms, the result in this paper would stillhold.
10
Equities are the only ﬁnancial asset in the economy.
8

6

ri

r1 (x)r∗r2 (x)x∗
Figure 1: ri − x planex

Although the rent on physical capital in Barro’s model, which is for one
region with no distribution policy, could be derived in the form of simple
AK technology, in our model this depends on the GDP ratio x. Figure 1
illustrates the rent on physical capital in each region, and it indicates that r1
is monotonically increasing in x and r2 is monotonically decreasing in x.11
As mentioned above, because the no-arbitrage condition always holds,
r1 = r2 ⇔ k1 = k2 holds in equilibrium, and then, the equilibrium ratio x
can be determined as follows.
x∗ = (1 − β)L2 /βL1 ,


1
∗
−1
k = f
,
τ [βL1 + (1 − β)L2 ]r ∗ = f  (k ∗ ),ω ∗ = f (k ∗ )/k ∗ − f  (k ∗ ),

(16)
(17)
(18)
(19)

where the asterisk denotes an equilibrium value. From (16) through (19), weshow that the equilibrium value r∗ and ω ∗ are time-invariant since x∗ and k ∗are time-invariant.
11

The second order condition, the sign of ∂ 2 ri /∂x2 depends on the functional form.

7

2.5

Steady-growth equilibrium

We let γ ∗ be a steady-growth rate and in a steady-growth equilibrium, theeconomy is characterized by

γ∗ ≡

Ẏi
Ċi
K̇i
Ȧ
Ġi
=
= =
=
,
Ci
Ki
A
Yi
Gi

where all variables are at the aggregate level, and A ≡ A1 + A2 denotes
the economy-wide assets. Let us examine the existence and stability of thesteady-growth equilibrium. From (10), (12), (13) and recalling that Y =(r + ω)A, we obtainȦ = (1 − τ )(r∗ + ω ∗ )A − C,(20)

where C ≡ C1 + C2 represents the economy-wide consumption level. Deﬁnea new variable z as C/A, then from (8) and (20) we have the dynamics withrespect to z:ż = χ(z)


1
∗
∗
∗
((1 − τ )r − ρ) − (1 − τ )(r + ω ) + z .
≡ z
σWe can easily verify that χ(z) = 0 has a unique positive solution, denotedby z ∗ , if and only ifρf (k ∗ ) f  (k ∗ )−+ (1 − τ )
> 0.
(21)
σ
k∗
σ
Figure 2 depicts the graph of χ when (21) holds. In this graph, it is shownthat z ∗ is globally unstable and hence the steady-growth equilibrium in thismodel uniquely exists. This implies that the economy has no transitional
dynamics and is always in a steady-growth equilibrium.

3

Comparative statics

In this section, we consider the impacts of an unanticipated permanent increase in β and τ . The eﬀects of changes in β and τ are summarized asfollows:

8

ż

o

z∗

z

Figure 2: Graph of χ

dγ ∗
dβ
dx∗
dβ
dγ ∗
dτ
dx∗
dτ

1
f  (k ∗ )L1 − L2= − (1 − τ )  ∗,σf (k ) τ [βL1 + (1 − β)L2 ]2
L2= − 2 < 0,
β L1


1
f  (k ∗ )1∗=−(1 − τ )  ∗ 2−r ,σf (k ) τ [βL1 + (1 − β)L2 ]

(23)

= 0.

(25)

(22)

(24)

Proposition 1. It is true thata. by increasing the distribution rate to the more populated region, thesteady-growth rate can be enhanced.
b. since the central government levies the same tax rate on both regions,regional inequalities cannot be reduced by a tax policy.
Proof. See the equations from (22) to (25).

94

Welfare analysis (Pareto improving policy)

In this section, we examine the welfare eﬀect in changing the distribution rate.
The redistribution policy between regions would be justiﬁed if it enhanced
the welfare of residents in less populated regions. We ﬁrst predict the levelof the consumption at time t in each region.
The budget constraint of households in the aggregate in region i isȦi (t) = (1 − τ )(r∗ Ai (t) + wi (t)Li ) − ci (t)Li .
By solving the ﬁrst order diﬀerential equation with respect to Ai (t), theinitial consumption in region i is given by12
 ∞−(1−τ )r∗ t
ci (0) = −φ ai (0) + (1 − τ )wi (t)e
dt ,

(26)

0

where φ ≡ (1/σ − 1)(1 − τ )r∗ − ρ/σ < 0.
Substituting (3), (12) and (19) into (26), we obtain:13
1
1(1 − τ )ω ∗ A(0),∗1 + x L1x∗ 1
(1 − τ )ω ∗ A(0).
c2 (0) = −φa2 (0) +1 + x∗ L2
c1 (0) = −φa1 (0) +

(27)
(28)

Thus the initial consumption depends on the initial asset (the ﬁrst termin (27) and (28)) and the lifetime wage (the second term in (27) and (28)).
In other words, the greater the initial asset or the lifetime wage is, the more
the initial consumption is. Furthermore, the lifetime wage depends on GDP
ratio (capital ratio) since it increases with capital invested where they live inan endogenous growth model.14
For the sake of simplicity, suppose that the rate of the relative risk aversion σ = 1, that is, the utility function is logarithmic. Further suppose thatthe production technology of the ﬁrms takes a Cobb-Douglas form, hence
Yi = Kiα (Gi Li )1−α , α ∈ (0, 1),
12

See appendix in detail.
The derivation of (27) and (28) is in the appendix.
14
In equilibrium, GDP ratio equals to capital ratio completely.
13

10

where α represents the distribution rate of capital.15 Let us now deﬁne thewelfare function in each region. The welfare over an inﬁnite planning horizonof the household in region i (Wi ) can be written as:
 ∞
Wi =

0

ln ci (t)e−ρt dt,

11
ln ci (0) + 2 γ ∗ .
=
ρ
ρ

(29)

Let us now diﬀerentiate (29) with respect to β, in other words, we consider
the welfare eﬀects when the distribution rate to region 1 rises. For thisoperation the welfare eﬀects on rising β are given bydWi
1 1 dci(0)1 dγ ∗=+ 2.
dβ
ρ ci (0) dβ
ρ dβ

(30)

Equation (30) reveals that the welfare eﬀect of a rise in β includes two distinct components. One, the ﬁrst term on the right-hand side, is the initialconsumption eﬀect. It reﬂects whether the initial consumption of the residents in each region increases or not, by raising the distribution rate to region
1. The other, the second term on the right-hand side, is the economy-wide
growth eﬀect. It represents the eﬀect of a rise in β on the steady-growth rate.
As mentioned in proposition 1, this eﬀect is always positive if and only if
the labor population in region 1 is greater than that of region 2. Therefore,
we have only to focus on the initial consumption eﬀect. It can be derived asfollows:
1−αdc1 (0)=A(0)ψ [(1 − α)β(L1 − L2 ) + αL2 ] ,dβα1−αdc2 (0)=A(0)ψ [(1 − α)(1 − β)(L1 − L2 ) − αL1 ] ,
dβ
α
1−α

1−3α

(31)
(32)

where ψ ≡ (1−τ )τ α B α , B ≡ βL1 +(1−β)L2 , respectively. Furthermore,in the case of the log-linear utility function, the initial consumption eﬀect isdivided into two components, both of which are linked to the lifetime wage.
One, the ﬁrst term in the square bracket on the right-hand side of equations(31) and (32), is the wage growth eﬀect. It captures the eﬀect of the wage
increases with an economy-wide growth rate. The other, the second term in
15

In the Cobb-Douglas case, the variables in equilibrium are given in appendix.

11

the bracket on the right-hand side, is the productivity eﬀect. It means thata rise in the distribution rate (a rise in β) makes region 1 more attractive
for investment since it enhances the productivity of capital in region 1. The
wage growth eﬀect is positive in both regions because we suppose L1 > L2 .
However, the productivity eﬀect is positive in region 1 but negative in region
2. From the above argument we have the following proposition.
Proposition 2. In the case that region 1 is more populous than region 2, a
policy which increases the distribution rate in region 1 (region 2) is Pareto
improving (Pareto worsening) if and only if L 2 /L1 ≤ Φ(L2 /L1 ).
Proof. We ﬁrst consider the eﬀect of a steady-growth rate. By assuming
region 1 is more populous, a rise in β enhances the steady-growth rate, thatis dγ ∗ /dβ > 0. Next, the two eﬀects on the initial consumption in region 1
have the same sign (both positive), hence dW1 /dβ > 0 from (31). On theother hand, those in region 2 have the opposite sign; that is, the wage growtheﬀect is positive and the productivity eﬀect is negative. Hence, the sign of
dW2 /dβ is ambiguous because of this trade-oﬀ. The necessary and suﬃcientcondition for a Pareto improving policy (dW2 /dβ ≥ 0) is as follows:
 
L2
L2
α


≡Φ
≤1−
.
L1
L1
(1 − α)(1 − β) 1 + 1 (1 − τ )r∗ + a2 (0) αB
ρ

(33)

A(0)

In the equation (33), they are variables r∗ , A(0), B that depends on L2 /L1 ,and they are monotonically increasing in L2 /L1 . Therefore, the sign ofdΦ(·)/d(L2 /L1 ) is ambiguous, but as a result of the numerical analysis, unlessthe parameters are extraordinary, Φ is increasing in L2 /L1 .16
Then it is useful for us to investigate the suﬃcient condition for the Pareto
improving policy, which is dc2 (0)/dβ ≥ 0 if and only if17
L2
α
.
≤1−
L1
(1 − α)(1 − β)(34)

From the equation (33), we derive the following proposition.
Proposition 3. The possibility of a Pareto improving policy is greater in thefollowing cases:
16
17

See appendix.
Figure 4 shows the region for the Pareto improving policy.

12

RHS or LHS

LHS

1necessary and suﬃcientO suﬃcient
Figure 3: Region for a Pareto improving policy

RHS

L2
L1

a. when the disparity of the labor population between regions (L 1 /L2 ) is
greater.
b. when the tax rate is closer to the growth-maximizing tax rate.
Proof. The proof of item a. is shown in Figure 3, as a result of the numericalanalysis.18 The proof of item b. is that the growth-maximizing tax rate isgiven by ∂γ ∗ /∂τ = 0 ⇔ ∂(1 − τ )r∗ /∂τ = 0. In other words, when τ = τ ∗ ,the right hand side of the equation (33) is the maximum, where τ ∗ is thegrowth-maximizing tax rate.
Let us next consider a more generalized case, σ = 1, in which the utility
function takes the CRRA form. The welfare functions in each region are
given by:18

We assumed that L1 > L2 . Hence higher disparity of a labor population corresponds
to a lower L2 /L1 .

13

 ∞

ci (t)1−σ −ρt
e dt,
1−σ
0
ci (0)1−σ=.
(1 − σ)[ρ − (1 − σ)γ ∗ ]

Wi =

(35)

Partially diﬀerentiating (35) with respect to β, we obtain


1
1−α
dc1 (0)= ψ
1−A(0) [(1 − α)β(L1 − L2 ) + αL2 ] ,a1 (0)(L1 − L2 )B +
dβ
σ
α


1−α
1
dc2 (0)= ψ1−
a2 (0)(L1 − L2 )B +A(0) [(1 − α)(1 − β)(L1 − L2 ) − αL1 ] .
dβσαThere is a slight diﬀerence from the log-linear utility function case in that theother eﬀect, the ﬁrst term on the right hand side, exists. We call this eﬀect
the consumption propensity eﬀect which has the same sign as the economywide growth eﬀect. Hence, the possibility of a Pareto improving policy isgreater than in the log-linear utility function case.

5

The model with lump-sum transfers

In this section, preserving the basic structure of the model, we consider asimpler distribution form, which is a lump-sum income transfer. Supposethat the transfer T of the aggregate amount is done from region 2 to region
1. As is well known, the subjective equilibrium condition is not aﬀected by
the lump-sum transfers. The aggregate budget constraints of the householdin each region change as follows:Ȧ1 (t) = (1 − τ )(r∗ A1 (t) + w1 (t)L1 ) − c1 (t)L1 + T,
Ȧ2 (t) = (1 − τ )(r∗ A2 (t) + w2 (t)L2 ) − c2 (t)L2 − T.
If we solve the ordinary diﬀerential equation for ci (0), we can derive


1
T
1−α 1
(1 − τ )r∗ A(0),c1 (0) = −φ a1 (0) ++∗∗(1 − τ )r L1
α 1 + x L1T
1 − α x∗ 1
(1 − τ )r∗ A(0).
c2 (0) = −φ a2 (0) −+(1 − τ )r∗ L2
α 1 + x∗ L2
14

The initial consumption in each region depends on initial assets, the amountof the lump-sum transfer, and the lifetime wage. Then, we examine the eﬀects
on welfare of changes in the lump-sum transfer. For simplicity, assume thatσ = 1, and the following equation is obtained:
1 1 dci(0)1 dγ ∗
dWi
=
+ 2
.
dT
ρ ci (0) dT
ρ dT

(36)

From the equation (36) we can state the following proposition.
Proposition 4. There is no possibility of a Pareto improving policy in thecase of a lump-sum income transfer.
Proof. The eﬀects on welfare are of two types, which are the initial consumption eﬀect and the economy-wide growth eﬀect. The latter eﬀect is zero, hencedγ ∗ /dT = 0, since the eﬀect of the lump-sum transfer on the steady-growthrate is neutral. Let us next examine its eﬀect on the composition of initialconsumption. Although it is easily seen that there is no wage growth eﬀect
and no productivity eﬀect, a new eﬀect of the lump-sum transfer emerges.
Therefore, we have only to contemplate this eﬀect. The equation (36) changes
into
dW1 (0)
φ
= −
> 0,
dT
(1 − τ )r∗ L1
φdW2 (0)=< 0.
dT(1 − τ )r∗ L2

(37)
(38)

From equations (37) and (38), we have shown proposition.

6

Multi-regional extension

This model can easily be extended to a multi-region model. Consider an
economy with n regions: i = 1, · · · , n. We assume that βi denotes a shareof government expenditure to region i, then the budget constraints of eachgovernment are changed as follows:
n

Yi ,

G = τ

(39)

i=1

Gi = βi G,
15

(40)

where ni=1 βi = 1. Equations (39) and (40) respectively represent the budget constraints of the central government and the local government in region
i. The proﬁt maximization of ﬁrms and the utility maximization of households remain basically unchanged, and each market equilibrium condition is
given by
n

n

Ai =
i=1
n

Ki ,

(41)

i=1
n

n

Yi =
i=1

K̇i + G,

Ci +
i=1

(42)

i=1

where (41) and (42) denote the integrated asset market and goods market,
respectively.
Let us next deﬁne the new variable si which represents the GDP share(Yi / nk=1 Yk ) in region i, then from (40) the quantity of capital per unit ofeﬀective labor in region i is as follows:ki = f−1si
τ βi Li.

From the no-arbitrage condition, ki = kj holds for all i, j, i = j. Hence,the variables in equilibrium are given by
s∗i =
k

∗

r∗
ω∗

= f

βi Li
,
n
k=1 βk Lk−11τ nk=1 βk Lk= f  (k ∗ ),= f (k ∗ )/k ∗ − f  (k ∗ ).

,

As in the two-region model, we turn to examine the comparative statics. But
in advance, we must determine the rule of the redistribution policy.
Assumption 1. The increase in the share of region i on a minute scaleequals the decrease in that of other regions, that is, Δβi = − nj=i Δβj . For
simplicity, we assume that the decrease in the share is the same amount in
all regions except for region i, hence Δβ j = Δβ̄ for all j = i.

16

Owing to the above assumption, we can derive the results on the comparative statics with respect to βi as follows:19n1
Lj , (43)Li −n2n − 1 j=i
k=1 βk Lk )
nsi
Lj > 0,(1 − s∗i )Li +(44)
n − 1 j=i

n
1
Lj
− s∗j Li −Lj−
< 0. (45)n−1n − 1 j=if  (k ∗ )dγ ∗1= − (1 − τ )  ∗dβi
σf (k ) τ (
ds∗i
=
dβi

n
k=1 βk Lk

ds∗j
=
dβi

n
k=1 βk Lk

1

1

1

Thus, we have the following proposition.
Proposition 5. The steady-growth rate can be enhanced if and only if theshare of government expenditure to the region which is more populous than
the average for the remaining regions is increased.
Proof. According to the equation (43), dγ ∗ /dβi > 0 ⇔ Li >1).

i=j Lj /(n −The equation (43) represents a multi-region case of the equation (22), andthe equation (44) and (45) mean that an increase in the distribution rate of
the region i leads to an increase (decrease) in the GDP share of the region i(region j).
As in the two-region model, we specify the production function of ﬁrms
as the Cobb-Douglas.20 Let us next derive the initial consumption by using
the asset market clearing condition. The same operation in the two-region
model leads to the following equation.
ci (0) = −φai (0) +1−α 1(1 − τ )r∗ s∗i A(0),
α Li

(46)

which holds for all i. Thereafter we diﬀerentiate (46) with respect to βi , inother words, we examine the eﬀects of an increase in the share of region i on
the welfare within the region and in the other regions. Suppose again thatσ = 1, then we obtaindWi
1 1 dci(0)1 dγ ∗=+ 2.
dβi
ρ ci (0) dβi
ρ dβi
19
20

The derivation of equations (43), (44) and (45) is in the appendix.
Each variable in equilibrium is given in the appendix.

17

(47)

We have already investigated the economy-wide growth eﬀect (dγ∗ /dβi ) in
the above comparative statics. Therefore we have only to consider the two
initial consumption eﬀects. (dci (0)/dβi and dcj (0)/dβi, j = i) They can be
rewritten as:


 

1−3α
dci (0)1−α1 − s∗iΨβi ζ α=αLi + L̄ + (1 − α)(Li − L̄)
dβi
α
s∗i




1−3α
1−α
1 ζ
dcj (0)αΨβj ζ=−α (Li − L̄) ++ (1 − α)(Li − L̄)dβi
αn − 1 βj1−αn1where Ψ ≡ (1 − τ )τ α A(0), ζ ≡ nk=1 βk Lk , L̄ ≡ n−1
j=i Lj , respectively.
Thus we can ﬁnd the following proposition similar to the two-region model.

Proposition 6. In the multi-region model, the policy that increases the shareof government expenditure to region i may be Pareto improving if region i ismore populous than the average of the other regions.
Proof. We follow almost the same procedure as in the two-region model.21
By assuming that region i is more populous than the average, a rise in βienhances the economy-wide growth rate, that is dγ∗ /dβi > 0. Next, thetwo eﬀects on the initial consumption in region i have the same sign (bothpositive), hence dWi /dβi > 0. On the other hand, those in region j have theopposite sign, that is, the wage growth eﬀect is positive and the productivity
eﬀect is negative. Hence, the sign of dWj /dβi is ambiguous because of thistrade-oﬀ. The necessary and suﬃcient condition for the Pareto improving
policy (dWj /dβi ≥ 0) is as follows:
Lj
L̄
α
≤1−
,
Li
[(n − 1)(1 − 2α) + D] sj
where
D≡

11−α
aj (0) Lj(1 − τ )r∗ .
+A(0) sjρ αThen it is useful for us to investigate the suﬃcient condition for the Pareto
improving policy, that is, dcj (0)/dβi ≥ 0 if and only if
21

We pay attention to analyzing the Pareto improving policy in the multi-region model
because we must check welfare in all regions. However, supposing that all regions are
symmetric except for the labor population and GDP share, it is a suﬃcient conditionof the possibility for the Pareto improving policy to assume that the region j is theleast populous and has the biggest GDP share, which is L j = min{L1 , L2 , · · · , Ln } andsj = max{s1 , s2 , · · · , sn }.

18

α
L̄
1 Lj
≤1−
.
Li
(n − 1)(1 − 2α) sj Li

7

Concluding remarks

In this paper, we have constructed a two-region endogenous growth modelwith government expenditure, and theoretically explored the relationship between the regional redistribution policy of public input and the welfare of theresidents in each region. We have shown that there is a possibility of Pareto
improvement if the policy is for redistribution from the less populous region
to the more populous region because it raises the equilibrium growth rate.
Furthermore, this paper has extended a two-region model to a multi-region
one, and has been successful in providing an analytical solution. This extension is highly fruitful and rewarding when we make policy recommendations.
In spite of our contributions, the possibility of further extension remains
for future research. First, we have assumed no congestion costs of government expenditure. If these are taken into consideration, our results may be
modiﬁed. Second, we have taken the regional share of government expenditure as given. It is much more fruitful to incorporate a framework of politicaldecisions into our model.

A

Appendix

A.1

Derivation of (26)

We solve the ordinary diﬀerential equation with respect to Ai (t).
∗∗e−(1−τ )r t [Ȧi (t) − (1 − τ )r∗ Ai (t)] = e−(1−τ )r t [(1 − τ )wi (t)Li − ci (t)Li ](A.1)
Integrating both sides within [0, ∞), we derive∞ −(1−τ )r∗ t
Ai (t) 0 =e−(1−τ )r∗ t

lim e

t→∞

Ai (t) − Ai (0) =

 ∞
0 ∞
0

∗

e−(1−τ )r t [(1 − τ )wi (t)Li − ci (t)Li ]dt(A.2)∗

e−(1−τ )r t [(1 − τ )wi (t)Li − ci (t)Li ]dt(A.3)
19

The ﬁrst term on the left hand side is 0 from the transversality condition.
∗
Then, using the Euler equation, we ﬁnd ci (t) = ci (0)eγ t . Substituting this
into (A.3), we can get
 ∞
 ∞
∗
[γ ∗ −(1−τ )r∗ ]tci (0)edt = ai (0) + (1 − τ )e−(1−τ )r t wi (t)dt
(A.4)
0

0

Let us now deﬁne the new variable φ as follows:φ ≡ γ ∗ − (1 − τ )r∗1[(1 − τ )r∗ − ρ] − (1 − τ )r∗
=
σ


1
ρ
=
− 1 (1 − τ )r∗ − < 0
σ
σ

(A.5)
(A.6)
(A.7)

Then ci (0) can be rewritten by ∞−(1−τ )r∗ t
wi (t)edt
ci (0) = −φ ai (0) + (1 − τ )

(A.8)

0

A.2

Derivation of (27) and (28)

Because of no transitional dynamics, the second term in the right hand side(wi (t)) of equation (26) is
wi (t) = ω ∗Ki (t).
Li(A.9)

Using the asset market equilibrium condition (12), the equation (A.9) canbe rewritten as

K1 (t) =1∗
A(0)eγ t .
∗1+x

By the same operation, the condition in region 2 is given by

K2 (t) =1 + x∗
γ∗t
A(0)e
.
x∗

Thus, we can derive the equilibrium wage as follows:
20

1 ∗
1
γ∗t
ω
A(0)e
,
1 + x∗ L1
x∗ 1 ∗∗
w2 (t) =ω A(0)eγ t .
∗1 + x L2
w1 (t) =(A.10)(A.11)

Substituting equations (A.10) and (A.11) into the initial consumption (26),
we obtain (27) and (28).

A.3

Cobb-Douglas specification (two-region model)x∗ =(1 − β)L2
βL1

1−α

1−α

r ∗ = ατ α [βL1 + (1 − β)L2 ] α

1−α
1−α
1
γ∗ =
(1 − τ )ατ α [βL1 + (1 − β)L2 ] α − ρ
σ
1 − α Ki (t) ∗wi (t) =
r
α
Li




ρ
1
1
1−α 1
∗
+ 1−
c1 (0) =(1 − τ )r∗ A(0)(1 − τ )r a1 (0) +∗σσα 1 + x L1




1
ρ
1 − α x∗ 1∗
+ 1−
(1 − τ )r∗ A(0)c2 (0) =(1 − τ )r a2 (0) +σσα 1 + x∗ L2

A.4

Cobb-Douglas specification (multi-region model)
s∗i =

βi Li
n
k=1 βk Lkr∗ = α τ

βk Lk
k=1

γ

∗

1
=
σ

 1−α
α

n



n

(1 − τ )α τ

βk Lk

 1−α
α


−ρ

k=1

1 − α Ki (t) ∗
r
α
L
 i



ρ
1
1−α 1
∗
+ 1−
ci (0) =(1 − τ )r∗ s∗i A(0)(1 − τ )r ai (0) +
σ
σ
α Li

wi(t) =21

A.5

Derivation of (43), (44) and (45)

Because of the Euler equation the steady-growth rate is given by γ∗ =1/σ((1 − τ )r∗ − ρ). Then the comparative statics with respect to βi is
dγ ∗
dr ∗
1
= (1 − τ )
.
dβi
σ
dβi

(A.12)

According to the equations (43) and (43), the equation (A.12) changes intof  (k ∗ )dγ ∗1= − (1 − τ )  ∗dβi
σf (k ) (τBy using assumption 1, d(as:n2
k=1 βk Lk )d(n
k=1 βk Lk )dβi.

(A.13)nk=1 βk Lk )/dβi in the equation (A.13) is rewritten



n

βk Lk

d

τ

= Li dβi +k=1

Lj dβ̄,
j=i

=

1
Li −
Lj
n − 1 j=i


dβi .

(A.14)

Hence we can derive the equation (43) by substituting (A.14) into (A.13). 22
Let us next derive the equation (44). Totally diﬀerentiating the equation(43), we obtain:
ds∗i
=
dβi

1
n
k=1 βk Lkd(Li − s∗ink=1 βk Lk )dβi

(A.15)

Substituting (A.14) into (A.15), we then derive the equation (44). We canobtain the equation (45) in a similar way.

22

To derive (A.14) we use the conditionk βk = 1 ⇔22k dβk = 0.

A.6

Numerical examples

This appendix shows the relationship between the possibility of a Pareto
improving policy and the labor population ratio L2 /L1 in a two-region modelwith numerical calculations. That is to say, we investigate the equation (33).
As mentioned above, on the right hand side, although dΦ(L2 /L1 )/d(L2 /L1 )is ambiguous, Φ almost always increases in L2 /L1 .
The speciﬁcation of parameters is as follows:
α = 0.4, ρ = 0.01, a1 (0) = 2, a2 (0) = 1, τ = 0.5 in ﬁgure 4-6
α = 0.4, ρ = 0.01, a1 (0) = 2, a2 (0) = 1, β = 0.5 in ﬁgure 7-9

1
0.9
0.8
0.6
0.5

RHS
LHS

0.4
0.3
0.2
0.1
0

0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1

LHS or RHS

0.7

L2/L1

Figure 4: β = 0.3

23

0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1

LHS or RHS
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1

LHS or RHS
0.9
0.8

1

0.7

0.6

0.5
0.4
RHS
LHS

0.3

0.2
0.1

0
L2/L1

Figure 5: β = 0.6

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
RHS
LHS

L2/L1

Figure 6: β = 0.9

24

0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1

LHS or RHS

0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1

LHS or RHS
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
RHS
LHS

L2/L1

Figure 7: τ = 0.3

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
RHS
LHS

L2/L1

Figure 8: τ = 0.6

25

RHS
LHS

0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1

LHS or RHS

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
L2/L1

Figure 9: τ = 0.9

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27