Modern Economy, 2010, 1, 171-179
doi:10.4236/me.2010.13020 Published Online November 2010 (http://www.SciRP.org/journal/me)
National a n d R egional Eco nomic Growth with Fisca l
Policies: Congested Public Goods and Endogenous Labor
Supply of a Small-Open Economy
Wei-Bin Zhang
Ritsumeikan Asia Pacific University, 1-1 Jumonjibaru, Beppu-shi, Oita-ken, Japan
Received August 12, 2010; revised September 16, 2010; accepted Sept em b er 20, 2010
Abstract
The purpose of this study is to build a two-regional growth model with capital accumulation, endogenous
time distribution between leisure and labor, and regional public goods with fiscal policies. We emphasize
dynamic interactions among capital accumulation, externalities, supply of public good with different fiscal
policies, congestion of public good, endogenous time, and economic geography. The economy consists of
two regions and each region consists of the industrial sector and public sector. First, we develop the
two-region growth model with public goods and fiscal policies. Second, we show how to find equilibrium
values of the dynamic system. Then, we simulate model with specified parameter values. Finally, we carry
out comparative statics analysis with regard to parameter changes in tax rates and congestion. Our compara-
tive statics analysis provides so me important insights. For instance, a main difference between the effects of
increasing the two regions’ tax rates on the output is that as region 1’s (2’s) tax rate on the industrial sector is
increased, the national industrial output, national capital employed by the economy, and the national wealth
are increased (reduced).
Keywords: Small-Open Economy, Interregional Economy, Capital Accumulation, Endogenous Leisure and
Work Time, Fiscal Policies, Public Goods
1. Introduction
Interactions between economic growth, public good sup-
ply and environmental changes have caused a great at-
tention from economists. Nevertheless, it is argued that a
few theoretical models have been proposed to deal with
these interactions within a interregional framework. As
argued by [1,2], a main obstacle to properly modeling
regional economic dynamics is that the traditional ap-
proaches to consumer behavior over time makes it ana-
lytically intractable to model interregional grow th on the
basis of profit- and utility-maximization. The productiv-
ity advantages of one region may be offset to some ex-
tent by the higher wages that must be paid in a system
where people are free to choose where they work and
live. Higher wages are often associated with some kinds
of disamenities (such as noise, pollutants, and densely
populated neighborhood) and high living costs. Labor
and capital are easily mobile between regions in indus-
trialized economies. As capital mobility becomes high
and costs associated with capital movement among re-
gions become low, it is reasonable to assume that capital
movement tends to equalize marginal productivities of
capital among regions within a national economy. But
there are different principles for analyzing temporary
equilibrium conditions for labor movement in a dynamic
regional framework. In this study, we determine popula-
tion distribution by the condition of equalizing utility
level. This paper is a generalization of the two-region
growth model proposed by Zhang [3]. This paper gener-
alizes the previous model in introducing different fiscal
policies and endogenous time distribution between lei-
sure and work into the regional dynamic model. We also
take account of congestion into consideration. This paper
is organized as follows. Section 2 defines the two-region
model with capital accumulation, endogenous time dis-
tribution, and public goods. Section 3 shows how to de-
termine equilibrium of variables. Section 4 examines
effects of changes in some parameters upon long-term
national economic growth and economic geography.

172 W. B. ZHANG
Section 5 concludes the study.
2. The 2-Region Trade Model with Capital
Accumulation
This paper builds a dynamic one-commodity and
two-region trade model to examine interdependence be-
tween regional trades and national growth with regional
public goods and congestion. We analyze trade issues
within the framework of a simple international macro-
economic growth model with perfect capital mobility.
This model is influenced by the neoclassical trade theory
with capital accumulation. Many one-commodity trade
models with capital accumulation have been proposed,
for instance, by [4-10]. It is assumed that the regions
produce a homogenous commodity and public goods. It
is assumed that there is only one (durable) good in the
national economy under consideration. Households own
assets of the economy and distribute their incomes to
consume and save. Industrial sectors or firms use capital
and labor. Exchanges take place in perfectly competitive
markets. Industrial sectors sell their product to house-
holds or to other sectors and households sell their labor
and assets to industrial sectors. Factor markets work well;
factors are inelastically supplied and the available factors
are fully utilized at every moment. Saving is undertaken
only by households, which implies that all earnings of
firms are distributed in the form of payments to factors
of production. We omit the possibility of hoarding of
output in the form of non-productive inventories held by
households. All savings volunteered by households are
absorbed by firms. We require saving and investment to
be equal at any point of time.
The system consists of two regions, indexed by
Each region has industrial and public sectors,
indexed by i and respectively. Perfect competi-
tion is assumed to prevail in good markets both within
each region and between the regions, and commodity is
traded without any barriers such as transport costs or
tariffs. The labor markets are perfectly competitive
within each region and between the regions. Let prices
be measured in terms of the commodity and the price of
the commodity be unity. We denote wage and interest
rates by and respectively, in the th
region. The interest rate is equal throughout the national
economy, i.e., j where is the rate of in-
terest fixed in the international economy. We assume a
homogenous population,
1, 2.j,p
(
j
*
r
()
j
wt ),rt
,
j
()rt *
r
,N in the economy. A person
is free to choose his residential location within the coun-
try. We assume that any person chooses the same region
where he works and lives. Each region has fixed land.
Land quality, climates, and environment are homogenous
within each region, but they may vary among the regions.
We neglect transportation cost of commodities between
and within regions. As amenity and land are immobile,
wage rates and land rent may vary between the regions.
Let


j
Nt and


j
Nt stand for respectively the
population and (qualified) labor force of region We
introduce ,
ij
.j

,
rj

and wj

to stand for, respectively,
the fixed tax rates on the industrial output, interest in-
come and wage income in region Let
.j( )
mj
K
t
and mj (),
mj
Nt ( )
F
t
,.mip
stand for the capital stocks, (quali-
fied) labor force employed by, and output level of region
’s sector j.m

2.1. Behavior of Producers
We assume that each firm chooses two productive fac-
tors, capital, j( ),
K
t and labor, j at each point
of time to maximize its profit, with the level of public
goods in the region as given. The production functions
are given by

,Nt



 
1,
ii
j ij
i
tN




,
1,
t
j,0,
j

2
ij i
i
tK


ii
Ft

(1)
where


jt

is a function of externalities, public ser-
vice and congestion. We specify as follows

jt

 

,,
ij
A,
c
p
Kt

 ,
e
 0
c
p
j
F

e
pjij
Kt

pj
ij
Kt
ji
tA






where

p
pj
F
t

measures the effect of public service on
the region’s productivity,

e
ij
K
t

the effect of external-
ities, and
 


/c
pj ij
K
tKt

the effect of congestion of
public goods. Similar to [11], we interpret that when
ec
0,



 there is no congestion and no externality.
The nonrival and nonexcludable public service is avail-
able equally to each agent, independent of the usage of
others. Obviously this is a limited case as most of public
services are subject to some degree of congestion. We
take account of congestion effects by the term,
 


c
t

r
/K,
*,
pj ij implying that for a fixed level of
public capital, a rise in the private capital tends to reduce
the efficiency of public services. There are different
ways of describing congestion (see [12]). Here we ne-
glect possible congestion effects due to the region’s
population and consumption activities.
Kt
Markets are competitive; thus labor and capital earn
their marginal products, and firms earn zero profits. The
rate of interest, and wage rates, are de-
termined by markets. Hence, for any individual firm
and

,wt
j*
r


t
j are given at each point of time. According
to the neoclassical growth theory as in the Solow model
[1], the marginal conditions are given by
w
*,
iij ijiij ij
kj
ij ij
F
F
rw
KN
 
  (2)

W. B. ZHANG
173
where k

is the depreciation rate of ph ysical cap ital and
1.
ij ij

 
2.2. Behavior of Consumers
Each worker may get income from land ownership,
wealth ownership and wages. In order to define incomes,
it is necessary to determine land ownership structure. It
can be seen that land properties may be distributed in
multiple ways under various institutions. This study as-
sumes the absentee land ownership. Land is owned by
absentee landlords who spend their land incomes outside
the economic system. This study uses the approach to
consumers’ behavior proposed by Zhang in the early
1990s [see, 1,2]. Let

j
kt stand for the per capita
wealth in region Let h stand for the level of hu-
man capital and the work time in region
Each consumer of region obtains income
.j
j
T

t.j
j
 


*,1,
jrjjwjjj
ytrkthTtwtj   2 (3)
from the interest payment, *,
rj j
rk  and the wage
payment, ,
wjj j
hTw

where 1
rj rj

  and
1
wjwj .

  The disposable income is given by
 
ˆ.
jjj
ytyt kt At each point of time, a con-
sumer distributes the total available budget among hous-
ing, saving,

,
j
lt ( ),
j
s
t consumption of goods,
The budget constraint is given by ()
j
ct .
*
ˆ
j
jjjjrjjwjjj
Rlc syrkhTw k  
j
where is land rent in region Let 0 stand
for the (fixed) available time for work and leisure. The
time constraint is expressed by .
( )
j
Rt .j
T
T
0
T
 
Tt t
jhj
Substituting this function into the budget constraint
yields
*0
j
jwjhjjjj
rj jwjjj
RlhT wcsy
rkhTwk



 
j

(4)
We assume that utility level that the consumers obtain
is dependent on the lot size, the leisure time,

,
j
lt


,
hj
Tt the consumption level of commodity,


,
j
ct
and the saving,

.
j
s
t

,
j
Ut
v
The utility level of the consumer
in region is specified as follows
,j
  
,
,,,, 0
hhhhh
jj pjjhjjj
hhhhh
UttFltTtc ts t
v






(5)
in which
,
h
v,
hh
,

 and h

are a typical person’s
elasticity of utility with regard to lot size, commodity
and savings in region We call .j,
hh
,

 and h

propensities to consume lot size, to consume goods, and
to hold wealth (save), respectively. We assume that
households would like to have more public goods with
the other things fixed, that is, In (5), 0.
h
v


jt
 is
called region ’s amenity level. In this study, we spec-
ify j
j

by




,
b
jjj
tNt   where (0)
j
  and
are parameters. We don’t specify the sign of as the
population may have either positive or negative effects
on regional attractiveness.
b
b
Maximizing


j
Ut subject to the budget constraints
(4) yields





 


  
,,
,
jjj wjhjjj
jj
Rt ythTtwtt
yt styt




jj
lt
ct
y 
(6)
in which
,,,,
h
1
hhh
hhhh





 

According to the definitions of

,
j
s
t the wealth
accumulation of the representative person in region
is given by j
 
jjj
kt stkt
 (7)
As households are assumed to be freely mobile be-
tween the regions, the utility level of people should be
equal, irrespective of in which region they live, i.e.,




.Ut
1
Ut2 The public sector maximizes the level
of public services by choosing capital,


,
pj
K
t labor
force,


,
pj
Nt as follows

pj
Ft
 
00 00
,,,
pp
pj pjpjpppj
AK tN tA

 0

 (8)
Let


pj
Yt stand for government ’s tax income.
Then we have j





 


*
pjijijrjj jwjj j
tNYF trktNtwt    t(9)
where ,
ij ij
F

*
rjjj
rkN  and wj jj wN

are respectively
the tax incomes from the production sector’s output, the
households’ interest payments and the households’ wage
incomes.
The public sector in region is faced with the fol-
lowing budget constraint j







*()
jpjk pjpj
wtNtrK tY   

t
(10)
Maximization of public services under the budget con-
straint yields


*,
kpjppjjpj
rKYwN   
ppj
Y  (11)
in which
00
0000
,
pp
pp
p
pp


p




The total capital stock employed by the economy,
(),
K
t is equal to the total capitals employed by all the
regions. That is

W. B. ZHANG
174
2
1
 


2
1
ij pjj
jj
K
tKtKtK



t

(12)
where
 


.
jij pj
K
tKtKt The assumption that
labor force and land are fully employed is represented by
 
2
1,,
jjjj
j
NtNltNtL j


1,2 (13)
where
j
L is the given (residential) area of region
We also have
.j
 


ijpjjj j
NtN tNthTtNt (14)
The total wealth of the national economy is the sum of
the wealth owned by all the households
 
2
1jj j
K
tktN

t (15)
We introduce as the value of the economy’s
net foreign assets at The income from the net foreign
assets,

Bt
.t


,Et which may be either positive, zero, or
negative, is equal to According to the defini-
tions of the national wealth, the capital stocks employed
by the economy and the net foreign assets, we have

*.rBt
 
.
K
tKt

0,

Bt

0,
Bt
0,
A country’s current balance at
time is the change in the value of its net claims over
the rest of the world – the change in its net foreign assets.
If the economy as a whole is lending (in this
case we say that the current account balance is in sur-
plus); if the economy as a whole is borrow-
ing (the current account balance is in deficit); and if
the economy as a whole is neither borr owing
nor lending (the current account balance is in balance).
We have thus built the model with endogenous capital
accumulation and regional capital and labor distribution.
We now examine spatial equilibrium and effects of
changes in different conditions upon the economic geog-
raphy.
t
Bt



Bt


3. Economic Equilibrium
As it is difficult to conduct dynamic analysis, we are
only concerned with steady states. In equilibrium the
change rates of all the variables are equal to zero. By (7),
we have
j
j
s
k at steady state. From this equation and
,
j
j
s
y

 we have /.
jj
yk

 From (2) and (11), we
solve //,
p
jij pjij
K
KNN

where /.
p
iip






From this equation,
j
ij pj
NNN and jij KK


,
p
j
K
we solve
,,
,
11
jj
ijpj j
j
j
jj
ij pj
jj
Nk
NN
kk
KkK
KK
kk






j
N



.

(16)
where /
j
pj ij
kKK
 From the definition of


,
jt
(8) and (16), we can express the production functions as
follows

1
jj j j
ij
jj
Ak KN
Fkk








(17)
where
1
101 101 10
10
,,
,pi
p
p cppi epp
ippjijpj
AAA


 
 
 
 
From (1), (16) and (17), we have

 

1
*
1
*
,
1
1
iij jjjj
k
j
kjij
j
ij j
Ak KN
rkk
rk
wNk
 
j
K





 











(18)
From the marginal conditions for capital in (18), we
have 2,
K1
K
K

where


 1/ 1
111 122
1
222 21
1.
1
i
K
i
Ak Nkk
k
Ak Nk

















 


From the definition of
j
y and /,
jj
yk

 we get
,
j
jj
wk  where


*0
1/ 1/.
jrjwj
From rhT  
j
jj
w  k and the marginal conditions in labor market
in (18), we get

1
1
1
iijjjjj
j
jj j
Ak KN
kkk



 







(19)
From
j
jj
NhTN in (14) and we
have 0,
jhj
TT T


0
/
j
jh
NNhTT
j
. From /,
hjjj wj
Tywh

 
/
jj
yk

 and ,
j
jj
wk   we get
hj j
j
wj
Ta h

 
 
We thus determined the time distribution. Inserting
/.
hjj wj
Th

 

in

0
/
jj hj
NNhTT

, we have
,
j
jj
NaN where

0
1// .
j
jwj
ahT   Insert
/
j
jj
lLaN
j

, /,
jj
ck

  and
j
j
s
k in the util-
ity function
0
00
00 1
1
hhhhh
hph
h
hp hp
hphp hh
vb
bv
pjj jjj
jj
vv
vv j
jj
jj
AL aa
UN
K
kk
kk
 











 
 
 

 



(20)
From (19), (20) and 1 UU
2

, we have 21N NN

,

W. B. ZHANG
175
where








000 0
000 00
1/1
2
12
1
/1
1
2
ˆ
,
,
hp hp
hphp hp
vv
N
vvv
k
kk k
k
k





 

















 

00
0
0
/1
11111222
11 222 2 21
0
0000
ˆ,
,
1
1/1
hp
hhhh
hhhh
v
vb
pi
vb
i p
hh
hp hhp
ALaa A
A ALaa
bv v


 
 





 











 


From 12
NNN,
j
jj
, and NaNN
21N
N

,
we solve
12
12 12
,N
N
N
N
N
NN
aa aa


 
(21)
We see that the labor distribution is uniquely deter-
mined as functions of 1 and From the definition
of k
2.k 
K
 and (21), we have
 

 1/ 1
111 22
12
1
222 1
1
,1
i
K
iN
Ak kk
kk k
Ak k

















 



From 21K
K
K and 12 ,
K
KK

 we have
12
,
11
K
K
K
K
K
KK


 (22)
We see that the capital distribution is uniquely deter-
mined as functions of and
1
From (11) and (1 6), we have
k
2.k 


*
1.
pj
j
pj
kj
k
K
Y
rk





 (23)
From the definition of we have
,
pj
Y

*
ij j
p
jjrj
iij j
Yar
k
 
 






wjjjj
kN



(24)
where we use /,
ijjiji ij
FwN

  ,
j
jj
wk  
,
j
jj
NaN and the equation for ij in (16). From
the marginal conditions for labor markets in (18) and
N
,
j
jj
wk   we obtain



*
1
kji
j
ij jj
rk
kNk
 
 




j
K
(25)
Substituting this equation and (23) and (25) into (24)
yields
1
**
ij jj
jjrjwj jj rjwj j
iij p
kar ar
 
 
 










(26)
The above equations determine the equilibrium values
of .
j
k
 The following lemma describes a procedure to
determine the equilibrium values of all the variables.
Lemma 1
For a given rate of interest in the global market, the na-
tional economy has a unique equilibrium point. The
equilibrium values of all the variables are given by the
following procedure:
j
k
 (26) →
j
N by (21) →
and ij
N
p
j
N by (16) →
j
jj
→ hj j NaNTa

 →
0hj j
TTT

 →
j
K
by the marginal conditions for
capital in (18) → 21
K
KK

 →
j
w by (18) →
ij
K
and
p
j
K
by (16) →
j
k by (25) →
p
j by (24)
→ Y
ij
F
by (17) →
p
j
F
by (8) →
j
y by (4) →
/
j
j
lL j N→
K
by (15) → BKK →
/
j
jj
ylR


→
j
c and
j
s
by (6) →
j
jj
NCc
and
j
jj
SsN →
j
U by (5).
Lemma 1 shows how to determine the values of all the
variables in equilibrium. As the expressions are compli-
cated, it is difficult to explicitly interpret the eq uilibrium
conditions. For illustration, we specify the parameter
values as follows
*00
12
12 12
12
11 12 22
0.05, 1,0.3,0.3,0.4,
0.05,0.04, 0.03,1.2,1,
1.1,1,0.1, 3,4,
0.09,0.3,0.05,0.05,0.09,
0.8, 10,3,4,
0.04,0.03.
ipp
cpeii
pp
hhhkh
h
irwirw
rh
AA
AAb
v
NLL
 


 

 






 

(27)
The rate of interest is fixed at 5 percent in interna-
tional market. The total population is 10 with human
capital level being unit. The produ ctivity para meter, 1i
A
,
is higher than region 2, the produ ctivity parameter of the
public sector in region 1 is higher than that in region 2.
We consider that region 1 is technologically more ad-
vanced than region 2. Region 1’ and 2’ amenity parame-
ters, 1

and 2,

are different, the value in region 1
being lower than region 2. Region 1’s land for housing is
less than Region 2’ land. The marginal propensity to
consume public goods is lower than the propensity
to consume the industrial goods,
,
h
v,
h

and th e propen sity
to consume the lot size, .
h

The tax rates on the output
level, the income from wealth and wage income are the
same within each region and the tax rates in region 1 are
higher than the tax rates region 2. The externality and
congestion parameters, e

and ,
c

are positive. It

176 W. B. ZHANG
should be remarked that although the specified values are
not based on empirical observations, the choice does not
seem to be unrealistic. For instance, some empirical
studies on the US economy demonstrate that the value of
the parameter, ,

in the Cobb-Douglas production is
approximately equal to 0.3 (for instance, [13]). With re-
gard to the technological parameters, what are important
in our interregional study are their relative values. The
presumed productivity differences between the regions
are not very large. This similarly holds for the specified
differences in the amenity parameters between the re-
gions.
Following the procedure in Lemma 1, we calculate the
equilibrium values of all the variables. We list the simu-
lation results as follo ws
12
4.64,14.79, 18.09,3.30,
2.07,1.55,
FK K B
kk
 

121
121
121
1.581.62 8.46
1.49 ,1.55 ,7.59,
0.100.07 0.86
ii i
pp p
NNK
NNK
NNK
  
 
  
 

  
 
 
  
 
  
1
1
22
2
21
1
22
2
4.93 2.64
6.34 5.07 2.01
5.88 ,,,
0.68 0.41
0.49 0.68 0.29
i
i
ip
h
pp
h
F
N
KF
N
KF
T
KF
T
 
 

 

 

 

 
 

 

 

 

 

  
 
 












111
222
111
222
0.531.19 0.23
0.400.88 0.17
,,
0.040.38 0.61
0.020.22 0.80
i
i
p
p
fwc
fwc
yRl
yRl
  
 
  
 
  
 

  
 
  
 

   

(28)
In (28), the variables, ij
f
and
p
j, are respectively
the output level per worker and th e exp end iture on pub lic
goods per resident in region defined as
y
j
/,
ijij ij
f
FN /,
p
jpjj
y
YN Less than half
of the national population but more than half of the total
capital are located in region 1. The per-capita levels of
wealth and consumption and wage rate in region 1 are
much higher than the corresponding variables in regions
2. The lot size of region 2 is larger than in region 1. The
consumption level per capita in region 1 is higher than in
region 2. We see that although workers can earn more
money in the advanced region than in the other region,
they have to pay much higher rent for housing than the
households in the other region. Expectably, the typical
household in the advanced region consumes more goods
and lives in a smaller house than the typical household in
region 2.
1,2 .j
4. Parameter Changes and Economic
Geography
It is important to ask questions such as how one region
may affect the national economy as its technology or
amenity is improved; or how the regional trade patterns
may be affected as the propensity or the total population
to save is increased. This section examines impact of
changes in some parameters on the national economy
and regional economic structures. As we have explicitly
provided the procedure to simulate the motion, it is
straightforward to make comparative analysis. First, we
examine effects of change in the total productivity of
region 2’s industrial sector, 2.
i
A
We increase 2i
A
from 1 to 1.1, keeping all the other parameter values as
specified in (27). The simulation results are given in
(29).
2:1 1.1,6.12,5.99,6.19, i
AFKK
12
7.05,0.67,16.29,Bk k 
12
12
12
7.29 7.10
7.29 ,7.10 ,
7.29 7.10
ii
pp
NN
NN
NN
 
 
 
 
 1
1
1
i
p
K
K
K

 
 
 
 
 
 
 
 
1
22
21
22
7.29
7.91 24.557.10
7.91 ,24.55,,
0
7.91 24.550
i
h
p
h
N
KN
KT
KT
  
 
 




 


 

 



 

 

 
 


11
22
11
22
7.91 0.67
24.55 16.29
,,
5.35 0.67
9.78 16.29
ii
ii
pp
pp
Ff
Ff
Fy
Fy
  


 


 


 



 

 


 






11
22
11
22
0.67 0.67
16.29 16.29
,
7.91 7.87
24.55 6.63
wc
wc
Rl
Rl

 

 
 
 

 
 

 
 


 
 
 
 
 
 


(29)
where

stands for the change rate of the variable in
percentage due to changes in parameter value. As region
2’s total productivity is increased, the national output,
wealth and capital are increased. The trade balance is
also improved. Hence, the national economy is improved
as a whole. However, there are interregional differences
in the impact due to the technological ch ange. The region
whose technology is improved attracts more households
to the region. Region 2’s rate is increased and the region
becomes more attractive. People immigrate to region 2
from region 1. The redistribution leads to fall in region

W. B. ZHANG
177
1’s land rent and rise a region 2’s land rent. The region
experiencing the technological change will increase its
wage rate, while the wage rate in the other region falls.
As the work time is not affected by the technological
change, the changes in the national labor supply are due
to regional reallocation of the households. As region 2
has more population, its lot size per household faces. We
thus see that as region 2 increases its technology, at a
new equilibrium its household will get more wage per
work hour and increase the consumption and wealth, but
the household will live in a smaller space.
In our dynamic system each region may conduct dif-
ferent fiscal policies. It is important to compare effects of
change in different taxes upon the economic system.
First, we increase the tax rate on region 1’s output from
0.04 to 0.05. The national output and wealth are in-
creased. The capital stock employed by the country is
increased. The trade balance is deteriorated. As region
1’s tax rate is increased, the expenditure on public good
is increased (given all the other conditions). The im-
proved infrastructure makes the region more attractive.
The region’s population is increased and produ ctivity per
work hour is increased. Region 1’s wage rate and con-
sumption are increased in association of rise in land rent;
region 2’s wage rate and consumption are reduced in
association of fall in land rent.
1
12
:0.040.05,0.23, 0.46, 0.12,
1.41, 0.10,0.05,
iFK K
Bkk
 

12
12
12
0.60 0.58
0.26 ,0.58,
13.89 0.55
ii
pp
NN
NN
NN
  

 
  
 

  
 
 
  

 
  
1
1
1
i
p
K
K
K


1
22
21
22
0.60
1.280.63 0.58
0.17,0.63,,
0
14.000.63 0
i
h
p
h
N
KN
KT
KT




 


 

 


 

 



 

 

 
 


11
22
11
22
0.88 0.28
0.63 0.05
,,
9.56 13.32
0.42 0.05
ii
ii
pp
pp
Ff
Ff
Fy
Fy
  
 
 
 
 
 
 

 
 

 
 
 
 
 
 






11
22
11
22
0.10 0.10
0.050.05
,
0.70 0.60
0.63 0.59
wc
wc
Rl
Rl
 
  
 
 


 
 

 
 


 
 
 
 
 
 






(30)
We now increase the tax rate on region 1’s wage from
0.04 to 0.05. Different from the increase in the tax rate
on output, the national output and the capital stock em-
ployed by the country are reduced. The trade balance is
improved and wealth is increased. As region 1’s tax rate
on the wage rate is increased, the expenditure on public
good is increased and work time is reduced (g iven all the
other conditions). The improved infrastructure makes the
region more attractive. Although the population of region
1 is increased, its labor force
The region’s population is increased and productivity
per work hour is increased. Region 1’s wage rate and
consumption are increased in association of rise in land
rent; region 2’s wage rate and consumption are reduced
in association of fall in land rent. As the tax rate on wage
is increased, the national supply of labor is reduced. Al-
though more peop le immigrate to more productive r egion,
the national output is still reduced. Because the output
and wage are reduced in region 2, the regional public
service and per capita expenditure on public good are
reduced.
1
12
: 0.040.05,0.75,0.16,
0.89,5.60,1.16,0.18,
wFK
KBkk
 

12
12
121
0.20 2.02
0.80 ,2.02 ,
9.07 2.02
ii
pp
NN
NN
NN
  
  
 
  
 1
1i
p
K
K
K

 
  
 
 
  

 
  

1
22
21
22
2.08
1.372.20 2.02
0.35,2.20,,
1.05
10.33 2.20 0
i
h
p
h
N
KN
KT
KT













 







 


 


11
22
11
22
0.35 1.69
2.20 0.18
,,
6.63 8.09
1,47 0.18
ii
ii
pp
pp
Ff
Ff
Fy
Fy
 


 
 
 


 
 

 
 

 
 
 


 
 






11
22
11
22
1.16 0.80
0.18 0.18
,
2.89 2.04
2.20 2.07
wc
wc
Rl
Rl
 
  
 
 


 
 

 
 
 
 
 
 
 
 
 


(31)
We now study the impact of ch anges in the cong estion
parameter, c

． We increase the parameter value from
0.05 to 0.06. The effects are listed in (32). The time dis-
tribution is not affected. As the congestion effect of pub-
lic goods becomes stronger, most of the economic vari-
ables are negatively affected. It should be noted that al-
though region 1’s population is increased, the region’s
total capital and output are reduced. The economic loss
caused by the strengthened congestion dominates the
economic gain brought about by the labor increase.

178 W. B. ZHANG
12
: 0.050.06,3.41,3.40,
3.41, 3.43,3.23,3.71,
cFK
Bk k
 
  
K
12
12
12
0.23 0.22
0.23 ,0.22 ,
0.23 0.22
ii
pp
NN
NN
NN
 

 
 
 

 
 
 
 

 
 
1
1
1
i
p
K
K
K



1
22
21
22
0.23
3.013.92 0.22
3.01 ,3.92,,
0
3.01 3.920
i
h
p
h
N
KN
KT
KT
 


 


 

 




 



 

 

 
 


11
22
11
22
3.01 3.23
3.92 3.71
,
0.823.3
1.28 3.71
ii
ii
pp
pp
Ff
Ff
Fy
Fy
   
 
 
 
 
 
 

 
 


 
 
 


 
 
,







11
22
11
22
3.23 3.23
3.71 3.71
,
3.01 0.23
3.92 0.22
wc
wc
Rl
Rl
  
  
 
 
 
 
 

 
 
 
 
 
 
 
 
 






(32)
We increase the externality parameter value, ,
e

from 0.03 to 0.04. The effects are listed in (33). As posi-
tive externalities become stronger, the industrial output,
total capital employed by the economy, and the total
wealth are increased. Region 1’s population is increased.
As region 2 loses labor and the time distribution is not
affected, the region’s output and capital stocks are re-
duced. Although region 1’s output is increased, as its
change rate is low, the net effect on the national indus-
trial good falls. Also the labor of each sector in region 1
is increased. The wage rate, consumption level and
wealth per capita in the two regions are all reduced. The
lot size in region 1 is reduced and the region’s housing
rent is increased.
12
: 0.030.04,0.52,0.51,
0.52, 0.57, 0.18,1.14,
e
F
K
Bk k
 
 
K
12
12
12
0.50 0.49
0.50 ,0.49 ,
0.50 0.49
ii
pp
NN
NN
NN
 


 

 
 


 


 
1
1
1
i
p
K
K
K



1
22
21
22
0.50
0.331.62 0.49
0.33 ,1.62 ,,
0
0.33 1.620
i
h
p
h
N
KN
KT
KT
 


 



 
 


 

 



 




 
 


11
22
11
22
0.33 0.18
1.62 1.14
,,
0.30 0.18
0.68 1.14
ii
ii
pp
pp
Ff
Ff
Fy
Fy
 
 
 
 
 
 
 
 

 
 


 
 
 


 
 
11
22
11
22
0.180.18
1.141.14
,
0.33 0.50
1.62 0.49
wc
wc
Rl
Rl
   
 
 
 
 
 
 

 
 
 
 
 
 


 
 






(33)
5. Conclusions
This paper proposed a two-region growth model with
capital accumulation, amenity and regional public goods
under assumptions of profit maximization, utility maxi-
mization, and perfect competition. We emphasize effects
of congestion and various fiscal policies on long-term
economic growth and economic geography. As the
model is structurally general, it is possible to deal with
various national as well as regional issues. Given the
contemporary literature on (spaceless) economic growth,
we may extend and generalize the model in different
ways. We may analyze behavior of the model with other
forms of production or utility functions. Other important
extensions in clude incorpor ating transportatio n costs and
endogenous population growth into the model. We may
also refine our model by dividing public expenditures on
capital into maintenance and new investment. As em-
pirically demonstrated in [14,15], maintenance expendi-
tures on public capital can account to 2 to 3 percent of
GDP in some economies like USA and Canada. As in
[16] in which the composition of public capital expendi-
tures under congestion is an important factor for optimal
fiscal policies, we may consider possible effects of pub-
lic maintenance expenditures on the depreciation of pub-
lic capital.
6. Acknowledgements
The author is grateful to an anonymous referee for the
constructive comments. Financial support under “the
Grant-in-Aid for Scientific Research (C), Project No.:
21530246, Japan Society for the Promotion of Science”
and “APU Academic Research Subsidy”, is gratefully
acknowledged. The usual disclaimer applies.
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179
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