Industrial Pollution and Economies of Agglomeration
Daisuke Ikazaki (Japan Women’s University)
Tohru Naito* (The University of Tokushima)
1. Introduction
This paper presents analysis of the effect of industrial pollution on economics agglomeration. The manufactured
goods sectors in many countries or regions often emit pollution of various kinds in their production. This
pollution lowers the utility level of households residing in the region where the manufactured goods sector emits
the pollution. Therefore, the environment for households is one factor under the decision of their residential
point. Our motivation is to clarify this relation with regional and environmental economics models.
Although Krugman (1991), Fujita, Krugman, and Venables (1999) and others have contributed importantly to
the “new economic geography", all such models are constructed using a nonlinear general equilibrium system.
Therefore, they derive an equilibrium solution that is achieved using numerical analyses with a computer under
appropriate given parameters. Ottaviano, Tabuchi, and Thisse (2002) explain derivation of the analytical
equilibrium solution using not a CES utility function, which is used in Dixit and Stiglitz (1978), but a
quasi-linear utility function.
We construct a model in which the manufactured goods sector emits pollution during production under a
framework with no CES utility function like that used by Krugman (1991); rather, this model uses a quasi-linear
utility function similar to that used by Ottaviano, Tabuchi, and Thisse (2002). Although Hosoe and Naito (2006)
consider that environmental damage affects the productivity of other sectors and the wage rate of households
indirectly, the environmental damage affects households’ utility directly in our model, which is similar to Stokey
(1998) and Ikazaki and Naito (2007).
This paper is organized along the following lines. First, we present a basic model of a two-sector – one-region
economy and describe the respective behaviors of the sectors. Section 3 analyzes the equilibrium in the short run
without labor mobility. Section 4 considers the population distribution between regions and its stability. Finally,
we conclude this paper and present some novel points. In addition, we suggest some remaining points as subjects
for future examination.
2. The model
We consider a two-region model with two sectors. Following the OTT model, we express these regions, which
are labeled respectively as H and F. We consider two kinds of goods: differentiated manufactured goods and
homogeneous agricultural goods. The labor in agricultural sector called A has no mobility between regions
because of binding force by land ownership. We assume that the agricultural goods are traded between regions
without transportation cost and the market of agricultural goods is perfectly competitive. We deal with these
goods as numeraire in our model. Therefore, we normalize the price of agricultural goods. Households in each
region have some agricultural goods as a positive initial endowment. Factor A is evenly distributed across
regions. It is assumed that the total supply of this factor is A and that each region has A/2.
On the other hand, we assume that the market of manufactured goods is monopolistic competition and that the
movement of these goods between regions necessitates a transportation cost. The transportation of a unit of those
goods between regions requires  unit of numeraire goods.
We assume that the input factor to produce manufactured goods is only labor, as in the OTT model and that L
denotes the number of laborers employed in the manufactured goods sector. Although the labor A employed in
the agricultural goods sector is immobile between regions, the labor employed in the manufactured goods sector
can move between regions. From this assumption, the labor employed in the manufactured goods sector moves
to the region, in which they can enjoy higher utility in the long run. Here, let   [01] represent the share of
region H of the total manufacturing labor. Therefore,  L and (1   )L respectively denote the numbers of
manufacturing laborers in region H and region F.
1. Household
We assume that all households are homogeneous and have a common preference in this model. We establish
the utility function of households in region l (l  H  F ) as follows.
N
Ul    q(i)di 
 
2
0
Therein, q (i ) and

N
0
 N
[q(i)]2 di    q(i)di   q0   3 Dl2

2  0
2
(1)
q0 respectively represent the consumption of variety i(i  (0 N )) and the endowment
of agricultural goods. Moreover, the parameters in (1) are satisfied with   0     0 . Here,  ,    ,
N , Dl ,  i (i  2 3) respectively signify the degree of preference for differentiated goods, the preference for
higher variety, the range of variety, the environmental pollution caused in region l (l  H  F ) , and the effects of
the damage on utility. Manufacturing production causes the environmental pollution and households cannot
control its level. Each household has a unit of labor and q 0 units of agricultural goods. When we define the
price of variety i of manufactured goods as p (i ) , the budget constraint of households is given as

N
0
p(i)q(i)di  q0  y  q 0
(2)
where y is the individual’s labor income. Therefore, we derive the demand function of manufactured goods
varieties i  (0 N ) as
N
q(i)  a  bp(i)  c  [ p( j )  p(i)]dj for i [0 N ]
0
a b and c respectively
c    (    ])[   ( N  1) ] .
where
signify
a    [   ( N  1) ] ,
b  1  [   ( N  1) ] ,
(3)
and
2. Production
1. Agricultural goods sector
Agricultural goods are produced using technology with constant returns to scale. Without losing generality,
we assume that the production of one unit of agricultural goods requires one unit of labor. The price of
agricultural goods is one because we defined the agricultural goods as numeraire. Therefore, if we describe
wHA  wFA as the wage rate of farmers in each region, respectively, wHA and wFA are equal to one in equilibrium.
2. Manufactured goods sector
We assumed that the market of agricultural goods is perfectly competitive and that the sole input of
agricultural goods production is labor. On the other hand, we assume that all manufactured goods are mutually
differentiated and that the market is one of monopolistic competition. Technology in manufacturing requires 
units of L to produce any amount of a variety, i.e. the marginal cost of production of a variety is set equal to zero.
This simplification is often performed in industrial organization theory. Therefore, economies of scale pertain
and  is an indicator to represent the degree of increasing returns to scale.
We refer to the number of manufacturing goods varieties in regions H and F. Let
nH and nF respectively
represent the number of manufactured goods varieties in regions H and F. Total manufacturing labor is given as
L. Therefore, the quantities of manufacturing laborers in regions H and F are given respectively as  L and
(1   )L . Considering that the only production factor of manufactured goods sector is labor, the following
quantities of manufacturing goods varieties in regions H and F,
nH , and nF are derived using labor market
clearing conditions.
(4)
nH   L   , nF  (1   ) L   ,
The number of varieties increases as  decreases or the number of manufacturing labor increases, from (4).
Next, we refer to the wage rate of manufacturing labor. Presuming that each variety is free to enter and leave the
monopolistic competitive market, each firm faces zero profit in equilibrium, as also posited by Dixit and Stiglitz
(1978). Considering the fact that the only input factor of the manufactured goods sector is labor, no firm receives
excess profit because of the zero-profit condition; its revenue reverts to the wage paid to manufacturing laborers.
Regarding the price of each variety, we assume that firms respectively carry out price discrimination for goods
sold in regions H and F. When we assume that the production technology of manufactured goods is symmetric
and that qHH and qHF denote the demands of variety i in regions H and F, respectively, then qHH and
qHF are as follows.
qHH (i)  a  (b  cN ) pHH (i)  cPH , qHF (i)  a  (b  cN ) pHF (i)  cPF 
In those equations, Pl (l  H  F ) is the price index of manufactured goods in region
PH  
nH
0
pHH (i)di  
nF
0
pFH (i)di, PF  
nH
0
pHF (i)di  
nF
0
(5)
l:
pFF (i)di
(6)
Here, we can describe the profit function of the firm to produce variety i in region H as follows.
 H  pHH qHH ( pHH )( A  2   L)  ( pHF  )  A  2  (1   ) L   wH 
(7)
Therein, wH denotes the wage rate of manufacturing laborers in region H.
3. Environment
In our model, the manufactured goods sector emits pollution, thereby bringing damage to households’ utility
in the production process. We assume that the emission function is convex with respect to the production variety.
Let Dl represent the pollution caused in region l (l  H  F ) ; it is given as
1
1
nH
nF
DH    {qHH (i)  qHF (i)}di  , DF    {qFH (i)  qFF (i)}di  
 0

 0

(8)
where the productions of variety in (8) are under the equilibrium price of the variety. Although the pollution
caused by the manufactured goods sector affects households’ utility as an external diseconomy, no firm
recognizes its pollution. Here, we assume that the parameter 1 is greater than one. Pollution increases
drastically as the production activities increase ( Dl (l  H  F ) is a convex function with respect to the total
output of manufactured goods in region l).
3. Equilibrium in the short run
We postulated three sectors in the previous section: household, production, and environment. In this section,
we analyze the short run equilibrium without migration between regions. Therefore, we take the distribution of
manufacturing laborers between regions,  , as given. Each variety producer can set her own price for each


region because each region is separated spatially. Therefore, let p HH
and pHF
represent the prices to



maximize (11) with respect to pHH and pHF . We can describe p HH
and pHF
as functions pHH
( PH ) and

pHF
( PF ) of PH and PF , which are the price index of manufactured goods in each region. The behavior of no
variety affects the price index of manufactured goods in each region because of the monopolistic competition
market. Assuming that the production of each variety is symmetric, we define the price index of the
manufactured goods sector in regions H and F as follows.




(9)
PH  nH pHH
( PH )  nF pFH
( PF ), PF  nH pFH
( PH )  nF pFF
( PF )
We analyze the equilibrium in the short run. Therefore, we can derive the price of a variety that is produced in
region l ( H  F ) and sold in region m(  H  F ) by dealing with (4) as given.

pHH

1 2a   c(1   ) N 
1 2a   c N 
 



, pFF 
, pHF  pFF
 , pFH
 pHH

2
2b  cN
2 2b  cN
2
2
(10)
Note that each price is the same for all intermediate goods i because the cost of production is not different

for all goods and each good enters symmetrically into the utility function. We understand that p HH
is a
decreasing function of  , meaning the share of manufacturing labor in region H. On the other hand, the
equilibrium price of the variety that is produced and sold in region F is an increasing function of  . The
equilibrium price also depends on the transportation cost of manufactured goods between regions. The prices
charged by both local and foreign firms fall when the mass of local firms increases, but the impact is weaker
when  is smaller. Moreover, the price of the variety with transportation cost, which is produced in region


and pFH
. We know that the
l ( H  F ) and sold in region m( H  F ) l  m , respectively, is given as pHF
difference of the selling price between the two regions is an increasing function of transportation cost.
Here,  HH and  HF represent the profit from the selling of the variety produced in region H and sold in
region Hand F, respectively.
A

A



(11)
 HH  (b  cN )( pHH
) 2    L  ,  HF  (b  cN )( pHF
  ) 2   (1   ) L  
2

2

The surplus coming from consumption in region H, S H ( ) , associated with the equilibrium prices is the
following.
a2 L aL  
cL2  
  (b  cL) L 

2
 2
 2
  pHH  (1   ) pFH

(
p
)

(1


)(
p
)

 pHH  (1   ) pFH




HH
FH 
2
2 



2b 
2
2
Moreover, taking account of symmetry, we can also derive S F ( ) as follows.
SH ( ) 
SF ( ) 
a2 L aL 
cL2 

  (b  cL) L 
 2
 2

 2
 (1   ) pHF
  pFF
(1


)(
p
)


(
p
)

(1   ) pHF
  pFF




HF
FF 
2
2 



2b 
2
2
(12)
(13)
We assume that the firms producing a variety are symmetric. Therefore, we define the total production of
manufactured goods in region H as follows.
QH  nH  qHH  qHF  
L
(b  cN )( pHH  pHF   )

From (10),
pHH  pHF   
is satisfied. Therefore, we can describe
(14)
2a  b
2b  cN
QH as follows.
L
 2a  b 
(b  cN ) 


 2b  cN 
Similar to region H, the total production in region F , QF is the following.
QH 
QF  nF (qFF  qFH ) 
(1   ) L

 2a  b 
(b  cN ) 

 2b  cN 
(15)
(16)
Next, we address the wage rate of manufacturing laborers. The zero profit condition is satisfied in equilibrium
because of the assumption that the manufactured goods market is one of monopolistic competition. Taking into
account that labor is the only input factor of manufactured goods, the equilibrium wage rates in each region are
derived. Consequently, we can describe the equilibrium wage rate in region H as the following function of  .
wH ( ) 
b  cL
2 A
2A


{ 2a   cL(1   )    L    2a  2 b   cL(1   )    (1   ) L }
2 2 
4(2b  cL) 
2

2

(17)

Similar to region H , wF is derived as follows.
wF ( ) 
b  cL
A

A

{[2a   cL ]2   (1   ) L   [2a  2 b   cL ]2    L }
2 2
4(2b  cL) 
2
2




(18)
4. Equilibrium in the long run
1. Equilibrium condition and damage caused by pollution
In the previous section, we determined the endogenous variables in the short run when we dealt with the
distribution of manufacturing labor between regions as given. Although we assume that the manufacturing labor
in each region cannot migrate among regions in the short run, we relax this assumption in this section. Therefore,
it is possible for manufacturing laborers to migrate to the region, which provides higher utility after comparing
utility levels in respective regions.  includes information related to how the household estimates environmental
pollution. Consequently, the value of  plays an important role in the decision of household’s resident location
in the long run. The indirect utility function in region H is as shown below when we deal with the distribution of
manufacturing laborers between regions as given.
(19)
VH ( )  S H ( )  wH ( )   3 ( DH ) 2  q 0
From (1) and (8), the damage to utility that is imparted by pollution in region H is
2
1
  L
 2a  b   

 3 ( DH )   3 
(b  cN ) 
    C2 
 2b  cN   
 
and  as the following.
(20)
2
where we define C2

L
 2a  b  
C2   3  (b  cN ) 
     1 2  1

 2b  cN  

Therefore, we can describe (19) as follows.
(21)
VH ( )  S H ( )  wH ( )  C2   q 0
(22)
From (1) and (8), the damage to utility that is imparted by pollution in region F is
 3 ( DF )
2
 (1   ) L
 2a  b
  3 
(b  cN ) 

 2b  cN




1  2



 C2 (1   )
(23)
We can describe the indirect utility function in region F as the following.
VF ( )  S F ( )  wF ( )  C2 (1   )  q 0
(24)
We next consider the distribution of manufacturing laborers between regions in the long run. The distribution
 is a spatial equilibrium when no manufacturing laborers may get a higher utility level by changing location.
Therefore,  is a partial equilibrium at   (01) when
(25)
V ( )  VH ( )  VF ( )  0
or at   0 when V (0)  0 , or at   1 when V (1)  0 . Next, we analyze the stability of a spatial
equilibrium. Presuming that V ( ) is positive within   (01) , ddt  0 (the notation t is time), i.e.,
manufacturing laborer will move to region H. On the other hand, presuming that V ( ) is negative within
  (01) , ddt  0 , i.e., manufacturing laborers will move to region F. More precisely, it is assumed
V ( )
if 0    1


    dt   min{0 V ( )}
if   1
max{0 V ( )} if   0

When there is

to satisfy V ( )  0 within
  (01) , laborers do not move between regions, i.e.,
 . Substituting (22) and (24) into (25), V ( )
equilibrium distribution is given by
(26)
is
 1
V ( )  S H ( )  S F ( )  wH ( )  wF ( )  C2   C2 (1   )  C (    )      C2     (1   )   (27)
 2
where
C and   are defined as follows.
C  2b (3b  3cL  cA)  c 2 L( A  L) 
L(b  cL)
4a (3b  2cL)
(28)
 0,  
2
2
2 (2b  cL)
2b (3b  3cL  cA)  c 2 L( A  L)
Here, we define g ( ) and f ( ) as follows.
1

g ( )  C (    )      f ( )     (1   ) 
2

Then we can rewrite (27) as follows.
(29)
V ( )  g ( )  C2 f ( )
Next we consider the shape of g ( ) and f ( ) in (29), respectively. We consider the homogeneous
regions. Therefore, we discuss

from
1
2
to one. Because (28) is positive, the first term of the right-hand side
in (29) is an increasing function with respect to  when  is larger than  . The slope of g ( ) depends
on transportation cost. We also consider the case in which the transportation cost is high. Actually, g ( ) is a

decreasing function with respect to  when
Differentiating f ( ) with respect to
 
. Moreover, we consider the shape of
f ( ) .
,
f ( )    1   (1   ) 1
(30)
f ( )   (  1)  2   (  1)(1   ) 2
(31)
Although g ( ) is the linear function with respect to  , the shape of f ( ) depends on the shape of
emission function of pollution with manufactured goods production ( 1 ), and the effect of pollution on
households (  2 ). First, we consider the case in which  is greater than two: f ( )  0 f" ( )  0 when
  2 is satisfied. We also consider the case in which 1    2 is 1–2. In this case, f ( ) is a concave
function with respect to  : f ( )  0 f" ( )  0 . Therefore, we can describe f ( ) . VH ( )  VF ( ) and
  0 when   1/ 2 . Therefore,   1  2 is always equilibrium. The distribution by which the
manufacturing laborers are distributed over both regions equally is the equilibrium in the long run. For that
reason, we designate the distribution by which the manufacturing laborers are distributed over both regions
equally as the dispersion equilibrium. We can know that the equilibrium in the long run depends on the value of
 and  . Thus, we know that it is not necessary to appear core periphery equilibrium when the transportation
cost is low relatively.
5. Concluding remarks
As described in this paper, we extend the core-periphery model by introducing the environmental damage
caused by the manufactured goods sector into Ottaviano, Tabuchi, and Thisse (2002), which uses the quasi-linear
utility function. Then, we analyze its properties.
The main results achieved through our analyses are the following. First, the traditional core-periphery model
determines the agglomeration or dispersion between regions by depending on the preference for variegation and
the transportation cost. Although the transportation cost engenders regional agglomeration in the traditional
core-periphery model, it is not necessary to have regional agglomeration under the low transportation cost by
introducing the environmental factors into the core-periphery model. On the other hand, results of these analyses
show that the manufacturing laborers disperse throughout both regions. This is identical to the result derived
using the traditional core-periphery model. However, we know that the stability of each equilibrium depends on
the relation between the agglomeration effect (forward and backward linkages) and the diseconomy by the
environmental damage when the pollution caused by manufactured goods production affects households’ utility
in our model. Particularly, when environmental pollution is an increasing function with respect to the production
of manufactured goods, equilibria of three kinds might appear as stable equilibria: the agglomeration equilibrium,
the dispersion equilibrium, and the imperfect agglomeration equilibrium.
We must address not only environmental problems themselves, but also the economic system in addition to
the environment simultaneously when we consider appropriate environmental policies. Because we take account
of spatial agglomeration, considered as one factor bringing environmental damage, in the model of this paper,
the results derived in this paper will contribute greatly to more appropriate environmental policy-making.
However, we presumed some factors to simplify the analysis in our model. For example, we do not consider
inequities such as production technology between firms. Although there is inequity of productivity between
regions in China, we do not describe this in our model. It is important for us to consider it in our model. It is
possible to extend our model in the future.
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