JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000, pp. 755–769
MULTI-FIRM CITY VERSUS COMPANY TOWN: A MICRO
FOUNDATION MODEL OF LOCALIZATION ECONOMIES*
Hesham M. Abdel-Rahman
Department of Economics and Finance, University of New Orleans, New Orleans, LA
70148, U.S.A. E-mail: [email protected]
ABSTRACT. When do we have a company town and when do we have a multi-firm city?
In this paper I analyze the impact of public infrastructure investment decisions on types
of cities in a decentralized urban system. This is done in a one-sector spatial general
equilibrium model of a closed economy. Investment in public infrastructures reduces the
fixed set up cost of all firms within the city resulting in multi-firm cities. Thus, in this
approach localization economies are modelled explicitly instead of assuming that larger
industrial size within the city enhances productivity. On the other hand, when the
infrastructure is not provided, a company town will be formed by a developer because of
the fixed cost required by each firm. The decision of whether to invest in the provision of
public infrastructures depends on the type of city that will provide households with the
highest utility. This paper characterizes the conditions that lead to each of the two
equilibrium configurations.
1.
INTRODUCTION
Public investment in infrastructures has been associated with higher urban
productivity.1 Furthermore, cities and states compete among themselves to
attract private investment to improve the welfare of their residents by increasing their provision of public infrastructures. This is particularly the case in most
of the developed countries. For example, the Clinton Administration has asserted
that the reason for slow growth at the national level is low investment in public
infrastructure. On the other hand, to my knowledge, the impact of public
infrastructure on urban productivity and city formation has not been modeled
explicitly in the literature despite the fact that infrastructure is a major cause
of localization economies.
In this paper I propose a spatial general equilibrium model of a system of
cities in a closed economy to explain the impact of investment in public infrastructure (water, electrical, sewer systems, and training facilities) on urban
*This research was supported by the University of New Orleans Summer Research Grant,
which is gratefully acknowledged. The comments of Alex Anas, Marcus Berliant, Masakisa Fujita,
Gerald Whitney, referees, participants of the RSAI Forty-Second North American Meetings in
Cincinnati, Ohio, and participants of the seminar at Kyoto University, Japan are appreciated.
Received November 1998; revised November 1999; accepted February 2000.
1
See Holtz-Eakin and Lovely (1996) and Moomaw, Mullen, and Williams (1995) among others.
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JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000
productivity, city formation, and urban industrial structure.2 Such an examination enables an understanding of the causes of localization economies instead
of relying on a vague concept that associates higher productivity with the size
of employment of a given industry in a given city. This approach takes into
consideration one main impact of infrastructure investment: namely, a reduction
in fixed cost required for production (set up cost) that would lead to improvement
of labor productivity. Empirical evidence suggests that improving some types of
infrastructure, such as water and sewer systems, has a significant impact on
productivity.3 In this paper I focus attention on the impact of infrastructure on
the firm, that is, urban productivity rather than household behavior.4
In modeling a system of cities, I develop a one-sector, spatial general
equilibrium model of a closed economy consisting of a homogeneous labor force.
I assume that the output markets in this economy are perfectly competitive. I
establish conditions under which the model generates multi-firm cities in which
the number of firms within each city is well defined. It is worth mentioning that
all city size models with localization economies, in which the output market is
perfectly competitive, result in an undefined number of firms within the city.5
Furthermore, the model provides an explanation for the conditions under which
there results a company town or multi-firm city. Indeed both types of cities
coexist in the United States, especially in the South, as well as in other countries.
Furthermore, it has been observed that the formation of multi-firm cities is a
result of incentives by local and state governments to attract firms to small
towns through the development of industrial parks or labor-training facilities.
Examples are the locations of the Saturn plant in Spring Hill, Tennessee and
Mercedes-Benz in Vance, Alabama.
In modeling city formation we typically consider two types of forces: forces
that lead to a concentration of population and economic activities, and forces
that lead to deconcentration (Mills, 1967). The equilibrium city size is determined when these forces are balanced at the margin. In the case of a multi-firm
city, the force that leads to concentration of firms in the city is the reduction in
fixed cost by each firm because of investment in public infrastructure. Furthermore, the average cost of infrastructure decreases with city size, another force
for agglomeration behavior. On the other hand, in the case of a company town
the city is formed because of indivisibility in production, that is, increasing
2
For a recent review of general equilibrium models of city formation see Quigley (1998) and
Abdel-Rahman (2000).
3
See Moomaw, Mullen, and Williams (1995) for empirical evidence of the effects of transportation, water, and sewer systems on urban productivity for the United States.
4
For the effect of transportation improvement on household behavior and city size see
Abdel-Rahman (1998).
5
See Henderson (1989) among others. Note that this is not the case in microeconomic theory.
For micro-foundation models of localization as a result of matching in the labor market see Helsley
and Strange (1990) and Abdel-Rahman and Wang (1995) and as a result of variety of intermediate
services see Abdel-Rahman and Fujita (1990).
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returns to scale resulting from fixed cost. The only deconcentration force in this
model is the high commuting cost resulting from the physical expansion of the
city.
The organization of this paper is as follows: In Section 2 I present the model.
In Section 3 I characterize the equilibrium configurations of a system of
company towns and a system of multi-firm cities. In Section 4 I present the
parameter range of each equilibrium configuration, and in Section 5 I present
my conclusions.
2.
THE MODEL
Consider a closed economy consisting of a system of cities spreading over a
flat, featureless plane. The total population of the economy is given by N. One
good x is produced in the economy. Good x can be used for consumption,
commuting, or for the production of infrastructure. Households in the model
economy are assumed to have identical preferences. Each household or worker
is endowed with one unit of labor which is supplied inelastically. The individuals
and households are perfectly mobile and free to reside in the city that maximizes
utility. For simplicity of calculation I postulate that each household consumes
one unit of land, that is, demand for land is perfectly inelastic. Cities are formed
by developers who maximize the surplus or profit from the development of the
city. Each developer creates one city in the economy where the number of cities
is large. Thus, each developer is assumed to be a local monopsony in the labor
market where the utility of each household is taken as given. The formation of
cities requires either public infrastructure for the multi-firm city or a fixed set
up cost for the company town. This model generates two possible equilibrium
configurations: (1) company towns, in which each city in the economy has only
one firm; (2) multi-firm cities, in which all cities in the economy have more than
one firm. Thus, this paper provides a framework in which sizes, types, and
number of cities in an economy can be explained.6
If public infrastructure is not provided only one firm will exist in each city
because the entry of a second firm to the city will reduce the utility of city
residents. This is because entry of the second firm will increase the city size and
consequently increase commuting costs without any benefit to city residents. In
the context of the model, the equilibrium configuration that emerges is the one
that provides a city’s residents with the highest utility level. The equilibrium
configuration that provides low utility is not sustainable in the economy because
developers will be able to attract households to cities that provide higher utility
and still be able to make a profit.
6
See Henderson (1974) for an initial model of a system of cities in which cities specialize in
the production of a single traded good.
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Household Behavior and City Formation
Each household in a given city can reside at only one location and can have
only a single job that requires commuting to the central business district (CBD)
in which all firms locate. For simplicity, I postulate that each household consumes one unit of land. In addition, all households in the economy are assumed
to have identical utility functions of the following form
(1)
u = Cx
where C is a positive scaling constant and x is the quantity of the good consumed
by a household. The budget constraint facing a household residing at distance r
from the CBD in a given city is
(2)
x(r) + tr + R(r) = W
where the price of good x is normalized to be one, t represents the amount of
good x used to commute a unit distance, R(r) represents the unit land rent at
distance r from a CBD of a given city, and W denotes the wage rate (in units of x).
Note that, in the context of the model the only type of commuting cost is monetary
cost in terms of good x.7
For simplicity, I assume that land in the city is owned by absentee landlords
who live outside the system. The developer of the city rents the land from
absentee landlords at the agricultural rent. For ease of calculation I assume that
the opportunity cost of land is zero because the agricultural sector is not
considered. The developer sublets the land to households at the market price
and provides the infrastructure required for the development of the city. The
total cost of public infrastructure is given by
(3)
TC(G) = Gβ
where G is the level of the infrastructure or the size of the facility and β > 1 is
the elasticity of the cost with respect to G.8 Thus, the average cost share of the
infrastructure decreases as city size increases.9 This benefit is one of the reasons
for agglomeration and thus the provision of infrastructure in large cities. On the
other hand, if it does not pay for the developer to provide infrastructure (the
utility level that can be maintained by households is negative or relatively low
compared to that of a company town), then a company town will result with the
developer providing the fixed set up cost for the city. In this case only one firm
will enter the city because entrance of other firms will increase the commuting
7
The introduction of time cost will not change the result of the model but makes the analysis
more cumbersome.
8
As will be seen in the Appendix this is required by the second-order condition on (9).
9
With the infrastructure cost function specified above, the model, in effect, exhibits increasing
returns in the context of city expansion. This assumption has been imposed in the literature, see
among others Helsley and Strange (1994). Nevertheless, the presence of transportation costs will
balance out this cost-reducing benefit at the margin.
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ABDEL-RAHMAN: A MODEL OF LOCALIZATION ECONOMIES
759
cost without any scale benefit. In this context, the set up cost may be thought of
as infrastructure privately provided by the developer of a company town.
Because each household consumes exactly one unit of land the total population of a representative local city is
b
z
n = 2πrdr = πb2
(4)
0
where b represents the distance to the urban fringe for each city. If N is the total
population of the economy then the number of cities in the system is m = N/n
because each equilibrium configuration will result in a system of identical cities.
Locational equilibrium requires that all workers in a given local city achieve
the same utility level. Hence, from Equations (1) and (2), at each r ≤ b it must
hold that tr + R(r) = tb + R(b). Recall that we normalize the opportunity cost of
land to be zero so that R(b) = 0. Thus, the land rent schedule under locational
equilibrium becomes R(r) = t(b – r). Using Equation (4) and integrating the above
land rent schedule the aggregate land rent ALR in each city is
b
ALR =
(5)
z bg
R r 2πrdr = µn 3 2
0
where µ ≡ t 3π 1 2 . The aggregate commuting cost ACC is given as
e
j
b
(6)
zb
g
ACC = t 2πr rdr = 2µn 3 2
0
Recall that the only commuting cost is a monetary cost in terms of good x. Given
this assumption, the per capita consumption cost c(n, U) necessary to attract n
households to the city such that each household achieves a given utility level U
can be derived from Equations (1) to (6) as
(7)
c n, U = UC −1 + 2µn 1 2
b g
Observe that the above function is increasing in n and U. This makes sense
intuitively because a larger city size implies higher commuting costs and thus
a higher consumption cost is required to maintain a given utility level. Furthermore, higher utility can only be achieved through larger consumption of x which
requires higher cost to the developer.
Production Sectors
The production of X requires a variable labor input X(L) and a fixed input
(set up cost) F, in terms of X given by
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JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000
X L = a −1 L
bg e
(8)
j
12
b g b
g
F G = f θG + 1
−1
where f, a, and θ are positive parameters and L is the amount of labor required
to produce X. The parameter θ is related to the elasticity of the function F with
respect to G. This parameter affects the rate at which an increase in infrastructure reduces the fixed set up cost of each firm in the city. One can interpret θ as
representing the utilization of infrastructure. The parameter f is the fixed set
up cost if G = 0. Thus, the firm technology can be represented by the cost function
g(X, G) = WaX2 + F(G), where W is the wage rate. From Equation (8) it can be
seen that F is a decreasing function of the level of infrastructure G. For example,
services provided by the developer of an industrial park do not have to be
provided by firms. Also, training programs provided by the developer reduce
on-the-job training by firms, as in the case of the location of the Mercedes-Benz
plant in Alabama.
3.
EQUILIBRIUM CONFIGURATIONS
Next, I define equilibrium. In this model cities are formed by a surplusmaximizing developer.10 Each developer can form only one city for which land is
rented at the opportunity cost (assumed to be zero). The developer then sublets
the land to households at the market rent. Developers maximize the surplus or
profit from development of a city. Because it is assumed that the number of cities
in the economy is large, each city to be formed is small compared to the national
economy. Furthermore households are free to choose the city in which to reside.
Thus, each developer is a local monopsony in the labor market and so will behave
as a utility taker. In addition, competition among developers will result in zero
surplus at equilibrium. At equilibrium all households must achieve a common
utility level. In the context of this model, developers form company towns or
multi-firm cities. Given this equilibrium concept, the resulting city system will
depend on which type provides households with the highest equilibrium utility.
This condition will be discussed further in Section 4.
Multi-firm City
First, the surplus of a developer of a multi-firm city is defined as the
difference between the total output net of fixed set up costs less the cost of
maintaining a national utility level U and the cost of provision of public
infrastructure. Assuming full employment in the city, then n = ML, where M is
the number of firms in a given city. Thus, from Equations (3), (4), and (8) the
surplus is given by
IJ
b g FGH nM
a K
S G, n =
(9)
10
12
−
LM Mf OP − nC
N bθG + 1g Q
U − 2µn 3 2 − G β
−1
See Fujita (1989), chapter 5 for this equilibrium concept.
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To make this problem consistent with the formation of a company town (as stated
later) the problem is solved in two steps. In the first step the developer chooses
the level of provision of infrastructure and the number of firms in the city. Then,
in the second step, city size is chosen. From the first-order conditions of the above
problem we have
∂S
Mf θ
β−1
=
− βG b g = 0
2
∂G
θG + 1
(10)
b g
∂S 1 F n I
f
= G
−
J
H
K
∂M 2 aM
bθG + 1g = 0
12
(11)
Equation (10) is the Samuelson condition for the optimal provision of a public
good. This condition requires that public infrastructure be provided up to the
point where the marginal benefit, that is, the reduction in the fixed set up cost,
equals the marginal cost of the public infrastructure. Equation (11) requires that
the marginal contribution of a firm to the city be equal to its marginal cost.11
Substituting Equations (10) and (11) into Equation (9) we have
F θ IJ
Sbn, U g = bβ − 1gG
H 4βaf K
(12)
β
β
β−1
n
β−1
− 2µn 3 2 +
n
nU
−
4 af
C
The above surplus function is a function of city size only. Given the following
assumption
ASSUMPTION 1: β > 3.12
the surplus function is characterized by the following Lemma.
LEMMA 1: Given Assumption 1, the surplus function S(n,U)13
(i)
is strictly concave in n,
(ii) is decreasing in a, f, t, and U,
(iii) is increasing in θ.
Observe that since the surplus is increasing in n up to a point which indicates
that the agglomeration economy dominates the diseconomy of city size. This
results in a unique optimal city size. This is an indication of the importance of
11
See the Appendix for the second-order condition.
This assumption implies that the cost function for infrastructure is sufficiently convex.
13
The second-order condition is given by
12
FG
H
β
θn
β − 1 4 afβ
IJ
K
β
β−1
−
3 32
µn
2
For the second-order condition to hold even in the case of zero surplus β has to be greater than three.
This is why Assumption 1 is imposed.
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JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000
the agglomeration force presented in this model, that is, the reduction in fixed
cost due to the provision of infrastructure, in generating cities. In other words,
this result indicates that infrastructure is a micro foundation for the existence
of localization economies. In the second step, developers choose the city size that
maximizes the surplus. The first-order condition for problem (12) is given by
1
(13)
FG
H
∂S
θ
= n β−1 β
∂n
4 afβ
IJ
K
β
β−1
b g
+ 4 af
−1
− 3µn 1 2 − UC −1 = 0
Equation (13) requires that city size be chosen such that the marginal benefit
of a household, given by the first term in Equation (13), is equal to the marginal
cost of a household given by the second and third terms in (13).14 Developers
form cities until the surplus is driven to zero. From this we have
µn3/2 = Gβ
(14)
which requires that the infrastructure be fully financed by aggregate land rent.
Thus Equation (14) is the Henry George Theorem. The equilibrium in this model
is an optimal solution. From Equations (10), (11), and (14) the equilibrium
provision of infrastructures may be derived as
(15)
G
*
LF 3π I F θ I
= MG
MNH t JK GH 4aβf JK
12
2
3
OP
PQ
b g
1 β−3
The equilibrium size of city can be obtained by substituting (15) into (10) and
(11) as
(16)
n
*
LF 3π I b g F θ I
= MG
MNH t JK GH 4afβ JK
12
β−1
β
OP
PQ
b g
2 β −3
It can be seen that for a given set of parameters {a , f, θ, t, N} the equilibrium
number of cities in the economy is given by m* = N/n*.
Observe that the solution to the problem (9) can be achieved by a decentralization mechanism in which the developer chooses the level of provision of
infrastructure and city size while competition in the output market determines
the wage rate and the number of firms. In this case production efficiency requires
that a representative firm in the industry employs labor in a given city at the
wage rate
14
The above problem has the same solution as the maximization of the average surplus.
However, in this case the Henry George Theorem results from the first-order condition not from the
zero-surplus condition as in the above problem. See the Appendix for the problem of maximization
of the average surplus.
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763
W = (2aX)–1
(17)
With unrestricted entry each firm in the industry will achieve zero profit in
equilibrium. Thus, from Equation (17) the equilibrium output by each firm in a
given city can be derived as
(18)
X*(G) = 2F(G)
The wage in a given city can be derived by substituting (18) and (8) into (17) as:
W(G) = (4af)–1(θG + 1)
(19)
Observe that the wage is increasing in G. In other words, this equation represents a micro foundation for a localization economy. Higher productivity is
associated with a larger provision of infrastructure, a result supported by
empirical evidence. Furthermore, this result explains the formation of cities as
a result of provision of infrastructure. Now the developer’s problem is to
maximize S(G, n).15 The surplus is defined as the total wage net of total
consumption costs and infrastructure costs. This problem is equivalent to the
ALR net of the cost of infrastructures given as16
(20)
−1
b g b g nbθG + 1g − 2µn
S G, n = 4 af
32
− nUC −1 − G β
Now I focus attention on the determination of the number of firms. Substituting (15) and (16) into (11) I derive the equilibrium number of firms M* in a
given multi-firm city as
(21)
bg
M* = µ
b g −3 −2bbβ2−β3−g3g bβ2−β3g L bββ−3g bβ 2−3g
MMθ + µ b4afβg bβ−33g OPP
β
b g b 4 ag f
−2 β − 1
β −3
N
2
Q
15
Another city-formation mechanism consistent with the one presented will occur, for example,
if a city government rents the land from an absentee landlord at the agriculture rent and then sublets
the land to households at the market rent. In this case the local government will supply the
infrastructure required for city formation. Thus the problem of the local government will be to
maximize the representative household utility subject to the resource constraint, ALR ≥ TC(G). Note
that in this case the constraint will be satisfied with equality. Therefore, if the constraint is
substituted into the objective function we obtain
−1
b g bθG + 1g − 3µ
U = C 4 af
23
Gβ 3
Observe that the above problem is a function of one choice variable G. Thus, the local government
chooses the level of provision of infrastructure and leaves everything else to the market. The
second-order condition for the problem is given by
∂ 2U
bβ 3g − 2
= − β 3 − 1 βµ 2 3G
∂G 2
which is negative if β > 3.
16
See the Appendix for the solution to the problem.
b g
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From Equation (21) we can conclude the following:
Result 1. Given Assumption 1, the equilibrium number of firms M* is larger the
smaller the value of f, a, and t, and the greater the value of θ.17 Observe that low
values of the parameters a, f, and t will increase city size more than they increase
the equilibrium employment of each firm in the city, thus resulting in a larger
equilibrium number of firms. It can be seen from the above result that the main
determinants of the number of firms are: (1) traditional economic factors, in
particular, labor productivity, and fixed set up cost; (2) spatial factors such as
the cost of commuting; and (3) a public infrastructure factor represented by the
level of utilization of infrastructure. It is interesting to see why spatial factors
affect the number of firms in the city. Observe that the number of firms in the
city is given by m = n/L but because n is determined by spatial and infrastructure
factors the number of firms turn are affected by the same factors.
Substituting Equations (15), (16), (5), (6), (3), and (2) into (1) yields the
equilibrium utility level for a household in a given multi-firm city as
(22)
U
*
LM
F θ I b g F 3π I b g b4af g b
= C b4 af g M1 − b3 − βg G J
H β K GH t JK
MN
β
β−3
−1
12
2
β −3
−3
β−3
OP
gP
PQ
Observe that if Assumption 1 is not satisfied then the equilibrium utility level
given by Equation (22) can be negative.
Result 2. Given Assumption 1, the equilibrium utility U* is increasing in θ and
decreasing in t, f, and a. The intuition behind this result is that larger f and a
and smaller θ imply lower productivity and consequently lower wages, as can
be seen from (19), as well as lower incomes for households in the city. Furthermore, higher t implies higher commuting costs and, thus, less consumption and
lower utility.
Company Town
Given the national utility level U the developer of each city must choose a
population n that maximizes the net surplus from city development. If a
company town is to be formed it will have only a single firm because a second
firm will double the set up cost and increase the commuting cost without any
productivity gain. Now, the surplus from a city development is defined as the
difference between the output of the firm and the cost of maintaining a national
utility level U, which is given by
(23)
b g FGH na IJK
S n, U =
12
− f − nC −1U − 2µn 3 2
17
For the multi-firm city to have a meaningful solution M* must be greater than two. This can
be satisfied for a large value of θ. Furthermore, we ignore the condition that M must be an integer.
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765
Observe that maximizing S is equivalent to maximizing the ALR-f.18 Now
I show
LEMMA 2: The net surplus function S(n, U)19
(i)
is strictly concave in n,
(ii) is decreasing in a, f, t, and U
Observe that because the surplus is increasing in n up to a point this indicates
that agglomeration economies dominate the diseconomy of city size. This results
in a unique optimal city size. This is an indication that a fixed set up cost, that
is, increasing returns to scale, is sufficient to generating a company town. The
first-order condition for problem (23) with respect to n is given as
∂S
= 2 −1 an
∂n
b g
(24)
−1 2
− UC −1 − 3µn 1 2 = 0
The first term in Equation (24) represents the gain in output as a result of a
marginal increase in population. The second and third terms represent the
marginal cost of an increase in the city’s population. Developers enter the market
by forming company towns as long as the net surplus is positive. Competition
among developers results in the equilibrium utility level at which the net surplus
is driven to zero. From the zero surplus I obtain
2 −1 a − 1 2 n 1 2 + µn 3 2 = f
(25)
The first term on the left-hand side of the above equation represents the
difference between the average value product of labor and the marginal value
product. Thus, the city size will be chosen at the phase of increasing returns to
scale in production. Furthermore, Equation (25) implies that the set up cost will
be financed by the aggregate land rent and the profit from production, which is
a version of the Henry George Theorem.20 Observe that for each set of parameters {a, t, f} there exists a unique n** and U**. This result follows directly from
Lemma 2. Furthermore, it can be seen that a larger city size n** implies a lower
equilibrium utility level, U**.21
Result 3. If an equilibrium configuration of a system of company towns exists,
the equilibrium utility U** is decreasing in a, f, and t. The intuition for this
comparative static result is the same as in Result 1. This result is used later in
18
See Fujita (1989) for a similar problem with commuting costs in terms of time only.
The second-order condition for problem (10) is given by
1
3
− a − 1 2 n −3 2 − µn −1 2 < 0
4
2
20
See Stiglitz (1977) for this result.
21
A closed-form solution can not be obtained for n and U in this problem.
19
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characterizing the condition under which each equilibrium configuration will
emerge.
4.
PARAMETER RANGES
Given the two possible equilibrium configurations outlined above, the one
that is sustainable is the one that will provide the highest utility for household
in the system.22 Developers will be able to attract households to cities that
provide higher utility and still be able to make profit. I turn to the conditions
under which each equilibrium will emerge. First, consider the case in which all
cities that emerge are multi-firm cities.
THEOREM 1. Given Assumption 1, then there exists a set of parameters {a, θ, f,
t} under which the equilibrium configuration will be of a multi-firm city
Proof. For this equilibrium configuration to occur the following condition
must hold
(26)
U* – U** > 0
where U* is defined by (22).23 Recall that U** is independent of θ because G is
not provided in the case of a company town.24 Thus, for a given set of parameters
{a, f, t} the value of U** is uniquely determined. However, U* is increasing in θ
by Result 2. Thus, by the continuity of the parameter θ, a sufficiently large value
of the parameter is found such that the above condition is satisfied. Q. E. D.
Intuitively if the utilization of infrastructures is sufficiently large, then the
cost of forming a multi-firm city is relatively small compared to that of a company
town. Hence, developers can provide households in these type of cities with a
high utility level. Thus, company towns cannot be sustained in equilibrium.
Observe that the condition for the formation of a multi-firm city is consistent
with the requirement that M* has to be large as given by Equation (21). This
result sheds some light on the types of industries in which government investment in public infrastructure will result in the formation of a multi-firm city.
THEOREM 2: Given Assumption 1, there exists a set of parameters {a, f, θ ,t}
under which the equilibrium configuration is a system of company towns
Proof. For an equilibrium configuration of a system of company towns to
occur multi-firm cities must not be able to provide households with higher utility.
Thus, we require that
(27)
U** – U* > 0
22
It is possible to have a mixture of the two equilibrium configuration. However, this can
happen only on a set of parameters of measure zero, that is, U* = U**.
23
Note that Assumption 1 is sufficient for U* > 0, but not necessary for this equilibrium
configuration to emerge. For the above condition to be satisfied we need condition (26) to be satisfied.
24
Note that we do not have to solve for the value of U** to prove this theorem. All we need is
that value U** exist uniquely for a given set of parameter.
© Blackwell Publishers 2000.
ABDEL-RAHMAN: A MODEL OF LOCALIZATION ECONOMIES
767
Given that Assumption 1 is satisfied then U* is increasing in θ by Result 1.
However, for a given set of parameters {a, f, t}, U** is uniquely determined. Thus,
by continuity of θ a small value of θ such that condition (27) is satisfied can be
found. Q. E. D.
The intuition behind this result is that for a sufficiently small value of θ the
cost of forming multi-firm cities is relatively high. Thus, the developer of
company towns can make higher surpluses and can offer higher utility to
households than would be possible in multi-firm cities. It is possible to get a
mixture of both equilibrium configurations in this model. This happens for a
given θ at which U* = U**. However, the number of cities of each type cannot be
determined. Therefore, this corresponds to degenerated case. In other words,
this mixed system can be only on a set of the parameter space having zero
measure.
5.
CONCLUSION
In this paper I present a one-sector general equilibrium spatial model of a
system of cities. Cities in this model are formed because of the provision of
infrastructures, that is, water, electricity, and sewer systems. The model generates two equilibrium configurations: (1) multi-firm cities, in which all cities in
the system consist of more than one firm; and (2) company towns, in which each
city in the system is formed by a developer and has a single firm. Which of the
two equilibrium configurations emerges depends on the parameter value of the
infrastructures cost, the fixed set up cost for each firm, the marginal physical
product of labor, the unit transportation cost, and the parameter for the utilization of infrastructure.
The model presented may be extended into a two-sector general equilibrium
model. This allows for the possibility of generating more than two equilibrium
configurations: one where each city specializes in the production of one good, the
other where a diversified city may exist because of the provision of a shared
public good. In this framework, public infrastructure may be used as a major
cause of urbanization economies. This is my direction of future research on this
topic.
REFERENCES
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———. 1998. “Multiple Types of Households in a Decentralized System of Cities,” Journal of Regional
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Abdel-Rahman, Hesham M. and Masahisa Fujita. 1990. “Product Variety, Marshallian Externalities,
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Fujita, Masahisa. 1989. Urban Economic Theory: Land Use and City Size. London: Cambridge
University Press.
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JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000
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APPENDIX
The Second-Order Conditions for Problem (9)
∂2 S
2 Mfθ 2
=
−
− β β − 1 G β−2 < 0
2
3
∂G
θG + 1
b
b g
g
FG IJ
H K
∂2 S
1 n
=−
2
4 a
∂M
1
2
−3
M
if
β > 1.
<0
2
fθ
∂2S
=
>0
2
∂G∂M
θG + 1
b
g
The Hessian determinant evaluated at the first-order conditions is given by
θf 2
b
g
2G θG + 1
b g
θG β − 1 + 1 > 0
4
The Solution to Problem (20)
The first-order conditions for the problem are
∂S
= 4 af
∂G
b g θn − βG b
∂S
= b4 af g bθG + 1g − 3µn
∂n
−1
−1
g =0
β−1
12
− UC −1 = 0
The second-order conditions are
∂2S
∂G 2
© Blackwell Publishers 2000.
= − β β − 1 Gb
b
g
β−2
g <0
ABDEL-RAHMAN: A MODEL OF LOCALIZATION ECONOMIES
769
which is satisfied if β > 1 and
∂2S
∂n 2
=−
3 −1 2
µn
<0
2
which is satisfied at the optimal.
∂2 S
= θ 4 af
∂G∂n
b g
−1
>0
Furthermore, the Hessian determinant evaluated at the first-order conditions
is given by
β2G b
(A.1)
g n −2 o 3bβ − 1g
2β − 1
2β µn 3 2 − G β
t
It can be seen that Assumption 1 has to be satisfied for (A.1) to be positive if the
surplus is zero.
The Maximization of the Average Surplus
∂AS
= − µn −1 2 + G β n −2 = 0
∂n
∂AS
−1
β−1
= 4 af θ − βG b g n −1 = 0
∂G
b g
The second-order conditions are
∂ 2 AS
= 1 2 µn −3 2 − 2G β n −3 < 0
∂2n
b g
∂ 2 AS
β−2
= − β β − 1 G b g n −1 < 0
∂2G
b g
∂ 2 AS
β−1
= βG b g n −2 > 0
∂G∂n
The Hessian determinant is given as
βG b
β−2
g n −2
− 1 2 µ β − 1 n −1 2 + G β n −2 β − 2
b gb g
b
g
which is positive by the first-order conditions and Assumption 1.
© Blackwell Publishers 2000.