Chukyo University Institute of Economics
Discussion Paper Series
July 2024

No. 2402
The effect of the public sector on the optimal
number of cities

Akiyoshi Furukawa

The eﬀect of the public sector on the optimal
number of cities
Akiyoshi Furukawa ∗Abstract
This study presents a theoretical analysis examining whether
abundant small municipalities increases eﬃciency. In recent years,
large scale municipal consolidations have decreased the number ofmunicipalities; however, some small municipalities have not consolidated and remain viable. This study examines the value of thesecircumstances. In the model analysis, this study uses the land capitalization.
This study found the following results. The optimal number ofcities is larger than one only when a suﬃcient amount of land isneeded for housing production. In this case, as the eﬀect of local
public good on preference and scale economies in producing the local
public good are smaller, the optimal number of cities rises. When local public sector inﬂuence decreases, larger number of municipalities
is more eﬃcient and the economy has no need of consolidation.

JEL classiﬁcation: R23, R53, H41, H73
Keywords: regional population; local public good; land capitalization; municipal consolidation∗

School of Economics, Chukyo University 101-2 Yagoto Honmachi, Showa-ku,
Nagoya, 466-8666, Japan Tel.: +81-52-835-7111 Fax: +81-52-835-7197 e-mail:ecockawa@mecl.chukyo-u.ac.jp1

1

Introduction

In developed countries, municipal consolidation is promoted for anticipating economies of scale and reducing public expenditure; however, somesmall municipalities do not consolidate, resulting in a larger number of municipalities. In Japan, Weese (2015) asserts that the current number ofmunicipalities is larger than the optimal number. Nakagawa (2016) ﬁnds
that the municipal consolidation in recent years has had a minimal eﬀect on
small municipalities. Avellaneda and Gomez (2014) note that the numberof municipalities in developing countries has grown. This study analyzes
whether the existence of many municipalities is desirable. To do so, thisstudy theoretically examines the eﬀect of local public sector on the optimal
number of cities.
Previous studies indicate that municipal consolidation reduces local
public expenditure through economies of scale; however, some studies suchas Bless and Baskaran (2016) and Miyazaki (2018), show that municipal
consolidation may increase local public expenditure. Although economies
of scale are present, the eﬀects of municipal consolidation appear elsewhere.

2

In this analysis, many factors other than public sector expenditure aﬀect
the optimal city number. Previous studies use land prices to evaluate consolidation (for example, Hu and Yinger (2016), Dumconbe, Yinger andZhang (2016), Hayashi and Suzuki (2018) ). Therefore, this study uses theland prices to examine the optimal number of cities. This study references
Behrens, Kanemoto and Murata (2015), whose analysis uses the Henry
George Theorem in which product diﬀerentiation and increasing returns
cause agglomeration economies wherein second-best economies represent
the modiﬁed Henry George Theorem.
In urban economies, optimal city size is achieved through a balance
between agglomeration and dispersion forces. For the local public sector,
local public goods have an agglomerative force, and local public goods
congestion has a dispersive force. When the city population is larger, local
public goods congestion is essential. However, because of the declining
birthrate and an aging population, population growth has declined in somedeveloped economies. In this economy, the congestion generated by local
public goods consumption is not a signiﬁcant problem; therefore, this study
analyzes pure local public goods.
3When a local public good is pure, it only exerts agglomerative force inurban economies. Intuitively, for the public sector, a smaller number ofcities is optimal because the dispersive force (congestion) is not present. In
other words, consolidation is always more desirable. This study analyzes
whether the public sector increases the optimal number of cities.
The remainder of this paper is organized as follows. Section 2 presents
the research model. Section 3 analyzes the optimal number of cities. Section 4 examines the eﬀect of the public sector on the optimal number ofcities and Section 5 concludes.

2

The ModelThis study references Behrens, Kanemoto and Murata (2015), whose analysis uses the Henry George Theorem in which product diﬀerentiation and increasing returns cause agglomeration economies wherein second-best economies
represent the modiﬁed Henry George Theorem. This study reconstructs
Behrens et al.’s (2015) model by introducing absentee landlords and manufactured goods that can be transported between cities. Moreover, the study
focuses the scale economies to analyze local public goods production.

4

In this study’s model, many potential sites for city development areavailable, each site has a ﬁxed amount of land (X̄L ), and n cities are developed from these potential sites, where n is the number of cities and isendogenous to policies. One city receives a ﬁxed amount of land on one
site, and the total amount of land used in the economy is nX̄L . The model
ignores the city’s spatial structure, and no transport costs exist. For individuals, the place of residence is homogeneous within the city. Individuals
are homogeneous and can migrate between cities without cost. Therefore,in equilibrium, utility is equal across individuals. The total population of
the economy is L̄ . This study analyzes symmetric allocations; therefore,
the city’s population size is l ≡ L̄/n . The city’s land size is ﬁxed at X̄L .
When a new city is added to the economy, the total amount of land that isused in the economy increases, though the total population of the economy
does not increase. Housing in each city is produced with land and labor as
inputs, in addition to manufactured goods and local public goods.
This study analyzes the optimal number of cities by assuming that the
number of cities is ﬁxed to derive economic equilibrium conditions anddetermine the optimal number of cities.
5

2.1

Model

Individuals obtain utility from manufactured goods, local public goods, andhousing. Individuals in a city have the following utility:u(x, g) = x + α log g

where x is manufactured goods, and g is local public goods. For simplicity,
this study assumes that housing consumption (x̄H ) is ﬁxed when an individual relocates in a city; therefore, utility does not include this consumption.
The individual’s budget constraint is as follows:

w = Px x + PH x̄H +C(g)lwhere each individual in a city supplies one unit of labor and acquires wage
w, and Px and PH are the prices of manufactured goods and housing, respectively. C(g) is the cost function of local public goods, for which eachindividual must pay the per capita cost of this good, C(g)/l . In equilibrium, individuals have the same utility (ū) regardless of their location. The
study determines the price of housing based on the bid price (PH ), which
represents the maximum price that an individual is willing to pay at utility

6level ū . The bid price function is as follows:
[

1
c(g)
PH =
w − Px {ū − α log g} −
x̄H
l

]

(1)

In each city, manufactured goods are produced under constant returns
and oﬀered on the exogenous national market. These goods are produced
using intermediate goods as inputs, under increasing returns and monopolistic competition with labor as the input, and cannot be traded between
cities. The production function of the manufactured good is as follows:
{∫

X=
0

N

}1

qjρ djρ

0<ρ<1where qj denotes intermediate good j (j ∈ [0, N ]) and N is the variety ofintermediate goods in a city. Each intermediate good is produced by one
ﬁrm. The production function of each intermediate good is as follows:

Lqj = f + bqj

where Lqj is labor input, f is ﬁxed labor input and b is marginal labor
input.
Housing is produced by land and labor, and is produced under constantreturns. This sector is competitive with free entry. The total amount of7

land in a city is X̄L . Absentee landlords are assumed to own the land, andonly housing producers use the land for production. In equilibrium, thehousing producer pays to rent the land, which equals total revenue minus
labor cost. The housing production function is as follows:

YH = FH (YLH , YOH )where YH is the total amount of housing and YLH and YOH are land andlabor inputs. The housing producer’s proﬁt (Π) is determined as follows:Π = PH YH − pL YLH − wYOH

where pL is the land rent. Proﬁt maximizing conditions are as follows:

FHL =

∂FH
pL
=
,
H
∂YL
PH

FHO =

∂FH
w
=
H
∂YO
PH

Because of free entry, in equilibrium, Π = 0 .
In each city, the local government provides local public goods, which
are pure. Therefore, spillover eﬀects on other cities do not occur and public
goods do not have congestion eﬀects. The local public good is produced
using manufactured goods as inputs. The cost function C(g) is as follows:

C(g) = Px g γ
8where γ represents the scale economies producing the local public good.
The government imposes the cost on individuals, which is equal amongindividuals.

2.2

Equilibrium

This subsection analyzes the equilibrium conditions, where the number ofcities and the supply of local public goods are determined.
First, consider the housing market in one city. Because the individual’s
housing demand is ﬁxed (x̄H ) , the aggregate demand in the city is lx̄H .
The market clearing condition for housing is as follows:lx̄H =

L̄x̄H = YH = FH (YLH , YOH )n

Because the amount of land is ﬁxed, YLH = X̄L . Then, labor input (YOH )is determined as a function of n, YOH (n).
Concerning the manufactured good producer’s behavior, the following
condition holds:

pmj = Px X 1−ρ qjd

ρ−1

where pmj is the price of intermediate good j and qjd is the demand forintermediate good j. Because intermediate goods are produced under mo9

nopolistic competition, the ﬁrst order condition for proﬁt maximization is
as follows:ρ
w = pmjb
Moreover, based on monopolistic competition behavior, the output of goods
j and labor input are obtained as follows:

qj =

ρf
b(1 − ρ)

Lqj =

f
1−ρ

The market clearing condition for intermediate goods and labor are as
follows:

qjd = qjL̄
= l = N Lqj + YOH (n)nThese equations obtain the following:

qjd = qj =ρfb(1 − ρ)

Lqj =

f
1−ρ

The variety of intermediate goods is obtained as follows:
(

N=

)

L̄
1−ρ
− YOH (n)nf10The price of intermediate goods (pmj ) and associated wage (w) are as follows:{

pmj = Pxρw = Pxb1−ρ
f{} 1−ρ [
ρ

1−ρ
f

] 1−ρ

L̄
− YOH (n)n

} 1−ρ [
ρ

ρ

] 1−ρ

L̄
− YOH (n)nρThen, the resulting amount of the manufactured good is as follows:
{

ρ
X =
b
∗

1−ρ
f

} 1−ρ [
ρ]1

ρL̄
− YOH (n)n(2)

In the following section, this study analyzes the optimal number of cities
to maximize the aggregate land rent. In equilibrium, we assume that utility(ū) is given. Then, the bid price of housing is as follows:

Px  ρ
PH =
x̄H b

{

1−ρ
f

} 1−ρ {
ρ

} 1−ρ

L̄
− YOH (n)nρng γ 
− {ū − α log g} −L̄The aggregate land rent in the economy is as follows:[]L̄
AR = n PH x̄H − wYOH (n)nρ= Px 
b{1−ρ
f} 1−ρ {
ρ}1L̄
− YOH (n)
n

ρ

{

}



ng γ
L̄
n + α log g − ū −L̄(3)

Before focusing the optimal number of cities, this study analyzes theeﬀect of this number on the public sector. When the number is given, the11

optimal amount of local public goods is as follows:
(
∗

g =

αL̄
γn

)1
γ

Total public spending in the economy is as follows:αnC(g ∗ ) = nPx g γ = Px L̄
γWhen the number of cities (n) increases, the amount of local public goods
decreases, as total public spending does not change. In the public sector
context, it is optimal for the number of cities to be smaller; that is, n = 1 is
optimal. In other words, consolidation is always desirable from the public
sector perspective. The next section examines a case in which multiple
cities are more valuable than a single city.

3

Social optimumThis section examines the optimal number of cities, considering the problemof maximizing the aggregate land rent that incurs all individual’s ﬁxed
utility. In the social optimum, the social planner maximizes land rent withrespect to the number of cities and local public goods.
Because this study analyzes symmetric allocations, the social planner12

must solve the following problem:

max AR(4)g,nFirst-order conditions for g, n are as follows:
]

[

∂AR
α n γ−1= Px L̄
− γg
=0
∂g
g
L̄



∂AR
ρ
= Px
∂n
b

1

ρ

{

{

1−ρ
f

} 1−ρ

} 1−ρ {

L̄
− YOH (n)
n

ρ

(5)

ρ

×
}

{

L̄
L̄
− 2 − YOH (n)′ n +− YOH (n)
n
n

}1 
ρ


−Px g γ = 0(6)where YOH (n)′ is (dYOH )/(dn) . Equation (5) presents the optimal allocationof local public goods in each city, and equation (6) indicates the optimal
number of cities. In the following analysis, the model assumes that secondorder conditions are satisﬁed, from which, the following condition holds:∂ 2 AR∂g 2∂ 2 AR∂n∂g∂ 2 AR∂g∂n∂ 2 AR∂n2= |D| > 0(7)This study next examines the condition in which the optimal numberof cities is larger than one and it is not optimal for multiple cities to consolidate and only one city carries the economy as a result.
13

If n = 1, ∂AR/∂n is expressed as follows:
∂AR
ρ
= Px
∂n n=1
b
{

{

1−ρ
f

} 1−ρ

} 1−ρ

L̄ − YOH (1)

ρ

ρ

[

×
]

1
ρ−1
L̄ − YOH (1) − YOH (1)′ − Px g γ
ρ
ρ

(8)

where YOH (1)′ = −FH (X̄L , YOH )/FHO . The condition in which the optimal
number of cities (n∗ ) is larger than one is obtained as (8) > 0. This condition is satisﬁed only if |YOH (1)′ | is larger, indicating that the labor input
for housing production drastically decreases with respect to the number ofcities. This condition holds when the marginal product of labor in housing(FHO ) is suﬃciently small and the total amount of production (FH (X̄L , YOH ))is suﬃciently large.
It is worth examining a case in which the labor input is unnecessary for
housing production. In this case, ∂AR/∂n is as follows:
∂AR
ρ
= Px
∂n n=1b{1−ρ
f} 1−ρ {
ρ



L̄
n

}1

ρ



ρ − 1
− Px g γ < 0ρ

The number of cities should be smaller to maximize the aggregate land
rent; that is, n∗ = 1. This means that the optimal number of cities islarger than one only when labor input is necessary for housing production.
To summarize these results, the following proposition holds:
14

Proposition 1

The optimal number of cities is larger

than one only if the marginal product of labor in housing is
suﬃciently small and the total amount of housing production
is suﬃciently large; otherwise, one city is optimal.

When the demand for housing is larger, the economy should provide
suﬃcient input for subsequent production. In the case of low labor productivity, a suﬃcient amount of land should be provided. If the numberof cities is smaller, a suﬃcient amount of labor is required for housing production, although it decreases the production of manufactured goods that
promotes agglomeration economies. Thus, multiple cities must provide a
suﬃcient amount of land.

4

The public sector’s eﬀect on the numberof cities

This section analyzes the eﬀect of the public sector on the number of cities.
The model examines the eﬀect of local public goods on preference (α) andscale economies in producing local public good (γ) . One object of thisstudy is to examine a case in which the existence of many cities is desirable.

15

This section analyzes the impact of the public sector on these cities.
Before analyzing the public sector, the study examines the eﬀect of total
population (L̄) on the optimal number of cities as follows:
dn∗
n
= >0
dL̄
L̄
where an increase in the total population (L̄) increases the optimal numberof cities. Supporting many individuals requires many cities that provide
housing and local public goods.
The study next analyzes the public sector. First, the eﬀect of the preference for local public good α is as follows:dn∗−Px 2 g γ−2 γ L̄
=
<0
dα
|D|

(9)

where the preference for the local public good decreases the optimal numberof cities because cities need a larger population to provide adequate local
public goods.
The eﬀect of scale economies in producing local public good γ is asfollows:
Px 2 g γ−2 αL̄
dn∗
=
>0
dγ
|D|
16

(10)

An increase in γ indicates a decrease in scale economies in the productionof the local public good, which leads to an increase in the optimal numberof cities. When public sector productivity decreases, a city should reduce
the production of local public goods and stimulate the private sector. The
optimal number of cities should be increased to promote the private sector
because it will raise the labor input for private production.
These results derive the following proposition.

Proposition 2

If public sector inﬂuence decreases, theoptimal number of cities increases.

Proposition 2 posits that multiple cities are more desirable when public
sector inﬂuence decreases. In this case, the private sector is more important
than the public sector. Stimulating the private sector requires a larger
number of cities to increase the input for private production.

5

Conclusion

This study examines the eﬀect of the local public sector on the optimal
number of cities. In each city, the local government provides a pure local
public good that has agglomerative force. It is optimal for the public sector
17that the number of cities is smaller; therefore, consolidation should be promoted. In this circumstance, the study demonstrates a case in which somesmall municipalities do not consolidate and the number of municipalities islarger, which is desirable.
This study found the following results. The optimal number of cities islarger than one only when a suﬃcient amount of land is needed for housing
production. In this case, as the eﬀect of local public good on preference andscale economies in producing the local public good are smaller, the optimal
number of cities rises. When local public sector inﬂuence decreases, larger
number of municipalities is more eﬃcient and the economy has no need of
consolidation.

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18

[2] Behrens, K., Kanemoto, Y. and Murata, Y., 2015, The Henry George
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[3] Blesse, S. and Baskaran, T., 2016, Do municipal mergers reduce costs?
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[4] Duncombe, W.D., Yinger, J. and Zhang, P., 2016, How does school
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[5] Hayashi, M. and Suzuki, T., 2018, Municipal mergers and capitalization: evaluating the Heisei territorial reform in Japan, CIRJE Discussion Paper F Series CIRJE-F-1105.

[6] Hu, Y. and Yinger, J., 2008, The impact of school district consolidation
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[7] Miyazaki, T., 2018, Exchanging the relationship between municipal
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[9] Weese, E., 2015, Political mergers as coalition formation: An analysisof the Heisei municipal amalgamation, Quantitative Economics 6, 257307.

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