Chukyo University Institute of Economics
Discussion Paper Series
July 2022

No. 2202

The Value of Small Regions

Akiyoshi Furukawa

The Value of Small Regions
Akiyoshi Furukawa ∗Abstract
This paper presents a theoretical analysis of whether small municipalities’ existence increases eﬃciency. In recent years, large-scalemunicipal consolidations have decreased the number of municipalities: however, some small municipalities do not consolidate andremain alive. This paper utilizes land capitalization to present thevalue of these situations, determining the following results: Only
when the travel cost is suﬃciently high and the population differences in the non-consolidation case are suﬃciently small, nonconsolidation is more valuable than consolidation. That eﬀect ismire signiﬁcant when the private production technology is not large,
the scale economies in the production of the local public good are
low, and the total population is large. In this case, the existence ofsmall municipalities is more eﬃcient.

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

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

1

Introduction

This paper presents a theoretical analysis to determine whether the existence of some small municipalities increases the eﬃciency. In recent years,
large-scale municipal consolidations have decreased the number of municipalities: however, some small municipalities do not consolidate and remain
alive. For example, in Japan, Weese (2008, 2013) shows that the optimal number of municipalities is less than the current number. While somesmall municipalities should consolidate and disappear for optimality, they
do not consolidate and remain alive. Nakagawa (2016) demonstrates thatthe large scale of municipal consolidations during recent years had little
impact on some small municipalities. Moreover, Avellaneda and Gomes(2014) point out that municipal consolidation is promoted in developed
countries, though in some developing countries, the number of municipalities has grown in recent years.
This paper follows previous studies about municipal consolidation toevaluate small municipalities. Previously, analyses of municipal consolidations have focused on evaluating local governments policies and expendi-

2

ture. For example, Dur and Staal (2008) analyze the spillover eﬀects oflocal public good. Blume and Blume (2007) analyze economies of scale inproviding local public goods, while Sørensen (2006) examines cost-sharing.
Previous studies indicate that municipal consolidations should decrease thelocal public expenditure.
Still, it is diﬃcult to evaluate small municipalities to utilize the local
public expenditure. Some empirical studies (Miyazaki (2018), Bless andBaskaran (2016) ) point out that municipal consolidations may increase local public expenditures, though it is unclear whether these consolidations
are desirable. Municipal consolidations aﬀect public expenditures, regional
populations, and land areas; they are how municipalities expand populations and land areas. For evaluating the economic eﬀects of municipalities,
it is not suitable to utilize local public expenditures; therefore, this paper
utilizes land prices.
Some previous studies also utilize land prices. Hu and Yinger (2008)
and Duncombe, Yinger, and Zhang (2016) analyze the relationship between
school district consolidation and property values based on capitalizationeﬀects in New York. Hayashi and Suzuki (2018) analyze these eﬀects in
3

Japan. This paper utilizes a theoretical capitalization analysis and presents
the value of small municipalities compared to municipal consolidations.
In this paper’s, the centripetal force is the private and public sector production. For maximizing the production, it is desirable to foster agglomeration that leads to municipal consolidations. Conversely, the centrifugal
force is the travel cost for consuming the local public goods. Small municipalities are valuable in contrast to agglomeration when this eﬀect is larger.
Then, the land value is greater than that of municipal consolidations cases,
as shown in this paper.
This paper is organized as follows. Section 2 presents the model. Section 3 analyzes the social optimum in municipal consolidation and nonconsolidation cases. Section 4 compares land rents in each case, and Section
5 concludes.

2

The model

The economy has a one-dimensional space in which the land destiny isone. Consider one region in the economy. The region initially consistsof two districts (districts 1 and 2 ). The population size in district i is ni4(i = 1, 2), and the region’s total population is N = n1 +n2 . Each individual
can migrate across regions and districts; therefore, the utility is equal acrossindividuals in equilibrium. In the region, two districts can consolidate into
one district. When this consolidation arises, all populations are allocatedto the consolidation district.
Each individual in district i earns an income wi . They have a utility
composed of private and local public goods. Moreover, they consume one
unit of land for housing, meaning that the length of district i equals the
population. The utility function of an individual in district i is:u(xi , gi ) = xi + α log gi

where xi is the amount of private good consumption, the price of which is
1; gi is the amount of the local public good in district i.
Individuals must travel to where the local government makes the goods
available to consume the local public goods. When the individual is locatedat a distance r from the local government, the travel cost is mr, where mis the travel cost per unit of distance. The land is homogeneous except
for this travel cost. Moreover, the cost of the local public good must be5

imposed on each individual. Let C(gi ) is the cost function of the local
public good; each individual must pay the burden, C(gi )/ni . The budget
constraint of the individual residing at distance r is:

xi + R(r) +C(gi )+ mr = winiwhere R(r) is the land rent at r.
In the following analysis, the model utilizes the bid rent function R∗ (r, u),representing the maximum rent the individual is willing to pay at distancer from the local government when the utility level is given at u. From themodel, we haveR∗ (r, u) = βi − mr − (u − α log gi ) −C(gi )ni(1)

The private good is produced with the labor as the input. In districti, each individual supplies one unit of labor, which can produce βi unitsof the private good. We assume that β1 > β2 , meaning that district 1 ismore productive than the other district. To maximize the production of
the private good, it is optimum that all populations agglomerate in district
1. Other than the private production, the two districts are homogeneous.
Because of the private production, if two districts must consolidate, all6

populations should allocate to district 1 instead of district 2. The private
good is utilized for consumption and the production of the local public
good. Because the private good sector operated under perfect competition,
the wage in district i, wi , must satisfy wi = βi .
Initially, in each district, the local government provides the local public
goods where it is located. In equilibrium, the local government locatesin the district center over the interval [− n2i , n2i ] and the distribution of
consumers is uniform. The local public good is pure and does not havespillover eﬀects on the other district. Only individuals in district i utilize
its local public good, produced using the private good as the input. The
cost function C(gi ) is given as:

C(gi ) = giγ

where γ represents the scale economies in producing the local public good.
The government imposes this cost on residents, and the burden is equalamong individuals.
The land market is perfectly competitive and the opportunity cost ofland is assumed to be zero. Absentee landlords are assumed to own the7

land, and ﬁrms and governments do not consume the land; only residents
use the land for housing. In this model, each resident uses a ﬁxed unit of one
land size, and the land at each point is allocated through a bidding process.
In equilibrium, because residents are identical, the utility is equal acrosslocations. Therefore, the equilibrium land rent at distance r equals the bid
rent. At the district’s boundary, the land rent equals the opportunity cost,which is zero. In the following analysis, it is assumed that the equilibrium
utility level u is given. Then, the land rent at distance r is as follows:R∗ (r, u) = βi − mr − (u − α log gi ) −

giγ
ni

(2)

This paper analyzes the eﬃcient allocation that maximizes the aggregate
land rent.

3Social optimumThis section analyzes the social optimum that maximizes the aggregate land
rent in the region. For example, the problem of minimizing the resource
cost in the ﬁxed level of utility leads to the maximization problem, which
Fujita and Thisse (2002) analyze. First, this study analyzes the case thattwo districts exist in the region, and second, we consider the case that two8

districts consolidate into one district.

3.1

The case of two districts

Consider the case where two districts exist in the region. A social plannermaximizes the aggregate land rent in which the utility level is ﬁxed acrossindividuals. When the utility level is u, the land rent at distance r from
district i local government, leading to equation (2). In equilibrium, thelocal government locates in the center over the interval [− n2i , n2i ], and the
distribution of individuals is uniform. Then, the aggregate land rent in
district 1 (ALR1 ) is:∫ n1 [

ALR1 = 2

2

0

]

gγ
β1 − mr − {u − α log g1 } − 1 drn1

= {β1 − u + α log g1 }n1 − g1γ −mn21
4

Similarly, the aggregate land rent in district 2 (ALR2 ) is:
mn22
γ
ALR2 = {β2 − u + α log g2 }n2 − g2 −4

The aggregate land rent in the region is:

S = ALR1 + ALR2
= {β1 − u + α log g1 }n1 − g1γ −

mn21
mn22
+ {β2 − u + α log g2 }n2 − g2γ −449The central planner must solve the following problem:

max S

g1 ,g2 ,n1s.t. N = n1 + n2

(3)First-order conditions for g1 , g2 , n1 are as follows:n1= γg1γ−1
g1(4)(N − n1 )= γg2γ−1
g2

(5)

α
α

β1 − β2 + α (log g1 − log g2 ) +m(N − 2n1 ) = 0
2(6)Equation (4) and (5) show the optimal allocation of local public goods andequation (6) indicates the optimal allocation of population. From equations(4) and (5), the following conditions are derived:
(

g1 =

αn1
γ

(

)1
γ

g2 =

α(N − n1 )
γ

)1
γ

(7)

By substituting these equations into (6), the eﬃcient allocation conditionof the population is obtained as:

L(n1 ) = β1 − β2 +

α
n1
m
log
+ (N − 2n1 ) = 0γ
N − n1
2(8)

From (8), we obtain the eﬃcient allocation of the population: When
α 4> m, that is, the travel cost is small, the optimal population of districtγN

10

1 is smaller than the other district, and this allocation is a unique solution.
Conversely, when the travel cost is not small, the case arises where theoptimal population of district 1 is larger than the other district. Then, thefollowing lemma is derived.

Lemma 1

Consider a region where two districts exist.

In the optimal allocation of population, when the travel cost is
suﬃciently small, the population of the more productive districtis smaller than the other district, and this allocation is a uniquesolution. Conversely, if the travel cost is suﬃciently high, thecase arises where the population of the more productive districtis larger than the other district.

Individuals can consume more local public goods if the travel cost is sufﬁciently small. Then, the eﬀect of local public goods is more signiﬁcant.
In the low productive district, more local public good should be produced
through the larger population to compensate for the low income. Therefore, the low productive district should have more population; however,when the travel cost is higher, the eﬀect of the private production is more11

signiﬁcant. Then, more private goods must be produced. For producing
the good, the high productive district should have more population. In thefollowing analysis, the high productive district (district 1 ) is assumed tohave more population, though the population diﬀerence is small. These1)
populations are n1 ∗ and n2 ∗ and satisfy that n1 ∗ > n2 ∗ , ∂L(n≤ 0, and∂n1

the travel cost is not small. From the following analysis, in this case theexistence of the small district may be valuable.
The comparative statics for the optimal population n1 ∗ , n2 ∗ = N − n1 ∗are as follows. First, the increase in the production technology β1 , β2 has
the following eﬀect. Using (8), when the production technology in one
district rises, the population of that district increases, having a direct eﬀect.
Next, consider the eﬀect on the local public good. If the utility weight of
the local public good increases, the population of district 1 increases. In
district 1, the rising population may increase the production of local public
good through private production. In district 2, the decreasing population
improves the utility by decreasing the travel cost. Similarly, consider theeﬀect of scale economies in producing the local public good. When the scaleeconomies increase ( γ decreases), the population of district 1 increases.
12This rising population could utilize more scale economies through private
production.
Third, we analyze the eﬀect of the travel cost. Using (8), when thetravel cost per unit of distance m increases, the population of district 1
decreases. It is necessary to decrease the scale of district 1 to decrease thetotal travel cost because it is larger than the other district in the model
assumption.
Finally, consider the eﬀect of total population N . The impact on the
population level in each district is not obvious, though we can show theeﬀect on the diﬀerence between these population. If the total population
N increases, the diﬀerence of the population n1 − n2 decreases. Because
the population of district 1 is enough to utilize the scale eﬀect, the agglomeration in district 1 should be restricted to limit the travel cost.
We utilize the regions aggregate land rent in the following analysis toevaluate the social optimum. In this section, concerning the case where
two districts exist in the region, the aggregate land rent is:
[

]

{

mn∗1
m
m
α
α
α
+ n∗1 2 + (N − n1 )2 − uN + N log − 1
N β1 + log n∗1 −
γ
2
4
4
γ
γ

13

}

(9)

The following analysis compares equation (9) and the aggregate land rentin the consolidation case.

3.2

The consolidation caseIn this region, two districts can consolidate to one district. Then all populations allocate to district 1. In the consolidation district, only one local
government locates in the district center over the interval [− N2 , N2 ] and provides the local public good. All individuals must travel to that place toconsume the local public good. The aggregate land rent is:∫ N [SA = 2

2

0

]

gγ
dr
β1 − mr − {u − α log g} −N

= {β1 − u + α log g}N − g γ −mN 2
4where g is the amount of the local public good in the consolidation district.
Similar to the two-district case, the central planner maximizes the aggregate
land rent. From this maximization problem, the optimal amount of local
public good is as follows:
(

g=

αN
γ

14

)1
γ

Then, the aggregate land rent is:
[

]

{

α
mN
m
α
α
SA = N β1 + log N −+ N 2 − uN + N log − 1
γ
2
4
γ
γ

}

(10)

The next section compares the aggregate land rent in the two-district caseand that in the consolidation case.

4

The comparison of land rentThis section analyzes whether the existence of small districts increases efﬁciency relative to the consolidation case. First, we compare the land rentin each case and show that the two-district case is more eﬃcient than the
consolidation case. Second, we analyze the comparative statics to analyze
the social optimum.

4.1

The comparison of land rentThis section analyzes the eﬀect that two districts coexist in the model where
they can consolidate into one district that exploits economies of scale inproducing private good. This study utilizes land rent to evaluate thateﬀect.
From the analysis of the previous section, when two districts coexist, theaggregate land rent S is the equation (9). Conversely, if they consolidate
15into one district, the aggregate land rent SA is the equation (10). The
diﬀerence between these land rents is as follows:

SA − S = N

αNmlog ∗ − (N − n∗1 )2
γ
n1
2

(11)

When the equation (11) is positive, the land rent in the consolidation caseis more signiﬁcant than in the two-district case, meaning that the consolidation is valuable. Conversely, when the equation (11) is negative, the land
rent in the coexistence of two districts is worthwhile.
First, consider that the travel cost m is suﬃciently small, though it isassumed that this cost is not small in the model. Then, the equation (11) isalways positive. From these results and the analysis of the previous section,
the following lemma is derived:

Lemma 2 Consider the optimal allocation of the population in the two-district case. If the allocation of the populationof a more productive district is smaller than the other district
is a unique optimal solution, the consolidation is always morevaluable than the coexistence of two districts.

When the travel cost is suﬃciently small, in the eﬀect of local public good,16

the diseconomies of scale disappear. Then, the consolidation that can exploit economies of scale in producing the private good is a valuable policy.
Next, consider the case where the travel cost is not small. In the model,
we assume that n∗1 is larger than N2 and n∗1 − n∗2 is not signiﬁcant. In thisassumption, equation (11) is minimized when n∗1 is nearly N2 and increases
with an increase in n∗1 . Therefore, the following proposition is derived:

Proposition 1 When the travel cost per distance is suﬃciently high and the diﬀerence in population between the twodistrict case is suﬃciently small, the aggregate land rent in thetwo-district case is larger than that in the consolidation case.
The diﬀerence between these land rents increases with a decrease in the population diﬀerentials.

Regarding land rent, we show that the coexistence of the two-district ismore valuable than the consolidation when travel cost is suﬃciently high,and the diﬀerence in population between the two districts is suﬃciently
small. If these two districts consolidate, this consolidation doubles thetravel distance for consuming the local public goods because their scales

17are almost the same. Moreover, because the travel cost per distance is high,the negative eﬀect of consolidation is larger than the economies of scale.
Then, the relative value of coexistence is more signiﬁcant.

4.2

Comparative statics

Section 4.1 shows that the coexistence of the two districts is more valuable
than the consolidation. This section utilizes comparative statics to analyze
that case in detail.
First, the increase in the production technology β1 (β2 ) has the following
eﬀect. Using (8), when the production technology in district 1 increases,
the diﬀerence in land rents (11) increases. Because the population of district 1 increases, the diﬀerence between the two districts increases. Thiseﬀect decreases the relative land rent in the coexistence of two districts.
Otherwise, if the production technology in district 1 decreases, the population diﬀerence between the two districts decreases. Then, the relativevalue of coexistence is more considerable.
Next, consider the eﬀect on the local public good. If the scale economies
in the production of the local public good increases ( γ decreases ), the dif-18

ference in land rents (11) increases. From the analysis of the previous
section, the population in district 1 increases; thus, the diﬀerence between
the two districts increases, and the consolidation is more valuable. Conversely, if γ increases, the diﬀerence between the two districts decreases,
increasing the value of the coexistence between the two districts.
Finally, we analyze the eﬀect of total population N . From the equation(11), the following equation is derived:∂αNm{SA − S} = log ∗ − (N − n∗1 )
∂N
γ
n1
2

(12)

Consider the population n̂1 that satisﬁes the equation (12) is zero. Whenn∗1 < n̂1 , (12) < 0, meaning that the diﬀerence in land rents (11) decreases
with the increase of the total population N . Conversely, when n∗1 > n̂1 ,(12) > 0, meaning that (11) increases with the rise in the total population.
Section 4.1 shows that only when the travel cost per distance is suﬃciently
high and the diﬀerence in population in the two-district case is suﬃciently
small, the coexistence of the two districts is more valuable than the consolidation; therefore, we only analyze this case. In this case, n∗1 < n̂1 issatisﬁed, and the diﬀerence in land rents decreases with the increase of the19total population N . If the total population increases, the diﬀerence of the
population decreases and the relative land rent increases in the coexistencecase. Conversely, when the total population decreases, the agglomerationin district 1 is fostered, and the relative land rent decreases. Though theland rent in the consolidation case decreases, the negative eﬀect in the twodistrict case is more important, increasing the value of the consolidation.
In summation, the coexistence of the two districts is more valuable
than the consolidation when the production technology is not large, thescale economies in producing the local public good are insigniﬁcant, andthe total population is large.

5

Conclusion

This paper analyzes whether district consolidation is eﬃcient. In each district, the local government provides the local public good with a centrifugal
eﬀect because of the travel cost; however, consolidation is eﬃcient for private sector’s production. This paper utilizes the land price in the model
analysis. We compare land rents in the consolidation case and the divisionof two districts, showing that the division of two districts is more valuable20

than in the consolidation case. Additionally, we analyze when that casearises.
This paper found the following results. When the travel cost is suﬃciently high, and the diﬀerence in populations in the division case is suﬃciently small, the aggregate land rent in the division case is more signiﬁcantthan that in the consolidation case; the division of two districts is more valuable than the consolidation. That diﬀerence in land rents is more signiﬁcant
when the private production technology is not large, the scale economies
in producing the local public good are not large, and the total populationin the two districts is large. In this case, the small district division is more
eﬃcient and the consolidation of these district is not eﬃcient.

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[4] Duncombe, W.D., Yinger, J. and Zhang, P., 2016, How does school
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Discussion Paper 1022 Yale University.

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