2012-19
Estimation of Optimal Metropolitan Size in Japan
with Consideration of Social Costs
Fumitoshi Mizutani Tomoyasu Tanaka

Noriyoshi Nakayama

Estimation of Optimal Metropolitan Size in Japan
with Consideration of Social Costs
by
Fumitoshi Mizutani
Kobe University, Graduate School of Business Administration
2-1 Rokkodai, Nada-ku, Kobe 657-8501 JAPAN(E-mail) toshi@kobe-u.ac.jp

Tomoyasu Tanaka
Kinki University, Faculty of Business Administration
3-4-1 Kowakae, Higashi-Osaka, 577-8502 JAPAN(E-mail) tanakatomo@bus.kindai.ac.jp

Noriyoshi Nakayama
Nagoya City University, Graduate School of Economics
1 Mizuho-cho, AzaYamanohata, Mizuho-ku, Nagoya, 467-8501 JAPAN(E-mail) nakayama@econ.nagoya-cu.ac.jp

Correspondent Author: Fumitoshi Mizutani
toshi@kobe-u.ac.jp[Abstract]:

The main purpose of this study is to estimate the optimal city size which would attainmaximum total surplus and sustainability, or a city size in which total benefits would equal total
costs.

We apply regressions to the total benefit function and the total cost function for 269

employment metropolitan areas for the year 2000 in Japan.

Our study can be distinguished fromothers in that we include in total costs such social costs as environmental pollution.
are that the optimal city size is 393,151 persons.

Our findings

The sustainable limit for city size is 1,057,412.

[JEL Classification]: R10, R12, Q50[Key Words]: Optimal City Size, Total Surplus, Environmental Costs, Social Costs

1. Introduction
Urban economists have long been interested in determining the optimal size of a city.
Theoretical studies on optimal city size include those by Alonso (1971), Henderson (1974a), Arnott(1979), and Kanemoto (1980).

Optimal city size is defined as that which maximizes the potential

welfare of participants in the economy.

Henderson (1974a) determines equilibrium and optimalcity size by presenting a general equilibrium model of a city where production and consumption
occur.

Henderson (1974b) investigates whether market-achieved city size is greater or less than

optimal city size when externalities such as air pollution are considered.

Arnott (1979) contributes

theoretical work that includes spatial considerations and a utility-maximizing framework.
(1989) offers sound theoretical foundations for optimal city size.

FujitaAlthough definitions vary in theirdetails (e.g. see Nakamura and Kaneuchi (2001)), in essence optimal city size means that attaining
the maximum difference between the aggregate total benefits and total costs of a city.
Compared with the theoretical work that has been done on optimal city size, empiricalstudies have been less developed and seem to provide inconsistent results.
1

can be divided into three groups.

These empirical studies

The first group seeks optimal city size from several perspectives,to determine the relationship between city size and urban agglomeration economies.

For example,

Kelly (1977) estimates the relationship between wage, employment and urban dimensions such ascity population, urbanized population, and population density.

Although not an empirical study,

Moomaw’s (1981) work reevaluates previous studies concerning the relationship between
productivity and city size.

And Henderson (1986) investigates the relationship between efficient

resource usage and city size by using data from SMSA (Standard Metropolitan Statistical Area) in
the U.S.

Yezer and Goldfarb (1978) develop a model demonstrating the conditions necessary for

efficient distribution of labor across areas under the existence of agglomeration economies andcongestion. They also apply an indirect empirical test for efficient allocation. After estimating
the relationship between city size and wage in 90 U.S. cities, they conclude that there are specificcity size ranges where necessary conditions for efficient allocation of resources are not met,
particularly for cities in the 1.5 to 2.5 million population range.
The second group of empirical studies evaluates optimal city size by using the Henry
George Theorem.

Kanemoto et al. (1996) summarize the essence of the theorem as follows: If the

only agglomeration forces are the commuting costs of workers who work at the center of the city,then the optimal city size is achieved when the Pigouvian subsidy for the agglomeration externalities
1

Empirical studies related to city size include the following: an evaluation of Chinese city size by Au
and Henderson (2006); an exploration of the relationship between labor productivity and such structuralfeatures of a city as its size, sprawl and traffic speed by Prud’homme and Lee (1999); an investigation
into the effect of a compact city by Burton (2000); a study of factors associated with worker productivity
by Cervero (2001); and a study of the compact city as a sustainable urban form by Holden and Norland(2005).

equals the total differential urban rent.
and Kanemoto and Saito (1998).

There are two studies in this group: Kanemoto et al. (1996)

Kanemoto et al. (1996), basing their evaluation on the Henry

George Theorem, empirically investigate whether or not Japanese cities, especially Tokyo, exceed
optimal city size.

They estimate aggregate production functions for metropolitan areas in Japan inorder to derive the magnitudes of agglomeration economies.

Because of the difficulty of

converting land prices into land rents, Kanemoto et al. do not test the Henry George Theorem
directly.

Instead, they compare the ratio of the total land values to the total Pigouvian subsidies foreach metropolitan area, and they conclude that there is no evidence supporting the hypothesis thatTokyo is too large.

Kanemoto and Saito (1998)’s study also evaluates whether or not Tokyo is toolarge by using the same methodology as Kanemoto et al. (1996). However, this study differs fromthe previous one in the definition of metropolitan areas and in the estimation method of total land
values.

Kanemoto and Saito (1998) conclude from their results that Tokyo is too large, an outcome

opposite to Kanemoto et al. (1996).
The third group of empirical studies employs an approach that estimates benefit and costfunctions.

Studies in this group include Capello and Camagni (2000), Nakamura and Kanauchi(2001), and Zheng (2007). Capello and Camagni (2000)’s real purpose is to take a critical view oftheoretical works on city size.

In their study, they use 58 Italian cities, estimating the average

location benefit function and the average location cost functions, which consist of variables such as
the size of city (population), the type of urban functions developed, and the network integration level.
They show that the city size attaining the highest average location benefit and the lowest average
location cost is 361,000 and 55,500 population, respectively.

Nakamura and Kanauchi (2001)

investigate optimal city size by using Japanese city data for 1975, 1980, 1985, 1990, 1995 and 1997.
They select observations of between 666 and 693 cities for each year and estimate both the average
benefit function and the average cost function.

In their study, they define several kinds of city size,

including the social optimal city size, the market equilibrium city size, the minimum cost city size,and so on. Among the authors’ several findings are that the social optimal city size in Japan is
between 3.32 million to 5.21 million population, and that the magnitude of the social optimal city
size declined between 1975 and 1997. Finally, Zheng (2007) estimates optimal city size by using
43 Japanese metropolitan areas in the year 2000. He constructs urban theoretical models andestimates both the total benefit and total cost functions based on the models. Total benefit isdefined as total disposal income, and total cost is defined as total expenditures of households. The
optimal city is defined as the maximum total surplus, which is the difference between total benefits
and total costs.

Zhang concludes that a city of optimal size has a population of about 18 million.

While previous empirical studies have made important contributions, they have been
insufficient not only in their number but also in their failure to consider important factors.
many empirical studies do not have theoretical foundations.

First,

There are no explanations as to whycertain variables are selected.
Second, empirical results are inconsistent, with optimal size defined by the same measures
varying from study to study.

Nakamura and Kaneuchi (2001)’s results show that the optimal city

has a population of between 3.32 million and 5.21 million, but Zheng (2007)’s optimal city has 18
million people.
Last, and most importantly, there are no studies in which social costs such asenvironmental pollution have been skillfully taken into account. More attention should be paid tosocial costs. It is the purpose of this study to consider such costs as we estimate optimal city size.
As for our empirical approach, we follow Nakamura and Kaneuchi (2001) and Zheng(2007).

As Arnott (2004) implies, Kanemoto et al. (1996)’s indirect approach of using the Henry

George Theorem still has weaknesses and is difficult to execute in empirical investigation.
The basic framework of our study follows the pioneering work of Zheng (2007), but ourmodel differs from Zheng’s in several ways.
pollution in the model.

First, we include the social cost of environmental

Second, we consider the public sector (local government), which provides

public goods to reduce the city’s environmental pollution.

Third, we set up a model with moregeneral function forms, which are different from those of Zheng (2007).
real estate providers.
areas.

Fourth, we do not include

Last, we use a greater sample size, one that includes smaller metropolitan

Despite these differences, our study has similarities to Zheng’s.

First, the empirical models

are estimated by the function of population, with other parameters such as price of output, housing
price and utility level being fixed over metropolitan areas.

Second, using Japanese city data, weestimate both the benefit and cost functions of the city.

In Japan, as there is no definition of

metropolitan area, we opt to use the metropolitan employment areas defined by Kanemoto andTokuoka (2002).

We use data from the year 2000 in estimating optimal city size.

2. Model of Optimal City
2.1 Basic Structure
Before we explain the structure of the city model, we will summarize importantcharacteristics and basic assumptions.

First, in this model, we consider three important actors: thefirm, the household, and the local government.

The firm produces the single exported good. Inorder to produce the good, the firm employs labor from the household, which receives wages fromthe firm and consumes the imported good, as well as housing.
local government.

The household must remit tax to the

During the course of certain of these activities, both the firm and the household

generate environmental pollution, the negative effects of which decrease the household’s utility level.
The local government’s role is to provide for the public good, in this case by reducing environmental
pollution, but it must do so under the budget constraints of tax revenues.

2.2 Individual Sector(1) Firm
A firm is producing the exported good, QX, by using one input factor, population as labor N.
The firm decides the amount of labor (population) required according to how it can maximize its
profit. The wage rate at the maximum, w*, can be obtained as a function as follows:
QX = QX (N)
maxN

(1)

p・QX (N) – w・N(2)

w* = p・Q'X (N)
where

(3)

QX (.): production function,
QX: exported product,
N: population,
p: price of the exported product,
w: wage.

(2) Household
A household’s utility is obtained by consuming the imported good, z, and a house, s, but its
utility decreases when the environmental condition, e, worsens.

The household decides the amountof imported good and house according to how it can maximize its utility under budget constraints.
U = U (z, s) – e(4)p・z + r・s = w – t,

(5)

where

U (.): utility function,
z: imported goods,
s: housing,e: environment,p: price of imported goods
r: price of housing,t: tax.

Given the utility level, the optimization problem is to determine how the household can minimize its
expenditure when deciding how much imported good and housing it can consume.
minz,s p・z + r・s + t(6)

s.t U = U (z, s) – e(7)

By minimizing the household’s expenditure function, we can obtain each good’s compensateddemand function as follows:
z* = z (p, r, U, e)(8)s* = s (p, r, U, e).

(9)By substituting z* and s* into the household’s expenditure, p・z + r・s, we can obtain the following
expenditure function:
HC* = p・z (p, r, U, e) + r・s (p, r, U, e) = HC (p, r, U, e).

(10)(3) Externalities: Environmental Pollution
Environmental pollution as negative externality is considered.

Environmental conditions

deteriorate as both the firm’s production and the population increase.

The function ofenvironmental pollution in a city is expressed as follows:
CE = CE (QX, N),
where

(11)

CE: environment pollution.

This equation can be rewritten from equation (1):CE = CE (QX (N), N) = CE (N) .

(12)(4) Local Government
The role of local government is to provide for the public good by reducing theenvironmental pollution caused by the activities of both firms and households.

The local

government levies lump-sum tax on households, and total tax revenues are t・ N.

The local

government spends all tax revenues on public goods to reduce environmental pollution.

Therefore,

tax revenues equals production of public goods.
(13)t・N = q・g
where

q: price of public goods,
g : amount of public goods.

Environmental pollution after the production of public goods is expressed as follows:e = CE (N) – q・g = CE (N) – t・N = e (N, t).

(14)

By substituting equation (14) into equation (10), the following result is obtained:
HC* = HC (p, r, U, e(N, t))= HC (p, r, U, N, t).

(15)

2.3 Total surplus, Total Benefits and Total Costs of a City
Total benefits, TB, of a city are ultimately distributed to the total of households.

Thefigure for household’s total revenues is obtained as follows:
TB = w*・N= p・Q'X (N)・N.

(16)

On the other hand, the total cost, TC, of a city is the final expenditure of households and theenvironmental pollution level.
TC = HC*・N + e

It is obtained as follows:= HC (p, r, U, N, t)・N + e (N, t).

(17)

Finally, total surplus, TW, is the difference between total benefits and total costs, and is expressed asfollows:
TW = TB – TC
= p・Q'X (N)・N - HC (p, r, U, N, t)・N - e (N, t).

(18)

2.4 Empirical ModelIn the previous section, we saw that by estimating equation-(16), (17) and (18), we canobtain total benefits, costs, and surplus.
(TC) separately.

In this study, we estimate total benefits (TB) and total costs

First, total benefit is a function of the price of exported output (p), laborproductivity (Q'X (N)), and population (N).

Although labor productivity itself is a function of

population, N, productivity level might be affected by the infrastructure of a city, such as in relation
to transport conditions.

In fact, an empirical study by Cervero (2001) shows a positive relation.

Therefore, we include transport conditions as control variables.

From the sample distribution, in

this study, this function is specified as the log-linear function:ln (TB/p) =  + ln N + 1lnTR1+2lnTR2
where

(19)

TB: total benefit
p: price of exported output
N: population of a city
TR1: road density
TR2: ratio of public transportation.

Second, total cost is a function of household expenditure (HC), population (N), andenvironmental pollution (e).

Household expenditure is a function of the price of exported output(p), housing price (r), utility level (U), population (N), and tax rate (t).
pollution is a function of population (N) and tax rate (t).

Furthermore, environmental

In this study, we assume that only utility

level is constant among different metropolitan areas. Therefore, total cost function is specified as
the log-linear function as follows:ln (TC) =  + ln N + ln p +  ln r + ln t
where

TC: total costs
p: price of exported output
N: population of a city
r: housing pricet: tax rate.

As a result, we estimate the functions shown in equation (19) and (20) separately.

(20)

3 Data
3.1 Sample Selection
As for the analysis, data is collected based on metropolitan areas.

However, as there is no

official definition for metropolitan area in Japan, this study uses the definition of Kanemoto andTokuoka (2002), as do other studies such as Mizuno et al. (2006), and Zheng (2007).

We use 269

employment metropolitan areas, consisting of 113 large employment metropolitan areas and 156
small metropolitan areas, as defined by Kanemoto and Tokuoka (2002).

Data on these areas wascollected in 2000.

3.2 Definition of Variables(1) Total Benefits
First, total benefits (TB) are defined as the total incomes of a city.

One benefit is one

individual income because final benefits are received by each individual. The figure for the city’s
total incomes is obtained by multiplying the average individual income of a city by its population.
It is worth noting that commuting subsidies from companies are excluded from average individual
income, as in Zheng (2007).

Details of variables are explained as follows.

Total benefits are defined as total annual incomes after subtracting commuting subsidies:
TB = AR – COM
where

(21)

TB : total benefits
AR : total annual incomes
COM : commuting subsidies.

Total annual incomes are obtained as follows:
AR = (AHR / HP)・N
where

(22)

AHR : annual incomes per household
HP : persons per household
N : city population.

For some cities, data are not available for total annual incomes per household.
we estimate total annual incomes by using taxable incomes.
AR/N = 1217.31 + 0.747 (YT/N)(14.532) (11.010)
where

R2 = 0.205

YT: taxable incomes(): t-statistics
2

R : coefficient of determination
sample size: 471.

Therefore,

The estimated results are as follows:(23)

Because the portion of total annual incomes that must be estimated accounts for less than 10% in a
metropolitan area, we consider it acceptable to supply missing data by using this method.

We estimate commuting subsidies, following Zheng (2007):
CMS = CMSA・ [(ARU)1/2 / (ARAU)1/2]・L,
where

(24)

CMS: commuting subsidies,
CMSA: average commuting subsidies per person,
ARU: habitable areas in metropolitan area,
ARAU: average habitable areas in metropolitan area,
L: number of employees.

Population figures are obtained from National Census, annual incomes per household andperson per household are obtained from National Survey of Family Income and Expenditure(Zenkoku Shohi Jittai Chosa), and taxable income, habitable areas and number of employees are
obtained from the Municipality Basic Data (Shikuchouson Kiso Deta), all issued by the Ministry ofInternal Affairs and Communications.

Information on average commuting subsidies per person is

obtained from the General Survey on Working Conditions (Chingin Jijoutou Sogo Chosa), issued by
the Ministry of Health, Labor and Welfare.

(2) Total Costs
Total costs are defined as the sum of household expenditures and environmental pollution
costs as social costs:
TC = HC + SC
where

(25)

TC : total benefits
HC : household expenditure
SC : environmental pollution costs.

Household expenditures consist of general consumption expenditures and housing
expenditures.
HC = CNS + HEX
where

(26)

CNS : general consumption expenditure
HEX : housing expenditures.

General consumption expenditures and housing expenditures are obtained as follows:
CNS = 12 (MCNS / HP)・N
where

MCNS : monthly general consumption expenditure per household(27)HP : persons per household
N : city population
HEX = PLH + PRP
where

(28)

PLH : repayment for land and housing
PRP : purchasing payment for properties.

Again, for some cities, data are not available on monthly general consumption expenditure
per household.

Therefore, we estimate general consumption expenditure by using annual

commercial sales.

The estimated results are as follows:ln (CNS/N) = 6.729 + 0.034 ln (COM/N)(57.219) (2.031)
where

(29)

2

R = 0.009

COM: annual commercial sales(): t-statistics

R2: coefficient of determination
sample size: 471.
Because the portion of general consumption expenditures that must be estimated accounts for lessthan 10% in a metropolitan area, we consider it acceptable to supply missing data by using thismethod.
Again, information on population is obtained from National Census.

Monthly general

consumption expenditures per household and person per household are obtained from National
Survey of Family Income and Expenditure (Zenkoku Shohi Jittai Chosa), and annual commercial
sales figures are obtained from the Municipality Basic Data (Shikuchouson Kiso Deta), all issued by
the Ministry of Internal Affairs and Communications.
Environmental pollution costs consist of the social costs of industries and the social costsof transportation.
SC = INSC + TRSC
where

(30)

SC : environmental pollution costs
INSC : social costs of industries
TRSC : social costs of transportation.

The social costs of industries result from air pollution.

The social costs of CO2, NOx,

SOx and SPM emissions are obtained as follows:
INSCij = bi・j (ij・Yj)
where

INSCij : social costs of industry-j for emission type-ibi : unit cost of emission type-i(31)

ij : coefficient of emission type-i in industry-j
Yj : output of industrial products in industry-j
i : 4 kinds of emissions (CO2, NOx, SOx and SPM)
j : 11 industries (agriculture and fishing, mining, construction, utilities,
commerce and food, finance and insurance, real estate, transport andtelecommunications, services, public service).

Information on the output of industrial products in industry-i is obtained from Annual
Report on Prefectural Accounts (Kenmin Keizai Keisan Nenpo), issued by the Cabinet Office.
However, the statistics are based on prefectural units. Therefore, we allocate each prefecture’s
industrial product according to the proportion of the metropolitan areas’ number of employees. The
coefficient of emission type-i is obtained from Embodied Energy and Emission Intensity Data forJapan Using Input－Output Tables (Sangyo Renkanhyo ni yoru Kankyo Fuka Gentani Deta Bukku,
issued by the National Institute for Environmental Studies.
The social costs of transportation consist of air pollution costs and congestion costs,
obtained as follows:
TRSCi= bi・iV + ・RL
where

(32)

TRSCij : social costs of transportation for emission type-ibi : unit cost of emission type-i

i: coefficient of emission type-i for travel length
V : total travel length by cars

 : congestion costs per route-km
RL : route-km of roads
i : 4 kinds of emissions (CO2, NOx, SOx and SPM).
Figures on total travel length by cars are obtained with the following equation:
V = 365 TRP・AREA・ATRL
where

(33)

TRP: number of trip generations and attractions per area per day
AREA : total areas of a city
ATRL : average trip length per trip.

The coefficient of emission type-i for travel length is obtained from Namikawa et al.
(2003).

Figures for trip generation and attractions per area per day, and average trip length are

obtained from Road Transport Census (Doro Kotsu Sensasu), by the Ministry of Land, Infrastructure
and Transport.

To calculate these numbers, we assume that average traffic speed is 35 km/h.

Finally, data for congestion costs per route-km are obtained from Road Transport Census (DoroKotsu Sensasu), by the Ministry of Land, Infrastructure and Transport.

These values are adjusted

according to the wage level of each region.
The unit cost of emission type-i is shown in Table 1.

*******************
Table 1*******************(3) Other Variables
First, population of a metropolitan area (N) is the sum of municipalities’ populations,obtained from National Census.
Second, the price of exported goods (p) is the consumer price index for the generalcategory, and housing price (r) is the consumer price index for the housing category.

These are all

obtained from National Survey of Prices (Zenkoku Bukka Tokei Chosa) issued by the Ministry ofInternal Affairs and Communications.
Third, tax rate (t) is defined as general construction expenses per population. In thisstudy, we assume that public expenditures are used for protection against environmental pollution.
Therefore, general construction expenditures are considered to be encompassed in the concept ofprotection.

Therefore, this variable is a kind of proxy variable.

The figures for general

construction expenditures are obtained from Annual Statistics on Municipal Government Finance(Shi-cho-son betsu Kessan Jokyo Shirabe), issued by the Ministry of Internal Affairs andCommunications.
Road density (TR1) is defined as total road length per habitable area.

And ratio of public

transportation (TR2) is defined as the ratio of the number of employees commuting railway or bus to
the total number of employees.

These data are obtained from both the Municipality Basic Data

(Shikuchouson Kiso Deta) and National Census.

(4) Statistics of Variables
Based on the definition of variables explained above, we construct data.
used for the regression analysis are summarized in Table 2.

All statistics

*******************
Table 2*******************

4 Empirical Analysis of Optimal City Size
4.1 Regression Results
In this section, we apply regressions for the empirical models of total benefits and total
costs, shown in equations (19) and (20).

In addition to these models, we estimate the household

expenditure function in order to compare the differences of results with the total cost function. The
estimation method is the OLS.

Regression results are summarized in Table 3, and the distributionof the actual total benefits and total costs are shown in Figure 1.
From these results, the goodness-of-fit in the regressions is acceptably high for thesemodels because adjusted R2 is very high: these are more than 0.98.

Coefficients of these

explanatory variables seem reasonable.
First, coefficients for population (N) show the positive sign.
benefits and total costs increase as population increases.

Therefore, both total

However, the degree of the coefficientdiffers between the total benefit function, when it is less than one, and the total cost function, when it
is more than one.

These results indicate that at some degree of population, costs exceed benefits.

Second, as the transport conditions (i.e. road density (TR1) and ratio of public
transportation (TR2)) improve, total benefits increase.

However, road density is statisticallysignificant but the ratio of public transportation is not. As small metropolitan areas are included inthis study, most likely the ratio of public transportation, which has a small value in small
metropolitan areas, becomes insignificant.

Furthermore, the effect of road density to total benefits

is not large.
Third, the price of exported output (p), housing price (r), and tax rate (t) all show thepositive sign in the total cost function.

Therefore, as these variables increase, total costs increase.

Among these variables, the price of exported output and the housing price show larger effects on
total costs. As these prices increase by 10%, total costs increase by about 4.7 to 5.3%.
Last, we compare the household expenditure function with the total cost function.
coefficient of population in the household expenditure function is one.
household expenditure costs increase proportionally.

The

This means that the

Therefore, although at some point of

population size household expenditures exceed total benefits, the point in this case would be larger
than in the case of total costs.

*******************
Table 3**************************************
Figure 1*******************

4.2 Estimation of Optimal City Size and Sustainable Limit City Size
In this section, we estimate optimal city size and sustainable limit city size by using
regression results.
surplus (TW).

Optimal city population size is defined as that which attains the largest total

Total surplus (TW) of a city is obtained by subtracting a city’s total costs (TC) fromits total benefits (TB). That is, total surplus (TW) is obtained by TW = TB – TC, as shown in equation(17).

In this study, we use models for total benefits (TB) and total costs (TC), which are estimatedseparately. On the other hand, sustainable limit city population size is that in which a city’s total
benefits (TB) equal its total costs (TC).

If a city exceeds the sustainable limit city size, it is deemedthat the city’s costs, including social costs, are larger than its total benefits.

From a social point ofview, it is not desirable for a city to exceed sustainable size.
Next, we would like to explain how to obtain the maximum city size and the sustainable
limit city size empirically.

As for the optimal city size, from both empirical models shown inequation (19) and (20), total surplus is obtained as follows:
TW = EXP[ + ln N + 1lnTR1+2lnTR2 + ln p] –EXP[ + ln N + ln p +  ln r + ln t].

(34)

Strictly speaking, the total surplus of a city depends on these variables.
main focus is on the relationship between total surplus and city population size.

However, ourWhen we consider

total surplus a function of city population, the first order condition of maximization of total surplus
attains the following condition:
EXP[ + ln N + 1ln TR1+2ln TR2 + ln p] (/N) –EXP[ + ln N + ln p +  ln r + ln t] (/N) = 0.

(35)

From this equation, we can obtain the following equation on optimal city size (Nop):
Nop = EXP[(-  + ln -ln + G1- G2)/( - )]
where

(36)

G1 = 1lnTR1+2lnTR2 + ln p
G2 =ln p +  ln r+ ln t.

As for the sustainable limit city size, as total benefits (TB) equal total costs (TC), fromboth empirical models shown in equation (19) and (20), the following condition holds:
EXP[ + ln N + 1lnTR1+2lnTR2 + lnp] =EXP[ + ln N + ln p +  ln r + ln t].

(37)

By rearranging this equation, we can obtain the following equation on sustainable limit city size(Nsl):
Nsl = EXP[(-  + G1- G2)/( - )]
where

(38)

G1 = 1lnTR1+2lnTR2 + ln p
G2 =ln p +  ln r+ ln t.

In the empirical results shown in Table 3, as  is larger than , the sustainable limit cityslsize, N , is larger than the optimal city size, Nop.
Next, we calculate both optimal city size and sustainable limit city size by using equations(36) and (38).

These city sizes depend on the values of variables other than population, N.

However, as we focus on city size, we fix values of these variables at the sample mean.
Furthermore, the city size changes according to the magnitude of parameters, especially in the caseof the coefficient for population. Therefore, we estimate the possible range of the optimal city sizeby changing the degree of coefficients for population.
and sustainable limit city size are shown in Table 4.

The estimation results of optimal city size

Figure 2 shows the estimated total surplus.

First, the optimal city size, or that which attains the maximum surplus, is 393,151 persons.
This population size characterizes the following metropolitan areas: Hakodate (367 thousand),
Aomori (341 thousand), Takaoka (375 thousand) and Okazaki (379 thousand).

As shown in Table

4, the size range is between 1,639,427 and 120,318 persons, much smaller than in previous studies,for example, a study by Zheng (2007), which shows that optimal city size is about 18 million.
However, while Zheng’s study does not consider social costs such as environmental pollution, we
take these costs into account, so that in our estimation, optimal city size becomes smaller.

In fact,

when we observe only social costs, we see that average social costs increase more sharply as
population increases.
Second, the sustainable limit city population size, in which total benefits equal total costs,
is 1,057,412. This kind of city size locates between 4,410,401 and 323,532.

This kind of city size

seems more close to the real metropolitan areas in Japan.

According to our estimation, although thethree largest metropolitan areas—Tokyo (31.8 million), Osaka (12.1 million), and Nagoya (5.4
million)—are too large, other metropolitan areas such as Kumamoto (1,021 thousand), Shizuoka(999 thousand, Hamamatsu (920 thousand) and Niigata (950 thousand) qualify as sustainable city
populations.

*******************
Table 4**************************************
Figure 2*******************

5

Conclusions
While urban economists have carried out theoretical studies on optimal city size,

empirical studies have not been well developed and have produced inconsistent results.

In order forplanners and policy makers to create a sustainable society, more empirical work needs to be done totest the desirability of a shift toward compact cities.
The main purpose of this study was to investigate empirically the optimal size of a city.
Although we find an optimal city size attaining maximum total surplus (i.e. the difference between
total benefits and total costs), our study differs from others in that we include social costs such asenvironmental pollution in total costs. Another important characteristic of our study is that theempirical models are obtained from theoretical models.

In order to find the optimal city size, weused a data set of 262 Japanese metropolitan areas for the year 2000 and applied several regression
models for total benefits and total costs.

From our results, we pinpoint the following points asimportant.
First, from the regression analysis, coefficients of population (N) show the positive sign inboth the total benefit function and the total cost function.
costs increase as population increases.

Therefore, both total benefits and total

However, the degree of the coefficient between total benefit

function and total cost function is different.

As it is less than one in the total benefit function but

more than one in the total cost function, at some degree of population, costs overtake benefits.
Second, the optimal city size, that attaining the maximum surplus, is 393,151 persons,much smaller than in previous studies. Our study is most distinguishable from others in that social
costs are considered, which may account for our much smaller optimal city size.

In fact, when weobserve only social costs, the average social costs increase more sharply as population increases.
Last, the sustainable limit city size, in which total benefits equal total costs, is 1,057,412.
This kind of city size closely reflects real metropolitan areas in Japan: Kumamoto, Shizuoka,
Hamamatsu and Niigata, all of whose quality of life is considered relatively good.
In conclusion, a city’s benefits and costs are so complex and there are so many factors
affecting city size that it is difficult to pinpoint exactly what a city population should be.

However,

if we consider social costs, the ideal city has a much smaller population than many cities in theworld today. Currently, there is a trend in urban planning based on the concept of compact cities.
Our results might provide empirical evidence to show that this trend is a step in the right direction.

[2012.5.1 1086]
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Table 1 Unit Cost of Emission TypeType of emission

CO2

NOX

SOX

SPM

Unit

Yen/t-C

Thousand yen/tThousand yen/tThousand yen/tValues2,423

11,841

1,150

10,989(Note):
These values are obtained from original literature: Nakamura (1997) for CO2, Small and Kazimi(1995) for NOX, SOX and SPM.

Minimum

Maximum

Total Benefit (TB)

Table 2 Statistics for the Analysis
Unit
Mean
Standard
Deviation
million yen
908,610
4,493,090

28,233

68,405,200

Total Cost (TC)million yen

970,001

5,354,000

27,94382,548,300

HouseholdExpenditure (HC)Population (N)million yen

564,976

2,907,840

19,901

44,398,900

person435,344

2,107,368

16,18431,814,200

Price of exported
output (p)Housing price (r)

-

1.016

0.029

0.945

1.102

-

0.816

0.116

0.563

1.645

Tax rate (t)index

58.66119.34817.337137.459

Road density (TR1)km/km2

0.100

0.038

0.024

0.228

Ratio of public
transportation (TR2)

-

0.113

0.088

0.014

0.641

VariableTable 3 Regression Results: Coefficients and Standard Errors
Model TB
Model TCln(TB/p)ln TC
Population
0.985***
1.037***(ln N)(0.008)(0.011)
0.067***
Road Density(0.020)(ln TR1)
0.022
Ratio of Public(0.014)
Transportation (ln TR2)
Price of Exported Output
0.530***(ln p)(0.089)Housing Price
0.470***(ln r)(0.089)Tax Rate
0.015(ln t)(0.032)
Constant
7.992***
7.120***(0.137)(0.193)Adjusted R2
0.987
0.980

Model HC
ln HC
1.001***(0.008)0.836***(0.064)0.164***(0.064)0.027(0.023)7.014***(0.134)0.989(Note):(1) Significant at 1% (***), 5% (**), 10% (*).

Table 4 Estimated Optimal City Size and Sustainable Limit City Size
Case
Lower Limit
Most Likely
Upper Limit
Optimal City Size

120,318

393,151

1,639,427

Sustainable Limit City Size

323,532

1,057,412

4,410,401(Note):(1) Optimal city population size is that at which a city attains maximum net benefits.
Sustainable limit city population size is that at which a city’s total benefits equal total costs.
(2) The upper and lower limits are obtained from the upper and lower bounds of the coefficient ofa city’s population (N) for both the total benefit (TB) function and the total cost function (TC).
The upper limit is obtained by using the upper limit in the total benefit function and the lower
limit in the total cost function. The lower limit is obtained by using the lower limit in total
the benefit function and the upper limit in total cost function. The 20% confidence interval isused for both functions.

lnTB
lnTC

26
25

lnTC

24

lnTB

23
22

lnTB
lnTC

21
20
19
18
17
16
8

9

10

11

12

13

14

15

16

lnN

Figure 1 Distribution of the Actual Total Benefits and Total Costs

17

18

TW (billon yen)
50
40
30
20
10
0
-10

0

200

400

600

800

1,000

1,200

1,400
N (thousand)

-20
-30
-40
-50

Figure 2 Estimated Total Surplus