Department of Policy and Planning Sciences
Discussion Paper Series

No.1328

Relationship between urban population growthand utility level of workers with heterogeneity of labor
efficiency
byNoriyuki HIRAOKA, Mitsuru OTA, and Stephen J. TURNBULL

May 2015

UNIVERSITY OF TSUKUBA
Tsukuba, Ibaraki 305-8573
JAPAN

Relationship between urban population growthand utility level of workers with heterogeneity of labor efficiency
Noriyuki Hiraoka i, Mitsuru Ota ii and Stephen John Turnbull iii

Abstract
We analyze the impact of urban population growth on individual utility in a generalequilibrium model of monopolistic competition with product diversity, pro-competitive effectand heterogeneity of labor efficiency between workers. We assume a monocentric city where
high-income class resides in the center while low-income class resides in the suburbs. When
urban population grows, utility level increases through the mass and price of product varieties
and land rent earned, while it decreases through effective labor supply and land rent paid. EachLPSDFWRQZRUNHU¶VXWLOLW\WKURXJKthese five factors is investigated separately. Finally, it isshown that, in case where urban population increases without the reduction of commuting
cost and the increase in labor efficiency, middle-income class has the highest possibility tolose utility and high-income class also has the possibility, while low-income class always
gains.
Keywords: utility level, urban population growth, heterogeneity of labor efficiency, mass ofvarieties, price of varieties, effective labor supply, land rent earned, land rent paid

1. Introduction
One of the important issues in urban/spatial economics is whether the utility level ofcitizens in a city increases or decreases as urban population grows. Textbooks on urban
economics such as Mills and Hamilton (1997) and O¶Sullivan (2012) present that cities haveboth economy and diseconomy of scale. In general, in the range of smaller population, utility
increases as the urban population grows, while, in the range of larger population, utility
decreases. At the birth of a city, it grows in terms of the economies of scale at the city levelsuch as increasing returns to scale in production, face-to-face communication, transaction
costs and the mass of varieties in product. As the population grows, urban costs such as land
rent, commuting cost, congestion and pollution increase. In a model of an autarkic city the
optimal city size is determined at the population which equalizes the marginal economy andiGraduate School of Systems and Information Engineering, University of Tsukuba,
Mitsubishi Research Institute, Inc.
ii Faculty of Engineering, Information and Systems, University of Tsukuba
iii Faculty of Engineering, Information and Systems, University of Tsukuba
1diseconomy, and in a model of a system of cities the city sizes are determined at the levels of
population which equalize utilities of the cities.
Although most of papers in urban/spatial economics assume that workers are identical,
there exist a small number of papers made up of two groups of papers which introduce
workers with heterogeneity. One is the group which analyzes the impact of heterogeneity on acity or a system of cities. Mansoorian and Myers (1993) introduced individuals with different
degrees of attachment to homes and showed that an incentive to make interregional transfers
in purchasing is necessary to realize the preferred population distribution in a system of cities.
Tabuchi and Thisse (2002) concluded that taste heterogeneity acts as a strong dispersion force
and full agglomeration does not realize in a two-region model. Murata (2003) showed thatmarket-mediated product diversity yields an agglomeration force through the home market
effect, whereas taste heterogeneity due to non-market interaction induces a dispersion force.
Amiti and Pissarides (2005) introduced skill differentiation between workers and showed thatthe reduction of labor mismatch is one of agglomeration forces.
The other is the group which analyzes the impact of the change in a city or a system ofcities on workers with heterogeneity. Behrens and Murata (2012a) introduced heterogeneityof labor efficiency between workers with a variable-elasticity-of-substitution type utility
function. They concluded that, when an economy transitions from autarky to free trade, themass of varieties may shrink in a higher income country and the richer cons umers in the
country may lose.
This paper belongs to the latter group. We introduce heterogeneity of workers to a
monocentric city model and analyze the impact of urban population change on the utility ofworkers. There exist various kinds of heterogeneities between workers which include
attachment to homes, skill, taste and labor efficiency described above. At the first step, we
pick heterogeneity of labor efficiency for our subject of study since we can easily infer thatworkers who vary in income caused by labor efficiency are affected differently in change ofsocial environment and the income distribution is statistically measured. Figure 1 illustrates
the taxable income distribution per worker employed in the private sector in Tokyo
metropolitan area. It implies that for the most part the distribution is decreasing in income.

2

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2.'!

&.2

1.&

%.1

0.%

$.0

/.$

#./

'.#

()*+,-'

#!"
'&"
'%"
'$"
'#"
'!"
&"
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$"
#"
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Taxable income per taxpayer(Unit: million Japanese yen)
Figure 1. Taxable income distribution in Tokyo metropolitan area: prefectures of Chiba,
Tokyo, Kanagawa and Yamanashi (Source: National Tax Agency, Japan, 2012, Survey on

e mployee incom e in the private sector )
70km
60km
50km
40km
30km
20km
10km

Figure 2. Regional distribution of taxable income per taxpayer in Tokyo metropolitan area:
Prefectures of Tokyo, Saitama, Chiba and Kanagawa (Source: National Tax Agency, Japan,
2013, Survey on i m position of m unicipal tax )3

In general, high-income class in U.S. tends to reside in the suburbs, while in Asia,
Europe and Latin America, low-income class resides in the suburbs (Muth (1969) and
Hohenberg and Lees(1986)). Muth(1969) explained this tendency with the difference ofincome elasticities of land demand and commuting cost. Fujita (1989) introduced the value ofcommuting time in addition to pecuniary payment for commuting in order to explain thetendency that middle-class resides farthest in the San Francisco Bay Area, which is the resultof empirical research in Wheaton(1977). Bruckner et al. (1999) presented an amenity-based
theory for the relative location of different income classes in Paris. Tokyo is one of giant
metropolises in Asia. Figure 2 illustrates the regional distribution of the taxable income pertaxpayer. It implies that, in general, high-income class resides closer to the center of Tokyo
metropolitan area, and income level is decreasing with distance. We analyze the relationshipbetween urban population growth and utility level of workers in the urban areas like Tokyo
metropolitan area where high-income class resides in the center.
Although urban area is two-dimensional, one-dimensional models are frequently
employed for the analysis of urban structure such as Fujita and Ogawa (1980), Ota and Fujita(1993), and Fujita and Krugman (1995). We also apply one-dimensional model to the study.
In the model, when urban population grows, utility level increases through the mass and priceof product varieties and land rent earned, and decreases through effective labor supply at theCentral Business District and land rent paid. $IWHUHDFKLPSDFWRQZRUNHU¶VXWLOLW\WKURXJKILYH

factors is shown separately, the total impact on utility is investigated. The result indicates that,in case where urban population increases without the reduction of commuti ng cost and theincrease in labor efficiency, workers with middle labor efficiency have the highest possibility
to lose utility and workers with higher labor efficiency also have the possibility, while
workers with lower labor efficiency always gain.
This paper is in five parts. Section 2 lays out the assumptions of the model. We derive
endogenously determined variables under spatial equilibrium conditions in Section 3 andpresent the impact on workers with heterogeneity of labor efficiency in terms of population
growth in Section 4. Section 5 considers the implication for the bottom line and roughly
compares the result and Tokyo metropolitan area. Section 6 concludes.

2. ModelWe assume that a monocentric city on a large homogeneous land which stretches out
along one-dimensional space ॿ and the amount of land available at each location ‫ ॿ א ݔ‬isone. All firms in the city are located at a single dimensionless point, Central Business District(henceforth, CBD), hence they do not consume land. We label the location of the CBD as theorigin of ॿ. The land of the city is consumed only by workers to live on. Workers commute
to firms located at the CBD to earn wages and consume land and varieties produced by firms.
For land ownership, we employ the public land ownership model in Fujita (1989). We
4

consider the city as an individual jurisdiction that owns the land of its region, and there exists
no global government. Namely, each worker living in the city owns an equal share of land,
and receives an income from their land ownership in addition to their wage. For simplicity,
farmer¶s rent is assumed to be zero. Each worker needs a fixed living space, that is, he/she
consumes a unit of land inelastically. This implies that workers live symmetrically around the
CBD and that the commuting distance of workers who live farthest from the CBD is ‫ܮ‬Τʹ, thatis, the city covers the interval ሾെ ‫ܮ‬Τʹǡ ‫ܮ‬Τʹሿ, where ‫ ܮ‬൐ Ͳ is the urban population of workers
given exogenously.
Each worker is endowed with labor efficiency ݄ ൒ Ͳ and supplies it inelastically tofirms at the CBD. We introduce heterogeneity of labor efficiency between workers by putting
the distribution function of labor efficiency ݃ሺ݄ሻ and the cumulative distribution function‫ܩ‬ሺ݄ሻ as follows:݀‫ܩ‬ሺ݄ሻ ൌ ݃ሺ݄ሻ݄݀

,‫ܩ‬ሺ݄௠௔௫ ሻ ൌ ͳ.

‫ܩ‬ሺͲሻ ൌ ͲǡFunctions ݃ሺ݄ሻ and ‫ܩ‬ሺ݄ሻ are assumed to be continuously differentiable for ݄ ‫ ג‬ሺͲǡ ݄௠௔௫ ሻ.
The average labor efficiency ݄ is given as follows:
௛೘ೌೣ

݄ൌන

݄ ݃ሺ݄ሻ݄݀

.

଴We assume that commuting cost borne by workers is of iceberg type, following Murata andThisse (2005) and Behrens and Murata (2009). Discount factor߬ incurred by commuting
from the location ‫ ݔ‬to the CBD is߬ ‫ ͳ ؠ‬െ ʹߠȁ‫ݔ‬ȁ,!
where ߠ ൐ Ͳ is the parameter of labor efficiency loss caused by commuting. The effective
labor supply ‫ ݏ‬which is the net amount of labor supplied at the CBD by a worker with labor
efficiency ݄ living at a distance ‫ ݔ‬from the CBD is given by‫ ݏ‬ൌ ݄߬ ൌ ݄ሺͳ െ ʹߠȁ‫ݔ‬ȁሻ.!

(1)

Under this assumption of commuting costs, the maximum city size in the model is ͳΤߠ Ǥ
There exist three major utility functions employed in monopolistic competition models.
The utility function most frequently employed is the constant-elasticity-of-substitution (CES)
type utility function, which is based on horizontally differentiated varieties , introduced in
Dixit and Stiglitz (1977). Models with a CES type utility function have income effect but do
not have pro-competitive effect. Another frequently employed utility function is the quadratic5

utility function with a continuum of varieties, which is proposed in Vives (1985) andintroduced into spatial economics in Ottaviano et al . (2002). Models with a quadratic utility
function have pro-competitive effect but do not have income effect. The third is thevariable-elasticity-of-substitution（VES） type utility function which is introduced in Behrens
and Murata(2007). Models with a VES type utility function have both income andpro-competitive effects. Since we analyze the impact including income and price changes,
following Behrens and Murata (2007, 2009, 2012a, 2012b) and Behrens et al. (2008, 2013),
we employ a following VES type utility function:ܷ ‫ ؠ‬න ‫ݑ‬ሺ‫ݍ‬ሺ݅ሻሻ݀݅

,!

(2)

ఆ

where, ‫ݑ‬ሺ‫ݍ‬ሺ݅ሻሻ ‫ ͳ ؠ‬െ ݁ ିఈ௤ሺ௜ሻ , ‫ݍ‬ሺ݅ሻ is the quantity of variety ݅ consumed by a worker, ߗ isthe set of varieties produced in the city and ߙ ൐ Ͳ is a utility parameter.
All firms have the same increasing returns to scale technology, and require ሺܿܳሺ݅ሻ ൅݂ሻ݄଴ units of labor to produce ܳሺ݅ሻ units of varieties, where ܿ ൐ Ͳ is the marginal and ݂ ൐Ͳ is the fixed labor requirements. We assume that firms can costlessly differentiate theirproducts. This implies that, under the increasing returns to scale technology, there is
one-to-one correspondence between firms and varieties, so that the mass of varieties ܰ isequal to the mass of firms (Fujita et al. (1999), Fujita and Thisse (2013)). Transport costs ofvarieties in the city are assumed to be zero.

3. Spatial equilibrium
We label the wage which is paid for the labor supply ݄଴ at the CBD as ‫ݓ‬଴ . On theassumption of production technology, the contribution of each worke r to production is
proportional to effective labor supply, hence the wage ‫ ݓ‬for the effective labor ‫ ݏ‬is‫ ݓ‬ൌ ‫ݓ‬଴ ሺ‫ݏ‬Τ݄଴ ሻ.

Without loss of generality, we put ‫ݓ‬଴ Ȁ݄଴ ൌ ͳ. Hence we have ‫ ݓ‬ൌ ‫ݏ‬, and, hereafter, we canuse wage ‫ ݓ‬and effective labor supply ‫ ݏ‬interchangeably.
Assigning effective labor supply ‫ ݏ‬to wage ‫ ݓ‬in (1) yields‫ ݓ‬ൌ ݄ሺͳ െ ʹߠȁ‫ݔ‬ȁሻ.

(3)

Since the city is symmetric around the CBD, hereafter, we consider the positive half of thearea ‫ ݔ‬൒ Ͳ. We obtain the gradient of a bid-rent curve of each worker by differentiating thewage with respect to the location ‫ ݔ‬as follows:6݀‫ݓ‬ൌ െʹ݄ߠ
݀‫ݔ‬

.

(4)

As shown in Fujita (1989), a worker with the steeper gradient of a bid-rent curve lives closer
to the CBD. Since (4) shows that the absolute value of the gradient of a bid -rent curve is
proportional to the labor efficiency of each worker, a worker with higher labor efficiency
lives closer to the CBD. A worker lives at the position where the gradient of the land rent ܴis equal to that of her bid-rent curve as follows:ܴ݀ൌ െʹ݄ߠ
݀‫ݔ‬.!

With attention to the assumption that each worker consumes a unit of land inelastically,
integrating the above equation yields‫ ܮ‬௛೘ೌೣܴ ൌ ܴ଴ െ නʹߠ݄݃ሺ݄ሻ݄݀ʹ ௛௛೘ೌೣൌ ܴ଴ െ ߠ‫ ܮ‬න݄݃ሺ݄ሻ݄݀

௛,where ܴ଴ is land rent at the CBD. Since workers with labor efficiency ݄ ൌ Ͳ live at theurban boundary, ȁ‫ݔ‬ȁ ൌ ‫ܮ‬Τʹ, and their land rent ܴ௅Ȁଶ is zero on the assumption of that
farmer¶s rent is zero, we obtain the land rent at the CBD as follows:௛೘ೌೣܴ௅Ȁଶ ൌ ܴ଴ െ ߠ‫ ܮ‬න݄݃ሺ݄ሻ݄݀

଴,ൌ ܴ଴ െ ߠ‫ ݄ܮ‬ൌ Ͳ

hence ܴ଴ ൌ ߠ‫݄ܮ‬ത. The land rent expressed in terms of the labor efficiency ݄ of a worker is௛೘ೌೣܴ ൌ ߠ‫ ܮ‬ቊ݄ െ න݄݃ሺ݄ሻ݄݀ቋ

௛

.

(5)

Since each worker consumes a unit size of land and workers with higher labor efficiency
live closer to the CBD, we obtain the relationship between a place of residence ‫ ݔ‬and labor
efficiency ݄ as follows:7‫ܮ‬‫ ݔ‬ൌ ሼ‫ܩ‬ሺ݄୫ୟ୶ ሻ െ ‫ܩ‬ሺ݄ሻሽʹ‫ܮ‬ൌ ሼͳ െ ‫ܩ‬ሺ݄ሻሽ
ʹ

(6)
.

After a specific cumulative distribution function ‫ܩ‬ሺ݄ሻ is given, we can obtain therelationship between location and land rent by substituting the inverse function of (6) into
(5).
Expenditure for varieties by a worker is‫ ܧ‬ൌ ‫ ݏ‬െ ܴ ൅ ሺߎ ൅ ‫ܴܮܣ‬ሻȀ‫ܮ‬,(7)where the first term on the right hand is the wage earned by the effective labor supply, thesecond term is the land rent paid for a unit of land, and the third term is the total profit offirms and aggregate land rent per capita. The utility maximization problem of a worker isgiven by‫ݍ‬ሺ݅ሻǡ ݅ ‫ ＝ܷ ߗ ג‬න ‫ݑ‬ሺ‫ݍ‬ሺ݅ሻሻ݀݅ ‫ݏ‬Ǥ ‫ݐ‬Ǥ න ‫݌‬ሺ݅ሻ‫ݍ‬ሺ݅ሻ݀݅ ൌ ‫ܧ‬
ఆ

ఆ

,

(8)

where ‫݌‬ሺ݅ሻ is the price of variety ݅. Solving this maximization yields the demand for variety
݅ as follows:ͳ‫݌‬ሺ݅ሻ‫ ܧ‬െ ߙ ‫׬‬ఆ  ൬൰ ‫݌‬ሺ݆ሻ݆݀
‫݌‬ሺ݆ሻ
‫ݍ‬ሺ݅ሻ ൌ‫׬‬ఆ ‫݌‬ሺ݆ሻ݆݀

(9).

Here, we derive the aggregate labor supply at the CBD, the aggregate expenditure forvarieties and the aggregate demand for variety ݅ for future reference. The aggregate labor
supply at the CBD ܵ is as follows:
௅
ଶ

(10)

ܵ ൌ න ‫ݔ݀ݏ‬௅ିଶ.

The aggregate expenditure for varieties ‫ ܧܣ‬is related to the aggregate labor supply as
follows:8௅ଶ‫ ؠ ܧܣ‬න ‫ݔ݀ܧ‬
௅
ିଶ

(11)

௅
ଶ

ߎ ൅ ‫ܴܮܣ‬ൌ න ൜‫ ݏ‬െ ܴ ൅ൠ ݀‫ݔ‬
௅
‫ܮ‬
ି
ଶ

,

ൌܵ

where we apply the fact that ‫ ܴܮܣ‬is the total amount of the land rent ܴ over the city and thezero profit condition for firms which is assumed afterwards. The aggregate demand ܳሺ݅ሻ forvariety ݅ is related to the aggregate labor supply at the CBD ܵ and the prices of varieties‫݌‬ሺ݅ሻ as follows:

௅
ଶ

ܵെ

ܳሺ݅ሻ ‫ ؠ‬න ‫ݍ‬ሺ݅ሻ݀‫ ݔ‬ൌ
௅
ିଶ

‫ܮ‬
‫݌‬ሺ݅ሻ ൬൰ ‫݌‬ሺ݆ሻ݆݀ߙ ‫׬‬ఆ
‫݌‬ሺ݆ሻ
‫׬‬ఆ ‫݌‬ሺ݆ሻ݆݀

(12).

The price elasticity ߳ of the aggregate demand is߳ൌെ‫݌‬ሺ݅ሻ ߲ܳሺ݅ሻ
‫ܮ‬ൌܳሺ݅ሻ ߲‫݌‬ሺ݅ሻ ߙܳሺ݅ሻ

(13)

,

which is inversely proportional to ߙ and ܳሺ݅ሻ.
The profit of a firm is given by the following equation:ߨሺ݅ሻ ൌ ‫݌‬ሺ݅ሻܳሺ݅ሻ െ ሺܿܳሺ݅ሻ ൅ ݂ሻ݄଴

.

(14)

We assume free entry and exit of firms, hence the profit of firms is zero at the equilibrium.
Each firm maximizes its profit (14) with respect to ‫݌‬ሺ݅ሻ, taking expenditure for varieties andthe mass of firms given. Substituting (12) into (14) yields‫ܮ‬‫݌‬ሺ݅ሻܵ െ ߙ ‫׬‬ఆ  ൬൰ ‫݌‬ሺ݆ሻ݆݀
‫݌‬ሺ݆ሻ
ሺ‫݌‬ሺ݅ሻ െ ݄଴ ሻ െ ݂݄଴ߨሺ݅ሻ ൌ
‫׬‬ఆ ‫݌‬ሺ݆ሻ݆݀

.

The first-order condition for profit maximization with respect to ‫݌‬ሺ݅ሻ is given as follows:‫݌‬ሺ݅ሻ
‫݌‬ሺ݅ሻ െ ݄ܿ଴න  ቆቇ ‫݌‬ሺ݆ሻ݆݀ ൅න ‫݌‬ሺ݆ሻ݆݀ ൌ ߙ‫ܧ‬ത‫݌‬ሺ݆ሻ
‫݌‬ሺ݅ሻ
ఆ
ఆ

9

,

(15)

where ‫ ܧ‬ൌ ‫ܧܣ‬Ȁ‫ ܮ‬is the average expenditure for varieties by workers. We can prove that theprice equilibrium is symmetric and unique（see Appendix A）. Evaluating (15) at thesymmetric prices yields‫ ݌‬ൌ ݄଴ ൅

ߙ‫ܧ‬

ܰ

,

(16)

where ܰ is the mass of firms/varieties.
At the symmetric price equilibrium, demands for varieties are also symmetric. In (14),eliminating the suffix ݅ which identifies a specific firm and using the macroscopic budget
constraint ‫ ܧܣ‬ൌ ܰ‫ ܳ݌‬yields
ͳ ܵ
ܳൌ ൬
െ ݂൰
ܿ ݄଴ ܰ

(17).

The labor market clearing condition is given as follows:න ሺܿܳ ൅ ݂ሻ݄଴݀݅ ൌ ܵ.

ఆSubstituting ܳ ൌ ‫ܧܣ‬Ȁܰ‫ ݌‬into this equation and solving it for the mass of firms ܰ yields

ܰൌ

ܵ
݄ܿ଴
൬ͳ െ
൰
݂݄଴
‫݌‬

(18)

.

We obtain the mass of firms in terms of the population by substituting (16) into (18) andsolving the equation for the mass of firms ܰ as follows:ܰൌܵ ‫ܦ‬ሺ‫ܮ‬ሻ െ ߙ݂݄଴ ‫ܮ‬
ʹ݂ܿ

(19)

,

where ‫ܦ‬ሺ‫ܮ‬ሻ ൌ ඥͶߙ݂ܿ‫ ܮ‬൅ ሺߙ݂ሻଶ

.

There exists a critical urban population ‫ܮ‬෨ in (19). The mass of firms increases in the smallerurban population than ‫ܮ‬෨ and decreases in the larger urban population than ‫ܮ‬෨. In Behrens and
Murata (2012a), the reason why utility of richer workers in a higher income country becomes
lower is that, after the two countries transition from autarky to free trade, in tha case that the
total population becomes larger than the critical population ‫ܮ‬෨, the mass of varieties decreases
in the country.
Finally, under the symmetric quantity of varieties, indirect utility function of workers
can be obtained as follows:10 ൌ ܰ ቀͳ െ ݁ ିఈ௤ሺேǡ௣ǡோǡ௦ǡோሻ ቁ
where

,

‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻ ൌ(20)ͳ൫‫ ݏ‬െ ܴ ൅ ܴ൯
ܰ‫݌‬

,

(21)

where ܴ ൌ ‫ܴܮܣ‬Ȁ‫ ܮ‬is aggregate land rent per capita.

4. Relationship between urban population growth and utility level
When the urban population changes, as shown in (20), utility of a worker is affected
through five variables: the mass of varieties ܰ, the price of varieties ‫݌‬, land rent earned ܴ,
effective labor supply ‫ ݏ‬and land rent paid ܴ. The total differential of indirect utility
function with respect to population can be broken down into five partial differentials as
follows:
ܸ݀ ߲ܸ ݀ܰ ߲ܸ ݀‫ܴ݀ ܸ߲ ݏ݀ ܸ߲ ܴ݀ ܸ߲ ݌‬
ൌ
൅
൅
൅
൅
݀‫ܮ݀ ܴ߲ ܮ݀ ݏ߲ ܮ݀ ܴ߲ ܮ݀ ݌߲ ܮ݀ ߲ܰ ܮ‬

.

(22)

To take a step further, emulating the income distribution from statistics which is shownin Fig.1, we assume the distribution function of labor efficiency of workers as a linearlydecreasing function, as follows:ʹ݃ሺ݄ሻ ൌ ൝݄௠௔௫

൬ͳ െ
Ͳ

݄

Ͳ ൑ ݄ ൑ ݄௠௔௫൰݄௠௔௫  ൜݄ ൏ Ͳǡ݄௠௔௫ ൏ ݄(23)

The average labor efficiency under the distribution function ݄ത is ݄௠௔௫ Ȁ͵Ǥ
Figure 3 illustrates an example of the distribution function ݃ሺ݄ሻ.
Distribution
0.8
0.6

gh
2
3h

0.4
0.2

hmax

h
0

0.5

1.0

1.5

2.0

2.5

3.0

Labor Efficiency hFigure 3. The distribution function ݃ሺ݄ሻ (The set of parameters applied to all figures is ܿ ൌ11ͲǤͳ, ݂ ൌ ͲǤͳ, ߙ ൌ ͲǤʹ, ߠ ൌ ͲǤͷ, ‫ ܮ‬ൌ ͳ and ݄ത ൌ ͳ, unless otherwise noted.)
From (5) and (6), the land rent under the distribution function (23) isܴൌߠ‫݄ܮ‬ଶʹ݄൬ͳ െ ത ൰
ത
͵݄
ͻ݄
(24)

ଷ

ൌ ߠ‫݄ܮ‬ത ቐͳ െ ͵ ൬ʹ‫ݔ‬ʹ‫ ݔ‬ଶ൰ ൅ ʹ൬ ൰ ቑ‫ܮ‬‫ܮ‬Figure 4 illustrates an example of the land rent curve. It is strictly decreasing and convex indistance.
Land Rent R
0.5

R0

0.4
0.3
0.2
0.1
0

0.1

0.2

0.3

0.4

L2
0.5

Location x

Figure 4. Land rent curve (The set of parameters is shown at Fig.3.)

For future reference, we note a number of variables under the distribution function (23)
as follows:‫ ݏ‬ൌ ݄ ቊͳ െ ߠ‫ ܮ‬൬ͳ െ

݄ ଶ
൰ ቋ
͵݄ത

,

ܵ ൌ ݄ത‫ ܮ‬൬ͳ െ

͵
ߠ‫ܮ‬൰
ͳͲ

,

‫ ܧ‬ൌ ݄ത ൬ͳ െ

͵
ߠ‫ܮ‬൰
ͳͲ

,

ܴത ൌ

͵
݄തߠ‫ܮ‬
ͳͲ

.

(25)
(26)
(27)
(28)

As for the relationship between the population and the mass of varieties, examining(19) under the distribution function (23), the maximum of the mass of varieties is at the
population ‫ ܮ‬ൌ ‫ܮ‬෨, where12‫ܮ‬෨ ൌሺͳͲܿ െ ߙ݂ߠሻ ൅ ඥͳͲߙ݂ܿߠ ൅ ሺߙ݂ߠሻଶ
ͻܿߠ

.

Since we can prove that ‫ܮ‬෨ is larger than the maximum population ͳȀߠ (see Appendix B),
the mass of varieties is always increasing in population. Figure 5 illustrates an example of therelationship between the population and the masses of varieties.

Mass of Firms N
4

3

2

1

0

1

2

3

4

Populatin L

Figure 5. Relationship between the population and the masses of varieties (The set ofparameters is shown at Fig.3. ͳΤߠ ൌ ʹ and ‫ܮ‬෨ ൎ ʹǤͶʹ͵͵)In this section, at first, each impact on worker ¶s utility through five factors is examined
separately. After that, the study on the total effect which synthesizes these five impacts is
made.
4.1 Separated impacts through five factors
4.1.1 Impact through the mass of varieties
The impact on the indirect utility function through the change in the mass of varieties isderived by differentiating (20) with respect to ܰ as follows:μܸൌ ͳ െ ሺͳ ൅ ߙ‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻሻ݁ ିఈ௤ሺேǡ௣ǡ஺௅ோǡ௦ǡோሻ
μܰ,

which is positive when ‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻ is positive. Since demand for each varietyப௏‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻ is positive and increasing in labor efficiency（see Appendix C）, பே is positive
for all workers. The impact of population change through the mass of varieties is߲ܸ ݀ܰ݀ܰൌ ቄͳ െ ൫ͳ ൅ ߙ‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻ൯݁ ିఈ௤ሺேǡ௣ǡோǡ௦ǡோሻ ቅ߲ܰ ݀‫ܮ‬݀‫ܮ‬13.

which is positive, since the relationship between the population and the mass of varieties ispositive. Utility of every worker increases as the population grows. Differentiating thisequation with respect to labor efficiency yields
݀ ߲ܸ ݀ܰ
݀‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻ ݀ܰ൬൰ ൌ Ƚଶ ‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻ݁ ିఈ௤ሺேǡ௣ǡோǡ௦ǡோሻ݄݀ ߲ܰ ݀‫ܮ‬݄݀݀‫ܮ‬which is always positive. The larger labor efficiency is, the larger

డ௏ ௗே
డே ௗ௅

,

is. Figure 6aడ௏ ௗேillustrates an example of the relationship between labor efficiency and డே ௗ௅ .
4.1.2 Impact through the price of varieties
The impact on the indirect utility function through the change in the price of varieties isderived by differentiating (20) with respect to ‫ ݌‬as follows:μܸߙ‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻܰ ିఈ௤ሺேǡ௣ǡோǡ௦ǡோሻ
ൌെ
݁
μ‫݌‬
‫݌‬

,

which is always negative, hence utility of workers increases when the price of varieties falls.
From (16) and (27), the change in the price of varieties is

݀‫݌‬
ൌ
݀‫ܮ‬

݀ ቆ ݄଴ ൅
݀‫ܮ‬

Ƚ‫ܧ‬
ቇ
ܰ

ൌെ

ߙ݄ത
͵
݀ܰ ͵ߙ݄തߠ
൬ͳ
െ
ߠ‫ܮ‬൰
െ
ܰଶ
ͳͲ
݀‫ܮ‬
ͳͲܰ

,

which is negative because of ߠ‫ ܮ‬൏ ͳ. The impact of population change through the price ofvarieties is߲ܸ ݀‫ ߙ ݌‬ଶ ݄ത ͳ͵݀ܰ ͵ൌ൜ ൬ͳ െ ߠ‫ܮ‬൰൅ ߠൠ ‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻ݁ ିఈ௤ሺேǡ௣ǡோǡ௦ǡோሻ߲‫ܮ݀ ݌‬‫ܰ ݌‬ͳͲ݀‫Ͳͳ ܮ‬,which is positive, since the relationship between the population and the mass of varieties ispositive and ߠ‫ ܮ‬൑ ͳ. Utility of workers increases as the population grows. Differentiating
this equation with respect to labor efficiency yields

14݀ ߲ܸ ݀‫݌‬ߙ ଶ ݄ത ͳ
͵
݀ܰ ͵
൬
൰ൌ
൜ ൬ͳ െ ߠ‫ܮ‬൰൅ ߠൠ݄݀ ߲‫ܮ݀ ݌‬‫ܰ ݌‬ͳͲ݀‫Ͳͳ ܮ‬(29)݀‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻൈ ݁ ିఈ௤ሺேǡ௣ǡோǡ௦ǡோሻሺͳ െ ߙ‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻሻ
݄݀

.

௙
ସଽ
In the case of ߙሺ௖ ሻߠ ൐ ଷ଺଼, for a critical labor efficiency ݄෪௣ ‫ א‬ሺͲǡ ݄௠௔௫ ሻ, (29) is positive forడ௏ ௗ௣Ͳ ൑ ݄ ൏ ݄෪௣ , zero for ݄ ൌ ݄෪௣ and negative for ݄෪௣ ൏ ݄ ൏ ݄௠௔௫ , hence డ௣ ௗ௅ has a maximum௙ସଽat ݄ ൌ ݄෪௣ . In the case of ߙሺ௖ ሻߠ ൑ ଷ଺଼, there exists a critical population ‫ܮ‬෪௣ . When Ͳ ൏ ‫ ܮ‬൏ ‫ܮ‬෪௣ ,there exists a critical labor efficiency ݄෪௣ and

డ௏ ௗ௣
డ௣ ௗ௅

is maximum at ݄ ൌ ݄෪௣ as same as in the

௙
ସଽ
డ௏ ௗ௣
case of ߙሺ௖ ሻߠ ൐ ଷ଺଼. When ‫ܮ‬෪௣ ൑ ‫ ܮ‬൏ ͳȀߠ, (29) is always positive and డ௣ ௗ௅ is increasing inlabor efficiency (see Appendix D). Figure 6b illustrates an example of the relationship
డ௏ ௗ௣

௙

ସଽ

between labor efficiency and డ௣ ௗ௅ in the case of ߙሺ௖ ሻߠ ൐ ଷ଺଼.
4.1.3 Impact through land rent earned
The impact on the indirect utility function through the change in land rent earned isderived by differentiating (20) with respect to ܴ as follows:߲ܸ ߙ ିఈ௤ሺேǡ௣ǡோǡ௦ǡோሻൌ ߲ܴ݁ത ‫݌‬,which is always positive, hence utility of workers increases when land rent earned increases.
Differentiating (28) with respect to ‫ ܮ‬yields the change of land rent earned in terms of
population as follows:
ܴ݀ത
͵
ൌ
ߠ݄ത
݀‫Ͳͳ ܮ‬,which is always positive. The larger the population is, the larger land rent earned is.
The impact of population change through the land rent earned is߲ܸ ܴ݀ത ͵ߙߠ݄ത ିఈ௤ሺேǡ௣ǡோǡ௦ǡோሻ
ൌ
݁
ͳͲ‫݌‬
߲ܴത ݀‫ܮ‬15,

which is always positive and decreasing, because ݁ ିఈ௤ሺேǡ௣ǡோǡ௦ǡோሻ is decreasing for ݄ ‫א‬ሺͲǡ ݄௠௔௫ ሻ.
డ௏ ௗோത

Figure 6c illustrates an example of the relationship between labor efficiency and డோത ௗ௅ .
4.1.4 Impact through effective labor supply
Effective labor supply is the labor efficiency net of commuting costs. Since labor
efficiency is independent of urban population, the impact through effective labor supply
captures the impact through the change in commuting costs caused by urban population
growth.
The impact on the indirect utility function through the change in the eff ective labor
supply is derived by differentiating (20) with respect to ‫ ݏ‬as follows:߲ܸ ߙ ିఈ௤ሺேǡ௣ǡோǡ௦ǡோሻൌ ߲݁‫݌ ݏ‬,

which is always positive, hence utility of workers increases when effective labor supply
increases. Differentiating (25) with respect to‫ ܮ‬yields the change of effective labor supply interms of population as follows:݀‫ݏ‬݄ ଶൌ െߠ݄ ൬ͳ െ ൰
݀‫ܮ‬
͵݄ത

,

which is negative except for the case of ݄ ൌ Ͳ or ݄௠௔௫ . Since workers with labor efficiency݄ ൌ Ͳ do not supply labor and workers with labor efficiency ݄ ൌ ݄௠௔௫ always live at the
CBD, they are not affected in change of commuting costs. Effective labor supply by otherworkers decreases, since the commuting distances increase as the population grows. The
impact of population change through the effective labor supply is߲ܸ ݀‫ݏ‬ߙߠ݄݄ ଶ ିఈ௤ሺேǡ௣ǡோǡ௦ǡோሻൌെ൬ͳ െ ത ൰ ߲݁‫ܮ݀ ݏ‬
‫݌‬
͵݄

,

(30)

which is negative except for the case of ݄ ൌ Ͳ and ݄௠௔௫ . The utility of workers except thosewith the labor efficiency ݄ ൌ Ͳ and ݄௠௔௫ decreases as the population grows. For a critical෪௦ ‫ א‬ሺͲǡ ݄௠௔௫ ሻ, డ௏ ௗ௦ has a minimum at ݄ ൌ ݄
෪௦ . Figure 6d illustrates anlabor efficiency ݄
డ௦ ௗ௅
డ௏ ௗୱ

example of the relationship between labor efficiency and డ௦ ௗ௅.
4.1.5 Impact through land rent paid
16

The impact on the indirect utility function through the change in the land rent paid isderived by differentiating (20) with respect to ܴ as follows:߲ܸߙൌ െ ݁ ିఈ௤ሺேǡ௣ǡோǡ௦ǡோሻ
߲ܴ
‫݌‬

,

which is always negative, hence utility of workers decreases when land rent paid increases.
Differentiating (24) with respect to ‫ ܮ‬yields the change of land rent paid in terms of
population as follows:ܴ݀ ߠ݄ଶʹ݄ൌ ത ൬ͳ െ ത ൰
݀‫ܮ‬
͵݄
ͻ݄

,

which is positive except for the case of ݄ ൌ Ͳ. Since workers with labor efficiency ݄ ൌ Ͳalways live at the urban boundary and do not pay land rent, they are not affected. Land rent
paid by other workers increases, as the urban population grows. The impact of populati on
change through the land rent paid is
߲ܸ ܴ݀
ߙߠ
ʹ݄
ൌ െ ത ݄ଶ ൬ͳ െ ത ൰ ݁ ିఈ௤ሺேǡ௣ǡோǡ௦ǡோሻ߲ܴ ݀‫ܮ‬
͵‫݄݌‬
ͻ݄

,

which is negative except for the case of ݄ ൌ Ͳ. The utility of workers except those with thelabor efficiency ݄ ൌ Ͳ decreases as the population grows. Differentiating this equation withrespect to the labor efficiency yields
݀ ߲ܸ ܴ݀ߙߠ൬൰ ൌ െ ത ݁ ିఈ௤ሺேǡ௣ǡோǡ௦ǡோሻ ݄݄݀ ߲ܴ ݀‫ܮ‬͵‫݄݌‬ʹ݀‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻʹߙ݄ଶ ݀‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻൈ ቊʹ െ ቆ ൅ ߙ
ቇ݄ ൅
ቋ
݄݀
݄݀
͵݄ത
ͻ݄ത

(31)

For a critical labor efficiency ݄෪ோ ‫ א‬ሺͲǡ ݄௠௔௫ ሻ, (31) is negative for Ͳ ൑ ݄ ൏ ݄෪ோ , zero for ݄ ൌడ௏ ௗோ݄෪ோ and positive for ݄෪ோ ൏ ݄ ൏ ݄௠௔௫ , hence డோ ௗ௅ has a minimum at ݄ ൌ ݄෪ோ (see Appendix
డ௏ ௗோE). Figure 6e illustrates an example of the relationship between labor efficiency and డோ ௗ௅ .

17

V N dN dL

V p dp dL
0.25

0.4
0.20

0.3
0.15

0.2

0.10

0.1

0.05

hmax

h

0
0.0

0.5

1.0

1.5

2.0

2.5

3.0

Labor Efficiency h

0
0.0

!

1.0

1.5

2.0

2.5

3.0

Labor Efficiency h(b) Price of varieties ‫݌‬: ݄෪௣ ൌ ʹǤ͹ʹʹ(a) Mass of varieties ܰ
V R d R dL
0.20

hmax

h
0.5

V s ds dL
0.05
0

0.15

0.05
0.10

0.5

1.0
h

1.5

2.0

2.5

3.0
hmax

Labor Efficiency h

0.10
0.15

0.05

hmax

h

0
0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.20
Labor Efficiency h! 0.25෪௦ ൌ ͲǤͺͳ͸Ͳ(d) Effective labor supply ‫ݏ‬: ݄(c) Land rent earned ܴ
V R dR dL
0
0.05

0.5

1.0
h

1.5

2.0

2.5

3.0
hmax

Labor Efficiency h

0.10
0.15
0.20
0.25
0.30

(e) Land rent paid ܴ:݄෪ோ ൌ ʹǤ͵ͲͺͳFigure 6. Separated impacts through five factors (The set of parameters is shown at Fig.3.)

18

4.2 Aggregate impact of population growth
Summing up the separated impacts in Subsection 4.1, we can point out two possibilities
෪௦ and ݄෪ோ arefor the loss of utility. First, since workers with the labor efficiency near ݄
greatly affected by the loss of utility in terms of effective labor supply and land rent paid,
their utility may decrease. Second, since workers with higher labor efficiency are greatly
harmed by the increase of land rent paid in spite of the fact t hat their gain from the increaseof land rent earned is smaller than others, their utility may decrease.
Counting up the impacts through five factors shown in Subsection 4.1, the total impact
of population growth on a worker is as follows:ܸ݀݀ܰൌ ቄͳ െ ൫ͳ ൅ ߙ‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻ൯݁ ିఈ௤ሺேǡ௣ǡோǡ௦ǡோሻ ቅ
݀‫ܮ‬
݀‫ܮ‬
ߙ ଶ ݄തͳ͵݀ܰ ͵ߠ൅‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻ݁ ିఈ௤ሺேǡ௣ǡோǡ௦ǡோሻ ൜ ൬ͳ െ ߠ‫ܮ‬൰
൅ ൠ
‫݌‬
ܰ
ͳͲ
݀‫Ͳͳ ܮ‬͵ߙߠ݄ത ିఈ௤ሺேǡ௣ǡோǡ௦ǡோሻ
൅
݁
ͳͲ‫݌‬
െ

ߙߠ݄ ିఈ௤ሺேǡ௣ǡோǡ௦ǡோሻ݄ ଶ݁൬ͳ െ ത ൰‫݌‬͵݄െߙߠ ିఈ௤ሺேǡ௣ǡோǡ௦ǡோሻ ଶʹ݄݄݁ ൬ͳ െ ൰
ത
͵‫݄݌‬
ͻ݄ത

(32)

Figure 7 shows the relationship of (32). For workers with labor efficiency ݄ ൌ Ͳ, thefirst three terms in the right hand of (32) are positive and the other two terms are zero, henceௗ௏workers with lower labor efficiency always gain. When urban population increases, ௗ௅decreases except for workers with lower labor efficiency and becomes negative for workers
෪௦ and ݄෪ோ . At the maximum population of the parameter set
with the labor efficiency near ݄(‫ ܮ‬ൌ ʹ), workers with highest labor efficiency lose their utility. For workers with highest
labor efficiency, since the fourth term of (32) is zero, the fifth term, the increase of land rent
paid, overweighs the total of the other three terms, the effect of the mass and prices ofvarieties and land rent earned.

19

dV dL
L 1
0.4
L 1.2
L 1.4

0.2

L 1.6

h

L 1.8

0
0.5

1.0

1.5

2.0

2.5

3.0 L 2hmax

Labor Efficiency h

0.2

ௗ௏

Figure 7. Relationship between labor efficiency ݄ and ௗ௅ (Other parameters than ‫ ܮ‬aresame as those in Fig.3.)
Finally, we find out one of necessary conditions that workers with highest labor
efficiency lose their utility by population growth at the maximum population. Rearranging(32) yieldsܸ݀ߙ ଶ ݄ത͵݀ܰൌ ቈͳ െ ቊ൫ͳ ൅ ߙ‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻ൯ െ൬ͳ െ ߠ‫ܮ‬൰ ‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻቋ ݁ ିఈ௤ሺேǡ௣ǡோǡ௦ǡோሻ ቉
݀‫ܮ‬
ܰ‫݌‬
ͳͲ
݀‫ܮ‬
ଶ
ଷ
͵ߙߠ݄ത ͳ݄݄ͳ ݄൅ቊ ൫ͳ ൅ ߙ‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻ൯ െ൅ ൬ ൰ െ ൬ ൰ ቋ ݁ ିఈ௤ሺேǡ௣ǡோǡ௦ǡோሻ
‫݌‬
ͳͲ
͵ ͵݄ത
͵݄ത
͵݄ത

(33)

Substituting labor efficiency ݄ ൌ ݄௠௔௫ and population ‫ ܮ‬ൌ ͳȀߠ into (33), we can obtain
one of necessary conditions that (33) is negative as follows (see Appendix F):݂ʹͶͲͳߙ൬ ൰ߠ ൏
ܿ
ͳ͵ͺͲ

.

(34)

Inequality (34) holds when ߙ, ݂Ȁܿ or ߠ is small. When utility parameter ߙ is smaller,
varieties are closer substitutes, hence the mass of varieties is relatively less important than
the quantity of varieties. When the fixed cost ݂ is relatively smaller than marginal cost ܿ,
the increasing returns to scale in production is smaller. When the parameter of labor
efficiency loss caused by commuting ߠ is smaller, the commuting cost is smaller and the
maximum population is larger.

5. Discussion
In this section, we consider the implication for the bottom line in the previous section
and roughly relate the prediction of the present model to the Tokyo metropolitan area. First,
20when urban population grows larger than a certain critical level, utility of middle-income
class begins to decrease, and subsequently that of high-income class begins to decrease. Thisimplies a possibility that, if the reduction of commuting cost or the increase in labor
efficiency does not exist, only low-income population increases. When the income
distribution in a metropolitan area stays constant as urban population grows, the increase oflabor efficiency by educational investment or the reduction of commuting cost by public
investment to transportation are imperative.
Second, since the public land ownership model is employed in the model, every worker
receives income from their land ownership. Actually, landowners and non-landowners coexist
in a metropolitan area. Landowners obtain greater benefits of the appreciation of land values
caused by urban population growth than the result in the previous section, while
non-landowners suffer greater damage. Since most workers who migrate to a metropolitan
area are non-landowners, this can be one of factors to curb the growth of a metropolitan area.
Third, Tomioka and Ohtake (2005) have the data closely-related to the result of ourmodel. Their questionnaire LQYHVWLJDWLRQ LQFOXGHV WZR LWHPV 2QH LV µ'R \RX WKLQN that the
disparity in income levels expanded in the past ILYH\HDUV"¶DQGWKHRWKHULVµ'R\RXWKink thatthe disparity in income will expand in the QH[W ILYH \HDUV"¶ They show that middle-income
class (annual income from 5 million to 10 million Japanese yen) and residents in metropolitan
areas (prefectures of Tokyo, Kanagawa, Chiba, Saitama, Aichi, O saka and Fukuoka)
recognize these two items as true more than average. Their report is consistent with our resultthat the utility of middle-income class decreases as urban population grows.
Finally, making use of Tomioka and Ohtake (2005), the result of our model is projected
upon Tokyo metropolitan area. Although metropolitan areas in the U. S. are defined by thegovernment, there is no counterpart in Japan. The definitions are proposed by a few
researchers. Kawashima et.al. (1993) defines F unctional U rban Regions (FUR), and theradius of Tokyo FUR at 1990 is around 40-60 kilometers. Kanemoto and Tokuoka (2002)
defines U rban E m ploym ent Area (UEA), and the radius of Tokyo UEA at 1995 is around50-70 kilometers. The difference arises mainly because the definition of FUR employs amono-centric core while that of UEA a multi-centric core. Here, we put the radius of Tokyo
metropolitan area as 50 kilometers, because our model assumes a monocentric city. The
average annual income is about 4,715 thousand Japanese yen from National Tax Agency(2012), hence the middle-income class defined in Tomioka and Ohtake (2005) approximately
corresponds to workers with labor efficiency from ݄ത to ʹ݄ത . This implies that the
middle-income workers who lose most in our model reside typically in the region from 17
kolometers to 34 kilometers from the center of Tokyo metropolitan area (see Appendix G).

6. Concluding Remarks
To our knowledge, this paper is the first formal discussion of the impact of population
21

growth on utility level of workers with heterogeneity of labor efficiency in a city. We haveanalyzed the impact on individual utility from urban population growth in a generalequilibrium model of monopolistic competition featuring heterogeneity of labor efficiency
between workers. We have decomposed the cause of the change in utility level into fivefactors: the mass and price of product varieties, land rent earned, effective labor supply andland rent paid. The decomposition helps us to understand the relative contribution. After each

LPSDFWRQZRUNHU¶VXWLOLW\WKURXJKILYHIDFWRUVLVLQYHVWLJDWHGVHSDUDWHO\ total impact onutility level is studied. Our findings indicate that, in case where urban population increases
without the reduction of commuting cost and the increase in labor efficiency, workers withmiddle labor efficiency have the highest possibility to lose utility and workers with higher
labor efficiency also have the possibility, while workers with lower labor efficiency always
gain.
This paper limits its focus to autarkic monocentric city. Multicentric city may alleviate
commuting cost and land rent paid, or a system of city may provide optimal number of cities
and their sizes. These extensions are left for future research.
Since workers/households vary in diverse ways, we believe that the study of the impacts
on workers/household with various heterogeneities in terms of urbanization should be
encouraged further.

Acknowledgments
The authors are very grateful to Yasusada Murata, Shinsuke Tsuruta as well as
participants in the 2014 Annual Meeting of Applied Regional Science Conference and theurban economics workshop at National Graduate Institute for Policy Studies.

Appendix
Appendix A. The symmetry of price equilibrium
Behrens and Murata (2007) show that the price equilibrium is symmetric and unique.
Assuming Ͳ ൏ ‫݌‬௠௜௡ ‫݌  ؠ‬ሺ݅ሻ ൑ ‫݌‬ሺ݅ሻ ൑ ‫݌‬௠௔௫ ‫݌  ؠ‬ሺ݅ሻ, from (15),‫݌‬௠௜௡
‫݌‬௠௜௡ െ ݄ܿ଴න  ൬൰ ‫݌‬ሺ݆ሻ݆݀ ൅න ‫݌‬ሺ݆ሻ݆݀ ൌ ߙ‫ܧ‬ത‫݌‬ሺ݆ሻ
‫݌‬௠௜௡
ఆఆ‫݌‬௠௔௫
‫݌‬௠௔௫ െ ݄ܿ଴න  ൬൰ ‫݌‬ሺ݆ሻ݆݀ ൅න ‫݌‬ሺ݆ሻ݆݀ ൌ ߙ‫ܧ‬ത‫݌‬ሺ݆ሻ
‫݌‬௠௔௫
ఆఆ
must hold. Subtracting these equations, we obtain22‫݌‬௠௔௫
‫݌‬௠௔௫ െ ‫݌‬௠௜௡൜ ൬൰൅݄ܿ଴ ൠ න ‫݌‬ሺ݆ሻ݆݀ ൌ Ͳ‫݌‬௠௜௡
‫݌‬௠௔௫ ‫݌‬௠௜௡ఆ.

If ‫݌‬௠௜௡ ൏ ‫݌‬௠௔௫ , the left hand of this equation is positive. Hence, ‫݌‬௠௜௡ ൌ ‫݌‬௠௔௫ , and the pricesof varieties are equal.

Appendix B. Relationship between urban population and the mass of firms
We show here that, under the distribution function of (23), the critical population ‫ܮ‬෨ atwhich the mass of firms is maximum is larger than the maximum population ͳȀߠ.
Differentiating (19) with respect to ‫ ܮ‬yields
݀ܰ
ͳ
݄ത
͵
ͻ
͵
ൌ
ቆ ቇ ൜ ߙ݂ߠ‫ܦ‬ሺ‫ܮ‬ሻ െ ߙ݂ܿߠ‫ ܮ‬൅ ʹߙ݂ܿ െ ሺߙ݂ሻଶ ߠൠ .
݀‫ܦ݂ܿʹ ܮ‬ሺ‫ܮ‬ሻ ݄଴ ͳͲ
ͷ
ͳͲ

(B.1)

Define the terms in braces as ܰଵ as follows:ܰଵ ‫ؠ‬͵ͻ͵ߙ݂ߠ‫ܦ‬ሺ‫ܮ‬ሻ െ ߙ݂ܿߠ‫ ܮ‬൅ ʹߙ݂ܿ െ ሺߙ݂ሻଶߠ
ͳͲ
ͷ
ͳͲ

(B.2)

.

Differentiating ܰଵ with respect to ‫ ܮ‬yields
݀ܰଵ ͵ߙ ଶ ݂ܿ ଶ ߠ ͻൌെ ߙ݂ܿߠ݀‫ܮ‬ͷ‫ܦ‬ሺ‫ܮ‬ሻ
ͷ

,

(B.3)

which is negative since ‫ܦ‬ሺ‫ܮ‬ሻ ൒ ߙ݂. From (B.2), ܰଵ ൌ ʹߙ݂ܿ ൐ Ͳ at ‫ ܮ‬ൌ Ͳ and  ܰଵ ൌ െλ,௅՜ஶhence ܰଵ ൌ Ͳ holds for a value of ‫ܮ‬෨ ൐ Ͳ. Let ܰଵ ൌ Ͳ and solve for ‫ܮ‬, we obtain ‫ܮ‬෨ asfollows:‫ܮ‬෨ ൌ

ሺͳͲܿ െ ߙ݂ߠሻ ൅ ඥͳͲߙ݂ܿߠ ൅ ሺߙ݂ߠሻଶ
ͻܿߠ.

In order to compare ‫ܮ‬෨ and ͳȀߠ, taking the difference of them yields as follows:‫ܮ‬෨ െͳ ሺܿ െ ߙ݂ߠሻ ൅ ඥͳͲߙ݂ܿߠ ൅ ሺߙ݂ߠሻଶ
ൌ
൐Ͳ
ߠ
ͻܿߠ

.

Inequality holds because of ඥͳͲߙ݂ܿߠ ൅ ሺߙ݂ߠሻଶ ൐ ߙ݂ߠ, hence the critical population is larger
than the maximum population(ͳΤߠ ൏ ‫ܮ‬෨).

23

Appendix C. Relationship between labor efficiency and demand for each variety
We make clear the relationship between labor efficiency ݄ and the demand for a variety‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻ.
Differentiating (21) with respect to ݄ as follows:݀‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻͳ ݀‫ܴ݀ ݏ‬
ൌ
൜ െ ൠ
݄݀
ܰ‫݄݀ ݄݀ ݌‬

,

(C.1)

, and

(C.2)

.

(C.3)

Differentiating (24) and (25) with respect to ݄ as follows:
ଶ
ܴ݀
ʹߠ‫ܮ‬
ͳ
ͳ
ൌെ
ଶ ൬݄ െ ʹ ݄௠௔௫ ൰ ൅ ʹ ߠ‫ܮ‬݄݄݀௠௔௫݀‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻͳ ݀‫ܴ݀ ݏ‬
ൌ
൬ െ ൰
݄݀
ܰ‫݄݀ ݄݀ ݌‬

Substituting (C.2) and (C.3) into (C.1), we obtain the differential of the demand for eachvariety ‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻ with respect to labor efficiency as follows:݀‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻ
ͳ
ߠ‫ܮ‬
ଶ
ൌ
ቊെ
ଶ ሺ݄ െ ݄௠௔௫ ሻ ൅ ͳቋ
݄݀
ܰ‫݌‬
݄௠௔௫
ଵ

,

ଵ

which is always positive, and its value is ே௣ ሺͳ െ ߠ‫ܮ‬ሻ at ݄ ൌ Ͳ, ே௣ at ݄ ൌ ݄௠௔௫ .
‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻ is strictly increasing in ݄, hence varieties are normal goods. Since there exists
a flexion point at ݄ ൌ ݄௠௔௫ , ‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻ is convex for Ͳ ൏ ݄ ൏ ݄௠௔௫ . From (21), thevalue of ‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻ isఏ௅௛೘ೌೣ
ଵ଴ே௣at ݄ ൌ Ͳ,

௛೘ೌೣ
ே௣

଻

ቀͳ െ ଷ଴ ߠ‫ܮ‬ቁ at ݄ ൌ ݄௠௔௫ . The reason why

workers with labor efficiency ݄ ൌ Ͳ can afford varieties is that they earn land rent which isequally distributed to each worker. Figure C.1 illustrates the relationship between labor
efficiency and demand for a variety.

24

Demand for Variety q
5
4
3
2
1

hmax

h
0

0.5

1.0

1.5

2.0

2.5

3.0

Labor Efficiency hFigure C.1. Relationship between labor efficiency݄ and demand for a variety ‫( ݍ‬The set ofparameters is shown at Fig.3.)

Appendix D. Impact through the price of varieties
Investigating the properties of (29), we obtain the result shown in 4.1.2.
We define ݃௣ଵ ሺ݄ሻ ‫ ͳ ؠ‬െ ߙ‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻ. Since the other part of (29) than ݃௣ଵ ሺ݄ሻ ispositive, we check the sign of ݃௣ଵሺ݄ሻ. ‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻ is strictly increasing in labor efficiency
as shown in Appendix C, hence ݃௣ଵ ሺ݄ሻ ൌ Ͳ has a solution at most. The values of ݃௣ଵ ሺ݄ሻ atthe end points of the range in labor efficiency are͵ͻߠ‫ܮ‬ቁ ൅ ߙ݂ ቀͳ െ ߠ‫ܮ‬ቁͳͲͳͲ݃௣ଵሺͲሻ ൌ͵ሺߙ݂ ൅ ‫ܦ‬ሺ‫ܮ‬ሻሻ ቀͳ െ ߠ‫ܮ‬ቁͳͲ͵ͳͳ‫ܦ‬ሺ‫ܮ‬ሻ ቀͳ െ ߠ‫ܮ‬ቁ ൅ ߙ݂ ቀെͷ ൅ ߠ‫ܮ‬ቁͳͲͳͲ݃௣ଵ ሺ݄௠௔௫ ሻ ൌ͵ሺߙ݂ ൅ ‫ܦ‬ሺ‫ܮ‬ሻሻ ቀͳ െ ߠ‫ܮ‬ቁͳͲ‫ܦ‬ሺ‫ܮ‬ሻ ቀͳ െ

（D.1）

（D.2）

With attention to the fact that the maximum value of population is ͳȀߠ, both the numerator
and denominator in (D.1) are positive, hence ݃௣ଵሺͲሻ is always positive. Though the sign ofthe denominator in (D.2) is positive, the sign of the numerator is indeterminate, hence thesign of ݃௣ଵ ሺ݄௠௔௫ ሻ is also indeterminate. Defining the numerator of （D.2） as ݃௣ଶ ሺ‫ܮ‬ሻ andrearranging it yieldsͳͳ͵݃௣ଶ ሺ‫ܮ‬ሻ ‫ ؠ‬െͷߙ݂ ൅ ቆ ߙ݂ߠ‫ ܮ‬൅ ‫ܦ‬ሺ‫ܮ‬ሻቇ െ ‫ܦ‬ሺ‫ܮ‬ሻߠ‫ܮ‬ͳͲͳͲ

The first term on the right hand is constant, while the second term ,

25

.

ଵଵ
ଵ଴

ߙ݂ߠ‫ ܮ‬൅ ‫ܦ‬ሺ‫ܮ‬ሻ, and theଷthird term,ଵ଴‫ܦ‬ሺ‫ܮ‬ሻߠ‫ܮ‬, are both strictly increasing in labor efficiency, hence ݃௣ଶ ሺ‫ܮ‬ሻ ൌ Ͳ has asolution at most. The values of ݃௣ଶ ሺ݄ሻ at the end points of the range in population are݃௣ଶ ሺͳȀߠሻ ൌ݃௣ଶሺͲሻ ൌ െͶߙ݂ ൏ Ͳ

(D.3)

ͳ
ሺ͹ඥͶߙ݂ܿȀߠ ൅ ሺߙ݂ሻଶ ‐͵ͻߙ݂ሻ
ͳͲ

(D.4)

When ݃௣ଶ ሺͳȀߠሻ is negative, ݃௣ଶሺ‫ܮ‬ሻ is always negative for Ͳ ൑ ‫ ܮ‬൑ ͳȀߠ, hence ݃௣ଵሺ݄௠௔௫ ሻis negative. Paying attention to the fact that ݃௣ଵ ሺͲሻ is positive, there exist a critical labor
ௗ

డ௏ ௗ௣

efficiency ݄෪௣ ‫ א‬ሺͲǡ ݄௠௔௫ ሻ and ௗ௛ ቀడ௣ ௗ௅ ቁ is positive for Ͳ ൑ ݄ ൏ ݄෪௣ , zero for ݄ ൌ ݄෪௣ andnegative for ݄෪௣ ൏ ݄ ൑ ݄௠௔௫ .

డ௏ ௗ௣
డ௣ ௗ௅

has a maximum at ݄ ൌ ݄෪௣ . From (D.4), the condition that݃௣ଶ ሺͳȀߠሻ is negative is݂Ͷͻߙ൬ ൰ߠ ൐
ܿ
͵͸ͺ

௙

ସଽ

௖

ଷ଺଼

In the case of ߙ ቀ ቁ ߠ ൑.

, since ݃௣ଶ ሺͲሻ ൏ Ͳ and ݃௣ଶሺͳȀߠሻ ൒ Ͳ, the sign of ݃௣ଶ ሺ‫ܮ‬ሻchanges at ‫ܮ‬෪௣ ‫ א‬ሺͲǡͳȀߠሻ. When the population is smaller than ‫ܮ‬෪௣ , ݃௣ଶ ሺ‫ܮ‬ሻ is negative and
డ௏ ௗ௣
డ௣ ௗ௅

௙

ସଽ

has a maximum at ݄ ൌ ݄෪௣ as same as in the case of ߙ ቀ௖ ቁ ߠ ൐ ଷ଺଼. When the population
ௗ

డ௏ ௗ௣

is larger than ‫ܮ‬௣ , since ௗ௛ ቀడ௣ ௗ௅ ቁ is always positive, the higher the labor efficiency, theడ௏ ௗ௣larger డ௣ ௗ௅ is.

Appendix E. Impact through land rent paid
Investigating the properties of (31), we obtain the result shown in 4.1.5.
We define the terms in braces in (31) as ݃ோ ሺ݄ሻ as follows:ʹ݀‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻʹߙ݄ଶ ݀‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻ݃ோ ሺ݄ሻ ‫ ʹ ؠ‬െ ቆ ത ൅ ߙቇ݄ ൅ ത
݄݀
݄݀
͵݄
ͻ݄

ଶ

Since both the second and the third terms, ቀଷ௛ഥ ൅ ߙௗ௤ሺேǡ௣ǡோǡ௦ǡோሻௗ௛ቁ ݄ andଶఈ௛మ ௗ௤ሺேǡ௣ǡோǡ௦ǡோሻ
, are
ഥ
ଽ௛
ௗ௛

increasing for Ͳ ൑ ݄ ൑ ݄௠௔௫ , ݃ோ ሺ݄ሻ ൌ Ͳ has a solution at most. The signs of ݃ோ ሺ݄ሻ at the26

end points of the range in labor efficiency are݃ோ ሺͲሻ ൌ ʹ ൐ Ͳߙ݄ത݃ோ ሺ݄௠௔௫ ሻ ൌ െ൏Ͳܰ‫݌‬From the above and (32), there exists a critical labor efficiency ݄෪ோ ‫ א‬ሺͲǡ ݄௠௔௫ ሻ and
ௗ

డ௏ ௗோ

ቀ

ௗ௛ డோ ௗ௅
డ௏ ௗோ
డோ ௗ௅

෪௦ , zero for ݄ ൌ ݄෪ோ and positive for ݄෪ோ ൏ ݄ ൑ ݄௠௔௫ .
ቁ is negative for Ͳ ൑ ݄ ൏ ݄

has a minimum at ݄ ൌ ݄෪ோ .

Appendix F. Relationship between urban population and utility level
We find out one of necessary conditions that workers with highest labor efficiency lose
their utility by population growth at the maximum population.
Substituting ݄ ൌ ݄௠௔௫ into (33) yieldsܸ݀݀ܰ͵ߙߠ݄ത ିఈ௤ሺேǡ௣ǡோǡ௦ǡோሻ൧೓స೓೘ೌೣൌ ܸ௅ଵ ሺ‫ܮ‬ሻ൅ ܸ௅ଶ ሺ‫ܮ‬ሻ݁൨݀‫ ܮ‬௛ୀ௛೘ೌೣ
݀‫ܮ‬
ͳͲ‫݌‬

,

(F.1)

where
ܸ௅ଵ ሺ‫ܮ‬ሻ ‫ ͳ ؠ‬െ ቊቀͳ ൅ ߙ‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻ൧௛ୀ௛

೘ೌೣ

െ

ቁ

ߙ ଶ݄ത
͵
ିఈ௤ሺேǡ௣ǡோǡ௦ǡோሻ൧೓స೓
೘ೌೣ൬ͳ െ ߠ‫ܮ‬൰ ‫ݍ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻ൧௛ୀ௛ ቋ ݁
೘ೌೣ
ܰ‫݌‬
ͳͲ

ܸ௅ଶ ሺ‫ܮ‬ሻ ‫ݍߙ  ؠ‬ሺܰǡ ‫݌‬ǡ ܴǡ ‫ݏ‬ǡ ܴሻ൧௛ୀ௛

೘ೌೣ

െ

͹
͵

,

.

We investigate the signs of ܸ௅ଵ ሺ‫ܮ‬ሻ and ܸ௅ଶ ሺ‫ܮ‬ሻ at the maximum population ‫ ܮ‬ൌ ͳȀߠ.
Substituting ‫ ܮ‬ൌ ͳȀߠ into ܸ௅ଵ ሺ‫ܮ‬ሻ yields
ଶ
ഥ
ଶଷ ఈ௛
ͳ
ͳ͸ͳ ߙ݄തʹ͵ ߙ݄തି ൬ ൰ܸ௅ଵ ൬ ൰ ൌ ͳ ൅ ൝ቆ ቇ െ ቆ ቇ െ ͳൡ ݁ ଵ଴ ே௣ߠͳͲͲ ܰ‫݌‬ͳͲ ܰ‫݌‬

ഥ
ఈ௛

.

ഥ
ఈ௛

When ே௣ is zero, the value of this equation is zero. When ே௣ is in the range from zero to
ଵଵଵ଴
ଵ଺ଵ

ହଵ

యళ

ഥ
ఈ௛

, ܸ௅ଵ ሺͳΤߠሻ is increasing to a maximum ͳ ൅ ଶଷ ݁ ି ళ ሺൎ ͳǤͲͲͳͳʹሻ. When ே௣ is in the range

27

larger than

ଵଵଵ଴
ଵ଺ଵ

, ܸ௅ଵ ሺͳΤߠሻ is decreasing asumptotically to one. Hence, ܸ௅ଵ ሺͳΤߠሻ is positive.

Substituting ‫ ܮ‬ൌ ͳȀߠ into ܸ௅ଶ ሺ‫ܮ‬ሻ yields
ͳ
Ͷ͸
ߙ݂
͹
ܸ௅ଶ ൬ ൰ ൌെΤሻߠ͹ ‫ܦ‬ሺͳ ߠ ൅ ߙ݂ ͵.

Putting the right hand negative and solving the inequality yields as follows:݂ʹͶͲͳߙ൬ ൰ߠ ൏
ܿ
ͳ͵ͺͲ

(34)

Since ܸ௅ଵ ሺͳΤߠሻ is non-negative, (34) must hold when (G.1) is negative at ‫ ܮ‬ൌ ͳȀߠ. (34) is
one of necessary conditions that workers with highest labor efficiency lose their utility by
population growth at the maximum population.
Appendix G. Correspondence between labor efficiency and residential location
Putting the area inside a circle of radius ‫ ݎ‬as ܵሺ‫ݎ‬ሻ and the radii at which workers
with labor efficiency ݄ത and ʹ݄ത as ‫ݎ‬௛ഥ and ‫ݎ‬ଶ௛ഥ ,ܵሺ‫ݎ‬୫ୟ୶ ሻ െ ܵሺ‫ݎ‬௛ഥ ሻൌ ‫ܩ‬ሺ݄തሻ
ܵሺ‫ݎ‬୫ୟ୶ ሻ, andܵሺ‫ݎ‬୫ୟ୶ ሻ െ ܵሺ‫ݎ‬ଶ௛ഥ ሻൌ ‫ܩ‬ሺʹ݄തሻܵሺ‫ݎ‬୫ୟ୶ ሻ,

where ‫ݎ‬୫ୟ୶ is the distance from CBD to the urban boundary.
Substituting ܵሺ‫ݎ‬ሻ ൌ ߨ‫ ݎ‬ଶ , ‫ܩ‬൫݄ത൯ ൌ ͷȀͻ and ‫ܩ‬൫ʹ݄ത൯ ൌ ͺȀͻ, we obtainͳ‫ݎ‬௛ഥ ൌ ൬ ൰ ‫ݎ‬୫ୟ୶
͵

, and

ʹ
‫ݎ‬ଶ௛ഥ ൌ ൬ ൰ ‫ݎ‬୫ୟ୶͵

.

Putting ‫ݎ‬୫ୟ୶ as 50 kilometers, we obtain‫ݎ‬௛ഥ ൎ ͳ͹, and‫ݎ‬ଶ௛ഥ ൎ ͵Ͷ

.

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