Urban Studies, Vol. 35, No. 1, 45± 51, 1998
Preference for Product Variety and City Size
H ikaru Ogawa
[P aper ® rst received, A ugust 1996; in ® nal form , A pril 1997]
S um m ary. In this p ap er, we exam in e the effect on city size of household preferen ce for prod u ct
variety . There are m any theoretica l analyses that have attem pted to clarify the determ inants of
city size in the urban literatu re. Com parative static analyses are used to ® nd out what hap pens
to the equilibrium city size if the exogen ou s variab les change by a sm all am ount. U sing the m odel
of m onop olistic com petition , we provid e a sim ple m od el of prod uct variety to show that the
equilibrium city size d epend s on the degree of househ old preferen ce for prod u ct variety . The
m ain resu lt is that the stron g preferen ce for prod uct variety prom otes th e form ation of a large
city.
1. Introduction
T he form ation of large cities has been mainly
explained by supply- side phenom ena like
com parative advantages, scale econom ies
and agglom eration econom ies in production
(M ills, 1972). Except the discussions of local
public goods, only few attempts have so far
been made to explain urban agglom eration
from the viewpoint of dem and-side phenom ena. This paper considers product variety as
a key factor in household agglom eration and
examines the effect on city size of changes in
household preference for product variety.
Analytical models of city size (popul ation
concentration) are hardly new . For instance,
Henderson (1974) develops a model of a city
to show how the city size varies with exogenous variables, like production technology
and transport cost. The form ation of large
cities is explained in part by the com parative
statics of those exogenous variables. It is also
well-known that public policies have an
im pact on the form ation of large cities. Modelling the public sector explicitly, Henderson
(1982) studies the im pact of public policies
on the size distribu tion of cities in a small
open econom y. In this model, public policies
consist of im port restrictions, minim um wage
laws, capital market restrictions and central
government intervention in local affairs. An
empirical study by Alperovich (1992)
attempts to clarify the relationship between
econom ic developm ent and popula tion concentration. Extendin g the seminal work of
W illiam son (1965) , Alperovich shows that in
the initial phase of econom ic developm ent,
populat ion tends to concentrate in a few core
cities. This is follow ed by a second phase of
developm ent in which econom ic development leads to popula tion dispersal. 1
When we consider the relationship be-
Hikaru Ogawa is in the Graduate School of E conomics, Nagoya University, F uro-Cho, Chikusa-Ku, Nagoya 464-01, Japan. Fax:
1 81-52-789-4924. E -mail: [email protected]. The author would like to thank P rofessors Nobuhiro Okuno, Tadashi
Yagi and M asatsugu Tsuji for encouragement and helpful comments. The author also thanks the referees w ho made comments and
suggestions for which the author is especially grateful. None of the above bears any responsibility for errors, however.
0042-0980/98/010045-07 $7.00
fromEusj.sagepub.com
at PENNSYLVANIA
STATE UNIV on May 12, 2016
1998 The
ditors of Urban
Studies
Ó Downloaded
46
H IKA RU OG AW A
1/(1±
r
)
6
5
4
3
2
1
0
1971 1973 1975 1977 1979 1981 1983 1985 1987 1989 1991 1993 1995
®
F igure 1. T he preferen ce for product variety in Japan, 1971±95. Note: r
0 im plies that househo lds
derive m ore utility from product variety, and r
1 implies the converse . The preferen ce for product
variety increase s through tim e. Source : Japanese E conom ic P lanning Agency (1995).
®
tween econom ic developm ent and household
preferences, we ® nd that with developm ent,
throug h tim e, house hold consum ption patterns shift away from prim ary goods (such as
agricultural products and foods) tow ards
modern services such as high-tech manufacturing. As in Alperovich, while econom ic
developm ent has been described by the
im provem ent of production technology, we
can easily infer that econom ic developm ent
can be measured by the change in household
preferences for product varietyÐ that is, with
econom ic developm ent, househo lds prefer
greater variety. As shown in Figure 1, household preference for product variety varies
throug h tim e and the desire for produc t variety seems to increase with econom ic developm ent. This fact is of central relevance to a
conside ration of the relationship s between
the preference for product variety and city
size, since populat ion tends to concentrate in
a few core cities in this period.
In this paper, we provide a simple model
of product dive rsity to examine the relationship betw een househo ld preference for product variety and city size. There are many
examples of com parative statics exercises to
® nd out what happens to equilibrium city
sizes if the exogenous variables change.
However, almost all concentrate exclusively
on the production sector. 2 W e develop a
model of a city to show how city size varies
as household preferences change. W ith
monopol istic com petition in a simple spatial
econom y, we mainly show, in this paper, that
city size becom es large when households
have a strong preference for produc t variety.
This result can explain the fact of popula tion
concentration from the viewpoint of changes
in househo ld preference through tim e. In
addition, we will have the usual result that
low com muting costs and high productivity
lead to the form ation of a large city.
The organisation of this paper is as follows: in section 2, we present a basic model,
using the framework of monopoli stic com petition. In section 3, we analyse the relationship between a preference for product variety
and city size. Discussion about the effect of
com muting cost on city size is also provide d.
Section 4 contains the concluding remarks.
2. M odel
In order to analyse the relationship between
household preference for product variety and
city size, a model of any single city is
speci® ed, using the model of monopoli stic
com petition. Before presenting the model
used to solve equilibrium city size, we need
to examine the behaviour of econom ic
agents. T his section is divided into two parts.
In the ® rst part, the behaviour of the household is presented. The monopoli stic com petition model for the production sector is
developed in the second part.
2.1 Household
The city is circular and the radius of city j is
described as r *j . Households are perfectly
mobile across cities and the popula tion in
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47
PROD UCT V ARIET Y A ND CITY SIZE
city j is denoted by N j . W e assum e that each
household has one unit of hom ogeneous
labour which is supplied in the individ ual’ s
city of residence. All households are
assumed to have identical tastes. The preference of residents in city j is de® ned in the
Cobb±Douglas form ,
uj 5
xa j Z b j
(1)
where,
Zj ;
distance r jl from the CBD is assumed to
increase in a linear way and is given by t j r jl ,
where t j is the amount of the com posite good
used in com muting a unit distance. Com muting costs consist of train fares, car expenses,
tim e, fatigue and so on. W e assume that each
household occupies one unit of space for
housing , regardless of residential location,
hence
Nj 5
S O qD
nj
i5
r
r
1
(2)
ji
1
and a , b . 0, a 1 b 5 1, x j is the consum ption
of com posite goods by the house hold living
in city j; and q ji is househo ld consum ption of
differentiated services i in city j. As the
differentiated services, we focus on the local
consum er services such as bars, restaurants,
beauty shops and all the kind of personal
services. x j is chosen as the num eraire.
As is shown below, the num ber of services
available in the city depends on the size of its
popula tion. The total num ber of services produced in a city, n j, is external to households
but internal to the city. T he parameter r P
(0,1) can be interpreted to represent the
desire for product variety. As r
0, households derive more utility from produc t variety and as r
1 they derive less utility from
product variety. W e have an interest in the
magnitude of parameter r , and how r
changes through tim e. The relationship
between the degree of preference for product
variety and tim e is shown in Figure 1. As a
proxy, using the data of price elasticity of
dem and for durable goods and equation (11)
below , we estimate the degree of preference
for product variety, r , in Japan. 3 T he results
show a negative relationship betw een econom ic developm ent and househo ld preference
for product variety, r , in Japan from 1971 to
1995. This im plies that the desire for product
variety is increasing throug h tim e.
Production takes place in the central business district (CBD) of the city and households com mute to the CBD to work. W e
express r jl as the distance l from the CBD in
city j. The com muting cost for a household at
®
®
p r *j 2,
(3)
where, p is pi ( 5 3.14¼ .).
Because the land is used only for residential purpos es, com petition for sites adjusts
land rents to offset any difference in com muting cost. T hus, in equilibrium , rent per
unit of space at distance r jl from the CBD,
R jl (rjl), is given by
R jl 5
t j(r *j 2
rjl).
(4)
Equation (4) ensures equal utility regardless
of location within a city in an equilibrium .
Using equation (4), the total differential land
rents of the city, DLR j, can be obtained as
2p
DLR j 5
5
where k j ;
E
r*j
t j(r *j 2
r jl )rjldr jl
(5)
0
3
2
à Nj ,
2
3
tjp
2
Ã
.
Assum ing that the differential land rents
accrued in a city are distributed in a lum psum manner to its residents, each househo ld
receives à kjN Ãj , as a share in land rents. A
household budge t constraint is given by
aj 2
t j r *j 1
à kjN Ãj 5
xj 1
O
i5
nj
p jiq ji ,
(6)
1
where the wage rate is described as a j, and p ji
is the price of service i in city j. In equation
(6), t jr *j is the sum of the com muting cost,
t j r jl , and the rent per unit of space, t j(r *j 2 r jl),
so that it represents the cost of living in city
j.
Based on the above assum ptions, the
household utility, equation (1), is maxim ised
subject to the budget constraint, equation (6).
The dem and functions are obtained as
xj 5
a y j,
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(7)
48
H IKA RU OG AW A
b y j I j2 r ,
1
Zj 5
where,
Ij2 1
yj ;
FO
;
nj
i5
(8)
2
p ji
12
1
aj 2
r
r
G
12
r
(9)
à kj N Ãj 5
t jr *j 1
aj 2
k j N Ãj .
(10)
Using equation (8), we obtain the price elasticity of demand for q ji , h , as follow s:4
h 5
dq ji p ji
2
dp ji q ji
5
1
12
(11)
r
2.2 The Production Sector
In a city, there are two production sectors,
one of which produces com posite goods and
the other produces the differential services.
T he latter is usually know n as the service
sector. T he production technology associated
with com posite goods is such that one unit of
labour can be transformed into a j , a unit of
com posite goods.
The market structure of the service sector
is assumed to follow the framework of
monopol istic com petition. Follow ing Cham berlin, that ® rms differentiate their product
costlessly, each ® rm supplies a single service. Each of the differentiated services in
city j, q ji, is assumed to be produced with a
production technology which represents
increasing returns to scale in produc tion. T he
am ount of labour, L ji, required for production
of the differentiated service is assumed to be
given as
L ji 5
f1
cQ ji ,
(12)
where, f is the ® xed labour requirement; c is
the marginal labour requirement (f, c . 0);
and Q ji is the output level of differentiated
service i in city j.
The ® rms in the service sector decide price
and output so as to maxim ise pro® ts,
P
ji
5
p jiQ ji 2
a jL ji.
(13)
T hus, the price is set as follow s:
p ji 5
a jcp
2
1
.
market until pro® t is driven to zero in the
service sector. Hence, in the long run, the
output level produced by each ® rm at equilibrium is obtained by
c 12
Notice that each service is supplied at a
com mon price in the city. As long as excess
pro® t exists, ® rms will continu e to enter the
.
r
(15)
3. Equilibrium City Size
To solve for equilibrium city size, it is
necessary to solve for utility as a function of
city size and variables exogenous to the city.
Given utility- maxim ising behaviour by
households, substitution of equations (7) and
(8) into equation (1) yields the indirect utility
function as
b
a a b b y jI j2 r .
Vj 5
(16)
This will be used later to solve for utility as
a function of city popula tion. Since each ® rm
sets its price at the same level described by
equation (14), households each consum e the
same amount of each service in city j, q ji 5 q j .
Thus, using equation (14), we can rewrite
equation (9) as
I j2
b
5
r
b (1 2
r
nj
r )
b
c2
r b.
(17)
From equations (10), (16) and (17), we can
rewrite the indirect utility function as follows:
a a b b c2
Vj 5
b
b (1 2
r
r b (a j 2
r )
k jN Ãj )n j
,
(18)
where,
b (1 2
nj 5
r )
f
k j N Ãj )N j .
(a j 2
(19)
Notice that ­ n j /d r , 0. That is, a strong preference for product variety increases the num ber of services produced in a city.
Differentiate V j with N j, we obtain
­ Vj
a a b b c2
5
­ Nj
5
(14)
r
f
Qj 5
F a b (1 2
j
a a b b c2
r )2
b
b
S­N
­ yj
b
r
r
2
a
nj
à kjN Ãj { r 1
nj
b (1 2
r
r )
1
b (1 2
yj
j
b (1 2
r
r )
N j2
3 b (1 2
1
G
­ nj r
­ Nj
r )} .
r )
D
(20)
At the optim al popula tion in a city, ­ V j /
­ N j 5 0: 5
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49
PROD UCT V ARIET Y A ND CITY SIZE
V
j
Nj*
N
j
Figure 2. O ptim al city size.
a jb (1 2
r )2
à kj N Ãj { r 1
3 b (1 2
r )} 5
0 (21)
T herefore, the optim al popula tion, N *j , can be
expressed by
N 5
*
j
F
G
2
2a j b (1 2 r )
.
k j { r 1 3 b (1 2 r )}
(22)
T he optim al city size in this model depends
on the labour productivity, a j, the com muting
cost, k j, and the degree of househo ld preference for product variety, r . We can easily
obtain the indirect utility curve, as show n in
Figure 2. In Figure 2, the utility level in a
city rises to a peak at N *j and then declines.
Now it is easy to see that any populat ion size
between 0 and N* cannot be a stable equilibrium . In this paper, follow ing Henderson
(1988) , we assume that a stable city size is
characterised by the condition, ­ u j /­ N j 5 0.
T here are several ways to solve for equilibrium city size, involving setting the city size
at the level where ­ u j /­ N j 5 0. Considering a
com petitive land developm ent market, Henderson denotes that a stable equilibrium is
characterised by the condition that for every
city ­ u j/­ N j 5 0. Suppose there is a city that
is not a point where ­ u j /­ N j 5 0, then developers can earn temporary pro® ts by setting
up and selling housing in cities of a more
ef® cient
size.
Developers
play
an
entrepreneurial role that facilitates large
movem ents of people so that a new city can
form . For further details of de® nition of equilibrium city size, see Henderson (1988, pp.
38±39).
In the urban model, it is possible to vary
the com muting cost variable, k j , to determine
its effect on city size. W e differentiate equation (22) with k j and N *j to obtain
­ N *j
,
­ kj
0.
(23)
Thus, we have the follow ing propos ition
from equation (23):
Proposition 1: A City with Low er Commuting Costs has a Large Population
There have been many studie s of megalopolises like T okyo and New York. W e can see
in equation (23) that the com muting cost,
2
k j ; 3 t jp 2 1/2 , is de® nitely im portant for the
form ation of large cities. In sum mary, a
decrease in the com muting cost leads to an
increase in city popula tion. This result is
consistent with the traditional literatureÐ for
example, the model dem onstrated by Hatta
and Ohkawara (1994) . T hey conclude that
the reason why the Tokyo area contains
tw ice the popula tion of New York can be
attributed to the difference in com muting
costs. Com paring Tokyo and New York, they
consider that the cost of com m uting in the
form er area is much smaller than in the
latter.6 The network of railroads is well
developed, and reduces the com muting cost
in Tokyo. In the Tokyo metropoli tan area in
the 1970s, investm ent in transport led to a
massive im provem ent in mobility that was
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50
H IKA RU OG AW A
reinforced by an increase in train frequencies. T hese polices certainly brough t about a
reduction in com muting expenses, including
less congestion, less stress and shorter com muting tim es.
As the relationship between household
preference for product variety and city size,
from equation (22), we obtain
­ N *j
5
­ r
*
2
3
2kN j 2
,
(1 2 r ) 2
0,
(24)
which leads us to the follow ing proposition:
P roposition 2: The City Size Increase when
Households have a Strong P reference for
P roduct Variety
In addition to the effect of com muting cost
on city size, we can conclude that a preference for product variety is also a key factor
in the form ation of large cities. From equation (19), we know that r 2 1 is the parameter
representing the degree of agglom eration
econom ies. A decrease in r , in turn, enlarges
the capability to accept a large popula tion, so
that city size increases as the preference for
product variety becom es strong. 7
As stated in Fujita (1988) and RiveraBatiz (1988) , preference for variety is a
major determinant of spatial structure in
recent years. It is well know n that individu als
evaluate particular locations or areas based
on differences in the variety of products as
well as in climate or amenities available. Our
result clearly show s that popula tion tends to
concentrate in a few core cities with increasing preference for product variety.
As in propos ition 1, it is clear that city size
is closely related to com muting cost; how ever, a low com muting cost is not a suf® cient
reason for the existence of large cities.
Notice that, in equation (22),
r
lim N *j 5
®
r Ý1
0.
(25)
1 im plies that differentiated services are
close to a perfect substitute. In this case,
there is no force for househo ld agglom eration, and there exists no optim al city size.
Although the preference for product var-
iety plays a key role in the form ation of large
cities, we know that
lim N *j 5
kj Ý 0
` .
(26)
k j 5 0 im plies that there only exists an
agglom eration force, resulting in the agglom eration of the entire population in a single
city.
Sum marising the above discussions, we
obtain the follow ing:
Remark: The advanta ge of a low commuting
cost is not a suf® cient condition for the existence of large cities. Large cities are formed
when a strong preference for product variety
is combined with a low com muting cost.
The view point taken in this section is that the
basic characteristics of cities are to be unde rstood in terms of household responses to
opportu nities for consum ing product variety.
The num ber of services increases as the preference for product variety strengthens. A
preference for product variety plays a very
im portant role in the form ation of large cities. T his result also provid es a possible
explanation for international variations in
city size. A country where households have
small r will have a large city size; while a
country with large r have small cities. 8
4. Conclud ing Remarks
In this paper, we consider the effect of preference for product variety on city size. The
main conclusions are that, ® rst, when households have a strong preference for product
variety, city size becom es large. Secondly, as
already know n, com muting cost is an im portant factor in the determination of city size.
However, it is not a suf® cient reason for the
existence of large cities. A necessary condition is other features of agglom eration
econom ies, such as the preference for product variety.
Notes
1.
Alperovic h (1992) de® nes econom ic developm ent in term s of increasin g per capita
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PROD UCT V ARIET Y A ND CITY SIZE
2.
3.
4.
5.
6.
7.
8.
G NP. By contrast , Rosen and Resnick (1980)
® nd that the relation ship betw een econom ic
develop m ent and populat ion concentr ation
appears negative at all levels of develop m ent.
A s a represen tative work, Dixit (1973) exam ines optim al city size as determ ined by the
relation ship betw een transpor t costs and
econom ies of scale in product ion. By integrating both supply- and dem and-side
approac hes, Abdel-R ahm an (1988) presents
the concept of product variety as an im portant factor in the form ation of large cities.
T hen Abdel-R ahm an discusses a subsidy
schem e that leads a market equilibr ium to a
® rst-best optim um .
T he data are from the Japanese E conom ic
Planning Agency (1995).
Follow ing Dixit and Stiglitz (1977), w e
assum e that the num ber of services is
suf® ciently large that the effect of each p ji on
Z j can be ignored .
E valuated at ­ V j/ ­ N j 5 0, the second-o rder
conditio n ­ 2V j/ ­ N 2j , 0 is satis® ed.
T hey state that ª if the populat ion of New
Y ork were doubled , keeping the current
com m uting facilitie s intact, traf® c congesti on
w ould becom e prohibi tive. In this sense, the
availabi lity of a netw ork of well-deve loped
com m uter railroad s keeps the com m uting
cost in Tokyo low er than in N ew Y ork.º See
H atta and Ohkaw ara (1994, p. 110).
N otice that w e obtain this result with the
usual propert y of ­ N *j / ­ a j . 0.
Consider the follow ing num erical exam ple in
order to obtain explicit solution s with two
countrie s. T here are tw o countrie s in which
exogeno us param eters are given by k j 5 k,
a j 5 a and b 5 0.5. In this case, w e can easily
obtain the result that the city size in the
country w ith r 5 0.4 is 39 per cent larger
than the city size in the country with r 5 0.5.
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