Breaking the optimality barrier:
does neoclassical urban theory necessarily predict an optimal city size?
OSTAP P ETRASYUK*
Abstract: This paper is an attempt to find out if neoclassical urban system
models necessarily predict the existence of an optimum city size. Using an
existing model and empirical data, the paper finds out that optimum city
size is not an inevitable implication of the model, and though the concept is
apparently correct in general, there may be exceptions from inverted Ushaped utility functions. Namely, the aggregate utility of cities that
overcome a certain threshold in share and variety of services in their
production function may grow steadily without ever going down. Such
threshold may be called the optimality barrier.
Key words: urban theory, optimum city size, service sector, variety, public services
JEL classification: R11, R12, R32
___________________________
* A PhD student, Department of Regional Studies, Faculty of Economics, University of Economics, Prague, University
of Economics, Prague, email: [email protected]
1. Introduction
The concept of optimum city size is at least 2450 years old. Probably even more – in 1700
B.C., when Babylon apparently reached population of 300,000 (Bairoch, 1988), there were probably
people there who wondered if their city wasn’t too big. But those people haven’t left us any written
accounts. The first known work on an ideal city size was written by Hippodamus of Miletus in the
middle of 4th century B.C. According to Aristotle’s Politics, Hippodamus believed that the ideal
city size is 10000 citizens (1267b). As the median size of a household was somewhere between 4.3
and 5 persons (Hansen, 2006), the total population of the Hippodamus’s ideal city wouldn’t be more
than 50000 – about two times smaller than Athens, the biggest Greek city of the times, and about 20%
bigger than other big cities of Greece. Plato, one hundred years after Hippodamus, envisaged an
ideal city with a more modest size – 5040 households (The Laws, book 5, 737e), or about 2000
persons. Aristotle himself introduces in his Politics the concept of efficiency of a city, saying that
too small a city-state won’t be self-sufficient, and too big city-state, though self-sufficient in the
mere necessities, will be ungovernable (1326b). The exact numbers are given in Nicomachean
Ethics: “Ten people would not make a city, and with a hundred thousand it is a city no longer”
(1170b.20). Even if we consider a hundred thousand people as a hundred thousand citizens, with
families and slaves, the total number will be five hundred thousand. The population of Alexandria, a
city founded by Aristotle’s disciple Alexander the Great, probably reached this limit by the first
century B.C. (Kehoe, 2010). In the second century A.D., Rome has certainly surpassed it, counting
at least 800,000 and probably as many as 1,300,000 inhabitants (Bairoch, 1988, p.81).
After the decline of Rome and the following deurbanization there seemingly wasn’t much
interest in the optimum city size. The Aristotelian idea of efficient city size was recalled only in the
last century, when Lösch (1940) introduced the concept of inverted U-shaped utility function
resulting in a single optimum city size (Papageorgiu and Pines, 2000). Since then the concept is the
prevailing one in urban theory.
The existence of an optimum city size is explained by interplay of two factors: agglomeration
economies and urban diseconomies. Agglomeration economies could be divided in several
subcategories (Toth, 2011):
- Internal economies of firms working in the city due to increasing returns to scale of
production.
- Internal economies of firms working in the city due to access to new technologies.
- External localization economies due to concentration of many firms of similar specialization
in one place and corresponding increasing return to scale in inputs, bigger labor pooling, exchange
of supply, intra-industry knowledge spillover, etc.
- External urbanization economies due to concentration of many different firms in one place
and corresponding decrease of transportation, utility, administrative and educational costs, interindustry knowledge spillovers, etc.
Agglomeration diseconomies are appearing due to increasing costs of infrastructure
maintenance, deteriorating environment, increasing land rents and costs of consumer goods,
congestion, etc.
The prevailing view of the modern urban theory is that agglomeration economies increase
rapidly at first, but at some point start to decrease, while agglomeration diseconomies are
continuously increasing. So, there’s a point where agglomeration diseconomies surpass
agglomeration economies, therefore after reaching a certain population limits cities becomes
ineffective (Toth, 2011). The optimum city size is where economies are high and diseconomies are
relatively low. Or, in other words, where utility of the city inhabitants (usually expressed in their
real wages) is the highest.
The following two stages of the development of the optimum city concept can be named. The
first one, starting from Lösch (1940), is mostly logical and empirical. Toth (2011) is giving several
examples of such works such as Isard (1956), Harris and Wheeler (1972), Sale (1973) and
Rasmussen (1973). The second one, according to Abdel-Rahman and Anas (2003), can be traced to
Henderson (1974), who created a mathematical model of a city economy based on land use and
external economies. The latter modeling approach has developed into Neoclassical Urban Systems
Theory (Henderson, 1996) and can be considered mainstream in contemporary urban studies.
The optimum city size concept is intuitively appealing and seems to be correct. It’s easy to
remember examples of cities which were growing fast initially, but after reaching a peak began to
stagnate: Detroit, Liverpool, Ostrava… Many and many more can be named. On the other hand
everybody can recall cities that are growing more or less steadily through the centuries and do not
show signs of stagnation or insurmountable inefficiency. Global megalopolises such as London,
New York, Paris and Tokyo are example of such cities, but smaller examples do also exist – such as
Prague, which, being the biggest city in the Czech Republic, is growing quite healthily, and has
become one of the richest European regions, while many medium-sized Czech cities are stagnating.
So, is the theory of the optimum city size correct, as demonstrated by the examples of Detroit
or Ostrava, or is it wrong, as suggested by Prague or New York? This article, using an existing
neoclassical urban systems model, hypothesizes that the concept is generally correct, but it may
have an exception. Namely, aggregate utility of cities that overcome a certain threshold in number
and variety of services in their production function may grow steadily without ever going down.
Such threshold may be called the optimality barrier.
In Section 2, we consider the wok of Rasmussen (1973) as an example of an empirical work
published before the birth of the neoclassical urban systems theory. In Section 3, we will study
some implications of the probably most comprehensive and detailed treatment of neoclassical urban
systems model published to this day – Fujita (1989) – to find out if the theory necessarily predicts
the existence of an optimum city size. Section 4 is dedicated to some circumstantial evidence of the
existence of optimality barrier. Finally, Section 5 contains some concluding remarks.
2. Empirical approach and its limitations.
In one of the last works before the appearance of neoclassical urban theory in Henderson
(1974), Rasmussen (1973) defines the optimum city size as the size where the net gains from
agglomeration are the greatest. He demonstrates it with the following figure:
Figure 2.1 – Optimal city size according to Rasmussen (1973, p.153)
At the first stage (OA, where the net gains are growing and the curve is convex) the net
agglomeration benefits are increasing very fast. At the second stage (OB, where the net gains are
growing but the curve is concave) the net agglomeration benefits are also increasing, but the
increase is slowing down. Finally, at the third stage (B to infinity) the negative agglomeration
externalities are offsetting the agglomeration benefits, and the net agglomeration gains are
decreasing. The optimum city size is at the point B.
It's necessary to clarify that the figure is based neither on specific empirical data nor on
mathematical calculations. It's the author's assumption. In a footnote, Rasmussen acknowledges that
some economists (unfortunately, he doesn't give any names) argue that negative externalities rarely
offset the gains from the agglomerations even in largest cities and consequently deny existence of
the Stage 3 and the existence of the optimum city size.
Rasmussen brings forward two main counterarguments, both are empirical.
First, he provides numbers for earnings in manufacturing for metropolitan areas in the USA
non-South for 1970, collected by U.S. Bureau of the Census:
Table 2.1 – Average hourly wages in manufacturing for different city sizes (Rasmussen, 1973,
p.158)
Size
Average hourly wage median
Under 200,000
3.34
200,000 – 499,000
3.57
500,000 – 999,999
3.74
1 – 2 million
3.75
2 – 4 million
3.83
Over 4 million
3.74
But it’s known that the biggest cities specialize mostly not in manufacturing, but in services,
and city size is negatively correlated with share of manufacturing jobs (Holmes and Stevens, 2004;
Glaeser, Scheinkman and Shleifer, 1995). Therefore, wages in manufacturing may poorly reflect
real median wage in the biggest cities. Indeed, if we take the full data, we can see that the personal
income is increasing for the biggest cities:
Table 2.2 – Per capita income (full data) for different city sizes
Size
Per capita income
1 – 2 million
3509
2 – 4 million
3672
Over 4 million
3794
Data from U.S. Bureau of the Census, Supplementary Report “Per Capita Income, Median Family Money Income and
Low Income Status for States 1969, Standard Metropolitan Statistic Areas and Counties 1970”, Table 2., processed by
the author.
It's nevertheless clear, as Rasmussen correctly states, that cost of living is also rising with city
size.
Rasmussen support it with the following table based on the same 1970 U.S. Census data (an
abridged version here)
Table 2.3 – Cost of living and city size (Rasmussen, 1973, p.158)
Size
Cost index
Under 1 million
100.00
1 – 2 million
102.32
2 – 4 million
104.84
Over 4 million
105.09
But, as we can see from the Table 2.2 and 2.3, per capita income between cities of 1-2 million
and 2-4 million increases by 4.6%, and between 2-4 million and over 4 million by 3.3%, which
more than compensates the increase in the cost of living.
The notion that real wages grew with city size in 1970 is supported by research of Glaeser and
Gottlieb (2006). Hoch (1972) also states that there is a linear relationship between the logarithm of
city size and the logarithm of money income, corrected for prices (as quoted by Segal (1976)). On
the other side, Glaeser and Gottlieb (2006) state that in 2000, the relation of wages and city size
became negative for all city sizes. At the same time the rate of urbanization increased. Glaeser and
Gottlieb (2006) are explaining it with the fact that by 2000, crime rates and other city disamenities
had fallen and the presence of urban social amenities (museums, theaters, better shopping options,
wider social networks, etc.) had become sufficient to compensate the urban dwellers for the
decrease in real wages.
It also can be said that the inhabitants of bigger cities are compensated for the higher average
prices by a greater variety of options and a greater ability to change their consumption bundles
(DuMond, Hirsch and Macpherson, 1999), and correspondingly, adjusted real wages grow with city
sizes.
Another consideration would be that the differences in the cost of living between smaller and
bigger cities do not register the fact that the average price for a category of products in biggest cities
could be higher because of the higher average quality of the products available, and people in big
cities could buy the goods of the same brands actually cheaper than their small cities counterparts.
That hypothesis is supported by the research of Handbury and Weinstein (2011, p.26), who have
found that “most of the positive relationship between prices and city size in the unit value index
reflects the fact that people in larger cities buy different, higher-priced varieties of goods”, and the
variety-adjusted cost of grocery products actually decreases with city size
Net agglomeration benefit or loss is not restricted to the cost of living. Other factors can affect
the life quality in the city. Rasmussen names congestion, air and water pollution and high crime
rates among the negative externalities of big cities. Other externalities, such as longer commuting
times, worse access to the nature, overcrowding, noise and so on could also be added to the list.
As Rasmussen states, “optimum city size is usually defined in terms of maximizing the
benefits of agglomeration, rather than meeting the demands of consumers for city size”. (Italics by
Rasmussen, p.161). On this observation, Rasmussen builds his second counterargument against the
deniers of the existence of the optimum city size. He cites a number of polls in Australia, France
and the USA (including one conducted by himself) where residents of big cities express their
support for diminution of population. For instance, he cites the 1968 Gallup where 56% of New
York residents indicated that if suitable employment were available, they would prefer to live in a
small town or rural area.
In all the aforementioned countries, freedom of intra-state movement is not restricted and a
person can move from a big city to a nearby small town or rural area rather fast and cheaply. Taking
into account the higher cost of living and land rent in big cities, which Rasmussen mentions in his
work, the move can actually be financially beneficial, all other things equal. So, why don’t people
who express their wish to live in rural areas actually move there? An answer is contained in the
Gallup question. They can't find suitable employment there.
Big cities provide better employment (and also consumption) opportunities than smaller ones.
For those who chose to live in the cities, despite their legitimate complains about congestion,
pollution and high costs, these employment opportunities are of higher importance than the
agglomeration externalities, and therefore the net benefits of living in big cities are positive. Massscale internal movement in fact exists, but it has the opposite direction – from rural to urban cities
(UN data, retrieved at http://esa.un.org/unup/).
Moreover, according to data cited by Rasmussen, inhabitants of small cities would prefer to
live in bigger ones (though they actually also don't move there). Both contradictory phenomena
most likely can be better explained by the saying “grass is always greener on the other side of the
fence” than by suboptimal distribution of city sizes, except for the cases of states which restrict
freedom of internal movement, such of Cuba or China.
Summing up, Rasmussen’s arguments in support of the single-peaked curve of net
agglomeration benefits are either based on incomplete statistical data or on unconvincing
interpretations of public opinions surveys and can’t be used in support of the theory of the existence
of a single optimum city size. The net agglomeration benefits curve, according to empirical data,
shouldn’t necessarily be single-peaked. It can be also either endlessly growing without a peak, or
have a more complex shape with several maximums and minimums. For example, there are data
showing that net agglomeration economies are positive up to the city size of 2.5 million, negative in
the cities between 2.5 and 6 million, and once again increasing in the largest urban areas, even
taking into account pollution, congestion and crime. (Berry, 1974).
3. Neoclassical approach – picking up where Fujita has stopped
One of the most comprehensive and detailed models of neoclassical theory urban systems can
be found in Fujita (1989).
Fujita defines the optimum city size as such where the highest common utility is achieved. To
find this size, Fujita introduces several functions and variables, including (but not limited to):
u – Common utility of the city;
N – Population of the city;
g(N) – Private marginal product of labor representing external economies equally enjoyed by
all firms in the city;
F(N) – Aggregate production function of the city of N residents;
Y(u, N) – Supply-income function or inverse population supply function representing the
household income that is necessary to ensure that N households will be supplied from the national
economy to the city;
Fujita defines utility as
U(s, z) = α log z + β log s,
(3.1)
where z represents the amount of composite consumer good (chosen as numeraire), s - the
consumption of land, and α and β are constants, such that α > 0, β > 0 and α + β = 1.
Regarding g(N), Fujita makes the following assumptions:
i)
g(N) > 0 for all N > 0
(3.2)
ii)
g'(N) > 0 for N < Ñ and g'(N) > 0 for N > Ñ
(3.3)
where Ñ is a given constant such as that 0 < Ñ ≤ ∞
Regarding Y(u,N), Fujita, after a series of mathematical manipulations, comes to the
following conclusions:
i)Y(u, N) > 0, and limN → ∞ Y(u, N) > 0;
(3.4)
ii)
Y(u, N) is increasing in both N and u: ∂Y(u, N)/∂N > 0, ∂Y(u, N)/∂u > 0; (3.5)
iii)
i) limN
→ ∞ Y(u,
N) = ∞, limu
→ ∞ Y(u,
N) = ∞.
(3.6)
Fujita mathematically derives that in the situation of absentee landownership F(N) = g(N)N
and that an equilibrium city size can be determined by solving the equation g(N) = Y(u, N) for N,
which can be expressed by the following picture:
Figure 3.1 – g(N) under assumption 3.3 according to Fujita (1989, p.275)
Here ua0 > u1 > ua1 > u2. It's obvious from the picture that ua0 represents the highest
equilibrium utility level the city can attain, and the corresponding city population Na0 is the
optimum city population for households.
From the figure, we can make the following assumptions:
i)
There exists the highest utility ua0 < ∞
(3.7)
ii)
ua0 is achieved at a unique city size Na0, such that 0 < Na0 < ∞
(3.8)
The ua (national utility level) curve under the assumptions may look like this:
Figure 3.2 – Utility curve under assumption 3.8 according to Fujita (1989, p.276)
Na0 is the optimum city population for households, because at it they reach the maximum
possible utility.
Fujita explicitly states that 3.6 and 3.7 are just assumptions and “it may not always be the
case”.
But under these assumptions, he comes to the conclusion that optimum city size increases
with the increase of national utility. Indeed, it's obvious from the Figure 3.1 that at Na0 g(N) is
tangent to Y(ua0, N) from below in its (g(N)) increasing phase. Therefore, because Y(u, N) is
increasing in u, the bigger is the utility, the higher is the corresponding Y(u, N) curve, the closer is
the tangent point to 0, and the smaller is Na0.
It follows from here that an equilibrium city in a country with a lower national utility level
should be larger than an equilibrium city in a country with a higher national utility. But it clearly
contradicts empirical data. The highest increase in urbanization started after the Industrial
Revolution, which greatly increased the utility level of the affected regions, and the previous waves
of urbanization were connected to agricultural and commercial revolutions, which had also
substantially increased utility (Bairoch, 1988). According to Ades and Glaeser (1995), central cities
are bigger in countries with a higher national GDP per capita. Also, as we can see from the
following table, among the 10 biggest areas of the world there are cities from countries with very
different utility levels:
Table 3.1 – The biggest world cities
City
Country
Population
Tokyo
Japan
37,126,000
Jakarta
Indonesia
26,063,000
Seoul
South Korea
22,547,000
Delhi
India
22,242,000
Manila
Philippines
21,951,000
Shanghai
China
20,860,000
New York
United States
20,464,000
São Paulo
Brazil
20,186,000
Mexico City
Mexico
19,463,000
Cairo
Egypt
17,816,000
Data from Demographia World Urban Areas (World Agglomerations): 8th Annual Edition, April 2012, Wendell Cox
Fujita himself makes a more careful conclusion that “given two cities (with the same industry
structure) in two different countries, the city in the country with a lower living standard will be
larger,” but even this is still problematic. It's hard to find two cities with exactly the same industry
structure, but most of the modern megalopolises share more or less a similar industry structure with
tertiary industries prevailing. For example, in 2007, service sector accounted for 81,1% of the
metropolitan area GPD in Tokyo 1 and 82.14% in Delhi2, and the utility level of India is much lower
than that of Japan, yet the size of Tokyo is about 70% greater than that of Delhi.
One of the possible explanations for such contradiction between the theory and the empirical
data is that the assumption 3.2 is misleading and g'(N) > N for any N, or at least g'(N) = 0 when and
only when N = ∞, i.e. the private marginal product is growing for any N < ∞. In other words, the
production function grows with the increase of city size for all city sizes. It conforms to the
empirical data for the existing city sizes (Glaeser and Gottlieb, 2009).
In this case, one of the possible shapes of the g(N) curve can look as follows:
Figure 3.3 – a possible shape of g(N) without assumption 3.3
Here we can see that a stable equilibrium city size is unattainable and optimum city size with
of the maximum household utility also doesn't exist, as household utility grows with the city size.
________________
1
The
Financial
Position
of
the
TMG
and
TMG
Bonds
–
http://www.zaimu.metro.tokyo.jp/bond/en/tosai_news_topics/news_topics/Overseas_IR_Presentation_Document20071
023.pdf
2
Economic Survey of Delhi – http://delhiplanning.nic.in/Economic%20Survey/ES2007-08/ES2007-08.htm
Let’s try to figure out in which cases g(N) grows indefinitely and in which ones it has a single
peak. Fujita states that
g(N) = ANb
(3.9)
Where
�=
1−� �
(3.10)
�
ρ and ν are taken from the following production function of a traded good X
�
� = �� ( (∑��=1
1⁄�
�
�� )
�
) ,
(3.11)
which is a variant of Dixit-Stiglitz (1977) function n for monopolistic competition.
Here, X is the amount of traded good produced by a firm, Nx the amount of labor, qi - amount
of nontraded differential services i employed by a firm, and ν, η and ρ are positive constants such
that ν + η = 1 and 0 < ρ < 1.
Using several other formulas Fujita derives
ANb = D(N + E)βeu
(3.12)
Where A, D, E and positive constants and β is a positive coefficient from the utility formula
3.1
From here we can derive the equilibrium utility function:
�
� ⁄�
� � = ��� + ���� �+
(3.13)
To find the equilibrium population we should maximize it for N. The maximization condition
is
��
��
=
�
� �
−
�
�+
(3.14)
As Fujita notices, it yields that the � � curve is single-picked if and only if
b/β < 1 or b < β
(3.15)
That’s the only case that Fujita’s actually considering in his work. Let’s consider the other
case:
b/β ≥ 1 or b ≥ β
(3.16)
Figure 3.4 – Three possible shapes of utility function
The blue line on the Figure 2.4 represents the case where b < β and u curve is the fastest
growing in the initial stage, but then it reaches its maximum and starts to decline.
The red line represents the case where b = β and u curve is continuously growing approaching
�
the constant K = ��� , which is the optimum city utility that can never be reached. We can call K
the optimality barrier.
The green line represents the case where b > β and u curve is growing without a limit, though
with ever decreasing rate, as the function is concave.
Let’s see when such growth is possible.
From 3.9 and 3.14
1−� �
�
≥β
(3.17)
The left hand side of the equation is growing in ν and declining in ρ. The share and variety of
services in the production function 3.10 is also growing in ν and declining in ρ.
Also, the right hand side of the equation is growing in β, while the share of composite goods
in utility is declining in β.
So if the share of composite goods in overall consumption is sufficiently small and the
composite good consumption is sufficiently big, the utility of a city may be constantly growing, as
seen from the green line on Figure 2.4.
Hence we can suppose that the city with a big share of services would grow almost
indefinitely, while industrial cities would grow up to a certain point and afterwards stagnate or
decline. Comparison of the pairs New York – Detroit, London – Liverpool and Prague – Ostrava
apparently support this hypothesis.
The hypothesis seemingly conforms to the central place theory, because cities of higher rank
serve as marketplaces for the cities of lower rank and provide various services for them, therefore
the share and variety of services in such cities’ economies should be the bigger the higher is the
rank of the city.
4. What city size data say – some circumstantial evidence.
An indirect confirmation of the hypothesis of ever-increasing utility of the cities of a highest
order would be a distortion of distribution of actual city sizes that keeps increasing with time. If
optimality barrier exists for some cities but not the others, most of the cities must reach some
population level and stop growing, because such cities become inefficient, the utility of its
inhabitants starts to decline, causing the population to run from such cities. But the minority of
cities (the ones that have overcame the optimality barrier) would grow bigger and bigger, attracting
population from the less efficient cities.
Let’s have a look at distribution of the US urban population in 1940, 1970 and 2000:
Figure 4.1 – Distribution of US cities by population size
60
50
40
1940
30
1970
2000
% of total population
20
Data
10
0
< 0.1
0.1 - 0.25 0.25 - 0.5
0.5 - 1
1 - 2.5
2.5 - 5
>5
US metropolitan areas by population size, in mln
from
American
Demographic
History
Chartbook:
1790
to
2000
by
Campbell
Gibson.
(www.demographicchartbook.com/Chartbook/images/chapters/gibson03.pdf) as retrieved on May 14, 2012.
Figure 4.2 -– Growth of number of US cities of different population sizes from 1940 to 2000
10
9
8
7
6
US population
5
US cities 5 mln or more
US cities 0.5 - 1 mln (10-1)
4
3
2
1
0
Data
from
American
Demographic
History
Chartbook:
1790
to
2000
by
(www.demographicchartbook.com/Chartbook/images/chapters/gibson03.pdf)
as
retrieved
Campbell
on
US population is multiplied by 10 to scale, number cities 0.5 – 1 mln is multiplied by 10 to scale.
-8
-1
May
14,
Gibson.
2012.
Figure 4.1 demonstrates the distribution of US population by city size categories in 1940,
1970 and 2000. The share of each category of urban population smaller than 5+ million remains
more or less stable, except for the category 1 – 2.5 mln, which demonstrated an upward shift in
1970. But the share of population living in the biggest cities is steadily increasing.
Figure 4.2 demonstrates the number of cities in the categories 5+ million inhabitants and 0.5 –
1 mln inhabitants (the latter is divided by 10 to scale) from 1940 to 2000 in 10-year increments,
compared with dynamics of total US population (divided by 100 mln to scale). The number of cities
with 0.5 – 1 mln inhabitants is growing at about the same rate as total US population, and the
number of cities with 5+ mln inhabitants is growing about 3 times faster.
It can be considered as circumstantial confirmation that the biggest cities serving as market
places of the highest order do not have an equilibrium size unlike the rest of the flock. We can
suppose from Figure 4.1 that if a US city overpasses a threshold of about 2.5 mln, it will probably
grow further.
Another circumstantial evidence can be seen in the following tables of the biggest UK cities3
population change as compared to the share of services in their economy:
Table 4.2 – Rates of growth of the biggest British cities and proportion of business services
Population 1999
Pop. change
Employment
Employment
– 2002
over prev. 5
in financial
in commercial
years (%)
activities (%)
services (%)
London
7,172,091
0.77
30.0
57
Manchester
418,600
0.59
25.3
56
Edinburgh
448,624
0.17
29.3
54
Leeds
715,399
0.03
21.
50
Bradford
467,657
-0.05
14.8
44
Sheffield
513,231
-0.16
15.6
44
Bristol
380,616
-0.16
26.0
53
Birmingham
977,087
-0.43
19.6
45
Glasgow
577,869
-0.71
21.6
49
Liverpool
439,476
-0.84
17.8
45
Panel A
___________________
3
Data for US cities cannot be used for the purpose because of the changed definitions between censuses and because
data on some biggest metropolises are missing or incomplete.
Population
Finance
Commerce Pop.
change Finance
Commerce
2003 - 2006
1999 - 02
1999 - 02
5 y. 2003 - 06
2003 - 06
2003 - 06
Manchester
438,700
25.3
56
2.29
28.3
60
London
7,413,100
30.0
57
1.03
31.8
62
Bristol
397,900
26.0
53
0.77
29.6
56
Bradford
480,900
14.8
44
0.71
15.5
45
Leeds
734,800
21.8
50
0.59
24.9
52
Edinburgh
453,700
29.3
54
0.29
32.7
58
Birmingham
994,300
19.6
45
0.21
22.2
49
Sheffield
519,800
15.6
44
0.17
19.6
49
Liverpool
441,500
17.8
45
-0.17
19.2
47
Glasgow
577,700
21.6
49
-0.18
25.0
52
Panel B
Com-
Com-
Population
Finance
merce
Pop. change
Finance
merce
2007 - 2009
2003 - 06
2003 - 06
5 y. 2007 - 09
2007 - 09
2007 - 09
Manchester
473,200
28.3
60
1.82
30.0
60
Bristol
426,100
29.6
56
1.65
31.0
58
Leeds
779,300
24.9
52
1.45
27.8
56
Bradford
501,400
15.5
45
1.02
17.4
47
Edinburgh
471,700
32.7
58
1.02
28.8
57
Sheffield
539,800
19.6
49
0.93
18.5
47
London
7,668,300
31.8
62
0.77
32.1
66
Birmingham
1,019,200
22.2
49
0.57
21.9
50
Glasgow
584,200
25.0
52
0.24
26.4
53
Liverpool
441,100
19.2
47
0.04
20.1
48
the
Urban
Panel C
Data
from
Eurostat
Audit
Data
Collection
as
retrieved
14.05.2012
from
http://epp.eurostat.ec.europa.eu/portal/page/portal/region_cities/city_urban/data_cities/database_sub1.
Financial activities are as in NACE Rev. 1.1 K, Commercial services are as in NACE Rev. 1.1 G-K.
It can be seen that the cities with higher proportion of employment in commercial services in
general and in financial intermediation business activities in particular are growing, and those with
the lowest share of employment in commercial services are either declining or demonstrating
growth rates within statistical error. Bradford is the obvious exception, but it can be explained by
the fact that Bradford is a part of Leeds-Bradford metropolitan area, and its city center is just 10
miles away from the one of Leeds. The cities have combined labor pool and the general balance of
jobs in the metropolitan area is more service-oriented, taking into account a big proportion of
service jobs in Leeds.
It can be said, of course, that growth rates should be compared with utility of the cities. It’s
certainly true and should be done in the future. But even by itself, growth rate is a good proxy for
city utility – if people are willing to move to the city and unwilling to leave it, it means that their
utility in this city increases.
Looking into all three tables, we can conclude that the threshold where a city begins to grow
is at about 50% of service jobs. It conforms to the work of Capello and Camagni (2000), who after
studying data from 58 Italian cities, though supporting existence of efficient city sizes, concluded
that agglomeration economies (or “city effect”) decline at first with the increase of the share of
higher urban functions (tertiary sector), but then start to increase again.
Figure 4.2 (Capello and Camagni, 2000, p.1490)
It’s interesting to note that the point where the city effects begins to grow is 49% - almost
exactly the same proportion as is seen in Table 4.1.
Another indirect confirmation of the hypothesis can be found in Glaeser, Scheinkman and
Shleifer (1995), which cites empirical data confirming that income and population growth move
together and both are negatively relative to the initial share of employment in manufacturing. Catin
(1997) points at the positive feedback loop between the size of a city, its productivity and share of
tertiary sector (and especially business services) in its output. Segal (1976) also finds that
agglomeration economies and productivity grow with city size across the specter.
One more assumption that can be made is that in an economy where the provision of local
public goods is subcontracted to small private companies, the variety and share of intermediate
services should be bigger, and the probability of breaking the “optimality barrier” higher, then in an
economy where local public goods are provided publicly. Therefore, high proportion of publically
provided public goods may impede city growth.
Let’s test this assumption on the British cities. As an instrument, we can use the proportion of
employment in public administration, health and education (PAHE) – services that are provided in
the UK mostly publicly (data from Eurostat).
Table 4.3 – Rates of growth of the biggest British cities and proportion of public services
Change
Population
(%) over
PAHE
Change
PAHE
Change
PAHE
2007 – 2009
5 years
1999 – 02
2003 – 06
2003 – 06
2007 – 09
2007 -09
before
1999 - 02
London
7,668,300
0.77
30.0
1.03
29.5
0.77
27.4
Birmingham
1,019,200
-0.43
31.2
0.21
33.8
0.57
35.8
Leeds
779,300
0.03
28.5
0.59
31.3
1.45
29.4
Glasgow
584,200
-0.71
34.4
-0.18
36.5
0.24
35.5
Sheffield
539,800
-0.16
33.1
0.17
34.7
0.93
37.1
Bradford
501,400
-0.05
29.4
0.71
33.6
1.02
35.0
Manchester
473,200
0.59
31.8
2.29
32.2
1.82
33.6
Edinburgh
471,700
0.17
33.4
0.29
33.4
1.02
35.6
Liverpool
441,100
-0.84
41.1
-0.17
43.4
0.04
43.3
Bristol
426,100
-0.16
30.9
0.77
33.3
1.65
32.2
Data from the Eurostat Urban Audit Data Collection as retrieved 14.05.2012 from
http://epp.eurostat.ec.europa.eu/portal/page/portal/region_cities/city_urban/data_cities/database_sub1.
We can see a substantial, though not perfect, negative correlation between the proportion of
publicly provided services in an economy and the dynamics of city size. The cities with a high
proportion of publicly provided services demonstrate the lowest or negative growth rates. The cities
with a low proportion of PAHE have higher growth rates.
The correlation holds (though weaker) for German cities, but doesn’t hold for Italian ones
(and there’s no consistent data for France in the Eurostat database). It can be explained by the fact
that among the three countries, the UK has the highest rate of population mobility, and Italy has the
lowest one (Puhani, 1999), and also by the fact that the proportion of PAHE in Italian cities is very
low (8-12%).
The aforementioned data and research is certainly not enough to prove (or disprove) the
hypothesis of existence of the higher rank cities able to indefinitely increase utility. There are many
empirical studies claiming that agglomeration diseconomies start to prevail after some city size
exactly for the biggest cities. For instance, Herzog and Schlottmann (1993) calculate that in cities
bigger than 4.8 mln, urban disamenities oversize urban amenities, and Kanemoto, Ohkawara and
Suzuki (1996) give estimate of 18 mln for Japan, which is bigger than most of the world
megalopolises, but still smaller than 9 of them (see Table 3.1). Therefore, more detailed empirical
research is necessary to estimate if a proportion of ρ,ν and β sufficient to “break” the “optimality
barrier” really exists. There are two obvious directions for such research; the first would be
measuring the correlation between variety and proportion of different types of services in city
economy and quality of life in a city; the second, measuring the correlation between quality of life,
elasticity of substitution between consumption of composite goods and housing, and consumer
preferences in general.
5. Concluding remarks and policy implications.
Unlike Aristotle, we know beyond doubt that huge cities with populations much bigger that
500 million inhabitants can be governed. But can such metropolises be efficient? From
methodological individualism point of view, the answer is clear – as long as individuals choose to
live in a city by their own will, they consider the city as maximizing their utility, and therefore, the
city is optimal for its actual citizens. But if we consider a city as an urban company, managed by a
“board of directors” (City Hall) whose aim is to increase prosperity of its shareholders, the question
remains – can the biggest cities increase such prosperity ad infinitum, or mounting transportation
costs, congestion, higher land rents etc. inevitably create the “optimality barrier” that cannot be
broken? The present work demonstrates that neoclassical urban systems theory doesn’t necessarily
forbid the existence of cities with ever-increasing utility, and some empirical studies indirectly
demonstrate that the “optimality barrier” can indeed be broken, providing sufficiently high
proportion of services in city economy. There is also empirical evidence to the contrary. To solve
the question, more research, both empirical and theoretical, is needed.
This work is just a preliminary study, so it’s possibly too early to derive from it any policy
implications. Moreover, as Pumain (2010) notices, systems of cities are complex systems which are
difficult to control, and the question if state or local policy can meaningfully affect their dynamics is
still open. But some assumptions can be cautiously made nevertheless. Policies helping to break the
optimality barrier and ignite constant utility growth may possibly include elimination of entry
barriers for small businesses and tax preferences or vacations for big ones and privatization and demonopolization of public services delivery. Some policies may be added to this short list (or
removed from it) after further research.
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