IS THERE
C I T Y SIZE DISTRIBUTION
FOR
DENEIDPING COUNTRIES?
by
PETE33 SPENCE GUDSEON
B .A.
onou ours) , Victoria University of Manchester , 1965
Dip. i n Econ. Dev.., Victoria University of Manchester,
MASTER OF ARTS
i n t h e department
Economics and Commerce
@
PET^ SPEDTCEG U ~ E O N , 1968
SIMOK
F'RAsm
UNIVERSITY
Dec ember
, 1968
1967
D r . K . Okuda
Senior Supervisor
D r . D. Maki
Examining Comr.'Littee
--Dr.N. Robinson
Examining Committee
ABSTRACT
The d i s t r i b u t i o n of c i t y s i z e s i n terms of population i n
both underdeveloped c o u n t r i e s and developed c o u n t r i e s , has r e c e n t l y come
under c l o s e examination by s p e c i a l i s t s i n many academic f i e l d s .
The ad-
v e n t of g e n e r a l systems theory has proved t o be an invaluable a n a l y t i c a l
approach t o t h e study of c i t y s i z e d i s t r i b u t i o n s , i n t h a t it i n c o r p o r a t e s
s t o c h a s t i c growth theory, and t h e concept of entropy.
These two a s p e c t s
of g e n e r a l systems theory have been very u s e f u l i n explaining some of
t h e e m p i r i c a l r e g u l a r i t i e s observed of c i t y s i z e d i s t r i b u t i o n s , e s p e c i a l l y
t.he log-normal ( o r rank-si ze ) d i s t r i b u t i o n
.
This essay p r i m a r i l y examines t h e v a r i o u s t h e o r i e s t h a t
have been developed t o explain t h e rank-size d i s t r i b u t i o n of c i t i e s , and
r e l a t e s t h e s e t h e o r i e s t o t h e g e n e r a l systems approach.
It i s a l s o
hypothesized ( i n a somewhat t e n t a t i v e f a s h i o n ) t h a t t h e log-normal c i t y
s i z e d i s t r i b u t i o n i s an "optimum" equilibrium o r "steady-state" d i s t r i k
b u t i o n toward which c i t y s i z e d i s t r i b u t i o n s t e n d under c e r t a i n l a r g e comp e t i t i v e environmental conditions.
This essay i s not intended t o be an exhaustive study of t h e
l i t e r a t u r e , and i n c o r p o r a t e s only a b r i e f excursion i n t o p o s s i b l e p o l i c y
c o n s i d e r a t i o n s of t h e t h e o r e t i c a l and e m p i r i c a l f i n d i n g s on t h e c i t y s i z e
d i s t r i b u t i o ~ si n developing and developed c o u n t r i e s .
(iii)'
TABLE OF CONTENTS
Page
................................................
I.
C i t y S i z e D i s t r i b u t i o n s and Developing Countries .....
I1 .
C i t y S i z e D i s t r i b u t i o n and P a r e t o ' s Law ..............
I11.
Zipf and t h e Rank-Size Rule ..........................
I V.
' C h r i s t a l l e r and t h e S i z e C l a s s e s of C i t i e s ...........
The City-Size D i s t r i b u t i o n a s a S t o c h a s t i c Process ...
V.
VI.
City-Size D i s t r i b u t i o n s and P o l i c y Consideration .....
V I I . City-Size D i s t r i b u t i o n s and General Systems Theory ....
1
I n d i a n City-Size D i s t r i b u t i o n ........................
IX .
conclusion ...........................................
References ..................................................
Introduction
1
6
9
13
16
18
25
36
42
48
50
(iv)
I
LIST OF DIAGRAMS
PAGE
lc
The log-normal (rank-size ) d i s t r i b u t i o n
The p r j i a t e d i s t r i b u t i o n
The i n t m n e d i a t e d i s t r i b u t i o n
2'
P o s s i b l e e f f e c t s of a
3
P o s s i b l e e f f e c t s of r e s t r i c t i n g t h e s i z e
of the 1 z . r g e ~ tc i t y .
la
lb
.................... 7
r e d i s t r i b u t i o n p o l i c y . . . . . . 28
... . .. .. . .... . ........ . . .. . .. 28
INTRODUCTION
The existence of d i f f e r e n t s i z e d i s t r i b u t i o n s of c i t i e s i n
b o t h developing and developed countries has long been recognized by students
i n many d i s c i p l i n e s , and has r a i s e d some fundamental questions f o r planning i n underdeveloped countries.
For example, i s t h e r e any s i g n i f i c a n c e
i n t h e f a c t ' t h a t some c i t i e s have grown f a s t e r t h a n o t h e r c i t i e s ?
t h e causes of t h i s growth?
What a r e
What are t h e problems associated with d i f f e r -
e n t i a l growth and s i z e of c i t i e s ?
Much of t h e work on urban problems has been done i n rel a t i o n t o regional development programming i n t h e United S t a t e s , and
comparatively l i t t l e research has been undertaken on t h e f'undamental
problems of defining an optimum s i z e ; d i s t r i b u t i o n of c i t i e s f o r developing countries.
optimality?
Is t h e r e indeed an optimum? What a r e t h e c r i t e r i a f o r
1% t h e r e an optimum with respect t o maximization of economic
growth, o r i s it some broadly defined welfare optimum t h a t i s desired?
Obviously t h e optinium s i z e d i s t r i b u t i o n of c i t i e s i n underdeveloped
countries could be defined i n terms of economic, s o c i a l , p o l i t i c a l , o r
any o t h e r c r i t e r i a , and t h e choice of optimum w i l l depend very much on t h e
o b j e c t i v e f'unction of t h e planner and those of s o c i e t y a t l a r g e .
It i s worthwhile a t t h e o u t s e t t o give some consideration t o
t h e general problems of defining our c r i t e r i a f o r optimality.
approach t h e problem from maay d i f f e r e n t angles;
One can
one can define an
absolute l i m i t t o t h e s i z e of any one c i t y under d i f f e r e n t s e t s of i n i t i a l
conditions and
c o n s t r a i n t s o r one coclld search f o r an optimum f o r t h e
whole range of c i t y s i z e s i n any m e country.
Any c r i t e r i a f o r optimum population s i z e involves, imp l i c i t l y o r e x y l i c i t l y , two elements:
f i r s t the normative element, which
places a p o s i t i v e o r negative valuation on a p a r t i c u l a r situation;
and
second, a f a c t u a l element which has the force of a statement of empirical
r e l a t i o n s h i p s between v a r i a t i o n i n c i t y s i z e and v a r i a t i o n i n the s i t u a t i o n i n question.
One can a l s o a t t a c k t h i s problem from
(1)
the point of
view of the t h e o r i s t of c i t y planning i n t e r e s t e d i n s e t t i n g general standards f o r the over-all planning of c i t i e s , or
(2)
determine a l i s t of
s p e c i f i c c r i t e r i a f o r determining optimum c i t y s i z e , o r
(3) examine each
of the c r i t e r i a from the standpoint of observable r e l a t i o n s h i p between
c i t y s i z e and the variables involved i n the c r i t e r i a .
Consider f o r example t h a t a c r i t e r i o n of optimum c i t y s i z e
i s t h a t a c i t y ' s s i z e should be t h a t which i s favorable t o the health of
i t s population.
Hence:
Let good health = a p o s i t i v e value.
Let. ill health = a negative value.
IS t'nere some s i g n i f i c a n t c o r r e l a t i o n between c i t y s i z e and health?
If
there were no such correlation, there would obviously be no "most favorable" s i z e , i . e . no optimum.
Clearly then, examining c i t y s i z e from the point of view of
the c i t y planning t h e o r i s t provides only one i l l u s t r a t i o n of a procedure f o r
v a l i d a t i n g the concept of optknun c i t y s i z e .
H i s t o r i c a l l y the concept of optimm s i z e of c i t i e s has
Un-
d e r l a i n planning theory and practice, eithe'r i n e x p l i c i t or implicit f o m .
Among the c r i t e r i a t h a t have been examined in r e l a t i o n t o
the optimum c i t y s i z e i n both developed
developing countries i s :
(1)
City s i z e and physical planning of c i t i e s w i t h respect t o
the frequent demand t h a t c i t i e s be small enough t o enable
ready access t o the countryside and a resonably moderate
journey t o work., i . e . transporation problems,
(2)
City s i z e and health
diseases, e t c . )
(3)
City s i z e and public s a f e t y
f i r e hazards, e t c )
.
( m o r t a l i t y r a t e s , incidence of
..
(crime r a t e s , accident r a t e s ,
(4) City s i z e and municipal e f f i c i e n c y (highways, s a n i t a t i o n , public welfare, schools, e t c . ) .
( 5 ) City s i z e and education expenditures.
( 6 ) City s i z e and cost of l i v i n g .
(7) City s i z e and public r e c r e a t i o n
( a c c e s s i b i l i t y t o parks,
zoos, t h e a t r e s , e t c . ) .
(8)
City s i z e and r e t a i l f a c i l i t i e s .
( 9 ) City s i z e and churches and a s s o c i a t i o n s .
(10)
City s i z e .and family l i f e (degree of homeownership, divorce r a t e s , domestic f a c i l i t i e s , e t c , )
(11)
City s i z e and miscellaneous psychological and s o c i a l chara c t e r i s t i c s of urban l i f e (provincialism, f r i e n d l i n e s s ,
s o c i a l contentment, community p a ~ t i c i p a t i o n ) .
.
It i s immediately apparent t h a t one could extend t h i s l i s t indefinately
a t t h e r i s k of increasing the already considerable overlap i n the tyye of
r e l a t i o n s h i p s , and r e a l i z i n g a t the same time t h a t t h e question of the
casual significance of the r e l a t i o n s h i p s between c i t y s i z e and these phenomenesis subject t o serious problems of s t a t i s t i c a l i n t e r p r e t a t i o n .
1
A r a t h e r b i z a r r e example of planning optimality according
t o a d e f i n i t e c r i t e r i o n i n advanced i n d u s t r i a l countries i s the idea t h a t
c i t i e s should be small enough t o have a low prp3bability of nuclear des2
truction.
A s a general statement it would be t r u e t o say t h a t i n the
p a s t economists have not paid much a t t e n t i o n t o optimum c i t y s i z e , and
even l e s s a t t e n t i o n t o optimum c i t y s i z e d i s t r i b u t i o n s .
More recelri; r e -
search on the r e l a t i o n s h i p of c i t y s i z e t o economic policy, e s p e c i a l l y
publik investment decisions and growth theory i n r e l a t i o n t o urbanization
and i n d u s t r i a l i z a t i o n i n unaerdeveloped countries and developed countries,
has helped t o eliminate t h e i r shortcomings.
3
Yet t h e study of optimum
c i t y s i z e d i s t r i b u t i o n s has been meagre t o say t h e l e a s t , u n t i l the advent of a general systems approach t o the study of c i t y s i z e d i s t r i b ~ t i o n s .
This approach t o the study examines the r o l e , functions, and
spatial
d i s t r i b u t i o n of c i t i e s as a sub'system i n a whole i n t e g r a t e d system t o economic and s o c i a l development, and has been u s e f u l l y applied t o developed
and underdeveloped countries a l i k e .
4
'
The m u l t i p l i c i t y of d i f f e r e n t c r i t e r i a f o r optimal c i t y
s i z e o r c i t y s i z e d i s t r i b u t i o n s makes it impossible, i n my opinion, t o
a r r i v e a t some meaningfuloverall c r i t e r i o n .
conclle the d i f f e r e n t c r i t e r i a .
It would be impossible t o r e -
I n view of t h i s a deductive approach t o
t h e general problem appears more reasonable.
Can me a r r i v e a t any con-
c h s ions from examination of the many d i f f e r e n t
c i t y size distributions?
causes underlying e x i s t i n g
I s there a c i t y s i z e d i s t r i b u t i o n t h a t canes
c l o s e s t t o an a c t u a l o r t h e o r e t i c a l optimum?
Evidence suggests t h a t t h i s
may be so, i n t h e form of t h e log-normal c i t y s i z e d i s t r i b u t i o n .
Consequently much of t h i s essay w i l l be concerned with examination of the log-normal types of c i t y s i z e d i s t r i b u t i o n and t h e i r
relevance t o t h e problem of an optimum c i t y s i z e d i s t r i b u t i o n .
However be-
f o r e t h i s i s dore one must examine t h e empirical evidence on e x i s t i n g d i s t r i b u t i o n s i n developing countries.
I.
---
CITY SIZE DL STRIBUTIONS AND DEVELOPING COUNTRIES
.-
--P
Possibly the most complete and comprehensive empirical
study of the c i t y s i z e distributions i n developed and developing countries
i s t h a t by Berry,
5 who analyzed c i t y s i z e d i s t r i b u t i o n s and t h e i r r e l a -
tionship t o l e v e l s of economic developzlent i n thirty-eight countries.
He
found t h a t the d i s t r i b u t i o n s f a l l i n t o two major categories, namely the
Rank-Size Distribution
6
and the Primate Distribution. 7
The Rank-Size Distribution was revealed in both developed
and underdeveloped countries when the cumulative frequency of c i t i e s with
a population of greater than twenty thousand people was ranked against
the s i z e of c i t y on a log-normal s c a l e ,
Thirteen of the t'nirty-eight Coun-
t r i e s had log-normally d i s t r i b u t e d sizes.
8
( See Diagram l a ) .
The Primate Distribution which was c h a r a c t e r i s t i c of f i f t e e n
out of t h i r t y - e i g h t countries examined, i s observed when a stratum of small
towns and c i t i e s i s dominated by one or more very large c i t i e s and there
a r e deficiencies . i n t h e number of c i t i e s of intermediate s i z e .
gram l b )
9
(See Dia-
Berry's study tended t o support the hypothesis t h a t Primate City
Distributions a r e associated with over-urbanization and superimposed colo n i a l economies i n underdeveloped countries o r with political-administrat i v e controls i n indigenous subsistence and peasant economies.
Furthermore
it has been argued t h a t primate c i t i e s have p a r a l y t i c e f f e c t s upon the
development of s x a l l e r urban places and tend t o be p a r a s i t i c i n r e l a t i o n t o
the remainder of t h e n a t i o n a l economy.
Nine out of t h i r t y - e i g h t of the countries examined had d i s t r i b u t i o n s intermediate between t h e log-normal (rank-size) and the primate
No.of C i t i e s
The LOG-NORMAL (=-SIZE)
DISTRIBUTION.
a:
b:
c:
d:
United S t a t e s
China
Korea
E l Salvador
Size of City. (10,000 i n h a b i t a n t s )
. No.of
Cities
DIAGRAM l b
601I
&
50.
*5
4039-
Tne PRIMATE DISITiiIBUTION.
u a~ 3.6.
%
rl
I
a~ rd
PI; ' 0 -
a:
a,
S M
-4 0
a
*riA
0
'J%
b:
Thailand
Denmark
b
u
0 1 ,
IC
I
50
'
ze of City. (10,000 i n h a b i t a n t s )
loo
DIAGRAM l c
No.of C i t i e s
60 1
50
4t
30
The INTERMEDIATE DISTRIBUTION.
-
a:
b:
Portugal
Australia
c : England and Wales
2.0
to
-
6.1
1- r
5
'0
50
A
'00
Size of City. (10,000 i n h a b i t a n t s )
distribution.
10
( See Diagram l c )
A s i g n i f i c a n t conclusion of Berry's study however was t h a t :
Different c i t y s i z e d i s t r i b u t i o n s a r e i n no way r e l a t e d t o the
r e l a t i v e economic development of countries. Rank-size i s not
t h e culmination of a process i n which n a t i o n a l unity i s expressed
i n a system of c i t i e s .ll
I n order t o appreciate t h e significance of ~ e r r y ' sconclusions on the r e l a t i o n s h i p of c i t y s i z e d i s t r i b u t i o n t o l e v e l of economic
development l2 and the r e l a t i o n t o the general problem of an optimal c i t y
s i z e d i s t r i b u t i o n , one must examine the various t h e o r i e s attempting t o
explain t h e empirical r e g u l a r i t i e s manifested i n t h e rank-size (log-normal)
distribution.
11.
CITY SIZE DISTRIBUTION AND PARETO' S LAW
Among t h e f i r s t economists t o recognize c e r t a i n regulari t i e s i n t h e c i t y s i z e r e l a t i o n s h i p s was H. W. Singer, l3 who compared
t h e c i t y s i z e d i s t r i b u t i o n s f o r seven countries with t h e Pareto type of i n come d i s t r i b u t i o n , and found them both t o be of a s i m i l a r shape.
The Pareto curve i s of t h e form y = A
a
4
o r y = Ax
where x = income l e v e l .
y =.number of persons with t h a t
l e v e l of income o r over.
I n logarithmic form,
l o g y = l o g A - d l o g x, and t h e Pareto
curve takes on a l i n e a r form t h a t can be conveniently p l o t t e d on a double
log scale.
Now - d i s t h e e l a s t i c i t y of t h e function of d i s t r i b u t i o n of
incomes, t h e r e f o r e o( = -d log y
d log
and i s constant.
:
Hence
can be i n t e r -
p r e t e d a s t h e e l a s t i c i t y of decrease i n t h e number of persons whkn passing
t o a higher income.
By increasing t h e income
x by dx
=
1,000, from 10,000 t o
11,000, we g e t a r e l a t i v e decrease i n t h e number of persons by approximately
dy =
-
Y
Whereas, i f dx = 1,000 and x = 11,000
then
g =- 6(
. ,
y
. &
61,000
.
-
1,000 = -oC which i s l e s s i n terms of percentage
11
than t h e r e l a t i v e decrease obtained a t t h e t r a n s i t i o n from t h e income of
10,000 t o an income of 11,000.
Therefore, t h e r e l a t i v e decrease (screening)
in-,the number of persons a s t h e income increases i s smaller and smaller and
- 10 diminishes i n proportion t o the income, therefore
-4dx
dy = Y
X
Hence, the advance t o a higher c l a s s of incomes i s e a s i e r
f o r persons who have already reached a higher income than f o r persons with
lower incomes.
The reduction in the r e l a t i v e decrease of the number of
persons, during t r a n s i t i o n t o higher and higher incomes, i n proportion t o
t h e income, c o n s t i t u t e s the essence of P a r e t o ' s Law.
14
The Pareto formula has been given a probability i n t e r p r e t a t i o n by Champernowne, l5 We can regard the Pareto curve i n two ways,
e i t h e r a s representing the exact number of persons of an income not small e r than
x,
or i n terms of the number of persons with an income smaller
than ( o r not smaller than) i t s mean value (mathematical expectation).
Con-
sequently, according t o the Pareto formula, the mathematical expectation
of t h e r e l a t i v e screening decreases i n proportion t o the income.
The mat'n-
ematical expectation of a persods being t r a n s f e r r e d t o higher classes of
income w i l l , therefore, be proportional t o the given income,
s i d e r t h e c i t y s i z e d i s t r i b u t i o n i n a s i m i l a r manner.
We can con-
For example, t h e
mathematical expectation of a c i t y being t r a n s f e r r e d t o a higher c l a s s s i z e
of c i t y (through growth of the c i t y i n terms of population),
t i o n a l t o the given c i t y s i z e .
i s propor-
Thus one would expect, on average, the
l a r g e s t c i t i e s t o grow a t a f a s t e r absolute r a t e than the smaller c i t i e s
through t h i s screening process, and have a lower c l a s s mobility than c i t i e s
of smaller s i z e .
Chmpernowne has analyzed t h e development through t h e of
t h e d i s t r i b u t i o n of incomes between certain.income ranges a s being a s t o c h a s t i c process, so t h a t the income of any individual i n any one year may
- 11 $
depend on what it was i n the previous year and on a random process.
From
the r e g u l a r i t y t h a t Champernowne had established empirically, Pareto
t r i e d t o derive a general sociological "law" which he regarded a s a "natu r a l lawff t h a t held f o r a l l times and a l l s o c i e t i e s .
~t would appear
from t h i s "lawff, t h a t a11 s o c i a l reforms intended t o remove inequality
in
t h e d i s t r i b u t i o n of national income were doomed t o f a i l u r e from the
o u t s e t , since the law of nature about t h e d i s t r i b u t i o n of income acted i n
all conditions and the d i s t r i b u t i o n of income would always take t h e shape
indicated by the formula he established.
This conclusion of Pareto may
have s i g n i f i c a n t e f f e c t on the attempts t o r e d i s t r i b u t e c i t y s i z e d i s t r butions away from or towards a Pareto-type d i s t r i b u t i o n or t o vary the
16
value of& ( t h e slope of the r e l a t i o n under log-normal conditions).
Singer claimed t h a t :
... ....
i n the d i s t r i b u t i o n of population among urban agglomer( f o r seven countries)
t h e r e appears t o
ations
be a remarkable s t a t i s t i c a l r e g u l a r i t y , which besides being
i n t e r e s t i n g i n i t s e l f and affording a complete analogy t o
"
P a r e t o ' s Law of Income Distribution, y i e l d s an exact quantat a t i v e measure f o r the r e l a t i v e r o l e s of t h e smaller and
largertyyes
humanagglomerations,i.e. a n i n d e x o f m e t r o ; polization.
....
n'i
Neither Singer, Allen, nor Champernowne have drawn any
conclusioiis about the significance of the apparent r e l a t i o n s h i p between the
Pareto income d i s t r i b u t i o n and c i t y s i z e d i s t r i b u t i o n s , an6 no doubt f o r
very good reasons.
l8
However it could be suggested t h a t Pareto-type
d i s t r i b u t i o n s a r e "natural laws of d i s t r i b u t i o n " , conforming t o an optimum c i t y s i z e d i s t r i b u t i o n under open systems and where t h e r e a r e l a r g e
and complex competitive forces operating on t h e c i t y s i z e d i s t r i b u t i o n .
"Optimum" i s defined i n a very r e s t r i c t e d sense, t o r e f e r t o a d i s t r i b u t i o n
towards which any d i s t r i 3 u t i o n tends under c e r t a i n complex competitive
conditions ( i . e , t h e Pareto d i s t r i b u t i o n ) .
This assumes t h a t s o c i a l , ec-
onomic, p o l i t i c a l , and other forces a r e minimized when t h i s d i s t r i b u t i o n
i s a t t a i n e d and it i s i n essence an equilibrium d i s t r i b u t i o n where forces
a c t i n g t o maintain t h i s equilibrium d i s t r i b u t i o n dominate forces a c t i n g
'
t o d i s t o r t t h i s d i s t r i b u t i o n away from equilibri~un. A t e n t a t i v e hypot h e s i s might be that:
the more a c i t y s i z e d i s t r i b u t i o n conforms t o the
Pareto d i s t r i b u t i o n , the more it represents an optimum d i s t r i b u t i o n under
a l a r g e competitive economid system.
.
Thus a measure of non-optimality
could be the deviations any one c i t y s i z e d i s t r i b u t i o n has, a t every
level from t h e Pareto d i s t r i b u t i o n .
These deviations would be very l a r g e
i n the case of t h e primate d i s t r i b u t i o n , and the question therefore a r i s e s
of i d e n t i f y i n g those forces which have caused t h i s deviation from t h e
Pareto d i s t r i b u t i o n .
111. ZIPF AND TEE RANK-SIZE RULE
Zipf 19 has pro5ably the best presentation of the empiri c a l findings on rank and s i z e of c i t i e s .
The rank-size r u l e s t a t e s t h a t
f o r a group of c i t i e s , usually those exceeding some size i n a p a r t i c u l a r
country, the r e l a t i o n s h i p between s i z e and rank of c i t i e s i s of the form:
where
P,* = population of a c i t y of rank
3
q
=
=
T'
population of l a r g e s t o r first-ranking c i t y
ccmstailt
Zipf's rank-size r u l e i s a s p e c i a l case of the Pareto-type of d i s t r i bution with q = 1 (and where q conforms t
i n logarithms:
l o g r = l o g pt-q 1.06
+ , SO
o i n~ the Pareto d i s t r i b u t i o n ) .
t h a t a p l o t t i n g of rank against
s i z e should give a s t r a i g h t l i n e with a slope of -q.
Zipf explains t h e
f a c t t h a t the exponent q equals unity i n the rank-size r u l e , i n terms of
the e q ~ l i t yof the forces of d i v e r s i f i c a t i o n and unification i n t h e won20
Diversification tends t o minimize the d i f f i c u l t y of rwring raw
O v a
m a t e r i a l s t o t h e places where they a r e t o be pr&essed.
u n i f i c a t i o n tends
t o minimize t h e d i f f i c u l t y of moving processed materials t o t h e ultimate
consluning populace.
Thus i f a l l persons were located a t the same point then
m.Ximum u n i f i c a t i o n would be achieved.
'fiere both the forces of diver-
s i f i c a t i o n and u n i f i c a t i o n are a t work, a d i s t r i b u t i o n of population i s Presumed t o occur t h a t i s a t optimum with reference t o both f o r c e s .
Since
the force of d i v e r s i f i c a t i o n makes f o r a l a r g e r (n) number of smaller P
.
communities, whereas the f o r c e of unification makes f o r a smaller number
of P communities, then an "optimum" c i t y s i z e d i s t r i b u t i o n e x i s t s when the
d i s t r i b u t i o n follows the rank-size rule and the opposing forces of divers i f i c a t i o n and unification are eqmlized.
However :
...
it i s c e r t a i n l y not clear what are the l o g i c a l l i n k s betwee:~the scheme proposed by Zipf t o explain rank-size regul a r i t y and observed rank-size regularity. 21
I s a r d , w r i t i n g i n 1956, a l s o has considerable scepticism about the ranks i z e rule:
...
how much v a l i d i t y and universality should be a t t r i b u t e d
t o t h i s rank-size r u l e i s , a t t i s stage, a matter of i n d i vidual opinion and judgement. 28
A number of other authors have a l s o questioned the v a l i d i t y of the rank-
size rule.
Stewart argues t h a t :
...
The so-called rank-size r u l e
i s an empirical finding not
a l o g i c a l s t r u c t u r e . Nevertheless, it p a r t i a l v e r i f i c a t i o n
suggests an underlying l o g i c a l b a s i s . $3
Furthermore stewart decided t h a t large heterogeneous areas f i t the model
b e t t e r than small r e l a t i v e l y homogeneous areas, and t h a t t h e rank-size r u l e
(1
breaks down i n many areas a t both extremes-&he
l a r g e s t and smallest
towns" and t h a t well-structured areas of urloan dominance tend t o have an
S-shaped r a t h e r than a l i n e a r logarithmic d i s t r i b u t i o n of towns by s i z e . 24
It cannot be clenied, however, t h a t t o a limited extent t h e r e
i s some b a s i s f o r t h e formulation of hypotheses and a d d i t i o n a l exploration
on the nature of t h e r e l a t i o n s h i p between the rank-size r u l e and an optimlm
c i t y size 6istribution.
I n terms of the forces of unification and diver-
s i f i c a t i o n it could be argued t h a t i n those developing countries i n which
there is a predominance of small towns, there is also a predominance of
the forces of diversification over those of unification.
Iv. ClRI STALLEX AND
THE SIZE CLASSES O F CITIES
A w e l l known a l t e r n a t i v e i n t e r p r e t a t i o n t o t h a t of Zipf,
both regarding t h e s i z e of c i t i e s and t h e processes cauing s i z e r e g u l a r i t i e s , i s t h a t of Walter C h r i s t a l l e r . 25
C h r i s t a l l e r a r e s i m i l a r i n many respects.
,
The schemes of Zipf and
~ 0 t hu t i l i z e notions of t h e do-
main of c i t i e s (domain of goods) f o r t h e performance of various economic
a c t i v i t i e s and t h e r u l e s of behavior leading t o the s p a t i a l s y s t c l of
c e n t r a l p l a c e s and associated arrangements of c i t y s i z e s
(diversifica-
t i o n and u n i f i c a t i o n ) a r e q u i t e s i m i l a r .
However f o r comparative.purposes C h r i s t a l l e r only present e d a formal theory of c i t y s i z e s and t h e i r d i s t r i b u t i o n f o r h i s k=3 n e t work of c i t i e s i n homogeneous space. 26
hierarchy of centres be taken a s ranks
r
~ n t h e k=3 network, l e t . t h e
= 1, 2,
3
....
and t h e pop-
u l a t i o n of l a r g e s t c i t y (primate c i t y ) = K, second l a r g e s t r a n k a g c i t i e s
K/3, e t c .
=
A rank-size d i s t r i b u t i o n i n t h e manner of Zipf i s formed if
t h e exponent :
thus where
= 2
and i f q l o g P = l o g ( K/p )
K
then l o g P = l o g ( 1 2 )
.
and P = K/3 a s required by C h r i s t a l l e r ' s theory. 27
C h r i s t a l l e r , i n f a c t , t r e a t e d h i s formulation of t h e b a s i s
of a h i e r a r c h i c a l system of c i t i e s a s an a n a l y t i c a l problem of determining
a r a t i o n a l or "optimum" s p a t i a l o r i e n t a t i o n of c i t i e s .
In Christaller's
case we should be very cautious about i n t e r p r e t i n g h i s system of spacing
and s i z e d i s t r i b u t i o n of c i t i e s as i n any sense optimal, 28 although it
i s e s s e n t i a l l y a deductive theory from general assumptions, since it i s a
r e l a t i o n s h i p between rank and s i z e of the c i t y ' s t r i b u t a r y areas and. not
i t s population.
29
V. THE, CITY -SIZE
---- DISTRIBUTION AS A
STOCHASTIC PROCESS
The general shape of the observed c i t y - s i z e d i s t r i b u t i o n
has l e d many recent students t o consider t h e d i s t r i b u t i o n a s generated by
,
- stocahstic
growth processes.
An e a r l y attempt t o formulate such an ap-
prdach was by Chanpernowne, 30 with regards t o t h e Pareto income d i s t r bution, and by Stouffer i n r e l a t i o n t o migration p a t t e r n s , 3'
who derived
a Yule-type d i s t r i b u t i o r , f o r mobility and distance of migration. 32
Simon 33 has argued t h a t t h e d i s t r i b u t i o n of c i t y - s i z e s
wag one of a family of d i s t r i b u t i o n s which have t h e following general
c h a r a c t e r i s t i c s i n common:
(a)
They a r e J-shaped, o r a t l e a s t highly skewed, with very long
t a i l s which can be approximated c l o s e l y by a function of t h e
t h e form:
f ( i ) = (a/ik)bi, where a , b, and k a r e constants,
and t h e convergence f a c t o r b i s so c l o s e t o l t h a t it o f t e n
may be dis?..egarded. Thus, f o r example, t h e ) number of . c i t i e s
t h a t have a population i i s approximately a/ik.
(b) The exponent, k i s of t h e form 1<k C 2 .
(c)
The function describes t h e d i s t r i b u t i o n , not merely i n t h e
t a i l , b u t a l s o f o r small value of i.
These t h r e e p r o p e r t i e s j u s t i d e n t i f i e d , define t h e c l a s s of functions which
Simon terms t h e Yule D i s t r i b u t i o n .
S t a t e d i n t h e s e terms, t h e d i s t r i b u t i o n f o r c i t y s i z e s i s
evolved under roughly t h e following notions.
k
d i s t r i b n t e d i n c i t i e s , with a c i t y considered t o be an aggregate of
population l a r g e r than some t h r e s h c l d s i z e .
(k
Consider a t o t a l population
The p r o b a b i l i t y t h a t t h e
+ 1 ) s t person being found i n c i t i e s of s i z e i i s assumed t o be pro-
portional t o i f ( i , k ) .
It i s a l s o assumed t h e r e i s a constant p r o b a b i l i t y
3
t h a t t h e ( k = l ) s t person w i l l be i n c i - t i e s not previously of threshold
,size when the t o t a l p q u l a t i o n was
k.
Thus a model f o r expected- c l t y
s i z e d i s t r i b u t i o n s can be calculated a s follows from a s e t of equetions
derived from Simon' s o r i g i n a l model.: 34
Let ( i )
-3(
=
Wlk
1
(ii)
f(l)=n~/~-o(
where K i s the t o t a l urban population i n the n~ c i t i e s of greater
than threshold s i z e .
where n i s the number of c i t i e s equal t o o r greater than threshold s i z e
of population k.
and f ( i ) = number of c i t i e s of population i.
From equations ( i ) and ( i i ) and the successive' application
of equation ( i i i ) , the expected d i s t r i b u t i o n of c i t y s i z e s can be constructed.
Since the s i t u a t i o n w i l l scarcely, i f ever be found i n which d76
(where i s an extremely small nmber) , we may w r i t e t h i s system i n the
more simplified form:
then from ( i i ) * and by successive application of (ij.i)*, the expected
d i s t r i b u t i o n of c i t y s i z e s may be constructed.
The d i s t r i b u t i o n of f ( i ) , t h e number of c i t i e s of s i z e i,
may be r e a d i l y converted i n t o the rank-size d i s t r i b u t i o n , whefi'e b i s the
number of c i t i e s of s i z e equal t o o r g r e a t e r than s i z e i, by the use of
t h e following tr3.nsPrmation:
wherek i s the number of centres of population equal t o o r g r e a t e r than
s i z e i,
nK i s a s before.
and
f ( i j ) i s the t o t a l number of centres of population l e s s than i.
.
Another s t o c h a s t i c theory of c i t y s i z e d i s t r i b u t i o n i s
t h a t of Thomas. 35
Thomas notes t h a t J e f f e r s o n ' s formulation of the
q u a l i t a t i v e "law of the primate city" evidences recognition t h a t the unequal population s i z e s of c i t i e s presents problems "worthy of investigation"
. 36
Jefferson' s notion t h a t a country' s leading c i t y i s always d i s -
proportionately l a r g e and exceptionally expressive of n a t i o n a l capacity
and f e e l i n g implies t h a t functional changes i n the nature of the c i t y
accompanies changes i n population s i z e .
However because of the q u a l i t a -
t i v e nature of the "~aw", it i s not possible t o a s c e r t a i n what i s meant by
"disproportionately large".
This term, when attached t o a p a r t i c u l a r ob-
servation, i n d i c a t e s t h a t the magnitude of the observation exceeds some
expected value,
but no precise expected value i s provided by Jefferson.
I n Chris'caller's scheme of c i t i e s the l a r g e s t c i t y i n an a r e a was not
"disproportionately large", but merely "as l a r g e as it should be."
Thomas' purpose was therefore t o develop a model of c i t y
d i s t r i b u t i o n s which was consistent with the observed f a c t s and could poss i b l y provide an "expected" c i t y s i z e d i s t r i b u t i o n , and t h a t t h i s "ideal"
. c i t y s i z e d i s t r i b u t i o n was based on the log-normal d i s t r i b u t i o n o r v a r i a -
t i o n s of it.
This log-normal d i s t r i b u t i o n ( o r "steady state'' d i s t r i b u t i o n )
w i l l occur under c e r t a i n basic assumptions.
These a r e fourfold:
MODEL 1
(i) No c i t y has a l o c a t i o n a l advantage i n r e l a t i o n t o
physical and c u l t u r a l c h a r a c t e r i s t i c s of the area,
thus the population of c i t i e s d i f f e r s only by chance.
(ii) X l a r g e number of independent forces determine t h e
populztion s i z e o r changes i n the population s i z e of
cities.
The change i n population s i z e of a c i t y i n r e l a t i o n t o
i t s i n i t i a l s i z e i s very small during any one period.
(iii)
Growth of c i t y s i z e i s proportional t o c i t y s i z e . 37
(iv)
The development of a log-normal frequency d i s t r i b u t i o n of
c i t y s i z e o c c u ~ sa s f o l l o ~ ~ s :
let X
0
= population s i z e a t i n i t i a l time period ( t o )
X1 = population s i z e a t end of time period ( t l )
hence growth of c i t y from tl
b u t r e l a t i v e growth
G = X1
- Xo
-t
.O
'
=
- Xo
.. .
X1
Xo
1x0
thus
X1
and
X1 = GIXo s Xo = x 0 ( G 1
= GIXo
+ 1)
n.b. Magnitude of G1 i s independent of Xo ( ~ s s u m p t i o n( i v ) )
Therefore over
n
.
time i n t e r v a l s
However
.
/X
=
logex
'Xn=logeN - logeXo
Xo
Therefore s u b s t i t u t i n g i n (1.5) gives
M
G = l o g X - logeXo
e n
K4
2
o r l o g e s = logeXo
Thus i f G1,
+
G1
+
G2
+
... .Gn
,
+
Gn
8
are stochasticallyindependent,then-G
K=C
normal d i s t r i b u t i o n a s n->G2....Gn
Hence, logeXn
i s normally dlst r i b u t e d ,
and
i s log-normally d i s t r i b u t e d .
Xn
leads t o a
MODEL 2 The development of a log-normal frequency d i s t r i b u t i o n of
c i t y size.
(i)
Physical and c u l t u r a l v a r i a b l e s which a f f e c t c i t y s i z e
a r e unevenly d i s t r i b u t e d over t h e area. Thus c e r t a i n
c i t i e s w i l l have more favourable l o c a t i o n s than o t h e r s .
The e f f e c t of differences i n q u a l i t y of l o c a t i o n i s t o
d i f f e r e n t i a t e c i t y populattion s i z e s , t h a t i s , even a t
t h e o u t s e t , population s i z e cannot be t r e a t e d a s a
normally d i s t r i b u t e d "error" term.
( i i ) t o ( i v ) a s before.
Assuming t h a t t h e i n i t i a l s i z e d i s t r i b u t i o n w i l l be log-normal i n form;
what s o r t of d i s t r i b u t i o n w i l l a r i s e i f a new s e t of f o r c e s ( o r o l d ones
strengthened) a c t on t h e d i s t r i b u t i o n ?
l e t Xo = i n i t i a l population s i z e a t to,
X1 = population s i z e a t tl
Yo = transformation of X
0
i n t o n a t u r a l logarithms, t h e r e f o r e
Yo = logeXo
n.b. Yo i s normally d i s t r i b u t e d (See Model 1 )
Therefore absolute change i n Yo i n period = Y1
- Yo
r e l a t i v e change i n Yo i n period = B 1 = Y 1
Theref ore f o r tn we have
n z B = & ~k - Yk
&I
J=I
Yk
-
-
YO/yO
-1
AS before
n.b. Since logeY, i s normally d i s t r i b u t e d , and Y, = logeXn t h e l o g a r ithms of t h e l o g a r i t h m of Xn a r e normally distributed.
Therefore transforining t o anti-logarithms, we may say t h a t t h e frequency
d i s t r i b u t i o n of c i t y population s i z e assumes
very skew log-lognormal form.
The Law of Proportionate Effect i s a l s o a l t e r e d :
a
Since El = Y1
-
YO/yo
Tnen Y
1
= BIYo
i.e . log,xl
=
+ Yo
log,^,
(2.6)
-i
logexo
which gives X1 = XoBXo = X,(B
(2.7)
+ 1)
(2.8)
Thus w i t h i n a s t o c h a s t i c framework, we can s t a t e t h a t t h e
log-lognormal d i s t r i b u t i o n occurs when growth a t any s t e p of t h e process i s
a random power of t h e ' p r e v i o u s population s i z e .
Thomas t e s t e d the s i z e d i s t r i b u t i o n of eighty-nine c i t i e s
f o r Iowa in 1930 and f o m d t h a t t h e d i s t r i b u t i o n approximated. t h e loglognormal d i s t r i b u t i o n , r a t h e r than t h e log-normal. 38
Kalecki, has shown
however, t h a t over a long time t h e skewedness. of successive frequency d i s t r i b u t i o n s i n c r e a s e s . 39
porate t h i s tendency
over a
.
Thomas has developed a separate model t o i n c o r 40
Madden,
i n a study of t h e growth of U.S. c i t i e s
100 year period from 1790 t o 1950 has demonstrated the s t a b i l i t y of
t h e rank-size (hg-normal) d i s t r i b u t i o n over time, despite t h e f a c t t h a t
i n d i v i d u a l c i t i e s moved a3out over f a i r l y wide ranges within t h e rarkage.
The changes i n rank by s i z e of t h e d i f f e r e n t l a r g e c i t i e s of t h e n a t i o n
presumably r e f l e c t t h e changing r o l e s played by c i t i e s i n t h e population
system;
t h a t i s , they r e f l e c t t h e charging shares of t h e t o t a l urban ec-
onomic a c t i v i t y obtained by these c i t i e s a t various decades.
Similarly, it
was shown t h a t t h e percentage growth of c i t i e s ( i n terms of a mean growth
f o r d i f f e r e n t s i z e groups), i s unrelated t o t h e p o s i t i o n of t h e c i t y i n
t h e general d i s t r i b u t i o n , confirming a previous formulation
by Vining,
41
which i s very s i m i l a r t o t h a t derived by Thomas.
The empirical evidence from both developed and developing
c o r n t r i e s seems t o suggest t h a t t h e log-nmmal d i s t r i b u t i o n o r v a r i a t i o n s
of it f o r c i t y s i z e s i s f a i r l y common.
It has been demonstrated t h a t t h e
log-ncrmal d i s t r i b u t i o n and the log-lognormal d i s t r i b u t i o n ( classes of
Yule d i s t r i b u t i o n s ) =ise,
growth processes over time.
under c e r t a i n conditions, out of s t o c h a s t i c
These d i s t r i b u t i o n s are "steady s t a t e " o r
" s t a t i s t i c a l equilibrium", or "natural law" d i s t r i b u t i o n s
.
The log-
normal d i s t r i b u t i o n appears t o a r i s e out of s t o c h a s t i c growth processes
i n an open system and where the p r o b a b i l i t y of growth of an individual
c i t y i s simply proportional t o the c i t y s i z e .
The log-lognormal d i s t r i -
bution a r i s e s out of s t o c h a s t i c growth processes i n a closed system, and
growth i s a random power of s i z e of c i t y .
42
-25IV.
CITY SIZE DISTRIBUTIONS AND POLICY CONSIDEMTIONS
The questions now a r i s e , o f which d i s t r i b u t i o n b e s t describes
t h e c i t y - s i z e d i s t r i b u t i o n i n developing countries, and what a r e t h e
causes behind t h e v a r i a t i o n s between t h e c i t y s i z e d i s t r i b u t i o n ?
What, f o r
example would b e . t h e e f f e c t on national economic growth of attempts t o
change a c i t y s i z e d i s t r i b u t i o n from a log-normal t o a non log-normal
distribution?
This has considerable relevance f o r c e n t r a l i z a t i o n o r de-
c e n t r a l i z a t i o n policies-
i s a d e c e n t r a l i z a t i o n p o l i c y under c e r t a i n con-
d i t i o n s going t o l e a d t o considerable changes i n t h e whole of t h e dist r i b u t i o n such t h a t it : i s no longer i n a s t a t e of s t a t i s t i c a l equilibrium
o r i n a steady s t a t e ?
Assuming t h a t t h e log-normal d i s t r i b u t i o n i s an optimum
d i s t r i b u t i o n i n a s i t u a t i o n where t h e r e a r e a largenunber of random and
competitive f o r c e s operating i n t h e socio-economic and p o l i t i c a l systems,
i s t h e r e l i k e l y t o be a most e f f i c i e n t a l l o c a t i o n of c i t y s i z e s w h e n t h e
d i s t r i b u t i o n conforms t o t h e log-normal d i s t r i b u t i o n ?
If t h e answer i s i n
t h e a f f i r m a t i v e , can one a l s o assume t h a t econoraic growth (and growth i n
t h e average s i z e o f c i t i e s ) i s ~naximizedwith t h i s type of d i s t r i b u t i o n ?
P o l i c i e s f o r decentralizing economic a c t i v i t i e s have been
c r i t i c i z e d on t h e b a s i s t h a t economic growth i s maximized when it i s conc e n t r a t e d i n c e r t a i n faxourable l a r g e urban areas, i . e .
t h e growth pole
theory. 43 The argument i s t h a t by v e r t i c a l and h o r i z o n t a l linkage e f f e c t s
and a spi.llover of growth from one region o r c i t y t o another, t h e whole
system w i l l grow a t a f a s t e r r a t e than under any o t h e r conditions.
Critics
of d e c e n t r a l i z a t i o n p o l i c i e s have c o n s i s t e n t l y argued t h i s point f o r many
years.
Furthermore i n terms of t h e attempts t o impose maximum l i m i t s t o
the s i z e of t h e l a r g e s t c i t y , it i s i n t e r e s t i n g t o conjecture about t h e
possible e f f e c t s such a policy would have under d i f f e r e n t i n i t i a l cond i t i o n s , on t h e whole d i s t r i b u t i o n of c i t i e s .
Randomness i s postulated a s a convenient technique f o r i n v e s t i gating the o v e r a l l properties of the settlement f i e l d .
44 Such a form-
ulation i s n e u t r a l as t o r a t i o n a l i t y whether s o c i a l l y or econonically
oriented:
every decision may be optimal from a p a r t i c u l a r point of view
and y e t t h e r e s u l t i n g actions as a whole may appear a s random.
Lack of
information, s o c i a l t i e s and so on w i l l change an economizing optimizing
problem but not the randomness formulation.
Since it i s not possible t o i d e n t i f y and measure a l l the forces
t h a t c o n t r o l t h e c i t y s i z e d i s t r i b u t i o n , t h e observation t h a t c i t y s i z e
d i s t r i b u t i o n s tend towards the log-normal under competitive forces may
provide a clue to the formulation of policy decisions under c z r t a i n c i r cumstances.
A town planner, f o r example, t r i e s t o reconcile a whole s e r i e s
of opposing f o r c e s , t o achieve some form of s o c i a l optimum c i t y s i z e and
d i s t r i b u t i o n of c i t i e s , and often finds it impossible t o a r r i v e a t some
unique solution.
Yet empirical evidence shows t h a t under c e r t a i n con-
d i t i o n s , t h a t of a large competitive system, c i t y s i z e d i s t r i b u t i o n s tend
towards the log-normal o r log-lognormal.
I t could be argued therefore t h a t
any d i s t r i b u t i o n t h a t does not conform t o the log-normal d i s t r i b u t i o n i s
non-optimal a t t h a t point i n time.
Thus primate c i t y d i s t r i b u t i o n s , wi~ich
have a r i s e n out of peculiar h i s t o r i c a l , geographical, sociological, poli t i c a l , o r economic forces are non-optimal.
This hypothesis would suggest
t h a t a policy of e i t h e r heavily investing i n intermediate s i z e citie-s o r
of reducing the primacy of the l a r g e s t c i t y w i l l encourage a move toward
an optimum.
( See Diagram 2)
A policy of r e s t r i c t i n g the absolute growth
of t h e l a r g e s t c i t y i n a country, could have some undesirable a f f e c t s on
the o v e r a l l r a t e of growth of a l l other c i t i e s , e s p e c i a l l y if the e x i s t i n g
d i s t r i b u t i o n of c i t i e s i s the log-nornel.
I n t h i s case, t h e r e would be a
growth of intermediate s i z e c i t i e s and f u r t h e r deviation from the lognormal d i s t r i b u t i o n .
(see Diagram 3)
I n t e r e s t i n g analogies t o the problem of policy considerations
with respect t o c i t y s i z e d i s t r i b u t i o n s have a r i s e n i n the case of t h e
s i z e d i s t r i b u t i o n of firms.
Simon and Bonini, 45 have noted t h a t t h e
s i z e d i s t r i b u t i o n of firms (whether within a single industry or a whole
country), i s almost always highly skewed, and t h a t i t s upper t a i l resembles
the Pareto ( o r log-normal) d i s t r i b u t i o n .
Attempts a t economic explanation
of t h e observed f a c t s about concentration of industry have always assumed
t h a t t h e b a s i c causal mechanism was the slope of the long-run average cost
curve;
but t h e r e was l i t t l e discussion of why t h i s mechanism should produce,
even occasionally, the p a r t i c u l a r highly skewed d i s t r i b u t i o n s t h a t are
observed.
The s t a t i c c o s t curve analysis y i e l d s no explanation a s t o why
t h e observed d i s t r i b u t i o n s approximate the Pareto d i s t r i b u t i o n , it only
shows a c r i t i c a l minimum s i z e of f i r m i n an industry.
Simon and Bonini
argued t h a t p l a n t cost clnves a r e generally J-shaped, and below sone c r i t i c a l
s i z e u n i t c o s t s r i s e rapidly,above the c r i t i c a l s i z e
c o s t s vary only
s l i g h t l y with s i z e of f i r m :
We can Eay then, t h a t , t h e c h a r a c t e r i s t i c cost curve f o r the
No.of Cities
smallest cities
- Heavy investment in inter-
mediate size cities.
f
'.~estrictionin size of
largest city
DIAGRAM 2: POSSIBLE EFFECTS of a FZDISTRIBUTION POLICY.
No.of Cities
; Growth of intermediate cities with
[the restriction of size,oflargest
Restriction of the la-rgest
city.
deal" LOG-NOML (RANKSIZE) DISTRIBUTION.
Size of City (log scale)
FOSSIBLE EFFECTS of RESTRICTING the SIZE of the L4XGEST CITY.
f i r m shows v i r t u a l l y constant returns t o s c a l e f o r s i z e s above
some c r i t i c a l minimum. Under these circumstances the s t a t i c
a n a l y s i s may p r e d i c t the minimum s i z e of firm i n an i@ustry b a t
it w i l l not predict the s i z e d i s t r i b u t i o n of firms.
Simon and Bonini attempt t o explain the s i z e d i s t r i b u t i o n of
firms on the b a s i s of a s t o c h a s t i c model.
They postulate (from an
assumption) t h a t t h e d i s t r i b u t i o n of percentage changes i n s i z e of the
firms (over a year) in. a given c l a s s s i z e i s the same f o r a l l c l a s s s i z e s . 47
-
THE MODEL
Assume there i s a minimum s i z e , Sm, of f i r m i n an industry.
t h i s minimum s i z e , u n i t costs a r e constant.
Above
Individual firms i n the
industry w i l l grow ( o r shrink) a t varying r a t e s depending on such
f a c t o r s a s (a)
ment,
(d)
profits,
(b)
dividends policy,
(c)
new invest-
mergers.
These f a c t o r s i n turn, may depend upon the e f f i c i e n c y of the 'individual firm, exclusive access t o p . w t i c u l a r f a c t o r s of production, consumer brand preference, the growth and decline of a
p a r t i c u l a r industry, product i n .which it specializes, and
numerous other conditions. 4!
me operation of a l l these forces w i l l generate a probability d i s -
t r i b u t i o n f o r the changes i n s i z e of firms of a given s i z e .
Thus the f i r s t b a s i c assumption ( t h e Law of Proportionate ~ f f e c t )i s
t h a t t h i s p r o b a b i l i t y d i s t r i b u t i o n i s the same f o r a l l s i z e c l a s s of
firms t h a t a r e w e l l above 9
,
.
The second basic assumption d i s t r i b u t i o n i s t h a t new firms a r e being
"born" i n the smallest s i z e c l a s s a t a r e l a t i v e l y constant r a t e .
It has been shown t h a t under these assumptions the Yule Distribution
+
w i l l be t h e steady-state d i s t r i b u t i o n of t h e process.
Then t h e
Let f ( s ) d s be the p r o b a b i l i t y density of firms by s i z e s .
Yule D i s t r i b u t i o n i s given by f ( s )
Where ~ ( s p,
=
K B ( S ,+
~ 1).
+ 1) i s a Beta function of
K i s a normalizing constant,
p i s a parameter.
It i s easy t o show t h a t a s S--tW,
f (s)
Pareto D i s t r i b u t i o n .
s
Ms
-(P
and ( p
+
1)
+
1).
, which
Thus i n considering any observed d i s t r i b u t i o n of f i r m
i s the
s i z e s above
t h e c r i t i c a l minimum s i z e , ahy s u b s t a n t i a l deviation of t h e r e s u l t s from
those p r e d i c t e d from t h e s t o c h a s t i c model i s a r e f l e c t i o n of some depart u r e fromthe Law of &oportionate Effect o r from one of t h e assumptions o f
t h e model.
Having observed such a departure, we can then t r y . t o provide f o r
it some reasonable economic i n t e r p r e t a t i o n .
Such a deviation may be a
.measure of concentration, a s Aitchison and Brown argue. 5O
The p a r m e t e r
( p ) can a l s o be used t o account f o r t h e d i s t r i b u t i o n of f i r m s i z e s , hence
in t h i s p a r t i c u l a r model, t h e concentration i n an industry i s not i n dependently determined, but i s a function of t h e r a t e of new entry. (where
p = 1 t h e r a t e of e n t r y cf new firms i n t o t h e system = 0 . )
51
If f i r m s i z e s a r e determined by a s t o c h a s t i c process, then t h e
appropriate way t o t h i n k about public policy i n t h i s area i s t o consider
t h e means by which t h e s t o c h a s t i c process can be a l t e r e d , and t h e consequences of employing these means.
As a very simple example, i f t h e
r a t e of e n t r y i n t o t h e industry can be increased, t h i s w i l l automatically
reduce t h e l e v e l of c o n c e n t r ~ t i o n , a s measured by the usual indices.
Similarly, i f , through t a x p o l i c i e s o r o t h e r means, a s i t u a t i o n of sharply
increasing c o s t s i s created i n an industry, t h i s s i t u a t i o n should cause a
departure Trom the Yule Distribution, i n the d i r e c t i o n of lower concentration.
Furthermore the same equilibrium d i s t r i b u t i o n may be produced
with various degrees of mixing,
i.e.
reordering of t h e rank of firms
in an industry, according t o the d i c t a t e s of desirable public policy goals
( i . e . mobility of firms versus concentration of firms)
.
Simon and Bonini a l s o suggest t h a t when the environment i s
changizg a t a r a t e t h z t i s large compared w i t h the adaptive speeds of the
organisms :
.
--
we can never expect t o observe the system i n the neighbourhood of equilibrium, and we must invoke some s u b s t i t u t e f o r the
s t a t i c equilibrium if we wish t o p r e d i c t behaviour. 52
; ;
The problems of a r r i v i n g a t an optimum c i t y s i z e d i s t r i b u t i o n may be considered ?rum the same point .of view as t h a t adopted by Simon and Bonini.
Much of the e x i s t i n g l i t e r a t u r e on c i t i e s , only considers the optimum s i z e
of c i t i e s , and t h e r e i s l i t t l e reference t o the optimum s i z e d i s t r i b u t i o n
of c i t i e s .
I n considering, f o r example, the economic case f o r and a g a k s t
a p o l i c y of encouraging the growth of l a r g e or small centres, o r t o see
whether t h e r e i s any economic "virtue" i n a policy of decentralization, the
approach suggested by Simon and Bonini may be useful.
A decentralization plan
has been followed i n Australia, f o r example, since a t l e a s t World War 11.
On purely economic grounds it appears on f i r s t s i g h t t h a t it i s an unjusti f i a b l e policy, i n terms of e f f i c i e n t resource allocation, but on broader
socio-ecmomic grounds it ;may be the reverse.
According t o Berry's study 53
A u s t r a l i a has a c i t y s i z e d i s t r i b u t i o n t h a t i s intermediate between t h e log-
I
-
This i s confirmed by Neutze 54 who remarks on:
normal and the primate.
...
the absence of medium s i z e high concentration of population
i n the s t a t e c a p i t a l s and a dearth of medium sized centres.
The absence of medium sized regional centres may have been a major f a c t o r
causing new growth i n recent years t o crowd i n t o the s t a t e c a p i t a l s of
Australia.
Thus the appropriate policy might be t o promote some of these
centres and provide a wider choice f o r firms t h a t f i n d location i n small
centres unprofitable.
55
The argument f o r a decentralization policy i s u s u a l . 1 ~s t a t e d i n
terms of n e t s o c i a l c o s t s and b e n e f i t s of r e d i s t r i b u t i o n of industry,
people, employment, investment, e t c . , taking i n t o account e x t e r n a l i t i e s
a n a t h e r e l a t i o n of the policy t o o v e r a l l growth.
Among the external e f f e c t s
of p r i v a t e (and public) l o c a t i o n decisions a r e those of t r a f f i c ' congest i o n and public expenditure on road building, etc., i n r e l a t i o n t o c i t y
s i z e . 56
The general conclusion i s t h a t the e x t e r n a l costs of t r a f f i c
growth w i l l be greater i n l a r g e r a t h e r than i n small centres.
The same
analysis can be applied t o parking f a c i l i t i e s , f a r e s i n public transport,
length of journey t o work, and cost of public services, e t c . ,
e x t e r n a l diseconomies of growth.
i n terms of
Account must a l s o be taken of economies
of s c a l e i n public services, government administration, the differences i n
cost of p r i v a t e goods and services i n terms of v a r i a t i o n of p r i c e s with
c i t y size.
Intervention i n the market a l l o c a t i o n mechanism w i t h respect
t o c i t i e s i s only j u s t i f i e d i f the pecuniary and non-pecuniary e x t e r n a l
econonies and diseconomies a r e so large t h a t they cannot be ignored and i f
t h e government intervention produces a more s o c i a l l y optimum fiistribution,
- 33 t
which it may w e l l not do under conditions of imperfect f o r e s i g h t . 57
One possible e x t e r n a l e f f e c t of growth of c i t i e s stems from
s c a l e economies o r diseconomies i n the provision of public services,
If public services a r e more expensive i n large c i t i e s , growth brings ex-
t e r n a l diseconomies.
Conversely, if c o s t s f a l l with growth there a r e
e x t e r n a l economies.
If we can measure the e f f e c t of s i z e of centre on
these costs we can show the d i r e c t i o n and the magnitude of the e x t e r n a l
e f f e c t s and a l s o t h e way they a f f e c t the r e l a t i v e a t t r a c t i v e n e s s of l a r g e
and small centres.
If we p l o t cost against s i z e and get a U-shaped
average cost curve then the e x t e r n a l e f f e c t s w i l l be causing unduly
large c i t i e s .
They w i l l be doing so as long a s c o s t s r i s e more r a p i d l y
or f a l l more slowly a s ropulation grows.
Tne major problems, e s p e c i a l l y
in developing countries, a r e not only t h e l a c k of data, but a l s o d i f -
ferences i n q u a l i t y of serviees which may a f f e c t costs. 58
K.S.
oma ax, 59
in an e a r l y study of B r i t i s h towns, found t h a t costs were lowest, i n terms
of expenditure per head in centres of from 50,000 t o 100,000 people, however no account was taken of q u a l i t y differences i n services.
Other
studies ( e s p e c i a l l y i n ,$he U. S. ) have found very l i t t l e r e l a t i o n s h i p be
-
tween population of municipality and costs of local. government service,
with the exception of whter, sewage, and education.
I sard,
61
60
f o r example, has discussed the emergence of the s p a t i a l
p a t t e r n of c i t i e s of d i f f e r e n t s i z e s i n terms of economies and diseconomies
of s c a l e a r i s i n g from a c t i v i t i e s p o s i t i v e l y associated w i t h s i z e of c i t y
(agglomeration and deglomeration economies).
out of:
Urbanization economies a r i s e
...
a higher l e v e l of use of the general apparata of an urban
s t r u c t u r e (such a s transpcirtation f a , c i l i t i e s , gas and water maims,
and t h e l i k e ) and from a f i n e r a r t i c u l a t i o n of economic a c t i v i t i e s .
( d a i l y , seasonally, and i n t e r -indus tr i a l l y )
.
Urbanization diseconomies
a r e engendered by r i s e s i n the cost of l i v i n g and money wages, in the
costs of l o c a l materials produced under conditions of diminishing r e t u r n s ,
i n time-cost and other costs of transportation, and i n land value and
rents.
C i t i e s a l s o a t t r a c t o r r e p e l unLts of production i n accordance
with t h e urbanization ( f o r the most p a r t , external) economies o r d i s economies relevant t o each u n i t of production.
I n considering the optimum s p a t i a l d i s t r i b u t i o n and hierarchy
of d i f f e r e n t c i t i e s within a given technological and resource environment, such urban economies and diseconomies a r e important.
It i s
tempting t o define an o v e r a l l index of economy o r function of economy
by summing a s e r i e s of n e t economies f o r d i f f e r e n t c i t y s i z e s t o a r r i v e
a t an optimal s i z e of c i t y .
I s a r d r e j e c t s t h i s procedure on the grounds
t h a t t h e r e a r e t o o many " l o g i c a l objections".
62
Also:
...
standardization of c i t i e s i s subject t o serious c r i t i c i s m .
There a r e no standard c i t i e s , each i s unique.
There i s a l s o a problem of weighting the importance of individual n e t
economies curves.
Another objection, perhaps the most serious of a l l ,
stems from the neglect of inter-dependence among the s e t s of n e t economies
curve s :
The above considerations are s u f f i c i e n t i n themselves t o i n v a l i d a t e the use, even i n an approximate fashion, of a s i m p l e
t o t a l curve .or index of economies and diseconomies i n the
functioning of c i t i e s of various s i z e s .
63
It appears t h a t we are thrown back t o cansidering the approach advocated
by Simon and Bonini ( f o r f i r m s i z e d i s t r i b u t i o n s ) for the c i t y s i z e
distribution also.
Since we have so many competitive s o c i a l , economic,
p o l i t i c a l , and other forces operating on the c i t y s i z e d i s t r i b u t i o n and
it i s d i f f i c u l t t o i s o l a t e what may be t h e c r i t i c a l f a c t o r s determining
t h e c i t y s i z e s and t h e i r d i s t r i b u t i o n , it would be more sound t o analyze
t h e problem of optimum c i t y s i z e d i s t r i b u t i o n within the context of the
t o t a l n a t i o n a l environment.
-
I n order t o f a c i l i t a t e our understanding of
the integrated nature of the whole c i t y s i z e d i s t r i b u t i o n it; would be
--
convenient and illuminating t o consicler the d i s t r i b u t i o n as a subsystem
-
i n a l a r g e r system of the functioning of society,
I
.
-
CIW SIZE DISTRIBUTIONS AND GENERAL SYSTEMS THEORY
The s t a b i l i t y of the rank-size formulation over time
tween nations 65 has been explained by various authors &
steady s t a t e s t o c h a s t i c growth processes.
64
and be-
i n terms of
The tendency f o r l i v i n g systems
t o maintain steady s t a t e s of many v a r i a b l e s which keep a l l subsystems
i n order of balance both with one another and with t h e i r environment i s
t h e essence of general systems theo~.y, These steady-states a r e described
i n terms of entropy, i n accordance with the second Law of Thermodynamics,
i n which maximum entropy i s a s t a t e of randomly d i s t r i b u t e d energy and
e s s e n t i a l l y a normal o r average s t a t e of equilibrium.
That t h e rank-size
d i s t r i b u t i o n i s a random s t a t e i s borne out by Simon and as such it i s
a proper subject o f systems theory.
Thus c i t y s i z e d i s t r i b u t i o n s may
be treate.d i n terms of average conditions of maximum entropy and i n the
more general context of the development of systems theory. 67
A broad review of the unifying nature of general systems theory
i n r e l a t i o n t o what has been discussed so f a r i s i n order a t t h i s juncture.
A system i s a s e t of 'objects ( f o r example, c e n t r a l places), a t t r i b u t e s of
the objects (population, establishinents
, business
types, t r a f f i c generated),
i n t e r - r e l a t i o n s h i p s among the objects (mid-point locations f o r lower l e v e l
centres, uniform spacing a t any given l e v e l ) , and among the a t t r i b u t e s
( t h e c e n t r a l place hierarchy).
Systems may be closed, i . e . e n t i r e l y self-contained,or open, i n
the sense t h a t they exchange energy (materials, messages, and ideas) with
a surrounding environment.
available t o do work.
Closed systems have a given energy supply
As work i s performed the energy i s d i s s i p a t e d and
w i l l eventually become randomly distr-ibuted t h r o u & o ~ t the system.
Using
the terminology of the second Law of Thermodynamics, the systein w i l l then
have reached a condition of maximum e n t r ~ p y . I n terms of c e n t r a l place
theory, a c e n t r a l place system, if it were closed and had run down t o a
s t a t e of maximum entropy then populztion and other a t t r i b u t e s of centres
would be completely u n r e l a t e i t o l e v e l of centres i n the hierarchy.
In
f a c t any t r a c e of hierarchy would vanish.
With r e l a t i v e constancy i n energy inputs and approximate balance
.
of inputs and outputs, open systems s e t t l e i n t o an organized equilibrium
between the tendency t o move toward maximum entropy and t h e need f o r
organization t o pcrform work.
steady-state.
Such an organized equilibrium is c a l l e d a
A central-place system i s open.
The central place h i e r -
archy i s a form of organization t h a t performs the work involved as
e f f i c i e n t l y a s possible, and the rank s i z e r e g u l a r i t y i s a manifestation
of a steady-state equilibrium.
A steady-state balances
(1)
organization i n t o a hierarchy to'perforrn the work e f f i c i e n t l y ,
randomization due t o chance l o c a l differences.
the need f o r
and
(2)
Any decrease i n energy i n -
puts increases the entropy i n an open system, and causes adjustments
changing the form of the steady-state.
By the same token, increasing energy
inputs cause form adjustments leading t o 3 . t h e . r organization ( o r negative
entropy).
Open syst&ms a l s o contain feedback mech~nismst h a t a f f e c t growth
even under conditions of constant energy inputs.
positive feedback would
tend t o decrease the randomizing e f f e c t s of l o c a l v a r i a b i l i t y , ant! negative
feedback t o increase them, thus respectively increasing e i t h e r organization
o r entropy.
One conclusion of t h e general systems t h e o r i s t s is worthy of- ncLe:
The steady s t a t e i n an open system i s one t h a t obeys p r i n c i p l e s of equifinality.
Whatever t h e i n i t i a l s i z e of t h e c e n t r a l places, t h e sane c i t y
s t e a d y - s t a t e w i l l be achieved provided t h e energy flows a r e t h e same.
The
steady-state r e s u l t s s o l e l y from energy flows, independ-ent of t h e i n i t i a l
s i z e conditions.
Thus a rank-size r e l a t i o n s h i p w i l l r e s u l t s o l e l y from t h e
b a l a n c e , o f l o c a l v a r i a b i l i t y and t h e o r g a n i z a t i o n a l needs f o r a hierarchy
under a given s e t of demand and supply c m d i t i o n s .
This i s a l s o a char-
a c t e r i s t i c of t h e competitive model used t~ d e r i v e a central-place system.
Barry has attempted t o i n t e g r a t e many of t h e 'elements of systems of cent r a l places i n t o t h e general systems approach, t h u s incorporating c e n t r a l -
68 Berry
place theory.
s t a t e s t h a t c i t i e s and s e t s of c i t i e s a r e systems
s u s c e p t i b l e t o tile same kinds of g e n e r a l i z a t i o n s , constructs, and models,
a s i n general systems theory since c i t y systems ,incorporate t h e two
complementary ideas of entropy and information.
b t r o p y i s achieved i n
t h e s t e a d y - s t a t e of a sfuochastic process and i s , a s has been s t a t e d bef o r e , a t i t s maximum , i f
,
t h i s proce'ss i s uncofistrained
( t h e rank-size
r u l e ) . 69
Consider t h e case where t h e aim i s t o divide N people among
two s e t t l e m e n t s , each hdving an equal c h n c e of a t t r a c t i n g a given population.
Let t h e number of settlements 3aving a population of
i
persons
N
be
Zi,
and
5 zi
= 1.
The numbex of. r a y s i n which people can be d i s -
is0
t r i b u t e d mong t h e set,tlemeni;s, ncgiec-king t h e s p a t i a l aspect and cons i d e r i n g only t h e frequericy d i s t r i b u t i o n i s g = '!/N
TT Zit.
(05
i5N)
When t h e system i s l a r g e i t s entropy i s :
Where H
=
entropy.
-i/n
H i s maximized when Zi = (Z/N)
N
Where n = /Z,
t h e mean population per settlement.
This exponential d i s t r i b u t i o n can be w r i t t e n a s a cumulative d i s t r i b u t i o n
function:
Where T = s i z e of l a r g e s t c i t y .
Under t h e s e circumstances, entropy i s maximized when
Hmax =
log(^^),
i.e.
t h e most probable s t a t e of t h e system i s t h a t which gives mcaximum entropy,
o r when the sum of logarithms i s a maximum.
This corresponds -Lo a s i t -
uation i n which, given t h e s i z e of t h e l a r g e s t c i t y , t h e p r o b a b i l i t y t h a t
the (u
+ 1 ) s t c i t y has a population P(U + 1 ) i s equal t o q.
Where P = P("
with ~ ( u )= t h e p o p h a t i o n of t h e l a r g e s t c i t y .
+
p( u)
The r a t i o
I n t h i s sense
q
=
i s a constant.
log(^^)
becomes s i m i l a r t o t h e r a n k - s i z e r u l e :
It a l s o follows t h a t a system of c i t i e s obeying t h e rank-size r u l e i s i n
a s t a t e of equilibrium i n which entropy has been maximized.
t h i s reason t h a t Berry and Garrison 70
and Curry 71
It i s f o r
argue t h a t systems
which deviate from the rank-size r u l e a r e more worthy of a t t e n t i o n than
a r e s y s t e m ~ M c hfollow.
It i s i n t h i s sense t h a t t h e prima.te c i t y d i s -
t r i b u t i o n h&s such i n t e r e s t i n r e l a t i o n t o developing countries.
Are
they a l s o a d i s t r i b u t i o n i n disequilibrium i n terms of t h e entropy concept?
Maruyama s t a t e s t h a t e q u i l i b r i a t i n g theory ( c y b e r n e t i c s ) may have
a c o n t r a d i c t i o n i n t h e form of a d i s e q u i l i b r i a t i n g system.
Many i n s t a n c e s
can be c i t e d i n which feedback does not l e a d t o s e l f - c o r r e c t i o n s towards
some pre-set e q u i l i b r i u n (morphostasis), 729 73
state distribution.
i n t h e form of t h e steady-
Rather, progressively g r e a t e r c o n t r a s t s appear a s
f o r example, between Myrdal's
Rich Lands and Poor Lands, T4
o r with pro-
g r e s s i v e l y g r e a t e r c e n t r a l i z a t i o n of functions i n fewer l a r g e c i t i e s , o r
when "the growth of a c i t y increases t h e i n t e r n a l structuredness of t h e
city itself.
11
75
Maruyama's t h e s i s i s r e l e v a n t t o t h e problem of t h e f a n t a s t i c
growth of primate c i t i e s i n many underdeveloped countries.
Could t h e s e
primate c i t i e s be examples of a deviation amplifying mutual c a u s a l process,
which w i l l take c e r t a i n underdeveloped c o u n t r i e s , further away from t h e
h y p o t h e t i c a l optimum equilibrium c i t y s i z e d i s t r i b u t i o n .
That this i s a
s e r i o u s p o s s i b i l i t y should not be discounted.
I n an economically underdeveloped s b c i e t y , on t h e o t h e r hand
under the laissez-faire policy and free play of market f o r c e s ,
t h e few p r i v i l e g e d people accumulate more power and we
while t h e l i v i n g standards of t h e poor tends t o f a l l .
?kth
Berry argues t h a t t h i s deviation (and t h e r e f o r e s t r u c t u r e )
amplifying t r e n d i n a system i s a tendency towards maxiram information and
reordering, and away from maxiinum entropy.
Thus t h e two f o r c e s a r e essen-
t i a l l y i n opposition. 77
A gl.ance a t t h e d a t e f o r t h e U.S.
shows it t o be more o r d e ~ ~ 2
than I n d i a , a n d l e s s than A u s t r a l i a , f o r example. 78
The evidence presented by Berry on c i t y s i z e d i s t r i b u t i o n s and economic
development, suggests t h a t although c i t y s i z e d i s t r i b u t i o n s a r e essent i a l l y uncorrelated t o l e v e l s of development, they a r e c o r r e l a t e d with
o t h e r f a c t o r s of a country's development, such a s s i z e of t h e country,
age of urbanization, e t c .
The log-normal d i s t r i b u t i o n i s hypothesized
a s r e p r e s e n t i n g a s t e a d y - s t a t e condition of maximum entropy, whereas a
primate d i s t r i b u t i o n may i n d i c a t e a simpler, p a t t e r n e d s t r u c t u r e .
Pro-
gression from the primate t o the log-normal s t a g e i s reached when t h e
urban s o c i e t y i s o l d and complex and has been influenced by ,large numbers
of f o r c e s i n many ways such t h a t t h e p a t t e r n i n g e f f e c t s of any of t h e s e
forces are l o s t .
Evidence t o support t h i s h y p t h e s i s was found i n t h e
f a c t t h a t t h e advanced c o u n t r i e s with primate dLstributions were very
small, which l i m i t e d possible complexitie,t e n t e r i n g int,o t h e urban scene,
and t h e l e s s e r developed c o u n t r i e s with 1-og-cormal d i s t r i b u t i o n s were
g e n e r a l l y very l a r g e and with long h i s t o r i e s of urbanization which i n creased t h e p o s ' s i b i l i t y of t h e urban p a t t e r n being a f f e c t e d by many f o r c e s .
However it i s d i f f i c u l t t o be c e r t a i n t h a t those countries which
do d i s p l a y a log-normal d i s t r i b u t i o n have i n f a c t an equilibrium c i t y s i z e
d i s t r i b u t i o n i n t h e sense t h a t t h e r e i s no pressure t o move away from t h i s
equilibrium d i s t r i b u % i o n . I n order t o exaliine whether t h e r e i s a sodioeconomic end p o l i t i c a l optimum i n a country with a log-normal d i s t r i b u t i o n ,
t h e c i t y s i z e d i s t r i b n t . i o n of I n d i a w i l l be examined.
VIII.
INDIAN CITY SIZE DISTRIBUTION
The r a t e of iwban population growth i n India i n the l a s t Tew
decades has been phenomenal.
From 1941 t o 1951 the growth was from 13.5%
t o 17.3% of t h e t o t a l population, representing an urban growth of 34.8%.
The.growth of urban places i n and by i t s e l f would be of l e s s concern i f
it d i d not produce a t t h e sane time a number of s o c i a l
a r e new and demand constructive solutiibns.
which
Unemployment i n Indian c i t i e s
i s high, e s p e c i a l l y among educated persons, and t h i s c r e a t e s s o c i a l and
p o l i t i c a l problems,
The growth of c i t i e s i s due i n large p a r t t o r u r a l -
urban migration, the conditions and causes of which w e s t i l l l i t t l e understood.
Moreover, housing, water supply, and s z n i t a r y services a r e
s o r e l y lacking i n . Indian c i t i e s , and the r a p i d growLh of population
c r e a t e s increasing pressures t o supply even minimum f a c i l i t i e s of t h i s
kind.
Viis makes necessary some action i n the d i r e c t i o n of urban planning
i n order t o b e t t e r balance short run and long run needs.
Much of the
present growth of c i t i e s takes place by the building of shanty towns,
which a r e being b u i l t on any piece of land t h a t happens t o be available,
I n d i m c i t i e s , and e s p e c i a l l y the smaller towns, s u f f e r from a plethora
of slums.
Thus it appears t h a t even though it has a log-normal d i s t r i -
bution, Indian c i t i e s are not without t h e i r problems.
I n a s much a s these urban centres ( e s p e c i a l l y those over 100,000)
form a system, and i n t h a t capacity, a f f e c t t h e urban and economic s t r u c t u r e
of India, it i s of i n t e r e s t t o analyze the properties of the urban centres
comprising t h e system.
Along what dimensions of v a r i a t i o n can Indian c i t i e s
be arranged? &t a r e the major s i m i l a r i t i e s and differences t h a t chara c t e r i z e these r e l a t i v e l y large, urban agglomerations?
Harris and Ullman, 79
summarized t h e c l a s s i c a l p r i n c i p l e s of
urbanization by recognizing three d i f f e r e n t types of c i t i e s :
1. C i t i e s a s c e n t r a l places p e r f o r m i ~ gcoinpreheneive services
f o r a surrounding area.
2.
C i t i e s a s transport f o c i and break of bulk points.
3.
Specialized function c i t i e s performing one servicc such
a s mining, maufacturing, or recre?.<ion f o r large &reas.
More r e c e n t l y Redfield and Singer 80. introduced another c l a s -
s i f i c a t i o n of c i t i e s .
Discussing the c u l t u r a l r o l e of c i t i e s , they rec-
ognized two types of c i t i e s :
1. C i t i e s of orthogenetic transformation. These are of the
r u r a l order
of culture c a r r i e d forward.
....
2.
C i t i e s of heterogenetic transformation. These are c i t i e s
of t h e t e c h n i c a l order, where l o c a l cultures are d i s i n t e g r a t e d and new integrations of society a r e developed
....
I n addition, these authors recognized two p a t t e r n s of urbanization, primary
and secondary.
I n the primary phase a p r e c i v i l i z e d folk society i s t r a n s -
formed by w b m i z a t i o n i n t o a peasant s o c i e t y w i t h correlated urban
centres.
This process takes place almcst e n t i r e l y within the framework
of a core c d t u r e t h a t develops i n an indigen&s c i v i l i z a t i o n .
Secondary
urbanizaticjn follows primary wbanization when a f o l k society, p r e c i v i l i z e d ,
peasant, or p a r t l y urbsnized, i s f u r t h e r urbanized by contact with peoples
of widely different cultures from t h a t of i t s own members.
- 44 ,
Hoselitz, 81
r e c o ~ i z e sy e t another s e t of c i t i e s on the b a s i s
. o f t h e i r r o l e i n the economic development of an area.
According t o him
a c i t y i s generative i f i t s continued existence and gro&h i s one of the
f a c t o r s accountable f o r the economic development of the area i n which it
It i s p a r a s i t i c i f it e x e r t s ail opposite impact.
i s located.
.
I
Further i n
an attempt t o t i e together h i s ideas with those of Redfield and Singer, he
observes t h a t although orthogenetic c i t i e s tend t o l i m i t i f not impede
c u l t u r a l change, t h i s does not mean t h a t orthogenetic c i t i e s axe neces s a r i l y p a r a s i t i c with regard t o economic growth.
Further, the process of primary urbanization, though leading t o
a reinforcement of e x i s t i n g c u l t u r a l p a t t e r n s , may be generative of economic growth.
I n t h e same way c i t i e s i n c e r t a i n stages of secondary urban-
i z a t i o n may e x e r t an unfavora,ble e f f e c t upon economic growth of the
wider geographical u n i t of,which they form a part.
of colonial c i t i e s .
An example i s t h a t
I n a similar vein, Berry points out t h a t the process
of secondary urbanization e x e r t s i t s e l f when an integrated system of c i t i e s
develops,
usually under t h e influence of forces external t o the l o c a l cul-
ture.
...
i n complex nodal systems of
Heterogenetic c i t i e s r e s u l t
economic organization characterized by r a p i d s o c i a l change. 82
How relevant are these notions t o an understanding of Indian c i t i e s ?
Do Indian c i t i e s f a l l i n t o one of the four possible classes a s mentioned by
Hoselitz in h i s discussion of generative and p a r a s i t i c c i t i e s ? 83 I n t h e
opinion of many students of Indian urbanization, Indian c i t i e s , l i k e t h e i r
counterparts i n t h e Western world, a r e centres of heterogenetic transformations and a r e generative of economic growth.
Also, the-prevelent pro-
cesses of urban growth i n India display a p a t t e r n which i s considered as
secondary urbanization.
This of course i s not intended t o imply t h a t there a r e no d i f ferences between the p a t t e r n of Indian urbanization, what ever it may be,
and the urban p a t t e r n s t h a t e x i s t i n the Western world.
During the past
decade a number of studies have appeared which deal s p e c i f i c a l l y with
urbanization i n the non-Western world.
,
84 The authors of these works
suggest t h a t urbanization i n Asia may involve q u i t e d i f f e r e n t patterns
of development and inter-relationships with economic development than
those observed i n
the West.
Some of the major differences indicated
by them are:
1. Urban deirelopnent i n many countries of Asia i s l s r g e l y an'
outgrc~wthof colonialism.
2.
There i s an increasing r o l e of c e n t r a l planning and governmental interventionism i n Asian economic development.
3.
There are g r e a t differences i n b a s i c outlook and value systems
between Asia and the West. 85
Although Indian urbanization shares most of the c h a r a c t e r i s t i c s
common t o other developing nations, it nevertheless has some d i s t i n c t i v e
features.
To begin with, India has a long urban t r a d i t i o n which goes
back more than a thousand years.
Unlike many countries of the non-Western
world, I n d i a has a well-developed urban hierarchy so f a r as c i t y s i z e d i s -
t r i b u t i o n i s concerned.
,There i s ample evidence t o the e f f e c t t h a t r u r a l -
urban migration i s the most important f a c t o r contributing t o urbanization
. i n I n d i a and t h i s migration i s d i r e c t e d not only towards t h e very l a r g e
c i t i e s but a l s o t o hundreds of medium-sized and smaller c i t i e s i n almost
a l l regions.
Furthermore it has been arbued t h a t urbanization i s n e i t h e r
a necessary nor a s u f f i c i e n t condition f o r economic growth.
For these
reasons and many others a policy of decentralization has been advocated
i n India.
The c m f l i c t between c e n t r a l i z a t i o n and a l t e r n a t i v e forms of
decentralization i s a t the moment a very r e a l issue i n India.
86 One
aspect of t h i s i s the posing of t h e problem a s a choice between two
d i f f e r e n t p s t t e r n s of economic development, one v i l l a g e based and t h e
other urban based.
The balance between i n d u s t r i a l ' growth and urban devel-
opment, the postulates of e q u a l i t y between r u r a l and urban standards of
d
costs and b e n e f i t s of regional d i s p e r s a l of industries,
l i v i n g , a ~ the
a r e the elements t h a t make decentralization a r e a l but s t i l l undecided
issue.
Due t o r a p i d population
which has l e d t'o pressure on t h e
urban centres and stagnation of the r u r a l economy, r a p i d urbanization has
tended t o precede r a t h e r than follow i n d u s t r i a l i z a t i o n , leading t o a wide
"development gap".
"over urbanization"
It appears i n two d i f f e r e n t forms, i n the form of
i.e
. urbanization
exceeding the range of economic
development, and i n t h a t of a marked deficiency of urban f a c i l i t i e s and
services.
87 I t i s f o r these reasons t h a t Indian planning p o l i c i e s h m e
tended t o concentrate on decentralization and f o s t e r the growth of medium
s i z e d towns.
88 J . P . Lewis, i n a study of the o v e r a l l economic problems
of Indian d e v e l h e n t argues t h a t there i s not:
...
a s usual discussions sometimes seem t o suggest, any lack
of medium s i z e d centres lying between v i l l a g e s , on the one
hand, and t h e l a r g e r c i t i e s on the other. Instead, throughout
t h e c i t y s i z e d i s t r i b u t i o n there presently are bases upon which
f u r t h e r concentrations of a c t i v i t i e s of population could be
b u i l t . 89
I
India, it appears, although it does have a log-normal d i s -
t r i b u t i o n , s u f f e r s from many serious s o c i a l and economic urban problems.
Planning p o l i c i e s have been.directed.towards reducing the s i z e of the
l a r g e s t c i t i e s , and f o s t e r i n g the growth of the medium sized towns.
If
the hyyothesis i s correct, t h a t the log-normal d i s t r i b u t i o n i s an optimm
c i t y s i z e d i s t r i b u t i o n , then t h i s policy would oniy l e a d t o a non-optimal
d i s t r i b u t i o n of c i t y s i z e .
Lewis' observation on the c i t y s i z e d i s t r i b u t i o n
in I n d i a would tend t o support t h i s hypothesis,
It i s impossible t o a r r i v e
a t any d e f i n i t e conclusions on the v a l i d i t y of a decentralization policy
i n r e l a t i o n t o the log-normal d i s t r i b u t i o n without f u r t h e r analysis, so
a t t h e present time the question of whether o r not the e x i s t i n g d i s t r i bution i s an opt5murn.i~s t i l l open.
It could be argued t h a t the present
d i s t r i b u t i o n minimizes the socio-economic forces acting on it t o change
despite the f a c t t h a t these forces appear t o be very large.
Any other
d i s t r i b u t i o n may conceivably, increase the serious s o c i a l and economic problems i n t h e urban centres.
This conclusion i s a l s o unverifiable a t the
present moment, but it i s i n t e r e s t i n g t o keep i n mind.
--
IX.
CONCLUSION
There i s considerable t h e o r e t i c a l support i n the l i t e r a t u r e f o r
t h e hypothesis t h a t t h e log-normal ( o r rank-size) d i s t r i b u t i o n i s an
"optimum", steady-state or s t a t i s t i c a l equilibrium d i s t r i b u t i o n , under cert a i n conditions ( c f .
the Pareto d i s t r i b u t i o n ) .
The existence of the log-
normal d i s t r i b u t i o n f o r c i t y s i z e s i n both underdeveloped and developed
countries, appears t o be the r e s u l t of the i n t e r a c t i o n of numerous comp e t i t i v e f o r c e s over a long period of time.
I n India, f o r exmple, the
log-normal d i s t r i b u t i o n has evolved during a long h i s t o r y of urbanization.
The' primate d i s t r i b u t i o n , which shows the g r e a t e s t deviation from the lognormal, r e s u l t s from the dominance of a small number of very large forces
c i t i e s s u f f e r f r o x problems E imilar
t o produce t h i s d i s t r i b u t i o n . Pr-%n&e.
t o the l a r g e s t c i t i e s i n the log-narmal d i s t r i b u t i o n , but t o a much
g r e a t e r extent.
The r e j e c t i o n of an approach based or-,net urban economies t o
analyze t h e optimum c i t y s i z e , i s reasonable when taken i n the context of
a systems approach t o c i t y s i z e d i s t r i b u t i o n s .
General systems theory,
incorparating s t o c h a s t i c growth theory, appears t o be a meaningffil way of'
examining the problem of an optimum c i t y s i z e d i s t r i b u t i o n , since it
focuses a t t e n t i o n away from single optimizing c r i t e r i a , and more tmiard the
whole of the d i s t r i b u t i o n .
It i s evident t h a t t h e r e a r e many problem i n
t e s t i n g the hypothesis t h a t the log-no:(-ma1 d i s t r i b u t i o n i s an optimum.
In
t h e case of India, with a log-normal c i t y s i z e d i s t r i b u t i o n , many s o c i a l
and economic problems face i t s c i t i e s , which appear t o be almost insoluble.
Considerable work must be undertaken before it can be f i r m l y
e s t a b l i s h e d t h a t any one type of d i s t r i b u t i o n i s an optimim under a w2de
v a r i e t y of conditions, although it appears t h a t a t h e o r e t i c a l f'ramework
has evolved which may be a useful a n a l y t i c a l approach t o t h e general problem
of optimum c i t y s i z e d i s t r i b u t i o n .
REFERENCES
E. Howard, Garden C i t i e s of Tomorrow o on don: Faber and Faber
Ltd., 1946).
F.J. Osborn, Transport, Town Development and T e r r i t o r i a l Planning
of Industry No. 20 (London: The New Fabian Research Bureau, Vict o r Gollancz, Ltd., 1934), p . 20.
Lewis Mumford, The Culture of C i t i e s ( ~ e wYork: Harcourt Brace,
1938), p . 488.
' ~ r a c B.
~ Augur, "The Dispersal of C i t i e s a s a Defense Measure",
l e t i n of t h e Atomic S c i e n t i s t s , Vol. IV, May, 1948, pp. 131
% a l t e r Tsard, Location and Space Econoq ( ~ e wYork:
Sons, Inc., 1956), esp. pp. 182 - 188.
Bul-
- 134.
John Wiley and
,
4 ~ o r r e s tP i t t s , Urban Systems and Economic Development ( ~ u ~ e n eOregon:
University of Oregon, School of Business Administration, 1962).
5 ~J..L. Berry,
'see
"City Size Distributions and Economic Development",
Economic Development and Cultural Change, Vol. 9, 1961, pp. 537 588. Reprinted i n J. Friedmann and W i l l i a m Alonso, Regional
Development and Planning (cambridge, Mass. : The M. I. T. Press,
1964). Berry's work was based on data p r e s e n t e d i n N . Ginsburg,
A t l a s of Economic Development (Chicago: 1961).
George K . Zipf, Human Behavior and t h e Principle of Least E f f o r t
(cambridge: Addison-Wesley Press, 1949).
Jefferson, "The I a w of t h e Primate c i t y " ,
Review, XXXM, 1939.
The Geographical
8 ~ t a l y , Finland, Belgium, U. S .A. , Poland, West Germany, Switzerland, I n d i a ,
Korea, China, B r a z i l , E l Salvador, Union of South Africa.
9 ~ r e e c e ,Austria , Spain, Denmark, Netherlands, Sweden, Portugal, Uruguay,
Peru, Guatemala, Mexico, Dominican Republic, Thailand, Jtipan, Ceylon.
1•‹parkis t a n , Malaya, Nicaragua, Ecuador, Yugoslavia, ~ o r w & , Englsnd and
Wales, A u s t r a l i a , Canada, New Zealand.
l l g e r r y , op. c i t . , p. 585.
12'I'he other major findings of Berry ' s study were:
I
On t h e log-normal d i s t r i b u t i o n .
m ~
i s tnoticeable t h a t countries with long urban tradTtions such a s
India and China, and highly developed countries such a s the United
S t a t e s and Germany have very s i m i l a r distributions."
'On t h e primate & s t r i b u t i o n .
h ~ fli l
f t e e n of the countries with primate c i t y d i s t r i b u t i o n were
small and they range from underdeveloped Thailand through countries
with dual and peasant economies t o Denmark and t h e Netherlands with
highly specialized a g r i c u l t u r a l economies."
Primacy was considered t h e simplest c i t y s i z e d i s t r i b u t i o n and
affected by only a few simple but dominant forces. The rank s i z e
d i s t r i b u t i o n however a r i s e s out of a compl-exity of competing economic, s o c i a l , and p o l i t i c a l forces over a long period of time.
11
There i s no r e l a t i o n s h i p between type of c i t y s i z e d i s t r i b u t i o n
and t h e degree t o which a country i s urbanized. Countries which
have u n t i l recently been p o l i t i c a l l y and/or economicalii dependent
upon some outside country tend t o have primate c i t i e s >~hicha r e
t h e n a t i o n a l c a p i t a l s , c u l t u r a l and economic centres, often the
chief p o r t , and t h e focus of n a t i o n a l consciousness and feelLng."
The model i n e f f e c t proposes a major hypothesis, t h a t i s , i r ~ c r e a s e d
entropy i s accompanied by a c l o s e r approximation of a c i t y s i z e
d i s t r i b u t i o n t o log-normality. This leads t o f u r t h e r sub-hypotheses
namely t h a t (1) t h e smaller t h e country i s and ( 2 ) t h e s h o r t e r
i s t h e h i s t o r y of urban l i f e i n t h a t country and ( 3 ) t h e lover
and hence simpler i s the l e v e l of economic and p o l i t i c a l development i n the country, t h e fewer w i l l be t h e forces a c t i n g on t h a t
country's c i t i e s .
Hence a s t h e l e v e l of economic development increases, t h e l e v e l
of complexity of c i t y d i s t r i b u t i o n increases. This l e d B e r ~ yt o
consider t h e r e l a t i o n s h i p between r e l a t i v e l e v e l s of economic
development and c i t y s i z e d i s t r i b u t i o n s .
Forty-five indices of economic development, based on transportation,
energy, a g r i c u l t u r a l y i e l d s , communications, G.R.P., trade,demographic and other data, were reduced t o four basic ~1.9tternsof
economic development, based on ( i ) technological, ( i i ) demographic, ( i i i ) trading, and ( i v ) s i z e of country c h a r a c t e r i s t i c s .
(i)and ( i i ) were f u r t h e r reduced t o a s c a l e of economic-demographic
development, which was r e l a t e d t o %he types of c i t y s i z e d i s t r i bution. Furthermore it was found t h a t "clifferent c i t y s i z e d i s t r i butions a r e i n no W ~ J T r e l a t e d t o t h e r e l a t i v e economic development
of countries". Hence primacy i s not r e l a t e d t o l e v e l of urbaniza t i o n of a country, since l e v e l of urbanization and l e v e l of economic development a r e highly correlated. There i s no ready explana t i o n i n t h e l e v e l of economic development, f o r v a r i a t i o n s i n
c i t y s i z e d i s t r i b u t i o n s on a scale from simple s t r u c t u r e (primary)
op. c i t . , p. 572 e t seq.
t o entropy (log-normality).
,
'%.w.
Singer, "The 'Courbe des Populations1: a P a r a l l e l t o Pareto's
Law1', Economic Journal, Vol. 46, 1936, pp. 254 - 263.
14'lhis screening process i s a l s o known a s t h e "Law of P r ~ p o r t i o n a t e~ f f e c t " .
15see D.G. Champernowne, "A Model of Income ~ i s t r i b u t i o n ,
" Econoluic
Journal, Vol. 63, pp. 318 e t seq.
...
16Champernowne mentions t h e "
impossibility of drawing any simp l e conclusions about the e f f e c t on P a r e t o t s 6 ( o f various r e d i s ", I b i d . , p. 319.
tributive policies
....
.
17singer, op. c i t , p. 260.
See a l s o G.R. Allen, "The 'Courbe des Populations'. -9 Further
Analysis", B u l l e t i n of t h e Oxford I n s t i t u t e of S t a t i s t i c s , Vol. 18,
1954, PP 179 - 189.
1 8 ~ l l e nf i t t e d t h e Pareto formula t o 44 countries and finds:
It
The main conclusions a r e therefore t h a t t h e Pareto law can be
u s e d w i t h much success t o summarize t h e r e l a t i o n s h i p between s i z e
of towns and number of towns above a s p e c i f i e d s i z e ( 2 , 0 0 0 ) " ~
Ibid - 9 P * 184.
l9zcPf,
op. c i t . ,
*Ozipf,
I b i d . , p. 376.
p . 264.
2$. J. L. Berry and W. L. Garrison, "Alternate Explanations of Urban
Rank-Size Relationships1', Annals of t h e Association of American
Geographers, Vol. 47, 1958, pp. 83 - 91.
2 2 ~ s a r d ,op. c i t . , p. 57.
23~.~
Stewart,
.
"The Size and Spacing of c i t i e s " ,
Review, Vol. 48, 1958, pp. 222-242.
Ihe Geographical
2 4 ~ e ea l s o Kenneth E. Rosing , "A Rejection of the Zipf Model ( ~ a n k Size ~ u l )e i n elation-to City s i z e " , The ~ r o f e s s i o n a lGeogrzpher
v o l . 18, NO. 2, 1966, pp. 75 - 82.
,
2h. Chri s t a l l e r ,
Die Zentralen Orte i n Suddeuhschland ( ~ e n:a Gustav
Fischer, 1933, t r a n s l a t e d i n 1954 a t t h e Bureau of Population and
Urban Research, University of Virginia, by C Baskin. )
.
.
2 6 C h r i s t a l l e r ' s k = 3 network was t h e optimum p a t t e r n of hexagonal networks organized according t o a marketing p r i n c i p l e . For t h e t r a n s p o r t p r i n c i p l e t h e optimum network was k = 4 , f o r t h e administrative
' p r i n c i p l e k = 7.
2 7 ~ e r r yand Garrison, op. c i t . , p . 86.
2 8 ~ e eRutledge v i n i n g , "A Description of Certain S p a t i a l Aspects of
an Economic System", Economic Development and Cultural Change,
Vol. 3, 1955, pp. 147 - 195. Vining argues t h a t "The a n a l y t i c a l
problem i s not t h a t of determining an 'optimum' o r 'economical'
. s p a t i a l o r i e n t a t i o n of c i t i e s . Rather it i s t h a t of analyzing
a p r o c e s s i m p l i c i t i n a sequence of i n d i v i d u a l a c t i o n s performed
w i t h i n a s p e c i f i e d system of c o n s t r a i n t s . The c h a r a c t e r i s t i c s of
t h i s process a r e among t h e r e l e v e n t f e a t u r e s t o be considered."
2%"I'e
considerable degree of empirical correspondence t o t h e rank: s i ze
w suggest t h e p r o p o s i t i o n t h a t t h e populations
r u l e f o r population m
of c i t i e s t e n d t o be d i r e c t l y r e l a t e d t o t h e i r maximum h i n t e r l a n d s .
.,
30~hampernowne, op. c i t
p.
318.
3 1 ~. A . S t o u f f e r ,
" ~ n t e r v e n i n gOpportunities : A Theory R e l a t i n g Mobi l i t y and Dlstance", American S o c i o l o g i c a l R'eview, Vol. 5, 1940,
pp. 845 - 868.
3*The Yule D i s t r i b u t i o n i s a g e n e r a l term f o r t h e log-normal types o f
d i s t r i b u t i o n , and includes t h e log-lognormal d i s t r i b u t i o n .
3 3 ~ e ea l s o H.A. Simon, "On a Class of Skew D i s t r i b u t i o n F'unctions",
Biometrika, Vol. 42, 1955, pp. 425 - 440. Also r e p r i n t e d a s
Chapter 9 i n Models of Man (New York: John Wiley and Sons, Inc
1957)
34The n o t a t i o n used i s t h a t of Berry
and Garrison, op. c i t
. , pp.
86
.,
-
89.
~ E . N . Thomas, "Additional Comments on Populatibn-Si ze Relationshius
&d
f o r sets-of Citiest1~.n
Q u a n t i t a t i v e ~ e o ~ r a W~.L.
h ~Garrison
,
D.F. Marble (eds
(hranston, Ill. : Northwestern University
S t u d i e s i n Geography #3, 1967)
.
37This i s t h e "Law of Proportionate ~ f f e c t " ,mentioned e a r l i e r ( s e e p. l o ) ,
i.e. 'A v a r i a t e s u b j e c t t o a process o f change i s s a i d t o obey t h e
Law of Proportionate E f f e c t i f t h e change i n t h e variaJie a t any s t e ~
of t h e process i s a random proportion of t h e previous value of t h e
v a r i a t e . ' ' See J. Aitchison and J.A.C. Brown, The Log-Normal Dist r i b u t i o n (cambridge: The University Press, 1967), p . 22.
3 8 ~ o m a s , op. c i t . , p.
179.
39M. Kalecki , "On t h e Gibrat D i s t r i b u t i o n " , Econometrica, Vol. 13, 1945,
pp.. 161 - 170.
4 0 ~ . ~Madden,
.
"On Some I n d i c a t i o n s of S t a b i l i t y i n t h e Growth of C i t i e s
i n t h e U.S.",
Economic Development and C u l t u r a l Change, Vol. 4,
1956, pp. 236 - 253.
1
41~.
Vining, "Regional V a r i a t i o n i n C y c l i c a l F l u c t u a t i o n Viewed a s a
Frequency D i s t r i b u t i o n " , Econometrica, J u l y 1945, Vol. 4 , ~ p ~. 8 -7
188.
42The log-normal d i s t r i b u t i o n i s a l s o shown t o h o l d f o r t h e p o ~ u l s t i o n
d e n s i t y p a t t e r n of c i t i e s . .See B.J.L. Berry, "The I n t e r n a l S t r u c Lure
o f t h e c i t y " i n Urban Prospects and Problems, Law and Conte;nporary
Problems Vol. XXX, 1965 (school of Law, Duke University, North
~ a r o l i n ja pp 111 - 119.
See a l s o H.H. Winsborough, "City Growth and City s t r u c t u r e " ,
J o u r n a l of Regional Sciences, Vol. 4, 1963, who argued t h a t i e a s u r e s
of concentration of t h e c i t y ' s population and of congestion c f
c i t i e s can be derived from t h e log-normal d i s t r i b u t i o n a s a p p l i e d
t o c i t y s i z e s and t o urban population d e n s i t i e s .
.
,
4 3 ~ l b e r t0. Hirschman, The S t r a t e g y of Economic Development
and London: Yale University Press, 1958).
( ~ e wHaven
'
4 4 ~ e eL. Curry, "Explorations i n Settlement Tneory: The Random S p a t i a l
Econoqy, P a r t l", Annals of t h e Association of American Geographers,
Vol. 54, 1964, pp. 138 - 146.
4 % . ~ . Simon
and C.P. Bonini, "The S i z e D i s t r i b u t i o n of Business Firms",
American Economic Review, Vol. 48, 1958, pp. 607 - 617.
47This assumption i s j u s t i f i e d on t h e b a s i s t h a t s i n c e t h e r e e x i s t s cons t a n t r e t u r n s t o s c a l e above a c r i t i c a l minimum s i z e of firm, it i s
'!natural t o expect t h e firms i n each c l a s s s i z e t o have t h e same
chance on t h e average of i n c r e a s i n g o r decreasing i n s i z e i n proport i o n t o t h e i r p r e s e n t s i z e . " This i s t h e Law of Proportionate E f f e c t
P - 609.
a s mentioned before. I b i d
-,
4 9 ~ e eSimon,
0 .
c i t . ; pp. 427
-
4.30.
5 0 ~ i t c h i s o nand Brown, op. c i t . , pp. 111 -
5'1t
116.
ti~
veness
i s a l s o apparent t h a t p i s a measure of t h e degree of ~ o mop+.i +
i n t h e system, a s p - 3 2 , t h e competitiveness a l s o i n c r e a s e s . Analogies with c i t y s i z e d i s t r i b u t i o n s a r e immediately apparent.
52~imonand Bonini
5 3 ~ e r r y , op. c i t . ,
5 4 ~ . ~Neutze,
.
, op.
cit
., p.
616.
p. 575.
Econonic Policy and t h e S i z e of C i t i e s
t r a l i a n National University, 1965).
( ~ a n b e r r a :The Aus-
,
5 5 ~ b i d . pp. 37
- 38.
5 6 ~..Scitovsky , "Two Concepts of External ~ c b n o m i e s ~ Journal
,
oS Poli t i c a l Econoqy, Vol. 63, 1955, pp. 446 - 449.
5 7 ~ . ~Stockfish,
.
" ~ x t e r n a lEconomies, Investment and Foresight",
Journal of P o l i t i c a l Econoqy, Vol. 62, 1954, pp. 143 - 151.
%.z.
Hirsch, " ~ e a s u r i n gFactors Affecting Expenditure Levels of
Local Goverment Services", Metropolitan S t . Louis Survey, S t .
Louis, 1957.
5 9 ~ . ~Lomax,
.
"The Relationship Between Expenditure Per Head and Size
of Population of County Boroughs i n England and Wales", Journal
of Royal S t a t i s t i c a l Society, Vol. 106, 1943, pp. 51 - 59.
60There have been many studies of t h e r e l a t i o n s h i p of expenditures t o
c i t y s i z e s especially i n t h e U.S. Among t h e more important are:
W.Z. Hirsch, "Expenditure Implications of Metropolitan Growth and
Consolidation", The Review of Economics and S t a t i s t i c s , Vol. 41,
1959, pp. 232 241. H.E. Brazer,
City Expenditure i n t h e U . S . ,
(Few York: National Bureau of Economic Research, 1959). J. Marg o l i s (ed.) The Public Econoqy of Urban Communities ( ~ a l t i m o r e :
Resources f o r t h e Future Inc
John Hopkins Press, 1965).
-
.,
,
61~sard, op. ci?:.. pp. 57
'%adden,
op. c i t
. , pp.
- 58.
247
-
249.
6 5 ~ e r w , op. c i t . , p. 582.
%r
example, Simon, op. c i t . , pp. 427
P * 319.
-
428 and Champernowne, op. c i t . ,
6 7 ~ .Vining,
.
"A Description of Certain S p a t i a l Aspects of an Economic
syster;l", ~conomic-Development and c u l t u r a l Change, Vol. 3, 1954- 5,
pp. 147 - 195. E.M. Hoover, "The Concept of a System of C i t i e s :
A Comefit on Rutledge ~ i n i n g ' sPaper", Economic Development and
. ~ u l t u r i Change,
l
Vol. 3, 1954-5, pp. 196 - 198.
@B.J.L. Berry, " c i t i e s a s Systems Within Systems of c i t i e s " , Pa e r s of
t h e Regional -Science Association, Vol. 13, 1964, pp. 147%@The concept of ectropy originates i n the second Lair of Thermo@namics.
If two systems with entropy S1 and S2 a r e u n i t e d then S1 = S2 = S.
The entropy of a closed p h y s i c a l system tends t h e r e f o r e t o increase
a s l o n g a s t h e system has not y e t reached equilibrium and entropy
may t h u s be taken a s a measure of t h e degree of e q u i l i z a t i o n reached
w i t h i n a system. Entropy of a closed system i s maximized when t h e
system i s i n equilibrium. Boltzmann's Law of Entropy argues t h a t
lower temperature should be understood a s a s t a t i s t i c a l e q u a l i z a t i o n
of d i f f e r e n c e s i n molecular speed.
Let p h y s i c a l entropy S = T o g , + R / m l o g ~ --------------. (1
Where
.
.
V = Volume of gas whose mass equals 1 u n i t .
C, = S p e c i f i c heat; at constant volume.
T = Absolute temperature.
m = Molecular weights of t h e gas.
R = General gas constant.
f-l
r\
Where pi = p r o b a b i l i t y of f i n d i n g an i d e a l i z e d physical system
i n t h e s t a t e i of n p o s s i b l e s t a t e s , and k = a c o n s t a ~ t .
-
Now t h e p r o b a b i l i t y t h a t molecules i n a gas container can be
d i s t r i b u t e d among n p o s s i b l e s t a t e s i n p ways i s :
Thus i f t h e system i s l a r g e t h e n :
s
\
= l o g p = plogp - .
n
Thus s i n c e S
=
-k
pilogpi
S = -k(plogp
------------------ (4)
i=l
L pilogpi-k(logpi)
/=I
Then
2
n
-2
pilogpi)
------------------ ( 5
+
C
-------------- ( 6 )
i=
1
(4) and (5) both hold only i f
2k
= 1
70 Berry and Garrison, op. &it
., p. 89.
71 ~ u r m ,op. c i t ., p . 144.
T 2 M. Maruyama, "The Second Cybernetics : Deviation Amplifying Mutua3.
Causal Processes", American S c i e n t i s t , V O ~ . 51, 1963, pp. 164-179.
73 Morphostasis i-s a
ships
... .
I1
"...
d e v i a t i o n counteracing mutual causal r e l a t i o n See Marujrama, op. c i t ., p . 1 6 h -
7 4 ~ .I@rdal,
~ c o n o m i cTheory and Underdeveloped Regions
75
Maruyama, op. c i t . ,
% b i d . , p . 165
-
pp. 167
-
on don:
1954).
168.
166. C f . Myrdal's v i c i o u s c i r c l e concept.
7 7 measure
~
of order may be R
H'
/H) where H = maximum entropy i n
a system under no s t r u c t u r a l r e l a t i o n s ( i . e . i n d i v i d u a l s i n i s o l a t i o n ) . H ' = maximum entropy i n a system with a given l e v e l of
s t r u c t u r a l r e l a t i o n s . Hence 05R 51 a r e two l i m i t i n g cases. When
R = 0 then H ' = H ( i . e . no s t r u c t u r e exists)', when R = 1 then H ' = 0
and a system i s developed i n t o a s i n g l e l i g h t l y c l u s t e r e d web of
a c t i v i t i e s . Now i n a c l o s e d p h y s i c a l system, entropy w i l l i n c r e a s e
with time and order decrease. However, a s e l f - o r g a n i z i n g system w i l l
by d e f i n i t i o n be i n c r e a s i n g order and decreasing entropy. See Curry,
op. c i t . , p . 145.
7 8 ~ u r r y , op. c i t . , p .
= 1-(
145.
7 9 ~ . ~H a. r r i s and E.L. Ullman,
"The Nature of C i t i e s " , Annals of t h e
American A c a d e q of P o l i t i c a l and S o c i a l Science, V a l . 242, Nov. 1945,
PP. 7 - 17.
8 0 ~ o b e r tR e d f i e l d and Milton Singer, "The C u l t u r a l Role of C i t i e s " ,
Economic Development and C u l t u r a l Change, Vol. 3, 1954,.pp. 53
81J3.~.
-
73.
H o s e l i t z , "Generative and P a r a s i t i c C i t i e s " , Economic Development
and C u l t u r a l Change, Vol. 3, A p r i l 1955, pp. 278 - 294.
8 2 ~J.. L. Berry,
"Urban Growth and Economic Development of Ashanti", i n
F o r r e s t P i t t s (ed. ), Urban Systems and Economic Development, pp. 52
54
8%Ioselitz, op. c i t . , p . 280.
8 4 ~ a r s i c u l a r l yi n t e r e s t i n g , a s w e l l a s of g r e a t informative value a r e t h e
proceedings of t h e two seminars on urbanization i n Asia and L a t i n
America sponsored j o i n t l y by t h e United Nations and i t s agencies.
See P.M. Hauser (ed. ), Urbanization i n Asia and t h e F a r East (ECAFE
Region, Bangkok, 8 - 18 August, 1956, C a l c u t t a , 1957). P.M. Hauser
( e d . ) , Urbanization i n L a t i n America (UNESCO,Santiago, Chile, 6 18 July, 1959, New York, 1961).
8 5 ~ . ~McGee,
.
The Southeast Asian City ( ~ e wYork: F.A. Praeger, 1967).
See a l s o P.M. Hauser, Urbanization i n Asia and t h e Far E a s t , p. 31.
8 6 ~ .H
~ o. s e l i t z ,
"A Survey of t h e L i t e r a t u r e of Urbanization i n ~ n d i a " ,
:
of
i n Roy Turner (ed. ) I n d i a ' s Urban Future ( ~ e r k e l e ~ University
C a l i f o r n i a Press, 1962), pp. 425 - 443.
,
-
8 7 ~ e eN.V. Sovani, "The Analysis of Over-Urbanization" , EconmZc Development and Cultural Change, Vol. 12, No. 2, 1964. See a l s o N.V.
Sovani, Urbanization and Urban India ( ~ e wYork: Asia Pu3lishing
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