urban structure and rank-size rule of cities · the rank-size rule means that

Urban Structure and Rank-Size Rule of Cities-Anexamination of cases in Japan from 1975 to 1995-

Hiroshi YOSHIMURA, ProfessorFaculty of Economics , Yamaguchi University, Yamaguchi-shi 753 Japan

SUMMARY: The aim of this paper is to examine the rank-size rule of cities or the constant elasticity ofcity's rank to city's size in the case of Japan from 1975 to 1995. The main results obtained are as follows.(D The Pareto distribution indicates the good fitness for the relation between the city's rank and city's size.(2) However, the value of elasticity of city's rank to city's size, i.e. Pareto exponent, considerably dependson the definition of city and the sample size (the number of cities adopted as samples in order of city'srank).(3) When we increase the sample size, the value of elasticity increases at first and soon reaches the peakand decreases gradually, without regard to the definition of city adopted in this paper.CD When we adopt two criteria, i.e. the weak criterion (with the range of 0.5 in the elasticity of city's rankto city's size) and the strong criterion (with the range of 0.2 in the elasticity), neither the rank-size rule northe constant elasticity is valid even on the basis of the weak criterion for the first category of city, i.e."cities, towns and villages". For the second category, i.e. "cities", the rank-size rule is not valid but theconstant elasticity is valid on the basis of the weak criterion, whereas neither the rank-size rule nor the con-stant elasticity is valid on the basis of the strong criterion. For the third category of city, i.e. "areas", bothof the rank-size rule and the constant elasticity are almost valid on the basis of the weak criterion, althoughneither the rankTsize rule nor the constant elasticity is valid on the basis of the strong criterion. For theforth category, i.e. "prefectures", neither the rank-size rule nor the constant elasticity is valid even on thebasis of the weak criterion, so this is similar to the first category.© Generally speaking, we can say that the rank-size rule and the constant elasticity are apt to be valid forthe middle category of city which is formed from point of economic activities, whereas neither the rank-sizerule nor the constant elasticity is valid for the small or large category of city which is the district formedfrom point of administration.© The facts from (D to © mentioned above are all valid for the period of 1975-95 in Japan.© For the first, second and forth categories of city which are administrative districts, the values of elasticityof city's rank to city's size increase among the "upper ranking" members ("large" cities) of each category,whereas the values decrease among "all" members, as time passes. For the third category (areas), thevalues decrease both among the "upper ranking" members and among "all" members, as time passes. Inother words, the small cities have turned out to be smaller relatively, and the cities of upper-middle sizehave grown larger, therefore differences of city's size as a whole increased from 1975 to 95.

1. INTRODUCTION

One of the main themes of urban economics is toprove the real state of urbanization or agglomera-t ion of cities and its mechanism. As concerns the

mechanism, we have the theory of transport cost,the theory of spatial competition, the central placetheory, the spatial production and consumer theory,and the theory of agglomeration economies (exter-

nalities in an urban context), and so on. (See, for

43

example, Hotelling2' and Mills31.) On the other

hand, there is the empirical studies of urban sizeand structures of urban areas from point of realstate of urbanization. The typical one of them is

the rank-size rule or the constant elasticity of city'srank to city's size. (See, for example, Beckman1'

and Rosen4).) Here, the rank-size rule means that

the product of the city's rank and its size is a con-

stant.

(Preto distribution) We can derive the rank-sizerule from the Pareto distribution like this,

R (S)=AS- (1),

where R(S) is the number of urban areas (cities)

with at least S-city' size (S or more population),and two coefficients (A and a) are constants whichwill be estimated from the data. So R(S) means

the rank of the city with S people. Here, theranking is formed in order of large population. Inthis paper, the city's size is measured by the

population, and I showed its validity in my previouspaper12'. We use the data of population in BasicData of Residents1®.

R: city's rankS : city's sizeA and a: constants

If we take logs of both sides of the above ex-

pression (1), we have the following log-linear type

equation.

logR=-alogS+logA •Eå

So, we have this expression,

Y=-aX+b

(2)

(3),

where F=log i?, X=log S, and b=\ogA.

In this case, the coefficient (a) of Xis Pareto ex-ponent or the elasticity of city's rank (R) to city'ssize (S), i.e.,

a=-(dR/R) /(dS/S).

(constant elasticity and rank-size rule) There-fore, the constant elasticity of city's rank to city's

size means that the coefficient (a) is constant, andthe rank-size rule means that

2, which includes some cities, towns and villages.Category 4 has 47 members, and the number ofmembers is constant for the period considered

here.We have to note two points. The first concerns

the special ward. In addition to cities, towns andvillages, we have 23 special wards in Tokyo metro-politan area as basic administrative unit in Japan.

We will put these 23 special wards together intoone, i.e. Tokyo-special-ward, and category 1 and2 include this Tokyo-special-ward as one of mem-

bers. The second concerns category 4. Category4 includes not only 43 prefectures but also fourmembers, i.e. Tokyo metropolitan area, Osaka

metropolitan prefecture , Kyoto metropolitan prefec-ture and Hokkaido special prefecture.

Category 3, "area", is not the administrative dis-trict, but an economic distribution area. "Area"usually has one central city, and the central city has

powerful urban functions in the area from points ofadministration, education, culture, health and wel-fare as well as economic, distributive and industrialactivities.

We have no clear and official definition of "area",because it is not the administrative unit. We willadopt the concept in Minryoku15) as "area" here, assame as my previous papers10)Fll)'12) and13). The

number of areas and the members of an area inMinryoku sometimes vary slightly as time passes.But we will adopt 110 areas in Minryoku15) through

this paper.As we can know through the explanation above,

the region of city becomes wider, according as we

movefrom category 1 to 4.

(sample size) Now, as concerns the other prob-

lem, i.e. the sample size, we could not decide apriori the size (the number of cities) which weshould adopt. Therefore we will examine many

cases depending on necessity where the number ofcities to be adopted spreads from 5, 10, 20, 30, -,100, -", 500, -, 1000, -, to all samples. Through

this examination of many cases, we can understandthe whole feature oi relation between city's rank

and city's size.

3. CATEGORY 1 (CITIES, TOWNS

AND VILLAGES)

(relation between city's rank and city's size)When we adopt category 1 (cities, towns and vil-lages) as the definition of city, we can find the

whole feature of the relation between city's rankand city's size at Fig. 1-1. From this figure, wecan find out the first important observation that the

relation between city's rank and city's size showsalmost the straight line (log-linear line), if we ex-clude the exceptional part close to 8 on the horizon-

tal axis. We can verify the good fitness to the log-linear relation by the high value of coefficients ofdetermination adjusted for the degree of freedom inAppended Table 1. And we can know that the do-

main of exceptional part is the part with about 8.04

or more value of log of city's rank, by observingFig. 1-6. (Figures from Fig. 1-2 to Fig. 1-6 are

the expanded ones of a part in Fig. 1-1.) Here,8.04=log 3100, so we can understand that the ex-ception is about 140 samples which are small townsand villages, by the simple algorism : the number of

all samples (about 3240) minus 3100. The excep-tion is about 4.3% of all samples.

We are able to know details of Fig 1-1 through

fromFig. 1-2 to Fig. 1-6. As the value of log ofcity's rank on the horizontal axis grows from zero,the value of log of city's size (population) on the

vertical axis decreases considerably at first, thendecreases gently, and decreases steeply at last.

(elasticity of city's rank to city's size) As con-cerns such relation, the coefficients (a) of X areuseful, as we show them in Fig 2-1, Fig 2-2 and

Appended Table 1. Fig 2-2 is the expanded figureof a part of Fig2-1. If we neglect the small de-cline of the elasticity at the first stage, the elasticity

of city's rank to city's size (a: coefficient of X) onthe vertical axis grows steeply at first, and itreaches the peak at the neighbourhood of sample

size 100, then declines gradually, as the sample

-45-

Fig. 1-1 City's rank and its population for 1975-95 in Japancities, towns and villages--1

1 2 3 4 S 6 7 8

Log of city's rank

-'75(S50) - '80(S55) -- '85(S60) '9O(HO2) -'96(H7)

Fig. 1-2 City's rank and its population for 1975-95 in Japancities, towns and villages-2

Log of city's rank

'75(S50) -'80(565) --'85CS60) - '90(H02) '95(H


14

13 . S

13

ｫｰ; - l . -

; 2 s ^

Log of city's rank

-'75(S50) -'80(S55) --'85(S60) - '90(H02) -- '95(H7)

-46


*S ll-6[-

Log of city's rank

75(S50) -*8O(S55) --'85(S60) - '90(H02) -~ '95(H7)

Fig. 1-5 City's rank and its population for 1975-95 in ]apancities, towns and villages---5

"S 9-5

7.S6.5 T

Log of city's rank

-'75(S50) -'80(S55) --*85(S60) -- '90(H02) - '95(H7)


7.9

Log of city's rank

-*75(S50) -'80(S55) ~'86(S60) --- '900102) - '96

Fig. 2-1 The number of samples and the size elasticity of dty's rankcities, towns and villages-1

1000 2000 3000Number of samples(cities, towns and villages) taken in order of large population

-'7E(SBO) -'80(S55) --'85(S60) -'90(H02) '95(H7)

Fig. 2-2 The number of samples and the size elasticity of city's rankcities, towns and villages-2

1Is1I*B

^ > , 蝣> - ̂ "

¥ V ;* * S ^ ^ = ;

200 400 600

Number of eamples(citie6, towns and villages) taken in order of large population

-'75(S50) -'80(S55) --"85(S60) -*90(H02) -'95(H7>

size (the number of samples) on the horizontal axisincreases. The sample size shows the city's rank,because the samples are taken in order of city's

rank. In other words, for example, the samplesize 100 means that we take 100 cities from city's

rank 1 to city's rank 100 as the samples to estimatethe relation between the city's rank and city's size.

(relation between the rank-size and the elasticity)Fig.3-1 and Fig. 3-2 show such a relation. Fig.

3-1 corresponds to Fig. 2-1, and Fig. 3-2 corres-ponds to Fig. 1-1. We should note that the ver-tical axis in Fig. 1-1 and Fig. 3-2 is X in equation(3), on the other hand the horizontal axis is Y in

Equation (3). In addition to this, we also note thatthe horizontal axis in Fig. 1-1 has the log-values,but this is not so in the horizontal axis in Fig. 3-2.

Fig. 3- 1 and Fig. 3- 2 summarize observations

appropriately which are stated above about the rela-tion between city's rank and city's size and theelasticity of city's rank to city's size on the basis ofFig. 1-1, Fig. 2-1 and Appended Table 1. There

are two observations. First, the city's size(population) decreases considerably at first, thendecreases gently, and decreases steeply at last, as

the city's rank drops. Second, the elasticity of ci-ty's rank to city's size grows steeply at first, and itreaches the peak at the neighbourhood of samplesize 100, then declines gradually, as the sample

size increases.

(change of the relation between city's rank and

city's size) We can observe the change of the re-lation between city's rank and city's size, as time

-48-

Fig. 3-1 The elasticity of city's rank to city's size and the Fig. 4-1 Change of the relation between the elasticity andsample size the sample size

ebsticity of city's rankto city's size

a1=a3

Fig. 3-2city's size(population)

elasticity of city's rankto city's size

19751975

1995

sample size(city's rank)

Fig. 4-2 Change of the relation between the elasticity andthe sample size

elasticity of city's rankto city's size

sample size{city's rank)

passes from 1975 to 95. According to figures form

Fig. 1-2 to Fig. 1-6, populations of cities with up-per ranks have grown, on the other hand, popula-tions in small towns and villages with lower ranks

have shrunk, for the period from 1975 to 95. Thelog-value of critical rank is about 7.4 as we can findit in Fig. 1-5. Therefore cities with upper ranksthan 1650 are expanding, and the other small cities

with lower ranks than 1650 are shrinking, because7.4 %log 1650. This means that half members ofcities, towns and villages are expanding and the

other half are shrinking, for there are about 3240cities, towns and villages in Japan.

(change of the elasticity) As concerns the elas-ticity, the critical sample size is about 450 as we

see Fig. 2-2. The elasticity for sample sizes with

less than 450 samples is apt to increase, on theother hand, the elasticity for more sample sizesthan 450 is apt to decrease.

Wewill show this change of the elasticity in Fig.

4-1. The elasticity of city's rank to city's sizemeans how many percent the city's rank changeswhen the city's size changes by one percent.

Therefore, the high elasticity means the largechange of rank for a fixed change of size. In otherwords, the high elasticity means that the differ-

ences of sizes of members are small, that is to say,the equalization of size. We can know the tenden-cy of equalization of size of cities with upper ranks

and the tendency of expansion of differences of allsamples.

-49-

(rank-size rule and constant elasticity) As con-

cerns the rank-size rule and the constant elasticity,we will show criteria of the validity of the rule and

the constancy at first.

criteria for the rank-size rule :

strong criterion: If all the elasticity is in thedomain from 0.9 to 1.1, that is to say, in the do-main of width 0.2 with the central value 1.0, the

rule is valid.weak criterion: If all the elasticity is in the

domain from 0.75 to 1.25, that is to say, in the do-main of width 0.5 with the central value 1.0, therule is valid.

criteria for the constant elasticity :

strong criterion: If all the elasticity is in thedomain of width 0.2, the constancy is valid.

weak criterion: If all the elasticity is in the do-

main of width 0.5, the constancy is valid.

On the basis of these criteria, we can say as fol-lows.

(category 1) When we think of all samples fromrank 5 to the bottom in 1975 at Appended Table 1,the maximum of the elasticity is 1.392, which cor-

responds to sample size 110 and 120, and the mini-mumis 0.853, which corresponds to all samples.The difference is 0.539, so the constancy is not

valid even for the weak criterion.If we restrict these samples to the case where

samples from rank 3000 to the bottom are ex-

cluded, the minimum is 0.927, which correspondsto the sample size 5, while the maximum is thesame value, i.e. 1.392, so the difference is 0.465.

Therefore, the constancy is valid for the weakcriterion, whereas not for the strong criterion:

Now we think of all samples from rank 10 to thebottom in the same year, i.e., 1975 in Appended

Table 1, the maximum and the minimum of theelasticity are the same as the first case, where wehave all samples from rank 5 to the bottom. So,

the constancy is not valid even for the weak crite-

rion, either.

On the other hand, if we restrict these samplesto the case where samples from rank 3000 to thebottom are excluded, the maximum and the mini-

mumof the elasticity are the same as the secondcase, where we have all samples from rank 5 to

3000. Therefore, the constancy is valid for theweak criterion, whereas not for the strong crite-rion.

If we exclude the domain with sample size fromabout 50 to 300, both of the constant elasticity and

the rule are valid for the weak criterion. Furth-ermore, if we restrict the sample size to the do-main from about 1000 to 2500, both are valid forthe strong criterion.

As concern 1980, the results are the same as thecases in 1975, as we showed just above. We willshow these situations in Table 1.

However, in 1985,90 and 95, the constancy is in-valid for all cases even for the weak criterion, asshowed in Table 1.

4. CATEGORY2 (CITIES)

(relation between city's rank and city's size) As

concerns category 2 (cities) of city, we can find thewhole feature of the relation between city's rankand city's size at Fig. 5-1. From this figure, we

can find out almost the same observation as thecase of category 1, where the relation between ci-ty's rank and city's size shows almost the straightline (log-linear line), if we exclude the exceptional

part close to 6.5 on the horizontal axis. We canverify this by looking at the high value of coef-ficients of determination in Appended Table 2.

And we can know that the domain of exception islocated in the neighbourhood of 6.48 in the horizon-tal axis by observing Fig. 5- 5. (Figures fromFig. 5-2 to Fig. 5-5 are the expanded ones of Fig.

5-1.) Here, 6.48=log 650, so we can understandthat the exception is about 5 or 14 samples, be-

-50

Table

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92

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0. 9

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71

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39

1.0

39

0. 6

32

0.

63

20

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81

0.

78

11

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65

1.3

59

1.2

65

1.3

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1. 3

92

1.

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21

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92

1.

392

1. 3

92

1.

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21

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92

1.3

92

1.1

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1.

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2.1

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52

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0. 5

39

0.4

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0.5

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0.4

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0. 4

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0.4

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0.3

53

0.3

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0.5

59

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55

90

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410

0.8

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0. 7

93

0.7

70

0.6

76

re

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0. 63

70

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783

1.2

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1. 24

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max

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0.9

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69

0. 9

69

1.

10

41

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0.6

35

0.

63

50

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max

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91

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1.4

19

1.4

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1.4

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1.4

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1.4

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1.4

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1.4

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1.4

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1.4

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esul

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that

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valid,

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therul

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A"in

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eans

that

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le(of

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se,

theco

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cy,

too)

isval

idfor

thew

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rion.

hiall

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,ne

ither

theco

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forthe

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Fig. 5-1 City's rank and its population for 1975-95 in Japancities- - -!

Log of city's rankf75(S50) ~'80(S55) --'85(S60) -'90(H02) --'95HO7)

Fig. 5-2 City's rank and its population for 1975-95 in Japancities -2

bo

3

Log ofcity's rankå 7B(SB0) -'80(555) --'85(560) -190(H02) - '95H07)

Fig. 5-3 City's rank and its population for 1975-95 in Japancities- - -3

Log ofcity's rank"75(S50) - "8O(SS5) -- "85CS60) - "90(1102) "95H07)

-52-


Log ofcity's rank-p75(S50) -P80(S5B) --'8S(S60) -'90(1102) -å '95K07)


6. a e.4

Log of city's rank-'75CS5O) - f80(S55) -- '85(S60> - '90(H02) '9BH07)

cause the number of all samples is about 655 or

664. The exception is about 1.5% of all samples.According to figures from Fig5-2 to Fig. 5-5, as

the value of log of city's rank on the horizontal axis

grows from zero, the value of log of city's size(population) on the vertical axis decreases consider-ably at first, then decreases gently, and decreases

steeply at last. This is the same as category 1.

(elasticity of city's rank to city's size) We will

show the elasticity in Fig 6 and Appended Table 2.If we neglect the small decline of the elasticity atthe first stage, the elasticity grows steeply at first,and it reaches the peak at the neighbourhood of

sample size 100, then declines gradually, as the

sample size increases. This is the same as categ-ory 1, too. However, all the values of elasticityare larger than 1.0. This point is different from

category 1.

(relation between the rank-size and the elasticity)As concerns the relation between the rank-size andthe elasticity, the relation showed in Fig. 3-1 and

Fig. 3-2 is valid in category 2, too. Therefore,two observations stated in category 1 are also valid

here.

(change of the relation between city's rank andcity's size) According to figures form Fig. 5-2 toFig. 5- 5, populations of cities with upper ranks

-53-

Fig. 6 The number of samples and the size elasticity of city's rankcities

1 00 200 300 -100 500 600

Number of samples(cities) taken in order of city's size

-'7S(S50) -'K0(SS5> - '85(560) -- '90(H02) -'95H(07)

have grown, on the other hand, populations in smallcities with lower ranks have shrunk, for the periodfrom 1975 to 95. The log-value of critical rank is

about 6.35 as we can find it in Fig. 5-5. There-fore cities with upper ranks than 570 are expand-

ing, and the other small cities with lower ranks than570 are shrinking, because 6.35 = log570. Thismeans that 85% of cities is expanding and the other

15% is shrinking.

(change of the elasticity) As concerns the elas-

ticity, the same things as category 1 are valid, asweshow in Fig. 4-1. Therefore, we can find out

that the difference of size of cities with upper ranksis apt to become smaller and the difference of sizeof all cities is apt to become larger.

(rank-size rule and constant elasticity) As con-

cerns the constant elasticity, the situation is diffe-rent from category 1, as we are able to see in Fig.

6, Table 1 and Appended Table 2.In category 2, let's consider the case where

there are samples from rank 5 to the bottom in

1975. The maximum is 1.392, which correspondsto sample size 110 and 120, on the other hand, theminimum is 0.927, which corresponds to sample

size 5. So the difference is 0.465, and the con-stancy is valid for the weak criterion, whereas not

for the strong criterion.

At all cases of each year in category 2, the con-

stancy is valid for the weak criterion, whereas notfor the strong criterion, as we can see at Table 1and Appended Table 2.

As concerns the whole period from 1975 to 95,

at the cases with samples from rank 10 to the bot-tom or to rank 600, the constancy is valid for theweak criterion, whereas not for the strong crite-

rion. However, at the cases with samples fromrank 5 to the bottom or to rank 600, the constancy

is invalid even for the weak criterion, because themaximumis 1.434, and the minimum is 0.927, sothe difference is 0.507, in these cases.

As concerns the rank-size rule, it is invalid evenon the basis of the weak criterion at all cases in

category 2.

5. CATEGORY3 (AREAS)

(relation between city's rank and city's size)Fig. 7- 1 shows the whole feature of the relationbetween city's rank and city's size in category 3.From this figure, we can find out almost the same

observation as the case of category 1. The rela-tion between city's rank and city's size in Fig. 7-1shows almost the straight line (log-linear line), if we

exclude the exceptional part close to 4.5 on thehorizontal axis. We can verify this by looking at

54

Fig. 7-1 City's rank and its population for 1975-95 in Japanareas---1

1 2 3 4Log of city's rank

-'75(S5O) -'80(555) - '85(S60) -- '90(H02> ~ '95(HO7)

Fig. 7-2 City's rank and its population for 1975-95 in Japanareas- --2

å " 16.5

a isoS 1S.S

IS

14.5

^ ^ ^ ^ 蝣1^ ^ ^ ^ Hg｣ 5��

^x ^ �

î ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ mî ^ ^ ^ ^ m ^ ^ ^ ^ ^ m ^ ^ ^ ^ mL o g o f c i t y ' s r a n k

' 7 5 ( S S 0 ) - ' 8 0 ( S S 5 ) - ' S S t S e O ) - " 9 0 ( H 0 2 ) ' 9 S C H O 7 )

Fig . 7 -3 Ci ty ' s r ank and i t s popu la t ion fo r 1975-95 in Japana reas - --3

1 4 . 5

M

1 3 . 6

i" - 1 s ? i-i- k

"* * & - .

- S J E = =3 ｻ. ..

Log of city's rank-'75(S50) -'80(S55> --'85(560) -'90(H02) - '9B(H07)

-55-

Fig. 7-4 City's rank and its population for 1975-95 in Japan

areas-4

,2 12.5 |-

4.25 4.5

Log of city's rank-"75(S50) -"80(855) --'8S(S60) -"90(H02> ' 95 (H07)

the high value of coefficients of determination inAppended Table 3. And we can know that the do-

main of exception is located in the neighbourhood of4.6 in the horizontal axis by observing Fig. 7- 4.Here, 4.6=?log 100, so we can understand that the

exception is about 10 samples, because the numberof all samples is 110. The exception is about 10%of all samples.

According to figures from Fig7-2 to Fig. 7-4, asthe value of log of city's rank on the horizontal axis

grows from zero, the value of log of city's size(population) on the vertical axis decreases consider-ably at first, then decreases gently, and decreases

faster at last. This is almost the same as categoryl and 2.

Fig. 3-2 is valid in category3, too.


city's size) According to figures form Fig. 7-2 toFig. 7- 4, populations of areas with upper rankshave grown, on the other hand, populations in small

areas with lower ranks have shrunk, for the periodfrom 1975 to 95. This is the same as category 1

and 2. The log-value of critical rank is about 4.25as we can find it in Fig. 7-4. Therefore areaswith upper ranks than 70 are expanding, and the

other small areas with lower ranks than 70 areshrinking, because 4.25=?log 70. This means thatabout three fourth of areas are expanding and theother one fourth are shrinking.

(elasticity of city's rank to city's size) Let's turnour eyes to the elasticity in Fig 8 and AppendedTable 3. Here we find a little difference from Fig.2-1 in category 1 and Fig. 4-1 in category2. Asthe sample size increases, the elasticity grows uni-formly at first, and it reaches the peak at the neigh-

bourhood of sample size 60, then declines gradu-ally. This shape is the upper-wards convex with-out twists. Furthermore, almost the values of

elasticity are rather near to 1.0.

(relation between the rank-size and the elasticity)As concerns the relation between the rank-size andthe elasticity, the relation showed in Fig. 3-1 and

(change of the elasticity) As concerns the elas-ticity, the situation is a little different from category

1 and 2. We show Fig. 4-2 which corresponds tocategory 3 and is different from Fig. 4- 1 corres-ponding to category 1 and 2. For all sample sizes,the elasticity decreases from 1975 to 95. That is

to say, the difference of sizes (population) amongareas have uniformly increased. We can think that

the difference of city's size is apt to increase whenweconsider the city form point of economic activi-ties.

(rank-size rule and constant elasticity) When wethink of all samples from rank 5 to the bottom in

-56

Fig. 8 The number of samples and the size elasticity of city's rank

areas

20 1(1 K(> HO 100

Number of samples(areae) taken in order of city's size

-'75(350) - '80(MS) -- 'S5{S6O) - '%(JID2) '950)07)

1975 at category 3, the maximum of the elasticity is1.191, which corresponds to sample size 60, andthe minimum is 0.632, which corresponds to sample

size 5. The difference is 0.559, so the constancyis not valid even for the weak criterion.

However, if we restrict these samples to thecase with samples from rank 10 to the bottom, theminimum is 0.781, which corresponds to the sample

size 10, while the maximum is the same value, i.e.1.191, so the difference is 0.410. Therefore, theconstancy is valid for the weak criterion, whereas

not for the strong criterion.In addition to this, the rank-size rule is also valid

for the weak criterion, whereas not for the strong

criterion, because all the elasticity is in the domainfrom 0.75 to 1.25, that is to say, in the domain ofwidth 0.5 with the central value 1.0.

As showed in Table 1, for all samples which in-clude upper ranks than 10 (smaller numerical values

than 10), the constancy is invalid even for the weakcriterion. On the other hand, for samples whichexclude upper ranks than 10, the rank-size rule (of

course, the constancy, too) is valid for the weakcriterion, whereas not for the strong criterion.

Here, we should note two points. First, as con-cerns the constant elasticity, to be exact, "rank 10"stated just above is able to be altered to "rank 8".

Because the constancy is valid for samples withsample size 8 and 9, too, from point of the weakcriterion, as we can see in Appended Table 3.

Second, as concerns the rank-size rule, to be ex-act, "rank 10" stated just above is able to be

altered to "rank 9". Because the rule is valid forsamples with sample size 9, too, from point of theweak criterion, as we can see in Appended Table 3.

6. CATEGORY 4 (PREFECTURES)

We know sufficiently that there is some questionabout dealing with prefectures as "cities". Gener-ally speaking, prefectures are too large to regard ascities. However, we will deal with prefectures

here, because they are very important local govern-ments functioning really.

(relation between city's rank and city's size)Fig. 9- 1 shows the whole feature of the relationbetween city's rank and city's size in category 4.From this figure, we can find out almost the same

observation as the case of category 1, 2, and 3.

The relation between city's rank and city's size inFig. 9-1 shows almost the straight line (log-linearline). We can verify this by looking at the highvalue of coefficients of determination in Appended

Table 4. However, the lines in Fig. 9- 1 arecurved a little compared with Fig. 2-1, Fig. 5-1and Fig. 7-1. So the values of coefficient of deter-

mination in Appended Table 4 are slightly smallerthan those in Appended Table 1, 2 and 3.

-57

Pig. 9-1 City rank and its population for 1975-95 in Japanprefectures- !

i

Î ^ ^ B ^ S ^ ^ S ^ ^ ^ ^ ^ ^ ^ ^ H-^・=5 5 = **

&--ss.

Log of city'eOprefecture's) rank-P75(S50) -'80(SB5J --'85(S60) -'M(H02) -'95 (H07)

Fig. 9-2 City rank and its population for 1975-95 in Japanprefectures- --2

3

Log of city's(prefecture's) rank-'75(S50> -P80(S55) --'85{S60) -'90(H02) '95 (H07)

Fig. 9-3 City rank and its population for 1975-95 in Japanprefectures-3

$

IS

14.5

14

^蝣s ^

Î ^ ^ B ^ l ^ ^ ^ ^ ^ ^ ^ ^ ^ l

^

= 5 t *

Fig. 9-4 City rank and its population for 1975-95 in Japanprefectures-4

* ***蝣

蝣5 ^

^ ^ H S? T !

^ ~- ^ b *

3. 2 3.4 3. fi

Log of

Fig. 10-3 The number of samples and the size elasticity of city's rankprefectures- --3

Number of eamplee(prefectures) taken in order of city's size

-p75(SS0) -'80(S5E) --'85CS6O) -~'90(H02) -å f95 (H07)

(elasticity of city's rank to city's size) We canknow the situation of the elasticity through Fig. 10-1 and Appended Table 4. Here we find some dif-

ference from category 1, 2 and 3. As the samplesize increases from 5, the elasticity grows at first,and it reaches the peak at the neighbourhood of

sample size 8, then declines very sharply, and forsamples with more than 15, the values of elasticityhardly vary. In addition to this, all the values of

elasticity are larger than 1.0, and they are consider-ably larger than those in category 1, 2 and 3.

(relation between the rank-size and the elasticity)

As concerns the relation between the rank-size andthe elasticity, the relation showed in Fig. 3-1 andFig. 3-2 is basically valid in category 4, but the

shape is fairly skew, as we can see in Fig. 10-1.


city's size) According to figures form Fig. 9-2 toFig. 9-4, populations of prefectures have increasedfor almost all city's ranks during the period from

1975 to 95. The growth is faster for the domainfrom about 1.0 to 2.0 of log-value of city's rank.As log3 = 1.1 and log8 = 2.1, prefectures with

high growth are ones with city's ranks from rank 3to 8. They have big cities, i.e. ordinance-desig-nated cities, which are poles of "poly-pole" in

Japan.

(change of the elasticity) As concerns the elas-ticity, the situation of category 4 is the same as

category1 and 2, as we can see in Fig. 10-1. Thetype of change is one of Fig. 4-1. The criticalvalue is ll or 12 of the horizontal axis. The differ-ences among prefectures with ranks of upper than12 is shrinking, on the other hand, it is expanding

among domain including more than 12 prefectures.

(rank-size rule and constant elasticity) At all

cases in category 4, the constancy is invalid evenfor the weak criterion, because the values of max-imum are too high, as we can see through Fig 10-1, Table 1 and Appended Table 4.

7. CONCLUSIONS

In Table 1, we will show conclusions of examina-tion of the rank-size rule and the constant elasticityof city's rank to city's size in Japan from 1975 to

95. And we will add Table 2 which is the sum-mary of Table 1. The main results are showed inSUMMARYfrom CD to ® at the beginning of this

paper.

-60-

Table 2 Summary of resultsth e co n sta n t e la sticity (a )

o f c ity 's rank to city 's s iz eth e ra nk -size ru le o f c iti e s

c rite rio n

str o n g w ea k s tr o n g w e ak

m ax . - m m . m a x . - m m . 0 .9 < all a 0 .7 5 < all a

o f a { 0 .2 o f a ( 0 .5 < 1 . 1 { 1 .2 5

c a te g o ry 1

c itie s ,

to w n s , a n d

v illa g e s

x l X

(n o te 1 )

X X

cate g o ry 2 citie sX o

(n o te 2 )

X X

ca te g o ry 3 ar e asX A

(n o te 3 )

X A

(n o te 4 )

ca teg o ry 4 p re fe ctur e s X X X X

"X" means "to be invalid for all or almost cases examined"."A"means "to be valid for almost cases examined"."O" means "to be valid for all cases examined".(note 1) The constant elasticity is valid for cases in 1975 and 1980 of samples 5-3000 of category 1.(note 2) As concerns each year, the constant elasticity is valid for all cases.

But, as concerns whole period 1975-95, it is valid except for a case with rank 5 in 1975.In fact, it is valid for samples from rank 6 to the bottom, as we can see throughAppended Table 2.

(note 3) The constant elasticity is valid for all cases with samples from 10 to the bottom.

To be exact, "rank 10" is able to be altered to "rank 8", as we can see throughAppended Table 3.

(note 4) The rule is valid for all cases with samples from rank 10 to the bottom.To be exact, "rank 10" is able to be altered to "rank 9", as we can see throughAppended Table 3.

-61-

REFERENCES

1) Beckman, M. and McPherson, J., "City Size Dis-tributions in a Central Place Hierarchy: An AlternativeApproach," Journal of Regional Science 10, pp. 25-33,1970.

2) Hotelling. H., "Stability in Competition," EconomicJournal, Vol. 39, pp. 41-57, 1929.

3) Mills, E. D. and Hamilton, B. W., Urban Econo-mics, Glenview, Scott, Foresman and Company, 1984.

4) Rosen, K. T. and Resnick, M., "The Size Distribu-tion of Cities : An Examination of the Pareto Law andPrimacy," Journal of Urban Economics 8, pp. 165-

186, 1980.5) Yoshimura, H. , "Urban Size and Serviee Industries",

Yamaguchi Journal of Economics, Business Aministra-tions & Laws, Vol. 36-112, pp. 1-40, 1986.

6) Yoshimura, H., "The Urban Factor of the Disparityof Wages among Cities", Gendai Keizaigaku no Ten-kai, Shunju-sha, pp. 303-315, 1987.

7) Yoshimura, H. , "A Measurement of AgglomerationEconomies", Yamaguchi Journal of Economics, Busi-ness Administrations & Laws, Vol. 37-3/4, pp. 59-98,1990.

8) Yoshimura, H., "Urban Size and New-service Indus-

tries", Yamaguchi Journal of Economics, BusinessAdministrations & Laws, Vol. 39-314, pp. 1-36, 1995.

9) Yoshimura, H., "Service Economies and the Con-centration of Economic Power to big City Areas", Stu-dies on Regional Economics (ffiroshima University),

Vol. 2, pp. 57-78, 1991.10) Yoshimura, H., "The Region and Size of Cities",

Studies on Regional Economics {Hiroshima University) ,Vol. 5, pp. 25-41, 1994.

ll) Yoshimura, H., "Economies of agglomeration inJapan," Yamaguchi Journal of Economics, Businessadministrations & Laws, Vol. 42-516, pp. 1-30, 1995.

12) Yoshimura, H., "On the Rank-Size Rule of Cities: Acase of Japan in 1990," Studies on Regional Economics(Hiroshima University), Vol. 6, pp. 37-42, 1995.

13) Yoshimura, H., "Agglomeration Economies andHouse Rent," Yamaguchi Journal of Economics, Busi-ness Administrations and Laws, Vol. 43-112, pp. 1-21, 1995.

14) Basic data of Residents (Jumin Kihon Daicho), theMinistry of Home Affairs, 1975, 1980, 1985, 1990,1995.

15) Power of People 1989 (Minryoku), Asahi Shinbun-sha, 1989.

(1996. 1.6.)

-62-

Appended Table 1(cities, towns and villages)

Relation between city's rank and city's size

s a m p l e

s iz e

1 9 7 5 ( S 5 0 ) 1 9 8 0 ( S 5 5 ) 1 9 8 5 ( S 6 0 ) 1 9 9 0 ( H O 2 ) 1 9 9 5 ( H O 7 )

e la s . ( a ) d e t e r . t - v a l u e e la s . ( a ) d e t e r . t - v a h i e e la s . a ) d e t e r . t - v a l u e e l a s . ( a ) d e t e r . t - v a lu e e l a s . ( a ) d e t e r . t - v a h i e

5 0 . 9 2 7 0 . 9 0 3 6 . 2 0 . 9 4 8 0 . 9 1 1 6 . 5 0 . 9 6 9 0 . 9 4 1 8 . 1 1 . 0 2 2 0 . 9 5 9 9 . 7 1 . 0 6 0 0 . 9 6 4 1 0 . 5

6 0 . 9 6 5 0 . 9 2 6 8 . 0 0 . 9 8 9 0 . 9 3 1 8 . 3 1 . 0 2 1 0 . 9 4 6 9 . 4 1 . 0 5 4 0 . 9 6 8 1 2 . 3 1 . 0 8 6 0 . 9 7 3 1 3 . 5

7 0 . 9 9 8 0 . 9 3 9 9 . 7 1 . 0 4 1 0 . 9 3 0 9 . 0 1 . 0 7 2 0 . 9 4 3 1 0 . 0 1 . 1 0 0 0 . 9 6 4 1 2 . 7 1 . 1 2 5 0 . 9 7 3 1 4 . 6

8 1 . 0 2 1 0 . 9 4 9 l l . 4 1 . 0 5 5 0 . 9 4 4 1 0 . 9 1 . 0 8 6 0 . 9 5 4 1 2 . 1 1 . 1 2 0 0 . 9 7 0 1 5 . 0 1 . 1 4 9 0 . 9 7 6 1 6 . 8

9 1 . 0 3 7 0 . 9 5 6 1 3 . 3 1 . 0 8 0 0 . 9 5 0 1 2 . 3 1 . 1 0 8 0 . 9 6 0 1 3 . 8 1 . 1 4 6 0 . 9 7 1 1 6 . 3 1 . 1 7 7 0 . 9 7 6 1 7 . 9

1 0 1 . 0 5 4 0 . 9 6 1 1 5 . 0 1 . 1 1 0 0 . 9 5 0 1 3 . 2 1 . 1 3 6 0 . 9 6 0 1 4 . 7 1 . 1 6 8 0 . 9 7 2 1 7 . 8 1 . 1 9 8 0 . 9 7 7 1 9 . 4

2 0 1 . 0 3 9 0 . 9 8 4 3 4 . 2 1 . 0 9 1 0 . 9 7 8 2 8 . 8 1 . 1 0 6 0 . 9 7 4 2 6 . 5 1 . 1 2 3 0 . 9 7 9 2 9 . 4 1 . 1 5 3 0 . 9 8 3 3 3 . 3

3 0 1 . 1 1 6 0 . 9 7 8 3 6 . 2 1 . 1 5 4 0 . 9 7 9 3 7 . 2 1 . 1 6 0 0 . 9 7 9 3 7 . 1 1 . 1 7 1 0 . 9 8 2 4 0 . 3 1 . 1 8 8 0 . 9 8 6 4 5 . 3

4 0 1 . 1 8 3 0 . 9 7 2 3 6 . 8 1 . 2 2 5 0 . 9 7 1 3 6 . 4 1 . 2 3 4 0 . 9 7 0 3 5 . 4 1 . 2 4 5 0 . 9 7 2 3 7 . 0 1 . 2 6 0 0 . 9 7 6 4 0 . 2

5 0 1 . 2 4 1 0 . 9 6 6 3 7 . 4 1 . 2 8 5 0 . 9 6 6 3 7 . 3 1 . 2 9 3 0 . 9 6 6 3 7 . 1 1 . 3 0 4 0 . 9 6 8 3 8 . 3 1 . 3 1 6 0 . 9 7 2 4 1 . 3

7 0 1 . 3 2 2 0 . 9 6 4 4 3 . 2 1 . 3 5 4 0 . 9 6 9 4 6 . 7 1 . 3 6 7 0 . 9 6 8 4 5 . 6 1 . 3 7 9 0 . 9 6 9 4 6 . 3 1 . 3 9 1 0 . 9 7 2 4 9 . 1

9 0 1 . 3 7 1 0 . 9 6 8 5 1 . 8 1 . 3 9 9 0 . 9 7 3 5 6 . 4 1 . 4 1 2 0 . 9 7 2 5 5 . 4 1 . 4 2 0 0 . 9 7 3 5 7 . 1 1 . 4 3 4 0 . 9 7 6 5 9 . 6

1 0 0 1 . 3 8 5 0 . 9 7 0 5 7 . 0 1 . 4 0 8 0 . 9 7 5 6 2 . 3 1 . 4 1 9 0 . 9 7 4 6 1 . 4 1 . 4 2 2 0 . 9 7 6 6 3 . 6 1 . 4 3 3 0 . 9 7 8 6 6 . 4

n o 1 . 3 9 2 0 . 9 7 3 6 2 . 9 1 . 4 0 5 0 . 9 7 8 6 9 . 0 1 . 4 1 2 0 . 9 7 7 6 7 . 7 1 . 4 1 8 0 . 9 7 8 7 0 . 4 1 . 4 2 9 0 . 9 8 0 7 3 . 4

1 2 0 1 . 3 9 2 0 . 9 7 6 6 8 . 9 1 . 4 0 3 0 . 9 8 0 7 5 . 7 1 . 4 0 5 0 . 9 7 9 7 3 . 9 1 . 4 0 9 0 . 9 8 0 7 6 . 5 1 . 4 1 6 0 . 9 8 1 7 8 . 9

1 3 0 1 . 3 8 3 0 . 9 7 7 7 4 . 4 1 . 3 9 3 0 . 9 8 1 8 1 . 3 1 . 3 9 4 0 . 9 8 0 7 9 . 1 1 . 3 9 5 0 . 9 8 1 8 1 . 3 1 . 4 0 0 0 . 9 8 2 8 3 . 3

1 4 0 1 . 3 7 3 0 . 9 7 8 7 9 . 4 1 . 3 8 1 0 . 9 8 2 8 6 . 0 1 . 3 7 6 0 . 9 8 0 S O 3s i . o 1 . 3 8 0 0 . 9 8 1 8 5 . 3 1 . 3 8 7 0 . 9 8 2 8 7 . 7

1 5 0 1 . 3 5 7 0 . 9 7 9 8 2 . 8 1 . 3 6 9 0 . 9 8 2 9 0 . 5 1 . 3 6 4 0 . 9 8 1 6 . 7 1 . 3 6 7 0 . 9 8 2 9 . 5 1 . 3 7 6 0 . 9 8 3 Q 9 7

2 0 0 1 . 3 0 2 0 . 9 8 1 1 0 0 . i 1 . 3 1 8 0 . 9 8 4 1 0 9 . 9 1 . 3 1 9 0 . 9 8 3 1 0 8 . 2 1 . 3 2 7 0 . 9 8 5 1 1 3 . 0 1 . 3 3 5 0 . 9 8 5 1 1 5 . 7

2 5 0 1 . 2 6 4 0 . 9 8 2 1 1 7 . 4 1 . 2 7 8 0 . 9 8 4 1 2 5 . 3 1 . 2 8 2 0 . 9 8 4 1 2 5 . 5 1 . 2 9 2 0 . 9 8 6 1 3 0 . 7 1 . 3 0 4 0 . 9 8 7 1 3 6 . 5

3 0 0 1 . 2 3 5 0 . 9 8 3 1 3 2 . 9 1 . 2 4 4 0 . 9 8 5 1 3 8 . 0 1 . 2 5 0 0 . 9 8 5 1 3 8 . 7 1 . 2 6 0 0 . 9 8 6 1 4 4 . 1 1 . 2 7 5 0 . 9 8 7 1 5 1 . 3

4 0 0 1 . 2 1 4 0 . 9 8 7 1 7 3 . 8 1 . 2 1 9 0 . 9 8 7 1 7 7 . 4 1 . 2 1 6 0 . 9 8 7 1 7 3 . 6 1 . 2 2 0 0 . 9 8 7 1 7 4 . 6 1 . 2 3 1 0 . 9 8 8 1 7 9 . 4

5 0 0 1 . 2 0 0 0 . 9 8 9 2 1 3 . 3 1 . 1 9 8 0 . 9 8 9 2 1 2 . 2 1 . 1 9 5 0 . 9 8 9 2 0 7 . 5 1 . 1 9 5 0 . 9 8 8 2 0 5 . 6 1 . 2 0 2 0 . 9 8 8 2 0 6 . 2

w o 1 . 1 8 5 0 . 9 9 0 2 4 8 . 0 1 . 1 7 8 0 . 9 9 0 2 4 0 . 7 1 . 1 7 2 0 . 9 8 9 2 3 2 . 0 1 . 1 6 8 0 . 9 8 8 2 2 6 . 2 1 . 1 7 2 0 . 9 8 8 2 2 1 . 3

7 0 0 1 . 1 6 7 0 . 9 9 1 2 7 1 . 6 1 . 1 5 9 0 . 9 9 0 2 6 3 . 7 1 . 1 5 1 0 . 9 8 9 2 5 1 . 9 1 . 1 4 7 0 . 9 8 8 2 4 5 . 1 1 . 1 4 8 0 . 9 8 8 2 3 8 . 0

1 0 0 0 1 . 1 1 5 0 . 9 9 0 3 0 9 . 2 1 . 1 0 7 0 . 9 8 9 3 0 1 . 7 1 . 0 9 8 0 . 9 8 8 2 9 0 . f 1 . 0 9 1 0 . 9 8 8 2 8 1 . 8 1 . 0 8 7 0 . 9 8 6 2 6 8 . 4

1 5 0 0 1 . 0 7 0 0 . 9 9 1 3 9 5 . 4 1 . 0 5 5 0 . 9 8 9 3 6 9 . 1 1 . 0 4 2 0 . 9 8 8 3 5 1 . 3 1 . 0 2 9 0 . 9 8 6 3 3 0 . 2 1 . 0 1 8 0 . 9 8 5 3 0 9 . 1

2 0 0 0 1 . 0 4 6 0 . 9 9 2 4 8 4 . 7 1 . 0 2 3 0 . 9 9 0 4 3 6 . 4 1 . 0 0 6 0 . 9 8 8 4 0 6 . 8 0 . 9 8 8 0 . 9 8 6 3 7 7 . 6 0 . 9 7 3 0 . 9 8 4 3 4 7 . 6

2 3 0 0 1 . 0 3 2 0 . 9 9 2 5 2 2 . 2 1 . 0 0 5 0 . 9 8 9 4 6 0 . i 0 . 9 8 6 0 . 9 8 8 4 2 7 . 3 0 . 9 6 7 0 . 9 8 6 3 9 6 . 0 0 . 9 4 8 0 . 9 8 3 3 6 1 . 7

2 5 0 0 1 . 0 2 1 0 . 9 9 1 5 3 4 . 9 0 . 9 9 0 0 . 9 8 8 4 6 1 . 8 0 . 9 7 0 0 . 9 8 6 4 2 6 . 0 0 . 9 5 0 0 . 9 8 4 3 9 4 . 8 0 . 9 3 0 0 . 9 8 1 3 6 1 . 0

2 8 0 0 0 . 9 9 8 0 . 9 8 9 4 9 9 . 6 0 . 9 6 4 0 . 9 8 5 4 3 3 . 8 0 . 9 4 1 0 . 9 8 3 4 0 0 . ( 0 . 9 1 9 0 . 9 8 0 3 7 2 . 2 0 . 8 9 7 0 . 9 7 6 3 4 0 . 2

3 0 0 0 0 . 9 6 8 0 . 9 8 2 4 0 8 . 9 0 . 9 3 2 0 . 9 7 8 3 6 4 . 8 0 . 9 0 8 0 . 9 7 5 3 4 1 . 8 0 . 8 8 5 0 . 9 7 2 3 2 0 . 1 0 . 8 6 1 0 . 9 6 7 2 9 5 . 5

3 1 0 0 0 . 9 4 1 0 . 9 7 4 3 3 8 . 0 0 . 9 0 5 0 . 9 6 8 3 0 8 . 5 0 . 8 8 1 0 . 9 6 5 2 9 3 . 0 0 . 8 5 8 0 . 9 6 2 2 7 9 . 0 0 . 8 3 3 0 . 9 5 6 2 6 0 . 6

3 2 0 0 0 . 8 9 8 0 . 9 5 5 2 5 9 . 2 0 . 8 6 1 0 . 9 4 9 2 4 2 . 8 0 . 8 3 9 0 . 9 4 5 2 3 5 . 3 0 . 8 1 7 0 . 9 4 2 2 2 7 . 6 0 . 7 9 0 0 . 9 3 5 2 1 4 . 3

a l l 0 . 8 5 3 0 . 9 2 6 2 0 2 . 2 0 . 8 1 8 0 . 9 2 0 1 9 3 . 6 0 . 7 9 7 0 . 9 1 8 1 9 0 . 7 0 . 7 7 6 0 . 9 1 5 1 8 6 . 7 0 . 7 5 8 0 . 9 1 3 1 8 3 . 8

city's size (S) : the number of population in cities, towns and villages

city's rank (R) : the ranking of the city's sizeregression equation: Y= -aX+b, where X=log S, Y=log R, b=log A, (R=AS"a)elas. (a) : the elasticity of city's rank to city's size: Pareto exponent

: the absolute value of the coefficient of X in the regression equationdeter. : the coefficient of determinnation adjusted for the degree of freedomsample size: the number of samples (cities, towns and villages) taken in order of city's rank

all: all samples: all of cities, towns and villages, including Tokyo-special-ward: 3244 samples in 1975 and '80, 3245 samples in '85 and '90, and 3235 samples in '95

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Appended Table 2 Relation between city's rank and city's size(cities)

s a m p l e

s iz e

1 9 7 5 ( S 5 0 ) 1 9 8 0 ( S 5 5 ) 1 9 8 5 ( S 6 0 ) 1 9 9 0 < H O 2 ) 1 9 9 5 ( H O 7 )

e la s . ( a ) d e t e r . t - v a i u e e l a s . ( a ) d e t e r . t - v a l u e e la s . ( a ) d e t e r . t - v a l u e e l a s . ( a ) d e t e r . t - v a lu e e l a s . ( a ) d e t e r . t - v a l u e

5

6

1 5 0

C o n t e n t s in t h i s p la c e a r e t h e s a m e a s A p p e n d e d T a b le 1 .

2 0 0 1 . 3 0 2 0 . 9 8 1 1 0 0 . i 1 . 3 1 1 0 . 9 8 3 1 0 8 . 1 1 . 3 1 9 0 . 9 8 3 1 0 8 . 2 1 . 2 6 0 0 . 9 8 6 1 4 4 . 1 1 . 3 3 5 0 . 9 8 5 1 1 5 . 7

3 0 0 1 . 2 3 5 0 . 9 8 3 1 3 2 . 9 1 . 2 3 9 0 . 9 8 4 1 3 7 . 5 1 . 2 5 0 0 . 9 8 5 1 3 8 . 7 1 . 2 6 0 0 . 9 8 6 1 4 4 . 1 1 . 2 7 5 0 . 9 8 7 1 5 1 . 3

3 5 0 1 . 2 2 2 0 . 9 8 5 1 5 2 . 9 1 . 2 2 6 0 . 9 8 6 1 5 7 . 7 1 . 2 3 0 0 . 9 8 6 1 5 5 . 5 1 . 2 3 5 0 . 9 8 6 1 5 7 . 3 1 . 2 4 8 0 . 9 8 7 1 6 3 . 0

4 0 0 1 . 2 1 4 0 . 9 8 7 1 7 3 . 8 1 . 2 1 6 0 . 9 8 8 1 7 8 . 0 1 . 2 1 6 0 . 9 8 7 1 7 3 . 6 1 . 2 1 9 0 . 9 8 7 1 7 4 . 2 1 . 2 3 1 0 . 9 8 8 1 7 9 . 3

4 5 0 1 . 2 0 7 0 . 9 8 8 1 9 4 . 3 1 . 2 0 6 0 . 9 8 9 1 9 7 . 3 1 . 2 0 7 0 . 9 8 8 1 9 2 . 4 1 . 2 0 5 0 . 9 8 8 1 8 9 . 4 1 . 2 1 6 0 . 9 8 8 1 9 3 . 9

5 0 0 1 . 2 0 0 0 . 9 8 9 2 1 3 . 3 1 . 1 9 5 0 . 9 8 9 2 1 2 . 9 1 . 1 9 1 0 . 9 8 8 2 0 3 . 1 1 . 1 8 5 0 . 9 8 7 1 9 5 . 0 1 . 1 9 4 0 . 9 8 7 1 9 6 . 0

5 5 0 1 . 1 9 2 0 . 9 9 0 2 3 1 . 3 1 . 1 8 2 0 . 9 8 9 2 2 4 . 3 1 . 1 7 3 0 . 9 8 7 2 0 7 . 8 1 . 1 6 1 0 . 9 8 5 1 9 3 . 2 1 . 1 6 7 0 . 9 8 5 1 8 9 . 1

6 0 0 1 . 1 8 3 0 . 9 9 0 2 4 5 . 6 1 . 1 6 6 0 . 9 8 9 2 2 7 . 7 1 . 1 5 1 0 . 9 8 6 2 0 5 . 0 1 . 1 3 5 0 . 9 8 3 1 8 5 . 9 1 . 1 3 7 0 . 9 8 2 1 7 8 . 7

6 1 0 1 . 1 8 1 0 . 9 9 0 2 4 7 . 7 1 . 1 6 2 0 . 9 8 8 2 2 7 . 4 1 . 1 4 7 0 . 9 8 6 2 0 3 . 6 1 . 1 2 8 0 . 9 8 2 1 8 2 . 8 1 . 1 3 0 0 . 9 8 0 1 7 4 . 4

6 2 0 1 . 1 7 8 0 . 9 9 0 2 4 8 . 7 1 . 1 5 8 0 . 9 8 8 2 2 5 . 9 1 . 1 4 1 0 . 9 8 5 2 0 0 . 8 1 . 1 2 2 0 . 9 8 1 1 7 9 . 2 1 . 1 2 2 0 . 9 7 9 1 7 0 . 3

6 3 0 1 . 1 7 4 0 . 9 9 0 2 4 7 . 1 1 . 1 5 2 0 . 9 8 7 2 2 1 . 4 1 . 1 3 5 0 . 9 8 4 1 9 6 . 7 1 . 1 1 4 0 . 9 8 0 1 7 4 . 5 1 . 1 1 3 0 . 9 7 8 1 6 5 . 4

6 4 0 1 . 1 7 0 0 . 9 8 9 2 4 3 . 0 1 . 1 4 6 0 . 9 8 6 2 1 4 . 0 1 . 1 2 7 0 . 9 8 3 1 9 0 . 2 1 . 1 0 5 0 . 9 7 8 1 6 8 . 7 1 . 1 0 4 0 . 9 7 6 1 5 9 . 9

6 5 0 1 . 1 6 3 0 . 9 8 8 2 3 4 . 6 1 . 1 3 7 0 . 9 8 4 2 0 2 . * 1 . 1 1 7 0 . 9 8 0 1 8 0 . 1 1 . 0 9 3 0 . 9 7 5 1 5 9 . 1 1 . 0 9 2 0 . 9 7 3 1 5 2 . 8

* 6 5 5

6 6 0

* 6 6 4

1 . 1 5 4 0 . 9 8 5 2 0 6 . 7 1 . 1 2 0 0 . 9 7 7 1 6 8 . 2 1 . 1 0 4 0 . 9 7 5 1 6 1 . 0 1 . 0 8 4 0 . 9 7 2 1 5 0 . 7 1 . 0 8 5

1 . 0 7 7

1 . 0 6 3

0 . 9 7 1

0 . 9 6 9

0 . 9 6 2

1 4 8 . 0

1 4 2 . 6

1 2 9 . 6

city's size (S) : the number of population in a citycity's rank (R) : the ranking of the city's sizeregression equation: Y=-aX+b, where X=log S, Y=log R, b-log A, (R=AS"ielas. (a) : the elasticity of city's rank to city's size: Pareto exponent

: the absolute value of the coefficient of X in the regression equationdeter.: the coefficient of determination adjusted for the degree of freedomsample size: the number of samples (cities) taken in order of city's rank

*: all samples: all of cities, including Tokyo-special-ward: 655 samples in 1975, '80 and '90, 656 samples in '85, and 664 samples in '95

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(areas)Appended Table 3 Relation between city's rank and city's size

s a m p le

siz e

1 9 7 5

Appended Table 4 Relation between city's rank and city's size(prefectures)

s a m p le

s iz e

1 9 7 5 ( S 5 0 ) 1 9 8 0 ( S 5 5 ) 1 9 8 5 ( S 6 0 ) 1 9 9 0 ( H O 2 ) 1 9 9 5 ( H O 7 )

e la s . ( a ) d e t e r . t - v a l u e e l a s . ( a ) d e t e r . t - v a l u e e la s . ( a ] d e te r . t - v a l u e e la s . a ) d e t e r . t - v a lu e e la s . ( a ) d e t e r . t - v a l u e

5 1 . 8 9 4 0 . 9 5 8 9 . 6 2 . 1 9 0 0 . 9 9 6 3 0 . 4 2 . 3 2 8 0 . 9 9 8 4 2 . 5 2 . 5 6 9 0 . 9 8 1 1 4 . 3 2 . 7 9 6 0 . 9 5 5 9 . 3

6 2 . 0 0 2 0 . 9 5 4 1 0 . 2 2 . 2 8 0 0 . 9 8 9 2 1 . 4 2 . 4 1 8 0 . 9 9 2 2 4 . 5 2 . 5 2 1 0 . 9 8 6 1 8 . 5 2 . 6 5 6 0 . 9 6 3 l l . 4

7 2 . 0 9 4 0 . 9 5 2 l l . 0 2 . 3 5 8 0 . 9 8 5 2 0 . 0 2 . 4 4 8 0 . 9 9 3 ｣ 3 . 0 2 . 5 5 4 0 . 9 8 8 9 9 7 . 6 8 2 0 . 9 7 2 1 4 . 4

8 2 . 1 2 2 0 . 9 6 1 1 3 . 2 2 . 3 7 6 0 . 9 8 8 2 4 . 1 2 . 4 9 9 0 . 9 9 2 ? Q 9 2 . 6 0 6 0 . 9 8 8 2 4 . 4 2 . 7 0 7 0 . 9 7 7 1 7 . 3

9 2 . 1 5 2 0 . 9 6 7 1 5 . 4 2 . 4 2 6 0 . 9 8 7 2 4 . 5 2 . 4 8 7 0 . 9 9 3 3 4 . 2 2 . 5 4 7 0 . 9 8 8 2 5 . 7 2 . 6 2 4 0 . 9 7 7 1 8 . 5

1 0 2 . 0 3 5 0 . 9 6 0 1 4 . 8 2 . 1 5 9 0 . 9 5 1 1 3 . 2 2 . 1 7 1 0 . 9 4 6 1 2 . 6 2 . 1 8 8 0 . 9 3 3 l l . 3 2 . 2 1 1 0 . 9 1 5 9 . 9

l l 1 . 8 0 5 0 . 9 2 5 l l . 1 1 . 7 8 7 0 . 8 8 1 8 . 7 1 . 8 0 8 0 . 8 8 1 8 . 7 1 . 7 8 9 0 . 8 6 2 8 . 0 1 . 8 1 3 0 . 8 4 9 7 . 6

1 2 1 . 6 6 3 0 . 9 1 1 1 0 . 6 1 . 6 4 0 0 . 8 7 3 8 . 7 1 . 6 6 6 0 . 8 7 4 8 . 8 1 . 6 5 8 0 . 8 6 0 8 . 3 1 . 6 6 3 0 . 8 4 5 7 . 8

1 3 1 . 6 0 5 0 . 9 1 5 l l . 4 1 . 5 7 3 0 . 8 7 9 9 . 4 1 . 5 8 2 0 . 8 7 7 9 . 3 1 . 5 5 3 0 . 8 6 0 8 . 6 1 . 5 4 3 0 . 8 4 4 8 . 1

1 4 1 . 5 7 6 0 . 9 2 1 1 2 . 4 1 . 5 3 9 0 . 8 8 9 1 0 . 2 1 . 5 3 4 0 . 8 8 5 1 0 . 0 1 . 4 9 7 0 . 8 6 8 9 . 3 1 . 4 8 4 0 . 8 5 3 8 . 7

1 5 1 . 5 1 9 0 . 9 2 1 1 2 . 8 1 . 4 7 4 0 . 8 9 0 1 0 . 7 1 . 4 6 8 0 . 8 8 6 1 0 . 5 1 . 4 4 1 0 . 8 7 3 9 . 9 1 . 4 3 4 0 . 8 6 1 9 . 4

1 6 1 . 4 8 7 0 . 9 2 5 1 3 . 6 1 . 4 3 9 0 . 8 9 6 l l . 4 1 . 4 3 4 0 . 8 9 2 l l . 2 1 . 4 0 7 0 . 8 8 1 1 0 . 6 1 . 3 9 9 0 . 8 6 9 1 0 . 0

1 7 1 . 4 6 7 0 . 9 2 9 1 4 . 5 1 . 4 1 2 0 . 9 0 2 1 2 . 2 1 . 4 1 4 0 . 9 0 0 1 2 . 0 1 . 3 8 7 0 . 8 8 9 l l . 4 1 . 3 7 8 0 . 8 7 8 1 0 . 8

1 8 1 . 4 5 2 0 . 9 3 4 1 5 . 5 1 . 3 9 6 0 . 9 0 8 1 3 . 0 1 . 4 0 2 0 . 9 0 7 1 2 . 9 1 . 3 7 6 0 . 8 9 7 1 2 . 2 1 . 3 6 8 0 . 8 8 7 l l . 6

1 9 1 . 4 4 5 0 . 9 3 8 1 6 . 6 1 . 3 8 9 0 . 9 1 5 1 3 . 9 1 . 3 8 9 0 . 9 1 3 1 3 . 7 1 . 3 6 6 0 . 9 0 4 1 3 . 0 1 . 3 5 8 0 . 8 9 5 1 2 . 4

2 0 1 . 4 3 9 0 . 9 4 2 1 7 . 6 1 . 3 8 7 0 . 9 2 0 1 4 . 8 1 . 3 8 5 0 . 9 1 8 1 4 . 7 1 . 3 6 1 0 . 9 1 0 1 3 . 9 1 . 3 5 4 0 . 9 0 2 1 3 . 2

:' S 1 . 4 4 5 0 . 9 5 7 2 3 . 1 1 . 4 0 5 0 . 9 4 0 1 9 . 4 1 . 3 8 3 0 . 9 3 9 1 9 . 3 1 . 3 5 7 0 . 9 3 3 1 8 . 2 1 . 3 4 4 0 . 9 2 6 1 7 . 4

:; o 1 . 4 3 8 0 . 9 6 5 2 8 . 3 1 . 3 9 5 0 . 9 5 1 2 3 . 8 1 . 3 6 6 0 . 9 5 1 2 3 . 6 1 . 3 3 9 0 . 9 4 6 2 2 . S 1 . 3 2 4 0 . 9 4 1 2 1 . 5

3 5 1 . 3 8 6 0 . 9 6 6 3 1 . 2 1 . 3 5 6 0 . 9 5 7 2 7 . 6 1 . 3 3 4 0 . 9 5 7 2 7 . 6 1 . 3 1 1 0 . 9 5 4 2 6 . 4 1 . 2 9 9 0 . 9 5 0 2 5 . 3

4 0 1 . 3 5 9 0 . 9 6 9 3 5 . 1 1 . 3 3 7 0 . 9 6 3 3 1 . 7 1 . 3 1 7 0 . 9 6 3 3 1 . 8 1 . 2 9 5 0 . 9 6 0 3 0 . 5 1 . 2 8 3 0 . 9 5 6 2 9 . 3

4 5 1 . 3 0 0 0 . 9 6 5 3 5 . 1 1 . 2 7 8 0 . 9 5 9 3 2 . 2 1 . 2 5 6 0 . 9 5 9 3 2 . 0 1 . 2 3 6 0 . 9 5 7 3 1 . 1 1 . 2 2 4 0 . 9 5 4 3 0 . 1

4 7 1 . 2 6 5 0 . 9 5 9 3 2 . 9 1 . 2 4 3 0 . 9 5 3 3 0 . 7 1 9 9 91 . u u u 0 . 9 5 3 3 0 . 7 1 . 2 0 3 0 . 9 5 1 3 0 . 0 1 . 1 9 0 0 . 9 4 8 2 9 . 1

city's size (S) : the number of population in a prefecturecity's rank (R) : the ranking of the city's sizeregression equation: Y=-aX+b, where X=log S, Y=log R, b=log A, (R=AS~

elas. (a) : the elasticity of city's rank to city's size: Pareto exponent: the absolute value of the coefficient of X in the regression equation

deter. : the coefficient of determination adjusted for the degree of freedomsample size: the number of samples (prefectures) taken in order of city's rank

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urban structure and rank-size rule of cities · the rank-size rule means that

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