都市人口の分布の型とその発生機構について : 都市人口密度への対数正規分布のあてはめ

概要

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Many discussions on the regularity of the population of cities have been done. R. Gibrat (1931) showed graphically that the distribution of population of cities approximately lognormal distribution. G. K. Zipf (1941) found that the relation between the size P and the rank R of population of cities is expressed by the equation (1. 1)or (1. 2), where A and a are parameters. And W. Christaller discussed on the hierarchical structure of the cities and insisted that the number of the cities which belong to the ith class of cities is K/3^i where K is a constant, and the size (Population) of the cities of the ith class is 3^<i-1>. Moreover, in 1958, M. Beckmann analysed the construction of the populations in urban and rural areas by a model, and obtained the population of cities which is shown by equation (1.5), where p_m is the population of cities which belong to the mth class counted from the class of smallest size of cities, and k, s and r are constants. On the other hand, B. J. L. Berry insisted that when the distribution of population of cities is truncated lognormal distribution, we can get the Zipf's rank size rule. Berry and W. L. Garrison showed that the Christaller's law and Zipf's rank-size rule are substantially equal to each other. And John B. Parr discussed on the relation between Beckmann's model and Zipf s rank size rule adding his comment to the Beckmann's model. In this paper, the present author discussed on the type of distribution of population of cities using the Japanese data of 1968. If the mechanism of growing of population of cities is expressed by the equation (2.1) where P_K is the population of cities at the beginning of the kth period of time and ΔPk is the increment of the population of cities during the kth period (k=0,1,2……,n), then the distribution of population of cities would be normal distribution. But, we could not find the normal distribution of population of cities in Japanese data, as shown in Figure 1. The distribution is skewed very strongly to right. If the mechanism is expressed by the equation (2.4) where p_i is the increase rate of population of cities during the ith period (i=1,2,……, n),the distribution of logarithm of population of cities would be normal distribution, in other words, the distribution would be lognormal distribution, as suggessed by Gibrat. But, unfortunately the actual distribution of population of cities in Japan was not expressed by a lognormal distribution as shown by Figure 2. Incidentally, the Zipf's rank-size rule was found in Japanese data, as shown in Figure 3. Moreover, we could also find the rule in two parts of Japan, the East and West Japan which are defined in Figure 4 and 5. Therefore, we can get the conclusion that "if the distribution of logarithm of population of cities is unimodal distribution (Figure 2, 7, and 8), we would be able to find the Zipf's rank-size rule". This means that we can get the Zipf's rank-size rule under the more relaxed condition than Berry's one according to which the distribution of population must be truncated lognormal distribution. Although the Zipf s rank-size rule was successfully applied to the distribution of population of cities in Japan, the mechanism of growing of population of cities are not explained so clearly. As a matter of fact, the Mandelbrot's model and Miller's model for the existence of the Zipf's rank-size rule are not adequate for the explanation of the existence of the rule in population of cities. And the Simon's model is also inadequate, because the frequency distribution of logarithm of the population of cities should be J-shaped distribution according to the Simon's model, while the actual distribution was not a J-shaped distribution but a unimodal distribution. Therefore, the present author constructed the model of changing of population density of cities expressed by the equation (4.6) instead of constructing the model of growing of population of cities itself. In equation (4.6) D_k is the population density of a city at the beginning of the kth period (k=0,1,2,……, n), A_k is the area of a city at the begining of the kth peiod (k=0,1,2,……, n) P_i is the increase rate of the population of a city during the ith period(i=1,2,……, n) and at is the increase rate of the area of a city during the ith period. According to this model, the distribution of population density of cities should become lognormol distribution by the central limit theorem. However, the actual distribution of population density of cities in Japan was not lognormal distribution (Figure 9). But, fortunately, when we divided the cities of all Japan into 4 groups ; (1)the cities whose populations are less than 50,000 in the East Japan, (2)the cities whose populations are 50,000 and over in the East Japan, (3)the cities whose populations are less than 50,000 in the West Japan, and (4)the cities whose population are 50,000 in the West Japan, we could find the fact that the lognormal distribution was successfully applied to each distribution of the 4 groups of population density of cities (Figure 14-17). This fact shows that the mechanism of the changing of population density of cities of the 4 groups is explained by the equation (4.6) which shows that the population of a city at the beginning of the nth period P_n will be expressed by P_oПp_i and the area of a city at the begining of the nth period A_n will be expressed by A_o Пa_i. It would be very interesting that the mechanism of changing of population density of cities is expressed by the central limit theorem, a general principle in statistics. After it is explained that the distribution of population density of cities of the 4 groups in Japan, in the last part of this paper, the reconstruction of the distribution of population density of all Japan was tried by summing up all the theoretical frequency distributions of population density of the 4 groups of cities (Figure 19). This compound theoretical distribution was successfully applied to the actual distribution Figure 21). This fact is also interesting from the statistical point of view.
流通経済大学の論文

著者

鈴木啓祐
流通経済大学

都市人口の分布の型とその発生機構について : 都市人口密度への対数正規分布のあてはめ

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