Predicting the Growth Pattern of Cities in India : Application of Statistical Models

Several probabilistic and deterministic (including economic) models have been proposed to predict the growth and pattern of distribution of cities by their population size. Among them Pareto family of distributions have shown a remarkable empirical regularity and the city size distribution in many countries is well approximated by these distributions. However, these models are useful to study the distribution of larger size cities satisfying certain regularity conditions. In the Indian context also Pareto distribution fits well only to the distribution of cities with a population size of million plus. No such exercise has been made for other cities which are populated less than a million. In the present paper, an attempt has been made to characterize cities of all sizes through appropriate models applied to predict the growth pattern of cities in Indian context.


Introduction
The study of growth and distribution of cities by their size has gained an increasing importance in the characterization of urban process.Several probabilistic and deterministic (including economic) models have been proposed to study the distribution of cities by their size.Among them, Zipf law, Pareto distribution, Gibrat's law and Rank Size Rule, all belonging to a single family, have shown a remarkable empirical regularity.In many countries, the city size distribution is well approximated by these laws of distributions.However, the empirical evidences show that the applications of these models are restricted to study the distribution of cities with population size more than a million, satisfying certain regularity conditions.No effort has been made to study the cities with population less than a million.

Concepts and Defintions
It is important for the data users to familiarize themselves with the concepts and the definitions of the terms used for proper evaluation of the data contained in this publication.At the same time, it is all the more important to understand the implications of the terms used at die Census of India 2001, for making meaningful comparisons of the similar data generated by various other agencies within the country and with the data produced by other countries in the world.The concepts and definitions adopted at die Census of India, 2001 (Registrar General andCensus Commissioner India, 2001) are as given below:

Rural-Urban Areas
The final population data are presented separately for rural and urban areas.The unit of classification in this regard is 'town' for urban areas and 'village' for rural areas.In the Census of India 2001, die definition of urban area is defined as follows: a) All places with a municipality, corporation, cantonment board or notified town area committee, etc., (Called Statutory Towns).

A. S. Kadi andB. I. Halingali
b) A place which satisfies the following three criteria simultaneously (Called Census Towns): i) a minimum population of 5,000 ii) at least 75 per cent of male working population engaged in non-agricultural pursuits; and iii) a density of population of at least 400 persons per sq.km (1,000 per sq.mile).
For identification of places, which would qualify to be classified as 'urban', all villages, which, as per the 1991 census had a population of 4,000 and above, a population density of400 persons per sq.km. and having at least 75 per cent of male working population engaged in non-agricultural activity, were considered.
To work out the proportion of male working population referred to above against b) (ii), the data relating to main workers were taken into account.
Apart from these, the outgrowths (OOs) of cities and towns have also been treated as urban under 'Urban Agglomerations': examples of out-growths are railway colonies, university campuses, port areas, military camps, etc., that may have come up near a statutory town or city but within the revenue limits of a village or villages contiguous to town or city.Each such individual area by itself may not satisfy the demographic criteria laid down at (b) above to qualify it to be treated as mi independent urban unit, but may deserve to be clubbed with the towns as a continuous urban spread.Thus, die town level data, wherever presented, includes the data for outgrowths separately of such towns and also of town plus outgrowth (s) population.

Classification of Cities
Similar to 1991 Indian Census and in 2001 Censuses also the urban units have been classified into the following six categories based on the size of the towns.
Towns with population of 100,000 and above are called Cities, and towns with a population size one million and above are called Primate Cities.In the Indian context, the census data on number of cities and growth pattern classified into six categories are based on their size.The growth and pattern of distribution of cities differ with categories.Hence this leads to formulate different models to describe their growth pattern and their prediction.Before taking up this problem of formulation of models for analysis, we shall consider a brief account of growth in a number of cities and their size in India from 1901 to 2001.

Growth and Distribution of Cities
Before 1950, that is, before the Independence of India, the size and the growth rate of the urban population and the transfer of individuals from rural to urban areas were found to be very slow (Mohan, 1985;Moonis Raza et al., 1981).Table 1 gives the number of towns in each census during 1901-2001.The table shows an increasing trend in the number of towns and total urban population living in these towns from 1921 to 2001.We find fluctations in the number of towns from 1961 onwards.This is because of the declassification of some towns and the addition of some other (Bose, 1978, Megeri, 2002).From Table 1, we find that, in spite of discontinuity in the growth of total population at some points during 1901-2001, the size of the urban population increased continuously from 25.85 millions to 285.35 millions.
In Indian Census, the urban units have been classified into six categories-Class-I to Class-VI based on the size of the towns (For details of classification and their demerits see Source of Table 2 and Nanda 1991).In 1901, about 1917 cities/towns accommodated all urban dwellers and then in 2001, the number of cities increased to 5151.In 1901, out of 1917 different city size classes, about 1701 belonged to the city size of class IV, V and VI, which contained 90 percent of the total urban population.During die last 50 years, hardly 1000 cities were added at the rate of 20 cities per annum to the number of 1901, and not much variation was found in the percentage share of urban population of each city size category.Until 1951, small towns of category IV, V and VI were found to be large in number.In 1951, the number of cities of these categories altogether was found to be 2499; out of total 3060 cities of all categories, residence facility is given to 18 million urban populations, covering only 29 percent of total urban population.Most of these towns cater such service as materials for their daily need, schooling, health, market for their products, etc., to the surrounding villages.It is surprising to note that the trend of increase in the number of cities   1901-2001, New Delhi On the other hand, the number of cities of smaller size IV, V and VI classes and their percentage share of urban population have shown a steady decline as a result of promotion of these cities to the higher categories, while cities in other size classes have shown a greater tendency of increase in their number as well as in their population size and percentage share of urban population (See Tables 2 and 3).For instance, the number of cities and towns of class-VI size category increased from 503 to 629 during 1901 and 1951, while during 1951 and 2001, the number decreased to 227.More or less, a similar trend was noticed with respect to towns of class-V size.On die other hand, die number of class-I cities have shown a steady and slow increase in their number from 25 with 1.33 percentage share of urban population to 49 in number with 2.02 percent urban population between 1901 and 1941.By 2001, their number had increased to 423, accounting for 138.8 million urban populations, which formed 62 percent of the urban population.Similar pattern has been observed with Class II, III and IV cities/towns (See Table 2 and Table 3 In India, the process of urbanization has been completely governed by class-I cities.This clearly indicates the spatial direction of developmental process that is taking place over the decades.Class-I cities alone account for 57 (See Table 4) percent of the total urban population in 1991.Thus, class-I cities are growing at the cost of other size classes, and their growth is much faster than other cities and towns in the country.Most of the development that has taken place in India is almost strengthening the urban economy, and is concentrated largely in cities of larger size, specifically in Metropolitan cities with a population size of more than a million.In 1901, there was only one such city, namely, Calcutta and in 1911 one more city Mumbai (Its Earlier name was Bombay) had been added.Till 1941 only these two were the cities of this category.In 1981, their number increased to 12, and by 1991 the number of such cities increased to 23, accounting for one thirds of the country's urban population, and 8.33 percent of total National population.The exponential growth rates of these cities for four successive decades are given in Table 2.
During 1901-2001, irrespective of the state of the country, cities under Categories-1 to IV have shown consistent increase in their number, whereas in the other last two categories, cities in category-V have increased in number after greater set back during the period 1941-51, owing to displacement of population from Pakistan at the time of independence and reorganization of state boundaries, and application of rural-urban concept during 1951 census and subsequent modifications in the defmitions have affected the number of cities, specifically in Category-V and VI, as a result of which there was a break in their trend.But cities of Category-I, II and III have shown slow but steady increase in their number during the period from 1901 to 1941, and in the later period, a large number of such cities have come up as a result of upward transition of cities from lower category to upper categories.
Figure 1 clearly shows how city size and growth rate had touched ground level after 1941 census.Cities of other categories, except class-I, have shown continuous fall in their growth rate.Hence after the independence, the process of urbanization has been completely governed by class-I cities. Class-I cities alone account for 62 percent of the total urban population in 2001.Thus, class-I cities are growing at the cost of other size classes, and their growth is much faster than that of other cities mid towns in the country.
Another important major change noticed is that gradual shift in the number of cities from lower size categories to cities of categories in lager size.More number of cities of size class-VI, class-V and class-IV have been transferred to the next higher size categories.Some cities have jumped more than two steps due to rapid growth of their population.This phenomenon was also true in cities of other size categories (See Tables 2 and 3).Predicting the Growth Pattern of Cities in India; Application of Statistical Models of higher order have increased in number at the cost of cities of lower category, and cities of smaller size, specifically class-VI cities, have consistently declined in number (See Table 4).However, the pattern of increase in die number of class-1 cities and their population increases over a decade at a greater speed, while the percentage of urban population in cities of other categories shows moderate increase during the period from 1901 to 1941, and during the later period, cities of other categories have shown, more or less, a similar pattern of decline in the percentage of urban population.Again, among these, cities of class-VI size have shown greater fall in their number and percentage of urban population after 1941, which may be due to greater number of transition or out migration to cities of other categories.To describe this differential pattern in the increase or decrease in the size or number of cities in different categories, different models are needed.

City Growth Models
Formulation of models to study city growth and dynamics is an important area of research.Considerable literature has been accumulated to generalize the growth of cities, but so far no success has been made to predict the city growth process (Gibrat, 1931;Zipf, 1949;Champemowne, 1953;Simon, 1955;Rosen and Resnick, 1980;Henderson, 1982;Mills and Becher, 1986;Gabaix, 1999).Gibrat Law (1931) is the earliest literature which states that, for a fixed number of cities, over a period of time, their size grow stochastically with common mean and variance, the growth rate equal to mean city size growth rate of cities of different sizes.
Zipf model (1949) for cities is another most conspicuous fact having a strong empirical support, and it constitutes a minimum criterion of admissibility for any model of local growth, and is described as: the probability that the size-S of a city, greater than some fixed size of population of a city x and is proportional to 1/x, P (S > x) = A/x" with A as constant.
that is, in all empirical findings Zipf uses a =1 as a bound for the distribution and it leads to rank size distribution (See Lange, 1962).A remarkable empirical regularity is that the rank size distribution is well approximated by the Pareto Distribution (1897), and is given by the distribution function: F(x) = 1-(s/x)a, s>0, a>0 and x> 0 (1) where F(x) = P (X < x) = P (rank of a city < x) On log scale cumulated pareto probabilities (proportions) show a linear model implying increases in the x values will show further reduction in log(s(x)) (See Lange, 1962).Therefore, when the rank size distribution is applied to cities, it fits only the cities of large population size having urban maturity, and is not applicable to cities with small size of population.Thus many of the leading researchers (Champemowne, 1953;Rosen and Resnick, 1980;Henderson, 1982) on the topic have concluded that this model is not successful in providing satisfactory economic explanation for the regularity.Cordoba (2003), in his theoretical work on the distribution of city size, suggested that only a specific Markov Process can generate Pareto distribution for city size, and give economic explanation for the regularity.
Intrestingly, enough in a theoretical investigation Champemowne (1953), Simon (1955) and Gabaix (1999) have shown, by using central limit theorem, that Gibrat Law, or proportional growth, Zipf law and rank size rule-all can lead to Pareto distribution under certain conditions.More precisely, if a stochastic variable follows a growth process that is independent of die position of the variable, then its limiting distribution follows Pareto distribution.
David Cuberes (2004), made a remark that most of the empirical evidence has looked at a few countries and used limited data set, leading to controversial conclusions about the good fit of Gibrat's Law to city size distribution.The method used to fit the distribution is also found to be misleading.Alperovich (1988Alperovich ( , 1993) ) applied rank size mle by estimating the coefficients by using Ordinary Least Square (OLS) method, and examined the coefficients by using ttest, and ultimately found that the distribution pattern of most of the countries did not support the rank size rule.Dobkins and Ioammides (2000) recommended, by using maximum likelihood estimator proposed by Hill (1975), instead of OLS, but Soo's (2005) results did not support this.Rosen and Resnick (1980) have experimented this model on 50 largest administrative urban areas in 44 countries, and Soo (2005) also made an international comparison by using updated data of 73 countries, but the estimate of a exceeded unity.Hence they question the validity of rank size rule as well as the family of Zipf s model.In another study, Nishiyama and Osada (2004) examined statistical properties of least square estimators for rank size rule and obtained exact bias and variance of OLS estimators.They have shown that the tstatistics does not have t-distribution, resulting in severe size distortion, if t-test is blindly applied.
Finally, Krugman (1996) and Fujita et al. (1999) conclude that most of the city growth models are deterministic and cannot account for observed change in the population size of a city.These models usually predict the equilibrium size of cities as a result of the interplay between positive and negative externalities.The models also predict that urban growth mainly occurs through increase in the number of cities.This prediction conflicts both with the A. S. Kadi andB.I. Halingali idea of proportional growth and stabilization of urban system as tbe number of cities stabilizes.

Applications of the Specific Models
It is clear from Section 3 that most of the existing models are related to Pareto Distribution, and fit only the cities of larger size, satisfying certain regularity conditions.Formulation of generalized model to fit cities of any population size seems to be a rather complex exercise when cites are governed by different regularities.The growth of some cities may be governed by their growth in economy, that of some is governed simply by their service to the surrounding villages and other may be due to historical reasons.Thus, the study of city growth models remain incomplete until we develop appropriate model (s) to account for differential growth pattern, prediction of future size and city size transition over a period of time.
As we noted in Section 3, in Indian context, cities have been classified into six categories based on their population size.After independence (Year 1947), concentration of urban population in Class-I cities is increasing consistently, and they are growing at the cost of other size cities.As a result, the growth of other cities declined over a period.Some of the cities (Class I to Class V) have similar pattern of growth, though they differ in their growth rate, while very small population size eities-Class VI have experienced severe set back in terms of their number as well growth rate.Thus, we consider in this section to formulate suitable model to predict the present and future trend and pattern of cities in each of the class under study separately.

Model for Class-I Cities
As we discussed in Section 4, Pareto distribution fits well to a particular class of cities, namely, those cities tending to attain urban maturity.In Indian context, class-I are of this nature.More or less, they have attained urban maturity, and are about to reach equilibrium conditions as their growth in population size is more or less governed by the economic development (Megeri, 2002). Zipf Model (1949), which belongs to Pareto family, based on rank size rule, well approximated by Pareto distribution (Rosen and Resnick, 1980), has been chosen to fit the city size distribution of class-I cities. Zipf model is defined as:

Model Specification for Class-VI Cities
For cities of size class-VI, none of the above models fits well since the pattern of growth rate of these cities differs from that of the cities of other classes.From Table 2 we may observe that, during the period from 1921 to 2001, the growth rate and also the numbers of such cities (See Table 3) are declining consistently, and the data clearly shows that the cities of other categories growing at the cost of these cities.Moreover, they exhibit a declining trend and are more close to linear form.Therefore, we prefer the fourth degree polynomial model, which is specified as Where, Yk is the population size of a city with rank k, and ai, i=0,l,2,3,4 are coefficients to be determined.The model fits well to the observed distribution of city sizes.The fitted graphs are given in figure 7 for the 1991 and 2001 census data.

Results and Discussion
In urban research, the study of city size distribution using mathematical models largely deals with cities of large size with million plus population, which have urban maturity, that is constant growth rate and their growth is more influenced by economic development.Other cities with population less than a million have received less attention, though their number is large, covering the major portion of urban population.These small cities have differentetial growth pattern, and are not correlated with their economic development, and their growth depends on the type of the service they provide to the surrounding villages, such as: marketing, administrative, health, education facities.Thus, the growth of class-I cities in India are governed by economic development and generating employment opportunities, and hence attract migrants from other cities of smaller size within the state and other state as well, and are growing at the cost of cities of other small size (Kadi and Sivamurthy, 198S;Megeri, 2002;Kadi and Megeri, 2009).
Considering differential growth pattern of India cities of different class, we have chosen models to predict their growth pattern.5.
The value of the parameter z is found to be 0.77 for 1991 and 0.74 for 2001, both are less than one.The computed adjusted value of R2 is significantly high, explaining more than 97 percentage of the total variation.The model fits well to die data (See graphs presented in Figure 2).Further the estimated value of z declines over a period of decade under study, implying the slope of the rank size gets flatter and is viewed as-in Indian context, concentration of urban population is increasing over a period of time.
Similarly, the performance of the exponential model fits well for cities of Class-II to Class-V (See Figure 3 to Figure 6).The adjusted value of R2 is found to be significantly high and explains more than 9S percentage variations (See Table 5).In fact this variation increases from one decade to other, implying the consistent increasing performance of the model over a period of time Finally, the distribution of Class VI cities has been fitted far 1991 and 2001 census data by using the 4th degree polynomial equation.Interestingly enough, the expected distribution exactly matches with the observed city size distribution.The observed urban population and expected urban population are almost closer to each other and foe model explains 99 percentage of foe total variation (See Table 5; Figure 7).
In foe Indian context, foe chosen models Pareto distribution for Cities of Class-I size, Exponential model for Class-II to Class-V and for class-VI size cities Polynomial model of foe 4th degree have shown a remarkable performance and foe results are encouraging.The estimated Pareto parameter z indicates increasing concentration of urban population in cities of population size more than a million (Class-I cities), and it will continue to increase in foe future also as foe necessary stage in terms of capital investment and industrial activities has been set.This leads to further deterioration of quality of life in great cities of India (See Kadi and Halingali, 2007).On foe other hand, foe growth rate of cities other than Class-I have shown consistent decline during 1991-2001, and they simply exist as a service center without much economic activites.Hence national planners and policy makers have to take necessary steps to introduce developmental policies which discourage migration from smaller cities to class-I cities.

Predicting the Growth Pattern of Cities in India:
Application of Statistical Models and size of population growth of class-I, class-II and class-III cities since 1901 till recent census, was not affected.

A
During 1901During  - 1941, more than 82 percent of urban population concentrated in cities of classill and below category, whereas in die later period the momentum of concentration increased towards class-I, class-II and class-III cities.These citiesCSP2010, 37.1-2: 133-159 each class of cities.The fitted graphs are presented in the figures 3 to 6 respectively for cities of class-II to class-V.
An attempt is made to study the concentration of urban population in class-I cities in Indian context by using Pareto distribution.The estimated parameter values for 1991 and 2001 CSP2010, 37.1-2: 133-159 Predicting the Growth Pattern of Cities in India: Application of Statistical Models census data on class -I cities and corresponding values of adjusted R2 values are given in Table

Figure
Figure 4. Fitting of Class-Ill Cities in India: 1991 and 20*1

Table 1 Urbanization Trends in India by Census Year: 1901 -2001
Source: Provisional population totals: Rural-Urban Distribution, Census of India,

Table 2 . Number of Cities/Towns by Size Class in India
: 1901-2001 A. S. Kadi andB.I. Halingali