3.2.4 Statistical Analysis of Income Distribution
In the first place, there has been an attempt to establish a functional relationship between the size of income and the number of recipients. Secondly, an attempt has been made to summarise the income distribution by a single measure of the inequalities of income. The most famous attempt in the first direction is the Pareto's law. The law, in its most dogmatic form, states that the distribution of income in the upper ranges of income tax payers shows a linear relationship. Mathematically, the Pareto's law may be stated as:
logN logA-c………………………………….
where / is the income size, N is the number of individuals having an income equal to or larger than X and A and a are constants (found from empirical statistics by fitting the data to the straight line given by the above equation).
It has been found that the constant (x (the slope of the straight line) is approximately equal to 1.5 in all countries at present. In addition, all ranges of income distribution follow the same linear relationship for all countries at present. It follows, therefore, that because of the unchanging and unchangeable nature of the whole range of income frequency distribution, economic welfare can be increased only through an increase in the total amount of income. It is obvious that the Pareto's law is of great importance for major questions of economic theory as well as economic policy. Many economists and statisticians have directed their attention towards testing of its validity. Results of such cumulative analysis have shown that the Pareto's law is quite inadequate as a mathematical generalisation. Because of the heterogeneity of the frequency distribution curve (due to grouping together of income from various economic categories), it is unlikely that any mathematical law describing adequately the entire distribution of income can ever be formulated. Other attempts to substitute for the linear distribution with another mathematical expression have also been found unsatisfactory for describing the distribution of income. However, a French economist, R. Gibrat, has obtained successful descriptions of a large number of frequency distributions of income by using a modification of the normal distribution curve of errors. The curve employed by Gibrat is:
Y-r""
Vii
Z=aloste-x„)+b………………………………………….
where y is the number of income recipients, X is the variable size of income, (X - x3 is a selected income constant, while a and b are constants to be found from empirical statistics. The assumption in which eq. 3.2.4.2 differs from the normal distribution curve is that the effect of each of the numerous contributory factors is not independent but proportional to the effect of other factors.
3.2.5 Measures of Inequality of Income
The more fruitful developments in the direction of summarising inequality of income by a single measure has yielded numerous measures. Such single measures of inequality of income broadly fall into four groups:
(1) The measures derived from a specific type of mathematical equation and hence, contingent upon the goodness of fit of the curve implied by that equation.
• The measures of the mean deviation type, available in the statistical theory of frequency distribution and applicable to diverse types of ----------------- .
• The measures of mean difference types.
• The measures constructed by using definite theoretical criteria in regard to welfare equivalents of individual income. In the first group, there are three important measures of inequality. The coefficient a of Pareto's equation has been employed as a measure of inequality. The steeper the slope (the larger the numerical value of a), the smaller the inequality of income. A second measure of inequality is Gini's index of concentration 5, which is derived from another equation of income distribution: log N=8 logS-log K where N is the number of individuals whose income is above a certain size, S is the sum of incomes (each greater than the certain size), and 5 and K are constants to be determined from empirical data. It may be noted that N is a function of the sum of incomes greater than a certain size rather than a function of that income size itself, as in the case of the Pareto's law. The relationship between Pareto's measure a and Gini's measure S can be eunr~cprl by the pniiatinn IX third measure of inequality of income may be derived from the curve employed by Gibrat. This measure is taken to be equal to 100/a. Of the dispersion measures developed in the statistical theory offrequency distribution, the average and the standard deviation naturally suggest themselves as indices of the inequality of income. The resulting relative measures of dispersion can be obtained from a frequency distribution in which the class intervals of income size are taken in absolute figures or in logarithms. The advantage of the latter procedure arises from the fact that the positive skewness characterizing frequency distributions of income is reduced by taking the income variable in terms of logarithms.The mean difference of incomes is given by the arithmetic average of differences(taken without regard to their positive or negative signs) between all possible pairs of incomes. This measure was suggested by Gini and it is known as the "ratio of concentration". Another widely known measure of inequality, which is related to Gini's ratio of concentration, is known as "Lorenz curve". In the Lorenz curve, the cumulative percentages of total income (lc) are plotted along theX-axis, while the cumulative percentages of population (PC), from the poorest to the richest, are plotted along the Y-axis. In the Lorenz curve, an equal distribution of income (total absence of inequality) is represented by a straight line passing through the origin and having a slope eaua] tn unity as shown by the straight Imp A in the fig. .
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