Different methods of measurement of inequalities
The Lorenz curve and Gini coefficient are typically used to measure inequality. A different way to measure inequality is introduced here: I = CN, the product of concentration and number of units.
The resultant index can be interpreted with reference to an inequality base where one unit owns all and the rest nothing. This inequality index also integrates the measurement of inequality, concentration, and diversification into one system, where diversification is measured as the inverse of concentration.
The Tideman-Hall concentration index also provides indexes of concentration, diversification and inequality as functions of Gini. As one application, the inequality index can be used to provide an index of economic development.
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Inequality can be ordinal or cardinal. An ordinal ranking of distributions in order of inequality implies some cardinal method of measuring or judging each distribution in relation to the others.
A cardinal ranking can be relational, defined with reference to other variables, or monadic, its magnitude being defined without reference to other variables. This paper is concerned with the measurement of monadic, cardinal inequality.
Inequality is the degree to which the units of a distribution have shares of some attribute which are unequal in quantity. This meaning of inequality is completely general for any type of distribution. In order to measure inequality, one needs to determine the meaning of the degree of being unequal.
These various inequality measures, along with the many others that have been proposed, yield different inequality rankings, and, applying yield diverse measurements of concentration. The literature on measurements of inequality thus also provides ways of measuring concentration and diversification.
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Michael Todaro has noted that the concept of economic development has come to be defined as a function of inequality as well as of per-capita GDP. If two countries A and B have an equal per-capita GDP, but A has a much more unequal distribution of income, due to a larger less-developed sector, then it can be regarded as overall less developed than B.
Hence, an increase in GDP in a small sector of an economy would not, by-this criterion, count as much as a broader-based increase, even if the average increase were the same for both.
Todaro (1994) proposes a “distributive share index” or a “poverty-weighted index” as an alternative to the plain GDP or GNP, and the U.N. Human Development Report calculates “human development indicators.” The inequality index IF, such as calculated in (3) or (13), can also be used to measure the degree of economic development YD as a function of GDP, population, and inequality:
Where YG is GDP. For a given per-capita GDP, the degree of economic development declines with increasing inequality.
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If IT is used, as an example, in 1991 Mexico and Brazil had per- capita incomes of $2870 and S2930, with Gini coefficients .50 and .57, respectively. Hence, using IT, YD for Mexico and Brazil are 1435 and 1260, Mexico having a higher development index despite its lower GDP due to the greater inequality in Brazil.
The equations I = CN and D = 1/C provide an integrated method of computing indexes for inequality, concentration, and diversification, given some distribution with N items. If C is an independently computed variable, then its use to compute inequality places a constraint on C, since certain intuitive constraints on I need to be met. Given N, I should increase with increasing concentration, and given C, I should increase with increasing N.
For equal-share distributions, increasing N should decrease C proportionately so as to leave I equal. The Herfindahl- Hirschman and Tideman-Hall indexes satisfy these criteria. It is to be calculated independently and C is the dependent variable, then the Tideman-Hall- Foldvary inequality index as a function of the Gini coefficient (14) has the desirable qualities needed for I = CN. As functions of Gini, the Tideman-Hall- Foldvary indexes for concentration (15), inequality (14) and diversification (16) may have many useful applications.
An inequality index IF calculated from CH or CT in (1) or from G in (14) has some more consistent properties in relationship to N and C than straight Gini, so it merits empirical investigation as to its usefulness in computing inequality in income and wealth as a perhaps superior substitute for the raw Gini Coefficient.