The problems of numerical inequalities estimates

The purpose of this paper is to compare the shortcomings of the widely used inequality coefficients that appear when working with real (ie, knowingly incomplete) data and searching for alternative quantitative methods for describing inequalities that lack these shortcomings.

Research methods:

– consideration of an extensive range of as full as possible real data on the population distribution by income, expenditure, property (ie data on the economic structure of society);

– revealing the specific shortcomings of these data on the economic structure of society, finding out which information is missing or presented disproportionately;

– comparison of the values of the most widely used indices of inequality calculated on real data on the economic structure, with a view to establishing the suitability of these indicators for problems of inequality estimation;

– development of an index of inequality that adequately describes the real economic structure of society.

Research data:

– official data of Rosstat and the Federal Tax Service on incomes of Russian citizens;

– specialized sites of announcements about the prices for real estate and cars;

– Credit Suisse Research Institute data on the distribution of Russian citizens by property level;

– Forbes data on income and wealth of the richest people in Russia.

It is shown that the income data are essentially incomplete and fragmentary – the width of the income range (i.e., the income of therichest member of society) is known, but the filling of rich cohorts is not known, since the incomes of the richest members of society are hidden.

We proposed the next (criteria as) requirements for an inequality index:

–  possibility of calculating the index of inequality for arbitrary quantization;

– invariance of the value of the inequality index for different quantization of the same data;

– sensitivity of the index to the width of the income range.

It is noted that only the exponential function describes societies with high social inequality enough well (the intensity of the exponential distribution is more than 10).

For the presented population distributions, the next indices of inequality are calculated:

– decile coefficient of funds;

– Gini coefficient;

– Pareto index;

– indicators of total entropy (zero, first or Tayle index, and second orders);

– the ratio of maximum income (property value) to the modal;

– intensity of exponential distribution.

It is shown, that:

– the value of the Pareto index does not have a unique relationship with the inequality;

– the coefficients of the funds (decile, quintile, etc.) are not computable for arbitrary quantization, and therefore are unsuitable for comparing data from various sources and have different quantization;

– The Gini index requires complete data on the rich;

– from all considered criteria of inequality the first three indicators of the total entropy, as well as the ratio of maximum income (property) to the modal strongly depend on data quantization.

Therefore they are unsuitable for comparison data from various sources with different quantization. It is concluded that the intensity of the exponential distribution does not possess the listed disadvantages and can be recommended as an index of inequality.

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