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Measuring Inequality An examination of the purpose and techniques of inequality measurement

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in·equal·i·ty Function: noun 1 : the quality of being unequal or uneven: as a : lack of evenness b : social disparity c : disparity of distribution or opportunity d : the condition of being variable : changeableness in·equal·i·ty Function: noun 1 : the quality of being unequal or uneven: as a : lack of evenness b : social disparity c : disparity of distribution or opportunity d : the condition of being variable : changeableness 2 : an instance of being unequal

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If a single person holds all of a given resource, inequality is at a maximum. If all persons hold the same percentage of a resource, inequality is at a minimum. Inequality studies explore the levels of resource disparity and their practical and political implications.

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Physical attributes – distribution of natural ability is not equal Physical attributes – distribution of natural ability is not equal Personal Preferences – Relative valuation of leisure and work effort differs Social Process – Pressure to work or not to work varies across particular fields or disciplines Public Policy – tax, labor, education, and other policies affect the distribution of resources

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Why measure Inequality? Measuring changes in inequality helps determine the effectiveness of policies aimed at affecting inequality and generates the data necessary to use inequality as an explanatory variable in policy analysis.

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How do we measure Inequality? Before choosing an inequality measure, the researcher must ask two additional questions: Does the research question require the inequality metric to have particular properties (inflation resistance, comparability across groups, etc)? What metric best leverages the available data?

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Choosing the best metric Range Range Ratio The McLoone Index The Coefficient of Variation The Gini Coefficient Theil’s T Statistic

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Range

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Range Pros Easy to Understand Easy to Compute

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Range Ratio

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Range Ratio Pros Easy to understand Easy to calculate Not skewed by severe outliers Not affected by inflation

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The McLoone Index

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The McLoone Index Pros Easy to understand Conveys comprehensive information about the bottom half

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The Coefficient of Variation

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The Coefficient of Variation Pros Fairly easy to understand If data is weighted, it is immune to outliers Incorporates all data Not skewed by inflation

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The Gini Coefficient The Gini Coefficient has an intuitive, but possibly unfamiliar construction. To understand the Gini Coefficient, one must first understand the Lorenz Curve, which orders all observations and then plots the cumulative percentage of the population against the cumulative percentage of the resource.

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A – Equality Diagonal Population = Income A – Equality Diagonal Population = Income B – Lorenz Curve C – Difference Between Equality and Reality

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The Gini Coefficient Mathematically, the Gini Coefficient is equal to twice the area enclosed between the Lorenz curve and the equality diagonal. When there is perfect equality, the Lorenz curve is the equality diagonal, and the value of the Gini Coefficient is zero. When one member of the population holds all of the resource, the value of the Gini Coefficient is one.

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The Gini Coefficient Pros Generally regarded as gold standard in economic work Incorporates all data Allows direct comparison between units with different size populations Attractive intuitive interpretation

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Theil’s T Statistic Theil’s T Statistic lacks an intuitive picture and involves more than a simple difference or ratio. Nonetheless, it has several properties that make it a superior inequality measure. Theil’s T Statistic can incorporate group-level data and is particularly effective at parsing effects in hierarchical data sets.

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Theil’s T Statistic Theil’s T Statistic generates an element, or a contribution, for each individual or group in the analysis which weights the data point’s size (in terms of population share) and weirdness (in terms of proportional distance from the mean). When individual data is available, each individual has an identical population share (1/N), so each individual’s Theil element is determined by his or her proportional distance from the mean.

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Theil’s T Statistic Mathematically, with individual level data Theil’s T statistic of income inequality is given by: where n is the number of individuals in the population, yp is the income of the person indexed by p, and µy is the population’s average income.

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Theil’s T Statistic The formula on the previous slide emphasizes several points: The summation sign reinforces the idea that each person will contribute a Theil element. yp/µy is the proportion of the individual’s income to average income. The natural logarithm of yp /µy determines whether the element will be positive (yp /µy > 1); negative (yp /µy < 1); or zero (yp /µy = 0).

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Theil’s T Statistic – Example 1

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Theil’s T Statistic Often, individual data is not available. Theil’s T Statistic has a flexible way to deal with such instances. If members of a population can be classified into mutually exclusive and completely exhaustive groups, then Theil’s T Statistic for the population (T ) is made up of two components, the between group component (T’g) and the within group component (Twg).

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Theil’s T Statistic Algebraically, we have: T = T’g + Twg When aggregated data is available instead of individual data, T’g can be used as a lower bound for Theil’s T Statistic in the population.

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Theil’s T Statistic The between group element of the Theil index has a familiar form: where i indexes the groups, pi is the population of group i, P is the total population, yi is the average income in group i, and µ is the average income across the entire population.

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Theil’s T Statistic – Example 2

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Group analysis with Theil’s T Statistic: As Example 2 hints, Theil’s T Statistic is a powerful tool for analyzing inequality within and between various groupings, because: The between group elements capture each group’s contribution to overall inequality The sum of the between group elements is a reasonable lower bound for Theil’s T statistic in the population Sub-groups can be broken down within the context of larger groups

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Theil’s T Statistic Pros Can effectively use group data Allows the researcher to parse inequality into within group and between group components

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Next Steps Those interested in a more rigorous examination of inequality metrics with several numerical examples should proceed to The Theoretical Basics of Popular Inequality Measures. Otherwise, proceed to A Nearly Painless Guide to Computing Theil’s T Statistic which emphasizes constructing research questions and using a spreadsheet to conduct analysis.