measuring inequality

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

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Measuring Inequality. An examination of the purpose and techniques of inequality measurement. What is inequality?. From Merriam-Webster:. - PowerPoint PPT Presentation

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Page 1: Measuring Inequality

Measuring InequalityMeasuring Inequality

An examination of the An examination of the purpose and techniques of purpose and techniques of inequality measurementinequality measurement

Page 2: Measuring Inequality

in·equal·i·tyin·equal·i·ty Function: Function: nounnoun11 :: the quality of being unequal or the quality of being unequal or uneven: as uneven: as aa :: lack of evenness lack of evenness bb :: social social disparity disparity cc :: disparity of distribution or disparity of distribution or opportunity opportunity dd :: the condition of being the condition of being variable : changeablenessvariable : changeableness22 :: an instance of being unequal an instance of being unequal

What is inequality?What is inequality?From Merriam-Webster:From Merriam-Webster:From Merriam-Webster:From Merriam-Webster:

Page 3: Measuring Inequality

Our primary interest is in economic Our primary interest is in economic inequality.inequality.

In this context, inequality measures the In this context, inequality measures the disparity between a percentage of disparity between a percentage of population and the percentage of population and the percentage of resources (such as income) received by resources (such as income) received by that population. that population.

Inequality increases as the disparity Inequality increases as the disparity increases. increases.

Our primary interest is in economic Our primary interest is in economic inequality.inequality.

In this context, inequality measures the In this context, inequality measures the disparity between a percentage of disparity between a percentage of population and the percentage of population and the percentage of resources (such as income) received by resources (such as income) received by that population. that population.

Inequality increases as the disparity Inequality increases as the disparity increases. increases.

Page 4: Measuring Inequality

If a single person holds all of a given If a single person holds all of a given resource, inequality is at a maximum. resource, inequality is at a maximum. If all persons hold the same If all persons hold the same percentage of a resource, inequality is percentage of a resource, inequality is at a minimum. at a minimum.

Inequality studies explore the levels of Inequality studies explore the levels of resource disparity and their practical resource disparity and their practical and political implications.and political implications.

Page 5: Measuring Inequality

• Physical attributes – distribution of natural Physical attributes – distribution of natural ability is not equalability is not equal

• Personal Preferences – Relative valuation of Personal Preferences – Relative valuation of leisure and work effort differs leisure and work effort differs

• Social Process – Pressure to work or not to Social Process – Pressure to work or not to work varies across particular fields or work varies across particular fields or disciplines disciplines

• Public Policy – tax, labor, education, and other Public Policy – tax, labor, education, and other policies affect the distribution of resourcespolicies affect the distribution of resources

Economic Inequalities can occur for Economic Inequalities can occur for several reasons:several reasons:

Economic Inequalities can occur for Economic Inequalities can occur for several reasons:several reasons:

Page 6: Measuring Inequality

Why measure Inequality?Why measure Inequality?

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

Page 7: Measuring Inequality

How do we measure How do we measure Inequality?Inequality?

Before choosing an inequality measure, Before choosing an inequality measure, the researcher must ask two the researcher must ask two additional questions:additional questions:

• Does the research question require Does the research question require the inequality metric to have the inequality metric to have particular properties (inflation particular properties (inflation resistance, comparability across resistance, comparability across groups, etc)?groups, etc)?

• What metric best leverages the What metric best leverages the available data?available data?

Page 8: Measuring Inequality

Choosing the best metricChoosing the best metric

• RangeRange

• Range RatioRange Ratio

• The McLoone Index The McLoone Index

• The Coefficient of VariationThe Coefficient of Variation

• The Gini CoefficientThe Gini Coefficient

• Theil’s T StatisticTheil’s T Statistic

Some popular measures include:Some popular measures include:Some popular measures include:Some popular measures include:

Page 9: Measuring Inequality

RangeRangeThe range is simply the difference between the The range is simply the difference between the highest and lowest observations.highest and lowest observations.

Number of employeesNumber of employeesNumber of employeesNumber of employees SalarySalarySalarySalary

22 $1,000,000$1,000,000

44

66

88

1212

66

$200,000$200,000

$100,000$100,000

$45,000$45,000

$24,000$24,000

$60,000$60,000

In this example, the Range = $1,000,000-$24,000In this example, the Range = $1,000,000-$24,000In this example, the Range = $1,000,000-$24,000In this example, the Range = $1,000,000-$24,000

= 976,000= 976,000= 976,000= 976,000

Page 10: Measuring Inequality

RangeRange

ProsPros

• Easy to Easy to UnderstandUnderstand

• Easy to ComputeEasy to Compute

ConsCons

• Ignores all but two Ignores all but two of the observationsof the observations

• Does not weight Does not weight observationsobservations

• Affected by inflationAffected by inflation

• Skewed by outliersSkewed by outliers

The range is simply the difference between the The range is simply the difference between the highest and lowest observations.highest and lowest observations.

Page 11: Measuring Inequality

Range RatioRange RatioThe Range Ratio is computed by dividing a value at The Range Ratio is computed by dividing a value at one predetermined percentile by the value at a lower one predetermined percentile by the value at a lower predetermined percentile.predetermined percentile.

95 percentileApprox. equals36th person

95 percentileApprox. equals36th person

5 percentileApprox. equals2nd person

5 percentileApprox. equals2nd person

In this example, the Range Ratio=200,000/24,000 In this example, the Range Ratio=200,000/24,000 =8.33=8.33

In this example, the Range Ratio=200,000/24,000 In this example, the Range Ratio=200,000/24,000 =8.33=8.33

Note: Any two percentiles can be used in producing a Range Ratio. Note: Any two percentiles can be used in producing a Range Ratio. In some contexts, this 95/5 ratio is referred to as the Federal Range In some contexts, this 95/5 ratio is referred to as the Federal Range Ratio.Ratio.

Note: Any two percentiles can be used in producing a Range Ratio. Note: Any two percentiles can be used in producing a Range Ratio. In some contexts, this 95/5 ratio is referred to as the Federal Range In some contexts, this 95/5 ratio is referred to as the Federal Range Ratio.Ratio.

Number of employeesNumber of employeesNumber of employeesNumber of employees SalarySalarySalarySalary

22 $1,000,000$1,000,000

44

66

88

1212

66

$200,000$200,000

$100,000$100,000

$45,000$45,000

$24,000$24,000

$60,000$60,000

Page 12: Measuring Inequality

Range RatioRange Ratio

ProsPros

• Easy to understandEasy to understand

• Easy to calculateEasy to calculate

• Not skewed by Not skewed by severe outlierssevere outliers

• Not affected by Not affected by inflationinflation

ConsCons

• Ignores all but two Ignores all but two of the observationsof the observations

• Does not weight Does not weight observationsobservations

The Range Ratio is computed by dividing a value at The Range Ratio is computed by dividing a value at one predetermined percentile by the value at a one predetermined percentile by the value at a lower predetermined percentile.lower predetermined percentile.

Page 13: Measuring Inequality

The McLoone IndexThe McLoone IndexThe McLoone Index divides the summation of all The McLoone Index divides the summation of all observations below the median, by the median observations below the median, by the median multiplied by the number of observations below multiplied by the number of observations below median.median. Number of employeesNumber of employeesNumber of employeesNumber of employees SalarySalarySalarySalary

22 1,000,000.001,000,000.00

44

66

88

1212

66

200,000.00200,000.00

100,000.00100,000.00

45,000.0045,000.00

24,000.0024,000.00

60,000.0060,000.00Observations

below median

Observationsbelow median

In this example, the summation of observations below the In this example, the summation of observations below the median = 603,000, and the median = 45,000median = 603,000, and the median = 45,000Thus, the McLoone Index = 603,000/(45,000(19)) = .7053Thus, the McLoone Index = 603,000/(45,000(19)) = .7053

In this example, the summation of observations below the In this example, the summation of observations below the median = 603,000, and the median = 45,000median = 603,000, and the median = 45,000Thus, the McLoone Index = 603,000/(45,000(19)) = .7053Thus, the McLoone Index = 603,000/(45,000(19)) = .7053

Page 14: Measuring Inequality

The McLoone IndexThe McLoone Index

ProsPros

• Easy to understandEasy to understand

• Conveys Conveys comprehensive comprehensive information about information about the bottom halfthe bottom half

ConsCons

• Ignores values Ignores values above the medianabove the median

• Relevance depends Relevance depends on the meaning of on the meaning of the median valuethe median value

The McLoone Index divides the summation of all The McLoone Index divides the summation of all observations below the median, by the median observations below the median, by the median multiplied by the number of observations below multiplied by the number of observations below median.median.

Page 15: Measuring Inequality

The Coefficient of VariationThe Coefficient of VariationThe Coefficient of Variation is a distribution’s The Coefficient of Variation is a distribution’s standard deviation divided by its mean.standard deviation divided by its mean.

Both distributions above have the same mean, 1, but the Both distributions above have the same mean, 1, but the standard deviation is much smaller in the distribution on the standard deviation is much smaller in the distribution on the left, resulting in a lower coefficient of variation. left, resulting in a lower coefficient of variation.

Both distributions above have the same mean, 1, but the Both distributions above have the same mean, 1, but the standard deviation is much smaller in the distribution on the standard deviation is much smaller in the distribution on the left, resulting in a lower coefficient of variation. left, resulting in a lower coefficient of variation.

Page 16: Measuring Inequality

The Coefficient of VariationThe Coefficient of Variation

ProsPros• Fairly easy to Fairly easy to

understandunderstand• If data is weighted, it If data is weighted, it

is immune to outliersis immune to outliers• Incorporates all dataIncorporates all data• Not skewed by Not skewed by

inflationinflation

ConsCons

• Requires Requires comprehensive comprehensive individual level individual level datadata

• No standard for an No standard for an acceptable level of acceptable level of inequalityinequality

The Coefficient of Variation is a distribution’s The Coefficient of Variation is a distribution’s standard deviation divided by its mean.standard deviation divided by its mean.

Page 17: Measuring Inequality

The Gini CoefficientThe Gini Coefficient

The Gini Coefficient has an intuitive, but The Gini Coefficient has an intuitive, but possibly unfamiliar construction.possibly unfamiliar construction.

To understand the Gini Coefficient, one To understand the Gini Coefficient, one must first understand the Lorenz must first understand the Lorenz Curve, which orders all observations Curve, which orders all observations and then plots the cumulative and then plots the cumulative percentage of the population against percentage of the population against the cumulative percentage of the the cumulative percentage of the resource.resource.

Page 18: Measuring Inequality

• A – Equality Diagonal A – Equality Diagonal Population = IncomePopulation = Income

• B – Lorenz Curve B – Lorenz Curve • C – Difference C – Difference

Between Equality Between Equality and Reality and Reality

AA

BB

CC

Cumulative PopulationCumulative Population

Cum

ula

tive

Inco

me

Cum

ula

tive

Inco

me

The Gini CoefficientThe Gini CoefficientThe Gini CoefficientThe Gini Coefficient

An equality diagonal represents perfect An equality diagonal represents perfect equality: at every point, cumulative population equality: at every point, cumulative population equals cumulative income.equals cumulative income.

An equality diagonal represents perfect An equality diagonal represents perfect equality: at every point, cumulative population equality: at every point, cumulative population equals cumulative income.equals cumulative income.

The Lorenz curve measures the actual The Lorenz curve measures the actual distribution of income.distribution of income.

The Lorenz curve measures the actual The Lorenz curve measures the actual distribution of income.distribution of income.

Page 19: Measuring Inequality

The Gini CoefficientThe Gini Coefficient

Mathematically, the Gini Coefficient is equal to Mathematically, the Gini Coefficient is equal to twice the area enclosed between the Lorenz twice the area enclosed between the Lorenz curve and the equality diagonal.curve and the equality diagonal.

When there is perfect equality, the Lorenz curve When there is perfect equality, the Lorenz curve is is the equality diagonal, and the value of the the equality diagonal, and the value of the Gini Coefficient is zero.Gini Coefficient is zero.

When one member of the population holds all of When one member of the population holds all of the resource, the value of the Gini Coefficient is the resource, the value of the Gini Coefficient is one.one.

Page 20: Measuring Inequality

The Gini CoefficientThe Gini Coefficient

ProsPros

• Generally regarded Generally regarded as gold standard in as gold standard in economic workeconomic work

• Incorporates all dataIncorporates all data

• Allows direct Allows direct comparison between comparison between units with different units with different size populationssize populations

• Attractive intuitive Attractive intuitive interpretationinterpretation

ConsCons

• Requires Requires comprehensive comprehensive individual level individual level datadata

• Requires more Requires more sophisticated sophisticated computationscomputations

Twice the area between the Lorenz curve and the Twice the area between the Lorenz curve and the equality diagonal.equality diagonal.

Page 21: Measuring Inequality

Theil’s T StatisticTheil’s T Statistic

Theil’s T Statistic lacks an intuitive picture Theil’s T Statistic lacks an intuitive picture and involves more than a simple and involves more than a simple difference or ratio.difference or ratio.

Nonetheless, it has several properties that Nonetheless, it has several properties that make it a superior inequality measure. make it a superior inequality measure.

Theil’s T Statistic can incorporate group-Theil’s T Statistic can incorporate group-level data and is particularly effective at level data and is particularly effective at parsing effects in hierarchical data sets.parsing effects in hierarchical data sets.

Page 22: Measuring Inequality

Theil’s T StatisticTheil’s T Statistic

Theil’s T Statistic generates an element, or a Theil’s T Statistic generates an element, or a contribution, for each individual or group in the contribution, for each individual or group in the analysis which weights the data point’s size (in analysis which weights the data point’s size (in terms of population share) and weirdness (in terms of population share) and weirdness (in terms of proportional distance from the mean). terms of proportional distance from the mean).

  When individual data is available, each individual When individual data is available, each individual

has an identical population share (1/N), so each has an identical population share (1/N), so each individual’s Theil element is determined by his individual’s Theil element is determined by his or her proportional distance from the mean.or her proportional distance from the mean.

Page 23: Measuring Inequality

Theil’s T StatisticTheil’s T Statistic

Mathematically, with individual level data Mathematically, with individual level data Theil’s T statistic of income inequality is Theil’s T statistic of income inequality is given by:given by:

  

  where where nn is the number of individuals in the is the number of individuals in the

population, population, yypp is the income of the person is the income of the person indexed by indexed by pp, and , and µµyy is the population’s is the population’s average income. average income.

n

p y

p

y

p yy

nT

1

ln**1

Page 24: Measuring Inequality

Theil’s T StatisticTheil’s T Statistic

The formula on the previous slide emphasizes The formula on the previous slide emphasizes several points:several points:

• The summation sign reinforces the idea that The summation sign reinforces the idea that each person will contribute a Theil element. each person will contribute a Theil element.

• yypp//µµyy is the proportion of the individual’s is the proportion of the individual’s income to average income.income to average income.

• The natural logarithm of The natural logarithm of yyp p //µµyy determines determines whether the element will be positive (whether the element will be positive (yyp p //µµyy > > 1); negative (1); negative (yyp p //µµyy < 1); or zero ( < 1); or zero (yyp p //µµyy = 0). = 0).

Page 25: Measuring Inequality

Theil’s T Statistic – Example Theil’s T Statistic – Example 11The following example assumes that exact salary The following example assumes that exact salary

information is known for each individual.information is known for each individual.

Number of employeesNumber of employeesNumber of employeesNumber of employees Exact SalaryExact SalaryExact SalaryExact Salary

22 $100,000$100,000

44

66

2244

$80,000$80,000

$60,000$60,000

$20,000$20,000$40,000$40,000

For this data, Theil’s T Statistic = 0.079078221For this data, Theil’s T Statistic = 0.079078221

Individuals in the top salary group contribute large positive elements. Individuals in the top salary group contribute large positive elements. Individuals in the middle salary group contribute nothing to Theil’s T Statistic Individuals in the middle salary group contribute nothing to Theil’s T Statistic because their salaries are equal to the population average. Individuals in the because their salaries are equal to the population average. Individuals in the bottom salary group contribute large negative elements.bottom salary group contribute large negative elements.

For this data, Theil’s T Statistic = 0.079078221For this data, Theil’s T Statistic = 0.079078221

Individuals in the top salary group contribute large positive elements. Individuals in the top salary group contribute large positive elements. Individuals in the middle salary group contribute nothing to Theil’s T Statistic Individuals in the middle salary group contribute nothing to Theil’s T Statistic because their salaries are equal to the population average. Individuals in the because their salaries are equal to the population average. Individuals in the bottom salary group contribute large negative elements.bottom salary group contribute large negative elements.

Page 26: Measuring Inequality

Theil’s T StatisticTheil’s T Statistic

Often, individual data is not available. Theil’s Often, individual data is not available. Theil’s T Statistic has a flexible way to deal with T Statistic has a flexible way to deal with such instances.such instances.

If members of a population can be classified If members of a population can be classified into mutually exclusive and completely into mutually exclusive and completely exhaustive groups, then Theil’s T Statistic exhaustive groups, then Theil’s T Statistic for the population (for the population (T T ) is made up of two ) is made up of two components, the between group component components, the between group component ((T’gT’g) and the within group component () and the within group component (TTwwgg). ).

Page 27: Measuring Inequality

Theil’s T StatisticTheil’s T Statistic

Algebraically, we have:Algebraically, we have:

T = T’T = T’g g + T+ Twwgg

When aggregated data is available When aggregated data is available instead of individual data, instead of individual data, T’T’gg can be can be

used as a lower bound for Theil’s T used as a lower bound for Theil’s T Statistic in the population. Statistic in the population.

Page 28: Measuring Inequality

Theil’s T StatisticTheil’s T Statistic

The between group element of the Theil index The between group element of the Theil index has a familiar form:has a familiar form:

where where i i indexes the groups, indexes the groups, ppi i is the is the population of group population of group ii, , P P is the total is the total population, population, yyii is the average income in is the average income in group group ii, and , and µµ is the average income across is the average income across the entire population. the entire population.

m

i

iiig

yy

P

pT

1

ln**'

Page 29: Measuring Inequality

Theil’s T Statistic – Example Theil’s T Statistic – Example 22Now assume the more realistic scenario where a Now assume the more realistic scenario where a

researcher has average salary information across researcher has average salary information across groups.groups.

Number of employees in groupNumber of employees in groupNumber of employees in groupNumber of employees in group Group Average SalaryGroup Average SalaryGroup Average SalaryGroup Average Salary

22 $95,000$95,000

44

66

2244

$75,000$75,000

$60,000$60,000

$25,000$25,000$45,000$45,000

For this data, For this data, T’T’gg = 0.054349998= 0.054349998

The top salary two salary groups contribute positive elements. The middle The top salary two salary groups contribute positive elements. The middle salary group contributes nothing to the between group Theil’s T Statistic salary group contributes nothing to the between group Theil’s T Statistic because the group average salary is equal to the population average. The because the group average salary is equal to the population average. The bottom two salary groups contribute negative elements.bottom two salary groups contribute negative elements.

For this data, For this data, T’T’gg = 0.054349998= 0.054349998

The top salary two salary groups contribute positive elements. The middle The top salary two salary groups contribute positive elements. The middle salary group contributes nothing to the between group Theil’s T Statistic salary group contributes nothing to the between group Theil’s T Statistic because the group average salary is equal to the population average. The because the group average salary is equal to the population average. The bottom two salary groups contribute negative elements.bottom two salary groups contribute negative elements.

Page 30: Measuring Inequality

Group analysis with Theil’s T Group analysis with Theil’s T Statistic:Statistic:As Example 2 hints, Theil’s T Statistic is a As Example 2 hints, Theil’s T Statistic is a

powerful tool for analyzing inequality within powerful tool for analyzing inequality within and between various groupings, because: and between various groupings, because:

• The between group elements capture each The between group elements capture each group’s contribution to overall inequalitygroup’s contribution to overall inequality

• The sum of the between group elements is a The sum of the between group elements is a reasonable lower bound for Theil’s T reasonable lower bound for Theil’s T statistic in the populationstatistic in the population

• Sub-groups can be broken down within the Sub-groups can be broken down within the context of larger groupscontext of larger groups

Page 31: Measuring Inequality

Theil’s T StatisticTheil’s T Statistic

ProsPros

• Can effectively use Can effectively use group datagroup data

• Allows the Allows the researcher to parse researcher to parse inequality into inequality into within group and within group and between group between group componentscomponents

ConsCons

• No intuitive No intuitive motivating picturemotivating picture

• Cannot directly Cannot directly compare compare populations with populations with different sizes or different sizes or group structuresgroup structures

• Comparatively Comparatively mathematically mathematically complexcomplex

Page 32: Measuring Inequality

Next StepsNext Steps

• Those interested in a more rigorous Those interested in a more rigorous examination of inequality metrics with examination of inequality metrics with several numerical examples should proceed several numerical examples should proceed to to The Theoretical Basics of Popular The Theoretical Basics of Popular Inequality Measures.Inequality Measures.

• Otherwise, proceed to Otherwise, proceed to A Nearly Painless A Nearly Painless Guide to Computing Theil’s T StatisticGuide to Computing Theil’s T Statistic which which emphasizes constructing research questions emphasizes constructing research questions and using a spreadsheet to conduct and using a spreadsheet to conduct analysis.analysis.