The Gini Index: Using calculus to measure inequity

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The Gini Index Using calculus to measure inequity

• Christine Belledin NCSSM
  belledin_at_ncssm.edu

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DATA USED TO QUANTIFY DISTRIBUTION OF income
Percent distribution of aggregate income for
sample data
Fifths of Families Percent of Income
Lowest fifth 4
Second fifth 10
Third fifth 13
Fourth fifth 21
Highest fifth 52
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DATA USED TO QUANTIFY DISTRIBUTION OF income
Cumulative percent distribution of aggregate
income for sample data
Fifths of Families Percent of Income
Lowest one-fifth 4
Lowest two-fifths 14
Lowest three-fifths 27
Lowest four-fifths 48
Lowest five-fifths 100
Cumulative proportion of aggregate income
Proportion of population
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Perfect Equity and Perfect Inequity

• What would the cumulative graph look like if the
  distribution was perfectly equitable? Perfectly
  inequitable?

Perfect Equity
Perfect Inequity
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The Gini Index
The ratio of the areas shown below.
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The Gini Index

• The ratio can have a value anywhere from 0 to 1.
• A Gini index of 0 represents perfect equity.
• A Gini index of 1 represents perfect inequity.
• The larger the ratio, the more inequitable the
  distribution of income.

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Finding the Lorenz Curve using Least Squares

• Since (0, 0) and (1, 1) are always points on the
  curves, a reasonable model for this data is a
  power function of the form y xn, with n gt 1.
• We choose not to use a power least squares
  procedure to fit a power function to the data
  because a Lorenz curve must contain the point (1,
  1), which is not guaranteed by this method.
• We will use the fact that a log-log re-expression
  linearizes data that is modeled by a power
  function.
• We now use our knowledge of calculus to find a
  least-squares estimate of n.

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• Consider the linear equation
• In our case, and
• We want to minimize
• This is a 1-variable optimization problem.

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Finding n
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Another Option for n
Your students may make another choice for the
method used to find the exponent. As long as
they are consistent in their procedure, important
comparisons can me made.
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Calculating the Gini Index
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Comparison of Methods 1 and 2 for sample data

• Method 1 n 2.0886
• Gini index 0.3525

Method 2 n 2.4956 Gini index 0.4278
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Student investigations

• Comparison of student measures to traditional
  Gini index.
• Relative values of the Gini indices for years
  when the president is Democrat and for years when
  the president is Republican.
• Investigating the historical events leading to
  the most drastic changes in the Gini index.
• Comparison of Gini indices for different
  countries around the world.