1. Christine Belledin NCSSM belledin@ncssm.edu

2. Fifths of Families Percent of Income Lowest fifth4 Second fifth10 Third fifth13 Fourth fifth21 Highest fifth52 Christine Belledin TCM 2010 Percent distribution of aggregate income for sample data

3. Christine Belledin TCM 2010 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 percent distribution of aggregate income for sample data Proportion of population Cumulative proportion of aggregate income

4.  What would the cumulative graph look like if the distribution was perfectly equitable? Perfectly inequitable? Christine Belledin TCM 2010 Proportion of population Cumulative proportion of aggregate income Proportion of population Cumulative proportion of aggregate income Perfect EquityPerfect Inequity

5. Christine Belledin TCM 2010 The ratio of the areas shown below.

6. 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. Christine Belledin TCM 2010

7.  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 = x n, with n > 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. Christine Belledin TCM 2010

8.  Consider the linear equation  In our case, and  We want to minimize  This is a 1-variable optimization problem. Christine Belledin TCM 2010

10. 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.

11. Christine Belledin TCM 2010

12. Method 1: n = 2.0886 Gini index = 0.3525 Method 2: n = 2.4956 Gini index = 0.4278

13.  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. Christine Belledin TCM 2010