Christine Belledin NCSSM belledin@ncssm.edu

DATA USED TO QUANTIFY DISTRIBUTION OF INCOME
Percent distribution of aggregate income for sample data
Fifths of
Families
Lowest fifth
Second fifth
Third fifth
Fourth fifth
Highest fifth
Percent of Income
4
10
13
21
52
Christine Belledin TCM 2010

DATA USED TO QUANTIFY DISTRIBUTION OF INCOME
Cumulative percent distribution of aggregate income for sample data
 y Fifths of
Families
Lowest onefifth
Lowest twofifths
Lowest threefifths
Lowest fourfifths
Lowest fivefifths
Percent of
Income
4
14
27
48
100

Cumulative proportion of aggregate income



  
Proportion of population
  x
Christine Belledin TCM 2010


What would the cumulative graph look like if the distribution was perfectly equitable? Perfectly inequitable?
Perfect Equity Perfect Inequity
 y

Cumulative proportion of aggregate income



 y

Cumulative proportion of aggregate income



  
Proportion of population
  x
    x

Proportion of population
Christine Belledin TCM 2010

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

•
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

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 = 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.
y
 x n ln y
 n ln x

We now use our knowledge of calculus to find a least-squares estimate of n.
Christine Belledin TCM 2010


Consider the linear equation Y
 nX .
 X
 ln .

We want to minimize
S
 i
4 

1

Y i
 nX i

2
.

This is a 1-variable optimization problem.
Christine Belledin TCM 2010

4 𝑑𝑆 𝑑𝑛
= 2 𝑌 𝑖 𝑖=1
− 𝑛𝑋 𝑖
∙ (−𝑋 𝑖
) 𝑑𝑆
If 𝑑𝑛
= 0, then
4
𝑋 𝑖
𝑌 𝑖 𝑖=1
4
= 𝑛 𝑋 𝑖
2 𝑖=1 and 𝑛 =
4 𝑖=1
4 𝑖=1
𝑋
𝑋 𝑖
𝑌 𝑖 𝑖
2
Since 𝑋 𝑖
= ln 𝑥 𝑖
and 𝑌 𝑖
= ln 𝑦 𝑖
, we have 𝑛 =
4 𝑖=1 ln(𝑥
4 𝑖=1 𝑖
) ln(𝑦 𝑖
ln 𝑥 𝑖
2
)
.
Christine Belledin TCM 2010

𝑛 = 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 ln 𝑦 ln 𝑥 𝑖 𝑖
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.
Christine Belledin TCM 2010

Area bounded by Lorenz curve and 𝑦 = 𝑥:
1
𝐴𝑟𝑒𝑎 𝐴 = 𝑥 − 𝑥 𝑛
0
𝑑𝑥 =
1
2
−
1 𝑛 + 1
Area of triangle for perfect equity:
𝐴𝑟𝑒𝑎 𝐵 =
1
2
𝐴𝑟𝑒𝑎 𝐴
Gini Index =
𝐴𝑟𝑒𝑎 𝐵
= 1 −
2 𝑛+1
.
Christine Belledin TCM 2010

COMPARISON OF METHODS 1 AND 2 FOR SAMPLE DATA


 y


     x
Method 1: n = 2.0886
Gini index = 0.3525
 y




     x
Method 2: n = 2.4956
Gini index = 0.4278
Christine Belledin TCM 2010


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