Transcript The Gini Index: Using calculus to measure inequity

THE GINI INDEX: USING CALCULUS TO
MEASURE INEQUITY
Christine Belledin
NCSSM
[email protected]
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
Christine Belledin
TCM 2010
DATA USED TO QUANTIFY DISTRIBUTION OF INCOME
Cumulative percent distribution of aggregate income for
sample data
Fifths of
Families
Percent of
Income

Lowest onefifth
4

Lowest twofifths
14
Lowest threefifths
27
Lowest fourfifths
48
Lowest fivefifths
100
y
Cumulative 
proportion of
aggregate

income

x





Proportion of population
Christine Belledin
TCM 2010
PERFECT EQUITY AND PERFECT INEQUITY

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






x
x



Proportion of population







Proportion of population
Christine Belledin
TCM 2010
THE GINI INDEX
The ratio of the areas shown below.
Christine Belledin
TCM 2010
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.
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 = xn, 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  xn
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 .

In our case, Y  ln y and X  ln x.

We want to minimize S 
4
 Y  nX  .
i 1

2
i
i
This is a 1-variable optimization problem.
Christine Belledin
TCM 2010
FINDING N
𝑑𝑆
=
𝑑𝑛
4
2 𝑌𝑖 − 𝑛𝑋𝑖 ∙ (−𝑋𝑖 )
𝑖=1
If
𝑑𝑆
𝑑𝑛
= 0, then
4
4
𝑋𝑖2
𝑋𝑖 𝑌𝑖 = 𝑛
𝑖=1
𝑖=1
and
𝑛=
4
𝑖=1 𝑋𝑖 𝑌𝑖
4
2
𝑖=1 𝑋𝑖
Since 𝑋𝑖 = ln 𝑥𝑖 and 𝑌𝑖 = ln 𝑦𝑖 , we have
𝑛=
4
𝑖=1 ln(𝑥𝑖 ) ln(𝑦𝑖 )
.
4
2
ln
𝑥
𝑖
𝑖=1
Christine Belledin
TCM 2010
ANOTHER OPTION FOR N
ln 𝑦𝑖
𝑛 = 𝐴𝑣𝑒𝑟𝑎𝑔𝑒
ln 𝑥𝑖
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
CALCULATING THE GINI INDEX
Area bounded by Lorenz curve and 𝑦 = 𝑥:
1
𝐴𝑟𝑒𝑎 𝐴 =
0
1
1
𝑥 − 𝑥 𝑑𝑥 = −
2 𝑛+1
𝑛
Area of triangle for perfect equity:
𝐴𝑟𝑒𝑎 𝐵 =
1
2
Gini Index =
𝐴𝑟𝑒𝑎 𝐴
𝐴𝑟𝑒𝑎 𝐵
=1−
2
𝑛+1
.
Christine Belledin
TCM 2010
COMPARISON OF METHODS 1 AND 2 FOR SAMPLE DATA

y







y


x
x




Method 1: n = 2.0886
Gini index = 0.3525






Method 2: n = 2.4956
Gini index = 0.4278
Christine Belledin
TCM 2010
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.

Christine Belledin
TCM 2010