christine belledin ncssm [email protected]. fifths of families percent of income lowest fifth4...

THE GINI INDEX: USING CALCULUS TO MEASURE INEQUITY

Christine Belledin NCSSM [email protected]

DATA USED TO QUANTIFY DISTRIBUTION OF INCOME

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

Percent distribution of aggregate income for sample data

DATA USED TO QUANTIFY DISTRIBUTION OF INCOME


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

x

y

Proportion of population

Cumulative proportion of aggregate income

PERFECT EQUITY AND PERFECT INEQUITY What would the cumulative graph look like if the

distribution was perfectly equitable? Perfectly inequitable?


x

y





x

y

Perfect Equity Perfect Inequity

THE GINI INDEX


The ratio of the areas shown below.

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.


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.

We now use our knowledge of calculus to find a least-squares estimate of n.


ny x

ln lny n x

Consider the linear equation

In our case, and

We want to minimize

This is a 1-variable optimization problem.


.Y nX

lnY y ln .X x

4

2

1

.i ii

S Y nX

FINDING N


𝑑𝑆𝑑𝑛 = 2ሺ𝑌𝑖 − 𝑛𝑋𝑖ሻ∙(−𝑋𝑖)4𝑖=1

If 𝑑𝑆𝑑𝑛 = 0, then

𝑋𝑖𝑌𝑖4

𝑖=1 = 𝑛 𝑋𝑖24

𝑖=1

and

𝑛 = σ 𝑋𝑖𝑌𝑖4𝑖=1σ 𝑋𝑖24𝑖=1

Since 𝑋𝑖 = lnሺ𝑥𝑖ሻ and 𝑌𝑖 = lnሺ𝑦𝑖ሻ, we have

𝑛 = σ ln(𝑥𝑖) ln(𝑦𝑖)4𝑖=1σ ሾlnሺ𝑥𝑖ሻሿ24𝑖=1 .

ANOTHER OPTION FOR N


𝑛 = 𝐴𝑣𝑒𝑟𝑎𝑔𝑒ቆlnሺ𝑦𝑖ሻ lnሺ𝑥𝑖ሻቇ

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.

CALCULATING THE GINI INDEX


Area bounded by Lorenz curve and 𝑦= 𝑥: 𝐴𝑟𝑒𝑎 𝐴= න 𝑥− 𝑥𝑛1

0 𝑑𝑥= 12− 1𝑛+ 1

Area of triangle for perfect equity:

𝐴𝑟𝑒𝑎 𝐵= 12

Gini Index = 𝐴𝑟𝑒𝑎 𝐴𝐴𝑟𝑒𝑎 𝐵 = 1− 2𝑛+1.

COMPARISON OF METHODS 1 AND 2 FOR SAMPLE DATA


x

y

x

y

Method 1: n = 2.0886 Gini index =

0.3525

Method 2: n = 2.4956 Gini index =

0.4278

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 ncssm [email protected]. fifths of families percent of income lowest fifth4...

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