# The Gini Index

Document Sample

```					The Gini Index

Paige Stillwell
and
Tanya Picinich
Overview
• Lorenz Curve
– Perfect Income Equality / Complete Income Inequality
• Gini Index
– Calculation Examples
•   United States Gini Index
•   Riemann Sum and Trapezoidal Rule
•   Potential Issues with the Gini Index
•   Why is the Gini Index Important?
•   Gini Index Comparison Across Countries
•   United States Gini Index Over Time
The Lorenz Curve
• Shows the share of total income of the population
from 0 to t where t is the rank of a household’s income
as a percentage of the total population
Lorenz Curve for 2007 United States

1.2

1
Income Quintiles

0.8

0.6

0.4

0.2

0
0   0.2      0.4       0.6       0.8        1   1.2
t values
Perfect Income Equality: Utopia
• 20% of the population makes 20% of the income and so on
• Lorenz curve has the equation y=x
Perfect Income Equality

1.2

1
Income Quintiles

0.8

0.6

0.4

0.2

0
0   0.2    0.4         0.6          0.8   1   1.2
t values
Complete Inequality
• One person makes all the money. Everyone else
makes nothing
Total Inequality of Income

1.2

1
Income Quintiles

0.8

0.6

0.4

0.2

0
0    0.2     0.4          0.6           0.8   1   1.2
t values
Gini Index
• Gives information about the income inequality of a country in
one number
• Ranges from 0 to 1
• Calculated as the area between perfect equality (y=x) and the
Lorenz curve
1
G(t )  2 *  (t  Lorenz (t )) dt
0
Perfect Income Equality

1.2

1
• The Gini Index for
Income Quintiles

0.8

0.6

0.4
perfect equality is 0
0.2

0
0   0.2    0.4         0.6          0.8   1   1.2
t values

• The Gini Index for
total inequality is 1
Example: L(t) = t2
• What’s the Gini Index for a country that has a
Lorenz curve of L(t) = t2 ?
1
G  2 *  (t  t 2 )dt
0

 t 2 t3  1
G  2*  
 2 3 0
        
 1 1
G  2*    0
 2 3
1
G   .333
3
Example: L(t) = t3
• Will the Gini Index increase or decrease from the
previous example if the equation changes to L(t) = t3?
1
G  2 *  (t  t 3 )dt
0

 t 2 t4  1
G  2*  
2 4 0
        
1 1
G  2*    0
2 4
1
G   .50
2
Reality: How the Gini Index is
Calculated
• In real life we are not given functions
• We must use data points to find the Gini Index
• We use income quintiles which are made available by
the U.S. Census Bureau
United States Income Quintiles 2007
Income Lower Second Middle Fourth Highest
Quintiles 5th        5th       5th        5th    5th
% of
total       55.3     25.9      11.3       4.4    3.1
income
The United States Gini Index 2007
• The United States Gini
Index for 2007 is quoted
by the U.S. Census
Bureau as G(t)=.463
• This value may vary
depending on what is
considered as income
and whether individuals
or households are
examined
Riemann Sum and
Trapezoidal Rule
• Approximates an integral when the equation of the function is
unknown
• Using the 2007 data we can approximate the Gini Index

United States Income Quintiles 2007
t     0       .20      .40      .60      .80      1

L(t)    0       .553     .812     .925     .969     1

• Riemann sums use either right or left endpoints to form rectangles
• The trapezoidal rule is the average of the right and left endpoint
approximations
Riemann Sum
• The definition of an integral of f from a to b is:

 f ( x)dx  lim  f x * x
a                               n
*
i
b                       n i 1

• Using right end points:                                       • Using left end points:
xi*  xi                                                       xi*  xi 1

 f ( x )dx  lim  f x * x                                  f ( x )dx  lim  f x          * x
a                                                              a                      n
n

i                                                              i 1
n  i 1                                      b               n  i 1
b

 x f ( x1 )  f ( x 2 )  f ( x3 )  f ( x4 )  f ( x5 )    x f ( x 0 )  f ( x1 )  f ( x2 )  f ( x3 )  f ( x 4 )
Riemann Sum
• Using right end points:                 •Using left end points:

R5 
1
.553  .812  .925  .969  1   L5 
1
0  .553  .812  .925  .969
5                                        5
        1                                      1
 .8518 -                              .6518 - 
        2                                      2
 .3518                                  .1518
     1                                     1
G  2 * R 5                           G  2 * L5  
     2                                     2
G  .7036                                G  .3036
Trapezoidal Rule
• Basically an average of the right and left endpoints from
the Riemann Sum
x
b

   f ( x )dx       *  f ( x0 )  2 f ( x1 )  2 f ( x2 )  ...  2 f ( xn 1 )  f ( xn )
a
2
ba
where x                    and xi  a  ix
n

• Using the 2007 Income Quintiles as data points we get:
T5 
.2
0  2(.553)  2(.812)  2(.925)  2(.969)  1
2
T5  .7518
Trapezoidal Rule
1         1
T5 -  .7518 - 
2         2
 .2518

G  2 * Area
G  2 * .2518
G  .5036

Actual Gini Index = .463
Potential Issues
• The Gini Index glosses over many details
• Gives a more accurate picture of the relationship
between the upper class and middle class than
the relationship between the upper class and
lower class
• Does not reflect unreported income and money
Ways to calculate a more accurate
Gini Index:
• Jackknife                       • Bootstrap
– Calculate the Gini Index       – Calculate the Gini Index
many times, but remove           from a random sample of
one data point each time         the income data many
times
– Produces a mean
distribution and a             – Produces a mean
standard deviation for the       distribution and a
Gini Index                       standard deviation for the
Gini Index
Why is the Gini Index so important?
• Compiles information about income inequality
into 1 number
• Allows for comparisons with other countries
• Shows how income inequality changes over time
• This information has great social, political, and
economic implications
Comparisons Across Countries
Countries With Low Gini Indices             Countries With High Gini Indices
Namibia                    70.7
Sweden                     23
Lesotho                    63.2
Denmark                    24                Central African Republic   61.3

Finland                    26                Haiti                      59.2
Bolivia                    59.2
Brazil                     56.7

• The Gini index for the entire world = 56 - 66
• Tolerance for inequality of income varies between countries
– US = higher tolerance for income inequality
– European countries = lower tolerance for income inequality
– Underdeveloped countries have a higher income inequality
United States Over Time
United States Gini Index over Time

50

40
Gini

30

20

10

0
1920   1930   1940   1950   1960   1970   1980   1990   2000   2010   2020
Year
Questions?

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 177 posted: 2/19/2012 language: English pages: 22
How are you planning on using Docstoc?