Income Inequality
Here we see the idea of income
distribution and the Lorenz Curve
1
Quintile
% of income
cumulative %
Bottom 20%
4
4
Second 20%
10
14
Middle 20%
16
30
Fourth 20%
23
53
Fifth 20%
47
100
Here we have a table of income as distributed across the economy
for a given year. Grouping is done on quintiles – population is
divided into fifths in a special way – here by income. Put people
in order, from low to high income. Then starting from the lowest
income person, take the first 20% of people and see what income
% they have. Then go to the next 20% and so on.
2
Cumulative %
of income
1
.8
Lorenz curve
.6
A
.4
.2
B
.2
.4
.6
.8
1
Cumulative % of
households
3
The Lorenz curve is a graph of the cumulative percent of
income by quintiles. Actually, in the graph each percent is
expressed in decimal form. So, for example, 20% is .2
The degree to which the curve is bowed out is an indication
of the income inequality in the economy, with a larger bow
meaning more inequality.
The Gini coefficient is the ratio of area A to area A + B. The
Gini has a coefficient value between 0 and 1, with 0 meaning
perfect equality of income.
Note that area A + B = .5(1)(1) = .5 because A + B is a
triangle and the area of a triangle is ½ of base times height.
Area A is found by working with areas of rectangles and
triangles.
4
Table 7-2 in the book has US income distribution for 2006. I
reproduce that table here (note it is in decimal form – what
the author calls “shares of income”).
Quintile, by the way, meansbreak up into groups of 20%.
Quintile
First 20%
Second
Third
Fourth
Fifth
Share of Income
.034
.086
.145
.229
.505
Cumulative Share
.034
.120
.265
.494
1.00
The 1.00 in the cumulative share looks like it should be .999,
but that is due to rounding the values in the real data to 3
decimals in the table here.
5
1
This is
the
Lorenz
curve.
.8
.6
.4
.2
.2
.4
.6
.8
1
1
.8
.6
.4
.2
.2
.4
a quintile at a time.
.6
.8
We
need
this
area
for
the
calcul
ation
of the
Gini
coeffi
cient.
Let’s
1 do it
1
.8
.6
.4
.2
.2
.4
.6
.8
1
In the first quintile we want this
part of the triangle made up of the
boundary of the quintile.
The area we want is then =
.5(.2)(.2) - .5(.2)(.034) = .02 .0034 = .0166
This area is made up of the whole
triangle and then subtracting out
the smaller triangle
Remember the area of a triangle is ½ of the base
times the height.
9
1
.8
.6
.4
.2
D
E
F
In the
second
quintile,
from .2
to .4 we
need to
add
areas D,
E, and
F. (in
class I
made a
mistake
here)
.2
.4
.6
.8
1
D = .5(.2)(.2), E = .2(.2 - .12), F = .5(.12 - .034)(.2), so D+E+F=.0446
1
.8
.6
.4
.2
G
H
I
In the
third
quintile,
from .4
to .6 we
need to
add
areas G,
H, and
I.
.2
.4
.6
.8
1
G = .5(.2)(.2), H = .2(.4 - .265), I = .5(.265 - .12)(.2), so G+H+I=.0615
1
.8
.6
.4
J
K
L
In the
fourth
quintile,
from .6
to .8 we
need to
add areas
J, K, and
L.
.2
.2
.4
.6
.8
1
J = .5(.2)(.2), K = .2(.6 - .494), I = .5(.494 - .265)(.2), so J+K+L=.0641
1
.8
.6
.4
.2
.2
.4
.6
.8
1
M
In the 5th quintile I really only want area N
but it may be easier to calculate M + N +
O and then subtract out M and O. So
N
.2(1 - .494) - .5(.2)(.2) - .5(1 - .494)(.2) =
.0306
O
The Gini coefficient here is (.0166 + .0446 + .0615 + .0641 +
.0306)/.5 = .2174/.5 = .4348
Note the Gini coefficient ranges from 0 to 1 and the closer to 1
the more unequal is the income across the community.
14