Srinivasulu Rajendran
Centre for the Study of Regional Development (CSRD)
Jawaharlal Nehru University (JNU)
New Delhi
India
r.srinivasulu@gmail.com
Objective of the session
1. To measure probability
distribution of wealth &
inequality through Lorenz
Curve and Gini Coefficient
1. What is the procedure to
measure wealth inequality
through Lorenz Curve & Gini
Coefficient?
2. How do we interpret results?
What is Lorenz Curve
 the Lorenz curve is a graphical representation of the
cumulative distribution function of the empirical
probability distribution of wealth
 To measure wealth inequality.
 The Lorenz curve can give a clear graphic
interpretation of the Gini coefficient. Let’s make the
Lorenz curve of per capita FOOD expenditure
distribution of Bangladesh.
Step 1
 we draw a set of axes in which the cumulative
percentage of wealth is measured along the y-axis
while the cumulative percentage of households is
measured along the x-axis. Usually, the graph’s axes are
closed off to form a box
Step 2
 to order the distribution from the smallest through
to the largest, thereby enabling us to answer the
following sequential questions:
A. what proportion of wealth is owned by the poorest
10 percent of the population?
B. what proportion of wealth is owned by the poorest
20 percent of the population?
C. what proportion of wealth is owned by the poorest
30 percent of the population?
This process continues until we reach the
point where 100 per cent of wealth is owned
by 100 per cent of the population.
Step 3
Assume that we live in a truly
equal society
 If this were to be the case, the relationship would be
such that as we move along the x-axis, each 10 per cent
increment of households would own an additional 10
per cent of wealth.
 In this case, the line we would draw would be a straight
line emanating from the origin. This is known as the
line of absolute equality and will have a slope of 45
degrees.
Step 4
 Finally, we can insert a line that is based on the data
set available to us. In this case, the line will bow away
from the line of absolute equality. The more unequal
society is, the further it will deviate away from the line
of absolute equality. It is this line which is known as
the Lorenz Curve.
Step 5
STATA Program
For District 1
glcurve pcmfx if district ==1, gl(gl2) p(p2) lorenz
twoway line gl2 p2 , sort || line p p , ///
xlabel(0(.1)1) ylabel(0(.1)1) ///
xline(0(.2)1) yline(0(.2)1)
///
title("Lorenz curve") subtitle("Monthly Per Capita Food Expenditure - Manikganj")
///
legend(label(1 "Lorenz curve") label(2 "Line of perfect
equality")) ///
plotregion(margin(zero)) aspectratio(1) scheme(economist)
1
2
Lorenz curve
Monthly Per Capita Food Expenditure - Manikganj
Lorenz curve
Monthly Per Capita Food Expenditure - Mymensingh
Lorenz curve
Line of perfect equality
1
.9
.8
.7
.6
.5
.4
.3
.2
.1
0
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1
Cumulative population proportion
Lorenz curve
A
O
3
A
O
4
Lorenz curve
Monthly Per Capita Food Expenditure-Kishoreganj
Line of perfect equality
1
.9
.8
.7
.6
.5
.4
.3
.2
.1
0
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1
Cumulative population proportion
Lorenz curve
Monthly Per Capita Food Expenditure-Jessore
Lorenz curve
Lorenz curve
A
O
Line of perfect equality
1
.9
.8
.7
.6
.5
.4
.3
.2
.1
0
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1
Cumulative population proportion
Line of perfect equality
1
.9
.8
.7
.6
.5
.4
.3
.2
.1
0
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1
Cumulative population proportion
A
O
Step 6
Interpretation
 In above figures, the line of absolute equality is
labelled OA. However, the Lorenz curve assumes a
different shape in the four diagrams. In Figure 2,
Mymensingh District, it can be seen that the poorest
sections of society command a very small proportion
of the country’s wealth.
 In Figure 4 Jessore District, societal wealth remains
unevenly distributed, but the poorer households are
(on average) better off as compare to Mymensingh
District.
 One of the advantages of using the Lorenz Curve is
that it provides a visual representation of the
information we wish to consider, in this case the
inequality of wealth prevailing in society.
 We could superimpose several Lorenz Curves onto the
same diagram to show changes in the way in which
wealth has been distributed across society at various
points in time.
 Even if the shape of the Lorenz Curve is not changing
significantly, poorer members of society may still be
much better off in terms of what they can afford to
buy.
 In other words, they are relatively no better off, but in
terms of spending power, they have the opportunity to
enjoy a wider range of luxury items. Commodities
which were considered to be luxuries fifty years ago
(for example, televisions and telephones) are now
taken for granted by most people.
Hands-on exercises
 Now repeat this exercise based on per capita total
expenditure for village adopted technology and not
adopted technology and compare its Lorenz curve
with the Lorenz curve for the whole area. What
conclusions emerge?
 Now repeat this exercise per capita total expenditure
for female head and male head household and
compare its Lorenz curve with the Lorenz curve for the
whole area. What conclusions emerge?
The Gini Coefficient
 The
Gini coefficient is to
measure
the
degree
of
concentration (inequality) of a
variable in a distribution of its
elements. It is the ratio of the
area between the Lorenz Curve
and the line of absolute
equality (numerator) and the
whole area under the line of
absolute
equality
(denominator).
Based
on
Figure Four, it can be seen that
the Gini Coefficient = C/0AB.
Lorenz curve
Monthly Per Capita Food Expenditure-Jessore
Lorenz curve
O
Line of perfect equality
1 A
.9
.8
.7
.6
.5
C
.4
.3
.2
.1
0
B
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1
Cumulative population proportion
Interpretation for Gini Coefficient
 The extreme values of the Gini Coefficient are 0 and 1.
 These are often presented in statistical publications as
percentages. Hence, the corresponding extreme values
are 0% and 100%.
 The former implies perfect equality (in other words,
everyone in society has exactly the same amount of
wealth) whereas the latter implies total inequality in
that one person has all the wealth and everyone else
has nothing.
 Clearly, these two extremes are trivial;
The key thing to bear in mind is
that the lower the figure that
Gini Coefficient takes (between
0% and 100%), the greater the
degree of prevailing equality.
STATA Program
 Atkinson, inequal, lorenz, relsgini
 These four ado-files provide a variety of measures of
inequality.
 atkinson computes the Atkinson inequality index using
the inequality aversion para-meter(s) specified in the
parameter list.
 inequal displays the following measures: relative mean
deviation, coefficient of variation, standard deviation of
logs, Gini index, Mehran index, Piesch index, Kakwani
index, Theil entropy index, and mean log deviation.
 lorenz displays a Lorenz curve.
 relsgini computes the Donaldson-Weymark relative SGini using the distributional sensitivity parameters
specified in the parameter list.
“inequal” command












inequality measures of pcmfx
relative mean deviation
coefficient of variation
standard deviation of logs
Gini coefficient
Mehran measure
Piesch measure
Kakwani measure
Theil entropy measure
Theil mean log deviation measure
.16517609
.47881288
.41682198
.23579119
.32276405
.19230475
.05184636
.09620768
.09104455
Hands-on Exercise
Let’s continue using the per capita total expenditure to
calculate inequality measures:
i. Compute the Gini coefficient, the Theil index and the
Atkinson index with inequality aversion
parameter equal to 1 for the four districts.
Gini
Theil
Atkinson
All regions
________ ________
________
District wise:
________ ________
________
Hands-Exercise
ii. Now repeat the above exercise using two
decile dispersion ratios and the share of
consumption of poorest 25%. STATA
command xtile is good for dividing the
sample by ranking. For example, to calculate
the consumption expenditure ratio between
richest 20% and poorest 20%, you need to
identify those two groups.
Reference for inequality
 http://web.worldbank.org/WBSITE/EXTERNAL/TOPI
CS/EXTPOVERTY/EXTPA/0,,contentMDK:20238991~
menuPK:492138~pagePK:148956~piPK:216618~theSiteP
K:430367,00.html