Transcript Measuring Income Inequality

Measuring Income Inequality
Measuring Income Inequality
Income Inequality
• The extent of income concentration within a country or group.
• In many countries in the Middle East there is a high income per
capita relative to other countries in the world. However, whilst
the average person is better off such a statistic tells you nothing
of the distribution of income within the country.
• Principles of Inequality Measurement (Ray, 1998, pp. 174178)
Measuring Income Inequality
(1) Anonymity Principle: permutations of incomes amongst ‘n’
people should not matter for inequality measures, so that
y  y  y ........  y
1
2
3
n
(2) Population Principle: the size of a country’s population is
unimportant but what is important is the proportions of the
population that earn different levels of income.
(3) Relative Income Principle: it is not the absolute level of
income that is important to inequality but the relative size of
incomes.
Measuring Income Inequality
E.G.If Thabo has $1 and Tshepo has $2, then Tshepo has twice
as much as Thabo or has 2/3rds of all the income. If Thabo has
$1500 and Tshepo has $2000 then Tshepo has 4/7th of total
income and income inequality between the two has declined.
The Dalton Principle: If Distribution ‘A’ can be achieved from
Distribution ‘B’ by constructing a sequence of regressive (from
poor to rich) transfers then A is more unequal than B.
The issue of whether income inequality changes with growth and
development or whether initial income inequality is important to
sustainable growth is discussed in the Pro-Poor growth lecture.
Measuring Income Inequality
The Gini Coefficient:
• The ‘Gini’ is based on the income levels of individuals.
• Assume there are ‘m’ distinct income groups, each income group
is denoted by ‘j’ but there are ‘m’ such groups.
• Within each income group ‘j’ there are a number of individuals
earning that income level. The total number of people ‘n’ is
equal to
m
 nj
j 1
Measuring Income Inequality
• The average/mean of any income (y)
distribution is denoted by 
This average is simply the total income of all
individuals divided by the number of individuals.
Hence,
m
 1/ n  n j y j
j 1
Measuring Income Inequality
• The Gini coefficient does not take the difference between individual income
and the mean income as would be done if estimating the coefficient of variation.
• Instead the Gini calculates the income differences between all pairs of
incomes. These differences are then summed together with absolute values
being used so that information is not lost through values being both positive
and negative.
• The income differences are paired and are counted twice. (Q) Why? Well, as
well as taking the difference between income of individual ‘j’ and individual
‘k’, so that
( y j  yk )
= income difference
Measuring Income Inequality
• Clearly the differences in income will be identical with opposite signs. Since
we take absolute values of income differences, we could estimate one of the
differences and multiply by a factor of 2.
• The Gini coefficient is represented by,
m m
G  1/ 2n  (   n j nk y j  yk )
2
j 1 k 1
• There are 2 summations because firstly we sum over all the ks holding each j
constant, and then we do the same for the js, summing over all the js holding
each k constant.
• Essentially we are summing every single income differential in the sample. So
everything inside the brackets represents the sum of the income differentials
for the whole sample.
Measuring Income Inequality
•
This large number is then divided by 2n 2  . The ‘2’ comes from counting income
differentials twice when summing over ‘ks’ and then ‘js’. The and the mean income
terms are included so as to normalize the Gini coefficient.
•
The Gini coefficient is good at picking up increasing or decreasing income inequality.
For example, transfers of income from a low-income person to a high-income
person would mean that the income differential between these two persons would
increase, meaning the Gini coefficient would increase reflecting increasing income
inequality.
•
The Gini coefficient is also related to the Lorenz curve in a diagrammatical way. The
Gini is actually equivalent to the area between the 45 degree line and the Lorenz curve
divided by the entire area beneath the 45 degree line. Hence, the higher the Gini the
more ‘bowed’ the Lorenz curve and the higher the degree of income inequality.
Measuring Income Inequality
Example of calculating a Gini Coefficient.
South Africa is assumed to have 6 people earning respectively,
R1000, R100, R15000, R150, R50, R100
The Gini Coefficient here is?
Firstly we must calculate the income differences for each j and k..
Thus,
(R1000-R100)=R900
(R1000-15000)= -R14000
(R1000-R150)=R850
(R1000-R50)=R950
(R1000-R100)=R900
for
y j  R1000
•
Then do the same for each other income differential and sum for both j and k. The
summation term should be equal to R147,200. The mean income for SA is R2733.33.
•
The Gini coefficient is equal to 0.748. This illustrates a high degree of income
inequality within SA. (Whilst these numbers are made up, in reality the income
distribution in South Africa is amongst the highest in the world as we will see later).
Measuring Income Inequality
The Lorenz Curve
• Is a diagram to explain income inequality in a country.
• Is based on two pieces of information, income and population.
• Information is required on both and then formed into two variables that
reflect the cumulative value of income and the population.
• On the horizontal axis we sort the cumulative population in the ascending
order of income, with the lowest income first followed by the second lowest
and so on. Hence the first 20% of the population will necessarily be the
poorest 20% of the entire population.
• NOTE: Of importance here is to understand that incomes of peoples must
be placed in ascending order with the poorest first, followed by second
poorest………..up to the richest household/family/person in the country.
Measuring Income Inequality
• In a perfectly egalitarian society there would be no difference in
cumulative income levels between the bottom 20% and the top
20% or any other 20% since all income is distributed equally.
• This is a kind of utopia, and is represented in Figure 1 by the
straight line which is a 45 degree line, or put another way has a
gradient of +1, a 1% increase in the population is everywhere
equal to a 1% increase in cumulative income.
• In reality every country has a different Lorenz curve with none
being egalitarian!!!!
Measuring Income Inequality
•
In reality the Lorenz curve is away and to the right of the egalitarian curve.
•
The further the curve is away from the 45 degree line the more unequal the country’s
distribution of income is.
•
The Lorenz curve shown in Figure 1 illustrates that the poorest 40% of the population
instead of having 40% of the country’s income actually has just 9% of the country’s
income!
•
By comparison, when we look at the top 20% of the country’s population (the richest
20%) we see that instead of them having 20% of income they actually control 65% of
the country’s income!
•
Put another way you can say the poorest 80% of the country’s population control 35%
of the income.
•
Note that the area between the 45 degree line and the Lorenz curve forms the
numerator of the Gini coefficient, with the denominator being equivalent to the area
below the 45 degree line.
Measuring Income Inequality
• The issue to be aware of in estimating Lorenz curves is that for different
years for the same country the curves can cross, making interpretation
problematic.
Example from Ray (1998, pp. 183).
•
•
Group 1: 6 people with incomes, 25, 175, 300, 350, 600, 1500 = 2950
Group 2: 6 people with incomes, 50, 80, 200, 600, 820, 1200 = 2950
•
You can plot these distributions and calculate Lorenz curves for both groups, by
calculating the cumulative percentage of each income value in terms of the total
income for each group.
•
The Lorenz curves cross. Means that we can get from Group 2 to Group 1 by both
progressive (from rich to poor) and regressive (from poor to rich)
travels……..interpretation and hence policy becomes problematic and if this does arise
in a country then must investigate at a more micro level!!!
Measuring Income Inequality
Other Income Inequality Indices
Entropy Measures
E.g. Theil Index
See www.worldbank.org/poverty/inequal/dddeisqo.htm.
The entropy measures have the desirable property of being decomposable so
that inequality within a group (intra-group inequality) and between groups
(inter-group inequality) can be estimated. This can have important
implications for policy makers.
Measuring Income Inequality
Other Income Inequality Measures
Theil Index – Entropy Measure.
Entropy means ‘disorder’ – deviations from perfect income
equality.
The basic form of the Thiel Index is
1  yi   yi 
E (1)     ln 
n i y y
Measuring Income Inequality
Theil Index cont…
Where yi is income of individual ‘i’, and y is
the average income of the population, ‘n’.
Can calculate the Theil Index from given income
per capita data.
Measuring Income Inequality
Theil Index cont…
Individual (1)
Income
(2)
1
Average
Income (3)
Ratio of
income to
average
income (4)
Log (ratio of
income to
average
income) (5)
(6)=(4) x (5)
300
0.170
-0.768
-0.131
2
500
0.284
-0.547
-0.155
3
1000
0.568
-0.246
-0.139
4
2000
1.136
0.056
0.063
5
5000
2.841
0.453
1.288
1760
Sum of values =
0.926
Theil Index is
sum of 6
divided by
observations
0.185
Measuring Income Inequality
Theil Index cont…
Can decompose the Theil index into between group inequality and within group inequality.
E.g. Look at income inequality within racial groups and then between racial groups.
Practically this entails estimating Theil index for each of the racial groups and then summing
them according to some weight (e.g. population weight) to give total within-race inequality,
Tw.
Then calculate the Theil using the ratio of average income for each racial group/average
income of entire population to give between race income inequality, Tb.
Add the two together for overall Theil index = Tw+Tb
See “Describing Income Inequality Theil Index and Entropy Class Indexes” by Lorenzo
Giovanni Bellù, and Paolo Liberati, Food and Agriculture Organization of the United Nations,
FAO.
References
Bellu, L., and Liberati, P., (2006), “Describing Income Inequality Theil Index and Entropy
Class Indexes”, Food and Agriculture Organization of the United Nations, FAO.
Ray, D., (1998), Development Economics, Princeton University Press.
Foster, Greer and Thorbecke (1984), “A class of decomposable poverty measures”,
Econometrica, Vol 52(3), pp. 761-6.
Foster and Shorrocks (1988), “Poverty Orderings”, Econometrica, Vol 56, pp. 173-7.
Atkinson (1987), “On the Measurement of Poverty”, Econometrica, Vol 55(4), pp. 749-64.