gini coefficient

© 2003 By Default!Slide 1

Inequality Measures Inequality Measures

Celia M. ReyesCelia M. Reyes

Introduction to Poverty AnalysisIntroduction to Poverty AnalysisNAI, Beijing, ChinaNAI, Beijing, China

Nov. 1-8, 2005Nov. 1-8, 2005


Chapter 6Chapter 6

Inequality MeasuresInequality Measures

Celia M. ReyesCelia M. Reyes

July 6, 2004July 6, 2004


Definition of InequalityDefinition of Inequality

Poverty measures previously discussed depend Poverty measures previously discussed depend on the average level of income or consumption on the average level of income or consumption in a country, and the distribution of income or in a country, and the distribution of income or consumption. consumption.

Based on these two elements, poverty Based on these two elements, poverty measures then focus on the situation of those measures then focus on the situation of those individuals or households at the bottom of the individuals or households at the bottom of the distribution.distribution.



Measure of poverty focus on the situation of Measure of poverty focus on the situation of individuals or households who find themselves individuals or households who find themselves at the bottom of the income distribution.at the bottom of the income distribution.

Inequality is a broader concept than poverty in Inequality is a broader concept than poverty in that it is defined over the entire population, not that it is defined over the entire population, not only for the population below a certain poverty only for the population below a certain poverty line. line.



Most inequality measures do not depend on the Most inequality measures do not depend on the mean of the distribution, and this property of mean of the distribution, and this property of mean independence is considered to be a mean independence is considered to be a desirable property of an inequality measure.desirable property of an inequality measure.

Inequality measures can be calculated for any Inequality measures can be calculated for any distribution, not just for consumption, income or distribution, not just for consumption, income or other monetary variables, but also for land and other monetary variables, but also for land and other continuous and cardinal variables.other continuous and cardinal variables.



The simplest way to measure inequality is by The simplest way to measure inequality is by dividing the population into fifths (quintiles) from dividing the population into fifths (quintiles) from poorest to richest, and reporting the levels or poorest to richest, and reporting the levels or proportions of income (or expenditure) that proportions of income (or expenditure) that accrue to each level. accrue to each level.

Table 6.1 shows the level of expenditure per Table 6.1 shows the level of expenditure per capita, in ‘000 dong per year, for Vietnam in capita, in ‘000 dong per year, for Vietnam in 1993, based on data from the Vietnam Living 1993, based on data from the Vietnam Living Standards Survey.Standards Survey.


Lowest Low-mid Middle Mid-upper Upper Overall

Per capita expenditure('000 dong/year)

518 756 984 1,338 2,540 1,227

% of expenditure 8.4 12.3 16 21.8 41.4 100

Expenditure quintiles

Source: Vietnam Living Standards Survey 1993.

Table 6.1: Breakdown of expenditure perTable 6.1: Breakdown of expenditure per capita by quintile, Vietnam 1993 capita by quintile, Vietnam 1993


Gini Coefficient of InequalityGini Coefficient of Inequality

The Gini coefficient is the most commonly used The Gini coefficient is the most commonly used measure of inequality.measure of inequality.

The coefficient varies between 0, which reflects The coefficient varies between 0, which reflects complete equality and 1, which indicates complete equality and 1, which indicates complete inequality, i.e., one person has all the complete inequality, i.e., one person has all the income or consumption and all others have no income or consumption and all others have no income or consumption.income or consumption.

6.2 Commonly used measures of inequality



The Gini coefficient is based on the Lorenz curve, a The Gini coefficient is based on the Lorenz curve, a cumulative frequency curve that compares the cumulative frequency curve that compares the distribution of a specific variable (eq. Income) with the distribution of a specific variable (eq. Income) with the uniform distribution that represents equality.uniform distribution that represents equality.

To construct the Gini coefficient, graph the cumulative To construct the Gini coefficient, graph the cumulative percentage of households (from poor to rich) on the percentage of households (from poor to rich) on the horizontal axis and the cumulative percentage of horizontal axis and the cumulative percentage of expenditure (or income) on the vertical axis. This gives expenditure (or income) on the vertical axis. This gives the Lorenz curve as shown in figure 1.the Lorenz curve as shown in figure 1.


Figure 6.1. Lorenz CurveFigure 6.1. Lorenz Curve

0102030405060708090

100

0 20 40 60 80 100Cumulative % of population

Cum

ulat

ive

% o

f exp

endi

ture


.))((11

11

N

iiiii yyxxGini

When there are N equal intervals on the X-axis this simplifies to

.)(111

1

N

iii yy

NGini


Let xi be a point on the X-axis, and yi a point on the Y-axis. Then



The Lorenz curve maps the cumulative The Lorenz curve maps the cumulative income/consumption share on the vertical axis income/consumption share on the vertical axis against the distribution of the population on the against the distribution of the population on the horizontal axis. In this example, 40 percent of horizontal axis. In this example, 40 percent of the population obtains around 20 percent of the the population obtains around 20 percent of the total income.total income.

The Gini coefficient is calculated as the area A The Gini coefficient is calculated as the area A divided by the sum of the areas A & B.divided by the sum of the areas A & B.




The Gini coefficient is not entirely satisfactory. The Gini coefficient is not entirely satisfactory. Consider the criteria that make a good measure Consider the criteria that make a good measure of income inequality, namely:of income inequality, namely:

– Mean independence. This means that if all Mean independence. This means that if all incomes were doubled, the measure would incomes were doubled, the measure would not change. The Gini satisfies this.not change. The Gini satisfies this.




– Population size independence. If the Population size independence. If the population were to change, the measure of population were to change, the measure of inequality should not change, ceteris paribus. inequality should not change, ceteris paribus. The Gini satisfies this too.The Gini satisfies this too.

– Symmetry. If you and I swap incomes, there Symmetry. If you and I swap incomes, there should be no change in the measure of should be no change in the measure of inequality. The Gini satisfies this.inequality. The Gini satisfies this.




– Pigou-Dalton Transfer sensitivity. Under this Pigou-Dalton Transfer sensitivity. Under this criterion, the transfer of income from rich to criterion, the transfer of income from rich to poor reduces measured inequality. The Gini poor reduces measured inequality. The Gini satisfies this too. satisfies this too.

– Decomposability. This means that inequality Decomposability. This means that inequality may be broken down by population groups or may be broken down by population groups or income sources or in other dimensions. The income sources or in other dimensions. The Gini index is not decomposable or additive Gini index is not decomposable or additive across groups. That is, the total Gini of across groups. That is, the total Gini of society is not equal to the sum of the Ginis for society is not equal to the sum of the Ginis for its subgroups.its subgroups.




– Statistical testability. One should be able to Statistical testability. One should be able to test for the significance of changes in the test for the significance of changes in the index over time. This is less of a problem index over time. This is less of a problem than it used to be because confidence than it used to be because confidence intervals can typically be generated using intervals can typically be generated using bootstrap techniques. bootstrap techniques.



Generalized Entropy measuresGeneralized Entropy measures

There are a number of measures of inequality There are a number of measures of inequality that satisfy all six criteria. Among the most that satisfy all six criteria. Among the most widely used are the Theil indexes and the mean widely used are the Theil indexes and the mean log deviation measure. Both belong to the family log deviation measure. Both belong to the family of generalized entropy inequality measures. of generalized entropy inequality measures.

The general formula is given by:The general formula is given by:

N

i

i

yy

NGE

1

11)1(

1)(




Where is the mean income. Where is the mean income.

GE measures vary between 0 and GE measures vary between 0 and , with zero , with zero representing an equal distribution and higher representing an equal distribution and higher value representing a higher level of inequality. value representing a higher level of inequality.

Represents the weight given to distances Represents the weight given to distances between incomes at different parts of the between incomes at different parts of the income distribution, and can take any real value.income distribution, and can take any real value.

y




For lower values of For lower values of , GE is more sensitive to , GE is more sensitive to changes in the lower tail of the distribution, and changes in the lower tail of the distribution, and for higher values GE is more sensitive to for higher values GE is more sensitive to changes that affect the upper tail. The changes that affect the upper tail. The commonest values of commonest values of used are 0,1 and 2. used are 0,1 and 2.

GE(1) is Theil’s T index, written asGE(1) is Theil’s T index, written as

N

i

ii

yy

yy

NGE

1

)ln(1)1(




GE(0), also known as Theil’s L, is called mean GE(0), also known as Theil’s L, is called mean log deviation measure because it gives the log deviation measure because it gives the standard deviation of log(y):standard deviation of log(y):

N

i iyy

NGE

1

)ln(1)0(



Atkinson’s inequality measuresAtkinson’s inequality measures

This class also has a weighting parameter ε This class also has a weighting parameter ε (which measures aversion to inequality) and (which measures aversion to inequality) and some of its theoretical properties are similar to some of its theoretical properties are similar to those of the extended Gini Index. those of the extended Gini Index.

The Atkinson class is defined as:The Atkinson class is defined as:

)1(1

1

111

N

i

i

yy

NA



Decile Dispersion RatioDecile Dispersion Ratio

The decile dispersion ratio, which presents the The decile dispersion ratio, which presents the ratio of the average consumption or income of ratio of the average consumption or income of the richest 10% of the population divided by the the richest 10% of the population divided by the average consumption or income of the bottom average consumption or income of the bottom 10%. 10%.

This ratio can also be calculated for other This ratio can also be calculated for other percentiles. For instance, dividing the average percentiles. For instance, dividing the average consumption of the richest 5% – the 95th consumption of the richest 5% – the 95th percentile – by that of the poorest 5% – the 5th percentile – by that of the poorest 5% – the 5th percentile. percentile.



Decile Dispersion RatioDecile Dispersion Ratio

The decile ratio is readily interpretable, by The decile ratio is readily interpretable, by expressing the income of the top 10% (the expressing the income of the top 10% (the “rich”) as a multiple of that of those in the “rich”) as a multiple of that of those in the poorest decile (the “poor”). poorest decile (the “poor”).

However, it ignores information about incomes However, it ignores information about incomes in the middle of the income distribution, and in the middle of the income distribution, and does not even use information about the does not even use information about the distribution of income within the top and bottom distribution of income within the top and bottom deciles. deciles.



Inequality ComparisonsInequality Comparisons

One could draw a profile of inequality, which would One could draw a profile of inequality, which would look at the extent of inequality among certain groups look at the extent of inequality among certain groups of households. This informs on the homogeneity of of households. This informs on the homogeneity of the various groups, an important element to take into the various groups, an important element to take into account when designing interventions.account when designing interventions.

One may also analyze the nature of changes in One may also analyze the nature of changes in inequality over time. One could focus on changes for inequality over time. One could focus on changes for different groups of the population to show whether different groups of the population to show whether inequality changes have been similar for all or have inequality changes have been similar for all or have taken place, say, in a particular sector of the taken place, say, in a particular sector of the economy. economy.



In rural Tanzania, although rural incomes increased In rural Tanzania, although rural incomes increased substantially between 1983 and 1991, inequality substantially between 1983 and 1991, inequality increased (with the Gini coefficient increasing from increased (with the Gini coefficient increasing from 0.52 to 0.72), especially among the poor. 0.52 to 0.72), especially among the poor.

This can be linked to important reforms that took This can be linked to important reforms that took place in agricultural price policy, which intensified place in agricultural price policy, which intensified inequalities, with the poor and less-efficient farmers inequalities, with the poor and less-efficient farmers unable to participate in the growth experienced by unable to participate in the growth experienced by wealthier, more efficient farmers (Ferreira, 1996). wealthier, more efficient farmers (Ferreira, 1996).



It is often instructive to analyze other It is often instructive to analyze other dimensions of inequality. dimensions of inequality.

For instance, in a country where public health For instance, in a country where public health provision is well developed and reaches all provision is well developed and reaches all strata of the population, one could expect to see strata of the population, one could expect to see lower levels of inequality in health outcomes lower levels of inequality in health outcomes than in income levels. than in income levels.



Table 6.2, presents measures of inequality for Table 6.2, presents measures of inequality for rural Egypt. The analysis could also focus on rural Egypt. The analysis could also focus on the inequality of different consumption the inequality of different consumption categories or income sources. In Egypt, it was categories or income sources. In Egypt, it was found that agricultural income represented the found that agricultural income represented the most important inequality-increasing source of most important inequality-increasing source of income, while non-farm income has the greatest income, while non-farm income has the greatest inequality-reducing potential. inequality-reducing potential.


Table 6.2: Decomposition of income Table 6.2: Decomposition of income inequality in rural Egypt, 1997 inequality in rural Egypt, 1997

Non- farm 61 42 0.63 30Agricultural 67 25 1.16 40Transfer 51 15 0.85 12Livestock 70 9 0.94 6Rental 32 8 0.92 12All sources 100 100 100

Percentage contribution to Income Source Percentage of

households Share in total income (% )

Concentration index for the


Decomposition of income inequalityDecomposition of income inequality

In static decompositions, household and In static decompositions, household and personal characteristics, such as education, personal characteristics, such as education, gender, occupation, urban and rural, and gender, occupation, urban and rural, and regional location, are determinants of household regional location, are determinants of household income.income.

At least part of the value of any given inequality At least part of the value of any given inequality measure must reflect the fact that people have measure must reflect the fact that people have different educational levels, occupations, different educational levels, occupations, genders, and so on. This inequality is referred to genders, and so on. This inequality is referred to as the “between-group” component. as the “between-group” component.



But for any such partition of the population, But for any such partition of the population, some inequality will also exist among those some inequality will also exist among those people within the same subgroup; this is the people within the same subgroup; this is the “within-group” component.“within-group” component.

The Theil indexes, and those of the Generalized The Theil indexes, and those of the Generalized Entropy class, can be decomposed across Entropy class, can be decomposed across these partitions in an additive way, but the Gini these partitions in an additive way, but the Gini index cannot.index cannot.



Some examples of different measures of Some examples of different measures of inequality are found in Tables 6.3. inequality are found in Tables 6.3.

And for a typical decomposition of inequality, And for a typical decomposition of inequality, consider example, for Vietnam in 1993 set out in consider example, for Vietnam in 1993 set out in Table 6.4. Table 6.4.

© 2003 By Default!Slide 32Table 6.3: Expenditure Inequality inTable 6.3: Expenditure Inequality in Selected Less Developed Selected Less Developed Countries Countries

Country Gini coefficient

Theil T Theil L

Côte d'Ivoire, 1985- 86 0.435 0.353 0.325Ghana, 1987- 88 0.347 0.214 0.205J amaica, 1989 n/a 0.349 0.32Peru, 1985- 86 0.43 0.353 0.319Vietnam, 1992- 93 0.344 0.2 0.169Source: Reported in Dollar and Glewwe (1999), p.40.


Theil T Between-group inequality

Population share (% )

All Vietnam 0.2 100 Urban 0.196 0.044 (22% of total) 20 Rural 0.136 80

Table 6.4: Decomposition of expenditure Table 6.4: Decomposition of expenditure inequality by area, Vietnam inequality by area, Vietnam

Source: Dollar and Glewwe (1999), p.41.


6.4 Decomposition of income inequality6.4 Decomposition of income inequality

Of equal interest is which of the different income Of equal interest is which of the different income sources, or components of a measure of well-sources, or components of a measure of well-being, are primarily responsible for the observed being, are primarily responsible for the observed level of inequality. For example, if total income level of inequality. For example, if total income can be divided into self-employment income, can be divided into self-employment income, wages, transfers, and property income, one can wages, transfers, and property income, one can examine the distribution of each income source. examine the distribution of each income source.


6.4 Decomposition of income inequality6.4 Decomposition of income inequality

Raising one of the income sources by 1 percent, Raising one of the income sources by 1 percent, what would happen to overall inequality?what would happen to overall inequality?

Table 6.5 shows the results for the Gini Table 6.5 shows the results for the Gini coefficient for both income and wealth sources coefficient for both income and wealth sources in Peru (1997).in Peru (1997).

© 2003 By Default!Slide 36Table 6.5: Peru: Expected change in income Table 6.5: Peru: Expected change in income inequality resulting from a one percent change in inequality resulting from a one percent change in income source, 1997 (as percentage of Gini change)income source, 1997 (as percentage of Gini change)

Income source Expected change

Wealth sources

Expected change

Self- employment income - 4.9 Housing 1.9Wages 0.6 Durable goods - 1.5Transfers 2.2 Urban property 1.3

Property income 2.1Agricultural

property - 1.6Enterprises 0

gini coefficient

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