Measuring Poverty: Inequality Measures

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Measuring Poverty Inequality Measures

• Charting Inequality
• Share of Expenditure of Poor
• Dispersion Ratios
• Lorenz Curve
• Gini Coefficient
• Theil Index
• Comparisons
• Decomposition

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Poverty Measures, Lao PDR
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Income Distribution

• Types of analysis
• Functional distribution
• Size distribution
• Functional distribution
• income accrued to factors of production such as
  land, labor, capital and entrepreneurship
• Size distribution
• income received by different households or
  individuals

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What is Inequality?

• Dispersion or variation of the distribution of
  income/consumption or other welfare indicator
• Equality everyone has the same income
• Inequality certain groups of the population have
  higher incomes compared to other groups in the
  population

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Why measure inequality? (1)

• Indicator of well-being
• Position of individual relative to rest of
  population
• Position of subgroup relative to other
  subgroups
• Different measures, different focus
• Poverty measures (HC, PGI, SPGI, etc) focus on
  the situation of individuals who are below the
  poverty line the poor.
• Inequality is defined over the entire population,
  not only for the population below a certain
  poverty line.

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Why measure inequality? (2)

• Inequality is measured irrespective of the mean
  or median of a population, simply on the basis of
  the distribution (relative concept).
• Inequality can be measured for different
  dimensions of well-being consumption/expenditure
  and income, land, assets, and any continuous and
  cardinal variables.

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Charting Inequality Histogram

• Divide population into expenditure categories
• Example 20 of households are in category 4

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Example Income Classes
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Example Bar Chart, Income Classes

• Percentage of families falling in each class

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Example CDF of Per Capita Expenditure
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Distribution Quintile and Deciles
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Expenditure/Income-iles

• Divide population into groups ranked from
  poorest to richest based on expenditure (or
  income)
• Divide into 5 groups income or expenditure
  quintiles
• Lowest 20 or first quintile poorest
• Highest 20 or fifth quintile richest
• Divide into 10 groups income or expenditure
  deciles

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Expenditure per capita by Quintile, Viet Nam
(1993)
Quintile Per Capita Expenditure of Total Expenditure
First Lowest 518 8.4
Second Low-middle 756 12.3
Third Middle 984 16.0
Fourth Mid-upper 1,338 21.8
Upper Fifth 2,540 41.4
All 1,227 100.0
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Share of Income of Poorest, Korea
Income decile 2000 2001 2002 2003
1st 2.9 2.9 3.0 2.7
2nd 4.7 4.6 4.7 4.8
3rd 5.8 5.7 5.8 6.1
4th 6.9 6.8 6.9 7.1
5th 7.9 7.8 7.9 8.1
6th 9.1 9.1 9.2 9.3
7th 10.5 10.5 10.5 10.7
8th 12.2 12.3 12.4 12.5
9th 14.7 15.0 15.1 15.0
10th 25.4 25.4 24.6 23.8
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Inequality Measures Based on -iles

• Share of income/consumption of lowest ile
• Dispersion ratios

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Share of Consumption of the Poorest

• Definition Total consumption/income of the
  poorest group, as a share of total
  consumption/income in the population.
• Where
• N is the total population
• m is the number of individuals in the lowest x
  .

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Poorest Quintiles Share in National Income or
Consumption (UNSD, 2005)
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Dispersion Ratio

• Definition measures the distance between two
  groups in the distribution of expenditure (or
  income or some other characteristic)
• Distance average expenditure of the richest
  group divided by the average expenditure of the
  poorest group
• Example

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Dispersion Ratios Examples
Expenditure decile Median
1st 37,324
2nd 47,289
3rd 54,397
4th 62,929
5th 74,775
6th 89,478
7th 108,633
8th 129,890
9th 172,011
10th 267,214
(1) 10th1st
(2) 10th 1st 2d(Kuznets ratio)
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Lorenz Curve and Gini Ratio
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Lorenz Curve
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Lorenz Curve Interpretation (1)

• If each individual had the same consumption
  (total equality), Lorenz curve would be the line
  of total equality.
• If one individual had all the consumption, Lorenz
  curve would be the curve of total inequality.

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Lorenz Curve Interpretation (2)

• The further away from the line of total equality,
  the greater the inequality.
• Example Inequality is greater in country D than
  in country C.

C
D
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Comparing Lorenz Curves

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Lorenz Criterion

• Whenever one Lorenz curve lies above another
  Lorenz curve the economy with the first Lorenz
  curve is more equal, and the latter more unequal
• e.g. A is more equal D is more unequal
• When 2 curves cross, the Lorenz criterion states
  that we need more information (or additional
  assumptions) before we can determine which of the
  underlying economies are more equal
• e.g. curves B and C

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Constructing Lorenz Curve, Example (1)
Quintile Cumulative Share of Population (p) of Total Expenditure Cumulative share of expenditure (e)
First 20 8.4 8.4
Second 40 12.3 20.7
Third 60 16.0 36.7
Fourth 80 21.8 58.5
Fifth 100 41.4 100.0
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Constructing Lorenz Curve, Example (2)
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Gini Coefficient Definition

• Measure of how close to or far from a given
  distribution of expenditure (or income) is to
  equality or inequality
• Varies between 0 and 1
• Gini coefficient ? 0 as the expenditure/income
  distribution ? absolute equality
• Gini coefficient ? 1 as the expenditure/income
  distribution ? absolute inequality

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Gini Coefficient Lorenz Curve (1)
Area between line of equality and Lorenz Curve
(A) If A0 then G0 (complete equality).
A
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Gini Coefficient Lorenz Curve (2)
Area below Lorenz Curve (B) If B0 then G1
(complete inequality).
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Gini Coefficient Lorenz Curve (3)

• Gini coefficient (G) is the ratio of the area
  between the line of total equality and the Lorenz
  curve (A) to the area below the line of total
  equality (AB)

A
B
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Lorenz Curve and Gini Coefficient
e
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Gini Coefficient A Formula

• Heres one. (There are other formulations.)

• Where
• N is population size
• y is expenditure of individual
• f is rank of individual in the distribution

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Gini Coefficient s and s

• () Easy to understand, in light of the Lorenz
  curve.
• (-) Not decomposable the total Gini of the total
  population is not equal to the sum of the Ginis
  for its subgroups.
• (-) Sensitive to changes in the distribution,
  irrespective of whether they take place at the
  top, the middle or the bottom of the distribution
  (any transfer of income between two individuals
  has an impact, irrespective of whether it occurs
  among the rich or among the poor).
• (-) Gives equal weight to those at the bottom and
  those at the top of the distribution.

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Measures of Inequality, Example
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Poor people in Senegal get bigger share of income
than poor people in the US
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General Entropy Indexes

• ? represents the weight given to distances
  between incomes at different parts of the income
  distribution
• Sensitive to changes at the lower end of the
  distribution if a is close to zero
• Equally sensitive to changes across the
  distribution if a is 1 (Theil index)
• Sensitive to changes at the top of the
  distribution if a takes a higher value.

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GE(1) and GE(0)

• GE(1) is Theils T index
• GE(0), also known as Theils L, is called mean
  log deviation measure

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The Theil Index Definition

• Varies between 0 (total equality) and 1 (total
  inequality). The higher the index, the more
  unequal the distribution of expenditure (or
  income).

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Theil Index s and s)

• () Gives more weight to those at the bottom of
  the income distribution.
• () Can be decomposed into sub-groups the
  population Theil is the weighted average of the
  index for each sub-group where the weights are
  population shares of each sub-group
• (-) Difficult to interpret
• (-) Sensitive to changes in the distribution,
  irrespective of whether they take place at the
  top, the middle or the bottom of the distribution
  (any transfer of income between two individuals
  has an impact, irrespective of whether it occurs
  among the rich or among the poor).

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Atkinsons Index

• This class also has a weighting parameter e
  (which measures aversion to inequality)
• The Atkinson class is defined as
• Ranges from 0 (perfect equality) to 1

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Criteria for Goodness of Measures

• Mean independence If all incomes are doubled,
  measure does not change.
• Population size independence If population size
  changes, measure does not change.
• Symmetry If two individuals swap incomes, the
  measure does not change.
• Pigou-Dalton transfer sensitivity Transfer of
  income from rich to poor reduces value of
  measure.
• Decomposability It should be possible to break
  down total inequality by population groups,
  income source, expenditure type, or other
  dimensions.

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Checklist of Properties
Property Dispersion Gini Theil
Mean independence
Population size independence
Symmetry
Pigou-Dalton Transfer Sensitivity
Decomposability
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Inequality Comparisons

• Extent and nature of inequality among certain
  groups of households. This informs on the
  homogeneity of the various groups, an important
  element to take into account when designing
  interventions.
• Nature of changes in inequality over time. One
  could focus on changes for different groups of
  the population to show whether inequality changes
  have been similar for all or have taken place,
  say, in a particular sector of the economy.
• Other dimensions of inequality land, assets, etc

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Example Inequality Changes over Time
Year Poverty Rate Gini Coefficient
1985 48 0.4466
1988 40 0.4446
1991 40 0.4680
1994 36 0.4507
1997 32 0.4872
2000 34 0.4818
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Example Gini Ratios, Indonesia
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Decomposition of Inequality
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At One Point in Time (1)

• Inequality decompositions are typically used to
  estimate the share of total inequality in a
  country which results from different groups, from
  different regions or from different sources of
  income.
• Inequality can be decomposed into between-group
  components and within-group components. The
  first reflects inequality between people in
  different sub-groups (different educational,
  occupational, gender, geographic
  characteristics). The second reflects inequality
  among those people within the same sub-group.

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Example, Viet Nam (1993)
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Decomposition of Inequality, Egypt
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At One Point in Time (2)

• Inequality decompositions can be calculated for
  the General Entropy indices, but not for the Gini
  coefficient. For future reference, the formula
  is
• where fi is the population share of group j
  (j1,2, k),
• vj is the income share of group j
• yj is the average income in group j.

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Changes over Time (1)

• Changes in the number of people in various groups
  or allocation effects
• Changes in the relative income (expenditure) of
  various groups or income effects
• Changes in inequality within groups or pure
  inequality effects.

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Changes over Time (2)

• The formula can get complicated, and is typically
  used for GE(0) only, as follows

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Poverty Changes over Time (1)

• Poverty is fully determined by the mean income or
  consumption of a population, and the inequality
  in income or consumption in the population.
• Changes in poverty can result from changes in
  mean income/consumption growth or from
  changes in inequality.

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Poverty Changes Over Time (2)
Growth effect Inequality
effect
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Poverty Changes Over Time (3)

• Decomposition can be done as follows

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Conclusions Recommendations

• Inequality is a difficult concept to measure.
• For analysis, use several measures
• Lorenz curve
• Gini coefficient
• Dispersion ratios
• Share of expenditure of the poorest x
• Theil Index
• Analysis
• Comparisons across subgroups
• Comparisons over time