Module 5: Measuring (step 3) Inequality Measures

Module 5: Measuring (step 3) Inequality Measures

Topics 1. Why measure inequality? 2. Basic dispersion measures 1. Charting inequality for basic dispersion measures 2. Basic dispersion measures (dispersion ratios and share of consumption of the poorest) 3. Aggregate measures 1. Qualities of aggregate inequality measures. 2. Charting inequality for aggregate measures 3. Aggregate inequality measures (the Gini coefficient and the Generalized Enthropy Measures) 4. Decomposition of Inequality 5. Conclusions Module 5: Inequality Measures 2

1. Why measure inequality? (1) The relative position of individuals or households in society is an important aspect of their welfare. In addition, the overall level of inequality in a country, region or population group is also in itself an important summary indicator of the level of welfare in that group. Poverty measures focus on the situation of individuals at the bottom of the distribution. Inequality is a broader concept. It is defined over the entire population, not only for the population below a certain poverty line. Module 5: Inequality Measures 3

1. 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, but also land, assets, and any continuous and cardinal variables. Module 5: Inequality Measures 4

2. Charting inequality. (1) T Putting information about expenditure, consumption or income distribution into diagrammatic form is often a good way to present the information. T Representations for basic dispersion measures: 1. Pen s parade 2. Frequency distribution T Representations for aggregate inequality measures: T Cumulative frequency distribution T Lorenz curve Module 5: Inequality Measures 5

2. Charting inequality. (2) Illustrating income distribution with a Pen s parade approach: ranking persons by their incomes/consumption Height/incomes richest poorest average median Persons ranked Module 5: Inequality Measures 6

2. Charting inequality. (3) Height/incomes Characteristics of the distribution Steep curve: few persons, large differences Flat curve: many persons, small differences Persons ranked Module 5: Inequality Measures 7

2. Charting inequality. (4) Pen s parade Frequency distribution Not normal: skewed to the left, resembles lognormal distribution Module 5: Inequality Measures 8

2. Charting inequality. (5) Frequency distribution: The graph (histogram) plots the percentage of households in specific expenditure groups. For instance, the following frequency distribution shows that 20% of individuals are in category 4 [ or f(4)=0.2]. Percentage of population 40 35 30 25 20 15 10 5 0 Density function f(y) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Expenditure categories Module 5: Inequality Measures 9

2. Charting inequality. (6) Cumulative frequency distribution: This graph plots the cumulative frequency the percentage of households with expenditure at or below a level. In relation to the previous graph, F(y) is the total area below and to left of f(y). [F(4) = f(4)+f(3)+f(2)+f(1) = 20+35+12+4=71%] cumulative frequency (%) 100 90 80 70 60 50 40 30 20 10 0 Cumulative density function F(y) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Expenditure category Module 5: Inequality Measures 10

3. Dispersion ratios. (1) Definition: The dispersion ratios measure the distance between two groups in the distribution of expenditure (or income). Typically, they measure the average expenditure/income of the richest x% divided by the average expenditure/income of the poorest x%. There are different alternatives, the most frequently used are for deciles and quintiles. (a decile is a group containing 10% of the total population, a quintile is a group containing 20% of the population). Module 5: Inequality Measures 11

3. Dispersion ratios. (2) Definition of dispersion ratios: Average income of top group i Decile ratio= Average income of bottom group j Percentile ratio= Lower cutpoint for top group i Upper cutpoint for bottom group j Group i and group j can be defined as deciles (1/10), quintiles (1/5), quartiles (1/4), etc. Module 5: Inequality Measures 12

3. Dispersion ratios. (3) Advantages: (+) The decile ratio and percentile ratio are readily understandable. Disadvantages: (-) The value of decile ratio is very much vulnerable to extreme values and outlayers, especially in case of estimates from small samples (-) No axiomatic basis: it is not derived from principles about equity Module 5: Inequality Measures 13

4. Share of consumption of the poorest. (1) This measure presents the total income consumption of the poorest group, as a share of total income/consumption in the population. C ( x) m i= 1 = N i= 1 y y i i Where N is the total population m is the number of individuals in the lowest x %. Module 5: Inequality Measures 14

4. Share of consumption of the poorest. (2) Advantages: (+) The share is readily understandable. (+) If a society is most concerned about the way the poor are living, it might be a better measure. Indeed, the share of consumption of the poorest is insensitive to changes at the top or the middle of the distribution. Module 5: Inequality Measures 15

Incomes 5. Summary of some basic measures of distribution Note: high dispersion in the extreme deciles lower dispersion elsewhere Decile ratio Percentile ratio Lowest decile 2nd decile Decile share... Highest decile Persons ranked Module 5: Inequality Measures 16

6. Qualities of aggregate measures of inequality. (1) Before discussing the various aggregate measures, we list the qualities that we could expect from inequality measures: 1. Mean independence: if all expenditure or income were doubled, the measure should not change. 2. Population size independence: if the population were to change, the measure should not change (if we merged identical distributions, inequality would be the same). Module 5: Inequality Measures 17

6. Qualities of inequality measures. (2) 3. Symmetry: if two individuals were to swap income or expenditure, the measure should not change. 4. Principle of transfers (Pigou-Dalton): if rich households were to transfer income or expenditure to poorer households, the measure should be reduced. 5. Decomposability: It should be possible to break down total inequality by population groups, income source, expenditure type, or other dimensions. Module 5: Inequality Measures 18

7. Charting aggregate inequality. (1) The Lorenz curve: The most frequently used chart. The curve maps the cumulative expenditure share on the vertical axis against the cumulative distribution of the population on the horizontal axis. In this example, 40 percent of the population obtains just under 20 percent of the total consumption. Cumulative % of consumption 100 90 80 70 60 50 40 30 20 10 0 0 20 40 60 80 100 Cumulative % of population Module 5: Inequality Measures 19

7. Charting aggregate inequality. (2) If each individual had the same consumption, or total equality, the Lorenz curve would be the line of total equality. If one individual had all the consumption (or income), the Lorenz curve would pass through the points (0,0), (100,0) and (100,100), the curve of total inequality. Cumulative % of consumption 100 90 80 70 60 50 40 30 20 10 0 Line Line Line Line of of of of total total total total equality equality equality equality Lorenz curves Curve of total inequality 0 20 40 60 80 100 Solid line: smaller inequality Dashed line: larger inequality Cumulative % of population Module 5: Inequality Measures 20

8. The Gini coefficient (1) Definition: The Gini coefficient is the most commonly used measure of inequality. It is defined as half the average of all pairwise absolute deviations between people, relative to the mean income/consumption Its value varies between 0, which indicates complete equality, and 1, which indicates complete inequality. It measures the extent to which the distribution is far from that of total equality. Module 5: Inequality Measures 21

8. The Gini coefficient (2) Formal definition: There are different formulae, the classical is as follows: Gini = n n yi y 1 j 1 2n( n 1) y i= = j Where y i and y j are individual income/consumption with a mean of, n is the total number of observations, y Module 5: Inequality Measures 22

8. The Gini coefficient (3) The Gini coefficient can be related to the Lorenz curve: It is calculated as the area A divided by the sum of the areas A and B. On the graph earlier, the Gini is 0 if there is total equality, and 1 if there is total unequality Cumulative % of consumption 100 90 80 70 60 50 40 30 20 10 0 A A B 0 20 40 60 80 100 Cumulative % of population Module 5: Inequality Measures 23

Gini 8. The Gini coefficient (4) An easily computable formula that creates a link to graphical presentation: 2 Cov( y 1 N i = N i= 1, y f i i ) Gini = 2 n i csum csum ( fi ) ( f ) n N csum csum ( yi ) ( y ) n Where (c)sum=(cumulative) sum, y i =income or consumption of the i th person, all incomes are in increasing rank order (f is for the rank), n=number of persons (in the sample) Module 5: Inequality Measures 24

8. The Gini coefficient (5) Advantages (+) and disadvantages (-) : (+) The coefficient is easy to understand, in light of the Lorenz curve. (-) The coefficient is not additive: the total Gini of the total population is not equal to the sum of the Ginis for its subgroups. Module 5: Inequality Measures 25

8. The Gini coefficient (6) (-) The coefficient is 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). (-) The coefficient gives equal weight to those at the bottom and those at the top of the distribution. Module 5: Inequality Measures 26

9. The Generalized Entrophy indices (1) The general formula for the General Entropy indices is: GE( α) α N 1 1 y = i α 2 α N i= 1 y 1 Where y i = the income/expenditure and N=the number of the individuals and α is a parameter representing the weight given to levels of well being at different parts of the distribution Module 5: Inequality Measures 27

9. The Generalized Entrophy indices (2) Depending on the value of the GE(0) = GE(2) = MLD GE(1) = Theil CV 2 = = = 1 1 N 1 N y i= 1 i= 1 1 N N N log yi y N i= 1.log y α i y y i parameter: y i y y 2 Module 5: Inequality Measures 28 1 2

9. The Generalized Entrophy indices (3) Characteristics of the measures parameter α α index sensitivity = 0 MLD lower end (mean log deviation) = 1 Theil index middle α = 2 CV upper end (coefficient of variation) Module 5: Inequality Measures 29

9. The Generalized Entrophy indices (4) Advantages and disadvantages: (+) The value of the indices (like in case of Gini), varies between 0 and 1 (+) different weights can be assigned to various levels of distribution, depending on value preferences (+) GE(a) indices can be decomposed into subgroups : the population GE(a) is the weighted average of the index for each sub-group where the weights are population shares of each sub-group, unlike the Gini coefficient. Module 5: Inequality Measures 30

9. The Generalized Entrophy indices (5) (-) It is difficult to interpret, unlike the Gini coefficient. (-) it is sensitive to changes in the distribution, irrespective of whether they take place at the top, the middle or the bottom of the distribution, unlike the share of the poorst from incomes/consumption (any transfer of income between two individuals has an impact, irrespective of whether it occurs among the rich or among the poor). Module 5: Inequality Measures 31

10. Decomposition of inequality. (1) T 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. T Inequality can be decomposed into between-group components and within-group components. The first reflects inequality between people in different subgroups (different educational, occupational, gender, geographic characteristics). The second reflects inequality among those people within the same subgroup. Module 5: Inequality Measures 32

10. Decomposition of inequality. (2) Inequality decompositions can be calculated for the General Entropy indices, but not for the Gini coefficient. For future reference, the formula is: I k k = + = α 1 α 1 I.. ( ) + W I B v j f j GE α j 2 j= 1 α α j= 1 where f i is the population share of group j (j=1,2, k), v j is the income share of group j; y j is the average income in group j. f j. y y j α 1 Module 5: Inequality Measures 33

11. Conclusions. T T T Inequality is a difficult concept to measure. As for poverty, each measure has its own advantages and disadvantages. In order to get a balanced picture, various measures should be computed and used in the analysis Module 5: Inequality Measures 34