1. Chapter 6
Economic Inequality

2. Preview What is Economic Inequality? Measurement of Inequality
Anonymity, Population, Relative Income, and Dalton Principles
The Lorenz Curve
Complete Measures: Coefficient of Variation and the Gini Coefficient

3. What is Economic Inequality?
Economic inequality refers to the distribution of an economic attribute, such as income or wealth, across citizens within a country or across countries themselves.
For example, how is the total income in a country distributed across its citizens? What proportion of total wealth is held by the richest? the poorest?
Economists study inequality for
intrinsic reasons (reducing inequality can be seen as an objective in itself)
functional reasons (inequality may affect other indicators of economic performance, such as growth).
The first step in understanding economic inequality is to know how to measure it.

4. Measurement of Inequality
Suppose there are n individuals in a society, indexed by i = 1,2,3,…,n
An income distribution describes how much income is received by each individual i:
We are interested in comparing “relative inequality” between two such distributions (over time, or between regions/countries, etc.)

5. Four Criteria for Measuring Inequality
The Anonymity Principle
Names do not matter, incomes can always be ranked without reference to who is earning it
The Population Principle
As long as the composition of income classes remain unchanged, changing the size of the population does not matter for inequality
What matters are the proportions of the population that earn different levels of income

6. The Anonymity and Population Principles: An Illustration

7. Four Principles (Continued)
The Relative Income Principle
Only relative income matters, and not levels of absolute income
Scaling everyone’s income by the same percentage should not affect inequality

8. The Relative Income Principle: An Illustration

9. Four Measures (Continued)
The Dalton Principle
If a transfer is made from a relatively poor to a relatively rich individual, inequality must increase
“Regressive” transfers (taking from poor and giving to the rich) must worsen inequality

10. Measurement of Inequality: A Formal Summary
An inequality index is a function of the form
A higher value of this measure I(.) indicates greater inequality
The Anonymity Principle: the function I(.) is insensitive to all permutations of the income distribution among the individuals

11. Measurement of Inequality: A Formal Summary
The Population Principle: For every distribution ,
“cloning” has no effect on inequality
The Relative Income Principle: For every positive number ,

12. Measurement of Inequality: A Formal Summary
The Dalton Principle: The function I(.) satisfies the Dalton Principle, if, for every distribution
and every transfer

13. The Lorenz Curve
The Lorenz curve illustrates how cumulative shares of income are earned by cumulatively increasing fractions of the population, arranged from the poorest to the richest.
A graphical method for measuring inequality

14. The Lorenz Curve
Interpretation: point A: poorest 20% earn 10% of total income; point B: poorest 80% earn 70% of total income: in other words, richest 20% earn 30% of total income.
Connecting points such as A and B gives the Lorenz Curve
Lorenz curve begins and ends with 450 line: poorest 0% earn 0%, and poorest 100% is the whole population
If there is no inequality, then the Lorenz curve is the 450 line

15. The Lorenz Curve: Properties
If everyone has the same income, then the Lorenz curve is the 450 line
The slope of the Lorenz curve is the contribution of the person at that point to the cumulative share of national income
The “distance” between the 450 line and the Lorenz curve indicates the amount of inequality in the society
The greater is inequality, the further will the Lorenz curve be from the 450 line

16. Measuring Inequality using the Lorenz Curve

17. The Lorenz Criterion
The previous graph gives us a measure of inequality called the Lorenz Criterion
An inequality measure I is Lorenz-consistent if, for every pair of income distributions ,
whenever the Lorenz curve of lies to the right of

18. Lorenz Curves for Different Countries

19. Lorenz Curves for Different Countries

20. Complete Measures of Inequality
Can we summarize inequality by a number?
Attractive for policymakers and researchers
When Lorenz curves cross, we cannot rank inequality across two distributions
A numerical measure of inequality helps rank distributions unambiguously

21. Measuring Inequality
Let there be m distinct incomes, divided into j classes
In each income class j, the number of individuals earning that income is
The total population is then given by
The mean or average of the distribution is given by

22. Measures of Inequality
Range
Kuznets Ratio
Mean Absolute Deviation
Coefficient of Variation
Gini Coefficient

23. The Range
Difference in the incomes of the richest and the poorest individuals, divided by the mean
Very crude measure of inequality
Does not consider people between the richest and poorest on the income scale
Fails to satisfy the Dalton Principle (why?)

24. The Kuznets Ratio
The ratio of the share of income of the richest x % to the poorest y % where x and y represent population shares
Example: share of income of the richest 10% relative to the poorest 60%
These ratios are basically “snapshots” of the Lorenz curve
Useful when detailed inequality data in not available

25. The Mean Absolute Deviation
The sum of all income distances from average income, expressed as a fraction of total income
The idea: inequality is proportional to distance from mean income
May not satisfy the Dalton Principle, if regressive transfers are made between income classes that are all above or below the mean

26. The Coefficient of Variation
Essentially the standard deviation(sum of squared deviations from the mean), divided by the mean
Gives greater weight to larger deviations from the mean
Lorenz-consistent (satisfies the four principles)

27. The Gini Coefficient
Sum of the absolute differences between all pairs of incomes, normalized by (squared) population and mean income
Takes the difference between all pairs of income and sums the absolute differences
Inequality is the sum of all pair-wise comparisons of two-person inequalities
Double summation: first sum over all k’s, holding each j constant. Then, sum over all the j’s.
Most commonly used measure of inequality
The Gini is normalized by dividing by population squared (because all pairs are added and there are n2 such pairs) and mean income
Since each pair (yj-yk) is counted twice, the whole expression is divided by 2

28. The Gini Coefficient (continued)
Satisfies all four principles: Lorenz-consistent

29. Which Measure is More Appropriate?