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Chapter 6
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
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.
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.)
iy
1 2, ,...., ny y y
1.1. The Anonymity PrincipleThe Anonymity Principle Names do not matter, incomes can always be
ranked without reference to who is earning it
2.2. The Population PrincipleThe 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
1 2 3,..., ny y y y
3.3. The Relative Income PrincipleThe 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
4.4. The Dalton PrincipleThe 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
An inequality index inequality index is a function of the form
A higher value of this measure I(.) indicates greater inequality
The Anonymity PrincipleThe Anonymity Principle: the function I(.) is insensitive to all permutations of the income distribution among the individuals
1 2( , ,..., )nI I y y y
1 2, ,...., ny y y
1,2,..., .n
The Population Principle: The Population Principle: For every distribution ,
“cloning” has no effect on inequality
The Relative Income Principle:The Relative Income Principle: For every positive number ,
1 2, ,...., ny y y
1 2 1 2 1 2, ,...., , ,...., ; , ,....,n n nI y y y I y y y y y y
1 2 1 2, ,...., , ,....,n nI y y y I y y y
The Dalton PrincipleThe Dalton Principle: The function I(.) satisfies the Dalton Principle, if, for every distribution and every transfer
1 2, ,..., ny y y
0, 1 2 1( , ,..., ) ( ,..., ,..., ,..., )
wherever
n i j n
i j
I y y y I y y y y
y y
The Lorenz curve 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
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
The previous graph gives us a measure of inequality called the Lorenz CriterionLorenz Criterion
An inequality measure I is Lorenz-Lorenz-consistentconsistent if, for every pair of income distributions
,
whenever the Lorenz curve of lies to the right of
1 2 1 2( , ,..., ) and , ,...,n my y y z z z
1 2 1 2, ,..., , ,...,n mI y y y I z z z
1 2( , ,..., )ny y y 1 2, ,..., mz z z
Can we summarize inequality by a numbernumber? Attractive for policymakers and researchers
When Lorenz curves crosscross, we cannot rank inequality across two distributions
A numerical measure of inequality helps rank distributions unambiguously
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 mean or average of the distribution is given by
jn
m
jjnn
1
j
m
jj ynn
1
1
1. Range
2. Kuznets Ratio
3. Mean Absolute Deviation
4. Coefficient of Variation
5. Gini Coefficient
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?)
1
1yyR m
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
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
j
m
jj ynn
M1
1
Essentially the standard deviationstandard 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)
2
1
1
j
m
j
j yn
nC
Sum of the absolute differences between all pairs of incomes, normalized by (squared) population and mean income
Takes the difference between allall pairs of income and sums the absolute differences
Inequality is the sum of all pair-wisepair-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
kj
m
j
m
kkj yynn
nG
1 1221
Satisfies all four principles: Lorenz-consistent