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