#### Transcript Slide 1

```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 yi is received by each individual i:
 y1 , y2 ,...., yn 

We are interested in comparing “relative
inequality” between two such distributions (over
time, or between regions/countries, etc.)
1.
The Anonymity Principle
 Names do not matter, incomes can always be ranked
without reference to who is earning it
y1  y2  y3 ,...,  yn
2.
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
The Relative Income Principle
3.


Only relative income matters, and not levels of
absolute income
Scaling everyone’s income by the same
percentage should not affect inequality
The Dalton Principle
4.


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 is a function of the form
I  I ( y1 , y2 ,..., yn )
 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  y1 , y2 ,...., yn  among the
individuals 1,2,..., n.

The Population Principle: For every
distribution  y1 , y2 ,...., yn  ,
I  y1 , y2 ,...., yn   I  y1 , y2 ,...., yn ; y1 , y2 ,...., yn 
 “cloning” has no effect on inequality

The Relative Income Principle: For every
positive number ,
I  y1 , y2 ,...., yn   I   y1 ,  y2 ,....,  yn 

The Dalton Principle: The function I(.)
satisfies the Dalton Principle, if, for every
distribution  y1 , y2 ,..., yn 
and every transfer   0,
I ( y1 , y2 ,..., yn )  I ( y1 ,..., yi   ,..., y j   ,..., yn )
wherever yi  y j

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

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 Criterion

An inequality measure I is Lorenz-consistent if,
for every pair of income distributions
( y1 , y2 ,..., yn ) and  z1 , z2 ,..., zm ,
I  y1 , y2 ,..., yn   I  z1 , z2 ,..., zm 
whenever the Lorenz curve of ( y1 , y2 ,..., yn ) lies to
the right of  z1 , z2 ,..., zm 

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
Let there be m distinct incomes, divided into j
classes
 In each income class j, the number of individuals
earning that income is n j
 The total population is then given by

n

m
 nj
j 1
The mean or average of the distribution is given by
1m
   nj yj
n j 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
R
1

 ym  y1 
 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?)

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
1 m
 nj y j  
M
n j 1
 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

Essentially the standard deviation(sum of squared
deviations from the mean), divided by the mean
1
C

nj
2
 y j   
j 1 n
m
 Gives greater weight to larger deviations from the mean
 Lorenz-consistent (satisfies the four principles)

Sum of the absolute differences between all pairs of
incomes, normalized by (squared) population and mean
income
1 mm
G  2   n j nk y j  yk
2n  j 1k 1
 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

Satisfies all four principles: Lorenz-consistent
```