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
Economists study inequality for
 intrinsic reasons (reducing inequality can be seen as an objective in
 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.)
The Anonymity Principle
 Names do not matter, incomes can always be ranked
without reference to who is earning it
y1  y2  y3 ,...,  yn
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
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
If a transfer is made from a relatively poor to a
relatively rich individual, inequality must
“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
 In each income class j, the number of individuals
earning that income is n j
 The total population is then given by
 nj
j 1
The mean or average of the distribution is given by
   nj yj
n j 1
Kuznets Ratio
Mean Absolute Deviation
Coefficient of Variation
Gini Coefficient
Difference in the incomes of the richest and the
poorest individuals, divided by the mean
 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
The sum of all income distances from average
income, expressed as a fraction of total income
1 m
 nj y j  
n j 1
 The idea: inequality is proportional to distance from mean
 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
 y j   
j 1 n
 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
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
 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