Inequality: A Dive into the Gini Index and Axioms of Measurement

Part 2: Measurement 2
Chapter 5 EOP
Martin Ravallion
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Lecture notes to accompany Ravallion’s 
The Economics of Poverty.
2
5
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What properties should a measure of
“inequality” have? Five standard axioms
(Review Box 5.3 of EOP)
Axiom 1
: 
Anonymity
 (or Symmetry): it does not matter who
has which income level. If we shuffle the incomes
around then inequality is unchanged. 
Axiom 2
: 
The
 
Transfer Principle
: transferring income from
the poor to the rich must increase inequality.
Axiom 3
: 
Income Scale Independence
: multiplying all
incomes by a constant does no change the measure
of inequality.
3
Axiom 4
: 
Population replication independence
; simply
replicating the original population cannot increase
inequality.
Axiom 5
: 
Decomposability
: total inequality is the sum of
between-group and within-group components.
Five axioms cont.,
4
Understanding the Gini index
(also called Gini coefficient)
How much “inequality” between
(say) three people?
We know each of their “incomes”
(or whatever variable we are using).
So we can easily calculate all the
gaps between their incomes.
5
 
To assure “scale independence” we need to divide by the
mean (making this a relative inequality” measure).
The Gini index is based on the average of all these
normalized gaps.
6
 Formula for the Gini index
= half the average difference between all pairs of incomes, 
normalized by the mean.
 
Note: Gini index satisfies Axioms 1-4 but not 5
 
y
1
 
y
2
 
y
3
 
n=
3
Area A
Area B
 
Gini index=A/(A+B)
Gini index graphically
7
Gini index = area between Lorenz Curve (
L
(
p
)) and 
diagonal as proportion of area under the diagonal
 
Note: Gini index=2 times Area A when the Lorenz Curve is
scaled in fractions rather than %
Software for calculating Gini index
Gini calculator 
(Buck Shlegeris); very easy to use; just
enter your numbers, separated by commas.
Also gives you the frequency distribution and Lorenz curve
When you have  a large data set and/or you are doing
other calculations it is better to use a standard statistical
software package like Stata. Try the  Stata command:
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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Note: if we ask this in an exam question you will only be
asked to calculate Gini index for a few easy numbers.
We only test your understanding of the concept.
8
Examples: (1,2,x)
Suppose that the distribution of income is (
1,2,3
). Then
the Gini index is:
The Gini index in the US is 
0.49 (2018)
. Think of this as
three incomes (
1,2,
x
). What would 
x
 have to be to get a
Gini index of 
0.49
?
Answer: 
x
=12. The Gini index for (
1,2,12
) is 
0.49
.
9
Note
: the Gini index for the distribution (1,2,
x
) is
 
        implying that                  for 
G
 
< 2/3.
 
 
 
Mean=2
N=3 (squared)
Example: World Gini index
 
The world Gini index is about 
0.65
. Think of this as three
incomes (
1,2,x
). What is 
x
 to get a Gini index of 
0.65
?
The answer: 
x=157
! The Gini index for (
1,2,157
) is 
0.65
.
10
 
More realistic version:
In $/day: (
0.75, 1, 1.5,
2, 2.5, 3.5, 5, 9, 20, 50
);
this gives Gini=
0.65
.
A common mistake about the maximum
Gini index
It is often said that the Gini index is 1 whenever the
richest person has all the income.
This is wrong. The index will have some upper bound
no greater than unity, but the size of that upper bound
depends on specific income distribution.
In a  population of size 
N
 in which the richest person
has all the income the Gini index reaches is maximum
value given by:
   
Exercise: Verify that the Gini index for an income distribution across two
people of (0, 
x
) is 0.5 no matter what the value of 
x
11
Gini can be insensitive,
esp., in small samples!
When inequality is high, the Gini index
can be quite unresponsive to change in
the incomes of the very poor or very
rich.
Example: Suppose that the distribution
of income is (1,2,157) giving a Gini
index of 0.65.
Doubling the income of either of the
poorest reduces the index to 0.64.
While doubling the income of the
rich increases the index to 0.66 (its
maximum with n=3)
12
L
(
p
)
p
(You can easily check
using 
Gini calculator
.)
Mean/median
Not as popular as Gini, but getting there!
See for example the 
Forbes article 
on rising inequality in
US cities.
But this is totally insensitive to (mean-preserving) income
transfers on either side of the median.
13
Mean log deviation (MLD)
(EOP, Box 5.4)
14
This simple measure satisfies all five axioms 
(including decomposability)
(1,2,3): MLD=0.10
(1,2,157): MLD=2.06
15
The standard axioms for an inequality measure
are not universally accepted
Axiom 1: Anonymity
: it does not matter who has which income
level.
Axiom 2: The (Pigou-Dalton) Transfer Principle
: transferring
income from the poor to the rich must increase inequality.
Axiom 3: Income scale independence
: multiplying all incomes by
a constant does not change the inequality measure.
Axiom 4: Population replication independence
; simply replicating
the original population cannot increase inequality.
Axiom 5: Decomposability
: total inequality = inequality between
groups + inequality within groups.
 
16
The standard axioms for an inequality measure
are not universally accepted
Axiom 1: Anonymity
: it does not matter who has which income
level.  
But it does matter in reality!
Axiom 2: The (Pigou-Dalton) Transfer Principle
: transferring
income from the poor to the rich must increase inequality.
Axiom 3: Income scale independence
: multiplying all incomes by
a constant does not change the inequality measure. 
=>
Relative inequality measures; but absolute gaps also matter!
Axiom 4: Population replication independence
; simply replicating
the original population cannot increase inequality.
Axiom 5: Decomposability
: total inequality = inequality between
groups + inequality within groups. 
Group identities may
matter more than this allows.
 
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Relative inequality is about 
ratios
; absolute
inequality is about 
differences
.
State A
: two incomes $1,000 and $10,000 per year
State B
: these rise to $2,000 and $20,000
Ratio is unchanged but the 
rich
 can buy 10 times more
from the income gains in state B than can the 
poor
Relativist sees no difference; absolutist sees higher
inequality in state B.
One is not right and the other wrong. Amiel and
Cowell found that 40% of (student) participants in
experiments view inequality in absolute terms.
Whether one thinks about inequality as absolute or
relative matters greatly to one
s views on the
distribution of the gains from economic growth.
Class survey: 
Which has higher “inequality”?
18
(Anonymity axiom)
(Transfer axiom)
(Transfer axiom)
(Transfer axiom)
(Transfer axiom)
Survey results 2020 (N=83)
19
Class survey results: 2014-20 (N=601)
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Class survey: Summary
Almost all of the class agrees with the 
anonymity axiom
.
Virtually all agree with the 
transfer axiom
.
Interesting that this is less strong for (2,4,6,10) => (2,5,5,10)
“Polarization” is penalized? Extremes further from the middle.
Scale independence 
is far more contentious: indeed,
majority are absolutists.
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Key difference: 
relative Gini is 
normalized by 
the current mean
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Normalized by a fixed mean
Note on the literature: The French economist Serge Kolm 
called the relative Gini the “rightest index” and the absolute 
Gini the ”leftist index.” This related to the debates around 
the protests in the late 1960s. The students rejected 
proposals for an equal % increase in all incomes.
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Consider:
State A
: two incomes $1,000 and $10,000 per year
State B
: these are swapped to $10,000 and $1,000
Both relative and absolute inequality is the same.
Yet the poorest person in State A has gained
enormously, while the richest has become the
poorest.
Relaxing anonymity typically calls for 
longitudinal
(“panel” or retrospective) data observing incomes
over time for the same people.
Inequality versus poverty
Consider two income distributions:
 
 
A: (1,1,1) 
 
B: (2,3,10)
Everyone would probably agree that B is more unequal.
But everyone has a higher income in B than A.
If we focus on the income of the poorest then we will
prefer B to A; the greater inequality in B has benefited
the poor.
This is an application of John Rawls’ “
difference
principle
”: “inequality is acceptable if poorest benefit.”
Income poverty is lower in B, but inequality is higher.
However (as Rawls and Sen emphasized), income is not
a complete metric of welfare.
25
(EOP Box 3.5)
What if 
relative
 income matters?
 
Suppose that welfare is own income normalized by the
mean:
The distributions of welfare are then:
 
A: (1,1,1) 
 
B: (0.4, 0.6, 2)
“A” now has both lower inequality and lower welfare
poverty (for all poverty lines less than 2).
More generally, suppose that your welfare is:
 
For some critical value of     welfare poverty is lower
under B.
Exercise: Verify that the poorest person is better off in state
B as long as
26
Which situation would you prefer?
 
Sate A.
 Having an annual income of $50,000, while the
others earn $25,000.
State B.
 Having an annual income of $100,000, while the
others earn $250,000.
This was asked of a sample of students and lecturers at
Harvard University. Over 50 percent opted for A (Solnick
and Hemenway).
27
What do Capuchin’s
monkeys think about
fairness?
Two monkeys. Each trades a rock for food. One gets
cucumber in return for a rock, the other gets grapes.
They see each other.
Look how the one who gets cucumber reacts (Frans de
Waal): 
https://www.youtube.com/watch?v=gOtlN4pNArk
Is this about unequal outcomes or fairness in trade?
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1.
Focus
: A small change in economic welfare for the
non-poor cannot change the measure of poverty
2.
Monotonicity
: A small increase (decrease) in
economic welfare for the poor must reduce
(increase) the measure of poverty
3.
Sub-group monotonicity
: An increase (decrease) in
poverty for any sub-group of the population must
increase aggregate measure of poverty.
4.
Transfer
: A small transfer of income from a poor
person to someone who is poorer must decrease
the measure of poverty (also called “weak transfer
axiom”)
Desirable properties of a poverty
measure
30
Head-count index
q
 
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Advantage
: easily understood
Disadvantages
: insensitive to distribution 
below
 the
poverty line
The headcount index fails both the Monotonicity and
Transfer Axioms
 
 
Question
: What happens to 
H
 if a poor person becomes poorer?
31
Large gains to the 
poor but no change 
in the headcount index
32
Quantile functions
Poverty line
Poverty gap index
  
y
1
,...., y
q
 <  z  <   y
q+1
, ..., y
n
Advantages
 of 
PG
: reflects 
depth
 of poverty
Disadvantages
: insensitive to 
severity
 of poverty
Example
:   A: (1, 2, 3, 4)    B: (2, 2, 2, 4)
Let z = 3.  
H
A
 = 0.75 = 
H
B
; 
PG
A
 = 0.25 = 
PG
B
.
The Poverty Gap Index fails the Transfer Axiom
 
 
Question
: How is PG affected by an inequality-increasing redistribution
of income amongst the poor?
33
Redistribution amongst the 
poor, but no change in the 
poverty-gap index
34
Interpretation of PG
 
The 
minimum
 cost of eliminating poverty is to fill all
poverty gaps exactly: that costs 
(
z-
z
)
q
  => perfect
targeting. (
z
=
mean for poor
)
The 
maximum
 cost of eliminating poverty is when
we know nothing about incomes: cost= 
z.n
=> No targeting (UBI = 
z
).
Ratio of minimum cost of eliminating poverty to the
maximum cost with no targeting:
Poverty gap index indicates the potential saving to
the poverty alleviation budget from using
information to target poor people.
 
35
Don’t be fooled by the aggregate poverty
gap!
Global PG (2013; $1.90) = 3.23%
Aggregate poverty gap 
$160 billion
. The conclusion
drawn is that it should be easy to eliminate poverty.
“..
about what the world spends on foreign aid
.”
This entirely ignores all the constraints that confront
finely targeted transfers.
Imperfect information
Incentive to escape poverty
Political economy
At the other extreme: the cost of a Basic Income of $1.90
a day is about 
$5,000 billion-
-over 30 times more.  (Of
course, the blog post would have lost some of its punch.)
36
Squared poverty gap index
            
 
 
   
The SPG satisfies all five core axioms
Example: A = (1, 2, 3, 4)    B = (2, 2, 2, 4)
  
   z = 3    SPG
A
 = 0.14;   SPG
B
 = 0.08
 
Advantage
 of SPG: sensitive to differences in
both depth and severity of poverty.
Disadvantage
: difficult to interpret
Question
: How is SPG affected by an inequality-increasing redistribution
of income amongst the poor?
37
Foster-Greer-Thorbecke class of
measures
            
 
 
P
0 
= H= Headcount index
P
1 
= PG= Poverty gap index
P
2 
= SPG= Squared poverty 
gap index
38
Individual poverty measures
39
For H every poor person gets the same weight. 
For PG every poverty gap gets the same weight. 
For SPG, larger gaps are given higher weight.
(H)
(PG)
(SPG)
 
“convex”
0
1
The Watts index
 
The Watts index is the mean proportionate poverty
gap(counting non-poor as having zero gap)
 
This satisfies all accepted axioms for poverty
measurement (including some we have not covered).
 
        
  
       
 
 
40
A thought experiment:
You can make a single small gift.
Who should get it?
Consider a small transfer payment that is not enough for
anyone to escape poverty.
You can only give it to 
one person
; who should it be?
If you use H as your measure then it does not matter
who gets the transfer, even if it goes to the non-poor!
If you use PG then you will give it to a poor person, but it
does not matter which one!
Using SPG or the Watts index you will give it to the
poorest person you can find.
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The disconnect between how economists
measure poverty and public discourse
Economists count poor people, possibly with
distributional weights.
Yet much of the public discourse focuses on living
standards of the poorest.
43
The “counting” vs. “Rawlsian”
approaches
The measures so far are examples of the 
counting
approach
 to poverty measurement.
An approach inspired by Rawls would focus instead on
the 
consumption floor
—the lowest level of living.
If the poorest person sees a gain (loss) then (by
definition) the consumption floor must rise (fall).
(EOP, pp. 238-40)
However, if we find that any poverty measure (by the
counting approach) shows a decrease that does not tell
us that the floor has risen!
44
Same reduction in the poverty count but
different implications for the poorest
45
 
Floor stays put
 
Rising floor
Other arguments for studying the floor
Rights-based approaches to justice
Justice must be concerned with each
 
citizen 
not averages
Rights must be secured for all; 
none left behind
.
Either we all live in a decent world, or nobody does.
George Orwell
Ma
hatma Gandhi’s talisman:
Recall the face of the poorest and weakest person you have
seen and ask if the step you contemplate is going to be any
use to them
.”
In setting the new Sustainable Development Goals, it
has been argued that: “
the indicators that track them
should be disaggregated to ensure 
no one is left behind
.”
Social policies also aim to raise the floor above the
biological minimum for survival.
46
Anecdotal judgements “on the ground”
often look to the poorest
In an article “
Just a little
bit richer
” the 
Economist
(4/4/15) asked how
much China’s poor area
programs have helped
reduce poverty.
47
The article points to a poor village in NE Shanxi that has
been left behind:
They laugh in unison when asked if they receive subsidies. The
arrival of electricity 30 years ago was a vast improvement. But
little else has changed in their lives since
.”
Safety net as a consumption floor
Statutory minimum wage rates:
 first appeared in late 19
th
century in an effort to help raise the consumption floor.
Universal Basic Income (UBI)
/Basic-income guarantee
(BIG): From the 1970s, we started to see arguments in
support of a fixed cash transfer to every adult. A firm
floor.
Social policy as a “
right of citizenship
” rather than
something to be targeted based on “need.”
e.g., demands for universal health insurance.
The ILO calls for a comprehensive 
Social Protection
Floor:
nationally defined sets of basic social security
guarantees
.
48
SSNs attempt to raise the floor
The term “
safety net
” evokes the idea of some sort of
floor, and some of the programs can be interpreted as
efforts to raise the floor above some biological level
This includes the two largest programs to date:
China’s 
Di Bao
 
program aims to assure that no family in
urban China falls below a stipulated minimum income.
India’s 
National Rural Employment Guarantee Scheme
interpretable as an attempt to enforce the minimum wage rate
in an informal economy.
Raising the floor is a common motivation for SSNs.
But is this being achieved in practice?
49
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In practice, command over market
goods is an incomplete welfare metric
 
Household consumption or income leaves out access to
non-market goods
 and 
intra-household inequality
.
With ideal poverty lines we could address these
deficiencies. But practice is far from ideal.
This has motivated efforts to develop multiple indices to
form a 
dashboard
: income poverty + child mortality +
schooling rates +…..
The dashboard approach identifies multiple indices.
Proposals to use instead a single 
composite index
.
But how do we do this? What weights to use? How to
add up “apples” and “oranges” without prices?
 
51
Example 1: Human Development Index
52
The implicit $ value of life in a composite
index
To illustrate, consider a simplified linear “HDI-type” index:
 
Index
 = 
w.Y
 + (1-
w
).
LE
 
w
=weight on income (
Y
)
 
(1-
w
)=weight on life expectancy (
LE
)
Now ask: if life expectancy falls by one year, how much extra
income will compensate? This is the implicit value of life in
the index. We find the answer this way:
 
w.
(
Y+x
) + (1-
w
).(
LE-
1) = 
w.Y
 + (1-
w
).
LE
           x=
implicit value of one year’s
 LE
On solving we find that 
x=
(1
-w
)
/w
(The calculation is a little more complex for actual HDI but
principle is the same)
53
Troubling tradeoffs in the HDI
(EOP: Box 5.20)
The implied value of life varies from very low levels in
poor countries—the lowest value of $0.51 per year is for
Zimbabwe, representing—to almost $9,000 per year in
the richest countries. =>
Can this be justified on ethical grounds?
What signal does it give to a poor country trying to
increase its HDI?
Key point: To make a composite index credible it is
crucial that you make the relative weights explicit, 
and
that you can defend them!
54
What $ value does the HDI attach to an
extra year of life?
55
 
(In 2013 the web site of the Bill and Melinda Gates
foundation said that “
All lives have equal value
.”)
 
Is this ethically
acceptable?
Example 2: Alkire and Santos (2010)
Two variables for 
health
 (malnutrition, and child mortality),
Two for 
education
 (years of schooling and school enrolment),
Six for deprivation in 
living standards
 (namely cooking with
wood, charcoal or dung; not having a conventional toilet; lack
of easy access to safe drinking water; no electricity; dirt, sand
or dung flooring and not owning at least one of a radio, TV,
telephone, bike or car).
Poverty is measured separately in each of these 10
dimensions.
The equally-weighted aggregate poverty measures for each of
these three main headings are then weighted equally (one-
third each) to form the composite index.
A household is identified as being poor if it is deprived across
at least 30% of the weighted indicators.
AS construct their index for more than 100 countries.
56
Aggregating over multiple dimensions
Almost every poverty measure found in practice is
multidimensional.
Exception: Historical rice-based poverty measures  in Asia (no
longer used)
The difference lies in the way in which one chooses to
aggregate across multiple dimensions., esp., the role played
by 
prices
.
Using the prices faced by people living at the poverty line
guarantees that the poverty bundle accords with choices of
people living at the poverty line.
Advocates of MIPs argue that 
prices do not exist or are
unreliable for valuation.
But prices do exist for most of the things people consume.
Prices can sometimes be imputed for other goods.
57
Are prices valid for aggregation?
Some degree of aggregation will be needed.
MIP advocates reject the use of prices in aggregation, either
actual prices or shadow prices. 
Why?
1.
“Loss of information on dimension-specific shortfall”
; but this
is not typically data so how can we lose it?
2.
“Collapses the problem into one dimension.” 
But so too does
the MIP.
3.
“Prices are missing or unreliable.” 
But how do we know that
the (ad hoc) MIP weights are any better?
58
We can agree that markets and prices are not perfect, but
can we ignore them?
Consistency with consumer choice =>
aggregation using prices
Aggregation without prices cannot in general yield poverty
measures that are consistent with the welfare of someone
living at the poverty line. 
This method ignores all implications of consumer choice in a market
economy.
The approach will identify some people as poor because they are lacking
in one or more things that they can afford, but do not want.
Everyone may agree that they are better off in situation A than B, yet the
MIP shows higher poverty in A.
Poverty comparisons between people and over time (which are never
easy) could be especially problematic.
When calibrated correctly, using actual or suitable imputed
prices guarantees that poor people would accept the tradeoffs
built into the poverty measure. 
(EOP: * Box 5.20.)
59
 
Could a single index ever be credible?
Critics of the HDI and Alkite-Santos index want other
dimensions added: “conflict, security, empowerment,…”
Where do we stop, and are we being too ambitious?
Political bias: Hide your poor performance in a composite
index
60
Yet for most purposes of poverty measurement we do not
want to form a single composite index.
The actionable things are not typically found in the composite
index but in its components.
Then the obvious first step for a “mashup index” is to un-pack it.
Some of the mashup indices found in practice can be readily
un-packed, though it remains unclear what policy purpose
was served by adding them up in the first place.
Dashboard vs composite index
We need to recognize the inadequacy of measuring poverty
solely in terms of command over market goods.
But are we asking too much of one measure of “poverty” to
include things like child mortality, schooling, violence or
empowerment, on top of material living standards?
We will inevitably be drawn to multiple indices in assessing
poverty and informing policies.
We need to focus our efforts on developing the best possible
distinct measures of the various dimensions of poverty
deemed relevant to a given setting, i.e., a
 better dashboard.
This is required for both the dashboard and composite index.
61
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Exploring the core concepts of inequality measurement, this content delves into the essential properties a measure of inequality should have, including the five standard axioms. It discusses the Gini index in detail, elucidating its formula and graphical representation. An informative guide for comprehending inequality metrics.


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  1. Georgetown University ECON 156: Poverty and Inequality Lecture notes to accompany Ravallion sThe Economics of Poverty. Part 2: Measurement 2 Chapter 5 EOP Martin Ravallion

  2. 5. Measuring inequality (EOP Ch.5) 2

  3. What properties should a measure of inequality have? Five standard axioms (Review Box 5.3 of EOP) Axiom 1: Anonymity (or Symmetry): it does not matter who has which income level. If we shuffle the incomes around then inequality is unchanged. Axiom 2: The Transfer Principle: transferring income from the poor to the rich must increase inequality. Axiom 3: Income Scale Independence: multiplying all incomes by a constant does no change the measure of inequality. 3

  4. Five axioms cont., Axiom 4: Population replication independence; simply replicating the original population cannot increase inequality. Axiom 5: Decomposability: total inequality is the sum of between-group and within-group components. 4

  5. Understanding the Gini index (also called Gini coefficient) How much inequality between (say) three people? We know each of their incomes (or whatever variable we are using). So we can easily calculate all the gaps between their incomes. To assure scale independence we need to divide by the mean (making this a relative inequality measure). The Gini index is based on the average of all these normalized gaps. 5

  6. Formula for the Gini index n n 1 = i = j = Gini y y i j 2 2 n y 1 1 = half the average difference between all pairs of incomes, normalized by the mean. y1-y2 y1-y3 y1-y1 y2-y1 y2-y2 y2-y3 n=3 y3-y1 y3-y2 y3-y3 y1 y2 y3 Note: Gini index satisfies Axioms 1-4 but not 5 6

  7. Gini index graphically Gini index = area between Lorenz Curve (L(p)) and diagonal as proportion of area under the diagonal Figura 2. Brasil 1981-1995: Curvas de Lorenz dedeLorenz lLLorenz deLorenz Curves 100 90 80 70 Gini index=A/(A+B) Renda Acumulada % 60 1981 50 Area A 1990 1995 40 30 Area B 20 10 0 0 10 20 30 40 50 60 70 80 90 100 Popula o Acumulada % Note: Gini index=2 times Area A when the Lorenz Curve is scaled in fractions rather than % 7

  8. Software for calculating Gini index Gini calculator (Buck Shlegeris); very easy to use; just enter your numbers, separated by commas. Also gives you the frequency distribution and Lorenz curve When you have a large data set and/or you are doing other calculations it is better to use a standard statistical software package like Stata. Try the Stata command: ssc describe fastgini Note: if we ask this in an exam question you will only be asked to calculate Gini index for a few easy numbers. We only test your understanding of the concept. 8

  9. Examples: (1,2,x) Suppose that the distribution of income is (1,2,3). Then the Gini index is: 1 ( ) 2 1 ( + + = x + + ) 1 + ) 1 2 x ( 2 = . 0 22 Gini 2 3 2 Mean=2 N=3 (squared) The Gini index in the US is 0.49 (2018). Think of this as three incomes (1,2,x). What would x have to be to get a Gini index of 0.49? Answer: x=12. The Gini index for (1,2,12) is 0.49. + ) 1 2 9 G ( 2 x = = Note: the Gini index for the distribution (1,2,x) is implying that for G< 2/3. ) 3 + x x G 2 3 G ( 3 9

  10. Example: World Gini index The world Gini index is about 0.65. Think of this as three incomes (1,2,x). What is x to get a Gini index of 0.65? The answer: x=157! The Gini index for (1,2,157) is 0.65. Income per capita ($ per day) More realistic version: In $/day: (0.75, 1, 1.5, 2, 2.5, 3.5, 5, 9, 20, 50); this gives Gini=0.65. 60 50 40 30 20 10 0 1 2 3 4 5 6 7 8 9 10 10

  11. A common mistake about the maximum Gini index It is often said that the Gini index is 1 whenever the richest person has all the income. This is wrong. The index will have some upper bound no greater than unity, but the size of that upper bound depends on specific income distribution. In a population of size N in which the richest person has all the income the Gini index reaches is maximum value given by: 1 1 N Exercise: Verify that the Gini index for an income distribution across two people of (0, x) is 0.5 no matter what the value of x 11

  12. Gini can be insensitive, esp., in small samples! L(p) When inequality is high, the Gini index can be quite unresponsive to change in the incomes of the very poor or very rich. Example: Suppose that the distribution of income is (1,2,157) giving a Gini index of 0.65. Doubling the income of either of the poorest reduces the index to 0.64. While doubling the income of the rich increases the index to 0.66 (its maximum with n=3) p (You can easily check using Gini calculator.) 12

  13. Mean/median Not as popular as Gini, but getting there! See for example the Forbes article on rising inequality in US cities. But this is totally insensitive to (mean-preserving) income transfers on either side of the median. 13

  14. Mean log deviation (MLD) (EOP, Box 5.4) n 1 = i = log( ) MLD n iy 1 This simple measure satisfies all five axioms (including decomposability) (1,2,3): MLD=0.10 (1,2,157): MLD=2.06 14

  15. The standard axioms for an inequality measure are not universally accepted Axiom 1: Anonymity: it does not matter who has which income level. Axiom 2: The (Pigou-Dalton) Transfer Principle: transferring income from the poor to the rich must increase inequality. Axiom 3: Income scale independence: multiplying all incomes by a constant does not change the inequality measure. Axiom 4: Population replication independence; simply replicating the original population cannot increase inequality. Axiom 5: Decomposability: total inequality = inequality between groups + inequality within groups. 15

  16. The standard axioms for an inequality measure are not universally accepted Axiom 1: Anonymity: it does not matter who has which income level. But it does matter in reality! Axiom 2: The (Pigou-Dalton) Transfer Principle: transferring income from the poor to the rich must increase inequality. Axiom 3: Income scale independence: multiplying all incomes by a constant does not change the inequality measure. => Relative inequality measures; but absolute gaps also matter! Axiom 4: Population replication independence; simply replicating the original population cannot increase inequality. Axiom 5: Decomposability: total inequality = inequality between groups + inequality within groups. Group identities may matter more than this allows. 16

  17. Relaxing scale independence: Relative vs. absolute inequality Relative inequality is about ratios; absolute inequality is about differences. State A: two incomes $1,000 and $10,000 per year State B: these rise to $2,000 and $20,000 Ratio is unchanged but the rich can buy 10 times more from the income gains in state B than can the poor Relativist sees no difference; absolutist sees higher inequality in state B. One is not right and the other wrong. Amiel and Cowell found that 40% of (student) participants in experiments view inequality in absolute terms. Whether one thinks about inequality as absolute or relative matters greatly to one s views on the distribution of the gains from economic growth. 17

  18. Class survey: Which has higher inequality? A B A B Neither Relativist Absolutist A&R Relativist Absolutist (1,2,3) (2,4,6) Absolutist A&R Absolutist A&R (1,2,3) (2,3,4) Relativist Relativist A&R A&R A&R (1,2,3) (3,1,2) (Anonymity axiom) (1,2,3) (1,2,4) (2,4,6) (4,8,12) (2,4,6) (4,6,8) (2,4,6) (3,4,5) (Transfer axiom) (2,4,6,10) (3,4,6,9) (Transfer axiom) (2,4,6,10) (1,4,6,11) (Transfer axiom) (2,4,6,10) (2,5,5,10) 18 (Transfer axiom)

  19. Survey results 2020 (N=83) A B A 7% B Neither Relativist 49% Absolutist 52% A&R 89% 1% Relativist 42% Absolutist 51% 13% 2% 2% 36% (1,2,3) (2,4,6) Absolutist 43% 4% 6% A&R 96% Absolutist 49% 5% 2% 2% A&R 96% 24% (1,2,3) (2,3,4) Relativist 45% 5% 2% 8% Relativist 45% A&R 84% A&R 95% 1% A&R 40% (1,2,3) (3,1,2) (1,2,3) (1,2,4) (2,4,6) (4,8,12) (2,4,6) (4,6,8) (2,4,6) (3,4,5) (2,4,6,10) (3,4,6,9) (2,4,6,10) (1,4,6,11) (2,4,6,10) (2,5,5,10) 19

  20. Class survey results: 2014-20 (N=601) Distribution A (1,2,3) Which has higher inequality? A B Absolutist 5% 53% Relativist 43% 7% 3% 5% A and R 3% 95% Absolutist 5% 55% Relativist 46% 9% A and R 90% 3% B Neither Relativist 42% Absolutist 50% A and R 92% 2% Relativist 40% Absolutist 45% 7% (2,4,6) (1,2,3) (2,3,4) (1,2,3) (3,1,2) (1,2,3) (1,2,4) (2,4,6) (4,8,12) (2,4,6) (4,6,8) (2,4,6) (3,4,5) 20

  21. Class survey: Summary Almost all of the class agrees with the anonymity axiom. Virtually all agree with the transfer axiom. Interesting that this is less strong for (2,4,6,10) => (2,5,5,10) Polarization is penalized? Extremes further from the middle. Scale independence is far more contentious: indeed, majority are absolutists. 21

  22. Two Gini indices of inequality Relative Gini index 1 y y j r i = G 2 y y 2 n Key difference: relative Gini is normalized by the current mean Absolute Gini index 1 a = G y y i j 2 2 n 22

  23. Two Gini indices 1988 0.72 2008 0.71 Relative Gini 1 ??? ??? 2?2?? Absolute Gini 0.72 0.90 1 2?2 ? ??? ??? Normalized by a fixed mean Note on the literature: The French economist Serge Kolm called the relative Gini the rightest index and the absolute Gini the leftist index. This related to the debates around the protests in the late 1960s. The students rejected proposals for an equal % increase in all incomes. 23

  24. Relaxing anonymity is harder Consider: State A: two incomes $1,000 and $10,000 per year State B: these are swapped to $10,000 and $1,000 Both relative and absolute inequality is the same. Yet the poorest person in State A has gained enormously, while the richest has become the poorest. Relaxing anonymity typically calls for longitudinal ( panel or retrospective) data observing incomes over time for the same people. 24

  25. Inequality versus poverty (EOP Box 3.5) Consider two income distributions: A: (1,1,1) Everyone would probably agree that B is more unequal. But everyone has a higher income in B than A. If we focus on the income of the poorest then we will prefer B to A; the greater inequality in B has benefited the poor. This is an application of John Rawls difference principle : inequality is acceptable if poorest benefit. Income poverty is lower in B, but inequality is higher. However (as Rawls and Sen emphasized), income is not a complete metric of welfare. B: (2,3,10) 25

  26. What if relative income matters? Suppose that welfare is own income normalized by the mean: The distributions of welfare are then: A: (1,1,1) B: (0.4, 0.6, 2) A now has both lower inequality and lower welfare poverty (for all poverty lines less than 2). More generally, suppose that your welfare is: ) / )( 1 ( + y y y i i yi/ y 1 0 For some critical value of welfare poverty is lower under B. Exercise: Verify that the poorest person is better off in state B as long as 375 . 0 26

  27. Which situation would you prefer? Sate A. Having an annual income of $50,000, while the others earn $25,000. State B. Having an annual income of $100,000, while the others earn $250,000. This was asked of a sample of students and lecturers at Harvard University. Over 50 percent opted for A (Solnick and Hemenway). 27

  28. What do Capuchins monkeys think about fairness? Two monkeys. Each trades a rock for food. One gets cucumber in return for a rock, the other gets grapes. They see each other. Look how the one who gets cucumber reacts (Frans de Waal): https://www.youtube.com/watch?v=gOtlN4pNArk Is this about unequal outcomes or fairness in trade? 28

  29. 6. Measuring poverty (EOP Ch.5) 29

  30. Desirable properties of a poverty measure Core axioms 1. Focus: A small change in economic welfare for the non-poor cannot change the measure of poverty 2. Monotonicity: A small increase (decrease) in economic welfare for the poor must reduce (increase) the measure of poverty 3. Sub-group monotonicity: An increase (decrease) in poverty for any sub-group of the population must increase aggregate measure of poverty. 4. Transfer: A small transfer of income from a poor person to someone who is poorer must decrease the measure of poverty (also called weak transfer axiom ) 30

  31. Head-count index q H = n q = no. people deemed poor; n = population size Advantage: easily understood Disadvantages: insensitive to distribution below the poverty line The headcount index fails both the Monotonicity and Transfer Axioms Question: What happens to H if a poor person becomes poorer? 31

  32. Quantile functions Poverty line Large gains to the poor but no change in the headcount index 32

  33. Poverty gap index q z 1 z y = i = i PG n 1 y1,...., yq < z < yq+1, ..., yn Advantages of PG: reflects depth of poverty Disadvantages: insensitive to severity of poverty Example: A: (1, 2, 3, 4) B: (2, 2, 2, 4) Let z = 3. HA= 0.75 = HB; PGA= 0.25 = PGB. The Poverty Gap Index fails the Transfer Axiom Question: How is PG affected by an inequality-increasing redistribution of income amongst the poor? 33

  34. Redistribution amongst the poor, but no change in the poverty-gap index 34

  35. Interpretation of PG The minimum cost of eliminating poverty is to fill all poverty gaps exactly: that costs (z- z)q => perfect targeting. ( z=mean for poor) The maximum cost of eliminating poverty is when we know nothing about incomes: cost= z.n => No targeting (UBI = z). Ratio of minimum cost of eliminating poverty to the maximum cost with no targeting: q z z 1 ( ) = = z H PG zn z Poverty gap index indicates the potential saving to the poverty alleviation budget from using information to target poor people. 35

  36. Dont be fooled by the aggregate poverty gap! Global PG (2013; $1.90) = 3.23% Aggregate poverty gap $160 billion. The conclusion drawn is that it should be easy to eliminate poverty. ..about what the world spends on foreign aid. This entirely ignores all the constraints that confront finely targeted transfers. Imperfect information Incentive to escape poverty Political economy At the other extreme: the cost of a Basic Income of $1.90 a day is about $5,000 billion--over 30 times more. (Of course, the blog post would have lost some of its punch.) 36

  37. Squared poverty gap index 2 q z 1 y = i = i SPG n z 1 The SPG satisfies all five core axioms Example: A = (1, 2, 3, 4) B = (2, 2, 2, 4) z = 3 SPGA = 0.14; SPGB = 0.08 Advantage of SPG: sensitive to differences in both depth and severity of poverty. Disadvantage: difficult to interpret Question: How is SPG affected by an inequality-increasing redistribution of income amongst the poor? 37

  38. Foster-Greer-Thorbecke class of measures P0 = H= Headcount index P1 = PG= Poverty gap index P2 = SPG= Squared poverty gap index q z 1 n z y = i ( 0 ) P = 1 i 38

  39. Individual poverty measures For H every poor person gets the same weight. For PG every poverty gap gets the same weight. For SPG, larger gaps are given higher weight. 1 (H) =0 Individual poverty measure 0 (PG) =1 (SPG) =2 convex income (y) Z 39

  40. The Watts index The Watts index is the mean proportionate poverty gap(counting non-poor as having zero gap) q 1 = i = W ln( / ) P z y i n 1 This satisfies all accepted axioms for poverty measurement (including some we have not covered). 40

  41. A thought experiment: You can make a single small gift. Who should get it? Consider a small transfer payment that is not enough for anyone to escape poverty. You can only give it to one person; who should it be? If you use H as your measure then it does not matter who gets the transfer, even if it goes to the non-poor! If you use PG then you will give it to a poor person, but it does not matter which one! Using SPG or the Watts index you will give it to the poorest person you can find. 41

  42. 7. A Rawlsian approach: Monitoring progress in assuring that no one is left behind (EOP, pp. 87-91 and pp. 238-40) 42

  43. The disconnect between how economists measure poverty and public discourse Economists count poor people, possibly with distributional weights. Yet much of the public discourse focuses on living standards of the poorest. 43

  44. The counting vs. Rawlsian approaches The measures so far are examples of the counting approach to poverty measurement. An approach inspired by Rawls would focus instead on the consumption floor the lowest level of living. If the poorest person sees a gain (loss) then (by definition) the consumption floor must rise (fall). (EOP, pp. 238-40) However, if we find that any poverty measure (by the counting approach) shows a decrease that does not tell us that the floor has risen! 44

  45. Same reduction in the poverty count but different implications for the poorest Cumulative % of population Cumulative % of population Rising floor Measure of welfare Measure of welfare Poverty line Poverty line Floor stays put Poorest left behind Same reduction in the incidence of poverty but without leaving the poorest behind 45

  46. Other arguments for studying the floor Rights-based approaches to justice Justice must be concerned with eachcitizen not averages Rights must be secured for all; none left behind. Either we all live in a decent world, or nobody does. George Orwell Mahatma Gandhi s talisman: Recall the face of the poorest and weakest person you have seen and ask if the step you contemplate is going to be any use to them. In setting the new Sustainable Development Goals, it has been argued that: the indicators that track them should be disaggregated to ensure no one is left behind. Social policies also aim to raise the floor above the biological minimum for survival. 46

  47. Anecdotal judgements on the ground often look to the poorest In an article Just a little bit richer the Economist (4/4/15) asked how much China s poor area programs have helped reduce poverty. The article points to a poor village in NE Shanxi that has been left behind: They laugh in unison when asked if they receive subsidies. The arrival of electricity 30 years ago was a vast improvement. But little else has changed in their lives since. 47

  48. Safety net as a consumption floor Statutory minimum wage rates: first appeared in late 19th century in an effort to help raise the consumption floor. Universal Basic Income (UBI)/Basic-income guarantee (BIG): From the 1970s, we started to see arguments in support of a fixed cash transfer to every adult. A firm floor. Social policy as a right of citizenship rather than something to be targeted based on need. e.g., demands for universal health insurance. The ILO calls for a comprehensive Social Protection Floor: nationally defined sets of basic social security guarantees . 48

  49. SSNs attempt to raise the floor The term safety net evokes the idea of some sort of floor, and some of the programs can be interpreted as efforts to raise the floor above some biological level This includes the two largest programs to date: China s Di Bao program aims to assure that no family in urban China falls below a stipulated minimum income. India s National Rural Employment Guarantee Scheme interpretable as an attempt to enforce the minimum wage rate in an informal economy. Raising the floor is a common motivation for SSNs. But is this being achieved in practice? 49

  50. 8. Multiple and composite indices of poverty and human development (EOP: Ch.5, Section 5.10) 50

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