ECONOMICS OF THE WELFARE
STATE
Sponsored by a Grant TÁMOP-4.1.2-08/2/A/KMR-2009-0041
Course Material Developed by Department of Economics,
Faculty of Social Sciences, Eötvös Loránd University Budapest (ELTE)
Department of Economics, Eötvös Loránd University Budapest
Institute of Economics, Hungarian Academy of Sciences
Balassi Kiadó, Budapest
Authors: Róbert Gál, Márton Medgyesi
Supervised by Róbert Gál
June 2011
Week 1
Measuring social inequality
Topics
What is the subject of inequality measurement?
Inequality indices
•
Basic indicators of dispersion
•
Graphical representation of inequalities
•
Basic indicators of dispersion
•
Representation of inequalities by the Lorenz curve
•
The Gini coefficient
•
•
Axiomatic approach to inequality measurement
•
Attributes of aggregate inequality indicators
•
Generalized Entropy indices
Decomposing inequalities
2
What is the subject of the measurement of
inequality?
Basically we are interested in the distribution of material living standards (consumptionpossibilities) among individuals.
Inequality of what?
The best basis of measuring consumption-possibilities could be wealth in the broad
sense: everything that produces income in the present or the future:
•
financial wealth: bank deposit, securities, etc.,
•
material wealth: durable consumer goods, real estate, etc.,
•
human capital: inherited abilities and learnt skills, knowledge
•
entitlements to government transfers: e.g. to social security pension income.
All types of wealth result in a flow of income.
In what form?
Inequality of what?
YF = YM+YN
YF = total income
YM= monetary income: earnings, capital-income, financial transfers from the government
YN = non-monetary income: job satisfaction, leisure time, service of material wealth,
value of self-produced consumption, non-financial transfers from the government
YF the measure of individual consumption-possibilities
YF however is not a proper measure of individual well-being: e.g. it does not take into
account uncertainty
In practice there are difficulties of measurement!
In case of non-monetary incomes: in almost every types.
3
In case of financial incomes, measurement of capital income (e.g. unrealized gain on
securities) and the entrepreneurial income is difficult.
Inequality among individuals?
We measure income on household level even though we are interested in the
distribution of living standards among individuals!
Solution: income per capita?
Income per capita is NOT a proper measure
•
Household public goods
•
Distribution within the household: e.g. needs differ according to ages
Equivalent income =total household income/number of consumption units in the
household
4
OECD II scale:
•
first adult: 1 consumption unit,
•
further adults: 0.5 consumption units
•
children (below 15) 0.3 consumption units
Graphical representation of a distribution
Representation of information on expenditure, consumption or income in the form of
diagrams is often very useful in the analysis of inequalities.
Representations of basic indicators of
dispersion
•
Pen’s parade
•
Frequency distribution
•
Cumulative frequency distribution
•
Lorenz curve
5
Charting the income distribution
6
Pen’s parade in Hungary: income of people ranked by their per capita income in
1992
income
650000
600000
550000
500000
450000
400000
350000
300000
250000
200000
150000
100000
50000
Persons (ranked)
0
1
18 35 52 69 86 10 12 13 15 17 18 20 22 23 25 27 29 30 32 34 35 37 39 40 42 44 46 47 49 51 52 54 56
3 0 7 4 1 8 5 2 9 6 3 0 7 4 1 8 5 2 9 6 3 0 7 4 1 8 5 2
Source: Tóth, 2005
(histogram) illustrates the relative
frequency of individuals sorted into
different categories of expenditure.
For example the enclosed frequency
distribution shows that 20% of individuals
fall into the fourth category. [i.e. f(4)=0.2].
Percentage of population
40
Frequency distribution: The diagram
35
30
F re q ue n cy dist ribu tio n f(y)
25
20
15
10
5
0
1
2 3
4 5
6 7
8
9 10 11 12 13 14 15
Categories of expenditure
Source: Tóth, mimeo.
7
Income distribution in 1992, illustration from the Hungarian Household Panel1
Source: Tóth, mimeo.
1
Number of people in the HHP sample
8
Income distribution in Hungary, 1992–19962
Number of persons
(1000)
1200
1000
92
93
94
800
95
96
600
400
200
Income (1000 Ft)
0
5
15
25
35
45
55
65
75
85
95
105 115
125 135
145 155 165
175 185
195 205
Source: Tóth, mimeo.
Cumulative frequency distribution:
frequency – percentage of households
on or below a given level of expense/income.
Compared to the previous graph F(y) is the
area below f(y) on the left side.
[F(4) = f(4)+f(3)+f(2)+f(1) = 20+35+12+4=71%]
cumulative frequency (%)
This graph illustrates cumulative
Cumulativ e fre que ncy function
F(y)
100
90
80
70
60
50
40
30
20
10
0
1
2 3
4
5 6
7
8 9 10 11 12 13 14 15
Categories of expenditure
Source: Tóth, mimeo.
2
equivalent incomes deflated to 1992
9
Ratios of dispersion
Definition:
Ratios of dispersion measure the distance between two groups in the income
distribution. Typically the average income of the richest x% of the population is divided
by the average expenses/income of the poorest x%.
Different alternatives exist. Most frequently it is based on the decile or the quintile of the
distribution (decile includes 10% of the total population, quintile includes 20% of it).
10
Advantages:
(+) The group-average ratios and percentile ratios are easy to interpret.
Disadvantages:
(–)
The value of the group-average-ratio is highly sensitive to extreme incomes,
particularly in case of small-sample estimations.
(–)
No axiomatic basis, not derived from principles of equity.
Representation of inequalities by the Lorenz
curve
100
Lorenz curve: The most common
the cumulative ratio of expenditure
on the vertical axis and the cumulative
ratio of population on the horizontal axis.
In this example 40% of the population
possess less than 20% of the total
consumption expenditure.
90
Cumulative % of consumption
representation. The curve illustrates
80
70
60
50
40
30
20
10
0
0
20
40
60
80
100
Cumulative % of population
11
If all individuals had the
Cumulative decile
shares (income)
same income, i.e. the distribution
of incomes was perfectly equal,
the Lorenz curve would be identical to
E: Line of
equality
Gini: surface
between E and S
divided by
surface of the
lower triangle
the diagonal (E: line of equality).
S: Lower inequality
If one person had the total income,
the Lorenz curve would pass through
L:Higher inequality
points (0,0), (100, 0), and (100,100).
This is the curve of „perfect inequality”.
Cumulative decile shares (population)
Line (S): lower inequality
Line (L): higher inequality
Source: Tóth, 2005
What if they intersect?
Indices of aggregate inequality: the Gini
coefficient
The Gini coefficient is related to the
The value of Gini equals the ratio of the
area A and area A+B.
In the previous figure Gini equals 0 in the
case of perfect equality and 1 in the case of
perfect inequality.
90
Cumulative % of consumption
Lorenz curve representation.
100
80
70
60
50
40
30
20
10
0
0
20
40
60
80
100
Cumulative % of population
12
Definition:
The Gini coefficient is the most commonly used indicator of inequality.
The definition of Gini: ratio of the average absolute income difference between every
pair of the sample and the average income.
The Gini coefficient can range from 0 to 1. The value of 0 expresses total equality and
the value of 1 maximal inequality. It measures the „deviation” from total equality.
Formal definition:
Different formulae exist; the classical formula of Gini is :
n
n
∑∑ y
Gini =
i
− yj
i =1 j =1
2n(n − 1) y
Where yi and yj stand for individual income/consumption values, is the average, and n is
the number of observations.
Advantages (+) and disadvantages (–) :
(+) The coefficient is easy to understand because of its connection to the Lorenz curve.
(–) The coefficient is not additively separable: the Gini of the total population is not
equal to the (weighted) sum of the Ginis of population subgroups.
The coefficient is sensitive to income-changes irrespective of whether the change is
taking place at the top, the middle, or the bottom of the distribution (all transfers of
income between two individuals have an effect independently of their financial situation).
13
Axiomatic approach to the measurement of
inequalities
In which distribution do you think inequality is higher?
1. A(5,8,10)
B(10,16,20)
2. A(5,8,10)
B(10,13,15)
3. A(5,8,10)
B(5,5,8,8,10,10)
4. A(1,4,7,10,13) B(1,5,6,10,13)
5. A(4,8,9)
vs B(5,6,10) ?
A’(4,7,7,8,9) vs B’(5,6,7,7,10) ?
See: Amiel és Cowell, 1999
What kind of attributes should we expect of this kind of index?
1. Scale independence: if all incomes are multiplied by constant k, the inequality index
should not change.
2. Population independence: if population increases in all income categories by the
same ratio, the inequality index should not change.
3. Symmetry: if two individuals transpose their income the value of the inequality index
should not change.
4. Axiom of transfers (Pigou–Dalton): if income is redistributed from a richer individual
to a poorer one (progressive transfer), so that their ranking does not change, the
inequality index should decrease.
5. Decomposability: requirement of a coherent relationship between inequality in the
total population and inequality in subgroups. If inequality in one subgroup increases
(all other things unchanged) total inequality should not decrease. Special type:
additive separability.
14
Indices of aggregate inequality:
the generalized entropy indices
Question: which axioms do or do not selected indices fulfill?
Theorem:
An index is consistent simultaneously with the axiom of scale independence, population
independence, axiom of transfers, and the axiom of additive decomposability if and only
if it is member of the generalized entropy family of indices.
The formula of the Generalized Entropy Indices:
1
GE (α ) = 2
α −α
 1

 N
α

 yi 
  − 1
∑
i =1  y 

N
Where yi = income/consumption,
N = number of individuals, and
α
is a parameter which weights the individuals of
different levels of distribution.
Depending on the value of
α
parameter:
 y
 
log
∑
y 
i =1
 i
y 
1 N yi
GE (1) = Theil = ∑ . log i 
N i =1 y
 y
1
GE (0) = MLD =
N
GE (2) =
N
CV 1  1
= 
2
y  N

 yi
∑
i =1 
N

y 

2



1
2
15
Attributes of certain indicators
Indicators of aggregate inequality:
the standardized entropy indices
Advantages and disadvantages:
(+) Axiomatic basis: we know its attributes.
(+) GE(α) indices can be separated to ”subgroups” : the GE(α) index calculated on the
total population is the weighted average of the indices calculated on its subgroups,
where weights are the proportions of subgroups in the total population (this is not
possible in the case of Gini).
(–) Difficult to interpret (in contrast to Gini)
16
Decomposition of inequalities
•
Inequalities are decomposed when one is curious about the extent to which
inequalities among various social groups, regions or income components are
responsible for the total inequalities in a country.
•
Inequalities can be separated to ”between group” and ”within group” components.
The first one shows the difference between the averages of people from different
subgroups, and the second one shows the differences within the groups.
Decomposition of inequalities:
distribution of income in total population,
1987 and 2001
Source: Tóth, 2005
17
Decomposition of inequalities: frequency
distribution at different levels of education
Source: Tóth, 2005
Decomposition of inequalities
Decomposition of an additively separable index (MLD):
MLD=
Within group
inequality
Where
Σk vkMLDk
+
Σk vk log (1/λk),
Between group
inequality
vk =nk/n and λk=µk/µ
18