Different Concepts of Welfare: Indices and complete orderings

Inequality, Poverty, Polarization measurement

Inequality

• At the heart of the Pigou-Dalton Principle, its very essence is interpersonal comparison, if you’ve got more than me we’re unequal.

• How to measure it?

• Relative or absolute?

• What particular features (groups).

Measures of Inequality: Relative Range, Relative Mean Deviation

and Variance.

In the following the income of agent i from population n will be represented by xi i = 1,..,n which without loss of generality will be ranked in order, i.e. x1 < x2 <..< xn-1 < xn . I(xi,z) will be an indicator function such that if xi z, I(xi,z) = 1 and is 0 otherwise. Relative Range .

Relative Mean Deviation D

Variance V

x - x = R 1n

| |ni

i=1

- xD = n

n

) - x( = V

2i

n

=1i

Coef of Var C, Log St. Dev. L, Gini G, Theil’s Entropy T, Shutz Coef S

V

= C

n

)x - ( = L

2i

n

=1i

lnln

1

1| |

n nn-1

i i j2 2ji=1 i=1

2x xG = 1 + .(n x

n n(n+1-i))

)x(.nx = T ii

n

=1i ln

)x - (.n

1 = S i

n

=1i

Poverty Measurement: The Axioms

Poverty indices have been developed over the years on an axiomatic basis. To fix ideas a brief discussion of the more important axioms is provided here, Zheng (1998) provides an excellent survey of the plethora of axioms that have been proposed and the extent to which various poverty measures are in accord with those axioms. Focus The focus axiom simply requires that the poverty measure should be independent of the income distribution of the non-poor essentially it argues that only the status of the poor should matter. Symmetry Sometimes called the anonymity axiom this requires that the measure be independent of who the poor are and that should a poor person and a non poor person switch places the measure should not alter. Replication Invariance This axiom requires that measures should admit the comparison of different sized distributions. Continuity Small increments of income changes of the poor should not result in large changes in the poverty measure.

Further Poverty Measure Axioms

Increasing Poverty Line. In comparing two identical societies the one with the higher poverty line should have the higher poverty index. Regressive Transfer Poverty measures should be sensitive to redistribution amongst the poor thus a regressive transfer should increase the poverty index. Transfer Sensitivity Requires that greater weight be given to income transfers low in the income distribution. Subgroup Consistency The poverty measure should reflect efforts to reduce poverty in subgroups. See Bohung Zheng

Absolute Poverty Measures

The Headcount H.

The Income Gap Ratio Y

The Poverty Gap Ratio P

n

z),xI( = H

i

n

=1i

,

n

i ii=1

n

ii=1

I( ,z)x x1

Y = ( ).(z - )z

I( z)x

nz

)x -z)(z ,xI( = P

ii

n

=1i

More Absolute Poverty Measures

Modified Poverty Gap Ratios (P*)

b)x -z)(z ,xI(nza

1 = P iii

n

=1i

*

where: a b i Sen (S) q+1 q+1-i Kakwani ik (q+1-i)k Thon n+1 (n+1-i) Note that the Sen Index (S) may be written as:

where Gp corresponds to the Gini coefficient for the poor.

)n + H

H.(GP) - (H + P = S

1-p

Relative Poverty Measures

• Rather than think about an absolute poverty cut-off “z” we may substitute some other measure which is “relative” to the population of interest (e.g. some function of median income in the population).

• Can insert this formulation of the cut-off in all of the preceding formulae.

• Presents difficulties with inter-temporal comparisons.

• Presents difficulties with inferences.

A Digression: The LICO and other sample dependent measures

• Canadian Council on Social Development (1992) claimed that 1.1 million children were living below the poverty line.

• LICO “Low Income Cut Off” = income level below which a household unit will spend on average at least 56.2% on food, shelter and clothing.

• Cut off changes over time with tastes, prices etc.• Problem if the income elasticity around the

poverty cut off is greater than one.

Overall Income Elasticities

Table 1. Aggregate Expenditure shares and income elasticities 1982-1992

Food Shelter Clothing

means 0.2028 0.2553 0.0759

medians 0.1933 0.2395 0.0681

minimum 0.0123 0.0086 0.0027

maximum 0.5719 0.8881 0.4256

elasticity (nc)(“t’s”(h0:e=1))

0.5985 0.7338 1.1843 (-33.1166) (-18.2072) (8.7282)

elasticity(“t’s”(h0:e=1))

0.5928 0.7898 1.2195 (-107.1506) (-36.4342) (88.5580)

Income Elasticities by Income Quantile

Table 2. Income Elasticities (“t’s”(h00:e=1)) for selected quantiles (nochildren)

Income Quantile Food Shelter Clothing

0 < .125 0.7232 1.0323 0.8545 (2.8010) (0.2025) (0.5341)

.125 < .25 0.9806 0.6501 1.8579 (0.0611) (0.7690) (1.1860)

.25 < .5 0.5650 0.6709 1.2931 (7.8121) (4.6928) (2.7441)

.5 < .75 0.3477 0.5687 1.0551 (6.3771) (3.6648) (0.3410)

.75 < 1.0 0.6414 0.6984 1.0301 (8.2658) (6.9676) (0.5266)

Table 3. Income Elasticities (“t’s”(h00:e=1)) for selected quantiles(children)

Income Quantile Food Shelter Clothing

0 < .125 0.8632 0.8144 0.9255 (2.1056) (1.7685) (0.4158)

.125 < .25 1.0641 0.4037 1.6311 (0.1918) (1.2418) (0.8268)

.25 < .5 0.6112 0.8019 1.1497 (6.8509) (2.7709) (1.3750)

.5 < .75 0.3264 0.6843 0.9745 (5.2136) (2.1237) (0.1252)

.75 < 1.0 0.6744 0.9061 1.0998 (2.2625) (0.6540) (0.5269)

Polarization

Polarization means the tendency of economic agents to form different groups and acquire identities

that enhance differences from other groups. It is both cause and consequence of much economic

behaviour. It has been employed, for example, in describing the diminution of the middle class in

wage, income and wealth distributions, in studying growth and convergence issues, and in

examining the plight of the poor. Although polarization is closely associated with trends in

inequality, increased polarization can correspond to an increase, a reduction or no change in

inequality.

The concept need not be confined to the study of changes within a population’s distribution of a

particular characteristic, but can be used in assessing the relative movements of two or more

distributions as they evolve (for example, polarization between ethnic groups, genders, and

nations). In this context polarization takes the form of distributions becoming ‘less alike’ in a

particular fashion; as such it involves comparison of complete distributions, not just their location

or scale characteristic.

Polarization Measurement: The Axioms

Polarization: an axiomatic foundation

Indices of polarization were formulated in Esteban and Ray (1994) and Duclos, Esteban and Ray

(2004) (see also Wang and Tsui, 2000) by positing a collection of axioms whose consequences

should be reflected in a Polarization measure. The axioms are founded upon a so-called

Identification-Alienation nexus wherein notions of polarization are fostered jointly by an agent’s

sense of increasing within-group identity and between-group distance or alienation. The four

axioms may be loosely summarized as follows:

Axiom 1: A mean preserving reduction in the spread of a distribution cannot increase polarization.

Axiom 2: Mean preserving reductions in the spread of sub-distributions at the extremes of a density

cannot reduce polarization.

Axiom 3: Separation of two sub-densities towards the extremes of the distributions range must

increase polarization.

Axiom 4: Polarization measures should be population-size invariant.

Polarization: The Measures

The polarization index developed for discrete distributions as a consequence of these axioms

(Esteban and Ray, 1994) may be written as:

1

1 1

| |n n

i j i ji j

P K x x

(1)

Here K is a normalizing constant, πi is the sample weight of the i’th observation and where α ≥ 0 is

a polarization sensitivity factor chosen by the investigator. It may readily be seen that α = 0 yields a

sample weighted Gini coefficient.

The continuous distribution analogue (Duclos, Esteban and Ray, 2004) may be written as:

( ) ( ) | | ( ) ( )y x

P F f y y x dF x dF y (2)

Again, α is the polarization sensitivity factor which in this case is confined to [0.25,1].

Other Polarization and Alienation Measures.

• Focusing on just two groups or clubs Alienation measures such as the Gini based pooled mean normalized difference in subgroup means, and the distributional overlap measure are candidates for distributional overlap (Anderson (2006)).

• Recently a new measure, the Bipolar Trapezoid, particularly useful in multi-dimensional environments has been developed (Anderson (2008).

AGINI

The classic Gini inequality coefficient which, with xi being the income of the i’th agent for agents i

= 1,..,n (where incomes are arranged in ascending rank order), may be written as:

2

1 1

1| | [1]

2

n n

i ji j

Gini x xn

where μ is the mean of the x’s. Suppose the poverty cut-off, defining perfectly segmented rich and

poor clubs, is somewhere between xp and xp+1 where p < n, then Gini is the average mean

normalized differences between agents within the poor club, agents within the rich club and

between poor and rich club agents. For measuring alienation only the last group of comparisons

that are relevant, in which case a new statistic “AGini” could be written as:

1 1

1 1

1 1( ) ( ) [2]

( )

n p

j ipnj p i

j ij p i

x xAGini x x

p n p n p p

Indeed the formulae in [2] can be generalized to general group differences where segmentation is

imperfect. The income distribution is now presumed to be a mixture of two sub-group population

distributions (Poor and Non Poor), where relationship to the poverty line is no longer the defining

feature of the clubs. Using Ipoor(i) as an indicator function equaling 1 when the i’th person is from

the Poor club and 0 otherwise AGini becomes:

1 1

( ) 1 ( )1( ( ) [3]

( ) (1 ( ))

poor poori n n

poor poorj j

I i I iAGini x

I j I j

The perfect segmentation case can be linked to the well known family of poverty and welfare

indices introduced by Foster, Greer and Thorbecke (1984) the representation of which is given by:

0

( , ) ( ) ( ) [4]z z x

P x z dF xz

where F(x) is the cumulative density function (with p.d.f. f(x)) describing the population of

incomes, z is the maximum of the poor and α is a parameter reflecting the degree of poverty

aversion so that since P0 corresponds to the proportion in poverty Pi/P0 may be construed as the

expected value of a weighted function of the normalized income deficiency where the weights are

the (i-1)’th power of the normalized income deficiency itself. Thus increasing i increases the

weights attached to those furthest from the poverty line. Along similar lines W(x,z), an index of

weighted relative distances of incomes above the poverty line, may be contemplated of the form:

( , ) ( ) ( ) [5]z

x zW x z dF x

z

In this case W0 corresponds to the proportion of the population above the poverty line W1 is a

normalized measure of relative well being of the non-poor, W2 is a measure of the intensity of the

relative well being of those above the poverty line and so on. In this case as α becomes very large

the index becomes almost entirely focused on the richest person, W1/W0 corresponds to the

expected normalized income surplus over the maximum poverty income etc. The population

analogue of AGini can be shown to be a special case (α = 1) of a general class of alienation

measures which, for pre-specified α ≥ 1, is given by:

0 0

( ) ( ) [6]W Pz

W P

An Overlap Measure.

The extent to which two distributions f(x) and g(x) overlap is given by:

min( ( ), ( ))OV f x g x dx

Clearly it is a number between 0 and 1 with 0 corresponding to no overlap and 1 to the perfect

matching of the two distributions. It follows that AOVER = 1-OV is a measure of the extent to

which the distributions do not match or are alienated. When f(x) and f(y) are specified to the extent

that all of their parameters can be estimated and the intersection points of f(x) and g(x) calculated

OV can be estimated parametrically. When f( ) and g( ) are unknown, given independent samples

from f( ) (represented by x) and g( ) (represented by y) of sizes nx and ny respectively, its empirical

counterpart may be implemented by choosing K + 1 partitions of the range of x defined by xk, k =

1, …, K and calculating:

1 1

1

( ) ( ) ( ) ( )( ) ( )max( , ) min(( ), ( ))

Kk k k kK K

kx y x x y y

I x x I x x I y x I y xI x x I y xOVEST

n n n n n n

Where I(z) is an indicator function equal to 1 when z ≤ 0 and 0 otherwise. The estimator and

associated tests are most effective when the intersection points of the unknown distributions

correspond with the chosen partition points. Since f(x) and g(x) are unknown so will their

intersection points be, however they could be estimated by adapting kernel estimation techniques.

The Trapezoidal Measure

The Trapezoid measure (details)

Formally when the poor and non-poor distributions are separately identified in K dimensions the

indicator BIPOL may be written as:

2

1

( )10.5( ( ) ( ))

Kmpk mrk

p mp r mrk k

x xBIPOL f x f x

K

When the groups are not separately identified (NI) and the index is calculated from the modal

points of the mixture distribution, noting that the poor and rich modes may be written in terms of

the underlying distributions as:

( ) ( ) ( ( ) ( ))

( ) ( ) ( ( ) ( ))

mp r mp p mp r mp

mr r mr p mr r mr

f x f x w f x f x

f x f x w f x f x

The index may also be written as:

2

1

( )10.5[ ( ) ( )]

Kmpk mrk

NI mp mrk k

x xBIPOL f x f x

K