Session 5

ReviewInequality measuresFour basic axiomsLorenzLorenz inconsistent measures

TodayAdditional PropertiesLorenz consistent measures

Inequality Measures

Notation x is the income distributionxi is the income of the ith personn=n(x) is the population size.D is the set of all distributions of any population size

DefinitionAn inequality measure is a function I from D to R which, for each distribution x in D indicates the level I(x) of inequality in the distribution.

Four Basic Properties

DefinitionWe say that x is obtained from y by a

permutation of incomes if x = Py, where P is a permutation matrix.

Ex

Symmetry (Anonymity) If x is obtained from y by a permutation

of incomes, then I(x)=I(y).

Idea All differences across people have been

accounted for in x

€

x = Py =

010

100

001

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

6

1

8

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥=

1

6

8

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

DefWe say that x is obtained from y by a

replication if the incomes in x are simply the incomes in y repeated a finite number of times

Ex

Replication Invariance (Population Principle)If x is obtained from y by a replication, then

I(x)=I(y).

Idea Can compare across different sized populations

€

x = (y1, y1, y2, y2,......, yn , yn )

€

x = (6,6,6,1,1,1,8,8,8)

Def We say that x is obtained from y by a

proportional change (or scalar multiple) if x=αy, for some α > 0.

Ex

Scale Invariance (Zero-Degree Homogeneity)If x is obtained from y by a proportional

change, then I(x)=I(y).

Idea Relative inequality

€

y = (6,1,8)

€

x = (12,2,16)

DefWe say that x is obtained from y by a (Pigou-

Dalton) regressive transfer if for some i, j:i) yi < yj

ii) yi – xi = xj – yj > 0

iii) xk = yk for all k different to i,j

Ex

Transfer Principle If x is obtained from y by a regressive

transfer, then I(x) > I(y).

Idea Mean preserving spread increases measured

inequality

€

y = (2,6,7)

€

x = (1,6,8)

Def Any measure satisfying the four basic properties (symmetry, replication invariance, scale invariance, and the transfer principle) is called a relative inequality measure.

Lorenz Curve (1905)

1. Order incomes lowest to highest to obtain ordered version

Ex

2. Find the cumulative share of population up to each i = 1,…,n:

p = i/n

3. Find the cumulative share of the income up to each i

4. Plot pairs and connect the dots

CumulPopp

CumulIncLx(p)

1/3 1/15

2/3 7/15

1 15/15

Ex Lorenz Curve

€

x = (8,6,1)

€

ˆ x = ( ˆ x 1,..., ˆ x n )

€

ˆ x = (1,6,8)

€

Lx ( p) = Σi=1k ˆ x i / Σi=1

n ˆ x i

Note At any population share p, the Lorenz curve Lx gives the share of income received by the lowest p of income recipients (often stated in % terms)

Lorenz Curve

DefLet F(s) be a cumulative distribution function (cdf) and let QF(p) = F-1(p) be its quantile function. Then the Lorenz Curve associated with F is given by:

€

LF ( p) =1

μQF (t )dt

0

p

∫

Lorenz Curve

€

ˆ x = (1,6,8)

Source: Foster (1985), p. 17

€

L(x, p) =1

x Fx

−1(t)dt0

p

∫

Characteristics

1. Starts at (0,0); ends at (1,1).2. Always convex (because population is

ordered from poorest to richest) and increasing (if incomes positive).

Q/ Lorenz curve of a perfectly equal distribution?

Q/ Lorenz curve of a perfectly unequal distribution?

The Lorenz Curve of a Perfectly Equal Distribution

0.0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1

p

L(p)

Lorenz Criterion

Def Given two distributions x and y, we say that x Lorenz-dominates y if and only if:

Lx(p) > Ly(p) for all p, with > for some p

Note x is unambiguously less unequal than y

Ex

€

y = (1,5,9)

€

x = (1,6,8)

Lorenz Curves for Two Distributions

0.00

0.20

0.40

0.60

0.80

1.00

0 0.2 0.4 0.6 0.8 1p

L(p)

X Y Equal Distribution

NoteLorenz criterion generates a relation that is incomplete: when Lorenz curves cross, the Lorenz criterion cannot decide between the two distributions.

But when it does hold, the ranking is considered unambiguous

Shorrocks and Foster (1987) provide additional conditions by which two distributions can be ranked when Lorenz curves cross. Still it does not eliminate all the incompleteness.

Q/ How does the Lorenz criterion fare with respect to the four basic axioms?

A/ Satisfies them all!

The Lorenz Curve and the Four Axioms

Symmetry and Replication invariance satisfied since permutations and replications leave the curve unchanged.

Proportional changes in incomes do not affect the LC, since it is normalized by the mean income. Only shares matter. So it is scale invariant.

A regressive transfer will move the Lorenz curve further away from the diagonal. So it satisfies transfer principle.

Lorenz Curves for Two Distributions

0.00

0.20

0.40

0.60

0.80

1.00

0 0.2 0.4 0.6 0.8 1p

L(p)

X Y Equal Distribution

€

y = (1,5,9)

€

x = (1,6,8)

Lorenz Consistency (Foster, 1985)

Def An inequality measure I: D→R is Lorenz consistent whenever the following hold for any x and y in D: (i) if x Lorenz dominates y, then I(x) < I(y), and (ii) if x has the same Lorenz curve as y, then I(x) = I(y).

Theorem An inequality measure I(x) is Lorenz consistent if and only if it satisfies symmetry, replication invariance, scale invariance and the transfer principle, ie, if it is a relative inequality measure.

NoteIf Lorenz curves don’t cross, then all relative measures follow the Lorenz curve.

If Lorenz curves cross, then some relative measure of inequality might be used to make the comparison. But the judgment may depend on the chosen measure.

Classifying Inequality Measures

Note Various ways to classify

By motivationPositive vs. normativeStatistical

By propertiesLorenz consistent vs. inconsistentDecomposable

Note Some well known measures are not Lorenz

consistent

Lorenz inconsistent measures

Violate one or more of the four basic axioms

UsuallyTransferScale invariance

Range

Exx = (8,6,1) I(x) = (8-1)/5 = 7/5

IdeaGap between the highest and the lowest income measure in number of means.

NoteIgnores the distribution between the extremes: Violates transfer principle

€

I( x) =1

μ( xmax−xmin)

90/10 ratio

Exx = (8,6,1) I(x) = 8

IdeaRatio of the 90th percentile income and the 10th percentile income.

NoteIgnores the distribution apart from the two points: Violates transfer principle

€

I( x) = Qx (.90) / Qx (.10)

Kuznets Ratio

I(x;p,r) = Sr(x)/sp(x)

wheresp(x) = Lx(p) share of income earned by the poorest p share of the population

Sr(x) = 1 – Lx(1-r) share of income earned by the richest r share of the population

Ex If p = r = 1/3, x = (8,6,1). Then I(x) = 8

NoteThe ratios derived from points along the

Lorenz Curve.Like the range, it ignores the distribution

between the cutoffs and therefore violates the transfer principle

Relative Mean Deviation

IdeaThe average departure of incomes from the mean, measured in means

NoteNot sensitive to certain transfers. Violates transfer principle.

€

I(x) =

|μ − xi |i=1

n

∑nμ

Variance

NoteBy squaring the gaps of each income to the mean, the bigger gaps receive a higher weight; it satisfies the transfer principle

However, the variance goes up four times when incomes are doubled; it violates scale invariance.

n

)x()x(V

2n

1ii

Variance of Logarithms

where

NoteApplies variance to the distribution of log-incomes.

It is scale invariant (why?)But violates transfer principle when relatively high incomes are involved.

Can be incredibly Lorenz inconsistent (See Foster and Ok, 1998).

€

˜ x = (ln(x1),.......,ln(xn ))

€

VL(x) = V(˜ x ) =

(μ ˜ x − ˜ x i)2

i=1

n

∑n

Additional Properties

Q/ How to discern among Lorenz consistent (or relative) measures?

A/ Additional axioms for inequality measures

DefLet = (μ,…, μ) denote the smoothed distribution which gives everyone in x the mean μ of x.

NormalizationIf x = , then I(x) = 0.

IdeaIf all incomes are same, then the measure of inequality is zero.

€

x

€

x

ContinuityFor any sequence xk, if xk

converges to x, then I(xk) converges to I(x)

Idea

I has no “jumps”

A small change in any income should not result in an abrupt change in inequality

MotivationHow should these progressive transfers (from x to x’, and from x to x”) be regarded by an inequality measure?x=(2,4,6,8) x' =(3,3,6,8) x" =(2,4,7,7)

Transfer sensitivityA transfer-sensitive inequality measure places greater emphasis on transfers at the lower end of the distribution; so I(x’) < I(x”)

Formalized by Shorrocks and Foster (1987).

Subgroup Consistency Suppose that x’ and x share means and population sizes, while y’ and y also share means and population sizes. If I(x’) > I(x) and I(y’) = I(y), then I(x’,y’) > I(x,y).

NoteImportant for policy design & evaluation, regional vs. national

Yet, there are arguments against; see, for example Foster & Sen (1997), p.160.

Additive Decomposability Suppose that x and y are any two distributions, then

where the final term is the inequality of the smoothed distribution in which persons in x receive the mean of x and persons in y receive the mean of y.

NoteVery useful for policy design & evaluation, regional vs. national vs. global.

Implies subgroup consistency (why?)

),()()(][][),( yxIyIwxIwBIWIyxI yx

Session 5

ReviewInequality measuresFour basic axiomsLorenzLorenz inconsistent measures

TodayAdditional PropertiesLorenz consistent measures

Lorenz Consistent Inequality Measures

We now consider measures that satisfy the four basic axioms

First positive, then normative.

Squared Coefficient of Variation

NoteThe variance of the (mean-)normalized distribution, thus it is scale invariant.

Not transfer sensitive. A transfer has the same impact regardless where it takes place.

Is additively decomposable, with weights on within-group term: 2)/)(/( xxx nnw

€

C2(x) = V(x /μ) =

(xi −μ

μi=1

n

∑ )2

n=

V(x)

μ 2

Gini Coefficient Several equivalent definitions

IdeaG = E|s – t|/(Es + Et) or the expected difference between two incomes drawn at random, over twice the mean

Ex differencesx = (6,1,8) Then μ = 5 and G = (28/9)/10 = .311

€

G(x) =1

2n2μ| xi − x j |

i= j

n

∑i=1

n

∑

€

0 5 7

5 0 2

7 2 0

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

Gini Coefficient

IdeaG = (μ – S)/μ where S = ΣΣmin(xi,xj)/n2 is the expected value of the min of two incomes drawn randomly.

Ex min’s

x = (6,1,8). Then μ = 5 and S = 31/9G = (5-S)/5 = 14/45 = .311

€

G(x) =1−1

n2μmin(xi,x j)

j=1

n

∑i=1

n

∑

€

1 1 1

1 6 6

1 6 8

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

Gini Coefficient

Noteincomes weighted by ranks; also:

Ex x = (6,1,8). Then μ = 5 and S = (1*8 + 3*6 + 5*1)/9

G = (5-S)/5 = 14/45 = .311

€

G(x) =1+1

n−

2

n2μ

⎛

⎝ ⎜

⎞

⎠ ⎟( x 1 + 2

( x 2 + .......+ n

( x n[ ]

€

where( x 1 ≥

( x 2 ≥ ..... ≥

( x n

€

G(x) =μ

μ−

−nμ

n2μ

⎛

⎝ ⎜

⎞

⎠ ⎟−

1

n2μ

⎛

⎝ ⎜

⎞

⎠ ⎟ 2

( x 1 + 4

( x 2 + .......+ 2n

( x n[ ] =

μ

μ−

1( x 1 + 3

( x 2 + .......+ (2n −1)

( x n[ ]

n2μ=

μ −S

μ

NoteRelationship with the Lorenz Curve

OC

D

A

AOCD

AG 2

NoteNot subgroup consistent (examples in OEI)

Not decomposable in ‘traditional way’. Instead it has an alternative ‘breakdown’ into three terms:

where R is a non-negative residual term that balances the equation. It indicates the extent to which the subgroups’ distributions overlap.

RyxGyGn

nxG

n

nRBGWGyxG

yyxx

),()()(][][),(

2

2

2

2

Theil’s first measure

Q/ What does this mean?

where |x| = Σxi and si = xi/|x| is i’s income share

Shannon’s entropy measureNote Se(1,0,…,0) = 0 Se(1/n,…,1/n) = ln(n)

So T(x1,…,xn) = max Se – Se(s1,…,sn)

€

T(x) =1

n

xi

μln

xi

μ

⎛

⎝ ⎜

⎞

⎠ ⎟

i=1

n

∑

€

T(x) =xi

| x |ln

nxi

| x |

⎛

⎝ ⎜

⎞

⎠ ⎟

i=1

n

∑ = ln(n) +xi

| x |ln

xi

| x |

⎛

⎝ ⎜

⎞

⎠ ⎟

i=1

n

∑ = ln(n) + si ln(si

i=1

n

∑ )

€

Se(p1,...,pn ) = − pi ln pi( )i=1

n

∑

PropertiesLorenz consistentAdditively decomposable with weights being

or income share of subgroup

€

w x = (nx /n)(μ x /μ)

Theil’s second measure

Q/ What does this mean?

where g = (Πxi)1/n is the geometric mean

NoteIf all incomes are equal, then D = 0 since μ = gOtherwise D > 0 since μ > g

€

D(x) =1

nln

μ

xi

⎛

⎝ ⎜

⎞

⎠ ⎟

i=1

n

∑

€

D(x) =1

nln

μ

xi

⎛

⎝ ⎜

⎞

⎠ ⎟

i=1

n

∑ =1

nln(μ)

i=1

n

∑ −1

nln xi( )

i=1

n

∑ = ln(μ) − ln(g) = ln(μ /g)

PropertiesLorenz consistentAdditively decomposable with weights being

or population share of subgroup

€

w x = (nx /n)

Generalized Entropy Measures

€

Iα (x) =

1

α (1− α )

1

n1−

xi

μ

⎛

⎝ ⎜

⎞

⎠ ⎟

α ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥i=1

n

∑ α ≠ 0,1

I1(x) = T(x) =1

n

xi

μln

xi

μ

⎛

⎝ ⎜

⎞

⎠ ⎟ α =1

i=1

n

∑

I0(x) = D(x) =1

nln

μ

xi

⎛

⎝ ⎜

⎞

⎠ ⎟ α = 0

i=1

n

∑

⎧

⎨

⎪ ⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪ ⎪

When α = 1, it is the Theil first measure.When α = 0, it is the Theil second measure, also

known as Mean Logarithmic Deviation

When α = 2, it is a multiple of the Squared Coefficient of Variation.

NoteLorenz Consistent.

Parameter α is indicator of ‘inequality aversion’ (more averse as α falls). It also indicates the measure’s sensitivity to transfers at different parts of the distributionWith α =2, it is ‘transfer neutral’.With α <2, it favours transfers at the lower end of the distribution (includes both Theil’s measures).

With α >2 it shows a kind of ‘reverse sensitivity’ stressing transfers at higher incomes. (These are less used)

They are additively decomposable with weights being:

Only Theil’s Measures (α=1 and α=0) have weights that sum up to 1.

)/)(/( xxx nnw

Normative Measures

Dalton (1920), Kolm (1968), Atkinson (1970), Sen (1973)

Core IdeaInequality represents the loss in welfare from unequal incomes

Q/Given a social welfare function W(x), how to measure inequality?

Dalton (1920) Used utilitarian social welfare with identical strictly concave, increasing utilities.

Measured inequality as maximum welfare minus actual welfare, all over maximum welfare.

Atkinson (1970)Noted the dependence on the way utility is measured

noted it need not be scale invariantdefined a new approach that did not have these problems

Atkinson (1970)Core ConceptEqually Distributed Equivalent Income (EDE) The income level which, if assigned to all individuals produces the same social welfare than the observed distribution.

Atkinson’s ApproachJ = Total incomeJK is set of all

possible distributions

I1, I2, I3 ane three social welfare levels

A = Actual distribution

D = Equally Distributed Equivalent Income

E = Mean Income

I = (E – D)/E

Atkinson’s normative measure

Interpretation€

I =1−xEDE

μ

Atkinson’s Class

All the members in the family are Lorenz consistent.

Parameter α is a measure of ‘inequality aversion’ or relative sensitivity of transfers at different income levels. The lower is α, the higher is the aversion to inequality and more weight is attached to transfers at the lower end of the distribution.

Each member is subgroup consistent but it is not additively decomposable.

01

0,11

1

)(/1

1

/1

1

n

n

i

i

n

i

i

x

x

nxA

EDEx

A 1

Atkinson’s Class

0xln

0,1x

)x(u

i

i

i

n

iixu

nxW

1

)(1

)(

0

0,11

1

/1

/1

1

n

i

ni

n

ii

EDE

x

xnx

Note

where

and