1. Session 5 Review Today Inequality measures Four basic axioms Lorenz
Lorenz inconsistent measures
Today
Additional Properties
Lorenz consistent measures

2. Inequality Measures Notation Definition x is the income distribution
xi is the income of the ith person
n=n(x) is the population size.
D is the set of all distributions of any population size
Definition
An 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.

3. Four Basic Properties Definition Ex Symmetry (Anonymity) Idea
We 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

4. Replication Invariance (Population Principle)
Def
We 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

5. Scale Invariance (Zero-Degree Homogeneity)
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

6. Def Ex Transfer Principle Idea
We 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

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

8. Lorenz Curve (1905) Lorenz Curve Cumul Pop p Inc Lx(p) 1/3 1/15 2/3Ex
Lorenz Curve
1. Order incomes lowest to highest to obtain ordered version Ex
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
Cumul
Pop p
Inc
Lx(p)
1/3
1/15
2/3
7/15 1
15/15

9. 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)

10. Lorenz Curve
Def
Let 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:

11. Lorenz Curve
Source: Foster (1985), p. 17

12. Characteristics Starts at (0,0); ends at (1,1).
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?

15. Lorenz Criterion Def Note Ex
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

17. Note
Lorenz 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.

18. Q/ How does the Lorenz criterion fare with respect to the four basic axioms?
A/ Satisfies them all!

19. 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.

20. 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.

21. Note
If 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.

22. Classifying Inequality Measures Note By motivation
Various ways to classify
By motivation
Positive vs. normative
Statistical
By properties
Lorenz consistent vs. inconsistent
Decomposable
Some well known measures are not Lorenz consistent

23. Lorenz inconsistent measures
Violate one or more of the four basic axioms
Usually
Transfer
Scale invariance

24. Range Ex x = (8,6,1) I(x) = (8-1)/5 = 7/5 Idea
Gap between the highest and the lowest income measure in number of means.
Note
Ignores the distribution between the extremes: Violates transfer principle

25. 90/10 ratio Ex x = (8,6,1) I(x) = 8 Idea
Ratio of the 90th percentile income and the 10th percentile income.
Note
Ignores the distribution apart from the two points: Violates transfer principle

26. Kuznets Ratio where I(x;p,r) = Sr(x)/sp(x)
sp(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
Note
The ratios derived from points along the Lorenz Curve.
Like the range, it ignores the distribution between the cutoffs and therefore violates the transfer principle

27. Relative Mean Deviation
Idea
The average departure of incomes from the mean, measured in means
Note
Not sensitive to certain transfers. Violates transfer principle.

28. Variance
Note
By 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.

29. Variance of Logarithms
where
Note
Applies 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).

30. Additional Properties
Q/ How to discern among Lorenz consistent (or relative) measures?
A/ Additional axioms for inequality measures

31. Def
Let = (μ,…, μ) denote the smoothed distribution which gives everyone in x the mean μ of x.
Normalization
If x = , then I(x) = 0.
Idea
If all incomes are same, then the measure of inequality is zero.

32. I has no “jumps” Continuity
For 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

33. Motivation Transfer sensitivity
How 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 sensitivity
A 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).

34. Subgroup Consistency Note
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).
Note
Important for policy design & evaluation, regional vs. national
Yet, there are arguments against; see, for example Foster & Sen (1997), p.160.

35. 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.
Note
Very useful for policy design & evaluation, regional vs. national vs. global.
Implies subgroup consistency (why?)

36. Session 5 Review Today Inequality measures Four basic axioms Lorenz
Lorenz inconsistent measures
Today
Additional Properties
Lorenz consistent measures

37. Lorenz Consistent Inequality Measures
We now consider measures that satisfy the four basic axioms
First positive, then normative.

38. Squared Coefficient of Variation
Note
The 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:

39. Gini Coefficient Several equivalent definitions
Idea
G = E|s – t|/(Es + Et) or the expected difference between two incomes drawn at random, over twice the mean
Ex differences
x = (6,1,8)
Then μ = 5 and
G = (28/9)/10 = .311

40. Gini Coefficient
Idea
G = (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 = 31/9
G = (5-S)/5 = 14/45 = .311

41. Gini Coefficient Note Ex incomes weighted by ranks; also:
x = (6,1,8). Then μ = 5 and S = (1*8 + 3*6 + 5*1)/9
G = (5-S)/5 = 14/45 = .311

42. Relationship with the Lorenz Curve
Note
Relationship with the Lorenz Curve O C D A

43. Note Not 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.

44. Theil’s first measure Q/ What does this mean?
where |x| = Σxi and si = xi/|x| is i’s income share
Shannon’s entropy measure
Note
Se(1,0,…,0) = Se(1/n,…,1/n) = ln(n)
So T(x1,…,xn) = max Se – Se(s1,…,sn)

45. Properties Lorenz consistent
Additively decomposable with weights being
or income share of subgroup

46. Theil’s second measure Q/
What does this mean?
where g = (Πxi)1/n is the geometric mean
Note
If all incomes are equal, then D = 0 since μ = g
Otherwise D > 0 since μ > g

47. Properties Lorenz consistent
Additively decomposable with weights being
or population share of subgroup

48. Generalized Entropy Measures
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.

49. Note Lorenz 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 distribution
With 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)

50. They are additively decomposable with weights being:
Only Theil’s Measures (α=1 and α=0) have weights that sum up to 1.

51. Normative Measures
Dalton (1920), Kolm (1968), Atkinson (1970), Sen (1973)
Core Idea
Inequality represents the loss in welfare from unequal incomes Q/
Given a social welfare function W(x), how to measure inequality?

52. 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 invariant
defined a new approach that did not have these problems

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

54. Atkinson’s Approach J = Total income
JK 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

55. Atkinson’s normative measure
Interpretation

56. 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.

57. Atkinson’s Class
Note
where
and