Inequality Measurement

Inequality Measurement
Inequality measurement Measurement Universitat Autònoma de Barcelona Frank Cowell December 2006

Issues to be addressed Builds on lecture 3
“Income Distribution and Welfare” Extension of ranking criteria Parade diagrams Generalised Lorenz curve Extend SWF analysis to inequality Examine structure of inequality Link with the analysis of poverty

Major Themes Contrast three main approaches to the subject
intuitive via SWF via analysis of structure Structure of the population Composition of Inequality measurement Implications for measures The use of axiomatisation Capture what is “reasonable”? Use principles similar to welfare and poverty

Overview... Relationship with welfare rankings Inequality measurement
Inequality rankings Inequality measures Relationship with welfare rankings Inequality axiomatics Inequality in practice

Inequality rankings Begin by using welfare analysis of previous lecture Seek an inequality ranking We take as a basis the second-order distributional ranking …but introduce a small modification Normalise by dividing by the mean The 2nd-order dominance concept was originally expressed in a more restrictive form.

Yet another important relationship
The share of the proportion q of distribution F is given by L(F;q) := C(F;q) / m(F) Yields Lorenz dominance, or the “shares” ranking G Lorenz-dominates F means: for every q, L(G;q) ³ L(F;q), for some q, L(G;q) > L(F;q) The Atkinson (1970) result: For given m, G Lorenz-dominates F Û W(G) > W(F) for all WÎW2

For discrete distributions
All the above has been done in terms of F-form notation. Can do the almost same in Irene-Janet notation. Use the order statistics x[i] where is the ith smallest member of… …the income vector (x1,x2,…,xn) Then, define Parade income cumulations GLC LC

The Lorenz diagram L(.; q) q L(G;.) L(F;.) proportion of income
1 0.8 L(.; q) 0.6 L(G;.) proportion of income Lorenz curve for F 0.4 L(F;.) 0.2 practical example, UK 0.2 0.4 0.6 q 0.8 1 proportion of population

Application of ranking
The tax and -benefit system maps one distribution into another... Use ranking tools to assess the impact of this in welfare terms. Typically this uses one or other concept of Lorenz dominance.

Official concepts of income: UK
original income + cash benefits gross income - direct taxes disposable income - indirect taxes post-tax income + non-cash benefits final income What distributional ranking would we expect to apply to these 5 concepts?

Impact of Taxes and Benefits. UK 2000/1. Lorenz Curve

Assessment of example We might have guessed the outcome…
In most countries: Income tax progressive So are public expenditures But indirect tax is regressive So Lorenz-dominance is not surprising. But what happens if we look at the situation over time?

“Final income” – Lorenz

“Original income” – Lorenz
0.5 0.6 0.7 0.8 0.9 1.0 Lorenz curves intersect 0.0 0.1 0.2 0.3 0.4 0.5 Is 1993 more equal? Or ?

Inequality ranking: Summary
Second-order (GL)-dominance is equivalent to ranking by cumulations. From the welfare lecture Lorenz dominance equivalent to ranking by shares. Special case of GL-dominance normalised by means. Where Lorenz-curves intersect unambiguous inequality orderings are not possible. This makes inequality measures especially interesting.

Overview... Three ways of approaching an index Inequality measurement
Inequality rankings Inequality measures Intuition Social welfare Distance Three ways of approaching an index Inequality axiomatics Inequality in practice

Inequality measures What is an inequality measure?
Formally very simple function (or functional) from set of distributions… …to the real line contrast this with ranking principles Nature of the measure? Some simple regularity properties… …such as continuity Beyond that we need some theory Alternative approaches to the theory: intuition social welfare distance Begin with intuition

Intuitive inequality measures
Perhaps borrow from other disciplines… A standard measure of spread… variance But maybe better to use a normalised version coefficient of variation Comparison between these two is instructive Same iso-inequality contours for a given m. Different behaviour as m alters.

Another intuitive approach
Alternative intuition based on Lorenz approach Lorenz comparisons (second-order dominance) may be indecisive Use the diagram to “force a solution” Problem is essentially one of aggregation of information It may make sense to use a very simple approach Try something that you can “see” Go back to the Lorenz diagram

The best-known inequality measure?
1 0.8 proportion of income 0.6 Gini Coefficient 0.5 0.4 0.2 0.2 0.4 0.6 0.8 1 proportion of population

The Gini coefficient (1)
Natural expression of measure… Normalised area above Lorenz curve Can express this also in Irene-Janet terms for discrete distributions. But alternative representations more useful each of these equivalent to the above expressible in F-form or Irene-Janet terms

Normalised difference between income pairs: In F-form: In Irene-Janet terms:

Finally, express Gini as a weighted sum In F-form Or, more illuminating, in Irene-Janet terms Note that the weights k are very special depend on rank or position in distribution will change as other members added/removed from distribution perhaps in interesting ways

Intuitive approach: difficulties
Essentially arbitrary Does not mean that CV or Gini is a bad index But what is the basis for it? What is the relationship with social welfare? The Gini index also has some “structural” problems We will see this later in the lecture Examine the welfare-inequality relationship directly

SWF and inequality Issues to be addressed: Begin with the SWF W
the derivation of an index the nature of inequality aversion the structure of the SWF Begin with the SWF W Examine contours in Irene-Janet space

Equally-Distributed Equivalent Income
The Irene &Janet diagram A given distribution Distributions with same mean xi xj Contours of the SWF Construct an equal distribution E such that W(E) = W(F) EDE income Social waste from inequality Curvature of contour indicates society’s willingness to tolerate “efficiency loss” in pursuit of greater equality E F O x(F) m(F)

Welfare-based inequality
From the concept of social waste Atkinson (1970) suggested an inequality measure: Ede income x(F) I(F) = 1 – —— m(F) Mean income Atkinson assumed an additive social welfare function that satisfied the other basic axioms. W(F) = ò u(x) dF(x) Introduced an extra assumption: Iso-elastic welfare. x 1 - e – 1 u(x) = ————, e ³ 0 1 – e

The Atkinson Index Given scale-invariance, additive separability of welfare Inequality takes the form: Given the Harsanyi argument… index of inequality aversion e based on risk aversion. More generally see it as a statement of social values Examine the effect of different values of e relationship between u(x) and x relationship between u′(x) and x

Social utility and relative income
4  = 0 3  = 1/2 2  = 1 1  = 2  = 5 1 2 3 4 5 x / m -1 -2 -3

Relationship between welfare weight and income
U' e =2 e =5 4 3 2 e = 0 1 e =1/2 e=1 x / m 1 2 3 4 5

A further look at inequality
The Atkinson SWF route provides a coherent approach to inequality. But do we need to use an approach via social welfare? An indirect approach Maybe introduces unnecessary assumptions Alternative route: “distance” and inequality Consider a generalisation of the Irene-Janet diagram

The 3-Person income distribution
x j Income Distributions With Given Total ray of Janet's income equality k x Karen's income Irene's income i x

Inequality contours x x x j k i Set of distributions for given total
Set of distributions for a higher (given) total Perfect equality Inequality contours for original level Inequality contours for higher level k x i x

A distance interpretation
Can see inequality as a deviation from the norm The norm in this case is perfect equality Two key questions… …what distance concept to use? How are inequality contours on one level “hooked up” to those on another?

Another class of indices
Consider the Generalised Entropy class of inequality measures: The parameter a is an indicator sensitivity of each member of the class. a large and positive gives a “top -sensitive” measure a negative gives a “bottom-sensitive” measure Related to the Atkinson class

Inequality and a distance concept
The Generalised Entropy class can also be written: Which can be written in terms of income shares s Using the distance criterion s1−a/ [1−a] … Can be interpreted as weighted distance of each income shares from an equal share

The Generalised Entropy Class
GE class is rich Includes two indices from Henri Theil: a = 1:  [ x / m(F)] log (x / m(F)) dF(x) a = 0: –  log (x / m(F)) dF(x) For a < 1 it is ordinally equivalent to Atkinson class a = 1 – e . For a = 2 it is ordinally equivalent to (normalised) variance.

Inequality contours Each family of contours related to a different concept of distance Some are very obvious… …others a bit more subtle Start with an obvious one the Euclidian case

GE contours: a = 2

GE contours: a < 2 a = 0.25 a = 0 a = −0.25 a = −1

GE contours: a limiting case
Total priority to the poorest

GE contours: another limiting case
Total priority to the richest

Overview... A fundamentalist approach Inequality measurement
Inequality rankings Inequality measures A fundamentalist approach Inequality axiomatics The approach Inequality and income levels Decomposition Results Inequality in practice

Axiomatic approach Can be applied to any of the three version of inequality Reminder – what makes a good axiom system? Can’t be “right” or “wrong” But could be appropriate/inappropriate Capture commonly held ideas? Exploit similarity of form across related problems inequality welfare poverty

Axiom systems Already seen many standard axioms in terms of W
anonymity population principle principle of transfers scale/translation invariance Could use them to characterise inequality Use Atkinson type approach But why use an indirect approach? Some welfare issues don’t need to concern us… …monotonicity of welfare? However, do need some additional axioms How do inequality levels change with income…? …not just inequality rankings. How does inequality overall relate to that in subpopulations?

Inequality and income level
The Irene &Janet diagram xj A distribution Possible distributions of a small increment ray of equality Does this direction keep inequality unchanged? Janet's income Or this direction? Consider the iso-inequality path. Also gives what would be an inequality-preserving income reduction See Amiel-Cowell (1999) C A B xi Irene's income

Scale independence xj Example 1.
xi xj Example 1. Equal proportionate additions or subtractions keep inequality constant Corresponds to regular Lorenz criterion

Translation independence
xi xj x 2 Example 2. Equal absolute additions or subtractions keep inequality constant

Intermediate case xj Example 3.
xi xj Example 3. Income additions or subtractions in the same “intermediate” direction keep inequality constant

Dalton’s conjecture x xj
xi xj x 2 Amiel-Cowell (1999) showed that individuals perceived inequality comparisons this way. Pattern is based on a conjecture by Dalton (1920) Note dependence of direction on income level

Inequality and income level
Three different standard cases scale independence translation independence intermediate (affine) Consistent with different types of measure relative inequality absolute intermediate Blackorby and Donaldson, (1978, 1980) A matter of judgment which version to use

Inequality decomposition
Decomposition enables us to relate inequality overall to inequality in constituent parts of the population Distinguish three types, in increasing order of generality: Inequality accounting Additive decomposability General consistency Which type is a matter of judgment Each type induces a class of inequality measures The “stronger” the decomposition requirement… …the “narrower” the class of inequality measures first, some terminology

A partition pj sj Ij population share (4) (3) (6) (5) (2) (1) income
The population Attribute 1 Attribute 2 One subgroup population share (1) (2) (3) (4) (5) (6) pj (ii) (i) (iii) (iv) income share sj Ij subgroup inequality

Type 1:Inequality accounting
This is the most restrictive form of decomposition: accounting equation weight function adding-up property

Type 2:Additive decomposability
As type 1, but no adding-up constraint:

Type 3: Subgroup consistency
The weakest version: population shares increasing in each subgroup’s inequality income shares

What type of partition? General Non-overlapping in incomes
The approach considered so far Any characteristic used as basis of partition Age, gender, region, income Non-overlapping in incomes A weaker version Partition just on the basis of income Distinction between them is crucial

Partitioning by income...
Non-overlapping income groups Overlapping income groups N1 N2 N1 x* x** x

A class of decomposable indices
Given scale-independence and additive decomposability, Inequality takes the Generalised Entropy form: Just as we had earlier in the lecture. Now we have a formal argument for this family. The weight wj on inequality in group j is wj = pj1−asja Weights only sum to 1 if a = 0 or 1 (Theil indices).

Another class of decomposable indices
Given translation-independence and additive decomposability, Inequality takes the Kolm form (Kolm 1976) Another class of additive measures But these are absolute indices There is a relationship to Theil indices (Cowell 2006 ).

Generalisation (1) Suppose we don’t insist on additive decomposability? Given subgroup consistency… …with scale independence: transforms of GE indices moments, Atkinson class ... …with translation independence: transforms of Kolm But we never get Gini index Gini is not decomposable! i.e., given general partition will not satisfy subgroup consistency to see why, recall definition of Gini in terms of positions:

Partitioning by income...
Overlapping income groups Consider a transfer:Case 1 Consider a transfer:Case 2 N1 N2 N1 x* x** x x x x' x' Case 1: effect on Gini is proportional to [i-j]: same in subgroup and population Case 2: effect on Gini is proportional to [i-j]: differs in subgroup and population

Generalisation (2) Relax decomposition further
Given nonoverlapping decomposability… …with scale independence: transforms of GE indices moments, Atkinson class + Gini …with translation independence: transforms of Kolm + absolute Gini

Gini contours Not additively separable

Gini axioms: illustration
Distributions for n=3 An income distribution Perfect equality Contours of “Absolute” Gini Continuity Continuous approach to I = 0 Linear homogeneity Proportionate increase in I Translation invariance I constant x2 x* • 1 • x3 x1

Overview... Performance of inequality measures Inequality measurement
Inequality rankings Inequality measures Performance of inequality measures Inequality axiomatics Inequality in practice

Why decomposition? Resolve questions in decomposition and population heterogeneity: Incomplete information International comparisons Inequality accounting Gives us a handle on axiomatising inequality measures Decomposability imposes structure Like separability in demand analysis

Non-overlapping decomposition
Can be particularly valuable in empirical applications Useful for rich/middle/poor breakdowns Especially where data problems in tails Misrecorded data Incomplete data Volatile data components

Choosing an inequality measure
Do you want an index that accords with intuition? If so, what’s the basis for the intuition? Is decomposability essential? If so, what type of decomposability? Do you need a welfare interpretation? If so, what welfare principles to apply? What difference is it make? Example 1: Absolute/Relative for world Example 2: recent US experience

Absolute vs Relative measures
Atkinson and Brandolini. (2004)

Inequality measures and US experience

References (1) Amiel, Y. and Cowell, F. A. (1999) Thinking about Inequality, Cambridge University Press, Cambridge, Chapter 7. Atkinson, A. B. (1970) “On the Measurement of Inequality,” Journal of Economic Theory, 2, Atkinson, A. B. and Brandolini. A. (2004) “Global World Inequality: Absolute, Relative or Intermediate?” Paper presented at the 28th General Conference of the International Association for Research on Income and Wealth. August 22. Cork, Ireland. Blackorby, C. and Donaldson, D. (1978) “Measures of relative equality and their meaning in terms of social welfare,” Journal of Economic Theory, 18, 59-80 Blackorby, C. and Donaldson, D. (1980) “A theoretical treatment of indices of absolute inequality,” International Economic Review, 21,

References (2) Cowell, F. A. (2000) “Measurement of Inequality,” in Atkinson, A. B. and Bourguignon, F. (eds) Handbook of Income Distribution, North Holland, Amsterdam, Chapter 2, Cowell, F. A. (2006) “Theil, Inequality Indices and Decomposition,” Research on Economic Inequality, 13, Dalton, H. (1920) “Measurement of the inequality of incomes,” The Economic Journal, 30, Kolm, S.-Ch. (1976) “Unequal Inequalities I,” Journal of Economic Theory, 12, Theil, H. (1967) Economics and Information Theory, North Holland, Amsterdam, chapter 4,

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