Frank Cowell TU Lisbon Inequality Poverty July 2006

Frank Cowell: TU Lisbon – Inequality & Poverty July 2006 Inequality Measurement Inequality measurement Measurement Technical University of Lisbon Frank Cowell http: //darp. lse. ac. uk/lisbon 2006

Frank Cowell: TU Lisbon – Inequality & Poverty Issues to be addressed n Builds on lecture 3 u n Extension of ranking criteria u u n n n “Income Distribution and Welfare” Parade diagrams Generalised Lorenz curve Extend SWF analysis to inequality Examine structure of inequality Link with the analysis of poverty

Frank Cowell: TU Lisbon – Inequality & Poverty Major Themes n Contrast three main approaches to the subject u u u n Structure of the population u u n intuitive via SWF via analysis of structure Composition of Inequality measurement Implications for measures The use of axiomatisation u u Capture what is “reasonable”? Use principles similar to welfare and poverty

Frank Cowell: TU Lisbon – Inequality & Poverty Overview. . . Inequality measurement Inequality rankings Relationship with welfare rankings Inequality measures Inequality axiomatics Inequality in practice

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

Frank Cowell: TU Lisbon – Inequality & Poverty Yet another important relationship n. The share of the proportion q of distribution F is given by L(F; q) : = C(F; q) / m(F) n. 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) n The Atkinson (1970) result: For given m, G Lorenz-dominates F Û W(G) > W(F) for all WÎW 2

Frank Cowell: TU Lisbon – Inequality & Poverty For discrete distributions n n n 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 u u n is the ith smallest member of… …the income vector (x 1, x 2, …, xn) Then, define u Parade u income cumulations u GLC u LC

1 0. 8 L(. ; q) 0. 6 L(G; . ) Lorenz curve for F 0. 4 L(F; . ) 0. 2 0. 4 0. 6 proportion of population q 0. 8 1 0 proportion of income Frank Cowell: TU Lisbon – Inequality & Poverty The Lorenz diagram practical example, UK

Frank Cowell: TU Lisbon – Inequality & Poverty Application of ranking n n n 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.

Frank Cowell: TU Lisbon – Inequality & Poverty Official concepts of income: UK original income + cash benefits What distributional ranking would we expect to apply to these 5 concepts? gross income - direct taxes disposable income - indirect taxes post-tax income + non-cash benefits final income

Frank Cowell: TU Lisbon – Inequality & Poverty Impact of Taxes and Benefits. UK 2000/1. Lorenz Curve

Frank Cowell: TU Lisbon – Inequality & Poverty Assessment of example n n We might have guessed the outcome… In most countries: u u u n n 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?

Frank Cowell: TU Lisbon – Inequality & Poverty “Final income” – Lorenz

Frank Cowell: TU Lisbon – Inequality & Poverty “Original income” – Lorenz 1. 0 0. 9 0. 8 0. 7 §Lorenz curves intersect 0. 6 0. 5 0. 0 0. 1 0. 2 0. 3 0. 4 0. 5 §Is 1993 more equal? §Or 2000 -1?

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

Frank Cowell: TU Lisbon – Inequality & Poverty Overview. . . Inequality measurement Inequality rankings Three ways of approaching an index Inequality measures Inequality axiomatics Inequality in practice • Intuition • Social welfare • Distance

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

Frank Cowell: TU Lisbon – Inequality & Poverty Intuitive inequality measures n n Perhaps borrow from other disciplines… A standard measure of spread… u n But maybe better to use a normalised version u n variance coefficient of variation Comparison between these two is instructive u u Same iso-inequality contours for a given m. Different behaviour as m alters.

Frank Cowell: TU Lisbon – Inequality & Poverty Another intuitive approach n n Alternative intuition based on Lorenz approach Lorenz comparisons (second-order dominance) may be indecisive u u n 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 u u Try something that you can “see” Go back to the Lorenz diagram

1 0. 8 0. 6 Gini Coefficient 0. 5 0. 4 0. 2 0. 4 0. 6 proportion of population 0. 8 1 0 proportion of income Frank Cowell: TU Lisbon – Inequality & Poverty The best-known inequality measure?

Frank Cowell: TU Lisbon – Inequality & Poverty The Gini coefficient (1) n n Natural expression of measure… Normalised area above Lorenz curve Can express this also in Irene-Janet terms u for discrete distributions. But alternative representations more useful each of these equivalent to the above u u expressible in F-form or Irene-Janet terms

Frank Cowell: TU Lisbon – Inequality & Poverty The Gini coefficient (2) n Normalised difference between income pairs: In F-form: u u In Irene-Janet terms:

Frank Cowell: TU Lisbon – Inequality & Poverty The Gini coefficient (3) n n Finally, express Gini as a weighted sum u In F-form u Or, more illuminating, in Irene-Janet terms Note that the weights k are very special u u u depend on rank or position in distribution will change as other members added/removed from distribution perhaps in interesting ways

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

Frank Cowell: TU Lisbon – Inequality & Poverty Overview. . . Inequality measurement Inequality rankings Three ways of approaching an index Inequality measures Inequality axiomatics Inequality in practice • Intuition • Social welfare • Distance

Frank Cowell: TU Lisbon – Inequality & Poverty SWF and inequality n Issues to be addressed: u u u n n 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

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

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

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

Frank Cowell: TU Lisbon – Inequality & Poverty Social utility and relative income U =0 4 3 = 1/2 2 =1 1 0 -1 -2 =2 =5 1 2 3 4 5 x/m

Frank Cowell: TU Lisbon – Inequality & Poverty Relationship between welfare weight and =1 income U' =2 =5 4 3 2 =0 1 0 =1/2 =1 0 1 2 3 4 5 x/m

Frank Cowell: TU Lisbon – Inequality & Poverty Overview. . . Inequality measurement Inequality rankings Three ways of approaching an index Inequality measures Inequality axiomatics Inequality in practice • Intuition • Social welfare • Distance

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

xj Janet's income Frank Cowell: TU Lisbon – Inequality & Poverty The 3 -Person income distribution Income Distributions With Given Total of y ra lity ua eq me aren K 0 co 's in Irene's xk income xi

Frank Cowell: TU Lisbon – Inequality & Poverty Inequality contours § 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 xj xk 0 xi

Frank Cowell: TU Lisbon – Inequality & Poverty A distance interpretation n n 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?

Frank Cowell: TU Lisbon – Inequality & Poverty Another class of indices n Consider the Generalised Entropy class of inequality measures: n The parameter a is an indicator sensitivity of each member of the class. u u n a large and positive gives a “top -sensitive” measure a negative gives a “bottom-sensitive” measure Related to the Atkinson class

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

Frank Cowell: TU Lisbon – Inequality & Poverty The Generalised Entropy Class n n GE class is rich Includes two indices from Henri Theil: u n [ x / m(F)] log (x / m(F)) d. F(x) – log (x / m(F)) d. F(x) a = 0: For a < 1 it is ordinally equivalent to Atkinson class u a = 1 – . For a = 2 it is ordinally equivalent to (normalised) variance. u n a = 1:

Frank Cowell: TU Lisbon – Inequality & Poverty Inequality contours n n 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 u the Euclidian case

Frank Cowell: TU Lisbon – Inequality & Poverty GE contours: a = 2

a = 0. 25 a=0 a = − 0. 25 a = − 1 Frank Cowell: TU Lisbon – Inequality & Poverty GE contours: a < 2

Frank Cowell: TU Lisbon – Inequality & Poverty GE contours: a limiting case a = −∞ n Total priority to the poorest

Frank Cowell: TU Lisbon – Inequality & Poverty GE contours: another limiting case a = +∞ n Total priority to the richest

Frank Cowell: TU Lisbon – Inequality & Poverty Overview. . . Inequality measurement Inequality rankings A fundamentalist approach Inequality measures Inequality axiomatics Inequality in practice • The approach • Inequality and income levels • Decomposition • Results

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

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

Frank Cowell: TU Lisbon – Inequality & Poverty Overview. . . Inequality measurement Inequality rankings A fundamentalist approach Inequality measures Inequality axiomatics Inequality in practice • The approach • Inequality and income levels • Decomposition • Results

§ The Irene &Janet diagram § A distribution xj lit y § Possible distributions of a small increment of eq ua § Does this direction keep inequality unchanged? § Or this direction? ra y Janet's income Frank Cowell: TU Lisbon – Inequality & Poverty Inequality and income level C A 0 l § Consider the isoinequality path. §Also gives what would be an inequality-preserving income reduction B xi Irene's income §See Amiel-Cowell (1999)

Frank Cowell: TU Lisbon – Inequality & Poverty Scale independence xj § Example 1. § Equal proportionate additions or subtractions keep inequality constant xi §Corresponds to regular Lorenz criterion

Frank Cowell: TU Lisbon – Inequality & Poverty Translation independence x 2 xj § Example 2. § Equal absolute additions or subtractions keep inequality constant xi

Frank Cowell: TU Lisbon – Inequality & Poverty Intermediate case xj § Example 3. § Income additions or subtractions in the same “intermediate” direction keep inequality constant xi

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

Frank Cowell: TU Lisbon – Inequality & Poverty Inequality and income level n n n Three different standard cases u scale independence u translation independence u intermediate (affine) Consistent with different types of measure u relative inequality u absolute u intermediate u Blackorby and Donaldson, (1978, 1980) A matter of judgment which version to use

Frank Cowell: TU Lisbon – Inequality & Poverty Overview. . . Inequality measurement Inequality rankings A fundamentalist approach Inequality measures Inequality axiomatics Inequality in practice • The approach • Inequality and income levels • Decomposition • Results

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

Frank Cowell: TU Lisbon – Inequality & Poverty • The population • Attribute 1 • Attribute 2 • One subgroup A partition population share pj income share (3) (1) (i) sj (ii) Ij subgroup inequality (iii) (iv) (2) (4) (5) (6)

Frank Cowell: TU Lisbon – Inequality & Poverty Type 1: Inequality accounting This is the most restrictive form accounting equation of decomposition: weight function adding-up property

Frank Cowell: TU Lisbon – Inequality & Poverty Type 2: Additive decomposability As type 1, but no adding-up constraint:

Frank Cowell: TU Lisbon – Inequality & Poverty Type 3: Subgroup consistency The weakest version: increasing in each subgroup’s inequality population shares income shares

Frank Cowell: TU Lisbon – Inequality & Poverty What type of partition? n General u u u n Non-overlapping in incomes u u n The approach considered so far Any characteristic used as basis of partition Age, gender, region, income A weaker version Partition just on the basis of income Distinction between them is crucial

Frank Cowell: TU Lisbon – Inequality & Poverty Partitioning by income. . . § Non-overlapping income groups § Overlapping income groups N 1 0 x* N 2 x** N 1 x

Frank Cowell: TU Lisbon – Inequality & Poverty Overview. . . Inequality measurement Inequality rankings A fundamentalist approach Inequality measures Inequality axiomatics Inequality in practice • The approach • Inequality and income levels • Decomposition • Results

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

Frank Cowell: TU Lisbon – Inequality & Poverty Another class of decomposable indices n Given translation-independence and additive decomposability, Inequality takes the Kolm form (Kolm ( 1976 ) n Another class of additive measures n u u But these are absolute indices There is a relationship to Theil indices (Cowell 2006 ).

Frank Cowell: TU Lisbon – Inequality & Poverty Generalisation (1) n n n Suppose we don’t insist on additive decomposability? Given subgroup consistency… …with scale independence: u u n …with translation independence: u n transforms of GE indices moments, Atkinson class. . . transforms of Kolm But we never get Gini index u u u 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:

Frank Cowell: TU Lisbon – Inequality & Poverty Partitioning by income. . . § Overlapping income groups § Consider a transfer: Case 1 § Consider a transfer: Case 2 N 1 0 x x* x N 2 x** x' N 1 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

Frank Cowell: TU Lisbon – Inequality & Poverty Generalisation (2) n n n Relax decomposition further Given nonoverlapping decomposability… …with scale independence: u u u n transforms of GE indices moments, Atkinson class + Gini …with translation independence: u u transforms of Kolm + absolute Gini

Not additively separable n Frank Cowell: TU Lisbon – Inequality & Poverty Gini contours

Frank Cowell: TU Lisbon – Inequality & Poverty Gini axioms: illustration • D for n=3 • An income distribution • Perfect equality • Contours of “Absolute” Gini • Continuity • Continuous approach to I = • Linear homogeneity • Proportionate increase in I • Translation invariance • I constant x 2 x* • 1 0 • x 1 x 3

Frank Cowell: TU Lisbon – Inequality & Poverty Overview. . . Inequality measurement Inequality rankings Performance of inequality measures Inequality axiomatics Inequality in practice

Frank Cowell: TU Lisbon – Inequality & Poverty Why decomposition? n Resolve questions in decomposition and population heterogeneity: u u u n Incomplete information International comparisons Inequality accounting Gives us a handle on axiomatising inequality measures u u Decomposability imposes structure. Like separability in demand analysis

Frank Cowell: TU Lisbon – Inequality & Poverty Non-overlapping decomposition Can be particularly valuable in empirical applications n Useful for rich/middle/poor breakdowns n Especially where data problems in tails n Misrecorded data u Incomplete data u Volatile data components u

Frank Cowell: TU Lisbon – Inequality & Poverty Choosing an inequality measure n Do you want an index that accords with intuition? u n Is decomposability essential? u n If so, what’s the basis for the intuition? If so, what type of decomposability? Do you need a welfare interpretation? u If so, what welfare principles to apply?

Frank Cowell: TU Lisbon – Inequality & Poverty Absolute vs Relative measures Atkinson and Brandolini. (2004)

Frank Cowell: TU Lisbon – Inequality & Poverty Inequality measures and US experience

Frank Cowell: TU Lisbon – Inequality & Poverty References n n n n 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, 244 -263 Atkinson, A. B. and Brandolini. A. (2004) “Global World Inequality: Absolute, Relative or Intermediate? ” Paper presented at the 28 th General Conference of the International Association for Research on Income and Wealth. August 22. Cork, Ireland. Cowell, F. A. (2000) “Measurement of Inequality, ” in Atkinson, A. B. and Bourguignon, F. (eds) Handbook of Income Distribution, North Holland, Amsterdam, Chapter 2, 87 -166 Cowell, F. A. (2006) “Theil, Inequality Indices and Decomposition, ” Research on Economic Inequality, 13, 345 -360 Piketty, T. and E. Saez (2003) “Income inequality in the United States, 1913 -1998, ” Quarterly Journal of Economics, 118, 1 -39. Theil, H. (1967) Economics and Information Theory, North Holland, Amsterdam, chapter 4, 91 -134

Frank Cowell: TU Lisbon – Inequality & Poverty References n n n n 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, 244 -263 Atkinson, A. B. and Brandolini. A. (2004) “Global World Inequality: Absolute, Relative or Intermediate? ” Paper presented at the 28 th 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, 107 -136 Cowell, F. A. (2000) “Measurement of Inequality, ” in Atkinson, A. B. and Bourguignon, F. (eds) Handbook of Income Distribution, North Holland, Amsterdam, Chapter 2, 87 -166 Cowell, F. A. (2006) “Theil, Inequality Indices and Decomposition, ” Research on Economic Inequality, 13, 345 -360

Frank Cowell: TU Lisbon – Inequality & Poverty References (2) Dalton, H. (1920) “Measurement of the inequality of incomes, ” The Economic Journal, 30, 348 -361 n Kolm, S. -Ch. (1976) “Unequal Inequalities I, ” Journal of Economic Theory, 12, 416442 n Theil, H. (1967) Economics and Information Theory, North Holland, Amsterdam, chapter 4, 91 -134 n