Frank Cowell: Oviedo – Inequality & Poverty Inequality Measurement March 2007 Inequality, Poverty and Income Distribution University of Oviedo Frank Cowell

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

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

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

Frank Cowell: Oviedo – Inequality & Poverty Inequality rankings Begin by using welfare analysis of previous lecture Begin by using welfare analysis of previous lecture Seek an inequality ranking Seek an inequality ranking We take as a basis the second-order distributional 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. The 2nd-order dominance concept was originally expressed in a more restrictive form.

Frank Cowell: Oviedo – Inequality & Poverty Yet another important relationship The share of the proportion q of distribution F is given by The share of the proportion q of distribution F is given by L(F;q) := C(F;q) /  (F) Yields Lorenz dominance, or the “shares” ranking Yields Lorenz dominance, or the “shares” ranking For given , G Lorenz-dominates F  W(G) > W(F) for all W  W 2 The Atkinson (1970) result: The Atkinson (1970) result: G Lorenz-dominates F  means:   for every q, L(G;q)  L(F;q),   for some q, L(G;q) > L(F;q)

Frank Cowell: Oviedo – Inequality & Poverty All the above has been done in terms of F-form notation. All the above has been done in terms of F-form notation. Can do the almost same in Irene-Janet notation. Can do the almost same in Irene-Janet notation. Use the order statistics x [i] where Use the order statistics x [i] where  is the ith smallest member of…  …the income vector (x 1,x 2,…,x n ) Then, define Then, define  Parade  income cumulations  GLC  LC For discrete distributions

Frank Cowell: Oviedo – Inequality & Poverty The Lorenz diagram proportion of income proportion of population L(G;.) L(F;.) L(.; q) q Lorenz curve for F practical example, UK

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

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

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

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

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

Frank Cowell: Oviedo – Inequality & Poverty “Original income” – Lorenz   Lorenz curves intersect   Is 1993 more equal?   Or ?

Frank Cowell: Oviedo – Inequality & Poverty Inequality ranking: Summary Second-order (GL)-dominance is equivalent to ranking by cumulations. Second-order (GL)-dominance is equivalent to ranking by cumulations.  From the welfare lecture Lorenz dominance equivalent to ranking by shares. 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. Where Lorenz-curves intersect unambiguous inequality orderings are not possible. This makes inequality measures especially interesting. This makes inequality measures especially interesting.

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

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

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

Frank Cowell: Oviedo – Inequality & Poverty Another intuitive approach Alternative intuition based on Lorenz approach Alternative intuition based on Lorenz approach Lorenz comparisons (second-order dominance) may be indecisive 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 It may make sense to use a very simple approach  Try something that you can “see”  Go back to the Lorenz diagram

Frank Cowell: Oviedo – Inequality & Poverty proportion of income proportion of population Gini Coefficient The best-known inequality measure?

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

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

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

Frank Cowell: Oviedo – Inequality & Poverty Intuitive approach: difficulties Essentially arbitrary 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? What is the relationship with social welfare? The Gini index also has some “structural” problems The Gini index also has some “structural” problems  We will see this later in the lecture What is the relationship with social welfare? What is the relationship with social welfare?  Examine the welfare-inequality relationship directly

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

Frank Cowell: Oviedo – Inequality & Poverty SWF and inequality Issues to be addressed: Issues to be addressed:  the derivation of an index  the nature of inequality aversion  the structure of the SWF Begin with the SWF W Begin with the SWF W Examine contours in Irene-Janet space Examine contours in Irene-Janet space

Frank Cowell: Oviedo – Inequality & Poverty Equally-Distributed Equivalent Income O xixi xjxj   The Irene &Janet diagram   A given distribution   Distributions with same mean   Contours of the SWF E (F)(F) (F)(F) F   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

Frank Cowell: Oviedo – Inequality & Poverty Atkinson assumed an additive social welfare function that satisfied the other basic axioms. Atkinson assumed an additive social welfare function that satisfied the other basic axioms.  (F) I(F) = 1 – ——  (F) Mean income Ede income Welfare-based inequality x  1 -  – 1 u(x) = ————,   1 –  Introduced an extra assumption: Iso-elastic welfare. Introduced an extra assumption: Iso-elastic welfare. From the concept of social waste Atkinson (1970) suggested an inequality measure: From the concept of social waste Atkinson (1970) suggested an inequality measure: W(F) =  u(x) dF(x)

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

Frank Cowell: Oviedo – Inequality & Poverty Social utility and relative income      U x / 

Frank Cowell: Oviedo – Inequality & Poverty Relationship between welfare weight and income       U' x / 

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

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

Frank Cowell: Oviedo – Inequality & Poverty The 3-Person income distribution 0 Irene's income Janet's income Karen's income i x k x x j ray of equality Income Distributions With Given Total

Frank Cowell: Oviedo – Inequality & Poverty Inequality contours 0 i x k x x j   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

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

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

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

Frank Cowell: Oviedo – Inequality & Poverty The Generalised Entropy Class GE class is rich GE class is rich Includes two indices from Henri Theil: Includes two indices from Henri Theil:   = 1:  [ x /  (F)] log (x /  (F)) dF(x)   = 0: –  log (x /  (F)) dF(x) For  < 1 it is ordinally equivalent to Atkinson class For  < 1 it is ordinally equivalent to Atkinson class   = 1 – . For  = 2 it is ordinally equivalent to (normalised) variance. For  = 2 it is ordinally equivalent to (normalised) variance.

Frank Cowell: Oviedo – Inequality & Poverty Inequality contours Each family of contours related to a different concept of distance Each family of contours related to a different concept of distance Some are very obvious… Some are very obvious… …others a bit more subtle …others a bit more subtle Start with an obvious one Start with an obvious one  the Euclidian case

Frank Cowell: Oviedo – Inequality & Poverty GE contours:  2

Frank Cowell: Oviedo – Inequality & Poverty GE contours:  2  25   −  − 

Frank Cowell: Oviedo – Inequality & Poverty GE contours: a limiting case  −∞ Total priority to the poorest Total priority to the poorest

Frank Cowell: Oviedo – Inequality & Poverty GE contours: another limiting case Total priority to the richest Total priority to the richest  +∞

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

Frank Cowell: Oviedo – Inequality & Poverty Axiomatic approach Can be applied to any of the three version of inequality Can be applied to any of the three version of inequality Reminder – what makes a good axiom system? 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 Exploit similarity of form across related problems  inequality  welfare  poverty

Frank Cowell: Oviedo – Inequality & Poverty Axiom systems Already seen many standard axioms in terms of W Already seen many standard axioms in terms of W  anonymity  population principle  principle of transfers  scale/translation invariance Could use them to characterise inequality Could use them to characterise inequality  Use Atkinson type approach But why use an indirect 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 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?

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

Frank Cowell: Oviedo – Inequality & Poverty Inequality and income level B C Irene's income Janet's income xixi xjxj 0 ray of equality   The Irene &Janet diagram   A distribution   Possible distributions of a small increment   Does this direction keep inequality unchanged?   Or this direction?   Consider the iso- inequality path.   Also gives what would be an inequality- preserving income reduction   See Amiel-Cowell (1999) ll ll A

Frank Cowell: Oviedo – Inequality & Poverty xixi xjxj Scale independence   Example 1.   Equal proportionate additions or subtractions keep inequality constant   Corresponds to regular Lorenz criterion

Frank Cowell: Oviedo – Inequality & Poverty xixi xjxj x 2 Translation independence   Example 2.   Equal absolute additions or subtractions keep inequality constant

Frank Cowell: Oviedo – Inequality & Poverty xixi xjxj Intermediate case   Example 3.   Income additions or subtractions in the same “intermediate” direction keep inequality constant

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

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

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

Frank Cowell: Oviedo – Inequality & Poverty Inequality decomposition Decomposition enables us to relate inequality overall to inequality in constituent parts of the population Decomposition enables us to relate inequality overall to inequality in constituent parts of the population Distinguish three types, in increasing order of generality: Distinguish three types, in increasing order of generality:  Inequality accounting  Additive decomposability  General consistency Which type is a matter of judgment 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

Frank Cowell: Oviedo – Inequality & Poverty A partition population share subgroup inequality income share j j s j I j (ii) (i) (iii) (iv) The population The population Attribute 1 Attribute 1 One subgroup One subgroup Attribute 2 Attribute 2 (1) (2) (3) (4) (5) (6)

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

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

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

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

Frank Cowell: Oviedo – Inequality & Poverty Partitioning by income... x*x* N1N1 N2N2 0 x ** N1N1   Non-overlapping income groups   Overlapping income groups x

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

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

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

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

Frank Cowell: Oviedo – Inequality & Poverty x*x* N1N1 N2N2 0 x ** N1N1 x' x   Case 2: effect on Gini is proportional to [i-j]: differs in subgroup and population x'x   Case 1: effect on Gini is proportional to [i-j]: same in subgroup and population x Partitioning by income...   Overlapping income groups   Consider a transfer:Case 1   Consider a transfer:Case 2

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

Frank Cowell: Oviedo – Inequality & Poverty Gini contours Not additively separable Not additively separable

Frank Cowell: Oviedo – Inequality & Poverty Gini axioms: illustration x1x1 x3x3 x2x2 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 0 1 x *

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

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

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

Frank Cowell: Oviedo – Inequality & Poverty Choosing an inequality measure Do you want an index that accords with intuition? Do you want an index that accords with intuition?  If so, what’s the basis for the intuition? Is decomposability essential? Is decomposability essential?  If so, what type of decomposability? Do you need a welfare interpretation? Do you need a welfare interpretation?  If so, what welfare principles to apply? What difference does it make? What difference does it make?  Example 1: recent US experience  Example 2: relative measures and world inequality  Example 3: Absolute/Relative for world

Frank Cowell: Oviedo – Inequality & Poverty Example 1: US Re-examine the data from Lecture 1 Re-examine the data from Lecture 1   DeNavas-Walt et al. (2005) DeNavas-Walt et al. (2005) Recall the impression of rising inequality Recall the impression of rising inequality  “fanning-out” of quantile ratios  increasing disparity of income shares Is this borne out by inequality measures? Is this borne out by inequality measures?  Gini  Atkinson indices – does it matter what  ?  GE indices – does it matter what  ?

Frank Cowell: Oviedo – Inequality & Poverty Inequality measures and US experience Source: Source: DeNavas-Walt et al. (2005)DeNavas-Walt et al. (2005)

Frank Cowell: Oviedo – Inequality & Poverty Example 2: International trends Recent debate on “convergence” / “divergence” Traditional approach takes each country as separate unit   shows divergence – increase in inequality   but, in effect, weights countries equally   it is obviously debatable that huge countries like China…   …get the same weight as very small countries New conventional view: New conventional view:  within-country disparities have increased  not enough to offset reduction in cross-country disparities.  (Sala-i-Martin 2006) (Sala-i-Martin 2006) (Sala-i-Martin 2006) Components of change in distribution are important Components of change in distribution are important  “correctly” compute world income distribution  decomposition is then crucial  what drives cross-country reductions in inequality?  Large growth rate of the incomes of the 1.2 billion Chinese

Frank Cowell: Oviedo – Inequality & Poverty Inequality measures and World experience Source: Sala-i-Martin (2006) Source: Sala-i-Martin (2006)Sala-i-Martin (2006Sala-i-Martin (2006

Frank Cowell: Oviedo – Inequality & Poverty Inequality measures and World experience: breakdown Source: Sala-i-Martin (2006) Source: Sala-i-Martin (2006)Sala-i-Martin (2006Sala-i-Martin (2006

Frank Cowell: Oviedo – Inequality & Poverty Example 3: World inequality again All the previous used relative inequality measures All the previous used relative inequality measures  Gini  Atkinson indices  GE indices What would happen if we switchewd to absolute measures? What would happen if we switchewd to absolute measures?  absolute Gini  Kolm indices Important role for changes in mean income Important role for changes in mean income

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

Frank Cowell: Oviedo – Inequality & Poverty References (1) Amiel, Y. and Cowell, F. A. (1999) Thinking about Inequality, Cambridge University Press, Cambridge, Chapter 7. Amiel, Y. and Cowell, F. A. (1999) Thinking about Inequality, Cambridge University Press, Cambridge, Chapter 7. Amiel, Y. and Cowell, F. A. (1999) Amiel, Y. and Cowell, F. A. (1999) Atkinson, A. B. (1970) “On the Measurement of Inequality,” Journal of Economic Theory, 2, Atkinson, A. B. (1970) “On the Measurement of Inequality,” Journal of Economic Theory, 2, Atkinson, A. B. (1970) Atkinson, A. B. (1970) 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. 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. Atkinson, A. B. and Brandolini. A. (2004) Atkinson, A. B. and Brandolini. A. (2004) Blackorby, C. and Donaldson, D. (1978) “Measures of relative equality and their meaning in terms of social welfare,” Journal of Economic Theory, 18, Blackorby, C. and Donaldson, D. (1978) “Measures of relative equality and their meaning in terms of social welfare,” Journal of Economic Theory, 18, Blackorby, C. and Donaldson, D. (1978) Blackorby, C. and Donaldson, D. (1978) Blackorby, C. and Donaldson, D. (1980) “A theoretical treatment of indices of absolute inequality,” International Economic Review, 21, Blackorby, C. and Donaldson, D. (1980) “A theoretical treatment of indices of absolute inequality,” International Economic Review, 21, Blackorby, C. and Donaldson, D. (1980) Blackorby, C. and Donaldson, D. (1980) 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. (2000) “Measurement of Inequality,” in Atkinson, A. B. and Bourguignon, F. (eds) Handbook of Income Distribution, North Holland, Amsterdam, Chapter 2, Cowell, F. A. (2000) Cowell, F. A. (2000)

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