Frank Cowell: EC426 Public Economics MSc Public Economics 2010/ Poverty, Inequality and Redistribution 7 February 2011

Frank Cowell: EC426 Public Economics Overview... Inequality and structure Poverty Welfare and needs Examine the composition of the population Inequality and Poverty

Frank Cowell: EC426 Public Economics Approaches to Inequality 1: Intuition 1: Intuition  example: Gini coefficient  but intuition may be unreliable guide 2 Inequality as welfare loss 2 Inequality as welfare loss  example: Atkinson’s index  1   (F)  1 [  x 1  dF(x) ] 1/ [1   but welfare approach is indirect  maybe introduces unnecessary assumptions 3: Alternative route: use distributional axioms directly 3: Alternative route: use distributional axioms directly  see  see Cowell (2007)Cowell (2007)

Frank Cowell: EC426 Public Economics Axioms: reinterpreted for inequality Anonymity Anonymity  permute individuals – inequality unchanged Population principle Population principle  clone population – inequality unchanged Principle of Transfers Principle of Transfers  poorer-to-richer transfer –inequality increases Scale Independence Scale Independence  multiplying all incomes by (where > 0) leaves inequality unchanged  relative inequality measures (Blackorby and Donaldson 1978) Blackorby and Donaldson 1978Blackorby and Donaldson 1978 (Alternative: Translation Independence) (Alternative: Translation Independence)  adding a constant  to all incomes leaves inequality unchanged  absolute inequality measures (Blackorby and Donaldson 1980) Blackorby and Donaldson 1980Blackorby and Donaldson 1980 Decomposability Decomposability  independence: merging with an “irrelevant” income distribution does not affect welfare/inequality comparisons  but here it is more instructive to look at decomposability interpretation

Frank Cowell: EC426 Public Economics Structural axioms: illustration xixi xkxk xjxj 0  x *   Set of distributions, n=3   An income distribution   Perfect equality   Inequality contours   Anonymity   Scale independence   Translation independence   Irene, Janet, Karen   Inequality increases as you move away from centroid   What determines shape of contours?   Examine decomposition and independence properties

Frank Cowell: EC426 Public Economics 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  Incomplete information  International comparisons Distinguish three types, in decreasing order of generality: Distinguish three types, in decreasing order of generality:  General consistency  Additive decomposability  Inequality accounting 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: EC426 Public Economics A partition population share subgroup inequality income share j j s j I j (ii) (i) (iii) (iv)   The population   Attribute 1   One subgroup   Attribute 2 (1) (2) (3) (4) (5) (6)

Frank Cowell: EC426 Public Economics Partition types and inequality measures General Partition General Partition  the approach considered so far  any characteristic used for partition (age, gender, region, income…) Non-overlapping Partition Non-overlapping Partition  weaker version: partition based on just income  scale independence: GE indices + Gini  translation independence:  indices + absolute Gini Can express Gini as a weighted sum Can express Gini as a weighted sum    (x) x dF(x)  where  (x) = [2F(x)  1] /   for absolute Gini just delete the symbol  from the above 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

Frank Cowell: EC426 Public Economics Partitioning by income... Gini has a problem with decomposability Gini has a problem with decomposability Type of partition is crucial for the Gini coefficient Type of partition is crucial for the Gini coefficient Case 1: effect on Gini is proportional to [rank(x)  rank(x')] Case 1: effect on Gini is proportional to [rank(x)  rank(x')]  same in subgroup and population Case 2: effect on Gini is proportional to [rank(x)  rank(x')] Case 2: effect on Gini is proportional to [rank(x)  rank(x')]  differs in subgroup and population What if we require decomposability for general partitions? What if we require decomposability for general partitions? x*x* N1N1 N2N2 0 x ** N1N1 x x' x x   A transfer: Case 1   A transfer: Case 2   Non-overlapping groups   Overlapping groups

Frank Cowell: EC426 Public Economics Three versions of decomposition General consistency General consistency  I(F) =  (I 1, I 2,… ;  1,  2,… ; s 1, s 2,…)  where  is increasing in each I j  I j = I(F (j) )  and F (j) is income distribution in group j Additive decomposability Additive decomposability  specific form of   I(F) =  j  j I(F (j) ) + I(F between )  where  j is a weight depending on population and income shares   j = w(  j, s j ) ≥ 0  F between is distribution assuming no inequality in each group Inequality accounting Inequality accounting  as above plus   j  j = 1

Frank Cowell: EC426 Public Economics A class of decomposable indices Given scale-independence and additive decomposability I takes the Generalised Entropy form: Given scale-independence and additive decomposability I takes the Generalised Entropy form:  [  2   ]  1  [[ x/  (F)]   1] dF(x) Parameter  indicates sensitivity of each member of the class. Parameter  indicates sensitivity of each member of the class.   large and positive gives a “top -sensitive” measure   negative gives a “bottom-sensitive” measure Includes the two Theil (1967) indices and the coeff of variation: Includes the two Theil (1967) indices and the coeff of variation:1967   = 0: –  log (x /  (F)) dF(x)   = 1:  [ x /  (F)] log (x /  (F)) dF(x)   = 2: ½  [[ x/  (F)]   1] dF(x) For  < 1 GE is ordinally equivalent to Atkinson (  = 1 –  ) For  < 1 GE is ordinally equivalent to Atkinson (  = 1 –  ) Decomposition properties: Decomposition properties:  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: EC426 Public Economics Inequality contours Each  defines a set of contours in the Irene, Karen, Janet diagram Each  defines a set of contours in the Irene, Karen, Janet diagram  each related to a concept of distance For example For example  the Euclidian case  other types  25   −   −   2

Frank Cowell: EC426 Public Economics Application: International trends Break down overall inequality to analyse trends:   I =  j  j I j + I between   given scale independence I must take the GE form   what weights should we use? Traditional approach takes each country as separate unit   shows divergence – increase in inequality   but, in effect, weights countries equally   debatable that China gets the same weight as very small countries New conventional view (Sala-i-Martin 2006) New conventional view (Sala-i-Martin 2006) (Sala-i-Martin 2006) (Sala-i-Martin 2006)  within-country disparities have increased  not enough to offset reduction in cross-country disparities. Components of change in distribution are important Components of change in distribution are important  “correctly” compute world income distribution  decomposition within/between countries is then crucial  what drives cross-country reductions in inequality?  Large growth rate of the incomes of the 1.2 billion Chinese

Frank Cowell: EC426 Public Economics Inequality measures and World experience Source: Sala-i-Martin (2006) Source: Sala-i-Martin (2006)Sala-i-Martin (2006Sala-i-Martin (2006

Frank Cowell: EC426 Public Economics 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: EC426 Public Economics Another class of decomposable indices Given translation-independence and additive decomposability: Given translation-independence and additive decomposability: Inequality takes the form Inequality takes the form    1  [e  [ x   (F)]  1] dF(x) (  ≠ 0)  or  [ x    (F)  ] dF(x) (  = 0) Parameter  indicates sensitivity of each member of the class Parameter  indicates sensitivity of each member of the class   positive gives a “top -sensitive” measure   negative gives a “bottom-sensitive” measure ()   negative gives a “bottom-sensitive” measure (Kolm 1976)Kolm 1976 Another class of additive measures Another class of additive measures  These are absolute indices  See Bosmans and Cowell (2010) Bosmans and Cowell (2010)Bosmans and Cowell (2010)

Frank Cowell: EC426 Public Economics Absolute vs Relative measures Is inequality converging? ( Sala-i- Martin 2006 ) Is inequality converging? ( Sala-i- Martin 2006 ) Sala-i- Martin 2006 Sala-i- Martin 2006 Does it matter whether we use absolute or relative measures? Does it matter whether we use absolute or relative measures? In terms of inequality trends within countries, not much In terms of inequality trends within countries, not much But worldwide get sharply contrasting picture But worldwide get sharply contrasting picture  Atkinson, and Brandolini (2010) Atkinson, and Brandolini (2010) Atkinson, and Brandolini (2010)  World Bank (2005) World Bank (2005) World Bank (2005)

Frank Cowell: EC426 Public Economics Overview... Inequality and structure Poverty Welfare and needs …Identification and measurement Inequality and Poverty

Frank Cowell: EC426 Public Economics Poverty measurement How to break down the basic issues How to break down the basic issues Sen (1979): Two main types of issues Sen (1979): Two main types of issues Sen (1979) Sen (1979)  Identification problem  Aggregation problem Jenkins and Lambert (1997): “3Is” Jenkins and Lambert (1997): “3Is” Jenkins and Lambert (1997) Jenkins and Lambert (1997)  Incidence  Intensity  Inequality Present approach: Present approach:  Fundamental partition  Depends on poverty line z  Exogeneity of partition?  Asymmetric treatment of information  Individual identification  Aggregation of information population non-poor poor

Frank Cowell: EC426 Public Economics Counting the poor Use the concept of individual poverty evaluation Use the concept of individual poverty evaluation  applies only to the poor subgroup  i.e. where x ≤ z Simplest version is (0,1) Simplest version is (0,1)  (non-poor, poor)  headcount Perhaps make it depend on income Perhaps make it depend on income  poverty deficit Or on distribution among the poor? Or on distribution among the poor?  can capture the idea of deprivation  major insight of Sen (1976) Sen (1976)Sen (1976) Convenient to work with poverty gaps Convenient to work with poverty gaps  g (x, z) = max (0, z  x) Sometimes use cumulative poverty gaps: Sometimes use cumulative poverty gaps: G (x, z) :=  x g (t, z) dF(t) x z 0 poverty evaluation xixi xjxj gigi gjgj

Frank Cowell: EC426 Public Economics Poverty evaluation g 0 poverty evaluation poverty gap x = 0 Non-Poor Poor gigi A gjgj B   the “head-count”   the “poverty deficit”   sensitivity to inequality amongst the poor   Income equalisation amongst the poor

Frank Cowell: EC426 Public Economics Brazil 1985: How Much Poverty? Rural Belo Horizonte poverty line Rural Belo Horizonte poverty line Brasilia poverty line Brasilia poverty line compromise poverty line compromise poverty line   A highly skewed distribution   A “conservative” z   A “generous” z   An “intermediate” z   The censored income distribution $0$20$40$60$80$100$120$140$160$180$200$220$240$260$280$300

Frank Cowell: EC426 Public Economics The distribution of poverty gaps $0$20$40$60 gaps

Frank Cowell: EC426 Public Economics Additively Separable Poverty measures ASP approach simplifies poverty evaluation ASP approach simplifies poverty evaluation Depends on own income and the poverty line: p(x, z) Depends on own income and the poverty line: p(x, z) Overall poverty is an additively separable function Overall poverty is an additively separable function  P =  p(x, z) dF(x)  Assumes decomposability amongst the poor  Analogy with decomposable inequality measures ASP leads to several classes of measures ASP leads to several classes of measures Take important special case: Take important special case:  make poverty evaluation depends on poverty gap.  normalise by poverty line  Foster-Greer-Thorbecke class (Foster et al 1984 ) Foster et al 1984Foster et al 1984  P =  [g(x, z)/z] a dF(x)  parameter a determines the sensitivity of the index

Frank Cowell: EC426 Public Economics Poverty evaluation functions p(x,z) z-x

Frank Cowell: EC426 Public Economics Poverty rankings Recall that we can use a version of second-order dominance to give inequality orderings Recall that we can use a version of second-order dominance to give inequality orderings  related to welfare orderings Foster and Shorrocks (1988a, 1988b) have a similar approach to orderings by P Foster and Shorrocks (1988a, 1988b) have a similar approach to orderings by P1988a1988b1988a1988b Concentrate on the FGT index’s particular functional form: Concentrate on the FGT index’s particular functional form: Theorem: Poverty orderings are equivalent to Theorem: Poverty orderings are equivalent to  first-order welfare dominance for a = 0  second-degree welfare dominance for a = 1  (third-order welfare dominance for a = 2)

Frank Cowell: EC426 Public Economics TIP / Poverty profile F(x)F(x) G(x,z) 0   Cumulative gaps versus population proportions   Proportion of poor   TIP curve (Jenkins and Lambert 1997)(Jenkins and Lambert 1997)   TIP curves have same interpretation as GLC   TIP dominance implies unambiguously greater poverty

Frank Cowell: EC426 Public Economics Views on distributions Do people make distributional comparisons in the same way as economists? Do people make distributional comparisons in the same way as economists? Summarised from Amiel-Cowell (1999) Summarised from Amiel-Cowell (1999)Amiel-Cowell (1999)Amiel-Cowell (1999)  examine proportion of responses in conformity with standard axioms  in terms of inequality, social welfare and poverty InequalitySWFPoverty NumVerbalNumVerbalNumVerbal NumVerbalNumVerbalNumVerbal Anonymity83%72%66%54%82%53% Population58%66%66%53%49%57% Decomposability57%40%58%37%62%46% Monotonicity--54%55%64%44% Transfers35%31%47%33%26%22% Scale indep.51%47%--48%66% Transl indep.31%35%--17%62%

Frank Cowell: EC426 Public Economics Empirical robustness Does it matter which poverty criterion you use? Does it matter which poverty criterion you use? Look at two key measures from the ASP class Look at two key measures from the ASP class  Head-count ratio  Poverty deficit (or average poverty gap) Use two standard poverty lines Use two standard poverty lines  $1.08 per day at 1993 PPP  $2.15 per day at 1993 PPP How do different regions of the world compare? How do different regions of the world compare? What’s been happening over time? What’s been happening over time? Use World-Bank analysis ( Ravallion and Chen 2006) Use World-Bank analysis ( Ravallion and Chen 2006) Ravallion and Chen 2006) Ravallion and Chen 2006)

Frank Cowell: EC426 Public Economics Poverty rates by region 1981,2001 HeadcountPov Gap $1.08$2.15$1.08$2.15 China East Asia India South Asia Sub-Saharan Africa All Regions Latin-America, Caribbean M. East, N. Africa E. Europe, Central Asia Sub-Saharan Africa India South Asia All Regions China East Asia Latin-America, Caribbean E. Europe, Central Asia M. East, N. Africa

Frank Cowell: EC426 Public Economics Poverty: East Asia

Frank Cowell: EC426 Public Economics Poverty: South Asia

Frank Cowell: EC426 Public Economics Poverty: Latin America, Caribbean

Frank Cowell: EC426 Public Economics Poverty: Middle East and N.Africa

Frank Cowell: EC426 Public Economics Poverty: Sub-Saharan Africa

Frank Cowell: EC426 Public Economics Poverty: Eastern Europe and Central Asia

Frank Cowell: EC426 Public Economics Overview... Inequality and structure Poverty Welfare and needs Inequality and Poverty Extensions of the ranking approach

Frank Cowell: EC426 Public Economics Social-welfare criteria: review Relations between classes of SWF and practical tools Relations between classes of SWF and practical tools Additive SWFs Additive SWFs  W : W(F) =  u(x) dF(x) Important subclasses Important subclasses  W 1  W : u() increasing  W 2  W 1 : u() increasing and concave Basic tools : Basic tools :  the quantile, Q(F; q) := inf {x | F(x)  q} = x q  the income cumulant, C(F; q) := ∫ Q(F; q) x dF(x)  give quantile- and cumulant-dominance Fundamental results: Fundamental results:  W(G) > W(F) for all W  W 1 iff G quantile- F  W(G) > W(F) for all W  W 1 iff G quantile-dominates F  W(G) > W(F) for all W  W 2 iff G F  W(G) > W(F) for all W  W 2 iff G cumulant-dominates F

Frank Cowell: EC426 Public Economics Applications to redistribution GLC rankings GLC rankings  Straight application of welfare result  Recall UK application from ONS  Does “final income” 2-order dom “original income”? [No,GLCs cross] LC rankings LC rankings  Welfare result applied to distributions with same mean  Does “final income” Lorenz dominate “original income”?  For UK application – yes Tax progressivity Tax progressivity  Let two tax schedules T 1, T 2 have disposable income schedules c 1 and c 2  Then T 1 is more progressive than T 2 iff c 1 Lorenz-dominates c 2 (Jakobsson 1976) Jakobsson 1976Jakobsson 1976  For this there must be a single-crossing condition on c 1 and c 2 (Hemming and Keen 1983) Hemming and Keen 1983Hemming and Keen 1983 All these based on the assumption of homogeneous population All these based on the assumption of homogeneous population  no differences in needs

Frank Cowell: EC426 Public Economics Income and needs reconsidered Standard approach using “equivalised income” assumes: Standard approach using “equivalised income” assumes:  Given, known welfare-relevant attributes a  A known relationship  (a)  Equivalised income given by x = y /  Equivalised income given by x = y /  is the "exchange-rate" between income types x, y Set aside the assumption that we have a single (). Set aside the assumption that we have a single (). Get a general result on joint distribution of (y, a) Get a general result on joint distribution of (y, a)  This makes distributional comparisons multidimensional  Intrinsically difficult To make progress: To make progress:  simplify the structure of the problem  again use results on ranking criteria  Atkinson and Bourguignon (1982, 1987)Cowell (2000)  see Atkinson and Bourguignon (1982, 1987), also Cowell (2000)1982Cowell (2000)1982Cowell (2000)

Frank Cowell: EC426 Public Economics Alternative approach to needs Sort individuals be into needs groups N 1, N 2,… Sort individuals be into needs groups N 1, N 2,…  a proportion  j are in group N j.  social welfare is W(F) =  j  j  a  Nj u(y) dF(a,y) To make this operational… To make this operational…  Utility people get from income depends on needs:  W(F) =  j  j  a  Nj u(j, y) dF(a,y) Consider MU of income in adjacent needs classes: Consider MU of income in adjacent needs classes: ∂u(j, y) ∂u(j+1, y) ────  ────── ∂y ∂y ∂y ∂y  “Need” reflected in high MU of income?  If need falls with j then MU-difference should be positive Let W 3  W 2 be the subclass of welfare functions for which MU- diff is positive and decreasing in y Let W 3  W 2 be the subclass of welfare functions for which MU- diff is positive and decreasing in y

Frank Cowell: EC426 Public Economics Main result Let F (  j) mean distribution for all needs groups up to and including j. Let F (  j) mean distribution for all needs groups up to and including j.  Distinguish this from the marginal distribution F (j) Theorem (Atkinson and Bourguignon 1987) Theorem (Atkinson and Bourguignon 1987)  W(G) > W(F) for all W  W 3 if and only if G (  j) cumulant-dominates F (  j) for all j = 1,2,... To examine welfare ranking use a “sequential dominance” test To examine welfare ranking use a “sequential dominance” test  check first the neediest group  then the first two neediest groups  then the first three, … etc Recently extended by Fleurbaey et al (2003) Recently extended by Fleurbaey et al (2003) Fleurbaey et al (2003) Fleurbaey et al (2003) Apply this to household to household types in Economic and Labour Market Review… Apply this to household to household types in Economic and Labour Market Review…

Frank Cowell: EC426 Public Economics Impact of Taxes and Benefits. UK Sequential GLCs 3+ adults with children 2+ adults with 3+children 2 adults with 2 children 2 adults with 1 child 1 adult with children 3+ adults 0 children 2+ adults 0 children 1 adult, 0 children

Frank Cowell: EC426 Public Economics Brief conclusion Framework of distributional analysis covers a number of related problems: Framework of distributional analysis covers a number of related problems:  Social welfare and needs  Inequality  Poverty Commonality of approach can yield important insights Commonality of approach can yield important insights Ranking principles provide basis for broad judgments Ranking principles provide basis for broad judgments  May be indecisive  specific indices could be used But convenient axioms may not find a lot of intuitive support But convenient axioms may not find a lot of intuitive support

Frank Cowell: EC426 Public Economics 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. and Bourguignon, F. (1982) “The comparison of multi-dimensional distributions of economic status,” Review of Economic Studies, 49, Atkinson, A. B. and Bourguignon, F. (1982) “The comparison of multi-dimensional distributions of economic status,” Review of Economic Studies, 49, Atkinson, A. B. and Bourguignon, F. (1982) Atkinson, A. B. and Bourguignon, F. (1982) Atkinson, A. B. and Bourguignon, F. (1987) “Income distribution and differences in needs,” in Feiwel, G. R. (ed), Arrow and the Foundations of the Theory of Economic Policy, Macmillan, New York, chapter 12, pp Atkinson, A. B. and Bourguignon, F. (1987) “Income distribution and differences in needs,” in Feiwel, G. R. (ed), Arrow and the Foundations of the Theory of Economic Policy, Macmillan, New York, chapter 12, pp Atkinson, A. B. and Brandolini. A. (2010) “On Analyzing the World Distribution of Income,” The World Bank Economic Review, 24 Atkinson, A. B. and Brandolini. A. (2010) “On Analyzing the World Distribution of Income,” The World Bank Economic Review, 24 Atkinson, A. B. and Brandolini. A. (2010) Atkinson, A. B. and Brandolini. A. (2010) 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) Bosmans, K. and Cowell, F. A. (2010) “,” Economics Letters, 109, Bosmans, K. and Cowell, F. A. (2010) “The Class of Absolute Decomposable Inequality Measures,” Economics Letters, 109, Bosmans, K. and Cowell, F. A. (2010) Bosmans, K. and Cowell, F. A. (2010) Cowell, F. A. (2000) “Measurement of Inequality,” in Atkinson, A. B. and Bourguignon, F. (eds) Handbook of Income Distribution, North Holland, Amsterdam, Ch 2, Cowell, F. A. (2000) “Measurement of Inequality,” in Atkinson, A. B. and Bourguignon, F. (eds) Handbook of Income Distribution, North Holland, Amsterdam, Ch 2, Cowell, F. A. (2000) Cowell, F. A. (2000) Cowell, F.A. (2007) “Inequality: measurement,” The New Palgrave, 2nd edn Cowell, F.A. (2007) “Inequality: measurement,” The New Palgrave, 2nd edn Cowell, F.A. (2007) Cowell, F.A. (2007) *Fleurbaey, M., Hagneré, C. and Trannoy. A. (2003) “Welfare comparisons with bounded equivalence scales” Journal of Economic Theory, –336 *Fleurbaey, M., Hagneré, C. and Trannoy. A. (2003) “Welfare comparisons with bounded equivalence scales” Journal of Economic Theory, –336Fleurbaey, M., Hagneré, C. and Trannoy. A. (2003) Fleurbaey, M., Hagneré, C. and Trannoy. A. (2003) Foster, J. E., Greer, J. and Thorbecke, E. (1984) “A class of decomposable poverty measures,” Econometrica, 52, Foster, J. E., Greer, J. and Thorbecke, E. (1984) “A class of decomposable poverty measures,” Econometrica, 52, Foster, J. E., Greer, J. and Thorbecke, E. (1984) Foster, J. E., Greer, J. and Thorbecke, E. (1984) Foster, J. E. and Shorrocks, A. F. (1988a) “Poverty orderings,” Econometrica, 56, Foster, J. E. and Shorrocks, A. F. (1988a) “Poverty orderings,” Econometrica, 56, Foster, J. E. and Shorrocks, A. F. (1988a) Foster, J. E. and Shorrocks, A. F. (1988a)

Frank Cowell: EC426 Public Economics References (2) Foster, J. E. and Shorrocks, A. F. (1988b) “Poverty orderings and welfare dominance,” Social Choice and Welfare, 5, Foster, J. E. and Shorrocks, A. F. (1988b) “Poverty orderings and welfare dominance,” Social Choice and Welfare, 5, Foster, J. E. and Shorrocks, A. F. (1988b) Foster, J. E. and Shorrocks, A. F. (1988b) Hemming, R. and Keen, M. J. (1983) “Single-crossing conditions in comparisons of tax progressivity,” Journal of Public Economics, 20, Hemming, R. and Keen, M. J. (1983) “Single-crossing conditions in comparisons of tax progressivity,” Journal of Public Economics, 20, Hemming, R. and Keen, M. J. (1983) Hemming, R. and Keen, M. J. (1983) Jakobsson, U. (1976) “On the measurement of the degree of progression,” Journal of Public Economics, 5, Jakobsson, U. (1976) “On the measurement of the degree of progression,” Journal of Public Economics, 5, Jakobsson, U. (1976) Jakobsson, U. (1976) *Jenkins, S. P. and Lambert, P. J. (1997) “Three ‘I’s of poverty curves, with an analysis of UK poverty trends,” Oxford Economic Papers, 49, *Jenkins, S. P. and Lambert, P. J. (1997) “Three ‘I’s of poverty curves, with an analysis of UK poverty trends,” Oxford Economic Papers, 49, Jenkins, S. P. and Lambert, P. J. (1997)Jenkins, S. P. and Lambert, P. J. (1997) Kolm, S.-Ch. (1976) “Unequal Inequalities I,” Journal of Economic Theory, 12, Kolm, S.-Ch. (1976) “Unequal Inequalities I,” Journal of Economic Theory, 12, Kolm, S.-Ch. (1976) Kolm, S.-Ch. (1976) Ravallion, M. and Chen, S. (2006) “How have the world’s poorest fared since the early 1980s?” World Bank Research Observer, 19, Ravallion, M. and Chen, S. (2006) “How have the world’s poorest fared since the early 1980s?” World Bank Research Observer, 19, Ravallion, M. and Chen, S. (2006) Ravallion, M. and Chen, S. (2006) *Sala-i-Martin, X. (2006) *Sala-i-Martin, X. (2006) “The world distribution of income: Falling poverty and... convergence, period”, Quarterly Journal of Economics, 121Sala-i-Martin, X. (2006)Sala-i-Martin, X. (2006) Sen, A. K. (1976) “Poverty: An ordinal approach to measurement,” Econometrica, 44, Sen, A. K. (1976) “Poverty: An ordinal approach to measurement,” Econometrica, 44, Sen, A. K. (1976) Sen, A. K. (1976) Sen, A. K. (1979) “Issues in the measurement of poverty,” Scandinavian Journal of Economics, 91, Sen, A. K. (1979) “Issues in the measurement of poverty,” Scandinavian Journal of Economics, 91, Sen, A. K. (1979) Sen, A. K. (1979) Theil, H. (1967) Economics and Information Theory, North Holland, Amsterdam, chapter 4, Theil, H. (1967) Economics and Information Theory, North Holland, Amsterdam, chapter 4, Theil, H. (1967) Theil, H. (1967) The World Bank (2005) 2006 World Development Report: Equity and Development. Oxford University Press, New York The World Bank (2005) 2006 World Development Report: Equity and Development. Oxford University Press, New York The World Bank (2005) The World Bank (2005)