. Introduction Inequality: Methods and Tools Julie A. Litchield March 999 Text or World Bank s Web Site on Inequality, Poverty, and Socio-economic Perormance: http://www.worldbank.org/poverty/inequal/index.htm Inequality means dierent things to dierent people: whether inequality should encapsulate ethical concepts such as the desirability o a particular system o rewards or simply mean dierences in income is the subect o much debate. Here we will conceptualise inequality as the dispersion o a distribution, whether that be income, consumption or some other welare indicator or attribute o a population. We begin with some notation. Deine a vector y o incomes, y, y.y i.y n ; y i R, where n represents the number o units in the population (such as households, amilies, individuals or earners or example). Let F(y) be the cumulative distribution unction o y, and I(y) an estimate o inequality. Inequality is oten studied as part o broader analyses covering poverty and welare, although these three concepts are distinct. Inequality is a broader concept than poverty in that it is deined over the whole distribution, not only the censored distribution o individuals or households below a certain poverty line, y p. Incomes at the top and in the middle o the distribution may be ust as important to us in perceiving and measuring inequality as those at the bottom, and indeed some measures o inequality are driven largely by incomes in the upper tail (see the discussion in section below). Inequality is also a much narrower concept than welare. Although both o these capture the whole distribution o a given indicator, inequality is independent o the mean o the distribution (or at least this is a desirable property o an inequality measure, as is discussed below in section ) and instead solely concerned with the second moment, the dispersion, o the distribution. However these three concepts are closely related and are sometimes used in composite measures. Some poverty indices incorporate inequality in their deinition: or example Sen s poverty measure contains the Gini coeicient among the poor (Sen, 976) and the Foster- Greer-Thorbecke measure with parameter α weights income gaps rom the poverty line in a convex manner, thus taking account o the distribution o incomes below the poverty line (Foster et al, 984). Inequality may also appear as an argument in social welare unctions o the orm W=W(µ(y), I(y)): this topic is discussed more ully below under the subect o stochastic dominance.. Measuring Inequality There are many ways o measuring inequality, all o which have some intuitive or mathematical appeal. However, many apparently sensible measures behave in perverse ashions. For example, the variance, which must be one o the simplest measures o inequality, is not See Atkinson (983) or a brie summary. Cowell (995) contains details o at least summary measures o inequality.
independent o the income scale: simply doubling all incomes would register a quadrupling o the estimate o income inequality. Most people would argue that this is not a desirable property o an inequality measure and so it seems appropriate to conine the discussion to those that conorm to a set o axioms. Even this however may result in some measures ranking distributions in dierent ways and so a complementary approach is to use stochastic dominance. We begin with the axiomatic approach and outline ive key axioms which we usually require inequality measures to meet 3... The Axiomatic approach. The Pigou-Dalton Transer Principle (Dalton, 90, Pigou, 9). This axiom requires the inequality measure to rise (or at least not all) in response to a mean-preserving spread: an income transer rom a poorer person to a richer person should register as a rise (or at least not as a all) in inequality and an income transer rom a richer to a poorer person should register as a all (or at least not as an increase) in inequality (see Atkinson, 970, 983, Cowell, 985, Sen, 973). Consider the vector y which is a transormation o the vector y obtained by a transer δ rom y to y i, where y i >y, and y i +δ>y -δ, then the transer principle is satisied i I(y ) I(y). Most measures in the literature, including the Generalized Entropy class, the Atkinson class and the Gini coeicient, satisy this principle, with the main exception o the logarithmic variance and the variance o logarithms (see Cowell, 995). Income Scale Independence. This requires the inequality measure to be invariant to uniorm proportional changes: i each individual s income changes by the same proportion (as happens say when changing currency unit) then inequality should not change. Hence or any scalar λ>0, I(y)=I(λy). Again most standard measures pass this test except the variance since var(λy)= λ var(y). A stronger version o this axiom may also be applied to uniorm absolute changes in income and combinations o the orm λ y+λ (see Cowell, 999). Principle o Population (Dalton, 90). The population principle requires inequality measures to be invariant to replications o the population: merging two identical distributions should not alter inequality. For any scalar λ>0, I(y)=I(y[λ]), where y[λ] is a concatenation o the vector y, λ times. Anonymity. This axiom sometimes also reerred to as Symmetry - requires that the inequality measure be independent o any characteristic o individuals other than their income (or the welare indicator whose distribution is being measured). Hence or any permutation y o y, I(y)=I(y ). Decomposability. This requires overall inequality to be related consistently to constituent parts o the distribution, such as population sub-groups. For example i inequality is seen to rise amongst each sub-group o the population then we would expect inequality overall to also increase. Some measures, such as the Generalised Entropy class o measures, are easily decomposed and into intuitively appealingly components o within-group inequality and between-group inequality: I total = 3 See Cowell (985) on the axiomatic approach. Alternative axioms to those listed below are possible and the appropriateness o these axioms has been questioned. See Amiel (998), Amiel and Cowell (998), Harrison and Seidl (994a, 994b) amongst others or questionnaire experimental tests o the desirability o these axioms, and Cowell (999), or an introduction to alternative approaches to inequality.
I within + I between. Other measures, such as the Atkinson set o inequality measures, can be decomposed but the two components o within- and between-group inequality do not sum to total inequality. The Gini coeicient is only decomposable i the partitions are non-overlapping, that is the sub-groups o the population do not overlap in the vector o incomes. See section 3 or ull details o decomposition techniques. Cowell (995) shows that any measure I(y) that satisies all o these axioms is a member o the Generalized Entropy (GE) class o inequality measures, hence we ocus our attention on this reduced set. We do however also present ormula or the Atkinson class o inequality measures, which are ordinally equivalent to the GE class, and the popular Gini coeicient.... Inequality Measures Members o the Generalised Entropy class o measures have the general ormula as ollows: α n y i GE( α) = α α n i= y where n is the number o individuals in the sample, y i is the income o individual i, i (,,...,n), and y = (/n) y i, the arithmetic mean income. The value o GE ranges rom 0 to, with zero representing an equal distribution (all incomes identical) and higher values representing higher levels o inequality 4. The parameter α in the GE class represents the weight given to distances between incomes at dierent parts o the income distribution, and can take any real value. For lower values o α GE is more sensitive to changes in the lower tail o the distribution, and or higher values GE is more sensitive to changes that aect the upper tail. The commonest values o α used are 0, and : hence a value o α=0 gives more weight to distances between incomes in the lower tail, α= applies equal weights across the distribution, while a value o α= gives proportionately more weight to gaps in the upper tail. The GE measures with parameters 0 and become, with l'hopital's rule, two o Theil s measures o inequality (Theil, 967), the mean log deviation and the Theil index respectively, as ollows: GE(0) = n n i= log y yi n yi GE() = log n y i= y i y With α= the GE measure becomes / the squared coeicient o variation, CV: CV n = y n i= ( y i y) 4 In the presence o any zero income values GE(0) will always attain its maximum,. Negative incomes restrict the choice o α to values greater than.
The Atkinson class o measures has the general ormula: ε ε = n yi A n i= y where ε is an inequality aversion parameter, 0<ε< : the higher the value o ε the more society is concerned about inequality (Atkinson, 970). The Atkinson class o measures range rom 0 to, with zero representing no inequality 5. Setting α=-ε, the GE class becomes ordinally equivalent to the Atkinson class, or values o α< (Cowell, 995). The Gini coeicient satisies axioms -4 above, but will ail the decomposability axiom i the sub-vectors o income overlap. There are ways o decomposing the Gini but the component terms o total inequality are not always intuitively or mathematically appealing (see or example Fei et al, 978, and an attempt at a decomposition with a more intuitive residual term by Yitzhaki and Lerman, 99). However the Gini s popularity merits it a mention here. It is deined as ollows (Gini, 9): ( ε ) Gini = n n y Σ i n y i= = y The Gini coeicient takes on values between 0 and with zero interpreted as no inequality 6... An Alternative Approach: Stochastic Dominance. Although the measures discussed above generally meet the set o desirable axioms it is possible that they will rank the same set o distributions in dierent ways, simply because o their diering sensitivity to incomes in dierent parts o the distributions. When rankings are ambiguous, the alternative method o stochastic dominance can be applied. We discuss three types o stochastic dominance below. The irst two are sensitive to the mean o the distribution, and are thereore not applicable to establishing inequality rankings. As the discussion below suggests, irst- and second- order stochastic dominance are undamentally o use in comparisons o social welare. They are presented irst, however, because they are logically prior to the dominance category which is associated with unambiguous comparisons o inequality across distributions: mean-normalised second-order dominance, or Lorenz dominance. 7 First order stochastic dominance. Consider two income distributions y and y with cumulative distribution unctions (CDFs) F(y ) and F(y ). I F(y ) lies nowhere above and at least somewhere below F(y ) then distribution y displays irst order stochastic dominance over distribution y : F(y ) F(y ) or all y. Hence in distribution y there are no more individuals with income less than a given income level than in distribution y, or all levels o income. We can 5 In the presence o any zero incomes A() always attains its maximum value,. 6 Zero incomes pose no problem or the Gini. However it may take a negative value i mean income is negative or a value greater than i there are some very large negative incomes (see Scott and Litchield, 994). 7 See Ferreira and Litchield (996) or an application o stochastic dominance analysis to Brazil.
express this in an alternative way using the inverse unction y=f - (p) where p is the share o the population with income less than a given income level: irst order dominance is attained i F - (p) F - (p) or all p. The inverse unction F - (p) is known as a Pen s Parade (Pen, 974) which simply plots incomes against cumulative population, usually using ranked income quantiles 8. The dominant distribution is that whose Parade lies nowhere below and at least somewhere above the other. First order stochastic dominance o distribution y over y implies that any social welare unction that is increasing in income, will record higher levels o welare in distribution y than in distribution y (Saposnik, 98, 983). Figure : First Order Stochastic Dominance Brazil 98-995: Pen s Parades Source: Ferreira and Litchield, 999, "Inequality, Poverty and Welare, Brazil 98-995". London School o Economics Mimeograph. Second order stochastic dominance. Consider now the deicit unctions (the integral o the CDF) y k o distributions y and y : G( y ) = F( y dy i=,. I the deicit unction o distribution y lies i, k i ) 0 nowhere above and somewhere below that o distribution y, then distribution y displays second order stochastic dominance over distribution y : G(y,k ) G(y,k ) or all y k. The dual o the deicit 8 The original Pen s Parades o Jan Pen were conceptualised by comparing the incomes o every individual in a population. The example that Pen gave was o lining up individuals in ascending order o income and re-scaling their heights to represent their income level. I these individuals were to be paraded past an observer she would typically see a large number o dwarves (poor people), eventually ollowed by individuals o average height (income) and inally ollowed by a small number o giants (very rich people). In practice comparing incomes at every income level proves too laborious, hence some degree o aggregation is usually employed and quantiles are compared.
curve is the Generalised Lorenz curve (Shorrocks, 983) deined as GL( p) = ydf( y), which plots cumulative income shares scaled by the mean o the distribution against cumulative population, where the height o the curve at p is given by the mean o the distribution below p. As Atkinson and Bourguignon (989) and Howes (993) have shown, second order dominance established by comparisons o the deicit curves or complete, uncensored distributions implies and is implied by Generalised Lorenz dominance: GL (p) GL (p) or all p. Second order dominance o distribution y over distribution y implies that any social welare unction that is increasing and concave in income will record higher levels o welare in y than in y (Shorrocks, 983). It should now be apparent that second order stochastic dominance is thereore implied by irst order stochastic dominance, although the reverse is not true. y k 0 Figure : Second Order Stochastic Dominance Brazil 98-995: Generalised Lorenz Curves Source: Ferreira and Litchield, 999, "Inequality, Poverty and Welare, Brazil 98-995". London School o Economics Mimeograph. Mean-normalised second order stochastic dominance. In order to rank distributions in terms o inequality alone, rather than welare, a third concept (also known as Lorenz dominance) is applied. I the Lorenz curve, the plot o cumulative income shares against cumulative population shares, o distribution y lies nowhere below and at least somewhere above the Lorenz curve o distribution y then y Lorenz dominates y. Any inequality measure which satisies anonymity and the Pigou-Dalton transer principle will rank the two distributions in the same way as the Lorenz curves (Atkinson, 970).
Figure 3: Mean-normalized second order stochastic dominance Brazil 98-995: Lorenz curves Note: 995 Lorenz dominates 990, but 98 Lorenz dominates 990 and dominates 995. Source: Ferreira and Litchield, 999, "Inequality, Poverty and Welare, Brazil 98-995". London School o Economics Mimeograph..3. Statistical Inerence and Sampling Variance. In order to make meaningul comparisons between estimates o inequality o dierent distributions and o the rankings implied by stochastic dominance we need to examine the statistical signiicance o the results. In the case o summary inequality measures we need to examine the standard errors o the estimates. There are a number o ways this can be done, depending on the degree o sophistication one wishes to apply and the measure o inequality in question. Cowell (995) lists some rough approximations or a number o inequality measures i the sample size is large and i one is prepared to make assumptions about the underlying distribution rom which the sample is taken. For example the coeicient o variation CV (which is a simple transormation o GE() see above) has standard error CV ([+CV ]/n) i we assume the underlying distribution is normal. The Gini coeicient, again assuming a normal distribution, has standard error 0.8086CV/ n. However this may be too rough or some purposes in which case a more accurate method can be applied. Cowell (989) shows that many inequality measures can be expressed in terms o their sample moments about zero. For example the Atkinson class can be written as A = ε ' [ µ ] r r ' µ
where µ r is the r th moment about zero, r=-ε and µ = the mean o the distribution, and the GE class as: GE( α) = α α where µ να are the moments about zero deined as: α α [ µ [ µ ] [ µ ], ] α 0, α 0 µ να ν α = z y df( z, y) where z is household size (or some other weighting variable), v=, and - <α <. The sample moments m να can be expressed as m να =/n z i ν y i α. I both mean household size and mean income are known, then Var(GE(α)) is relatively easy to derive: ^ V = [ n ][ α α α ] [ m0 ] [ m, α m α [ m α] ] For the ull details o the method and or the cases where α=0 and, and where the population mean income and household size are not known see Cowell (989, 995). It is also possible to test the statistical signiicance o any stochastic dominance results. Howes (993) describes one test based on a simple test o sample mean dierences: z i ^ ^ * i ξ i ξ = ^ Cii C + N N ^ * ii * where ξ=(ξ, ξ w ) is a vector o heights o curves (Pen s Parades, Lorenz or Generalized Lorenz curves), C ii is the relevant element in the diagonal o the variance-covariance matrix associated with ξ and N is the sample size. Z i is asymptotically normally distributed. See Howes (993) or uller details and an empirical application to China, and Ferreira and Litchield (996) or an application to Brazil. 3. The Determinants o Inequality The preceding discussion o measurement and comparisons o inequality should have suiced to establish what a complex and multiaceted phenomenon it is. Because it is inluenced by the welare o any individual or household in a society, and because welare itsel is aected by so many actors, and determined in general equilibrium, the study o causation or determination o inequality is a perilous ield. Authors aware o the muddy waters in which they wade qualiy their every statement with cautionary remarks about how results are merely indicative or suggestive.
They are oten at pains to point out that decomposition results are descriptive, and inerences o causation are merely suggestive. The analytical techniques are in their inancy, and such caution is both warranted and necessary. Nevertheless, once the appropriate caution o interpretation is internalised, some techniques do allow us to glimpse interesting patterns. In the absence o more deinitive inerence methods, some o these decomposition and regression analyses are oten worthwhile exercises. We begin with decomposition techniques. 3.. Decomposition techniques Decomposability is desirable or both arithmetic and analytic reasons. Economists and policy analysts may wish to assess the contribution to overall inequality o inequality with and between dierent sub-groups o the population, or example within and between workers in agricultural and industrial sectors, or urban and rural sectors. Decompositions o inequality measures can shed light on both its structure and dynamics. Inequality decomposition is a standard technique or examining the contribution to inequality o particular characteristics and can be used to assess income recipient characteristics and income package inluences. The ield was pioneered by Bourguignon (979), Cowell (980), and Shorrocks (98a, 98b, 984). For more details on the methodologies, see Deaton (997), and Jenkins (995). Fields (980) provides summaries o applications to developing countries. Decomposition by population sub-group. The point o this decomposition is to separate total inequality in the distribution into a component o inequality between the chosen groups (I b ), and the remaining within-group inequality (I w ). Two types o decomposition are o interest: irstly the decomposition o the level o inequality in any one year, i.e a static decomposition, and secondly a decomposition o the change in inequality over a period o time, i.e. a dynamic decomposition. The static decomposition. When total inequality, I, is decomposed by population subgroups, the Generalised Entropy class can be expressed as the sum o within-group inequality, I w, and between group inequality, I b. Within-group inequality I w is deined as: I w = k = α w = v α w GE( α ) where is the population share and v the income share o each partition, =,,..k. In practical terms the inequality o income within each sub-group is calculated and then these are summed, using weights o population share, relative incomes or a combination o these two, depending on the particular measure used. Between-group inequality, I b, is measured by assigning the mean income o each partition, y, to each member o the partition and calculating: I b = α α k = y y α
Cowell and Jenkins (995) show that the within- and between-group components o inequality, deined as above, can be related to overall inequality in the simplest possible way: I b + I w = I. They then suggest an intuitive summary measure, R b, o the amount o inequality explained by dierences between groups with a particular characteristic or set o characteristics, Rb = I b / I. Hence we can conclude that x% o total inequality is explained by between group inequalities, and (00-x)% is accounted or by inequalities within groups. By increasing the number o partitions we can account or the eect o a wider range o structural actors 9. The dynamic decomposition. Accounting or changes in the level o inequality by means o a partition o the distribution into sub-groups must entail at least two components o the change: one caused by a change in inequality between the groups and one by a change in inequality within the groups. The second one is the pure inequality eect, but the irst one can be urther disaggregated into an eect due to changes in relative mean incomes between the subgroups - an income eect - and one due to changes in the size o the subgroups - an allocation eect. Hence we can decompose the change in total inequality into three components: an allocation eect arising rom changes in the number o people within dierent partitions, an income eect arising rom changes in relative incomes between partitions, and inally a pure inequality eect arising rom changes in inequality within partitions (Mookeree and Shorrocks, 98). The arithmetic becomes complicated or some measures, so this is usually only applied to GE(0), as ollows: k GE( 0) = k k GE( 0) = + GE( 0) + log( ) = = k ( ) [ λ λ ] + v log( µ ( y) = where y is income, is the dierence operator, λ is the mean income o group relative to the overall mean, i.e. µ(y )/µ(y) and the over-bar represents a simple average. The irst term captures the pure inequality eect, the second and third term capture the allocation eect and the inal term the income eect. By dividing both sides through by the initial value o GE(0) proportionate changes in inequality can be compared to proportionate changes in the individual eects 0. Decomposition by income source Total income is usually made up o more than one source: labour earnings, income rom capital, private and public transers, etc; and so it is useul to express total inequality I as the sum o actor contributions, where each contribution depends on the incomes rom a given actor source,, i.e. 9 The Atkinson measure can also be decomposed by population sub-group see Cowell and Jenkins (995). 0 This is actually an approximation o the true decomposition, but both Mookheree and Shorrocks (98) and, later, Jenkins (995) argue that or computational purposes this approximation is suicient.
I = where S depends on incomes rom source. Factor income source provides a disequalising eect i S >0, and an equalising eect i S <0. Now deine so s s S S I =. S is the absolute contribution o actor to overall inequality, while s is the proportional actor contribution. The exact decomposition procedure depends on the measure o inequality used, but whichever measure is used must naturally be decomposable and, given the large number o income sources, it must be deined or zero incomes. In practice the easiest measure to decompose in this way is GE(). In that case: S = s GE( ) = ρ χ GE(). GE() where ρ is the correlation between component and total income and χ = µ / µ is s actor share. A large value o S suggests that actor is an important source o total inequality. For the dynamic decomposition we can write GE( ) = GE( ) t+ GE( ) t = S = [ ρ χ GE( ). E( ) ] and proportionate inequality changes as % GE( ) GE( )/ GE( ) s % S = = t A large value o s % S suggests that changes in actor have a large inluence in changes in total inequality. See Jenkins (995) or the complete methodology and an application to the UK, and Theil (979). 3.. Regression analyses The decomposition techniques described above are very suitable or assessing the contribution o a set o actors (household-speciic attributes or income sources) to inequality. However one drawback is that the importance o a particular attribute will vary depending on the measure o inequality that is decomposed. Fields (997) proposes an alternative decomposition technique, which allows one to assess the importance o household speciic attributes in explaining the level STATA routines or decompositions by population sub-group and by income source are available at the Boston College Department o Economics. See above or details and conditions o use.
o inequality, where the amount explained by each actor is independent o the inequality measures used. The method involves running a standard set o regressions o the orm: ln( y i ) = α + β X + ε where the subscript i reers to the individual, reers to the population sub-group and X is a vector o explanatory variables. Then the relative contributions, s, o each actor can then be estimated as: s = cov[ a Z, Y ] / σ ( Y ) = a * σ ( Z )* cor[ Z, Y] / σ ( Y ) where a is the vector o coeicients (α, β i ), Z is the vector o explanatory variables plus a constant (, x i ), and Y is log income. The change in inequality over time can also be decomposed using the s s estimated above, although the estimates are sensitive to the inequality measure used. See Fields (997) or ull details and an application to data rom Bolivia and Korea. An alternative approach is the quantile regression methodology, where instead o estimating the mean o a dependent variable conditional on the values o the independent variables, one estimates the median: minimising the sum o the absolute residuals rather than the sum o squares o the residuals as in ordinary regressions. It is possible to estimate dierent percentiles o the dependent variables, and so to obtain estimates or dierent parts o the income distribution. Furthermore, it is possible to use dierent independent variables or dierent quantiles, relecting the view that data may be heteroskedastic with dierent actors aecting the rich and poor. See Deaton (997) or an introduction to quantile regressions and some applications to developing country data. Regression techniques are also applied when we want to model the eect o aggregate actors rather than speciic attributes o a household. One method is to regress the level o inequality in each year (or each country, population sub-group etc.) on a set o explanatory variables, such as the rates o unemployment (UE) and inlation (INF), as ollows: I ( y) β INF + u t = α + βuet + t t A second method that is oten applied in this macro-economic context is to regress a set o income shares on the independent variables, as ollows: s INF + u it = α i + β iuet + β i t it where s it denotes the income share o the ith quantile group in year t 3. The i quantile share regressions are a set o seemingly unrelated regressions (see Zellner, 96) but since the right hand side variables are the same in each equation, the SURE estimation technique suggested by Zellner is equivalent to a set o simple OLS regressions. See Blinder and Esaki (978) or an The result holds or all inequality measures that are symmetric and continuous, e.g. all members o the Generalised Entropy class, the Atkinson class and the Gini. 3 See Blinder and Esaki (978) or the original speciications.
application to the USA, Buse (98) on Canadian inequality, Ferreira and Litchield (999) on Brazil and Nolan (987) on UK inequality.
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