The bidimensional decomposition of inequality: A nested Theil approach

Transcription

1 The bidiensional decoposition of inequality: A nested Theil approach MICHELE GIAMMATTEO * Departent of Public Econoics University of Roe La Sapienza.giaatteo@uniroa.it July, 2007 Abstract In this paper we propose a nested inequality decoposition by incoe sources and population subgroups derived by the Theil index. We firstly otivate our preference for its associated decoposition by incoe sources with respect to the axio-based proposal of Shorrocs (982) and the Gini-based decoposition of Leran and Yitzhai (985). Then we enhance the set of desirable proprieties able to sustain that choice with the additional requireent of subgroup decoposability. The nested decoposition of the Theil index allows the overall level of inequality to be function of only three types of factors: source-group incoe and population shares; source-group inequality. Finally, using LIS icro data on incoes, we apply it to the case of geographical disaggregation of inequality in Italy between 989 and Keywords: Inequality; Incoe sources; Decoposition; Italy; JEL Classification: D3, O5. * I would lie to express y deep gratitude to y supervisor, Prof. Maurizio Franzini, for his fundaental support throughout y Ph.D. studies. I also than Joan Esteban for his detailed and constructive coents received during the otivating discussions we had on inequality decoposition issues. The financial support of the CRISS (Networ on the Econoics of the Welfare State - is gratefully acnowledged.

2 . Introduction In inequality decoposition studies we can distinguish two fundaental approaches. The traditional and larger applied technique is to identify the influence coing fro specific population subgroups. A copleentary - rather than alternative - approach is to establish how different types of incoe affect total inequality: an exaple could be to detect the relative contribution of incoes fro financial investents with respect to wages, capital profits, rents or state factors (transfers and taxes). Our view is that an exhaustive analysis should include a ixture of explanatory factors based on subgroups as well as on incoe sources. In this paper we propose a nested-theil decoposition of inequality, which allows us to identify siultaneously incoe sources and sub-population deterinants of the overall level of inequality. We cobine into a unique approach the standard decoposition by population subgroups, that separates total inequality in within-group and between-group coponents, and the decoposition by incoe sources, which divides overall inequality into proportional factor contributions. The first crucial theoretical result is given by Shorrocs (982), in which the objective of dividing total inequality into the partial contribution of each incoe coponent is dealt by starting with the definition of few fundaental axios and then deriving the decoposition rule that satisfies the. The second possibility is suggested by Leran and Yitzhai (985), who derive a decoposition rule that, following atheatically fro the disaggregation of the inequality index taen as reference (the Gini coefficient), shows properties which depend strictly on that initial choice. 2 In what follows, I refer to these alternative ethods as those generating axio-based and natural decoposition rules, respectively. For the forer, I also use the attribute of desirable. Moreover, the ter natural is used not to say that decoposition is the only possible and unaniously acceptable approach, but only because it is obtained by arithetical derivation fro the index of reference. The fundaental difference between the two ethods is that, while the axioatic approach yields a decoposition rule which respects the assuptions directly iposed upon it, in the second case the properties that one can derive ust just be accepted, as a consequence of their Decoposition by population sub-groups has been the leading approach followed fro any researchers to quantify how education, age, sex and other individual characteristics, affect inequality. The approach consists in dividing a saple into discrete categories (rural and urban residents, individuals with priary or secondary school or higher education, etc) and then calculating the level of inequality within each sub-saple and between the eans of the subsaples. Aong the others, one iportant liitation of this ind of analysis is the lac of control for the endogeneity of soe explicative variables that ay theselves be partly deterined by incoe patterns. This and other probles have been overpassed adopting regression technique in decoposition analysis (Oaxaca 973, Fields 998, Bourgignon et al. 998), which allows continuous variables being perissible, too. 2 See also Pyatt et. al. (980), Star et. al. (986), Leibbrandt et. al. (996).

3 coplete dependence on the global inequality index. 3 Despite this, the natural decoposition derivable by the Theil index of inequality is shown to be the ost suitable and consistent ethod to ipleent in epirical application. This is established on the basis of the satisfaction of three very iportant properties, which allow us to identify such decoposition as the only well-behaved ethod aong those considered. It is worth ephasising that another iportant technique of decoposition, based on the concept of the Shapley values, 4 is not considered in this paper because of the negation of a very fundaental property: the independence of the level of disaggregation. 5 Despite the great variety of possible applications it could reproduce 6, our view is that it cannot be included in the following analysis because of our ain objective: to see the ost consistent, satisfactory, not abiguous (or contradictory) decoposition ethod. The second, and decisive, step is to enhance the set of desirable proprieties able to sustain the choice of the Theil-based decoposition with the additional requireent of subgroup decoposability. This is done putting in evidence the different advantages coing fro the use of the Gini and Theil indices when a two-way decoposition of overall inequality is ipleented. Many authors have faced the proble of ultidiensional decoposition (see Aita, 2003; Conceição et.al., 2000; Wodon, 999), but only considering the hierarchical structure of their population attributes of reference (territorial, sectorial, and so on). The only attept of providing a ixed approach of inequality decoposition by population subgroups and incoe sources is given by Mussard (2004). Considering the Dagu (997) decoposition of the Gini coefficient and the additional disaggregation of total incoe into its ain coponents, he proposes a bidiensional approach of study which is shown to possess less properties (and appealing structure) of an equivalent ethod derivable fro the Theil index of inequality. The paper is organized as follows. In section 2 we introduce in details the Shorrocs s axio-based approach (982) as well as two aong all the possible natural decopositions of the Gini and Theil indices. Firstly, we ephasize the satisfaction of a fundaental static property (of unifor addition, 3 In this respect, Shorrocs (982) writes: Using the natural decoposition of the Gini [ ] could be justified by arguing both that the Gini coefficient should be used as the easure of inequality (which is an acceptable position to tae), and that we ust choose the decoposition rule that follow naturally fro the conventional way in which the Gini forula is written. This latter position is siply untenable. Just a few years later Leran and Yitzhai (985) answer: The approach based on the Gini is worth pursuing for three reasons. First, its use is desirable, because perits one to for the necessary conditions for (second order) stochastic-doinance. Second, our decoposition yields an intuitive interpretation of the eleents aing up each source s contribution to inequality. Third, it gives the advantage of exaining the arginal changes in the size of an incoe source on overall inequality. 4 See Shorrocs (999) and Sastre and Trannoy (200) for a detailed analysis of the Shapley-based ethodology. 5 In order to understand the iportance of this property, observe that if it were not satisfied the contribution, let us say, of earnings ight change if capital incoes were partitioned into rent, interest, and dividends, or transfer payents were split into (private and public) pensions, uneployent subsidies, and so on. 6 In particular, it could be of fundaental iportance in the bidiensional decoposition of inequality analysed in this paper.

4 i.e. negative contribution of equally distributed sources); secondly, we propose two new dynaic principles which any suitable rule of decoposition should plausibly satisfy. In particular, the following three aspects are highlighted 7 : i) different rules of decoposition show divergent responses to unifor source variations, which ultiately depend on the level and spread of the source interested by the change and those corresponding to the overall distribution; ii) the positive or negative input of initial contribution and the share of the sources of total incoe constrain the range of those variations (both in ters of sign and level); iii) the Theil-based rule is found to be the only decoposition which fully satisfies both our desirable dynaic principles. In order to underline the iportance of such ind of analysis, note that Leran and Yitzhai (985) ephasize: how percentage changes in particular taxes or transfer influence the distribution of incoe are iportant policy issues. 8 This is because any public policies concern directly types of incoe rather than typologies of individuals: different tax or transfer prograes, iniu wage schees, or pension refors can be better evaluated in ters of distribution effects if the ere disaggregation of incoe by incoe recipients is overcoe. Section 3 presents a short overview of the Theil (967) ain result about subgroup decoposability, with particular attention devoted to its appealing functional for. In fact, it can be seen as a function of three very siple eleents: population and incoe shares and subgroup inequality. In section 4 we derive the nested (or bidiensional) decoposition of the Theil index by incoe sources and population subgroups. Finally, using LIS icro data on incoes, section 5 presents the results of an application of that nested rule of decoposition to the case of geographical disaggregation of inequality in Italy between 989 and We are thus able to separate each source contribution to total inequality into two additional ters: the fractions affecting between and within-group regional inequality. 2. Inequality decoposition by incoe sources In this section we propose a coparative analysis of three largely applied rule of decoposition by factor coponents. We firstly analyse the axioatic derivation of Shorrocs (982) in which the objective of dividing total inequality into the partial contribution of each incoe coponent is dealt by starting with the definition of few fundaental axios and then deriving the decoposition rule 7 They are analytically derived in Giaatteo (2007). 8 Leran and Yitzhai (985), Star et. al. (986), Leibbrandt et. al. (996) proposed a siilar study (only in the case of the Gini decoposition), providing results on the sign of the global index variation. Se also Paul S. (2004), for an extension over a larger set of inequality indices.

5 that satisfies the. Then we ove on the ground of the natural decoposition 9 considering those of two well nown inequality easures: the Gini coefficient and the Theil index 0. The fundaental differences between the two ethods is that, while the axioatic approach yields a decoposition rule which respects the assuptions directly iposed upon it, in the second case the properties that one can derive ust just be accepted, as a consequence of their coplete dependence on the global inequality index. Our ain point is that if an axioatic approach of study has to be generally preferred because of the possibility of ensuring the satisfaction of desirable (decoposition) properties, on the other hand the existing natural decopositions cannot be ignored, ainly because of the consistencies they ay show with respect to the standard inequality theory. We propose to evaluate the appropriateness of the three decoposition ethods said above on the basis of the satisfaction of one iportant static property (unifor addition) and two additional dynaic principles. These are introduced in order to evaluate the decopositions behaviour which follows fro a siple variation in the incoes of only one source. Our opinion is that the dynaic perspective proposed in the following analysis should deserve ore attention aong researchers. In fact, if the ain objective of every decoposition study is that of looing for the inequality deterinants over tie, the essential requireent should be that of using ethodologies which consistently respond to source distribution changes. More precisely, one should expect that a very siple variation in the incoes of a particular source should cause its associated contribution to change without generating perverse results. Since incoe coponents are usually distributed with different degrees of inequality and different agnitude, every decoposition analysis should be able to deterine in a clear and consistent way how those variations affect decoposition outcoes and, as consequence, total inequality. This objective is central for policy scheduling: odd decoposition behaviours could iply erroneous identification of the ain inequality causes, with the consequent invalidation of analysis interested in estiating the distribution effects of specific policies. In this respect, looing for reasonable relations between source variations and associated contribution effects can be used as a reference fraewor for the identification of well-behaved decoposition rules. 9 Note that the ter natural is used not to say that decoposition is the only possible and unaniously acceptable, but only because it is obtained by arithetical derivation fro the index of reference. 0 See Leran and Yitzhai (985) and Paul (2004), respectively. For the decoposition of the Gini index by incoe coponents see also Pyatt et. al. (980), Star et. al. (986), Leibbrandt et. al. (996). In the following sections the concept of dependence of a decoposition procedure fro an index of inequality will be explored in depth. Right now, suffice it to say that soe decoposition properties could be inconsistent with fundaental inequality principles.

6 2. Axio-based and natural decopositions Let us denote with n D the class of all incoe distributions Y coposed of n units (individuals or households). Suppose that total incoe is also divisible into M different incoe sources, such that n n M M i i i= i= = = [] Y = y = y = Y Given a general inequality index I( Y ) we can define the absolute contribution to total inequality of the -th coponent of incoe as the generic function C C( Y Y) =, with ; n Y D and i 0 y at C = ; =. I ( Y ) least for one i 2. Therefore, we define the -th proportional contribution as c c( Y Y) Hereafter, we identify it also as the generic rule of decoposition. Note also that the c is usually a function of the total index of reference in the case of natural decopositions; it is not so when the Shorrocs rule is adopted. To conclude this first set of definitions, in what follows we denote the functions C and c with different notations, ore precisely: W and w in the case of the Shorrocs proposal; G and g for Gini s natural decoposition ; T and t for the Theil-based ethod. The starting point of Shorrocs (982) is to suggest six fundaental assuptions (axios) which every ethod of decoposition should plausibly satisfy 3. These allow hi to deonstrate that, W ( Y ; Y) Cov( Y, Y) w ( I) = = for all Y μe [2] 2 IY ( ) σ ( Y) is the only rule of decoposition satisfying the six axios and such that the relative iportance of different incoe coponents with respect to total inequality is independent of the choice of the easures. 4 A first ethod followed by any authors to derive (one of the possible) natural decoposition of the Gini coefficient is based on a atheatical derivation which initially considers this index as defined by 2 Note that the contribution shown by each source Y is usually different fro its own inequality. 3 See Shorrocs (982) pp Note also that it corresponds to that of the natural decoposition of the variance.

7 G( Y) = ( 2cov [ Y, F( Y) ]) μ( Y) where F ( Y ) denotes the cuulative distribution corresponding to the density function f ( y ), defined as the ran of y i in Y divided by the nuber of observations. Leran and Yitzhai (985) show that the Gini coefficient can be decoposed as the su of the absolute contribution coing fro every incoe coponent as, i M G( Y) = R Gini S [3] = Every absolute contribution ( RGiniS ) is the product of three easures: R is the Gini correlation between the incoe coponent and total incoe; 5 Gini is the relative Gini of coponent ; and S is the coponent share of total incoe. Using [3] we can also define the proportional Gini contribution coing fro the incoe source, RGiniS g = =,..., M [4] G( Y) Several researchers have tried to study ore systeatically the above decoposition. Aong others, Podder (993) claied that this interpretation of the Gini decoposition can be shown to be «wrong and totally isleading». To understand why, he taes as an exaple a constant coponent of incoe. In this case the concentration coefficient of the coponent ( Gini ) is zero. This necessarily eans that its absolute contribution to the overall level of inequality is also zero. Despite this, Podder ephasizes that «[ ] it is reasonable to thin the addition of a constant to all incoes leading to a reduction in inequality if we accept relative easures». Note that this undesirable property of zero contribution for equally distributed sources is also satisfied by the Sharocs derivation [2]: ore specifically, it is directly required by one of his six axios. Finally, let us introduce the natural decoposition of the Theil index. His well-nown forula is 5 cov ( Y, F[ Y ]) R =, where FY [ ] = ( f ( y ),..., f( y )), and f ( y n i ) is equal to the ran of cov ( Y, F[ Y] ) the nuber of observations. y i in Y divided by

8 y T( Y) = ln i yi [5] nμ μ i Siply taing into account the basic relation [] and applying it to the generic y i in [5], we can derive the following (natural) decoposition t T ( Y) = = T ( Y) i i y i ln yi μ y i ln yi μ [6] yi where T ( ) ln Y = yi nμ i μ is the generic absolute contribution. 6 Contrary to the Shorrocs proposal [2] and the Gini based decoposition [4], t satisfies a very iportant property: it is strictly negative for any equally distributed source of incoe. 7 The iportance of this property of unifor addition for an inequality index coes fro the fact that it is directly iplied whenever the transfer axio and the scale invariance axio are satisfied. That said, is it also paraount to extend this requireent to the decoposition ground? To phrase it differently, should the unifor addition property set out a reference axio which would constrain the consistent derivation of a decoposition rule? Let us underline how for the class of the natural decopositions the satisfaction of the property by an inequality index does not guarantee that the associated rule does the sae. 8 Thus, even if the corroboration of one property by an overall index does not bind its natural decopositions, could soe inconsistencies arise as well? We believe that one useful way to follow could be that of focusing the attention on the dynaic behaviour of the existing decoposition rules, ensuring 6 y It is not the factor s Theil, but the pseudo-theil. In fact, the weights for the -th factor incoes log i nμ μ are based on the ran of the total distribution. 7 If yi = μ i, the nuerator of T siplifies to μ ultiplied by the negative of the Theil-L index of overall n inequality, n ln( μ yi ). i= 8 This is the case, for exaple, of the Gini-based rule [4] and the natural decoposition of GE(2) given by the Shorrocs Theore: while these two indices satisfy the property of unifor addition, the corresponding decopositions RGS neglect it: g = δ δ δ δ 0 GY ( ) = and Cov( δ e, Y) sδ = = 0. Var( Y )

9 (avoiding) the application of appropriate (iproper) ethods as a consequence of the satisfaction (negation) of reasonable requisites. 2.2 Unifor variations of incoes and well-behaved decoposition rules In the previous section we have introduced three rules of decoposition which allow us to distinguish how different incoe coponents contribute to the total level of inequality in a given distribution. We have also introduced the iportant property of unifor addition, which we propose as a first criterion of identification of suitable decoposition. If the acceptance (rejection) of that property cannot constitute the only and decisive principle for evaluating the existing ethodologies, what is evident fro uch of the available epirical literature is the high frequency of divergent results that the applications of different ethods produce. 9 The ordinary decoposition of the Gini coefficient, for exaple, ay indicate that wage incoes have a very sall positive influence on overall inequality, while that of the Theil index ay show the opposite. Moreover, there are no theoretical foundations on the basis of which to explain such conflicting epirical findings. The objective of this section is to loo for a theoretical fraewor able to support the existing decoposition ethods in ters of their consistent dynaic behaviour. We propose to follow an approach of study founded on the arginal variations of incoe sources in order to identify wellbehaved decoposition rules. To this end, we copare the different responses that the rule proposed by Shorrocs, the natural decopositions of the Gini and Theil indices provide as a result of a unifor proportional change in all incoes of one coponent. Basically, we tae as starting points the following two general dynaic principles: PRINCIPLE A (for inequality increasing sources). Consider a source of incoe contributing positively to total inequality ( C + ). When all the incoes in Y + increase uniforly, the proportional contribution of the sae source has to get bigger, independently of the initial share of Y. PRINCIPLE B (for inequality decreasing sources). Consider a source of incoe contributing negatively to total inequality ( C ). When all the incoes in Y increase uniforly, the proportional contribution of the sae source has to get saller (bigger) if its initial share of Y is sall (big) enough. 9 See, for exaple, Morduch and Sicular (2002) for an epirical application to rural China.

10 They found on the fact that increasing uniforly a source also increases, but less than proportionally, the aggregate aount of incoe. Thus, holding constant the other sources in ters of levels and spread, the increased weight of the varied source on total inequality should have different effects, depending on its initial character (inequality increasing or decreasing). More specifically, in the first case it should always reinforce its positive contribution (PRINCIPLE A), while in the second case the effect should not be independent of its initial share (PRINCIPLE B). The idea behind the first principle is that if a coponent contribution is initially positive, and we increase proportionally its share (driving it closer to the whole distribution), then one would expect to observe a greater ipact on total inequality, since the relative weight of its own inequality with respect to the other sources has risen. On the other hand, if a source initially contributes negatively to total inequality, one would expect that, above a specific level of Y (with μ close enough to μ ), the character of the coponent should show a positive change. In other words, the expected behaviour of any source contributing negatively to total inequality should not be onotonic: its negative contribution should be reinforced when the share is sall, and weaened for shares bigger than a specified threshold. 20 To better understand the point, one can iagine the coponent Y increasing (continuously) until it can be considered close enough to Y. It sees clear that, accepting the continuity for the function characterising the decoposition rule, alost the overall inequality would be ascribed to the incoe source subject only to a change of its character fro negative to positive influence. Principles A and B allow us to derive strong conclusions about the appropriateness of the three decopositions described in the previous section. Maing explicit the response of [2], [4] and [6] to a unifor variation incoes of just one source 2, it is possible to bind the sign of their arginal behaviours. They result to be function of: i) the initial character of the coponent (inequality increasing or decreasing) subject to the scale transforation; ii) the incoe source (and overall) inequality; iii) the source share of total incoe. Despite this, each decoposition shows very peculiar, and conceptually not expected, behaviour: the Gini decoposition constrains the initial contributions to increase (decrease), whenever they are initially positive (negative). This effect disagrees with the expected not onotonic behaviour of inequality decreasing sources (above Principle B). Moreover, the initial share of the altered source does not play any role in defining the sign and agnitude of the proportional contribution. In the Shorrocs axio-based proposal other conditional factors play a crucial role in defining the sign of the arginal variations, even if these 20 This threshold cannot be established in general: it will depend on the specific aount, spread and ran of the incoes in Y and Y. 2 The corresponding analytical results are explicitly derived in Giaatteo (2007).

11 are still independent of the source shares. The ratio of partial to total variance aes the prediction about the decoposition behaviour also ore coplex and, in soe cases, unjustifiable. Finally, the Theil-based decoposition is the only one which perfectly satisfies both the Principles A and B. As it should be expected, all the inequality increasing sources enhance their positive effect on inequality as a consequence of an increase of their share on total incoe; conversely, if a source shows a negative effect on total inequality, increasing its relative size can iply different perforances dependently of its initial relative weight on total incoe: the higher the -th source share of Y, the ore liely is a positive variation (i.e. a fall in its absolute equalising effect) The corroboration of the two dynaic principles A and B, in addition to the satisfaction of the property of unifor addition, should suggest the Theil decoposition [6] as the best (aong the three considered) well-behaved 22 rules of inequality decoposition by incoe sources It can be consistently applied with the objective of explaining the total patterns of inequality through the distribution of the incoe coponents. The Gini-based rule and the Shorrocs axio-based proposal are two less desirable decopositions to ipleent. Their use, in fact, should iply the unpleasant occurrence of obscure reaction to changes in the source distributions, which in turn could cause the overall inequality to respond perversely to such changes. 3. Theil index and subgroup decoposability «[ ] We now find that this easure has a siple interpretation in ters of incoe shares and population shares; oreover, that it can be aggregated in a straightforward anner. In that respect it is ore attractive that ost well-nown inequality easures such as Gini s concentration ratio.» (Theil, 967, pp ) The ain otivation of decoposing inequality by population subgroups is given by the possibility of exaining the relationship between the deographic structure of a population and the associated incoe distribution. As well nown, the Theil (967) axioatic easures properly achieve this objective; as a consequence, they are often used in epirical wors in order to provide eys of understanding for the observed patterns of inequality. This section briefly suarises the ain decoposability results derivable by the Theil index forulation, in order to underline the appealing characteristics also shown by the nested (group and source-based) decoposition of inequality proposed in the following section. 22 In pure atheatics, "well-behaved" objects are those that can be proved or analyzed by elegant eans to have elegant properties. In both pure and applied atheatics, well-behaved also eans not violating any assuptions needed to successfully apply whatever analysis is being discussed.

12 Consider each individual in the total population characterized by the general pair ( y ) total incoe i, = y of y Y and one attribute =,..., K. Suppose that this attribute divides the total i population into K utually exclusive and exhaustive groups. Then, we can define i = K n n n ; π = n = ; and n Y = n μ = y, =,..., K i i= where n and μ represent the nuber of individuals and the -group ean, respectively. The iniu requireent for population decoposability is that if inequality increases in a population subgroup then, other things being equal, inequality increases overall (property of subgroup consistency). Given the generic index characterization ( ) (,,..., ; πμ, ) I Y =Φ I I I K, it requires that the function Φ has to be strictly increasing in each of its first K arguents = ( ) 2 I I Y 23. The aggregation proble is solved by Theil (967) providing a breaing down rule ade of two coponents: the first identifies the distance between hoogeneous groups of units, while the second incorporates the dispersion within each group. In forula, we have T ( Y ) K K n n μ μ n y μ y i i = ln ln + n μ μ n μ n μ μ = = i= K = π s ln s + π s T = Tb+ Tw = = K [7] n μ Y where πs = = n μ Y is the total incoe share held by subpopulation. The between-group (Tb ) and the within-group (Tw ) coponents easure the inequality contribution coing, respectively, fro the differences in subgroup eans ( μ ) and the incoe differences inside each population subgroup. Note that the first ter contributes nothing only if s =,. In all other cases it will be strictly positive. The second ter, which corresponds to the weighed ean of the K y sub-indices = ln y i i T, is also never negative and reaches its iniu (zero) in case of n μ μ n i= equally distributed incoes inside each subpopulation. 23 See Bourguignon (979), Cowell (980), Shorrocs (980, 984) for an exhaustive treatent of the subgroups decoposition and the class of additive inequality easure.

13 An equivalent expression of [7] possesses particular appeal because of its interpretation as function of incoe and population shares, T ( Y) y y = + yi n K K n Y ln Y n Y i ln i = Y Y n = Y i= yi i i [8] where Y = nμ. [8] says us that: Y i) when the incoe shares Y each =,..., K (,..., ) = y y i i i n equal the corresponding population shares n for i n, then the between (within) coponent contributes nothing to overall inequality; ii) the bigger the discrepancy between group (individual) relative incoe and group (individual) population shares, the greater the contribution of the between (within) coponent of inequality; iii) the Theil index is a function of three siple eleents: subgroup incoe and population shares, and subgroups inequality. n Soe wors have tried to extend this fundaental one-diensional result to the case of two (or ore) attributes. Aita (2003), for instance, considers the three-level hierarchical structure of a country into regions, provinces and districts in order to derive a nested geographical decoposition given by, TR = TwP + TbP+ TbR The overall regional incoe inequality is thus decoposed into the within-province ( Tw ), the between-province ( Tb ), and the between-region ( Tb ) coponents. The within-province P contribution to inequality is a weighted average of Theil indices at the province level (the weights being the shares of province incoe in the region), while the overall-between-province coponent is a weighted average of between-province incoe inequalities within each region 24. Conceição, et. al. (2000) also discuss the iplications of the Theil index decoposition into a sequence of nested and hierarchic group structures but, differently fro the Aita analysis which focuses on the geographical disaggregation, they apply the ultilevel decoposition of the Theil R P 24 See Aita (2003) for a detailed description of the procedure and the results of his epirical application to China and Indonesia.

14 index on wages and eployent, operating disaggregation by industrial classification. Their analysis presents two interesting peculiarities: the first consists of introducing the ulti-sequence decoposition of the Theil index in order to identify the inforation gain of oving towards a higher nuber of SIC 25 digits in explaining wages evolution; as second, they study the deterinants of between-group inequality variation over tie as a function of the two ain index constituents: incoe and population shares. Wodon (999), instead, provides a ultidiensional extension of the (one-level) group decoposition of the Gini index proposed by Yitzhai and Leran (YL, 99). A first strategy consists of taing into account g utually exclusive groups obtained by the cobination of the two attribute diensions and h (i.e. g=,..., K H ). Then, he applies the one-diensional YL decoposition along the g categories. A ore interesting approach to the bivariate proble is to proceed sequentially (i.e. to operate the disaggregation of the K groups into H sub-groups). The derived Gini decoposition is thus given by the su of the following eleents: a) the within groups coponent of total inequality 26 ; b) two ters identifying the first and the second order stratification coponent 27 ; c) two other ters identifying the first and second order between-group coponents of inequality 28. In what follows we go through the nested decoposition of inequality by incoe sources and population subgroups. In particular, we copare the possibilities which the Gini coefficient provides (Mussard, 2004) with an alternative derivation based on the Theil index of inequality. 4. Nested decoposition rules Theoretical and epirical literature on inequality decoposition has ainly developed independent analysis for sub-populations and incoe sources disaggregation. Given the theoretical results of section 2 about the possible ways of identifying the inequality contribution coing fro the various incoe sources, the appropriateness of the Theil index is reinforced by the possibility of extending its appealing properties to the case of a nested (bidiensional) rule of decoposition. This perits to account in a straightforward anner for subgroup and incoe source distribution structure, providing group-source inequality contributions 25 Standard Industrial Classification. 26 It is the result of two within group expansions, starting with the diension and following with diension h. 27 The first (second) order stratification ter easures the (YL) stratification within the overall population (within the group ). 28 The first order between groups ter easures the inequality between groups according to diension, while the second order between group ter easure the extent of the inequality, within group, between the households with different characteristic h.

15 which can be expressed as a function of incoe shares, population shares and group-source specific inequality. Mussard (2004) proposes an interesting attept of providing theoretical bases of a unified decoposition approach 29. His starting point consists of considering the Gini index as the reference easure of inequality. If on one hand this choice has, obviously, robust foundation because the well-nown Gini relation with the Lorenz curve and deprivation theory, on the other hand it is not so appropriate when the objective of the analysis is to decopose total inequality. The three ain reasons of this clai are represented by: i) the Gini interaction (third) ter of its subgroup decoposition 30 ; ii) on the ground of incoe source disaggregation, the not satisfaction of the property of unifor addition and of the dynaic principle B treated in section 2; iii) its functional final structure, which give clear but not useful (or functioning) indications about the eleentary factors driving source and subgroup inequality contributions. In the rest of this section we will try to validate, instead, the Theil index appropriateness. As noted above: i) it iplies an easy and suitable inequality interpretation as function of siple incoe and population shares; ii) it also iplies a natural decoposition by incoe sources which satisfies very attractive proprieties; iii) it is perfectly decoposable by population subgroups. As specified below, iv) the bidiensional (or nested) decoposition of the Theil index suggests a very appealing opportunity of linage between functional and personal incoe distribution analysis; v) it also constitutes a useful tool of policy evaluation: crucial governent chooses such as those concerning labour aret refors, transfers and taxes schedule, policy decentralisation, etc., could be properly evaluated in ters of inequality effect. Consider the total distribution of incoe Y coposed of n units (individuals or households) receiving incoe fro M different sources of incoe Y, such that [] is still true and y 0 with i y > 0 at least for one j. Mussard (2004) shows how using one of the possible Gini coefficient j decoposition by incoe sources and the Dagu (997) disaggregation by population subgroups it is possible to derive the following bidiensional decoposition, 29 See also Rao (969) for an attept of providing a unified solution to the decoposition issue. 30 Mussard (2004) uses the Dagu (997) result about the possibility of decoposing of the Gini ratio into the following three coponents:. the contribution of the within groups incoe inequalities; 2. the net contribution of the extended Gini inequality between subpopulations taing into account variations in ean, standard deviation, asyetry; 3. the between groups inequalities of the transvariazione. See also Labert and Aronson (993) for the relation between the Gini residual ter and the Lorenz curve.

16 G * ( ( yi 2 )), + yj, yij, K n n M = = i= j= 2 = 2μn K n nh ( ( y )) * i, + yj, h yij, h 2 2 M + = 2 h= i= j= 2 = 2μn [9] where y i, is the generic -th incoe source in group and y * ij, h is the -th source of the iniu between y i, and y, jh. Fro expression [9] is possible to divide the generic source contribution to total inequality G into the two coponents of within and between-group inequality. A siilar derivation can be obtained for the Theil index. In section 2 we have already shown how to decopose it by incoe sources, that is y y i i y y i i TY ( ) = ln ln = = n μ μ n μ μ n M n M i= = i= = T( ) [0] where T ( ) is the generic pseudo-theil for the source. The next fundaental step is to ipleent the source-based decoposition [0] into the subgroup disaggregation of total inequality given by [7]. Keeping in ind the basic incoe distribution structure [] and the sub-eans additivity, μ j M = μ [] = j we are able to divide the between-group coponent of total inequality Tb into M source contributions as following, M K K n n μ μ = Tb ln μ = = ln μ n μ μ n μ μ = = M K M μ = πs ln = Tb( ) = = μ [2] =

17 K where Tb( ) s ln μ = π μ = represents the -th source contribution to the overall between- n μ group inequality, and the generic weight πs = is the share of total incoe held by the -th n μ subgroup and corresponding to the incoe source. Consistently with the definition of the pseudo- Theil T ( ), we can define Tb( ) as the generic between-group pseudo-theil (or ean pseudo- Theil). Note that Tb( ) is zero whenever Tb is zero (i.e. s =, ) but, differently fro the standard subpopulation decoposition, the partial contribution of the source (to the overall between-group inequality) can also be negative. Following a siilar procedure, but considering the individual incoe relations y i M = y instead than [], we can disaggregate by incoe sources the within-group coponent of the Theil index as, = i K n n μ y y i i Tw = ln n μ n μ μ = i= M K y n i = s = y π i = ln n i= μ μ = y y = M K n M i i s ln π = = n i= μ μ = Tw( ) [3] K n y y i i where Tw( ) = πs ln = n i= μ μ j is the weighted su of the K group-pseudo-theil n y y i i T ( ) = ln n, and represents the -th source contribution to the overall within-group = μ μ j i inequality. Also in this case, we can define Tw( ) the generic within-group pseudo-theil. Expression [2] and [3] allow us to derive the following subgroup-source nested decoposition of the Theil index,

18 T( Y) = Tb + Tw = Tb( ) + Tw( ) M M [4] = = where Tb( ) and Tw( ) represent, respectively, the contribution to between and within-group inequality which can be assigned to the -th incoe coponent. The Theil bidiensional decoposition [4] naturally increases the set of inequality deterinants, which are not directly observable when subgroup and incoe source decopositions are ipleented separately. Even if this opportunity is also ade possible by the existing ultidiensional decoposition of Aita (2003), Conceição, et. al. (2000) and Wodon (999), the chance of introducing incoe source deterinants of overall inequality, ust be seen as the ajor iproveent of [4]. It provides the chance of inquiring any econoic aspects (basically given by the functional structure of total incoe, separable into wages, profits and rents - as well as state factors) with other iportant econoic and social issues. The gender (or ethnic) discriination issue, the geographical uneven distribution of resources and the analysis of the ipact of deographic structure on the overall level of inequality are only three aong the possible fraewors of ipleentation of a bidiensional decoposition approach. Moreover, the possibility of controlling for fundaental state factors (such as transfers and taxes) provides a useful tool for a ore accurate analysis of target policy effects on total inequality. To conclude, let us note how the bidiensional decoposition [4] perits to achieve these objectives reducing the basic inequality deterinants to very few and siple factors: the incoe and population shares and the group-source inequality. This is clearly not the case of the Gini bidiensional decoposition [9], which shows source-group contributions only partially functional to this role. 5. Epirical analysis 5.. Data and ethodology In this paper we focus on disposable incoe, which is obtained by the su of all wor and transfer incoes of all individuals in each household. It is also an equivalent easure, being divided by the square root of the nuber of household eber in order to tae into account econoies of scale. For what concerns outlier observations, the LIS recoendation is to ipose botto and top codes redefinition of incoes, so that all the observations below the % of equivalent ean incoe and all those above 0 ties the edian unequivalised incoe are substituted by the respective thresholds. In our case, the su of all incoe sources has to equal the corresponding

19 individual (household-based) total incoe, the redefinition of code incoes into two thresholds would generate inconsistencies between the (adjusted) total incoe and the su of its factor coponents. As a consequence, we have just erased the top and botto codes of each distribution (0.% at the botto and at the top of the disposable incoe distribution). Despite the SHIW Ban of Italy survey ipleented in the LIS database covers a very long period of tie 3, the analysis has been restricted to the period , in order to derive epirical findings able to be ore easily extended with those of other countries present in the sae database. In the following application we eployed a slight different forulation of the nested Theil decoposition [4], because of the need of using weighted procedure which corrects for saple selection. The adjusted weighted procedure 32 for [4] is thus given by M K M K n ( w) ( w) ( w) y y i i TY ( ) μ μ P ln μ = P pi ln + = = ( w) ( w) = = ( w) i= ( w) ( w) μ μ μ μ μ [5] where p i represent the individual weights noralised for household ebers, P the su of the s group siple weights p i ( i=,..., n ), while μ ( w), μ ( w), and μ ( w) are the weighted eans for total, -th subgroup, and -th source of the -th subgroup distributions Epirical findings The study of the relationship between deographic structure and econoic inequality has been largely investigated in the recent literature. Many are the diensions which one can exaine in order to loo for inequality deterinants. Age distribution aong population (or ore detailed) household ebers, gender or regional factors, attained education, and worer/job types are only soe aong the individual characteristics usually taen into account in epirical wors. Independently of the decoposition scope, eaning and possible interpretation 34, it is here interesting to note how the fundaental requireent is the possibility of splitting inequality as the su of between-group-source and within-group-source contributions. This involves in turn the use 3 The survey started in 965, even if only in the early Eighties becae an iportant source of Italian households inforation on incoe and consuption. Brandolini and D Alessio (200), for instance, proposed inequality analysis between 977 and 995. After 989 onward the sapling ethodology did not change, the diension of the saple does not vary uch until now 32 The weights are proportional to the actual population of the strata fro which the saple observations are drawn fro. 33 Note that when the standard (no weighted) forulation is adopted we siply have p i = n and P = n n. 34 See Kanbur (2006) for a discussion of these points.

20 of easures which support additive decoposability, and as aply verified in the previous paragraph of this Chapter, Theil indices fully satisfy this property. In what follows we apply the nested inequality decoposition rule [5] to the Italian incoe distributions between 989 and 2000, taing disposable incoe as the reference econoic diension, and considering the partition of the overall population into four subgroups defined by their geographical location 35. Brandolini and D Alessio (200) exaine the effects of deographic structure on the evolution of inequality in Italy between 977 and 995 applying ean logarithic deviation decoposed. Their epirical result can be synthesized in the following three points: ) inequality in disposable incoes between persons (disaggregated by several diensions) showed considerable fluctuations but no particular ediu-ter tendency; 2) in the id-990s Italy was, together with the United Kingdo, the EU country with the highest inequality (this result has been aply confired, aong the others, by the Italian Statistical Institute ISTAT, 2005); 3) Deographic effects on inequality appeared having been sall: they played only a secondary role in defining the evolution of inequality in Italy, as well as in explaining the deviations fro the levels recorded for other EU countries. Table 5. shows the Theil index pattern for the period , as well as its disaggregation into the within and between regional coponents. Total inequality decreased of 7% between 989 and 99, but rapidly increased of nearly 40% during the next two years. This rise was followed by a first oderate reduction until 995. Starting fro this year inequality turned to rise until the decade highest level of in 998. Finally, in 2000 it went bac to the iddle decade level. Table 5. - Disposable incoe inequality in Italy, : total patterns and subgroup decoposition by geographical location. Disposable incoe Absolute values Row % Tb Tw Theil Tb Tw Theil Source: Own calculations on LIS database. 35 The acro-regions coposition considered in the elaboration is the following: (North-West) Pieeonte, Lobardia, Liguria, Trentino; (North-East) Veneto, Friuli Venezia Giulia, Eilia Roagna; (Centre) Toscana, Ubria, Marche. Lazio; (South and Islands) Abruzzo, Molise, Capania, Puglia, Basilicata, Calabria, Sicila, Sardegna.

21 Table 5. also points out the contribution of the within and between territorial coponents. Quite differently fro the coonly shared idea of an increasing Italian regional dualis, the within regions coponent of inequality sees to have ore decisively driven the overall trend. More precisely, within-region inequality slightly increased its percentage contribution between 989 and 993 eroding part of the inequality influence iputable to the between factor. Thereafter, the two relative contributions reained nearly constant around the 2% (between coponent) and 88% (within coponent) in explaining total inequality. Despite this, a different ey of interpretation arises if one goes through the specific coponents patterns. In fact, while the diinishing trend of total inequality observed in the first years of the decade can be ascribed to the joined contraction of both between and within coponents, the two ost iportant inequality upward variations (between and ) are alost fully iputable to the widening within regions contribution. In su, Italian inequality resulted to be increased of 28.8% between 989 and This overall trend was the result of a oderate increase in between-region inequality (5% over the decade) and a stronger ipact of the within coponent (+33%). Saying it differently, total inequality in 989 was for a 3.4% due to differences between geographical areas, as for a 86.6% to the level of their own unequal distributions; after years the sae percentages were % and 89% for the between and within coponents, respectively. The territorial analysis of the Italian inequality fails to shed light on the econoic deterinants underlying overall trend. One could, surely, interprets the results of Tables 4. taing into account the regional characteristics of labour and capital arets, their specific productive structures, as well as the decentralised governent interventions. All these eys of explanation are not, certainly, worthless. Despite this, the incoe sources decoposition allows such aspects to be better detected in a clear and consistent fraewor. In order to ephasize the added value that would coe fro a joined subpopulations-incoe sources analysis, let us briefly go through the following questions. Does the population age structure atter because the different source coposition of personal incoes (fro wor, property and transfer) in the households 36? Does the standard analysis of the incoe gender gap hide very different feale and ale types of incoe received (for exaple, fro wor or capital activities)? How uch of the regional deterinants on national incoe inequality could be associated to the different ipact of dependent (private or public) wors, autonoous wors, financial incoes or different ipact (because the diverse faily or activity 36 Brandolini and D Alessio (200) noted how the incoes of heads of household below the age of 40 worsened between 977 and 995, while it iproved for heads aged over 65. Could be possible to strictly lin this trend to the ipact of different incoe sources within failies?