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32 Volume LXIII nn. 3-4 – Luglio-Dicembre 2009 welfare. In this study the idea is to study the dynamic of the inequality of the income distribution also in the time dimension by considering macrocohorts. A macrocohort can be defined as a group with fixed membership, formed by individuals, that can be identified as they show up in the survey (Deaton 1985). Grouping individuals according to a common characteristic gives the possibility to observe macrocohorts over time. We grouped the households on eight indipendent cross-sections of Italian housholds between 1997-2004 on the basis of the head's year of birth. We defined eleven groups by a five-years band except for the first and the last macrohorts, where the age brackets are extended due to the small cell size 3 . We decided to focus our attention on those macrocohorts with the haed aged 20-79 in 1997. After grouping data, we adjusted expenditure according to the Carbonaro's equivalence scale, this allows us to compare households of different size, netted out of economies of scale effects. If we consider expenditure levels as proxy for income, we are able to compute the Gini index for macrocohorts and year and estimate the contribution of several factors to income dispersion. In the next section we will present the methodology. 3. The Shapley value decomposition The Shapley value decomposition approach is a regression-based methodology that permits to quantify the impact on inequality of several causal determinants (Shorrocks, 1982, 1999). The idea is to specify an income generating model that includes what we consider contributory factors and subsequetly decomposes the Gini index isolating the effect of each factor. The methodology can be used to decompose any inequality index using any income general funtional form. Shorrocks (1999) proposed a general application of the Shapley value method to decompose income inequality, this procedure treats symmetrically all factors and leads to an exact additive decomposition of inequality index into contributory factors. The methodology has the advantage that groups of factors may be considered as a single entity without affecting their total contribution. 3 It is necessary that each macrocohort size is sufficienty large in order to obtain consistent estimates.Verbeek and Nijman (1992) showed that fairly large cohort sizes, i.e. 100 individuals, are needed to validly ignore the cohort nature of the data.
Rivista Italiana di Economia Demografia e Statistica 33 To clarify the procedure, we start from our income generating model, where Y denotes the macrocohorts expenditure, X Occ , X Edu and X Dem are groups of variables related respectively to occupation, education and demographic characteristics of households. The variables related to macrocohorts and months are collected in Z . We use a linear generating model for income. Y = β0 + β1XOcc + β2XEdu + β3XDem + β4Z + ε t. (1) The fitted values of the regression will be computed as follows, ˆ = βˆ βˆ X + βˆ X + βˆ X βˆ Z, Y 0 + 1 Occ 2 Edu 3 Dem + 4 (2) they will be used to obtain the Gini index for macrocohort and year, we call it Ĝ TOT , i.e. the total predicted income inequality. The Shapley value decomposition allows us to quantify the factor contributions by removing βˆ k Xk from the total income and measuring the extent by which Ĝ TOT changes. There are many possible sequences in which k X may be eliminated, therefore we have to compute, for the same group, several contributions according to all the possible elimination sequences. The k -th factor contribution to inequality will be given by the average of those computed in each sequence. In our model with three explanatory groups of variables, there will be three rounds of elimination, the computation becomes more complicated the larger is the set of groups considered. In the first round we start eliminating one group of variables at a time. If we drop the occupation group in the income generating model, we will obtain a predicted income given by the following ˆ = βˆ βˆ X + βˆ X βˆ Z. Y 0 + 2 Edu 3 Dem + 4 (3) With the new predicted values, netted out of the occupation effect, we compute the Gini index, G ˆ 1 Edu , Dem . Now we can obtain the occupation contribution, C , where 1 denotes the round number, as the difference between the total predicted income inequality and the inequality computed dropping occupation 1 C Gˆ − Gˆ . (4) Occ = TOT Edu, Dem
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INFORMAZIONI PER GLI AUTORI REGOLE
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SOCIETÀ E RIVISTA ADERENTI AL SIST
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