Stata Technical Bulletin
STB-48
indices related to the Atkinson inequality indices, namely equally-distributed-equivalent income l⅛e(e), social welfare indices
W(e), and the Sen welfare index; see below for details.
Calculations for ineqdeco exclude zero and negative income values since not all the indices are defined in such cases.
ineqdecθ is a stripped-down version of ineqdeco for situations when users wish to include zero and negative incomes in
calculations, but estimates are provided for the Gini and GE(2) indices only in this case. Some programs for inequality indices
have been provided in an earlier STB: see inequal and rspread in STB-23 (Whitehouse 1995, Goldstein 1995). These provide
estimates for additional inequality indices. But weights cannot be used in all the programs and none of them provides full
decompositions by population subgroup or estimates welfare indices.
The inequality indices differ in their sensitivities to differences in different parts of the distribution. The more positive a
is, the more sensitive GE(α) is to income differences at the top of the distribution; the more negative a is the more sensitive
it is to differences at the bottom of the distribution. GE(G) is the mean logarithmic deviation, GE(1) is the Theil index, and
GE(2) is half the square of the coefficient of variation. The more positive e > 0 (the inequality aversion parameter) is, the more
sensitive A(e) is to income differences at the bottom of the distribution. It is readily confirmed that for each member of the
Atkinson class e = eŋ, there is a corresponding ordinally-equivalent member of the Generalized Entropy class with a = 1 — eŋ.
The Gini coefficient is most sensitive to income differences about the middle (more precisely, the mode).
ineqdeco has been designed not to estimate indices which are more “top-sensitive” or “bottom-sensitive” than those
provided because experience shows that these can be very sensitive to the presence of just one or two very large or small income
outliers.
A more detailed description is as follows. Consider a population of persons (or families or households, etc.,), i = 1,..., n,
with income ya and weight u⅛. Let fi = wwN, where N = ɪʃ'ɪi wW When the data are unweighted, u⅛ = 1 and N = n.
Arithmetic mean income is m. Suppose there is an exhaustive partition of the population into mutually exclusive subgroups
к = 1,..., K.
The Generalized Entropy class of inequality indices is given by
GE(α) = -ɪ-
α(1 — a)
г n
∑fi(yi∕m}a
-i=ι
— 1 ,α≠0, α≠1
GE(I) = ∑∕i(⅜∕m)log(⅜∕m)
i=l
GE(O) = ∑fi⅛g(m∕yt)
i=l
Each GE (a) index can be additively decomposed as
GE(α) = GEw (a) ⅛ GEs (a)
where GEw (a) is within-group inequality and GEs (α) is between-group inequality; see Shorrocks (1984),
GEw (α)
Σ
fe=ι
Vk~aSkGEk(a)
where Vk = Aζ⅛/N is the number of persons in subgroup d divided by the total number of persons (subgroup population share),
and Sfs is the share of total income held by fc’s members (subgroup income share).
GE⅛(α), inequality for subgroup k, is calculated as if the subgroup were a separate population, and GEs (α) is derived
assuming every person within a given subgroup fc received fc’s mean income, m⅛.
Define the equally-distributed-equivalent income
ɪede(ɛ) —
- n
∑fi(yi)
-i=ι
1—e
1/1—e
, e > 0, e 1 1