Stata Technical Bulletin
ɪede(ɪ) — / j,
i=l
fi ⅛g(yt)
The Atkinson indices (Atkinson 1970) are defined by
A(e) = 1 - [yede(e)∕nι]
These indices are decomposable but not additively decomposable (Blackorby, Donaldson, and Auersperg 1981):
A(e) = Aw(a) + AB(a) - [A1,y(α)]. [As(α)]
where
and
Aw (ɑ) = 1
fe=l
Ab(o) = 1 -
У de
Social welfare indices (Jenkins 1997) are defined by
We = -ɪ- [yβde(e)]1 e, 0 0 0, e ≠ 1
1 — e
W1 = log[yβde(1)]
Each of these indices is an increasing function of a generalized mean of order (1 — e). All the welfare indices are additively
decomposable:
W(e)
= ∑VkWk(e)
k=l
The Gini coefficient is given by
G = 1 + (1∕2V) -
(ɪ) ∑(jv-i + 1b
∖mJ∖zJ
where persons are ranked in ascending order of yB
The Gini coefficient (and the percentile ratios) are not properly decomposable by subgroup into within- and between-group
inequality components.
Sen’s (1976) welfare index is given by
5 = m(1-G)
Syntax
ineqdeco varname [weight] [if exp] [in range] [, bygroup(groupvar) w summ]
fweights and aweights are allowed.
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