Decomposing Household Income by Source and Subgroup - Alex Eble

Decomposing Household Income by Source and Subgroup
III. Theil’s T
Theil’s T index, the member of the general entropy family of inequality indices
corresponding to = 1, is the sum of each individual’s contribution to total inequality.
Theil’s T index weights a data point’s (individual’s) population share and distance
from the mean through the following equation:
T
1
n
p
p
* * ln
p1 n y y
y
In this index, a data point gives a contribution to the overall index based on a
decreasing function of the probability of its occurrence. In other words, given a
normal distribution, the further from the mean an individual’s income is the greater is
that individual’s contribution to the inequality index. (Theil 1967)
Another interesting characteristic is Theil’s T’s non-linearity. As the richest half of the
population’s share of income increases linearly Theil’s T index increases at a more
than linear rate. This phenomenon is due to the decreasing nature of the negative
contribution to overall inequality. When there is total equality in the group, Theil’s T
reaches its minimum, zero. (Conceição and Ferreira, 2000) 3
According to scholars at the University of Texas Inequality Project, (UTIP) a think
tank focusing primarily on the measure and analysis of inequality, the main advantage
of Theil’s T index is the facility with which it decomposes inequality into between and
within group components. Another strength of Theil’s T is its capacity to analyze
inequality from aggregated data is this manner. Several other indices, including the
Gini Coefficient and the CV, require comprehensive individual-level data which is
often unavailable to social scientists. (UTIP 2005) Sicular and Morduch (2002)
show that Theil’s T index can also readily be decomposed among factor incomes. A
prior complaint about Theil’s T was that due to its logarithmic nature, it would be
undefined under negative and zero incomes. Sicular and Morduch show that Theil’s T
can be decomposed for income components and furthermore that Theil’s T is in fact
defined for zero and negative factor income values in the following equation:
s
k
TT
1
n
1
n
n
p1
n
p1
y
y p
ln
y
y p
ln
y
They go on to laud the benefits of Theil’s T as compared to other more frequently
used inequality measures: “The Gini coefficient falls if an income source is increased
by a constant amount for all members of a population”, a desirable characteristic, “but
none of the components of the standard decomposition of the Gini are affected,”
ignoring what we hope to measure as a decrease in income inequality for the given
3 For graphical representation of this phenomenon, please also refer to Conceição and Ferreira, 2000.
k
p
p
(2)
(3)
3