The E ect on Inequality of Changes in an
Income Component
Hilde Bojer
Department of Economics
University of Oslo
POBOX 1095 Blindern
0317 Oslo 3
Norway
email: hilde.bojer@econ.uio.no
December 28, 1998
Abstract
The e ect of a proportionate change in one of two income components is discussed by decomposing the coecient of variation. The
results will hold for any inequality measure when there is Lorenz
dominance. It is shown that the e ect of such a change depends on the
size of the income component in question as well as its distributional
properties, and that increasing a component with a given distribution
may sometimes augment, sometimes diminish, the inequality of total
income.
1
We wish sometimes to study the e ect on income inequality of increasing or
decreasing one of several income components. Aslaksen and Koren (1993) and
Helpful comments from Arne Strm are most gratefully acknowledged.
0
Bonke (1992) discuss the e ect of including the value of household production
in household income. We may want to know how increased labour force
participation of married women a ects the inequality of married couples'
income, or it may be desirable to compute the e ect on inequality of increased
child bene ts.
In such analyses, the decomposition by income component of a given
measure of inequality is of very limited value, since there is no clear connection between pseudo-Ginis, interaction coecients or contributions to inequality on the one hand, and the e ect on inequality of a proportionate
increase in the corresponding income component on the other hand.1
Take for example the case of an income component which is given in the
same amount to each recipient. As is well known, a proportionate increase in
this component will reduce total inequality, but the contribution to inequality
of the income component equals zero however calculated.2
I shall discuss the e ect of a proportionate change in one income component by decomposing the coecient of variation. The coecient of variation
is only one of many possible inequality measures. But where there is Lorenz
dominance, all inequality measures will give the same ordering of distributions. Results that hold for the coecient of variation must therefore hold
for all other inequality measures when Lorenz dominance is present.
Also, the decomposition of the coecient of variation has certain desirable
properties of symmetry, as shown by Shorrocks (1982).
For simplicity of presentation I shall use the square of the coecient of
variation, v2, rather than the coecient of variation itself. This does not
a ect the ordering properties of the measure.
Let Z = Z1 + Z2 = Z1 + kZ~2 be total income, consisting of two components, and let m , m1 and m2 denote their respective means. The squared
coecient of variation, v2, of total income is de ned as
j
j
j
j
j
z
1
m )2
v2 = (Zm;
(1)
2
Choose the scale such that m1 = 1 and m~ 2 = 1. Then m2 = k and
m = 1 + k. Let v2 be the square of the coecient of variation of component
i, and note that v2 is independent of the mean. Let s12 be the empirical
n
j
j
z
z
z
i
i
1
2
For an overview of decomposition formulas, see Shorrocks (1982) or (1988)
See Podder (1993) for other examples and a further discussion.
1
covariance of the two components de ned by
s12 = n1 (Z1 ; 1)(Z2 ; k) = n1 (Z1 ; 1)k(Z~2 ; 1) = ks~12
(2)
Then
v2 = f (k) = (1 +1 k)2 [v12 + k2v22 + 2ks~12]
(3)
We can now study the e ect on inequality of proportional variations in
component no 2 by letting k vary from 0 to +1.
As k increases, v2 obviously tends toward v22.
To study the e ect of k in more detail, we compute the derivative of
2
v = f (k).
f 0(k) = (1 +2 k)3 [k(v22 ; s~12) ; (v12 ; s~12)]
(4)
As we see, the sign of the derivative depends not only on the coecients
of variation and the covariance, but also on k itself.
Since the covariance matrix is positive semide nite we have s~212 v12v22.
Therefore [k(v22 ; s~12) and (v12 ; s~12) cannot both be negative.
The following cases may occur:
1. s~12 < v12 and s~12 < v22. This includes all cases of negative covariance.
Both [k(v22 ; s~12) and (v12 ; s~12) are positive, and
2
s~12
f 0(k) = 0 for k = k = vv12 ;
~12
2;s
We see that f 0(k) < 0 when 0 < k < k and f 0(k) > 0 when k > k.
Furthermore,
v12 > v22 , k > 1
.
The value k is the point of minimum of inequality.
j
j
j
j
j
2. v12 s~12 < v22 , f 0 > 0 for all k > 0:
3. v22 s~12 < v12 , f 0 < 0 for all k > 0:
4. v12 = s~12 = v22 , f 0 = 0 for all k > 0:
2
j
2 Summary
When the covariance between component 1 and component 2 is either negative or positive but smaller than both squared coecients of variation,
the introduction of a new income component will always at rst decrease
inequality. As the size of the new income component increases, inequality
will decline to a minimum, and then increase until it reaches the inequality
of the new income component.
When the covariance is positive and greater than the coecient of variation of one of the components, the e ect of an increase in component 2 will
be a reduction of inequality if v22 < v12 and an increase in inequality if v22 > v12.
3 References
Aslaksen, Iulie and Charlotte Koren (1993) :
A Women's perspective on Income Distribution: Reconsidering the value of Household
Work. Paper presented to Out of the Margin, Amsterdam, June 1993.
Bonke, Jens (1992) : Distribution of Economic Resources: Implications
of Including Household Production Review of Income and wealth, 38,
281{293.
Podder, Nripesh (1993) : The Disaggregation of the Gini Coecient by
Factor Components and its Application to Australia. Review of Income
and wealth, 39, 51{61.
Shorrocks (1988) : Aggregation Issues in Inequality Measurement, in W.
Eichorn (ed):Measurement in Economics, Heidelberg: Physics Verlag.
Shorrocks (1982) : Inequality Decomposition by Factor Components, Econometrica, 50, 193{211.
3