14 Equity indicators
Women earning twice as much as men is equally bad as men earning twice as much as women.
The next property is the Pigou-Dalton property (the transfer principle). It says that if you take an amount from a richer person and give it to a poorer person, inequality should diminish as long as the poorer person is still poorer than the rich after the transfer.
These properties seem uncontroversial. The population principle is also perhaps uncontroversial. It says that if we replace each income earner by the same number of clones, the inequality measure should not change. The controversial properties, however, are mainly two. Scale invariance says that if you multiply each income by the same positive constant, inequality is unchanged. That is often felt to be a rightist view. On the other hand, translation invariance says that if you add the same amount to each income, inequality is unchanged. This is often felt to be a leftist view. A compromise between these principles – a centrist view – is possible but probably mathematically cumbersome.
The Gini coefficient is the most commonly used income inequality measure. It can be explained with reference to Figure 14.1 below. On the horizontal axis, a population is ordered by income from the lowest to the highest. On the vertical axis is the cumulative share of total income. If everybody had the same income, any ten per cent of the population would have ten per cent of the income, and the straight line “Equity”
would be produced. In reality, the twenty per cent with the lowest income has only about 3 per cent of total income, the forty per cent with the lowest income has only about 25%, etc. This is shown by the “Empirical distribution” curve. This curve is called a Lorentz curve. (In actual fact, the depicted Lorentz curve shows the income distribution of Norwegian taxpayers in 1995). Obviously, the area between the two curves is an indicator of income inequality, ranging from 0 for perfectly equal distributions to 0.5 for distributions where one person earns all income. The Gini coefficient is twice this area to get a measure of inequality between 0 and 1.
Figure 14.1. Lorenz curve for the taxpayer population of Norway 1995.
0.4 0.2 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Share of income
A Lorentz curve
Empirical distribution Equality
0.6 0.8 1
Share of population
For our purposes, probably the most useful formulation of the Gini coefficient is:
(14.1)
∑∑
= =
−
⋅
= n
i n
j
j
i x
x x G n
1 1
2 | |
2 1
Here we have assumed a population of n individuals with incomes x = (x1,x2,…,xn).
The average income is x. Suppose however that there are instead n income groups with incomes x = (x1,x2,…,xn), ni members of group i, i = 1, …,n and ∑ini =N. Then
(14.2)
∑∑
= =
−
⋅
= n
i n
j ninj xi xj
x G N
1 1
2 | |
2
1 .
The Gini obeys the first three principles and scale invariance, and consequently does not exhibit translation invariance.
The Gini coefficient is not additively decomposable. Additive decomposability means that if the population consists of groups, the inequality measure can be decomposed into a term showing inequality within groups and a term showing inequality between groups. This is obviously useful for our purposes. For instance, our population belong to different zones, and it might be interesting to see to what extent the unequal distribution of benefits among income groups is due to the unequal spatial distribution.
The class of additively decomposable inequality measures was characterised by Shorrocks (1980). It turns out that the members of this class that exhibit the properties of symmetry, the Pigou-Dalton transfer principle, the population principle and scale invariance are of the following form:
(14.3)
∑
∑
∑
=
=
=
=
=
≠
−
= −
n
i
i i n
i i
n
i
c i c
x x x x S n
x x S n
or x c
x c
S nc
1 1
1 0
1
1 log ) (
1 log ) (
1 0 for ) 1
1 ( ) 1 (
x x x
where x = (x1,x2,…,xn) > 0 is the distribution of income among the n members of the population, and xis the mean income. The constant c can take all real values. This class of functions Sc is called the class of generalised entropy measures. For some values of c, they behave rather oddly as measures of income inequality. For instance, for c > 1, the measure is very sensitive to transfers of income among the rich, while for c < 0, it is very sensitive to transfers of income among the poor. Furthermore, only S0
will have the property that when decomposed, the weights on the within-group terms are constants and sum to 1. Thus S0 seems a very good candidate for our inequality measure.28,29
28 The weights on the within-group terms of S1 will also sum to 1, but will be functions of between-group inequality. On the other hand, S1 (and all measures with c > 0) has the property that there is an upper limit to inequality, given by log n in the case of S1. This allows for a normalisation of the measure and is obviously convenient for expressing targets.
29 The S0 and S1 measures are originally due to Theil (1967). Theil measures used to be denoted by T, but since they are special cases of the Shorrock measures, we denote them by S.
For an application of entropy measures to residential location, see Hårsman and Quigley (1998).
Decomposition of S0 takes the form:
(14.4)
∑
∑ ∑
∑ ∑ ∑
=
+
= +
= +
=
=
=
g g
g
g n
i g
i g
g g
n
i g
i g g
g g
x n x B n
x x B n
x x n
n B n
S w B
S
g g
1 log where
1 log 1 log
) (
1 1
0
0 xg
Here, the groups are indexed by g, the population in group g is ng and average income in group g is xg. B is the across-groups inequality measure, resulting from abstracting from all income differences inside groups. (ng/n) is the weight of the inequality inside group g in the total measure S0.
All of the measures treated so far exhibit scale invariance. For political balance and for technical reasons, we will also have a need for inequality measures displaying translation invariance. Of course, if we are not certain which of our inequality measure embody the norms and values of the decision makers, there is a third option, namely to present the distributional impacts of a strategy in a raw form, for the decision makers themselves to pass judgement on whether or not inequality has decreased.
The Kolm measure (Kolm 1976) obeys the first three principles and translation invariance. It is
(14.5)
( ( ) )
−
=
∑
= n
i
i
a a x x
n K a
1
1 exp 1log ) (x
where a > 0 is a transfer sensitive parameter.
The technical reason for applying (14.5) is that it allows some (or all) xi’s to be negative, whereas (14.3) does not.