Properties of Lorenz Curves for Transformed Income Distributions ()
1. Introduction
Redistributions of income according to tax or transfer policies can be considered as variable transformations of the initial income. The transformation is usually assumed to be positive, monotone-increasing and continuous. The initial results are given in Theorem 1. [1-3] Consider a nonnegative random variable X with the distribution function
, mean 
and Lorenz curve
. Let 
be a continuous monotone increasing function and assume that 
exists. Then Lorenz curve 
for 
exists and 1) 
if 
is monotone decreasing2) 
if 
is constant and 3) 
if 
is monotone increasing.
The importance of case (1) is that it gives the inequality effect of progressive taxation. The case (2) corresponds to flat taxes. The last case (3) is of minor economic importance, but it is included in order to complete the theorem. Recently, Fellman [4,5] has also discussed discontinuous transformations. If the transformation is considered as a tax or a transfer policy, the transformed variable is either the post-tax or the post-transfer income. Under the assumption that Theorem 1 should hold for all income distributions, the conditions are both necessary and sufficient [2,4]. Hemming and Keen [6] have given an alternative version of the conditions. In this study we consider other general properties of the transformed Lorenz curves.
2. Background
Consider income X, defined on the interval
, where
, with the distribution function
, density function
, mean
, percentile 
defined as 
and Lorenz curve
. The general formulae are
(1)
and
(2)
where
.
We consider the transformation
, where 
is non-negative, continuous and monotone-increasing. Since the transformation can be considered as a tax 
or a transfer policy
, the transformed variable is either the post-tax or the posttransfer income.
The mean and the Lorenz curve for variable Y are
(3)
and
(4)
A fundamental theorem concerning Lorenz dominance is [2,4].
Theorem 2. Let 
be an arbitrary, non-negative, random variable with the distribution
, mean 
and Lorenz curve
. Let 
be a nonnegative, monotone-increasing function, let 
and let 
exist. The Lorenz curve 
of Y exists and the following results hold:
1) 
if and only if 
is monotone-decreasing
2) 
if and only if 
is constant and
3) 
if and only if 
is monotone-increasing.
In the following, we consider additional properties of the Lorenz curve
. If
is constant, then according to Theorem 1 (2), 
and the transformed Lorenz curve is identical with the initial one, a case which will be ignored.
3. Results
3.1. The Ratio 
Is Monotonically Decreasing
According to Theorem 1 
Lorenz dominates
. We introduce the values M and m such that
and
.
Consequently,
.
Let
,
. Assume that 
and that 
and consequently,
.
Note that points 
and 
are chosen arbitrarily and that the equality signs cannot be ignored because we also include the functions
which are not uniformly strict decreasing in the class of transformations. Hence, we have to include members for which equalities hold for almost the whole range and, in addition, sub-intervals in which strict inequalities hold can be chosen arbitrarily short and located arbitrarily within the range
. If one pursues general conditions, the inequalities (8) and (9) obtained below cannot be improved. If we assume that
is monotonically decreasing, then 
must be continuous, otherwise
should have positive jumps [1].
From
it follows that
. The integration over the interval 
yields
(5)
and
.
Analogously, it follows from
that
, and we obtain
. (6)
Consequently,
. (7)
When 
in (7), then
and one obtains
. (8)
The lower bound gives an evaluation of how much the Lorenz curve has increased. The upper bound is of minor interest and is commented on later.
When 
in (7), then
and one obtains
.
In order to compare these inequalities with the inequalities in (8), we change the argument from p to q, and the inequalities are
(9)
The lower bound gives an evaluation of how much the Lorenz curve has increased. The upper bound is of minor interest and is discussed later.
Inequality (8) is applicable to small values and inequality (9) to large values of q. For small values of q, we consider the difference
(10)
and for large q we consider the difference
. (11)
In general,
and
.
The ratio
is decreasing and consequently
.
Now we differentiate 
and obtain
Consequently 
is increasing from zero at 
to a maximum 
for 
(say).
Now we differentiate 
and obtain
Consequently 
is decreasing from 
to zero when
. The point
, at which the shift from (10) to (11) is performed, is chosen so that
. Now,
;
that is,
.
Consequently,
Since the ratio
is decreasing, the difference
shifts its sign from plus to minus at point
. Hemming and Keen ([6]) gave the condition for Lorenz dominance that
crosses the
level once from above. Our results above have shown that the crossing point is
. The condition obtained can also be otherwise explained. If we write it as
we obtain the formula
that is, the Lorenz curves 
and 
have parallel tangents and the distance 
between the Lorenz curves is maximal for
.
We define the difference function as
, (12)
and the lower bound of 
is
. (13)
Figure 1 shows the Lorenz curves
, 
, the lower bound 
and the difference 
between 
and the lower bound
.
Remarks. The variable Y Lorenz dominates X, and the upper bounds in (8) and (9) tells us nothing about the reductions in the inequality. The upper bound contains the maximum value 
and one has to take it for granted that it is also inaccurate when M is finite. In addition, there may be situations in which
. The minimum value m can be zero, and in this case the upper bound is one and the obvious inequality 
is obtained.
3.2. The Ratio 
Is Monotonically Increasing
The analysis of this case follows similar traces to the earlier study and the results are analogous to our earlier results, but in this case 
may be discontinuous. Only the inequality signs have changed their directions. We introduce the values
and 
such that
and 
and consequently
.
Note, that in this case the points 
and 
are also chosen arbitrarily and that the equality signs cannot be ignored because we also include functions
which are not uniformly strictly increasing in the class of transformations. Hence, we have to include members for which equalities hold for almost the whole range and, in addition, the subintervals where strict inequalities hold can be arbitrarily short and can be located arbitrarily within the range. If one pursues general conditions, the inequalities (17) and (18) obtained below cannot be improved.
If 
is discontinuous, the discontinuities can only
be a countable number of finite positive jumps. Under such circumstances 
is still integrable.
We use the same notations as above and assume that
, that 
and consequently that
.
Now,
.
Consider
. The integration over the interval 
yields
(14)
and
.
Analogously, if we consider 
we obtain
(15)
and
.
Hence,
. (16)
When 
in (16), then
and one obtains
. (17)
Now, the initial variable X Lorenz dominates the transformed Y and the upper bound is the interesting case.
When 
in (16), then
one obtains
After a shift from p to q, we obtain
(18)
Now the upper bound is of interest. Formula (17) is applicable for small values and formula (16) for large values of q. In the following, we consider the difference between the upper bound and the Lorenz curve
, that is, for small values of q
. (19)
For large values of q, we consider the difference
. (20)
In general,
.
The ratio
is increasing and consequently,
.
Now we differentiate 
and note that
is increasing and obtain
.
Consequently 
is increasing from zero to a maximum for
.
Now we differentiate 
and obtain
.
Consequently 
is decreasing from a maximum to zero. The point denoted
, at which the shift from 
to 
is performed, satisfies 
.
Now,
that is,
.
This condition is identical with the condition, given above, in which
is decreasing.
Again, the condition
can be written
and we obtain the formula
that is, the Lorenz curves 
and 
have parallel tangents and the distance between the Lorenz curves is maximal.
We define the difference function as
, (21)
and the upper bound of 
is
(22)
In Figure 2, we sketch the Lorenz curves
, 
, the upper bound 
and the difference 
between the upper bound 
and
.
Now the lower bounds are of minor interest because the initial variable X Lorenz dominates Y. Note that 
is possible in some situations and the lower bound in (17) can be zero. Note that M can be great and even 
is possible in some situations and the lower bound in (18) can be even negative.
Example 1. The Pareto distribution. Consider income X with the Pareto distribution 
and
, where 
and
. Now,
and the Lorenz curve
.
From 
we obtain
. Let the transformation be 
so that the function 
is decreasing. We obtain
, the Lorenz curve
,
and
For
, the ratio
is decreasing, this case being sketched in Figure 1, and if 
the ratio
is increasing, this case being sketched in Figure 2.
4. Conclusion
Redistributions of income have commonly been defined as variable transformations of the initial income variable. The transformations are mainly considered as tax or transfer policies yielding post-tax or post-transfer incomes and therefore, the transformations are usually assumed to be positive, monotone-increasing and continuous. Recently, discontinuous transformations have also been discussed. The fundamental concern has been the Lorenz ordering between the initial and the transformed income. In this study we constructed limits for he transformed Lorenz curves. We considered the optimal cases that the transformed variable Lorenz dominates the initial one and the initial variable Lorenz dominates the transformed one. In applications, the first case is more important than the second, because it yields policies which reduce the inequality. The case (2) in Theorem 2 is not included in this study because the initial and the transformed Lorenz curves are identical. The limits obtained hold generally for all distributions and a broad class of transformations. If one pursues general conditions the inequalities obtained cannot be improved.
5. Acknowledgements
We are grateful to an anonymous referee for comments and suggestions on a previous version of the manuscript. This study was in part supported by a grant from the “Magnus Ehrnrooths Stiftelse” Foundation.