Theoretical Economics Letters
Vol.06 No.03(2016), Article ID:67115,8 pages
10.4236/tel.2016.63050
Transformations and Lorenz Curves: Sufficient and Necessary Conditions
Johan Fellman
Hanken School of Economics, Helsinki, Finland
Copyright © 2016 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Received 10 May 2016; accepted 3 June 2016; published 6 June 2016
ABSTRACT
In this study, we reconsider the effect of variable transformations on income inequality. Under the assumption that the theorems should hold for all income distributions, earlier given sufficient conditions are also necessary. Different versions of the conditions are compared. Furthermore, one can prove that the assumption of continuity of the transformations can be implicitly included in the necessary and sufficient conditions, and hence, it can be dropped from the assumptions. The effects of two transformations on income inequality are compared.
Keywords:
Discontinuity, Income Distribution, Income Inequality, Lorenz Dominance, Tax Policy, Transfer Policy
1. Introduction
It is a well-known fact that variable transformations are valuable in considering the effect of tax and transfer policies on income inequality. The transformation is usually assumed to be positive, monotone increasing and continuous. Under the assumption that the theorems should hold for all income distributions, conditions given earlier are both necessary and sufficient [1] [2] . Hemming and Keen [3] have given an alternative version of the conditions. Recently, Fellman [2] [4] also discussed discontinuous transformations. One general result is that continuity is a necessary condition if the transformation should preserve or reduce income inequality. 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. In this study, we reconsider the effect of variable transformations on the redistribution of income. Two transformations are studied and their effects on income inequality are compared.
2. Properties of a Transformed Variable
Consider the income X with the cumulative distribution function, the frequency distribution
, the mean
, and the Lorenz curve
. We assume that X is defined for
and that
is continuous. Furthermore, we consider the transformation
, where
is non-negative and monotone increasing. A fundamental theorem concerning the effect of income transformations on Lorenz curves was first given by Fellman [5] , Jakobsson [1] , and Kakwani [6] and later by Fellman [7] [8] . Hemming and Keen [3] gave a new condition for the Lorenz dominance. We have
Theorem 1. Let be a random variable with an arbitrary continuous frequency distribution
, mean
, and the Lorenz curve
. Let
be positive, continuous, and monotone increasing, let
, and let
exist. Then, the Lorenz curve
of Y exists and the following results hold:
1) if
is monotone decreasing
2) if
is constant
3) if
is monotone increasing.
Proof: From the fact that
,
it follows that exists.
The case 2) follows immediately from the fact that the Lorenz curve remains when linear transformation is performed. Consider the difference
(1)
By definition,. First, we assume that
is continuous and monotone decreasing for
. Then
attains zero only once, being first positive and then negative. Hence, the difference
and the case 1) is proved.
For the case 3), is monotone increasing for
. Also in this case
. The difference
attains zero only once, being first negative and then positive. Hence,
and the case 3) is proved.
If we consider tax policies, x is the pre-tax income and the function is the after-tax income and the ratio
is the relative tax. If the ratio
is monotonically decreasing,
is monotone increasing and the tax policy is progressive. Hence, Theorem 1 1) states the well-known result that progressive taxes reduce income inequality.
In addition, if we consider income increases and that is the increased income and that 1) holds then the income increase reduces the income inequality.
According to Theorem 1, we obtain in 1) a sufficient condition that the transformation g(x) results in a new income distribution, which Lorenz dominates the initial one. What can be said about necessary conditions? If we analyze the proof of Theorem 1, we observe that the difference
(2)
plays a central role. For a transformation for which the quotient
is not monotone decreasing for all
, an income distribution
can be chosen so that the result in the proof holds, i.e. dominance is obtained. We have only to choose
and
so that
is non-negative for all p. For example if the quotient
is both increasing and decreasing we choose the distribution
so that
is positive only in an interval where
is monotone decreasing.
The sufficient condition of Hemming and Keen [3] is (with our notations) that for a given distribution the function
crosses the line
once from above. The Hemming-Keen condition is equivalent with the condition that
crosses the level
from above, which is easier to compare with ours. We ob-
serve that if their condition holds then the integrand in (2) starts from positive values, changes its sign once, and ends up with negative values.
If we demand necessary conditions, they must be formulated as a condition that holds for all income distributions. The condition of Hemming and Keen must be that
must satisfy the condition “crossing once from above for all distributions
” [3] . We start with the condition in Theorem 1 1) and prove that it is also necessary. This can be proved in the following way ( [1] [9] , p. 189). Let a transformation
satisfy the initial conditions (positive, continuous, and monotone increasing) and let
be increasing within some interval (
). Now, we prove that there exists an income distribution
such that the transformed variable
does not Lorenz dominate the initial variable X.
Consider an income distribution
(3)
For the pair, Theorem 1 3) holds and the transformation results in a new variable Y, which is Lorenz dominated by the initial variable X. This result indicates that if
is monotone increasing even in a short interval, then there are income distributions such that the transformation
cannot result in Lorenz dominance. Hence, if we demand that, for all distributions
, the transformed variable
shall Lorenz dominate X then the condition in Theorem 1 1) is necessary. In the example considered above, the Hemming-Keen condition is not satisfied. Consequently, if
is not monotone decreasing then there are distributions for which the Hemming-Keen condition does not hold. On the other hand, if we assume that
is monotone decreasing then
satisfies the condition “crossing once from above for every distribution
”. Hence, our condition and the Hemming-Keen condition are equivalent as necessary conditions. In a similar way, we can prove that if the other results in Theorem 1 should hold for every income distribution the conditions in 2) and in 3) are also necessary.
Now, we follow [8] and drop the assumption that is continuous and consider discontinuous functions. What can be said about the case that
is discontinuous? Assume that
is still positive and monotone increasing and satisfies the condition that
exists for every stochastic variable X, whose distribution
satisfies the general conditions given above, then the discontinuities can only consist of denumerable finite positive jumps. Now we will prove that if there exists one such jump there exists at least one distribution
such that the transformation
does not Lorenz dominate the initial variable X.
Let a be a discontinuity point such that and
where
. If
should be monotone increasing, we have to assume that
. Let
and
, then
(4)
Hence, we note that the quotient cannot be monotone decreasing within a short interval
. Choose
so small that the point a is the only discontinuity point within the interval
(later we may reduce h even more).
Consider the uniform distribution
(5)
For this variable X, the mean is. For the transformed variable Y = g(X), the mean is
(6)
where and
.
If then
,
,
,
, and consequently,
.
Assume that we choose h so small that. Consider now
(7)
To obtain Lorenz dominance, the integrand must start from positive (non-negative) values and then change its sign once and become negative in such a manner that the difference D (p) starts from zero and then attains positive values, whereupon it decreases back to zero.
The sign of the integrand depends on the factor, which starts from the value
(8)
If we assume that h satisfies the earlier conditions and furthermore, the integrand in (7) starts
from negative values, and consequently, the whole integrand is negative and the difference starts from negative values. For the corresponding income distribution, the transformed variable Y does not Lorenz dominate the initial variable X. Hence, the continuity of is also a necessary condition if we demand that the transformed variable should Lorenz dominate the initial variable irrespectively of the distribution fx(x). However, we noted already that the continuity is a necessary condition for the monotone decreasing assumption in 1). From this, it follows that the condition in Theorem 1 1) implies continuity, and hence, the explicit assumption of continuity can be dropped. In a similar way, we can obtain the same result if we study the condition in 2). However, in the case 3) the discontinuity does not jeopardize the monotone increasing property of the quotient
, and the result in Theorem 1 3) holds even if the function is discontinuous. Therefore, also in this case we can drop the explicit continuity assumption.
Summing up, for arbitrary distributions, , the conditions in Theorem 1 1), 2), and 3) are both necessary and sufficient for the dominance relations and the additional assumption about the continuity of the transformation
can be dropped. We obtain a generalized theorem ( [1] [2] [6] - [8] ).
Theorem 2. Let be a random variable with an arbitrary continuous distribution
, mean
, and the Lorenz curve
. Let
be a positive, monotone increasing function, let
, and let
exist. Then 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
3) if and only if
is monotone increasing.
Remark. From the discussion above, it follows that only in the case 3) can the transformation be discontinuous.
If we apply these results on income raise policies and on tax policies the transformed variable is the income after the income raise or after the taxation (cf. e.g. [5] [10] - [12] ). We obtain that only income raise policies that (with respect to the initial income) give decreasing relative salary increments result in a decreased income inequality for all initial income distributions. An analogous result holds for progressive tax policies.
3. Comparison of Two Transformed Variables
Theorem 1 can be used when the effect of a given tax or salary policy is studied. If several policies are to be compared, the following theorems, which are generalizations of Theorem 1 and Theorem 2, will prove valuable. The generalization of Theorem 1 was first presented by Fellman [10] and proved in [13] . Wilfling [14] later regenerated this theorem. The Hemming-Keen theorem was primarily given in this context. Consider two policies (transformations) and
. Following Fellman ( [10] [13] ), we have
Theorem 3. Let X be a continuous and non-negative random variable with an arbitrary distribution, mean
, and the Lorenz curve
. Let
and
be continuous, non-negative and monotone increasing, let
and
, and let
and
exist. If the Lorenz curves of Y and Z are
and
, respectively, then the following results hold:
1) if
is monotone decreasing
2) if
is constant
3) if
is monotone increasing.
Proof: If (constant), then
and the case 2) follows immediately from Theorem 1. If we assume that
is monotone decreasing for x > 0, then
attains the value zero only once, be-
ing first positive and then negative. Hence, and the case 1) is proved. The case 3) can be proved if we let
and
exchange their roles and the proof of the case 1) is performed.
Now we study two different salary increase policies.
Example ( [13] )
1. The salary increases are of the same size regardless of the previous salary
In this case, and the ratio
is strictly decreasing.
2. The salary increases are of the same size up to a certain salary level, thereafter they are strictly proportional. Now the transformation function is
(9)
The continuity of for
demands that
and
. The ratio
is
(10)
and is monotone decreasing.
In both cases, the ratio is monotone decreasing and the policies reduce the income inequality. Now we compare the two policies under the assumption that both give the same increase of the initial mean from
to
. For the increased means, we obtain
and
(11)
If the two increase means should be identical, we obtain the relation
and
.
If we apply Theorem 3 on our two policies, we obtain
(12)
Hence, the ratio is monotone decreasing for all x and the transformation
reduces the inequality more than the transformation
.
If we assume that the conditions in Theorem 3 should hold for every income distribution, we can drop the condition that and
are continuous and we can prove in a similar way as above that the conditions are also necessary. We obtain
Theorem 4. Let X be a continuous and non-negative random variable with an arbitrary distribution, mean
, and the Lorenz curve
. Let
and
be non-negative and monotone increasing, let
and
, and let
and
exist. If the Lorenz curves of Y and Z are
and
, respectively, then the following results hold:
1) if and only if
is monotone decreasing
2) if and only if
is constant
3) if and only if
is monotone increasing.
In a similar way as above, we obtain that the discontinuities in and
can only be finite positive jumps. If the condition in Theorem 4 1) holds, then
can be discontinuous, but
can be discontinuous only at such points where
is discontinuous, and additionally, the corresponding jumps must
be such that is monotone decreasing. In 2)
and
can be discontinuous only at the same points, and additionally, the corresponding jumps must be such that
remains constant. In 3)
can be discontinuous, but
can be discontinuous only at such points where
is discontinuous, and additionally, the corresponding jumps must be such that
is monotone increasing.
Remark. Theorems 3 and 4 are generalized versions of Theorems 1 and 2, respectively. This is clear if we introduce the simplified condition.
4. Conclusions
Redistributions of income have commonly been defined as 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 been discussed. Particularly, we were interested in determining if one can drop the assumptions of continuity of the transformations.
In this study, we considered the effect of variable transformations on the redistribution of income. The aim was to compare and generalize the conditions considered in earlier papers. The fundamental concern has been the Lorenz ordering between the initial and transformed income. We have obtained that, if we demand sufficient and necessary conditions, theorems earlier obtained still hold and the continuity assumption can be implicitly included in the general conditions. Especially, we have considered the optimal cases that the transformed variable Lorenz dominates the initial one. In applications, this case is important because it yields policies which reduce income inequality. The main result is that continuity is a necessary condition if income inequality should remain or be reduced.
Empirical applications of the optimal policies of classes of transfer policies and of tax policies considered here have been discussed in Fellman et al. [12] [15] . There we developed “optimal yardsticks” to gauge the effectiveness of given real tax and transfer policies in reducing inequality.
Acknowledgements
This study was supported by grants from the Finnish Society of Sciences and Letters and Magnus Ehrnrooth Foundation.
Cite this paper
Johan Fellman, (2016) Transformations and Lorenz Curves: Sufficient and Necessary Conditions. Theoretical Economics Letters,06,442-449. doi: 10.4236/tel.2016.63050
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