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]. In this study, we reconsider the effect of variable transformations on the redistribution of income. Different versions of the conditions are compared [1,3-5]. One main result is that continuity is a necessary condition if one assumes that income inequality should remain or be reduced. In addition, in our earlier studies of classes of tax policies, the results were based on the assumption that the transformations were differentiable and satisfies a derivative condition [6,7].
2. Basic Properties of Income Transformations
Consider income X with the distribution function
, the mean
, and the Lorenz curve
. We assume that X is defined for 
and that 
is continuous.
A fundamental theorem concerning the effect of income transformations on Lorenz curves and Lorenz dominance was given by Fellman [3] and Jakobsson [1] and later by Kakwani [4]. We have.
Theorem 1. [1,3,4]. Let X be an arbitrary non-negative, random variable with the distribution
, mean
, and Lorenz curve
. Let 
be nonnegative, continuous and monotone-increasing and let 
exist. Then the Lorenz curve 
of 
exists and the following results hold 1) 
if 
is monotone decreasing;
2) 
if 
is constant and;
3) 
if 
is monotone increasing.
Following Fellman [2], we obtain in 1) a sufficient condition that the transformation 
generates a new income distribution which Lorenz dominates the initial one. The analysis should be based on the difference
(1)
where 
; that is, 
[2]. Furthermore,
. In order to obtain Lorenz dominance, the difference 
in Equation (1) must start from zero, attain positive values and then decrease back to zero. Consequently, the difference
(2)
must start from positive (non-negative) values and then change its sign and become negative. If 
is exceptionally increasing within the interval
, then a variable X with a distribution 
defined in the interval 
exists such that 3) holds and 
. Consequently, the condition that 
is decreasing is necessary if the rule holds for all income distributions 
[1,2]. Analogously, if the other results in Theorem 1 hold for every income distribution, the conditions in 2) and 3) are also necessary.
Hence, the continuity of 
is a necessary condition if we demand that the transformed variable should Lorenz dominate the initial variable for every distribution. From this it follows that if the condition in Theorem 1 1) has to be necessary, it implies continuity and hence an explicit statement of continuity can be dropped. Considering the condition in 2), we observe that 
and 
consequently is continuous.
However, in case 3) discontinuities do not jeopardize the monotone increasing property of the quotient 
and the result in Theorem 1 3) holds even if the function is discontinuous. Therefore, Fellman [2] dropped the explicit continuity assumption in this case as well.
Summing up, for arbitrary distributions, 
, the conditions 1), 2), and 3) in Theorem 1 are both necessary and sufficient for the dominance relations and an additional assumption about the continuity of the transformation 
can be dropped. We obtain the more general theorem [2].
Theorem 2. Let X be an arbitrary non-negative, random variable with the distribution 
, mean 
and Lorenz curve 
, let 
be a non-negative, monotone increasing function and let 
and 
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. It follows from the discussion above that the transformation 
can be discontinuous only in case 3).
Hemming and Keen [5] gave an alternative condition for Lorenz dominance. Their condition, with our notations, is that for a given distribution
, 
crosses the 
level once from above. Consequently, 
in (2) starts from positive valueschanges its sign once and ends up with negative values. Hence, their condition is equivalent to our condition.
Furthermore, if we assume that 
is monotonedecreasing (non-increasing), then 
satisfies the condition “crossing once from above for every distribution
”. Hence, both conditions, the HemmingKeen condition and ours, are also equivalent as necessary conditions. Recently, Fellman [8] obtained limits for the transformed Lorenz curves. These limits are related to the results given by Hemming and Keen.
3. Properties of Tax Policies
If we apply the results above to tax policies, the transformed variable 
is the income after the taxation (cf., e.g., [6,7,9,10]). In order to obtain a realistic class of policies, Fellman [6,7] assumed continuous transformations and included the additional restriction
. This condition indicates that the tax paid is an increasing function of the income x. In order to generalize the results and allow that the function 
is not uniformly differentiable everywhere, we replace the derivative restriction in this study by the more general condition
. According to this restriction, the tax is an increasing function of the income x. In fact, the tax is 
and the increment in the tax is 
and a positive increment 
yields the restriction
. If 
holds, it follows that
(3)
but the condition 
is more general and does not imply uniform differentiability. Both restrictions imply that the transformation 
is continuous. We intend to show that the assumption 
is sufficient for the whole theory.
Now, the class of tax policies is U:
(4)
We consider the extreme policies
(5)
and
(6)
It is apparent that while function (5) is not differentiable at point 
and (6) at point
, the condition 
holds for all x. The Lorenz curve corresponding to (5) is
(7)
where 
and the Lorenz curve corresponding to (6) is
(8)
where 
([6]).
Policy (5) is optimal, that is, it Lorenz dominates all the policies in class U, and policy (6) is Lorenz dominated by all policies in U [6,7].
In the following, we show how the main result in [7] can be obtained when we replace the restriction 
by the more general restriction
. The function 
may be piecewise differentiable like transformations (5) and (6). We consider post-tax income distributions with the mean
. Without the restriction
, the necessary and sufficient condition that a given Lorenz curve 
of the distribution 
corresponds to a member of class U is that the initial distribution 
stochastically dominates
. The inclusion of the restriction 
results in the stochastic dominance being only necessary; that is, the transformed distribution 
must satisfy additional conditions.
Assume a given differentiable Lorenz curve 
with a continuous derivative. These conditions can be assumed because the corresponding transformation 
has to continuously satisfy the condition
. Starting from
, the connection between 
and the post-tax distribution 
with the mean 
is that
, where 
is the inverse function of
. The corresponding transformation is 
The condition 
can be written as
(9)
where 
and 
On the other hand, we can write
(10)
and define 
and 
.
If we assume that 
is piecewise differentiable, then 
and 
are piecewise differentiable.
If we assume that the density functions 
and 
exist, we obtain
(11)
where 
and
(12)
where 
and
.
Consequently, 
and 
From 
and from the condition 
it follows that
(13)
and, consequently,
. If we let
, then 
and 
and we obtain 
for all p. This condition can also be written as 
or 
(14)
when
. We can reverse the steps from (14) to (9) and all the results in Fellman [7] still hold, but the proof had to be slightly modified.
4. Conclusions
In this study we reconsidered the effect of variable transformations on the redistribution of income. The aim was to generalise the conditions considered in earlier papers. We were particularly interested in whether we could drop the assumption of differentiability of the transformations when tax policies are considered. The main result is that with a slight modification of the proof the additional condition 
is obtained.
We have also seen that if we demand sufficient and necessary conditions, theorems obtained earlier still hold and the continuity assumption can be included in the general conditions. The main result is that continuity is a necessary condition if one maintains that the income inequality should remain or be reduced.
The study of the class of tax policies indicated that the differentiability assumed earlier, can be dropped but, if one wants to retain the realism of the class, the transformations should still be continuous and satisfy the restriction
. The previous results in Fellman [6,7] still hold.
Empirical applications of the optimal policies of a class of transfer policies and the class of tax policies considered here have been discussed by Fellman et al. [9,10], where “optimal yardsticks” to gauge the effectiveness of given real tax and transfer policies in reducing inequality were developed.