Properties of Lorenz Curves for Transformed Income Distributions

Redistributions of income can be considered as variable transformations of the initial income variable. The transformation is usually assumed to be positive, monotone-increasing and continuous, but discontinuous transformations have also been discussed recently. If the transformation is a tax or a transfer policy, the transformed variable is either the post-tax or the post-transfer income. A central problem has been the Lorenz dominance between the initial and the transformed income. This study considers analyses of other properties of the transformed Lorenz curves, especially its limits. We take in account mainly two cases (a) the transformed variable Lorenz dominates the initial one and (b) the initial Lorenz dominates the transformed one. For applications, the first case is more important than the second. The limits obtained are not accurate for a specific transformation, but do hold generally for all distributions and a broad class of transformations so that, if one pursues general conditions the inequalities obtained cannot be improved.


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][2][3] Consider a nonnegative random variable X with the distribution function   X F x , mean X  and Lorenz curve .Let be a continuous monotone increasing function and assume that exists.Then Lorenz curve x is constant and x 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.

Background
Consider income X, defined on the interval   , a b , where 0 a x b      , with the distribution function where A fundamental theorem concerning Lorenz dominance is [2,4].
Theorem 2. Let X be an arbitrary, non-negative, random variable with the distribution of Y exists and the following results hold: x is monotone-increasing.
In the following, we consider additional properties of the Lorenz curve .If x 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.

   
Y X L p L p 

  u x x
Is Monotonically

Decreasing
According to Theorem 1

F
. We introduce the values M and m such that x q F  .Assume that p q  and that p a x q x x b     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   , a b .If one pursues general conditions, the inequalities (8) and (9) obtained below cannot be improved.If we assume that   u x x is monotonically decreasing, then must be continuous, otherwise x should have positive jumps [1]. From and Analogously, it follows from x u x xu x  , and we obtain Consequently, and one obtains 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 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 and for large q we consider the difference The ratio x is decreasing and consequently . Now we differentiate and obtain D q for (say).0 Now we differentiate and obtain Consequently, shifts its sign from plus to minus at nt .Hemming d poi 0 q nz and Keen ( [6]) gave the condition for Lore ominance that level once from above.Our results above have shown that the crossing point is 0 q .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 maxim l for q q We define the difference function as and the lower bound of is Figure 1 shows the Lorenz curves the lower bound   L q and the differe  nce

 
D q be-, and and the lower bound   L q  .Remarks.The variable Y Lorenz dominates the upper b in (8) and (9) tells us ing a ounds noth bout the reductions in the inequality.The upper bound contains the maximum value M 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 M   .The minimum value m can be zero, and in this case the upper bound is one and the obvious inequality

Is Monotonically
The is case follows similar traces to the earli he results are analogous to our earlier Increasing analysis of th er study and t results, but in this case   u x may be discontinuous.Only the inequality signs have changed their directions.We introduce the values Note, that in this case the points are also chosen arbitrarily and that the equality signs cannot be p and q ignored because we also include functions   u x x which are not uniformly strictly ncreasing in the class of transformations.Hence, we have to include members for i 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   u x is discontinuous, the discontinuities can only , that p q  and consequently th and one obtains Now, the initial variable X Lorenz domina formed Y and the upper bound is the interesting case.
After a shift from p to q, we obtain Now the upper bound is of interest.Formula ( 17) is applicable for small values and formula (16) for larg values of q.In the following, we consider the differ be e ence tween the upper bound and the Lorenz curve For large values of q, we consider the difference The ratio x is increasing and consequently, Now we differentiate and note that x is increasing and obtain is increasing from zero to a maximum for Now we differentiate and obtain Consequently is decreasing from a maximum to zero.The po ted , at which the shift from to rfor d, satisfies This condition is identical with the condition, given above, in which and we obtain the formula and that is, the Lorenz curves   for In Figure 2, we sketch the Lorenz curves   X L q , the upper bound and the difference and the Lorenz curve

Conclusion
Redistributions of income have commonly been defined as variable transformations of the initial income variable.The transformations are mainly considered as tax 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 initial and the transform in .In t constructed limits for he transformed Lorenz curves.We considered the optimal cases that the variable Lorenz dominates the i ions on Lorenz ol.44, No. 4, 1976, pp.823-formed 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.

u
 is non-negative, continuous and monotone-increasing.Since the transformation can be considered as a tax either the post-tax or the posttransfer income.The mean and the Lorenz curve for variable Y are

Figure 1 .
Figure 1.A sketch of the Lorenz curves