Properties of Non-Differentiable Tax Policies

In this study, we reconsider the effect of variable transformations on the redistribution of income. We assume that the density function is continuous. If the theorems should hold for all income distributions, the conditions earlier given are both necessary and sufficient. Different conditions are compared. One main result is that continuity is a necessary condition if one demands that the income inequality should remain or be reduced. In our previous studies, of tax policies the assumption was that the transformations were differentiable and satisfy a derivative condition. In this study, we show that it is possible to reduce this assumption to a continuity condition.


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][4][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].

Basic Properties of Income Transformations
Theorem 1. [1,3,4].Let X be an arbitrary non-negative, random variable with the distribution   X F x , mean X  , and Lorenz curve .Let be nonnegative, continuous and monotone-increasing and let  exists and the following results hold 1) Consider income X with the distribution function   X F x , the mean X  , and the Lorenz curve .We assume that X is defined for and that where theorem concerning the effect of income transformations on Lorenz curves and Lorenz dominance was given by Fellman [3] and Jakobsson [1] [1,2].Analogously, if the other results in Theore old for every income distribution, the conditions in 2) and 3) are also necessary.
Hence, the continuity of m 1 h

 
u x is a necessary condition if we demand that the tran ormed 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 sf   u x kx  and   u x consequently is continuous.wever, in case 3) discontinuitie Ho s do ize the m not jeopard onotone increasing property of the quotient   u x x and the result in Theorem 1 3) holds even if the fu nction is discontinuous.Therefore, Fellman [2] dropped the explicit continuity assumption in this case as well.
Summing up, for arbitrary distributions,   X F x h nece , the co .Let X be an arbitrary non-negative, rando nditions 1), 2), and 3) in Theorem 1 are bot ssary and sufficient for the dominance relations and an additional assumption about the continuity of the transformation   u x can be dropped.We obtain the more general theorem ].

 
X F x ".Hence, both conditions, the Hemming-Keen on 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.conditi

Properties of Tax Policies
policies, the trans-If we apply the results above to tax formed variable

 
Y u X  is the income after the taxation (cf., e.g., [6 n order to obtain a realistic class of policies, Fellman [6,7] assumed continuous transformations and included the additional restriction ,7,9,10]).I This condition indicates that the tax paid is an i function of the income x.In order to generalize the results and allow that the function   u x is not uniformly differentiable everywhere, we replace the derivative restriction in this study by the more general condition ncreasing   u x x    .According to this restriction, the tax is an unction of the income x.In fact, the tax is increasing f

 
x u x  and the increment in the tax is ositive increment and a p x  yields the restriction nd does not imply uniform ility.Both restrictions imply that the transformation   u x is continuous.We intend to show that the assum  is sufficient for the whole theory.Now, the class of tax polic ption The condition can be written as u (5) and It is apparent that while function ( 5) is not differentiable at point and ( 6) at point , the condition lds for all x.The L 0 a ho (5) is where and the Lorenz curve corresponding to where ( [6]).Po al, that is, it Lorenz dominates all the policies in class U, and policy ( 6) is Lorenz nated by all policies in U [6,7].
e following, how the main result in [7]  Starting from   L p , the connection between   L p and the post-tax distribution where On the other ha and , we where f y  for all p.This condition can also be written as 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.

Conclusions ed in earlier
In this study we reconsidered the effect of variable transformations on the redistribution of income.The aim was to generalise the conditions consider papers.We were pa Social Choice and Welfare, 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   doi:10.1007/s00355-008-0362-4 [3] J. Fellman, "The Effect of Transformations on Lorenz Curves," Econometrica, Vol.44, N 824.
  y is a cessary 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 restrictio we obtain in 1) a sufficient condition that the transformation   u x generates a new income distribution which Lorenz dominates the initial one.The analysis should be based on the difference be piecewise differentiable like transformations (5) and(6).We consider post-tax income distributions with the mean to a member of class U is that the initial distribution ) Hemming and M. J. Keen, "Single Crossing Conditions in Comparisons of Tax Progressivity," Journal of Public Economics, Vol. 20, No.