_{1}

^{*}

In earlier papers, classes of transfer policies have been studied and maximal and minimal Lorenz curves obtained. In addition, there are policies belonging to the class with given Gini indices or passing through given points in the plane. In general, a transformation describing a realistic transfer policy has to be continuous. In this paper the results are generalized and the class of transfer policies is modified so that the members may be discontinuous. If there is an optimal policy which Lorenz dominates all policies in the class, it must be continuous. The necessary and sufficient conditions under which a given differentiable Lorenz curve can be generated by a member of a given class of transfer policies are obtained. These conditions are equivalent to the condition that the transformed variable stochastically dominates the initial variable <i>X</i>. The theory presented is obviously applicable in connection with other income redistributive studies such that the discontinuity can be assumed. If the problem is reductions in taxation, then the reduction for a taxpayer can be considered as a new benefit. The class of transfer policies can also be used for comparisons between different transfer-raising situations.

Lorenz curves were initially introduced for comparison and analysis of income distributions in a country in different times or in different countries in the same era. Later it has been widely applied in different contexts. Especially, classes of transfer and tax policies have been studied and maximal and minimal Lorenz curves

We use similar notations as in my previous papers. Let the income be X with the distribution function

density function

We introduce the transformation

tax or post-transfer income. The mean and the Lorenz curve for the variable Y are

A general theorem concerning Lorenz dominance ( [

Theorem 1. Let X be an arbitrary non-negative, random variable with the distribution

1)

2)

3)

Classes of transfer policies. The class of transfer policies

H:

where

H^{*}:

where

If an optimal policy exists which Lorenz dominates all policies in H^{*}, then according to Theorem 1, it must be continuous because

Consequently, although class (2), also contains discontinuous policies in comparison with initial class H, the policy

being optimal among all continuous policies, is still optimal, having the Lorenz curve

The inferior Lorenz curve can be obtained from the sequence [

These policies give no benefits to the poorest sector of the population (_{S} Í H^{*} and that their Lorenz curves converge towards an

inferior Lorenz curve. If we define

and monotone increasing: _{S} Í H^{*} and the corresponding Lorenz curve is

where

Assume that

The Lorenz curve is inferior because we can prove [

Theorem 2. The Lorenz curve ^{*}.

Proof. Consider an arbitrary, continuous or discontinuous policy ^{*}. Using the condition

This inequality holds for all^{*} of transfer policies containing discontinuous policies satisfies the same properties as the initial class discussed in [

A policy with a given Lorenz curve. In Fellman [

Now we generalise the results, for discontinuous transformations as well. We have stressed above that

One has to assume that the Lorenz curve

with the exception of a countable number of cusps. The corresponding distribution

In general, when the Lorenz curve ^{*}; that is, we will characterise attainable Lorenz curves, although they are not universally differentiable.

The crucial part of this proof is to show that ^{*} of transfer policies containing discontinuous policies satisfies the same properties as the initial class dis-

cussed in [

has a cusp for

choose a δ > 0 so small that

Now, the transformation

We have studied the effects of transfer policies in this paper. In general, a transformation describing a realistic transfer policy has to be continuous. However, the theory presented is obviously applicable in connection with other income redistributive studies such that the discontinuity cannot be excluded. If the problem is reductions in taxation, then the tax reduction for a taxpayer can be considered as a new benefit [^{*} can consequently be used for comparisons between different tax-reducing policies. If changes of transfers are of interest, then the transfer policies can also be applied in transfer-raising situations. If transfers are increased, the effect of increases on a receiver can be considered through transfer policies belonging to H^{*}. In general, the changes may be mixtures of several different components and discontinuity cannot be excluded. The continuity assumption can be dropped and the class H^{*} of transfer policies containing discontinuous policies satisfies the same properties as the initial class discussed in ( [

Empirical applications of the optimal policies among a class of tax policies and the class of transfer policies considered here have been discussed in ( [

We have studied the effects of discontinuous transfer policies. The theory presented is applicable in connection with income redistributive studies such that the discontinuity cannot be excluded. A tax reduction for a taxpayer or a transfer increase on a receiver can be considered as new benefits. In general, such changes may be mixtures of different policy components and discontinuity cannot be excluded. However, one main result is still that continuity is a necessary condition if income inequality should remain or be reduced.

This work was supported in part by a grant from the Magnus Ehrnrooths Stiftelse foundation.

JohanFellman, (2016) Transfer Policies with Discontinuous Lorenz Curves. Journal of Mathematical Finance,06,28-33. doi: 10.4236/jmf.2016.61003