Transformations and Lorenz Curves: Sufficient and Necessary Conditions

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.


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 posttransfer 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.

Properties of a Transformed Variable
Consider the income X with the cumulative distribution function ( ) X F x , the frequency distribution ( ) X f x , the mean X µ , and the Lorenz curve ( ) X L p .We assume that X is defined for 0 x ≥ and that ( ) X f x is conti- nuous.Furthermore, we consider the transformation ( )

Y g X =
, where ( ) g ⋅ 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 0 X ≥ be a random variable with an arbitrary continuous frequency distribution Y L p exists.The case 2) follows immediately from the fact that the Lorenz curve remains when linear transformation is performed.Consider the difference ( ) By definition, ( ) ( ) First, we assume that ( ) x is continuous and monotone decreasing for attains zero only once, being first positive and then negative.Hence, the difference ( ) 0 D p ≥ and the case 1) is proved.
For the case 3), ( ) x is monotone increasing for 0 x > .Also in this case ( ) ( ) attains zero only once, being first negative and then positive.Hence, ( ) 0 D p ≤ and the case 3) is proved.
If we consider tax policies, x is the pre-tax income and the function ( ) g x is the after-tax income and the ra- tio ( ) is the relative tax.If the ratio ( ) 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 ( ) g x 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 plays a central role.For a transformation ( ) g x for which the quotient ( ) g x x is not monotone decreasing for all 0 x > , an income distribution ( ) X f x can be chosen so that the result in the proof holds, i.e. dominance is obtained.We have only to choose ( ) X f x and ( ) g x so that ( ) D p is non-negative for all p.For example if the quotient ( ) g x x is both increasing and decreasing we choose the distribution ( ) X f x so that ( ) X f x is positive only in an interval where ( ) g x x is monotone decreasing.The sufficient condition of Hemming and Keen [3] is (with our notations) that for a given distribution ( ) once from above.The Hemming-Keen condition is equivalent with the condition that ( ) from above, which is easier to compare with ours.We observe 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 ( ) X f x .The condition of Hemming and Keen must be that ( ) g x must satisfy the condition "crossing once from above for all distributions ( ) X f x " [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 ( ) g x satisfy the initial conditions (positive, continuous, and monotone increasing) and let ( ) g x x be increasing within some interval ( 0 a x b < < < < ∞ ).Now, we prove that there exists an income distribution ( ) For the pair ( ) ( ) ( ) , f x g x , 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 ( ) g x x is monotone increasing even in a short interval, then there are income distributions such that the transformation ( ) g x cannot result in Lorenz dominance.Hence, if we demand that, for all distributions ( ) f x , the transformed variable ( ) Y g X = 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 ( ) g x x 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 ( ) g x x is monotone decreasing then ( ) g x satisfies the condition "crossing once from above for every dis- tribution ( ) f x ".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 ( ) g x is continuous and consider discontinuous functions.
What can be said about the case that ( ) g x is discontinuous?Assume that ( ) g x is still positive and monotone increasing and satisfies the condition that ( ) ( ) E g x exists for every stochastic variable X, whose distribution ( ) f x satisfies the general conditions given above, then the discontinuities can only consist of denumerable fi- nite positive jumps.Now we will prove that if there exists one such jump there exists at least one distribution Hence, we note that the quotient ( ) g x x cannot be monotone decreasing within a short interval . Choose 0 h > 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 ( ) For this variable X, the mean is where .
Assume that we choose h so small that 0 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 0 0 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 ( ) g x is also a necessary condition if we demand that the transformed variable should Lorenz dominate the initial variable irrespectively of the distribution f x (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 ( ) g x x , 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, ( ) X f x , the conditions in Theorem 1 1), 2), and 3) are both neces- The continuity of ( ) and is monotone decreasing.
In both cases, the ratio ( ) g x x 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 0 µ to 1 µ .For the increased means, we obtain Hence, the ratio Let ( ) g x be positive, continuous, and monotone increasing, let( ) dominate the initial variable X.Let a be a discontinuity point such that the two increase means should be identical, we obtain the relation