The name is absent



{T2 (xi , yi)}in=1 the tax liability distribution. The connection between the Lorenz dominance
criterion and the dual progressivity measures proposed in this paper is shown in the next
Theorem.

Theorem 1 Given a (finite) set of tax-payers with labor incomes x = (x1 ... xn) and
capital incomes y = (y
1 ... yn) and two arbitrary quasi-progressive dual tax schedules
T
1(, ) and T2(, ) that are labor and capital yield-equivalent, then

-→       -→

1. ψ1(x,y) < ψ2(x,y) for all (x,y) implies that V2(x,y) is LD by V1(x,y). However, the
reciprocal does not hold.

2.(x,y) (x,y) for all (x,y) implies that T1(x,y) is LD by T2(x,y). However, the
reciprocal does not hold.

Proof.

-→


-→


1. By definition, if ψ1 (x, y) < ψ2 (x, y) for all (x, y) we have that for all x


(9)


ψ1L(x) < ψ2L (x)

and for all y

(10)


ψ1K(y) < ψ2K(y).

From Proposition 1 in Jakobbson (1976) -see Proposition 4 in the Appendix-, we have
that (9) and (10) are equivalent to
VL(x) = {xi L2(xi)}n=1 LD by VL(x) = {xi
L1(xi)}n=1 and '2K(y) = {yi K2(yi)}‰1 LD by VK(y) = {yi K2(yi)}n=1. On the
other hand, from hypothesis
8 and Lemma 4 in the Appendix we have that V2(x,y) =
{xn L2(xn) + УпK2(yn)}n=1 is LD by V1(x,y) = {xi L1(xn) + y K1(yi)}n=1.

8It is important to take into account that both L(x) and K(y) as well as both Li (x) y Kj (y) for all
i = 1, 2, 3 and j = 1, 2, 3 are non-decreasing functions and therefore do not alter the relative order of the
income distributions. The same can be said about both
V L (x) and V K (y) as well as ViL(x) and VjK (y) for all
i = 1, 2, 3 and j = 1, 2, 3.

11



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