{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
T1(∙, ∙) 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 hypothesis8 and Lemma 4 in the Appendix we have that V2(x,y) =
{xn — L2(xn) + Уп — K2(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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