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 ≤ ... ≤ x_n) and
capital incomes y = (y 1 ≤ ... ≤ y_n) 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.

ψ₁^L(x) < ψ₂^L (x)

and for all y

ψ₁^K(y) < ψ₂^K(y).

From Proposition 1 in Jakobbson (1976) -see Proposition 4 in the Appendix-, we have
that (9) and (10) are equivalent to VL(x) = {x_i — L₂(x_i)}n=₁ LD by VL(x) = {x_i —
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 V_i^L(x) and Vj^K (y) for all
i = 1, 2, 3 and j = 1, 2, 3.

11

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