_____s__-'
for p = 1, ..., n. Analogously, the Lorenz curve of x2 + y2 = (x21 + y12 ≤ ... ≤ x2n + yn2 ) is
defined by
Lx2+y2 (p) =
∑i=1 (x? + y?)
Σ (*? + y?)
for p = 1, ..., n. Then, from hypothesis, for p = 0, n,
pp
Lx1+y1 (p) > Lx2+y2 (p) '⇔ ( ( (xi + yi ) > ( ( (xi + yi ) .
t=1 t=1
Finally, given 0 < p < n,
p ppppp
∑ (x1 + y1) = ∑4 + ∑y1 >∑χ? + ∑y? = Σ (x? + y?)
t=1 t=1 t=1 t=1 t=1 t=1
for X2 is LD by x1, y2 is LD by y1 and the hypothesis of the lemma. ■
Proof. [Proof of Lemma 1] We have
(16)
R(kZ ) = ( L(kxi) + K (kyi).
i=1
If we take the derivative of (16) with respect to k and evaluate it in k = 1 we obtain
R(Z )Z = ∑ L(Xi)Xi + K '(yi)yi*
i=1
On the other hand, from (7),
Z?A(Z) + R(Z)
Z?A(Z)
Z?A(Z)
n
∑ L(Xi)Xi + K'(yi)yi ⇔
i=1
nn
( (L'(Xi)Xi + K'(yi)yi) — ( (L(Xi) + K(yi)) ⇔
n
Σ
i=1
L(Xi)Xi - L(Xi) 2
™2 Xi
Xi
K'(yi)yi - K(yi)
y?
yi? .
Example 1
Consider the quasi-progressive dual tax schedule T (X, y) = L(X) + K(y)where L(X) and
K(y) are unidimensional tax schedules, on the labor income and on the capital income re-
spectively, that are applied to the following income distribution z = {(2, 1), (3, 4), (6, 5)} in
the way defined in the table below:
33