xo
T (xo)
= T'(x0) ∙ τxχι0) + o(x1 — x0). Therefore, if a income distribution x is dense, i.e. any
xi is close enough to xi+1 then Jakobsson assumptions are correct. Finally notice that this
final assumption would not hold for the highest tail of the income distribution. However, in
such cases the tax schedule is almost proportional and both measures coincide.
Lemma 3 (Pfahler) Given a unidimensional tax schedule T(x) and given the three Pfahler-
based unidimensional tax cuts
X S t 1(g) = Xg ⇒ V2(x) ^ V3(x) ^ V1(x)
and
-1 dV2(X) dV3(X) dV1(X)
X S t (g) = Xg ⇒ —; ^ ;^ —;
dX dX dX
where Vi (X) is the post-tax income of a tax-payer with pre-tax income X when reform i is
applied to T(X) for all i = 1, 2, 3.16
Pfahler (1984) shows that the magnitude t 1(g) = xg - break-even pre-tax income - is
important for two reasons:
• Consider the proportional tax schedule proportional Tp(X) = gX. Then,
T(X)
T(x) ≥ Tp(x) = gx ⇔---- ≥ g ⇔ t(x) ≥ g ⇔ x ≥ xg
• xg is the same for the three reforms as it only depends on g = TT.
Lemma 4 Let X1 = (X11 ≤ ... ≤ X1n), X2 = (X21 ≤ ... ≤ X2n), y1 = (y11 ≤ ... ≤ yn1) and
y2 = (y12 ≤ ... ≤ yn2) be income distributions such that X2 is LD by X1 and y2 is LD by y1 . If
---—S -- _____s --
∑n=1 X1 = ∑n=1 X2 and ∑n=ι y11 = ∑n=1 У2 then x2 + y2 is LD by x1 + y1.
Proof. The Lorenz curve of X1 + y1 = (X11 + y11 ≤ ... ≤ X1n + yn1) is defined by
Lx1+y1 (p) =
∑j=1 (x1 + i⅛)
∑n=1 (Xi1 +
16We rewrite and extend what Pfahler (1984) says in his paper.
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