The name is absent

= T'(x₀) ∙ τ^xχ^ι₀) + 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) = x_g - 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 ≥ x_g

• x_g is the same for the three reforms as it only depends on g = TT.

Lemma 4 Let X¹ = (X₁¹ ≤ ... ≤ X¹_n), X² = (X²₁ ≤ ... ≤ X²_n), y¹ = (y₁¹ ≤ ... ≤ y_n¹) and

y² = (y₁² ≤ ... ≤ y_n²) 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 ≤ ... ≤ X¹_n + y_n¹) is defined by

∑j=1 (x1 + i⅛)

∑n=1 (Xi¹ +

¹⁶We rewrite and extend what Pfahler (1984) says in his paper.

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