The name is absent

As t(x) = (∑m+1 T_i(x)} \x we have

d ∙ t(x) + h

m+1

^ d ∙ T_i(x) + h ∙ x
i=0

x

Ti(x)

m+1   z------------------^------------------∖

5 d ∙ max {0, min{x_i — x_i-1, x — x_i-1}} ∙t_i
i=0

/                     ”                      ∖

m+1

h ∙ У? max{0, min{x_i — x_i-1, x — x_i-1}}
i=0

x

^{m +} 1                                                                / >i ∖

max max {0, min{x_i — χ_i-1,χ — x_i-1}} ∙ (dt_i + h)

i=0

Lemma 2 (Pfahler) Let T(x) a unidimensional tax schedule and T_i(x),i = 1,2,3 the Pfahler-
based linear cuts with the same aggregate total tax liability. Then α(x) = α1(x) < α3(x) <
α2(x) and ψ(x) = ψ₂ (x) < ψ₃ (x) < ψ₁(x).

Proposition 4 (Jakobsson) Let T1(x) and T2(x)be two dual tax schedules. For any pre-tax
income distribution x = (x₁, ...,x_n) of a (finite) set of tax-payers,

1. T₁(x) = (T₁(x₁),..., T₁ (x_n)) is LD by T₂(x) = (T₂(x₁),..., T₂(x_n)) if and only if α₁(x) >
α2(x) for all x ≥ 0.

2. Vι(x) = (xι — Tι(xι), ...,x_n — Tι(x_n,))is LD by ½(x) = (xι — T2(x1), ...,xn — T2(xn)) if
and only if ψ₁(x) > ψ₂(x) for all x ≥ 0.

It is important to point out that Jakobsson’s proof (1976) assumes implicitly that a

more restrictive constraint than α2 (x0) < α2(x0), for all x0, holds. In fact, he assumes that

α₂(x₀, x₁) < α₂(x₀, x₁) for all x₀ < x₁, given an income distribution x = {x₀ ≤ x₁ ≤ x₂ ≤ ...}.

However, when x0 ≈ x1 both conditions almost coincide. Indeed, α(x0, x1) =

31

<<   29   30   31   32   33   34   35   >>

More intriguing information