The name is absent

It is important to point out that the definition of Liability Progression for a unidimensional
tax schedule T (z) was introduced by Musgrave and Thin (1948) as the limit of the elasticity
of the tax liability with respect to pre-tax incomes x0 < x1

when xi approximates xo . We introduce this definition because discrete definitions will be
needed for some proofs (see Proposition 4 in the App endix).

Finally, the larger αT (x) is, the more progressive T(x) is in x according to the liability
progression criterion. Therefore, we say that T(x,y) is more progressive than T(x,y) accord-
ing to the dual liability progression criterion if -→(x,y) ≥ -→(x,y) for all (x,y). Analogous
comments can be made about the other two dual measures.

3 Linear reforms and their effect on the inequality of the post-
tax income distribution

This section is devoted to the study of the relative effect that each of the reforms proposed has
on the inequality of the post-tax income distribution. This effect can be studied 1) globally,
i.e. understanding what happens to the whole post-tax income distribution for the different
cuts proposed, according to some global criteria and 2) locally, i.e. comparing relative net
gain (or loss) across reforms for each tax-payer.

3.1 Global effects

First we focus on the global effects of linear dual tax reforms. We specifically prove that,
provided some constraints to be introduced below hold, Pfahler-based cuts T_i,_j∙ can be com-
pared according to the (restrictive) criterion of Lorenz domination⁶ . A symmetrical study

⁶Given an income distribution x = (x₁ ≤ ... ≤ x_n), the Lorenz curve L_x is defined by the following ordered
pairs (2,Lχ(p)), for p = 1, ...,n, where

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