The name is absent

Proof. According to Pfahler (1984), β₁(x) < β(x), β₂(x) > β(x) and β₃(x) = β(x), where
β(x), β₁(x), β₂ (x) and β₃(x) are the average rate progression of the unidimensional taxes
T (x), T1 (x), T2 (x) and T3(x), res-

pectively. Then, the proof is immediate from Lemma 1 and the definition of dual average
rate progression. ■

Again, notice that there is no restriction over the relative order on the capital and labor
income distributions. That is, Condition 2 is not needed. Moreover, Figure 1 also repre-
sents Proposition 3. Observe that, as in the unidimensional tax case, there is no trade-off
between the preferences of the Government concerning the aggregate total tax liability and
the redistribu-
tion of the post-tax income.

5 An empirical application: the case of the Spanish Income
dual Tax

This section describes an empirical application related to the dual income tax introduced
in Spain since 2007. By means of a static micro-simulation model SIMESP (Arcarons and
Calonge, 2008) and a very large data set of taxpayers drawn from 2004 Spanish Income Tax
Returns population we assess the redistributive and progressivity effects of linear tax reforms.
We consider two empirical applications.

First, yield-equivalent tax cuts applied on the dual income tax are compared. Specifically,
T1,1, T2,2 and T3,3 tax cuts defined in Section 2 are simulated to analyze their progressivity
and redistributive effects on income distribution. Elasticity coefficients of the simulated tax
functions are also estimated.

In our second empirical exercise, the current dual tax applied since 2007 is compared to
the income tax which had been applied before that date. As we shall comment later on,
2007 dual tax reform introduced significant changes in the fiscal law. According to Spanish

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