The name is absent

2 Dual taxes, tax cuts and progressivity measures

The current structure of the Spanish Personal Income Tax is a dual one. There are two
different tax schedules applied separately onto two different income bases, henceforth called
labor and capital income bases. Regarding the analysis of tax reforms, there are two matters
of significance: 1) the effect on the income inequality and 2) the degree of progressivity of
the post-reform tax. Pfahler studied these issues in the context of three types of linear tax
reforms for unidimensional taxes. We shall substantiate Pfahler’s work in the context of dual
taxes. This extension could be useful in other scenarios, for instance, the one in which a
government intends to carry out a reform on (unidimensional) personal income tax and on
value-added tax simultaneously.

Pfahler assumed that a tax function consisted only of a tax schedule which, applied to
the pre-tax income, provided the post-tax income. Given any progressive tax schedule, he
proposed three different (unidimensional) tax reforms to be defined respectively either as
fraction a of the tax liability T (x), as a fraction b of the post-tax income V (x) = x - T (x)
or as a fraction c of the pre-tax income x. Each one of these reforms is neutral with respect
to local measure of tax progressivity: liability progression, residual progression and average
rate progression (see Musgrave and Thin, 1948). If t(x) = T (x)/x is the average tax rate of
the current tax schedule T (x), let T1(x), T2(x) and T3(x) be the three reforms:

T₁(x) = T (x) — aT (x) ^⇒    t₁(x) = (1 — a)t(x)

T2(x) = T (x) — bV (x) ^⇒     t₂(x) = (1 + b)t(x) — b

T3(x) = T (x) — cz ^⇒    t₃(x) = t(x) — c.

Generally, the derivative of the tax schedule is a sum of step functions defined by marginal
tax rates and by income thresholds. In this particular case, it is easy to prove that applying
the above linear cuts on the whole tax schedule is equivalent to transforming the marginal
tax rates according to the linear function that defines the tax cut². The three above trans-
formations of the original tax are neutral-revenue if and only if b = ag/ (1 — g) and c = ag,
²See Remark 1 in the Appendix

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