where g = T is the quotient between the average tax liability T and the average pre-tax
income z. Observe that g is the proportion of the tax burden in terms of pre-tax incomes,
whereas g/ (1 — g) = T/V is the proportion of the tax burden in terms of post-tax incomes.
Hence, if we assume the yield-equivalent condition holds, there is only one design parameter,
let us say a, whereas g is an ex-ante parameter and c and b are obtained multiplying a by g
and g/ (1 — g) respectively.
A tax function maps tax-payers’ pre-tax income to net post-tax income. In the case of the
Spanish PIT, the tax function is organized as follows. First of all, the pre-tax income of each
tax-payer is divided into labor and capital pre-tax income respectively. Then, three similar
sequential steps are applied separately to both labor and capital incomes. In the first one,
using tax-payer’s monetary and non-monetary information, allowances are deducted from
gross income to obtain the taxable labor and the taxable capital incomes respectively. In the
second step, using only tax-payer’s monetary information, two different tax rate schedules
are applied to each of the taxable incomes to obtain the gross tax liability. The difference
between the taxable income and the gross tax burden is called gross post-tax income. Finally,
the net post-tax income is obtained from the gross post-tax income after applying a series of
tax credits.
Because of the difficulty of introducing non-monetary information into an analytical model
in the theoretical part of this paper and for the sake of simplicity, we do not consider the
role of allowances and tax credits on the tax. This approach is the same that Pfahler
(1984) implicitly followed. When it comes to real tax functions, allowances can be roughly
approximated by linear transformations of the pre-tax income (see for instance Fries et al,
1982). If we assume that allowances and tax credits steps are essentially a homothety, then
taxable incomes can be seen as re-scaled pre-tax incomes, whereas net post-tax incomes can
be seen as re-scaled gross post-tax incomes and none of our theoretical results has to be
abandoned when we consider the real input and output of the tax function. Simulation
results in Section 5 are obtained according to our theoretical predictions.