The name is absent

where δ = x/ (x + y) and the relative change ∆R on the aggregate total tax liability with
respect to the aggregate total pre-tax income does not depend either on i or on j . The above
expression generalizes the yield-equivalent condition introduced by Pfahler to the dual tax
case.

An example of the usefulness of the above expression is provided next. When carrying
out a tax reform, the policymaker usually decides which extra aggregate amount of money
the tax-payers can keep (or the Govern-

ment not get), in comparison to the current tax. For instance, the policyma-

ker wants to carry out a cut T1,2 that supposes that each tax-payer keeps on average 10%
more of the pre-tax income compared to the pre-reform tax, i.e. ∆R = 0.1. Notice that
gL and gK depend only on the current dual tax, whereas the proportion of the average pre-
tax labor income with respect to the pre-tax average income, δ, depend only on the income
distribution, i.e. neither gL, gK nor δ depend on the reforms. For instance, gL = 0.3, gK = 0.2
and δ = 0.8. In such case, (4) results on 24 ∙a_L + 16 ∙b_κ = 10. That is, there is only one degree
of freedom: either aL or bK. If for some reason bK has to be equal to 0.05, then aL = 0.383.

Observe that a particular and more restrictive case is obtained where ∆L_i and ∆K_j∙
are both equal across reforms. In such case, (4) reduces to the yield-equivalent condition
introduced by Pfahler, separately applied on the labor and on the capital parts of the dual
tax.

Regarding the measuring of the degree of progressivity of a dual tax, the first approach
is to consider a unidimensional measure. Given a dual tax schedule T (x, y), we could 1)
apply the definition of liability progression directly to T (x, y) seen as a unidimensional tax
on the total income x + y and 2) consider a weighted mean of αL(x) and αK(y), the liability
progressions associated to labor and capital tax schedules respectively.

However, the two above approaches are not without their flaws. The first option is not
valid as T(x, y) need not be a function on x + y. Indeed, it is not difficult to find two tax
schedules L(x) and K(y) and two pairs of incomes (x,y) y (x',y') such as x + y = X + y'

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