Proposition 2 Let x = (x1,...,xn) and y = (y1,...,yn) be the labor and capital income
distributions of a (finite) set of tax-payers and let (xg, yg) = (l-1(gL), k-1(gK)), where
T (x, y) = L(x) + K(y) is the current quasi-progressive dual tax schedule. Then,
1. If (x, y) ≤ (xg, yg), the chosen tax reform by a tax-payer with incomes (x, y) is T2,2 .
2. If (x, y) ≥ (xg, yg), the chosen tax reform by a tax-payer with incomes (x, y) is T1,1 .
Notice that there is no restriction on the relative order on the capital and labor income
distributions. That is, unlike in Theorem 1, we only require Condition 1 and not Condition 2.
Also observe that a (finite) set of tax-payers can be divided into three subsets. Indeed, given
a tax-payer with a pair of incomes (x, y) we will say he or she is rich if (x, y) ≥ (xg, yg), poor if
(x, y) ≤ (xg , yg ) and middle-class otherwise. Then, if poor tax-payers account for more than
half the population, the T2,2 reform would be chosen in an election to decide which reform
out of the nine proposed should be carried out, provided that all tax-payers voted rationally.
In such a case, these interests would be aligned with those of a Government concerned about
the redistribution of post-tax incomes.
4 Revenue elasticity
Progressivity and redistribution of taxes are relevant equity issues to be considered when
adopting tax reforms. Related to this analysis, there is also an empirical issue which is
important not only from an equity perspective but from a government one when preparing
a Budget, which is the elasticity of tax revenue with respect to income. With progressive
taxation, revenue is elastic with respect to a proportional growth of all incomes, and the
amount of this elasticity is also crucial for macroeconomic projections.
Hutton and Lambert (1979) show that increasing the average rate progres-
sion β(x) of a tax at every point of the income distribution raises the elasticity of the
revenue function. Considering this result, Pfahler-based Ti,j∙ tax cuts for dual taxes can
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