The name is absent

and T(x,y) = T(x',y'). On the other hand, in the second option a more progressive labor
tax could be shadowed by a regressive -or less progressive- capital tax. The drawbacks of
both unidimensional approximations are the same as those arising when we try to extend the
complete order structure of R to R².

On the basis of what have been stated, the next step would be to establish bidimensional
definitions of the progressivity of dual taxes⁵ . Such definitions should be used to compare,
whenever possible, the degree of progressivity of any pair of dual tax schedules.

Next we define the labor post-tax income VL(x) = x-L(x) ≥ 0, the capital post-tax income
VK(y) = y-K(y) ≥ 0, the total post-tax income V (x, y) = x+y-T (x, y) = VL(x)+VK (y) ≥ 0,
the average labor type tL(x) = L(x), the average capital type tK(y) = K(^y) and the average
xy

total type t(x,y) = T(^χy). Let also > be the ordinary partial order on R²: (x₁,y₁) > (x₂,y₂)
x+y

if x1 > x2 and y1 > y2 . Then, the following three measures are proposed:

• Dual Liability Progression: the elasticities of labor liability and capital liability
with respect to the pre-tax labor income and the pre-tax capital income respectively
are defined as

⁽⁵⁾          → (x, y) = (^αL(^x),^αK ⁽y)) = ( .   LXX), d^κKy^yl ⅛ ).

• Dual Residual Progression: the elasticities of both the post-tax labor income and
the post-tax capital income with respect to the pre-tax labor income and the pre-tax
capital income respectively are defined as

(6)      Ψ⅛,y) = (Ψ⅛),ΨK (y))= ( dVLXx ɪ     y V ).

• Dual Average Rate Progression: the changes of both the labor average type and

the capital average type with respect to the pre-tax labor income and the pre-tax capital

income respectively are defined as

L(x) _d K ⁽У⁾
x У
dx ^, dy

⁵For a further justification see Example 1 in the Appendix.

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