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), dκKyyl 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
(7)
β-→T (x, y) = (βL (x), βK (y)) =
L(x) d K (У)
x У
dx , dy
5For a further justification see Example 1 in the Appendix.