also be partially ordered by the responsiveness of the aggregate tax liability with respect
to income. Indeed, let Z = in=1 (xi + yi) be the pre-tax aggregate total income and
R = in=1 (L(xi) + K(yi)) the aggregate total tax liability. If Z changes only due to equipro-
portionate changes in x and y 10, R can be viewed as function of Z.11 Suppose that, at some
initial point, Z = Z0. Then, R = R(kZ) with R(Z) = R0 = R(Z0).
The average-rate responsiveness and the elasticity of the aggregate total tax liability with
respect to the aggregate total income are defined respectively as
(14)
R(Z)Z - R(Z)
(Φ)'
(15)
E(Z) =
R'(Z )Z
RZ.
Next, we obtain the following result, that is proved in the appendix12.
Lemma 1 Let x = (x1,...,xn) and y = (y1,...,yn) be the labor and capital income distri-
butions respectively of a (finite) set of tax-payers and T (x, y) = L(x) + K(y) is the current
quasi-progressive dual tax schedule. Then,
A(Z) =
∑n=1 (βL(xβ ∙ x2 + βK(yi) ∙ У2}
Z 2
E(Z) = 1 +
∑n=1 (βL(χi) ∙ x2 + βK(yi) ∙ Vi}
∑n=ι T (χi,yi)
Hence, we are in a position to extend Pfahler’s result (1984).
Proposition 3 Let x = (x1,...,xn) and y = (y1,...,yn) be the labor and capital income
distributions respectively of a (finite) set of tax-payers. Let also T(∙, ∙) be a quasi-progressive
dual tax schedule and (γ,^) ∈ R+. Then, Γ(γ, φ,T, x, y) is a lattice endowed with a partial
order ^ defined by the elasticity E(Z) of the aggregate total tax liability and the supremum
and infimum operators ∨, ∧ are defined in (11).
10 r = (r1,...,rn) is obtained from a equiproportionate change of s' = (s1,..., sn) if ri = ksi for all i = 1,..., n.
11 Hutton and Lambert (1979) make the same assumption.
12The proof, although similar to Hutton and Lambert (1979), is made for discrete distributions.
16
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