The name is absent

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

Next, we obtain the following result, that is proved in the appendix¹².

Lemma 1 Let x = (x₁,...,x_n) and y = (y₁,...,y_n) 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,

∑n=1 (^βL(^xβ ∙ ^x2 + ^βK⁽yi) ∙ У2}
Z ²

∑n=1 (^βL(^χi) ∙ ^x2 ^{+ β}K⁽yi) ∙ Vi}
∑n=ι T (^χi^,yi)

Hence, we are in a position to extend Pfahler’s result (1984).

Proposition 3 Let x = (x₁,...,x_n) and y = (y₁,...,y_n) 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,...,r_n) is obtained from a equiproportionate change of s' = (s₁,..., s_n) if ri = ks_i 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.

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