The name is absent

Γ^v (γ, φ, T, x, y) and ∨T, ∧T : ΓT (γ, φ, T,x,y) × ΓT (γ, φ, T,x,y) → ΓT (γ, φ, T, x, y) by:

(¹2)               LVi,j ∨^v LVk,ι = LViφk,jφi,  LVi,j ∧V LVk,ι = L‰k_jθ_i

LTi,j ∨^t LTk,ι = LTi_ek,j_θl,  LTi,j ∧^t LTk,ι = LT_φk,j_φι .

Next we prove that linear dual tax cuts form a lattice.

Proposition 1 Let T(∙, ∙) be a quasi-progressive dual tax schedule, (γ,φ) ∈ R+ and a (finite)
set of tax-payers with income distributions x = (x₁ ≤ ... ≤ x_n) and y = (y₁ ≤ ... ≤ y_n). Then

1. Γ^v(γ, φ, T, x,y) is a lattice endowed with the partial order ^ defined by the Lorenz Dom-
inance and the supremum and infimum operators ∨^V, ∧^V defined in (12). Moreover,
the Lorenz curve of x + y = {x_n + y_n}_n≥₁ is LD by the infimum of the lattice.

2. Γ^t(γ,φ,T,x,y) is a lattice endowed with the partial order ^ defined by the Lorenz
Dominance and the supremum and infimum operators ∨^T, ∧^T defined in (13). Moreover,
the supremum of the lattice is LD by the Lorenz curve of x + y = {x_n + y_n}_n≥₁.

Proof.

1. Part 1 can be easily proved from Pfahler (1984), Lemma 4 and Theorem 1.

2. Part 2 can be proved analogously to Part 1.

Corollary 1 Given (γ, φ) ∈ R+, x and y labor and capital pre-tax income distributions
respectively and T(∙, ∙) a quasi-progressive dual tax schedule, Γ(γ,φ,T) is a lattice endowed
with the partial order ^ defined by the Lorenz Dominance of the post-tax income distributions
and the supremum and infimum operators ∨, ∧ defined in (11). Moreover, T₂,₂(∙, ∙) is the most
progressive reform and T₁,₁(∙, ∙) the least progressive.

The lattice structure is shown in Figure 1 below. In a lattice it is only possible to compare
elements bearing a vertical relationship.

13

<<   11   12   13   14   15   16   17   >>

More intriguing information