Γv (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‰kjθi
(13)
LTi,j ∨t LTk,ι = LTiek,jθl, LTi,j ∧t LTk,ι = LTφk,jφι .
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 ≤ ... ≤ xn) and y = (y1 ≤ ... ≤ yn). Then
1. Γv(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 = {xn + yn}n≥1 is LD by the infimum of the lattice.
2. Γt(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 = {xn + yn}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