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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}n1 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}n1.

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



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