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To check that the reciprocal does not hold it is sufficient to take any two income

distributions r = (r1 ... rn) and s = (s1 ... sn) so that ∑n=1 ri = ∑nsi,
V1L(F) = F, V1K(y) = F, V2L(X) = F, V2K(X) = X and F LD by F. Then V2(X,X) = Vι(X,y)
-→       -→         -→       -→

holds but neither ψ1 (x, y) < ψ2(x, y) nor ψ1(x, y) > ψ2(x, y) do hold.

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

We are now in a position to prove that the nine linear reforms proposed can be compared
in a specific way: they form a lattice
9 . However, before doing so we need to introduce some
definitions.

Let Γ(γ,^>,T) be the set of nine linear dual tax reforms of a quasi-progressive dual tax
T(x,y) with aggregate labor tax liability γ and aggregate capital tax liability φ. We also
denote by
Γv (γ, φ,T, X, y) (resp. Γt (γ,φ,T, X, y)) the set of nine curves LVij = LVij (F, y)
(resp.
LTi,j = LTi,j(X,y)) where Tij(∙, ∙) Γ(γ,φ,T) and X and y are respectively the labor
and capital pre-tax income distributions.

For any (γ, φ) R+, given two reforms Ti,j(∙, ∙), Tk,l(, ∙) Γ(γ, <^), we define the following
operations
, : Γ(γ, φ, T) × Γ(γ, φ, T) → Γ(γ, φ, T):

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Ti,j Tk,l = Tiφk,jφl,  Ti,j Tk,l = Tiφk,jφl,

where the commutative operators φ, θ : {1,2,3} × {1,2,3} → {1,2,3} are defined by 1 φ
1 = 1
, 1φ2 = 2, 1φ3 = 3, 2φ2 = 2, 2®3 = 2, 3φ3 = 3 and 1θ1 = 1, 1θ2 = 1, 1θ3 = 1,
2 = 2
, 2 θ 3 = 3, 3 θ 3 = 3. We define as well v, v : Γv(γ, φ, T,X,y) × Γv(γ, φ, T,X,y)

9A given set S is a partially ordered set (poset) if there is a reflexive, antisymmetric and transitive binary
relation
that orders some pair of elements in S. We say a poset S is a lattice if any element in S has

supremum and infimum, i.e, there are operations , : S × S S so that for any x, y S there is x y S

(resp. x y S) so that x,yx y (resp. x,y x y ) and for all z S\ (x y) (resp. z S\ (x y)) so

that x, y z (resp. z x, y), x y ^ z (resp. z x y).

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