The name is absent

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

distributions r = (r₁ ≤ ... ≤ r_n) and s = (s₁ ≤ ... ≤ s_n) so that ∑n=₁ ri = ∑n=ι si,
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 lattice⁹ . 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 Γ^v (γ, φ,T, X, y) (resp. Γ^t (γ,φ,T, X, y)) the set of nine curves LVij = LVij (F, y)
(resp. LT_i,j = LT_i,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 T_i,_j∙(∙, ∙), T_k,_l(∙, ∙) ∈ Γ(γ, <^), we define the following
operations ∨, ∧ : Γ(γ, φ, T) × Γ(γ, φ, T) → Γ(γ, φ, T):

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, 2θ
2 = 2, 2 θ 3 = 3, 3 θ 3 = 3. We define as well ∨^v, ∧^v : Γ^v(γ, φ, T,X,y) × Γ^v(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,y √ x ∨ 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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