The name is absent

In what follows let x ≥ 0 be taxable labor income before tax and y ≥ 0 taxable capital
income before tax. Then, a dual tax schedule is defined by

(1)                          T(x,y) = L(x) + K(y) ≤x+y

where L(∙), K(∙) are unidimensional tax schedules. When both L(∙),K(∙) are progressive³, we
say that T (x, y) is quasi-progressive.

Consider a (finite) set of tax-payers with pre-tax labor incomes and pre-tax capital incomes
given by distributions x = (x₁, ...,x_n) and y = (y₁, ...,y_n) respectively, where (x_k,y_k) are the
incomes of the tax-payer k. Then, let x be the average pre-tax labor income, y the average
pre-tax capital income, L be the average labor liability and K the average capital liability.
We also introduce the following initial rates, g_L = L and g K = K.

xy

We represent the dual tax schedule obtained from a dual tax schedule T (x, y) applying a
reform of type i on the labor tax schedule and a reform of type j on the capital tax schedule
by

Li(x)                  Kj (y)

^,--------------^--------------4     ,---------------^---------------4

(2)                     Ti,j(x, y) = ρ_i^LL(x) + σ_i^Lx+ρ_j^KK(y) +σ_j^Ky,

for i = 1, 2, 3 and j = 1, 2, 3. As an example, in the case T1,2 the parameters are ρ₁^L = 1 - aL,
σ_i^L = 0, ρ₁^K = 1 + bK and σ_i^K = -bK. By means of simple algebra, it is proven that the
positive (resp. negative) change ∆Ri,j on the aggregate total ⁴ post-tax income when a tax
cut (resp. a tax increase) of type Ti,j (x, y) is applied to T (x, y) is given by

∆Ri,j = ((1 - PL) L - σLx + (1 - PK) K - σKy) ∙ n.

Dividing equation (3) by (x + y) ∙ n, the yield-equivalent condition for the above tax cuts

³That is, -d (L(^x)) ≥ 0, -d (K(^y)) ≥ 0.

dx x       dy y

⁴We use total to refer to both labor and capital.

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