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, ...,xn) and y = (y1, ...,yn) respectively, where (xk,yk) 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, gL = 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) = ρiLL(x) + σiLx+ρjKK(y) +σjKy,
for i = 1, 2, 3 and j = 1, 2, 3. As an example, in the case T1,2 the parameters are ρ1L = 1 - aL,
σiL = 0, ρ1K = 1 + bK and σiK = -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
(3)
∆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
is
(4)
∆R = δ ∙
∆ ∆Li ∖
,---------------λ---------------4
(1 - PL) gL - σL
∖ /
+ (1 - δ) ∙
/ ∆Κj ʌ
(1 - PjK) gκ - σK
∖ /
3That is, -d (L(x)) ≥ 0, -d (K(y)) ≥ 0.
dx x dy y
4We use total to refer to both labor and capital.