|
X |
У |
x + У |
T(x, У) |
t(x, y) |
L(x) |
l(x) |
K(y) |
k(y) |
|
2 |
1 |
3 |
1.05 |
0.35 |
0.80 |
0.40 |
0.25 |
0.25 |
|
3 |
4 |
7 |
3.25 |
0.46 |
1.25 |
0.42 |
2.00 |
0.50 |
|
6 |
5 |
11 |
5.60 |
0.51 |
2.60 |
0.43 |
3.00 |
0.60 |
Table A.1. Income distributions and quasi-progressive dual tax schedule.
Consider also the following two reforms: T1,3(x,y) = T(x,y) — 0.1 ∙ L(x) — 0.24 ∙ y and
T3,2(x,y) = T(x,y) — 0.232 ∙ x — 0.6585 ∙ (y — K(y)), which applied to the above tax and
income distributions result in the table below:
|
x |
У |
L1,3(x) |
K1,3(y) |
T1,3(x, У) |
L3,2(x) |
К3,2(у) |
T3,2(x, У) |
|
2 |
1 |
0.72 |
0.01 |
0.73 |
0.34 |
0.20 |
0.54 |
|
3 |
4 |
1.13 |
1.04 |
2.17 |
0.55 |
1.87 |
2.42 |
|
6 |
5 |
2.34 |
1.80 |
4.14 |
1.21 |
2.87 |
4.08 |
Table A.2. Dual tax cuts.
Observe that the two reforms proposed are not separately labor and capital yield-equivalent,
although they are globally yield-equivalent -the aggregate total tax liability is approximately
equal to 7.04-. That is, Condition 1 does not hold.
On the other hand, if we take into account the discrete definition of the residual progres-
sion it can be obtained17 that, given an income distribution z = (z1 ≤ ... ≤ zn), and two
reforms T1(z) y T2(z), if ψT1 (zi, zi+1) > ψT2 (zi, zi+1) for some zi, i ∈ {1, ..., n — 1}, then
(17)
V1(zi+1) > V2(zi+1)
Vι(zi) V2(zi) .
Hence, it can be checked that, by (17),
V1L3(3,4) < V3,2(3,4) V1K3(3,4) > V3K2(3,4) V1,3(3,4) > V3,2(3,4)
V1L3(2,1) < V3L2(2,1), V^D > V3,2(2,1) , V1,3(2,1) > V3,2(2,1) ,
viL3(6,5) < v3L2(6,5) V1K3(6,5) > v3K2(6,5) V1,3(6,5) < V3,2(6,5)
V1L3(3,4) < V3l2(3,4) , V K3 3,∣ > V3k2(3,4) , V1,3(3,4) < V3,2(3,4),
17 V1(zi+1)-V1(zi) . Zi 3 V2(zi+1)-V2(zi) . Zi ,. V1(zi+1)-V1(zi) 3 V2(zi+1)-V2(zi) ,. V1 (zi+1) _ -∣ 3
V1(zi) zi+1-zi V2(zi) zi+1-zi V1(zi) V2(zi) V1(zi)
V2(zi+1) _ -∣ ,. V1 (zi+1) 3 V2(zi+1)
V2(zi) V1(zi) V2(zi)
34
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