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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
T
3,2(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 obtained
17 that, given an income distribution z = (z1 ... zn), and two
reforms T
1(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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