IMPACT OF TAXES AND TRANSFERS

Enami, Lustig, Aranda, No. 25, November 2016
n
⟺ g 1 > П K T1 + ∑ i=1 g iП Ti + ∑j=1
b j ρ Bj
1 − ∑n
i=1 g i + ∑m
j=1 b j
K
m
K
(53) ⟺ g 1 > П K T1 + П X−∑
n Ti
RS
m
i=1 +∑j=1
B j
Formula 52 for the simple case of one tax and one transfer is
= gП T K +bρ B
K
1−g+b
M T = G X+B − G X−T+B =⏟
− ρ B RS j
.
Assuming no reranking
.
RS
П X−T+B
The derivatives with respect to progressivity and size are shown as follows:
and
∂M T
K
∂П = g
T
1 − g + b
∂M T
∂g = [П T K (1 − g + b)] + [gП K T + bρ K B ]
(1 − g + b) 2
= ПT K + П RS
X−T+B
1 − g + b
RS
− ρ Bj
Equation 53a shows the condition under which the derivative of the marginal contribution of a
tax with respect to its progressivity would be greater than the derivative with respect to its size:
⟺
∂M T
K
∂П > ∂M T
T
∂g
g
1 − g + b > [П T K (1 − g + b)] + [gП K T + bρ K B ]
(1 − g + b) 2
(53a)
⟺ g > П K RS
T + П X−T+B
2.3.2. The Derivatives for the Case of a Marginal Change in Transfers
The marginal contribution M Bi of a transfer B i (B i = B 1 is chosen without the loss of generality)
in the case of multiple taxes and benefits can be similarly written in this format as
or
M B1 = G N\B1 − G N
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