IMPACT OF TAXES AND TRANSFERS

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Enami, Lustig, Aranda, No. 25, November 2016 M B1 = G X−∑ n m i=1 Ti +∑j=2 B j − G X−∑ n m i=1 Ti +∑j=1 B j n = (G X − G X−∑ n Ti m i=1 +∑j=1 B j ) − (G X − G X−∑ n Ti RS =⏟ П X−∑ n m i=1 Ti +∑j=1 B j Assuming no reranking RS m RS RS − П X−∑ m i=1 +∑j=2 B j ) n m i=1 Ti +∑j=2 B j = ∑ i=1 (1 − g i)П Ti + ∑ j=1(1 + b j )ρ Bj 1 − ∑n i=1 g i + ∑m − ∑ i=1 (1 − g i)П Ti + ∑ j=2(1 + b j )ρ Bj j=1 b j 1 − ∑n i=1 g i + ∑m j=2 b j n K m K = ∑ i=1 g iП Ti + ∑j=1 b j ρ Bj 1 − ∑n i=1 g i + ∑m − ∑ i=1 g iП Ti + ∑j=2 b j ρ Bj j=1 b j 1 − ∑n i=1 g i + ∑m j=2 b j or n n K RS m m K RS (54) = b n 1[(1−∑ g i m i=1 +∑j=2 b j )ρ K n K m K B1 −(∑i=1 g i П Ti +∑j=2 b j ρ Bj )] (1−∑n i=1 g i +∑m j=1 b j )(1−∑n i=1 g i +∑m j=2 b j ) . The derivatives with respect to progressivity and size are expressed in equations 55 and 56 respectively: (55) ∂M B 1 b ∂ρK = 1 B 1−∑n g 1 i=1 i +∑m j=1 b j . Note that the derivative 36 is always positive given the usual assumption about the total size of n m taxes and transfers, that is, 1 − ∑i=1 g i + ∑j=1 b j > 0 ∂M B1 = [ρ B K n m n K m K (1−∑ g 1 i=1 i +∑j=1 b j )]−[(+1)(∑i=1 g i П Ti +∑j=1 b j ρ Bj )] ∂b 1 (1−∑n i=1 g i +∑m j=1 b j ) 2 (56) = [ρ B 1 K n m n K m K (1−∑i=1 g i +∑j=1 b j )]−(∑i=1 g i П Ti +∑j=1 b j ρ Bj ) . (1−∑n i=1 g i +∑m j=1 b j ) 2 48
Enami, Lustig, Aranda, No. 25, November 2016 To sign the preceding derivative, please note that it is equal to = ρ K RS B1 −П X−∑ n i=1 Ti +∑ m j=1 B j 1−∑n i=1 g i +∑m j=1 b j . Because the denominator is always positive, the sign depends only on the numerator, which is the Kakwani index of transfer ( ρ K B1 ) and R-S index of the net system with B 1 RS (П X−∑ n m i=1 Ti +∑j=1 B j ). The following condition assures the derivative is positive. Condition MB1: K > П X−∑ n Ti ρ B1 RS m i=1 +∑j=1 B j Table 15 shows what the ultimate sign will be. Here, we assume that there is no reranking, so the R-S index being positive is equivalent to the fiscal system being equalizing. Table 15. The Sign of the Derivative of the Marginal Contribution of a Transfer with Respect to its Relative Size The Transfer of Interest: B 1 Regressive ρ K B1 < 0 Neutral ρ K B1 = 0 Progressive K > 0 ρ B1 Unequalizing RS П X−∑ n m i=1 Ti +∑j=1 B j < 0 Positive (more equalizing), if and only if condition MB1 holds Positive (more equalizing) Positive (more equalizing) The Whole System (including B 1 ) Neutral RS П X−∑ n m i=1 Ti +∑j=1 B j = 0 Negative (more unequalizing) Zero Positive (more equalizing) Equalizing RS П X−∑ n m i=1 Ti +∑j=1 B j > 0 Negative (more unequalizing) Negative (more unequalizing) Positive (more equalizing), if and only if condition MB1 holds 49
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IMPACT OF TAXES AND TRANSFERS

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