IMPACT OF TAXES AND TRANSFERS

Enami, Lustig, Aranda, No. 25, November 2016
explain the change in the value function between V(0) and V(Ḭ i ). The second term is the
difference between the Shapley value of the aggregated source Ḭ i and the value of function V
when only aggregated source Ḭ i is added. It is clear to see that
k
(A-4) ∑j=1 NSh Iij = Sh Ii .
The proof is as follows:
k
∑ NSh Iij
j=1
k
(s!) × ((k − s − 1)!)
= ∑ { ∑ ( (V(S ∪ Ḭ
k!
ij ) − V(S)))
j=1
S∈NS Iij
}
k
+ ∑ 1 k (Sh I i
+ V(Ḭ i ) − V(0))
j=1
k
k
→ ∑j=1 NSh Iij = ∑ {∑ ( (s!)×((k−s−1)!)
j=1
(V(S ∪ Ḭ ij ) − V(S)))
V(0).
S∈NS Iij
}
k!
+ Sh Ii + V(Ḭ i ) −
Note that in the second term, the summation over k and (1/ k) cancel each other. Now note
that the term inside the braces is equal to Sh Iij if one decomposes the change in V between
V(0) and V(Ḭ i ). The summation over the Shapley value of all j income concepts that are part of
Ḭ i is simply equal to the total change in the value function between V(0) and V(Ḭ i ). This means
the preceding equation could be written as follows:
and therefore,
k
→ ∑ NSh Iij
j=1
= V(Ḭ i ) − V(0) + Sh Ii + V(Ḭ i ) − V(0)
k
→ ∑j=1 NSh Iij = Sh Ii .
Note that the value of j has to be at least 1 (that is, one income inside each income group) and
if all income groups have j = 1, then the nested Shapley is reduced to the simple Shapley.
This nested Shapley formula, however, suffers from a few theoretical problems. First, the
choice of decomposing V(Ḭ i ) − V(0) between sub-elements of Ḭ i (the first term in equation
A-3) is arbitrary. One can choose any element of set S Ii . Let’s call it O j and then decompose
V(Ḭ i ∪ O j ) − V(O j ) between elements of Ḭ i and the decomposition also satisfies equation A-4.
Equation A-3 can then be generalized as
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