Employing the first-order conditions for the optimal tax problem (see
appendix C for details), we can re-write equation (15) as follows:
∂L∖
****
∂s I s=0,t ,T ,g ,z
(16)
w
μ1 ∙ Z ∙ [1 - F(w)] - z ∙ ʃ[^' [VS (w)] ∙ λs (w'^F(w)
w
144444444424444444443
Redistributive Term
w
z ∙ ʃ[^'[VS (w)] ∙ λs (w)]^F(w)
w
-( μ- μ2) ∙ z ∙[1 - F( w)]
14444244443
Pigouvian Term
1444442444443
Signaling Correction Term
Equation (16) decomposes the effect of a small tax on contributions into three
terms. The first term on the right-hand side of equation (16) captures the redistributive
effect of a unit increase in the tax on charitable contributions. To see this, note that the
term Z ∙ [1 - F(w)] is the additional amount of revenues raised by a unit increase in
the tax on contributions (at s=0). Multiplying it by μ1, the marginal social benefit of a
unit increase in the transfer (T), yields the effect of the extra revenues on social
welfare. As the burden of this unit increase on each status-signaling individual (that
is, each individual with innate ability exceedingιw) is z, then, indeed, the first term
on the right-hand side of equation (16) captures the redistributive effect of a tax on
contributions. This effect is positive and works in the direction of taxing "charitable"
contributions, when the social welfare function exhibits a sufficiently large degree of
11
inequality aversion.
The second term, which also works in the direction of levying a tax on
contributions, measures the corrective effect which offsets the wasteful status-
signaling costs. To see this, fully differentiate the signaling constraint in (5) with
11 For instance, when the social planner is Rawlsian the second expression in the first set of brackets
disappears and, clearly the re-distributive term is positive. Notably, in such a case, also the signaling
correction term vanishes, as the contributors obtain zero weight in the social welfare measure.
17