On the Desirability of Taxing Charitable Contributions

Appendix B: Derivation of Equation 15

Employing the envelope theorem, the effect of a small tax levied on contributions is
given by the partial derivative of the Lagrangean in equation (14) evaluated at s=0
given the optimal levels of the set of control variables: t, T, g and z . Note that the
(uniform) level of contribution in the pure status signaling case, z , is chosen as a
control variable in the optimization, as we incorporate the signaling constraint in
equation (5) into the Lagrangean [the last expression on the right-hand side of
equation (14)], so that its partial derivative with respect to s is zero.

Differentiation yields thus the following expression:

∂ V^s ( w)  ∂ V^ns ( w) '

∂s       ∂s

In the pure status signaling case only individuals whose innate ability exceeds the
threshold W choose to contribute (the amount of z ), whereas all other individuals set
their contribution level at zero and are thus unaffected by the tax, s. Thus, it follows
∂V ^Ns (w) ∂l^Ns (w)

that-----=-0- =----=0- = 0 . By virtue of the individual optimization (employing the

∂s        ∂s

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