On the Desirability of Taxing Charitable Contributions

the term on the right-hand side. Suppose, by negation, that the inequality is strict, that
is the individual strictly prefers to engage in signaling. By continuity considerations,
an individual with ability slightly lower than w will then also choose to signal, which
would violate the definition of our equilibrium. Thus, we obtain a contradiction.
Clearly, the equality in (5) implies that the signaling constraint in (4) has to bind,
which in turn implies that z^s (w) = Z > z^NS (w). Mild parametric restrictions on the
utility function in (1) guarantee the existence of a threshold level of contributions, Z ,
which satisfies the equality in (5). For example, when the functions u, h and v are
logarithmic, it is straightforward to verify that the equality in (5) is satisfied.⁸ As
∂V^s(τw)/∂Z < 0 whereas by construction, ∂V^ns(τw)/∂Z = 0, there exists a unique
value of Z for which the equality in (5) is satisfied (for any given set of tax
parameters). To establish the existence of an equilibrium, we need to verify that for
any ability w < w, individuals choose not to engage in signaling; whereas, for any
ability level w > w, individuals do engage in signaling. There are two cases to
consider. Consider first the case where z^NS (w) ≥ z. In such a case, V^s (w) > V^ns (w),
by construction. By virtue of the strict concavity of the functions u, h and v, charitable
contribution, z, is a normal good; hence, as Z > z^NS(w), this may only hold true for
w > w, which is consistent with the separating equilibrium presumption. Consider
next the case where z' (w) < Z. In order to prove the result, it suffices to show that
the following condition holds:

(6)    ∂V^s(w)/∂w-∂V^ns(w)/∂w ≥ 0.

⁸ To see this, note, that whenZ = z^NS, it follows that V^ns(w) < V^s (w) ; whereas, when
Z = [w (1 -1) + T]/(1 + .s`), namely the individual spends her entire potential income on charitable
contributions, consumption and leisure drop to zero, hence V^s ( w) = -∞, which obviously implies that
V^ns (w) > V^s (w). Existence follows by the intermediate value theorem.

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