On the Desirability of Taxing Charitable Contributions

To see why the equality in (6) holds, note that, by employing the envelope theorem,
the inequality in (6) implies:

(7)     λ^s(w)∙(1 -1)∙ [1 -l^s(w)]-λN(w)∙(1 -1)∙[1 -Γ'(w)] ≥ 0,

where λ^S and λ^NS are, correspondingly, the Lagrange multipliers in the individual
maximization for the 'signaling' and 'non-signaling' cases. By virtue of the strict
concavity of the functions u, h and v, both consumption, c, and leisure, l, are normal
goods. Thus, l^S < l^NS and λ^S > λ^NS (as c^S < c^NS), which implies that the inequality in
(7) is satisfied. This concludes the proof and the characterization of the equilibrium.
To summarize:

c^s (w)   if z¹N^s (w) < z and w ≥ w

l^s (w)   if z^NS (w) < z and w ≥ w

z^s (w)   if z(w) < z and w ≥ w

V ^* (w) = max{V ^NS (w),V ^S (w)}

Note also that z itself is determined in equilibrium, so as to make all individuals (and
only these individuals) with innate ability above the threshold w contribute an
amount (weakly) exceeding z. Formally z is defined implicitly by equation (5).

4.The Tax-Treatment of Contributions

The government is seeking to maximize some egalitarian social welfare function by
choosing the fiscal instruments t,T,s and g, subject to a revenue constraint, taking

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