On the Desirability of Taxing Charitable Contributions



Appendix A: Proof of Proposition 1

We assume that second order conditions are satisfied, thus it suffices to show that
there exists a marginal welfare gain by slightly
decreasing the tax rate on charitable
contributions from
s=0. Differentiating the Lagrangean in (12) with respect to t, T
and g yields the following first-order conditions (suppressing the tax parameters to
abbreviate notation):

(A1)


(A2)


W '[V ( w )] λ( w ) w


ww

+μ ∙  [w[1 -l(w)]]dF(w)+(μ-μ2)


ww

⅛ =0 = [WT(w)]λ(W)]dF(W)-μtf[wdlw

∂Tj                               JL ∂T

WW


+ (μ1 μ2)


w


dF ( w ) - μ = 0,


[1 -1 ( w )]] dF ( w ) - μ1 t


dF ( w )


dF ( w ) = 0,


dF( w )


w

L

(A3) — , =0 =α∙r∙(g)l[W,[V(W)]]dF(w)-(μ -⅛) = 0.

g

w

By virtue of the homotheticity assumption, one can write the optimal choice of an
individual of ability
w as follows:

(A4)   z(w) = δ[(1 -1) w [1 -1(w )] + T] /(1 + ,),

where 0 δ< 1 and δ is independent of w.

Substituting for z(w) from (A3) into (13), following some algebraic manipulations
employing (A1) and (A2), and re-arranging, yields the following simplified form of
the derivative in (13):

22



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