On the Desirability of Taxing Charitable Contributions



(A5)

^∑-L=0 =μιδ (1 - t) t
s


- μ1 ∙ δ∙ T t


dF ( w ) + ( μ - μ) ∙ δ∙T


μι t


dF(w) + (μ -μ1)

dF ( w ).


dF (w ) - ( μ - μ1) δ(l-1 )

w


dF ( w )


dF ( w )


We need to prove that the sign of the derivative in (A5) is negative. We first turn to
further simplify the expression on the right-hand side of (A5). Consider the following
optimization problem, where an individual, given some labor/leisure choice,
l, is
choosing how to allocate the net income across consumption good,
c, and charitable
contribution,
z. Formulating the Lagrangean yields:

(A6)   L(w, l, t,τ, s) max[u(c) +v(z)+ν[(1- t)w(1 -l)+ T- c- (1 +s)z]],

where ν denotes the Lagrange multiplier. Now consider a small change in the tax
system around the optimal linear labor income tax system (set for
s=0), which is
defined as follows:
ds = ∆,dt = -∆ ∙δ(1 -t) and dT = ∆∙ δT , where ∆ > 0 and is
arbitrarily small. Fully differentiating the
Lagrangean in (A6), using the envelope
theorem, then yields:

(A7) dL,=o,,T =∆∙v[-z+ δ[(1 -1 ) w(1 -1 ) + T ]] = 0,

where the last equality holds for any 1, by virtue of the homotheticity [see (A4)] and
the
separability assumptions. It follows that the optimal labor/leisure choice of an
individual of ability
w (for all w) is unaffected by the suggested small perturbation in
the tax system around the optimum. Thus, for all
w, it follows that:

(A8) - δ(1 -1 )dw +dw + δT.≡1*w* = 0.

t    ∂s        ∂T

23



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