(A5)
^∑-L=0 =μι ∙ δ (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 = ∆∙ δ∙ T , 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 + δ∙T.≡1*w* = 0.
∂t ∂s ∂T
23