Appendix B: Derivation of Equation 15
Employing the envelope theorem, the effect of a small tax levied on contributions is
given by the partial derivative of the Lagrangean in equation (14) evaluated at s=0
given the optimal levels of the set of control variables: t, T, g and z . Note that the
(uniform) level of contribution in the pure status signaling case, z , is chosen as a
control variable in the optimization, as we incorporate the signaling constraint in
equation (5) into the Lagrangean [the last expression on the right-hand side of
equation (14)], so that its partial derivative with respect to s is zero.
Differentiation yields thus the following expression:
(B1)
⅛=0 =∫ W,PVN (w)] ∙
∂5 _
∂V NS (w)
∂5
dF(w) + ∫ W'[Vs(w)] ∙
W _
dF ( w )
W'
μ ∙ t ∙∫
w
dF ( w ) - μ ∙ t ∙∫
∂ls(w)
w----
∂s
dF(w) + μ ∙ z ∙ [1 - F(W)] +
∂ Vs ( w) ∂ Vns ( w) '
∂s ∂s
In the pure status signaling case only individuals whose innate ability exceeds the
threshold W choose to contribute (the amount of z ), whereas all other individuals set
their contribution level at zero and are thus unaffected by the tax, s. Thus, it follows
∂V Ns (w) ∂lNs (w)
that-----=-0- =----=0- = 0 . By virtue of the individual optimization (employing the
∂s ∂s
envelope theorem) it follows that
∂ Vs ( w )
∂s
= -λ(w) ∙ z . Substitution into equation (B1)
yields equation (15) in the main text.
26
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