To see why the equality in (6) holds, note that, by employing the envelope theorem,
the inequality in (6) implies:
(7) λs(w)∙(1 -1)∙ [1 -ls(w)]-λN(w)∙(1 -1)∙[1 -Γ'(w)] ≥ 0,
where λS and λNS 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, lS < lNS and λS > λNS (as cS < cNS), which implies that the inequality in
(7) is satisfied. This concludes the proof and the characterization of the equilibrium.
To summarize:
(8a)
c *( w ) = <
cs (w) if z1Ns (w) < z and w ≥ w
cNS (w) otherwise
(8b)
(8d)
l*(w)=<
ls (w) if zNS (w) < z and w ≥ w
lNS (w) otherwise
(8c)
z*(w) = <
zs (w) if z(w) < z and w ≥ w
zNS (w) otherwise
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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