then for every immigration quota there is a majority of young in every period,
and thus the young decisive voter in every period will be indifferent between all
possible immigration quota levels. ■
7.2 Proposition II:
Proof. We must show that the vector of policy decision rules, Ψ = (T, G),
satisfies the following equilibrium conditions:
1. Ψ(Yt-1) = argmax∏t Vi(Yt-1, ∏t, ∏t+1), subject to ∏t+1 = Ψ(γt).
2. Ψ(Yt-1) = Ψ(Yt-ι).
3. S(∏t,kt) = ɪɪ (1+γt)wtlt(i-τtt)(1-f(τt+1)), with τt+1 = T(γt).
. t , t 1+β Ψ+1 1+n+γt (1+m) , t+1 γ t .
Consider first the case where there is a majority of old in period t, i.e. ut ≥ 1.
Using the fact that,
wtlt(1 - τt) = ((ι- α)kα(1 + γt) (1 - τtɔ (39)
1 — α
1 + rt = α ((1 - α)k-ψ(1 + Yt)ψ(1 - τt)) ψ+α (40)
the utility of the old voter can be rewritten as:
τt((1-a)(1+γt)-akα)ψ+α (1-τt) ψ+α [(1+n)+γt-1(1+m)](1+γt)
(1+Yt-1)
V o(γt-1, kt) =
(41)
+α ((1 - α)k-ψ(1 + Yt)ψ(1 - τt)) ψ+α kt √— ' )
It is can be proved that Vo(γt-1, kt) is maximized by setting πt = (ψ+y ,1).
Consider next the case where there is a majority of young in period t, i.e.
ut < 1. Substituting for wtlt (1 - τt) and 1 + rt+1 from equations (39) and (40),
the utility of the young voter subject to: πt+1 = Ψ(γt), can be written in the
Lagrangian form, in the following way:
L =f L(kt) with ∏t+1 = (0,Min[γ*, - m ]) if Ut+1< 1
(42)
1 L(kt) with ∏t+1 = (ψ+1,1) otherwise
where A = (1 + β)L°g (1++β ψ+1) +
and L(kt) is defined as follows:
βLog (β), λ1 is the Lagrangian multiplier,
A + (1 + β)Log[ ((1 - α)kα(1+ γt) α(1 - τt)^ (1 + βf (τt+1)]
L(kt) =
1 — α
+βLog (α(1 - α)kt++1(1 + γt+1)ψ (1 - τt+1)) ψ+α
, ,/.........1+Ψ, , ,
λ ∕k β Ψ (1+Yt)((1-α)ktα(1+Yt)-α(1-τt)) ψ+α (1-f (τt+ι).
λ1(kt+1 1+β Ψ+1 1+n+γt(1+m) )
(43)
26
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