As a first step, it is easy to prove that the indirect utility of the young subject
to constant next period policy variables, is maximized by setting: πt = (0,γ*),
where γ* ∈ [0,1] is defined as follows:
* β(1 — α)Ψ(n — m) + α(1 + Ψ)(1 + n)x
(44)
-α(1 + Ψ)(1 + m)x
We will prove that in the case where m + n > 0 and n < 0, the indirect
utility of the young Vy(γt-1,kt) is maximized by πt = (0,Min[γ*, — —]) 8. If
γ* ≤ — mm , then it is sufficient to prove that the indirect utility of the young is
higher by setting πt+1 = (ψ+-1-,1) than by setting πt+1 = (0, γ*). It is easy to
see that the higher is next period immigration quota the higher is the indirect
utility of the young since it increases next period interest rate. Regarding the
next period tax rate, it is sufficient to prove that:
. β Ψ ψ ∖ 1+β / ψ ∖β ψ+α
0 = Log[(1 + βf (0))1+β] ≤ Log[ Ç1 + βf (ψ+1)J Ç1 — ψ+1J ]
(45)
due to the fact that the following holds,
ψ, l-а ψ Ψ ∖ -ψβ ψ+α
0 = Log[(1 — f(0))-ψβψ+α] ≤ Log[(j — f(-)J ] (46)
/ ∖ι+β / ∖ βΨ+α
Define the function: d(Ψ) = LogH 1 + βf (ψψ+ι H (1 — ψψ+∏ °]∙ The
derivative of d(Ψ) is the following expression:
Ψ+α+β-αβ
1 1 ∖ Ψ+α
kΨ+1J
(α - l)β(l + β)2 ( . )β-1
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(Ψ(1 -α)(Ψ + α) + (Ψ + 1)(Ψ + α+β + αβ + αβΨ)Log(ɪ ))
(Ψ + α + β+αβ+αβΨ)2
Since this derivative is positive for every Ψ > 0, and for Ψ = 0 the function is
equal to zero (d( Ψ = 0)= 0), then d(Ψ) is positive for every Ψ > 0.
Otherwise, if γ* > — —, we must prove that the following holds,
a 1 — α
Log ((1 + γ')-α)ψ+ψ(1+β) + Log ((2+ЩЦ")-*(1——)ψ) '+“
≤ (1+ β)Log[(1 — — )-αψ+α (1+ βf ( ψ+ι)]+
β 1 — α
τnn (√1-mm)((1-mm)-α)ψ+α(1-f(ψ+)ʌ-ψ2ψ (1__ψ.ψ+α
Log l( 1+n-mm(1+—) ) 2 ^1 Φ+1J)
(48)
8 If the population growth rates are both positive, m, n > 0, then it is straightforward to
see that Vy (Yt-ι, kt) is maximized by ∏t = (0, Min[γ*, — mn]).
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