which can be rewritten in the following way:
(U
7m
^ -α(1+β) ⅜++α
(ι-mm
ι+n-mm (ι+m)
1∙'
1+n+γ* (1+m)
1
V + T7
-α(1+β) -1++α
β I-α
-Ψ β Ψ+α
(90)
>1
Since γ * ≤1, it is enough to prove that:
(U
^ -α(1+β) ψ++α
(1+Y )
m
(1-n
1+n-mm (1+m)
1+γ •
1+n+γ* (1+m)
fl—ɪ A
<1 + -I
α(1+β) ⅛++α
Д 1-α
Ψ β Ψ+α
>1
(91)
Denote by k(y) the following function:
1 1 ∖ -α(1+β)⅜+α / - Z - ∖ -α(1+β)⅛++α∖
kfe)=U; ) Г . U ) ) I
(92)
Since the derivative of k(y) by y is positive for — mm ≤ y ≤ 1, it means that
k(γ*) > k(— mm). Thus, the optimal strategy of the young in that case is to set
the immigration quota to be γt = — mm, which completes the proof. ■
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38