The Role of Immigration in Sustaining the Social Security System: A Political Economy Approach

setting: γ_t = γ*, where γ* ∈ [0,1]. Define γ to be the immigration quota which
satisfies the following property:

V y (kt^,γt=- mm ^,γt+ι=1)=V y(kt^,γt=^γ,γt+ι=^γ)        (⁸4^a)

The indirect utility of the young voter subject to: γ_t+1 = Ψ(γ_t), can be written
in the following way:

where L(kt) is defined as follows:

A+(1+β)Log   (1 - α)kt^α(1 + γt)^-α

+βLog (α ((1 - α)(ι+β                ((1 - α)kα(1+ Yt)^-α)^ψ+^ψ

(86)

It is easy to see that if γ* ≤ -mm, then the young decisive voter will set the
current immigration quota to be γ_t = γ*. Otherwise γ* > -mm. If additionally
the optimal immigration quota satisfy the property: γ < γ*, we have to prove
that the optimal strategy of the young is to set according to: γ_t = γ*, which will
induce a young decisive voter in the next period voting for the same strategy,
meaning:

V^y (kt^,Yt = - mm^,Yt+ι = 1) < V ^y(kt^,Yt = γ*^,Yt+ι = γ*)       (⁸7^a)

Using the fact the indirect utility of the young is higher the higher is next
period immigration quotas (as it increases the next period interest rate), we
derive, from equation (93), the following inequality:

V^y(kt^,Yt = -zn^,Yt+ι = 1) = V^y(kt^,Yt = γ^,Yt+ι = γ) < V^y(kt^,Yt = γ*^,Yt+ι = γ*)
(88a)

Thus, the optimal strategy of the young when γ* > - mm and γ < γ* is to set
the immigration quota to be: γ_t = γ*. Otherwise, if γ* > - mm, but the optimal
immigration quota satisfies the property: γ ≥ γ*, we must prove that the young
will set the immigration quota to be: γ_t = - mm, which will induce an old decisive
voter in the next period voting for γ_t+1 = 1, meaning:

V^y (kt^,Yt = - m^,γt+ι = 1) > V ^y(kt^,Yt = γ*^,Yt+ι = - mm )      (⁸9^a)

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