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

Since as was proved d(Ψ) is positive for every Ψ > 0, and for γ* > — mm, the
following holds,

1+Ψ                                                   1-α

(1 + β)Log[ ((i±⅞)^-α) ^ψ+α ]+βLog ((i±⅞)^-ψ(^1-α)^ψ+^α(1—2m)^ψ)    ≤ 0

^{1 -} m/             ^{1 -} m             ²

(49)

it is sufficient to prove that,

1-α                                         1-α

/1 + n — ⁿ (1 + m)∖ ^{-ψ ψ}+^α        Ψ , Ψ Δ^{-ψ ψ}+^α   , _λ

βLog χ⅛χ ≤ βLog 1 — f ⅛rr)        (50)

∖1 + n + γ*(1 + m) J              ∖ Ψ + 1 J

Substituting γ* from equation (44) into equation (50), we can rewrite the in-
equality in the following way,

Λ 1 — α 1 Ψ ʌ ≥ β(1 — α)Ψ
∖ ^*^ α 1+ β Ψ + 1J ~ α(1 + Ψ)x^

Since this expression is positive, it completes the proof that V ^y(γ_t-1, , k_t) is
maximized by setting π_t = (0, Min[γ*, — mn]). ■

7.3 Proposition III:

Proof. The proof will consist of two parts. The first part will prove that
when there is a majority of young voters the policy decisions for the tax rate
and immigration quotas stated maximizes the young indirect utility function,
under the assumption that next period decisive voter is young. The second part
will complete the proof and show that under certain conditions on the models
parameters, the vector of policy decision rules as defined in the proposition,
satisfies the equilibrium conditions.

The first part of the proof:

We follow the proof of Forni (2004) to derive the policy decision rules. The
policy decision rules are obtained by using as a constraint the first derivative
with respect to the policy variables of the logarithm of the capital accumulation
equation. The policy decision rules are the following:

γ_t = 1

where x = 1 + (1ψ+αα^β ,and c is a positive constant of integration. The policy
decision rule of the immigration quotas is at its maximal value, and the policy
decision rule of the tax rate is implicitly given in equation (52). Define the

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