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

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)w^tl^t(i^-τtt)(^1-f(^τt+1)), with τt+1 = T(γ_t).

^{. t , t} 1+β Ψ+1        1+n+γ_t (1+m)        ^{,        t+1}      γ t ^.

Consider first the case where there is a majority of old in period t, i.e. u_t ≥ 1.
Using the fact that,

wtlt(1 ^{- τ}t) = ((ι^{- α})k^α(1 ⁺ γ_t)   ^{(1 - τ}tɔ                ⁽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:

+α ((1 - α)k-^ψ(1 + Yt)^ψ(1 - τt)) ^ψ+^α kt      √— '   )

It is can be proved that V^o(γ_t-1, k_t) is maximized by setting π_t = (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(γ_t), 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

1      L(k_t) with ∏t+1 = (ψ+1,1)       otherwise

where A = ⁽¹ + β)L°g (1++β ψ+1) +
and L(k_t) is defined as follows:

A ^{+ (1 + β})L^og[ ((^{1 - α})k^α(^{1+ γ}t) ^α(^{1 - τ}t)^    ⁽¹ + ^βf (^τt+1^)]

1 — α

^+βL^og (^α(^{1 - α})^kt++1^{(1 + γ}t+1)^{ψ (1 - τ}t+1)) ^ψ+^α
,     ,/.........1+Ψ,     ,     ,

_λ ∕_k         β Ψ (1+Yt)((1-α)k_t^α(1+Yt)^-α(1-τt)) ^ψ+^α (1-f (τt+ι).

^λ1(^kt+1 1+β Ψ+1                1+n+γ_t(1+m)                )

(43)

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