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) = argmaxt 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)wtlt(itt)(1-f(τt+1)), with τt+1 = T(γt).

. t , t 1+β Ψ+1        1+n+γt (1+m)        ,        t+1      γ t .

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

wtlt(1 - τt) = (- α)kα(1 + γt)   (1 - τtɔ                (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:

τt((1-a)(1+γt)-akα)ψ+α (1-τt) ψ+α [(1+n)+γt-1(1+m)](1+γt)
(1+Yt-1)


V ot-1, kt) =


(41)


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

It is can be proved that Vo(γt-1, kt) is maximized by setting πt = (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(γt), 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

(42)


1      L(kt) with t+1 = (ψ+1,1)       otherwise

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

βLog (β), λ1 is the Lagrangian multiplier,


A + (1 + β)Log[ ((1 - α)kα(1+ γt) α(1 - τt)^    (1 + βf (τt+1)]

L(kt) =


1 — α

+βLog (α(1 - α)kt++1(1 + γt+1)ψ (1 - τt+1)) ψ+α
,     ,/.........1+Ψ,     ,     ,

λ k         β Ψ (1+Yt)((1-α)ktα(1+Yt)(1-τt)) ψ+α (1-f (τt).

λ1(kt+1 1+β Ψ+1                1+n+γt(1+m)                )

(43)

26



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