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



where L(kt) is as defined in equation (43). The first part of this proposi-
tion, proved that if next period decision rules are set according to the "de-
mographic steady" strategy, and the capital per (native-born) worker is in the
range:
[F(τ 1), F(τ 1)], then the optimal solution for the young is πt = (τ(kt), 1).
In addition, we have shown in proposition II, that under the assumption that
next period policy decision rules are given according to:

π =f (0,Min[γ*, - mm ]) if      ut+1< 1

(69)


t+1 ɪ ( ψψ+1,1)       otherwise

the young voter’s indirect utility function is maximized by the "demographic
switching" strategy:
πt = (0, Min[γ*, mm]). Therefore we must show that if
kt [F(T 1),F(τ 1)], the value of the young voter’s indirect utility function is
higher under the "demographic steady" strategy. Since the value of the young
voter’s indirect utility function under the "demographic steady" strategy is con-
stant in
kt [F(Tι),F(τ 1)], and the value of the young voter’s indirect util-
ity function under the "demographic switching" strategy is increasing in
kt ,
the value of the young voter’s indirect utility function under the "demographic
steady" strategy must not be lower than the "demographic switching" strategy
for
kt = F(τ1):

β β β β ,T, ʌ -(1+β) Ψβ(1-α)

L°g[(2+⅛m 1+βψ+i)      2 ψ+α c)

(1 + β)Log ɑ (1 α)F(τ 1)α(1 + γt)) ψfc (1+ βf (ψψ+-1 )ʌ +

/                                     Z                   ^J+Ψ                  ∖β

Log ( α((1 α)(1 ɪ)2φ(ɪ                                    ' )-Ф)    )

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

(70)

In addition, we must require that if kt [F(τ 1), F(τ 1)], then also the aggregate
saving decision rule,
S(kt, πt = (τ(kt),1), τt+1 = τ(kt+1)) [F(τ 1), F(τ 1)]. The
derivative of next period capital per (native born) worker (defined according
to this aggregate saving decision rule) by
kt is positive for kt [F(τ 1),F(τ 1)].

Denote by h1 (y) and h2 (y) the following functions:

ft1(y)= У-Γ⅛ ψ + 12 + n + m ((1 α)(F(τ 1))α2(1 - τ 1))ψ+α(1-f (τ(y)))
, .       (71)

ft2(y)= У1⅛ ψ+12 + n2 + m ((1 α)(F (τ 1))α 2(1 τ 1))'' (1f (τ (y)))
_                                                _   (72)

Denote by k1,k1 [F(τ 1),F(τ 1)], the solutions of equations: h2(k1) = 0
and h1(k1) = 0 respectively. Thus, the required condition is that [k1,k1]

32



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