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

where L(k_t) 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 π_t = (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

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 k_t = 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)^-α) ^ψfc (1+ βf (ψψ+-1 )ʌ +

/                                     Z                   ^J+Ψ                  ∖β

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

Ψ+1     1+β Ψ+1              1+n+γ_t(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^-α(1 - τ 1))^ψ+α(1-f (τ(y)))
, .       (71)

^ft2(y)= У—1⅛ _ψ+_{12 + n}² _{+ m} ((1 — α)(F (τ 1))^α 2^-α(1 — τ 1))'' (1—f (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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