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

[F(τ 1 ), F(τ 1)]. Under this condition, if kt ∈ [F(τ 1 ), F(τ 1)], the indirect utility
of the young voter is maximized by the "demographic steady" strategy and the
aggregate saving decision rule follows: S(k_t , π_t = (τ (k_t),1), τ_t+₁ = τ (k_t+₁)) ∈
[F (τ 1), F (τ 1)]. For kt ∈ [F (T 1 ), F (τ 1)]^c, the value of the young voter’s indirect
utility function is not lower under the "demographic switching" strategy than
under the "demographic steady" strategy (since for τt ∈ [τ 1,τ 1]c, the solution
would imply setting the constrained). Thus, if kt ∈ [F(τ 1 ),F(τ 1)]^c, the in-
direct utility of the young voter is maximized according to the "demographic
switching" strategy: πt = (0,Min[γ*, — mm]) and the aggregate saving decision
rule follows: S(kt,πt = (0,Min[γ*, — mn]),τt+1 = ψψ+1 ). It should be noted that
since the optimal solution changes next period decisive voter from young to old,
for all values of k_t+₁(defined according to this aggregate saving decision rule:
S(kt,πt = (0,Min[γ*, — mm]),τt+1 = ψ+1 )), there are no additional conditions
on k_t+1. These conditions are sufficient to assure that the equilibrium conditions
are satisfied when: m + n > 0 and n < 0.

If the population growth rates satisfy the properties: n, m > 0, there is a
majority of young in every period.

If kt ∈ [F(τ 1),F(τ 1)], we must prove that the indirect utility of the young
voter is maximized by setting: Ψ(τ (kt),1) and the aggregate saving decision rule
follows: S(kt,πt = (τ (kt),1),τ t+1 = τ (kt+1)). Otherwise, If kt ∈ [F (τ 1),F (τ 1)]^c,
we must prove that the indirect utility of the young voter is maximized by set-
ting: πt = (0,7*) and the aggregate saving decision rule follows: S(kt,πt =
(0,Y*)^,τ t+1 = 0). The young voter’s indirect utility function can be written in
its Lagrangian form as follows:

L = L(kt) with πt+1 = (τ(kt+1),1)   if kt+1∈[F(τ 1),F(τ 1)]

L(kt) with πt+1 = (0,γ*)      otherwise
where L(k_t) is as defined in equation (43). Note that the immigration quota
is not restricted (γ_t = γ*), since the young decisive voter cannot change next
period decisive voter from young to old. According to proposition II, the in-
direct utility of the young subject to constant next period policy variables, is
maximized by setting: πt = (0, γ*). Thus, similarly to the previous case, we will
require that the value of the young voter’s indirect utility function under the first
decision rule (π_t = (τ (k_t),1)), should not be lower than the value of the young
voter’s indirect utility function under the second decision rule (πt = Ψ(0,γ*))

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