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

If the population growth rates satisfy the properties: m + n > 0 and n < 0.

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

The utility of the old voter is the same as in the previous proposition and thus
V^o(γt_-1,kt) is maximized by setting: π_t = (ψψ+ι,1). But unlike the previous
proposition the saving of the young in period t also depends on next period policy
variables. Thus, for k_t ∈ [g(F(τ ₁)), g(F(τ 1))], the aggregate saving decision rule
should follow: S(k_t,π_t = (ψψ+ι ,1),τt+1 = τ(kt+1)). Otherwise, the aggregate
saving decision rule should follow: S(k_t,π_t = (ψψ+ι,1),τt+1 = 0). Since the
derivative of next period capital per (native born) worker (defined according to
the first aggregate saving decision rule) by kt can be either negative or positive,
the first condition will require that the derivative will be positive, meaning that

the tax rate should be in the range τ ∈ [τ ₁,τ ₁] ⊆ [τ ₁,τ], where τ ₁ =
Denote by g(y) : kt+1- > k_t the following function:

Thus, for k_t ∈ [g(F(τ ₁)),g(F(τ ₁))], next period policy variables are set ac-
cording to: πt+1 = (τ (kt+1),1) and the aggregate saving decision rule follows:
S(kt,πt = (_ψψ+₁,1),τt+1 = τ(kt+1)). Note that for kt ∈ [g(F(τ 1)),g(F(τ 1))]^c
the decision rule of the tax rate τ_t(k_t) is outside the relevant range, [τ ₁,τ ₁].

Therefore the solution would imply setting the constrained, meaning either

τ = τ ₁ or τ = τ ₁.The required condition is that setting the constrained would
yield the aggregate saving decision rule: S(kt,πt = (ф+у,1),τt+1 = 0).

Consider next the case where there is a majority of young in period t, i.e.
ut < 1. If kt ∈ [F(τ ₁),F(τ ₁)], we must prove that the indirect utility of
the young voter is maximized by the "demographic steady" strategy, mean-
ing: πt = Ψ(τ (kt),1) and the aggregate saving decision rule follows: S(kt, πt =
(τ(k_t),1),τt+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 the "demographic
switching" strategy, meaning: πt = (0, Min[γ*, — mm]) and the aggregate saving
decision rule follows: S(kt,πt = (0, Min[γ*, — mn]),τt+1 = -jɪɪ). Substituting
for wtlt (1 - τt) and 1 + rt+1 from equations (39) and (40), 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   L(kt) with πt+1 = (0, Min∖γ*, — mn])      otherwise

₄                   L(kt) with ^πt+1 = ⁽Ψ+i ^,1)

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