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
ot-1,kt) is maximized by setting: πt = (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(F1))], the aggregate saving decision rule
should follow:
S(ktt = (ψψ+ι ,1),τt+1 = τ(kt+1)). Otherwise, the aggregate
saving decision rule should follow:
S(ktt = (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 k
t 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 11] 1,τ], where τ 1 =
Denote by g(y) : k
t+1- > kt the following function:

φ

Ψ+1.


g(y) = β -Ê— -Jψ--2----(1 - f (τ(y))) ((1 - α)2-α(1 -

1+β Ψ+12+n+m


1+Ψ

)Ψ+α ∖

)


Ψ+α
α(1 + Ψ)


(67)


Thus, for kt [g(F(τ 1)),g(F(τ 1))], next period policy variables are set ac-
cording to: π
t+1 = (τ (kt+1),1) and the aggregate saving decision rule follows:
S(ktt = (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 τt(kt) is outside the relevant range, [τ 11].

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(ktt = (ф+у,1),τt+1 = 0).

Consider next the case where there is a majority of young in period t, i.e.
u
t < 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: π
t = 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(ktt = (0, Min[γ*, mn]),τt+1 = -jɪɪ). Substituting
for w
tlt (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=

(68)

if      ut+1< 1

otherwise


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, Minγ*, mn])      otherwise

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

31



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