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
Vo(γt-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 kt ∈ [g(F(τ 1)), g(F(τ 1))], the aggregate saving decision rule
should follow: S(kt,πt = (ψψ+ι ,1),τt+1 = τ(kt+1)). Otherwise, the aggregate
saving decision rule should follow: S(kt,πt = (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- > 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: 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 τt(kt) 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(kt),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=
(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, Min∖γ*, — mn]) otherwise
4 L(kt) with πt+1 = (Ψ+i ,1)
31