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

at kt = F(τ₁):

_    2 2   _{2 β ψ} ∖ ^-(¹+^β) Ψβ(1-α) ∖

Log Ç ( 2+n+m 1+β Ψ+1J      ^{2 ψ}+^α cj ≥

(1 + β)Log ɑ (1 - α)F(τι)^α(1+ γ*)^-α) ^ψfc) +

2                                       I                     1 1+Ψ          ∖ ^β

Loo ( α((1 - α)(1+ ₇*)^φ(ɪɪ (¹+Y*)((^1-α)F(^τι)^α(¹÷^γ*)^-α) ^ψ+^α )-Ψ)ψ^-α ʌ
'-oguf(l^{l   α})(J-^{+ γ} ) ⁽ 1+e Ψ+1           ι+n+₇*(ι+m)          )   )

_             (74)

Thesameconditionsasbeforearerequired, meaning that [k₁,k₁] ⊆ [F (τ ₁ ),F (τ 1)].
But, unlike the previous case, when kt ∈ [F(τ ₁ ),F(τ ₁)]c, the optimal strategy
do not change next period decisive voter from young to old. Therefore we
will additionally require that for kt ∈ [F(τ ₁ ),F(τ ₁)]c, the value of the young
voter’s indirect utility function under the corner solutions of the first decision
rule will be lower (or equal) to the value of the young voter’s indirect utility
function under the second decision rule, and that the aggregate saving decision
rule follows: S(kt,πt = (0,γ*),τt+1 = 0). Since for kt ∈ [F(τ ₁ ),F(τ 1)]^c, the
decision rule of the tax rate τt (kt) is a corner solution, the solution would imply
setting the constrained, meaning τ = τ ₁ or τ = τ ₁. Therefore the required
condition is that setting the constrained would yield that the value of the young
voter’s indirect utility function under the corner solutions of the first decision
rule is lower (or equal) and that the aggregate saving decision rule follows:
S(kt,πt = (0,γ*),τt+1 = 0). These conditions are sufficient to assure that the
equilibrium conditions are satisfied when: n > 0.

If the population growth rates satisfy the property: n + m < 0, there is a
majority of old in every period. It is straightforward to see that the old decisive
voter’s utility Vo(γt_-1,kt) is maximized by setting πt = ( ψ^ψ+1,1). Since the
identity of next period decisive voter’s do not change from old to young, the
aggregate saving decision rule follows: S(kt,πt = (-ψ^ψ+j-,1),τt+1 = -ψ^ψ+1 ). ■

7.4 A model without a social security system:

In order to emphasis the role of the social security system in the model, we next
consider a similar model with private saving, but without transfer payments
from the young to the old. We will prove that there is an equilibrium that
incorporates two strategies depending on the population growth rates. If the
immigration quota preferred by the young for its direct effects on wages, is low
enough, the young decisive voter will restrict immigration even further, in order
to change next period decisive voter from young to old, so that next period old
decisive voter will set no restrictions on immigration. This policy is favorable

34

<<   32   33   34   35   36   37   38   >>

More intriguing information