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



Substituting for λ1 from equation (57) into equations (55) and (56), we derive

the following equations:


∂L

(63)


=—= λ2+λ3= 0
∂τt

∂L (1 + β)      n + m

——=~------ I ^∣----------7τ-----ʌ ) —λ4+λ5= 0            (64)

dYt  1+ Yt V1 + n + Yt(1 + m))

Since we have assumed that m > n from equation (64) we derive that γt has a
corner solution. The solution for the tax rate, on the other hand,
τ t, may be
bounding or not, meaning that
τt = τ(kt) [0, 1]9 . Substituting the solutions
for the tax and openness rate into the indirect utility of the young, we obtain
that the optimal solution for the openness rate is
Yt = 1.

The optimal solutions should also satisfy the second order sufficient condi-
tion, meaning that the bordered Hessian of the Lagrangian should be negatively
defined. Since the solution of the immigration quotas is a corner solution where
the largest immigration quota maximizes the young voter’s indirect utility func-
tion, the bordered Hessian of the Lagrangian is equal to:

2L        ∂2L               ∂2L       2L

-gτ (gτ2kt+1-gk5kt+13T?) + gk (g'∂τt∂kt+1 -gk2τt) (65)
where gτ and gk are the derivatives of the constraint of the capital per (native-
born) worker from equation
(58) with respect to τt and kt+1 respectively. The
bordered Hessian can be rewritten in the following way:

(I+.!)2

Ψ+α


1__2x(1 + 1-ατt)(1 - τt) 1______________

2 2 ∖2 ^7        :              2τr~[      :      Γ2      ∙     . (66)

(1 - τt) ((1 + β) 1-α(1 τ.) ',(1 + 1-ατt)) (1 + 1-αɪτt)2
α       t    Ψ+α       α t         α 1+β t

( ((1 + β)1 (1


x(1 + 1τt)(1 Tt) (⅛α) +

τt)(1+β)


τ t)-β(1+αα) (1+1 τ t)) (1+1 ι+β

Denote by [τ1, τ2] the range of the tax rate for which the bordered Hessian of
the Lagrangian is negatively defined. The optimal solution for the tax rate,
τ(kt), is in the range kt [F(τ 1), F(τ 1)], where the function F(τ) is decreasing
in
τ.

The second part of the proof:

As in the proposition II, we must show that the vector of policy decision
rules,
Ψ = (T, G), satisfies the equilibrium conditions (the only difference is
that the policy decision rules,
Ψ(Yt-1 , kt), depend not only on the previous
immigration policy but also on the current capital per (native born) worker).

9 Note that the utility with τt = 1 is equal to minus infinity. Thus, the range for the tax
rate is [0, 1).

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