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

∂L

=—= —λ2+λ3= 0
∂τt

∂L (1 + β)      —n + m

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

^dYt  ¹⁺ Yt V^{1 + n +} Yt(^{1 + m}))

Since we have assumed that m > n from equation (64) we derive that γ_t 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 Y_t = 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

^-g^τ (^gτ∂2kt+1^-g^k5kt+13T?) ⁺ g^k (^g'∂τt∂kt+1 ^-g^k∂2τt) (⁶5)
where g_τ and g_k 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:

1__2x(^{1 + 1-α}τt)(^{1 -} τt) 1______________

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

^{(1 - τ}t) ((1 + β) ¹-^α(1 — τ.) —',(1 + ¹-^ατt)) (1 + ¹-^αɪτt)²
α       t    Ψ+α       α ^t         α 1+β ^t

x(¹ + ^1-ατt)(¹ — Tt) (⅛^α) +

—τ t)-^β(1+αα) (1+1^-α τ _t)) (1+1^-α 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 k_t ∈ [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(Y_t-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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