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 (gτ∂2kt+1-gk5kt+13T?) + gk (g'∂τt∂kt+1 -gk∂2τt) (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-α (1
x(1 + 1-ατt)(1 — Tt) (⅛α) +
τt)(1+β)
—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 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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