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

following function: F(τ) = ((1 + ^1-_α^ατ)^1+β(1 - τ)^β⅛^α) C) x
rewrite the policy decision rule of the tax rate as: F(τ_t) = k_t.

F(τ) is decreasing in τ, for τ ∈ [0, τ], where τ = ^1++++0^, and increasing in
τ, for τ ∈ [τ, 1]. Thus, according to equation (52), for every value of capital per
(native-born) worker, kt, there are two solutions for τ(kt) in the range [0, 1).

The solution which satisfies the equilibrium conditions, which is denoted by
τ(k_t), is decreasing in kt for kt ∈ [F(τ),F(0)].

The solution for the policy variables given in equations (52) and (53), will
be proved to satisfy the first order conditions of the problem. Substituting for
wtlt(1 -τt) and 1 +rt+1 from equations (39) and (40), the young voter’s indirect
utility function under the assumption that next period decisive voter is young,
which sets next period policy decision rules for the tax rate and immigration
quotas to be τt+1 = τ (kt+1),and γ_t+1 = 1 respectively, can be written in its
Lagrangian form as follows:

L(kt) = A +(1 + β)Log ((1 - α)kt^α(1+ γ_t)^-α(1 - τ t))' +

(1 + β)Log[(1 + βf (τ(kt+ι)) + βLogα ((1 - α)k^-+ψ2^ψ (1 - τ (kt+ι)))'
_λ ∕_k         β Ψ (1+Yt ) ((1 - α)k? (1+γ t )^{- α} (1 — τ t)) ^ψ+^α (1-f(τ(k_t+1)),

^λ1(^{kt + 1}   ι+β Ψ + 1                   1+n+γt(1+m)                  )

-λ2(τt - 1) - λ3(-τt) - λ4(γ_t - 1) - λ5(γ_t)

The Kuhn-Tucker conditions are:

∂L       1 + Ψ 1 + β    1 + Ψ kt+1

=— = ⁰= ^- 7Γ^-— --^λl7Γ^-— τ--^λ2+^λ3

∂τt       Ψ + α 1 - τt    Ψ + α 1 - τt

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