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

As a first step, it is easy to prove that the indirect utility of the young subject
to constant next period policy variables, is maximized by setting: π_t = (0,γ*),
where γ* ∈ [0,1] is defined as follows:

*   β(1 — α)Ψ(n — m) + α(1 + Ψ)(1 + n)x

-α(1 + Ψ)(1 + m)x

We will prove that in the case where m + n > 0 and n < 0, the indirect
utility of the young V^y(γ_t-1,k_t) is maximized by π_t = (0,Min[γ*, — —]) 8. If
γ* ≤ — mm , then it is sufficient to prove that the indirect utility of the young is
higher by setting πt+1 = (ψ+-1-,1) than by setting πt+1 = (0, γ*). It is easy to
see that the higher is next period immigration quota the higher is the indirect
utility of the young since it increases next period interest rate. Regarding the
next period tax rate, it is sufficient to prove that:

. _β        Ψ         ψ ∖ ^1+β / ψ ∖^{β ψ}+α

0 = Log[(1 + βf (0))^1+β] ≤ Log[ Ç1 + βf (_ψ+₁)J    Ç1 — _ψ+₁J    ]

(45)

due to the fact that the following holds,

_ψ, l-а         ψ         Ψ   ∖ -^{ψβ ψ}+α

0 = Log[(1 — f(0))^-ψβψ+^α] ≤ Log[(j — f(-)J       ]     (46)

/             ∖ι+β /        ∖ βΨ+^α

Define the function: d(Ψ) = LogH 1 + βf (ψψ+ι H    (1 — ψψ+∏   °]∙ The

derivative of d(Ψ) is the following expression:

(α - l)β(l + β)2 (     .            )^β-1

(Ψ(1 -α)(Ψ + α) + (Ψ + 1)(Ψ + α+β + αβ + αβΨ)Log(ɪ ))
(Ψ + α + β+αβ+αβΨ)2

Since this derivative is positive for every Ψ > 0, and for Ψ = 0 the function is
equal to zero (d( Ψ = 0)= 0), then d(Ψ) is positive for every Ψ > 0.

Otherwise, if γ* > — —, we must prove that the following holds,

a 1 — α

Log ((1 + γ')^-α)^ψ+ψ(1^+β) + Log ((2+ЩЦ")-*(1——)^ψ) '+“

≤ (1+ β)Log[(1 — — )^-αψ+α (1+ βf ( ψ+ι)]+

β 1 — α

_τnn (√1-mm)((1-mm)^-α)^ψ+^α(1-f(ψ+)ʌ-ψ₂ψ ⁽1__ψ.^ψ+^α

Log l⁽         1+n-mm(1+—)         )   ² ^¹ Φ+1J)

(48)

⁸ If the population growth rates are both positive, m, n > 0, then it is straightforward to
see that V^y (Yt_-ι, kt) is maximized by ∏t = (0, Min[γ*, — mn]).

27

<<   25   26   27   28   29   30   31   >>

More intriguing information