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

Proposition 7 The equilibrium can be specified as follows:

Yt = Mⁱn[γ*, -mm] if γ* ≤ γ
γ_t = γ*        otherwise

γ_t = 1                otherwise

(81)

S(^γt,k^t) = ι+ββψ+ι i+J+YtYti+m)^wt^lt                    (⁸2)

where γ* is defined as in the proposition II, and γ is given implicitly by the
following equation:

The equilibrium paths depends on the population growth rates and on the
initial amount of capital per (native-born) worker the economy is endowed with.
There are four types of equilibrium paths: 1. if n > 0, there are some re-
strictions on immigration. 2. if m+ n < 0, there are no restrictions on
immigration. 3. if n < 0 and m+ n > 0, there are two possible equilibrium
paths: if γ* ≤ γ, there is a "demgraaphCc switching" equilibrium path, where
some level of immigration always prevails: in periods where the decisive voter
is old, there are no restrictions on immigration; and in periods where the deci-
sive voter is young, there are some restrictions on immigration. Otherwise, the
decisive voter is always young, and there are less restrictions on immigration
than in the "demographic switching" equilibrium path when the decisive voter is
young.

Proof. As in proposition II, we must show that the immigration policy decision
rule, G, satisfies the equilibrium conditions.

Consider first the case where there is a majority of old in period t, i.e. ut ≥1.
It is easy to see that the V ^o(γ_t-i, kt) is maximized by setting γ_t =1.

Consider next the case where there is a majority of young in period t, i.e.
ut <1. 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 γ_t = Min[γ*, — mm], ifY* ≤ Y
and by γ_t = γ* otherwise¹⁰. As was proved in proposition II, the indirect utility
of the young subject to constant next period policy variables, is maximized by

¹⁰ 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 γ_t = γ*.

36

<<   34   35   36   37   38   39   40   >>

More intriguing information