Proposition 7 The equilibrium can be specified as follows:
G(γt-1) =
Yt = Min[γ*, -mm] if γ* ≤ γ
γt = γ* otherwise
if ut(γt-1) < 1
γt = 1 otherwise
(81)
S(γt,kt) = ι+ββψ+ι i+J+YtYti+m)wtlt (82)
where γ* is defined as in the proposition II, and γ is given implicitly by the
following equation:
(⅞+m )
-α(ι+β) ψ+α
(1¾ Г
(i-mm
i+n-mm (i+m)
1+Y
1+n+γ(1+m)
(≡ )
-α(1+β) ψψ++α
(83)
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(γt-i, kt) is maximized by setting γt =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, Vy (γt-1, kt), is maximized by γt = Min[γ*, — mm], ifY* ≤ Y
and by γt = 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 Vy(Yt-ι,kt) is maximized by γt = γ*.
36