Since immigrants gain the right to vote only in the second period of their
life in the host economy, the next period ratio of old to young voters who are
allow to vote, denoted by ut+1, is given by:
u = (1+ γt)
(13)
ut+1 (1 + n) + Yt(1 + m)
Assuming that in case of a tie the old will be the decisive, the condition, ut+1 <
1, assures a majority of young individuals in the next period, while the condition,
ut+1 ≥ 1, assures a majority of old individuals. Therefore, the state variable of
the economy, affects the next period ratio of young to old voters, ut+1 , which
sets the profile of the next period decisive voter.
The Markov Perfect political equilibrium of the baseline model and its pos-
sible equilibrium paths, which depend on the population growth rates of the
native-born and immigrant populations, can be formalized as follows:
Proposition 2 There exists an equilibrium with the following feature :
τt = 0
if ut(γt-1) < 1
otherwise
if ut(γt-1) < 1
otherwise
(14)
(15)
T(Yt-1) = t_ ɪ
t τ t = Ψ+1
G(Yt-ι) = I Yt 1m
γt = 1
where γt is restricted to be between zero and one. Under the assumption that the
native-born population growth rate is lower than that of the immigrant’s, there
are three possible equilibrium paths, depending on the population growth rates of
the native-born and immigrant population, as follows: 1. if n > 0, there is no
taxation/social security benefits; 2. if m + n < 0, migration quota is set at its
maximum, and there is a positive level of taxation/social security benefits (the
"Laffer point" tax rate). 3. if n < 0 and m + n > 0, there is a "demographic
switching" equilibrium path, where some positive level of immigration always
prevails while there is an alternate taxation/social security policy; in periods
where the decisive voter is old, the economy is fully opened to immigration and
there is a positive level of taxation (the "Laffer point" tax rate); whereas in
periods where the current decisive voter is young, there is no taxation/social
security benefits and a more restrictive policy towards immigration.
The proposition is proved in the appendix.
The interpretation of the proposition is as follows.
If the old-young ratio is smaller than one (ut < 1), the decisive voter in the
current period is a young voter. The young decisive favors naturally a zero tax
11