πt = (τt , γt) is defined as the vector of policy platform, and V i is the indirect
utility of the current decisive voter.
2. The fixed-point condition requires that if next period policy outcome is
derived by the vector of policy decision rules- Ψ, the maximization of the indi-
rect utility of the current decisive voter will reproduce the same law of motion,
Ψ(γt-1) = Ψ(γt-1), as in 1.
The policy variables, which are the tax rate, τt, and the immigration quo-
tas, γt , have to maximize the decisive voter’s indirect utility function, while
taking into account that next period political-economy policy rules depend on
the current state variable, i.e. the current immigration quotas. Current and
future political economy policy rules , as a function of state variables must be
identical. Thus, the subgame-perfect Markov equilibrium notion states that the
expected political-economy policy function, which depends on the current state
variables, must be self-fulfilling.
The subgame-perfect Markov equilibrium is characterized by a "demographic
switching" strategy. Assuming that immigrants enter the country while young
and gain the right to vote only in the next period when they are old, voters
take into account the effect of admitting a certain number of immigrants on the
composition of voters and their voting preferences in the next period. Moreover,
when the number of young exceed the number of old in the population, the young
decisive voter admits a limited number of immigrants, in order to change the
decisive voter’s identity from young to old in the next period and maximize the
next period benefits she receives.
The equilibrium path depends on the native-born and immigrant’s popu-
lation growth rates. If the population growth rates of the native-born and
immigrants are both positive, there is a steady state with no taxation/social
security benefits. If alternatively, the sum of the population growth rates is
negative, there is also another steady state, but with a certain positive level
of taxation/social security benefits (the "Laffer point" tax rate) and full open-
ness to immigration. Otherwise, the sum of the population growth rates can be
positive and the native-born population’s population growth rate negative. In
this case, there is a "demographic switching" equilibrium path where some quo-
tas on immigration always prevails while there is an alternate period by period
taxation/social security policy, depending on the identity of the decisive voter.
In a given period there is a certain amount of taxation/social security benefits
(the "Laffer point" tax rate) and no restrictions on immigration, while in the
next there is no taxation/social security benefits and a more restrictive policy
towards immigration.
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