is the immigration policy rule, γt = G(γt-1 , kt), and S : [0, ∞)- > [0, ∞), is
the saving decision rule, kt+1 = S(πt , kt), such that the following functional
equations hold:
1. Ψ(γt-i,kt) =argmax∏t Vi(γt-ι, ∏t, ∏t+ι) subject to ∏t+ι = Ψ(‰ S(∏t, kt)).
β β(qr β ∖ — β Ψ (1+γt)wtlt(1 -τt)(1 -f(τt+1)) ,,,,'+A ʃ , — 'l‰ Cllqτ I- И
2. S(πt,kt) ι+β Ψ+1 1+n+γt(1+m) ,withτ t+1 T (γt, S(πt, kt)).
3. 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 indirect
utility of the current decisive voter subject to the law of motion of the capital
stock, will reproduce the same law of motion, Ψ(γt-1,kt) = Ψ(γt-1,kt), as in
1.
Policy variables have to maximize the decisive voter’s indirect utility func-
tion, while taking into account the law of motion of capital and the fact that
next period decision rules depend on the state variables, i.e. the current period
immigration quotas and next period capital per (native-born) worker. Equilib-
rium paths depend on the native-born and immigrant population growth rates
(as in the baseline model) and on the initial stock of capital per (native-born)
worker.
There are two types of equilibria.
The first type, is characterized by a "demographic switching" strategy, sim-
ilarly to the base-line model. When the decisive voter is young, she admits
a limited number of immigrants in order to change the decisive voter’s iden-
tity from young to old in the next period. The additional effect caused by the
existence of savings and the endogeneity of factor price determination, is only
quantitative.
The other equilibrium type is however different from the base-line model.
The additional state variable, the stock of capital per (native-born) worker,
plays now a crucial role. Rational voters take into account that the current
policy variables can affect next period policy variables not only through the
composition of old to young voters, but also through the effect on next period
capital per (native-born) worker; the additional state variable. There is another
possible strategy of the young; a "demographic steady" strategy, where the equi-
librium tax rate is a decreasing function of the capital per (native-born) worker,
and migration quota is set at its maximum level. This level of immigration
quotas renders a majority for the young in every period. The new equilibrium
of the extended model, combines strategies concerning both the old-young com-
position in the population, and the level of capital: there is a range of values of
the capital per (native-born) worker, for which the "demographic steady" strat-
egy dominates; while for values outside this range, the "demographic switching"
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