ko4, for which there is a "demographic steady" equilibrium path, characterized
by a positive tax rate which depends on the capital per (native-born) worker state
variable and no restrictions on immigration. If ko is not in this range, there
are at least few periods in which there is a "demographic switching" equilibrium
path, characterized by an alternate taxation/social security policy where some
positive level of immigration always prevails; 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 additional equilibrium path in the extended model are characterized by
different optimal strategy of the young, depending on the values of the capital
per (native-born) worker: for a range of values of the capital per (native-born)
worker state variable the decision rules of the young decisive voter do not change
the next period decisive voter’s identity, while for other values of the capital per
native-born work force the "demographic switching" strategy is still optimal.
When the "demographic steady" strategy is optimal, the tax rate depends neg-
atively on the amount of capital per (native-born) worker and there are no
restrictions on immigration.
The equilibrium tax rate of the "demographic steady" strategy, is a decreas-
ing function of capital per (native-born) worker. This is due to the fact that
there are two conflicting forces of the effect of the next period tax rate on next
period capital per native-born workers. On one hand, a higher tax rate in the
next period raises future social security benefits. Larger benefits, tend to reduce
current savings. This would cause a reduction in next period capital per native-
born work force ("Effect One"). On the other hand, a higher next period tax
rate tend to decrease the amount of hours worked next period which lowers the
next period interest rate and social security benefits. The consequent fall in the
current young future income induces more savings. This tends to increase next
period capital per (native-born) worker ("Effect Two"). Since "Effect One" is
stronger than "Effect Two", the tax rate is decreasing in the amount of capi-
tal per (native-born) worker 5 . There are no restrictions on immigration since
larger immigration quotas has an additional positive effect on the indirect utility
4 The range is defined as follows: if there is a ma jority of old (i.e. uo ≥ 1) the range of
the initial capital per (native born) worker is: [g(F(τι)),g(F(τ 1))]; otherwise, the range is:
[F (τ 1),F (τ 1)].
5 This equilibrium prop erty of the tax function, is already noted by Forni (2005).
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