in all periods so as to finance welfare arrangements of the form (8), (9) and (10)
and that the intertemporal budget constraint for the public sector is fulfilled,
i.e.
∞ 1i
∑ (ι+r )bt+i=0 (13)
Initial debt is assumed to be zero. By use of (11 and (13), it is found that the
sustainable tax rate τb, satisfies
[b - αι - (1 + ρt)(γ + α2)] y1t + 1 + r £b - α1 - (1 + ρt+ι)(γ + α2)] y1t+1
+t⅛ )’
τb - α1 - (1 + ρt+2)(γ + α2) y1t+2
or
(ι+ρt)+ι+g (ι+ρt+ι)+μ ι+r ) (ι+ρt+2)
0 = [b - αι] ---1+g-(Y+α2)
1 1+r
(14)
Consider the special case, where ρ is constant over time. In this case the sus-
tainable tax rate is
τb = α1 +(γ + α2)(1 + ρ)
It is seen to coincide with the tax rate under PAYG-financing, cf (7). This
indicates that differences between the two modes of financing arise when the
demographic dependency ratio is not constant over time. It is an implication
that the sustainable tax in this case is independent of the interest rate and the
growth rate.
Trend changes in demographic dependency ratio
Consider a stylized situation with a trend change in the demographic depen-
dency ratio given as (1 + ρt+i) = (1 + ρ)i+1. This implies that the sustainable
tax (14) becomes
= α1 +(γ + α2)(1 + ρ)
(1 + r) - (1 + g)
(1 + r) - (1 + g)(1 + p)
It follows that
db
dp
(γ + α2)
(1 + r) - (1 + g) ) (1 + r) - (1 + g)
(1 + r) - (1 + g)(1 + p)+( + P)( + g)((1 + r) - (1 + g)(1 + p))2
>0
db
dg
(γ + α2)(1 + p)
(1 + r)p
[(1 + r) - (1 + g)(1 + p)]2
>0
db
dr
-(γ + α2)(1 + p)
_________(1 + g)P________
[(1 + r) - (1 + g)(1 + p)]2
<0
where it is assumed that (1 + r) > (1 + g)(1 + p).
36
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