in a pay-as-you-go system3 .
The utility of the representative young individual, as before, is logarithmic.
lΨ+1
Uy(wt,τt,st,rt+1,bt+1) = Log(wtlt(1 - τt) - st - ψt + 1) +
βLog(bt+1 + (1 + rt+1)st) (16)
U o(st-1, rt, bt) = bt + (1 + rt)st-1 (17)
where rt is the interest rate, and st is the savings of the young at period t.
The production function is a Cobb-Douglas production function which is
assumed to use both labor and capital as its factors of production:
Yt = Nt1-aKtα (18)
where Kt is the aggregate amount of capital and Nt is defined as in the previ-
ous section. The wage rate and interest rate are determined by the marginal
productivity conditions (capital is assumed to depreciate completely at the end
of the period):
wt = (1 - a)(1 + γt)-alt-aktα
rt = α(1 + γt)1-alt1-aktα-1 - 1
(19)
(20)
where kt is capital per (native-born) worker. The balanced government budget
constraint is derived as in the previous section:
bt+1 =
τ t+ιwt+ιlt+ι[(1 + n) + Yt (1 + m)](1 + γt+ι)
(1 + Yt)
(21)
The saving-consumption decision of young individuals are made by maxi-
mizing their utility while taking the prices and policy choices as given, and the
labor-leisure decision is given as in the previous section:
st = ɪ ββ ψ wtlt(1 — τt) +—-ʌ (22)
t 1+ β V Ψ + 1 t tV tJ 1 + rt+1) V 7
ltΨ = wt (1 - τt)
(23)
The market clearing condition requires that the net domestic saving generates
net domestic investment:
3 To isolate the unique role of social security, the reader can compare the equilibrium types
derived in appendix 7.4. The appendix assumes a similar model, but without a social security
system.
15
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