max
0∞W(c,n,
k, Φ)e-ρtdt
(10a)
subject to
.
k= F [k, 1 - T (n)] - c - (δ + n)k - g. (10b)
We show that:
Proposition 1 In a closed economy model of capital accumulation with en-
dogenous population growth and infinitely-lived consumers, tax efficiency re-
quires the subsidization of capital in the long-run; this implies that it is op-
timal to tax labor income in order to finance a given stream of government
spending and the capital subsidy.
Proof. The first-order conditions for the ”Ramsey optimum” (10) are
Wce-ρt = Γ,
(11a)
Wne-ρt = Γ(FιT ' + k), (11b)
Γ=-Wke-ρt-Γ(Fk-δ-n), (11c)
where Γ is the co-state variable on the feasibility constraint, Wc = Uc [1 +
(1 - T)
Φ(1+ηc)], Wn = Un[1+Φ(1+ηn)], and Wk = Φ ——— Uc. Пс and Пп represent
general equilibrium elasticities for consumption and fertility, respectively.13
13 These elasticities are defined as ηc
=(c
ηn
=(c
Ucn
q) UT +
n
(1 - T)T''
T/2
kUc [1 +
Un [
Ucc
q) π
(1 - T)T"
T/2
(1 - T) (Unc
kUcc)
(1 - T) (Unn
Uc
kUnc)
Un
12
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