Optimal Taxation of Capital Income in Models with Endogenous Fertility



max


0W(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 +
U
n [


Ucc
q) π


(1 - T)T"


T/2


(1 - T) (Unc


kUcc)


(1 - T) (Unn


Uc


kUnc)


Un


12




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