Optimal Taxation of Capital Income in Models with Endogenous Fertility

U_ce^-ρt = λe^- ʃo [(1^-τ k )(r^-δ)^-n]^ds

Une^-ρt = λe^- ^t[(^1-τ k)(^r-δ)^-n]^ds[(1 — τ ι)wT ' + kɑe ‰t[(^{1-τ k} )(^r-δ)^-n]^ds],    (4b)

where λ = Uc [c(0), n(0)] = Uc [0] is the Lagrange multiplier on the intertem-
poral budget constraint (2).

Equation (4b) can be rewritten as

Un = (1 ^{- τ} 1 )wT' + k^,
U_c

while from (4a), we obtain the following Euler equation

-— lnUc = (1 - τk)(r - δ) - n - ρ.               (4c)

Equation (4b’) asserts that the marginal rate of substitution of consump-
tion for fertility must equal the opportunity cost of one unit of fertility, given
by the after-tax wage times the marginal time-cost of child-rearing plus the
per capita capital stock. Equation (4c) ensures that in the intertemporal
equilibrium the rate of return on consumption, i.e. ρ - dt ln U_c, is equal to
the after-tax return on per capita capital, namely, (1 - τk)(r - δ) - n.

Production is carried out by many competitive firms. The production
function, which is given by y = F (k, l), satisfies the usual neoclassical prop-
erties of regularity and is linearly homogeneous in k and l.

Maximum profits requires

Fk (k, l)=r,

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