Uce-ρt = λe- ʃo [(1-τ k )(r-δ)-n]ds
(4a)
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
(4b’)
Un = (1 - τ 1 )wT' + k,
Uc
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 Uc, 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,
(5a)