Optimal Taxation of Capital Income in Models with Endogenous Fertility

U(c, n)e^-ρtdt,

where c is per capita consumption, n the fertility rate, and ρ the exogenous
rate of time preference.⁷ The instantaneous utility function U (c, n), which
is strictly increasing in c and n, satisfies the usual properties of regularity. c
and n are assumed to be normal goods.

Two constraints must be respected when (1) is maximized. One is the
intertemporal budget constraint, given by

where w represents the wage rate, l labor hours, q lump-sum transfers from
the government (in per capita terms), r the before-tax interest rate, δ the
constant capital depreciation rate, and k the per capita capital stock (k0 is k
at time 0). τ k and τl indicate ad valorem capital and labor income tax rates,
respectively; capital depreciation allowances are permitted.

Moreover, the time allocation constraint

l+T(n)=1,

must also be considered in addition to (2) when (1) is maximized. According

to (3), the fixed time endowment (normalized to one) can be used either for
working or for raising children. T ( ∙ ) denotes the amount of time devoted
to child-rearing (with T' > 0 and T'' ^ 0).

The maximization of (1) subject to (2) and (3) yields the following first-
order conditions

⁷ Note that n also represents the population growth rate as the mortality rate is zero
and the economy is closed.

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