The above equations may be written in implicit form as9
n = n(c, k; τl), nc > 0; nk < 0; nτl > 0; (8a’)
k = k(n; τk), kn < 0; kτk < 0;
(8b’)
(8c’)
c = c(k,n),ck > 0; cn < 0.
By reducing the after-tax wage and hence the opportunity cost of fertility,
an increase in τl stimulates the fertility rate (for a given level of consump-
tion).10 Hence, more time for child rearing is required; a reduction in labor
hours worked takes place. The capital stock is driven down because there
is an inverse relationship between per capita capital and population growth;
this is a sort of reverse Malthus law implied by the ”modified golden rule”
(8b). Consumption is reduced by the lower capital stock and the higher
fertility rate.
Ariseinτk pulls the capital stock down as, for a given fertility rate, the
after-tax marginal product of capital is lowered.11 The fall in k generates two
contrasting effects on fertility; it stimulates the demand for fertility as the
nc
kn
9The expressions of the partial derivatives of relationships (8’) are
UnUcc — UcUcn
UcΛ
> 0; nk =
Uc[(1 - τι)T'Fik + 1]
ck = Fk
[1 + (1 - τk)T'Fkl] < k
(1 - τk)Fkk T Tk
- δ - n> 0; cn
(FiT' + k) < 0;
Λ
(Fk - δ) < 0;
(1 - τ k)Fkk
< 0; n
UcT 'Fl
c ʌ l > 0;
Λ
where Λ = Unn - UnUcn + (1 - τι)Uc(T'2Fu - FiT'') < 0.
Uc
10 The basic steady state effects of labor taxation are given by:
dn nτ l dk knnτ l dc
dTl = ^Ω^ > 0; dTl = ω < ; dTl
(ckkn + cn)n
< 0;
where Ω = 1 — kn(¾ + ncCk) - cnnc > 0.
11Ariseinτk has the following long-run effects
dn
dτ k
(ncck + nk)k
(1 - cnnc)kτk
0;
where Ω > 0 has been defined in footnote 10.
dc
dτ k
(ck + nkcn)kτk
< 0;
10
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9. The name is absent
10. The name is absent