Optimal Taxation of Capital Income in Models with Endogenous Fertility

instead, lowers the capital intensity and the capital stock, since the fertility
rate remains constant. The wage rate falls. Consumption, nonhuman wealth
and gross national income are reduced as well.¹⁸ The stock of net foreign
assets is, on the contrary, increased.

3.3 Normative analysis

Residence-based regime

In the small open economy, the implementability constraint is given by¹⁹

¹⁸To work out the effects on consumption and financial wealth, write (18a) and (18c)
as follows

c = x(a; τ_k), x_a > 0; x_τk > 0;                           (18a’)

c = m(a,τk),ma > 0; mτ_k < 0;                      (18c’)

where

Uc²       .n          Uc² (1 - τ 1)T'w'

χa = -,--------^c-------? > 0; χ_τι. = - т~~—------------?

^a   (UcUcn - Un Ucc)       ^τk     (UcUcn - UnUcc)

.∩         ^τ kr*(^f '^{- δ})   ∩

ma = ρ > 0; mτk = (1 - _τk)2_f '' < 0.

from (18’)- are

^dc = (^xamτk ^- ma^xτk ) < 0; ^da = (mτk ^{- x}τk ) < 0;

dτk (Xa - ma) _      ; dτk (Xa - ma)     ;

where saddle-point stability requires x_a >m_a .

¹⁹ Equation (19) is obtained by integrating the consumers’ flow budget constraint
(12) forward, incorporating the condition preventing Ponzi games —that is, limt_→∞ a
e^-∫o[(^1-τa)^{r -n}]^ds=0- and using the relationship U_ce^-ρt = U_c[0]e^- Jo[(^1-τa)^{r -n}]^ds. The
implementability constraint (19) remains unaltered under the source-based regime of cap-
ital taxation even if the relationship (14), the proper ”no Ponzi game” condition and the
relationship U_ce^-ρt = U_c[0]e^-ʃo(r ^-n)^ds are now employed.

18

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