This note studies optimal taxation of income in a growth model with endogenous fertility proposed by Barro and Becker(1989). It is found that the optimal tax rate on capital income converges to zero after one transition period, and the government should not tax labor income in period 1 and thereafter. These results are obtained for a general period utility function.

Takeo Kozu

English

Research Papers in Economics

Optimal factor income taxation in a neo−classical growth
model with endogenous fertility 
Takeo Kozu
Tokyo Metropolitan University
Abstract
This note studies optimal taxation of income in a growth model with endogenous fertility
proposed by Barro and Becker(1989). It is found that the optimal tax rate on capital income
converges to zero after one transition period, and the government should not tax labor income
in period 1 and thereafter. These results are obtained for a general period utility function.
I would like to thank Sadao Kanaya, Inyong Shin, Kouji Sasaki and Kiyoshi Yonemoto for helpful comments. All remaining
errors are mine.
Citation: Kozu, Takeo, (2005) "Optimal factor income taxation in a neo−classical growth model with endogenous fertility."
Economics Bulletin, Vol. 5, No. 9 pp. 1−7
Submitted: April 11, 2005.  Accepted: October 25, 2005.
URL: http://www.economicsbulletin.com/2005/volume5/EB−05E60001A.pdf1 Introduction
The seminal studies by Chamley (1986) and Judd (1985) characterize the
optimal path of tax on capital income in an optimal growth model. They
show that in a steady state, a government that commits itself to its policy
should set the tax rate on capital income to be zero. Several researchers have
made extensions in vatious environments and examined whether the zero-tax
result is appropriate.1
This note extends the analysis to another direction. Many empirical stud-
ies show that fertility is related to some economic variables. Thus, the tax
system may aﬀect the agents’ fertility behavior. If so, how should a govern-
ment design its policies to maximize the agents’ welfare? In particular, is
the zero-tax result for capital income applicable? To investigate the optimal
paths of tax rates in such a situation, this note follows the model proposed
by Barro and Becker (1989), which provides a standard framework to incor-
porate fertility behavior.
We ﬁnd that the optimal tax rate on capital income is equal to zero not
only in the steady state but during the transitional periods. In addition, it
is found that the optimal tax rate on the labor income is zero for t ¸ 1.
2 The Model
2.1 The Economy
In a non-stochastic and closed economy, we investigate the optimal level
of tax rates on capital and labor income in an optimal growth model with
fertility choice proposed by Barro and Becker (1989). Following Barro and
Becker (1989), it is assumed that the utility of an adult in the generation t,
Ut, can be written as
Ut = u(ct;xt) + ¯(nt)ntUt+1:
The utility of an adult in generation t consists of his/her own consumption(ct),
leisure(xt)2 and the number of children(nt). The function u(ct;xt) represents
the utility from consumption and leisure. The parents are altruistic, so that
they are concerned about the welfare of each child(Ut+1). ¯(nt) is a function
1See for survey Chari and Kehoe (1999).
2We assume that each adult is endowed with one unit of time.
1that measures the degree of altruism of an adult toward each child. We as-
sume that 0 < ¯(nt) < 1;¯0(nt) > 0 and ¯00(nt) < 0. Then, the welfare of a
dynastic head (generation 0) is given by
U0 =
1 X
t=0
BtNtu(ct;xt); (1)
where Bt ´
Qt¡1
i=0 ¯(ni) and
Nt ´
t¡1 Y
i=0
ni = nt¡1Nt¡1:3 (2)
Each adult spends time, lt, in the labor market and earns ¯ wtlt, where
¯ wt(´ (1¡¿l
t)wt) is the after-tax wage rate and ¿l
t is wage tax rate. Assets are
held by an adult (denoted at and consist of physical capital kt and government
bonds bt) and earn ¯ rtat. ¯ rt(´ (1¡¿k
t )rt) represents the the after-tax interest
rate and ¿k
t represents the tax rate on capital income. An adult spends
income and non-depreciable inheritance on consumption and a bequest to
children ntat+1. To raise nt children, parents need °(nt) units of time. It is
assumed that °0(nt) > 0 and °00(nt) > 0. Then, the constraints on time and
goods for an adult are written as
xt + lt + °(nt) = 1; (3)
and
ct + ntat+1 = (1 + ¯ rt)at + ¯ wtlt:
As we assume the transversality condition holds, the dynastic budget con-
straint on goods can be written as
1 X
t=0
dtNt(¯ wtlt ¡ ct) + (1 + ¯ r0)a0 = 0; (4)
where dt ´ (1 + ¯ r0)
Qt
i=0
1
1+¯ ri.
The optimal behavior of the dynasty can be described as the solution to
the problem that maximizes (1) subject to (2)-(4). The ﬁrst-order conditions
are (2)-(4),
u2t = u1t ¯ wt; (5)
u1t = u1t+1¯(nt)(1 + ¯ rt+1); (6)
3N0 and B0 are normalized to unity.
2and
BtNt+1u2t°
0(nt) =
µ
1 +
nt¯0(nt)
¯(nt)
¶ 1 X
i=t+1
BiNiui+u10
1 X
i=t+1
diNi( ¯ wili¡ci); (7)
where ujt represents the partial derivative of u(ct;xt) with respect to j-th
(j = 1;2) argument.
In a competitive market, ﬁrms maximize proﬁt. The aggregate produc-
tion function is given by F(Ntkt;Ntlt), which is assumed to be constant-
returns-to-scale and to satisfy the Inada conditions.4 The conditions for the
optimality are
@Ft
@kt
(´ F1t) = rt;
@Ft
@lt
(´ F2t) = wt: (8)
2.2 The Optimal Taxation Problem
The government ﬁnances the exogenous sequance of expenditure, fGtg1
t=0,
by taxes on capital and labor income f¿k
t ;¿l
tg1
t=0
5 and bonds fbtg1
t=0. The
government determines the ﬁnancing strategy so that the competitive equi-
librium described below exists.
A competitive equilibrium is an allocation, fct;lt;xt;nt;kt;Nt;Gtg1
t=0, a
sequence of prices, frt;wtg1
t=0, and a sequence of policies, f¿k
t ;¿l
t;btg1
t=0 such
that the following conditions are satisﬁed. First, given the sequences of the
policies and the prices, the allocation maximizes the welfare of the dynasty
subject to the budget constraint (4). Second, given the sequences of the
policies and the prices, the allocation solves ﬁrm’s maximization problem.
Finally, given the allocation and the sequence of prices, the government’s
budget constraint
Gt + (1 + rt)Ntbt = Nt¿
k
t rt(kt + bt) + Nt¿
l
twtlt + Nt+1bt+1; t = 0;1;:::; (9)
is satisﬁed.
Note the resource constraints of the economy for goods is written as
Ntkt + F(Ntkt;Ntlt) = Ntct + Nt+1kt+1 + Gt; t = 0;1;:::: (10)
4We assume that there is no technological progress for simplicity.
5The capital taxation in period zero must be restricted, since it would be equivalent to
lump-sum taxation. For simplicity, we set ¿k
0 = 0.
3To derive the optimal policy,6 we follow the primal approach in Lucas and
Stokey (1983). We can use household’s ﬁrst-order conditions, the household’s
present-valued budget constraint and ﬁrm’s ﬁrst-order conditions to get the
implementability constraints, which are
BtNt+1u2t°
0(nt) =
µ
1 +
nt¯0(nt)
¯(nt)
¶ 1 X
i=t+1
BiNiui+
1 X
i=t+1
BiNi(u2ili¡u1ici);t ¸ 0
(11)
and
1 X
t=1
BtNtu(ct;xt) = W0; (12)
where
W0 =
¯0
¯0 + n0¯0
0
[u20fn0°
0(n0) + l0g ¡ u10fc0 ¡ (1 + F10)a0g]:
The implementability constraints need to include (11) because fertility be-
havior occurs outside of the market and cannot be taxed.
The allocations in the competitive equilibrium satisfy (11) and (12) be-
cause the implementability constraints are derived from the ﬁrst-order con-
ditions of consumers and ﬁrms.
Also we can show that any allocations satisfying the implementability con-
straints and resource constraints are competitive equilibrium allocations. For
any allocation which satisﬁes the implementability constraints and resource
constraints, denoted by fc¤
t;l¤
t;x¤
t;n¤
t;k¤
t;N¤
t ;G¤
tg1
t=0, we can show that the al-
location constitutes a competitive equilibrium deﬁned above if the sequences
of prices and of policies are deﬁned by
rt ´ F1(N
¤
t k
¤
t;N
¤
t l
¤
t); (13)
wt ´ F2(N
¤
t k
¤
t;N
¤
t l
¤
t); (14)
¿
l
t ´ 1 ¡
u2(c¤
t;x¤
t)
u1(c¤
t;x¤
t)F2(N¤
t k¤
t;N¤
t l¤
t)
; (15)
¿
k
t+1 ´ 1 ¡
u1(c¤
t;x¤
t) ¡ ¯(n¤
t)u1(c¤
t+1;x¤
t+1)
¯(n¤
t)u1(c¤
t+1;x¤
t+1)F1(N¤
t+1k¤
t+1;N¤
t+1l¤
t+1)
; (16)
6To avoid any issue of time inconsistency, it is assumed that the government can commit
to future policies implied by the optimal tax problem on initial period.
4and
bt+1 ´
P1
i=t+1 B¤
i N¤
i fu1(c¤
i;x¤
i)c¤
i ¡ u2(c¤
i;x¤
i)l¤
ig
B¤
tN¤
t+1u1(c¤
t;x¤
t)
¡ k
¤
t+1:
The optimal tax problem for our model is to maximize (1) subject to (2),
(3) and (10)-(12).7 We call this problem as (P). As in Jones et al. (1997), we
consider the modiﬁed problem (P 0), which is (P) without constraints (11).
Then, under the assumption that each probrem has a unique solution path,
we can verify that, for t ¸ 1, the solution path to (P 0) conincides with the
one to (P).
For t ¸ 1, the ﬁrst-order conditions to (P 0) are
´tNt+1 = ´t+1Nt+1(1 + F1t+1); (17)
(1 + ¸)BtNtu1t = ´tNt; (18)
(1 + ¸)BtNtu2t = ´tNtF2t; (19)
(1 + ¸)BtNtu2t°
0
t = (1 + ¸)
¯0
t
¯t
1 X
i=t+1
BiNiui + ¹tNt (20)
and
(1 + ¸)Bt+1ut+1 ¡ ¹t + ¹t+1nt+1 ¡ ´tkt+1
+ ´t+1(kt+1 + F1;t+1kt+1 + F2;t+1lt+1 ¡ ct+1) = 0; (21)
where ¹t, ´t and ¸ denote the Lagrange multipliers on (2), (10) and (12),
respectively.
It is easy to verify that (17)-(21) lead to (11). Thus the ﬁrst-order con-
ditions to (P 0) satisfy the missing constraints (11). Since each problem is
assumed to have a unique optimal path, it follows that the solution paths to
(P) and (P 0) are the same for t ¸ 1. Therefore, for t ¸ 1, the optimal tax
rates for capital and labor income can be obtained by using (17)-(21).
From (17) and (18), we get
u1t = ¯(nt)u1t+1(1 + F1t+1); t ¸ 1: (22)
This equality, along with (16), implies that the optimal tax rate for capital
income in generation t is zero for t ¸ 2. We also derive the optimal path of
7The equation (9), the government’s budget constraint, is not necessarily included in the
Ramsey problem because it is implied by the resource constraint (10) and the household’s
budget constraint (4), which is associated with (12) in the optimal taxation problem.
5tax rate on labor income. Using (15), (18) and (19), the optimal tax rate on
labor income in generation t is zero for t ¸ 1.
There are several points worth noting. First, previous literature have
referred to steady states, while Chamley (1986) and Judd (1999) identify a
class of the utility functions that cause the optimal tax rate on capital income
to be zero except for the initial period.8 In our derivation, there is no need
to specify the form of the period utility function to obtain the same result.9
Second, our result characterizes the behavior of the optimal labor income
tax rate not only in the steady state but on the transitional periods. It is
diﬀerent from the result of Jones et al. (1997), which show that the optimal
tax rates on capital and labor income should be zero along the balanced
growth path in an endogenous growth model with human capital.
Finally, while in Barro and Becker (1989), the economy may converge to
the steady state within a single generation, Benhabib and Nishimura (1989)
consider the extended model and show that the economy does not necessarily
jump to the steady state in one generation. Speciﬁcally, they show that the
property on the convergence depends on the functional form of ¯(nt).10 Since
we do not specify the form of ¯(nt), there can exist transitional periods in
the model above.
3 Summary
This paper investigates the dynamic optimal taxation problem in a growth
model with endogenous fertility. To describe the fertility decision, we adopt
the model proposed by Barro and Becker (1989).
We show that the optimal level of tax rate on capital income converges to
zero after one transition period(generation), and the government should not
tax on labor income in period 1 and thereafter. In the previous literature,
the form of the utility function has been restricted in a particular class to
obtain the similar result. These results are obtained using a general form of
period utility function on consumption and leisure.
8This result depends on the assumption that none of the tax rates have upper bounds.
If tax rates are assumed to be bounded from above, tax rates will be at thier bounds for
some periods.
9Unfortunately, however, it is diﬃcult to interpret some economic intuitions indepen-
dently of a speciﬁc functional form.
10Kanaya (2002, 2003) examines economic implication of Benhabib and Nishimura
(1989) and provides some further results.
6References
Barro, R. J. and G. S. Becker (1989): “Fertility Choice in a Model of
Economic Growth,” Econometrica, 57, 481–501.
Benhabib, J. and K. Nishimura (1989): “Endogenous Fluctuations in the
Barro-Becker Theory of Fertility,” in Demographic Change and Economic
Development, ed. by A. Wening and K. Zimmerman, Springer-Verlag, 29–
41.
Chamley, C. (1986): “Optimal Taxation of Capital Income in General
Equilibrium with Inﬁnite Lives,” Econometrica, 54, 607–622.
Chari, V. V. and P. J. Kehoe (1999): “Optimal Fiscal and Monetary
Policy,” NBER Working papers 6891, National Beureau of Economic Re-
search, Inc.
Jones, L. E., R. E. Mannuelli, and P. E. Rossi (1997): “On the
Optimal Taxation of Capital Income,” Journal of Economic Theory, 73,
93–117.
Judd, K. L. (1985): “Redistributive Taxation in a Simple Perfect Foresight
Model,” Journal of Public Economics, 28, 59–83.
——— (1999): “Optimal Taxation and Spending in General Competitive
Growth Models,” Journal of Public Economics, 71, 1–26.
Kanaya, S. (2002): “A Graphical Analysis of Dynastic Population
Model(1),” Unpublised manuscript, Tokyo Metropolitan University, (in
Japanese).
——— (2003): “A Graphical Analysis of Dynastic Population Model(2),”
Unpublised manuscript, Tokyo Metropolitan University, (in Japanese).
Lucas, R. E. and N. Stokey (1983): “Optimal Fiscal and Monetary
Policy in an Economy without Capital,” Journal of Monetary Economics,
12, 55–93.
7

Optimal factor income taxation in a neo-classical growth model with endogenous fertility

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Optimal factor income taxation in a neo-classical growth model with endogenous fertility

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