Child policy ineffectiveness in an overlapping generations small open economy with human capital accumulation and public education
Luciano Fanti* and Luca Gori**
Department of Economics, University of Pisa, Via Cosimo Ridolfi, 10, I–56124 Pisa (PI), Italy
Abstract Motivated by the recent decrease in the number of children experienced in several developed countries, in this paper we consider a small open economy model with overlapping generations, endogenous fertility and human capital formation through public education, and look at the role the government can play in affecting fertility through the widely used child allowance policy. Contrary to conventional view, we show that the public provision of child allowances is fertility-neutral in the long run, that is it is not effective as a pro-natalist policy, while also reducing human capital accumulation. In contrast, the financing of the public education system is beneficial to both fertility and human capital. These results hold in the cases of both fixed and time cost of children.
Keywords Child allowance; Fertility; Public education; Small open economy
JEL Classification I28; J13
We thank an anonymous referee for useful comments. Usual disclaimers apply.
* E-mail address: lfanti@ec.unipi.it; tel.: +39 050 22 16 369; fax: +39 050 22 16 384.
1. Introduction
In the recent decades the literature on economic development highlighted the prominent roles played by human capital formation as well as by the endogenous demographic behaviours of individuals in the economic system, focusing in particular on the interaction between them (e.g. Becker et al., 1990; Galor and Weil, 1996).
Human capital accumulation is widely considered as being one of the most important sources of economic development, and it may occur through different channels: for instance, educational attainment and the learning by doing with the former being mainly reached through formal schooling.1 In this paper we focus on the formation of human capital through formal schooling, in particular public schooling, motivated by the fact that, historically, Governments have been the main providers of education especially in European countries, but even in North-American ones.2 The effects of public provision of education3 on growth have been explored by Glomm and Ravikumar (1992), Eckstein and Zilcha (1994), Benabou (1996), Fernandez and Rogerson (1996) and many others. However these literature ignores the effects of public education on (endogenous)
1 For instance, Mankiw et al. (1992) found that school enrolment is positively correlated with per working-age person
GDP.
2 Indeed, for instance, Glomm and Ravikumar (2001, p. 808) argue that during the last century the fraction of students
at the elementary and secondary level attending public schools has been above 95 per cent in Canada and 86 per cent in the U.S.
3 For the sake of completeness, we note that recent economic literature argued that in the process of human capital
formation the private input may complement the public education input (for instance, through effective parental time). As noted by Glomm and Kaganovich (2008, p. 1012) “It is believed that private parental inputs at a pre-school stage as well as parent’s time spent helping the child in school related activities such as homework, reading and field trips play a fundamental role in the formation of human capital.” The relationship between parental inputs and public education inputs in human capital formation has been explored by Glomm and Kaganovich (2003), Viaene and Zilcha (2003) and Houtenville and Smith Conway (2003).
fertility. The widely recognised prominent role of the human capital formation in the economic development has important implications for the economy, and can therefore significantly affect the results of analyses concerning for instance fertility and family policies.
The recent demographic behaviour in several developed economies is characterised by a sharp decrease in the number of children as well as by an increasing longevity. As regards the first aspect, since a low fertility rate seems to be the result of a rational choice that individuals make, it is prominent in the political agendas the use of adequate family policies to affect individual fertility. A widespread instrument used is the provision of child allowances to parents (e.g. Neyer, 2003): the reduction in the cost of child rearing due to the existence of a per child subsidy is expected to positively affect the choice of how many children to raise.
There are two important papers by Zhang (1997) and de la Croix and Doepke (2003) that jointly analyse the interplay between fertility, education and economic growth. Both are framed in the literature of the quantity-quality trade-off (along the line of Becker and Barro, 1988) and assume time cost of children in an overlapping generations (OLG) model. Moreover, the former author, different from the latter, assumes that parents care about the utility of their descendants (i.e. complete or strong altruism towards children (see, Razin and Ben-Zion, 1975; Zhang, 1995). In both papers investments in education are privately provided by parents and agents choose the number of children as well as the amount of goods to be invested in the education of each child (Zhang, 1997) or the time should be devoted to schooling activities per child (de la Croix and Doepke, 2003) in a context of endogenous growth. However, while de la Croix and Doepke (2003) adopt the standard growth model à la Diamond (1965), Zhang (1997) does not consider the production side of the economy.
In particular, Zhang (1997) looked at the role of education and fertility subsidies finding that a rise in the education subsidies reduces fertility and speeds up economic growth because the relative cost of education shrinks and the educational spending relative to per capita family income increases. In contrast, raising the child allowance increases fertility and depresses economic growth
because the relative cost of education raises and the educational spending relative to per capita family income lowers.
De la Croix and Doepke (2003), instead, mainly focused on the inequality issue and showed that fertility differentials between the poor and the rich matter for inequality and growth. However, they did not deal with public policies.4
Different from the previous literature, in this paper we analyse the effects of child and education policies in an economy with endogenous fertility, public education and fixed cost of children, and look at the role the government can play in affecting fertility rates, by developing a small open economy model where individuals derive utility from material consumption over the life cycle and the number of children they have (i.e. the so called weak form of altruism towards children, see Zhang and Zhang, 1998),5 and assuming, in particular, the standard OLG model of neoclassical growth à la Diamond (1965).6 It is shown that human capital accumulation is reduced by child allowances, while their overall effect on the choice of the number of children crucially depends on the relative strength of the substitution effect due to the reduced cost of children and the income
4 A crucial hypothesis of their model is that teachers instead of parents provide education. This creates, in turn, a
positive externality on the accumulation of human capital, while also implying that the cost of education is fixed for parents, so that the cost of educating children is higher for poor parents. They found that the poorer the parents, the higher the demand for children and the lower their education level. The fertility differential depends on the initial distribution of income. The larger such a differential the lower average education.
5 Notice that we did not choose a class of model with complete or strong altruism (Barro, 1974; Becker and Barro,
1988; Ehrlich and Lui, 1991), where parents internalise the utility of their descendants, because we concentrate on the effects of taxes and expenditures. Therefore, such public policies, as known, tend to be ineffective in that class of models because there is in practice an individuals’ infinite horizon. In fact, “in these models the effects of changes in taxes are negated via changes in bequests, and so are ill suited to analyzing social security or publicly funded education.” (Pecchenino and Pollard, 2002, p. 149).
6 Other papers, such as van Groezen et al. (2003) and Fenge and Meier (2005) adopted the same framework (i.e., for
instance, weak form of altruism and small open economy in an OLG Diamond’s model), but, different from ours, they abstract from human capital formation and focus essentially on the interaction between family and pension policies.
effect due to the negative change in the endowment of human capital.7 Therefore we argue, rather unexpectedly, that a child allowance policy is fertility-neutral. Indeed the individual demand for children ultimately depends only – in a positive way – on the educational contribution rate. These results hold even when child rearing activities are time consuming and thus reduce the time spent working. Our results contribute to shed new light on the relationship between fertility, education, economic growth and public policies in the OLG literature.
In particular, the results by Zhang (1997) are the most related to ours. Let us now briefly explain the reasons why while Zhang (1997) concluded that the public provision of child allowances pulls up fertility and education subsidies decrease fertility in the long run, in this paper the opposite result is obtained. In fact, while in the model by Zhang individuals face a trade-off when choosing between material consumption, education expenditure and the quantity (i.e. the number) of children, in our model the trade-off is only between material consumption and the quantity of children. This is because while in the former model education is privately chosen by individuals, in the present context it is publicly provided and thus it does not represent an individual choice variable. Moreover, while the results by Zhang are in line with the commonplace, that is child allowances shift private expenditures from education (and material consumption) to the number of children, and education subsidies, instead, make the opposite, in our model: (i) as regards child allowances, in the long run the substitution effect towards the number of children (which is the prevailing effect in the Zhang’s model) is exactly counterbalanced by the negative income effect8 induced by a reduced human capital accumulation due to the increased number of children that attend school over which
7 Indeed, the idea that a child allowance policy should act as a fertility-enhancing device is essentially based on the
effect of a strong substitution effect through a reduced cost of child rearing. However, Fanti and Gori (2007) showed, in an OLG closed economy à la Diamond, that such an idea is valid only in either the short-run or in a partial equilibrium context with fixed factor prices, but it may not hold in a closed economy general equilibrium context.
8 We recall that in this type of models with endogenous fertility and fixed cost of children the fertility rate is positively
the public schooling expenditure must be split; (ii) as regards policies supporting education, in our model, where, different from the Zhang’s model, there is no trade off between private expenditure in education and the quantity of children, the effects go as follows: a rise in the financing of public education requires an increase in the educational contribution rate, which implies, in turn, a reduction in the disposable income of the young workers. However, this negative effect is overweighed by the positive effect on the working income due to the increased human capital accumulation. Therefore, the overall effect of a rise in the educational contribution rate is a stimulus to fertility.
Moreover, in the Zhang’s model there exists an additional effect that tends to reinforce his conventional results on economic growth: that is, the assumption of time costs of children. In such a case in fact a rise in the child allowance contributes to decrease the expenditure in education and increase the quantity of children, while also causing a reduction in the supply of labour as the time needed to care about children increases (in contrast, when education subsidies shift from the number of children to the educational expenditure also cause a rise in the supply of labour).
In order to check for the robustness of our findings, we have also introduced the hypothesis of time costs of children following Zhang (1997). It is shown that even by considering such a child cost structure, although a multiplicity of equilibria and an obvious negative relationship between fertility and income emerge, the “neutrality” result of the effect of child allowances as well as the positive effect of the public education policy on fertility are confirmed.
To sum up, our findings suggest that the public provision of child allowances misses their pro-natalist purpose while also dampening the formation of human capital. In contrast, an enlargement of the public provision of education monotonically increases both human capital and fertility and can therefore be used with pro-natalist purposes.
The remainder of the paper is organised as follows. Section 2 develops the model and analyses how child allowance and public education policies affect fertility. Section 3 concludes. Moreover,
in the Appendix we relax the assumption of fixed cost of children and introduce the hypothesis of time cost of children showing that the main results of the paper still hold.
2. The model
2.1. Firms
Consider a small open economy with perfect capital mobility that faces an exogenously given (constant) interest rate r . Production takes place according to a standard neoclassical constant-returns-to-scale technology f
kt,ht
, where k and t h represent the per worker stock of physical t capital and the endowment of human capital of each individual (i.e. efficient labour), respectively. Since capital is perfectly mobile, both the capital-labour ratio and the wage per unit of human capital w are fixed and constant.2.2. Government
At time t the government runs two distinct balanced budget policies: (i) a public education plan such that the per worker schooling expenditure at time t , g , is entirely financed with a constant t wage income tax at the rate 0 1, that is
t
t wh
g , (1)
where w is the working income of each member of the young adult (child bearing) generation and
t
h represents the endowment of human capital of each young person, and (ii) a child allowance policy: t t t wh n , (2)
where the left-hand side represents the per worker child benefit expenditure and the right-hand side the per worker tax receipts, where 0 is the benefit per child, n is the number of children at t t
and 0t 1 is the wage tax rate adjusted from period to period to balance out the budget.
2.3. Individuals
The representative individual entering the working period at t is endowed with a homothetic and separable utility function U and choose the consumption profile t c1,t and c2,t1, savings s and the t
number of children n to maximise the following lifetime logarithmic utility function (see, e.g., t Eckstein and Wolpin, 1985, Galor and Weil, 1996):
t
t
t t c c n U ln 1, ln 2,1 ln , (3) subject to
t t t t t t t s r c h w n p s c 1 1 1 , 2 , 1 , (4)where p0 is the (fixed) cost of child rearing,9 0 1 is the individual subjective discount factor, 0 captures the relative taste for children and 0 1 is the constant probability of surviving at the end of the working period.
Moreover, it is assumed the existence of a perfect annuity market where savings are intermediated through mutual funds. Therefore, old survivors will benefit not only from their own past saving plus interest, but also from the saving plus interest of those who have deceased.
The first order conditions for an interior solution are:
r c c t t 1 1 , 1 1 , 2 , (5)
9 This assumption will be later relaxed in the model presented in the Appendix, where a time cost of children is
p n c t t , 1 . (6)
Eq. (6) implies that p must hold to ensure the existence of a finite positive number of children, that is the child allowance must not be fixed at too high a level.
Combining Eqs. (5) and (6) with Eqs. (2) and (4), the demand for children is:
p
p h w n t t 1 1 . (7)2.4. Human capital formation
We assume that human capital evolves according to the following Cobb-Douglas learning technology:10 1 1 t t t BH G H , (8)
where H and G represent the aggregate level of human capital and the aggregate public schooling expenditure, respectively, B0 is a scale parameter and 01.11 Therefore, the accumulation of human capital in per worker terms reads as:
t t t t n g Bh h 1 1 , (9)
Exploiting Eqs. (1), (7) and (9) human capital at the steady state is:
w p p B h 1 1 1 . (10)From Eq. (10) the following remark holds:
10 See, for instance, Glomm and Ravikumar (1997) and Zhang et al. (2003), which however assume a stationary
population normalised to one and therefore in their models aggregate and per worker human capital coincide.
Remark 1. The steady-state human capital is: (i) a negative monotonic function of the child allowance; (ii) a positive monotonic function of the educational contribution rate.
While the latter claim trivially derives from the fact that human capital is accumulated through public schooling financed with a constant educational contribution rate, the former is explained by the fact that, since the impact effect of child allowances tends to increase the individual demand for children, then, loosely speaking, there exists a “congestion” effect that affects in a negative way the acquisition of human capital per child.
Below we analyse the effects of child allowances on the individual demand for children in the long run. The rate of fertility as a generic function of the child allowance can be written as follows:
*
*
* n , h
n . (11)
Totally differentiating Eq. (12) with respect to yields: * * * * * h h n n d dn . (12)
It is easy to see that, in the absence of human capital, the conventional wisdom (e.g. van Groezen et al. 2003, Fenge and Meier, 2005) according to which a child allowance policy stimulates fertility rates, holds. However, in line with the recent theoretical and empirical literature, when the realistic role of the human capital accumulation is taken into account, things are indeed different. In particular, Eq. (12) reveals that the final effect of a child allowance policy depends crucially on the relative strength of the substitution effect due to the reduced cost of children, and the income effect due to a change in the endowment of human capital.
Combination of (7) with (10) gives the equilibrium rate of fertility, that is:
B w 1
n . (13)
Proposition 1. The child policy is fertility-neutral.
Proof. The proof is straightforward since n/0. Q.E.D.
Proposition 2. An enlargement of the public education system always increases fertility.
Proof. The proof uses the following derivative:
1
1 0 Bw n . (14) Q.E.D.Proposition 3. The rate of fertility is independent of both the parents’ taste for children and rate of longevity.
Proof. The proof is obvious since n/ 0 and n/ 0. Q.E.D.
Therefore, from Eq. (13) and Propositions 1–3, we may derive the interesting conclusion that the long-run rate of fertility depends positively only on the human capital formation, while it is kept unaltered by a change in the child allowance.
Moreover, in the Appendix we introduce time cost of children to show that child allowances still remain fertility-neutral while the financing of public education can still be used to raise fertility, confirming that the main results of the present paper (e.g., Propositions 1 and 2) are a robust feature of OLG economies with endogenous fertility.
This paper contributes to highlight the link between child allowance policy, public education policy and the individual demand for children in a small open economy with overlapping generations. It is shown that the child allowance policy dampens the formation of human capital and its final effect on fertility depends crucially on the relative strength of the substitution effect due to the reduced cost of children and of the income effect due to the induced reduction in the endowment of human capital.
Contrary to the conventional views, child allowance policies have no effects on the individual choice of fertility, while public education can be used to enhance fertility. These results are robust regardless of whether child rearing costs are fixed or time consuming. Moreover the analysis has shown that the rate of fertility is independent of both the parents’ taste for children and the rate of longevity.
In conclusion the essential message is that the fertility rate in the long run positively depends only on human capital and, hence, an enlargement of the supply of public education always promotes the individual demand for children. The policy implication is that child allowances are not effective as a pro-natalist policy while also reducing the accumulation of human capital. In contrast, the provision of public education can be used as an instrument to increase both human capital and fertility.
Appendix
In this appendix we introduce time cost of children to investigate whether and how the results of the main text, i.e. Propositions 1, 2 and 3, are changed. We show that child allowances and public education policies still remain fertility-neutral and fertility-enhancing, respectively.
Define 0 z1 as the fraction of the time endowment each young adult devotes to raise a child. This hypothesis results in a trade-off between working in the labour market and raising children.
Indeed, if an individual has n children at t , she works only the fraction t 1znt of her time
endowment with z being the share of time spent raising her descendants. The higher z , the nt
lower the time spent working.
The government budget constraints of both the public education system and child allowance system modify, respectively, to become:
t
t t wh zn g 1 , (1’)
t
t t t wh zn n 1 , (2’)The representative individual chooses savings and fertility to maximise the lifetime utility function Eq. (3) subject to
t t t t t t t t s r c n n z h w s c 1 1 1 1 , 2 , 1 . (4’)The first order conditions for an interior solution gives Eq. (5) and
t t t t whz n c 1 , 1 . (6’)Eq. (6’) implies that whtz
1t
must hold to guarantee the existence of a finite positivenumber of children.
Combining Eqs. (5) and (6’) with (2’) and (4’) we get:
t t t t t t t t t n z h w n z h w n z h w n h w n 1 1 1 1 1 . (7’)If 0 the fertility rate were constant and independent of w . The existence of child allowances, ht
therefore, introduces an interesting feature: two positive real solutions for n exist in that case. In t
fact, with some algebraic manipulations from Eq. (7’) we find the following quadratic equation in
t
0 3 2 2 1 nt nt , (7’’) where :
1
2
1
0 1 whtz , 2:
1
whtz
1
2whtz
1
0 and
1
0 : 3 wht . Applying the Descartes’ rule of sign from Eq. (7’’), we find that two positive
roots for n do exist. In particular, they are given by: t
0 2 4 1 3 1 2 2 2 , 1 t n . (7’’.1) 0 2 4 1 3 1 2 2 2 , 2 t n . (7’’.2) Of course 4 1 3 0 2 2
must hold to ensure that n1,t and n2,t are two positive real numbers. Otherwise two complex conjugate roots would exist. Moreover, the second order conditions for a maximum are always satisfied.
Now, combining Eqs. (1’), (9) and using, alternatively, Eqs. (7’’.1) and (7’’.2) we find two different human capital accumulation functions, which imply, in turn, the existence of two different values of the steady state stock of human capital, h and 1 h .2 12 Below, we show numerically that the
steady state number of children corresponding to h and 1 h (that is, 2 n and 1 n ) is exactly the same 2 in both cases and it is independent of the preference parameters and , and the rate of longevity,
. Moreover, and most important, the two main results enunciated in the main text in the case of fixed children costs, as regards the effects of the two public policies on fertility (i.e., Propositions 1 and 2), are also confirmed in the case in which child rearing activities are time consuming: that is,12 Notice that in this paper we are not interested in analysing the dynamical properties of the human capital
accumulation functions corresponding to n1,t and n2,t, but only to the analysis of the effects of child and education
policies along the stationary path. Therefore, it is important to note that, although the steady state h1 is always unstable, while h2 can be either stable or unstable and the dynamics of human capital is non-monotonic in that case, the comparative statics analysis as regards the relationship child and education policies in the long run holds regardless of the stability properties of equilibria.
child allowances are fertility neutral, while the financing of public education is fertility-enhancing. In addition, we also study how the steady state human capital reacts when the policy variables vary. In order to show these results, we now take, only for illustrative purposes, the following configuration of parameters: B10, 0.5, w1, r2, z0.25, 0.3 and 1. Then in Tables 1.A and 1.B (2.A and 2.B) we choose 0.1 ( 0.1) and analyse how the two steady state values of both the fertility rate and human capital react to a rise in the child allowance (the educational contribution rate ) by assuming alternatively 0.8 and 0.3.13
Table 1.A. Human capital and fertility in the long run when varies ( 0.8). 05 . 0 0.08 0.1 0.12 0.15 0.2 1 h 1.021 1.633 2.042 2.45 3.063 4.084 1 n 2.15 2.15 2.15 2.15 2.15 2.15
Table 1.B. Human capital and fertility in the long run when varies ( 0.3). 05 . 0 0.08 0.1 0.12 0.15 0.2 2 h 0.599 0.959 1.199 1.439 1.798 2.398 2 n 2.15 2.15 2.15 2.15 2.15 2.15
From Tables 1.A and 1.B it is easy to see that a rise in the child allowance, for a given value of the educational contribution rate, is fertility neutral in both scenarios. Therefore, while the
13 Notice that we construct tables only for different values of the taste for children to show that child allowances are
fertility-neutral and education policies are fertility-enhancing, as in the model in the main text. We do not show the fertility neutrality result of both the psychological subjective discount factor and rate of longevity for the sake of brevity. However, numerical simulations as regards these two cases are available on request.
introduction and the rise in child allowances determine two different human capital equilibrium values, they preserve the uniqueness and the invariance of the fertility equilibrium value.
Table 2.A. Human capital and fertility in the long run when varies ( 0.8). 05 . 0 0.08 0.09 0.1 0.11 0.12 1 h 4.44 2.26 2.121 2.042 1.997 1.976 1 n 1.696 2 2.079 2.15 2.215 2.274
Table 2.B. Human capital and fertility in the long run when varies ( 0.3). 08 . 0 0.1 0.15 0.2 0.25 0.3 2 h 1.13 1.199 1.407 1.653 1.934 2.255 2 n 2 2.15 2.427 2.623 2.771 2.887
As can be seen from Tables 2.A and 2.B, a rise in the educational contribution rate, for a given value of the child allowance, promotes fertility. In particular, the fertility rate moves from 2 to 2.887 when 0.3, and from 1.696 to 2.274 when 0.8, as long as the educational tax rate increases from 8 per cent to 30 per cent of wage income). Therefore, even in the case of time cost of children the result presented in the main text for which the higher the educational contribution rate, the higher the fertility rate is preserved.
Moreover, it is worth noting that Tables 1 and 2 above also show that: (i) a rise in the child allowance, for a given value of the educational contribution rate, promotes human capital accumulation; (ii) a rise in the educational contribution rate, for a given value of the child allowance, deteriorates (promotes) human capital accumulation in the case of Table 2.A (Table 2.B). The results of changing the policy variables on human capital may be different from those obtained under the assumption of fixed cost of children. However, since in this paper we focused on
the relationship between child and education policies and fertility, a deeper investigation of their effects on human capital formation in the cases of fixed and time cost of children is left for future research.
To sum up, the main results of the present paper as regards the effects of child and education policies on fertility also hold with a different specification of the cost of children, confirming that they are a robust feature of an OLG economy with endogenous fertility when parents care about the number of children they have (i.e. weak form of altruism).
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