## Abstract

We examine the effects of child policies on both transitional dynamics and long-term demo-economic outcomes in an overlapping-generations neoclassical growth model à la Chakraborty (J Econ Theory 116(1):119–137, 2004) extended with endogenous fertility under the assumption of weak altruism towards children. The government invests in public health, and an individual’s survival probability at the end of youth depends on health expenditure. We show that multiple development regimes can exist. However, poverty or prosperity does not necessarily depend on the initial conditions, since they are the result of how a child policy is designed. A child tax, for example, can be used effectively to enable those economies that were entrapped in poverty to prosper. There is also a long-term welfare-maximising level of the child tax. We show that a child tax can be used to increase capital accumulation, escape from poverty and maximise long-term welfare also when (a) a public pay-as-you-go pension system is in place and (b) the government issues an amount of public debt. Interestingly, there also exists a couple child tax–health tax that can be used to find the second-best optimum optimorum. In addition, we show that results are robust to the inclusion of decisions regarding the child quantity–quality trade-off under the assumption of impure altruism. In particular, there exists a threshold value of the child tax below (resp. above) which child quality spending is unaffordable (resp. affordable) and different scenarios are in existence.

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## Notes

See also de la Croix et al. (2012) who revisit the serendipity theorem of Samuelson (1975) with fertility and longevity. In addition, another strand of literature deals with the effects of infectious diseases on life expectancy, fertility and economic and development (Young 2005; Chakraborty et al. 2010; Kalemli-Ozcan and Turan 2011; Kalemli-Ozcan 2012).

In a model with educational investments and endogenous fertility, Chen (2010) developed an OLG model showing that (a) with exogenous lifetime, multiple development regimes with club convergence exist when mortality rates are large, and (b) with endogenous lifetime, a unique stable steady state exist when mortality rates are small.

While public child support programmes have been extensively examined in the economic literature (Peters 1995; Momota 2000; van Groezen et al. 2003; Apps and Rees 2004; van Groezen and Meijdam 2008), the theoretical analysis of the effects of child taxes on long-term demo-economic outcomes is, to the best of our knowledge, relatively scarce. For instance, in the literature with endogenous fertility, Bental (1989) represents one of the first attempts to discuss the effects of child taxes in a model where children are considered as a capital good (old-age security hypothesis). He finds that a tax on children can achieve the optimal capital–labour ratio but fails to realise the optimal population growth. Recently, Fanti and Gori (2009) have shown that a child tax can be used to actually raise population growth in the long run, while also raising per worker GDP.

For instance, van Groezen et al. (2003, p. 237) argued that “The rate of fertility should therefore be treated as an endogenous variable, that is, as the result of a rational choice which is influenced by economic constraints and incentives. Economic theory can thus help in explaining why the observed decline in the (desired) number of children would occur.”

The way of modelling children as a desirable good that directly enters the parents’ utility is called a weak form of altruism towards children (Zhang and Zhang 1998).

For instance, the tax penalties imposed by the Chinese birth planning programme on parents with more than one child are currently computed as a fraction of either the disposable income of people living in urban areas or cash income (estimated by the local authorities) of people living in rural areas. In general, they are proportional to the number of children that exceeds the quota planned by the government.

A value of the output elasticity of capital (

*α*) of one third is usual to represent developed economies (Gollin 2002). According to Zhang et al. (2001), a value of the taste for the number of children included in the range 0.8 =*γ*= 1.5 is reasonable to capture the parents’ taste of children relative to material consumption in the utility function (according to the specification of preferences given by Eq. 5). Moreover, the values of both the scale parameter*A*and percentage of child cost on working income*q*, are chosen to get a value of long-run fertility close enough to unity to be as much as realistic as possible in representing actual developed economies. The value*δ*= 10 follows that of de la Croix and Ponthière (2010). Finally,*τ*= 0. 1 implies a ratio of health expenditure to per worker GDP of almost 7 % (which is an average value for developed countries; see World Health Organization 2010) when*b*= 0. We note that similar results (not reported in the paper for economy of space) can be found with different parameter values.Policies consisting in cash subsidies for children are largely adopted in several countries. As an example, in Italy, a 1,000 euro child grant for each newborn was introduced in the year 2005, while in Poland, every woman will benefit from a one-off 258 euro payment for every child, and women from poorer families will receive double the previous amount. Evidence of a positive impact of family policy programmes (national expenditure for child allowances, maternity and so on) on fertility and women labour participation in Western European countries can be found in Kalwij (2010).

For instance, the one-child policy had the effect of reducing the

*total fertility rate*in China from more than five births per woman in the 1970s to slightly less than two births per woman in recent years.In Section 5.1, we discuss the use of both health and child taxes as a second-best optimal policy.

Result 7 holds for several parameter constellations (with and without pensions) that are reported to save space.

To clarify this assumption, we note that utility function (25) can be rewritten as \(U_t =\ln ( {c_{1,t} } )+\pi _t \ln ( {c_{2,t+1} } )+( {\gamma _1-\gamma _2 })\ln ( {n_t })+\gamma _2 \ln ( {( {q+Q_t } )w_t n_t } )\). For

*γ*_{1}<*γ*_{2}, a sequence of feasible allocation may exist such that utility diverges to positive values. Then,*γ*_{1}>*γ*_{2}is required.

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## Acknowledgments

The authors gratefully acknowledge Riccardo Cambini, Giam Pietro Cipriani, Davide Fiaschi, Erasmo Papagni, Neri Salvadori and seminar participants at the International Conference in Honour of Salvatore Vinci: Poverty Traps: An Empirical and Theoretical Assessment, and the University of Siena, for stimulating discussions and valuable comments on an earlier draft. Special thanks go to Mauro Sodini for invaluable comments. We acknowledge that this work has been performed within the activity of the PRIN-2009 project “Structural Change and Growth”, MIUR (Ministry of Education), Italy. The authors are also indebted to two anonymous reviewers and the editor Prof. Alessandro Cigno for insightful comments and suggestions on earlier drafts.

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## Additional information

*Responsible editor*: Alessandro Cigno

## Appendices

### Appendix 1. Proof of Proposition 1

### Lemma 1

*Define the right-hand side of* Eq. 10.2
*as*
*G*(*k*). *Then, we have (1.i)*
*G*(0) = 0, (1.ii) *G*
^{″}(*k*) > 0 *for any*
*k* > 0, (1.iii) \(\lim _{k\to 0^{+}} G^{\prime } (k)=+\infty \), (1.iv) lim\(_{k\to +\infty } G^{\prime } (k)=0\) and (1.v) \(G^{\prime \prime }(k)\)
*admits at most three roots and*
\(G^{\prime \prime }(0)\ne 0\).

From Eq. 10.2, property (1.i) is straightforward. Differentiating the right-hand side of Eq. 10.2 with respect to *k* gives

By defining \(k^{\alpha \delta }:=x\) as a new supporting variable, (33) can be transformed to

Since no positive real roots of Eq. 34 exist, then (33) implies that \(G^{\prime }(k)>0\) for any *k* > 0. This proves (1.ii).

Moreover,

and

which prove (1.iii) and (1.iv), respectively. Now, differentiating (33) with respect to *k* gives

where \(\Lambda _{1}:=(1-\alpha ){\mathrm {B}}F>0\) and \(\Lambda _{4}:=\pi _{0} (1-\alpha )>0\). Knowing that \(k^{\alpha \delta }:=x\), Eq. 35 can be rewritten as follows:

From Eq. 36, it is clear that *f*(*k*, *x*) admits at most three roots for *x* and *f*(*k*, 0) ≠ 0. Hence, from Eq. 35, *G*
^{″}(*k*) admits at most three roots for *k* and \(G^{\prime \prime }(0)\ne 0\) for any *k* > 0. This proves (1.v).

Proposition 1 therefore follows. In fact, by properties (1.i) and (1.iii), zero is always an unstable steady state of Eq. 10.2. By (1.ii)–(1.iv), *G*(*k*) is a monotonic increasing function of *k* and eventually falls below the 45° line, so that at least one positive stable steady state exists for any *k* > 0.

Now, assume ad absurdum the existence of an odd number of equilibria. By (1.ii)–(1.iv), there cannot be an odd number of inflection points for any *k* > 0. By property (1.v), therefore, the number of inflection points of *G*(*k*) is either zero or two for any *k* > 0. Since at least one positive stable steady state exists, then for any *k* > 0 the phase map *G*(*k*) may intersect the 45° line from below *at most* once before falling below it. Hence, an even number of equilibria must exist. There are either two steady states, with the positive one being the unique asymptotically stable equilibrium, or *at most* one positive steady state separates the lowest asymptotically stable steady state from the highest asymptotically stable one, and, thus, the number of equilibria is four.

In addition, from Eqs. 35 and 36, we observe that if Λ_{2} > 0 and Λ_{3} > 0, then no inflection points of *G*(*k*) exist for any *k* > 0 and *G*
^{″}(*k*) < 0, and two steady states exist in that case. In contrast, *G*(*k*) has two inflection points for any *k* > 0 if at least either Λ_{2} < 0 or Λ_{3} < 0 is fulfilled, and, hence in this case, four steady states *can* exist.

### Appendix 2. Proof of Proposition 2

Differentiating Eq. 10.2 with respect to *b* gives

where \(\partial D/\partial b=(1-{\alpha })A>0\) for any \(b\in [0,+\infty )\). Since *G*
*b*
*″*(*k*, *b*) > 0 for any *k* > 0 and *b* ∈ 0, + ∞), then for any *k*
^{∗} > 0 such that *G*(*k*
^{∗}, *b*
_{1}) < *k*
^{∗}with *b*
_{1} ∈ 0, + ∞), a threshold value *b*
^{∗} > *b*
_{1} exists such that *G*(*k*
^{∗}, *b*
^{∗}) = *k*
^{∗}. Therefore, *G*(*k*
^{∗}, *b*) > *k*
^{∗}holds for any *b* > *b*
^{∗}.

### Appendix 3. Public debt

In this appendix, we show that the main results of this paper also hold when the government at each date *t* issues an amount *Z*
_{
t
} of public debt and levies lump sum taxes (\((\tau_{t}^{z})\)) on the young workers (Diamond 1965; Jaeger and Kuhle 2009; Spataro and Fanti 2011).

The level of national debt evolves according to the following equation \(Z_{t+1}=Z_{t}R_{t}-\tau_{t}^{z}N_{t}\), which can be transformed in per-worker terms as follows:

where *z*
_{
t
} = *Z*
_{
t
} / *N*
_{
t
}. Following Diamond (1965), we assume that the (non-negative) level of debt is constant over time, i.e. *z*
_{
t + 1} = *z*
_{
t
} = *z*. Thus, (38) becomes the following:

The maximisation of expected utility function (5) subject to the lifetime budget constraint

gives the following demand for children and savings (upon substitution of Eq. 39 for \(\tau_{t}^{z}\)), respectively:

Market clearing can now be expressed as *n*
_{
t
}(*k*
_{
t + 1} + *z*) = *s*
_{
t
}. Then, by using (1), (2), (9), (41) and (42), we get

Since the capital accumulation equation is the same as in the basic model of Section 2 minus a constant, the main findings of the paper also hold qualitatively when the government issues a constant (non-negative) amount of public debt.

We have not pursued this analysis further to save space. However, numerical simulations (available on request) show that a child tax can be used to permanently escape from poverty and maximise steady-state welfare, and their values are higher than when *z* = 0. Unlike the basic model, a child tax now increases the debt per child (*z* / *n*
_{
t
}) due to a lower population growth in the short run. However, the increase in capital accumulation implies that the ratio of debt per young person over GDP per young person goes down in the long term.

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Fanti, L., Gori, L. Endogenous fertility, endogenous lifetime and economic growth: the role of child policies.
*J Popul Econ* **27**, 529–564 (2014). https://doi.org/10.1007/s00148-013-0472-x

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DOI: https://doi.org/10.1007/s00148-013-0472-x

### Keywords

- Child policy
- Endogenous fertility
- Health
- Life expectancy
- OLG model

### JEL Classifications

- I1
- J13
- O4