Child mortality and fertility: public vs private education

Abstract

How does the effect of child mortality reductions on fertility and education vary across educational systems? To answer this question, we develop an overlapping-generations model where altruistic parents care about both the number and human capital of their surviving children. We find that, under a private education system, if income is low initially, the economy converges to a Malthusian stagnation steady state. For a high level of initial income, the economy reaches a growth path in which children’s education rises and fertility decreases with income. In the growth regime under private education, exogenous shocks that lower child mortality are detrimental for growth: fertility increases and education declines. In contrast, under a public education system, the stagnation steady state does not exist, and health improvement shocks are no longer detrimental for growth. We therefore offer a new rationale for the introduction of public education.

Appendix

1.1 A Optimal conditions with private education

Given agents maximization problem by Eqs. 10, 11, and 12, the first-order conditions yield Eqs. 13 and 14 for the optimal number of children and the optimal education spending, respectively. Substituting Eqs. 13 and 14 into the budget constraint, we obtain the optimal consumption as follows:

$$c_{t}^{i}=h_{t}^{i}\left( 1-\gamma \right) ,$$

from which, consumption is above the subsistence level, i.e., $\tilde{c},$ if:

$$h_{t}^{i}\geq \frac{\tilde{c}}{\left( 1-\gamma \right) },$$

(38)

where we define the human capital level $\tilde{h}=\tilde{c}/\left( 1-\gamma \right)$ such that $c_{t}^{i}=\tilde{c}.$

When $c_{t}^{i}>\tilde{c}$, a fraction 1 − γ of $h_{t}^{i}$ is devoted to the consumption and a fraction γ of $h_{t}^{i}$ is devoted to children, that is:

$$\begin{array}{rll}c_{t}^{i}&=&h_{t}^{i}\left( 1-\gamma \right) ,\\[5pt] \left( \frac{\phi h_{t}^{i}}{\pi (h_{t}^{i})}+e_{t}^{i}\right) n_{t}^{i}&=&\gamma h_{t}^{i}, \end{array}$$

where $n_{t}^{i}=\pi (h_{t}^{i})N_{t}^{i}$ is the number of surviving children.

When $c_{t}=\tilde{c},$ the difference between income and the subsistence consumption is devoted to children, that is:

$$\begin{array}{rll} c_{t}&=&\tilde{c},\nonumber\\ [5pt] \left(\frac{\phi h_{t}^{i}}{\pi (h_{t}^{i})}+e_{t}^{i}\right) n_{t}^{i}&=&h_{t}^{i}-\tilde{c}.\end{array}$$

(39)

Given Eq. 14, there is a corner solution for education if:

$$h_{t}^{i}\leq \frac{\pi \left( h_{t}^{i}\right) \theta }{\phi \alpha },$$

(40)

where, using Eq. 6, we obtain the following solutions for $h_{t}^{i}$:

$$h_{1}=\frac{-\left( \phi \alpha -\overline{\pi }\theta \right) +\sqrt{\left( \phi \alpha -\overline{\pi }\theta \right) ^{2}+4\phi \alpha \underline{\pi } \theta }}{2\phi \alpha }>0,$$

(41)

$$h_{2}=\frac{-\left( \phi \alpha -\overline{\pi }\theta \right) -\sqrt{\left( \phi \alpha -\overline{\pi }\theta \right) ^{2}+4\phi \alpha \underline{\pi } \theta }}{2\phi \alpha }<0, $$

(42)

where h ₁ > 0 and h ₂ < 0.

We define the human capital $h_{1}=\underline{h}$ such that, when $h_{t}^{i}< \underline{h}$, the optimal choice for education is zero. Given the human capital level $\tilde{h}$, it is possible to distinguish two cases depending on if $\tilde{h}<\underline{h}$ or $\tilde{h}>\underline{h}$. We assume that is $\tilde{h}<\underline{h}.$ Hence, when $\tilde{c} < h_{t}^{i} < \tilde{h},$ the optimal choice for education is zero, that is:

$$e_{t}^{i}=0,$$

substituting this solution into Eq. 39, the optimal number of children, i.e., $N_{t}^{i}$, is given by:

$$N_{t}^{i}=\frac{1}{\phi }\left( 1-\frac{\tilde{c}}{h_{t}^{i}}\right) .$$

(43)

When $h_{t}^{i}=\tilde{h},$ it follows that:

$$N_{t}^{i}=\frac{\gamma }{\phi }.$$

When $\tilde{h}<h_{t}^{i}\leq \underline{h}$, consumption is above the subsistence level $c_{t}>\tilde{c}$ and the optimal spending in education is zero. Hence, agents maximize the following utility function:

$$U=(1-\gamma )\log \left[ h_{t}^{i}\left( 1-\frac{\phi n_{t}^{i}}{\pi \left(h_{t}^{i}\right)}\right) \right] +\gamma \log \left(n_{t}^{i}h_{t}^{i}{}^{1-\alpha }\theta ^{\alpha }\right), $$

which yields the following optimal decision rule for the number of children:

$$N_{t}^{i}=\frac{\gamma }{\phi }.$$

When $h_{t}^{i}>\underline{h}$, the optimal number of children and the optimal choice for education are given by Eqs. 13 and 14, respectively.

1.1.1 A.1 Optimal fertility

The optimal fertility $N_{t}^{i},$ given by Eq. 17, when $\tilde{c} < h_{t}^{i} < \tilde{h},$ increases in human capital and has a concave shape with respect to $h_{t}^{i},$ that is:

$$\frac{\partial N_{t}^{i}}{\partial h_{t}^{i}}=\frac{\tilde{c}}{\phi h_{t}^{2} }>0,$$

(44)

and:

$$\frac{\partial ^{2}N_{t}^{i}}{\partial (h_{t}^{i})^2}=-\frac{2\tilde{c}}{\phi h_{t}^{3}}<0.$$

(45)

When $h_{t}^{i}>\bar{h}$, the optimal number of children decreases in $h_{t}^{i},$ that is:

$$\frac{\partial N_{t}^{i}}{\partial h_{t}^{i}}=-\frac{\gamma \left( 1-\alpha \right) \theta \left[ \pi \left( h_{t}^{i}\right) -h_{t}^{i}\pi ^{\prime }\left( h_{t}^{i}\right) \right] }{\left[ \phi h_{t}^{i}-\pi \left( h_{t}^{i}\right) \theta \right] ^{2}}<0,$$

(46)

since $\pi \left( h_{t}^{i}\right) -h_{t}^{i}\pi ^{\prime }\left( h_{t}^{i}\right) >0.$ Now, we proceed to analyze the optimal number of surviving children $n_{t}^{i}$, that is:

$$n_{t}^{i}=\left\{ \begin{array}{ll} \qquad0&\qquad\qquad\text{if }h_{t}^{i}\leq \tilde{c}, \\ \displaystyle\frac{\pi \left( h_{t}^{i}\right) }{\phi }\left( 1-\displaystyle\frac{\tilde{c}}{h_{t}^{i} }\right)& \qquad\qquad\text{if \ }\tilde{c}<h_{t}^{i}< \tilde{h},\\ \displaystyle\frac{\pi \left( h_{t}^{i}\right) \gamma }{\phi }&\qquad\qquad\text{if \ }\tilde{h}\leq h_{t}^{i}\leq \underline{h},\\ \qquad\displaystyle\frac{\pi \left( h_{t}^{i}\right) \gamma h_{t}^{i}\left( 1-\alpha \right) }{ \phi h_{t}^{i}-\pi \left(h_{t}^{i}\right)\theta }& \qquad\qquad\text{if \ }h_{t}^{i}>\underline{h}. \end{array} \right. $$

(47)

Given Eq. 44, when $\tilde{c}\leq h_{t}^{i}\leq \underline{h}$, the optimal number of surviving children increases in human capital, that is:

$$\frac{\partial n_{t}^{i}}{\partial h_{t}^{i}}=\frac{\partial \pi \left( h_{t}^{i}\right) }{\partial h_{t}^{i}}N_{t}^{i}+\frac{\partial N_{t}^{i}}{ \partial h_{t}^{i}}\pi \left( h_{t}^{i}\right) >0. $$

(48)

When $h_{t}^{i} < \underline{h}$, the optimal number of surviving children decreases in $h_{t}^{i},$ that is:

$$\frac{\partial n_{t}^{i}}{\partial h_{t}^{i}}=\frac{\gamma \left( 1-\alpha \right) \left\{ \phi \left( h_{t}^{i}\right) ^{2}\pi ^{\prime }\left( h_{t}^{i}\right) -\theta \left[ \pi \left( h_{t}^{i}\right) \right] ^{2}\right\} }{\left[ \phi h_{t}^{i}-\pi \left(h_{t}^{i}\right)\theta \right] ^{2}}<0, $$

(49)

if:

$$\phi <\frac{\theta \left( \underline{\pi }\right) ^{2}}{\overline{\pi }- \underline{\pi }}. $$

1.1.2 A.2 Optimal education

Given the optimal education choice in Eq. 16, when $h_{t}^{i}>\underline{h}$, the spending in education of each child increases in $h_{t}^{i}$ and has a concave shape with respect to $h_{t}^{i},$ that is:

$$\frac{\partial e_{t}^{i}}{\partial h_{t}^{i}}=\frac{\phi \alpha \left[ \pi \left(h_{t}^{i}\right)-h_{t}^{i}\pi ^{\prime }\left(h_{t}^{i}\right)\right] }{\left[ \pi \left(h_{t}^{i}\right)\right] ^{2}\left( 1-\alpha \right) }>0, $$

since $\pi (h_{t}^{i})-h_{t}^{i}\pi ^{\prime }(h_{t}^{i})>0.$

The second derivative is given as follows:

$$ \frac{\partial ^{2}e_{t}^{i}}{\partial \left( h_{t}^{i}\right) ^{2}}=-\frac{ 2\phi \alpha \underline{\pi }\left( \overline{\pi }-\underline{\pi }\right) }{\left( 1-\alpha \right) \left( h_{t}^{i}\overline{\pi }+\underline{\pi } \right) ^{3}}<0. $$

1.2 B Human capital

1.2.1 B.1 Private education

Given human capital accumulation in Eq. 24, the economy shows multiple development paths if:

$$h^{i}_{t+1}\left(\underline{h}\right)<\underline{h},$$

(50)

which is satisfied if:

$$\phi <\frac{1}{\alpha }\left( \frac{\underline{\pi }+\overline{\pi }\theta }{ 1+\theta }\right)$$

(51)

When $h_{t}^{i}\leqslant \underline{h},$ the economy shows the stable steady state h _L , that is:

$$h_{L}=\theta ,$$

where:

$$h^{\prime}_{t+1}(h_{L})=\left( 1-\alpha \right) <1.$$

When $h_{t}>\underline{h},$ the economy grows in the long run at a constant rate:

$$\lim\limits_{h_{t}\rightarrow \infty }\frac{h_{t+1}^{i}}{h_{t}^{i}}=\frac{\alpha \phi }{\overline{\pi }\left( 1-\alpha \right) }, $$

if:

$$\phi >\frac{\overline{\pi }\left( 1-\alpha \right) }{\alpha }.$$

(52)

1.2.2 B.2 Public education

1.2.2.1 B.2.1 Relative human capital

We define the dynamic of the relative human capital of a poor agent given by Eq. 35 as:

$$\hat{h}_{t+1}^{P}=\Psi \left(\hat{h}_{t}^{P}\right).$$

(53)

It increases with respect to $\hat{h}_{t}^{P}$, that is:

$$\Psi ^{\prime }\left(\hat{h}_{t}^{P}\right)=\frac{4\left(1-\alpha \right)}{\left[ 1+\left( \frac{2 }{\hat{h}_{t}^{P}}-1\right) ^{1-\alpha }\right] ^{2}\left(\hat{h} _{t}^{P}\right)^{2}\left( \frac{2}{\hat{h}_{t}^{P}}-1\right) ^{\alpha }}>0, $$

(54)

and $\Psi ^{\prime \prime }(\hat{h}_{t}^{P})<0$ for $\hat{h}_{t}^{P}\leq 1$ and $\Psi ^{\prime \prime }(\hat{h}_{t}^{P})>0$ for $\hat{h}_{t}^{P}>1$, that is (see Fig. 6):

$$\Psi ^{\prime \prime }\left(\hat{h}_{t}^{P}\right)\!=\!\frac{8(1\!-\!\alpha )\left( \frac{2}{ \hat{h}_{t}^{P}}\!-\!1\right) ^{\alpha }\left\{ \left( 2\!-\!\hat{h}_{t}^{P}\right) \left( \hat{h}_{t}^{P}\!-\!\alpha \right) \!+\!\left( \frac{2}{\hat{h}_{t}^{P}} \!-\!1\right) ^{\alpha }\hat{h}_{t}^{P}\left( 2\!-\!\hat{h}_{t}^{P}\!+\!\alpha \right) \right\} }{\left( 2\!-\!\hat{h}_{t}^{P}\right) \hat{h}_{t}^{P}\left\{ 2\!+\!\left[ \left( \frac{2}{\hat{h}_{t}^{P}}\!-\!1\right) ^{1\!-\!\alpha }\!-\!1\right] \hat{h} _{t}^{P}\right\} ^{3}} $$

Equation 53 shows two steady states 0 and 1. The steady state 0 is unstable, that is:

$$\Psi ^{\prime }(0)=\infty$$

(55)

and the steady state 1 is stable, that is:

$$\Psi ^{\prime }(1)=(1-\alpha )<1.$$

(56)

1.2.2.2 B.2.2 Average human capital

The average human capital of children given by Eq. 36 increases with respect to average human capital of parents. In particular, we define $e(\overline{h}_{t})=\overline{e}_{t}$:

$$ \frac{\partial \overline{h}_{t+1}}{\partial \overline{h}_{t}}=\left[ \theta +e\left(\overline{h}_{t}\right)\right] ^{\alpha }\left[ \frac{\alpha \overline{h} _{t}e^{\prime }\left(\overline{h}_{t}\right)}{\theta +e\left(\overline{h}_{t}\right)}+1\right] >0, $$

where $e^{\prime }(\overline{h}_{t})>0,$ that is:

$$ e^{\prime }\left(\overline{h}_{t}\right)=\frac{\left( 1-\gamma \right) \phi \tau }{ \gamma }\left[ \frac{\pi \left(\overline{h}_{t}\right)-\pi ^{\prime }\left(\overline{h}_{t}\right) \overline{h}_{t}}{\left[ \pi \left(\overline{h}_{t}\right)\right] ^{2}}\right] >0, $$

since, from Eq. 6, we have that $\pi (\overline{h}_{t})/ \overline{h}_{t}\pi ^{\prime }(\overline{h}_{t})>1.$

Finally, the economy grows in the long run, that is:

$$\lim\limits_{\overline{h}_{t}\rightarrow \infty }\frac{\overline{h}_{t+1}}{ \overline{h}_{t}}=\frac{\tau (1-\gamma )\phi }{\overline{\pi }\gamma }, $$

(57)

if the following condition holds:

$$\phi >\frac{\overline{\pi }\gamma }{\tau (1-\gamma )}.$$

(58)