Childlessness and Economic Development: A Survey

Abstract

This chapter shows why, beyond average fertility, childlessness matters in itself. Childlessness reacts to economic incentives in a peculiar way, which cannot be described by usual economic models of fertility. We make a distinction between childlessness due to poverty and childlessness due to economic opportunities. It allows to better understand the dynamics of childlessness along the history but also why the extensive margin of fertility (childlessness) does not adjust to economic shocks and development policies in the same way as the intensive margin (number of children of mothers). Introducing marriage into this framework provides new insights: higher educational homogamy per se decreases childlessness as it favors marriage. Nevertheless, because educational homogamy is most of the time accompanied by a rise in female education, it may also be associated with increases in childlessness in the data.

Appendix: Proof of Proposition 3.1

The aim of this appendix is not to determine how people behave for any couple {w,  Ω} but to show that there exists a non-empty state space $\mathcal {H}$ such that agents behave the way described in Proposition 3.1. To do so, we first solve the maximization problem described in the core of the chapter under the following constraint: $n\in ]0,\frac {1}{\tau }[$ and e > 0. In this pure interior regime, we get that:

$$\displaystyle \begin{aligned} \begin{array}{rcl} c^*&\displaystyle = &\displaystyle \displaystyle\frac{(1+\tau\nu)w+\rho\pi+\Omega}{1+\alpha+\phi\beta} {} \end{array} \end{aligned} $$

(3.7)

$$\displaystyle \begin{aligned} \begin{array}{rcl} n^*&\displaystyle =&\displaystyle \displaystyle\frac{\alpha}{1+\alpha+\phi\beta}\frac{(1+\tau\nu)w+\rho\pi+\Omega}{\tau w}-\nu \end{array} \end{aligned} $$

(3.8)

$$\displaystyle \begin{aligned} \begin{array}{rcl} e^*&\displaystyle =&\displaystyle \displaystyle\frac{\phi\beta}{1+\alpha+\phi\beta}\frac{(1+\tau\nu)w+\rho\pi+\Omega}{\rho}-\pi {} \end{array} \end{aligned} $$

(3.9)

From this, we can deduce the level of wages under which such an interior regime may exist and prevail. We get that n^∗ > 0 ⇔ $w< \frac {\alpha (\rho \pi +\Omega )}{(1+\phi \beta )\tau \nu -\alpha }\equiv w_n$ ; e^∗ > 0 ⇔ $\frac {(1+\alpha )\rho \pi -\phi \beta \Omega }{\phi \beta (1+\tau \nu )}\equiv w_e$.

Simple computations indicate that w_n > w_e ⇔ $\Omega <\frac {(\tau \nu -\alpha )\rho \pi }{\tau \phi \beta \nu }\equiv \hat {\Omega }$. Let’s now assume that $\Omega <\hat {\Omega }$, we get that ∀w ∈ ]w_e, w_n[, the purely interior regime as described in Eqs. (3.7)–(3.9) prevails. For any w ≥ w_n, n^∗ = 0 implying e^∗ = 0 by definition and so c^∗ = w +  Ω.

We now have to analyze cases where w ≤ w_e. In this situation, we have to solve the maximization problem under the constraint that e^∗ = 0 and $n^*\in ]0,\frac {1}{\tau }[$. In this situation, we obtain that:

$$\displaystyle \begin{aligned} \begin{array}{rcl} c^*&\displaystyle = &\displaystyle \displaystyle \frac{(1+\tau\nu)w+\Omega}{1+\alpha} {} \end{array} \end{aligned} $$

(3.10)

$$\displaystyle \begin{aligned} \begin{array}{rcl} n^*&\displaystyle =&\displaystyle \displaystyle\frac{\alpha}{1+\alpha}\frac{(1+\tau\nu)w+\Omega}{\tau w}-\nu {} \end{array} \end{aligned} $$

(3.11)

$$\displaystyle \begin{aligned} \begin{array}{rcl} e^*&\displaystyle =&\displaystyle 0 {} \end{array} \end{aligned} $$

(3.12)

A simple inspection of Eq. (3.11) indicates that $n^*\leq \frac {1}{\tau }$ ⇔ $w\geq \frac {\alpha \Omega }{1+\tau \nu }\equiv \underline {w}$. A condition of existence for this regime is that $\underline {w}<w_e$ what is satisfied if $\Omega >\frac {\rho \pi }{\phi \beta }\equiv \tilde {\Omega }$. Let’s assume that this condition on Ω is fulfilled too. We then get that $\forall w\in ] \underline {w},w_e]$, the optimal behavior of our representative agent is represented by the set of Eqs. (3.10)–(3.12). The last situation we have to explore is $w\leq \underline {w}$. In this situation, as $n^*=\frac {1}{\tau }$, e^∗ = 0, we get:

$$\displaystyle \begin{aligned} \begin{array}{rcl} c^*&\displaystyle = &\displaystyle \Omega \\ n^*&\displaystyle =&\displaystyle \frac{1}{\tau} \\ e^*&\displaystyle =&\displaystyle 0 \end{array} \end{aligned} $$

The advised reader would have noticed that we had to assume $\Omega >\tilde {\Omega }$ and $\Omega <\hat {\Omega }$. For Proposition 3.1 to be valid, there should exist a state-space allowing the condition $\tilde {\Omega }<\hat {\Omega }$ to be fulfilled. We obtain that:

$$\displaystyle \begin{aligned} \tilde{\Omega}<\hat{\Omega} \iff \frac{\tau\nu-\alpha}{\tau\nu}<1 \end{aligned}$$

what is always satisfied. We then get that ∀ $\Omega \in ]\tilde {\Omega },\hat {\Omega }[$, the representative agent behaves as described in Proposition 3.1 what validates this latter.