2.1 Demographics and endowments

Individuals live for three periods: childhood, young adulthood, and old adulthood. Children do not make any economic decisions, but they can acquire skills. Young adults have one unit of productive time and are endowed with skills that they acquire during their childhood. They make the relevant economic decisions, including investment decisions. Old adults do not work and simply consume their savings.

2.2 Production

The consumption good is produced with a technology that uses capital, K, and efficiency units of labor, L, as inputs, such that:Footnote 4

(1) $$\begin{equation}Y=AK^{\alpha}L^{1-\alpha}, \ \alpha\in(0,1), \ A>0.\end{equation}$$

Capital depreciates fully after use. Let w be the wage rate and let R be the rental price of capital. Profit maximization implies that inputs are paid according to their marginal productivity, such that:

(2) $$\begin{eqnarray}w & = & (1-\alpha)AK^{\alpha}L^{-\alpha},\end{eqnarray}$$

(3) $$\begin{eqnarray}R & = & \alpha AK^{\alpha-1}L^{1-\alpha}.\quad\end{eqnarray}$$

2.3 Households

Fertility: Couples can have up to $N>0$ children, and they can control their family size, n, by investing in contraceptive use, such that:

(4) $$\begin{equation}n=N-\theta q, \ \theta>0,\end{equation}$$

where $q\geq0$ is the investment in contraception and $\theta$ is related to the efficiency of contraception on birth control. Contraception is costly and the relative price of contraception is $\phi_q\geq0$ .

Human capital: Parents invest in the education of their children, $e\geq0$ , such that the human capital of their children is given by:

(5) $$\begin{equation}h^{\prime}=h(e)=e^{\zeta}, \ \zeta\in(0,1).\end{equation}$$

Investment in education is in terms of the consumption good. Children are also time-consuming. Each child takes a fraction $\chi\in(0,1)$ of her parents’ time endowment. We assume that parents are able to provide some hours in the labor market even when they have the maximum amount of children, that is, $\chi N<1$ .

Preferences and optimal decisions: Consumption of couples during the young adulthood period is denoted by $c_{y}$ , while $c^{\prime}_{o}$ denotes consumption of the couple in the next period, when old. Preferences of households are represented by:

(6) $$\begin{equation}U(c_{y}, c^{\prime}_o, n, h^{\prime})=\log(c_{y})+\beta\log(c^{\prime}_{o})+\gamma\log(n)+\xi\log(h^{\prime})+\psi\left(\frac{n-\bar{n}}{\bar{n}}{}\right),\end{equation}$$

where $\beta$ , $\gamma$ , $\xi$ , and $\psi$ are positive numbers.Footnote 5 The variable $\bar{n}$ is the average fertility in society and describes social norms in reproduction behavior.Footnote 6

Let s denote savings during the young adulthood period. The problem of the couple is to choose $c_{y}$ , $c^{\prime}_{o}$ , q, s, and e to maximize (6) subject to (4) and (5), and the following budget constraints:

(7) $$\begin{eqnarray}c_{y}+s+\phi_qq+en=wh(1-\chi n),\end{eqnarray}$$

(8) $$\begin{eqnarray}c^{\prime}_{o}=R^{\prime}s.\end{eqnarray}$$

Equation (7) states that consumption plus savings and expenditures on contraceptionFootnote 7 and education equals income. Equation (8) implies that old couples consume their savings from the young adulthood period. Whenever $q>0$ , then the equations which describe the solution of this problem after imposing the symmetric equilibrium condition that $n=\bar{n}$ areFootnote 8

(9) $$\begin{eqnarray}c_{y}=\frac{1}{(1+\beta+\gamma+\psi)}\left(wh-\frac{\phi_q}{\theta}N\right),\end{eqnarray}$$

(10) $$\begin{eqnarray}s=\frac{\beta}{(1+\beta+\gamma+\psi)}\left(wh-\frac{\phi_q}{\theta}N\right), \mbox{ and } c^{\prime}_o=R^{\prime}s, \end{eqnarray}$$

(11) $$\begin{eqnarray}e=\frac{\xi\zeta}{(\gamma+\psi-\xi\zeta)}\left(wh\chi -\frac{\phi_q}{\theta}\right),\end{eqnarray}$$

(12) $$\begin{eqnarray}q=\frac{N}{\theta}-\frac{(\gamma+\psi-\xi\zeta)}{\theta(1+\beta+\gamma+\psi)}\left(\frac{wh-\frac{\phi_q}{\theta}N}{wh\chi-\frac{\phi_q}{\theta}}\right),\end{eqnarray}$$

(13) $$\begin{eqnarray}n=\frac{(\gamma+\psi-\xi\zeta)}{(1+\beta+\gamma+\psi)}\left(\frac{wh-\frac{\phi_q}{\theta}N}{wh\chi-\frac{\phi_q}{\theta}}\right). \end{eqnarray}$$

We make the following assumption:

The assumption that $N\chi<1$ implies that even when fertility is at its maximum ( $q=0$ ), couples still supply a positive number of hours to the labor market. The second part of the assumption implies that when the price of modern contraceptive methods is zero ( $\phi_q=0$ ), then fertility is lower than the case in which there is no investment in modern contraceptive methods ( $q=0$ ).

Observe that when $\phi_q$ goes to zero, then fertility does not depend on labor income (wh). This is because when income rises the opportunity cost (time cost) of having more children rises (substitution effect), but since children are a normal good, then the income effect induces parents to have more children. With log-utility these two effects cancel each other out, and when $\phi_q=0$ then fertility does not depend on income—see equation (13).Footnote 9This is explained in Doepke (Reference Doepke2015) and Jones et al. (Reference Jones, Schoonbroodt, Tertilt and Shoven2010). When $\phi_q$ is positive, then there is a negative association between fertility and income, as reported in the data. In this case, richer parents can increase the intensity of their use of contraceptive methods in order to control family size.

Notice that whenever the price of modern contraceptives falls, then fertility decreases and this relationship is stronger for a higher $\psi$ , which measures how social norms affect reproduction behavior.

Proof. See Appendix A.

When the price of contraception falls, it becomes cheaper to lower the number of children. As more people do this, others follow suit due to the social norms. With stronger social norms (higher $\psi$ ), this feedback becomes more powerful.

One can argue that it is not necessary to explicitly add investment in contraceptives into a standard quantity–quality fertility model because parameter $\chi$ , which corresponds to the time cost of children, could capture that investment. Better access to contraceptives could be translated into a rise in parameter $\chi$ such that it would raise the quality of children (e) as well as reduce their quantity (n). In fact, the proportional changes in n and e due to a proportional variation in $\chi$ have opposite signs but equal magnitude. A fall in the price of contraceptives ( $\phi_q$ ) generates not only different quantitative but also qualitative effects. Indeed, a fall in $\phi_q$ also increases e and reduces n but observe that parameter $\chi$ does not affect the consumption-saving decision, while the price of contraceptives does. In addition, family planning interventions which reduce the price of contraceptives have strong effects on the quantity and quality of children when income levels are low. Proposition 2 summarizes these findings.

Proof. See Appendix A.

2.4 Closing the model

Let P denote the number of young adult households such that $P^{\prime}=nP$ . In equilibrium, demand equals supply in all markets. In the labor market, this means that $L=P(1-\chi n)h$ , and in the capital market, $K^{\prime}=Ps$ . Let k denote physical capital per young household. In equilibrium with $q>0$ , it can be shown that $h^{\prime}=Dk^{\prime\zeta}$ with $D=\left(\frac{\xi\zeta}{\beta}\right)^{\zeta}>0$ , and $w(k)=(1-\alpha)D^{-\alpha}Ak^{\alpha(1-\zeta)}(1-\chi n(k))^{-\alpha}$ . When $q=0$ , we also have that $h^{\prime}=Dk^{\prime\zeta}$ , and $w(k)=(1-\alpha)D^{-\alpha}(1-\chi N)^{-\alpha}k^{\alpha(1-\zeta)}$ . In addition,

(14) $$\begin{equation}n(k)=\min\left\{N,\frac{(\gamma+\psi-\xi\zeta)}{(1+\beta+\gamma+\psi)}\left(\frac{(1-\alpha)D^{-\alpha}Ak^{\alpha+\zeta(1-\alpha)}(1-\chi n(k))^{-\alpha}-\frac{\phi_q}{\theta}N}{(1-\alpha)D^{-\alpha}Ak^{\alpha+\zeta(1-\alpha)}(1-\chi n(k))^{-\alpha}\chi-\frac{\phi_q}{\theta}}\right)\right\}.\end{equation}$$

Then the following proposition summarizes the fertility choice and our main results.

Proof. See Appendix A.

Part (i) of Proposition 3 shows that, for extremely low levels of development, individuals are too poor to use contraception and fertility is at its maximum. After a certain threshold of development is reached ( $\underline{k}$ ), people start using contraception and fertility declines. The richer the economy becomes, the lower is the fertility level (part (ii)). This happens since, as people become richer, contraception becomes relatively cheaper. Finally, part (iii) shows that family planning interventions are more powerful in the presence of social norms due to the feedback from one individual’s decisions onto others.

We can also see these results graphically. Figure 1 displays how the fertility rate changes with physical capital stock in an economy without reproductive externality (gray solid line) and with reproductive externality (black solid line). Fertility is higher in an economy with reproductive externalities for any level of capital stock above $\underline{k}(\psi=0)$ . When fertility starts to fall with capital levels, then the slope is steeper in an economy with a positive $\psi$ . Therefore, for any $k\geq\underline{k}(\psi)$ , family planning interventions that decrease the price of contraceptives have stronger effects on fertility in the presence of social norms in fertility, as can be seen by the vertical difference between the solid line and the dashed line in these two economies—one displayed by gray lines ( $\psi=0$ ) and the other displayed by black lines ( $\psi>0$ ).

Figure 1. Fertility and capital levels.

Note: Fertility and physical capital stock for two economies: one without reproductive externality ( $\psi=0$ )—gray lines; and another with reproductive externality ( $\psi>0$ )—black lines. The dotted lines correspond to the case of family planning interventions which reduce $\phi_q$ for these two economies.

We can now move to finish the solution of the model. The condition that equilibrates the capital market implies that

(15) $$\begin{equation}k^{\prime}=G(k)=\left \{\begin{array}{cc}\frac{\beta(1-\alpha)D^{-\alpha}A(1-\chi N)^{-\alpha}k^{\alpha+\zeta(1-\alpha)}}{(1+\beta+\xi\zeta)N} &\mbox{for $k\leq \underline{k}$}, \\& \\\frac{\beta\left((1-\alpha)D^{-\alpha}Ak^{\alpha+\zeta(1-\alpha)}(1-\chi n(k))^{-\alpha}\chi-\frac{\phi_q}{\theta}\right)}{\gamma+\psi-\xi\zeta} &\mbox{for $k> \underline{k}$}.\end{array}\right.\end{equation}$$

We also have that

(16) $$\begin{equation}h^{\prime}=Dk^{\prime\zeta},\end{equation}$$

and human and physical capital are positively related.

Proof. See Appendix A.

Given the path for n, k, and h, we can find consumption and investment decisions (9)–(11), as well as investment in contraceptive methods. Asymptotically, the system may diverge to infinity, converge to a zero, or converge to a nonzero steady-state equilibrium. Observe that when $k<\underline{k}$ , we have that $\frac{\partial G(k)}{\partial k}>0$ , $\frac{\partial^2 G(k)}{\partial k^2}<0$ , and $lim_{k\rightarrow0}\frac{\partial G(k)}{\partial k}=\infty$ . Therefore, the system does not converge to a zero steady state. If $\underline{k}$ is sufficiently large,Footnote 10 then there will be a locally stable steady state $k^*_N=G(k^*_N)$ in which $n(k)=N$ . In this case, there is no investment in modern contraceptive methods ( $q=0$ ), and therefore family planning interventions do not have any effect on the long-run level of the capital stock, that is, $k^*_N$ is independent of $\phi_q$ . However, whenever $\underline{k}<k^* _N$ , then it can be shown that there exists a locally stable steady-state equilibrium $k^*>\underline{k}$ such that fertility decreases with capital accumulation, and family planning interventions have long-run effects on capital accumulation and output. This is summarized in the following proposition.

Proof. See Appendix A.

Since human capital and physical capital are associated by equation (16), then it is trivial to show the following result.

Proof. See Appendix A.

Notice that the steady-state level of capital per young household, $k^*$ , depends on the parameters of the model, and therefore it can lie at any point on the horizontal axis in Figure 1. Hence, the impact of family planning interventions (the gap between the solid and dotted lines) also depends on the exogenous parameters that contribute to determine how rich the economy is.

Taking Proposition 5 and Corollary 1 together, we have that economies with a higher degree of social norms in fertility end up with lower physical and human capital levels. Decreasing the price of contraception leads to lower fertility and more accumulation of both human and physical capital. This latter effect is quantitatively more powerful when social norms are stronger. As contraception becomes cheaper, individuals lower fertility. Due to social norms, others follow suit.