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Key forces behind the decline of fertility: lessons from childlessness in Rouen before the industrial revolution

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Abstract

To better understand the forces underlying fertility decisions, we look at the forerunners of fertility decline. In Rouen, France, completed fertility dropped between 1640 and 1792 from 7.4 to 4.2 children. We review possible explanations and keep only three: increases in materialism, in women’s empowerment, and in returns to education. The methodology is one of analytic narrative, bringing together descriptive evidence with a theoretical model. We accordingly propose a theory showing that we can discriminate between these explanations by looking at childlessness and its social gradient. An increase in materialism or, under certain conditions, in women’s empowerment, leads to an increase in childlessness, while an increase in the return to education leads to a decrease in childlessness. Looking at the Rouen data, childlessness was clearly on the rise, from 4% in 1640 to 10% at the end of the eighteenth century, which appears to discredit the explanation based on increasing returns to education, at least for this period.

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Fig. 1

Sources: Fertility rates and mortality rates are from Greenwood et al. (2005). Gross domestic product per capita is from Maddison (2010) and average years of education are from Maddison (2001). Contraception effectiveness is from Greenwood and Guner (2010)

Fig. 2

Note: The top left panel represents the three regimes. In the three other panels, the solid line represents the initial situation and the dashed line represents the situation after a rise in materialism/women empowerment/rise in return to education

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Notes

  1. Very similar pictures could be presented for other developed countries.

  2. This trade-off between the number of children and their survival was recognized as early as 1837: “ces essaims d’enfans ne rendent que plus impossibles les soins indispensables pour leur assurer pleine vie.” (D’Ivernois 1836).

  3. It was overtaken by Lyon and Marseilles around 1700, see data in Bairoch et al. (1988).

  4. The aggregate results are detailed in the book by Bardet (1983), but the original individual data are not made available by the author.

  5. Natural fertility does not tend to vary much across populations (see Werner (1986) and Coleman (1996)). However, sterility can be affected by venereal diseases or health problems due to abortion (Szreter 1996) Consequently, a share of sterility due to venereal diseases could come from prostitutes and could be higher in cities with numerous soldiers or sailors.

  6. A very high childlessness rate is found among the English upper class (Gobbi and Goñi (2016) and de la Croix and Schneider (2017) for the same period, but without the same time trend.

  7. While that decrease in child mortality does reduce the number of births, it does not necessarily reduce the number of surviving children, depending on how uncertainty about child survival affects household preferences. This is discussed further in Doepke (2005), Baudin (2012).

  8. The number of deaths of children under one year old observed in the city divided by the number of births.

  9. From page 255 of Bardet (1983), the mean age at first marriage for women with a known birth certificate is: 1670–1699: 24.4, 1700–1729: 26.2, 1730–1759: 24.9, and 1760–1789: 26.1.

  10. Marie-Anne Lepage, born in Rouen, married to Fiquet du Bocage, in 1768; Madame de l’Etoile (born in Rouen) in 1770 and 1771; Madame de Courcy in 1774; and Julie d’Assier de la Chassagne comtesse de Laurencin in 1774 and 1777.

  11. Jeanne Bisson de la Courdraye, Anne de la Roche Guilhem, Jeanne Marie le Prince de Beaumont, Marie Caroline Delabarre, and Louise Cavelier. The first four can be found in the Index Biobliographicus Notorum Hominum, the last one in Briquet (1804).

  12. Divorce, however, became illegal again in 1816, and people wanting to divorce had to wait until 1884 when a new divorce law was passed. See de la Croix and Mariani (2015) for a political economy theory of the adoption of divorce laws in Western Europe.

  13. See Doepke (2015) for a survey on the emergence of the concept of the quality-quantity trade-off, and Klemp and Weisdorf (2016) and Galor and Klemp (2017) for evidence of the mechanism on historic parish reconstitution data from England and Quebec.

  14. Until the seventeenth century, noblemen assumed positions of command (regardless of their competence). Over the period 1600–1700, armies grew considerably in size, requiring more competent officers. For instance, in 1675, Louis XIV made power dependent on merit and seniority (rather than on social class or birth).

  15. Superior goods make up a larger proportion of spending as income rises. Here, because of the constant term \({\bar{x}}\), \(x_{ij}\) is not consumed at all for low level of incomes, while becomes consumed once income is above a certain threshold. Notice, however, from "Appendix  A” that the income-elasticity of its demand is not strictly larger than one for any level of income.

  16. In general, this is true as long as substitution effects dominate income effects.

  17. Such a postponement-driven childlessness has an involuntary component in the sense that the couple would have preferred to still be fertile, but has a voluntary component too, as, by postponing, the couple accepted to lower its probability to be able to procreate.

  18. It is interesting to note that in de la Croix (2012), p. 48–65, the early drop in fertility in Rouen is related to the rise in the return to schooling. Yet, at the time, de la Croix did not look at childlessness, which would have discredited this mechanism.

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Acknowledgements

The authors acknowledge the financial support of the project ARC 15/19-063 of the Belgian French-speaking Community. We thank Jean-Pierre Bardet, Paula E. Gobbi, and two referees for comments on an earlier draft.

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Correspondence to David de la Croix.

Appendices

Appendix A: Solving the model

Using \(s_{ij}=n_{ij} h_{ij}\), the utility can be rewritten:

$$\begin{aligned} \ln ({\bar{x}} +x_{ij}) + \gamma _{i} \frac{n^{1-\beta }_{ij}}{1-\beta } \; \frac{s_{ij}^\beta }{\beta } \end{aligned}$$
(1)

The household maximizes utility subject to her budget constraint,

$$\begin{aligned} p_x x_{ij}+ p_n(\omega _j) n_{ij} + s_{ij}=\omega _j \end{aligned}$$
(2)

and the non-negativity constraints:

$$\begin{aligned} x_{i,j},n_{ij}\ge 0. \end{aligned}$$

Three regimes are possible: the interior regime with \(x_{ij},\;n_{ij}>0\), a Malthusian regime with no luxuries and \(x_{ij}=0,\;n_{ij}>0\), and a child-free regime \(x_{ij}>0,\;n_{ij}=0\).

Interior Regime We start with the interior regime with both \(x_{ij}\) and \(n_{ij}\) positive. In that case, the first-order conditions can be solved as:

$$\begin{aligned} \displaystyle s_{ij}=\, & {} \beta \left( {p_x} \bar{x}+\omega - \varrho _i \right) \end{aligned}$$
(3)
$$\begin{aligned} \displaystyle n_{ij}&= {} (1-\beta ) \;\frac{ {p_x} \bar{x}+\omega - \varrho _i }{p_n}, \end{aligned}$$
(4)
$$\begin{aligned} \displaystyle x_{ij}= & {} \frac{ \varrho _i }{ p_x}-{\bar{x}}, \end{aligned}$$
(5)
$$\begin{aligned} \displaystyle h= & {} \frac{\beta p_n}{1-\beta }, \end{aligned}$$
(6)

where

$$\begin{aligned} \varrho _i=\frac{(1-\beta )^\beta \beta ^{1-\beta } p_n^{1-\beta }}{\gamma _i} \end{aligned}$$
(7)

is a monotonic transformation of the relative cost of quantity \( p_n\) over quality.

The demand for superior goods \(x_i\) is decreasing in its price \(p_x\) and increasing in the cost of children quantity \( p_n\). The demand for quality \(h_i\) only depends on its cost relative to quantity and is thus a positive function of the price \(p_n\). It does not depend on the price of luxuries \(p_x\).

Fertility, i.e., the demand for quantity of children \(n_i\), depends positively on income \(\omega _j\) for given \(p_n\). If the cost of children is a time cost, \( p_n\) is proportional to income, say \( p_n=\phi \omega _j\). In that case, fertility varies negatively with income. Moreover,

Proposition 1

Fertility in the interior regime decreases in its own price \(p_n\), increases with the price of the superior good \(p_x\)and decreases with the return to education  \(\beta \) if and only if

$$\begin{aligned} \omega >\left( 3-\frac{1}{\beta }\right) -p_x {\bar{x}}+(1-\beta )\varrho _i \ln \frac{\beta p_n}{1-\beta } \end{aligned}$$

The indirect utility is this regime is:

$$\begin{aligned} V_{ij}=\frac{p_x{\bar{x}}+\omega _j}{\varrho _i}+\ln \varrho _i-\ln p_x -1. \end{aligned}$$
(8)

Child-free regime Now consider the regime where the household is childless and consumes superior goods. In that case, \(x_i=\omega _j/p\), \(n_i=0\) and \(h_i=0\). The indirect utility is:

$$\begin{aligned} W_j=\ln \left( \frac{\omega _j}{ p_x}+{\bar{x}}\right) . \end{aligned}$$
(9)

Malthusian regime Finally, consider the regime where the household does not consume superior goods: \(x_i=0\). In that case, the first-order conditions can be written as:

$$\begin{aligned} \displaystyle s_j=\, & {} \beta \omega _j, \end{aligned}$$
(10)
$$\begin{aligned} \displaystyle n_j=\, & {} (1-\beta ) \frac{\omega _j}{ p_n}, \end{aligned}$$
(11)
$$\begin{aligned} \displaystyle h=\, & {} \frac{\beta p_n}{1-\beta }. \end{aligned}$$
(12)

And the indirect utility function is:

$$\begin{aligned} Z_{ij}=\ln {\bar{x}} + \frac{\omega _j}{\varrho _i} \end{aligned}$$
(13)

In this regime, we have:

Proposition 2

Fertility in the Malthusian regime is unaffected by the price of the superior goods  \(p_x\)  and decreases with its own price  \(p_n\)  and with the return to education  \(\beta \).

From the comparisons of the indirect utilities, we can infer bounds on \(\gamma _i\) delimiting the different regimes.

Proposition 3

There exists

$$\begin{aligned} {\tilde{\gamma }}_j=\frac{(1-\beta )^\beta (\beta \; p_n)^{1-\beta }}{(p_x\;{\bar{x}} +\omega _j)}, \;\;\;\;\;\; \text{ and } \;\;\;\;\;\; {\hat{\gamma }}=\frac{(1-\beta )^\beta (\beta \; p_n)^{1-\beta }}{p_x\;{\bar{x}} }, \end{aligned}$$

with  \({\tilde{\gamma }}_j<{\hat{\gamma }}\)such that:

  1. 1.

    if  \(\gamma _i<{\tilde{\gamma }}_j\), \(x_{ij}>0\) and  \(n_{ij}=0\)(child-free regime),

  2. 2.

    if  \({\tilde{\gamma }}_j \le \gamma _i\le {\hat{\gamma }}\), \(x_{ij}>0\)  and  \(n_{ij}>0\)  (interior regime),

  3. 3.

    if  \({\hat{\gamma }}<\gamma _i\), \(x_{ij}=0\)  and  \(n_{ij}>0\)(Malthusian regime).

Proof

From Eq. (5), \(\gamma _i\) needs to be larger than \( {\tilde{\gamma }}\) for \(x_i\) to be positive. Moreover, the difference \(V_{ij}-W_j\) is given by:

$$\begin{aligned} V_{ij}-W_j=\ln \left( \frac{\varrho _i}{p_x{\bar{x}} +\omega _j}\right) +\frac{p_x{\bar{x}} +\omega _j}{\varrho _i}-1 \end{aligned}$$

The derivative of this difference with respect to \(\gamma _i\) is

$$\begin{aligned} \frac{\partial (V_{ij}-W_j)}{\partial \gamma _i}= \frac{p_x{\bar{x}} +\omega _j-\varrho _i}{\gamma _i \varrho _i} \end{aligned}$$

At the point \(\gamma _i={\tilde{\gamma }}\), the indirect utilities are equal: \(V_{ij}=W_j\). When \(\gamma _i\) increases above \({\tilde{\gamma }}\), \(V_{ij}\) increases and \(W_j\) stays constant; hence, the interior regime dominates the corner regime \(x_i=0\) for all \(\gamma _i>{\tilde{\gamma }}\).

From Eq. (4), \(\gamma _i\) needs to be larger than \( {\hat{\gamma }}\) for \(n_i\) to be positive. Moreover, the difference \(V_{ij}-Z_{ij}\) is given by:

$$\begin{aligned} V_{ij}-Z_{ij}=\ln (\varrho _i p_x {\bar{x}}) +\frac{p_x {\bar{x}}}{\varrho _i}-1 \end{aligned}$$

The derivative of this difference with respect to \(\gamma _i\) is

$$\begin{aligned} \frac{\partial (V_{ij}-Z_{ij})}{\partial \gamma _i}=\frac{p_x{\bar{x}}}{(1-\beta )^\beta (\beta p_n)^{1-\beta }} -\frac{1}{\gamma _i} \end{aligned}$$

At the point \(\gamma _i={\hat{\gamma }}\), the indirect utilities are equal: \(V_{ij}=Z_{ij}\). When \(\gamma _i\) decreases below \({\tilde{\gamma }}\), \(V_{ij}-Z_{ij}\) increases, hence the interior regime dominates the corner regime \(n_i=0\) for all \(\gamma _i<{\hat{\gamma }}\). \(\square \)

Figure 2 plots the two thresholds \({\hat{\gamma }}\) and \({\tilde{\gamma }}\) as a function of \(\gamma _i\) and \(\omega _j\).

The childlessness rate \(\chi \) in the economy is given by:

$$\begin{aligned} \chi = F({\tilde{\gamma }}), \end{aligned}$$

where \(F(\cdot )\) is the cumulative distribution function of \(\gamma _i\). Hence, when there exogenous changes in parameters, childlessness varies in the same direction as \({\tilde{\gamma }}\).

Proposition 4

The childlessness rate  \(F({\tilde{\gamma }})\)  increases with the price  \(p_n\)  and decreases with the price of luxuries  \(p_x\). It decreases with the return to education  \(\beta \)  if and only if:

$$\begin{aligned} 1-2\beta -(1-\beta )\beta \ln \left( \frac{\beta p_n}{1-\beta }\right) <0 \end{aligned}$$

i.e., when  \(p_n\) is not too large.

Proof

Let us compute

$$\begin{aligned} \frac{\partial {\tilde{\gamma }}}{\partial \beta }=\frac{p_n^{1-\beta } \left( 1-2\beta -(1-\beta )\beta \ln \left( \frac{\beta p_n}{1-\beta }\right) \right) }{(1-\beta )^{1-\beta } \beta ^\beta (p_x {\bar{x}} +\omega _j)}. \end{aligned}$$

The proposition follows. \(\square \)

Appendix B: Robustness to the introduction of a basic consumption good

We introduce a consumption good \(c_{ij}\) and show how the main equations are altered.

Utility is now:

$$\begin{aligned} \ln ({\bar{x}} +x_{ij}) + \gamma _{i} \frac{n^{1-\beta }_{ij}}{1-\beta } \; \frac{s_{ij}^\beta }{\beta }+\ln c_{ij} \end{aligned}$$
(1)

The budget constraint is

$$\begin{aligned} p_x x_{ij}+ p_n(\omega _j) n_{ij} + s_{ij}+c_{ij}=\omega _j \end{aligned}$$
(2)

Interior regime

The solution is:

$$\begin{aligned} \displaystyle s_{ij}= {} \beta \left( {p_x} \bar{x}+\omega - 2 \varrho _i \right) \end{aligned}$$
(3)
$$\begin{aligned} \displaystyle n_{ij}= {} (1-\beta ) \;\frac{ {p_x} \bar{x}+\omega - 2 \varrho _i }{p_n}, \end{aligned}$$
(4)
$$\begin{aligned} \displaystyle x_{ij}= {} \frac{ \varrho _i }{ p_x}-{\bar{x}}, \end{aligned}$$
(5)
$$\begin{aligned} \displaystyle h= {} \frac{\beta p_n}{1-\beta } \end{aligned}$$
(6)
$$\begin{aligned} , \displaystyle c_{ij}= {} \varrho _i \end{aligned}$$
(7)

where

$$\begin{aligned} \varrho _i=\frac{(1-\beta )^\beta \beta ^{1-\beta } p_n^{1-\beta }}{\gamma _i} \end{aligned}$$
(8)

The indirect utility is this regime is:

$$\begin{aligned} V_{ij}=\frac{p_x{\bar{x}}+\omega _j}{\varrho _i}+2\ln \varrho _i-\ln p_x -2. \end{aligned}$$
(9)

Child-free regime

The indirect utility is:

$$\begin{aligned} W_j=2\ln \left( \frac{\omega _j}{ p_x}+{\bar{x}}\right) -\ln (4 p_x). \end{aligned}$$
(10)

Malthusian regime

The first-order conditions can be written as:

$$\begin{aligned} \displaystyle s_{ij}=\, & {} \beta (\omega _j-\varrho _i), \end{aligned}$$
(11)
$$\begin{aligned} \displaystyle n_{ij}=\, & {} (1-\beta ) \frac{\omega _j-\varrho _i}{ p_n}, \displaystyle h = \frac{\beta p_n}{1-\beta }, \end{aligned}$$
(12)
$$\begin{aligned} \displaystyle c_i=\, & {} \varrho _i \end{aligned}$$
(13)

And the indirect utility function is:

$$\begin{aligned} Z_{ij}=\ln {\bar{x}} + \frac{\omega _j}{\varrho _i}+ \ln \varrho _i-1 \end{aligned}$$
(14)

It remains true that there exists

$$\begin{aligned} {\tilde{\gamma }}_j=\frac{2(1-\beta )^\beta (\beta \; p_n)^{1-\beta }}{(p_x\;{\bar{x}} +\omega _j)}, \;\;\;\;\;\; \text{ and } \;\;\;\;\;\; {\hat{\gamma }}=\frac{(1-\beta )^\beta (\beta \; p_n)^{1-\beta }}{p_x\;{\bar{x}} }, \end{aligned}$$

with \({\tilde{\gamma }}_j<{\hat{\gamma }}\), such that:

  1. 1.

    if \(\gamma _i<{\tilde{\gamma }}_j\), \(x_{ij}>0\) and \(n_{ij}=0\) (child-free regime),

  2. 2.

    if \({\tilde{\gamma }}_j \le \gamma _i\le {\hat{\gamma }}\), \(x_{ij}>0\) and \(n_{ij}>0\) (interior regime),

  3. 3.

    if \({\hat{\gamma }}<\gamma _i\), \(x_{ij}=0\) and \(n_{ij}>0\) (Malthusian regime).

And it remains true that the childlessness rate \(F({\tilde{\gamma }})\) increases with the price \(p_n\) and decreases with the price of luxuries \(p_x\). It decreases with the return to education \(\beta \) if and only if:

$$\begin{aligned} 1-2\beta -(1-\beta )\beta \ln \left( \frac{\beta p_n}{1-\beta }\right) <0 \end{aligned}$$

i.e., when \(p_n\) is not too large.

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Brée, S., de la Croix, D. Key forces behind the decline of fertility: lessons from childlessness in Rouen before the industrial revolution. Cliometrica 13, 25–54 (2019). https://doi.org/10.1007/s11698-017-0166-9

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