Many people think that lower child mortality causes fertility to decline.

One prominent theory for this relationship, as described by Our World in Data¹, is that “infant survival reduces the parents’ demand for children”². (Infants are children under 1 years old).

In this article, I want to look at how we can precisify that theory, and what magnitude the effect could possibly take. What fraction of the decline in birth rates could the theory explain?

Important. I don’t want to make claims here about how parents actually make fertility choices. I only want to examine the implications of various models, and specifically how much of the observed changes in fertility the models could explain.

Constant number of children

One natural interpretation of “increasing infant survival reduces the parents’ demand for children” is that parents are adjusting the number of births to keep the number of surviving children constant.

Looking at Our World in Data’s graph, we can see that in most of the countries depicted, the infant survival rate went from about 80% to essentially 100%. This is a factor of 1.25. Meanwhile, there were 1/3 as many births. If parents were adjusting the number of births to keep the number of surviving children constant, the decline in infant mortality would explain a change in births by a factor of 1/1.25=0.8, a -0.2 change that is only 30% of the -2/3 change in births.

The basic mathematical reason this happens is that even when mortality is tragically high, the survival rate is still thankfully much closer to 1 than to 0, so even a very large proportional fall in mortality will only amount to a small proportional increase in survival.

Some children survive infancy but die later in childhood. Although Our World in Data’s quote focuses on infant mortality, it makes sense to consider older children too. I’ll look at under-5 mortality, which generally has better data than older age groups, and also captures a large fraction of all child mortality³.

England (1861-1951)

England is a country with an early demographic transition and good data available.

Doepke 2005 quotes the following numbers:

Fertility fell by 57%, while survival to 5 years rose by 32%. Hence, if parents aim to keep the number of surviving children constant, the change in child survival can explain 43%⁴ of the actual fall in fertility. (It would have explained only 23% had we erroneously considered only the change in infant survival.)

Sub-Saharan Africa (1990-2017)

If we look now at sub-Saharan Africa data from the World Bank, the 1990-2017 change in fertility is from 6.3 to 4.8, a 25% decrease, whereas the 5-year survival rate went from 0.82 to 0.92, a 12% increase. So the fraction of the actual change in fertility that could be explained by the survival rate is 44%. (This would have been 23% had we looked only at infant survival).

Source data and calculations

So far, we have seen that this very simple theory of parental decision-making can explain 30-44% of the decline in fertility, while also noticing that considering childhood mortality beyond infancy was important to giving the theory its full due.

However, in more sophisticated models of fertility choices, the theory looks worse.

A more sophisticated model of fertility decisions

Let us imagine that instead of holding it constant, parents treat the number of surviving children as one good among many in an optimization problem.

An increase in the child survival rate can be seen as a decrease in the cost of surviving children. Parents will then substitute away from other goods and increase their target number of surviving children. If your child is less likely to die as an infant, you may decide to aim to have more children: the risk of experiencing the loss of a child is lower.⁵

For a more formal analysis, we can turn to the Barro and Becker (1989) model of fertility. I’ll be giving a simplified version of the presentation in Doepke 2005.

In this model, parents care about their own consumption as well as their number of surviving children. The parents maximise⁶

where

• $n$ is the number of surviving children and $V$ is the value of a surviving child
• $\epsilon$ is a constant $\in (0,1)$
• $u(c)$ is the part of utility that depends on consumption⁷

The income of a parent is $w$, and there is a cost per birth of $p$ and an additional cost of $q$ per surviving child⁸. The parents choose $b$, the number of births. $s$ is the probability of survival of a child, so that $n=sb$.

Consumption is therefore $c=w-(p+qs)b$ and the problem becomes $\max_{b} U = u(w-(p+qs)b) + (sb)^\epsilon V$

Letting $b^{*}(s)$ denote the optimal number of births as a function of $s$, what are its properties?

The simplest one is that $sb^*(s)$, the number of surviving children, is increasing in $s$. This is the substitution effect we described intuitively earlier in this section. This means that if $s$ is multiplied by a factor $x$ (say 1.25), $b^*(s)$ will be multiplied more than $1/x$ (more than 0.8).

When we looked at the simplest model, with a constant number of children, we guessed that it could explain 30-44% of the fall in fertility. That number is a strict upper bound on what the current model could explain.

What we really want to know, to answer the original question, is how $b^*(s)$ itself depends on $s$. To do this, we need to get a little bit more into the relative magnitude of the cost per birth $p$ and the additional cost $q$ per surviving child. As Doepke writes,

If a major fraction of the total cost of children accrues for every birth, fertility [i.e. $b^*(s)$] would tend to increase with the survival probability; the opposite holds if children are expensive only after surviving infancy⁹.

This tells us that falling mortality could actually cause fertility to increase rather than decrease.¹⁰

To go further, we need to plug in actual values for the model parameters. Doepke does this, using numbers that reflect the child mortality situation of England in 1861 and 1951, but also what seem to be some pretty arbitrary assumptions about the parent’s preferences (the shape of $u$ and the value of $\epsilon$).

With these assumptions, he finds that “the total fertility rate falls from 5.0 (the calibrated target) to 4.2 when mortality rates are lowered to the 1951 level”¹¹, a 16% decrease. This represents is 28% of the actually observed fall in fertility to 2.1.

Extensions of Barro-Becker model

The paper then considers various extensions of the basic Barro-Becker model to see if they could explain the large decrease in fertility that we observe.

For example, it has been hypothesized that when there is uncertainty about whether a child will survive (hitherto absent from the models), parents want to avoid the possibility of ending up with zero surviving children. They therefore have many children as a precautionary measure. Declining mortality (which reduces uncertainty since survival rates are thankfully greater than 0.5) would have a strong negative impacts on births.

However, Doepke also considers a third model, that incorporates not only stochastic mortality but also sequential fertility choice, where parents may condition their fertility decisions on the observed survival of children that were born previously. The sequential aspect reduces the uncertainty that parents face over the number of surviving children they will end up with.

The stochastic and sequential models make no clear-cut predictions based on theory alone. Using the England numbers, however, Doepke finds a robust conclusion. In the stochastic+sequential model, for almost all reasonable parameter values, the expected number of surviving children still increases with $s$ (my emphasis):

To illustrate this point, let us consider the extreme case [where] utility from consumption is close to linear, while risk aversion with regards to the number of surviving children is high. … [W]hen we move (with the same parameters) to the more realistic sequential model, where parents can replace children who die early, … despite the high risk aversion with regards to the number of children, total fertility drops only to 4.0, and net fertility rises to 3.9, just as with the benchmark parameters. … Thus, in the sequential setup the conclusion that mortality decline raises net fertility is robust to different preference specifications, even if we deliberately emphasize the precautionary motive for hoarding children.

So even here, the fall in mortality would only explain 35% of the actually observed change in fertility. It seems that the ability to “replace” children who did not survive in the sequential model is enough to make its predictions pretty similar to the simple Barro-Becker model.