I recently read Zvi's post on fertility rate. In it, they discuss the falling fertility rate around the world, and how to increase it. Before discussing whether or not we should increase the fertility rate (which I will not do), we need to establish whether or not it is decreasing, and, on a more fundamental level, what the fertility rate is. This is the question this post will tackle.

What is the fertility rate?

If your first instinct were to search on Wikipedia, you would notice that the website doesn't have a page called fertility rate. It has three pages ; fertility, total fertility rate and birth rate, discussing slightly different concepts. Still, the fertility page states:

The fertility rate is the average number of children born by a female during her lifetime and is quantified demographically.

This is the definition I will be using, because it answers the question "Will humanity go extinct because of a lack of pregnancy?". If each female has on average more than one female child, then no, humanity will not go extinct, since each generation is more numerous than the previous one.^{[1]} Now that we have a clear definition, we need to measure this number.

How to measure the fertility rate?

It is impossible to measure the fertility rate. This is a direct consequence of its definition: we would need to know how many children the current female population will have in the future. We can measure the fertility rate of dead individuals, yes, but we can only model the fertility rate of the current population. Nonetheless, we have several models.

Crude Birth Rate (CBR)

The crude birth rate is the simplest tool we have: divide the number of children born during a given year by the size of the population alive at that time. It is usually expressed in children per 1000 persons, or sometimes per 1000 females.

The problem with this simple indicator is that it ignores many factors. For example, an increase in life expectancy would result in a higher number of elderly, thus reducing the CBR. Although very simple, CBR isn't very helpful to detect a decreasing fertility rate.

Total Fertility Rate (TFR)

Total fertility rate is a way smarter indicator, similar to life expectancy. It works as follows:

Determine, for each age, the average number of children females of this age had this year

Derive the number of children a female would have, as the sum over all ages of the average number of children

In a nutshell, the TFR in 2022 is the number of children resulting from a female being 16 in 2022 (and potentially having a child), then going back in time one year and being 17 in 2022, then 18 in 2022, etc... until death. Here is what TFR in France looks like^{[2]}

TFR is smart because it deals with non-uniformity in the age distribution: if the number of females per age is not uniform, TFR will react very differently from CBR, for example.

The main problem with TFR is that it implicitly assumes that cohorts behave similarly. Indeed, a 20 year old female living today was born in 2003, while a 40 year old female was born in 1983. They may behave very differently, and it is not obvious that a 20 year old female now will behave in 20 years like a 40 year old female behaves now (in terms of number of children at least).

Mean Age of Birth (MAB)

One very important indicator is the mean age of birth. It gives the mean age of the mothers of the children born in a given year.

As we can see, it varies, which confirms the intuition that all cohorts do not behave similarly. Thus, TFR will not necessarily match the fertility rate.

In particular, an increasing MAB means that TFR underestimates the fertility rate. For example, let's consider the following toy model:

All females will have exactly 2 children.

Females born before 1992 will have 2 children before turning 30, thus before the year 2022.

Females born after 1992 will not have a single child before turning 30.

In this toy model, by definition, the fertility rate is exactly 2. But, if we look at the TFR in 2022, it will count

The number of children of females born before 1992, who are at least 30.

The number of children of females born after 1992, who are not 30 yet.

In this example, the TFR would thus be 0 in 2022.

Complete Cohort Fertility (CCF)

The intuition behind complete cohort fertility is very simple: just wait. If you want to measure the fertility of female born in 1950, just wait for all of them to die, and then count how many children they had. Easy.

Although CCF is the best indicator, because it measures exactly fertility, its obvious drawback is the lag: if we need to wait for a whole cohort to be dead before computing the CCF, we will never know if fertility rate is decreasing.

Actually, we usually do not wait for the whole cohort to be dead, but for them to be about 50 years old, assuming that the cohort will not have children anymore, or at least not a significative number. Here's what CCF looks like, compared to TFR.

A first very natural question: does it even make sense to compare the number of children born in 1960 with the number of children of females born in 1960? Aren't we comparing two very different things? The answer is yes, and, in some sense, this is the fundamental difference between TFR and CCF. TFR estimates fertility using only children born in a single year, while CCF computes fertility using children born from a given cohort, over its lifetime.

Nonetheless, in order to measure how good of a proxy TFR has been, and since MAB is about 30 years, we could shift CCF 30 years in the future, thus comparing the population of children born in a given year to the population of children born from a female cohort 30 years older.

So, for example, the TFR in 1980 was 1.946, while the CCF for females 30 years old in 1980 was 2.117. Journalists and politicians alike would have been very worried by the decline of fertility during the second half of the 20th century while, as far as we can tell, no cohort so far has had an average number of children below 2.0 (although the 1970 cohort had 2.005 children on average). We can see that the TFR started to strongly underestimate CCF between 1970 and 1980, which roughly corresponds to the moment when MAB started to increase.

CCF40

CCF is great, but it takes too long. An alternative is CCF40, which counts the number of children a female has before the age of 40. This indicator is thus available sooner.

As we can see, the data goes further. Obviously, CCF40 underestimates CCF because females can have children after 40. Also, the gap between the two seems to get bigger over time, which is in line with females having children later, and in particular having more children after 40.

Speculative future CCF

One thing we could do to estimate the CCF which isn't available yet is shift the CCF40 a little bit to match the CCF. This way, the 10 extra years of CCF40 we have should be closer to the CCF we don't yet have.

I did exactly that ; I added an affine function to CCF40 to make it fit CCF on two points. The two curves fit fairly well over the whole interval. I am not a demographer ; please do not put too much faith in this attempt.

Takeaways

The main takeaway is the difference between TFR and CCF. Despite TFR remaining consistently below 2 for almost 50 years in France, not a single cohort so far has averaged fewer than 2 children per female.

Next time you hear, on the news or elsewhere, that the fertility rate remains below 2.0, keep in mind that this rate, probably TFR, is an answer to the wrong question. In particular, if you know that pregnancy keeps happening later and later in life, you need to keep in mind that TFR underestimates the fertility rate.

^{^}

You may be surprised that we count the children per female and not per male, for example. This has likely to do with practical reasons, since it is easier (and historically has been easier) to count the number of children a female gave birth to, versus the number of children a male had.

^{^}

If you don't really care about France, feel free to look up the data for your favorite country yourself :)

I recently read Zvi's post on fertility rate. In it, they discuss the falling fertility rate around the world, and how to increase it. Before discussing whether or not we should increase the fertility rate (which I will not do), we need to establish whether or not it is decreasing, and, on a more fundamental level, what the fertility rate is. This is the question this post will tackle.

## What is the fertility rate?

If your first instinct were to search on Wikipedia, you would notice that the website doesn't have a page called

fertility rate.It has three pages ; fertility, total fertility rate and birth rate, discussing slightly different concepts. Still, thefertilitypage states:This is the definition I will be using, because it answers the question "Will humanity go extinct because of a lack of pregnancy?". If each female has on average more than one female child, then no, humanity will not go extinct, since each generation is more numerous than the previous one.

^{[1]}Now that we have a clear definition, we need to measure this number.## How to measure the fertility rate?

It is impossible to measure the fertility rate. This is a direct consequence of its definition: we would need to know how many children the current female population will have in the future. We can measure the fertility rate of dead individuals, yes, but we can only model the fertility rate of the current population. Nonetheless, we have several models.## Crude Birth Rate (CBR)

The crude birth rate is the simplest tool we have: divide the number of children born during a given year by the size of the population alive at that time. It is usually expressed in children per 1000 persons, or sometimes per 1000 females.

I extracted, as an example, the CBR for France from humanfertility.org.

The problem with this simple indicator is that it ignores many factors. For example, an increase in life expectancy would result in a higher number of elderly, thus reducing the CBR. Although very simple, CBR isn't very helpful to detect a decreasing fertility rate.

## Total Fertility Rate (TFR)

Total fertility rate is a way smarter indicator, similar to life expectancy. It works as follows:

In a nutshell, the TFR in 2022 is the number of children resulting from a female being 16 in 2022 (and potentially having a child), then going back in time one year and being 17 in 2022, then 18 in 2022, etc... until death. Here is what TFR in France looks like

^{[2]}TFR is smart because it deals with non-uniformity in the age distribution: if the number of females per age is not uniform, TFR will react very differently from CBR, for example.

The main problem with TFR is that it implicitly assumes that cohorts behave similarly. Indeed, a 20 year old female living today was born in 2003, while a 40 year old female was born in 1983. They may behave very differently, and it is not obvious that a 20 year old female now will behave in 20 years like a 40 year old female behaves now (in terms of number of children at least).

## Mean Age of Birth (MAB)

One very important indicator is the mean age of birth. It gives the mean age of the mothers of the children born in a given year.

As we can see, it varies, which confirms the intuition that all cohorts do not behave similarly. Thus, TFR will not necessarily match the fertility rate.

In particular, an increasing MAB means that TFR underestimates the fertility rate. For example, let's consider the following toy model:

In this toy model, by definition, the fertility rate is exactly 2. But, if we look at the TFR in 2022, it will count

In this example, the TFR would thus be 0 in 2022.

## Complete Cohort Fertility (CCF)

The intuition behind complete cohort fertility is very simple: just wait. If you want to measure the fertility of female born in 1950, just wait for all of them to die, and then count how many children they had. Easy.

Although CCF is the best indicator, because it measures exactly fertility, its obvious drawback is the lag: if we need to wait for a whole cohort to be dead before computing the CCF, we will never know if fertility rate is decreasing.

Actually, we usually do not wait for the whole cohort to be dead, but for them to be about 50 years old, assuming that the cohort will not have children anymore, or at least not a significative number. Here's what CCF looks like, compared to TFR.

A first very natural question: does it even make sense to compare the number of children born in 1960 with the number of children of females born in 1960? Aren't we comparing two very different things? The answer is yes, and, in some sense, this is the fundamental difference between TFR and CCF. TFR estimates fertility using only children born in a single year, while CCF computes fertility using children born from a given cohort, over its lifetime.

Nonetheless, in order to measure how good of a proxy TFR has been, and since MAB is about 30 years, we could shift CCF 30 years in the future, thus comparing the population of children born in a given year to the population of children born from a female cohort 30 years older.

So, for example, the TFR in 1980 was 1.946, while the CCF for females 30 years old in 1980 was 2.117. Journalists and politicians alike would have been very worried by the decline of fertility during the second half of the 20th century while, as far as we can tell, no cohort so far has had an average number of children below 2.0 (although the 1970 cohort had 2.005 children on average). We can see that the TFR started to strongly underestimate CCF between 1970 and 1980, which roughly corresponds to the moment when MAB started to increase.

## CCF40

CCF is great, but it takes too long. An alternative is CCF40, which counts the number of children a female has before the age of 40. This indicator is thus available sooner.

As we can see, the data goes further. Obviously, CCF40 underestimates CCF because females can have children after 40. Also, the gap between the two seems to get bigger over time, which is in line with females having children later, and in particular having more children after 40.

## Speculative future CCF

One thing we could do to estimate the CCF which isn't available yet is shift the CCF40 a little bit to match the CCF. This way, the 10 extra years of CCF40 we have should be closer to the CCF we don't yet have.

I did exactly that ; I added an affine function to CCF40 to make it fit CCF on two points. The two curves fit fairly well over the whole interval.

I am not a demographer ; please do not put too much faith in this attempt.## Takeaways

The main takeaway is the difference between TFR and CCF. Despite TFR remaining consistently below 2 for almost 50 years in France, not a single cohort so far has averaged fewer than 2 children per female.

Next time you hear, on the news or elsewhere, that the fertility rate remains below 2.0, keep in mind that this rate, probably TFR, is an answer to the wrong question. In particular, if you know that pregnancy keeps happening later and later in life, you need to keep in mind that TFR underestimates the fertility rate.

^{^}You may be surprised that we count the children per female and not per male, for example. This has likely to do with practical reasons, since it is easier (and historically has been easier) to count the number of children a female gave birth to, versus the number of children a male had.

^{^}If you don't really care about France, feel free to look up the data for your favorite country yourself :)