# How can the fertility rate be below 2 but the number of births is greater than deaths (South Korea)?

I read an article published in 2020 which claimed that the South Korean population had declined "for the first time", the number of deaths (~300,000) outpacing births (~275,000). However, this would seem to be mathematically impossible because the birth rate has been below 2 in South Korea for decades (see chart below from World Bank). How is it even remotely possible?

Update: Based on one answer there is the idea that the increase in life expectancy could be an explanation. This seemed improbable to me originally, so I didn't consider it. But in any case, let's consider it:

So we see the life expectancy has increased from say 80 to about 82 apparently between 2010 to the present time, while the birth rate has been well below 2 that whole time. So, I guess I don't know the math of this. How do you compute birth rate vs life expectancy?

• A population weighted toward older people will have more deaths than births: there's no mathematical problem with that. At an extreme, consider a population with no child-bearing people at all.
– whuber
Dec 4, 2022 at 16:46
• @whuber: I think the OP is surprised at the fact that the population has only now begun to shrink. Dec 4, 2022 at 16:49
• "birth rate has been below 2 in South Korea for decades" OK, but decades is not all that long. Most women will go on living for a long time after their last child is born. If it were below 2 for 200 years, sure that seems unlikely. But 40 years doesn't seem surprising. Dec 5, 2022 at 3:16
• Because population effects have a really long lag time. "Decades" is actually far too short for these effects to average out. A century or 2 is much closer to what you'd need.
– Kaz
Dec 5, 2022 at 8:42
• @whuber 70 years is already close to a century. And that’s for an incredibly extreme event. So anything more gradual would take longer. I don’t see the contradiction.
– Kaz
Dec 5, 2022 at 17:43

If you have a large number of young people in a country with a life expectancy in the 80s, almost all of them aren't going to die until their 70s.

But if they have an average of 1 child each (and by their 40s, this is pretty locked in), there is a long-term population decline locked in.

An easy way to calculate the fertility rate is to work out what percentage of women at each age have a child that year, work out a woman's death rate, and then integrate.

I'll make a toy country. In this country, people live exactly 80 years, then drop dead.

Women have a 1.5 children at age 40 (exactly), starting 40 years ago.

Prior to 40 years ago, women had 4.0 children at age 40 (exactly).

120 years ago, start out with 80,000 women evenly of every age (1000 < 1 etc).

So there are 4000 kids born, 2000 of them women. This happens for the next 40 years, at which point there are 8000 kids born, 4000 of them women. Another 40 years pass with that birth rate.

We are now 40 years ago, and fertility plummets.

Now, death rates are based on *how many people where born 80 years ago). This is 4000 people (2000 of them women).

There are 8000 women turning 40. They each have 1.5 children, having 12000 children, 6000 of them women.

Meanwhile, only 4000 old people die. We have fertility below replacement (1.5 children per woman per lifetime), but birth are greater than deaths!

This continues for the next 40 years. Each year, the 8000 women turning 40 all have 1.5 children (12000 total), 6000 of them are women, and 4000 80 year olds (2000 of them women) die.

We now hit year 0. Now, only 6000 women turn 40; so they have 9000 children, 4500 of them women. 80 years ago 8000 (4000 of them women) kids where born, so 8000 80 year olds die.

The gap between births and deaths remain, despite 40 years of low fertility.

40 more years pass at this rate - we are at year +40. Now only 4500 women turn 40, giving birth to 6750 children, 3375 of them women. Meanwhile 12000 people turn 80 and die.

It isn't until 80 years after fertility dropped that births finally fell behind deaths, despite the fact that nobody lives past 80.

Why does this happen?

Birth rates are based off of the birth rates of women over the fertile ages, very roughly 18 to 43, times the fertility per year.

Fertility rate is the integration of the current fertility rate for each year.

Death rates is mostly based off of how many 70+ people there are in the country; death rates are pretty low prior to 70.

So one tracks births 30ish years ago, the other tracks births 70ish years ago. So long as birth rates are going up 30-70 years ago sufficiently fast (to make up for the sup-1 fertility rate) this means births are greater than deaths.

The toy example made this really sharp, but the effect remains with less sharp situations.

• "starting 30 years ago... prior to 40 years ago". I don't think you meant to leave a 10-year gap there :) Dec 5, 2022 at 9:06
• I'm scared of the prospect of having 1.5 children... Dec 5, 2022 at 9:48
• @Stef nice. But I don't see +0 to +80 there, which is when the lines cross.
– Yakk
Dec 5, 2022 at 14:31
• @hobbs 30 replaced with 40.
– Yakk
Dec 5, 2022 at 14:57
• @Yakk Corrected version: imgur.com/a/CGp3LSi
– Stef
Dec 5, 2022 at 17:23

I struggled with this apparent paradox for a while before I discovered Hans Rosling's succinct 2-minute explanation for it. I link the video below, but I understand that this effect arises because of the demographic profile of the population.

The number of births/deaths can intuitively be seen as

$$\text{Number of Births} = \text{Fertility Rate} \times \text{Population of Women of Childbearing Age}$$

$$\text{Number of Deaths} = \text{Mortality Rate} \times \text{Population of Old People}$$

The reason we did not see the number of deaths overtake number of births despite a low fertility rate was because the population of women of childbearing age were relatively higher than the population of old people. This was because the people then in their 70s and 80s were born in the 1940's, when the population of South Korea was much lower:

Accordingly, over time we would expect South Korea's demographic profile to change to have more and more older people, which will cause the second term in the death equation and hence the number of deaths to rise. And indeed that is what we see:

Link to Hans Rosling's video: Hans Rosling Explains Population Growth with Toilet Paper!. Charts taken from South Korea Population (LIVE) and Age distribution in South Korea 2021.

• Ok, so the deaths will exceeed the births, its just that it will happen in the future, because there is a delay. Dec 5, 2022 at 20:42
• +1 "The reason we did not see the number of deaths overtake number of births despite a low fertility rate was because the population of women of childbearing age were relatively higher than the population of old people" It is also a decrease in death rate, but this effect alone will also explain it. At the moment that the birthrate drops below 2, the current population is still based on the old birthrates, which is a wide base population pyramid with more younger population than older population. Dec 6, 2022 at 19:20

If life expectancy rises quickly enough, the population can grow, with births outpacing deaths, even if the birth rate is below 2, and this can indeed continue for a long time (essentially, the length of time this can happen is not mathematically bounded, but only by a slowdown in the increase of life expectancy). Importantly, the "kind" of life expectancy which matters is not that at birth, but the probability of older people to die in an increment of time.

Another effect that could lead to growing populations in spite of low birth rates could be net immigration, but I would not expect that to be very important in South Korea.

• I was aware of this, however, it seemed improbable to me that increases in life expectancy could be significant enough to only now be overtaken by the falling birth rate. I guess we need to turn to math here. I have updated my question to reflect this idea. Dec 4, 2022 at 19:03

The other factor that can affect the birth rate is the age of women when they have children. According to this UN document Potential impact of later childbearing on future population, factors such as increased access to higher education and increased uptake of professional careers lead to women not only having fewer children but having their first child at an older age. This results in an increasing time between generations and thus a lower birth rate.

This is very similar to the question

Why does the US death rate not "match" life expectancy

In my answer there I simulated what would happen with the deathrate when a sudden change in the risk of death occurs. This gives a dip in deathrate that stabilizes later

The dip is because, the current distribution of the population is still based on the old risk of death. The population is not a representation of the current state of risk of death, but has a history with different risks of death (as it was in the past).

See also the development of the population pyramid. This is the case of the Netherlands, but you can make a similar graph for South Korea. The number of young people is decreasing (less births) but the population increases because people have stopped dying. The volume of the population increases by having more people of old age.