In an earlier post I discussed why international migration would not and could not stabilise Japan's population, at current rates of fertility. So, now let's look at the other side of the coin - fertility - and see if raising the birth rate could provide the solution that growth-oriented planners wish for.

Below is a 'back of the envelope' method for understanding why Japan can't stabilise its population by simply having more children. Data is taken from the Government of Japan's Ministry of Internal Affairs and Communications Statistics Bureau website. These calculations are indicative only and do not use the methodologies that demographers use to calculate population change.

In 1970 the population of Japan was 104.7 million people. The number of 20 year olds was 2.19 million persons, of whom 1.1 million were female. In that year the Total Fertility Rate (TFR), meaning the average number of children born to each woman in her lifetime, was 2.13 children per woman - which is very slightly above the rate necessary for population reproduction. Here's a little calculation.

If each 20 year old woman alive in 1970 had given birth to 2.13 children in her lifetime, then this cohort would have produced 2.34 million children. Allowing for deaths, infertility, and some outward migration, this translates to a stable population, if it persisted. So far so good.

But it didn't persist. In 1974 Japan began experiencing below replacement fertility, which has continued uninterrupted to this day. This means that every year since 1974 the population has produced fewer children than itself - which translates, eventually, to a shrinking population once tempo effects unwind.

In 1990 the population of 20 year olds was 1.91 million, of whom 932,000 were female. Precisely 20 years after 1970, this shows that the real average fertility rate among those women aged 20 in 1970 was below the 1970 fertility rate. If we assume that every child born to this cohort of women survived and none left Japan before each reached the age of 20, then we can calculate that those women aged 20 in 1970 theoretically reproduced at a rate of 1.72 children per woman. The actual rate of reproduction among this cohort was therefore somewhere between the 1970 TFR of 2.13 and the theoretically lowest TFR of 1.72, allowing for deaths and out-migration. For the sake of argument let's take the mid point between the two of 1.93 children per woman - below replacement.

So, although the 1970 TFR indicated that Japan had a stable population, in actual fact the country had already entered a long period of shrinkage.

Now, if the 1990 cohort of 20 year old women had reproduced at the 1990 TFR they would have given birth to 1.4 million children. However, the population of 20 year olds in 2010 was 1.2 million, a shortfall of 200,000 on the theoretical figure; which means that the 1990 cohort of 20 year olds, like their sisters from 1970, reproduced at a lower rate than the 1990 TFR - at the very least 1.28 children per woman. Using the same methodology for the 1970 cohort, the real rate of reproduction for this cohort was probably 1.34, being the mid point between 1.28 and the headline rate of 1.39 for that year.

Taking the 1970 and 1990 cohort data together, and assuming that both cohorts were typical of long term reproduction trends, we arrive at an interesting conclusion. For at least 40 years Japanese women reproduced at lower rates than real time data was suggesting would happen.

Now, let's come forward to the present. In 2013 the number of 20 year olds in Japan was 1.22 million, of whom 595,000 were female, and the TFR was 1.43. This means that the 2013 cohort of 20 year olds, at 2013 rates of fertility would theoretically produce 851,000 children. However, Japan is experiencing a fertility bounce back; so this generation may produce more children than my calculation suggests. Lets say they produce 1 million children, which would require a TFR among 20 year olds in 2013 of 1.68 children per woman. That's possible, but it's still under half the number of children produced by 20 year olds in 1970, and represents a big rise in the fertility rate from 1.43. And it's still well below the population replacement rate of about 2.1.

So, to repeat another way, if 2013's cohort of women reproduce at a TFR of 1.68 then they would give birth to 1 million children, yet 1990's cohort of 20 year olds produced 1.2 million children at a TFR of 1.34. Here we have an apparent contradiction; a higher fertility rate in 2013 than 1990, yet fewer children being delivered. How can that be? Well, it's because long term low fertility has produced a much smaller cohort of mothers in 2013 than in 1990 so that, even with a higher rate of fertility in 2013, there would still be fewer children being born. And that is indeed what is happening.

So, let's summarise thus far. The number of 20 year old Japanese women in 1970 was 1.1 million and they gave birth to around 1.91 million children at a real rate of approximately 1.93 children each. In 1990 the number of 20 year old women was 932,000, about 170,000 fewer than in 1970, and they gave birth to around 1.2 million, at a real rate of around 1.34 children each. The 2013 cohort of women numbered 595,000 which is 54 per cent of the number of 20 year old women alive in 1970. The headline fertility rate in 1970 was 2.13 and in 2013 it was 1.43. 1.1 million times 2.13 equals 2.34 million. 595,000 times 1.43 equals 851,000. The difference between the theoretical number of births for these cohorts is a shortfall in 2013 of 1.49 million children. The real shortfall is likely to be smaller; perhaps 1.1 million. Nevertheless, this is still a huge number, and represents a drop in the aggregated number of children being born to each cohort of between 40 and 50 per cent since 1970.

OK so far? Now let's try a little numbers game. Let's hypothesize that the 2013 cohort of 20 year olds reproduces at the 1970 TFR. They would then produce 1.28 million children, about 700,000 fewer births than the 1970 age 20 cohort. If in 2013 Japan would have the same number of theoretical births as in 1970, when Japan was more or less reproducing itself, then the TFR would need to be 3.93 children per 20 year old in 2013. The same calculation for 1990 would require a TFR of 2.99.

Even this would not keep Japan at a stable population, because the 1970 total population was 104 million, not the 127 million of today. To reproduce at a rate that keeps the current population stable, 2013's 20 year olds would have to reproduce at a rate higher than 4.0 children per woman; starting tomorrow! Any delay and the fertility rate necessary to keep the population stable would have to rise still further.

The last time that Japan had a TFR of 2.99/3.93 was in 1950/55, and the last time the TFR was higher than 4.0 was during the postwar baby boom prior to 1950; an exceptional time in Japanese history, bearing in mind the huge loss of life during World War Two. Personally, I don't see any chance of Japan being able to raise fertility to levels anywhere near population stabilisation levels.

If international migration cannot help Japan stabilise its population, and neither can an increase in fertility, then what is the prognosis for the future? My answer is simple.

Japan had better get used to having a shrinking population. It's happening right now, it's been going on since 2008, it's accelerating, and it's not going to end soon.