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I follow some Japanese based websites, and I've read that "Japan population is going to reduce by half in 50 years". The estimates in this article are more optimistic, according to the graph it seems to estimate around 30 millions lose in population in the same period of time.

Then, I looked into this chart in Wikipedia about countries population growths. Some European countries like Spain and Portugal seems to have much worse population growth than Japan. Japan -0.12 , Spain -0.21 , Portugal -0.45 . Then I read this article, and they say Spain will lose 5.6 millions inhabitants in the following 50 years. This is proportionally a much lower figure than Japan, in spite of it is stated by that Wikipedia article Spain decreased much more than Japan in the period of 2010-2015 . I suppose many factors are taken into account for calculating population growth aside births and deaths such as immigration, emigration, etc.

I would like to know if there are mathematical equations to calculate population growth or how are they predicted?

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It's possible that the stories about drastic drops in Japan refer to working-age population, not the population itself.

This is a link to the U.S. Census methodology.

I have only casually looked at these data over the years, but I would summarise the general idea as follows.

  1. We already have the current population distribution. We work with the data based on the number of people in each age cohort.
  2. We have estimates of life expectancy based on age, and actuaries spend a lot of time working on developing projections. (The defined benefit pension and life insurance industry depend upon such actuarial projections.)
  3. We usually have a good idea of the birth rate, based on the characteristics of under-30 population. Even if we miss, it takes a long time for the errors to really add up. (People have to reach puberty before the prediction error compounds.)
  4. This covers the existing population; what remains to do is estimate the effect of net migration.

For the developed countries, the big unknown is typically net immigration. In the euro area, the collapse of economies has often caused a considerable amount of emigration.

Outside of net immigration, the only thing that could cause the projections to off is a massive change in the death rate (or birth mortality). Birth rates tend to be slow-moving; the baby boom after World War II was a response to highly unusual earlier conditions (Great Depression, WWII). Life expectancy is another wild card, but it is unclear how much it will change.

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