Thought Toys · Cycles & change · Exhibit 118

The growth stops. The growing doesn't.

Suppose every family in a fast-growing country switched to replacement size tonight — just enough children to replace the parents, no more. Each generation now exactly replaces itself, so in the long run this country cannot grow. It keeps growing anyway, for another sixty-five years, and settles about a third larger. Nothing is wrong with the arithmetic. The country is simply too young to stop.

The age pyramid, and the total over time ● people alive now   ● the shape it is heading for

years since— population— where it settles— measured growth now—

—

↳ your turn — drag How fast it had been growing before the change down to nothing, and the whole effect vanishes

What you're seeing

A country that has been growing has far more children than grandparents. That is the wide triangle on the left. It is not an accident of this model; it is what sustained growth looks like from the side.

Now every family switches to replacement size. Each generation will exactly replace itself, so the growth built into family size is zero from tonight. Press Run the years. The total on the right goes up anyway, for decade after decade, and settles about a third higher.

Watch the fourth counter while it runs, because it is the whole exhibit in one number. It reads the growth you could actually measure in this country — births minus deaths, as a percentage a year — and it stays positive for decades after the switch. Two different rates share the one word. The rate that went to zero tonight is a property of family size. The rate you can measure is a property of the population, and it takes two generations to catch up.

The reason is on the left. Those enormous young cohorts have not had their children yet. For the next thirty years an unusually large slice of the country walks into its childbearing years, and the births keep outrunning the deaths — not because families are large, but because there are so many families. Growth stops only when the pyramid has squared off into the column drawn in cyan.

This is the difference between a rate and a stock. Setting the per-person growth rate to zero does not set the growth of the population to zero, because a population is not a single number. It has a shape, and the shape is not yet the one that goes with the new rate.

Two things worth finding with the dials. Drag the first slider down to no prior growth and the effect vanishes completely — same model, same switch, and the total does not move a hair. And press Find the size that stops it for five years: to hold the total still for even one period, families must be far below replacement, not at it — and holding that level is not a stable country, it is one shrinking almost 2% a year. Momentum can be cancelled at an instant. It cannot be wished away.

The rule, exactly. A one-sex Leslie projection on 5-year age bands, 0–4 up to 85+. Each period, everyone ages one band and a fraction survives; births come from the women in the childbearing bands.

n0(t+1) = Σi Fi ni(t)     ni+1(t+1) = si ni(t)

Replacement is the fertility at which the net reproduction rate — daughters per woman over a whole lifetime, allowing for those who do not live to have them — equals one. Both sliders scale F only; the life table never changes, so nothing here is a story about people living longer. A one-sex model with fixed mortality is the standard vehicle for this effect: adding two sexes, a birth sex ratio, or improving survival changes the numbers and does not create or remove the momentum.

Verified in node before this page existed (improve/verify/118-demographic-momentum.js, 37 checks):

  • The fertility scale derived from a net reproduction rate of one gives a dominant eigenvalue of exactly 1.000000000000 — the Euler–Lotka identity, checked rather than cited.
  • At the default settings the population settles 32.3% higher, is still rising 65 years after the switch, and reaches half of that eventual gain only in year 25.
  • That settling point is computed two independent ways — by projecting the country forward year by year, and in closed form from the eigenvectors — and the two agree to one part in a billion.
  • The negative control the whole exhibit rests on: start instead from a country that was already still, switch to the same replacement fertility, and the momentum is exactly 1.000000000000 with no drift over two thousand years. Same model, same switch. The growth comes from the shape of the population, not from anything in the arithmetic.
  • A second negative control, against a model that simply always grows: start from a country that had been shrinking and the momentum is 0.72. It keeps falling after its rate reaches zero, for the same reason in reverse.
  • Momentum arrives continuously, not as a jump: a country growing 0.1% a year carries 3% extra, and the figure rises monotonically with the prior growth rate.
  • To freeze the total on the spot takes 54.8% of replacement fertility. Held forever, that level is a country shrinking 1.85% a year.
  • With fertility set to zero the population falls monotonically to nothing, which is how the projection proves it is really running and can go down.

Also in Cycles & change: The epidemic threshold →

All 17 in Cycles & change
  1. 04Predator & prey
  2. 102Infinite fuel. Finite speed.
  3. 112The best moment to give up on a good patch
  4. 118The growth stops. The growing doesn't. — you are here
  5. 16The epidemic threshold
  6. 17Compound interest
  7. 53A feedback loop that overshoots
  8. 54Why planets speed up near the star
  9. 55The enzyme that hits a ceiling
  10. 65No spike, no matter how long you wait
  11. 68A perfect engine still throws most of it away
  12. 71Squeeze a reaction and it pushes back
  13. 73The switch that won't switch back
  14. 75Below a threshold, a population can't come back
  15. 80A bigger dose doesn't get there faster
  16. 83Come too close and a moon becomes a ring
  17. 95A system about to collapse gets slow before it goes.

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