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Most economic growth comes from innovation, not the accumulation of capital or labor. And innovation rates are mostly due to two competing factors. One the one hand, we pick the low hanging fruit of the easiest highest-payoff innovations to try first. On the other hand, we can more easily pursue innovation ideas when our world is richer, has better tech, and knows more.

If the first factor dominated, growth would slow down, while if the second dominated, growth would speed up. But in fact, outside of a few rare jumps to much faster growth modes, growth has been roughly exponential, neither accelerating nor decelerating. Thus these competing factors roughly balance. Each year we find on average ~N typical innovations, each of which makes an industry that is ~A% of the economy ~B% more valuable, with N,A,B staying roughly constant over time.

This view has a dramatic implication for a world of low fertility, and thus falling population. If the world economy reaches a peak and then falls to a level that is only X% of that peak, but stays within the same growth mode, then at that point the rate of innovation would be no more than X% of the prior peak rate. Fewer than X% of N typical innovations would be found per year.

That is, in a shrinking economy, innovation grinds to a halt! And if after a shrinking economy, economic growth should restart, then a restart of innovation would be delayed until the new economy rose to pre-population-fall levels. If a culture of innovation had withered in the meantime, such a restart might take even longer.

In the past, under a growing economy, we have happily invested in innovations that better achieve scale economies, often by decreasing marginal costs at the expense of increasing fixed costs. But a shrinking economy would achieve fewer scale economies, and seek to reverse such changes, lowering fixed costs by raising marginal costs. Complex systems that incur technical debt, which today are periodically remade wholesale, would in that world continue to be repaired at increasing costs, or dropped entirely. These effects would also shrink the economy.

Thus once we have entered into a shrinking world economy, we cannot put much hope on tech or innovation solutions. If such solutions are not found before the economic decline, they will probably never be found.

Yes, a shrinking economy could gain some advantages from larger levels of prior accumulated capital. But its workforce is also older on average, with more retired folks to support. Also, decline might cut the world’s mood, increasing conflict, the opposite of the usual effect claimed for today whereby growth raises our moods, decreasing conflict. On the other hand, a smaller world could inherit more capital from prior generations, save on congestion costs, and have less environmental impact.

**Added 25Aug:** (This corrects Aug23 added)

Chad Jones offers a simple math model of this situation. Let world product Y be given by Y = A^a *N, where A is tech level and N is population, and let A’s rate of change be proportional to N^b *A^c. Jones shows that if N falls exponentially. then so does the growth rate of A. That is, innovation comes to a halt, while individual consumption Y/N approaches a constant.

His model doesn’t include capital C, but we can do so via N = L^2/3 *C^1/3, and by assuming a constant marginal product of capital dY/dC, which is an interest rate. But doing this just gives us us another model of the same form as Jones, now in terms ofL instead of N, and with new values of the constants a,b,c. So again innovation comes to a halt.

In fact, we could generalize to Y = A^a *Product_i K_i^a_i, and ANY single K_i falling exponentially would be enough if all the others were set to have constant marginal product dY/dK_i.

(FYI, the last doubling of world product per person took ~19 years, while the average age of mothers at birth in rich nations is now ~30.)

**Added 5Oct:** Here is why innovation grinds to a halt with a falling popularion. Assume total innovation so far is proportional to an integral over all past econ activity. In an exponentially rising economy, econ activity in the last period is a substantial fraction of all econ activity so far. But in an exponentially falling economy econ activity in the last period becomes a much smaller fraction of all econ activity so far, a fraction that falls to zero as the size of the economy fall to zero.

## Shrinking Economies Don’t Innovate

I've just added to the post.

"Assume total innovation so far as an integral over all past econ activity" -- I don't find that assumption compelling. As an obvious counterexample, imagine several populations that don't communicate; innovation rate is likely to be highly sublinear in the number of such populations.

An alternate explanation I propose for the observation of roughly constant rates of innovation over time is a limit to how much innovation (change) a given person will accept. As in "science advances one funeral at a time". If that is true, the threat to future innovation isn't population decline, but long lifespans/immortality, for example by the invention of ems.