The rate at which we find new innovations is roughly proportional to the size of the world economy. Yes, there likely are other ways to promote innovation, but firms and governments have long been trying to promote local innovation, without much success. So this simple innovation-economy relation will probably long continue.

Our economy thus grows exponentially, and has done so for a very long time. (Since 1980, it has doubled on average every 24 years. Since 1961, that is 20 years.) However, this innovation rate is limited by population, and due to falling fertility, population is going to peak soon, plausibly in about thirty years or so, and then decline, And in a falling economy the innovation rate should fall proportionally, and thus “grind to a halt”, not restarting again until well after fertility rises again, plausibly many centuries later.

Of course if we can make cheap-enough human-level AI before this fall gets too deep, the world economy can continue to grow, and thus innovate well, with machines replacing the disappearing humans. But that possibility raises a key question: just how much more innovation we can expect to see before a coming centuries-long pause?

One simple first-cut way to estimate this is to simplify our future growth path, and assume that our economy keeps growing exponentially until it suddenly switches to falling exponentially forever. (Compared to a more realistic transitional economic deceleration, this simple model should *over-estimate* total economic product, and thus innovation, across this period.)

We can then ask: how long would the economy have to have instead kept growing at prior rates to create the same total product as we would have been seen by infinity in the declining economy scenario? Add that duration to the future date at which growth switches to get our key estimate: expect (a bit less than) as much innovation as you would have seen by that future date in a simple keeps-growing-at-same-prior-rate scenario.

For this, we need only solve a math problem: Imagine two economies start out the same size, but one grows with doubling time D, while the other shrinks with halving time H. Over an infinite future, the declining one makes a finite total product. In what finite time T does the growing one make that same total product? The answer: T = D*log_2(1+H/D). For D=H, then T=D.

To apply this formula, we need only estimate how fast the economy might fall. In a declining economy, product should eventually fall faster than population as we lose scale economies, but early on while innovation remains high the economy might fall slower than does population. But let’s take population as a proxy.

The big question is then: just how far will average world fertility (kids per woman lifetime) fall? If fertility were 1.48, then population would fall by a factor of two every two 30-year generations, for a halving time of 60 years. If fertility fell to only 1.68, the halving time would be four generations, or 120 years, while a fertility of 1.05 would give one generation, or 30 years. (World median fertility across nations is 2.0; South Korea and Hong Kong are below 0.8)

Putting this all together, if the switch from growth to decline happens in 30 years, then we should expect roughly 30 + D*log_2(1+H/D) more years worth of innovation. When D = 23.9 and H = 60, that gives 73.3 years from now: *~2097*. For H = 120, it gives 91.9 years, and for H = 30 it gives 58.0 years. *So we are talking less than roughly sixty to ninety more years worth of innovation. *(Note this isn’t literal years, but instead years-equivalent in our familiar growing world.)

Of course maybe population will peak later than I’ve said, because fertility will fall slower than I’ve estimated. The world economy might peak before or after population does, depending on many factors. Innovation is higher for the young, of which there will be fewer, and is likely lower in Africa, which is the last region with high fertility.

But though I look forward to seeing more careful analyses, these effects just can’t change the overall answer by that much. *Expect no more than a century’s worth of further innovation, before the great innovation pause*. *Maybe only a half century*. While I know many are excited by recent AI progress, I think it will actually be quite a challenge to achieve human level AI by this deadline. And of course adopting policies to slow down AI innovation on purpose will only make it harder to make this deadline.

Same goes if you were hoping for life or fertility extension, or artificial wombs. You’ve got less than a century to not only work out the basic concepts, but also all the messy details needed to translate abstract ideas in concrete practice, and then to lower costs through scaling up practice. And to get past regulatory obstacles, which many seem eager to add to your troubles.

This analysis seems to rely a great deal on a model in which causality runs pretty strictly from population growth to innovation. But I read history more the other way around. Innovation enabled population growth. Going forward, my intuition is that if innovators were surrounded by much better institutional support, we would see more innovation even if world population were an order of magnitude below what it is today.

edited Jan 2Consider the perspective of a randomly choosen person from all of human history (behind the veil of ignorance). Do they want to maximize innovation **per year**? Certainly not. They would want to maximize **Innovation per person year**.

You don't want to have more people existing at earlier, less productive, times because that makes the average human life less good. Since a smaller population both decreases innovation and person years that pass each year proportionally we should be indifferent to it.

To put it differently, If a demon caused literally everything in the universe to slow to half speed (according to his infernal clock) would that be bad because now it takes 2 years to increase productivity by 5% instead of one year? Surely not. What's relevant is the amount of human experience which occurs at each level of productivity not some arbitrary unit of time.