This would be an important update for me, so I’m excited to see people looking into it and to spend more time thinking about it myself.
High-level summary of my current take on your document:
I agree that the 1AD-1500AD population data seems super noisy.
Removing that data removes one of the datapoints supporting continuous acceleration (the acceleration between 10kBC − 1AD and 1AD-1500AD) and should make us more uncertain in general.
It doesn’t have much net effect on my attitude towards continuous acceleration vs discontinuous jumps, this mostly pushes us back towards our prior.
I’m not very moved by the other evidence/arguments in your doc.
Here’s how I would summarize the evidence in your document:
Much historical data is made up (often informed by the author’s models of population dynamics), so we can’t use it to estimate historical growth. This seems like the key point.
In particular, although standard estimates of growth from 1AD to 1500AD are significantly faster than growth between 10kBC and 1AD, those estimates are sensitive to factor-of-1.5 error in estimates of 1AD population, and real errors could easily be much larger than that.
Population levels are very noisy (in addition to population measurement being noisy) making it even harder to estimate rates.
Radiographic data often displays isolated periods of rapid growth from 10,000BC to 1AD and it’s possible that average growth rates were something like 2000 year doubling. So even if 500-2000 year doubling times are accurate from 1AD to 1500, those may not be a deviation from the preceding period.
You haven’t looked into the claims people have made about growth from 100kya to 10kya, but given what we know about measurement error from 10kya to now, it seems like the 100kya-10kya data is likely to be way too noisy to say anything about.
Here’s my take in more detail:
You are basically comparing “Series of 3 exponentials” to a hyperbolic growth model. I think our default simple hyperbolic growth model should be the one in David Roodman’s report (blog post), so I’m going to think about this argument as comparing Roodman’s model to a series of 3 noisy exponentials. In your doc you often dunk on an extremely low-noise version of hyperbolic growth but I’m mostly ignoring that because I absolutely agree that population dynamics are very noisy.
It feels like you think 3 exponentials is the higher prior model. But this model has many more parameters to fit the data, and even ignoring that “X changes in 2 discontinuous jumps” doesn’t seem like it has a higher prior than “X goes up continuously but stochastically.” I think the only reason we are taking 3 exponentials seriously is because of the same kind of guesswork you are dismissive of, namely that people have a folk sense that the industrial revolution and agricultural revolutions were discrete changes. If we think those folk senses are unreliable, I think that continuous acceleration has the better prior. And at the very least we need to be careful about using all the extra parameters in the 3-exponentials model, since a model with 2x more parameters should fit the data much better.
On top of that, the post-1500 data is fit terribly by the “3 exponentials” model. Given that continuous acceleration very clearly applies in the only regime where we have data you consider reliable, and given that it already seemed simpler and more motivated, it seems pretty clear to me that it should have the higher prior, and the only reason to doubt that is because of growth folklore. You can’t have it both ways in using growth folklore to promote this hypothesis to attention and then dismissing the evidence from growth folklore because it’s folklore.
On the acceleration model, the periods from 1500-2000, 10kBC-1500, and “the beginning of history to 10kBC” are roughly equally important data (and if that hypothesis has higher prior I don’t think you can reject that framing). Changes within 10kBC − 1500 are maybe 1/6th of the evidence, and 1⁄3 of the relevant evidence for comparing “continuous acceleration” to “3 exponentials.” I still think it’s great to dig into one of these periods, but I don’t think it’s misleading to present this period as only 1⁄3 of the data on a graph.
(Enough about priors, onto the data.)
I think that the key claim is that the 1AD-1500AD data is mostly unreliable. Without this data, we have very little information about acceleration from 10kBC − 1500AD, since the main thing we actually knew was that 1AD-1500AD must have been faster than the preceding 10k years. I’d like to look into that more, but it looks super plausible to me that the noise is 2x or more for 1AD which is enough to totally kill any inference about growth rates. So provisionally I’m inclined to accept your view there.
That basically removes 1 datapoint for the continuous acceleration story and I totally agree it should leave us more uncertain about what’s going on. That said, throwing out all the numbers from that period also removes one of the main quantitative datapoints against continuous acceleration [ETA: the other big one being the modern “great stagnation,” both of these are in the tails of the continuous acceleration story and are just in the middle of the constant exponentials in the 3-exponential story, though see Robin Hanson’s writeup to get a sense for what the series of exponentials view actually ends up looking like—it’s still surprised by the great stagnation], and comes much closer to leaving us with our priors + the obvious acceleration over longer periods + the obvious acceleration during the shorter period where we actually have data, which seem to all basically point in the same direction.
Even taking the radiocarbon data as given I don’t agree with the conclusions you are drawing from that data. It feels like in each case you are saying “a 2-exponential model fits fine” but the 2 exponentials are always different. The actual events (either technological developments or climate change or population dynamics) that are being pointed to as pivotal aren’t the same across the different time series and so I think we should just be analyzing these without reference to those events (no suggestive dotted lines :) ). I spent some time doing this kind of curve fitting to various stochastic growth models and this basically looks to me like what individual realizations look like from such models—the extra parameters in “splice together two unrelated curves” let you get fine-looking fits even when we know that the underlying dynamics are continuous+stochastic.
I currently don’t trust the population data coming from the radiocarbon dating. My current expectation is that after a deep dive I would not end up trusting the radiocarbon dating at all for tracking changes in the rate of population growth when the populations in question are changing how they live and what kinds of artifacts they make (from my perspective, that’s what happened with the genetics data, which wasn’t caveated so aggressively in the initial draft I reviewed). I’d love to hear from someone who actually knows about these techniques or has done a deep dive on these papers though.
I think the only dataset that you should expect to provide evidence on its own is the China population time series. But even there if you just take rolling averages and allow for a reasonable level of noise I think the continuous acceleration story looks fine. E.g. I think if you compare David Roodman’s model with the piecewise exponential model (both augmented with measurement noise, and allowing you to choose noisy dynamics however you want for the exponential model), Roodman’s model is going to fit the data better despite having fewer free parameters. If that’s the case, I don’t think this time series can be construed as evidence against that model.
I agree with the point that if growth is 0 before the agricultural revolution, rather than “small,” then that would undermine the continuous acceleration story. I think prior growth was probably slow but non-zero, and this document didn’t really update my view on that question.
I feel really confused what the actual right priors here are supposed to be. I find the “but X has fewer parameters” argument only mildly compelling, because I feel like other evidence about similar systems that we’ve observed should easily give us enough evidence to overcome the difference in complexity.
This does mean that a lot of my overall judgement on this question relies on the empirical evidence we have about similar systems, and the concrete gears-level models I have for what has caused growth. AI Impact’s work on discontinuous vs. continuous progress feels somewhat relevant and evidence from other ecological systems also seems reasonably useful.
When I try to understand what exactly happened in terms of growth at a gears-level, I feel like I tend towards more discontinuous hypotheses, because I have a bunch of very concrete, reasonably compelling sounding stories of specific things that caused the relevant shifts, and while I have some gears-level models for what would cause more continuous growth, they feel a lot more nebulous and vague to me, in a way that I think usually doesn’t correspond to truth. The thing that on the margin would feel most compelling to me for the continuous view is something like a concrete zoomed in story of how you get continuous growth from a bunch of humans talking to each other and working with each other over a few generations, that doesn’t immediately abstract things away into high-level concepts like “knowledge” and “capital”.
because I have a bunch of very concrete, reasonably compelling sounding stories of specific things that caused the relevant shifts
Be careful that you don’t have too many stories, or it starts to get continuous again.
More seriously, I don’t know what the small # of factors are for the industrial revolution, and my current sense is that the story can only seem simple for the agricultural revolution because we are so far away and ignoring almost all the details.
It seems like the only factor that looks a priori like it should cause a discontinuity is the transition from hunting+gathering to farming, i.e. if you imagine “total food” as the sum of “food we make” and “food we find” then there could be a discontinuous change in growth rates as “food we make” starts to become large relative to “food we find” (which bounces around randomly but is maybe not really changing). This is blurred because of complementarity between your technology and finding food, but certainly I’m on board with an in-principle argument for a discontinuity as the new mode overtakes the old one.
For the last 10k years my impression is that no one has a very compelling story for discontinuities (put differently: they have waytoo many stories) and it’s mostly a stylized empirical fact that the IR is kind of discontinuous. But I’m provisionally on board with Ben’s basic point that we don’t really have good enough data to know whether growth had been accelerating a bunch in the run-up to the IR.
To the extent things are discontinuous, I’d guess that it’s basically from something similar to the agricultural case—there is continuous growth and random variation, and you see “discontinuities” in the aggregate if a smaller group is significantly outpacing the world, so that by the time they become a large part of the world they are growing significantly faster.
I think this is also reasonably plausible in the AI case (e.g. there is an automated part of the economy doubling every 1-2 years, by the time it gets to be 10% of the economy it’s driving +5%/year growth, 1-2 years later it’s driving +10% growth). But I think quantitatively given the numbers involved and the actual degree of complementarity, this is still unlikely to give you a fast takeoff as I operationalized it. I think if we’re having a serious discussion about “takeoff” that’s probably where the action is, not in any of the kinds of arguments that I dismiss in that post.
I find the “but X has fewer parameters” argument only mildly compelling, because I feel like other evidence about similar systems that we’ve observed should easily give us enough evidence to overcome the difference in complexity.
I mean something much more basic. If you have more parameters then you need to have uncertainty about every parameter. So you can’t just look at how well the best “3 exponentials” hypothesis fits the data, you need to adjust for the fact that this particular “3 exponentials” model has lower prior probability. That is, even if you thought “3 exponentials” was a priori equally likely to a model with fewer parameters, every particular instance of 3 exponentials needs to be less probable than every particular model with fewer parameters.
The thing that on the margin would feel most compelling to me for the continuous view is something like a concrete zoomed in story of how you get continuous growth from a bunch of humans talking to each other and working with each other over a few generations, that doesn’t immediately abstract things away into high-level concepts like “knowledge” and “capital”.
As far as I can tell this is how basically all industries (and scientific domains) work—people learn by doing and talk to each other and they get continuously better, mostly by using and then improving on technologies inherited from other people.
It’s not clear to me whether you are drawing a distinction between modern economic activity and historical cultural accumulation, or whether you feel like you need to see a zoomed-in version of this story for modern economic activity as well, or whether this is a more subtle point about continuous technological progress vs continuous changes in the rate of tech progress, or something else.
I mean something much more basic. If you have more parameters then you need to have uncertainty about every parameter. So you can’t just look at how well the best “3 exponentials” hypothesis fits the data, you need to adjust for the fact that this particular “3 exponentials” model has lower prior probability. That is, even if you thought “3 exponentials” was a priori equally likely to a model with fewer parameters, every particular instance of 3 exponentials needs to be less probable than every particular model with fewer parameters.
Thanks, this was a useful clarification. I agree with this as stated. And I indeed assign substantially more probability to a statement of the form “there were some s-curve like shifts in humanity’s past that made a big difference” than to any specific “these three specific s-curve like shifts are what got us to where we are today”.
As far as I can tell this is how basically all industries (and scientific domains) work—people learn by doing and talk to each other and they get continuously better, mostly by using and then improving on technologies inherited from other people.
It’s not clear to me whether you are drawing a distinction between modern economic activity and historical cultural accumulation, or whether you feel like you need to see a zoomed-in version of this story for modern economic activity as well, or whether this is a more subtle point about continuous technological progress vs continuous changes in the rate of tech progress, or something else.
Hmm, I don’t know, I guess that’s just not really how I would characterize most growth? My model is that most industries start with fast s-curve like growth, then plateau, then often decline. Sure, kind of continuously in the analytical sense, but with large positive and negative changes in the derivative of the growth.
And in my personal experience it’s also less the case that I and the people I work with just get continuously better, it’s more like we kind of flop around until we find something that gets us a lot of traction on something, and then we quickly get much better at the given task, and then we level off again. And it’s pretty easy to get stuck in a rut somewhere and be much less effective than I was years ago, or for an organization to end up in a worse equilibrium and broadly get worse at coordinating, or produce much worse output than previously for other reasons.
Of course enough of those stories could itself give rise to a continuous growth story here, but there is a question here about where the self-similarity lies. Like, many s-curves can also give rise to one big s-curve. Just because I have many s-curve doesn’t mean I get continuous hyperbolic growth. And so seeing lots of relative discontinuous s-curves at the small scale does feel like it’s evidence that we also should expect the macro scale to be a relatively small number of discontinuous s-curves (or more precisely, s-curves whose peak is itself heavy-tail distributed, so that if you run a filter for the s-curves that explain most of the change, you end up with just a few that really mattered).
My model is that most industries start with fast s-curve like growth, then plateau, then often decline
I don’t know exactly what this means, but it seems like most industries in the modern world are characterized by relatively continuous productivity improvements over periods of decades or centuries. The obvious examples to me are semiconductors and AI since I deal most with those. But it also seems true of e.g. manufacturing, agricultural productivity, batteries, construction costs. It seems like industries where the productivity vs time curve is a “fast S-curve” are exceptional, which I assume means we are somehow reading the same data differently. What kind of industries would you characterize this way?
(I agree that e.g. “adoption” is more likely to be an s-curve given that it’s bounded, but productivity seems like the analogy for growth rates.)
it seems like most industries in the modern world are characterized by relatively continuous productivity improvements over periods of decades or centuries
This agrees with my impression. Just in case someone is looking for references for this, see e.g.:
Nagy et al. (2013) - several of the trends they look at, e.g. prices for certain chemical substances, show exponential growth for more than 30 years
Farmer & Lafond (2016) - similar to the previous paper, though fewer trends with data from more than 20 years
Bloom et al. (2020) - reviews trends in research productivity, most of which go back to 1975 and some to 1900
Some early examples from manufacturing (though not covering multiple decades) are reviewed in a famous paper by Arrow (1971), who proposed ‘learning by doing’ as a mechanism.
Note: Actually looking at the graphs in Farmer & Lafond (2016), many of these do sure seem pretty S-curve shaped. As do many of the diagrams in Nagy et al. (2013). I would have to run some real regressions to look at it, but in particular the ones in Farmer & Lafond seem pretty compatible with the basic s-curve model.
Overlapping S-curves are also hard to measure because obviously there are feedback effects between different industries (see my self-similarity comment above). Many of the advances in those fields are driven by exogenous factors, like their inputs getting cheaper, with no substantial improvements in their internal methodologies. One of my models of technological progress (I obviously also share the model of straightforward exponential growth and assign it substantial probability) is that you have nested and overlapping S-curves, which makes it hard to just look at cost/unit output of any individual field.
For analyzing that hypothesis it seems more useful to hold inputs constant and then look at how cost/unit develops, in order to build a model of that isolated chunk of the system (and then obviously also look at the interaction between industries and systems to get a sense of how they interact). But that’s also much harder to do, given that our data is already really messy and noisy.
Thanks for poking at this, it would be quite interesting to me if the “constant exponential growth” story was wrong. Which graphs in Farmer & Lafond (2016) are you referring to? To me, the graph with a summary of all trends only seems to have very few that at first glance look a bit like s-curves. But I agree one would need to go beyond eyeballing to know for sure.
I agree with your other points. My best guess is that input prices and other exogenous factors aren’t that important for some of the trends, e.g. Moore’s Law or agricultural productivity. And I think some of the manufacturing trends in e.g. Arrow (1971) are in terms of output quantity per hour of work rather than prices, and so also seem less dependent on exogenous factors. But I’m more uncertain about this, and agree that in principle dependence on exogenous factors complicates the interpretation.
To me, the graph with a summary of all trends only seems to have very few that at first glance look a bit like s-curves. But I agree one would need to go beyond eyeballing to know for sure.
Yeah, that was the one I was looking at. From very rough eye-balling, it looks like a lot of them have slopes that level off, but obviously super hard to tell just from eye-balling. I might try to find the data and actually check.
Thanks for your super detailed comment (and your comments on the previous version)!
You are basically comparing “Series of 3 exponentials” to a hyperbolic growth model. I think our default simple hyperbolic growth model should be the one in David Roodman’s report (blog post), so I’m going to think about this argument as comparing Roodman’s model to a series of 3 noisy exponentials.
I think that Hanson’s “series of 3 exponentials” is the neatest alternative, although I also think it’s possible that pre-modern growth looked pretty different from clean exponentials (even on average / beneath the noise). There’s also a semi-common narrative in which the two previous periods exhibited (on average) declining growth rates, until there was some ‘breakthrough’ that allowed the growth rate to surge: I suppose this would be a “three s-curve” model. Then there’s the possibility that the growth pattern in each previous era was basically a hard-to-characterize mess, but was constrained by a rough upper bound on the maximum achievable growth rate. This last possibility is the one I personally find most likely, of the non-hyperbolic possibilities.
(I think the pre-agricultural period is especially likely to be messy, since I would guess that human evolution and climate/environmental change probably explain the majority of the variation in population levels within this period.)
It feels like you think 3 exponentials is the higher prior model. But this model has many more parameters to fit the data, and even ignoring that “X changes in 2 discontinuous jumps” doesn’t seem like it has a higher prior than “X goes up continuously but stochastically.” I think the only reason we are taking 3 exponentials seriously is because of the same kind of guesswork you are dismissive of, namely that people have a folk sense that the industrial revolution and agricultural revolutions were discrete changes. If we think those folk senses are unreliable, I think that continuous acceleration has the better prior. And at the very least we need to be careful about using all the extra parameters in the 3-exponentials model, since a model with 2x more parameters should fit the data much better.
I think this is a good and fair point. I’m starting out sympathetic toward the breakthrough/phase-change perspective, in large part because this perspective fits well with the kinds of narratives that economic historians and world historians tend to tell. It’s reasonable to wonder, though, whether I actually should give much weight to these narratives. Although they rely on much more than just world GDP estimates, their evidence base is also far from great, and they disagree on a ton of issues (there are a bunch of competing economic narratives that only partly overlap.)
A lot of my prior comes down to my impression that the dynamics of growth just *seem * very different to me for forager societies, agricultural/organic societies, and industrial/fossil-fuel societies. In the forager era, for example, it’s possible that, for the majority of the period, human evolution was the main underlying thing supporting growth. In the farmer era, the main drivers were probably land conversion, the diffusion and further evolution of crops/animals, agricultural capital accumulation (e.g. more people having draft animals), and piecemeal improvements in farming/land-conversion techniques discovered through practice. I don’t find it difficult to imagine that the latter drivers supported higher growth rates. For example: the fact that non-sedentary groups can’t really accumulate capital, in the same way, seems like a pretty fundamental distinction.
The industrial era is, in comparison, less obviously different from the farming era, but it also seems pretty different. My list of pretty distinct features of pre-modern agricultural economies is: (a) the agricultural sector constituted the majority of the economy; (b) production and (to a large extent) transportation were limited by the availability of agricultural or otherwise ‘organic’ sources of energy (plants to power muscles and produce fertiliser); (c) transportation and information transmission speeds were largely limited by windspeed and the speed of animals; (d) nearly everyone was uneducated, poor, and largely unfree; (e) many modern financial, legal, and political institutions did not exist; (f) certain cultural attitudes (such as hatred of commerce and lack of belief in the possibility of progress) were much more common; and (g) scientifically-minded research and development projects played virtually no role in the growth process.
I also don’t find it too hard to believe that some subset of these changes help to explain why modern industrialised economies can grow faster than premodern agricultural economies: here, for example, is a good book chapter on the growth implications of relying entirely on ‘organic’ sources of energy for production. The differences strike me as pretty fundamental and pretty extensive. Although this impression is also pretty subjective and could easily amount to seeing dividing lines where they don’t exist.
Another piece of evidence is that there’s extreme between-states variation in the growth rates, in modern times, which isn’t well-explained by factors like population size. We’ve seen that it is possible for something to heavily retard/bottleneck growth (e.g. bad political institutions), then for growth to surge following the removal of the bottleneck. It’s not too hard to imagine that pre-modern states had lots of blockers. They were in some way similar to 20th/21st century growth basket cases, only with some important extra growth retardants—like a lack of fossil fuels and artificial fertilizer, a lack of knowledge that material progress is possible, etc. -- thrown on top.
There may also be some fundamental meta-prior that matters, here, about the relative weight one ought to give to simple unified models vs. complex qualitative/multifactoral stories.
On top of that, the post-1500 data is fit terribly by the “3 exponentials” model. Given that continuous acceleration very clearly applies in the only regime where we have data you consider reliable, and given that it already seemed simpler and more motivated, it seems pretty clear to me that it should have the higher prior, and the only reason to doubt that is because of growth folklore.
I don’t think the post-1500 data is too helpful help for distinguishing between the ‘long run trend’ and ‘few hundred year phase transition’ perspectives.
If there was something like a phase transition, from pre-modern agricultural societies to modern industrial societies, I don’t see any particular reason to expect the growth curve during the transition to look like the sum of two exponentials. (I especially don’t expect this at the global level, since diffusion dynamics are so messy.)
The data is also still pretty bad. While, I think, we can be pretty confident that there was a lot of growth between 1500 and 1800 (way more than between 1200 and 1500), the exact shape of this curve is still really uncertain. The global population estimates are still ‘guesstimates’ for most part of the world, throughout this period. Even the first half of the twentieth century is pretty sketchy; IIRC, as late as the 1970s, there were attempts to estimate the present population of China that differed by up to 15%. (I think the Atlas of World Population History mentions this.) We shouldn’t read too much into the exact curve shape.
A further complication is that there’s a pretty unusual ecological event at the start of the period. Although this is pretty uncertain, the pretty abrupt transfer of species from the New World to the Old World (esp. potatoes and corn) is thought to be a major cause of the population surge. This strikes me as a sort of flukey one-off event that obscures the ‘natural’ growth dynamics for this period; although, you could also view it as endogenous to technological progress.
In particular, although standard estimates of growth from 1AD to 1500AD are significantly faster than growth between 10kBC and 1AD, those estimates are sensitive to factor-of-1.5 error in estimates of 1AD population, and real errors could easily be much larger than that.
I wouldn’t necessarily say they were significantly faster. It depends a bit on exactly how you run this test, but, when I run a regression for “(dP/dt)/P = a*P^b” (where P is population) on the dataset up until 1700AD, I find that the b parameter is not significantly greater than 0. (The confidence interval is roughly -.2 to .5, with zero corresponding to exponential growth.)
Of course, though, the badness of the data cancels out this finding; it doesn’t really matter if there’s not a significant difference, according to the data, if the data isn’t reliable.
Even taking the radiocarbon data as given I don’t agree with the conclusions you are drawing from that data. It feels like in each case you are saying “a 2-exponential model fits fine” but the 2 exponentials are always different. The actual events (either technological developments or climate change or population dynamics) that are being pointed to as pivotal aren’t the same across the different time series and so I think we should just be analyzing these without reference to those events (no suggestive dotted lines :) )
The papers typically suggest that the thing kicking off the growth surge, within a particular millennium, is the beginning of intensive agriculture in that region — so I don’t think the pivotal triggering event is really different. Although I haven’t done any investigation into how legit these suggestions are. It’s totally conceivable that we basically don’t know when intensive agriculture began in these different areas, or that the transition was so smeared out that it’s basically arbitrary to single out any particular millennium as special. If the implicit dotted lines are being drawn post-hoc, then that would definitely be cause for suspicion about the story being told.
I currently don’t trust the population data coming from the radiocarbon dating. My current expectation is that after a deep dive I would not end up trusting the radiocarbon dating at all for tracking changes in the rate of population growth when the populations in question are changing how they live and what kinds of artifacts they make (from my perspective, that’s what happened with the genetics data, which wasn’t caveated so aggressively in the initial draft I reviewed). I’d love to hear from someone who actually knows about these techniques or has done a deep dive on these papers though.
I’m also pretty unsure of this. I’d maybe give about a 1⁄3 probability to them being approximately totally uninformative, for the purposes of distinguishing the two perspectives. (I think the other datasets are probably approximately totally uninformative.) Although the radiocarbon dates are definitely more commonly accepted as proxies for historic human population levels than the genetic data, there are also a number of skeptical papers. I haven’t looked deeply enough into the debate, although I probably ought to have.
I think that Hanson’s “series of 3 exponentials” is the neatest alternative, although I also think it’s possible that pre-modern growth looked pretty different from clean exponentials (even on average / beneath the noise). There’s also a semi-common narrative in which the two previous periods exhibited (on average) declining growth rates, until there was some ‘breakthrough’ that allowed the growth rate to surge: I suppose this would be a “three s-curve” model. Then there’s the possibility that the growth pattern in each previous era was basically a hard-to-characterize mess, but was constrained by a rough upper bound on the maximum achievable growth rate. This last possibility is the one I personally find most likely, of the non-hyperbolic possibilities.
It seems almost guaranteed that the data is a mess, it just seems like the only difference between the perspectives is “is acceleration fundamentally concentrated into big revolutions or is it just random and we can draw boundaries around periods of high-growth and call those revolutions?”
There may also be some fundamental meta-prior that matters, here, about the relative weight one ought to give to simple unified models vs. complex qualitative/multifactoral stories.
Which growth model corresponds to which perspective? I normally think of “‘industry’ is what changed and is not contiguous with what came before” as the single-factor model, and multifactor growth models tending more towards continuous growth.
A lot of my prior comes down to my impression that the dynamics of growth just *seem * very different to me for forager societies, agricultural/organic, and industrial/fossil-fuel societies.
I’m definitely much more sympathetic to the forager vs agricultural distinction.
Does a discontinuous change from fossil-fuel use even fit the data? It doesn’t seem to add up at all to me (e.g. doesn’t match the timing of acceleration, there are lots of industries that seemed to accelerate without reliance on fossil fuels, etc.), but would only consider a deep dive if someone actually wanted to stake something on that.
I don’t think the post-1500 data is too helpful help for distinguishing between the ‘long run trend’ and ‘few hundred year phase transition’ perspectives.
If there was something like a phase transition, from pre-modern agricultural societies to modern industrial societies, I don’t see any particular reason to expect the growth curve during the transition to look like the sum of two exponentials. (I especially don’t expect this at the global level, since diffusion dynamics are so messy.)
It feels to me like I’m saying: acceleration happens kind of randomly on a timescale roughly determined by the current growth rate. We should use the base rate of acceleration to make forecasts about the future, i.e. have a significant probability of acceleration during each doubling of output. (Though obviously the real model is more complicated and we can start deviating from that baseline, e.g. sure looks like we should have a higher probability of stagnation now given that we’e had decades of it.)
It feels to me like you are saying “No, we can have a richer model of historical acceleration that assigns significantly lower probability to rapid acceleration over the coming decades / singularity.”
So to me it feels like as we add random stuff like “yeah there are revolutions but we don’t have any prediction about what they will look like” makes the richer model less compelling. It moves me more towards the ignorant perspective of “sometimes acceleration happens, maybe it will happen soon?”, which is what you get in the limit of adding infinitely many ex ante unknown bells and whistles to your model.
The papers typically suggest that the thing kicking off the growth surge, within a particular millennium, is the beginning of intensive agriculture in that region — so I don’t think the pivotal triggering event is really different.
Is “intensive agriculture” a well-defined thing? (Not rhetorical.) It didn’t look like “the beginning of intensive agriculture” corresponds to any fixed technological/social/environmental event (e.g. in most cases there was earlier agriculture and no story was given about why this particular moment would be the moment), it just looked like it was drawn based on when output started rising faster.
I wouldn’t necessarily say they were significantly faster. It depends a bit on exactly how you run this test, but, when I run a regression for “(dP/dt)/P = a*P^b” (where P is population) on the dataset up until 1700AD, I find that the b parameter is not significantly greater than 0. (The confidence interval is roughly -.2 to .5, with zero corresponding to exponential growth.)
I mean that if you have 5x growth from 0AD to 1700AD, and growth was at the same rate from 10000BC to 0AD, then you would expect 5^(10,000/1700) = 13,000-fold growth over that period. We have uncertainty about exactly how much growth there was in the prior period, but we don’t have anywhere near that much uncertainty.
Doing a regression on yearly growth rates seems like a bad way to approach this. It seems like the key question is: did growth speed up a lot in between the agricultural and industrial revolutions? It seems like the way to pick that is to try to use points that are as spaced out as possible to compare growth rates in the beginning and late part of the interval from 10000BC to 1500AD. (The industrial revolution is usually marked much later, but for the purpose of the “2 revolutions” view I think you definitely need it to start by then.)
So almost all of the important measurement error is going to be in the bit of growth in the 0AD to 1500AD phase. If in fact there was only 2x growth in that period (say because the 0AD number was off by 50%) then that would only predict 100-fold growth from 10,000BC to 0AD, which is way more plausible.
The industrial era is, in comparison, less obviously different from the farming era, but it also seems pretty different. My list of pretty distinct features of pre-modern agricultural economies is: (a) the agricultural sector constituted the majority of the economy; (b) production and (to a large extent) transportation were limited by the availability of agricultural or otherwise ‘organic’ sources of energy (plants to power muscles and produce fertiliser); (c) transportation and information transmission speeds were largely limited by windspeed and the speed of animals; (d) nearly everyone was uneducated, poor, and largely unfree; (e) many modern financial, legal, and political institutions did not exist; (f) certain cultural attitudes (such as hatred of commerce and lack of belief in the possibility of progress) were much more common; and (g) scientifically-minded research and development projects played virtually no role in the growth process.
If you just keep listing things, it stops being a plausible source of a discontinuity—you then need to give some story for why your 7 factors all change at the same time. If they don’t, e.g. if they just vary randomly, then you are going to get back to continuous change.
So to me it feels like as we add random stuff like “yeah there are revolutions but we don’t have any prediction about what they will look like” makes the richer model less compelling. It moves me more towards the ignorant perspective of “sometimes acceleration happens, maybe it will happen soon?”, which is what you get in the limit of adding infinitely many ex ante unknown bells and whistles to your model.
I agree the richer stories, if true, imply a more ignorant perspective. I just think it’s plausible that the more ignorant perspective is the correct perspective.
My general feeling towards the evolution of the economy over the past ten thousand years, reading historical analysis, is something like: “Oh wow, this seems really complex and heterogeneous. It’d be very surprising if we could model these processes well with a single-variable model, a noise term, and a few parameters with stable values.” It seems to me like we may in fact just be very ignorant.
Does a discontinuous change from fossil-fuel use even fit the data? It doesn’t seem to add up at all to me (e.g. doesn’t match the timing of acceleration, there are lots of industries that seemed to accelerate without reliance on fossil fuels, etc.), but would only consider a deep dive if someone actually wanted to stake something on that.
Fossil fuels wouldn’t be the cause of the higher global growth rates, in the 1500-1800 period; coal doesn’t really matter much until the 19th century. The story with fossil fuels is typically that there was a pre-existing economic efflorescence that supported England’s transition out of an ‘organic economy.’ So it’s typically a sort of tipping point story, where other factors play an important role in getting the economy to the tipping point.
Is “intensive agriculture” a well-defined thing? (Not rhetorical.) It didn’t look like “the beginning of intensive agriculture” corresponds to any fixed technological/social/environmental event (e.g. in most cases there was earlier agriculture and no story was given about why this particular moment would be the moment), it just looked like it was drawn based on when output started rising faster.
I’m actually unsure of this. Something that’s not clear to me is to what extent the distinction is being drawn in a post-hoc way (i.e. whether intensive agriculture is being implicitly defined as agriculture that kicks off substantial population growth). I don’t know enough about this.
Doing a regression on yearly growth rates seems like a bad way to approach this.
I don’t think I agree, although I’m not sure I understand your objection. Supposing we had accurate data, it seems like the best approach is running a regression that can accommodate either hyperbolic or exponential growth — plus noise — and then seeing whether we can reject the exponential hypothesis. Just noting that the growth rate must have been substantially higher than average within one particular millennium doesn’t necessarily tell us enough; there’s still the question of whether this is plausibly noise.
Of course, though, we have very bad data here—so I suppose this point doesn’t matter too much either way.
If you just keep listing things, it stops being a plausible source of a discontinuity—you then need to give some story for why your 7 factors all change at the same time. If they don’t, e.g. if they just vary randomly, then you are going to get back to continuous change.
You don’t need a story about why they changed at roughly the same time to believe that they did change at roughly the same time (i.e. over the same few century period). And my impression is that that, empirically, they did change at roughly the same time. At least, this seems to be commonly believed.
I don’t think we can reasonably assume they’re independent. Economic histories do tend to draw casual arrows between several of these differences, sometimes suggesting a sort of chain reaction, although these narrative causal diagrams are admittedly never all that satisfying; there’s still something mysterious here. On the other hand, higher population levels strike me as a fairly unsatisfying underlying cause.
[[EDIT: Just to be clear, I don’t think the phase-transition/inflection-point story is necessarily much more plausible than the noisy hyperbolic story. I don’t have very resilient credences here. But I think that, in the absence of good long-run growth data, they’re at least comparably plausible. I think that economic history narratives, the fairly qualitative differences between modern and pre-modern economies, and evidence from between-country variation in modern times count for at least as much as the simplicity prior.]]
Economic histories do tend to draw casual arrows between several of these differences, sometimes suggesting a sort of chain reaction, although these narrative causal diagrams are admittedly never all that satisfying; there’s still something mysterious here.
Just to make this more concrete:
One example of an IR narrative that links a few of these changes together is Robert Allen’s. To the extent that I understand/remember it, the narrative is roughly: The early modern expansion of trade networks caused an economic boom in England, especially in textile manufacturing. As a result, wages in England became unusually high. These high wages created unusually strong incentives to produce labor-saving technology. (One important effect of the Malthusian conditions is that they make labor dirt cheap.) England, compared to a few other countries that had similarly high wages at other points in history, also had access to really unusually cheap energy; they had huge and accessible coal reserves, which they were already burning as a replacement for wood. The unusually high levels of employment in manufacturing and trade also supported higher levels of literacy and numeracy. These conditions came together to support the development of technologies for harnessing fossil fuels, in the 19th century, and the rise of intensive R&D; these may never have been economically rational before. At this point, there was now a virtuous cycle that allowed England’s growth—which was initially an unsustainable form of growth based on trade, rather than technological innovation—to become both sustained and innovation-driven. The spark then spread to other countries.
This particular tipping point story is mostly a story about why growth rates increased from the 19th century onward, although the growth surge in the previous few centuries, largely caused by the Colombian exchange and expansion of trade networks, still plays an important causal role; the rapid expansion of trade networks drives British wages up and makes it possible for them to profitably employ a large portion of their population in manufacturing.
It feels like you are drawing some distinction between “contingent and complicated” and “noise.” Here are some possible distinctions that seem relevant to me but don’t actually seem like disagreements between us:
If something is contingent and complicated, you can expect to learn about it with more reasoning/evidence, whereas if it’s noise maybe you should just throw up your hands. Evidently I’m in the “learn about it by reasoning” category since I spend a bunch of time thinking about AI forecasting.
If something is contingent and complicated, you shouldn’t count on e.g. the long-run statistics matching the noise distribution—there are unmodeled correlations (both real and subjective). I agree with this and think that e.g. the singularity date distributions (and singularity probability) you get out of Roodman’s model are not trustworthy in light of that (as does Roodman).
So it’s not super clear there’s a non-aesthetic difference here.
If I was saying “Growth models imply a very high probability of takeoff soon” then I can see why your doc would affect my forecasts. But where I’m at from historical extrapolations is more like “maybe, maybe not”; it doesn’t feel like any of this should change that bottom line (and it’s not clear how it would change that bottom line) even if I changed my mind everywhere that we disagree.
“Maybe, maybe not” is still a super important update from the strong “the future will be like the recent past” prior that many people implicitly have and I might otherwise take very seriously. It also leads me to mostly dismiss arguments like “this is obviously not the most important century since most aren’t.” But it mostly means that I’m actually looking at what is happening technologically.
You may be responding to writing like this short post where I say “We have been in a period of slowing growth for the last forty years. That’s a long time, but looking over the broad sweep of history I still think the smart money is on acceleration eventually continuing, and seeing something like [hyperbolic growth]...”. I stand by the claim that this is something like the modal guess—we’ve had enough acceleration that the smart money is on it continuing, and this seems equally true on the revolutions model. I totally agree that any specific thing is not very likely to happen, though I think it’s my subjective mode. I feel fine with that post but totally agree it’s imprecise and this is what you get for being short.
The story with fossil fuels is typically that there was a pre-existing economic efflorescence that supported England’s transition out of an ‘organic economy.’ So it’s typically a sort of tipping point story, where other factors play an important role in getting the economy to the tipping point.
OK, but if those prior conditions led to a great acceleration before the purported tipping point, then I feel like that’s mostly what I want to know about and forecast.
Supposing we had accurate data, it seems like the best approach is running a regression that can accommodate either hyperbolic or exponential growth — plus noise — and then seeing whether we can object the exponential hypothesis. Just noting that the growth rate must have been substantially higher than average within one particular millennium doesn’t necessarily tell us enough; there’s still the question of whether this is plausibly noise.
I don’t think that’s what I want to do. My question is, given a moment in history, what’s the best way to guess whether and in how long there will be significant acceleration? If I’m testing the hypothesis “The amount of time before significant acceleration tends to be a small multiple of the current doubling time” then I want to look a few doublings ahead and see if things have accelerated, averaging over a doubling (etc. etc.), rather than do a regression that would indirectly test that hypothesis by making additional structural assumptions + would add a ton of sensitivity to noise.
You don’t need a story about why they changed at roughly the same time to believe that they did change at roughly the same time (i.e. over the same few century period). And my impression is that that, empirically, they did change at roughly the same time. At least, this seems to be commonly believed.
I don’t think we can reasonably assume they’re independent. Economic histories do tend to draw casual arrows between several of these differences, sometimes suggesting a sort of chain reaction, although these narrative causal diagrams are admittedly never all that satisfying; there’s still something mysterious here. On the other hand, higher population levels strike me as a fairly unsatisfying underlying cause.
It looked like you were listing those things to help explain why you have a high prior in favor of discontinuities between industrial and agricultural societies. “We don’t know why those things change together discontinuously, they just do” seems super reasonable (though whether that’s true is precisely what’s at issue). But it does mean that listing out those factors adds nothing to the a priori argument for discontinuity.
Indeed, if you think that all of those are relevant drivers of growth rates then all else equal I’d think you’d expect more continuous progress, since all you’ve done is rule out one obvious way that you could have had discontinuous progress (namely by having the difference be driven by something that had a good prima facie reason to change discontinuously, as in the case of the agricultural revolution) and now you’ll have to posit something mysterious to get to your discontinuous change.
On the acceleration model, the periods from 1500-2000, 10kBC-1500, and “the beginning of history to 10kBC” are roughly equally important data (and if that hypothesis has higher prior I don’t think you can reject that framing). Changes within 10kBC − 1500 are maybe 1/6th of the evidence, and 1⁄3 of the relevant evidence for comparing “continuous acceleration” to “3 exponentials.” I still think it’s great to dig into one of these periods, but I don’t think it’s misleading to present this period as only 1⁄3 of the data on a graph.
I’m going to try and restate what’s going on here, and I want someone to tell me if it sounds right:
If your prior is that growth rate increases happen on a timescale determined by the current growth rate, e.g. you’re likely to have a substantial increase once every N doublings of output, you care more about later years in history when you have more doublings of output. This is what Paul is advocating for.
If your prior is that growth rate increases happen randomly throughout history, e.g. you’re likely to have a substantial increase at an average rate of once every T years, all the years in history should have the same weight. This is what Ben has done in his regressions.
The more weight you start with on the former prior, the more strongly you should weight later time periods.
In particular: If you start with a lot of weight on the former prior, then T years of non-accelerating data at the beginning of your dataset won’t give you much evidence against it, because it won’t correspond to many doublings. But T years of non-accelerating data at the end of your dataset would correspond to many doublings, so would be more compelling evidence against.
This would be an important update for me, so I’m excited to see people looking into it and to spend more time thinking about it myself.
High-level summary of my current take on your document:
I agree that the 1AD-1500AD population data seems super noisy.
Removing that data removes one of the datapoints supporting continuous acceleration (the acceleration between 10kBC − 1AD and 1AD-1500AD) and should make us more uncertain in general.
It doesn’t have much net effect on my attitude towards continuous acceleration vs discontinuous jumps, this mostly pushes us back towards our prior.
I’m not very moved by the other evidence/arguments in your doc.
Here’s how I would summarize the evidence in your document:
Much historical data is made up (often informed by the author’s models of population dynamics), so we can’t use it to estimate historical growth. This seems like the key point.
In particular, although standard estimates of growth from 1AD to 1500AD are significantly faster than growth between 10kBC and 1AD, those estimates are sensitive to factor-of-1.5 error in estimates of 1AD population, and real errors could easily be much larger than that.
Population levels are very noisy (in addition to population measurement being noisy) making it even harder to estimate rates.
Radiographic data often displays isolated periods of rapid growth from 10,000BC to 1AD and it’s possible that average growth rates were something like 2000 year doubling. So even if 500-2000 year doubling times are accurate from 1AD to 1500, those may not be a deviation from the preceding period.
You haven’t looked into the claims people have made about growth from 100kya to 10kya, but given what we know about measurement error from 10kya to now, it seems like the 100kya-10kya data is likely to be way too noisy to say anything about.
Here’s my take in more detail:
You are basically comparing “Series of 3 exponentials” to a hyperbolic growth model. I think our default simple hyperbolic growth model should be the one in David Roodman’s report (blog post), so I’m going to think about this argument as comparing Roodman’s model to a series of 3 noisy exponentials. In your doc you often dunk on an extremely low-noise version of hyperbolic growth but I’m mostly ignoring that because I absolutely agree that population dynamics are very noisy.
It feels like you think 3 exponentials is the higher prior model. But this model has many more parameters to fit the data, and even ignoring that “X changes in 2 discontinuous jumps” doesn’t seem like it has a higher prior than “X goes up continuously but stochastically.” I think the only reason we are taking 3 exponentials seriously is because of the same kind of guesswork you are dismissive of, namely that people have a folk sense that the industrial revolution and agricultural revolutions were discrete changes. If we think those folk senses are unreliable, I think that continuous acceleration has the better prior. And at the very least we need to be careful about using all the extra parameters in the 3-exponentials model, since a model with 2x more parameters should fit the data much better.
On top of that, the post-1500 data is fit terribly by the “3 exponentials” model. Given that continuous acceleration very clearly applies in the only regime where we have data you consider reliable, and given that it already seemed simpler and more motivated, it seems pretty clear to me that it should have the higher prior, and the only reason to doubt that is because of growth folklore. You can’t have it both ways in using growth folklore to promote this hypothesis to attention and then dismissing the evidence from growth folklore because it’s folklore.
On the acceleration model, the periods from 1500-2000, 10kBC-1500, and “the beginning of history to 10kBC” are roughly equally important data (and if that hypothesis has higher prior I don’t think you can reject that framing). Changes within 10kBC − 1500 are maybe 1/6th of the evidence, and 1⁄3 of the relevant evidence for comparing “continuous acceleration” to “3 exponentials.” I still think it’s great to dig into one of these periods, but I don’t think it’s misleading to present this period as only 1⁄3 of the data on a graph.
(Enough about priors, onto the data.)
I think that the key claim is that the 1AD-1500AD data is mostly unreliable. Without this data, we have very little information about acceleration from 10kBC − 1500AD, since the main thing we actually knew was that 1AD-1500AD must have been faster than the preceding 10k years. I’d like to look into that more, but it looks super plausible to me that the noise is 2x or more for 1AD which is enough to totally kill any inference about growth rates. So provisionally I’m inclined to accept your view there.
That basically removes 1 datapoint for the continuous acceleration story and I totally agree it should leave us more uncertain about what’s going on. That said, throwing out all the numbers from that period also removes one of the main quantitative datapoints against continuous acceleration [ETA: the other big one being the modern “great stagnation,” both of these are in the tails of the continuous acceleration story and are just in the middle of the constant exponentials in the 3-exponential story, though see Robin Hanson’s writeup to get a sense for what the series of exponentials view actually ends up looking like—it’s still surprised by the great stagnation], and comes much closer to leaving us with our priors + the obvious acceleration over longer periods + the obvious acceleration during the shorter period where we actually have data, which seem to all basically point in the same direction.
Even taking the radiocarbon data as given I don’t agree with the conclusions you are drawing from that data. It feels like in each case you are saying “a 2-exponential model fits fine” but the 2 exponentials are always different. The actual events (either technological developments or climate change or population dynamics) that are being pointed to as pivotal aren’t the same across the different time series and so I think we should just be analyzing these without reference to those events (no suggestive dotted lines :) ). I spent some time doing this kind of curve fitting to various stochastic growth models and this basically looks to me like what individual realizations look like from such models—the extra parameters in “splice together two unrelated curves” let you get fine-looking fits even when we know that the underlying dynamics are continuous+stochastic.
I currently don’t trust the population data coming from the radiocarbon dating. My current expectation is that after a deep dive I would not end up trusting the radiocarbon dating at all for tracking changes in the rate of population growth when the populations in question are changing how they live and what kinds of artifacts they make (from my perspective, that’s what happened with the genetics data, which wasn’t caveated so aggressively in the initial draft I reviewed). I’d love to hear from someone who actually knows about these techniques or has done a deep dive on these papers though.
I think the only dataset that you should expect to provide evidence on its own is the China population time series. But even there if you just take rolling averages and allow for a reasonable level of noise I think the continuous acceleration story looks fine. E.g. I think if you compare David Roodman’s model with the piecewise exponential model (both augmented with measurement noise, and allowing you to choose noisy dynamics however you want for the exponential model), Roodman’s model is going to fit the data better despite having fewer free parameters. If that’s the case, I don’t think this time series can be construed as evidence against that model.
I agree with the point that if growth is 0 before the agricultural revolution, rather than “small,” then that would undermine the continuous acceleration story. I think prior growth was probably slow but non-zero, and this document didn’t really update my view on that question.
I feel really confused what the actual right priors here are supposed to be. I find the “but X has fewer parameters” argument only mildly compelling, because I feel like other evidence about similar systems that we’ve observed should easily give us enough evidence to overcome the difference in complexity.
This does mean that a lot of my overall judgement on this question relies on the empirical evidence we have about similar systems, and the concrete gears-level models I have for what has caused growth. AI Impact’s work on discontinuous vs. continuous progress feels somewhat relevant and evidence from other ecological systems also seems reasonably useful.
When I try to understand what exactly happened in terms of growth at a gears-level, I feel like I tend towards more discontinuous hypotheses, because I have a bunch of very concrete, reasonably compelling sounding stories of specific things that caused the relevant shifts, and while I have some gears-level models for what would cause more continuous growth, they feel a lot more nebulous and vague to me, in a way that I think usually doesn’t correspond to truth. The thing that on the margin would feel most compelling to me for the continuous view is something like a concrete zoomed in story of how you get continuous growth from a bunch of humans talking to each other and working with each other over a few generations, that doesn’t immediately abstract things away into high-level concepts like “knowledge” and “capital”.
Be careful that you don’t have too many stories, or it starts to get continuous again.
More seriously, I don’t know what the small # of factors are for the industrial revolution, and my current sense is that the story can only seem simple for the agricultural revolution because we are so far away and ignoring almost all the details.
It seems like the only factor that looks a priori like it should cause a discontinuity is the transition from hunting+gathering to farming, i.e. if you imagine “total food” as the sum of “food we make” and “food we find” then there could be a discontinuous change in growth rates as “food we make” starts to become large relative to “food we find” (which bounces around randomly but is maybe not really changing). This is blurred because of complementarity between your technology and finding food, but certainly I’m on board with an in-principle argument for a discontinuity as the new mode overtakes the old one.
For the last 10k years my impression is that no one has a very compelling story for discontinuities (put differently: they have way too many stories) and it’s mostly a stylized empirical fact that the IR is kind of discontinuous. But I’m provisionally on board with Ben’s basic point that we don’t really have good enough data to know whether growth had been accelerating a bunch in the run-up to the IR.
To the extent things are discontinuous, I’d guess that it’s basically from something similar to the agricultural case—there is continuous growth and random variation, and you see “discontinuities” in the aggregate if a smaller group is significantly outpacing the world, so that by the time they become a large part of the world they are growing significantly faster.
I think this is also reasonably plausible in the AI case (e.g. there is an automated part of the economy doubling every 1-2 years, by the time it gets to be 10% of the economy it’s driving +5%/year growth, 1-2 years later it’s driving +10% growth). But I think quantitatively given the numbers involved and the actual degree of complementarity, this is still unlikely to give you a fast takeoff as I operationalized it. I think if we’re having a serious discussion about “takeoff” that’s probably where the action is, not in any of the kinds of arguments that I dismiss in that post.
I mean something much more basic. If you have more parameters then you need to have uncertainty about every parameter. So you can’t just look at how well the best “3 exponentials” hypothesis fits the data, you need to adjust for the fact that this particular “3 exponentials” model has lower prior probability. That is, even if you thought “3 exponentials” was a priori equally likely to a model with fewer parameters, every particular instance of 3 exponentials needs to be less probable than every particular model with fewer parameters.
As far as I can tell this is how basically all industries (and scientific domains) work—people learn by doing and talk to each other and they get continuously better, mostly by using and then improving on technologies inherited from other people.
It’s not clear to me whether you are drawing a distinction between modern economic activity and historical cultural accumulation, or whether you feel like you need to see a zoomed-in version of this story for modern economic activity as well, or whether this is a more subtle point about continuous technological progress vs continuous changes in the rate of tech progress, or something else.
Thanks, this was a useful clarification. I agree with this as stated. And I indeed assign substantially more probability to a statement of the form “there were some s-curve like shifts in humanity’s past that made a big difference” than to any specific “these three specific s-curve like shifts are what got us to where we are today”.
Hmm, I don’t know, I guess that’s just not really how I would characterize most growth? My model is that most industries start with fast s-curve like growth, then plateau, then often decline. Sure, kind of continuously in the analytical sense, but with large positive and negative changes in the derivative of the growth.
And in my personal experience it’s also less the case that I and the people I work with just get continuously better, it’s more like we kind of flop around until we find something that gets us a lot of traction on something, and then we quickly get much better at the given task, and then we level off again. And it’s pretty easy to get stuck in a rut somewhere and be much less effective than I was years ago, or for an organization to end up in a worse equilibrium and broadly get worse at coordinating, or produce much worse output than previously for other reasons.
Of course enough of those stories could itself give rise to a continuous growth story here, but there is a question here about where the self-similarity lies. Like, many s-curves can also give rise to one big s-curve. Just because I have many s-curve doesn’t mean I get continuous hyperbolic growth. And so seeing lots of relative discontinuous s-curves at the small scale does feel like it’s evidence that we also should expect the macro scale to be a relatively small number of discontinuous s-curves (or more precisely, s-curves whose peak is itself heavy-tail distributed, so that if you run a filter for the s-curves that explain most of the change, you end up with just a few that really mattered).
I don’t know exactly what this means, but it seems like most industries in the modern world are characterized by relatively continuous productivity improvements over periods of decades or centuries. The obvious examples to me are semiconductors and AI since I deal most with those. But it also seems true of e.g. manufacturing, agricultural productivity, batteries, construction costs. It seems like industries where the productivity vs time curve is a “fast S-curve” are exceptional, which I assume means we are somehow reading the same data differently. What kind of industries would you characterize this way?
(I agree that e.g. “adoption” is more likely to be an s-curve given that it’s bounded, but productivity seems like the analogy for growth rates.)
This agrees with my impression. Just in case someone is looking for references for this, see e.g.:
Nagy et al. (2013) - several of the trends they look at, e.g. prices for certain chemical substances, show exponential growth for more than 30 years
Farmer & Lafond (2016) - similar to the previous paper, though fewer trends with data from more than 20 years
Bloom et al. (2020) - reviews trends in research productivity, most of which go back to 1975 and some to 1900
Some early examples from manufacturing (though not covering multiple decades) are reviewed in a famous paper by Arrow (1971), who proposed ‘learning by doing’ as a mechanism.
Note: Actually looking at the graphs in Farmer & Lafond (2016), many of these do sure seem pretty S-curve shaped. As do many of the diagrams in Nagy et al. (2013). I would have to run some real regressions to look at it, but in particular the ones in Farmer & Lafond seem pretty compatible with the basic s-curve model.
Overlapping S-curves are also hard to measure because obviously there are feedback effects between different industries (see my self-similarity comment above). Many of the advances in those fields are driven by exogenous factors, like their inputs getting cheaper, with no substantial improvements in their internal methodologies. One of my models of technological progress (I obviously also share the model of straightforward exponential growth and assign it substantial probability) is that you have nested and overlapping S-curves, which makes it hard to just look at cost/unit output of any individual field.
For analyzing that hypothesis it seems more useful to hold inputs constant and then look at how cost/unit develops, in order to build a model of that isolated chunk of the system (and then obviously also look at the interaction between industries and systems to get a sense of how they interact). But that’s also much harder to do, given that our data is already really messy and noisy.
Thanks for poking at this, it would be quite interesting to me if the “constant exponential growth” story was wrong. Which graphs in Farmer & Lafond (2016) are you referring to? To me, the graph with a summary of all trends only seems to have very few that at first glance look a bit like s-curves. But I agree one would need to go beyond eyeballing to know for sure.
I agree with your other points. My best guess is that input prices and other exogenous factors aren’t that important for some of the trends, e.g. Moore’s Law or agricultural productivity. And I think some of the manufacturing trends in e.g. Arrow (1971) are in terms of output quantity per hour of work rather than prices, and so also seem less dependent on exogenous factors. But I’m more uncertain about this, and agree that in principle dependence on exogenous factors complicates the interpretation.
Yeah, that was the one I was looking at. From very rough eye-balling, it looks like a lot of them have slopes that level off, but obviously super hard to tell just from eye-balling. I might try to find the data and actually check.
This is one of my favorite comments on the Forum. Thanks for the thorough response.
Also want to second this! (This is a far more extensive response and summary than I’ve seen on almost any EA forum post.)
Seriously. This did an incredible job of crystallizing my own confusions.
Hi Paul,
Thanks for your super detailed comment (and your comments on the previous version)!
I think that Hanson’s “series of 3 exponentials” is the neatest alternative, although I also think it’s possible that pre-modern growth looked pretty different from clean exponentials (even on average / beneath the noise). There’s also a semi-common narrative in which the two previous periods exhibited (on average) declining growth rates, until there was some ‘breakthrough’ that allowed the growth rate to surge: I suppose this would be a “three s-curve” model. Then there’s the possibility that the growth pattern in each previous era was basically a hard-to-characterize mess, but was constrained by a rough upper bound on the maximum achievable growth rate. This last possibility is the one I personally find most likely, of the non-hyperbolic possibilities.
(I think the pre-agricultural period is especially likely to be messy, since I would guess that human evolution and climate/environmental change probably explain the majority of the variation in population levels within this period.)
I think this is a good and fair point. I’m starting out sympathetic toward the breakthrough/phase-change perspective, in large part because this perspective fits well with the kinds of narratives that economic historians and world historians tend to tell. It’s reasonable to wonder, though, whether I actually should give much weight to these narratives. Although they rely on much more than just world GDP estimates, their evidence base is also far from great, and they disagree on a ton of issues (there are a bunch of competing economic narratives that only partly overlap.)
A lot of my prior comes down to my impression that the dynamics of growth just *seem * very different to me for forager societies, agricultural/organic societies, and industrial/fossil-fuel societies. In the forager era, for example, it’s possible that, for the majority of the period, human evolution was the main underlying thing supporting growth. In the farmer era, the main drivers were probably land conversion, the diffusion and further evolution of crops/animals, agricultural capital accumulation (e.g. more people having draft animals), and piecemeal improvements in farming/land-conversion techniques discovered through practice. I don’t find it difficult to imagine that the latter drivers supported higher growth rates. For example: the fact that non-sedentary groups can’t really accumulate capital, in the same way, seems like a pretty fundamental distinction.
The industrial era is, in comparison, less obviously different from the farming era, but it also seems pretty different. My list of pretty distinct features of pre-modern agricultural economies is: (a) the agricultural sector constituted the majority of the economy; (b) production and (to a large extent) transportation were limited by the availability of agricultural or otherwise ‘organic’ sources of energy (plants to power muscles and produce fertiliser); (c) transportation and information transmission speeds were largely limited by windspeed and the speed of animals; (d) nearly everyone was uneducated, poor, and largely unfree; (e) many modern financial, legal, and political institutions did not exist; (f) certain cultural attitudes (such as hatred of commerce and lack of belief in the possibility of progress) were much more common; and (g) scientifically-minded research and development projects played virtually no role in the growth process.
I also don’t find it too hard to believe that some subset of these changes help to explain why modern industrialised economies can grow faster than premodern agricultural economies: here, for example, is a good book chapter on the growth implications of relying entirely on ‘organic’ sources of energy for production. The differences strike me as pretty fundamental and pretty extensive. Although this impression is also pretty subjective and could easily amount to seeing dividing lines where they don’t exist.
Another piece of evidence is that there’s extreme between-states variation in the growth rates, in modern times, which isn’t well-explained by factors like population size. We’ve seen that it is possible for something to heavily retard/bottleneck growth (e.g. bad political institutions), then for growth to surge following the removal of the bottleneck. It’s not too hard to imagine that pre-modern states had lots of blockers. They were in some way similar to 20th/21st century growth basket cases, only with some important extra growth retardants—like a lack of fossil fuels and artificial fertilizer, a lack of knowledge that material progress is possible, etc. -- thrown on top.
There may also be some fundamental meta-prior that matters, here, about the relative weight one ought to give to simple unified models vs. complex qualitative/multifactoral stories.
I don’t think the post-1500 data is too helpful help for distinguishing between the ‘long run trend’ and ‘few hundred year phase transition’ perspectives.
If there was something like a phase transition, from pre-modern agricultural societies to modern industrial societies, I don’t see any particular reason to expect the growth curve during the transition to look like the sum of two exponentials. (I especially don’t expect this at the global level, since diffusion dynamics are so messy.)
The data is also still pretty bad. While, I think, we can be pretty confident that there was a lot of growth between 1500 and 1800 (way more than between 1200 and 1500), the exact shape of this curve is still really uncertain. The global population estimates are still ‘guesstimates’ for most part of the world, throughout this period. Even the first half of the twentieth century is pretty sketchy; IIRC, as late as the 1970s, there were attempts to estimate the present population of China that differed by up to 15%. (I think the Atlas of World Population History mentions this.) We shouldn’t read too much into the exact curve shape.
A further complication is that there’s a pretty unusual ecological event at the start of the period. Although this is pretty uncertain, the pretty abrupt transfer of species from the New World to the Old World (esp. potatoes and corn) is thought to be a major cause of the population surge. This strikes me as a sort of flukey one-off event that obscures the ‘natural’ growth dynamics for this period; although, you could also view it as endogenous to technological progress.
I wouldn’t necessarily say they were significantly faster. It depends a bit on exactly how you run this test, but, when I run a regression for “(dP/dt)/P = a*P^b” (where P is population) on the dataset up until 1700AD, I find that the b parameter is not significantly greater than 0. (The confidence interval is roughly -.2 to .5, with zero corresponding to exponential growth.)
Of course, though, the badness of the data cancels out this finding; it doesn’t really matter if there’s not a significant difference, according to the data, if the data isn’t reliable.
The papers typically suggest that the thing kicking off the growth surge, within a particular millennium, is the beginning of intensive agriculture in that region — so I don’t think the pivotal triggering event is really different. Although I haven’t done any investigation into how legit these suggestions are. It’s totally conceivable that we basically don’t know when intensive agriculture began in these different areas, or that the transition was so smeared out that it’s basically arbitrary to single out any particular millennium as special. If the implicit dotted lines are being drawn post-hoc, then that would definitely be cause for suspicion about the story being told.
I’m also pretty unsure of this. I’d maybe give about a 1⁄3 probability to them being approximately totally uninformative, for the purposes of distinguishing the two perspectives. (I think the other datasets are probably approximately totally uninformative.) Although the radiocarbon dates are definitely more commonly accepted as proxies for historic human population levels than the genetic data, there are also a number of skeptical papers. I haven’t looked deeply enough into the debate, although I probably ought to have.
It seems almost guaranteed that the data is a mess, it just seems like the only difference between the perspectives is “is acceleration fundamentally concentrated into big revolutions or is it just random and we can draw boundaries around periods of high-growth and call those revolutions?”
Which growth model corresponds to which perspective? I normally think of “‘industry’ is what changed and is not contiguous with what came before” as the single-factor model, and multifactor growth models tending more towards continuous growth.
I’m definitely much more sympathetic to the forager vs agricultural distinction.
Does a discontinuous change from fossil-fuel use even fit the data? It doesn’t seem to add up at all to me (e.g. doesn’t match the timing of acceleration, there are lots of industries that seemed to accelerate without reliance on fossil fuels, etc.), but would only consider a deep dive if someone actually wanted to stake something on that.
It feels to me like I’m saying: acceleration happens kind of randomly on a timescale roughly determined by the current growth rate. We should use the base rate of acceleration to make forecasts about the future, i.e. have a significant probability of acceleration during each doubling of output. (Though obviously the real model is more complicated and we can start deviating from that baseline, e.g. sure looks like we should have a higher probability of stagnation now given that we’e had decades of it.)
It feels to me like you are saying “No, we can have a richer model of historical acceleration that assigns significantly lower probability to rapid acceleration over the coming decades / singularity.”
So to me it feels like as we add random stuff like “yeah there are revolutions but we don’t have any prediction about what they will look like” makes the richer model less compelling. It moves me more towards the ignorant perspective of “sometimes acceleration happens, maybe it will happen soon?”, which is what you get in the limit of adding infinitely many ex ante unknown bells and whistles to your model.
Is “intensive agriculture” a well-defined thing? (Not rhetorical.) It didn’t look like “the beginning of intensive agriculture” corresponds to any fixed technological/social/environmental event (e.g. in most cases there was earlier agriculture and no story was given about why this particular moment would be the moment), it just looked like it was drawn based on when output started rising faster.
I mean that if you have 5x growth from 0AD to 1700AD, and growth was at the same rate from 10000BC to 0AD, then you would expect 5^(10,000/1700) = 13,000-fold growth over that period. We have uncertainty about exactly how much growth there was in the prior period, but we don’t have anywhere near that much uncertainty.
Doing a regression on yearly growth rates seems like a bad way to approach this. It seems like the key question is: did growth speed up a lot in between the agricultural and industrial revolutions? It seems like the way to pick that is to try to use points that are as spaced out as possible to compare growth rates in the beginning and late part of the interval from 10000BC to 1500AD. (The industrial revolution is usually marked much later, but for the purpose of the “2 revolutions” view I think you definitely need it to start by then.)
So almost all of the important measurement error is going to be in the bit of growth in the 0AD to 1500AD phase. If in fact there was only 2x growth in that period (say because the 0AD number was off by 50%) then that would only predict 100-fold growth from 10,000BC to 0AD, which is way more plausible.
If you just keep listing things, it stops being a plausible source of a discontinuity—you then need to give some story for why your 7 factors all change at the same time. If they don’t, e.g. if they just vary randomly, then you are going to get back to continuous change.
I agree the richer stories, if true, imply a more ignorant perspective. I just think it’s plausible that the more ignorant perspective is the correct perspective.
My general feeling towards the evolution of the economy over the past ten thousand years, reading historical analysis, is something like: “Oh wow, this seems really complex and heterogeneous. It’d be very surprising if we could model these processes well with a single-variable model, a noise term, and a few parameters with stable values.” It seems to me like we may in fact just be very ignorant.
Fossil fuels wouldn’t be the cause of the higher global growth rates, in the 1500-1800 period; coal doesn’t really matter much until the 19th century. The story with fossil fuels is typically that there was a pre-existing economic efflorescence that supported England’s transition out of an ‘organic economy.’ So it’s typically a sort of tipping point story, where other factors play an important role in getting the economy to the tipping point.
I’m actually unsure of this. Something that’s not clear to me is to what extent the distinction is being drawn in a post-hoc way (i.e. whether intensive agriculture is being implicitly defined as agriculture that kicks off substantial population growth). I don’t know enough about this.
I don’t think I agree, although I’m not sure I understand your objection. Supposing we had accurate data, it seems like the best approach is running a regression that can accommodate either hyperbolic or exponential growth — plus noise — and then seeing whether we can reject the exponential hypothesis. Just noting that the growth rate must have been substantially higher than average within one particular millennium doesn’t necessarily tell us enough; there’s still the question of whether this is plausibly noise.
Of course, though, we have very bad data here—so I suppose this point doesn’t matter too much either way.
You don’t need a story about why they changed at roughly the same time to believe that they did change at roughly the same time (i.e. over the same few century period). And my impression is that that, empirically, they did change at roughly the same time. At least, this seems to be commonly believed.
I don’t think we can reasonably assume they’re independent. Economic histories do tend to draw casual arrows between several of these differences, sometimes suggesting a sort of chain reaction, although these narrative causal diagrams are admittedly never all that satisfying; there’s still something mysterious here. On the other hand, higher population levels strike me as a fairly unsatisfying underlying cause.
[[EDIT: Just to be clear, I don’t think the phase-transition/inflection-point story is necessarily much more plausible than the noisy hyperbolic story. I don’t have very resilient credences here. But I think that, in the absence of good long-run growth data, they’re at least comparably plausible. I think that economic history narratives, the fairly qualitative differences between modern and pre-modern economies, and evidence from between-country variation in modern times count for at least as much as the simplicity prior.]]
Just to make this more concrete:
One example of an IR narrative that links a few of these changes together is Robert Allen’s. To the extent that I understand/remember it, the narrative is roughly: The early modern expansion of trade networks caused an economic boom in England, especially in textile manufacturing. As a result, wages in England became unusually high. These high wages created unusually strong incentives to produce labor-saving technology. (One important effect of the Malthusian conditions is that they make labor dirt cheap.) England, compared to a few other countries that had similarly high wages at other points in history, also had access to really unusually cheap energy; they had huge and accessible coal reserves, which they were already burning as a replacement for wood. The unusually high levels of employment in manufacturing and trade also supported higher levels of literacy and numeracy. These conditions came together to support the development of technologies for harnessing fossil fuels, in the 19th century, and the rise of intensive R&D; these may never have been economically rational before. At this point, there was now a virtuous cycle that allowed England’s growth—which was initially an unsustainable form of growth based on trade, rather than technological innovation—to become both sustained and innovation-driven. The spark then spread to other countries.
This particular tipping point story is mostly a story about why growth rates increased from the 19th century onward, although the growth surge in the previous few centuries, largely caused by the Colombian exchange and expansion of trade networks, still plays an important causal role; the rapid expansion of trade networks drives British wages up and makes it possible for them to profitably employ a large portion of their population in manufacturing.
It feels like you are drawing some distinction between “contingent and complicated” and “noise.” Here are some possible distinctions that seem relevant to me but don’t actually seem like disagreements between us:
If something is contingent and complicated, you can expect to learn about it with more reasoning/evidence, whereas if it’s noise maybe you should just throw up your hands. Evidently I’m in the “learn about it by reasoning” category since I spend a bunch of time thinking about AI forecasting.
If something is contingent and complicated, you shouldn’t count on e.g. the long-run statistics matching the noise distribution—there are unmodeled correlations (both real and subjective). I agree with this and think that e.g. the singularity date distributions (and singularity probability) you get out of Roodman’s model are not trustworthy in light of that (as does Roodman).
So it’s not super clear there’s a non-aesthetic difference here.
If I was saying “Growth models imply a very high probability of takeoff soon” then I can see why your doc would affect my forecasts. But where I’m at from historical extrapolations is more like “maybe, maybe not”; it doesn’t feel like any of this should change that bottom line (and it’s not clear how it would change that bottom line) even if I changed my mind everywhere that we disagree.
“Maybe, maybe not” is still a super important update from the strong “the future will be like the recent past” prior that many people implicitly have and I might otherwise take very seriously. It also leads me to mostly dismiss arguments like “this is obviously not the most important century since most aren’t.” But it mostly means that I’m actually looking at what is happening technologically.
You may be responding to writing like this short post where I say “We have been in a period of slowing growth for the last forty years. That’s a long time, but looking over the broad sweep of history I still think the smart money is on acceleration eventually continuing, and seeing something like [hyperbolic growth]...”. I stand by the claim that this is something like the modal guess—we’ve had enough acceleration that the smart money is on it continuing, and this seems equally true on the revolutions model. I totally agree that any specific thing is not very likely to happen, though I think it’s my subjective mode. I feel fine with that post but totally agree it’s imprecise and this is what you get for being short.
OK, but if those prior conditions led to a great acceleration before the purported tipping point, then I feel like that’s mostly what I want to know about and forecast.
I don’t think that’s what I want to do. My question is, given a moment in history, what’s the best way to guess whether and in how long there will be significant acceleration? If I’m testing the hypothesis “The amount of time before significant acceleration tends to be a small multiple of the current doubling time” then I want to look a few doublings ahead and see if things have accelerated, averaging over a doubling (etc. etc.), rather than do a regression that would indirectly test that hypothesis by making additional structural assumptions + would add a ton of sensitivity to noise.
It looked like you were listing those things to help explain why you have a high prior in favor of discontinuities between industrial and agricultural societies. “We don’t know why those things change together discontinuously, they just do” seems super reasonable (though whether that’s true is precisely what’s at issue). But it does mean that listing out those factors adds nothing to the a priori argument for discontinuity.
Indeed, if you think that all of those are relevant drivers of growth rates then all else equal I’d think you’d expect more continuous progress, since all you’ve done is rule out one obvious way that you could have had discontinuous progress (namely by having the difference be driven by something that had a good prima facie reason to change discontinuously, as in the case of the agricultural revolution) and now you’ll have to posit something mysterious to get to your discontinuous change.
I’m going to try and restate what’s going on here, and I want someone to tell me if it sounds right:
If your prior is that growth rate increases happen on a timescale determined by the current growth rate, e.g. you’re likely to have a substantial increase once every N doublings of output, you care more about later years in history when you have more doublings of output. This is what Paul is advocating for.
If your prior is that growth rate increases happen randomly throughout history, e.g. you’re likely to have a substantial increase at an average rate of once every T years, all the years in history should have the same weight. This is what Ben has done in his regressions.
The more weight you start with on the former prior, the more strongly you should weight later time periods.
In particular: If you start with a lot of weight on the former prior, then T years of non-accelerating data at the beginning of your dataset won’t give you much evidence against it, because it won’t correspond to many doublings. But T years of non-accelerating data at the end of your dataset would correspond to many doublings, so would be more compelling evidence against.